# Feynman geometry

Sen Hu and Andrey Losev

###### Abstract

In this paper we introduce a notion of Feynman geometry on which quantum field
theories could be properly defined. A strong Feynman geometry is a geometry
when the vector space of $A_{\infty}$ structures is finite dimensional. A weak
Feynman geometry is a geometry when the vector space of $A_{\infty}$
structures is infinite dimensional while the relevant operators are of trace-
class. We construct families of Feynman geometries with ”continuum” as their
limit.

School of Mathematical Sciences, University of Science and Technology of China
Department of Mathematics, Moscow Higher School of Economics

††footnotetext: Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Non-Abelian gauge symmetry, as discovered by Yang-Mills, is a basic concept in
modern physics. There is a basic principle established by A. Einstein and C.
N. Yang: symmetry dictates interactions, e.g. Lagrangians are determined by
symmetries. Gauge invariance, conformal invariance and renormalizability of
the theory determine the Lagrangian for the standard model. Quantum Yang-Mills
theory is now the foundation of most of elementary particle theory, and its
predictions have been tested at many experimental laboratories, but its
mathematical foundation is still unclear.

Symmetries used by A. Einstein and C. N. Yang follow from differential
geometry. In case of A. Einstein theory of general relativity it is a
diffeomorphism symmetry that is a fundamental equivalence between smooth
manifolds. In case of C. N. Yang theory the gauge symmetry comes from the
geometry of principal fiber bundles, and may be understood as diffeomorphisms
that preserve bundle structure.

Would these symmetries persist in the quantum world or should they be
modified? The answer to this question depends on a more fundamental question:
should we consider the geometry of the space-time in QFT the same classical
differential geometry or it has to be modified? The following paper discusses
this issue and concludes that most probably the geometry has to be modified to
what we call Feynman geometry.

There are mainly two approaches in the construction of quantum field theories:
Dirac-Segal’s approach and Feynman’s approach. In Dirac’s approach one
constructs quantum theories by abandoning classical physical concepts while
keeping classical concept of space-time. This approach was pursued by many
prominent physicists and mathematicians, namely by Atiyah-Segal-Witten and by
Belavin-Polyakov-Zamalochikov, and many fruitful mathematical results were
obtained by this approach.

In Feynman’s approach it saves many classical concepts such as space-time,
trajectories, actions etc. It replaces the minimal action principle in
classical mechanics by path integral to calculate various correlation
functions. This approach is more familiar to physicists.

Here we propose a new approach to construct quantum gauge field theories by
constructing quantum many-body systems over some spaces we call it de Rham
type of DGA. As long as we have de Rham type of DGA we could construct all
kinds of actions. We demonstrate this in section 4.

The space of de Rham type of DGA are characterized by Feynman geometries. A
strong Feynman geometry is a geometry when the vector space of $A_{\infty}$
structures is finite dimensional. A weak Feynman geometry is a geometry when
the vector space of $A_{\infty}$ structures is infinite dimensional while the
relevant operators are of trace-class. This notion unifies several important
works, including lattice theory, fuzzy supersphere, Feynman geometry of Hodge
type (momentum cutoff), Mnev’s construction of $A_{\infty}$ structure via
Whitney forms, Costello’s construction of $A_{\infty}$ structures via homotopy
associativity, and Zwiebach’s string field theory. We construct families of
Feynman geometries with ”continuum” as their limit.

We propose to construct quantum field theories over an continuum as limit of a
family of theories of Feynman geometries. The advantage of such a construction
is its non-perturbative feature.

##  2 Dirac-Segal and Feynman approaches to QFT

The main question of theoretical physics is what is the definition of Quantum
Field Theory. Despite of many attempts this question still does not have a
completely satisfactory solution. These attempts may be separated into two
different approaches that we may call Dirac-Segal approach and Feynman
approach.

###  2.1 Dirac-Segal approach and its classical limit

In the Dirac-Segal approach the space-time is considered a very classical
Riemann geometry - smooth manifolds with the boundary, but ”physics” has
nothing to do with classical physics - instead of fields and integrals over
the space of fields we have functors from the category of cobordisms (equipped
with some geometrical structure) into category of vector spaces. We put Dirac
name here on the first place since he was first to understand that strongly
quantum mechanics where cobordisms are just intervals, geometrical data -
their lengths, the image of an interval is an evolution operator $I(t)$ acting
on the abstract space of states $V$, and functoriality is just a semigroup
condition:

$I(t_{1}+t_{2})=I(t_{1})I(t_{2}),\;\;\;I(t)\in End(V)$ (1)

that may be solved in an almost standard form:

$I(t)=\exp(tH)$ (2)

It is very important to note that there is no Plank constant in Dirac-Segal
approach (it is absolutely clear from the explicit solution - Dirac solution -
in one-dimensional case). The ”Plank constant” actually appears if we consider
a family of theories $Th_{h}$, depending on $h$, with the following
properties. For each $h$ there is an algebra $End(V_{h})$ is generated by the
set of operators $O(h)_{a}$, such that

$[O(h)_{a},O(h)_{b}]=hf(h)_{ab}^{c}O_{c}^{h}$ (3)

and $f(h)$ has a finite limit when $h$ tends to zero, and relations between
generators have a finite limit at $h=0$. Hamiltonian has the following form

$H(h)=\frac{H^{cl}(h)}{h},\;\;H^{cl}(h)=y^{a}(h)O_{a}(h),$ (4)

and coefficients $y^{a}(h)$ are regular at $h=0$. Let us call family of
observables $A(h)$ having a classical limit if it can be expanded through the
generators with coefficients $z^{a}(h)$

$A(h)=z^{a}(h)O_{a}(h)$ (5)

and $z^{a}(h)$ are regular at $h=0$. Then, at $h=0$ symbols $O_{a}(0)$ form a
commutative associative algebra, its spectrum may be considered as a
Poissonian manifold (may be with singularities), its Poisson structure is
determined by

$\\{O_{a}(0),O_{b}(0)\\}_{f}(0)=f_{ab}^{c}(0)O_{c}(0)$ (6)

and evolution is given by classical mechanics equation

$\frac{dA}{dt}=\\{H^{cl}(0),A(0)\\}.$ (7)

Warning. Note that there is no $h$ in the basic evolution equation, the
denominator $h$ appears only when evolution operator is written in terms of
operators that have regular classical limit, i.e.

$\exp(tH)=\exp(\frac{H^{cl}}{h}).$ (8)

Example: Fuzzy sphere.

Consider $N+1$ dimensional representation of $su(2)$,

$\\{T_{a},T_{b}\\}=\epsilon_{abc}T_{c},\;\;\sum_{a=1}^{3}T_{a}^{2}=N(N+1).$
(9)

Consider generating set

$O_{a}=T_{a}/N$ (10)

Then generators $O_{a}$ form an algebra

$\\{O_{a},O_{b}\\}=\frac{1}{N}\epsilon_{abc}O_{c},\;\;\sum_{a=1}^{3}O_{a}^{2}=1+\frac{1}{N}.$
(11)

If we consider $h=1/N$, then it is a particular case of fuzzy sphere.

##  3 Feynman approach and Feynman geometry

The alternative to Dirac-Segal approach is the Feynman approach. This approach
not just preserves but actively use the basic notions of classical physics
such as fields and Lagrangians. The only modification is the principal of
minimal (extremal) action. The condition of extremality is replaced by
functional integral.

However, such an integral in most cases does not exist as a mathematical
object. In early days of functional integral these difficulties were
considered as temporary ones, it was believed that heuristic manipulation
could produce correct results, and mathematical justification would come
later. However, in the last 65 years nobody provided such justification,
mathematicians do not even consider it as a problem, and physicists do not
want to think about it. We think that the main problem is that integral has to
be taken over infinite-dimensional space, and such integrals rarely exist.

The problem can be easily pointed out in perturbative approach. Infinite
dimensional space of integration in most cases leads to divergences.

The standard way out is to make regularization and then renormalization.
Regularization basically replaces the space of functions on the classical
space by another space, like space of function on the set of points (vertexes
of auxillary lattice) or by space of functions with limited derivative
(momentum cut-off). In the standard approach such replacement is often
considered as a technical tool, a trick for computation of ideal infinite-
dimensional integral (that does not exist). In dreaming about such ideal
integral it is mostly assumed that the property of the regularized integrals
should be similar to properties of ideal integrals, or at least the difference
in properties would somehow go away when regularized space of function would
tend to original infinite dimensional one.

In this way of thinking the ideal action often has geometrical meaning
formulated in terms of conventional differential geometry, and has symmetries
prescribed by differential geometry. People never even ask about what kind of
geometry corresponds to the regularized space of function.

Actually, the regularized space of function does not even form an associative
algebra. Consider as an example the space of function on a circle of length
$2\pi$. Let us make a momentum cut-off, restrict functions to be liner
combinations of $\exp(kx)$ with $|k|<3N$. Consider the following three
functions:

$f=\exp(2Nx),\;\;g=\exp(2Nx),\;\;h=\exp(-2Nx)$ (12)

and let us simply restrict the multiplication on the reduced space. Then
multiplying them in one way we get

$(f\cdot g)\cdot h=0\cdot h=0$ (13)

since the product of $f$ and $g$ before momentum cut-off equals to
$\exp(4Nx)$, is outside the allowed momentum range and has to be considered as
zero. At the same time multiplying in other order

$f\cdot(g\cdot h)=f\cdot 1=f$ (14)

so we have a violation of associativity. It was actually expected since the
space of functions of momentum higher than $3N$ does not form an ideal
(actually, it does not even form an algebra). Therefore, according to
traditional setup the regularized space of function does not form an algebra,
and, therefore, there is no geometrical object that may be attached to it
through conventional algebra-geometric correspondence. In particular, all
properties and symmetries of the classical action are violated and we do not
have any control of this violation, we may only hope that they will be
recovered after cut-off of the renormalized theory would be taken to infinity.
We think that such complete loss of geometry in regularized theory means
loosing control of symmetries and cannot be a part of proper definition of
functional integral.

Situation of lattice regularization is more tricky but the basic phenomena
persist. The algebra of functions on vertexes is an honest commutative and
associative algebra, but this is not enough to construct the action. What is
actually enough (as we will show in the following section) is the de Rham DGA
(differential graded algebra). It is a supercommutative associative algebra.
In the regularization this algebra should be replaced by an algebra of
cochains. However, there is no supercommutative associative algebra structure
on the space of cochains.

It is the moment of truth. In order to save the idea of functional integral we
have to find some kind of algebraic structure on the regularized space of
differentials forms (it does not matter, with momentum or lattice cut-off).
This structure should go to supercommutative DGA structure in the limit when
we go to ideal ”continuum” limit. One may expect that algebro-geometric
correspondence would somehow be extended to this structure, and we call
geometry, corresponding to such structure Feynman geometry.

The good news is that such algebraic structure already exists in mathematics,
and it is called $A_{\infty}$ structure. The brief outline of this structure
and its relation to BV (Batalin-Vilkovisky) language is presented in Appendix.

Let us call the $A_{\infty}$ structure with operations belonging to the trace
class Feynman geometry. Here we may consider two types of Feynman geometry.
First type is strong Feynman geometry when vector space of $A_{\infty}$
structure is finite dimensional. The second type is weak Feynman geometry,
when vector space is infinite-dimensional but operations are of trace-class,
so there is no divergences in Feynman diagrams computations.

Thus, our proposal is to develop Feynman approach to QFT by considering
quantum field theories over Feynman geometry instead of continuum geometry.

The deviations of Feynman geometry from conventional geometry (as we will see
below) happen in two directions. Feynman geometry can be nonsupercommutative
but associative DGA, or Feynman geometry may include higher operations.
General Feynman geometry deviates from conventional in both directions.

Therefore, our proposal is not to concentrate on each particular example of
Feynman geometry but rather consider QFT over general Feynman geometry. Then,
consider the variation of QFT when Feynman geometry is changing - a kind of
connection in the bundle of theories over parameters of Feynman geometry, and,
finally, define a Feynman QFT over continuum geometry as a limit of QFT over
Feynman geometry tending to continuum limit (kind of universal
renormalization).

Before we come to examples of Feynman geometry we would like to stress that
our world is actually described by Feynman geometry. Experiments can only
measure that our geometry is continuum to some accuracy, and they can only
improve this accuracy. If QFT may be defined over Feynman geometry we may only
ask why in our world this geometry is so close to continuum, but this question
looks like the question why cosmological constant is so small, and cannot be
answered with the present knowledge of physics.

##  4 All actions of traditional QFT written in terms of de Rham DGA

###  4.1 Pure gravity

The fundamental description of gravity is given by the following data -
D-dimensional smooth manifold with a principal $Spin(D)$ bundle $S$, equipped
with the connection $\nabla$ and a morphism $e$ from the tangent bundle to the
vector bundle $L$ (Lorentzian bundle, that is a vector bundle associated to
$S$ by D-dimensional (vector) representation of $Spin(D)$). In this way $e$
may be considered as a $L$-valued one-form, and connection $\nabla$ also may
be rewritten in terms of the 1-form $\omega$ with values in the second
external power of $L$ (in the case of trivialized bundle $L$)

$\nabla=d+\omega$ (15)

Thus, the Einstein-Cartan action

$S^{EC}=\int(e^{a_{1}}\ldots
e^{a_{D-2}}(d\omega^{a_{D-1}a_{D}}+\omega^{a_{D-1}b}\cdot\omega^{ba_{D}})\epsilon_{a_{1}\ldots
a_{D}}$ (16)

may be written in the language of de Rham DGA.

###  4.2 Chiral fields in 2d and self-dual Yang Mills

By chiral fields we mean $b-c$ or $\beta-\gamma$ systems. The easiest thing to
modify is the spin 0 chiral system with the action

$S^{\chi}=\int P\bar{\partial}X=\int PdX$ (17)

where $P$ is a $(1,0)$ form. We would like to express subdivision of
differential forms into (p,q)-types using only $e$-field. To do this we
propose to consider $P$ field as a composite field and rewrite it as

$P=e^{+}p$ (18)

where $e^{+}$ is and eigenvalue $+i$ eigenvector of $e^{a}$, $a=1,2$ with
respect to $SO(2)$ rotation, and $p$ is a complex scalar field. Thus, the
action written in terms of DGA (with an integral) looks as follows

$S_{DGA}^{\chi}=\int pedX$ (19)

The action of self-dual Yang-Mills may be rewritten in a similar fashion.
Namely, standard action of self-dual Yang-Mills has the form

$S_{SDYM}=\int TrP(dA+A^{2})$ (20)

where $P$ is a self-dual 2-form with values in adjoint representation of the
gauge group, $A$ is the connection form in the gauge bundle (we assume that
the gauge bundle is trivial and therefore connection may be considered as a
1-form with values in adjoint representation of the gauge algebra. Like in the
chiral field case we will consider field $P$ as a composite field, namely, we
introduce the field $p_{ab}$ with values in adjoint representation of the
gauge group times the self-dual 2-tensors of $L$, thus

$P=e^{a}e^{b}p_{ab}$ (21)

so the action takes the form that is written in the DGA language

$S_{DGA}^{SDYM}=\int e^{a}e^{b}p_{ab}(dA+A^{2})$ (22)

Remark 1. If we will not be interested in dynamical gravity, i.e. only
consider it as a fixed background, we can use just a two-form $E^{ab}$ with
values in the antisymmetric self-dual tensors of $L$. Then, the action would
take the form

$S_{DGA,E}^{SDYM}=\int E^{ab}p_{ab}(dA+A^{2})$ (23)

It is interesting to note that the $E$ field background is more general than
$e$ field since not all $E$ are decomposable into a product of two $e$. It
opens a previously unexplored possibility to study self-dual Yang-Mills and
(as we will see later) ordinary Yang-Mills theory in a novel background.

Remark 2. An alternative approach would be to introduce the new structure in
DGA, a polarization of middle degree elements, that is decomposition of this
subspace into a sum of two spaces, invariant under multiplication over degree
zero elements. Thus, we may consider the first order actions as actions
already written in the language of polarised DGA. Thus, actions (3) and (6)
may be considered as written already in terms of polarised DGA. The
disadvantage of this approach is that it is not clear how to write Einstein
action on the space of polarisations.

###  4.3 Spinor fields in arbitrary dimensions

The very notion of spin connection comes from the desire to couple spinors to
gravity. Therefore it is not surprising that there is such a coupling. What we
want to stress, is that this coupling also is written in terms of DGA. Namely,
spinor field should be considered as a section of the spinor bundle associated
with the $Spin(D)$ bundle, that basically means that for the case of trivial
bundles spinor fields $\psi_{\alpha}$ are zero forms with values in the spinor
representation of $Spin(D)$. The action for the Dirac spinor reads

$S=\int\bar{\psi}_{\alpha}(d\psi_{\beta}+\omega_{\beta}^{\delta}\psi_{\delta})\gamma_{a_{1}}^{\alpha\beta}e^{a_{2}}\ldots
e^{a_{D}}\epsilon_{a_{1}\ldots a_{D}}+m\bar{\psi}_{\alpha}\psi_{\alpha}vol$
(24)

where

$vol=e^{a_{1}}\ldots e^{a_{D}}\epsilon_{a_{1}\ldots a_{D}}$ (25)

and $m$ is the mass of the fermion field. Note the similarity of this action
to action for two-dimensional chiral field - really, this is how chiral
fermions are described in $D=2$. Similarly one can write the coupling of the
spinor field to the gauge field.

###  4.4 Sigma-models

There are two ways to write a free scalar field action in the language of DGA.

First way works only in dimension 2 and utilises the previously written action
for a chiral field. We consider two copies of chiral action (actually, chiral
and anti-chiral) and add a standard $P\bar{P}$ term that now looks as
$p\bar{p}e^{+}e^{-}$ term. This procedure may be generalized to the general
Riemannian metric and a $B$-field background using approach of [1].

The second way works in any geometry of the world-sheet and target, and is
based on the first order representation of the D-dimensional sigma-model.
Consider the field $P_{\mu}$ that is a $(D-1)$ -form with values in the
pullback of the cotangent bundle to the target manifold. Consider the action

$S=\int(P_{\mu}dX^{\mu}+P_{\mu}*P_{\nu}G^{\mu\nu}(X))$ (26)

Integrating the field $P$ out we get the standard sigma-model action. The
first term looks perfect from the DGA language point of view, however, the
second does not - it involves the Hodge star operation that cannot be
formulated in the DGA language. To deal with this problem we have to go to
composite fields like we did in the case of chiral field, namely, we use a
field $p_{a_{1}\ldots a_{D-1}}$ that is a section of the $D-1$ external power
of $L*$, and make a $D-1$ form out of it using $e$-fields (taking a pullback
to under the maps between $L^{*}$ and $T^{*}$):

$P_{\mu}=e^{a_{1}}\ldots e^{a_{D-1}}p_{a_{1}\ldots a_{D-1},\mu}$ (27)

Then action (12) becomes a legal expression in DGA language, i.e.

$S=\int e^{a_{1}}\ldots e^{a_{D-1}}p_{a_{1}\ldots
a_{D-1},\mu}dX^{\mu}+p_{a_{1}\ldots a_{D-1},\mu}p_{a_{1}\ldots
a_{D-1},\nu}G^{\mu\nu}(X))vol(e)$ (28)

where $vol(e)$ was defined in equation (11). It is clear that two ways of
writing down sigma-model in the first order formalism are classically
equivalent, but there is question if they are really equivalent on the quantum
level.

###  4.5 Yang-Mills theory

The Yang-Mills theory in arbitrary dimension can be written along the second
way. Namely, we write down the first order action

$S=Tr(P(dA+A^{2}))+g_{0}^{2}Tr(P*P)$ (29)

where $P$ is the $(D-2)$-form with values in the adjoint representation.
Similarly to sigma-model case this form can be expressed through the section
of the $(D-2)$-external power of $L*$

$P(p,e)=e^{a_{1}}\ldots e^{a_{D-2}}p_{a_{1}\ldots a_{D-2}}$ (30)

and Yang-Mills action takes the form

$S=\int Tr(P(p,e)(dA+A^{2}))+Tr(p_{a_{1}\ldots a_{D-2}}p_{a_{1}\ldots
a_{D-2}})vol(e)$ (31)

The case of $D=4$ is distinguished since there is another representation of
the Yang-Mills similar to the first way of describing the $D=2$ sigma model.
Namely, in this case we just add to the action od the self-dual Yang-Mills the
$E^{2}p^{2}$ term, namely

$S^{YM}=\int
E^{ab}Tr(p_{ab}(dA+A^{2}))+g_{0}^{2}Tr(p_{ab}E^{ab}p_{ce}E^{ce}).$ (32)

In order to have standard interaction with gravity we need to consider

$E^{ab}=e^{a}e^{b}.$

##  5 Noncommutative Feynman geometry of the DGA type

It seems impossible to have a family of finite dimensional supercommutative
DGA that have a de Rham DGA as a limit (may be is possible to have a rigorous
proof of this statement but we do not know such a proof). Therefore, to have
finite dimensionality we should sacrifice either supercommutativity or
associativity (associativity would be replaced by homotopical associativity).
In this section we explain what could be done if we sacrifice
supercommutativity condition, case of homotopical associativity will be
treated in the next section.

We know two apparently different way to get nonsupercommutative finite
dimensional DGA that are close in some sense to de Rham DGA. The first example
is quite known DGA that people use to explain the ring structure on
cohomology. We call it lattice DGA. The second is related to noncommutative
geometry of the manifold.

###  5.1 Lattice type

Consider simplicial complex. Physicists often consider it as a triangulation
of a manifold. Differential forms on the manifold may be considered as
cochains of simplicial complex since they can be integrated against simplexes.

Consider the set of simplicial complexes obtained by subsequent barycentric
subdivisions. Given two smooth differential forms we can always find such a
subdivision that would produce different cochains. Therefore, we would like to
equip cochains on simplicial complex with the structure of differential graded
algebra. The question of taking a limit when the number of barycentric
subdivision is going to infinity and its relation to de Rham DGA is subtle and
we will discuss it elsewhere.

It is easier to discuss the chains rather than cochains, and to give a formula
for comultiplication. The problem of finding of supercommutative coassociative
comultiplication is known as a Kolmogorov problem and is known to have a
negative answer.

It is enough to study the case of a complex associated to a single ordered
n-simplex. The complex is a set of linear combinations of basic elements that
are increasing sequences of integers between 1 and n. We will denote them as
$(n_{1},\ldots,n_{k})$. The boundary operator $\partial$ acts as

$\partial(n_{1},\ldots,n_{k})=-(n_{2},\ldots,n_{k})+\sum_{j=2}^{k}(-1)^{l}(n_{1},\ldots,n_{j-1}n_{j+1}\ldots
n_{k})+(-1)^{k}(n_{1},\ldots,n_{k-1})$ (33)

The associative comultiplication reads as

$\partial(n_{1},\ldots,n_{k})=\sum_{j=1}^{k}(n_{1},\ldots,n_{j})\otimes(n_{j},\ldots,n_{k})$
(34)

The associativity of this comultiplication can be explicitly checked, but it
is definitely not the cocommutative multiplication.

In the simplest example of 1-dimensional triangulation, namely, let us
subdivide a circle into set of intervals. Let us take orientation on the
circle and order end points belonging to each interval according to this
orientation. Then we have comultiplication on chains that induces
multiplication on cochains. In the continuum limit this multiplication tends
to standard de Rham multiplication, however, for the small size of the
interval (let as call it $a$), the nonsupercommutativity of multiplications is
reflected in the commutator

$f\cdot\omega-\omega\cdot f=\omega a\partial_{x}f$ (35)

###  5.2 Fuzzy supersphere

Consider polynomials of two even and two odd variables
$C[z_{0},z_{1},\theta_{0},\theta_{1}]$, where $z$ are even and $\theta$ -odd.
Assign to all of these variables grading 1, and consider a subspace $P_{N}$ of
$C[z_{0},z_{1},\theta_{0},\theta_{1}]$ of polynomials of degree $N$, i.e.
define an Euler vector field $E$

$E=\sum_{i=0}^{1}\theta_{i}\frac{\partial}{\partial\theta_{i}}+z_{i}\frac{\partial}{\partial
z_{i}}$ (36)

then

$(E-N)P_{N}=0$ (37)

Consider standard $\sigma$-matrixes $\sigma_{a}$ orthonormal basis in
traceless hermitian operators on $C^{2}$). Note, that the following operators
commute with $E$ and therefore act on $P_{N}$: Operators of noncommutative
coordinates $x$

$x_{a}=1/N\sum_{i,j}\sigma_{a,ij}z_{i}\frac{\partial}{\partial z_{j}}$ (38)

Noncommutative de Rham operator

$D=\sum_{i=0}^{1}\theta_{i}\frac{\partial}{\partial\theta_{i}}$ (39)

and operators of supercoordinates

$\psi_{a}=1/N\sum_{i,j}\sigma_{a,ij}\theta_{i}\frac{\partial}{\partial z_{j}}$
(40)

Note, that operators of supercoordinates are commutators of de Rham operator
and operators of coordinates

$\psi_{a}=[D,x_{a}]$ (41)

It is clear that the algebra generated by $x$ and $\psi$ is a subalgebra of
$End(P_{N})$ and thus is finite dimensional. Moreover, like in case of fuzzy
sphere, in the limit $N\rightarrow+\infty$, this algebra tends to the de Rham
algebra on the two-dimensional sphere.

Note, not only $x$ fail to commute just like in example of the ordinary fuzzy
sphere (with the standard poisson bracket), but also $x$ and $Dx$ are not
commuting:

$[x_{a},Dx_{b}]=1/N\epsilon_{abc}Dx_{c}$ (42)

while supercoordinates supercommute

$[Dx_{a},Dx_{b}]=0$ (43)

It would be interesting to find proper generalization of the superfuzzy sphere
construction - like fuzzy sphere can be generalized to finite-dimensional
algebra of operators acting on sections of the very ample line bundle on a
Kaeler manifold.

##  6 Strong Feynman geometry of A-infinity type on a subcomplex

###  6.1 Reminder on induction of A-infinity structure on a subcomplex from
DGA on the complex

In this section we consider the procedure that produces Feynman geometries
from infinite dimensional DGA’s with finite dimensional cohomology. This
construction was developed in parallel in mathematical physics and in
mathematics with different motivation and in different terms, but the
construction is basically the same.

Mathematicians were mostly interested in inducing the algebraic structure on
cohomology, and they found that besides the ring structure that comes from
restriction of multiplication to cohomology (that everybody knows) there are
also higher multiplications (Massey operations). If we combine these higher
operations with multiplications we get the $A_{\infty}$ structure on
cohomology with zero differential. Later mathematicians generalized these
constructions, and they start talking about $A_{\infty}$ structures obtained
by contraction of the acyclic subcomplex.

In mathematical physics people studied solutions to BV master equations. In
particular, they studied solutions to classical limit of BV equation. As it is
well-known, BV action may be considered as a polyvector field. In the special
case when BV action is a vector field classical BV equations lead to either
$L_{\infty}$ or $A_{\infty}$ structure. From this perspective the procedure of
contraction of acyclic subcomplex get the following ”physical” interpretation.

The main property of solutions to BV equation is the following. If we
decompose BV variables into background variables and dynamical variables, and
integrate out (against a Lagrangian submanifold) dynamical variables we will
get the effective action for background variables, and this effective action
would also satisfy BV equations.

Therefore, if DGA contains contractible subcomplex as a direct summand (as a
subcomplex) we may consider it as dynamical variables, and treat the rest as a
background. One can show that homotopy (that inverts differential on
contractable subcomplex) determines the Lagrangian submanifold, and variables
of this subcomplex may be integrated out. The effective action would be a sum
of Feynman diagramms, and in defining induced $A_{\infty}$ structure we have
to take only tree diagrams. The trivalent vertices in this diagrams correspond
to multiplication in the algebra and propagator is given by homotopy.

Let us reformulate this in the mathematical language. Suppose that there is a
DGA structure on the space $V$ with differential $D:V\rightarrow V$ and
multiplication $m_{2}:V\otimes V\rightarrow V$. Suppose that $V$ has a
decomposition as a complex

$V=V_{IR}\oplus V_{UV},$ (44)

with inclusion

$\iota:V_{IR}\rightarrow V$

and projection

$\pi:V\rightarrow V_{IR}$

and homotopy $h$

$h:V\rightarrow V$

with the properties

$Dh+hD=1-\iota\pi$ (45)

(note that the right hand side is the projection on $V_{UV}$)

$h^{2}=0$ (46)

Then there are induced operations

$M_{k}^{IR}:V_{IR}^{\otimes k}\rightarrow V_{IR}$

given as a sum over rooted flat trees with $k$ leaves and one root. For the
exact formula reader may consult [2], but we will give the simplest cases:

$M_{2}^{IR}(u,v)=\pi m_{2}(\iota u,\iota v)$ (47) $M_{3}^{IR}(u,v,w)=\pi
m_{2}(hm_{2}(\iota u,\iota v),\iota w)+\pi m_{2}(\iota u,hm_{2}(\iota v,\iota
w))$ (48)

###  6.2  A-infinity algebras of Hodge type as strong Feynman geometry

Suppose $X$ is a compact manifold with a metric. Then for any positive number
$\Lambda$ there is a Feynman geometry constructed as follows. Let us start
with de Rham DGA, and decompose it into direct sum of subcomplexes
$\Omega_{E}$ corresponding to Laplacian eigenvalues. Let us form $V_{UV}$ from
subcomplexes with eigenvalue larger than $\Lambda$ , and $V_{IR}$ from
subcomplexes with eigenvalues smaller than $\Lambda$.

$\Omega^{*}=(\oplus_{E\leq\Lambda}\Omega_{E})\oplus(\oplus_{E\geq\Lambda}\Omega_{E})$
(49)

The space

$V_{IR}=\oplus_{E\leq\Lambda}\Omega_{E}$

is definitely finite dimensional, and the space

$V_{UV}=\oplus_{E\geq\Lambda}\Omega_{E}$

is acyclic. In particular, we may take as a homotopy

$h=d^{*}/E$ (50)

Thus we have Feynman geometry depending on $\Lambda$ and we approach de Rham
geometry as $\Lambda$ goes to infinity.

The parameter $\Lambda$ resembles a momentum cutoff, however, there is a
fundamental difference between Feynman geometry approach and old fashioned
cutoff approach. In terms of algebraic structures ”old cutoff” means that we
are using the algebra with nonassociative multiplication, and therefore we are
breaking many symmetries like gauge symmetry (just recall that to protect
gauge symmetry people invented such things as Pauli-Willars regularization).
In the Feynman geometry approach we should properly take into account higher
multiplications and write action over a general Feynman geometry.

###  6.3 Strict A-infinity algebras of lattice type as strong Feynman geometry

In section () we studied noncommutative associative algebra of cochains
associated to simplicial complex. However, it is possible to get
supercommutative $A_{\infty}$ structure based on decomposition

$\Omega^{*}=KerI\oplus Cochains$ (51)

In order to do this we need to find an embedding of cochains into differential
forms compatible with the differential. This problem was solved by Whitney -
and corresponding forms are called Whitney forms. In particular, in the one
dimensional case the Whitney forms are

$C_{0}\rightarrow\omega_{0}(C_{0})=C_{0}(A)(1-t)+C_{0}(B)t$ (52)
$C_{1}\rightarrow\omega_{1}(C_{1})=C_{1}(BA)dt$ (53)

It is quite instructive to write $KerI$ explicitly. In degree 1 it consists of
1 forms $\omega(t)dt$ with zero integral :

$\int_{0}^{1}\omega(t)dt=0$ (54)

and in degree zero it consists of functions that may be represented as

$f_{\omega}=\int_{0}^{t}\omega(u)du$ (55)

It is clear that $f_{\omega}$ takes zero values at $0$ and at $1$ , thus
belongs to Ker I, and actually spans it. The homotopy operator is just

$h:\omega\rightarrow f_{\omega}$ (56)

and zero otherwise.

These structures were studied in [2].

In higher dimensions there is no distinguished homotopy, therefore there are
many $A_{\infty}$ structures.

##  7 Weak Feynman geometry

Weak Feynman geometry is an $A_{\infty}$ structure that may be not finite
dimensional as a vector space but whose operations are of trace class. Namely,
the $m_{k}$ should be such that

$m_{k}(v_{1},\ldots,v_{k-1},\cdot)$

considered for a given $v_{1},\ldots,v_{k-1}$ as a linear operator from $V$ to
$V$ should be of the trace class. Note, that the standard multiplication in
the de Rham complex is not weakly Feynman, say, multiplication by 1 is an
identity operator in the Hilbert space that is not of trace class.

###  7.1 Weak Feynman geometry of Costello type

The idea of Costello Feynman geometry is to do the two step procedure. First,
modify multiplication to make it trace class. Then, most probably, the
modified multiplication would not be associative. But it could, and we will
see actually would be homotopically associative, thus providing $A_{\infty}$
structure.

Let us consider definition of index of an operator as a prototype. Index was
actually a supertrace of multiplication by 1, that did not exist. So, the
result of multiplication was smoothed by

$\exp(-\beta\Delta).$

It gives the idea to replace $m_{2}$ of de Rham DGA by

$m_{2}^{\beta}(v,u)=\exp(-\beta\Delta)m_{2}(v,u)$ (57)

Then one can show that multiplication would not be associative. However, it
could be completed to $A_{\infty}$ structure by introducing the following
higher operation.

$m_{3}^{\beta}(u,v,w)=\exp(-\beta\Delta)m_{2}(\int_{0}^{\beta}\exp(-t\Delta)hdtm_{2}(u,v),w)+\pi
m_{2}(u,\int_{0}^{\beta}\exp(-t\Delta)hdtm_{2}(v,w))$ (58)

This $A_{\infty}$ structure was discovered by Costello [3] and for topological
strings by Costello-Li [4].

###  7.2 String theory in form of Zwiebach as weak Feynman geometry

The second example of the $A_{\infty}$ structure is given by string theory in
the works of Zwiebach [5]. Thus, string theory development is not orthogonal
to the programm we are proposing but rather can be included into it as an
important example.

##  8 Instead of conclusion: open questions

###  8.1 Promotion of actions from de Rham DGA to $A_{\infty}$ structures

We have shown in section 4 that all actions of classical field theory may be
formulated in terms of de Rham DGA. However, it is not clear a priori how to
promote them to general $A_{\infty}$ structure, specifically taking into
account possible noncommutativity of operations and existence of higher
operations. Proper promotion should take into account symmetries of the
theory, thus, rigorously the problem looks as promotion of BV action that
includes gauge symmetries to general $A_{\infty}$ structure. There is a well-
known case where such promotion always exist, this is the case of so-called
$BF$ theories.

Originally they were formulated as non-Abelian gauge D-dimensional theories
coupled to a field $B$ that is a $D-2$ form with values in adjoint
representation of the gauge group with the action

$S=\int Tr(B(dA+A^{2}))$ (59)

In terms of $A_{\infty}$ structures this may be reformulated as follows.
Consider two $A_{\infty}$ structures. First one would be de Rham DGA
(considered as $A_{\infty}$ structure), second - the matrix algebra considered
as $A_{\infty}$ structure with zero differential. Take a tensor product of
these two structures and form a new $A_{\infty}$ structure. Then write a
canonical action for this structure (see Appendix). The outcome would be
exactly the BF theory. The only subtle moment here is the issue of integral
and $Tr$ in the action. Actually, in canonical action for $A_{\infty}$
structure we have pairing with a dual field. In case of de Rham DGA fields
dual to differential forms may be represented also as differential forms with
canonical pairing replaced by an integral. In the case of matrix algebra the
dual object can be also considered as a matrix with pairing given by a trace
of multiplication.

This explains how to promote $BF$ theory over general Feynman geometry - we
should just replace de Rham DGA by a general Feynman geometry and write down
canonical action.

Therefore, we can cover in this way 2D Yang-Mills with zero coupling constant
and 3D gravity.

The most interesting question is whether it is possible to promote other
actions from the section 3 to general Feynman geometry. If it is not possible
in general what is the obstruction? Could it be a new selection rule on
classical actions?

If such obstruction may vanish on certain Feynman geometry what is a subclass
of Feynman geometry where such obstruction vanish?

These are quite new types of questions in QFT, we are going to address these
questions in the following publications.

###  8.2 Dirac-Segal approach to QFT versus Feynman geometry approach and the
dream of Ultimate QFT

In this paper we studied two approaches to QFT, Dirac-Segal and Feynman
approaches. One may think that they are orthogonal to each other. In Dirac-
Segal approach the geometry of the space-time is a standard continuum
geometry, but there is no Plank constant, and no classical fields and
Lagrangians. In the Feynman approach there are classical fields, Lagrangians
and Plank constant, but geometry of the space-time is not classical anymore.
It may look that these approaches exclude each other.

However, it is not the case. From our experience in dimensions 1 and 2 we know
that there are theories that have description in both approaches, and
resulting theory is the same. The best known example is the WZWN that can be
described and actually solved in Dirac-Segal approach through integrable
representations of affine lie algebras. At the same time this theory has a
functional integral representation.

We may conjecture that the Ultimate Definition of QFT would combine these two
paths in the following way - both geometry and physics would be nonclassical.
Namely, we expect that it would be possible to generalize Dirac-Segal axioms
to an arbitrary Feynman geometry and it would be the goal of axiomatic
development of QFT. At the same time Feynman integrals over fields on Feynman
geometry would serve as a constructive examples of solutions to these new
axioms.

##  9 Appendix: BV master equation and $A_{\infty}$ structures

For physicist the proper way to understand $A_{\infty}$ structures is through
the BV language. Consider the superspace (called BV space) of the form $\Pi
T^{*}X$ with coordinates $x^{a}$ on the base $X$ and $x_{a}^{*}$ on the
cotangent fiber with opposite parity. Functions on BV superspace may be
considered as polyvector fields on the space $X$. The main operator is the BV
operator on functions on BV space

$\Delta_{BV}=\sum_{a}\frac{\partial^{2}}{\partial x^{a}\partial x_{a}^{*}}$
(60)

The geometrical meaning of $\Delta_{BV}$ is the divergence of the polyvector
field; actually, if we apply it to a vector field

$v=x_{a}^{*}V^{a}(x)$ (61)

we would get

$\Delta_{BV}v=\sum_{a}\partial_{a}V^{a}$ (62)

The main object in BV language is a BV action $S(h)$ \- even function on the
BV space with values in formal power series in $h$. The BV master equation
reads

$\Delta_{BV}\exp S(h)/h=0$ (63)

If the BV action represents a vector field , then corresponding vector field
is odd and the classical limit of BV equation states that it is a homological
vector field:

$Q=V^{a}(x)\frac{\partial}{\partial x^{a}}\;\;Q^{2}=0$ (64)

expanding homological vector field in coordinates we get a set of operations

$V^{a}(x)=\sum_{k=1}^{\infty}\mu_{a_{1}\ldots a_{k}}^{a}x^{a_{1}}\ldots
x^{a_{k}}/k!$ (65)

where $\mu_{a_{1}\ldots a_{k}}^{a}$ is considered as an operation

$\mu_{k}:V^{\otimes k}\rightarrow V$

written in a particular basis. Set of quadratic equations on operations
$\mu_{k}$ is called set of conditions for $L_{\infty}$ structure. The first
condition states that $\mu_{1}$ is a differential, second - that $\mu_{1}$
together with $\mu_{2}$ form a Leibnitz rule, third - that Jacobi equation for
$\mu_{2}$ is true up to a commutator of $\mu_{1}$ and $\mu_{3}$ and so on.

Note, that $L_{\infty}$ structure is a structure on supercommutative
operations.

However, there is a possibility to encode nonsupercommutative operations in
homological vector field. Namely, let us consider take $X$ to be $C^{MN^{2}}$
with coordinates $x_{\alpha}^{i,\beta}$, that we will denote simply as
$\hat{x}^{i}$.

Consider the BV action that is still a vector field, but of a very special
form

$A=\sum_{k=1}^{\infty}m_{i_{1}\ldots
i_{k}}^{j}Tr(\hat{x^{*}}_{j}\hat{x}^{i_{1}}\ldots\hat{x}^{i_{k}})/k!$ (66)

If $N$ is big enough then conditions that $A$ solves classical master equation
is exactly the set of quadratic equations for operations $m$ that determine
$A_{\infty}$ structure.

They start with $m_{1}$ being a differential, Leibnitz rule for $m_{1}$ and
$m_{2}$, associativity of $m_{2}$ up to a commutator of $m_{1}$ and $m_{3}$
and so on.

## References

  * [1] A. Losev, A. Marshakov, A. Zeitlin, On First Order Formalism in String Theory, Phys.Lett.B633:375-381,2006, arXiv:hep-th/0510065.
  * [2] Anton Alekseev, Pavel Mnev: One-dimensional Chern-Simons theory, Commun.Math.Phys.307:185-227, 2011, arXiv:1005.2111.
  * [3] K. Costello: Renormalization and effective field theory, AMS, 2011.
  * [4] K. Costello and Si Li: Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model, arXiv:1201.4501.
  * [5] B. Zwiebach, CLOSED STRING FIELD THEORY: QUANTUM ACTION AND THE B-V MASTER EQUATION, hep-th/9206084.
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