aainstitutetext: Mathematical Institute, University of Oxford,
Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UKbbinstitutetext:
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

# Broken (super) conformal Ward identities at finite temperature

Enrico Marchetto b Alessio Miscioscia b and Elli Pomoni
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

When a (super) conformal field theory is placed on a non-trivial manifold, the
(super) conformal symmetry is broken. However, it is still possible to derive
broken Ward identities for these broken symmetries, which provide additional
constraints on the theory. We derive and apply the broken Ward identities
associated with the (super) conformal group on the thermal manifold
$\mathcal{M}_{\beta}=S_{\beta}^{1}\times\mathbb{R}^{d-1}$ and
$\mathcal{M}=T^{2}\times\mathbb{R}^{d-2}$. The novel constraints not only
systematically reproduce known results, including an implicit formulation of
the generalized Cardy formula, but also elegantly relate the thermal energy
spectrum with the conformal spectrum.

###### Keywords:

Conformal Field Theory, Finite Temperature, Ward Identity, Broken symmetries.

††preprint: DESY-23-083

##  1 Introduction and Summary

While there have been remarkable strides in our understanding of non-
perturbative effects in Quantum Field Theory (QFT), our current knowledge of
QFT at finite temperature remains relatively limited. In the space of QFTs, a
special role is played by Conformal Field Theories (CFTs), not only because
they correspond to the fixed points of the renormalization group flows, but
also because conformal invariance implies powerful constraints on the CFT data
studied by the conformal bootstrap program. Turning on a temperature $T$ can
be understood as placing the CFT on the thermal manifold
$\mathcal{M}_{\beta}=S_{\beta}^{1}\times\mathbb{R}^{d-1}$, where $\beta=1/T$
is the length of the circle. The thermal bootstrap, initiated by
Iliesiu:2018fao ; Iliesiu:2018zlz , and its relations to black hole physics
Witten:1998zw ; Horowitz:1999jd ; deHaro:2000vlm ; Son:2002sd ; Aharony:2003sx
; Casalderrey-Solana:2011dxg ; Ammon:2015wua ; Nastase:2015wjb ; Caron-
Huot:2022lff , is currently a very active field of research to which we wish
to contribute with this paper.

In QFT (continuous) symmetries are associated with differential equations
correlation functions must obey, known as Ward identities. When a symmetry is
broken we do not expect the corresponding Ward identity to hold, but we expect
it will be corrected by an additional breaking term. In this paper, we provide
a general way to compute the breaking term for any continuous symmetry, with
particular interest on the Conformal Group, on a generic manifold
$\mathcal{M}$. Focusing on the thermal manifold $\mathcal{M}_{\beta}$,
dilatations, special conformal transformations, and rotations including the
time direction (the compactified direction) are broken, while translations and
rotations including only space directions are preserved. This work is devoted
to the computation of the breaking terms for the broken Ward identities. Since
the currents associated with the Conformal Group generators are proportional
to the stress-energy tensor we show that every broken Ward identity is a non-
trivial identity between an $n$-point function and an integrated $(n+1)$-point
function including (some components of) the stress-energy tensor. We derive
systematically those equations and we start exploring some of their physical
implications. They have to be satisfied for every local CFT at finite
temperature and provide constraints which go beyond the familiar zero-
temperature conformal bootstrap conditions. We further show that some of them
are related to the Hamiltonian and the momentum on the thermal manifold
$\mathcal{M}_{\beta}$. In particular, the Hamiltonian is related to the
dilatation operator while the momentum is related to the boost operator. Those
relations are derived and discussed for the first time in this paper for
general dimension. Notably, the thermal energy spectrum is directly related to
the zero-temperature spectrum.

What is more, supersymmetry is expected to be broken at finite temperature due
to the anti-periodicity of fermions on the thermal circle $S_{\beta}^{1}$. We
therefore extend the study of broken and unbroken Ward identities to the case
of superconformal field theories, showing that the R-symmetry is preserved
while supersymmetry and superconformal transformations are broken but are
associated with equations that compare an $n$-point function with an
$(n+1)$-point function including the supercurrent $G_{\alpha}^{\mu I}$.

Because the geometry of the manifold $\mathcal{M}_{\beta}$ is conformally
flat, the two-point functions can be expanded in an operator product expansion
(OPE), which however will converge only in a suitable region Iliesiu:2018fao ;
Gobeil:2018fzy . Exploring broken Ward identities in this regime, we check
their consistency with the results of Iliesiu:2018fao . The broken Ward
identities have to be satisfied for any correlation function at finite
temperature and therefore could be used to enrich the bootstrap problem for
correlation functions at finite temperature Alday:2020eua ; Dodelson:2023vrw ;
Caron-Huot:2008vbk ; Dodelson:2022yvn . As a preliminary exercise in this
direction, we show (in Appendix C) that the bootstrap problem can be exactly
solved in the case of a free scalar field in four dimensions.

Finally, in order to show the generality of our methods, we compute the same
broken Ward identities for another interesting case, namely for the manifold
$\mathcal{M}=T^{2}\times\mathbb{R}^{d-2}$. This manifold is another
interesting example of a non-trivial manifold in which the CFT can be placed.
This geometry also attracted recent interest for different reasons
Benjamin:2023qsc ; Luo:2022tqy .

The outline of the paper is the following:

  * $\star$

In Section 2 the derivation of the broken Ward identities on a general
manifold is presented. We apply this method to the thermal case.

  * $\star$

In Section 3 we apply the identities computed above for the case of the
Conformal Group. We show that from those identities it is possible to read the
operatorial expression of the Hamiltonian and the momentum at finite
temperature (see Subsection 3.1). Finally, we combine broken Ward identities
and operator product expansion and we show the consistency of broken Ward
identities with known results.

  * $\star$

In Section 4 we enlarge the study to the case of superconformal quantum field
theories. We produce broken Ward identities and we show that R-symmetry is
preserved. We also discuss the $\mathrm{U}(1)_{Y}$ bonus symmetry in
$\mathcal{N}=4$ SYM at finite temperature and the corresponding non-
renormalization theorem.

  * $\star$

In Section 5 we repeat the discussion of broken Ward identities for the case
of the manifold $\mathcal{M}=T^{2}\times\mathbb{R}^{d-2}$ producing broken
Ward identities for this geometry.

A number of technical points and derivations are presented in the appendices.
In particular more comments on the dilatation broken Ward identity are
presented in Appendix A. Some explicit checks of the equations derived in the
paper can be found in Appendix B. The solution for the free scalar theory in
four dimensions obtained directly by solving the bootstrap problem is provided
in Appendix C. Finally some general comments on the broken Ward identities
when combined with the operator product expansion can be read in Appendix D.

##  2 Broken Ward identities at finite temperature

It is known that one-, two- and three-point functions of a generic CFT living
on the flat space-time are completely fixed, up to constants, by the
symmetries composing the Conformal Group. These symmetries include
translations, dilatations, rotations, and special conformal transformations.
At the quantum level, each of these symmetries is associated with a Ward
identity, which represents a specific differential equation that correlation
functions must satisfy. In the flat spacetime $\mathbb{R}^{d}$, these
differential equations are easily solvable and they return known results
Osborn:1993cr ; Rychkov:2016iqz ; Simmons-Duffin:2016gjk ; Gopakumar:2022kof .

In this Section, we extract a Ward identity (or a set of Ward identities) for
each symmetry transformation on the thermal manifold
$\mathcal{M}_{\beta}=S_{\beta}^{1}\times\mathbb{R}^{d-1}$. We do not expect
the symmetries to be preserved in general. For instance, we expect dilatations
to be broken because of the insertion of a scale in the theory, represented by
the temperature $T$ (in the following appearing trough its inverse
$\beta=1/T$111In the following we will use the units in which the Boltzmann
constant is set to $k_{B}=1$.). In the derivation procedure, the non-
triviality of the geometry leads to a modification of the standard flat space
Ward identity: we will interpret this modification as a _breaking_ of the Ward
identity. Although the primary focus of this paper is the thermal manifold,
this Section can be extended to the case of a theory on a generic manifold
$\mathcal{M}$. Therefore, we present the initial steps in this Section within
a general framework. In Section 5 we will study the case of a (super)
conformal field theory on the manifold
$\mathcal{M}=T^{2}\times\mathbb{R}^{d-2}$.
Let us consider a generic CFT in $d$ dimensions. We are interested in studying
the properties of the generic $n$-point correlation function

$\Braket{\mathcal{O}_{1}(x_{1})\dots\mathcal{O}_{n}(x_{n})}_{\mathcal{M}}\ ,$
(1)

where $\braket{\dots}_{\mathcal{M}}$ indicates the correlation function on the
manifold $\mathcal{M}$.

The derivation of zero temperature Ward identities, i.e. Ward identities in
the ambient flat spacetime manifold, is known in the literature (see for
example DiFrancesco:1997nk ; Papadodimas2010 ). We start from a theory
described by a formal action $S$ and we define an infinitesimal symmetry
transformation
$\varphi^{\prime}(x)=\varphi(x)-i\omega_{a}\bm{G}_{a}\varphi(x)$, where

$i\omega_{a}\bm{G}_{a}\varphi(x)=\omega_{a}\frac{\delta
x^{\mu}}{\delta\omega_{a}}\partial_{\mu}\varphi-\omega_{a}\frac{\delta\mathcal{F}}{\delta\omega_{a}}\
.$ (2)

Here $\mathcal{F}$ is a function that parameterizes the variation of the field
$\varphi$ under the symmetry generated by $\bm{G}_{a}$. We will focus in
Section 3 on the Conformal Group for which the action of the generators
$\bm{G}_{a}$ on a generic correlator are known and listed in Table 1. We
extend in Section 4 the results for the superconformal case.
The correlation function (1) can be represented as a path integration over the
fundamental fields

$\Braket{\mathcal{O}_{1}(x_{1})\dots\mathcal{O}_{n}(x_{n})}_{\mathcal{M}}=\frac{1}{\mathcal{Z}}\int[d\varphi]\
\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})e^{-S[\varphi]}\ ;$ (3)

since the fundamental fields $\varphi$ are integration variables, we can apply
the transformation (2) as a functional change of variables

$\Braket{\mathcal{O}_{1}(x_{1})\dots\mathcal{O}_{n}(x_{n})}_{\mathcal{M}}=\frac{1}{\mathcal{Z}}\int[d\varphi^{\prime}]\
\left(\mathcal{O}_{1}(x_{1})+\delta\mathcal{O}_{1}(x_{1})\right)\ldots\left(\mathcal{O}_{n}(x_{n})+\delta\mathcal{O}_{n}(x_{n})\right)e^{-S^{\prime}[\varphi^{\prime}]}\
,$ (4)

where $\delta\mathcal{O}_{i}(x_{i})$ denotes the effect of the transformation
(2) on the $i$-th local operator. If the theory is assumed to be invariant
under rigid symmetry transformations, the transformed action on a generic
manifold $\mathcal{M}$ can be written as

$S^{\prime}[\varphi^{\prime}]=S[\varphi]+\int_{\mathcal{M}}d^{d}x\sqrt{g}\
\nabla_{\mu}^{\phantom{\mu}}\omega_{a}^{\phantom{\mu}}(x)J_{a}^{\mu}(x)\ ,$
(5)

where $g$ is the metric on the manifold $\mathcal{M}$, and $J^{\mu}_{a}$ are
the current operators associated with the conservation of the symmetry
generated by $\bm{G}_{a}$. In a Lagrangian theory, the current operators admit
an explicit realization where the canonical stress-energy tensor appears

$J_{a}^{\mu}=\frac{\delta
x^{\mu}}{\delta\omega_{a}}\mathcal{L}-\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\varphi}\partial_{\nu}\varphi\frac{\delta
x^{\nu}}{\delta\omega_{a}}+\sum_{\varphi}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\varphi}\frac{\delta\mathcal{F}}{\delta\omega_{a}}=T_{\text{can.}}^{\mu\nu}\frac{\delta
x_{\nu}}{\delta\omega_{a}}+\sum_{\varphi}\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\varphi}\frac{\delta\mathcal{F}}{\delta\omega_{a}}\
.$ (6)

By focusing solely on the currents associated with the symmetries of the
Conformal Group, we can reshape them in terms of the symmetric, traceless
stress-energy tensor $T^{\mu\nu}$222The current associated with spacetime
translations is $a_{\nu}T_{\text{can}}^{\mu\nu}$, where
$T_{\text{can}}^{\mu\nu}$ is the canonical stress-energy tensor; however, as
in the case of a CFT at zero temperature DiFrancesco:1997nk , it can be shown
that the broken Ward identities are not altered by replacing the canonical
stress-energy tensor with the traceless, symmetric one: the only difference
that can appear are divergences that will be removed in the normal ordering
procedure., which is a well-defined operator in any local CFT, even if not
Lagrangian. As we will see in Section 4 for the superconformal case, the
charges are related to the stress-energy tensor $T^{\mu\nu}(x)$ and the
supercharge $G_{\alpha}^{\mu I}(x)$. The explicit realizations of the
canonical currents associated with the symmetries of the Conformal Group are
listed in Table 2.

By inserting the transformed action in the expression (4) and expanding at
first order in the infinitesimal parameters $\omega_{a}$ we get

$\begin{split}\sum_{i}\left\langle\mathcal{O}_{1}(x_{i})\ldots\delta\mathcal{O}_{i}(x_{i})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}}&=\left\langle\delta
S\
\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}}\\\
&=\int_{\mathcal{M}}d^{d}x\sqrt{g}\ \nabla_{\mu}\omega_{a}(x)\left\langle
J_{a}^{\mu}(x)\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}}\
.\end{split}$ (7)

Recalling that $\delta\mathcal{O}(x)=-i\omega_{a}(x)\bm{G}_{a}\mathcal{O}(x)$
and taking the functional derivatives in $\omega_{b}(y)$ both on the left and
on the right side of the equation (7), the expression reads

$-i\sum_{i}\delta(x_{i}-y)\left\langle\mathcal{O}_{1}(x_{1})\ldots\bm{G}_{a}\mathcal{O}_{i}(x_{i})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}}=\\\
=\int_{\mathcal{M}}d^{d}x\sqrt{g}\ \partial_{\mu}\delta(x-y)\ \left\langle
J_{a}^{\mu}(x)\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}}\
.$ (8)

The derivative in the $x$ coordinates can be replaced with a derivative in the
$y$ coordinates up to a sign. We, therefore, get the un-integrated broken Ward
identities

$i\sum_{i}\delta(x_{i}-y)\left\langle\mathcal{O}_{1}(x_{1})\ldots\bm{G}_{a}\mathcal{O}_{i}(x_{i})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}}=\nabla_{\mu}^{y}\left\langle
J_{a}^{\mu}(y)\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}}\
.$ (9)

It is crucial to notice that the current on the right-hand side of the
equation (9) iteratively acts on all the operators in the correlation
function. For example, if we consider, on the thermal manifold
$\mathcal{M}_{\beta}=S^{1}\times\mathbb{R}^{d-1}$, the dilatation broken Ward
identity for a one-point function the equation (9) specializes to

$i\delta(x-y)\left\langle[\bm{D},\mathcal{O}](x)\right\rangle_{\beta}=\frac{\partial}{\partial
y^{\mu}}\left[(y-x)_{\nu}\left\langle
T^{\mu\nu}(y)\mathcal{O}(x)\right\rangle_{\beta}\right]\ ,$ (10)

where here and in the following we use $\langle\ \cdot\
\rangle_{\beta}=\langle\ \cdot\ \rangle_{\mathcal{M}_{\beta}}$ to indicate
thermal correlators. This can be easily understood as the fact that we are
integrating on the boundary of an infinite-dimensional ball, having as a
center the position of the operator $\mathcal{O}$. On the other hand, this
intuition perfectly agrees with the definition of the operator as the
integration of the 0th component of the conserved current
Zamolodchikov:1989hfa ; Belavin:1984vu ; Mussardo:2020rxh ; e.g. for
dilatations

$\bm{D}\,\mathcal{O}(x)=\int d^{d-1}y\ (x-y)_{\nu}T^{0\nu}(y)\mathcal{O}(x)\
.$ (11)

The un-integrated broken Ward identities, even if very explicit, can be
simplified if we specify the manifold $\mathcal{M}$; as anticipated we are
mainly interested in the thermal case, therefore from now on we will focus on
this specific choice. A slightly different situation, namely
$\mathcal{M}=T^{2}\times\mathbb{R}^{d-2}$, is discussed in Section 5. In the
thermal case $\mathcal{M}_{\beta}=S_{1}^{\beta}\times\mathbb{R}^{d-1}$, after
integrating both the right and the left-hand sides of (7), we get

$i\sum_{i}\left\langle\mathcal{O}_{1}(x_{1})\ldots\bm{G}_{a}\mathcal{O}_{i}(x_{i})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=\int
d^{d-1}y\int_{0}^{\beta}dy_{0}\ \frac{\partial}{\partial y^{\mu}}\left\langle
J_{a}^{\mu}(y)\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
.$ (12)

By replacing in the equation (12) the thermal manifold $\mathcal{M}_{\beta}$
with the flat space $\mathbb{R}^{d}$, turning the volume integral into a
boundary integral sets the right-hand side to zero (with the correct boundary
conditions on the fundamental fields) and we recover the well known (un-
broken) Ward identities

$i\sum_{i}\left\langle\mathcal{O}_{1}(x_{1})\ldots\bm{G}_{a}\mathcal{O}_{i}(x_{i})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathbb{R}^{d}}=0\
.$ (13)

However, on a generic manifold $\mathcal{M}$ the integration over the boundary
$\partial\mathcal{M}$ does not return a trivial result, hence the right-hand
side of the equation (12) can be interpreted as a _breaking term_ correcting
the Ward identity. In order to allow for explicit computations, we will
parameterize the thermal circle $S^{1}_{\beta}$ with a coordinate
$\tau\in\left[0,\beta\right)$, i.e. we impose 333In the following, Greek
indices $\mu,\nu,\rho$ will run over all the directions of the thermal
manifold (from 0 to $d-1$), while Latin indices $i,j,k$ will run over the
spatial directions of the component $\mathbb{R}^{d-1}$ (from 1 to $d-1$). We
work in Euclidean signature, so there is no difference between raised and
lowered indices.

$\tau\equiv\tau+\beta\ .$ (14)

We have to further impose boundary conditions for the fields: periodic and
anti-periodic boundary conditions are assigned to bosons and fermions
respectively. Now we can evaluate the breaking term on the thermal manifold.
We can rewrite the right-hand side of the equation (12) as the sum of two
contributions: the first one is given by

$\int_{\mathbb{R}^{d-1}}d^{d-1}y\
\Braket{\left[J_{a}^{0}(\beta,\vec{y})-J_{a}^{0}(0,\vec{y})\right]\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})}_{\beta}\
,$ (15)

and the second one by the integral over the boundary parallel to
$S^{1}_{\beta}$

$\lim_{R\to\infty}\int_{0}^{\beta}d\tau\int_{S_{1}^{d-2}}d\Omega\
R^{d-2}n_{i}\left\langle
J_{a}^{i}(\tau,R,\Omega)\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
,$ (16)

where $n_{i}$ is the unit vector fixing the orientation of the boundary and
$d\Omega$ is the measure on the $(d-2)$-sphere of unitary radius. The term
that appears in equation (16) can be discussed by invoking the clustering
property

$\lim_{R\to\infty}\left\langle
J_{a}^{i}(\tau,R,\Omega)\mathcal{O}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=\lim_{R\to\infty}\left\langle
J_{a}^{i}(\tau,R,\Omega)\right\rangle_{\beta}\left\langle\mathcal{O}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
.$ (17)

To study the clustered correlator we can use translations invariance on the
thermal manifold: no point in the spatial component $\mathbb{R}^{d-1}$ nor on
the thermal circle $S^{1}_{\beta}$ should be regarded as preferred. Since the
invariance under space-time translations is unbroken, every one-point function
must be a constant. Consequently the term (17) is constant in $R$, $\tau$, and
$\Omega$ and therefore its integral over the thermal manifold is divergent
unless the one-point function of the current is exactly zero. Therefore
equation (17) encodes a possible source of infrared divergence in the
correlation function. The divergent term, if present, will be proportional to

$\langle\mathcal{O}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\rangle_{\beta}\times\text{Vol}\left[\mathcal{M}_{\beta}\right]\
.$ (18)

This suggests that the breaking term appearing in the broken Ward identities
should be renormalized by subtracting all the possible infrared divergences:
the immediate consequence is that the boundary term (16) is either zero or
infrared divergent and therefore it does not contribute to the breaking term.
This procedure is consistent (see Appendix B) and makes explicit some comments
already present in literature Papadodimas2010 . Let us now take into account
the other boundary term in equation (15). If we define the object

$\Gamma^{\beta}_{a}(\vec{y})=J^{0}_{a}(\beta,\vec{y})-J^{0}_{a}(0,\vec{y})\ ,$
(19)

we can rewrite the equation (15) as

$\int_{\mathbb{R}^{d-1}}d^{d-1}y\Braket{\Gamma^{\beta}_{a}(\vec{y})\mathcal{O}_{1}(x_{1})\dots\mathcal{O}_{n}(x_{n})}_{\beta}\
.$ (20)

From this definition it is already clear that translations and spatial
rotations are unbroken because of the periodicity of the current
$J^{\mu}(\beta,\vec{x})=J^{\mu}(0,\vec{x})$. However, dilatations, rotations
involving the time direction, and special conformal transformations can be
broken since their respective currents are not periodic. Supersymmetry
breaking is instead related to the anti-periodicity of the fermionic operators
(which makes the supercurrent antiperiodic).
The final form of the broken Ward identities is then given by

$i\sum_{i}\left\langle\mathcal{O}_{1}(x_{1})\ldots\bm{G}_{a}\mathcal{O}_{i}(x_{i})\ldots\
\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=\int d^{d-1}y\
\left\langle\Gamma_{a}^{\beta}(\vec{y})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
.$ (21)

Further observe that among all the possible symmetry, the dilatation broken
Ward identities gives information about the non-criticality of the theory and
can be interpreted as a non-perturbative version of the Callan-Symanzik
equation. Indeed it is easy to see that the dilatation broken Ward identities
at finite temperature can be corrected by taking into account also the other
scale present in the theory, namely the temperature. Indeed all correlators
have to satisfy the differential equation

$\left(\bm{D}+\beta\frac{\partial}{\partial\beta}\right)\left\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=0\
.$ (22)

The derivation of the equation above is provided in Appendix A and can be
thought of as a simple dimensional analysis statement.

Symmetry | Infinitesimal transformations: $i\omega_{a}\bm{G}_{a}\mathcal{O}_{i}(x_{i})$
---|---
Translations | $a^{\mu}\frac{\partial}{\partial x^{\mu}_{i}}\mathcal{O}_{i}(x_{i})$
Dilatations | $b\left(x^{\mu}_{i}\frac{\partial}{\partial x^{\mu}_{i}}+\Delta_{i}\right)\mathcal{O}_{i}(x_{i})$
Rotations | $c^{\mu\nu}\left(-x_{i,\mu}\frac{\partial}{\partial x^{\nu}_{i}}+x_{i,\nu}\frac{\partial}{\partial x^{\mu}_{i}}+i\bm{S}_{\mu\nu}\right)\mathcal{O}_{i}(x_{i})$
Special conf. tr. | $d^{\mu}\left(-x_{i}^{2}\frac{\partial}{\partial x^{\mu}_{i}}+2x_{i,\mu}^{\phantom{\nu}}x_{i}^{\nu}\frac{\partial}{\partial x^{\nu}_{i}}+2x_{i,\mu}\Delta_{i}-2ix^{\nu}_{i}\bm{S}_{\mu\nu}^{\phantom{\nu}}\right)\mathcal{O}_{i}(x_{i})$
Table 1: _Symmetries of the Global Conformal Group and their actions on the
operator $\mathcal{O}_{i}(x_{i})$ living in an irreducible representation of
the Lorentz Group _DiFrancesco:1997nk_. $a^{\mu}$ and $d^{\mu}$ are
infinitesimal vectors, $b$ is an infinitesimal scalar and $c^{\mu\nu}$ an
antisymmetric tensor; $\bm{S}_{\mu\nu}$ represents the spin operator._

Explicit checks of equation (21) with the breaking terms of Table 2 were
performed in two-dimensional CFTs and free theory in four space-time
dimensions. The most relevant (and non-trivial) checks are provided in
Appendix B.

##  3 Conformal Group at finite temperature

By plugging in the general broken Ward identity (21) the specific breaking
terms associated with the symmetries of the Conformal Group (whose associated
operators are listed in Table 2), it is possible to produce specific
constraints on the thermal correlation functions of a generic CFT at finite
temperature. In Table 2 we list for every symmetry in the Conformal Group the
current and the associated breaking term $\Gamma^{\beta}_{a}(x)$, which can be
computed by plugging the explicit expressions for the currents
$J^{\mu}_{a}(x)$ in equation (19). In the following, for the sake of notation
clarity, we will set

$\mathcal{O}_{1\cdots
n}(\bm{x})=\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{n}(x_{n})\ .$ (23)

Symmetry | Currents: $\omega_{a}J_{a}^{\mu}$ | Breaking terms: $\Gamma^{\beta}_{a}(0,\vec{x})$
---|---|---
Translations | $a_{\nu}T^{\mu\nu}$ | $0$
Dilatations | $bx_{\nu}T^{\mu\nu}$ | $\beta T^{00}(0,\vec{x})$
Spatial rot. | $-c_{\,i}^{j}x_{j}T^{\mu i}+c_{i}^{\,j}x_{j}T^{\mu i}$ | $0$
Boosts | $-c_{\,i}^{0}\tau T^{\mu i}+c_{i}^{\,0}\tau T^{\mu i}$ | $\beta T^{0i}(0,\vec{x})$
S.c.t. on $S^{1}_{\beta}$ | $d\left(-x^{2}\delta_{0\nu}+2\tau x_{\nu}\right)T^{\mu\nu}$ | $\beta\left[(\beta-2\tau)\,T^{00}(0,\vec{x})+2(x_{i}-y_{i})T^{0i}(0,\vec{x})\right]$
S.c.t. on $\mathbb{R}^{d-1}$ | $d^{i}\left(-x^{2}\delta_{i\nu}+2x_{i}x_{\nu}\right)T^{\mu\nu}$ | $\beta\left[(\beta+2\tau)\,T^{0j}(0,\vec{x})+2(x_{i}-y_{i})T^{ij}(0,\vec{x})\right]$
Table 2: _Explicit $\Gamma_{a}^{\beta}(x)$ terms for the symmetries of the
global Conformal Group. The $(\tau,\vec{y})$ coordinates locates the operator
to which the breaking term is applied. Notice that the thermal contributions
explicitly break the manifest $SO(d)$ invariance, but preserve the manifest
$SO(d-1)$ invariance._

##### Translations

The Ward identity associated with translations in every direction is left
unbroken by the introduction of a finite temperature. The constraints are
simply

$\sum_{r=1}^{n}\frac{\partial}{\partial\tau_{r}}\Braket{\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}=0\ ,\qquad\sum_{r=1}^{n}\frac{\partial}{\partial
x_{r}^{i}}\Braket{\mathcal{O}_{1\cdots n}(\bm{x})}_{\beta}=0\ .$ (24)

##### Dilatations

The Ward identity associated with dilatations is broken by the introduction of
a finite temperature. In fact, the theory ceases to be scale-less: the effect
is captured by the broken Ward identity

$\sum_{r=1}^{n}\left(\tau_{r}\frac{\partial}{\partial\tau_{r}}+x_{r}^{i}\frac{\partial}{\partial
x_{r}^{i}}+\Delta_{r}\right)\Braket{\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}=\beta\int d^{d-1}y\,\left\langle
T^{00}(0,\vec{y})\mathcal{O}_{1\cdots n}(\bm{x})\right\rangle_{\beta}\ ,$ (25)

where $\Delta_{r}$ indicates the classical conformal dimension of the $r$-th
operator.

##### Rotations

From the list of breaking terms in Table 2, we can notice that only boosts are
associated to a broken Ward identity. Spatial rotations are associated to

$\sum_{r=1}^{n}\left(x_{r,i}\frac{\partial}{\partial
x_{r}^{j}}-x_{r,j}\frac{\partial}{\partial
x_{r}^{i}}-i\bm{S}^{r}_{ij}\right)\Braket{\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}=0\ ,$ (26)

where $\bm{S}_{\mu\nu}^{r}$ denotes the application of the spin operator to
the $r$-th operator. Boosts are instead associated with the following broken
Ward identity

$\sum_{r=1}^{n}\left(\tau_{r}\frac{\partial}{\partial
x_{r}^{i}}-x_{r,i}\frac{\partial}{\partial\tau_{r}}+i\bm{S}_{i0}^{r}\right)\Braket{\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}=\beta\int
d^{d-1}y\,\Braket{T^{0i}(0,\vec{y})\mathcal{O}_{1\cdots n}(\bm{x})}_{\beta}\
.$ (27)

##### Special Conformal Transformations

We conclude the list with the broken Ward identities associated with the
special conformal transformations. The broken Ward identity associated with
special conformal transformations acting in the $S^{1}_{\beta}$ direction is

$\sum_{r=1}^{n}\left[\left(\tau_{r}^{2}-x^{i}_{r}x^{i}_{r}\right)\frac{\partial}{\partial\tau_{r}}+2\tau_{r}x^{i}_{r}\frac{\partial}{\partial
x_{r}^{i}}+2\tau_{r}\Delta_{r}+2ix_{r}^{i}\bm{S}^{r}_{i0}\right]\Braket{\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}=\\\ =\sum_{r=1}^{n}\beta\left(\beta-2\tau_{i}\right)\int
d^{d-1}y\left(\Braket{T^{00}(0,\vec{y})\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}+(x_{i}-y_{i})\Braket{T^{0i}(0,\vec{y})\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}\right)\ ,$ (28)

while the broken Ward identity associated with the special conformal
transformations acting in the $\mathbb{R}^{d-1}$ spacetime component is

$\sum_{r=1}^{n}\left[\left(\tau_{r}^{2}+x^{j}_{r}x^{j}_{r}\right)\frac{\partial}{\partial
x^{i}_{r}}-2x_{r,i}x^{\mu}_{r}\frac{\partial}{\partial
x^{\mu}_{r}}-2x_{r,i}\Delta_{r}+2i\tau_{r}\bm{S}_{i0}^{r}+2ix_{r}^{j}\bm{S}_{ij}^{r}\right]\Braket{\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}=\\\ =\sum_{r=1}^{n}\beta\left(\beta+2\tau_{i}\right)\int
d^{d-1}y\left(\Braket{T^{00}(0,\vec{y})\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}+(x_{i}-y_{i})\Braket{T^{0i}(0,\vec{y})\mathcal{O}_{1\cdots
n}(\bm{x})}_{\beta}\right)\ .$ (29)

###  3.1 Hamiltonian, momentum and (generalized) modularity

In this section we are going to derive the operatorial expression of the
Hamiltonian and the momentum at finite temperature. Those expression are
consequences of the broken Ward identities. The energy spectrum that can be
obtained from the diagonalization of such Hamiltonian corresponds to the
energy spectrum of the theories that leave on the vertical axis above the
critical point at zero temperature in Fig. 1.

$T$$g$$\left(g^{\star},0\right)$CFT at zero TThermal CFT Figure 1: _A
schematic representation of the $(g,T)$ plane, where $T$ is the temperature
and $g$ is a coupling constant of the theory. The zero temperature CFT sits on
the critical point $(g^{\star},0)$, represented by a blue dot. In this paper,
we study thermal CFT, which can be represented as the QFTs lying on the
vertical dashed red line above the critical point._

It can be observed that the newly found broken Ward identities (25) and (27)
assume a very interesting physical interpretation when we introduce the
Hamiltonian and spatial momentum444The minus sign is because we are using the
convention for Euclidean theories, where
$T_{\text{Lorenzian}}^{00}=i^{2}T_{\text{E
Euclidean}}^{00}=-T_{\text{Euclidean}}^{00}$. The same for the $i$ in front of
the definition of the momentum,
$T_{\text{Lorenzian}}^{0i}=iT_{\text{Euclidean}}^{0i}$.

$H=-\int_{\mathbb{R}^{d-1}}d^{d-1}x\
\left(T^{00}(\tau,\vec{x})+\frac{d-1}{d}\frac{b_{T}}{\beta^{d}}\right)\
,\qquad P^{i}=i\int_{\mathbb{R}^{d-1}}d^{d-1}x\ T^{0i}(\tau,\vec{x})\ ,$ (30)

where Iliesiu:2018fao

$\left\langle
T^{00}\right\rangle_{\beta}=-\frac{d-1}{d}\frac{b_{T}}{\beta^{d}}\ .$ (31)

The latter equation, regarding the one-point function of the stress-energy
tensor, can be obtained directly from equation (25) when specialized to the
case of one-point functions. By using the translational invariance of one-
point functions, equation (25) reads

$\bm{H}=-\frac{1}{\beta}\bm{D}-E_{0}\ ,$ (32)

where $\bm{H}$ is the Hamiltonian operator of the theory at finite
temperature. In general, this theory should be thought of as a
$(d-1)$-dimensional, interacting QFT at finite temperature Iliesiu:2018fao ;
Benjamin:2023qsc . This means that the energies of the $(d-1)$-dimensional,
interacting theory can be written in terms of the conformal spectrum of the
$d$-dimensional conformal field theory; in particular, after having chosen as
a basis $\ket{i}=\mathcal{O}_{i}\ket{0}$ (where $\mathcal{O}_{i}$ is a scaling
operator), it is possible to write down the matrix elements of the Hamiltonian
as

$\langle i|\bm{H}|j\rangle_{\beta}=-\frac{\Delta_{i}}{\beta}\delta_{ij}-E_{0}\
,$ (33)

where $\left\\{\Delta_{i}\right\\}$ is the spectrum of the dilatation operator
when applied to operators (inserted in the origin). Those eigenvalues are zero
temperature data and corresponds to the conformal dimensions of the operator
content of the zero-temperature CFT. Equation (33) shows that those zero
temperature data corresponds to the energy spectrum at finite temperature,
i.e. on the vertical dashed axis in Fig. 1. Indeed from the expression above
one can read the energy spectrum of the theory at finite temperature555 The
minus sign is due to the Euclidean signature. Switching to the Lorentzian one,
the first contribution turns positive.

$E_{n}=-\frac{\Delta_{n}}{\beta}-E_{0}\ ,$ (34)

where the $\Delta_{n}$ are ordered from the lowest to the highest one. The
unitarity of the original CFT implies that the minimum value of $\Delta_{n}$
is given by $0$: this corresponds to the identity operator. $E_{0}$ is the
energy of the thermal vacuum, which takes into account the Casimir energy of
the system666The Casimir energy of a thermal system can be understood as a
contribution given by the energy of the thermal gas: in fact, while at zero
temperature $\Braket{T^{00}}=0$, at finite temperature
$\Braket{T^{00}}_{\beta}=-\left(1-\frac{1}{d}\right)\frac{b_{T}}{\beta^{d}}$,
which returns the energy density
$\mathcal{E}^{\text{gas}}_{0}=-\frac{d-1}{d}\frac{b_{T}}{\beta^{d-1}}$
Iliesiu:2018fao ; PhysRevE.53.4414 ; Luo:2022tqy .. Equation (34) is perfectly
in agreement with a trivial dimensional analysis: since the only scale in the
theory is the temperature, i.e. $\beta$, then $E\propto 1/\beta$777$E_{0}$ is
an exception since
$E_{0}=\mathcal{E}^{\text{gas}}_{0}\times\text{Vol}\left[\mathcal{M}_{\beta}\right]$..
The non-triviality of equation (34) comes from the explicit coefficient in
front of the temperature. In a similar way, the boost operator $\bm{L}_{i}$,
is related to momentum operators via equation (27). Explicitly

$\bm{P}_{i}=\frac{i}{\beta}\bm{L}_{i}\ ,$ (35)

where $\bm{P}_{i}$ is the spatial momentum operator of the theory at finite
temperature. The physical interpretation of the equations (32) and (35) is
that the dilatation operator generates, at finite temperature, the time
translations (where here the time direction is the compactified direction on
$S^{1}_{\beta}$), whereas the boost operator generates the spatial
translations.
As already observed in Iliesiu:2018fao , the definitions of Hamiltonian and
momentum in (30) are the most natural ones, interpreting the system as a
finite temperature system. However, from the Kaluza-Klein perspective, this is
not the most natural way of defining Hamiltonian and momentum; in the latter
case, it is more natural to compactify a space direction and not the time
direction. Defining $x^{1}\in\mathbb{R}$ as the time direction

$H=-\int_{0}^{\beta}d\tau\int_{\mathbb{R}^{d-2}}d^{d-2}x\
\left(T^{11}(\tau,\vec{x})+\frac{1}{d}\frac{b_{T}}{\beta^{d}}\right)\ ,\quad
P^{i}=i\int_{0}^{\beta}d\tau\int_{\mathbb{R}^{d-2}}d^{d-2}x\
T^{1i}(\tau,\vec{x})\ ,$ (36)

where $i=0,2,3,4,\ldots$

There is also a third possibility we are not exploring in this paper
concerning the CFT at finite volume (the Conformal Group is broken also in
this case). In the language of this paper, it can be defined as a conformal
field theory on the manifold $\mathbb{R}\times S_{R}^{d-1}$, where
$\mathbb{R}$ represents the time direction and the space directions are
compactified on a $(d-1)$-sphere of radius $R$. In two dimensions, Hamiltonian
and momentum of finite volume theories are well known in the literature
Cardy:1989da ; Cardy:1991kr ; Mussardo:2020rxh  and can be checked explicitly,
by exploiting the conformal map between the plane
$\mathbb{R}^{2}\simeq\mathbb{C}$ and the cylinder $\mathbb{R}\times
S^{1}_{R}$. Generalizations to higher dimensions are also provided in
literature together with impressive applications Iliesiu:2018fao ;
Hogervorst:2013sma ; Hogervorst:2014rta . In this perspective, equations (32),
(35) are curiously and intriguingly similar (the only differences concern the
Euclidean/Lorentzian conventions) to the Hamiltonian and momentum at finite
volume, hinting at a possible relation between finite temperature and finite
volume systems. This fact is understood in two dimensions, since it is a
consequence of the modular invariance Cardy:1986ie ; DiFrancesco:1997nk ;
Mussardo:2020rxh , but it is not completely clear in higher dimensions. As
already proposed in literature Berg:2023pca ; Downing:2023uuc , we can
speculate on the possibility to generalize the modular invariance in higher
dimensions. One way is to consider the conformal field theory on
$S_{1}^{\beta}\times S_{d-1}^{R}$ in which we choose the time direction to be
defined by $0\leq\tau\leq\beta$ and the spatial direction to be $0\leq
x^{i}\leq R$. The latter geometry is related to $\mathbb{R}^{d}$ via conformal
maps; the thermal CFT can be derived by sending $R\to\infty$ while the finite
temperature version of the same theory can is describe by the limt
$\beta\to\infty$. Therefore the conformally equivalent geometry
$S_{1}^{\beta}\times S_{d-1}^{R}$ contains information both of the thermal and
the finite volume theory.
Furthermore the consistency of the theory on the thermal manifold can be
thought of as a generalized modular invariance El-Showk:2011yvt  and this fact
seems to be captured, for the first time from this computation, since the
Hamiltonian computed for the finite temperature system is the same Hamiltonian
(up to constant factors) known for the finite volume system.

The Hamiltonian of a finite temperature CFT can be written in terms of the
dilatation operator $\bm{D}$ of the zero-temperature CFT, and how the momentum
operator of the finite temperature system can be expressed in terms of the
boost operator of the zero-temperature CFT. This simple observation seems to
suggest that some finite temperature quantities could be recovered by
considering zero-temperature CFT data, through the, previously derived, broken
Ward identities.
One very important finite temperature quantity in which we are interested is
the _thermal mass_ of the theory. This mass is not a physical mass, but it can
be defined as the exponential factor that rules the behavior of the two-point
functions at spatial infinity, i.e. in the limit $|\vec{x}|\to\infty$

$\left\langle\phi(\tau,\vec{x})\phi(0)\right\rangle_{\beta}\simeq\left\langle\phi(0)\right\rangle_{\beta}^{2}+O\left(e^{-m_{th}|\vec{x}|}\right)\
,$ (37)

where $m_{th}$ is the thermal mass. When $\vec{x}=(x,0,0,\ldots)$, we have

$\left\langle\phi(0)\phi(\tau,x)\right\rangle_{\beta}=\left\langle\phi(0)e^{i\bm{H}\tau-\bm{P}_{1}x}\phi(0)\right\rangle_{\beta}\
,$ (38)

where $\bm{H}$ and $\bm{P}_{1}$ are the operators defined in the previous
Section. Observe that the definition of the thermal mass is simply the first
non-zero eigenvalues of $\bm{P}_{1}$, which is a well-defined differential
operator. In particular, it is possible to choose the basis of the Hilbert
space $\ket{\mathcal{O}_{\Delta,J}}=\mathcal{O}_{\Delta,J}(0)\ket{0}$ thanks
to state-operator correspondence. Then the thermal mass is given by

$m_{\text{th}}=\frac{1}{\beta}\min_{\Delta,J}\left\langle\mathcal{O}_{\Delta,J}|i\bm{L}_{1}|\mathcal{O}_{\Delta,J}\right\rangle=\frac{1}{\beta}\min_{\Delta,J}\lim_{y\to\infty}y^{2\Delta}\left\langle\mathcal{O}_{\Delta,J}(y)i\bm{S}_{01}\mathcal{O}_{\Delta,J}(0)\right\rangle\
,$ (39)

where it is understood that the minimum excludes zero eigenvalues and the
correlators are zero temperature correlators.

####  3.1.1 An implicit version of the Cardy formula

In Appendix A we presented a different derivation of broken Ward identities,
leading to equation (22), and we observed that when applied to one point
functions, dilatations Ward identities imply

$\frac{\partial}{\partial\beta}\left\langle\mathcal{O}(0)\right\rangle_{\beta}=-\frac{1}{\beta}\int
d^{d-1}x\left\langle T^{00}(0,\vec{x})\mathcal{O}(0)\right\rangle_{\beta}\ .$
(40)

This formula already appeared in literature El-Showk:2011yvt  and it has a
very concrete simple interpretation. The temperature is related to the radius
of the thermal circle, or equivalently to the value of the metric component
$g_{00}$. Since we can think at $g_{00}$ as the source of $T^{00}$, then a
small change in the temperature is equivalent to the insertion of the zero-
zero component of the stress-energy tensor. This physical interpretation is
extremely useful, in particular considering the specific case in which

$\frac{\partial}{\partial\beta}\langle
T^{00}(0)\rangle_{\beta}=-\frac{1}{\beta}\int d^{d-1}x\left\langle
T^{00}(x)T^{00}(0)\right\rangle_{\beta}\ .$ (41)

It is natural to think (41) as the differential equation generalizing the
Cardy Formula in every spacetime dimension El-Showk:2011yvt . Indeed using
thermodynamic relations it is possible to compute the entropy of the system as
a function of the $b_{T}$ coefficient, which plays the same role of the
central charge $c$ in two dimensions888Indeed in two dimensions $b_{T}\propto
c$.. The $b_{T}$ coefficient is implicit in equation (41) and it is not known
in general, apart from exceptional cases, leaving the generalization of the
Cardy formula in higher dimensions an interesting problem to be addressed in
the future DiPietro:2014bca ; Assel:2015nca ; Mukhametzhanov:2019pzy ;
Carlip:2000nv ; Brevik:2004sd ; Wang:2001bf ; Benjamin:2023qsc .

###  3.2 Thermal one-point functions

We consider a generic thermal one-point function
$\langle\mathcal{O}(x)\rangle_{\beta}$, where $\mathcal{O}(x)$ is a local
operator. The unbroken Ward identities (24) are of fundamental importance: in
our case, they specialize to

$\frac{\partial}{\partial\tau}\left\langle\mathcal{O}(\tau,\vec{x})\right\rangle_{\beta}\left(x\right)=0\
,\qquad\frac{\partial}{\partial
x^{i}}\left\langle\mathcal{O}(\tau,\vec{x})\right\rangle_{\beta}=0\ .$ (42)

Thanks to these equations, we conclude that a thermal one-point function (of a
local operator) is a constant all over the thermal manifold. Once we
acknowledge this, the other Ward identities can be significantly simplified.
First of all, the unbroken Ward identity describing spatial rotations simply
becomes

$\left\langle\bm{S}_{ij}\mathcal{O}(x)\right\rangle_{\beta}=0\ ;$ (43)

dilatations and boosts are encoded into simplified broken Ward identities

$\displaystyle\Delta_{\mathcal{O}}\langle\mathcal{O}(x)\rangle_{\beta}$
$\displaystyle=\beta\int
d^{d-1}y\,\Braket{T^{00}(0,\vec{y})\mathcal{O}(0)}_{\beta}\ ,$ (44)
$\displaystyle i\langle\bm{S}_{i0}\mathcal{O}(x)\rangle_{\beta}$
$\displaystyle=\beta\int
d^{d-1}y\,\Braket{T^{0i}(0,\vec{y})\mathcal{O}(0)}_{\beta}\ .$ (45)

The broken Ward identity associated with special conformal transformations in
the $S^{1}_{\beta}$ direction simplifies to

$2\tau\Delta_{\mathcal{O}}\Braket{\mathcal{O}(\tau,x)}_{\beta}+2ix^{i}\Braket{\bm{S}_{i0}\mathcal{O}(\tau,x)}_{\beta}=\beta\left(\beta-2\tau\right)\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}(\tau,x)}_{\beta}+\\\ +\beta\int
d^{d-1}y\ (x_{i}-y_{i})\Braket{T^{0i}(0,\vec{y})\mathcal{O}(\tau,x)}_{\beta}\
,$ (46)

while the one in the $\mathbb{R}^{d-1}$ component is

$-2x_{i}\Delta_{\mathcal{O}}\Braket{\mathcal{O}(x)}_{\beta}+2i\tau\Braket{\bm{S}_{i0}\mathcal{O}(x)}_{\beta}=\beta\left(\beta+2\tau\right)\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}(\tau,x)}_{\beta}+\\\ +\beta\int
d^{d-1}y\ (x_{j}-y_{j})\Braket{T^{ij}(0,\vec{y})\mathcal{O}(\tau,x)}_{\beta}\
.$ (47)

To summarize, we conclude that the unbroken and broken Ward identities of the
Global Conformal Group constrain the thermal one-point functions to be
kinematically constant. From the point of view of their tensorial structure,
they must be $SO(d-1)$-invariant tensors. All the other features can be
extracted in principle from the equations (44) and (45). As anticipated, we
will now focus on the thermal one-point functions of scalar, vector, and rank-
two tensor local operators.

##### Scalar operators

Scalar operators are particularly easy to study since
$\Braket{\bm{S}_{0i}\mathcal{O}(x)}_{\beta}=\Braket{\bm{S}_{ij}\mathcal{O}(x)}_{\beta}=0$
identically. The rotation broken Ward identity simply becomes

$\int d^{d-1}y\,\Braket{T^{0i}(0,\vec{y})\mathcal{O}(0)}_{\beta}=0\ .$ (48)

which is satisfied, since for $SO(d-1)$ invariance the integrand has to be odd
in one spatial coordinate. The only meaningful constraint on the thermal one-
point function of a scalar operator $\mathcal{O}(x)$ is given by the broken
Ward identity (44)

$\Braket{\mathcal{O}(x)}_{\beta}=\frac{\beta}{\Delta_{\mathcal{O}}}\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}(0)}_{\beta}\ .$ (49)

As already noticed this equation is extremely interesting and can give, in
principle, non-trivial constraints on structure constants of the zero-
temperature theory.

##### Vector operators

When we consider a local vector operator $\mathcal{O}^{\mu}(x)$, the Ward
identities associated with rotations are not trivial anymore. We need to
evaluate explicitly the functions
$\left\langle\bm{S}_{0i}\mathcal{O}^{\mu}(x)\right\rangle_{\beta}$ and
$\left\langle\bm{S}_{ij}\mathcal{O}^{\mu}(x)\right\rangle_{\beta}$. First of
all, we specialize the generators $\bm{S}_{ij}$ and $\bm{S}_{0i}$ to the
vector representation of $SO(d)$

$\tensor{\left(\bm{S}_{ij}\right)}{{}^{\mu}_{\nu}}=i\left(\delta^{\mu}_{i}\delta^{\phantom{\mu}}_{j\nu}-\delta^{\mu}_{j}\delta^{\phantom{\mu}}_{i\nu}\right)\
,\qquad\tensor{\left(\bm{S}_{0i}\right)}{{}^{\mu}_{\nu}}=i\left(\delta^{\mu}_{0}\delta^{\phantom{\mu}}_{i\nu}-\delta^{\mu}_{i}\delta^{\phantom{\mu}}_{0\nu}\right)\
.$ (50)

We can now explicitly write down the unbroken and broken Ward identities

$\displaystyle\left\langle\bm{S}_{ij}\mathcal{O}^{\mu}(x)\right\rangle_{\beta}$
$\displaystyle=i\left(\delta^{\mu}_{i}\Braket{\mathcal{O}_{j}(x)}_{\beta}-\delta^{\mu}_{j}\Braket{\mathcal{O}_{i}(x)}_{\beta}\right)=0\
,$ (51)
$\displaystyle\left\langle\bm{S}_{0i}\mathcal{O}^{\mu}(x)\right\rangle_{\beta}$
$\displaystyle=i\left(\delta^{\mu}_{0}\Braket{\mathcal{O}_{i}(x)}_{\beta}-\delta^{\mu}_{i}\Braket{\mathcal{O}_{0}(x)}_{\beta}\right)=-i\beta\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}^{\mu}(0)}_{\beta}\ .$ (52)

Let us start by solving the equation (51). If we set $\mu=0$, then the
identity is trivially satisfied; if we set $\mu=k$, then we can multiply both
sides by $\delta^{i}_{k}$, obtaining

$\left(d-2\right)\Braket{\mathcal{O}_{j}(x)}_{\beta}=0\ .$ (53)

This equation is always satisfied in $d=2$, hence it doesn’t provide any
additional constraint; in $d>2$, it returns

$\Braket{\mathcal{O}_{i}(x)}_{\beta}=0$ (54)

which is also a solution of the unbroken Ward identity (51). This was to be
expected for symmetry reasons. Moreover, it simplifies the equation (52)

$\delta^{\mu}_{i}\Braket{\mathcal{O}^{0}(x)}_{\beta}=\beta\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}^{\mu}(0)}_{\beta}\ .$ (55)

By setting $\mu=0$, we get an equation satisfied for symmetry reasons;
instead, by setting $\mu=j$ and multiplying by $\delta^{i}_{j}$ we get an
explicit expression for the $0$-th component

$\Braket{\mathcal{O}^{0}(x)}_{\beta}=\frac{\beta}{d-1}\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}^{i}(0)}_{\beta}\ .$ (56)

The dilatations broken Ward identity is

$\Braket{\mathcal{O}^{\mu}(x)}_{\beta}=\frac{\beta}{\Delta_{\mathcal{O}}}\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}^{\mu}(0)}_{\beta}\ .$ (57)

If we consider $\mu=i$ we get an identity for symmetry reasons, while if we
consider $\mu=0$, we get a second constraint on the $0$-th component

$\Braket{\mathcal{O}^{0}(x)}_{\beta}=\frac{\beta}{\Delta_{\mathcal{O}}}\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}^{0}(0)}_{\beta}\ .$ (58)

In conclusion, the most general statement we can produce only by applying the
unbroken and broken Ward identities is

$\displaystyle\Braket{\mathcal{O}^{i}(x)}_{\beta}$ $\displaystyle=0\ ,$ (59)
$\displaystyle\Braket{\mathcal{O}^{0}(x)}_{\beta}$
$\displaystyle=\frac{\beta}{\Delta_{\mathcal{O}}}\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}^{0}(0)}_{\beta}=\frac{\beta}{d-1}\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}^{i}(0)}_{\beta}\ .$ (60)

In Section 3.4, we will show
$\left\langle\mathcal{O}^{0}(x)\right\rangle_{\beta}=0$ if $\mathcal{O}^{\mu}$
lives in the OPE of two scalar local operators.

##### Rank-two tensor operators

We focus on a local rank-two tensor operator $\mathcal{O}^{\mu\nu}(x)$. The
unbroken Ward identity associated with spatial rotations can be written as

$\left\langle\bm{S}_{ij}\mathcal{O}_{\mu\rho}(x)\right\rangle_{\beta}=i\left(\delta_{\mu
i}\Braket{\mathcal{O}_{j\rho}(x)}_{\beta}-\delta_{\mu
j}\Braket{\mathcal{O}_{i\rho}(x)}_{\beta}+\delta_{i\rho}\Braket{\mathcal{O}_{\mu
j}(x)}_{\beta}-\delta_{j\rho}\Braket{\mathcal{O}_{\mu i}(x)}_{\beta}\right)=0\
.$ (61)

If we set $\mu=\rho=0$, we get a trivial identity, so we simply set

$\Braket{\mathcal{O}_{00}(x)}_{\beta}=a\ ,$ (62)

where $a$ is a constant. By setting $\mu=k$ and $\rho=0$, we get an equation
similar to (53)

$(d-2)\Braket{\mathcal{O}_{j0}(x)}_{\beta}=0\ ,$ (63)

which again is trivially solved if $d=2$, while in higher dimensions it
produces the following constraint

$\Braket{\mathcal{O}_{i0}(x)}_{\beta}=0\ .$ (64)

We can get $\Braket{\mathcal{O}_{0j}(x)}_{\beta}=0$ in a similar way. Finally,
we want to constraint $\Braket{\mathcal{O}_{ij}(x)}_{\beta}$ by choosing
$\mu=k$ and $\rho=l$

$\langle\bm{S}_{ij}\mathcal{O}_{kl}\left(x\right)\rangle_{\beta}=i\left(\delta_{ki}\Braket{\mathcal{O}_{jl}(x)}_{\beta}-\delta_{kj}\Braket{\mathcal{O}_{il}(x)}_{\beta}+\delta_{il}\Braket{\mathcal{O}_{kj}(x)}_{\beta}-\delta_{jl}\Braket{\mathcal{O}_{ki}(x)}_{\beta}\right)=0\
.$ (65)

This equation is trivially satisfied in $d=2$ since the only existing rotation
is the one generated by $\bm{S}_{01}$. In higher dimension, the solution is
given by the only $SO(d-1)$-invariant tensor with two indices

$\Braket{\mathcal{O}_{ij}(x)}_{\beta}=b\delta_{ij}\ ,$ (66)

where $b$ is a constant.
We conclude that the thermal one-point function
$\Braket{\mathcal{O}^{\mu\nu}(x)}_{\beta}$ is completely determined by the
constants $a$ and $b$. We can use the other Ward identities to set constraints
on $a$ and $b$. In particular, by using the dilatation broken Ward identity,
the only non-trivial constraints are

$\displaystyle a$
$\displaystyle=\Braket{\mathcal{O}^{00}(x)}_{\beta}=\frac{\beta}{\Delta_{\mathcal{O}}}\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}^{00}(0)}_{\beta}\ ,$ (67)
$\displaystyle b$
$\displaystyle=\frac{\beta\delta_{ij}}{d-1}\Braket{\mathcal{O}^{ij}(x)}_{\beta}=\frac{\beta\delta_{ij}}{(d-1)\Delta_{\mathcal{O}}}\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}^{ij}(0)}\ .$ (68)

The broken Ward identity associated with boosts establishes a relationship
between the two constants: the identity reads

$\delta_{\mu 0}\Braket{\mathcal{O}_{i\rho}(x)}_{\beta}-\delta_{\mu
i}\Braket{\mathcal{O}_{0\rho}(x)}_{\beta}+\delta_{0\rho}\Braket{\mathcal{O}_{\mu
i}(x)}_{\beta}-\delta_{i\rho}\Braket{\mathcal{O}_{\mu 0}(x)}_{\beta}=-\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}_{\mu\rho}(0)}_{\beta}$ (69)

By choosing $\mu=0$ and $\rho=i$, we extract the only non-trivial identity

$b=a-\frac{1}{d-1}\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}^{0i}(0)}_{\beta}.$ (70)

In Section 3.4, we will show that, if $\mathcal{O}^{\mu\nu}$ lives in the OPE
of two scalar operators, $a$ and $b$ must comply to an additional constraint
that sets the thermal one-point function to be traceless.
We conclude by remarking how all the results we obtained are in agreement with
the general result of Iliesiu:2018fao

$\langle\mathcal{O}^{\mu_{1}\ldots\mu_{J}}(x)\rangle_{\beta}=\frac{b_{\mathcal{O}}}{\beta^{\Delta_{\mathcal{O}}}}\left(e^{\mu_{1}}\cdots
e^{\mu_{J}}-\text{traces}\right)\ ,$ (71)

where $e^{\mu}$ is a unit vector in the compactified direction of
$S^{1}_{\beta}$. The authors of Iliesiu:2018fao  got the result from a pure
symmetry-driven argument; in our work symmetries’ constraints are encoded in
the broken and unbroken Ward identities and, even if these differential
equations are in general complicated, in the OPE regime they return the same
results of Iliesiu:2018fao , as expected. If the broken Ward identities can
constrain the correlators away from the OPE regime it remains an open problem
we are not going to address in this paper.

###  3.3 Thermal scalar two-point functions

The natural step following the analysis of the thermal one-point functions is
to consider thermal two-point functions. Two-point functions, differently from
one-point functions, retain an explicit kinematic dependence. The goal of this
Section is to formulate a bootstrap problem for the two-point function of two
_scalar_ local operators $\mathcal{O}_{1}(x_{1})$ and
$\mathcal{O}_{2}(x_{2})$. Working with scalars, the set of broken and unbroken
Ward identities significantly simplifies, motivating our choice. In order to
employ a more compact notation, we set

$\Braket{\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})}_{\beta}=f_{\beta}\left(\tau_{1},\tau_{2},\vec{x}_{1},\vec{x}_{2}\right)\
.$ (72)

####  3.3.1 Broken and unbroken Ward identities constraints

The easiest constraint is given by the unbroken translations Ward identity

$\displaystyle\frac{\partial}{\partial\tau_{1}}f_{\beta}\left(\tau_{1},\tau_{2},\vec{x}_{1},\vec{x}_{2}\right)+\frac{\partial}{\partial\tau_{2}}f_{\beta}\left(\tau_{1},\tau_{2},\vec{x}_{1},\vec{x}_{2}\right)$
$\displaystyle=0\ ,$ (73) $\displaystyle\frac{\partial}{\partial
x_{1}^{i}}f_{\beta}\left(\tau_{1},\tau_{2},\vec{x}_{1},\vec{x}_{2}\right)+\frac{\partial}{\partial
x_{2}^{i}}f_{\beta}\left(\tau_{1},\tau_{2},\vec{x}_{1},\vec{x}_{2}\right)$
$\displaystyle=0\ .$ (74)

These equations are solved in full generality by

$f_{\beta}\left(\tau_{1},\tau_{2},\vec{x}_{1},\vec{x}_{2}\right)=f_{\beta}\left(\tau_{1}-\tau_{2},\vec{x}_{1}-\vec{x}_{2}\right)=f_{\beta}\left(\tau_{12},\vec{x}_{12}\right)\
.$ (75)

We can now consider the constraints coming from applying the broken and
unbroken rotations Ward identity. Since we are focusing on scalar local
operators, the equations simplify. The unbroken Ward identity (26) reads

$\left(x_{12,i}\frac{\partial}{\partial
x_{12}^{j}}-x_{12,j}\frac{\partial}{\partial
x_{12}^{i}}\right)f_{\beta}\left(\tau_{12},\vec{x}_{12}\right)=0$ (76)

and it is solved by

$f_{\beta}\left(\tau_{12},\vec{x}_{12}\right)=f_{\beta}\left(\tau_{12},\left|\vec{x}_{12}\right|\right)\
;$ (77)

The broken Ward identity (27) instead reads

$\left(\tau_{12}\frac{\partial}{\partial
x_{12}^{i}}-x_{12,i}\frac{\partial}{\partial\tau_{12}}\right)f_{\beta}\left(\tau_{12},\left|\vec{x}_{12}\right|\right)=\beta\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})}_{\beta}\
,$ (78)

which represents a new non-trivial constraint on the two-point function. The
dilatations broken Ward identity provides an additional non-trivial constraint

$\left(\tau_{12}\frac{\partial}{\partial\tau_{12}}+x_{12}^{i}\frac{\partial}{\partial
x_{12}^{i}}+\Delta_{1}+\Delta_{2}\right)f_{\beta}\left(\tau_{12},\left|\vec{x}_{12}\right|\right)=\beta\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})}_{\beta}\
.$ (79)

We complete the exposition of the complete set of constraints on the thermal
scalar two-point functions coming from the Ward identities with the special
conformal transformations. The equations (28) and (29) in this case specialize
in complicated equations, however the breaking terms of the special conformal
transformations can be written in terms of breaking terms of dilatations and
rotations. Therefore, also considering $\beta-\tau=\tau$, we do not expect any
additional constraints coming from special conformal transformations.

####  3.3.2 Additional consistency constraints

In the previous Section, we discussed how thermal scalar two-point functions
can be constrained by enforcing the broken and unbroken Ward identities. In
this Section, we recap the other important constraints already known and
discussed in the literature Iliesiu:2018fao ; Alday:2020eua , with the goal of
formulating a complete bootstrap problem.

##### KMS condition

The periodicity of correlations functions for finite temperature theories is
known in the literature as KMS condition Kubo:1957mj ; Martin:1959jp ;
Iliesiu:2018fao ; Alday:2020eua . This condition reads

$f_{\beta}(\tau,|\vec{x}|)=f_{\beta}(\tau+\beta,|\vec{x}|)\ .$ (80)

##### Analyticity

Let us define in the complex plane the variables
$\omega=\text{Arg}(\tau+i|\vec{x}|)$, and $r=\text{Abs}(\tau+i|\vec{x}|)$. In
these coordinates, the two-point function reads

$f_{\beta}(\tau,|\vec{x}|)=f_{\beta}\left(\frac{r}{\omega},r\omega\right).$
(81)

Then, the analiticity consistency condition states that the two-point function
can only have cuts starting in $\omega=\pm r$ and $\omega=\pm r^{-1/2}$;
moreover, it has poles in $\omega=0,\pm\tilde{\omega}$, where
$\tilde{\omega}=\sqrt{(\tau+i|\vec{x}|)/(\tau-i|\vec{x}|)}$. Outside of these
poles and cuts, the two-point function must be an analytic function
Iliesiu:2018fao . This consistency condition is illustrated in Figure 2.

$-1/r$$1/r$$-r$$r$$-\tilde{\omega}$$\tilde{\omega}$$\omega$ Figure 2: Scheme
of the analyticity condition of the two-point function
$f(\tau,|\vec{x}|)=f(r/\omega,r\omega)$ in the $\omega$ plane. As shown in
Iliesiu:2018fao , the two-point function must be analytic everywhere outside
the poles and the cuts drawn in the $\omega$ plane.

##### Regge limit

Let us define in the complex plane the variables $z=\tau+i|\vec{x}|$ and
$\bar{z}=\tau-i|\vec{x}|$. With this set of coordinates, we can introduce the
variables $\rho=\sqrt{z\overline{z}}$ and
$\eta=\frac{z+\overline{z}}{2\sqrt{z\overline{z}}}$. In these coordinates, the
two-point function reads $f_{\beta}\left(\rho,\eta\right)$. The Regge limit is

$\lim_{\eta\rightarrow\infty}f_{\beta}\left(\rho,\eta\right)\ ,\quad\rho\text{
fixed}\ .$ (82)

The two-point function must be polynomially bounded in the Regge limit
Alday:2020eua .

##### Large distance limit

As mentioned before, it is believed that the compactification on the thermal
manifold produces a massive theory with a mass gap $m_{\text{th}}>0$ 999By
dimensional analysis this mass has to be proportional to $\beta^{-1}$.
Moreover, notice that, as already suggested in Iliesiu:2018fao ;
Benjamin:2023qsc , the existence of a thermal mass is not always guaranteed:
in some cases (e.g. bosonic free theory) symmetries preserve the theory from a
mass gap. This could also be the case if the theory spontaneously breaks a
symmetry at finite temperature as in the case of Chai:2020onq ; Chai:2020zgq ,
however, we are not aware of any example in integer spacetime dimension.
Nevertheless, in a generic interacting CFT at finite temperature the existence
of such a mass gap is expected from the fact that thermal fluctuations make
correlations between local operators negligible at large distances.. Assuming
this folk-theorem, the two-point function $f_{\beta}(\tau,|\vec{x}|)$
approaches spatial infinity as Iliesiu:2018fao ; Caron-Huot:2022akb

$f_{\beta}(\tau,|\vec{x}|)\sim\Braket{\mathcal{O}_{1}}_{\beta}\Braket{\mathcal{O}_{2}}_{\beta}+\mathcal{O}\left(e^{-m_{\text{th}}|\vec{x}|}\right)\
.$ (83)

##### Zero temperature limit

Since a thermal two-point function can be defined on a thermal manifold with
an arbitrary value of $\beta$, we expect
$f_{\beta}\left(\tau,|\vec{x}|\right)$ to be a continuous function in $\beta$;
moreover, the physical expectation is for it to be smooth and to have a well-
defined limit for $\beta\rightarrow\infty$

$\lim_{\beta\rightarrow\infty}\Braket{\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})}_{\beta}=\Braket{\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})}_{\mathbb{R}^{d}}\
.$ (84)

##### OPE

When two local scalar operators $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, with
conformal dimensions $\Delta_{1}$ and $\Delta_{2}$ respectively, lie in a
sphere whose interior is flat, the OPE is exact and reads Iliesiu:2018fao

$\mathcal{O}_{1}(x_{1})\times\mathcal{O}_{2}(x_{2})\sim\sum_{\mathcal{O}\in\mathcal{O}_{1}\times\mathcal{O}_{2}}\frac{f_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+|\vec{x}|^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-\Delta_{1}-\Delta_{2}-J\right)}x_{\mu_{1}}\ldots
x_{\mu_{J}}\mathcal{O}^{\mu_{1}\ldots\mu_{J}}(x_{2})\ ,$ (85)

where $x^{\mu}=x_{1}^{\mu}-x_{2}^{\mu}$, $\Delta_{\mathcal{O}}$ is the
conformal dimensions of $\mathcal{O}$,
$f_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}}$ is the three-point function
OPE constant at zero temperature, and $c_{\mathcal{O}}$ is the normalization
coefficient appearing in the two-point function at zero temperature

$\langle\mathcal{O}^{\mu_{1}\ldots\mu_{J}}(x)\mathcal{O}_{\nu_{1}\ldots\nu_{J}}(0)\rangle=c_{\mathcal{O}}\frac{I_{(\nu_{1}\ldots\nu_{J})}^{\,(\mu_{1}\ldots\mu_{J})}}{|x|^{2\Delta_{\mathcal{O}}}}\
,\hskip
28.45274ptI_{\,\nu}^{\mu}=\delta_{\,\nu}^{\mu}-\frac{2x_{\nu}x^{\mu}}{x^{2}}\
.$ (86)

The geometry of the thermal manifold imposes a minimal radius of convergence
to the OPE since the largest radius for a sphere with a flat interior is
$\beta$. Consistency with the OPE regime requires the thermal two-point
function to read, inside the convergence radius,

$f_{\beta}(\tau,|\vec{x}|)=\sum_{\mathcal{O}\in\mathcal{O}_{1}\times\mathcal{O}_{2}}\frac{f_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+|\vec{x}|^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-\Delta_{1}-\Delta_{2}-J\right)}x_{\mu_{1}}\ldots
x_{\mu_{J}}\Braket{\mathcal{O}^{\mu_{1}\ldots\mu_{J}}}_{\beta}\ .$ (87)

Working in the OPE regime, it is possible to apply the constraints derived
from the broken and unbroken Ward identities to the expression (87): we will
explore this in the Section 3.4. At this stage let us comment that the two-
point function has to be consistent with the OPE decomposition and therefore
the anomalous dimensions are those at zero temperature Alday:2020eua .
The bootstrap problem for the thermal scalar two-point function
$\Braket{\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})}_{\beta}$ consists of
finding a function of the coordinates $x_{1}$, $x_{2}$ and the inverse of the
temperature $\beta$ such that it satisfies all the conditions listed in this
Section. First of all, the unbroken Ward identities set the dependence on the
coordinates to be like in Equation (77). Then:

  1. 1.

it must be periodic over the thermal circle (cfr. Equation (80));

  2. 2.

in the $\omega$ plane, it must be analytic everywhere apart from the poles and
cuts illustrated in Figure 2;

  3. 3.

its Regge limit must be polynomially bounded; its zero temperature limit must
return the same two-point function, but computed on the flat space; the large
distance limit must return a clustered structure like in Equation (83);

  4. 4.

it must be compatible with the OPE expansion in the OPE regime, i.e.
$\sqrt{\tau^{2}+\vec{x}^{2}}\leq\beta$;

  5. 5.

it must satisfy the constraints (78) and (79), imposed by the broken
symmetries of the Global Conformal Group.

The last point of this list represents the novelty of this work with respect
to the already existing literature on the subject; nevertheless, we can also
think of these new conditions as additional constraints to bootstrapping one-
point functions at the finite temperature given the two-point functions
(including the stress-energy tensor) or, even more intriguingly, as a
consistency condition for the zero-temperature conformal field theory. As
already anticipated, in Section 3.4 we will combine the OPE expansion with
these newly found constraints.

If the bootstrap problem set above returns unique quantities or not is an
unsolved problem; in particular, the minimal amount of conditions to impose to
uniquely determine the two-point functions is not known. Suppose the existence
of two different solutions $f^{1}$ and $f^{2}$ to the problem. Imagine first
that the two solutions differ by the contribution of a finite number of local
operators in the OPE regime: then, the large distance limit of the difference
$\Delta f=f_{1}-f_{2}$ is101010The limit holds only for even spin operators;
however it is sufficient because the one point function of an odd spin
operator is zero and their contribution to the two-point function thereof.

$\lim_{r\to\infty}\left|\Delta f(\tau,r)\right|=\infty\ ,$ (88)

and therefore if $f_{1}$ satisfies the clustering property, expressed in
equation (83), then $f_{2}$ does not satisfy it (and _vice-versa_), so only
one among $f_{1}$ and $f_{2}$ is a solution to the bootstrap problem. If
$f_{1}$ and $f_{2}$ differ for an overall factor then the limit (84) differ
for the same overall factor, which can be identified with the normalization of
the two-point function at zero temperature. If we fix this normalization
before solving the bootstrap problem, then it fixes the residue of the poles
of the two-point function at finite temperature. Two solutions $f_{1}$ and
$f_{2}$ that do not differ by an overall factor must have the same poles with
the same residues since the poles are given by the zero temperature
contribution and the limit $\beta\to\infty$ defines the coefficients in front.
We couldn’t find a way to discriminate between two solutions complying with
all the reasoning above but differing for one or more branch cuts. This
problem was already pointed out in Alday:2020eua  where the authors proved
unicity assuming some specific conditions111111The necessary condition for the
unicity proposed in Alday:2020eua  is $\text{Disc}(z,\overline{z})=0$,
$0\leq\overline{z}=\tau-ir\leq 1$ and $1\leq z=\tau+ir$., by using the Regge
boundedness of the thermal correlator and the subtracted thermal dispersion
relations. Observe that, however, in the case of a free scalar theory in four
spacetime dimensions, it is possible to reconstruct the thermal two-point
function of the free scalar theory in four dimensions by only using the
bootstrap conditions recapped above (cfr. Appendix C) . This method seems to
work only in the case of the free scalar and the deep reason for that is the
simplicity of the OPE expansion between two fundamental scalars that allows
for a factorization of the two-point function in terms of _holomorphic_
functions. In general, it is difficult to read this factorization from the OPE
(and we do not expect that every two-point function can be factorized in a
similar way), however, we do not exclude that a more general unicity can be
proven with more sophisticated tools. We leave this to future studies.

###  3.4 Broken Ward identities in the OPE regime

The constraints set up by the broken symmetries of the global Conformal Group
are difficult to solve by themselves since the structure of the thermal
breaking term requires the knowledge of a thermal $(n+1)$-point function in
order to study a thermal $n$-point function. However, in this Section we show
how the breaking term splits into lower-point functions in the OPE regime,
allowing for an explicit derivation of the solution to the bootstrap problem
and for additional constraints on the thermal one-point functions. We are
going to focus on (broken) dilatations and rotations; in the Appendix D, we
show instead that a generic broken Ward identity, when combined with the OPE,
always returns the Leibniz rule for the generators of the broken symmetry.
Let us start by focusing on the new constraint coming from the dilatations
broken Ward identity. We study the thermal scalar two-point function

$f_{\beta}(\tau,|\vec{x}|)=\Braket{\varphi(x)\varphi(0)}_{\beta}\ ,$ (89)

where both scalar operators now have conformal dimension $\Delta_{\varphi}$.
By introducing the radial coordinate $r=|\vec{x}|$, the dilatations broken
Ward identity (79) reads

$\left(\tau\frac{\partial}{\partial\tau}+r\frac{\partial}{\partial
r}+2\Delta_{\varphi}\right)f_{\beta}\left(\tau,r\right)=\beta\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\varphi(x)\varphi(0)}_{\beta}\ .$ (90)

If we consider the operators $\varphi$ in the OPE regime, the broken Ward
identity can be rewritten as

$\left(\tau\frac{\partial}{\partial\tau}+r\frac{\partial}{\partial
r}+2\Delta_{\varphi}\right)f_{\beta}\left(\tau,r\right)=\\\
=\beta\sum_{\mathcal{O}\in\varphi\times\varphi}\frac{f_{\varphi\varphi\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}-J\right)}x_{\mu_{1}}\dots
x_{\mu_{J}}\int
d^{d-1}y\Braket{T^{00}(0,\vec{y})\mathcal{O}^{\mu_{1}\dots\mu_{J}}\left(0\right)}_{\beta}\
.$ (91)

The right-hand side can be rewritten by plugging the equation (49), which is
the dilatations broken Ward identity applied to thermal one-point functions

$\left(\tau\frac{\partial}{\partial\tau}+r\frac{\partial}{\partial
r}+2\Delta_{\varphi}\right)f_{\beta}\left(\tau,r\right)=\\\
=\sum_{\mathcal{O}\in\varphi\times\varphi}\frac{f_{\varphi\varphi\mathcal{O}}\Delta_{\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}-J\right)}x_{\mu_{1}}\dots
x_{\mu_{J}}\Braket{\mathcal{O}^{\mu_{1}\dots\mu_{J}}}_{\beta}\ .$ (92)

The left-hand side of the equation (92) respects the OPE regime as well, so it
can be expanded in a similar fashion. We get

$\sum_{\mathcal{O}\in\varphi\times\varphi}\left(\tau\frac{\partial}{\partial\tau}+r\frac{\partial}{\partial
r}+2\Delta_{\varphi}-\Delta_{\mathcal{O}}\right)f_{\beta}^{(\mathcal{O})}(\tau,r)=0\
,$ (93)

where

$f_{\beta}^{(\mathcal{O})}(\tau,r)=\frac{f_{\varphi\varphi\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}-J\right)}x_{\mu_{1}}\dots
x_{\mu_{J}}\Braket{\mathcal{O}^{\mu_{1}\dots\mu_{J}}}_{\beta}\ .$ (94)

Since
$\langle\mathcal{O}^{\mu_{1}\ldots\mu_{J}}\rangle_{\beta}\propto\beta^{-\Delta_{\mathcal{O}}}$
by dimensional analysis, we can interpret the sum over the operators in the
OPE appearing in the equation (94) as a sum over powers of $\beta$121212This
requires additional care in the case of a degenerate spectrum. In the latter
case, we can either distinguish the two operators from the spin or from other
global symmetries; in the first case, the contribution of any operator is
always captured by equation (102) by the linearity of the differential
equation (94), in the second case the one-point function is zero because the
operator is not in the trivial representation of the global symmetry’s group..
Hence, each term of the series constitutes an independent differential
equation

$\left(\tau\frac{\partial}{\partial\tau}+r\frac{\partial}{\partial
r}+2\Delta_{\varphi}\right)f_{\beta}^{(\mathcal{O})}(r,\tau)=\Delta_{\mathcal{O}}f_{\beta}^{(\mathcal{O})}\
.$ (95)

If we consider the identity operator
$\mathds{1}\in\mathcal{O}_{1}\times\mathcal{O}_{2}$, then
$\Delta_{\mathds{1}}=0$ and the equation (95) returns the zero temperature
result. For all the other cases we can write down also the boosts broken Ward
identity

$\left(\tau r\frac{\partial}{\partial
r}-r^{2}\frac{\partial}{\partial\tau}\right)f_{\beta}\left(\tau,r\right)=x_{i}\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\varphi(x)\varphi(0)}_{\beta}\ .$ (96)

This equation can be expanded in OPE as well; we expand the right-hand side
first

$\left(\tau r\frac{\partial}{\partial
r}-r^{2}\frac{\partial}{\partial\tau}\right)f_{\beta}\left(\tau,r\right)=\\\
=\beta\sum_{\mathcal{O}\in\varphi\times\varphi}\frac{f_{\varphi\varphi\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}-J\right)}x_{\mu_{1}}\dots
x_{\mu_{J}}x_{i}\int
d^{d-1}y\Braket{T^{0i}(0,\vec{y})\mathcal{O}^{\mu_{1}\dots\mu_{J}}\left(0\right)}_{\beta}\
,$ (97)

and then we apply the broken Ward identity (45) to rewrite the integral as

$\left(\tau r\frac{\partial}{\partial
r}-r^{2}\frac{\partial}{\partial\tau}\right)f_{\beta}\left(\tau,r\right)=\\\
=i\beta\sum_{\mathcal{O}\in\varphi\times\varphi}\frac{f_{\varphi\varphi\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}-J\right)}x_{\mu_{1}}\dots
x_{\mu_{J}}x_{i}\Braket{\bm{S}_{i0}\mathcal{O}^{\mu_{1}\dots\mu_{J}}\left(0\right)}_{\beta}\
.$ (98)

The equation (95) together with the equation (98) can be interpreted as a
system of partial differential equations. In general, we have that the first
differential equation, i.e. (95), fixes

$f_{\beta}^{\mathcal{O}}(\tau,r)\propto\left(r^{2}+\tau^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}-J\right)}\tau^{J}g_{\mathcal{O}}\left(\frac{r}{\tau}\right)\
,$ (99)

where the function $g_{\mathcal{O}}(r/\tau)$ remains unfixed. The next step is
to plug this solution in the rotation broken Ward identities; order by order
in the OPE expansion equation (98) reads

$\frac{r}{\tau^{2}}\tau^{J}\left(-Jr\tau
g_{\mathcal{O}}\left(\frac{r}{\tau}\right)+\left(r^{2}+\tau^{2}\right)g_{\mathcal{O}}^{\prime}\left(\frac{r}{\tau}\right)\right)=x_{\mu_{1}}\ldots
x_{\mu_{J}}x_{i}\left\langle\bm{S}_{i0}\mathcal{O}^{\mu_{1}\ldots\mu_{J}}\right\rangle_{\beta}\
.$ (100)

For scalars, i.e. $J=0$, we get $g_{\mathcal{O}}(r/\tau)=\text{const}.$,
meaning that the one-point function of a scalar
$\left\langle\mathcal{O}\right\rangle_{\beta}$ does not have any kinematical
structure, as expected. For vectors, i.e. $J=1$, by imposing the reality
condition on the solution131313The differential equation is also solved by a
second function proportional to a hypergeometric function: however, this
function is not real for all the possible values of $r$ and $\tau$., we get
$\left\langle\mathcal{O}^{\mu}\right\rangle=0$ and only a term proportional to
an odd power of $\tau$ can contribute to the OPE. However, this contribution
is zero because of the symmetry $\tau\to-\tau$, which forbids odd functions of
$\tau$ to appear. For 2-rank tensors, i.e. $J=2$, we can recall the results of
Section (3.2) to see that the solution to the two differential equations is
given by

$f_{\beta}^{(\mathcal{O})}(\tau,r)\propto\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}\right)}C_{2}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)\
.$ (101)

This result can be extended to all values of $J$ by observing that the first
differential equation is solved by the ansatz

$f_{\beta}^{(\mathcal{O})}(\tau,r)=\frac{\mathcal{C}_{J,\nu}^{\mathcal{O}}}{\beta^{\Delta_{\mathcal{O}}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}\right)}C_{J}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)\
,$ (102)

where the $\beta^{-\Delta_{\mathcal{O}}}$ factor is introduced by dimensional
analysis and $\mathcal{C}_{J,\nu}^{\mathcal{O}}$ is an overall coefficient to
be fixed. We can fix it by imposing consistency of the thermal two-point
function with the OPE. By comparing the expressions (94) and (102) we identify

$\frac{\mathcal{C}_{J,\nu}^{\mathcal{O}}}{\beta^{\Delta_{\mathcal{O}}}}C_{J}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)=\frac{f_{\varphi\varphi\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{-\frac{J}{2}}x_{\mu_{1}}\dots
x_{\mu_{J}}\Braket{\mathcal{O}^{\mu_{1}\dots\mu_{J}}}_{\beta}\ .$ (103)

If $\nu=\frac{d-2}{2}$, the following identity holds Iliesiu:2018fao

$C_{J}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)=\frac{2^{J}(\nu)_{J}}{J!}\left(\tau^{2}+r^{2}\right)^{-\frac{J}{2}}x_{\mu_{1}}\ldots
x_{\mu_{J}}\left(e^{\mu_{1}}\ldots e^{\mu_{J}}-\text{traces}\right)\ ,$ (104)

leading to

$\frac{\mathcal{C}_{J,\nu}^{\mathcal{O}}}{b_{\mathcal{O}}}\frac{2^{J}(\nu)_{J}}{J!}\frac{b_{\mathcal{O}}}{\beta^{\Delta_{\mathcal{O}}}}\left(e^{\mu_{1}}\ldots
e^{\mu_{J}}-\text{traces}\right)=\frac{f_{\varphi\varphi\mathcal{O}}}{c_{\mathcal{O}}}\Braket{\mathcal{O}^{\mu_{1}\dots\mu_{J}}}_{\beta}\
,$ (105)

from which we can both recover the formula (71) and fix the overall
coefficient

$f_{\beta}^{(\mathcal{O})}(\tau,r)=\frac{J!}{2^{J}(\nu)_{J}}\frac{1}{\beta^{\Delta_{\mathcal{O}}}}\frac{f_{\varphi\varphi\mathcal{O}}b_{\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}\right)}C_{J}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)\
,$ (106)

which immediately leads to the final solution141414The procedure highlighted
in this short Section shows how it is possible to recover the results already
presented in Iliesiu:2018fao  by employing only the broken Ward identities and
the OPE. It must be noticed that strictly speaking, the formula (71) was not
recovered in full generality, but only for those operators living in the OPE
between two identical scalar operators; moreover, the second differential
equation has to be solved spin by spin.

$f_{\beta}(\tau,r)=\sum_{\mathcal{O}\in\varphi\times\varphi}\frac{J!}{2^{J}(\nu)_{J}}\frac{1}{\beta^{\Delta_{\mathcal{O}}}}\frac{f_{\varphi\varphi\mathcal{O}}b_{\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}\right)}C_{J}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)\
.$ (107)

Further, observe that the OPE decomposition of the two-point function implies
that odd spin operators’ one-point functions have to be zero; this can be
easily checked by considering the fact that
$G_{J}^{\nu}(\tau/\sqrt{r^{2}+\tau^{2}})=G_{J}^{\nu}(-\tau/\sqrt{r^{2}+\tau^{2}})$
only for even spin operator and therefore $\tau\to-\tau$ is a symmetry only if
the one-point functions of odd spin operator are zero.
In conclusion, working with the broken Ward identities in the OPE regime
allowed us to obtain already-known results about the structure of the one-
point functions and the OPE decomposition of the two-point functions. This was
expected since broken Ward identities contain symmetry information about
broken and unbroken symmetry. These results also suggest that broken Ward
identities could contain more information, possibly also outside the OPE
regime.

##  4 Superconformal broken Ward identities

Whether any trace of supersymmetry is preserved at finite temperature has been
a longstanding question Fuchs:1984ed ; Aoyama:1984bk ; Das:1978rx ;
Girardello:1980vv , and significant progress has been made recently with
important developments and applications Friess:2006fk ; Berenstein:2022nlj ;
Caron-Huot:2022akb ; Caron-Huot:2022lff ; Alday:2020eua ; Petkou:2021zhg ;
Karlsson:2019qfi ; Karlsson:2021duj ; Karlsson:2022osn ; Benjamin:2023qsc .
The discussion on the broken Ward identities (at finite temperature) presented
in Section 2 can be specialized to the case of superconformal field theories.
In the following, we will focus on supersymmetric and superconformal theories
in four dimensions; however, the arguments and the procedures can be adapted
to other dimensions.
The Superconformal Group extends the Conformal Group with supersymmetry
generators $\bm{Q}_{\alpha}^{I}$ (and $\bm{\overline{Q}}^{\dot{\alpha}I}$),
superconformal generators $\bm{S}_{\alpha}^{I}$ (and
$\bm{\overline{S}}^{\dot{\alpha}I}$) and R-symmetry generators $\bm{R}^{IJ}$
satisfying precise commutator relations producing specific decomposition of
the Hilbert space of the theory Eberhardt:2020cxo ; Dolan:2001tt ;
Dolan:2002zh .
Due to the anti-periodicity of the fermionic operators over the thermal circle
$S^{1}_{\beta}$, we expect supersymmetry to be broken; however, it is not
obvious _a priori_ if the R-symmetry is broken or unbroken. We derive in this
Section explicit broken Ward identities for all the superconformal generators.

Since the discussion in this Section is completely analogous to the one
presented in Section 2 we do not derive again step by step the (broken) Ward
identities, but we will simply discuss the differences between conformal
generators and superconformal ones.

###  4.1 Supersymmetry and superconformal symmetry

The procedure underlined in Section 2 can be followed in every detail, with a
few remarks. In particular, the first difference concerns the variation of the
action with respect to a symmetry with fermionic generators151515In the
following, the spinor indices $\alpha,\beta,\gamma$ will run over $\\{1,2\\}$;
the same thing applies to the dotted spinor indices
$\dot{\alpha},\dot{\beta},\dot{\gamma}$. The R-symmetry indices will be
denoted by the set $I,J,K$ and will run over $\\{1,\dots,\mathcal{N}\\}$, with
$\mathcal{N}$ representing the amount of supersymmetry.

$\delta S=\int_{\mathcal{M}}d^{d}x\sqrt{g}\ \nabla_{\mu}\xi^{\alpha}(x)\
G_{\alpha}^{\mu}(x)\ ,$ (108)

where $G_{\alpha}^{\mu}$ is the fermionic current, $g_{\mu\nu}$ is the metric
on the manifold $\mathcal{M}$ and $\xi^{\alpha}$ is an infinitesimal spinor.
From the above equation it is clear that, on a generic non-flat manifold, all
the fermionic generators are broken since there is not a covariantly constant
spinor. At finite temperature, i.e. when
$\mathcal{M}_{\beta}=\mathbb{R}^{d-1}\times S^{1}_{\beta}$, the spacetime is
locally flat and this argument does not prevent fermionic generators from
realizing quantum symmetries. We can then proceed straight-forwardly as in the
bosonic case. A second remark can be observed in the evaluation of the
following spatial boundary integration

$\lim_{R\to\infty}\left\langle
J_{a}^{i}(\tau,R,\Omega)\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=\lim_{R\to\infty}\left\langle
J_{a}^{i}(\tau,R,\Omega)\right\rangle_{\beta}\left\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=0\
.$ (109)

Differently from the bosonic case, this contribution does not return an
infrared divergence because the one-point function of the fermionic current is
necessarily zero: this could follow from the solution of the rotation broken
Ward identity, or from a symmetry argument Iliesiu:2018fao . The rest of the
procedure to compute broken Ward identities is not altered. In order to
discuss the breaking terms it is useful to write down the superconformal
current

$J_{SC}^{\mu}(x)=\psi_{I}^{\alpha}(x)G_{\alpha}^{\mu I}(x)\ ,$ (110)

where $\psi_{I}^{\alpha}(x)$ is the conformal Killing spinor

$\psi_{I}^{\alpha}(x)=\lambda_{I}^{\alpha}+x_{\nu}\left(\overline{\sigma}^{\nu}\right)^{\,\alpha\dot{\alpha}}\mu_{I\dot{\alpha}}\
,$ (111)

where $\mu_{I\dot{\alpha}}$ and $\lambda_{I}$ are constant spinors. At zero
temperature, we define the supercharges and the superconformal charges161616As
it was done in the example (11), the correct definition of the quantum
superconformal generators must take into account the correct spacetime
dependence
$\bm{\overline{S}^{\dot{\alpha}}}\,\mathcal{O}(x)=\int_{\mathbb{R}^{d-1}}d^{d-1}y\
\left(\overline{\sigma}^{\nu}\right)_{\,\alpha\dot{\alpha}}(y_{\nu}-x_{\nu})\
G^{0\alpha}(y)\mathcal{O}(x)\ .$

$Q^{\alpha}=\int_{\mathbb{R}^{d-1}}d^{d-1}x\ G^{0\alpha}(x)\ ,\hskip
28.45274pt\overline{S}^{\dot{\alpha}}=\int_{\mathbb{R}^{d-1}}d^{d-1}x\
\left(\overline{\sigma}^{\nu}\right)_{\,\alpha\dot{\alpha}}x_{\nu}\
G^{0\alpha}(x)\ ,$ (112)

When $\lambda_{I}^{\alpha}=0$, $\mu_{I\dot{\alpha}}\neq 0$ the current (110)
corresponds to the symmetry generated by superconformal generators; instead
when $\mu_{I\dot{\alpha}}=0$, $\lambda_{I}^{\alpha}\neq 0$ the current is
associated to supersymmetry generators.
Due to the fermionic nature of $G_{\alpha}^{\mu I}(\tau,\vec{x})$, and the
anti-periodicity of fermionic operators with respect to the thermal circle, it
is natural to write down the most general breaking term, generally defined in
(19), for the superconformal current (110), as

$\left[2\lambda_{I}^{\alpha}+\left((\beta-2\tau)\left(\overline{\sigma}_{0}\right)^{\alpha\dot{\alpha}}+2(x_{i}-y_{i})\left(\overline{\sigma}^{i}\right)^{\alpha\dot{\alpha}}\right)\mu_{I\dot{\alpha}}\right]G_{\alpha}^{0I}(\beta,\vec{y})\
,$ (113)

where the coordinates $(\tau,\vec{y})$ are the coordinates of the operator to
which the generator is applied. The expression above agrees with the physical
expectation according to which the fermionic nature of $\bm{Q}_{\alpha}^{I}$
and $\bm{S}_{\alpha}^{I}$ implies an explicit breaking of supersymmetry at
finite temperature. Leaving the spinor indices and the R-symmetry indices
implicit for the sake of notation clarity, the broken Ward identities read

$\sum_{r=1}^{n}\left\langle\mathcal{O}_{1}(x_{1})\ldots\left[\bm{Q},\mathcal{O}_{r}\right](x_{r})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=2\int
d^{d-1}y\ \left\langle
G^{0}(\beta,\vec{y})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
,$ (114)
$\sum_{r=1}^{n}\left\langle\mathcal{O}_{1}(x_{1})\ldots\left[\bar{\bm{S}},\mathcal{O}_{r}\right](x_{r})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=\\\
=\sum_{r=1}^{n}(\beta-2\tau_{r})\overline{\sigma}^{0}\int d^{d-1}y\left\langle
G^{0}(\beta,\vec{y})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}+\\\
+\sum_{r=1}^{n}\overline{\sigma}^{j}\int d^{d-1}y\ (y_{j}-x_{j})\left\langle
G^{0}(\beta,\vec{y})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
.$ (115)

These two equations encode the superconformal symmetry breaking at finite
temperature. Equation (114) rules the behaviour of correlation functions on
which we act with the supersymmetry generators. In particular the breaking
term contains a correlation function in which the supercurrent appear. In
deriving those equations it is completely clear that the fact that
supersymmetry is broken at finite temperature is therefore due to the anti-
periodic boundary conditions given to fermions. For completeness, we also
write down the broken Ward identities for right chiral supersymmetry
generators

$\sum_{r=1}^{n}\left\langle\mathcal{O}_{1}(x_{1})\ldots\left[\bar{\bm{Q}},\mathcal{O}_{r}\right](x_{r})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=2\int
d^{d-1}y\
\left\langle\bar{G}^{0}(\beta,\vec{y})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
,$ (116)
$\sum_{r=1}^{n}\left\langle\mathcal{O}_{1}(x_{1})\ldots[{\bm{S}},\mathcal{O}_{r}](x_{r})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=\\\
=\sum_{r=1}^{n}(\beta-2\tau_{r})\sigma^{0}\int
d^{d-1}y\left\langle\bar{G}^{0}(\beta,\vec{y})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}+\\\
+\sum_{r=1}^{n}\sigma^{j}\int d^{d-1}y\
(y_{j}-x_{j})\left\langle\bar{G}^{0}(\beta,\vec{y})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}\
\ ,$ (117)

###  4.2 The fate of R-symmetry at finite temperature

The Superconformal Group includes the R-symmetry which can be understood as a
rotation of the supercharges. Even if supersymmetry is broken at finite
temperature it is not immediately clear what happens to the R-symmetry. To
address this question and demonstrate the power and generality of the
discussion in Section 2, let us explicitly compute the breaking term that
appears in the (broken) Ward identities. In order to do that, we need to
recall that the R-symmetry generators can be defined as

$R^{IJ}=\int_{\mathbb{R}^{d-1}}d^{d-1}x\ J^{0,IJ}(x)\ ,$ (118)

where $J^{\mu,IJ}$ is a bosonic current. Therefore we must consider the
breaking term defined by the equation (19) in the case of the R-symmetry
current. Since there is no additional coordinate dependence (as in the case of
the dilatation broken Ward identity, for example) and the current is bosonic
(so it is periodic over the thermal circle), we conclude that the breaking
term is simply zero, the Ward identity is unbroken and R-symmetry is
preserved.
This result is noteworthy because it shows that in a
supersymmetric/superconformal field theory at finite temperature, although the
supersymmetry/superconformal symmetry is broken, the R-symmetry remains intact
as a global symmetry acting on the operators of the theory. This has important
consequences, as it implies that every operator charged under the Abelian
$\mathfrak{u}(1)$ sub-algebra of the R-symmetry at zero temperature must have
a zero one-point function: in formula

$\left\langle\mathcal{O}\right\rangle_{\beta}=0\ ,\hskip
28.45274pt\text{charged}\ .$ (119)

For instance, in the case of the $\mathcal{N}=4$ SYM theory, this means that
in the stress-tensor multiplet there are only two operators that are allowed
to have a non-zero one-point function: the Lagrangian operator and the stress-
energy tensor. All the other operators are charged under R-symmetry and
therefore their thermal vacuum expectation values are zero.

Another way to see that the R-symmetry is preserved is to consider the real-
time formalism, where an $n$-point function at finite temperature can be seen
as a weighted sum of an infinite number of $(n+2)$-point functions at zero
temperature Kapusta:1989tk . All the zero-temperature $(n+2)$-point functions
are invariant under the R-symmetry. Hence, it is expected that the R-symmetry
is preserved even at finite temperature. This holds true for any global
symmetry of the theory generated by a bosonic charge.

The preservation of the R-symmetry can also be understood holographically by
considering the case of the duality between type IIB superstring on
$\text{AdS}_{5}\times S^{5}$ and $\mathcal{N}=4$ SYM in four dimensions. In
this case, the R-symmetry algebra $\mathfrak{so}(6)$ is realized as the
isometry algebra of $S^{5}$, and these isometries remain unaltered even when
the non-compact component of the ten-dimensional spacetime is given by a black
hole solution. This heuristically happens because the black hole solution does
not affect the geometry of $S_{5}$. Therefore, the R-symmetry is preserved in
the holographic dual description, which in the case of a black hole background
corresponds to a finite temperature CFT.

All the currents and the breaking terms corresponding to supersymmetry,
superconformal, and R-symmetry are summarized in Table 3.

Charge | Current: $\omega_{a}J^{\mu}_{a}(x)$ | Breaking term: $\Gamma_{a}^{\beta}(\beta,\vec{x})$
---|---|---
$\bm{Q}_{\alpha}^{I}$ | $\xi G^{\mu}(x)$ | $2\,G_{\alpha}^{\,0}(\beta,\vec{x})$
$\overline{\bm{Q}}_{\dot{\alpha},I}$ | $\xi\overline{G}^{\mu}(x)$ | $2\,\overline{G}_{\dot{\alpha}}^{\,0}(\beta,\vec{x})$
$\bm{S}^{\alpha}_{I}$ | $\xi\sigma^{\nu}x_{\nu}\overline{G}^{\mu}(x)$ | $\left[(\beta-2\tau)\left(\sigma^{0}\right)^{\alpha}+2(x_{i}-y_{i})\left(\sigma^{i}\right)^{\alpha}\right]\overline{G}^{\,0}(\beta,\vec{x})$
$\bm{\overline{S}}_{\dot{\alpha}}^{I}$ | $\xi\,\overline{\sigma}^{\nu}x_{\nu}G^{\mu}(x)$ | $\left[(\beta-2\tau)\left(\overline{\sigma}^{0}\right)_{\dot{\alpha}}+2(x_{i}-y_{i})\left(\overline{\sigma}^{i}\right)_{\dot{\alpha}}\right]G^{\,0}(\beta,\vec{x})$
$\bm{R}^{IJ}$ | $J_{R}^{\mu,IJ}(x)$ | $0$
Table 3: _Explicit currents and breaking terms for the symmetries of the
global Superconformal Group. The $(\tau,\vec{y})$ coordinates locates the
operator to which the breaking term is applied. To unclutter the notation,
contracted spinor and R-symmetry indices have been left implicit._

###  4.3 $\mathrm{U}(1)_{Y}$ Bonus symmetry at finite temperature and non-
renormalization in $\mathcal{N}=4$ SYM

In $\mathcal{N}=4$ SYM it is known that some two- and three-point functions
satisfy a bonus $\mathrm{U}(1)_{Y}$ symmetry Intriligator:1998ig ;
Intriligator:1999ff . This symmetry is not exact since only a subsector is
invariant and therefore it produces selection rules for $n$-point functions of
short operators171717The selection rule is also satisfied by the three-point
functions of two short and one long operators. only when $n=2,3$ in the planar
limit.
It has been conjectured and argued in Brigante:2005bq  that the
$\mathrm{U}(1)_{Y}$ bonus symmetry is preserved at finite temperature in the
planar limit. By following the procedure adopted in Section 2, we do not find
any obstacles to this claim. In particular, the $\mathrm{U}(1)_{Y}$ bonus is a
bosonic symmetry and it acts trivially on the spacetime coordinates. It is
indeed a good opportunity to remind that continuous symmetries can be broken
at a finite temperature only if the currents depend explicitly, i.e. not only
through the fields, on the space-time coordinates (e.g. the case of the
dilatation operator) or if the symmetry is generated by fermionic charges,
which are anti-periodic on the thermal circle $S^{1}_{\beta}$ (e.g.
supersymmetry). Nevertheless, the bonus $\mathrm{U}(1)_{Y}$ symmetry is rather
special because it is not an exact symmetry of the theory and therefore the
procedure in Section 2 cannot be applied. In particular the main obstacle is
that it is not a symmetry of the action. In order to understand if this
symmetry is preserved at finite temperature, it can be tested on the
correlation functions between two short operators $\mathcal{O}_{S}$. In
general, in the OPE between two short operators long operators
$\mathcal{O}_{L}$ can appear as well, therefore, in the OPE regime

$\left\langle\mathcal{O}_{S}(x)\mathcal{O}_{S}(0)\right\rangle_{\beta}=\ldots+\frac{f_{\mathcal{O}_{S}\mathcal{O}_{S}\mathcal{O}_{L}}b_{\mathcal{O}_{L}}}{2^{J}c_{\mathcal{O}_{L}}\beta^{\Delta_{\mathcal{O}_{L}}}}|x|^{\Delta_{\mathcal{O}_{L}}-2\Delta_{\mathcal{O}_{S}}}C_{J}^{(1)}\left(\frac{\tau}{|x|}\right)+\ldots\
,$ (120)

where we isolated the contribution of a long operator with a non-zero one-
point function. The zero-temperature non-renormalization theorem tells us that
the scaling dimensions $\Delta_{\mathcal{O}_{S}}$ are protected quantities on
the conformal manifold. In the large $N$ limit, the coefficients
$f_{\mathcal{O}_{S}\mathcal{O}_{S}\mathcal{O}_{L}}$ are also protected (as a
consequence of the bonus symmetry). If the non-renormalization theorem hold at
finite temperature, the equation (120) would imply that either the scaling
dimensions $\Delta_{\mathcal{O}_{L}}$ are protected (which is not true in
general, since they are not protected at zero temperature), or
$b_{\mathcal{O}_{L}}=0$ for every long operator in the OPE between the two
short operators $\mathcal{O}_{S}$. The latter condition is true in the strict
large $N$ limit for the case of gauge invariant operators Brigante:2005bq ;
Furuuchi:2005qm . However, as already noticed in Brigante:2005bq , this is a
large $N$ statement that does not hold in the finite $N$ case. Since it is not
guaranteed that $b_{\mathcal{O}_{L}}=0$ at finite $N$, the non-renormalization
of the thermal two-point function
$\left\langle\mathcal{O}_{S}(x)\mathcal{O}_{S}(0)\right\rangle_{\beta}$ does
not hold in general.
On the other hand the $\mathrm{U}(1)_{Y}$ bonus symmetry is an approximate
symmetry at large $N$. This suggests that the $\mathrm{U}(1)_{Y}$ bonus
symmetry is indeed preserved at finite temperature (in the same approximation
and in the same limit of the zero temperature case) because all the one-point
functions that could spontaneously break the $\mathrm{U}(1)_{Y}$ are equal to
zero at large $N$.

##  5 (Super)Conformal Field theories on $T^{2}\times\mathbb{R}^{d-2}$

Gen. | $\Gamma_{a}^{0}$ | $\Gamma_{a}^{1}$
---|---|---
$\bm{P}_{\mu}$ | $0$ | $0$
$\bm{D}$ | $\beta T^{00}$ | $\beta^{\prime}T^{11}$
$\bm{L}_{ij}$ | $0$ | $0$
$\bm{L}_{0i}$ | $\beta T^{0i}$ | $0$
$\bm{L}_{1i}$ | $0$ | $\beta^{\prime}T^{1i}$
$\bm{L}_{01}$ | $\beta T^{01}$ | $\beta^{\prime}T^{01}$
$\bm{K}_{0}$ | $\beta\left[(\beta-2\tau_{0})\,T^{00}+2\tau_{1}T^{01}+2y_{i}T^{0i}\right]$ | $\beta^{\prime}\left[(\beta^{\prime}-2\tau_{1})\,T^{01}+2\tau_{0}T^{01}+2y_{i}T^{0i}\right]$
$\bm{K}_{1}$ | $\beta\left[(\beta-2\tau_{0})\,T^{10}+2\tau_{1}T^{11}+2y_{i}T^{1i}\right]$ | $\beta^{\prime}\left[(\beta^{\prime}-2\tau_{1})\,T^{11}+2\tau_{1}T^{11}+2y_{i}T^{1i}\right]$
$\bm{K}_{i}$ | $\beta\left[(\beta+2\tau_{0})\,T^{0j}+2\tau_{1}T^{1j}+2y_{i}T^{ij}\right]$ | $\beta^{\prime}\left[(\beta^{\prime}+2\tau_{1})\,T^{1j}+2\tau_{0}T^{0j}+2y_{i}T^{ij}\right]$
$\bm{Q}_{\alpha}$ | $2\ G^{0}$ | $2\ G^{1}$
$\overline{\bm{Q}}_{\alpha}$ | $2\ \overline{G}^{0}$ | $2\ \overline{G}^{1}$
$\overline{\bm{S}}^{\dot{\alpha}}$ | $\left[(\beta-2\tau_{0})\left(\overline{\sigma}^{0}\right)+2\tau_{1}\left(\overline{\sigma}^{0}\right)+2y_{i}\left(\overline{\sigma}^{i}\right)\right]G^{0}$ | $\left[(\beta^{\prime}-2\tau_{1})\left(\overline{\sigma}^{1}\right)+2\tau_{0}\left(\overline{\sigma}^{0}\right)+2y_{i}\left(\overline{\sigma}^{i}\right)\right]G^{1}$
$\overline{\bm{S}}^{\dot{\alpha}}$ | $\left[(\beta-2\tau_{0})\left(\overline{\sigma}^{0}\right)+2\tau_{1}\left(\overline{\sigma}^{0}\right)+2y_{i}\left(\overline{\sigma}^{i}\right)\right]G^{0}$ | $\left[(\beta^{\prime}-2\tau_{1})\left(\overline{\sigma}^{1}\right)+2\tau_{0}\left(\overline{\sigma}^{0}\right)+2y_{i}\left(\overline{\sigma}^{i}\right)\right]G^{1}$
$\bm{R}$ | 0 | 0
Table 4: _Explicit breaking terms for the symmetries of the (Super) conformal
Group on $T^{2}\times\mathbb{R}^{d-2}$. Spinor and R-symmetry indices are
suppressed for simplicity. The operators are computed in
$(\beta,\tau_{1},\vec{x}),(\tau_{0},\beta^{\prime},\vec{x})$ depending if they
appear in the $\Gamma_{a}^{0},\Gamma_{a}^{1}$ column. We indicate with
$\vec{y}$ the $\mathbb{R}^{d-2}$ distance between the insertion of the
operator to which the current is applied and
$(\tau_{0}/\beta,\beta^{\prime}/\tau_{1},\vec{x})$ indicates the position of
the operator. Modular invariance is encoded in the “symmetry” between the
$\Gamma_{a}^{0}$ and the $\Gamma_{a}^{1}$ columns. _

We discuss here the breaking of the (Super) conformal Group on the manifold
$T^{2}\times\mathbb{R}^{d-2}$. We derive all the conformal and superconformal
Ward identities for this geometry and we briefly discuss the immediate
consequences. The procedure we adopt here is exactly the same of Section 2,
showing the generality of this procedure; for this reason, we skip some
details, already discussed for the finite temperature case. The starting point
is that on the geometry
$\mathcal{M}_{\beta,\beta^{\prime}}=T^{2}(\beta,\beta^{\prime})\times\mathbb{R}^{d-2}=S_{\beta}^{1}\times
S_{\beta^{\prime}}^{1}\times\mathbb{R}^{d-2}$, the broken Ward identities take
the form

$i\omega_{a}\bm{G}_{a}\left\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}=\left\langle\delta
S\
\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}\
.$ (121)

The variation of the action is the integration of the derivative of the
current over the whole manifold and we have therefore to consider

$\begin{split}&\int_{T^{2}\times\mathbb{R}^{d-2}}d^{d}x\
\partial_{\mu}\left\langle
J_{a}^{\mu}(x)\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}=\\\
&\ \ \ \ \ =\lim_{R\to\infty}\int_{T^{2}\times
S_{1}^{d-3}}d\Omega\,d\tau_{0}\,d\tau_{1}\ R^{d-3}\left\langle
J_{a}^{i}(\Omega,R,\tau_{0},\tau_{1})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}+\\\
&\ \ \ \ \ \ \ \ \ \ \
+\int_{\mathbb{R}^{d-2}}d^{d-2}x\int_{0}^{\beta^{\prime}}d\tau_{1}\
\left\langle\Gamma_{a}^{0}(\tau_{1},\vec{x})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}+\\\
&\ \ \ \ \ \ \ \ \ \ \
+\int_{\mathbb{R}^{d-2}}d^{d-2}x\int_{0}^{\beta}d\tau_{0}\
\left\langle\Gamma_{a}^{1}(\tau_{0},\vec{x})\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}\
,\end{split}$ (122)

where the first boundary term vanishes (or leads to a renormalized infrared
divergence) because of clustering and the second term depends on the breaking
terms

$\Gamma_{a}^{0}(\tau_{1},\vec{x})=J^{0}(\beta,\tau_{1},\vec{x})-J^{0}(0,\tau_{1},\vec{x})\
,$ (123)
$\Gamma_{a}^{1}(\tau_{0},\vec{x})=J^{1}(\tau_{0},\beta^{\prime},\vec{x})-J^{1}(\tau_{0},0,\vec{x})\
.$ (124)

Similarly to the finite temperature case, these quantities can be explicitly
computed and the results for the (Super) conformal Group are provided in Table
4. Also in the case of the torus, it is clear that translations, a subset of
the rotations, R-symmetry, and any global symmetry are preserved. All the
results appearing in the table are in agreement with the expectations
Luo:2022tqy . In particular, modular invariance is explicit just by looking at
Table 4: when $\beta$ and $\beta^{\prime}$ are exchanged, the time direction
(chosen to be $x^{0}$) and the direction $x^{1}$ can be interchanged as well
to preserve the set of (broken) Ward identities.

As in Subsection 3.1, we can write down Hamiltonian and momentum operators.
However, in this case an interesting physical interpretation is missing.
Nonetheless it is possible to write the implicit version of the Cardy formula
as in the thermal case (cfr. equation (41)):

$\left\langle
T^{00}\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}=\frac{\beta}{d}\int
d^{d-2}x\ \int_{0}^{\beta^{\prime}}d\tau_{1}\ \left\langle
T^{00}(0,\tau_{1},\vec{x})T^{00}(0)\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}+\\\
+\frac{\beta^{\prime}}{d}\int d^{d-2}x\ \int_{0}^{\beta}d\tau_{0}\
\left\langle
T^{11}(\tau_{0},0,\vec{x})T^{00}(0)\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}\
.$ (125)

The above equation corresponds to the dilatation broken Ward identity and it
relates the free energy of the theory to the two-point function of the stress-
energy tensor. It can also be generalized for any operator $\mathcal{O}$

$\Delta_{\mathcal{O}}\left\langle\mathcal{O}\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}=\beta\int
d^{d-2}x\ \int_{0}^{\beta^{\prime}}d\tau_{1}\ \left\langle
T^{00}(0,\tau_{1},\vec{x})\mathcal{O}(0)\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}+\\\
+\beta^{\prime}\int d^{d-2}x\ \int_{0}^{\beta}d\tau_{0}\ \left\langle
T^{11}(\tau_{0},0,\vec{x})\mathcal{O}(0)\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}\
.$ (126)

An interesting application of those equations concerns the deconstructions of
six-dimensional theories (in particular the $\mathcal{N}=(2,0)$ theory)
Arkani-Hamed:2001wsh ; Hayling:2017cva ; Niarchos:2019onf ; Niarchos:2020nxk ;
Bourton:2020rfo ; Giedt:2003xr ; Kim:2018lfo . Furthermore, one can observe
that the dimensional analysis argument posed in Appendix A can be easily
adapted to the case of $\mathcal{M}=T^{2}\times\mathbb{R}^{d-2}$. In
particular the radii of both circles have to be taken into consideration and
the dilatation broken Ward identity can be written as

$\left(\bm{D}+\beta\frac{\partial}{\partial\beta}+\beta^{\prime}\frac{\partial}{\partial\beta^{\prime}}\right)\left\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\mathcal{M}_{\beta,\beta^{\prime}}}=0\
.$ (127)

##  6 Conclusions and future directions

In this paper, the general procedure for computing constraints from broken
symmetries on non-trivial manifolds is discussed, with a special focus on CFTs
on the thermal manifold
$\mathcal{M}_{\beta}=S_{\beta}^{1}\times\mathbb{R}^{d-1}$. The explicit
relations between correlation functions are systematically derived,
emphasizing that broken Ward identities typically relate an $n$-point function
to an $(n+1)$-point function, see e.g. equations (25), (27), (44).This work is
only the beginning of the exploration of the consequences of these novel
equations.

The implications of these equations in the OPE regime of are explored, where
they correctly reproduce known results established in previous works
Iliesiu:2018fao ; Benjamin:2023qsc . This demonstrates the consistency of the
derived broken Ward identities with existing knowledge in the field.
Furthermore, the paper highlights the precise connections between the broken
Ward identities of the Conformal Group and the Hamiltonian (energy spectrum)
and momentum at finite temperature, as well as the dilatation and boost
operators at zero temperature. These relations provide novel insights into the
behavior of the energy and the mass spectrum under the combined effects of
temperature and conformal transformations.

Using broken Ward identities we provide a clean derivation of an implicit
version of the Cardy formula in higher dimensions, expressed in equation (41).
We refer to this equation as implicit because it relates the free energy (or
equivalently the one-point function of the stress-energy tensor) with an
integrated two-point function of the stress-energy tensor. This equation has
already appeared in El-Showk:2011yvt  and was derived using the observation
that changing the temperature is equivalent to varying the zero-zero component
of the metric. Our derivation is a direct consequence of the breaking of
dilatation invariance and it comes elegantly as a specific application of
broken Ward identities. Moreover, broken Ward identities automatically allow
for the generalization of such a formula in the case of higher point functions
and different operators (not necessarily the stress-energy tensor).
Furthermore in (125) we provide a similar but novel implicit Cardy formula for
the geometry $\mathcal{M}=T^{2}\times\mathbb{R}^{d-2}$. This equation,
together with (41), provides further constraints on any local CFT at finite
temperature that go beyond the conformal bootstrap conditions El-Showk:2011yvt
. Last week, a completely different approach, based on the thermal effective
action, appeared in Benjamin:2023qsc  (see in particular equation (A.3)). In
the latter case an explicit version of the Cardy formula for large dimension
operators was derived.

In this paper we also extend our analysis to superconformal field theories at
finite temperature, demonstrating that even though superconformal symmetry is
explicitly broken, the R-symmetry is preserved in any CFT at finite
temperature. The special nature of the $\mathrm{U}(1)_{Y}$ bonus symmetry of
$\mathcal{N}=4$ SYM theory at zero temperature requires a different
discussion. Nonetheless we conclude that this symmetry is also preserved at
finite temperature and in the large $N$ limit, as already anticipated in
Brigante:2005bq .

In addition our methods can easily be generalized to other manifolds. We
explicitly discuss the case of a (super)conformal field theory on the manifold
$\mathcal{M}=T^{2}\times\mathbb{R}^{d-1}$. The latter manifold has many
interesting physical implications and it already appears in different contexts
in literature (see for example Luo:2022tqy ; Benjamin:2023qsc ). Furthermore,
the torus compactification is relevant for deconstructing six-dimensional to
four-dimensional superconformal field theories Arkani-Hamed:2001wsh ;
Hayling:2017cva ; Niarchos:2019onf ; Niarchos:2020nxk ; Bourton:2020rfo ;
Giedt:2003xr ; Kim:2018lfo .

The question regarding the uniqueness of the solution of the bootstrap problem
for thermal two-point functions is indeed an important one, and it has been
previously discussed in the literature, including Alday:2020eua . The
equations derived from Broken Ward identities in this paper provide additional
non-trivial constraints for both zero and finite temperature CFTs, which go
beyond the standard bootstrap constraints. They enrich the bootstrap problem
and open up new avenues for studying and understanding thermal CFTs.

In Appendix C, we present a detailed analysis of the exact computation of the
two-point function of a free scalar theory in four dimensions. This is done by
imposing only KMS invariance, the correct behavior in the Regge limit, and
using the OPE expansion with the input of operator dimensions. While this
method yields analytical results for the free scalar theory, difficulties
arise when attempting to apply it to more complicated cases where the OPE
expansion cannot be trivially factored into holomorphic functions. However, it
is important to note that the inability to factorize the OPE expansion in
holomorphic functions does not necessarily imply that the two-point functions
of more interesting theories cannot be computed using a similar approach. It
is indeed possible that the problem turns out to be just of technical nature.
The analysis in Appendix C serves as a starting point and highlights the
challenges that may arise in more general cases. Further investigations and
developments are needed to explore the possibilities of applying this method
to compute two-point functions in other theories of interest.

In yet a different strand of research it would be interesting to combine
broken Ward identities with different (other than the OPE) expansions. For
example using the Källén – Lehmann representation which holds for general
QFTs, we may be able to perturbe away from the vertical dashed red line in
Fig. 1.

Assuming the eigenstate thermalization hypothesis (ETH) a finite temperature
two-point function $\langle\phi(x)\phi(0)\rangle_{\beta}$ can be written as a
four-point function at zero temperature
$\langle\mathcal{O}_{\Delta}(\infty)\phi(x)\phi(0)\mathcal{O}_{\Delta}\rangle$,
in the limit $x\to 0$, $\Delta\to\infty$, where $|x|\Delta$ is fixed
Delacretaz:2022ojg ; Delacretaz:2021ufg ; Karlsson:2021duj ; Karlsson:2019qfi
; Fukushima:2023swb . Inserting such an equation in the broken Ward
identities, one-point functions of scalars can be computed as integrated four-
point functions at finite temperature.

Finally, it is also possible to use broken Ward identities to derive exact
constraints for non-local BPS observables such as Wilson loops in
supersymmetric theories generalizing known zero-temperature results. We are
currently pursuing this direction of research.

###### Acknowledgements.

It is a pleasure to thank António Antunes, Davide Cassani, Alejandra Castro,
Pietro Ferrero, Stefano Giusto, Apratim Kaviraj and Vasilis Niarchos for very
useful discussions at different stages of the work. AM and EP have benefited
from the German Research Foundation DFG under Germany’s Excellence Strategy –
EXC 2121 Quantum Universe – 390833306.

##  Appendix A A comment on the dilatation broken Ward identity

The dilatation broken Ward identity plays a crucial role in this paper as it
provides valuable insights into the theory and its embedding in theory space.
While dilatation invariance is exact at critical points, such as in zero-
temperature CFT, the violation of dilatation invariance carries important
physical information about the theory.
By considering the thermal manifold $\mathcal{M}_{\beta}$ and the broken
dilatation symmetry, we showed in Section 2 that the breaking term associated
with the dilatation operator can be explicitly computed. This breaking term
captures the thermal effects on the dilatation symmetry and provides a
correction that accounts for the presence of temperature.
We present here an alternative derivation of the dilatation broken Ward
identities, shedding light on the interplay between dilatation invariance,
thermal effects, and the underlying dynamics of the system. Let us start from
the simple dimensional analysis of a generic thermal correlator

$\left\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=\frac{1}{\beta^{a_{0}}}\frac{1}{x_{1}^{a_{1}}}\ldots\frac{1}{x_{n}^{a_{n}}}f\left(\frac{x_{1}}{\beta},\ldots,\frac{x_{n}}{\beta}\right)\
,$ (128)

where $f$ is an unknown function. Dimensional analysis fixes

$\sum_{i=0}^{n}a_{i}=\sum_{j=1}^{n}\Delta_{j}\ .$ (129)

The above expression for the correlator can be further simplified by using
$SO(d-1)$ invariance: however, this is not necessary for our purpose. We can
simply observe that the correlation function is invariant under the
transformations

$\mathcal{O}_{i}\to\lambda^{-\Delta_{i}}\mathcal{O}_{i}\ ,\hskip
28.45274ptx_{i}\to\lambda_{i}x_{i}\ ,\hskip
28.45274pt\beta\to\lambda_{i}\beta\ .$ (130)

The first two transformations are simply dilatations, whereas the last one is
a non-physical scaling of the temperature181818This transformation is not
physical because the temperature defines the theory in the theory space
parameterized by the couplings and the temperature (cfr. Fig. 1)..
Nevertheless, the fact that the dilatation invariance is compensated by a
scaling of the temperature implies that

$\left(\bm{D}+\beta\frac{\partial}{\partial\beta}\right)\left\langle\mathcal{O}_{1}(x_{1})\ldots\mathcal{O}_{n}(x_{n})\right\rangle_{\beta}=0\
.$ (131)

Observe that the dilatation breaking term in Table 2 can be derived from this
equation. To see this, we can consider that a scaling of the temperature is
equivalent to a scaling of the metric component $g_{00}$, which is “dual” to
$T^{00}$ in the sense that the stress-energy tensor is the source of $g_{00}$
Papadodimas2010 . Let us consider the case of a one-point function for
simplicity: then

$\frac{\partial}{\partial\beta}\langle\mathcal{O}\rangle_{\beta}=-\frac{1}{\beta}\int
d^{d-1}x\int_{0}^{\beta}d\tau\ \langle
T^{00}(\tau,\vec{x})\mathcal{O}(0)\rangle_{\beta}=-\int d^{d-1}x\ \langle
T^{00}(0,\vec{x})\mathcal{O}(0)\rangle_{\beta}\ ,$ (132)

where the last step is justified by the fact that the stress-energy tensor
component $T^{00}$ generates the Hamiltonian (and therefore it commutes with
it). Finally

$\Delta_{\mathcal{O}}\langle\mathcal{O}\rangle_{\beta}=\beta\int d^{d-1}x\
\langle T^{00}(0,\vec{x})\mathcal{O}(0)\rangle_{\beta}\ .$ (133)

##  Appendix B Explicit checks of broken Ward identities

###  B.1 Consistency with the identity operator $\mathds{1}$

The simplest consistency condition for a broken Ward identity is its good
behavior when a single copy of the identity operator $\mathds{1}$ is plugged
into the equation. Let us consider the dilatation broken Ward identity. Due to
the properties of the identity operator, the left-hand side of the dilatation
broken Ward identity trivially identical to zero. Hence, the broken Ward
identity reduces to

$\beta\int d^{d-1}y\Braket{\Gamma^{\beta}_{\text{dil.}}(\vec{y})}_{\beta}=0\
.$ (134)

This specific case highlights the importance of the renormalization procedure,
in particular, the renormalization of the IR divergence (18)

$\beta\int^{\text{ren.}}d^{d-1}y\Braket{\Gamma^{\beta}_{\text{dil.}}(\vec{y})}_{\beta}=\beta\int
d^{d-1}y\Braket{\Gamma^{\beta}_{\text{dil.}}}_{\beta}-\Braket{\Gamma^{\beta}_{\text{dil.}}}_{\beta}\times\text{Vol}\left[\mathcal{M}_{\beta}\right]\
,$ (135)

where we highlighted the difference between a renormalized integral (the one
commonly used in the formulas of this work) and a non-renormalized one.
Recalling that any thermal one-point function is constant, the one-point
function of $\Gamma^{\beta}_{a}$ can be extracted from the first integral,
making the whole expression identically equal to zero and proving the
consistency condition (134).

###  B.2 Two dimensions and broken Ward identities

Another sanity check is given by the fact that in two dimensions the
dilatation broken Ward identity gives the correct expression of the one-point
function of the stress-energy tensor. The broken Ward identities derived in
this work cannot be directly checked, since in order to say something about an
$n$-point thermal function, the information required includes a $(n+1)$-point
thermal function. In higher dimensions, the number of points of the
correlation functions has been reduced by studying the OPE regime. However, in
two dimensions the equations can be checked straightforwardly since the
thermal correlation functions are known. This comes from the equivalence
between the two-dimensional cylinder and the thermal manifold, and the
existence of a conformal map between the plane $\mathbb{R}^{2}\sim\mathbb{C}$,
where all the functions are known, and the cylinder $\mathbb{R}\times
S_{\beta}^{1}$. We recall that the conformal map is

$\mathbb{C}\ni\omega=\sigma+i\tau=\frac{\beta}{\pi}\ln z\ ,$ (136)

where we are using the convention $0\leq\tau\leq\beta$ and
$-\infty<\sigma<\infty$.

##### Dilatation broken Ward identity

Let us test, as an example, the one-point function of the stress-energy
tensor. The dilatation broken Ward identity reads

$\Delta_{T}\left\langle T^{00}\right\rangle_{\beta}=\beta\int dy\ \left\langle
T^{00}(0,y)T^{00}(0)\right\rangle_{\beta}\ .$ (137)

The stress-energy tensor on the complex plane can be written as $2\pi
T^{00}(z,\overline{z})=T(z)+\overline{T}(\overline{z})$, and it is known that

$\left\langle T(z)T(0)\right\rangle=\frac{c/2}{z^{4}}\ ,\hskip
28.45274pt\left\langle\overline{T}(\overline{z})\overline{T}(0)\right\rangle=\frac{c/2}{\overline{z}^{4}}\
.$ (138)

The application of the conformal map returns the two-point function on the
cylinder Datta:2019jeo ; DiFrancesco:1997nk ; Mussardo:2020rxh 191919Here we
are using the usual convention for two-dimensional CFT: in this convention
$b_{T}>0$ for unitary theories.

$\left\langle
T^{00}(\omega,\overline{\omega})T^{00}(0,0)\right\rangle_{\beta}=\left(\frac{\pi}{\beta}\right)^{4}\frac{c^{2}}{144\pi^{2}}+\frac{c}{8\pi^{2}}\left(\frac{\pi}{\beta}\right)^{4}\left[\text{csch}^{4}\left(\frac{\pi\omega}{\beta}\right)+\text{csch}^{4}\left(\frac{\pi\overline{\omega}}{\beta}\right)\right]\
$ (139)

which must be interpreted as the thermal two-point function to be inserted in
the integral (137). Up to IR divergences renormalization, the integration
returns

$\left\langle T^{00}\right\rangle_{\beta}=\frac{\pi}{3\beta^{2}}c\ ,$ (140)

which is correct in a free bosonic theory, i.e. when $c=1$ Iliesiu:2018fao ,
but it is also the starting point to compute the Cardy formula in two
dimensions Mukhametzhanov:2019pzy .
Further observe that for a one-point function, it is trivial to check that the
breaking term of the dilatation is given by $-\beta\partial_{\beta}$. For what
is explained in Appendix A this is already an explicit check of the broken
Ward identities.

##### Rotation broken Ward identity

The next check concerns the rotation broken Ward identity. In this case, we
have that

$-2\left\langle
T^{00}(0)\right\rangle_{\beta}=i\left\langle\bm{S}_{01}T^{01}(0)\right\rangle_{\beta}=\beta\int
dy\ \langle T^{01}(0,y)T^{01}(0)\rangle_{\beta}\ .$ (141)

All these correlators can be explicitly computed by using equation (138).
Further observe that

$\langle T^{01}(x)T^{01}(0)\rangle_{\beta}=-\langle
T^{00}(x)T^{00}(0)\rangle_{\beta}\ ,$ (142)

and therefore the rotation broken Ward identity is satisfied with the same
calculations we performed for the dilatation broken Ward identities. The
rotation broken Ward identity can be reconducted to a dilatation broken Ward
identity also in the case of the two-point function. This is not a generic
statement, but it is a consequence of the chirality of the correlation
functions (including the stress-tensor) in two dimensions. We do not expect
this to be true also in higher dimensions.

##### Special conformal transformations broken Ward identities:

The last check we have to perform regards the special conformal
transformations. In this case, we have to check that

$2\tau\Delta_{T}\left\langle
T^{00}(\tau,x)\right\rangle_{\beta}=(\beta-2\tau)\beta\int dy\ \left\langle
T^{00}(y)T^{00}(0)\right\rangle_{\beta}=\Delta_{T}(\beta-2\tau)\left\langle
T^{00}(\tau,x)\right\rangle_{\beta}\ ,$ (143)

which corresponds to the broken Ward identity for $\bm{K}_{0}$; the broken
Ward identity for $\bm{K}_{1}$ follows naturally. In this case we did not have
to compute anything explicitly, but we just applied the dilatation broken Ward
identity. In order to see that the above equation is correct we have to
consider that the one-point functions are invariant under time translations,
therefore it is possible to compute them in $2\tau$, so that, calling
$\tilde{\tau}=2\tau$ we can use the equivalence $\tau\equiv\tau+\beta$ to
explicitly see that the broken Ward identity is just an identity202020Observe
that the periodicity of the time coordinates is a feature of the time
coordinate of local operators in correlation functions. One may be tempted to
conclude that every $\beta$ factor in the broken Ward identities is equivalent
to zero, producing in this way some inconsistency; this is clearly not correct
since $\beta$ is a fixed scale of the theory..
When tested on the higher point functions, the simplest possible example is
the case of the two-point function of scalar primaries of weights $(h,h)$,
such that $2h=\Delta$; these two-point functions read

$\left\langle\phi(\tau,\sigma)\phi(0,0)\right\rangle_{\beta}=\left(\frac{\pi}{\beta}\right)^{\Delta}\operatorname{csch}^{\Delta}\left(\frac{\pi}{\beta}(\tau+i\sigma)\right)\operatorname{csch}^{\Delta}\left(\frac{\pi}{\beta}(\tau-i\sigma)\right)\
.$ (144)

Straightforward computations show that the breaking term of the Ward identity
corresponding to $\bm{K}_{0}$ is given by

$\beta\tau\int d\tilde{\sigma}\ \left\langle
T^{00}(0,\tilde{\sigma})\phi(\tau,\sigma)\phi(0,0)\right\rangle_{\beta}+\beta\sigma\int
d\tilde{\sigma}\ \left\langle
T^{01}(0,\tilde{\sigma})\phi(\tau,\sigma)\phi(0,0)\right\rangle_{\beta}\ ,$
(145)

whereas the breaking term corresponding to $\bm{K}_{1}$ is given by

$\beta\sigma\int d\tilde{\sigma}\ \left\langle
T^{00}(0,\tilde{\sigma})\phi(\tau,\sigma)\phi(0,0)\right\rangle_{\beta}-\beta\tau\int
d\tilde{\sigma}\ \left\langle
T^{01}(0,\tilde{\sigma})\phi(\tau,\sigma)\phi(0,0)\right\rangle_{\beta}\ .$
(146)

These breaking terms can be written as differential operators acting on the
two-point function in equation (144). However, in this way the broken Ward
identities simply trivialize, showing that dilatations, rotations, and
momentum (broken) Ward identities trivialize the special conformal
transformation broken Ward identity.
Similar checks can be also performed for the two-point function of the stress-
energy tensor (138). All the results are in agreement with our expectations.

##### Supersymmetry and superconformal broken Ward identities

In order to test the supersymmetry broken Ward identities we can use the OPE
212121We focus on the analytic sector, but analogous equations can be written
for the anti-analytic sector.

$G(z_{1})G(z_{2})=\frac{2c}{3(z_{1}-z_{2})^{3}}+\frac{2}{z_{1}-z_{2}}T(z)+\ldots$
(147)

to conclude that the superconformal descendent of the supercurrent is given by
the stress-energy tensor $T$. This is clear also from the super-Virasoro
algebra anti-commutator

$\\{G_{n},G_{m}\\}=2L_{n+m}+\frac{c}{3}\left(n^{2}-\frac{1}{4}\right)\delta_{n+m,0}\
,$ (148)

remembering that

$L_{n}=\oint\frac{dz}{2\pi i}z^{n+1}T(z)\ ,\hskip
28.45274ptG_{m}=\oint\frac{dz}{2\pi i}z^{m+1/2}G(z)\ .$ (149)

We have therefore to check that

$\left\langle T^{00}\right\rangle_{\beta}=4\int d\sigma\ \left\langle
G^{0}(\beta,\sigma)G^{0}(0,0)\right\rangle_{\beta}\ .$ (150)

It is easy to check that the above holds in any two-dimensional superconformal
theory; in particular, the coefficient of the one-point function of the
stress-energy tensor can be computed from the two-point function of the
supercurrent and it is given by equation (140).

##  Appendix C Free scalar theory in $d=4$

We are going to test the broken Ward identities in the free scalar theory in
four dimensions. In particular we will explicitly solve the bootstrap problem
posed in Subsection 3.3.2 for the two-point function of two fundamental scalar
fields $\varphi(x)$. We will also consider the bootstrapped two-point function
to extract several one-point functions, as it was done in Petkou:2021zhg ;
Iliesiu:2018fao . Focusing on the one-point function of the stress-energy
tensor $T^{\mu\nu}$, we will show that the relation (70) has a clear physical
interpretation, providing a strong non-trivial check for the broken Ward
identities.

##### Computation of $\Braket{\varphi(x)\varphi(0)}_{\beta}$

In Section 3.4 we computed the explicit expression of the two-point functions
in the OPE regime directly from the dilatations broken Ward identity

$f_{\beta}(\tau,r)=\sum_{\mathcal{O}\in\varphi\times\varphi}\frac{J!}{2^{J}(\nu)_{J}}\frac{1}{\beta^{\Delta_{\mathcal{O}}}}\frac{f_{\varphi\varphi\mathcal{O}}b_{\mathcal{O}}}{c_{\mathcal{O}}}\left(\tau^{2}+r^{2}\right)^{\frac{1}{2}\left(\Delta_{\mathcal{O}}-2\Delta_{\varphi}\right)}C_{J}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)\
.$ (151)

Specializing the result for a scalar free theory, we can use the explicit OPE
expansion

$\varphi(x)\times\varphi(0)\sim
1+a\varphi^{2}(0)+\underbrace{b_{\mu}\varphi(0)\partial^{\mu}\varphi(0)}_{\sim
V^{\mu}(0)}+\underbrace{c_{\mu\nu}\varphi(0)\partial^{\mu}\partial^{\nu}\varphi(0)}_{\sim
T^{\mu\nu}(0)}+\dots\ ,$ (152)

where the specific structure of the operators is not important for our
purpose. The crucial information we can extract from the OPE is a formula for
the conformal dimension of the operator $\mathcal{O}\in\varphi\times\varphi$
(excluding the identity operator $\mathds{1}$) as a function of its spin $J$.
Recalling that $\Delta_{\varphi}=\frac{d-2}{2}$,

$\Delta_{\mathcal{O}}=d-2+J\ ,$ (153)

Using this formula, we can turn the sum over the operators in the equation
(152) into a sum over the spins $J$

$f_{\beta}(\tau,r)=\frac{1}{\left(\tau^{2}+r^{2}\right)^{\frac{d-2}{2}}}+\sum_{J=0}^{\infty}k_{J}\frac{1}{\beta^{d-2+J}}\frac{J!}{2^{J}(\nu)_{J}}\left(\tau^{2}+r^{2}\right)^{\frac{J}{2}}C_{J}^{(\nu)}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right)\
,$ (154)

where we grouped all the dynamic information in the coefficients

$k_{J}=\frac{f_{\varphi\varphi\mathcal{O}}b_{\mathcal{O}}}{c_{\mathcal{O}}}\
.$ (155)

Since the equation (154) is missing manifest periodicity over the thermal
circle, it should be established by constraining the coefficients $k_{J}$.
Since the coefficients $k_{J}$ are functions of the $b_{\mathcal{O}}$ one-
point function data, imposing manifest periodicity to the two-point scalar
function will provide information about the one-point functions of the scalar
free theory at the same time Iliesiu:2018fao .
From now on, we will set $d=4$. Thanks to this choice, the equation (154) now
reads

$f_{\beta}^{d=4}(\tau,r)=\frac{1}{\tau^{2}+r^{2}}+\sum_{J=0}^{\infty}\frac{1}{\beta^{2+J}}\frac{k_{J}}{2^{J}}\left(\tau^{2}+r^{2}\right)^{\frac{J}{2}}U_{J}\left(\frac{\tau}{\sqrt{\tau^{2}+r^{2}}}\right),$
(156)

where we used the identity

$C_{J}^{(1)}\left(x\right)=U_{J}\left(x\right),$ (157)

with $U_{J}(x)$ being the $J^{\text{th}}$ Chebyshev polynomial of the second
kind. These polynomials admit a closed formula

$U_{J}\left(x\right)=\frac{\left(x+\sqrt{x^{2}-1}\right)^{J+1}-\left(x-\sqrt{x^{2}-1}\right)^{J+1}}{2\sqrt{x^{2}-1}}$
(158)

which can be plugged into the $d=4$ solution

$f_{\beta}^{d=4}(\tau,r)=\frac{1}{\tau^{2}+r^{2}}+\frac{1}{2\beta^{2}r}\sum_{J=0}^{\infty}k_{J}\left(\frac{i}{2\beta}\right)^{J}\bigg{[}(r-i\tau)^{J+1}+(-1)^{J}(r+i\tau)^{J+1}\bigg{]}\
.$ (159)

The structure of the solution can be further improved by noticing that the
flat space term can be rewritten as

$\frac{1}{\tau^{2}+r^{2}}=\frac{1}{2\beta^{2}r}\left(-\frac{1}{4}\right)\left(\frac{i}{2\beta}\right)^{-2}\left[\left(r-i\tau\right)^{-1}+(-1)^{-2}\left(r+i\tau\right)^{-1}\right]$
(160)

and by adding the following factor, identically equal to 0

$0=\frac{1}{2\beta^{2}r}k_{-1}\left(\frac{i}{2\beta}\right)^{-1}\left[\left(r-i\tau\right)^{0}+(-1)^{-1}\left(r+i\tau\right)^{0}\right]\
.$ (161)

After having defined $k_{-2}=-1/4$, the solution reads

$f_{\beta}^{d=4}(\tau,r)=\frac{1}{2\beta^{2}r}\sum_{J=-2}^{\infty}k_{J}\left(\frac{i}{2\beta}\right)^{J}\bigg{[}(r-i\tau)^{J+1}+(-1)^{J}(r+i\tau)^{J+1}\bigg{]}\
.$ (162)

The expression can be reshaped by relabeling $J=\ell-2$ and imposing the
parity $\tau\to-\tau$ to leave the two-point function invariant, forcing the
sum to run only over even integers $\ell=2n$: this can be achieved by setting
$k_{\text{odd}}=0$

$f_{\beta}^{d=4}(\tau,r)=\frac{\pi}{2\beta
r}\sum_{n=0}^{\infty}\frac{k_{2n-2}}{\pi^{2n}}\left(-\frac{1}{4}\right)^{n-1}\left\\{\left[\frac{\pi}{\beta}(r-i\tau)\right]^{2n-1}+\left[\frac{\pi}{\beta}(r+i\tau)\right]^{2n-1}\right\\}\
.$ (163)

By introducing the dimensionless complex variables

$w=\frac{\pi}{\beta}(r-i\tau)\ ,\qquad\overline{w}=\frac{\pi}{\beta}(r+i\tau)\
,$ (164)

we conclude that the two-point function can be reduced to