††institutetext: Stanford Institute for Theoretical Physics and Department of
Physics, Stanford University, Stanford, CA 94305, USA

# Superinvariants Below Critical Loop Order

Renata Kallosh

###### Abstract

Investigation of on-shell superinvariants is the standard way to identify the
candidate UV divergences in supergravity. Geometric on-shell superinvariants
in 4d supergravity at $\mathcal{N}\geq 4$ are available starting at the
critical loop order $L=\mathcal{N}$ Kallosh:1980fi ; Howe:1980th . However,
more than 10 years ago, it was conjectured that the UV divergencies in
supergravity might appear earlier, for $L=\mathcal{N}-1$, due to the existence
of harmonic superspace superinvariants based on Grassmann analytic superfields
Bossard:2011tq . Here we show that these harmonic superinvariants at the
nonlinear level are inconsistent, because Grassmann analyticity breaks local
$\mathcal{H}$ symmetry of the $\mathcal{N}\geq 4$ on-shell superspace.
Therefore, the “puzzling enhanced cancellation” of UV divergences for
$L=\mathcal{N}-1$ in superamplitude calculations for $L=3,\,\mathcal{N}=4$ and
$L=4,\,\mathcal{N}=5$ is explained by nonlinear local supersymmetry consistent
with unbroken local $\mathcal{H}$ symmetry.

##  1 Introduction

$\mathcal{N}\geq 4$ supergravities with physical scalars in
${\mathcal{G}\over\mathcal{H}}$ coset space in 4d have a formulation with
independent and linearly realized duality $\mathcal{G}$ symmetry and local
$\mathcal{H}$-symmetry Cremmer:1979up ; deWit:1982bul . Local
$\mathcal{H}$-symmetry is a $U(\mathcal{N})$ (or $SU(\mathcal{N})$) group.
These symmetries are also present in the on-shell superspace for these
theories Brink:1979nt ; Howe:1981gz .

In Kallosh:2023asd ; Kallosh:2023thj ; Kallosh:2023css  the assertion was made
that for $\mathcal{N}\geq 4$ supergravities111 $\mathcal{N}\leq 4$
supergravities in 4d have anomalies but no anomalies are known at
$\mathcal{N}\geq 5$. nonlinear supersymmetry invariants cannot be constructed
unless 1) they come from a whole superspace integral, not a subspace of the
whole superspace, which is necessary for the manifest local $\mathcal{H}$
symmetry of the measure of integration 2) the geometric superspace Lagrangian
respects both global $\mathcal{G}$-duality symmetry and local $\mathcal{H}$
symmetry. These conditions provide the definition of $L_{cr}=\mathcal{N}$ for
4d supergravities at which nonlinear superinvariants can be constructed, based
on dimensional analysis in on-shell superspace Kallosh:1980fi ; Howe:1980th .
Therefore if there is a UV divergence at $L<L_{cr}$ it means that nonlinear
supersymmetry is broken by this UV divergence.

Our goal here is to provide a detailed explanation of the statements made in
Kallosh:2023asd ; Kallosh:2023thj ; Kallosh:2023css  with regard to
$L=\mathcal{N}-1$ superinvariants222It is known that for $\mathcal{N}\geq 5$
UV divergences at $L\leq\mathcal{N}-2$ would mean that E7 type $\mathcal{G}$
symmetry is broken Beisert:2010jx , Freedman:2018mrv . However, the case of
$L=\mathcal{N}-1$ is not protected by the soft scalar limit theorem. For
example, the case L=7 for $\mathcal{N}=8$ or L=4 for $\mathcal{N}=5$ are not
protected by $E_{7(7)}$ or $SU(1,5)$, respectively. But since at L=4,
$\mathcal{N}=5$ UV divergences in 82 diagrams cancel Bern:2014sna , this
requires an explanation why harmonic $L=\mathcal{N}-1$ superinvariants
predicting this UV divergence are not valid at the nonlinear level. in an on-
shell harmonic superspace proposed in Bossard:2011tq . We argued in
Kallosh:2023asd ; Kallosh:2023thj ; Kallosh:2023css  that these
counterterms/superinvariants are valid at the linear level but break nonlinear
local supersymmetry.

Our argument in Kallosh:2023asd ; Kallosh:2023thj ; Kallosh:2023css  was based
on the fact that the proof of the consistency of the harmonic superspace
$(\mathcal{N},1,1)$ proposed in Hartwell:1994rp  was given for Yang-Mills
theory and for $\mathcal{N}=1,2,3,4$ conformal supergravity, but not for
$\mathcal{N}\geq 4$ Poincaré supergravity. Here we will study the relevant
$L=\mathcal{N}-1$ superinvariants of the harmonic superspace
$(\mathcal{N},1,1)$ and show directly that they are inconsistent at the
nonlinear level. This will explain why nonlinear supersymmetry invariants are
not available at $L<L_{cr}$.

An important issue to stress here is the closure of a local supersymmetry
algebra: for example, in the case of $\mathcal{N}=8$ supergravity the
classical action Cremmer:1979up ; deWit:1982bul  has local supersymmetry,
general covariance, local Lorentz symmetry, Maxwell symmetry and local
$SU(8)$-symmetry (local $\mathcal{H}$-symmetry). The commutator of two
nonlinear local supersymmetries involves, in addition to all other local
symmetries of the action, also a local $SU(8)$ symmetry, and a term
proportional to equations of motion Cremmer:1979up ; deWit:1982bul

$\\{\delta_{1s},\delta_{2s}\\}=\delta_{\rm
Diff}+\delta_{3s}+\delta_{SO_{(3,1)}}+\delta_{U(1)}+\delta_{SU(8)}+{\rm EOM}$
(1)

A detailed derivation of this commutator in the nonlinear case is presented in
Sec. 3.4 in Hillmann:2009zf . In the linear case, the term $\delta_{SU(8)}$ in
the RHS is absent. The reason we present here the local supersymmetry anti-
commutator in (1) is to show that breaking of local $SU(8)$ symmetry by
quantum corrections means that nonlinear local supersymmetry is inevitably
broken. This is also valid in case that equations of motion are satisfied,
i.e. on-shell, when terms ${\rm EOM}$ in (1) vanish. Thus

${\rm broken}\,\,{\rm local}\,\,\mathcal{H}\quad\Rightarrow\quad{\rm
broken}\,\,{\rm nonlinear}\,\,{\rm supersymmetry}$ (2)

In the unitary gauge, when local $SU(8)$ symmetry is gauge-fixed, the algebra
of two supersymmetries still includes a field-dependent nonlinearly realized
$SU(8)$ symmetry, with duality $\mathcal{G}$ symmetry parameters.

We will also show in detail why the whole superspace integral measure of
integration for $\mathcal{N}\geq 4$ respects local $\mathcal{H}$ symmetry. It
is broken at the nonlinear level if any subspace of the whole superspace is
used for superinvariants, unless new coordinates are added to a standard
superspace. If new coordinates, like harmonic ones, are used to define
superinvariants, there is no problem at the linear level. The problem only
appears at the nonlinear level at $\mathcal{N}\geq 4$ as we will show.

The off-shell harmonic superspace construction for SYM theories and
$\mathcal{N}=2$ supergravity was described in Galperin:2001seg . We present a
short description of these harmonic superspaces and projective superspaces
Kuzenko:2008ep ; Kuzenko:2008ry  in Appendix A.

The important difference between $\mathcal{N}=2$ and $\mathcal{N}\geq 4$ is
that at $\mathcal{N}\geq 4$ there is a ${\mathcal{G}\over\mathcal{H}}$ coset
structure and the relevant global $\mathcal{G}$ and local $\mathcal{H}$
symmetries are present. In $\mathcal{N}=2$ supergravity superspace,
originating from conformal $\mathcal{N}=2$ superspace, it is possible to
gauge-fix local tangent space $\mathcal{H}=U(2)$ symmetry of the standard
superspace, see Howe:1981gz , sec. 7 on “de-gauging” local $U(2)$ symmetry.
This local symmetry was present in $\mathcal{N}=2$ conformal supergravity but
was eliminated in Poincaré $\mathcal{N}=2$ supergravity. However, at higher
$\mathcal{N}$, the local $(U(\mathcal{N})$ $\mathcal{H}$ symmetry is a tangent
space symmetry of the superspace Howe:1981gz ; it is at the core of the
superspace construction.

Despite this fact, it was assumed in Bossard:2011tq  that to define on-shell
harmonic superspace, one only needs to satisfy integrability conditions for
Grassmann analyticity conditions. It was indeed shown in Bossard:2011tq  that
these are satisfied for $L=\mathcal{N}-1$ type supersymmetry invariants.

We will show here that these integrability conditions are necessary but not
sufficient for the consistency of the $\mathcal{N}\geq 4$ harmonic on-shell
superspace. They are actually sufficient in $\mathcal{N}=2$ case, see
Kuzenko:2008ep ; Kuzenko:2008ry  whose strategy was used in Bossard:2011tq  to
construct on-shell harmonic superspace at $\mathcal{N}\geq 4$.

But in $\mathcal{N}\geq 4$ there is a duality $\mathcal{G}$-symmetry which
enforces a compensating field dependent $(S)U(\mathcal{N})$ transformation, in
case that local $\mathcal{H}$-symmetry is gauge-fixed in the spacetime action
Cremmer:1979up ; deWit:1982bul ; Kallosh:2008ic ; Bossard:2010dq . This
compensating field-dependent $(S)U(\mathcal{N})$ transformation is
incompatible with harmonic coordinates. Therefore, the Grassmann analyticity
in on-shell harmonic superspace assumed in Bossard:2011tq  for
$\mathcal{N}\geq 4$ supergravities breaks local $(S)U(\mathcal{N})$ symmetry
and nonlinear supersymmetry.

In $d>4$ the relevant ${\mathcal{G}\over\mathcal{H}}$ supergravity
superinvariants below critical loop order are also known in harmonic
superspace at the linearized level Bossard:2009sy ; Bossard:2014lra ;
Bossard:2014aea ; Bossard:2015uga . These are useful to describe the UV
divergences which did occur at $d>4$. None of these harmonic superinvariants
with Grassmann-analytic superfields were claimed to exist at the nonlinear
level. In all cases, they are at $L<L_{cr}$ Kallosh:2023css  and show that
local $\mathcal{H}$ symmetry and nonlinear supersymmetry are broken by quantum
corrections. This means that these local symmetries are anomalous. We explain
in Appendix B how the arguments in case of 4d supergravities with
$L=\mathcal{N}-1$ work for $d>4$ supergravities and obstruct the nonlinear
generalization of the harmonic superinvariants constructed in Bossard:2009sy ;
Bossard:2014lra ; Bossard:2014aea ; Bossard:2015uga .

##  2 Local $\mathcal{H}$ symmetry of the whole curved superspace volume

The value of $L_{cr}$ in integer dimensions $d\geq 4$ was determined by
dimensional analysis in superspace, in 4d it is $L_{cr}=\mathcal{N}$
Kallosh:1980fi ; Howe:1980th . It was based on the on-shell superspace
construction in Brink:1979nt ; Howe:1981gz . The ones below $L_{cr}$ are
either defined in a subspace of the whole superspace, which breaks local
$\mathcal{H}$ symmetry or depend on Lagrangians breaking $\mathcal{G}$
symmetry. In $d>4$ the value of $L_{cr}$ was determined in Kallosh:2023css .

The whole superspace has the measure of integration, which has both curved
space symmetry as well as tangent space symmetry.

$dV=d^{d}x\,d^{4\mathcal{N}}\theta\,{\rm Ber}\,E\ .$ (3)

Here, Berezinian, denoted ${\rm Ber}\,E$, is a generalization of the
determinant of the curved space in the presence of Grassmann coordinates,
where it is defined as a superdeteminant. This is analogous to general
relativity where $dx^{m}e_{m}^{a}(x)=E^{a}$, and the volume of integration can
be given in the form with manifest curved space symmetry or in a form with
manifest tangent space Lorentz symmetry $SO(1.3)$

$\int d^{4}x\sqrt{g}\quad\Rightarrow\quad\int E^{a}\wedge E^{b}\wedge
E^{c}\wedge E^{d}\,\epsilon_{abcd}\ .$ (4)

Note that the Lorentz space covariant tensor $\epsilon_{abcd}$ needs four of
$E^{a}$ to be saturated. An example of a subspace of a whole space breaking
tangent space symmetry in pure gravity in 4d would be an integral over less
than four $E^{a}$’s. In this case, the local $SO(1.3)$ Lorentz symmetry of the
measure of integration would be broken.

The 4d curved superspace coordinates
$Z^{M}=(x^{m},\theta^{\underline{\mu}},\theta^{\underline{\dot{\mu}}})$
include in addition to spacetime coordinates $x^{m}$ also $4\mathcal{N}$
anticommuting coordinates
$\theta^{\underline{\mu}},\theta^{\underline{\dot{\mu}}}$. The relevant
preferred frames are defined as

$E^{A}=dZ^{M}E_{M}{}^{A}(Z)$ (5)

There are general coordinate transformations $Z^{M}\rightarrow
Z^{M}+\xi^{M}(Z)$ and tangent space rotations:

$\delta E^{A}=E^{B}L_{B}{}^{A}$ (6)

These tangent space symmetries play an important role in the construction of
superinvariants. Here $A=(a,{\underline{\alpha}},{\underline{\dot{\alpha}}})$
where

$E^{\underline{\alpha}}=E^{\alpha}_{i}\qquad
E^{\underline{\dot{\alpha}}}=E^{\dot{\alpha}\,i}\,,\qquad\alpha,\dot{\alpha}=1,2\,,\qquad
i=1,\dots,\mathcal{N}$ (7)

The tangent space group (also called structure group) is a product of a local
Lorentz group and local internal group $SL(2,\mathbb{C})\times\mathcal{H}$
where $\mathcal{H}$ is $U(\mathcal{N})$ or its subgroup. This means that
$SL(2,\mathbb{C})\times\mathcal{H}$ parameters in (6) are functions of the
superspace coordinates
$Z^{M}=(x^{m},\theta^{\underline{\mu}},\theta^{\underline{\dot{\mu}}})$.

$L_{B}{}^{A}=L_{B}{}^{A}(Z)\ .$ (8)

Let us also define local $\mathcal{H}$ tensors

$E_{ij}=E^{\alpha}_{i}\wedge
E^{\beta}_{j}\epsilon_{\alpha\beta}\,,\qquad\bar{E}^{kl}=\bar{E}_{\dot{\alpha}}^{k}\wedge\bar{E}_{\dot{\beta}}^{l}\epsilon^{\dot{\alpha}\dot{\beta}}$
(9)

and their products

$E_{i_{1}\dots i_{\mathcal{N}}j_{1}\dots
j_{\mathcal{N}}}=E_{i_{1}j_{1}}\wedge\dots\wedge
E_{i_{\mathcal{N}}j_{\mathcal{N}}}\ ,$ (10) $\bar{E}^{i_{1}\dots
i_{\mathcal{N}}j_{1}\dots
j_{\mathcal{N}}}=\bar{E}^{i_{1}j_{1}}\wedge\dots\wedge\bar{E}^{i_{\mathcal{N}}j_{\mathcal{N}}}\
.$ (11)

The integral of the whole superspace volume (3) can be presented in the form

$\displaystyle\int E^{a}\wedge E^{b}\wedge E^{c}\wedge
E^{d}\,\epsilon_{abcd}\,\,E_{i_{1}\dots i_{\mathcal{N}}j_{1}\dots
j_{\mathcal{N}}}\epsilon^{i_{1}\dots i_{\mathcal{N}}}\,\epsilon^{j_{1}\dots
j_{\mathcal{N}}}\,\,\bar{E}^{l_{1}\dots l_{\mathcal{N}}k_{1}\dots
k_{\mathcal{N}}}\epsilon_{l_{1}\dots l_{\mathcal{N}}}\,\epsilon_{k_{1}\dots
k_{\mathcal{N}}}\ ,$ (12)

where $SL(2,\mathbb{C})\times\mathcal{H}$ symmetries with parameters which are
functions of the superspace coordinates
$Z^{M}=(x^{m},\theta^{\underline{\mu}},\theta^{\underline{\dot{\mu}}})$ are
manifest.

Clearly, any attempt to integrate over less of $E^{\alpha}_{i}$,
$\bar{E}_{\dot{\alpha}}^{k}$ than $4\mathcal{N}$ will break $U(\mathcal{N})$
$\mathcal{H}$-symmetry: to saturate $\epsilon^{i_{1}\dots
i_{\mathcal{N}}}\,\epsilon^{j_{1}\dots j_{\mathcal{N}}}\epsilon_{l_{1}\dots
l_{\mathcal{N}}}\,\epsilon_{k_{1}\dots k_{\mathcal{N}}}$ one needs
$4\mathcal{N}$ of these. This is the same as in 4d: to preserve local Lorentz
symmetry, one needs 4 of $E^{a}$’s to saturate $\epsilon_{abcd}$.

As long as no new coordinates are introduced into superspace, the only
$SL(2,\mathbb{C})\times\mathcal{H}$ invariant measure of integration has
dimension $-d+2\mathcal{N}$, in 4d case

${\rm dim}[dV_{whole}]=-4+2\mathcal{N}\ .$ (13)

But when harmonic coordinates are added to a superspace, at the linear level,
this opens a possibility to have the dimension of the measure of integration
less than $-d+2\mathcal{N}$. Specifically, in the case of Bossard:2011tq  and
in 4d $\mathcal{N}\geq 4$, the Grassmann analytic subspace of the superspace
has a measure of dimension

$dim[dV_{harm}]=-4+2(\mathcal{N}-1)\ ,$ (14)

which explains the claimed existence of $L=\mathcal{N}-1$ harmonic
superinvariants. We will see that at the linear level, these are consistent;
however, we will show that the local $\mathcal{H}$-symmetry of the measure of
integration over the subspace is broken at the nonlinear level.

##  3  $\mathcal{N}\geq 4$ on-shell superspace geometry and its tangent space
$\mathcal{H}$ symmetry

The physical scalars in these theories are in the
${\mathcal{G}\over\mathcal{H}}$ coset space where where $\mathcal{H}$ is
$U(\mathcal{N})$ or its subgroup, e. g. $SU(8)$ for $\mathcal{N}=8$, $U(6)$
for $\mathcal{N}=6$, $U(5)$ for $\mathcal{N}=5$, $U(4)$ for $\mathcal{N}=4$.
The groups $\mathcal{G}$ are $E_{7(7)}$ for $\mathcal{N}=8$, $SO^{*}(12)$ for
$\mathcal{N}=6$ and $SU(5,1)$ for $\mathcal{N}=5$, $SU(4)\times SU(1,1)$ for
$\mathcal{N}=4$. These are known as groups of type E7.

Following Howe:1981gz  we consider a superspace with coordinates

$z^{\mathscr{M}}=(z^{M},y^{\underline{m}},y^{\underline{m^{\prime}}})$ (15)

where $z^{M}$ are the coordinates of ordinary superspace with four bosonic
coordinates $x^{m}$ and $2\mathcal{N}$ anticommuting coordinates
$\theta^{\underline{\mu}}$ and its complex conjugate
$\theta^{\underline{\dot{\mu}}}$. This is a standard superspace where
$\mathcal{N}=8$ on-shell supergravity was studied in Brink:1979nt . In
Howe:1981gz , the extra bosonic coordinates were added: $y^{\underline{m}}$
are ${1\over 2}\mathcal{N}(\mathcal{N}-1)$ complex coordinates with complex
conjugates $y^{\underline{m^{\prime}}}$. This version is given (mostly in Sec.
8) of Howe:1981gz  for all $\mathcal{N}$, and its geometry is consistent with
a normal superspace version with coordinates $z^{M}$ in Brink:1979nt  where
$\mathcal{N}=8$ was studied.

A tangent space vector is

$X^{\mathscr{A}}=(X^{A},X^{\underline{a}},X^{\underline{a^{\prime}}})\qquad
X^{A}=(X^{a},X^{\underline{\alpha}},X^{\underline{\alpha^{\prime}}})$ (16)
$X^{\underline{a}}=X_{ij},\qquad X^{\underline{a^{\prime}}}=\overline{X}^{ij}$
(17)

The tangent space group is $SL(2,C)\times\mathcal{H}$. Here $\mathcal{H}$ is
$SU(8)$ for $\mathcal{N}=8$, $U(\mathcal{N})$ for $\mathcal{N}=6,5,4$. The
matrix $L_{{\mathscr{A}}}{}^{\mathscr{B}}$ defining the tangent space rotation
on the tangent vector

$\delta
X^{\mathscr{A}}=X^{\mathscr{B}}L_{{\mathscr{B}}}{}^{\mathscr{A}}(z^{\mathscr{M}})$
(18)

is given in eqs. (2.5), (8.4) in Howe:1981gz . It is convenient to use this
version where all geometric fields depend trivially on extra coordinates

$D_{\underline{a}}=\partial_{\underline{m}}=0\ .$ (19)

The vielbein one-form is
$E^{\mathscr{A}}=dz^{\mathscr{M}}E_{\mathscr{M}}{}^{\mathscr{A}}$ and
${\mathscr{E}}^{A}=E^{A}$ due to (19). There is a standard general coordinate
transformation in a superspace. The superspace has a local tangent space
symmetry $SL(2,C)\times\mathcal{H}$ where $\mathcal{H}$ is $U(\mathcal{N})$ or
its subgroup.

$\delta
E^{\mathscr{A}}=E^{\mathscr{B}}L_{{\mathscr{A}}}{}^{\mathscr{B}}(z^{\mathscr{M}})$
(20)

Also, a one-form connection $\hat{\Omega}$ is introduced, which transforms
inhomogeneously

$\delta\hat{\Omega}_{{\mathscr{A}}}{}^{\mathscr{B}}=-dL_{{\mathscr{A}}}{}^{\mathscr{B}}-L_{{\mathscr{A}}}{}^{\mathscr{C}}\hat{\Omega}_{{\mathscr{C}}}{}^{\mathscr{B}}+\hat{\Omega}_{{\mathscr{A}}}{}^{\mathscr{C}}L_{{\mathscr{C}}}{}^{\mathscr{B}}=-DL_{{\mathscr{A}}}{}^{\mathscr{B}}\
.$ (21)

The torsion two-form is an exterior derivative of $E^{\mathscr{A}}$, and
$R_{{\mathscr{A}}}{}^{\mathscr{B}}$ is the curvature two-form

$T^{\mathscr{A}}=DE^{\mathscr{A}}\,,\qquad
R_{{\mathscr{A}}}{}^{\mathscr{B}}=d\hat{\Omega}_{{\mathscr{A}}}{}^{\mathscr{B}}+\hat{\Omega}_{{\mathscr{A}}}{}^{\mathscr{C}}\wedge\hat{\Omega}_{{\mathscr{C}}}{}^{\mathscr{B}}\
.$ (22)

This leads to Bianchi identities

$DT^{\mathscr{A}}=E^{\mathscr{B}}\wedge
R_{{\mathscr{A}}}{}^{\mathscr{B}}\,,\qquad
DR_{{\mathscr{A}}}{}^{\mathscr{B}}=0\ .$ (23)

The set of fundamental Bianchi identities (23) was solved in Brink:1979nt ;
Howe:1981gz  where the set of on-shell constraints was imposed on solutions.

An important point here is the role played by the global E7-type symmetry for
$\mathcal{N}\geq 4$. The tangent space symmetries in eq. (18) include local
Lorentz symmetry and local $(S)U(\mathcal{N})$ $\mathcal{H}$-symmetry. The
superspace connection $\hat{\Omega}$ is constructed from the superspace
vielbein

$\displaystyle{\cal
V}=\left(\begin{array}[]{cc}U^{IJ}{}_{ij}&\bar{V}^{IJij}\\\
V_{IJij}&\bar{U}_{IJ}{}^{ij}\\\ \end{array}\right)\ ,$ (26)

where ${\cal V}$ an element of the Lie group $\mathcal{G}$:
$(S)U(\mathcal{N})$ acts on the small indices $ij$ in a natural way with
superspace-depending parameters, while the global group $\mathcal{G}$ acts on
the capital indices by

$\delta{\cal V}=-\Lambda{\cal
V}\,,\qquad\Lambda=\left(\begin{array}[]{cc}\Lambda^{IJ}{}_{KL}&\,\,\bar{\Sigma}^{IJKL}\\\
\Sigma_{IJKL}&\,\,\bar{\Lambda}_{IJ}{}^{KL}\\\ \end{array}\right)\ ,$ (27)

The superspace connection $\hat{\Omega}=(d{\cal V}^{-1}){\cal V}$ is
$\mathcal{G}$-invariant and it is a connection one-form having zero curvature
$d\hat{\Omega}+\hat{\Omega}\wedge\hat{\Omega}=0$. This summarizes the status
of the $\mathcal{N}\geq 4$ on-shell superspace Brink:1979nt ; Howe:1981gz .

Is is possible to gauge-fix the tangent space $U(\mathcal{N})$ symmetry of the
superspace with $\mathcal{N}\geq 4$? We show how this is done by “de-gauging”
in $\mathcal{N}=2$ supergravity in Appendix A where there is no relevant
duality $\mathcal{G}$-symmetry.

Such a “de-gauging” was not done in $\mathcal{N}\geq 4$ superspace, but the
gauge-fixing of $\mathcal{H}$ local symmetry was performed in the unitary
gauge for the spacetime Lagrangian in Cremmer:1979up ; deWit:1982bul ;
Kallosh:2008ic ; Bossard:2010dq . It explains why the on-shell superspace
Brink:1979nt ; Howe:1981gz  needs to preserve the local $\mathcal{H}$ symmetry
as a tangent space symmetry.

##  4 Harmonic on-shell superspace of $\mathcal{N}\geq 4$ supergravities

The on-shell harmonic superspace counterterms for $\mathcal{N}\geq 4$
supergravities were constructed in Bossard:2011tq  and have a universal form
for $\mathcal{N}\geq 4$. The superinvariants in the harmonic superspace at
$d=4,L=\mathcal{N}-1$ are one loop order below $L_{cr}=\mathcal{N}$ and
represent a ${1\over\mathcal{N}}$ BPS invariants. This means that instead of
the whole superspace integrals $d^{4\mathcal{N}}\theta$, they involve
$d^{4(\mathcal{N}-1)}\theta$ integration.

In 4d $\mathcal{N}$-extended on-shell superspace was defined in Brink:1979nt ;
Howe:1981gz , and we described it shortly in Sec. 3. The purpose of
introducing additional coordinates, the harmonic ones, is to be able to
justify subspace integrals over a fraction of the whole superspace and to
define the so-called Grassmann analytic superfields, which depend on the
coordinates of the analytic subspace only.

The off-shell harmonic superspace is not known to exist for supergravity with
$\mathcal{N}>2$ Galperin:2001seg , but for the case of interest with
$\mathcal{N}\geq 4$, the hope was that the on-shell harmonic superspace could
be constructed. So the proposal for the on-shell harmonic superspace of
supergravity was to add harmonic coordinates $u^{I}{}_{i}$ and their inverse
$u^{i}{}_{I}$, $i=1,\dots,\mathcal{N},\,I=1,\dots,\mathcal{N}$. The role of
these coordinates is to represent the flag manifold $H\backslash K$

$\mathbb{F}_{1,1}(\mathcal{N})\cong\bigl{(}U(1)\times
U({\mbox{$\mathcal{N}$}}-2)\times U(1)\bigr{)}\backslash
U({\mbox{$\mathcal{N}$}})\ ,$ (28)

where $K\cong U(\mathcal{N})$ and the isotropy group is $H\cong U(1)\times
U({\mbox{$\mathcal{N}$}}-2)\times U(1)$. It is proposed in Bossard:2011tq ,
with reference to harmonic spaces in Galperin:2001seg , to work in
$\mathcal{N}\geq 4$ supergravities on such a coset space. The index $I$ is
split according to the structure of the isotropy group:

$I=(1,r,{\mbox{$\mathcal{N}$}})$ (29)

The element of $U({\mbox{$\mathcal{N}$}})$ is denoted by $u^{I}{}_{i}$, and it
is stated that the local gauge group acts to the right and the isotropy group
acts to the left. At this point, we need to clarify the meaning of the element
of $U({\mbox{$\mathcal{N}$}})$ called $u^{I}{}_{i}$ and defined in a proposal
made in Bossard:2011tq .

In the principal $U(\mathcal{N})$ bundle there is a Lie-algebra-valued one-
form $\omega$ that combines the Maurer-Cartan form on the group with the
$U(\mathcal{N})$ connection on the base:

$\omega=duu^{-1}+u\Omega
u^{-1}\qquad\Rightarrow\qquad\omega^{I}{}_{J}=du^{I}{}_{i}(u^{-1})^{i}{}_{J}+u^{I}{}_{i}\Omega^{i}{}_{j}(u^{-1})^{j}{}_{J}\
.$ (30)

Here $\Omega^{i}{}_{j}$ is the $U(\mathcal{N})$ connection of the standard
superspace Brink:1979nt ; Howe:1981gz . This form is split into its isotropy
$\mathfrak{h}$ and coset components $\mathfrak{f}$

$\omega=\omega_{\mathfrak{h}}+\omega_{\mathfrak{f}}\ .$ (31)

The curvature is introduced

$d\omega+\omega^{2}=uRu^{-1}\ ,$ (32)

and it is split into its $\mathfrak{h}$ and $\mathfrak{f}$ part:

$\displaystyle D\omega_{\mathfrak{f}}$ $\displaystyle=$
$\displaystyle-(\omega_{\mathfrak{f}}\wedge\omega_{\mathfrak{f}})_{\mathfrak{f}}+(uRu^{-1})_{\mathfrak{f}}$
$\displaystyle d\omega_{\mathfrak{h}}+\omega_{\mathfrak{h}}^{2}$
$\displaystyle=$
$\displaystyle-(\omega_{\mathfrak{f}}\wedge\omega_{\mathfrak{f}})_{\mathfrak{h}}+(uRu^{-1})_{\mathfrak{h}}\
,$ (33)

$D$ here is the exterior derivative covariant with respect to $H$. It is
suggested in Bossard:2011tq  that “in equations (33), the gauge was fixed with
respect to the isotropy group acting on $K$ so that $u$ are functions of local
coordinates, $t$ say, on $F$.” However, there is also a statement in
Bossard:2011tq : “there has not been a choice of $U(\mathcal{N})$ gauge”. We
will study below the consequences of the gauge-fixing performed in
Bossard:2011tq .

The standard superspace Brink:1979nt ; Howe:1981gz  has $\mathcal{N}^{2}$
local symmetries of $\mathcal{H}=U(\mathcal{N})$ symmetry in the tangent
space. The local tangent symmetry $SL(2,\mathbb{C})\times U(\mathcal{N})$
includes the local Lorentz symmetry as well as a local $U(\mathcal{N})$.

The claim in Bossard:2011tq  is that the tangent space local symmetry of the
harmonic space is $SL(2,\mathbb{C})\times H$, which means that the local
symmetries in the coset directions were gauge-fixed. Consider this in more
details.

The $\mathcal{H}$ symmetries are split in a harmonic superspace Bossard:2011tq
where harmonic coordinates are in a coset space

${\mathcal{H}\over H}={U({\mbox{$\mathcal{N}$}})\over U(1)\times
U({\mbox{$\mathcal{N}$}}-2)\times U(1)}$ (34)

There are $(\mathcal{N}-2)^{2}+2$ symmetries in $H$ and remaining

$\mathcal{N}^{2}-[(\mathcal{N}-2)^{2}+2]=4\mathcal{N}-6$ (35)

in the coset direction. Thus, in presence of harmonic coordinates the local
$\mathcal{H}$ symmetry of the normal on-shell superspace is represented by
$(\mathcal{N}-2)^{2}+2$ symmetries in $H$ and $4\mathcal{N}-6$ symmetries in
the coset space (34).

The number of directions in the coset space (34) is $4\mathcal{N}-6$. The
harmonic measure of integration given in Bossard:2011tq  is

$d\mu_{\scriptscriptstyle({\mbox{$\mathcal{N}$}},1,1)}:=d^{4}x\,d^{4{\mbox{$\mathcal{N}$}}-6}t\,d^{2({\mbox{$\mathcal{N}$}}-1)}\theta\,d^{2({\mbox{$\mathcal{N}$}}-1)}\bar{\theta}\,\mathcal{E}(\hat{x})V(t)\
.$ (36)

where additional $(4\mathcal{N}-6)$ bosonic $t$-coordinates are present in the
measure of integration and $\hat{x}$ stands for all the harmonic superspace
coordinates aside from 4 fermionic directions missing in (36) comparative to
whole $d^{4\mathcal{N}}\theta$.

It was suggested in Bossard:2011tq  to introduce a quantity $h(I)$ with
$I=(1,r,{\mbox{$\mathcal{N}$}})$ such that

$h(1)=1\ ,\quad h(r)=0\ ,\quad h({\mbox{$\mathcal{N}$}})=-1\ .$ (37)

The coset indices are then pairs $I,J$ such that $h(I)\neq h(J)$ and the coset
symmetry parameters are

$(\Lambda^{I}{}_{J})_{\mathfrak{f}}\,:\qquad h(I)\neq h(J)\ .$ (38)

while the $H$-indices are pairs $I,J$ with $h(I)=h(J)$ and the $H$ symmetry
parameters are

$(\Lambda^{I}{}_{J})_{\mathfrak{h}}\,:\qquad h(I)=h(J)\ .$ (39)

In standard on-shell superspace Brink:1979nt ; Howe:1981gz  with
$\mathcal{N}\geq 4$ all geometric torsion and curvatures depend on spinorial
superfield $\chi_{\alpha\,kij}(x,\theta,\bar{\theta})$, its conjugate, and
their superspace derivatives.

Now we are looking at curvature constraints in harmonic superspace
Bossard:2011tq . For example, in $\mathcal{N}=4,5,8$, certain components of
harmonic superspace curvature are quadratic in spinorial superfield
$\chi_{\alpha\,kij}(x,\theta,\bar{\theta})$ of the standard superspace and
quadratic in harmonic coordinates, and take the following form

$(R^{1}_{\alpha\,\dot{\beta}\mathcal{N}})^{1}{}_{1}=(R^{1}_{\alpha\,\dot{\beta}\mathcal{N}})^{\mathcal{N}}{}_{\mathcal{N}}={1\over
2}B_{\alpha\,\dot{\beta}}\neq 0\ .$ (40)

Here

$B_{\alpha\,\dot{\beta}}(x,\theta,\bar{\theta},u)=u^{1}{}_{l}\,\Big{(}\bar{\chi}_{\dot{\beta}}^{lij}(x,\theta,\bar{\theta})\chi_{\alpha\,kij}(x,\theta,\bar{\theta})\Big{)}\,u^{k}{}_{\mathcal{N}}\
.$ (41)

There is a subset of vector fields
$(\tilde{E}_{\alpha}^{1},\tilde{E}_{{\dot{\alpha}}\mathcal{N}}):=\tilde{E}_{\hat{\alpha}}$
and the corresponding curvature components of the harmonic superspace consists
of $H$ and coset directions

$(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}=(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{h}}+(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{f}}$
(42)

where $(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{h}}$ has
$h(I)=h(J)$ and $(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{f}}$
has $h(I)\neq h(J)$. The constraints on these curvature components are given
in Bossard:2011tq  in the form

$(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{f}}=0\,$ (43)

and

$(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{h}}\neq
0\qquad\Rightarrow\qquad(R^{1}_{\alpha\,\dot{\beta}\mathcal{N}})^{1}{}_{1}=(R^{1}_{\alpha\,\dot{\beta}\mathcal{N}})^{\mathcal{N}}{}_{\mathcal{N}}={1\over
2}B_{\alpha\,\dot{\beta}}$ (44)

based on the standard on-shell superspace structure in Howe:1981gz

We see, therefore, from (40), (41) that components of the on-shell harmonic
superspace curvature, which are not vanishing, are in the direction
$(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{h}}$ with $h(I)=h(J)$.
All other components in direction
$(R_{\hat{\alpha}\,\hat{\beta}})^{I}{}_{J}|_{\mathfrak{f}}$ with $h(I)\neq
h(J)$ have to vanish, according to (43).

A tangent space symmetry for a curvature 2-form $R^{I}{}_{J}$ is

$(R^{I}{}_{J})^{\prime}=\Lambda^{I}{}_{K}R^{K}{}_{L}\Lambda^{L}{}_{J}$ (45)

This splits into the coset part and the $H$ part. The coset part has the
following symmetries

$\Lambda^{1}{}_{r},\,\,\Lambda^{r}{}_{\mathcal{N}},\,\,\Lambda^{1}{}_{\mathcal{N}}$
(46)

and the complex conjugate ones are

$\Lambda^{r}{}_{1},\,\,\Lambda^{\mathcal{N}}{}_{r},\,\,\Lambda^{\mathcal{N}}{}_{1}$
(47)

Consider 3 cases

$(R^{1}{}_{r})^{\prime}=\Lambda^{1}{}_{K}R^{K}{}_{L}\Lambda^{L}{}_{r}=\Lambda^{1}{}_{1}R^{1}{}_{1}\Lambda^{1}{}_{r}+\Lambda^{1}{}_{\mathcal{N}}R^{\mathcal{N}}{}_{\mathcal{N}}\Lambda^{\mathcal{N}}{}_{r}+\dots\,,$
(48)

$\,\,\,(R^{r}{}_{\mathcal{N}})^{\prime}=\Lambda^{r}{}_{K}R^{K}{}_{L}\Lambda^{L}{}_{\mathcal{N}}=\Lambda^{r}{}_{1}R^{1}{}_{1}\Lambda^{1}{}_{\mathcal{N}}+\Lambda^{r}{}_{\mathcal{N}}R^{\mathcal{N}}{}_{\mathcal{N}}\Lambda^{\mathcal{N}}{}_{\mathcal{N}}+\dots\,,$
(49)

$\,\,\,(R^{1}{}_{\mathcal{N}})^{\prime}=\Lambda^{1}{}_{K}R^{K}{}_{L}\Lambda^{L}{}_{\mathcal{N}}=\Lambda^{1}{}_{1}R^{1}{}_{1}\Lambda^{1}{}_{\mathcal{N}}+\Lambda^{1}{}_{\mathcal{N}}R^{\mathcal{N}}{}_{\mathcal{N}}\Lambda^{\mathcal{N}}{}_{\mathcal{N}}+\dots\,.$
(50)

All terms with $\dots$ vanish since they include curvatures vanishing
according to (43). Since each of the vanishing coset curvature components is
transformed into a non-vanishing $H$ curvature components
$R^{1}{}_{1}=R^{\mathcal{N}}{}_{\mathcal{N}}\neq 0$, we find $2\mathcal{N}-3$
complex conditions on parameters $\Lambda^{I}_{J}$, which is a condition that
$\Lambda^{I}_{J}|_{\mathfrak{f}}=0$. It means that there are $4\mathcal{N}-6$
constraints on local $U({\mbox{$\mathcal{N}$}})$ symmetry of the tangent space
$\Lambda^{i}{}_{j}$

$\Lambda^{I}_{J}|_{\mathfrak{f}}=u^{I}{}_{i}\Lambda^{i}{}_{j}u^{j}{}_{J}=0$
(51)

To summarize, the standard superspace Brink:1979nt ; Howe:1981gz  has a local
$U({\mbox{$\mathcal{N}$}})$ symmetry of the tangent space with
$\mathcal{N}^{2}$ independent parameters
$\Lambda^{i}{}_{j}(x,\theta,\bar{\theta})$. But in the harmonic space with
$R^{I}{}_{J}|_{\mathfrak{f}}=0$ and
$R^{1}{}_{1}=R^{\mathcal{N}}{}_{\mathcal{N}}\neq 0$, there is a constraint
(51) which says that all $4\mathcal{N}-6$ coset symmetries are broken.

Curvatures in coset direction $R^{I}{}_{J}|_{\mathfrak{f}}=0$ have to vanish
to provide an integrability condition for Grassmann analyticity. The only way
to make this picture consistent is to break the coset symmetries associated
with the coset space in (34), i. e. to break local $U({\mbox{$\mathcal{N}$}})$
symmetry of the standard superspace of $\mathcal{N}\geq 4$ supergravities in
4d.

Note that in $\mathcal{N}\geq 4$ case there is no superspace with partially
gauge-fixed $U(\mathcal{N})$ symmetry. In the next section, we explain the
reason for this in the context of the ${\mathcal{G}\over\mathcal{H}}$ coset
space of $\mathcal{N}\geq 4$ supergravities 333 In $\mathcal{N}=2$
supergravity, this procedure of constructing harmonic superspace is consistent
since in standard $\mathcal{N}=2$ superspace, the local $U(2)$ symmetry can be
gauge-fixed, using the compensating multiplets, as it was done in Howe:1981gz
or in projective superspace in Kuzenko:2008ep ; Kuzenko:2008ry . We stress
here that in $\mathcal{N}=2$ there is no spinorial superfield
$\chi_{\alpha\,[kij]}(x,\theta,\bar{\theta})$ from which non-vanishing
curvature components are build as shown in eq. (44), and therefore all
components of curvature $R^{I}{}_{J}$ vanish in $\mathcal{N}=2$: local $U(2)$
is not a symmetry of the tangent space..

##  5 Unitary gauge in spacetime with gauge-fixed local $\mathcal{H}$-symmetry

In spacetime, local $\mathcal{H}$-symmetry was gauge-fixed for the classical
Lagrangian in the original papers Cremmer:1979up ; deWit:1982bul  where it was
explained that a field-dependent compensating $SU(8)$ transformation is
required to preserve the gauge. A detailed study of this compensating $SU(8)$
transformation was performed in Kallosh:2008ic ; Bossard:2010dq . Under the
combined action of local $SU(8)$ and global $E_{7(7)}$ the 56-bein transforms
as

${\cal V}\quad\rightarrow\quad{\cal V}^{\prime}=h(x){\cal
V}g^{-1}\,,\qquad\qquad h(x)\in SU(8)\;,\;\;g\in E_{7(7)}$ (52)

i. e. from the left by a local $SU(8)$ and from the right by a global
$E_{7(7)}$. Before gauge-fixing local $SU(8)$ the number of scalars is 133,
the number of the elements in ${\cal V}$.

The unitary gauge is defined as follows Cremmer:1979up ; deWit:1982bul

${\cal V}={\cal V}^{\dagger}$ (53)

which leaves us with 70 scalars $\phi^{ijkl}$ in the ${\bf 70}$ of $SU(8)$

${\cal
V}(x)\equiv\exp\Phi(x)=\exp\left(\begin{array}[]{cc}0&\phi_{ijkl}(x)\vspace{0mm}\\\
\phi^{ijkl}(x)&0\end{array}\right)$ (54)

with $\phi^{ijkl}=(\phi_{ijkl})^{*}$. The 133-70=63 local parameters of
$SU(8)$ were used to eliminate the unphysical 63 scalars. This gauge is known
as a unitary gauge, with only 70 physical scalars. The unitary gauge ${\cal
V}={\cal V}^{\dagger}$ is not preserved by a $\mathcal{G}$-symmetry
transformation since this symmetry involves all 133 scalars, elements of
unconstrained ${\cal V}$. However, if a $\mathcal{G}$-symmetry transformation
is complemented by a field-dependent $\mathcal{H}$-symmetry transformation, it
is possible to maintain the gauge. In such a case, in the unitary gauge, the
$\mathcal{H}$-symmetry transformation is nonlinearly realized and depends on
$x$-dependent scalar fields of the theory and on global parameters of
$\mathcal{G}$-symmetry.

The analysis in Kallosh:2008ic  was performed in terms of inhomogeneous
coordinates of the ${E_{7(7)}\over SU(8)}$ coset space defined as function of
the independent scalars fields $\phi,\bar{\phi}$ as follows:

$y_{ij,kl}(x)\equiv\phi_{ijmn}(x)\left({\tanh(\sqrt{\bar{\phi}\phi(x)}\over\sqrt{\bar{\phi}\phi(x)}}\right)^{mn}_{\;\;\;\;\;{kl}}$
(55)

Infinitesimal $E_{7(7)}$ transformation is

$\displaystyle E^{-1}=\left(\begin{array}[]{cc}1+\Lambda&-\Sigma\\\
-\bar{\Sigma}&1+\bar{\Lambda}\\\ \end{array}\right)\ ,$ (58)

In terms of the nonlinear scalar fields $y,\bar{y}$, the matrix $\mathcal{V}$
is given by the following expression

$\displaystyle\mathcal{V}(y,\bar{y})=\left(\begin{array}[]{cc}P^{-1/2}&-P^{-1/2}y\\\
-\bar{P}^{-1/2}\bar{y}&\bar{P}^{-1/2}\\\ \end{array}\right)\,,\qquad
P(y,\bar{y})_{ij}{}^{kl}\equiv(\delta_{ij}{}^{kl}-y_{ijrs}\bar{y}^{rskl})$
(61)

The unitary gauge condition (53), (54) is preserved if the transformation law
includes simultaneous  multiplication of the 56-bein from the right by an
arbitrary rigid $E_{7(7)}$ group element depending on 63 rigid parameters
$\Lambda_{J}{}^{L}$ and 70 rigid parameters $\Sigma_{IJKL}$ as well as a
multiplication from the left by a compensating $SU(8)$ transformation
depending on scalars as well as on 63 rigid parameters $\Lambda_{J}{}^{L}$ and
70 rigid parameters $\Sigma_{IJKL}$

$\displaystyle\mathcal{U}(y,\bar{y};\Lambda,\Sigma)\mathcal{V}(y^{\prime})=\mathcal{V}(y,\bar{y})E^{-1}(\Lambda,\Sigma)\
,$ (62)

The compensating $SU(8)$ transformation with parameters depending on scalars
and $E_{7(7)}$ parameters in eq. (58) is

${\cal U}(y(x),\bar{y}(x),\Lambda,\Sigma)=1+\frac{1}{2}\
P^{-1/2}\Delta(\Lambda,\Sigma)P^{-1/2}\equiv 1+\Upsilon\ ,$ (63)

where $\Delta(\Lambda,\Sigma)$ is given by

$\displaystyle\Delta(\Lambda,\Sigma)=\\{\Lambda,P\\}+(y\bar{\Sigma}-\Sigma\bar{y})-y(\bar{\Sigma}y-\bar{y}\Sigma)\bar{y}\
.$ (64)

In Bossard:2010dq  in a different setup, it was also found that the
compensating $SU(8)$ transformation induced and by $E_{7(7)}$

$g\equiv\exp{\Sigma}=\left(\begin{array}[]{cc}0&\Sigma_{ijkl}\vspace{0mm}\\\
\Sigma^{ijkl}&0\end{array}\right)\;\;.$ (65)

has the following scalar-dependent form

$\delta^{SU(8)}\Big{(}\tanh(\Phi(x)/2)*{\Sigma}\Big{)}$ (66)

where $\Phi(x)$ is defined in eq. (54) and the $*$ operation is associated
with the Baker-Campbell-Hausdorf formula for multiplication of two Lie algebra
elements $\exp\Phi(x)$ in (54) and $\exp{\Sigma}$ in (65).

Clearly, one can see in both cases in Kallosh:2008ic ; Bossard:2010dq  that
the compensating $SU(8)$ transformation in (63) and in (66) depend on scalars
fields $\phi_{ijkl}(x)$ and on $E_{7(7)}$ parameters.

The reason that there is no superspace available corresponding to the unitary
gauge with only physical scalars has to do with the fact that the on-shell
superspace at the point where all $\theta$’s vanish represents the $x$-space
supergravity equations of motion and their symmetries. These symmetries are
either local or global; local ones, like Lorentz symmetry and internal
symmetries, become symmetries of the tangent space. The scalar field dependent
$\mathcal{H}$ symmetries are not compatible with the concept of the
superspace, which has curved superspace general coordinate transformations and
tangent space structure group. Therefore, the unitary gauge in
$\mathcal{N}\geq 4$ supergravities cannot be represented in superspace.

The situation is different in $\mathcal{N}=2$ case, where local $U(2)$ or a
part of it can be removed from the structure group in superspace, see Appendix
A, where we present some known examples. Therefore $\mathcal{N}=2$
supergravity was considered in the harmonic or projective superspace, but not
in case $\mathcal{N}>2$ Galperin:2001seg ; Kuzenko:2008ep ; Kuzenko:2008ry .
An important difference between $\mathcal{N}=2$ case and $\mathcal{N}\geq 4$
supergravities is the presence of duality symmetry $\mathcal{G}$ in
$\mathcal{N}\geq 4$ and absence of it in $\mathcal{N}=2$.

In the on-shell harmonic superspace Bossard:2011tq , the additional
coordinates $u^{I}{}_{i}$ were nevertheless added to $\mathcal{N}\geq 4$
superspace, which, as we have shown, breaks local $\mathcal{H}$ symmetry, and
makes this superspace inconsistent.

##  6 Harmonic Lagrangian and G-analyticity break local $\mathcal{H}$ symmetry

The local $U({\mbox{$\mathcal{N}$}})$ symmetry acts linearly on harmonic
superspace superfields of a standard superspace Howe:1981gz

$\delta\bar{\chi}^{ijk}(x,\theta,\bar{\theta})=\Lambda^{[i}{}_{l}(x,\theta,\bar{\theta})\bar{\chi}^{ljk]}(x,\theta,\bar{\theta})\,,$
(67)

It is suggested in Bossard:2011tq  that the new coordinates $u^{I}{}_{i}(t)$
are transformed by a local $U({\mbox{$\mathcal{N}$}})$, which would mean that
the parameters of transformations depend on local coordinates
$(x,\theta,\bar{\theta})$. But in fact, this is inconsistent with the
constraints on the harmonic superspace curvatures (40), (43) and the fact that
$u^{I}{}_{i}$ depend only on local coordinates of the coset space.

Therefore the harmonic superspace superfields like

$\bar{\chi}^{Ijk}(x,\theta,\bar{\theta},t)\equiv
u^{I}{}_{i}(t)\bar{\chi}^{ijk}(x,\theta,\bar{\theta})$ (68)

are not covariant under local $U({\mbox{$\mathcal{N}$}})$ symmetry.

The harmonic superspace curvatures in Bossard:2011tq  are defined as a
contraction of a local $SU(\mathcal{N})$ tensor quadratic in spinorial
superfields $\chi_{ijk\alpha}(x,\theta)$ and its conjugate with harmonic
coordinates $u^{1}_{i}(t),u^{l}_{\mathcal{N}}(t)$. These curvatures are

$(R^{1}_{\alpha\,\dot{\beta}\mathcal{N}})^{1}{}_{1}=(R^{1}_{\alpha\,\dot{\beta}\mathcal{N}})^{\mathcal{N}}{}_{\mathcal{N}}={1\over
2}B_{\alpha\,\dot{\beta}}(x,\theta,\bar{\theta},u)={1\over
2}u^{1}{}_{l}\,\Big{(}\bar{\chi}_{\dot{\beta}}^{lij}(x,\theta,\bar{\theta})\chi_{\alpha\,kij}(x,\theta,\bar{\theta})\Big{)}\,u^{k}{}_{\mathcal{N}}\
.$ (69)

and the Lagrangian is

${\cal L}=B_{\alpha\dot{\beta}}B^{\alpha\dot{\beta}}(x,\theta,\bar{\theta},t)$
(70)

where

$(B_{\alpha\dot{\beta}})^{1}{}_{\mathcal{N}}\equiv
u^{1}_{i}u^{l}_{\mathcal{N}}\bar{\chi}^{ijk}_{\dot{\beta}}\chi_{\alpha
ljk}\,,\quad(B^{\alpha\dot{\beta}})^{1}{}_{\mathcal{N}}=\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}(B_{\beta\dot{\alpha}})^{1}{}_{\mathcal{N}}$
(71)

Under local $U(\mathcal{N})$ symmetry (67) with $\mathcal{N}^{2}$ independent
local parameters $\Lambda^{i}{}_{l}(x,\theta,\bar{\theta})$, the superfield
$B_{\alpha\,\dot{\beta}}$ is not covariant

$\delta_{U({\mbox{$\mathcal{N}$}})|_{\mathfrak{f}}}\,B_{\alpha\,\dot{\beta}}(x,\theta,\bar{\theta},t)=\delta_{U({\mbox{$\mathcal{N}$}})|_{\mathfrak{f}}}\Big{(}u^{1}{}_{l}(t)\,\Big{(}\bar{\chi}_{\dot{\beta}}^{lij}(x,\theta,\bar{\theta})\chi_{\alpha\,kij}(x,\theta,\bar{\theta})\Big{)}\,u^{k}{}_{\mathcal{N}}(t)\Big{)}\neq
0$ (72)

The reason for this is the fact that the variation of $u^{I}{}_{i}(t)$ is not
one which would cancel the variation of the superfield $\chi$. Namely, one
would need a transformation $\delta
u^{I}{}_{i}(t)=u^{I}{}_{k}(t)\Lambda^{k}{}_{i}(x,\theta,\bar{\theta})$ with
$\mathcal{N}^{2}$ independent parameters and linearly realized local
$U(\mathcal{N})$ symmetry. But this is not the case. In particular, the actual
transformation consistent with constraints on harmonic superspace curvatures
requires that these symmetries in the coset direction are broken.

The Lagrangian is a contraction of a local $U(\mathcal{N})$ tensor quartic in
superfields $\chi$, and the object depending on four $u^{I}_{i}(t)$’s

${\cal L}=B_{\alpha\dot{\beta}}B^{\alpha\dot{\beta}}=T^{nl}{}_{im}\times{\cal
L}^{im}{}_{nl}(x,\theta,\bar{\theta})\ .$ (73)

The local $U(\mathcal{N})$ tensor is

${\cal
L}^{in}{}_{ml}(x,\theta,\bar{\theta})=\bar{\chi}^{ijk}_{\dot{\beta}}\chi_{mjk\alpha}\,\bar{\chi}^{nop}_{\dot{\alpha}}\chi_{lop\beta}\,\epsilon^{\alpha\beta}\,\epsilon^{\dot{\alpha}\dot{\beta}}$
(74)

and the object quartic in $u$’s is

$T^{nl}{}_{im}=u^{1}{}_{i}u^{1}{}_{m}u^{n}{}_{\mathcal{N}}u^{l}{}_{\mathcal{N}}\
.$ (75)

It follows that the harmonic Lagrangian ${\cal L}$ breaks local
$U({\mbox{$\mathcal{N}$}})$ symmetry in the coset directions of
${U(\mathcal{N})\over U(1)\times U({\mbox{$\mathcal{N}$}}-2)\times U(1)}$ so
that

$\delta_{U({\mbox{$\mathcal{N}$}})|_{\mathfrak{f}}}{\cal
L}=\delta_{U({\mbox{$\mathcal{N}$}})|_{\mathfrak{f}}}B_{\alpha\,\dot{\beta}}B^{\alpha\,\dot{\beta}}(x,\theta,\bar{\theta},t)\neq
0$ (76)

The superfield $B_{\alpha\dot{\beta}}$ and all its powers are Grassmann
analytic according to Bossard:2011tq . It means that

$D_{\gamma}^{1}B_{\alpha\dot{\beta}}=\bar{D}_{\dot{\gamma}\mathcal{N}}B_{\alpha\dot{\beta}}=0\
,$ (77)

where

$D_{\gamma}^{1}=u^{1}{}_{i}D_{\gamma}^{i}\,\qquad\bar{D}_{\dot{\gamma}\mathcal{N}}=D_{\dot{\gamma}i}u^{i}{}_{\mathcal{N}}\
.$ (78)

The proof of integrability condition for each $\mathcal{N}\geq 4$ is given in
Appendix A of Bossard:2011tq  based on standard superspace geometry
Howe:1981gz  with local $U(\mathcal{N})$ symmetry of the tangents space. Here
$D_{\gamma}^{i}$ and $D_{\dot{\gamma}i}$ transform under a local
$U(\mathcal{N})$ symmetry according to positions of the indices $i$.

If we accept the claim in Bossard:2011tq  that $u^{1}_{i},u^{l}_{\mathcal{N}}$
transform under local $U(\mathcal{N})$, we would think that the G-analyticity
condition is invariant under local $U(\mathcal{N})$ symmetry. However, this is
inconsistent with constraints on curvatures in the harmonic superspace. It
means that the G-analyticity condition (77) breaks local $U(\mathcal{N})$
despite the integrability condition for the G-analyticity condition shown in
eq. (43) is satisfied.

Let us look at the G-analyticity of the Lagrangian (70). This Lagrangian is
claimed to be duality-invariant G-analytic scalar superfield. The fact that it
depends on spinorial superfield $\chi^{ijk}_{\alpha}(x,\theta)$ of the
standard superspace, where this superfield is a superspace torsion, is indeed
consistent with duality $\mathcal{G}$ symmetry, under the condition that the
local $U(\mathcal{N})$ structure group symmetry is valid. The G-analyticity of
the Lagrangian means that

$D_{\alpha}^{1}{\cal L}=\bar{D}_{\dot{\alpha}\mathcal{N}}{\cal L}=0\ .$ (79)

The analyticity condition of the Lagrangian can be presented in the form

$D_{\alpha}^{1}{\cal L}=F^{nl}{}_{qim}\times(D_{\alpha}^{q}{\cal
L}^{im}{}_{nl})(x,\theta,\bar{\theta})\ ,$ (80)

where

$F^{nl}{}_{qim}=u^{1}{}_{q}u^{1}{}_{i}u^{1}{}_{m}u^{n}{}_{\mathcal{N}}u^{l}{}_{\mathcal{N}}\
.$ (81)

It is also given by a product of local $U(\mathcal{N})$ tensor
$(D_{\alpha}^{q}{\cal L}^{im}{}_{nl})(x,\theta,\bar{\theta})$ times a product
of five $u$’s in eq. (81). It is, therefore, not invariant under a local
$U(\mathcal{N})$ symmetry.

If one would like to gauge-fix the local $U(\mathcal{N})$ symmetry in
superspace, following procedure known in space-time, see Sec. 5, without
manifest supersymmetry, one would have to build first a normal on-shell
superspace breaking its tangent space local $U(\mathcal{N})$ symmetry. The
compensating, superfield-dependent, local $U(\mathcal{N})$ symmetry
transformation would be required to preserve global $\mathcal{G}$ symmetry.
Even ignoring the fact that such an on-shell superspace with gauge-fixed local
$U(\mathcal{N})$ symmetry was not constructed, we observe that at $\theta=0$
the compensating $U(\mathcal{N})$ transformation depends on scalars
$\phi_{ijkl}(x)$. In superspace, it would depend on superfields, so the new
coordinates $u_{i}^{I}(t)$ would be required to transform on $U(\mathcal{N})$
symmetry with superfield dependent parameters. Therefore, if a corresponding
superspace with gauge-fixed local $U(\mathcal{N})$ symmetry would be
constructed, it would be inconsistent with harmonic coordinates
$u_{i}^{I}(t)$: these cannot accommodate at $\theta=0$ spacetime field
dependent $U(\mathcal{N})$ transformation. Thus, even in the unitary gauge,
there is a clear inconsistency of the nonlinear harmonic superspace
Bossard:2011tq  for $\mathcal{N}\geq 4$.

To summarize, the inconsistency with local $U(\mathcal{N})$ symmetry in
Bossard:2011tq  is that, on one hand, this local symmetry is claimed to be not
gauge-fixed. But actually, harmonic coordinates are treated by analogy with
harmonic superspace coordinates Galperin:2001seg  or projective superspace
coordinates in $\mathcal{N}=2$ case. For example, in Kuzenko:2008ep ;
Kuzenko:2008ry , whose strategy is used in Bossard:2011tq , it is explained
that the $u_{i}^{+}$ serve merely to totally symmetrize all $SU(2)$ indices
$i$. Also, they present the explicit dependence of $u^{+}{}_{i}$ on two
${SU(2)\over U(1)}$ coset coordinates but only after the local $SU(2)$
symmetry was gauge-fixed in the coset directions. And since there was no
$\mathcal{G}$ duality symmetry, no compensating field dependent $SU(2)$ was
required. It allowed a consistent construction of the harmonic or projective
superspace in $\mathcal{N}=2$ supergravity. This, however, is not possible in
$\mathcal{N}\geq 4$ supergravities.

It is not surprising therefore that 12 graphs in $\mathcal{N}=4$, $L=3$
Bern:2012cd  and 82 diagrams in $\mathcal{N}=5$, $L=4$ cancelled Bern:2014sna
: the $L=\mathcal{N}-1$ harmonic superinvariants in Bossard:2011tq  break
local $U(\mathcal{N})$ symmetry and, as a result of the algebra (1), also a
local nonlinear supersymmetry.

##  7 Discussion

In 4d supergravity at $\mathcal{N}\geq 4$ the critical loop order
$L_{cr}={\mathcal{N}}$ is the loop order where the geometric superinvariants
Kallosh:1980fi ; Howe:1980th  have both local $\mathcal{H}$ symmetry and
global duality $\mathcal{G}$ symmetry of the relevant on-shell superspace. Any
subspace of the whole superspace with less fermionic directions and no
additional coordinates, like harmonic ones, breaks the local $\mathcal{H}$
symmetry of the on-shell superspace.

In 4d $\mathcal{N}\geq 4$ supergravity, an attempt was made in Bossard:2011tq
to construct a harmonic on-shell superspace with the property that there are
${1\over\mathcal{N}}$ BPS superinvariants built in a subspace of the whole
superspace, where the number of fermionic directions instead of $4\mathcal{N}$
is equal to $4(\mathcal{N}-1)$. This corresponds to candidate counterterms at
$L=L_{cr}-1=\mathcal{N}-1$.

In an on-shell harmonic superspace Bossard:2011tq , the integrability
condition for the existence of Grassman analytic superfields, i.e.,
superfields independent on some of the fermionic directions, requires that
harmonic superspace curvature components in a coset direction
${H\backslash\mathcal{H}}$ to vanish, namely

$(R_{\hat{\alpha}\,\hat{\beta}})^{1}{}_{r}=(R_{\hat{\alpha}\,\hat{\beta}})^{r}{}_{\mathcal{N}}=(R_{\hat{\alpha}\,\hat{\beta}})^{1}{}_{\mathcal{N}}=0$
(82)

This is necessary for the G-analyticity condition

$D_{\gamma}^{1}B_{\alpha\dot{\beta}}=\bar{D}_{\dot{\gamma}\mathcal{N}}B_{\alpha\dot{\beta}}=0$
(83)

to be consistent with the harmonic superspace geometry.

However, harmonic superspace curvature components in $H$-direction do not
vanish

$(R_{\hat{\alpha}\,\hat{\beta}})^{1}{}_{1}=R_{\hat{\alpha}\,\hat{\beta}})^{\mathcal{N}}{}_{\mathcal{N}}\neq
0$ (84)

since the harmonic superspace Lagrangian is built from these non-vanishing
components of the curvature

${\cal
L}=(R_{\alpha\,\dot{\beta}})^{1}{}_{1}(R_{\beta\,\dot{\alpha}})^{1}{}_{1}\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}\,.$
(85)

But to keep some curvatures vanishing and some not vanishing is only possible
if local symmetries in the coset direction are broken: namely, there are
$4\mathcal{N}-6$ constraints on local $\mathcal{H}=U({\mbox{$\mathcal{N}$}})$
symmetry of the tangent space of the standard superspace Brink:1979nt ;
Howe:1981gz  with parameters $\Lambda^{i}{}_{j}(x,\theta,\bar{\theta})$

$u^{1}{}_{i}\Lambda^{i}{}_{j}(x,\theta,\bar{\theta})u^{j}{}_{r}=u^{r}{}_{i}\Lambda^{i}{}_{j}(x,\theta,\bar{\theta})u^{j}{}_{\mathcal{N}}=u^{I}{}_{1}\Lambda^{i}{}_{j}(x,\theta,\bar{\theta})u^{j}{}_{\mathcal{N}}=0$
(86)

And since local $U({\mbox{$\mathcal{N}$}})$ symmetry of the tangent space is
broken, it also means nonlinear local supersymmetry is broken, according to
the algebra (1).

At the linear level, in the unitary gauge with physical scalars only, harmonic
superspace $L=\mathcal{N}-1$ superinvariants, constructed in Bossard:2011tq ,
are eligible superinvariants. At the linear level, the compensating
$(S)U(\mathcal{N})$ transformation, required to preserve duality $\mathcal{G}$
symmetry, is just a global field independent $\mathcal{H}$ transformation.
Therefore, harmonic coordinates are consistent with global $\mathcal{H}$
symmetry. But at the nonlinear level, the field-dependent compensating
$SU(\mathcal{N})$ transformation is incompatible with harmonic coordinates,
and the $L=L_{cr}-1=\mathcal{N}-1$ superinvariants break nonlinear
supersymmetry.

The loop computation in $\mathcal{N}=4$, $L=3$ Bern:2012cd , and in
$\mathcal{N}=5$, $L=4$ Bern:2014sna  show that UV infinities supported by
$\mathcal{N}-1$ harmonic superinvariants cancel. We explained here that
$\mathcal{N}-1$ harmonic superinvariants break nonlinear $\mathcal{H}$
symmetry and nonlinear supersymmetry. Therefore these symmetries explain the
absence of UV divergences in these cases.

## Acknowledgement

I am grateful to A. Linde, H. Nicolai, R. Roiban, and Y. Yamada for useful
discussions related to the current work and to E. Ivanov and E. Sokatchev for
earlier discussions of harmonic superspace, in general. This work is supported
by SITP and by the US National Science Foundation grant PHY-2310429.

##  Appendix A $\mathcal{N}=2$ harmonic superspace and G-analyticity

Harmonic superspace constructions are known to be successful in super Yang-
Mills theory with $\mathcal{N}\leq 3$ Galperin:2001seg . Minkowski space is
the simplest example of a coset space ${G\over H}$ where $G$ is the Poincaré
group and $H$ is the Lorentz group. The coset parameters are coordinates
$x^{a}$, $a=0,1,2,3$

$M^{4}={G\over H}={\\{L_{ab},P_{a}\\}\over\\{L_{ab}\\}}=x^{a}$ (87)

The superspace is a space where Poincaré supersymmetry algebra can be realized
in the same way as the Poincaré algebra is realized in Minkowski space.
Poincaré superalgebra has an outer automorphism group $U(\mathcal{N})$ (also
known as R symmetry). The spinor generators $Q_{\alpha}^{i}$ and
$\bar{Q}^{\dot{\alpha}i}$ transform according to the fundamental
representations ${\bf\mathcal{N}}$ and ${\bf\bar{\mathcal{N}}}$ of
$U(\mathcal{N})$, respectively.

The real superspace has coordinates

$\mathbb{R}^{4|4\mathcal{N}}={G\over
H}={\\{L_{ab},su(\mathcal{N}),P_{a},Q_{\alpha}^{i},\bar{Q}^{\dot{\alpha}i}\\}\over\\{L_{ab},su(\mathcal{N})\\}}=(x^{a},\theta_{i}^{\alpha},\bar{\theta}^{\dot{\alpha}i})$
(88)

Here $H$ is the group of automorphisms $SL(2,C)\times SU(\mathcal{N})$.

For example, harmonic superspace for $\mathcal{N}=2$ theories is obtained by
keeping only the $U(1)$ part of the automorphism group $SU(2)$ in the subgroup
$H$ :

$\mathbb{HR}^{4+2|8}={G\over
H}={\\{L_{ab},P_{a},Q_{\alpha}^{i},\bar{Q}^{\dot{\alpha}i},su(2)\\}\over\\{L_{ab},u(1)\\}}=(x^{a},\theta_{i}^{\alpha},\bar{\theta}^{\dot{\alpha}i},u^{+}_{i})\equiv(X^{A})\,,\qquad
i=1,2$ (89)

Two new even coordinates $u^{+}_{i}$ correspond to the coset $SU(2)/U(1)$, a
two-dimensional sphere $S^{2}$ and $u_{i}^{-}$ is its inverse,
$u^{+i}u_{i}^{-}=1$. The spinor derivatives are defined as follows

$D_{\alpha}^{\pm}=u^{\pm}_{i}D_{\alpha}^{i}\,,\qquad\bar{D}_{\dot{\alpha}}^{\pm}=u^{\pm}_{i}\bar{D}_{\dot{a}}^{i}$
(90)

The main advantage of the harmonic superspace is the existence of a new
invariant subspace containing only half of the original Grassmann variables.
The analytic superspace is defined as follows

$\mathbb{HA}^{4+2|4}={G\over
H}={\\{L_{ab},P_{a},Q_{\alpha}^{i},\bar{Q}^{\dot{\alpha}i},T^{\pm\pm},T^{0}\\}\over\\{L_{ab},T^{0},Q_{\alpha}^{+},\bar{Q}_{\dot{\alpha}}^{+}\\}}$
(91)

Here $T^{\pm\pm},T^{0}$ are generators of $SU(2)$ and

$Q_{\alpha}^{+}=u^{+}_{i}Q_{\alpha}^{i}\,,\qquad\bar{Q}_{\dot{\alpha}}^{+}=u^{+}_{i}\bar{Q}_{\dot{\alpha}}^{i}\
$ (92)

The generators $L_{ab},T^{0},Q_{\alpha}^{+},\bar{Q}_{\dot{\alpha}}^{+}$ form a
closed algebra. Derivatives in the harmonic directions are introduced

$\partial^{\pm\pm}=u^{\pm i}{\partial\over\partial u^{\mp
i}}\,,\qquad\partial^{0}=u^{+i}{\partial\over\partial
u^{+i}}-u^{-i}{\partial\over\partial u^{-i}}\ \ $ (93)

Under $SU(2)$ the new coordinates transform as follows

$(u^{\pm}_{i})^{\prime}=\Lambda_{i}{}^{j}(u^{\pm}_{j})e^{\pm
i\psi(\Lambda,u)}$ (94)

where $\Lambda_{i}{}^{j}$ is an element of $SU(2)$ and $\psi$ is a local phase
of induced $U(1)$ transformation, depending on $\Lambda$ and $u^{+}_{i}$.

The condition of the Grassmann analyticity (G-analyticity) is now possible in
the analytic harmonic superspace.

$D_{\alpha}^{+}\Phi(x_{A},\theta^{\pm}_{A},\bar{\theta}^{\pm}_{A},u)=0\,,\qquad\bar{D}_{\dot{\alpha}}^{+}\Phi(x_{A},\theta^{\pm}_{A},\bar{\theta}^{\pm}_{A},u)=0\
.$ (95)

It means that there is a basis where the superfield depends only on 1/2 of
fermionic coordinates

$\Phi(x_{A},\theta^{+}_{A},\bar{\theta}^{+}_{A},u^{\pm}_{i})\ ,$ (96)

where

$x_{A}^{a}=x^{a}-2i\theta^{(i}\sigma^{a}\bar{\theta}^{j)}u^{+}_{i}u^{-}_{j}\,,\qquad\theta^{\pm}_{A\,\alpha}=u^{\pm}_{i}\theta^{i}_{\alpha}\,,\qquad\bar{\theta}^{\pm}_{A\,\dot{\alpha}}=u^{\pm}_{i}\bar{\theta}^{i}_{\dot{\alpha}}\,.$
(97)

Grassmann-analyticity defining derivatives $D_{\alpha}^{+}$ and
$\bar{D}_{\dot{\alpha}}^{+}$ form a commutative algebra with the the harmonic
derivative $D^{++}$.

Under $SU(2)$ transformations with parameters $\Lambda_{i}{}^{j}$ spinorial
derivatives in eq. (19), supersymmetry charges in eq. (92) and analytic space
coordinates in eq. (97) are $SU(2)$ invariant.

In $\mathcal{N}=2$ an example of the analytic superfield is an off-shell
hypermultiplet. In a superspace without harmonic variables it is $q^{i}(X)$.
on-shell $D_{\alpha}^{(i}q^{j)}(X)=0$, but off-shell there is a harmonic
superfield which is G-analytic

$q^{+}(X,u)=q^{i}(X)u_{i}^{+}\,,\qquad
D_{\alpha}^{+}q^{+}(X,u)=\bar{D}_{\dot{\alpha}}^{+}q^{+}(X,u)=0\ .$ (98)

Note that the superfield $q^{+}(X,u)$ and its G-analyticity conditions defined
in eq. (98) are invariant under $SU(2)$ transformations with superspace-
independent parameters $\Lambda_{i}{}^{j}$.

Conformal supergravities are associated with the superconformal algebra
$SU(2,2|\mathcal{N})$; at the nonlinear level, they are available only at
$\mathcal{N}\leq 4$. In superspace, conformal supergravities were described in
Howe:1981gz  for 4d $\mathcal{N}\leq 4$ by explicitly gauging
$SL(2,\mathbb{C})\times U(\mathcal{N})$ and identifying the relevant
constraints on the torsion of curved superspace. It is therefore not
accidental that, for example, conformal $\mathcal{N}=2$ supergravity has a
local in superspace $U(2)$ symmetry, which is in the structure group of the
conformal superspace.

In case of $\mathcal{N}=2$ Poincaré supergravity is obtained from conformal
one in Howe:1981gz , by de-gauging $U(2)$ symmetry, so that the tangent space
is just $SL(2,\mathbb{C})$. The following is a short version of a derivation
of Poincaré $\mathcal{N}=2$ supergravity superspace in Howe:1981gz . One
begins with $SL(2,\mathbb{C})\times U(2)$ connections in conformal
supergravity

$\hat{\Omega}_{A}{}^{B}=\Omega_{A}{}^{B}+\Sigma_{A}{}^{B}\ ,$ (99)

where $\hat{\Omega}_{A}{}^{B}$ is $SL(2,\mathbb{C})\times U(2)$ connection and
$\Omega_{A}{}^{B}$ is $SL(2,\mathbb{C})$ connection and proceed with de-
gauging by requiring restrictions on $\Sigma_{A}{}^{B}$ so that the superspace
has no internal symmetry anymore. First, the local $U(1)$ is gauge-fixed, and
afterwards also the local $SU(2)$. One of the important restrictions
explaining the difference between off-shell and on-shell superspace is

$\Sigma^{ij}_{\alpha\,k}=2\delta^{i}_{k}\rho^{j}_{\alpha}-\delta^{j}_{k}\rho^{i}_{\alpha}\
.$ (100)

When constraints on torsion were imposed, as well as de-gauging constraints,
the superspace Bianchi identities were solved consistently. In the final off-
shell superspace with the tangent group $SL(2,\mathbb{C})$ there are two
covariant superfields

$M_{\alpha\beta}(x,\theta,\bar{\theta})\qquad{\rm
and}\qquad\rho^{i}_{\alpha}(x,\theta,\bar{\theta})$ (101)

A superfield equation of motion is

$\rho^{i}_{\alpha}(x,\theta,\bar{\theta})=0$ (102)

Thus, all non-vanishing components of torsion and curvatures on-shell depend
on one superfield $M_{\alpha\beta}$. There is no obstruction to adding
harmonic coordinates to this superspace since there is no local $SU(2)$
symmetry, and all geometric Bianchi identities were solved consistently. This
will be different in the case of the on-shell $\mathcal{N}\geq 4$
supergravities, where the local $(S)U(\mathcal{N})$ has a different nature and
global duality symmetries $\mathcal{G}$ play an important role.

Harmonic $\mathcal{N}=2$ superspace for Poincaré supergravity is described in
Galperin:2001seg , starting with a conformal $\mathcal{N}=2$ superspace. To
gauge-fix extra local symmetries, the compensator superfields are introduced.
They are used to gauge fix the local $SU(2)$ so that the gauge-fixed local
$SU(2)$ is compatible with harmonic coordinates $u_{i}^{\pm}$, i=1,2 in
$\mathcal{N}=2$ Poincaré supergravity.

The construction of $\mathcal{N}\geq 4$ harmonic superspace in Bossard:2011tq
, where $L=\mathcal{N}-1$ counterterms are claimed to exist, follows the
strategy of $\mathcal{N}=2$ projective superspace in Kuzenko:2008ep ;
Kuzenko:2008ry . We will explain the reason why the construction in projective
superspace in Kuzenko:2008ep  and the consequent studies of the action of
$\mathcal{N}=2$ supergravity in Kuzenko:2008ry  are consistent with the
presence of isotwistor variables $u^{+}_{i}$, $i=1,2$. These isotwistor
variables $u_{i}^{+}$ are considered to be inert with respect to supergravity
gauge group $SU(2)$. It is stressed that $u_{i}^{+}$ are constant.

In Kuzenko:2008ep ; Kuzenko:2008ry  the fermionic derivatives are defined as

$D^{+}_{\alpha}:=u_{i}^{+}D_{\alpha}^{i},\,D^{-}_{\alpha}:=u_{i}^{-}D_{\alpha}^{i}$
(103)

But since $u^{\pm}_{i}$ are constant, they cannot transform under local
$SU(2)$. The reason why this is not a problem in $\mathcal{N}=2$ supergravity
in Kuzenko:2008ep ; Kuzenko:2008ry  is the following. The action derived in
Kuzenko:2008ep  is invariant under the super-Weyl transformations generated by
a covariantly chiral parameter $\sigma$ where
$\bar{D}_{\dot{\alpha}}^{i}\sigma=0$. It also has a local $U(2)$ symmetry. To
get $\mathcal{N}=2$ Poincaré supergravity, one must gauge-fix various extra
local symmetries, including the local $SU(2)$.

The action of conformal supergravity in eq. (1.2) of Kuzenko:2008ry  depends
on a superconformal compensator, an off-shell tensor multiplet described by a
symmetric real superfield $H^{ij}(x,\theta)$. When $H^{ij}$ is constrained

$H^{ij}=-{i\over
2}(\sigma_{1})^{ij}\,G\,,\qquad(\sigma_{1})^{ij}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)^{ij}$ (104)

the action does not have local $SU(2)$ symmetry anymore, only the local U(1)
is left. The local U(1) gauge connection is
$\Phi_{\alpha}^{i\,jk}=i\,\Sigma^{i}_{\alpha}(\sigma_{1})^{ij}$. One can
further use the de-gauging procedure for $\mathcal{N}=2$ as in Howe:1981gz ,
where one makes a choice $\Sigma^{i}_{\alpha}=0$ so that $SU(2)$ gauge
symmetry is fixed and only global $SU(2)$ remains. This is consistent with the
fact that the isotwistor coordinates $u_{i}^{+}$ and their inverse are inert
under local $SU(2)$.

Therefore in $\mathcal{N}=2$ supergravity a contraction of the type (103) is
acceptable, local $SU(2)$ is gauge-fixed in superspace.

##  Appendix B Superinvariants below critical loop order in $d>4$

Dimensional analysis in superspace to establish the value of $L_{cr}$ starting
from which the geometric superinvariants exist was performed in
Kallosh:2023css . Here we reproduce the result for $L_{cr}$ obtained in
Kallosh:2023css  in a slightly more convenient form. A dimension of the
gravitational coupling is ${\rm dim}\Big{[}{1\over\kappa^{2}}\Big{]}={\rm
dim}[M_{Pl}^{2}]=d-2$. For integer dimensions $d\geq 4$ we determine the
critical loop order defined by an on-shell counterterm which is a geometric
whole superspace integral with the Lagrangian invariant under global
$\mathcal{G}$ and local $\mathcal{H}$.

$S_{cr}=\kappa^{2\,(L_{cr}-1)}\int
d^{4\mathcal{N}}d^{d}x\,det\,E\,\mathcal{L}(x,\theta)\ .$ (105)

In such case, the minimal dimension superfield $\mathcal{H}$-invariant
Lagrangian depends on 4 spinor dimension 1/2 superfields and their superspace
derivatives.

${\rm dim}\Big{[}\mathcal{L}(x,\theta)\Big{]}=2+n\ ,\qquad n>0$ (106)

In the presence of $n$, one can always make $L_{cr}$ an integer. The reason
$n$ is positive is that all covariant superspace derivatives, as well as
superfield-dependent torsions and curvatures, have positive dimension. The
counting goes as follows

${\rm dim}[\kappa^{2}]\,(L_{cr}-1)+{\rm dim}\,[\rm full\,\,superspace]+{\rm
dim}\,[\mathcal{L}]=0\,,$ (107)

so that the action is dimensionless, and we find

$-(d-2)(L_{cr}-1)+2\mathcal{N}-d+2+n=0\ ,$ (108)

It follows that

$L_{cr}={2\mathcal{N}+n\over(d-2)}\ ,\qquad n\geq 0$ (109)

In each case, we need to find a minimal value of $n$, which makes $L_{cr}$ an
integer. The result for the case of maximal supergravity is

$\displaystyle d=4,\,L_{cr}=8:\quad$ $\displaystyle\kappa^{14}\int
d^{4}\,x\,D^{10}R^{4}+\dots\,\quad\quad\quad n=0$ (110) $\displaystyle
d=5,\,L_{cr}=6:\quad$ $\displaystyle\kappa^{10}\int
d^{5}\,x\,D^{12}R^{4}+\dots\,\quad\quad\quad n=2$ (111) $\displaystyle
d=6,\,L_{cr}=4:\quad$ $\displaystyle\kappa^{6}\int
d^{6}\,x\,D^{10}R^{4}+\dots\,\,\,\quad\quad\quad n=0$ (112) $\displaystyle
d=7,\,L_{cr}=4:\quad$ $\displaystyle\kappa^{6}\int
d^{7}\,x\,D^{14}R^{4}+\dots\,\,\quad\quad\quad n=4$ (113) $\displaystyle
d=8,\,L_{cr}=3:\quad$ $\displaystyle\kappa^{4}\int
d^{8}\,x\,D^{12}R^{4}+\dots\,\,\quad\quad\quad n=3$ $\displaystyle
d=9,\,L_{cr}=3:\quad$ $\displaystyle\kappa^{4}\int
d^{9}\,x\,D^{15}R^{4}+\dots\,\,\quad\quad\quad n=5$

In Bern:1998ug ; Bern:2007hh ; Bern:2008pv ; Bern:2009kd ; Bern:2012uf ;
Bern:2018jmv  the first divergences in maximal supergravities have been found
at $5\leq d\leq 9$

$\displaystyle d=9,\qquad L_{UV}=2<L_{cr}=3$ (115) $\displaystyle d=8,\qquad
L_{UV}=1<L_{cr}=3$ (116) $\displaystyle d=7,\qquad L_{UV}=2<L_{cr}=4$ (117)
$\displaystyle d=6,\qquad L_{UV}=3<L_{cr}=4$ (118) $\displaystyle d=5,\qquad
L_{UV}=5<L_{cr}=6$ (119)

It appears that in all cases where we have the information about UV
divergences in $d\geq 5$ from loop computations, the UV divergences were
discovered at the loop order below what we have defined as $L_{cr}$. According
to our assertion, the nonlinear supersymmetry invariants cannot be constructed
unless they come from a whole superspace integral. Therefore, it follows from
(119) that the UV divergences and the relevant CTs in all these maximal
supergravities above 4d break full nonlinear supersymmetry and local
$\mathcal{H}$-symmetry. At the linearized level in $d>4$, the relevant
${\mathcal{G}\over\mathcal{H}}$ supergravity superinvariants below critical
loop order were constructed in harmonic superspace in Bossard:2009sy ;
Bossard:2014lra ; Bossard:2014aea ; Bossard:2015uga . In all cases, they
represent the linearized version of the superinvariants at loop level below
$L_{cr}={2\mathcal{N}+n\over(d-2)},\,n>0$ The analysis of harmonic
superinvariants below critical order performed in this paper for $d=4$ can now
be applied to $d>4$. We focus on maximal supergravities in each dimension. In
all cases, the physical scalars are in the coset space

${\mathcal{G}\over\mathcal{H}}\quad:\quad{E_{6(6)}\over
USp(8)}\,,\quad{E_{5(5)}\over USp(4)\times USp(4)}\,,\quad{E_{4(4)}\over
USp(4)}\,,\quad{E_{3(3)}\over U(2)}\,,\quad{GL(2)\over SO(2)}$ (120)

for $d=5,6,7,8,9$ respectively. There are Lagrangians with local $\mathcal{H}$
symmetry:

$USp(8)\,,\quad USp(4)\times USp(4)\,,\quad USp(4)\,,\quad U(2)\,,\quad SO(2)$
(121)

for $d=5,6,7,8,9$ respectively. Harmonic variables in all cases parametrize
the coset space ${\mathcal{H}\over H}$ where $H$ is a subgroup of
$\mathcal{H}$.

The geometric whole superspace invariants at
$L_{cr}={2\mathcal{N}+n\over(d-2)}$ have volume of integration over all
fermionic directions with local $\mathcal{H}$-symmetry. Harmonic coordinates
allow a smaller subspace of fermion variables by imposing Grassmann
analyticity conditions on superfields.

For example, in maximal $\mathcal{N}=(2,2)$ supergravity in six dimensions the
relevant harmonic variables parametrize one of $USp(4)\times USp(4)$ as
follows Bossard:2014lra : ${USp(4)\over U(2)}$. This provides the Grassmann
analytic superfields and linearized superinvariant describing 3-loop UV
divergence of d=6 maximal supergravity. As in 4d case, the fact that harmonic
variables break local $USp(4)\times USp(4)$ symmetry can be established using
the superspace geometry. This means that there is no nonlinear generalization
of the ${1\over\mathcal{N}}$ BPS superinvariant presented in eq. (3.32) in
Bossard:2009sy . In fact, there is no claim that such a harmonic below
critical order superinvariant has a nonlinear generalization in any of the
papers in Bossard:2009sy ; Bossard:2014lra ; Bossard:2014aea ; Bossard:2015uga
. Moreover, concerning 6d and 8d maximal supergravities, it is explained that
there is an obstruction for the generalization of some of harmonic invariants
to a nonlinear level. One can also check, case by case, that all harmonic
superinvariants below critical order have no nonlinear generalization.

The reason why in 4d there was a claim in Bossard:2011tq  that superinvariants
below critical order exist in harmonic superspace compatible with nonlinear
supersymmetry is the fact that harmonic Lagrangian depended on geometric
spinorial superfields and had dimension 2. But in all $d>4$ cases, harmonic
Lagrangians for superinvariants below critical order in Bossard:2009sy ;
Bossard:2014lra ; Bossard:2014aea ; Bossard:2015uga  have dimension zero, they
are not geometric and clearly exist only at the linear level.

To summarize, harmonic superinvariants at $L<L_{cr}$ are useful to describe
the UV divergences that did occur at $d>4$. None of these harmonic
superinvariants with Grassmann-analytic superfields were claimed to exist at
the nonlinear level. In all cases, they are at $L<L_{cr}$ and show that local
$\mathcal{H}$ symmetry and nonlinear supersymmetry are broken by quantum
corrections. This means that computations in Bern:1998ug ; Bern:2007hh ;
Bern:2008pv ; Bern:2009kd ; Bern:2012uf ; Bern:2018jmv  have already
established that these local symmetries are anomalous, and therefore, all
$d>4$ maximal supergravities do not have consistent perturbative quantum field
theory corrections.

Only in 4d case it is not known yet if perturbative quantum field theory
corrections are consistent with all symmetries of the theory. The 4-loop
cancellation of UV divergence in $\mathcal{N}=5$ indicates a possibility that
4d maximal supergravity might be UV finite perturbatively. But more loop
computations will be necessary to find out the actual UV properties of higher
loop corrections in 4d maximal supergravity.

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