mnomx’164 mnomx’171 mnomx’245 mnomx’244

# Mysterious Triality and Rational Homotopy Theory

Hisham Sati, Alexander A. Voronov

###### Abstract

Mysterious Duality has been discovered by Iqbal, Neitzke, and Vafa [INV02] as
a convincing, yet mysterious correspondence between certain symmetry patterns
in toroidal compactifications of M-theory and del Pezzo surfaces, both
governed by the root system series $E_{k}$.

It turns out that the sequence of del Pezzo surfaces is not the only sequence
of objects in mathematics that gives rise to the same $E_{k}$ symmetry
pattern. We present a sequence of topological spaces, starting with the four-
sphere $S^{4}$, and then forming its iterated cyclic loop spaces
$\mathcal{L}_{c}^{k}S^{4}$, within which we discover the $E_{k}$ symmetry
pattern via rational homotopy theory. For this sequence of spaces, the
correspondence between its $E_{k}$ symmetry pattern and that of toroidal
compactifications of M-theory is no longer a mystery, as each space
$\mathcal{L}_{c}^{k}S^{4}$ is naturally related to the compactification of
M-theory on the $k$-torus via identification of the equations of motion of
$(11-k)$-dimensional supergravity as the defining equations of the Sullivan
minimal model of $\mathcal{L}_{c}^{k}S^{4}$. This gives an explicit duality
between algebraic topology and physics.

Thereby, we extend Iqbal-Neitzke-Vafa’s Mysterious Duality between algebraic
geometry and physics into a triality, also involving algebraic topology. Via
this triality, duality between physics and mathematics is demystified, and the
mystery is transferred to the mathematical realm as duality between algebraic
geometry and algebraic topology. Now the question is: Is there an explicit
relation between the del Pezzo surfaces $\mathbb{B}_{k}$ and iterated cyclic
loop spaces of $S^{4}$ which would explain the common $E_{k}$ symmetry
pattern?

To our teachers: Igor V. Dolgachev and Yuri I. Manin

###### Contents

  1. 1 Introduction
  2. 2 The 4-sphere and its cyclifications as the universal targets for M-theory and its reductions
    1. 2.1 The Sullivan minimal model
    2. 2.2 $S^{4}$ as the universal target for M-theory via Hypothesis H
    3. 2.3 The Quillen model and M-theory gauge structure
    4. 2.4 Dimensional reduction of M-theory on tori and iterated cyclifications of $S^{4}$
    5. 2.5 The Sullivan minimal model of the cyclic loop space
    6. 2.6 Type IIB and T-duality
  3. 3 Toroidal symmetries
    1. 3.1 Toroidal symmetries of minimal algebras
    2. 3.2 Toroidal symmetries of a cyclification
    3. 3.3 Toroidal symmetries of the minimal algebras of cyclifications of $S^{4}$
    4. 3.4 Topological interpretation of toroidal symmetries
    5. 3.5 Toroidal symmetries of type IIB
  4. 4 The $E_{k}$ symmetry of iterated cyclic loop spaces
    1. 4.1 Reminder: the combinatorial data from del Pezzo surfaces
    2. 4.2 The “Cartan subalgebra” and weight lattice
    3. 4.3 The “anticanonical class”
    4. 4.4 The $E_{k}$ root system and its Weyl group
    5. 4.5 The $E_{k}$ symmetry of type IIB
    6. 4.6 27 “Lines” in the cyclic loop space $\mathcal{L}_{c}^{6}S^{4}$
    7. 4.7 Cyclifying $\geq 9$ times and Kac-Moody algebras

##  1 Introduction

Mysterious Duality has been discovered by Iqbal, Neitzke, and Vafa [INV02] as
a remarkable, yet mysterious correspondence between certain symmetry patterns
in toroidal compactifications of M-theory and del Pezzo surfaces, both
governed by the root system corresponding to the exceptional series $E_{k}$,
$k\leq 8$.

##### Del Pezzo surfaces.

Consider the _del Pezzo surface_ $\mathbb{B}_{k}$ obtained as the blowup of
the complex projective plane $\mathbb{CP}^{2}$ at $k$ generic points
$x_{1},\dots,x_{k}$, $0\leq k\leq 8$. The Picard group ${\rm
Pic}(\mathbb{B}_{k})$ of isomorphism classes of line bundles is in this case
isomorphic to the divisor class group and the second cohomology group:
$\operatorname{Pic}(\mathbb{B}_{k})\cong H^{2}(\mathbb{B}_{k};\mathbb{Z})$.
This is a rank-$(k+1)$ lattice with a natural Lorentzian inner product given
by the intersection form. Another important feature of the del Pezzo surface
$\mathbb{B}_{k}$ is the _anticanonical class_ :
$-\mathcal{K}_{k}:=-\Omega^{2}_{\mathbb{B}_{k}}$, which is ample and defines a
map $\mathbb{B}_{k}\to\mathbb{C}\mathbb{P}^{9-k}$, also called
_anticanonical_. This map is an embedding for $k\leq 6$. The _degree of the
del Pezzo surface_ is the self-intersection number
$(-\mathcal{K}_{k})\cdot(-\mathcal{K}_{k})=\mathcal{K}_{k}\cdot\mathcal{K}_{k}=9-k$.

There is also an “outlier” del Pezzo surface
$\mathbb{B}_{1}^{\prime}:=\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$
of degree 8 with Picard group of rank 2. This surface is related to the del
Pezzo surfaces $\mathbb{B}_{k}$ by a single blowup: if we blow up a point in
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$, we will obtain a
surface isomorphic to $\mathbb{B}_{2}$.

The connection between algebraic geometry and Lie theory comes from the fact
that the Cartan matrices of the exceptional Lie algebras of type $E_{k}$ and
their root systems arise from the above data: a lattice with a distinguished
element and inner product (see [Ma74]).

In general, even for a fixed $k$, the surfaces $\mathbb{B}_{k}$ obtained from
varying the blowup points are not isomorphic as complex manifolds. However,
they are diffeomorphic, so these surfaces give rise to the same combinatorial
data, and we will just speak of “the” del Pezzo surface $\mathbb{B}_{k}$ for
each $k$.

##### The Mysterious Duality correspondence.

There is a correspondence between del Pezzo surfaces $\mathbb{B}_{k}$ and
M-theory “wrapped” on tori $T^{k}$, [INV02], as follows. Given an element
$\omega$ of $H^{2}(\mathbb{B}_{k};\mathbb{R})$, considered as a generalized
Kähler form on $\mathbb{B}_{k}$, the generalized volumes
$\omega(\mathcal{C}):=\omega\cdot\mathcal{C}=\int_{\mathbb{B}_{k}}\omega\cup\mathcal{C}$
of the standard basic classes
$\mathcal{C}=\mathcal{H},\mathcal{E}_{1},\dots,\mathcal{E}_{k}$, see Section
4.1, may be thought of as logarithms, up to certain constants, of the
coordinates $(\ell_{p},R_{1},\linebreak[0]\dots,\linebreak[1]R_{k})$ on the
moduli space of M-theory compactified on the flat $k$-torus $T^{k}$:

$\omega(\mathcal{H})=-3\ln\ell_{p};\qquad\omega(\mathcal{E}_{i})=-\ln(2\pi
R_{i})\;,\quad i=1,\dots,k,$

where $\ell_{p}$ is the Planck scale and the $R_{i}$’s are the radii of the
torus factors. The moduli space of M-theory compactified on $T^{k}$ is usually
taken to be the double quotient $K\backslash G/G(\mathbb{Z})$, where $G$ is
the U-duality group, which is the (real split form of the) Lie group $E_{k}$,
$G(\mathbb{Z})$ is its integral form, and $K$ is the maximal compact subgroup
of $G$ [HT95][OP99]. In [INV02] a simpler moduli space $\mathcal{M}_{k}:=A/W$
is used. Taking into account the Iwasawa decomposition $G=KAN$ with $A$ the
$\mathbb{R}$-split abelian factor and $N$ the unipotent factor and the
identification of the Weyl group as $W=\mathcal{N}(A)/A$, where
$\mathcal{N}(A)$ is the normalizer of $A$, one may think of passing from
$K\backslash G/G(\mathbb{Z})$ to $A/W$ as some sort of abelianization:

$K\backslash G/G(\mathbb{Z})=(AN)/G(\mathbb{Z})\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.93054pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\rm\bf\color[rgb]{0.05,0.25,0.65}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.25,0.65}abelianize}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 57.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}\ignorespaces{\hbox{\lx@xy@drawsquiggles@}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 57.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathcal{M}_{k}=A/W\;.$
(1)

Further, the Mysterious Duality correspondence has the following features
[INV02]:

  1. (i)

Automorphisms of $\mathbb{B}_{k}$ and $H^{2}(\mathbb{B}_{k};\mathbb{Z})$
correspond to the U-duality transformations of M-theory on $T^{k}$. The
U-duality group of M-theory on $T^{k}$, which for rectangular
compactifications with no $C$-field is given by the Weyl group of $E_{k}$, is
related to a subgroup of the automorphism group of
$H^{2}(\mathbb{B}_{k};\mathbb{Z})$ which preserves the intersection form and
canonical class.

  2. (ii)

The moduli space $A/W$ of compactified M-theory corresponds to the moduli
space $H^{2}(\mathbb{B}_{k};\mathbb{R})/W$ of generalized Kähler forms
$\omega$, regarded as metric/transcendental data on $\mathbb{B}_{k}$, up to
automorphisms of $\mathbb{B}_{k}$ considered as an algebraic surface.

  3. (iii)

Two classes of rational curves $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ related
as $\mathcal{C}_{1}+\mathcal{C}_{2}=-\mathcal{K}_{k}$ on $\mathbb{B}_{k}$
correspond to two ${\rm D}p_{1}$-branes with ${\rm D}p_{2}$-branes with
$p_{1}+p_{2}=7-k$, expressing electric-magnetic duality.

  4. (iv)

The $p$-branes of type IIA and type IIB string theory in 10 dimensions
correspond to classes of rational curves on $\mathbb{B}_{1}$ and
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$, respectively.

##### Our conceptual take.

We bring in algebraic topology, in the form of (rational) homotopy theory, and
propose a triality of the form:

Algebraic Geometry \raisebox{-.9pt} {3}⃝\raisebox{-.9pt} {1}⃝ Algebraic
TopologyPhysics\raisebox{-.9pt} {2}⃝ Figure 1: Mysterious Triality. The solid
arrow \raisebox{-.9pt} {2}⃝ stands for a relationship between physics and
algebraic topology which is made explicit, while the dotted arrows
\raisebox{-.9pt} {1}⃝ and \raisebox{-.9pt} {3}⃝ denote still mysterious
correspondences based on combinatorial coincidences. It is enough to solidify
one of the dotted arrows to remove the mystery veil off the face of Triality.

The remarkable fact is that the duality pictured by arrow \raisebox{-.9pt}
{2}⃝ is explicit. Let us explain the main idea behind it. Rational homotopy
theory associates to a topological space a certain algebra in two ways: the
Quillen model, which is a differential graded (dg) Lie algebra, and the
Sullivan model, which is a dg-commutative algebra. Roughly speaking, the
Sullivan model is a rational homotopy model based on cohomology, while the
Quillen model is based on homotopy, so they are dual in some sense; see
[Ta83][Maj00][FHT01].

In our context, the Sullivan minimal model of the 4-sphere $S^{4}$ captures
the dynamics of the fields in M-theory, as proposed in [Sa13], and developed
further under the name Hypothesis H in [FSS17][FSS19b][FSS19c][GS21] (and
applied in [Ro20]). Particularly, the generators of the Sullivan minimal model

$M(S^{4})=\mathbb{R}\big{[}g_{4},g_{7}\;|\;dg_{4}=0,dg_{7}=-\tfrac{1}{2}g_{4}^{2}\big{]}$

correspond to the basic supergravity fields $G_{4}$ and $G_{7}=*G_{4}$,
expressed as differential forms on 11d spacetime, whereas the differential of
$M(S^{4})$ corresponds to the equations of motion (EOMs):

$dG_{4}=0,\qquad dG_{7}=-\tfrac{1}{2}G_{4}\wedge G_{4}=0\;.$ (2)

Moreover, the algebra of gauge transformations for these fields is captured by
the Quillen model (see Example 2.2). A gentle introduction to the relationship
between M-theory and rational homotopy theory is given in [FSS19a].

Furthermore, we find a striking match between the equations for the Sullivan
minimal model of the iterated cyclic loop space (_cyclification_)
${\mathcal{L}_{c}^{k}S^{4}}$ of the four-sphere and the EOMs of $k$-fold
circle reduction of M-theory; see Section 2 (specifically, Section 2.1,
Examples 2.5 and 2.6, and Section 2.6). This is deeply rooted in the fact that
writing out the EOMs for the reduction of supergravity on a circle $S^{1}$ is
akin to the process of working out the Sullivan minimal model of the cyclic
loop space; see (13) and (16), cf. adjunction (9).

We may formulate the above more precisely as the main mathematical physics
result of the paper.

######  Theorem 1.1 (Supergravity dynamics from rational homotopy theory).

The Sullivan minimal model $M({\cal L}_{c}^{k}S^{4})$ determines the duality-
symmetric equations of motion of $(11-k)$-dimensional supergravity descending
from 11-dimensional supergravity section 2.5, with the gauge algebra structure
determined by the corresponding Quillen model (Examples 2.2, 2.3). There is
also a minimal Sullivan algebra corresponding to type IIB, with T-duality
between type II theories (Prop. 2.12) observed at the level of Sullivan
minimal models.

This matching also implies the following general philosophy (extending the
$k=1$ case in [FSS18a][FSS18b]):

> Any feature of or statement about the Sullivan minimal model of an iterated
> cyclic loop space $\mathcal{L}_{c}^{k}S^{4}$ (or the rational homotopy type
> thereof) may be translated into a feature of or statement about the
> compactification of M-theory on the $k$-torus.

Examples of such derive from the mathematical core of the paper: we find that
toroidal symmetries of the rational homotopy type of
$\mathcal{L}_{c}^{k}S^{4}$ lead naturally to the root system $E_{k}$, for each
$k\geq 0$. Combined with the theorem above, this explains the appearance of
the $E_{k}$ root system in toroidal compactifications of M-theory and the
connection of the Weyl group $W(E_{k})$ to U-duality. The split real torus
action on $M(\mathcal{L}_{c}^{k}S^{4})$ translates into trombone and rescaling
symmetries of $(11-k)$-dimensional supergravity (see section 3.3), the 27
exceptional vectors in the root system $E_{6}$ translate into a collection of
27 distinguished fields in 5d spacetime (see section 4.6). The luxury of the
principle above was not available within the duality between del Pezzo
surfaces and torodial compactifications of M-theory, as there was only a
collection of surprising coincidences, which were not based on an explicit
relation. The very lack of an explicit relation is the essence of the mystery
behind the duality proposed by [INV02]. This is the dotted arrow
\raisebox{-.9pt} {1}⃝ in Figure 1.

The associated series of mathematical results is collected in the following
metatheorem.

######  Theorem 1.2 (Metatheorem).

  1. (a)

The maximal $\mathbb{R}$-split torus of the real algebraic group
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ for $k\geq 0$ is a
$(k+1)$-dimensional torus $T^{k+1}$ canonically isomorphic to
$\mathbb{G}_{m}^{k+1}$ over $\mathbb{R}$ (Cor. 3.6).

  2. (b)

The action of the maximal split torus $\mathbb{G}_{m}^{k+1}$ on
$M(\mathcal{L}^{k}_{c}S^{4})$ may be lifted to an action on the space
$\mathcal{L}_{c}^{k}S^{4}$ in the rational homotopy category. In this way, the
last factor $\mathbb{G}_{m}$ of $\mathbb{G}_{m}^{k+1}$ acts via self-maps of
the target $S^{4}$ and the first $k$ factors act via self-maps of the source
$S^{1}$s (Props. 3.12 and 3.13).

  3. (c)

For type IIB, a maximal split torus $T^{B}=\mathbb{G}_{m}^{2}$ is identified
explicitly (Prop. 3.14).

  4. (d)

The $(k+1)$-dimensional real abelian Lie algebra
$\mathfrak{h}_{k}=\operatorname{Lie}(T^{k+1})\subseteq\operatorname{Der}M(\mathcal{L}_{c}^{k}S^{4})$,
which plays the role of a Cartan subalgebra, of the maximal $\mathbb{R}$-split
torus $T=T^{k+1}$ of the algebraic group
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ has an explicit canonical
basis. So does the linear dual $\mathfrak{h}_{k}^{*}$, which plays the role of
a weight space (Thm. 4.1).

  5. (e)

There is a unique element of the Lie algebra $\mathfrak{h}_{k}$, which acts on
the Quillen minimal model $Q(\mathcal{L}_{c}^{k}S^{4})$ as the degree operator
(Thm. 4.4).

  6. (f)

For each $k$, $0\leq k\leq 8$, the above bases (d) and element (e) give rise
to the exceptional root data $E_{k}$. This data, extracted from cyclic loop
spaces $M(\mathcal{L}^{k}_{c}S^{4})$, replicates the root data determined by
del Pezzo surfaces $\mathbb{B}_{k}$ (Thm. 4.6). The construction of the root
data of Theorem 4.6 extends to $k\geq 9$ (Remark 4.7).

  7. (g)

For type IIB, the exceptional root data from the rational homotopy model for
type IIB replicates the data determined by the del Pezzo surface
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$ and produces the root
system $A_{1}$ (Prop. 4.9).

  8. (h)

$27$ Lines via rational homotopy of $6$-fold cyclic loop space: In the weight
decomposition

$\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})=\bigoplus_{\alpha\in\mathbf{P}(\mathfrak{h}_{6})}\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})_{\alpha}$

corresponding to the $7$-torus action on the Quillen minimal model
$Q(\mathcal{L}_{c}^{6}S^{4})=\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})[1]$,
the $27$ exceptional vectors $\alpha_{i}\in\mathbf{P}(\mathfrak{h}_{6})$,
$i=1,\dots,27$, single out precisely the second real homotopy group
$\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$:

$\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})=\bigoplus_{i=1}^{27}\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})_{\alpha_{i}}\;.$

Moreover,

$\dim\pi_{\bullet}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})_{\alpha_{i}}=1$

for each $i=1,\dots,27$, which means there are $27$ canonically defined,
linearly independent lines in the $\mathbb{R}$-vector space
$\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})$ and
$\dim\pi_{2}^{\mathbb{R}}(\mathcal{L}_{c}^{6}S^{4})=27$ (Thm. 4.10).

Thus, our work provides an explicit, conceptual correspondence
\raisebox{-.9pt} {2}⃝ between physics and algebraic topology in the Triality
above and thereby uncovers the mystery of Mysterious Duality, if understood in
a broad sense as a duality between physics and mathematics. The other two
sides of the Triality, see Figure 1, still remain a mystery. Filling out
either of the mysterious sides of the triangle would complete the story and
resolve the Mysterious Duality conjecture of [INV02].

This leads to a new, conjectural duality within mathematics, a duality between
the algebraic geometry of del Pezzo surfaces and the algebraic topology of
cyclic loop spaces of the 4-sphere, formulated as:

######  Conjecture 1.3.

There must be an explicit relation between the series of del Pezzo surfaces
$\mathbb{B}_{k}$, $0\leq k\leq 8$, and
$\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$ on the one hand and
the series of iterated loop spaces $\mathcal{L}^{k}_{c}S^{4}$, $0\leq k\leq
8$, and the topological model $IIB$, see section 2.6, on the other hand. In
particular, blowing up a del Pezzo surface should correspond to taking a
cyclification of an iterated cyclification of $S^{4}$. This relation should
match the $E_{k}$ symmetry patterns occurring in both series, as well as
relate other geometric data, such as relate the volumes of curves on del Pezzo
surfaces with certain metric data on the iterated loop spaces.

##### The $E_{k}$ symmetry patterns.

The dimensional reduction of M-theory on a $k$-torus gives rise to a theory in
$D=11-k$ dimensions with the scalar fields with symmetry pattern [INV02] in
the five columns in Table 1 below, matching the familiar pattern for del Pezzo
surfaces [Ma74], to which we add the 6th column for cyclic loop spaces
(“cyclifications” $\mathcal{L}_{c}^{k}S^{4}$ of $S^{4}$), as well as the 7th
column corresponding torus symmetry, both of which we discover in this paper.
This highlights the interrelations among Lie theory (4th column), algebraic
geometry (5th column), and topology/physics (6th column), as appropriate by
the trichotomy/triality in Figure 1.

$D$ | $k$ | Type of $E_{k}$ | Lie algebra $\mathfrak{g}$ | del Pezzo | Model | Maximal Split Torus
---|---|---|---|---|---|---
11 | 0 | $A_{-1}$ | $\mathfrak{sl}_{0}=\varnothing$ | $\mathbb{C}\mathbb{P}^{2}$ | $S^{4}$ | $\mathbb{G}_{m}$
10 | 1 | $A_{0}$ | $\mathfrak{sl}_{1}=0$ | $\mathbb{B}_{1}$ | $\mathcal{L}_{c}S^{4}$ | $\mathbb{G}_{m}\times\mathbb{G}_{m}$
10 | 1 | $A_{1}$ | $\mathfrak{sl}_{2}$ | $\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$ | $IIB$ | $\mathbb{G}_{m}\times\mathbb{G}_{m}$
9 | 2 | $A_{1}$ | $\mathfrak{sl}_{2}$ | $\mathbb{B}_{2}$ | $\mathcal{L}_{c}^{2}S^{4}$ | $\mathbb{G}_{m}^{2}\times\mathbb{G}_{m}$
8 | 3 | $A_{2}\times A_{1}$ | $\mathfrak{sl}_{3}\oplus\mathfrak{sl}_{2}$ | $\mathbb{B}_{3}$ | $\mathcal{L}_{c}^{3}S^{4}$ | $\mathbb{G}_{m}^{3}\times\mathbb{G}_{m}$
7 | 4 | $A_{4}$ | $\mathfrak{sl}_{5}$ | $\mathbb{B}_{4}$ | $\mathcal{L}_{c}^{4}S^{4}$ | $\mathbb{G}_{m}^{4}\times\mathbb{G}_{m}$
6 | 5 | $D_{5}$ | $\mathfrak{so}_{10}$ | $\mathbb{B}_{5}$ | $\mathcal{L}_{c}^{5}S^{4}$ | $\mathbb{G}_{m}^{5}\times\mathbb{G}_{m}$
5 | 6 | $E_{6}$ | $\mathfrak{e}_{6}$ | $\mathbb{B}_{6}$ | $\mathcal{L}_{c}^{6}S^{4}$ | $\mathbb{G}_{m}^{6}\times\mathbb{G}_{m}$
4 | 7 | $E_{7}$ | $\mathfrak{e}_{7}$ | $\mathbb{B}_{7}$ | $\mathcal{L}_{c}^{7}S^{4}$ | $\mathbb{G}_{m}^{7}\times\mathbb{G}_{m}$
3 | 8 | $E_{8}$ | $\mathfrak{e}_{8}$ | $\mathbb{B}_{8}$ | $\mathcal{L}_{c}^{8}S^{4}$ | $\mathbb{G}_{m}^{8}\times\mathbb{G}_{m}$
Table 1:  The $E_{k}$ pattern in Lie theory, $(0\leq k\leq 8)$ del Pezzo
surfaces, and cyclifications of $S^{4}$.

Our formulation allows us to extend to higher ranks, $k=9$, $10$, and $11$,
corresponding to the infinite-dimensional cases. Indeed, since our discussion
extends beyond the Lie setting to the Kac-Moody setting, we have extensions of
the Triality in Figure 1 and of 1.3 that go beyond the Fano case on the
algebraic side in section 4.7. We also observe that cyclic loop spaces, when
we go beyond $k=8$, undergo a transition analogous to that on the del
Pezzo/root systems/Lie algebra side: the degree of the cyclification
$\mathcal{L}_{c}^{k}S^{4}$ in the sense of (46) ceases to be positive, the
corresponding root system becomes infinite, and the metric on the
$k$-dimensional real vector space holding the root system is no longer
Euclidean; see Remark 4.7. The surface $\mathbb{B}_{9}$ gives rise to a rank-9
parabolic lattice, while $\mathbb{B}_{10}$ and $\mathbb{B}_{11}$ correspond to
rank-10 and 11 hyperbolic lattices.

$D$ | $k$ | Type of $E_{k}$ | Kac-Moody algebra $\mathfrak{g}$ | Non-Fano Surface | Model | Maximal Split Torus
---|---|---|---|---|---|---
2 | 9 | $E_{9}=\widehat{E}_{8}$ | affine $\mathfrak{e}_{9}=\widehat{\mathfrak{e}}_{8}$ | $\mathbb{B}_{9}$ | $\mathcal{L}_{c}^{9}S^{4}$ | $\mathbb{G}_{m}^{9}\times\mathbb{G}_{m}$
1 | 10 | $E_{10}$ | hyperbolic $\mathfrak{e}_{10}$ | $\mathbb{B}_{10}$ | $\mathcal{L}_{c}^{10}S^{4}$ | $\mathbb{G}_{m}^{10}\times\mathbb{G}_{m}$
0 | 11 | $E_{11}$ | Lorentzian $\mathfrak{e}_{11}$ | $\mathbb{B}_{11}$ | $\mathcal{L}_{c}^{11}S^{4}$ | $\mathbb{G}_{m}^{11}\times\mathbb{G}_{m}$
Table 2:  The $E_{k}$ pattern in Kac-Moody theory $(k\geq 9)$, further blowups
of $\mathbb{C}\mathbb{P}^{2}$ and cyclifications of $S^{4}$.

Note that our approach can be made quite general by looking at other
topological spaces than $S^{4}$. This would then lose the connection to
M-theory, but the rational homotopy theory aspects would still be interesting
to explore. For instance, other spheres would have the same toric symmetries
of the rational homotopy type and produce the same root data.

##### Acknowledgments.

We are grateful to Alexey Bondal, Igor Dolgachev, Amer Iqbal, Mikhail
Kapranov, and Urs Schreiber for helpful discussions. We are also grateful for
the suggestion of the referee and editor to split the paper into a more
mathematical part, which is what this paper is, and a more physical follow-up
part [SV22]. We appreciate that the anonymous referee practically worked with
us on weeding out errors and restructuring the exposition to improve the
paper. The first author thanks the University of Minnesota, the Aspen Center
for Physics, and the Park City Mathematics Institute (IAS) for hospitality
during the work on this project, and acknowledges the support by Tamkeen under
the NYU Abu Dhabi Research Institute grant CG008. The second author thanks NYU
Abu Dhabi and Kavli IPMU for creating remarkable opportunities to initiate and
work on this project. His work was also supported by World Premier
International Research Center Initiative (WPI), MEXT, Japan, and a
Collaboration Grant from the Simons Foundation (#585720).

##  2 The 4-sphere and its cyclifications as the universal targets for
M-theory and its reductions

We will provide our main topological setting for the rest of the paper using
(rational) homotopy theory, along the lines outlined in the Introduction.

###  2.1 The Sullivan minimal model

Here we replace the notion of a _rational Sullivan minimal model_ of a
topological space with a less common notion of a _real Sullivan minimal model_
, given that real coefficients of physical fields could be a bit more natural
than rational ones (see the discussion in [FSS20]). We will therefore assume
that our algebraic models are defined over the reals $\mathbb{R}$ (see
[BS95][GM13]).

To every path-connected, _nilpotent_ (the fundamental group is nilpotent and
acts nilpotently on higher homotopy groups) topological space $Z$, rational
homotopy theory associates a minimal Sullivan algebra, called the _Sullivan
minimal model_ $M(Z)$ of $Z$, a differential graded commutative
$\mathbb{R}$-algebra (DGCA) $M=M(Z)=(S(V),d)$ of a certain type, called a
minimal Sullivan algebra, after [Su77]; see the definition below and standard
rational homotopy theory references, e.g., [FHT01][FOT08][GM13].

Here and henceforth, we will be restricting our attention to spaces which have
finite-dimensional real homology groups and, respectively, minimal Sullivan
algebras having _strong finite type_ , i.e., based on a graded vector space
$V$ of finite total dimension, $\dim V<\infty$. The spaces of interest below,
namely the four-sphere $S^{4}$ and its cyclifications, satisfy this condition.

######  Definition 2.1 (Sullivan minimal models).

  1. (i)

A _Sullivan algebra_ is a differential graded commutative $\mathbb{R}$-algebra
(DGCA) $(M,d)$ based on the free graded commutative algebra $M=S(V)$ on a
graded real vector space $V=\bigoplus_{n>0}V^{n}$ with a differential $d:M\to
M$ of degree 1, $d^{2}=0$, satisfying the following nilpotence condition,
known as the _Sullivan condition_ : $V$ is the union of an increasing series
of graded subspaces

$V(0)\subseteq V(1)\subseteq\dots$ (3)

such that $d(V(0))=0$ and $d(V(k))\subseteq S(V(k-1))$ for $k\geq 1$.

  2. (ii)

A _Sullivan model_ of a DGCA $A$ is a Sullivan algebra $M$ with a _quasi-
isomorphism_ $M\to A$, i.e., a homomorphism which induces an isomorphism on
cohomology.

  3. (iii)

We say that a Sullivan algebra is _minimal_ if

$d(M)\subseteq M^{+}\cdot M^{+},$

where $M^{+}:=\bigoplus_{n>0}M^{n}=S^{\geq 1}(V)$.

A minimal Sullivan model of a _connected_ (i.e., $A^{n}=0$ for $n<0$ and
$A^{0}=\mathbb{R}$) DGCA $A$ exists and is unique up to isomorphism.

To every topological space $Z$, Sullivan’s construction in rational/real
homotopy theory associates a DGCA $A_{\operatorname{PL}}(Z)$, called the
algebra of real polynomial differential forms on $Z$.111We clarify that, by a
little abuse of terminology, we will often say “rational” even when one should
more correctly say “real.” However, it will always be clear from the context
what field of coefficients we is working with. If $Z$ is a smooth manifold,
one can take the de Rham algebra of smooth differential forms on $Z$ instead.
(This example is the main reason why we prefer real homotopy theory to
rational one). If $Z$ is path-connected, nilpotent, and has finite-dimensional
real homology groups, then the DGCA $A_{\operatorname{PL}}(Z)$ gives rise to a
minimal Sullivan model $S(V)$, defined up to isomorphism, called the _$($
real$)$ Sullivan minimal model_ of $Z$.

###  2.2 $S^{4}$ as the universal target for M-theory via Hypothesis H

We adopt the perspective proposed in [Sa13] of viewing the 4-sphere $S^{4}$ as
the universal space of form fields in M-theory. The significance of this is
that $S^{4}$ encodes, entirely in its topology, the field $G_{4}$ and its dual
$G_{7}$ as well as their dynamics. This space is viewed as a universal space
in the sense that these field configurations are given at the homotopy level
by real homotopy classes of maps from spacetime $Y^{11}$ to $S^{4}$, and
whenever geometry is included, one would need all maps; see
[FSS15][FSS17][GS21] (but here we will concentrate on topology).

In 11-dimensional supergravity, which is the low-energy limit of M-theory, the
equations of motion (EOMs) are [CJS78]

$dG_{4}=0,\qquad d*G_{4}+\tfrac{1}{2}G_{4}\wedge G_{4}=0\;.$

When combined with the self-duality condition

$G_{7}:=*G_{4}\;,$ (4)

these may be rewritten as (2) by using $G_{4}$ and $G_{7}$, where the fields
$G_{4}$ and $G_{7}$ are represented by differential forms of degree 4 and 7,
respectively, on the 11-dimensional spacetime $Y^{11}$ of M-theory, and $*$
denotes the Hodge star operator, which captures the dependence on the metric
on $Y^{11}$. Note that locally we may write

$G_{4}=dC_{3},\qquad G_{7}=dC_{6}-\tfrac{1}{2}C_{3}\wedge G_{4}\;$

for some differential forms $C_{3}$ and $C_{6}$, viewed as the corresponding
potentials. However, we will use the duality-symmetric (doubled field)
formulation, where $G_{4}$ and $G_{7}$ are treated independently [BBS98] (see
also [MS03] [Sa06][ST17][BMSS18] for more global treatments). This will also
suppress any explicit dependence on the metric, suitable for our topological
perspective, which may be regarded as describing the topological background of
the full story. To get the full picture at the level of fields, one simply
adds metric data to spacetime and imposes the duality relation $*G_{4}=G_{7}$.
We have also found a way to add metric data to the universal real homotopy
model of $S^{4}$ (and its cyclifications) via introducing moduli parameters;
see the end of Section 4.4.

The topological aspects of the M-theory dynamics at the real homotopy level
are captured by the real homotopy theory description given by the Sullivan
minimal model of $S^{4}$, [Sa13], which we denote $M(S^{4})$:222We will use
lowercase letters for universal elements and uppercase letters to denote
spacetime fields.

$\displaystyle M(S^{4})=(\mathbb{R}[g_{4},g_{7}],d)\,,$ $\displaystyle
dg_{4}=0,\qquad dg_{7}=-\tfrac{1}{2}g_{4}^{2}\,,$ (5)

where the degree of each of the generators $g_{4}$ and $g_{7}$ is given by the
corresponding subscript: $\lvert g_{4}\rvert=4$, $\lvert g_{7}\rvert=7$. Here
we are choosing to include the factor of $-\tfrac{1}{2}$ in the expressions of
the model, as opposed to being absorbed by the generators (see [FSS17, Ex.
3.3]).

Comparing (2) with (5), we see that there exists a differential graded (dg)
algebra homomorphism

$\displaystyle M(S^{4})$
$\displaystyle\longrightarrow(\Omega^{\bullet}(Y),d),$ (6) $\displaystyle
g_{4}$ $\displaystyle\longmapsto G_{4},$ $\displaystyle g_{7}$
$\displaystyle\longmapsto G_{7},$

where $(\Omega^{\bullet}(Y),d)$ is the de Rham algebra of the 11-dimensional
spacetime $Y^{11}$. The de Rham algebra is, in fact, a real homotopy model of
the manifold $Y^{11}$, and this model could be different from the Sullivan
minimal model. Rational (or, actually, real [FSS20]) homotopy theory provides
a canonical continuous map

$Y\longrightarrow S^{4}_{\mathbb{R}},$ (7)

where $S^{4}_{\mathbb{R}}$ is the _rationalization over $\mathbb{R}$_ of
$M(S^{4})$, a certain universal topological space whose Sullivan minimal model
$M(S^{4}_{\mathbb{R}})$ is isomorphic $M(S^{4})$, such that the pullback map
from the Sullivan minimal model of $S^{4}_{\mathbb{R}}$ to the de Rham model
of $Y^{11}$ is given by (6). The space $S^{4}_{\mathbb{R}}$ has the same real
homotopy type as $S^{4}$ via a map $S^{4}\to S^{4}_{\mathbb{R}}$, but
$H_{n}(S^{4}_{\mathbb{R}};\mathbb{Z})$ is a real vector space for each $n\geq
1$. The rationalization may be obtained via a simplicial construction from the
dg-commutative algebra $M(S^{4})$; see [BS95][FSS20].

In [Sa13], it is suggested that there is actually a continuous map to the
honest-to-goodness 4-sphere $S^{4}$,

$Y\longrightarrow S^{4},$ (8)

that induces the homomorphism (6). Indeed, a comparison of this target to its
linearization, that is to say, the Eilenberg-MacLane classifying space
$K(\mathbb{Z},4)$, which encodes the C-field as captured by a degree 4 class,
with the 4-sphere is presented in [GS21] through a Postnikov tower analysis.
The nonabelian nature and the shift by Pontrjagin classes in the quantization
condition of the C-field are studied in [FSS19b].

The generator $g_{4}$ of $M(S^{4})$ may be realized as a volume form on the
sphere $S^{4}$, but in the de Rham algebra of $S^{4}$, there is no room for
$g_{7}$. One may desire to have a model of the universal target space for
M-theory, so that not only $g_{4}$, but also $g_{7}$ may be realized as a
differential form. One such model was suggested to us by M. Kapranov: it is
the complement
$\mathbb{H}\mathbb{P}^{\infty}\mathbin{\vbox{\hbox{$\scriptscriptstyle\mathchoice{\hbox
to0.0pt{$\displaystyle{\setminus}$\hss}}{\hbox
to0.0pt{$\textstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle{\setminus}$\hss}}{\hskip
0.2pt\setminus}$}}}\mathbb{H}\mathbb{P}^{\infty-2}=\bigcup_{N}\mathbb{H}\mathbb{P}^{N}\mathbin{\vbox{\hbox{$\scriptscriptstyle\mathchoice{\hbox
to0.0pt{$\displaystyle{\setminus}$\hss}}{\hbox
to0.0pt{$\textstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle{\setminus}$\hss}}{\hskip
0.2pt\setminus}$}}}\mathbb{H}\mathbb{P}^{N-2}$ of a codimension 2 plane
$\mathbb{H}\mathbb{P}^{\infty-2}$ in the quaternionic projective space
$\mathbb{H}\mathbb{P}^{\infty}$. The infinite-dimensional manifold
$\mathbb{H}\mathbb{P}^{\infty}\mathbin{\vbox{\hbox{$\scriptscriptstyle\mathchoice{\hbox
to0.0pt{$\displaystyle{\setminus}$\hss}}{\hbox
to0.0pt{$\textstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle{\setminus}$\hss}}{\hskip
0.2pt\setminus}$}}}\mathbb{H}\mathbb{P}^{\infty-2}$ is homotopy equivalent to
$\mathbb{H}\mathbb{P}^{1}\cong S^{4}$ and may prove to be useful in 11d
supergravity.

###  2.3 The Quillen model and M-theory gauge structure

The Sullivan minimal model $M(Z)$ (see section 2.1) of each of the spaces $Z$
we are considering has quadratic differential. This model is actually the
symmetric algebra on a space of generators having a certain homotopy-theoretic
meaning:

$M(Z)=(S(Q(Z)[-1]^{*}),d),$

where $Q(Z)$ is the _Quillen minimal model 333Here we abandon the traditional
notion of minimality, based on a free graded Lie algebra, in favor of a more
modern one: $Q(Z)$ is an $L_{\infty}$-algebra with the zero differential, see
[BFMT20]. The differential $d$ on $M(Z)$ may be identified as the Chevalley-
Eilenberg differential, but this is beside the point here. of_ $Z$ [Qu69],
which in our quadratic-differential case is given by the _graded Lie algebra
of real homotopy groups_

$Q(Z):=\pi_{\bullet}(Z)\otimes\mathbb{R}[1]$

of $Z$. Let us explain what this means. Consider the _real homotopy groups of
$Z$_, as a graded vector space over $\mathbb{R}$

$\pi_{\bullet}(Z)\otimes\mathbb{R}:=\bigoplus_{i\in\mathbb{Z}}\pi_{i}(Z)\otimes\mathbb{R},$

where

$\pi_{i}(Z)\otimes\mathbb{R}:=\left\\{\\!\\!\\!\\!\begin{array}[]{cl}0&\quad\text{for
$i\leq 0$},\\\
\big{(}\pi_{1}(Z)/[\pi_{1}(Z),\pi_{1}(Z)]\big{)}\otimes_{\mathbb{Z}}\mathbb{R}&\quad\text{for
$i=1$},\\\ \pi_{i}(z)\otimes_{\mathbb{Z}}\mathbb{R}&\quad\text{for $i\geq
2$}.\end{array}\right.$

If we shift the grading down by one and consider the natural isomorphism

$\pi_{\bullet}(Z)\otimes\mathbb{R}[1]\xrightarrow{\;\;\sim\;\;}\pi_{\bullet}(\Omega
Z)\otimes\mathbb{R}\;,$

where $\Omega Z:=\operatorname{Map}_{*}(S^{1},Z)$ is the based loop space of
$Z$, we will get a natural graded Lie-algebra structure with respect to the
_Samelson product_. This is induced on $\pi_{\bullet}(\Omega
Z)\otimes\mathbb{R}$ by the commutator map $[-,-]:\Omega Z\times\Omega
Z\to\Omega Z$ of the concatenation $\Omega Z\times\Omega Z\to\Omega Z$ of
based loops in $Z$:

$S^{a+b}=S^{a}\times S^{b}/S^{a}\vee
S^{b}\xrightarrow{\;\gamma_{\,1}\times\gamma_{\,2}\;}\Omega Z\times\Omega
Z\xrightarrow{\;[-,-]\;}\Omega Z\;,$

for $\gamma_{\,1}\in\pi_{a}(\Omega Z)$ and $\gamma_{\,2}\in\pi_{b}(\Omega Z)$,
where $S^{a}\vee S^{b}$ is the wedge sum of the pointed spaces $S^{a}$ and
$S^{b}$.

An equivalent, more standard description of the corresponding Lie bracket on
$\pi_{\bullet}(Z)[1]\otimes\mathbb{R}\cong\pi_{\bullet}(\Omega
Z)\otimes\mathbb{R}$, is known as _Whitehead product_ , which does not appeal
to based loop spaces; see, e.g., [FHT01]. We will not need it here (see
[FSS19c][FSS20, §3.2] for detailed discussion in this context).

Now let us return to the case $Z=\mathcal{L}_{c}^{k}S^{4}$. We will start with
$k=0$. The following is, in a sense, dual to the description of the fields via
the Sullivan minimal model of $S^{4}$ in (5).

######  Example 2.2 (M-theory gauge algebra via the Quillen model of $S^{4}$).

The Quillen model of $S^{4}$ is just the graded Lie algebra on two generators

$\displaystyle Q(S^{4})=\mathbb{R}e_{3}\oplus\mathbb{R}e_{6},$
$\displaystyle\lvert e_{3}\rvert=3,\quad\lvert e_{6}\rvert=6,$
$\displaystyle[e_{3},e_{3}]=e_{6},\quad[e_{3},e_{6}]=0,\quad[e_{6},e_{6}]=0,$

which is actually the free graded Lie algebra over $\mathbb{R}$ generated by
$e_{3}$. This captures the algebra of gauge transformation of the C-field and
its dual and, hence, also captures the Dirac quantization of the M-branes (see
[CJLP98b][LLPS99][KS03][Sa10]).

We will revisit the reduction of this algebra, corresponding to cyclic loop
spaces $\mathcal{L}_{c}^{k}S^{4}$ for $k>0$ in Example 2.3.

###  2.4 Dimensional reduction of M-theory on tori and iterated cyclifications
of $S^{4}$

##### Type IIA and looping.

The reduction from M-theory in 11 dimensions to type IIA string theory in 10
dimensions is captured by looping. Such a process has been utilized
topologically at the level of bundles leading to loop bundles in 10 dimensions
starting from an $E_{8}$ gauge bundle (capturing the purely topological
aspects of $G_{4}$) in 11 dimensions [MS03]. In our current case, we do this
at the level of universal target spaces, taking into account the rotation of
the circle. This leads to the concept of a cyclic loop space or cyclification,
advocated in [FSS18b] (see (11) below)

$\mathcal{L}_{c}S^{4}:=\mathcal{L}S^{4}/\\!\\!/S^{1}.$

##### Why cyclic loop space.

Let us provide mathematical justification of the appearance of the cyclic loop
space. The 10-dimensional spacetime of type IIA string theory is actually the
“11-dimensional spacetime $Y^{11}$ wrapped on $S^{1}$”, which translates to
the mathematical language simply as the 10-dimensional quotient $Y^{11}/S^{1}$
by a free action of $S^{1}$. Now, there is an adjunction

$\operatorname{Mor}_{/BS^{1}}(Y^{11}/S^{1},\mathcal{L}_{c}Z)\xrightarrow{\;\;\sim\;\;}\operatorname{Mor}(Y^{11},Z),$
(9)

where $Y^{11}$ is a space with a free action of $S^{1}$, $Z$ is another
topological space, the left-hand side is the set of morphisms in the category
$/BS^{1}$ of spaces over $BS^{1}$ (equivalent to the category of principal
$S^{1}$-bundles, such as $Y\to Y/S^{1}$ and
$\mathcal{L}Z\to\mathcal{L}_{c}Z$), and the right-hand side is the set of
continuous maps $Y\to Z$; see [FSS18b][BMSS18, Theorem 2.44]. This adjunction
produces a map

$Y/S^{1}\longrightarrow\mathcal{L}_{c}S^{4}$ (10)

from the map (8) (or using the corresponding rationalization
$S^{4}_{\mathbb{R}}$ of $S^{4}$ over $\mathbb{R}$ and the map (7)). If the
action of $S^{1}$ on $Y^{11}$ is not free, one has to replace the naive
quotient $Y^{11}/S^{1}$ by the homotopy quotient $Y^{11}/\\!\\!/S^{1}$ in the
above. Roughly speaking, thinking of $\mathcal{L}_{c}S^{4}$ as the space of
unparameterized (also known as equivariant) free loops in $S^{4}$, a map like
(8) from an $S^{1}$-space $Y^{11}$ will produce a map (10) which assigns to a
point $y\in Y^{11}/S^{1}$ the map that takes the $S^{1}$-orbit in $Y^{11}$
over $y$ to $S^{4}$ by the given map $Y^{11}\to S^{4}$.

We would like to make more precise what we mean by iterated cyclic loop
spaces. Let $\mathcal{L}Z=\operatorname{Map}(S^{1},Z)$ be the _free loop
space_ of a topological space $Z$, which we will assume to be path-connected.
The free loop space admits a natural (right) action of the group $S^{1}$ by
rotating loops

$(f\cdot z)(z^{\prime})=f(zz^{\prime})\,,$

for $f\in\mathcal{L}Z$ and $z,z^{\prime}\in S^{1}$, and we define the _cyclic
loop space_ or _cyclification_ $\mathcal{L}_{c}Z$ to be the _homotopy
quotient_

$\mathcal{L}_{c}Z:=\mathcal{L}Z/\\!\\!/S^{1}.$ (11)

One may construct it using the _Borel construction_ $(\mathcal{L}Z\times
ES^{1})/S^{1}$, where $ES^{1}$ is the universal space for $S^{1}$ bundles, as
the quotient by the diagonal action of $S^{1}$, i.e., the quotient by the
relation

$(f\cdot z,e)\sim(f,z\cdot e)\,,$

for $f\in\mathcal{L}Z$, $e\in ES^{1}$, and $z\in S^{1}$. We will use the
convention that _if $\,Z$ is not simply connected_, when $\mathcal{L}Z$
acquires path components, we _retain only the component of the constant loop_
and in that case use the same notation

$\mathcal{L}Z:=(\mathcal{L}Z)_{0}\;,$

which also makes our cyclic loop spaces _path-connected_.

##### M-theory on $T^{k},k\leq 8$ and higher cyclic loop spaces.

The reduction of the above system on tori $T^{k}=(S^{1})^{k}$ leads to a low-
dimensional system corresponding to duality-symmetric supergravity actions in
these dimensions matching the EOMs of M-theory compactified on a
$k$-dimensional torus $T^{k}$ in an iterative way, as we explain below.

Suppose the 10-dimensional spacetime $X^{10}=Y^{11}/S^{1}$ of type IIA string
theory (or the 10-dimensional spacetime $X^{10}$ of type IIB supergravity)
also has a free action of $S^{1}$. Then applying the adjunction (9) to the map
(10), we get a natural map

$X^{10}/S^{1}=(Y^{11}/S^{1})/S^{1}\longrightarrow\mathcal{L}_{c}^{2}S^{4}.$

This would be the second toroidal compactification of M-theory to a
9-dimensional spacetime. Iterating this process, for each $k$, $0\leq k\leq
11$, we will be getting a map

$\left(\dots((Y^{11}/S^{1})/S^{1})\dots/S^{1}\right)\longrightarrow\mathcal{L}_{c}^{k}S^{4},$

the quotient being by $k$ copies of $S^{1}$, from the $k$th toroidal
compactification of the 11-dimensional M-theory to the $k$-fold cyclification
$\mathcal{L}_{c}^{k}S^{4}$ of the four-sphere.

Hence, for $k\geq 0$, the _iterated cyclic loop space_ $\mathcal{L}_{c}^{k}Z$
is the $k$-fold iteration of the cyclic loop space construction:

$\displaystyle\mathcal{L}_{c}^{0}Z$ $\displaystyle:=Z,$
$\displaystyle\mathcal{L}_{c}^{k}Z$
$\displaystyle:=\mathcal{L}_{c}(\mathcal{L}_{c}^{k-1}Z)\qquad\text{for $k\geq
1$}.$

We will often refer to iterated loop spaces as _cyclifications_. We will be
interested mostly in the iterated cyclic loop spaces
$\mathcal{L}_{c}^{k}S^{4}$ of the 4-sphere $S^{4}$, both for $0\leq k\leq 8$
and for $k\geq 9$. For $k=1$ this is studied extensively in [FSS18a][FSS18b]
in relation to T-duality and the twisted K-theory description of the fields in
type II string theory (cf. section 2.6). The generalization is possible thanks
to the general construction in [BMSS18].

The above matching of the EOMs with the differential on the Sullivan model of
the $k$-fold cyclic loop space $\mathcal{L}_{c}^{k}S^{4}$ is another striking
phenomenon, which we observe in this paper; see section 2.5.

######  Example 2.3 (The brane/reduced M-theory gauge algebra via the Quillen
model of the cyclification of $S^{4}$).

We have seen the description of the M-theory gauge algebra via Quillen model
of $S^{4}$ is just the graded Lie algebra on two generators in Example 2.2.
The gauge algebra of the reduced fields in $11-k$ dimensions will correspond
to the Quillen model of $\mathcal{L}_{c}^{k}S^{4}$. We will not work out the
details, as it is clear that matching the fields and EOMs of reduced M-theory
with the generators and their differentials in the Sullivan minimal model of
the cyclification of $S^{4}$ implies similar matching between the gauge
algebra and the Quillen model. Thus, the gauge algebra formulas in
[CJLP98b][LLPS99] should be reproduced just by looking at the Quillen model of
$\mathcal{L}_{c}^{k}S^{4}$.

###  2.5 The Sullivan minimal model of the cyclic loop space

Our goal here is to describe the Sullivan minimal model of the cyclic loop
space $M(\mathcal{L}_{c}Z)$ in terms of the Sullivan minimal model $M(Z)=S(V)$
of the space $Z$. Along the way, we also provide a duality-symmetric reduction
of fields in M-theory on tori, extending and organizing partial/local results
in the supergravity and M-theory literature.

Suppose $Z$ is a path-connected, nilpotent space with finite-dimensional real
homotopy groups. With our convention in section 2.4, we claim the following:

######  Proposition 2.4 (Basic properties of the cyclic loop space).

  1. (i)

$\mathcal{L}_{c}Z$ is also path-connected, nilpotent, and has finite-
dimensional rational homotopy groups.

  2. (ii)

The Sullivan minimal model $M(\mathcal{L}_{c}Z)$ of $\mathcal{L}_{c}Z$ is
given by the extension (13) below.

###### Proof.

(i) Indeed, the path-connected free loop space $\mathcal{L}Z$ sits in a fiber
sequence

$\Omega Z\to\mathcal{L}Z\xrightarrow{\;{\rm ev}_{*}\;}Z,$ (12)

where ${\rm ev}_{*}$ is evaluation at the basepoint $*\in S^{1}$ and $\Omega
Z=\operatorname{Map}_{*}(S^{1},Z)$ is the _based loop space_. The constant-
loop section of ${\rm ev}_{*}$ splits the homotopy groups:
$\pi_{i}(\mathcal{L}Z)=\pi_{i}(Z)\oplus\pi_{i}(\Omega
Z)=\pi_{i}(Z)\oplus\pi_{i+1}(Z)$ for $i\geq 2$ and
$\pi_{1}(\mathcal{L}Z)=\pi_{1}(\Omega
Z)\rtimes\pi_{1}(Z)=\pi_{2}(Z)\rtimes\pi_{1}(Z)$. The action of
$\pi_{1}(\Omega Z)$ on $\pi_{i}(\mathcal{L}Z)$ for $i\geq 2$ is trivial, while
the action of $\pi_{1}(Z)$ on $\pi_{i}(\mathcal{L}Z)$ for $i\geq 2$ is the sum
of actions on $\pi_{i}(Z)$ and $\pi_{i+1}(Z)$. Therefore, the action of
$\pi_{1}(\mathcal{L}Z)$ on $\pi_{i}(\mathcal{L}Z)$, $i\geq 2$, is nilpotent.
The group $\pi_{1}(\mathcal{L}Z)$ itself is nilpotent as a semidirect product
of a nilpotent group acting nilpotently on an abelian group. Indeed, the
action of $\pi_{1}(Z)$ on $\pi_{1}(\Omega Z)$ defining the semidirect product
is compatible with the standard action of $\pi_{1}(Z)$ on $\pi_{2}(Z)$, which
is assumed to be nilpotent, see , e.g., [AFO17]. As concerns the homotopy
groups of the cyclic loop space $\mathcal{L}_{c}Z$, it just adds a copy of
$\mathbb{Z}$ to $\pi_{2}(\mathcal{L}Z)$, on which
$\pi_{1}(\mathcal{L}_{c}Z)=\pi_{1}(\mathcal{L}Z)$ acts trivially, as one can
see from the fiber sequence

$S^{1}\to\mathcal{L}Z\times ES^{1}\to\mathcal{L}_{c}Z,$

in which the inclusion of the fiber $S^{1}$ over the constant loop factors
through the contractible space $ES^{1}$.

(ii) Vigué-Poirrier and Burghelea [VPB85] show444Vigué-Poirrier and Burghelea
assume that $Z$ is simply connected, but their argument applies to the more
general nilpotent case verbatim, by taking $V[1]$ to be the truncated
desuspension (14) of $V$, given that Halperin’s theorem on fibrations [Hal83,
Theorem 20.3] is done in the nilpotent case. that the Sullivan minimal model
$M(\mathcal{L}_{c}Z)$ is isomorphic to $(S(V\oplus
V[1]\oplus\mathbb{R}w),d_{c})$, which is an _extension_ (in the sense of
Halperin [Hal83], also known as a _$\Lambda$ -extension_)

$(\mathbb{R}[w],0)\longrightarrow(S(V\oplus
V[1]\oplus\mathbb{R}w),d_{c})\longrightarrow(S(V\oplus V[1]),d_{f})$ (13)

corresponding to the homotopy fiber sequence

$\mathcal{L}Z\to\mathcal{L}_{c}Z\to BS^{1}.$

Here $V[1]=\bigoplus_{n>0}V[1]^{n}$ is the _truncated desuspension_ , the
graded vector space with components

$V[1]^{n}:=\begin{cases}V^{n+1}&\text{for $n>0$},\\\ 0&\text{for $n\leq
0$}.\end{cases}$ (14)

The truncation affects only the non-simply connected case, when $V^{1}\neq 0$,
as it truncates only this graded component. The differentials $d_{f}$ and
$d_{c}$ may be described as follows. Let

$\displaystyle s:V$ $\displaystyle\longrightarrow V[1],$ $\displaystyle v$
$\displaystyle\longmapsto\begin{cases}v&\text{for $v\in V^{n}$, $n>1$},\\\
0&\text{for $v\in V^{1}$},\end{cases}$

be the natural map, a surjection of degree $-1$. Then, for all $v\in V$,

$\displaystyle d_{f}v$ $\displaystyle:=dv,$ $\displaystyle d_{f}sv$
$\displaystyle:=-sdv,$ (15)

where $s$ is extended to a unique degree-$(-1)$ derivation of $S(V\oplus
V[1])$ such that $s(u)=0$ for all $u\in V[1]$. Since this derivation $s$ is a
version of a graded polynomial de Rham differential, we have $s^{2}=0$. Then,
for all $v\in V$, we define

$\displaystyle d_{c}v$ $\displaystyle:=dv+sv\cdot w,$ $\displaystyle d_{c}sv$
$\displaystyle:=-sdv,$ (16) $\displaystyle d_{c}w$ $\displaystyle:=0.\qed$

Similarly, the Sullivan minimal model of the free loop space $\mathcal{L}Z$ is
a Halperin extension

$(S(V),d)\longrightarrow(S(V\oplus V[1]),d_{f})\longrightarrow(S(V[1]),0)$
(17)

corresponding to the fiber sequence (12).

######  Example 2.5 (Reduction on a circle and the cyclification
$\mathcal{L}_{c}S^{4}$).

From $M(S^{4})$ in (5), the Sullivan minimal model $M(\mathcal{L}_{c}S^{4})$
of the cyclic loop space of $S^{4}$ can be presented as (cf. [FSS17, Ex.
3.3][FSS18b, Ex. 2.7])

$\displaystyle
M(\mathcal{L}_{c}S^{4})=(\mathbb{R}[g_{4},g_{7},sg_{4},sg_{7},w],d)\;,$
$\displaystyle dg_{4}=sg_{4}\cdot w,\qquad
dg_{7}=-\tfrac{1}{2}g_{4}^{2}+sg_{7}\cdot w\;,$ $\displaystyle dsg_{4}=0,\quad
dsg_{7}=sg_{4}\cdot g_{4},\quad dw=0\;.$

As in [FSS18b, Ex. 2.7], making the change of variables $f_{2}=w$,
$h_{3}=sg_{4}$, $f_{4}=g_{4}$, $f_{6}=sg_{7}$, and $h_{7}=g_{7}$, this can be
rewritten as

$M(\mathcal{L}_{c}S^{4})=\big{(}\mathbb{R}[f_{2},f_{4},f_{6},h_{3},h_{7}],\
df_{2}=0,\,dh_{3}=0,\,df_{4}=h_{3}f_{2},\,df_{6}=h_{3}f_{4},\,dh_{7}=-\tfrac{1}{2}{f_{4}}^{2}+f_{2}f_{6}\big{)}.$

Being defined for $\mathcal{L}_{c}S^{4}$, these equations are universal, and
we obtain the corresponding ones in spacetime by pulling back, giving the
datum of a closed 3-form $H_{3}$ and of 2-, 4- and 6-forms $F_{2}$, $F_{4}$
and $F_{6}$ on $X$ such that

$dF_{2}=0;\qquad dF_{4}=H_{3}\wedge F_{2};\qquad dF_{6}=H_{3}\wedge F_{4},$

together with a 7-form $H_{7}$, which is a potential for a certain 8-form:

$dH_{7}=-\tfrac{1}{2}F_{4}\wedge F_{4}+F_{2}\wedge F_{6};$

cf. [CW84][HN85][GP84] for the classical theory and [BNS04] for the duality-
symmetric formulations of type IIA $D=10$ supergravity. Hence, if $Y\to X$ is
rationally a principal $S^{1}$-bundle, then a map $Y\to S^{4}$ in the rational
category will induce, by the hofiber/cyclification adjunction (9), such a set
of differential forms on the base $X$. As explained in [FSS17][FSS18b], the
above equations for the differentials of the $F_{2n}$’s are precisely (a
subset of) the equations for a $H_{3}$-twisted cocycle $\sum_{n}F_{2n}u^{n}$
in $(\Omega^{\bullet}(X)(\\!(u)\\!),d_{H_{3}})$ with $F_{0}=0$, corresponding
to the EOMs and Bianchi identities captured by twisted (rational) even
K-theory, as appropriate for type IIA string theory. We consider the type IIA
and IIB perspectives in section 2.6.

Proposition 2.4 can be iterated to give the same result for
$\mathcal{L}_{c}^{2}Z$, so that we can now continue with further dimensional
reduction. 555 We will have multiple circle fiber directions and corresponding
labels on the contractions $s_{i}$ and the classes of the circles $w_{i}$. We
realize that the notation is not fully in parallel with the convention of
using such labels to indicate the degree, but choosing another notation such
as $s_{(i)}$ might overload the expressions when multiple such occur below. We
hope the distinction will be clear from the context.

######  Example 2.6 (Reduction on a 2-torus and the double cyclification
$\mathcal{L}_{c}^{2}S^{4}$).

For the Sullivan minimal model $M(\mathcal{L}^{2}_{c}S^{4})$ of the double
cyclification of the sphere $S^{4}$, we have

$\displaystyle M(\mathcal{L}^{2}_{c}S^{4})$
$\displaystyle=\left(\mathbb{R}[g_{4},g_{7},s_{1}g_{4},s_{1}g_{7},w_{1},s_{2}g_{4},s_{2}g_{7},s_{2}s_{1}g_{4},s_{2}s_{1}g_{7},s_{2}w_{1},w_{2}],d\right),$
$\displaystyle dg_{4}$ $\displaystyle=s_{1}g_{4}\cdot w_{1}+s_{2}g_{4}\cdot
w_{2},\qquad dg_{7}=-\tfrac{1}{2}g_{4}^{2}+s_{1}g_{7}\cdot
w_{1}+s_{2}g_{7}\cdot w_{2},$ $\displaystyle ds_{1}g_{4}$
$\displaystyle=s_{2}s_{1}g_{4}\cdot w_{2},\quad ds_{1}g_{7}=s_{1}g_{4}\cdot
g_{4}+s_{2}s_{1}g_{7}\cdot w_{2},$ $\displaystyle ds_{2}g_{4}$
$\displaystyle=-s_{2}s_{1}g_{4}\cdot w_{1}+s_{1}g_{4}\cdot s_{2}w_{1},\quad
ds_{2}g_{7}=s_{2}g_{4}\cdot g_{4}-s_{2}s_{1}g_{7}\cdot w_{1}-s_{1}g_{7}\cdot
s_{2}w_{1},$ $\displaystyle ds_{2}s_{1}g_{4}$ $\displaystyle=0,\quad
ds_{2}s_{1}g_{7}=-s_{2}s_{1}g_{4}\cdot g_{4}+s_{1}g_{4}\cdot s_{2}g_{4},$
$\displaystyle dw_{1}$ $\displaystyle=s_{2}w_{1}\cdot w_{2},\quad
ds_{2}w_{1}=0,\quad dw_{2}=0\;.$

These equations are again universal, and we again obtain the corresponding
ones in spacetime by pullback. These are the EOMs and Bianchi identities of
type II string theory at low energy, i.e., type II supergravity in 9
dimensions in the duality-symmetric formulation. A more common physics
notation for the above fields (once pulled back to the 9d spacetime
$(X/S^{1})/S^{1}$, but omitting the pullback notation) is

$\begin{array}[]{cccccc}g_{4}=F_{4},&g_{7}=H_{7},&s_{1}g_{4}=H_{3}^{(1)},&s_{1}g_{7}=F_{6}^{(1)},&s_{2}g_{4}=H_{3}^{(2)},&s_{2}g_{7}=F_{6}^{(2)},\\\
s_{2}s_{1}g_{4}=\mathcal{F}_{2},&s_{2}s_{1}g_{7}=\mathcal{F}_{5},&w_{1}=F_{2}^{(1)},&w_{2}=F_{2}^{(2)},&s_{2}w_{1}=F_{1}^{(2)}.\end{array}$

The classical EOMs are given in [BHO95][DR96], so the above can be viewed as a
duality-symmetric extension.

######  Remark 2.7 (Iterated $S^{1}$ vs. direct $T^{2}=S^{1}\times S^{1}$
reduction).

We compare the two settings:

(i) In the iterated case, $M(\mathcal{L}_{c}^{2}S^{4})$, notice the appearance
of the _axion_ $s_{2}w_{1}$, which would be absent in the direct reduction on
$T^{2}$ corresponding to the “toroidification”
$M(\operatorname{Map}(T^{2},S^{4})/\\!\\!/T^{2})$.

(ii) Note that $dw_{1}=s_{2}w_{1}\cdot w_{2}$ above, whereas in the direct
$T^{2}$-reduction, we would have $dw_{1}=dw_{2}=0$. Likewise, note the two
terms in the differential $ds_{2}g_{4}$ above, while in the direct reduction,
it would be on an equal footing with $ds_{1}g_{4}$, that is,
$ds_{2}g_{4}=s_{1}s_{2}g_{4}\cdot w_{1}$.

######  Example 2.8 (Reduction on a 3-torus and the triple cyclification
$\mathcal{L}_{c}^{3}S^{4}$).

Since the double cyclification $\mathcal{L}_{c}^{2}S^{4}$ is not simply
connected and its Sullivan minimal model has one generator, $s_{2}w_{1}$, of
degree one, for the triple cyclification $\mathcal{L}_{c}^{3}S^{4}$, a new
phenomenon happens: the desuspension $V[1]$ becomes truncated, see (14). We
will describe the Sullivan minimal model $M(\mathcal{L}_{c}^{3}S^{4})$. The
list of generators will include a new generator, $w_{3}$, and those of the
previous case, Example 2.6, as well as their desuspensions, $s_{3}g$, where
$g$ is a former generator of $M(\mathcal{L}_{c}^{2}S^{4})$, except for
$g=s_{2}w_{1}$, which gets truncated: we may just as well assume that

$s_{3}s_{2}w_{1}=0\,.$ (18)

This truncation will also affect the equations for the differentials in the
following way. In accordance with (16), the differentials of the former
generators $g$ of $M(\mathcal{L}_{c}^{2}S^{4})$ from Example 2.6 above will
all acquire an extra term $s_{3}g\cdot w_{3}$, such as

$dg_{4}=s_{1}g_{4}\cdot w_{1}+s_{2}g_{4}\cdot w_{2}+s_{3}g_{4}\cdot w_{3}\,,$

except for the equation

$ds_{2}w_{1}=0\,,$ (19)

which will remain intact. The differentials of the desuspensions $s_{3}g$ of
the former generators $g$ of $M(\mathcal{L}_{c}^{2}S^{4})$ will work by the
expression in (16), such as

$ds_{3}g_{4}=-s_{3}s_{1}g_{4}\cdot w_{1}+s_{1}g_{4}\cdot
s_{3}w_{1}-s_{3}s_{2}g_{4}\cdot w_{2}+s_{2}g_{4}\cdot s_{3}w_{2}\,,$

but those generators $g$ whose differentials contained $s_{2}w_{1}$ will be
affected in the way that in (16), we shall impose the relation (18):

$\displaystyle ds_{3}w_{1}=s_{2}w_{1}\cdot s_{3}w_{2},\qquad
ds_{3}s_{2}g_{4}=-s_{3}s_{2}s_{1}g_{4}\cdot w_{1}-s_{2}s_{1}g_{4}\cdot
s_{3}w_{1}-s_{3}s_{1}g_{4}\cdot s_{2}w_{1}\;,$ $\displaystyle
ds_{3}s_{2}g_{7}=-s_{3}s_{2}g_{4}\cdot g_{4}+s_{2}g_{4}\cdot
s_{3}g_{4}+s_{3}s_{2}s_{1}g_{7}\cdot w_{1}+s_{2}s_{1}g_{7}\cdot
s_{3}w_{1}+s_{3}s_{1}g_{7}\cdot s_{2}w_{1}\;.$

Let us list all the $d$-closure equations for the generators of
$M(\mathcal{L}_{c}^{3}S^{4})$, as this will be important for us later:

$ds_{3}s_{2}s_{1}g_{4}=0,\qquad ds_{2}w_{1}=0,\qquad ds_{3}w_{2}=0,\qquad
dw_{3}=0\;.$

The above equations are again universal, and we obtain the corresponding ones
in spacetime by pullback, and capture the equations of motion and Bianchi
identities of type II string theory at low energy, i.e., type II supergravity
in 8 dimensions in the duality-symmetric formulation, extending, for instance,
[AT85].

######  Example 2.9 (Reduction on $T^{k}$ and $k$-fold cyclifications
$\mathcal{L}_{c}^{k}S^{4}$ for $k\geq 3$).

The pattern of Example 2.8, as predicted by Equations (13)–(16), pertains. The
$d$-closed generators, which will play an important role later, consists of

  1. (i)

$k$ elements of degree one:

$s_{3}s_{2}s_{1}g_{4},\quad s_{2}w_{1},\quad s_{3}w_{2},\quad\dots,\quad
s_{k}w_{k-1}\,,$

  2. (ii)

and an element of degree two: $w_{k}$.

The number $n_{k}$ of generators does not follow the recursion
$n_{k}=2n_{k-1}+1$ seemingly suggested by (13), because of the truncations.
However, the set of generators is easy to account for:

$\displaystyle s_{i_{l}}\dots s_{i_{1}}g_{4},$ $\displaystyle\qquad\text{where
$0\leq l\leq 3$ and $1\leq i_{1}<\dots<i_{l}\leq k$},$ $\displaystyle
s_{i_{l}}\dots s_{i_{1}}g_{7},$ $\displaystyle\qquad\text{where $0\leq l\leq
6$ and $1\leq i_{1}<\dots<i_{l}\leq k$},$ $\displaystyle w_{i},$
$\displaystyle\qquad 1\leq i\leq k,$ $\displaystyle s_{j}w_{i},$
$\displaystyle\qquad 1\leq i<j\leq k.$

The pullbacks of their differentials to spacetime correspond, likewise, to the
EOMs and Bianchi identities of duality-symmetric low energy string
theory/supergravity in dimensions $11-k$. Supplying these leads to a duality-
symmetric extension of the non-duality symmetric versions studied, e.g., in
[LP96][LLP98], and surveyed in [CDF91][Ta98].

###  2.6 Type IIB and T-duality

The equivalence of real Sullivan minimal models $M(X)\cong M(Y)$ is in
correspondence with the underlying topological spaces $X$ and $Y$ being
rationally (in fact, under our formalism, “really”) homotopy equivalent. This
will allow us to obtain a candidate for the real model for type IIB without
having to immediately work out a topological space analogue for IIB of the
cyclification of $S^{4}$ on the type IIA side.

Starting with a topological model of the real homotopy type for type IIA, let
us call that $IIA$, and one for type IIB, let us call that $IIB$, it would be
ideal to have an equivalence upon dimensional reduction of both to nine
dimensions, i.e., in the spirit of the approach of [FSS18a], upon
cyclification, $M(\mathcal{L}_{c}IIA)\cong M(\mathcal{L}_{c}IIB)$. This would
correspond to a real homotopy equivalence
$\mathcal{L}_{c}IIA\sim\mathcal{L}_{c}IIB$, a universal version of _T-
duality_. The situation here is subtler, and this section is devoted to
discussing the intricacies of type IIB, versions of the equivalence
$\mathcal{L}_{c}IIA\sim\mathcal{L}_{c}IIB$ of real homotopy types, and the
discrepancy between them.

Now starting with $S^{4}$ as the universal space for M-theory (as in [Sa13]
and (5)), we get the cyclification $\mathcal{L}_{c}S^{4}$ as the model for
type IIA in ten dimensions (as in [FSS17] and Example 2.5). Dimensionally
reducing further amounts to double cyclification $\mathcal{L}^{2}_{c}S^{4}$ in
nine dimensions (as in Example 2.6). On the other hand, dimensionally reducing
type IIB to nine dimensions leads to the once-cyclified space
$\mathcal{L}_{c}IIB$. In terms of $S^{4}$, the expected equivalence in nine
dimensions would hence amount to the equivalence of Sullivan models

$M(\mathcal{L}_{c}IIB)\cong M(\mathcal{L}^{2}_{c}S^{4})$

and a real homotopy equivalence at the level of spaces

$\mathcal{L}_{c}IIB\sim\mathcal{L}^{2}_{c}S^{4}.$

While we will not work out the decyclification, we do have a path along which
to proceed:

---
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}_{c}}$$\scriptstyle{\mathcal{L}_{c}^{2}}$$\textstyle{IIB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}_{c}}$$\textstyle{IIA\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{L}_{c}}$$\textstyle{9d}$

This diagram is not only motivated by physics but also by Mysterious Duality
and the del Pezzo story, in which we have

---
$\textstyle{\mathbb{C}\mathbb{P}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{b^{2}}$$\textstyle{\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{\mathbb{B}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{\mathbb{B}_{2}}$

where $b$ denotes the process of blowing up (rather than the blowup map, which
would go in the opposite direction).

##### The Sullivan minimal model $M(IIB)$ of the real homotopy type $IIB$.

Consider the free graded commutative algebra
$\mathbb{R}[\omega_{1},h_{3},\omega_{3},\omega_{5},h_{7},\omega_{7}]$, the
subscripts denoting the degrees of the respective elements. Define a
differential by the following equations:

$\displaystyle d\omega_{1}$ $\displaystyle=0,\;\;\;\;\qquad\qquad\quad
dh_{3}=0,$ $\displaystyle d\omega_{3}$
$\displaystyle=h_{3}\omega_{1},\qquad\qquad\;\;d\omega_{5}=h_{3}\omega_{3},$
(20) $\displaystyle dh_{7}$
$\displaystyle=\omega_{3}\omega_{5}+\omega_{7}\omega_{1},\quad
d\omega_{7}=h_{3}\omega_{5}.$

Pulling back these universal differential forms to spacetime, we get the EOMs
of $D=10$ type IIB supergravity in the duality-symmetric formulation, and
without imposing self-duality; cf. [SW83][HW84][Sch83] for the classical
formulation and [DLS97][DLT98] for the duality-symmetric formulation. Note
that we do not include fields of degree greater than seven, as in the dual
picture of type IIA (see Example 2.6), this would require parametrized
homotopy theory [BMSS18]. Additionally, there are several subtleties in type
IIB which makes a purely topological perspective delicate due to the mixing
between geometry and topology, in the sense that some of the fields arise from
the metric, as we explain further below (see [BMSS18]).

##### T-duality: comparing $M(\mathcal{L}_{c}IIA)$ and
$M(\mathcal{L}_{c}IIB)$.

Computing $M(\mathcal{L}_{c}IIB)$ using the recipe of section 2.5 and changing
the variables $sh_{3},w,s\omega_{3},s\omega_{5},s\omega_{7}$ in the notation
of (13) and (16) to

$\displaystyle
c_{2}:=sh_{3},\qquad\tilde{c}_{2}:=w,\qquad\omega_{2}:=s\omega_{3},$
$\displaystyle\omega_{4}:=s\omega_{5},\qquad\omega_{6}:=s\omega_{7},$

more common in physics, we get the following presentation.

######  Proposition 2.10 (Cyclification of type IIB to $d=9$ ).

$\displaystyle
M(\mathcal{L}_{c}IIB)=\left(\mathbb{R}[\omega_{1},c_{2},\tilde{c}_{2},\omega_{2},h_{3},\omega_{3},\omega_{4},\omega_{5},sh_{7},\omega_{6},h_{7},\omega_{7}],d\right)$
$\displaystyle d\omega_{1}=0,\qquad dc_{2}=0,\qquad d\tilde{c}_{2}=0,$
$\displaystyle d\omega_{2}=-c_{2}\cdot\omega_{1},\quad
dh_{3}=c_{2}\cdot\tilde{c}_{2},\quad
d\omega_{3}=h_{3}\cdot\omega_{1}+\omega_{2}\cdot\tilde{c}_{2},$ $\displaystyle
d\omega_{4}=-c_{2}\cdot\omega_{3}+h_{3}\cdot\omega_{2},\qquad
d\omega_{5}=h_{3}\cdot\omega_{3}+\omega_{4}\cdot\tilde{c}_{2},$ (21)
$\displaystyle
dsh_{7}=-\omega_{2}\cdot\omega_{5}+\omega_{3}\cdot\omega_{4}-\omega_{6}\cdot\omega_{1},\quad
d\omega_{6}=-c_{2}\cdot\omega_{5}+h_{3}\cdot\omega_{4},$ $\displaystyle
dh_{7}=\omega_{3}\cdot\omega_{5}+\omega_{7}\cdot\omega_{1}+sh_{7}\cdot\tilde{c}_{2},\quad
d\omega_{7}=h_{3}\cdot\omega_{5}+\omega_{6}\cdot\tilde{c}_{2}\,.$

We would like to match it with
$M(\mathcal{L}_{c}IIA)=M(\mathcal{L}_{c}^{2}S^{4})$, see Example 2.6. Writing

$\displaystyle\omega_{2}:=w_{1},\qquad
h_{3}:=s_{1}g_{4},\qquad\omega_{4}:=g_{4},$
$\displaystyle\omega_{6}:=s_{1}g_{7},\qquad h_{7}:=g_{7},$
$\displaystyle\omega_{1}:=s_{2}w_{1},\qquad
c_{2}:=s_{2}s_{1}g_{4},\qquad\tilde{c}_{2}:=w_{2},$
$\displaystyle\omega_{3}:=s_{2}g_{4},\qquad\omega_{5}:=s_{2}s_{1}g_{7},\qquad
sh_{7}:=s_{2}h_{7}\,,$

we get a compatible presentation for the DGCA $M(\mathcal{L}_{c}IIA)$.

######  Proposition 2.11 (Cyclification of type IIA to $d=9$ ).

$\displaystyle
M(\mathcal{L}_{c}IIA)=\left(\mathbb{R}[\omega_{1},c_{2},\tilde{c}_{2},\omega_{2},h_{3},\omega_{3},\omega_{4},\omega_{5},sh_{7},\omega_{6},h_{7}],d\right),$
$\displaystyle d\omega_{1}=0,\qquad dc_{2}=0,\qquad d\tilde{c}_{2}=0,$
$\displaystyle d\omega_{2}=\omega_{1}\cdot\tilde{c}_{2},\quad
dh_{3}=c_{2}\cdot\tilde{c}_{2},\quad
d\omega_{3}=-c_{2}\cdot\omega_{2}+h_{3}\cdot\omega_{1},$ $\displaystyle
d\omega_{4}=h_{3}\cdot\omega_{2}+\omega_{3}\cdot\tilde{c}_{2},\qquad
d\omega_{5}=-c_{2}\cdot\omega_{4}+h_{3}\cdot\omega_{3},$ (22) $\displaystyle
dsh_{7}=\omega_{3}\cdot\omega_{4}-\omega_{5}\cdot\omega_{2}-\omega_{6}\cdot\omega_{1},\quad
d\omega_{6}=h_{3}\cdot\omega_{4}+\omega_{5}\cdot\tilde{c}_{2},$ $\displaystyle
dh_{7}=-\tfrac{1}{2}\omega_{4}^{2}+\omega_{6}\cdot\omega_{2}+sh_{7}\cdot\tilde{c}_{2}\,.$

Now it is time to compare $M(\mathcal{L}_{c}IIA)$ with
$M(\mathcal{L}_{c}IIB)$. Here is a striking conclusion:

######  Proposition 2.12 (Matching in 9 dimensions).

Up to replacement

$c_{2}\leftrightarrow-\tilde{c}_{2},$

all the equations for the differential of $M(\mathcal{L}_{c}IIA)$ in (22)
exactly match those of $M(\mathcal{L}_{c}IIB)$ in (21), except for the
following mismatch for the generators of degree $7$:

Model | $M(\mathcal{L}_{c}IIA)$ | $M(\mathcal{L}_{c}IIB)$
---|---|---
Generators | $h_{7}$ |  $h_{7}$, $\omega_{7}$
Differentials | $dh_{7}=-\tfrac{1}{2}\omega_{4}^{2}+\omega_{6}\cdot\omega_{2}+sh_{7}\cdot\tilde{c}_{2}$ |  $dh_{7}=\omega_{3}\cdot\omega_{5}+\omega_{7}\cdot\omega_{1}-sh_{7}\cdot c_{2}$,
$d\omega_{7}=h_{3}\cdot\omega_{5}-\omega_{6}\cdot c_{2}$

With relations imposed algebraically, this means that

$M(\mathcal{L}_{c}IIA)/(h_{7},dh_{7})\cong
M(\mathcal{L}_{c}IIB)/(h_{7},\omega_{7},dh_{7},d\omega_{7})\,,$ (23)

that is to say, the two DGCAs, $M(\mathcal{L}_{c}IIA)$ and
$M(\mathcal{L}_{c}IIB)$, are isomorphic modulo the differential ideals
generated by the generators of degree 7. Perhaps, topologically more
interesting is an isomorphism between the dg-subalgebras $M_{A}\subset
M(\mathcal{L}_{c}IIA)$ and $M_{B}\subset M(\mathcal{L}_{c}IIB)$ generated by
all the generators except those of degree 7:

$\textstyle{{}_{\phantom{A}{}_{\phantom{A}}}{M_{A}}\;\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\;\;\;M_{B}{}_{\phantom{A}{}_{\phantom{A}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(\mathcal{L}_{c}IIA)}$$\textstyle{M(\mathcal{L}_{c}IIB)\,.}$
(24)

These dg-subalgebras are minimal Sullivan, and their isomorphism morally
corresponds to the rational equivalence over $\mathbb{R}$ of quotient spaces
of $\mathcal{L}_{c}IIA$ and $\mathcal{L}_{c}IIB$.

Another way to evade the above mismatch is to introduce “stable” models of
types IIA and IIB, which turn out to be perfectly compatible with T-duality,
at the expense of using spectra in lieu of spaces. See details in [FSS18a]. We
plan to give a physical interpretation of the discrepancy in an upcoming paper
[SV22].

##  3 Toroidal symmetries

Here we describe the toroidal symmetry of the iterated cyclic loop spaces
$\mathcal{L}^{k}_{c}S^{4}$ for $k\geq 0$ from section 2.5, for which we
provide topological interpretation.

###  3.1 Toroidal symmetries of minimal algebras

We will be interested in real toroidal symmetries of minimal Sullivan algebras
$M$, i.e., diagonalizable actions $T\to\operatorname{Aut}M$ of a _real split
torus_ , an affine algebraic group $T$ isomorphic over $\mathbb{R}$ to the
group $\mathbb{G}_{m}^{k}$ for some $k\geq 1$, where
$\mathbb{G}_{m}=\operatorname{GL}(1)$ is the multiplicative group. Here
$\operatorname{Aut}M$ is the group of automorphisms of $M$ as a DGCA. When $M$
has strong finite type, $\operatorname{Aut}M$ is an affine algebraic group
over $\mathbb{R}$, because it is defined by the invertibility of the Jacobian
condition in the affine $\mathbb{R}$-variety $\operatorname{End}M$; see
[Su77][Re78]. Thus, by an _action_ $T\to\operatorname{Aut}M$ above, we mean a
morphism of algebraic groups defined over $\mathbb{R}$.

We will consolidate essentially all toroidal symmetries of $M$, which could be
done by considering a _maximal $\mathbb{R}$-split torus_
$T\subseteq\operatorname{Aut}M$. In our study, a maximal split torus will play
a role similar to that of a maximal torus in the theory of compact Lie groups
or that of a Cartan subalgebra in the theory of complex semisimple Lie
algebras. Indeed, a real split torus $T\subseteq\operatorname{Aut}M$ gives
rise to a _weight decomposition_ :

$M=\bigoplus_{\alpha\in\mathfrak{X}(T)}M_{\alpha}$ (25)

indexed by the _character group_
$\mathfrak{X}(T)=\operatorname{Hom}_{\mathbb{R}}(T,\mathbb{G}_{m})$ of real
algebraic group homomorphisms from $T$ to the multiplicative group
$\mathbb{G}_{m}$, so that $T$ acts on each _weight space_ $M_{\alpha}$ by the
character $\alpha$:

$M_{\alpha}=\big{\\{}m\in M\;|\;t\cdot m=\alpha(t)m\quad\text{for all $t\in
T$}\big{\\}}.$

If $T$ is a real split torus of dimension $n\geq 0$, then
$\mathfrak{X}(T)\cong\mathbb{Z}^{n}$; see [Bo91, Corollary 8.2 and Proposition
8.5]. Since automorphisms of $M$ have to respect the DGCA structure, i.e., the
$\mathbb{Z}$-grading, differential, and multiplication on $M$, the weight
decomposition is automatically compatible with it. That is, we have

  1. (1)

$M^{n}=\bigoplus_{\alpha\in\mathfrak{X}(T)}(M_{\alpha}\cap M^{n})$ for each
$n\geq 0$;

  2. (2)

$d(M_{\alpha})\subseteq M_{\alpha}$ for each $\alpha\in\mathfrak{X}(T)$;

  3. (3)

$M_{\alpha}\cdot M_{\beta}\subseteq M_{\alpha+\beta}$ for all
$\alpha,\beta\in\mathfrak{X}(T)$.

Here and henceforth we write the group law in the character group
$\mathfrak{X}(T)$ additively and employ the exponential notation:

$t^{\alpha}:=\alpha(t)\qquad\text{for $t\in T$, $\alpha\in\mathfrak{X}(T)$.}$

Specifically for minimal Sullivan algebras, weight decompositions have been
considered by L. Renner in his Master’s thesis [Re78] on automorphism groups
of minimal algebras.

Maximal split tori are unique up to conjugation in a real agebraic group; see
[Bo91, Theorem 15.14]. This implies that the weight decompositions
corresponding to different maximal split tori are related by automorphisms of
$M$.

Given an abstract $\mathbb{R}$-split torus $T$, a weight decomposition (25)
defines an obvious action of $T$ on $M$.

A weight decomposition of a minimal Sullivan algebra $M=S(V)$ induces a weight
decomposition

$V=\bigoplus_{\alpha\in\mathfrak{X}(T)}V_{\alpha}$

of the generating space $V$, because the latter may be canonically identified
with the _space of indecomposables_ : $I(M)=M^{+}/(M^{+})^{2}=S^{\geq
1}(V)/S^{\geq 2}(V)$, and $M^{+}$ and $(M^{+})^{2}$ are split into weight
spaces by definition, whereas a diagonalizable action diagonalizes on
invariant subquotients; cf. [Re78, Proposition 3.4.2]. This does not mean that
$V$ is necessarily a $T$-invariant subspace of $M$. For that matter, we will
distinguish the generating space $V\subset S(V)=M$ and the subquotient $I(M)$
of the indecomposables of $M$.

###  3.2 Toroidal symmetries of a cyclification

Here we apply the results of Section 3.1 to a “cyclification”, i.e., to the
cyclic loop space $\mathcal{L}_{c}Z$ of a space $Z$ considered in Section 2.4.
Let us start with a few observations.

######  Proposition 3.1 (Split tori of a Sullivan model).

Suppose an $\mathbb{R}$-split torus $T$ acts on a minimal Sullivan algebra
$M=(S(V),d)$ of strong finite type.

  1. (i)

Then the weights of the action, that is to say, the characters
$\alpha\in\mathfrak{X}(T)$ for which the weight space $M_{\alpha}$, see (25),
is nontrivial, are generated multiplicatively on

$P(M):=Z(M^{+})/Z(M^{+})\cap(M^{+})^{2}\subseteq I(M),$

where $Z(M^{+}):=\ker d|_{M^{+}}=\\{x\in M^{+}\;|\;dx=0\\}$.

  2. (ii)

The action of an $\mathbb{R}$-split torus on $M$ is determined by its action
on $P(M)$. In other words, if $T\subseteq\operatorname{Aut}M$ is an
$\mathbb{R}$-split torus, then the composition
$T\subseteq\operatorname{Aut}M\to\operatorname{GL}(P(M))$ is injective. In
particular, $\dim T\leq\dim P(M)$.

  3. (iii)

If $T$ is an $\mathbb{R}$-split torus of $\operatorname{Aut}M$ of dimension
$\dim T=\dim P(M)$, then $T$ is a maximal split torus of
$\operatorname{Aut}M$.

  4. (iv)

If the differential on $M=S(V)$ is quadratic, then the natural map $\ker d\cap
V=Z(M^{+})\cap V\to P(M)$ is an isomorphism.

###### Proof.

(i) This part generalizes Proposition 3.4.2 of Renner’s thesis [Re78], and so
does the proof. However, our generalization to the non-simply connected and
non-algebraically closed case requires new ideas and more work.

We will need to use a more invariant version of the Sullivan nilpotence
condition 2.1(i), due to Bousfield and Gugenheim [BG76]. Every minimal
Sullivan algebra $M$ admits a canonically defined double filtration

$\displaystyle M(0)\subseteq M(1)\subseteq M(2)\subseteq\dots,$ $\displaystyle
M(n,0)\subseteq M(n,1)\subseteq M(n,2)\subseteq\dots,$

$n\geq 1$, by dg-subalgebras, such that $M(0)=\mathbb{R}$,
$\bigcup_{n}M(n)=M$, and

$\bigcup_{p}M(n,p)=M(n)\qquad\text{for each $n\geq 1$}.$ (26)

The subalgebra $M(n)$ is defined as the dg-subalgebra generated by $M^{i}$ for
$1\leq i\leq n$. The subalgebra $M(n,p)$ is defined inductively, starting from
$M(n,0)=M(n-1)$, as the subalgebra generated by $M(n,p-1)$ and the elements
$x\in M^{n}$ such that $dx\in M(n,p-1)$. Conversely, every connected DGCA $M$
which is free as a graded commutative algebra and for which the subalgebras
defined above satisfy (26) is a minimal Sullivan algebra, see [BG76, Proof of
Proposition 7.5]. Bousfield and Gugenheim simply call connected, free as
graded commutative algebras DGCAs satisfying (26) minimal. The equivalence of
the two types of models, the minimal Sullivan model and Bousfield-Gugenheim’s
minimal model follows from the existence and uniqueness theorems for each
type, see [Su77] and [BG76], respectively.

The augmentation ideal $M^{+}\subset M$ and its powers $(M^{+})^{n}=S^{\geq
n}(V)$ are $T$-invariant. Moreover, the weight decomposition of $M$ maps
isomorphically to the weight decomposition of the associated graded algebra
$\operatorname{gr}M:=\bigoplus_{n\geq 0}(M^{+})^{n}/(M^{+})^{n+1}=S(I(M))$.
The subquotient $I(M)$ of $M$ is $T$-invariant and inherits a weight
decomposition. Thus, the weights of $M$ are generated multiplicatively by the
weights of $I(M)$. Similarly, $P(M)$ is a $T$-invariant subquotient of $M$ and
inherits a weight decomposition. Note that the subalgebras $M(n)$ and
$M(n,p)$, defined intrinsically by using the multiplicative and dg-structures
on $M$, are $T$-invariant and get a weight decomposition.

We will show that the weights of the subalgebras $M(n,p)$ are multiplicatively
generated by the weights of $P(M)$ and $M(n,p-1)$ and run a double induction
on $n$ and $p$. By definition, the subalgebra $M(n,p)$ is generated by
$M(n,p-1)$ and $x\in M^{n}$ such that $dx\in M(n,p-1)$. Its subspace
$Z(M^{n})\subseteq M^{n}$ of $n$-cycles fits into a short exact sequence of
$T$-invariant spaces:

$0\longrightarrow Z(M^{n})\cap(M^{+})^{2}\longrightarrow
Z(M^{n})\longrightarrow P(M^{n})\longrightarrow 0\,.$

Observe that $Z(M^{n})\cap(M^{+})^{2}\subseteq M(n-1)=M(n,0)$, because the
component of degree $n$ of $(M^{+})^{2}$ is spanned by products of elements in
$M^{i}$ for $1\leq i\leq n-1$. On the other hand, the weights of a
$T$-invariant complement $C(M^{n})$ to $Z(M^{n})$ in the space $\\{x\in
M^{n}\;|\;dx\in M(n,p-1)\\}$ are among the weights of $M(n,p-1)$, because for
$x\neq 0\in C(M^{n})$, $dx$ will be nonzero and have the same weight.

(ii) From Part (i), we can deduce that if an $\mathbb{R}$-split torus $T$ acts
faithfully on $M$, that is to say, $T\subseteq\operatorname{Aut}M$, then it
would also act faithfully on $P(M)$. Indeed, if there is a $t\in T$ acting on
$P(M)$ trivially, then by Part 1, for all weights $\alpha$ of $M$, we have
$\alpha(t)=0$, which means $t$ acts trivially on $M$. Thus, $T$ embeds into
the general linear group $\operatorname{GL}(P(M))$ of $P(M)$ regarded as a
vector space. Since the maximal torus of $\operatorname{GL}(P(M))$ has
dimension equal to $\dim P(M)$, we conclude that $\dim T\leq\dim P(M)$.

(iii) Follows from (ii).

(iv) Note that the composition $V\subseteq M^{+}\to I(M)$ is injective.
Therefore, $\ker d|_{V}=Z(M^{+})\cap V$ maps injectively to $P(M)$. To show
that this map is surjective, let $\bar{x}\in P(M)$ and $x\in M^{+}=S^{\geq
1}(V)$, $dx=0$, represent $\bar{x}$. Decompose $x=x_{1}+x_{2}+\dots$, with
$x_{i}\in S^{i}(V)$. Then $dx=0$ implies $dx_{1}=dx_{2}=\dots=0$, because when
the differential $d$ is quadratic, $dx_{i}\in S^{i+1}(V)$. Therefore,
$x_{1}\in V$ also represents $\bar{x}$. ∎

Within the context of Section 2.5, we now establish the following results. For
the rest of this section, let $Z$ be a path-connected and nilpotent space and
$M(Z)=(S(V),d)$ its real Sullivan minimal model, which we assume to have
strong finite type. Consider the real Sullivan minimal model
$M(\mathcal{L}_{c}Z)=(S(V\oplus V[1]\oplus\mathbb{R}w),d_{c})$ of the cyclic
loop space $\mathcal{L}_{c}Z$ of $Z$.

######  Lemma 3.2 (The growth of dimension of the space of closed generators).

If the differential $d$ on $M(Z)$ is quadratic, then

$\dim\left(\ker d_{c}\cap(V\oplus V[1]\oplus\mathbb{R}w)\right)=\dim(\ker
d\cap V)+1.$

###### Proof.

We will start with a simpler case of simply connected $Z$ and make necessary
adjustments in the more general case.

Simpler Case: $Z$ is simply connected. At the level of minimal Sullivan model
$M=M(Z)=S(V)$, this means that $V^{1}=0$.

We claim that

$\ker d_{c}\cap(V\oplus V[1]\oplus\mathbb{R}w)=(\ker d\cap
V)[1]\oplus\mathbb{R}w.$ (27)

Indeed, from formulas (16), we see that no nonzero element of $V=V^{>1}$ could
be in $\ker d_{c}$, whereas $w$ is always in $\ker d_{c}$. We claim that $\ker
d_{c}\cap V[1]=(\ker d\cap V)[1]$. Indeed, to justify the inclusion $(\ker
d\cap V)[1]\subseteq\ker d_{c}\cap V[1]$, we start with a $d$-closed element
$v\in V\subseteq M(Z)$ and observe that $d_{c}sv=-sdv=0$, again from (16).

Let us prove the opposite inclusion: $\ker d_{c}\cap V[1]\subseteq(\ker d\cap
V)[1]$. If $d_{c}sv=0$ for some $v\in V$, then $sdv=0$. Note that $dv$ is in
$S(V)$, which may be thought of as the algebra of polynomial functions on the
graded manifold $\mathbb{V}^{*}=\operatorname{Spec}S(V)$ or, passing to the
grading modulo 2, on the supermanifold $\mathbb{V}^{*}$, the affine superspace
associated with the super vector space $V^{*}$. The minimality condition on
$d$ moreover implies that $dv\in S^{\geq 2}(V)$. Note also that the
differential $s$ may be identified with the de Rham differential of the affine
superspace $\mathbb{V}^{*}$:

$s=d_{\operatorname{dR}}:\Omega^{0}(\mathbb{V}^{*})=S(V)\;\longrightarrow\;\Omega^{1}(\mathbb{V}^{*})=S(V)\otimes
V[1]\;.$

(In $\mathbb{Z}$-graded geometry, one usually has $V[-1]$ for the cotangent
space, but it is the same as $V[1]$ under mod 2 grading.) So, by the super
Poincaré lemma [Ma97, Proposition 3.4.5], $\ker s=S^{0}(V)=\mathbb{R}$.
Therefore, if $sdv=0$ then $dv\in S^{\geq 2}(V)\cap S^{0}(V)=0$, i.e.,
$v\in\ker d\cap V$. We conclude that $\ker d_{c}\cap V[1]=(\ker d\cap V)[1]$.
Summing up all the parts of $V\oplus V[1]\oplus\mathbb{R}w$, we see that (27)
holds.

General Case: $Z$ is not necessarily simply connected. We will adjust formula
(27). Now we claim that

$\ker d_{c}\cap(V\oplus V[1]\oplus\mathbb{R}w)=(\ker d\cap V^{1})\oplus(\ker
d\cap V^{\geq 2})[1]\oplus\mathbb{R}w.$

So, it is again clear from (16) that $w\in\ker d_{c}$. Given that $sv=0$ for
any $v\in V^{1}$, it is also clear that we have $\ker d_{c}\cap V=\ker d\cap
V^{1}$. What remains to be proven is that $\ker d_{c}\cap V[1]=(\ker d\cap
V^{\geq 2})[1]$.

Let us start with showing $(\ker d\cap V^{\geq 2})[1]\subseteq\ker d_{c}\cap
V[1]$. If $v\in V^{\geq 2}$ is such that $dv=0$, then $d_{c}sv=-sdv=0$.

For the opposite inclusion, $\ker d_{c}\cap V[1]\subseteq(\ker d\cap V^{\geq
2})[1]$, suppose that $d_{c}sv=0$ for some $v\in V^{\geq 2}$. (We ignore $v\in
V^{1}$, as in this case $sv=0$.) The second formula (16) then implies $sdv=0$.
Note that $dv\in S^{\geq 2}(V)$ because of the minimality of $(S(V),d)$.
Another observation is that since $sv=0$ for any $v\in V^{1}$, the
differential $s:S(V)\to S(V)\otimes V^{\geq 2}[1]$ may now be identified with
the relative de Rham differential of the relative affine superspace
$\mathbb{V}^{*}=(\mathbb{V}^{1})^{*}\times(\mathbb{V}^{\geq 2})^{*}$ over
$(\mathbb{V}^{1})^{*}$:

$s=d_{\operatorname{dR}}:\Omega^{0}_{\mathbb{V}^{*}/(\mathbb{V}^{1})^{*}}(\mathbb{V}^{*})=S(V)\;\longrightarrow\;\Omega^{1}_{\mathbb{V}^{*}/(\mathbb{V}^{1})^{*}}(\mathbb{V}^{*})=S(V)\otimes
V^{\geq 2}[1].$

As in Case 1, the super Poincaré lemma implies that $\ker s=S(V^{1})$.

Getting back to our $v\in V^{\geq 2}$ such that $d_{c}sv=0$, we see that
$dv\in S^{\geq 2}(V)\cap S(V^{1})=S^{\geq 2}(V^{1})$. Since the degree of $v$
is at least two, the degree of $dv$ must be at least three. This means that
$dv\in S^{\geq 3}(V^{1})$. But the differential is assumed to be quadratic:
$dv\in S^{2}(V)$, whence $dv=0$. Lemma is proven. ∎

######  Proposition 3.3 (Automorphisms of the Sullivan model of a
cyclification).

The automorphisms of $M(Z)$ extend naturally to automorphisms of
$M(\mathcal{L}_{c}Z)$. Moreover, one has a natural inclusion

$\operatorname{Aut}M(Z)\times\mathbb{G}_{m}\subseteq\operatorname{Aut}M(\mathcal{L}_{c}Z).$
(28)

###### Proof.

In general, knowing that the group of automorphisms of the Sullivan minimal
model of a space is isomorphic to the group of rational homotopy self-
equivalences thereof and the functoriality of the construction
$Z\mapsto\mathcal{L}_{c}Z$, we get a morphism
$\operatorname{Aut}M(Z)\to\operatorname{Aut}M(\mathcal{L}_{c}Z)$. This
morphism is injective, because if $g\in\operatorname{Aut}M(Z)$ acts trivially
on $M(\mathcal{L}_{c}Z)$, then it will act trivially on every subquotient,
including $M(Z)$, see (13) and (17).

It will be useful to have explicit formulas for this extension,
$\rho:\operatorname{Aut}M(Z)\hookrightarrow\operatorname{Aut}M(\mathcal{L}_{c}Z)$.
Suppose that $g:M(Z)\to M(Z)$ is an automorphism of $M(Z)=S(V)$. Then $g$
defines an automorphism $\rho(g)$ of $M(\mathcal{L}_{c}Z)=S(V\oplus
V[1]\oplus\mathbb{R}w)$ by acting on the free generators $V\oplus
V[1]\oplus\mathbb{R}w$ as follows:

$\displaystyle\rho(g)v$ $\displaystyle:=gv\qquad$ $\displaystyle\text{for
$v\in V$},$ $\displaystyle\rho(g)(sv)$ $\displaystyle:=s(gv)\qquad$
$\displaystyle\text{for $sv\in V[1]$},$ $\displaystyle\rho(g)w$
$\displaystyle:=w.$

Now, for $t\in\mathbb{G}_{m}$, define an action

$\displaystyle t\cdot v$ $\displaystyle:=v\qquad$ $\displaystyle\text{for
$v\in V$},$ $\displaystyle t\cdot(sv)$ $\displaystyle:=t(sv)\qquad$
$\displaystyle\text{for $sv\in V[1]$},$ (29) $\displaystyle t\cdot w$
$\displaystyle:=t^{-1}w.$

These formulas define an explicit inclusion (28). ∎

######  Theorem 3.4 (Maximal split tori of the Sullivan model of a
cyclification).

Let us also assume that the differential in the Sullivan minimal model
$M(Z)=(S(V),d)$ is _quadratic, i.e.,_ the restriction of $d$ to $V$ maps $V$
to $S^{2}(V)$: $d|_{V}:V\to S^{2}(V).$ Suppose there is an $\mathbb{R}$-split
torus $T\subseteq\operatorname{Aut}M(Z)$ such that $\dim T=\dim P(M(Z))$, see
3.1.

Then $T$ is a maximal split torus of $\operatorname{Aut}M(Z)$,
$\dim(T\times\mathbb{G}_{m})=\dim P(M(\mathcal{L}_{c}Z))$ and
$T\times\mathbb{G}_{m}$ is a maximal split torus of
$\operatorname{Aut}M(\mathcal{L}_{c}Z)$.

###### Proof.

By 3.1(iii), $T$ is a maximal split torus of $\operatorname{Aut}M(Z)$. 3.3
implies that $T\times\mathbb{G}_{m}$ is a split torus in
$\operatorname{Aut}M(\mathcal{L}_{c}Z)$. To prove that it is maximal, using
the 3.1(iii) once again, it is enough to show that $\dim
P(M(\mathcal{L}_{c}Z))=\dim P(M(Z))+1$, which is equal to
$\dim(T\times\mathbb{G}_{m})$ by assumption. Applying 3.1(iv), we see that
$\ker d\cap V\cong P(M(Z))$. Formulas (16) for the differential $d_{c}$ of
$M(\mathcal{L}_{c}Z)$ show that the differential is also quadratic. Therefore,
we have $\ker d_{c}\cap(V\oplus V[1]\oplus\mathbb{R}w)\cong
P(M(\mathcal{L}_{c}Z))$, and conclude with Lemma 3.2. ∎

Given the decomposition of the Sullivan minimal model into weight spaces of
the previous section, section 3.1, the toroidal symmetries of the theorem
yield the following statement.

######  Corollary 3.5 (Weight decomposition in Sullivan models).

If $M(Z)=\bigoplus_{\alpha\in\mathfrak{X}(T)}M_{\alpha}$ is the weight
decomposition corresponding to the action of a maximal split torus $T$, which
induces a weight decomposition
$I(M(Z))=\bigoplus_{\alpha\in\mathfrak{X}(T)}I(M(Z))_{\alpha}$ on the space
$I(M(Z))$ of indecomposables, then the weight decomposition of
$M(\mathcal{L}_{c}Z)$ corresponding to its maximal split torus
$T_{c}=T\times\mathbb{G}_{m}$ induces the following weight decomposition on
its space $I(M(\mathcal{L}_{c}Z))=I(M(Z))\oplus I(M(Z))[1]\oplus\mathbb{R}w$
of indecomposables:

  1. (i)

The weight of $w\in M(\mathcal{L}_{c}Z)$ is
$\epsilon_{1}=(0,-1)\in\mathfrak{X}(T_{c})=\mathfrak{X}(T)\times\mathfrak{X}(\mathbb{G}_{m})$;

  2. (ii)

If $v\in(I(M(Z))_{\alpha}$ for some weight $\alpha\in\mathfrak{X}(T)$, then
the image $v$ in $I(M(Z))\subseteq I(M(\mathcal{L}_{c}Z))$ has weight
$\alpha:=(\alpha,0)\in\mathfrak{X}(T)\times\mathfrak{X}(\mathbb{G}_{m})$;

  3. (iii)

If $v\in(I(M(Z))_{\alpha}$ for some weight $\alpha\in\mathfrak{X}(T)$, then
the image $sv$ in $(I(M(Z))[1]\subseteq I(M(\mathcal{L}_{c}Z))$ has weight
$\alpha-\epsilon_{1}\in\mathfrak{X}(T)\times\mathfrak{X}(\mathbb{G}_{m})$.

###  3.3 Toroidal symmetries of the minimal algebras of cyclifications of
$S^{4}$

Here we apply the results of the previous section to the “cyclifications,” the
iterated cyclic loop spaces $\mathcal{L}^{k}_{c}S^{4}$ of the 4-sphere
$S^{4}$, $k\geq 0$, from section 2.4. The resulting symmetries of the Sullivan
minimal model hold universally for fields of $k$-dimensional reductions of
M-theory and may be interpreted as trombone and torus rescaling symmetries
discussed in [CDF91][CLPS98] and [DS09]. Let us start with an immediate
consequence of Theorem 3.4.

######  Corollary 3.6 (Toroidal symmetries).

The maximal $\mathbb{R}$-split torus of the real algebraic group
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ for $k\geq 0$ is $T^{k+1}$,
isomorphic to $\mathbb{G}_{m}^{k+1}$ over $\mathbb{R}$. The structure of
$\mathcal{L}_{c}^{k}S^{4}$ as an iterated cyclification $($or the structure of
$M(\mathcal{L}_{c}^{k}S^{4})$ as a sequence of Halperin extensions (13) and
(17)$)$ determines a canonical splitting $T^{k+1}\cong\mathbb{G}_{m}^{k+1}$.

###### Proof.

The statement follows from Theorem 3.4 by induction. The base case $k=0$ is
done in Example 3.7 below. ∎

######  Example 3.7 ($k=0$: The 4-sphere $S^{4}$).

We start with the automorphism group $\operatorname{Aut}M(S^{4})$ of the
Sullivan minimal model $M(S^{4})$ of $S^{4}$; see (5). By the degree argument,
an automorphism of $M(S^{4})$ must take $g_{4}$ to a scalar multiple of
itself:

$g_{4}\mapsto tg_{4}$ (30)

for some $t\in\mathbb{G}_{m}(\mathbb{R})$, and this determines the action of
the automorphism on $g_{7}$:

$g_{7}\mapsto t^{2}g_{7},$

and thereby on the whole DGCA $M(S^{4})$. This gives an identification
$\operatorname{Aut}M(S^{4})\cong\mathbb{G}_{m}$ over $\mathbb{R}$, and,
therefore, $\operatorname{Aut}M(S^{4})$ automatically coincides with its
maximal split torus $T$ and $\dim T=\dim P(M(S^{4}))=1$, as
$P(M(S^{4}))\cong\ker
d\cap(\mathbb{R}g_{4}\oplus\mathbb{R}g_{7})=\mathbb{R}g_{4}$, see 3.1(iv).
Note that this identification is unique up to automorphism of
$\mathbb{G}_{m}$, which could only be $t\mapsto t^{-1}$ if not trivial. Thus,
we get a weight decomposition determined by

$g_{4}\in M(S^{4})_{\epsilon_{0}},\qquad g_{7}\in M(S^{4})_{2\epsilon_{0}},$

with the weights defined up to common sign, that is to say,

$t^{\epsilon_{0}}=t$ (31)

or $t^{\epsilon_{0}}=t^{-1}$. (Again, the weights are just determined by the
weight of $g_{4}$, as that is the only $d$-closed generator). We can always
normalize this ambiguity so as to have positive weights and assume (31) is
valid. This choice also has topological motivation, as we will see in section
3.4.1.

######  Example 3.8 ($k=1$: The cyclification $\mathcal{L}_{c}S^{4}$ of the
sphere $S^{4}$).

We now look at the maximal real split torus of
$\operatorname{Aut}M(\mathcal{L}_{c}S^{4})$ compatible with the structure of
$M(\mathcal{L}_{c}S^{4})$ as the Sullivan minimal model of the cyclic loop
space of $S^{4}$; see Example 2.5. 3.6 canonically identifies the maximal
split torus of $\operatorname{Aut}M(\mathcal{L}_{c}S^{4})$ as
$\mathbb{G}_{m}\times\mathbb{G}_{m}$, acting on
$M(\mathcal{L}_{c}S^{4})=\mathbb{R}[g_{4},g_{7},sg_{4},sg_{7},w]$ as follows:

$\displaystyle t\cdot g_{4}=t^{\epsilon_{0}}g_{4},\qquad t\cdot
g_{7}=t^{2\epsilon_{0}}g_{7},$ (32) $\displaystyle t\cdot
sg_{4}=t^{\epsilon_{0}-\epsilon_{1}}sg_{4},\qquad t\cdot
sg_{7}=t^{2\epsilon_{0}-\epsilon_{1}}sg_{7},$ $\displaystyle t\cdot
w=t^{\epsilon_{1}}w,$

where $t\in(\mathbb{G}_{m}\times\mathbb{G}_{m})(\mathbb{R})$. This corresponds
to a weight decomposition determined by

$\displaystyle g_{4}\in M(\mathcal{L}_{c}S^{4})_{\epsilon_{0}},\qquad g_{7}\in
M(\mathcal{L}_{c}S^{4})_{2\epsilon_{0}},$ $\displaystyle sg_{4}\in
M(\mathcal{L}_{c}S^{4})_{\epsilon_{0}-\epsilon_{1}},\qquad sg_{7}\in
M(\mathcal{L}_{c}S^{4})_{2\epsilon_{0}-\epsilon_{1}},$ $\displaystyle w\in
M(\mathcal{L}_{c}S^{4})_{\epsilon_{1}},$

as per Corollary 3.5.

######  Example 3.9 ($k=2$: The double cyclification
$\mathcal{L}^{2}_{c}S^{4}$).

We now consider Example 2.6. Again, iterating the application of Theorem 3.4,
the maximal torus of $\operatorname{Aut}M(\mathcal{L}^{2}_{c}S^{4})$ is
identified canonically with the product
$\mathbb{G}_{m}^{3}=(\mathbb{G}_{m}\times\mathbb{G}_{m})\times\mathbb{G}_{m}$,
where the first factor $\mathbb{G}_{m}\times\mathbb{G}_{m}$ refers to the
maximal torus of $\operatorname{Aut}M(\mathcal{L}_{c}S^{4})$ identified in the
previous example. Continuing the use of notation of Corollary 3.5, we obtain a
weight decomposition of $M(\mathcal{L}^{2}_{c}S^{4})$, which is determined on
its generators (see Example 2.6) as follows:

$\displaystyle g_{4}\in M_{\epsilon_{0}},\qquad g_{7}\in M_{2\epsilon_{0}},$
$\displaystyle s_{1}g_{4}\in M_{\epsilon_{0}-\epsilon_{1}},\qquad
s_{1}g_{7}\in M_{2\epsilon_{0}-\epsilon_{1}},$ $\displaystyle w_{1}\in
M_{\epsilon_{1}},$ (33) $\displaystyle s_{2}g_{4}\in
M_{\epsilon_{0}-\epsilon_{2}},\qquad s_{2}g_{7}\in
M_{2\epsilon_{0}-\epsilon_{2}},$ $\displaystyle s_{2}s_{1}g_{4}\in
M_{\epsilon_{0}-\epsilon_{1}-\epsilon_{2}},\qquad s_{2}s_{1}g_{7}\in
M_{2\epsilon_{0}-\epsilon_{1}-\epsilon_{2}},$ $\displaystyle s_{2}w_{1}\in
M_{\epsilon_{1}-\epsilon_{2}},\qquad w_{2}\in M_{\epsilon_{2}},$

where, for brevity, we have been writing $M$ for
$M(\mathcal{L}^{2}_{c}S^{4})$.

######  Example 3.10 ($k=3$: The triple cyclification
$\mathcal{L}^{3}_{c}S^{4}$).

Now we consider Example 2.8. Again the maximal torus of
$\operatorname{Aut}M(\mathcal{L}^{3}_{c}S^{4})$ splits canonically to become
$\mathbb{G}_{m}^{4}=(\mathbb{G}_{m}^{3})\times\mathbb{G}_{m}$, where the first
factor comes from the double cyclification. The resulting weight decompositon
of $M=M(\mathcal{L}^{3}_{c}S^{4})$ repeats the formulas (33) verbatim for the
weights of those generators which are the generators of
$M(\mathcal{L}^{2}_{c}S^{4})$. The weight of $w_{3}$ is $\epsilon_{3}$. For
the weights of generators of the type $s_{3}g$, where $g$ is a generator on
the list (33), the formulas are the same as (33), except that weight
$\epsilon_{3}$ gets subtracted, e.g.,

$\displaystyle s_{3}g_{4}\in M_{\epsilon_{0}-\epsilon_{3}},\qquad
s_{3}w_{2}\in M_{\epsilon_{2}-\epsilon_{3}}.$

The weight $\epsilon_{1}-\epsilon_{2}-\epsilon_{3}$ will not be present, as
$s_{3}s_{2}w_{1}$ gets truncated to zero.

######  Example 3.11 ($k\geq 3$: The $k$-fold cyclification
$\mathcal{L}^{k}_{c}S^{4}$).

Let us say a few words on the general pattern we see for $k\geq 3$. All the
weights of the generating space $V$ for $S(V)=M(\mathcal{L}^{k}_{c}S^{4})$
will be of the form

$\displaystyle\epsilon_{0}-$
$\displaystyle\sum_{j=1}^{l}\epsilon_{i_{j}},\qquad\text{where $0\leq l\leq 3$
and $1\leq i_{1}<\dots<i_{l}\leq k$},$ $\displaystyle 2\epsilon_{0}-$
$\displaystyle\sum_{j=1}^{l}\epsilon_{i_{j}},\qquad\text{where $0\leq l\leq 6$
and $1\leq i_{1}<\dots<i_{l}\leq k$},$
$\displaystyle\epsilon_{i},\;\qquad\qquad 1\leq i\leq k,$
$\displaystyle\epsilon_{i}$ $\displaystyle-\epsilon_{j},\qquad\quad 1\leq
i<j\leq k.$

Each of the corresponding weight spaces in $V$ will be one-dimensional; see
Example 2.9.

###  3.4 Topological interpretation of toroidal symmetries

In this section, we interpret the toroidal symmetries (29) and (30) as
rational (real) homotopy equivalences. The idea of doing that was suggested by
A. Bondal. These symmetries have physical interpretation of trombone and torus
rescaling symmetries, mentioned in section 3.3.

####  3.4.1 Toroidal symmetries coming from $S^{4}$

Let us start with the symmetry (30). Given an integer $n\in\mathbb{Z}$, define
$\varphi_{0}(n):S^{4}\to S^{4}$ to be any continuous map of degree $n$. Such a
map induces a homomorphism, given by multiplication by $n$:

$\displaystyle\varphi_{0}(n)_{*}:\pi_{4}(S^{4})$
$\displaystyle\longrightarrow\pi_{4}(S^{4})$ $\displaystyle x$
$\displaystyle\longmapsto nx$

on the degree-four homotopy group $\pi_{4}(S^{4})$ of $S^{4}$. It also induces
the identity morphism on $\pi_{0}(S^{4})$. Recall that
$\pi_{7}(S^{4})\cong\mathbb{Z}\oplus\mathbb{Z}_{12}$, where the free part may
be canonically identified with the subgroup
$\mathbb{Z}y\subset\pi_{7}(S^{4})$, $y$ being the class of the Hopf fibration
$S^{7}\to S^{4}$. We know how $\varphi_{0}(n)$ acts on $y$:

$\displaystyle\varphi_{0}(n)_{*}:y$ $\displaystyle\mapsto n^{2}y,$

because the Whitehead square $[x,x]$ of the generator
$x\in\pi_{4}(S^{4})\cong\mathbb{Z}x$, the homotopy class of
$\operatorname{id}:S^{4}\to S^{4}$, is twice the generator $y$ of
$\mathbb{Z}y\subset\pi_{7}(S^{4})$ (see [FSS19b] for explanation in this
context):

$[x,x]=2y.$

When we pass to rational homotopy groups, all torsion disappears, and we have
$\pi_{4}(S^{4})\otimes\mathbb{Q}=\mathbb{Q}x$ and
$\pi_{7}(S^{4})\otimes\mathbb{Q}=\mathbb{Q}y$. Since there are no other
rational homotopy groups $\pi_{i}(S^{4})\otimes\mathbb{Q}$, $i\geq 1$, we see
that, when $n$ is nonzero, $\varphi_{0}(n):S^{4}\to S^{4}$ is a rational
homotopy equivalence. Hence, it has an inverse $\varphi_{0}(n)^{-1}:S^{4}\to
S^{4}$ in the rational homotopy category. We may denote this morphism
$\varphi_{0}(1/n):=\varphi_{0}(n)^{-1}$, given that it acts on the generator
$x\in\pi_{4}(S^{4})\otimes\mathbb{R}$ as

$x\mapsto\frac{1}{n}x.$

Composing a map $\varphi_{0}(p):S^{4}\to S^{4}$ of degree
$p\in\mathbb{Z}\mathbin{\vbox{\hbox{$\scriptscriptstyle\mathchoice{\hbox
to0.0pt{$\displaystyle{\setminus}$\hss}}{\hbox
to0.0pt{$\textstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptstyle{\setminus}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle{\setminus}$\hss}}{\hskip
0.2pt\setminus}$}}}\\{0\\}$ and a map $\varphi_{0}(1/q)$ for $q\in\mathbb{N}$,
we are getting an automorphism $\varphi_{0}(p/q):S^{4}\to S^{4}$ in the
rational homotopy category. This way we get a group homomorphism

$\displaystyle\mathbb{Q}^{\times}$
$\displaystyle\longrightarrow\operatorname{Aut}_{\mathbb{Q}}S^{4},$
$\displaystyle p/q$ $\displaystyle\longmapsto\varphi_{0}(p/q),$

where $\operatorname{Aut}_{\mathbb{Q}}S^{4}$ stands for the automorphism group
in the rational homotopy category. Given our description of
$\operatorname{Aut}M(S^{4})$ in Example 3.7 and the fact that the rational
homotopy category (of rational, nilpotent, finite-type spaces) is equivalent
(via a contravariant functor) to the category of minimal Sullivan algebras
over $\mathbb{Q}$, we see that (34) actually defines an isomorphism

$\mathbb{Q}^{\times}\xrightarrow{\;\sim\;}\operatorname{Aut}_{\mathbb{Q}}S^{4}\xrightarrow{\;\sim\;}\operatorname{Aut}M(S^{4})(\mathbb{Q}),$

where $\operatorname{Aut}M(S^{4})(\mathbb{Q})$ is the group of rational points
of the algebraic group $\operatorname{Aut}M(S^{4})$. Since the action formulas
are polynomial, the group isomorphism defines an isomorphism of algebraic
groups over $\mathbb{Q}$:

$\displaystyle\mathbb{G}_{m}$
$\displaystyle\longrightarrow\operatorname{Aut}M(S^{4}),$ (34) $\displaystyle
r$ $\displaystyle\longmapsto\varphi_{0}(r).$

Via the action on the target $S^{4}$ of $\mathcal{L}^{k}_{c}S^{4}$, the
isomomorphism (34) may be canonically lifted, as in 3.3, to a
$\mathbb{Q}$-isomomorphism

$\mathbb{G}_{m}\xrightarrow{\;\sim\;}\operatorname{Aut}M(\mathcal{L}^{k}_{c}S^{4}),$

which is exactly the action of the multiplicative group $\mathbb{G}_{m}$ on
$M(\mathcal{L}^{k}_{c}S^{4})$ coming from the action of $\mathbb{G}_{m}$ on
$g_{4}\in M(\mathcal{L}^{k}_{c}S^{4})$, as in (30) and (32). Summarizing, we
have:

######  Proposition 3.12 (Toroidal symmetries from $S^{4}$).

Consider the degree $n$ maps $\varphi_{0}(n):S^{4}\to S^{4}$, for
$n\in\mathbb{Z}\setminus\\{0\\}$.

  1. (i)

These maps are invertible in the rational homotopy category.

  2. (ii)

The compositions of these maps with their inverses gives a group isomorphism
$\mathbb{Q}^{\times}\xrightarrow{\sim}\operatorname{Aut}M(S^{4})(\mathbb{Q})$,
which defines naturally an isomorphism of algebraic groups over $\mathbb{Q}$

$\mathbb{G}_{m}\xrightarrow{\;\sim\;}\operatorname{Aut}M(S^{4})\;.$

  3. (iii)

This lifts canonically to an algebraic-group homomorphism
$\mathbb{G}_{m}\xrightarrow{\;\sim\;}\operatorname{Aut}M(\mathcal{L}^{k}_{c}S^{4})$
for $k\geq 1$, which provides an action of $\mathbb{G}_{m}$ on the Sullivan
minimal models of the cyclifications of $S^{4}$.

####  3.4.2 Toroidal symmetries coming from $S^{1}$

The situation with the action (29) is subtler. Let us consider the general
case of $\mathcal{L}_{c}Z$. For $n\in\mathbb{Z}$, the $n$-fold winding map

$\displaystyle S^{1}$ $\displaystyle\longrightarrow S^{1}$ $\displaystyle z$
$\displaystyle\longmapsto z^{n}$ (35)

induces a continuous map $\psi(n):\mathcal{L}Z\to\mathcal{L}Z$:

$\psi(n)(f)(z):=f(z^{n})$

for $f\in\mathcal{L}Z=\operatorname{Map}(S^{1},Z)$ and $z\in
S^{1}\subset\mathbb{C}$. For $n\neq 0$, the map $\psi(n)$ is a rational (and
real) homotopy equivalence, because so is the power map (35). However,
$\psi(n)$ is not $S^{1}$-equivariant, unless $n=1$, as, for instance, for the
right action $(f\cdot z^{\prime})(z):=f(z^{\prime}z)$ of $S^{1}$ on
$\mathcal{L}Z$, we have

$\psi(n)(f\cdot z^{\prime})(z)=f(z^{\prime}z^{n})\neq
f((z^{\prime}z)^{n})=(\psi(n)(f)\cdot z^{\prime})(z).$

Moreover, we can say that

$\psi(n)(f\cdot z^{\prime})=\psi(n)(f)\cdot\root n \of{z^{\prime}}$

for any choice of the $n$th root, or, better, just using the rational homotopy
inverse of the rational equivalence (35). Accordingly, the map $\psi(n)$ would
not induce a map $\mathcal{L}_{c}Z\to\mathcal{L}_{c}Z$ of the homotopy
quotient $\mathcal{L}_{c}Z=\mathcal{L}Z\times_{S^{1}}ES^{1}$. Indeed, by
definition of the quotient $\mathcal{L}Z\times_{S^{1}}ES^{1}$, a point
$(f\cdot z^{\prime},e)\in\mathcal{L}_{c}Z$ is equivalent to the point
$(f,z^{\prime}\cdot e)$, but $(\psi(n)(f\cdot
z^{\prime}),e)=(\psi(n)(f)\cdot\root n \of{z^{\prime}},e)\sim(\psi(n)(f),\root
n \of{z^{\prime}}e)$, which is not equivalent to $(\psi(n)(f),z^{\prime}\cdot
e)$.

What saves the situation is that the map $\psi(n)$ extends in the rational
homotopy category to a morphism $\mathcal{L}Z\times
ES^{1}\to\mathcal{L}Z\times ES^{1}$ that respects the equivalence relation

$(f\cdot z^{\prime},e)\sim(f,z^{\prime}\cdot e)$ (36)

and thereby descends to the quotient $\mathcal{L}Z\times_{S^{1}}ES^{1}$.
Indeed, note that the topological group morphism (35) induces a continuous map
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n):ES^{1}\to
ES^{1}$ of the total space $ES^{1}$ of the universal bundle $ES^{1}\to BS^{1}$
by functoriality. In the standard simplicial model of $ES^{1}$, this map
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)$
can be expressed as $[z_{0},\dots,z_{p}]\mapsto[z_{0}^{n},\dots,z_{p}^{n}]$,
where $[z_{0},\dots,z_{p}]$, $z_{i}\in S^{1}$, is a $p$-simplex of $ES^{1}$.
The map
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n):ES^{1}\to
ES^{1}$ is not $S^{1}$-equivariant, either. Say, for the (left) action
$z^{\prime}\cdot[z_{0},\dots,z_{p}]:=[z^{\prime}z_{0},\dots,z^{\prime}z_{p}]$
of $S^{1}$ on $ES^{1}$, we have

${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)(z^{\prime}\cdot[z_{0},\dots,z_{p}])=[(z^{\prime}z_{0})^{n},\dots,(z^{\prime}z_{p})^{n}]\neq[z^{\prime}z_{0}^{n},\dots,z^{\prime}z_{p}^{n}]=z^{\prime}\cdot{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)([z_{0},\dots,z_{p}]).$

What we have is

${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)(z^{\prime}\cdot
e)=(z^{\prime})^{n}\cdot{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)(e),\qquad
e\in ES^{1}.$

For $n\neq 0$, the map
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)$
is a rational (and real) homotopy equivalence and therefore has a rational
homotopy inverse
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}$,
so that

${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}(z^{\prime}\cdot
e)=\root n
\of{z^{\prime}}\cdot{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}(e).$

Now, the morphism

$\psi(n)\times{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}:\mathcal{L}Z\times
ES^{1}\to\mathcal{L}Z\times ES^{1},$

which is invertible in the rational (real) homotopy category, respects the
equivalence relation (36):

$\psi(n)\times{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}(f\cdot
z^{\prime},e))=(\psi(n)(f)\cdot\root n
\of{z^{\prime}},{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}(e))\\\
\sim\;\;\;(\psi(n)(f),\root n
\of{z^{\prime}}\cdot{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}(e))=\psi(n)\times{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}(f,z^{\prime}\cdot
e),$

and therefore induces a rational automorphism of $\mathcal{L}_{c}Z$, which we
denote by

$\varphi_{1}(n):=\psi(n)\times{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}:\mathcal{L}_{c}Z\longrightarrow\mathcal{L}_{c}Z\,.$

As in the case of $\varphi_{0}(n)$ in section 3.4.1, the rational homotopy
equivalence
$\varphi_{1}(n):\mathcal{L}_{c}Z\xrightarrow{\sim}\mathcal{L}_{c}Z$ extends to
a group homomorphism

$\displaystyle\mathbb{Q}^{\times}$
$\displaystyle\longrightarrow\operatorname{Aut}_{\mathbb{Q}}\mathcal{L}_{c}Z,$
$\displaystyle r$ $\displaystyle\longmapsto\varphi_{1}(r).$

From this, we get a $\mathbb{Q}$-algebraic-group homomorphism

$\mathbb{G}_{m}\longrightarrow\operatorname{Aut}M(\mathcal{L}_{c}Z),$

so as $r\in\mathbb{G}_{m}$ acts on $w\in H^{2}(BS^{1};\mathbb{R})\subset
M(BS^{1})$ as $r^{-1}w$, because
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)$
induces the action $w\mapsto nw$ on degree-two cohomology and we used
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(n)^{-1}$,
and $r\in\mathbb{G}_{m}$ acts on $sv\in V[1]\subset M(\mathcal{L}Z)$ as
$r(sv)$. This motivates formulas (29). Summarizing, we have:

######  Proposition 3.13 (Toroidal symmetries from $S^{1}$).

The $n$-fold winding maps $S^{1}\to S^{1}$, $n\in\mathbb{Z}\setminus\\{0\\}$,
induce morphisms $\varphi_{1}(n):\mathcal{L}_{c}Z\to\mathcal{L}_{c}Z$ in the
rational homotopy category by the construction above.

  1. (i)

These morphisms are invertible, and the compositions of them with their
inverses give a group homomorphism
$\mathbb{Q}^{\times}\xrightarrow{\sim}\operatorname{Aut}M(\mathcal{L}_{c}Z)(\mathbb{Q})$,
which defines naturally a morphism of $\mathbb{Q}-algebraicgroups$

$\mathbb{G}_{m}\xrightarrow{\;\;\sim\;\;}\operatorname{Aut}M(\mathcal{L}_{c}Z)\;.$

  2. (ii)

For an iterated cyclic loop space $\mathcal{L}_{c}^{k}Z$, $k\geq 1$, the
algebraic-group morphisms corresponding to different iterations commute and
thereby define an algebraic-group morphism
$(\mathbb{G}_{m})^{k}\xrightarrow{\sim}\operatorname{Aut}M(\mathcal{L}^{k}_{c}Z),$
which provides an action of $(\mathbb{G}_{m})^{k}$ on the Sullivan minimal
model of the $k$-fold cyclification $\mathcal{L}_{c}^{k}Z$ of $Z$.

###  3.5 Toroidal symmetries of type IIB

Since the type IIB model (see section 2.6) falls out of the previous sequence
of cyclifications of $S^{4}$, we need to treat it separately. We will work
with our “unstable” model $M(IIB)=(S(V),d)$ of type IIB; see (20).

By 3.1, to identify a maximal $\mathbb{R}$-split torus $T^{B}$ of
$\operatorname{Aut}M(IIB)$, we need to start with computing $\ker d\cap
V=\mathbb{R}h_{3}\oplus\mathbb{R}\omega_{1}$ in the notation of the system
(20). Thus, a maximal $\mathbb{R}$-split torus is at most 2-dimensional. The
explicit formulas below identify a 2-dimensional split torus acting faithfully
on $M(IIB)$, which has to be maximal by the dimension argument.

Take the $\mathbb{R}$-split torus $T^{B}:=\mathbb{G}_{m}^{2}$ acting as the
group of diagonal matrices on the real plane spanned by $h_{3}$ and
$\omega_{1}$. Denote by $\beta_{0}=(1,0)$ the weight of $h_{3}$ and by
$\beta_{1}=(0,1)$ the weight of $\omega_{1}$, so that we have the action

$t\cdot h_{3}=t^{\beta_{0}}h_{3},\qquad
t\cdot\omega_{1}=t^{\beta_{1}}\omega_{1},\qquad\qquad t\in
T^{B}=\mathbb{G}_{m}^{2}.$ (37)

Then, with (20), we have the following:

######  Proposition 3.14 (Toroidal symmetry in type IIB).

The formulas

$\displaystyle t\cdot\omega_{\,3}=t^{\beta_{0}+\beta_{1}}\omega_{\,3},\qquad
t\cdot\omega_{\,5}=t^{2\beta_{0}+\beta_{1}}\omega_{\,5},$ (38) $\displaystyle
t\cdot h_{7}=t^{3\beta_{0}+2\beta_{1}}h_{7},\qquad
t\cdot\omega_{\,7}=t^{3\beta_{0}+\beta_{1}}\omega_{\,7}$

extend the action (37) to an action of the torus $T^{B}=\mathbb{G}_{m}^{2}$ on
$M(IIB)$. This identifies $T^{B}$ as a maximal $\mathbb{R}$-split torus of
$\operatorname{Aut}M(IIB)$.

##  4 The $E_{k}$ symmetry of iterated cyclic loop spaces

Here we unravel the $E_{k}$ symmetry of the iterated cyclic loop spaces
$\mathcal{L}^{k}_{c}S^{4}$, described in section 2.5, where $E_{k}$ for $k\geq
0$ is understood in the sense of Table 1 and Table 2 in the Introduction. Our
goal is to use the toroidal symmetries of the cyclic loop spaces
$\mathcal{L}_{c}^{k}S^{4}$ from section 3 and build certain canonical
combinatorial data: “a lattice $N_{k}$ with an inner product $(-,-)$ and a
distinguished element $K_{k}^{*}\in N_{k}$”, similar to the triple
$(N_{k},(-,-),\mathcal{K}_{k})$ in the theory of del Pezzo surfaces, see
below. This will automatically produce the $E_{k}$ root system, see Theorem
4.6.

###  4.1 Reminder: the combinatorial data from del Pezzo surfaces

Let us recall how the triple $(N_{k},(-,-),\mathcal{K}_{k})$ shows up in the
del Pezzo theory, for the sake of motivation and setting up notation. The
Picard group $\operatorname{Pic}\mathbb{B}_{k}$ happens to be isomorphic to
the 2nd cohomology group $H^{2}(\mathbb{B}_{k};\mathbb{Z})$. This is a lattice
with basis $\mathcal{H},\mathcal{E}_{1},\dots,\mathcal{E}_{k}$:

$H^{2}(\mathbb{B}_{k};\mathbb{Z})\cong\mathbb{Z}\mathcal{H}\oplus\mathbb{Z}\mathcal{E}_{1}\oplus\cdots\oplus\mathbb{Z}\mathcal{E}_{k},$

where $\mathcal{H}$ is the class of the proper transform of the line (here
also a hyperplane) $\mathbb{C}\mathbb{P}^{1}$ in $\mathbb{CP}^{2}$ and
$\mathcal{E}_{i}$ is the class of the exceptional divisor over the blowup
point $x_{i}\in\mathbb{CP}^{2}$. See [Ma74][De80][Be96][KSC04]. The 2nd
integral cohomology has a natural inner product given by the intersection
form:

$\mathcal{H}\cdot\mathcal{H}=1,\quad\mathcal{H}\cdot\mathcal{E}_{i}=0,\quad\mathcal{E}_{i}\cdot\mathcal{E}_{j}=-\delta_{ij},\qquad
1\leq i,j\leq k\;.$ (39)

Thus the intersection matrix of $\mathbb{B}_{k}$ is given by the Lorentzian
form $Q={\rm diag}(1,-1,-1,\dots,-1)$. Hence
$H^{2}(\mathbb{B}_{k};\mathbb{Z})\linebreak[0]\cong\mathbb{Z}^{1,k}$ and
$\mathbb{B}_{k}$ has Betti numbers $b_{2}^{+}=1$, $b_{2}^{-}=k$, with
signature $\sigma=1-k$.

The _canonical class_ of $\mathbb{B}_{k}$ may be expressed as

$\mathcal{K}_{k}:=\Omega^{2}_{\mathbb{B}_{k}}=-3\mathcal{H}+\mathcal{E}_{1}+\dots+\mathcal{E}_{k},$
(40)

while the ample _anticanonical class_ becomes

$-\mathcal{K}_{k}=3\mathcal{H}-\mathcal{E}_{1}-\dots-\mathcal{E}_{k}.$ (41)

The _degree of a divisor $\mathcal{D}\in\operatorname{Pic}(\mathbb{B}_{k})$_
is measured with respect to the anticanonical map as:

$\deg\mathcal{D}:=-\mathcal{K}_{k}\cdot\mathcal{D}\;.$ (42)

The “outlier” del Pezzo surface
$\mathbb{B}_{1}^{\prime}:=\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}$
of degree 8 has Picard group of rank 2:

$\operatorname{Pic}(\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1})\cong
H^{2}(\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1};\mathbb{Z})\cong\mathbb{Z}\mathfrak{l}_{1}\oplus\mathbb{Z}\mathfrak{l}_{2}\;,$

where $\mathfrak{l}_{1}$ and $\mathfrak{l}_{2}$ are the classes of the two
$\mathbb{C}\mathbb{P}^{1}$ factors. The intersection pairing works as follows
(cf. (39)):

$\mathfrak{l}_{1}\cdot\mathfrak{l}_{1}=\mathfrak{l}_{2}\cdot\mathfrak{l}_{2}=0,\quad\mathfrak{l}_{1}\cdot\mathfrak{l}_{2}=1\;.$

The anticanonical class is given by
$-\mathcal{K}_{\mathbb{B}_{1}^{\prime}}=2\mathfrak{l}_{1}+2\mathfrak{l}_{2}$.

###  4.2 The “Cartan subalgebra” and weight lattice

In this section, we will present the first element of the triple arising from
$\mathcal{L}_{c}^{k}S^{4}$, the lattice. The idea is to use the weight lattice
coming from the weight decompositions of Corollary 3.5 and of section 3.3 and
use the Lie algebra $\mathfrak{h}_{k}=\operatorname{Lie}(T^{k+1})$ of the
maximal real split torus $T^{k+1}$ of
$\operatorname{Aut}M(\mathcal{L}_{c}^{k+1}S^{4})$ with its canonical
factorization $T\xrightarrow{\sim}\mathbb{G}_{m}^{k+1}$, see 3.6. This Lie
algebra constitutes the infinitesimal symmetries corresponding to the toroidal
symmetries of section 3.2–section 3.3 and acts on the Sullivan minimal model
$M(\mathcal{L}_{c}^{k+1}S^{4})$ by derivations. That is, we have a Lie algebra
homomorphism:

$\mathfrak{h}_{k}\longrightarrow\operatorname{Der}M(\mathcal{L}_{c}^{k}S^{4}),$

which comes from taking the differential of the action

$T^{k+1}\longrightarrow\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4}).$

Under the action of the Lie algebra $\mathfrak{h}_{k}$ on
$M=M(\mathcal{L}_{c}^{k}S^{4})$, the weight decomposition of section 3.1 and
Corollary 3.5 becomes

$M=\bigoplus_{\alpha\in\mathbf{P}(\mathfrak{h}_{k})}M_{\alpha},$

where
$\mathbf{P}(\mathfrak{h}_{k})\subseteq\mathfrak{h}_{k}^{*}=\operatorname{Hom}_{\mathbb{R}}(\mathfrak{h}_{k},\mathbb{R})$
is the _weight lattice_ , the image of the character group

$\mathfrak{X}(T)=\operatorname{Hom}_{\mathbb{R}}(T,\mathbb{G}_{m})\linebreak[0]\cong\mathbb{Z}^{k+1}$

under the differential map

$\displaystyle\mathfrak{X}(T)$
$\displaystyle\longrightarrow\mathfrak{h}_{k}^{*},$ $\displaystyle\beta$
$\displaystyle\longmapsto d\beta.$

The Lie algebra $\mathfrak{h}_{k}$ acts on each weight space $M_{\alpha}$ with
the weight $\alpha$:

$M_{\alpha}=\\{m\in M\;|\;h\cdot m=\alpha(h)m\quad\text{for all
$h\in\mathfrak{h}_{k}$}\\}.$

As we will see soon, the Lie algebra $\mathfrak{h}_{k}$ is an avatar of the
Cartan subalgebra of the Lie algebra of type $E_{k}$ of “hidden” symmetries of
the cyclic loop spaces of the four-sphere.

######  Theorem 4.1 (Bases for the Lie algebra of symmetries and its dual).

  1. (i)

The $(k+1)$-dimensional real abelian Lie algebra
$\mathfrak{h}_{k}=\operatorname{Lie}(T^{k+1})\subseteq\operatorname{Der}M(\mathcal{L}_{c}^{k}S^{4})$
of the maximal $\mathbb{R}$-split torus $T=T^{k+1}$ of the algebraic group
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ has a canonical basis
$\\{h_{0},h_{1},\linebreak[0]\dots,\linebreak[1]h_{k}\\}$.

  2. (ii)

The weights $\epsilon_{0}$ of $g_{4}$ and $\epsilon_{i}$ of $w_{i}$ for $1\leq
i\leq k$ give a canonical basis
$\\{\epsilon_{0},\epsilon_{1},\dots,\epsilon_{k}\\}$ of the vector space
$\mathfrak{h}_{k}^{*}$. This is also a basis of the weight lattice
$\mathbf{P}(\mathfrak{h}_{k})\subseteq\mathfrak{h}_{k}^{*}$.

###### Proof.

(i) The factorization $T=\mathbb{G}_{m}^{k+1}$ of the maximal split torus of
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ was canonically defined from
compatibility with the iterated cyclic loop space structure (see 3.6):

$T=\\{(t_{0},t_{1},\dots,t_{k})\;|\;t_{i}\in\mathbb{G}_{m}\\}\;,$

with $t_{0}$ acting on $g_{4}$ by $t_{0}g_{4}$ and trivially on
$w_{1},\dots,w_{k}$ and $t_{i}$ acting on $w_{i}$ by $t_{i}^{-1}w_{i}$ and
trivially on the other $w_{j}$’s and $g_{4}$. This factorization implies a
canonical factorization of the tangent space $\mathfrak{h}_{k}$ at
$\operatorname{id}\in T$: $\mathfrak{h}_{k}=\mathbb{R}^{k+1}$. It determines a
basis $\\{h_{0},h_{1},\dots,h_{k}\\}$, the standard basis of
$\mathbb{R}^{k+1}$.

(ii) The identification of $T$ as $\mathbb{G}_{m}^{k+1}$ in 3.6 was derived
iteratively from Theorem 3.4 as coming from the action of the torus on the
generators $g_{4}$, $w_{1}$, …, and $w_{k}$ of $M(\mathcal{L}_{c}^{k}S^{4})$.
Therefore, the weights of these generators provide a natural set of weights
$\epsilon_{0},\epsilon_{1},\dots,\epsilon_{k}$. Differentiating the action of
$T$ on these generators (see the previous paragraph), we obtain

$\displaystyle h_{0}\cdot g_{4}$ $\displaystyle=g_{4},\qquad h_{0}\cdot
w_{i}=0\qquad\;\,\text{ for $i=1,\dots,k$},$ $\displaystyle h_{i}\cdot g_{4}$
$\displaystyle=0,\qquad\;\;h_{i}\cdot w_{j}=-\delta_{ij}\quad\text{ for
$i,j=1,\dots,k$}.$

This implies that

$\displaystyle\epsilon_{0}(h_{0})$
$\displaystyle=1,\qquad\epsilon_{i}(h_{0})=0\qquad\;\,\text{ for
$i=1,\dots,k$},$ $\displaystyle\epsilon_{0}(h_{i})$
$\displaystyle=0,\qquad\epsilon_{j}(h_{i})=-\delta_{ij}\quad\text{ for
$i,j=1,\dots,k$}.$

From these equations, we conclude that
$\\{\epsilon_{0},\epsilon_{1},\dots,\epsilon_{k}\\}$ form a basis of the dual
vector space $\mathfrak{h}_{k}^{*}$ as well as the weight lattice
$\mathbf{P}(\mathfrak{h}_{k})\subseteq\mathfrak{h}_{k}^{*}$. ∎

From section 3.3, we can compute the action of the Lie algebra
$\mathfrak{h}_{k}$ on the other generators of the minimal DGCA
$M(\mathcal{L}_{c}^{k}S^{4})$. For instance:

######  Example 4.2 (Action of the Lie algebra on the cyclic loop space in
type IIA).

For $k=1$, we have

$\displaystyle h\cdot g_{4}=\epsilon_{0}(h)g_{4},\quad h\cdot
g_{7}=2\epsilon_{0}(h)g_{7},\quad h\cdot
sg_{4}=(\epsilon_{0}(h)-\epsilon_{1}(h))sg_{4},$ $\displaystyle h\cdot
sg_{7}=(2\epsilon_{0}(h)-\epsilon_{1}(h))sg_{7},\quad h\cdot
w=\epsilon_{1}(h)w,$

where

$\displaystyle\epsilon_{0}(h_{0})$
$\displaystyle=1,\qquad\epsilon_{0}(h_{1})=0,$
$\displaystyle\epsilon_{1}(h_{0})$
$\displaystyle=0,\qquad\epsilon_{1}(h_{1})=-1.$

######  Corollary 4.3 (The dual lattices and inner product).

  1. (i)

The lattice
$\mathfrak{h}_{k}^{\mathbb{Z}}:=\mathbb{Z}h_{0}\oplus\mathbb{Z}h_{1}\oplus\dots\oplus\mathbb{Z}h_{k}\subset\mathfrak{h}_{k}$
is the dual of the weight lattice
$\mathbf{P}(\mathfrak{h}_{k})=\mathbb{Z}\epsilon_{0}\oplus\dots\oplus\mathbb{Z}\epsilon_{k}\subset\mathfrak{h}_{k}^{*}$.

  2. (ii)

The nondegenerate bilinear form
$\mathfrak{h}_{k}\otimes\mathfrak{h}_{k}\to\mathbb{R}$ determined by the
isomorphism

$\displaystyle\mathfrak{h}_{k}$ $\displaystyle\to\mathfrak{h}_{k}^{*}$
$\displaystyle h_{i}$ $\displaystyle\mapsto\epsilon_{i}$

provides the vector space $\mathfrak{h}_{k}$ with a canonical Lorentzian inner
product $(-,-)$. This inner product satisfies the formulas:

$(h_{0},h_{0})=1,\qquad(h_{i},h_{j})=-\delta_{ij}\quad\text{ for \; $i\geq 0$,
$j\geq 1$.}$

  3. (iii)

The inner product induced on the dual space $\mathfrak{h}_{k}^{*}$ is given by
the formulas:

$(\epsilon_{0},\epsilon_{0})=1,\qquad(\epsilon_{i},\epsilon_{j})=-\delta_{ij}\quad\text{
for \; $i\geq 0$, $j\geq 1$.}$

###  4.3 The “anticanonical class”

In this section, we identify a distinguished element
$-K_{k}\in\mathfrak{h}_{k}^{\mathbb{Z}}$, analogous to the anticanonical class
$-\mathcal{K}_{k}$ (see the Introduction, section 1, and section 4.1) of the
del Pezzo surface $\mathbb{B}_{k}$. Recall, that the anticanonical class
$-\mathcal{K}_{k}$ of the del Pezzo surface “acts” on the Picard group
$\operatorname{Pic}(\mathbb{B}_{k})\cong H^{2}(\mathbb{B}_{k};\mathbb{Z})$ by
degree:

$\deg\mathcal{D}:=-\mathcal{K}_{k}\cdot\mathcal{D},\qquad\mathcal{D}\in\operatorname{Pic}(\mathbb{B}_{k}).$
(43)

Here the “action” is understood as the intersection product
$\operatorname{Pic}(\mathbb{B}_{k})\otimes\operatorname{Pic}(\mathbb{B}_{k})\to\mathbb{Z}$.
The _degree of a divisor_ $\mathcal{D}\in\operatorname{Pic}(\mathbb{B}_{k})$
is defined using the anticanonical morphism
$f:\mathbb{B}_{k}\to\mathbb{C}\mathbb{P}^{d}$, and

$d=h^{0}(\mathbb{B}_{k},-\mathcal{K}_{k})-1=(-\mathcal{K}_{k})\linebreak[0]\cdot(-\mathcal{K}_{k})=9-k$

is known as the _degree of the del Pezzo surface $\mathbb{B}_{k}$_, whence
$-\mathcal{K}_{k}$ is the pullback $f^{*}\mathcal{H}$ of the hyperplane class
$\mathcal{H}\in\operatorname{Pic}(\mathbb{C}\mathbb{P}^{d})$ and formula (43)
makes sense; see, e.g., [Do12] and also section 4.1.

In the case of cyclic loop spaces, we have been dealing with one notion of
degree, in the sense of $\mathbb{Z}$-grading of the Sullivan minimal model
$M(\mathcal{L}_{c}^{k}S^{4})$. There is another one, natural for the Quillen
minimal model. The significance of that other notion of degree for us is
coming from the fact that it corresponds to the degree of the C-fields, i.e.,
the potentials $C_{3}$ and $C_{6}$ of the basic fields $G_{4}$ and $G_{7}$,
see (4), and thereby to the dimension of the corresponding branes, the M2- and
M5-branes, respectively. This other notion of degree just differs from the
degree we have been using on the generators of the Sullivan minimal model by
one, but has a homotopy-theoretic origin, as we now explain.

We have seen the Quillen model in section 2.3 and at the end of section 2.4.
The maximal split torus $T^{k+1}$ and its Lie algebra $\mathfrak{h}_{k}$ act
on the Quillen minimal model $Q(\mathcal{L}_{c}^{k}S^{4})$ with the same
weights as on the generators of the Sullivan minimal model
$M=M(\mathcal{L}_{c}^{k}S^{4})$. Indeed, the weights on the space
$V=M^{+}/(M^{+})^{2}$ of generators, its dual $V^{*}$ and its degree shift
$V^{*}[1]=Q(\mathcal{L}_{c}^{k}S^{4})$ will just be the same as those on $V$.
Let us denote the elements of the basis of
$V^{*}[1]=Q(\mathcal{L}_{c}^{k}S^{4})$ dual to the basis

$\\{g_{4},\,g_{7},\,w_{1},\,s_{1}g_{4},\,s_{1}g_{7},\,s_{1}w_{1},\dots,\,w_{k}\\}\qquad\text{by}\qquad
e_{3},\,e_{6},\,x_{1},\,s_{1}e_{3},\,s_{1}e_{6},\,s_{1}x_{1},\dots,\,x_{k},$

respectively. From the fact that $S(Q(Z)[-1]^{*}),d)$ is the Chevalley-
Eilenberg cochain complex of the graded Lie algebra $Q(Z)$, one can deduce
that the subspace dual to $\ker d\cap V[1]=\ker d\cap Q(Z)^{*}$ generates the
Quillen minimal model as a graded Lie algebra.

A remarkable fact is that the _degree operator_

$x\mapsto\lvert x\rvert\cdot x,\qquad x\in Q(\mathcal{L}_{c}^{k}S^{4}),$ (44)

in the Quillen minimal model singles out a distinguished element of the Lie
algebra $\mathfrak{h}_{k}$. Here is a more precise statement.

######  Theorem 4.4 (Degree in the $k$-fold cyclic loop space of $S^{4}$).

There is a unique element of the Lie algebra $\mathfrak{h}_{k}$, namely,

$-K_{k}:=3h_{0}-h_{1}-\dots-h_{k},$ (45)

which acts on the Quillen minimal model $Q(\mathcal{L}_{c}^{k}S^{4})$ as the
degree operator:

$-K_{k}\cdot x=\lvert x\rvert\cdot x.$

###### Proof.

Indeed, the operator $3h_{0}$ acts on $g_{4}$ and thereby $e_{3}$ with weight
3 and on $x_{i}$, $i=1,\dots,k$, with weight zero:

$3h_{0}\cdot e_{3}=3e_{3},\qquad 3h_{0}\cdot x_{i}=0,$

whereas the operator $-h_{i}$ acts on $g_{4}$, $e_{3}$, and $x_{j}$ by zero,
except for $x_{i}$, on which it acts by 1:

$-h_{i}\cdot x_{i}=x_{i}.$

Likewise, the element (45) acts on $e_{6}$ via multiplication by its degree,
which is 6. Also the degree of $s_{i}x$ for $x\in Q(Z)$ will be one less than
the degree of $x$ and the weight of $s_{i}x$ will be $\alpha-\epsilon_{i}$ for
$x$ of weight $\alpha$. Since $\epsilon_{i}(3h_{0})=0$ and
$\epsilon_{i}(-h_{j})=\delta_{ij}$, the element (45) will act on $s_{i}x$ by
its degree, provided we know that it acts on $x$ by the degree of $x$. This
way we get a complete matching between the action of $3h_{0}-h_{1}-\dots-
h_{k}$ and the degree operator on the Quillen minimal model
$Q(\mathcal{L}_{c}^{k}S^{4})$.

The uniqueness of an element of $\mathfrak{h}_{k}$ which acts on
$Q(\mathcal{L}_{c}^{k}S^{4})$ by degree comes from the fact that an arbitrary
element $a_{0}h_{0}+a_{1}h_{1}+\dots+a_{k}h_{k}\in\mathfrak{h}_{k}$ acts on
$e_{3}$ by $a_{0}$ and on each $x_{i}$ by $-a_{i}$, which forces $a_{0}$ to be
3 and each $a_{i}$ to be $-1$, because the degree of $e_{3}$ is 3 and the
degree of $x_{i}$ is 1. ∎

Recall from (43) that the anticanonical class $-\mathcal{K}_{k}$ of the del
Pezzo surface $\mathbb{B}_{k}$ acts on the Picard group via intersection
pairing by degree. Thus, it makes all sense to use the element

$K_{k}=-3h_{0}+h_{1}+\dots+h_{k}\;\in\;\mathfrak{h}_{k}$

as a distinguished element, the analogue of the canonical class. Extending the
analogy with del Pezzo surfaces,

######  Definition 4.5 (Degree of cyclification).

We define the _degree of the cyclic loop space $\mathcal{L}_{c}^{k}S^{4}$ of
the four-sphere_ as

$\deg\mathcal{L}_{c}^{k}S^{4}=(-K_{k},-K_{k})=9-k.$ (46)

###  4.4 The $E_{k}$ root system and its Weyl group

We now explain the role of the Weyl group and how we obtain the exceptional
root data from $\mathcal{L}_{c}^{k}S^{4}$.

######  Theorem 4.6 (Exceptional root data from cyclic loop spaces of the
4-sphere).

(i) For each $k\geq 0$, the data

$\big{(}\mathfrak{h}_{k}^{*},\linebreak[1]\\{\epsilon_{o},\epsilon_{1},\linebreak[0]\dots,\epsilon_{k}\\},(-,-),K_{k}^{*}\big{)}$

associated to the cyclic loop space $\mathcal{L}^{k}_{c}S^{4}$ and its
Sullivan minimal model $M(\mathcal{L}^{k}_{c}S^{4})$ consists of

  * (a)

a real vector space $\mathfrak{h}_{k}^{*}$ with a basis
$\\{\epsilon_{0},\epsilon_{1},\dots,\epsilon_{k}\\}$, which generates a
lattice $\mathbf{P}(\mathfrak{h}_{k})\subset\mathfrak{h}_{k}^{*}$;

  * (b)

a symmetric bilinear form
$\mathfrak{h}_{k}^{*}\otimes\mathfrak{h}_{k}^{*}\to\mathbb{R}$ given by

$(\epsilon_{0},\epsilon_{0})=1,\qquad(\epsilon_{i},\epsilon_{j})=-\delta_{ij},\qquad
i>0,j\geq 0;$

  * (c)

a distinguished element
$K_{k}^{*}=-3\epsilon_{0}+\epsilon_{1}+\dots+\epsilon_{k}$.

(ii) This data replicates the data

$\big{(}H^{2}(\mathbb{B}_{k};\mathbb{R}),\\{\mathcal{H},\mathcal{E}_{1},\dots,\mathcal{E}_{k}\\},(-,-),\mathcal{K}_{k}\big{)}$

determined by the rational surface $\mathbb{B}_{k}$, considered as the blowup
of $\mathbb{C}\mathbb{P}^{2}$ at $k$ points; see section 4.1 in the del Pezzo
case, when $k\leq 8$. For $k\leq 8$, the data produces the root system

$R_{k}:=\big{\\{}\alpha\in\mathbf{P}(\mathfrak{h}_{k})\;|\;(\alpha,K_{k}^{*})=0,(\alpha,\alpha)=-2\big{\\}}\subset(K_{k}^{*})^{\perp}\subset\mathfrak{h}_{k}^{*}$
(47)

of type 666True/genuine $E_{k}$ for $k=6,7$, and 8, and using the conventions
of Table 1 for $0\leq k\leq 5$. $E_{k}$ and the Weyl group $W(E_{k})$,
generated by the reflections in the hyperplanes orthogonal to the roots $r\in
R_{k}$, now in the context of cyclic loop spaces of $S^{4}$.

This result has an independent interest, apart from Mysterious Duality, as it
uncovers a new symmetry pattern for the series $\mathcal{L}^{k}_{c}S^{4}$,
$0\leq k\leq 8$, of cyclic loop spaces of the 4-sphere. This should have a
number of topological consequences shedding new light on these spaces. For