# Type-B Anomaly Matching and the 6D (2,0) Theory

Vasilis Niarchos a,♣,111On leave of absence from the Department of
Mathematical Sciences and Centre for Particle Theory, Durham University,
Durham DH1 3LE, UK, Constantinos Papageorgakis b,♢ and Elli Pomoni c,♠

###### Abstract

We study type-B conformal anomalies associated with $\frac{1}{2}$-BPS Coulomb-
branch operators in 4D $\mathcal{N}=2$ superconformal field theories. When the
vacuum preserves the conformal symmetry these anomalies coincide with the two-
point function coefficients in the Coulomb-branch chiral ring. They are non-
trivial functions of exactly-marginal couplings that can be determined from
the $S^{4}$ partition function. In this paper, we examine the fate of these
anomalies in vacua of the Higgs-branch moduli space, where conformal symmetry
is spontaneously broken. We argue non-perturbatively that these anomalies are
covariantly constant on conformal manifolds. In some cases, this can be used
to show that they match in the broken and unbroken phases. Thus, we uncover a
new class of data on the Higgs branch of 4D $\mathcal{N}=2$ conformal field
theories that are exactly computable. An interesting application of this
matching occurs in $\mathcal{N}=2$ circular quivers that deconstruct the 6D
(2,0) theory on a torus. In that context, we argue that 4D supersymmetric
localisation can be used to calculate non-trivial data involving
$\frac{1}{2}$-BPS operators of the 6D theory as exact functions of the complex
structure of the torus.

Contents

toc

##  1 Introduction and Summary of Results

In even spacetime dimensions a conformal field theory (CFT) can exhibit two
types of conformal anomalies, type-A and type-B [1]. The type-A conformal
anomalies are a close analogue of the chiral anomaly—since they do not
introduce a mass scale, they can be expressed in terms of a topological
invariant that integrates to zero in topologically-trivial spaces. An example
is the coefficient $a$ multiplying the Euler density in the Weyl anomaly of 4D
CFTs. In contrast, the type-B anomalies are associated with divergences that
require the introduction of a scale. They manifest themselves in the Weyl
transformation of the effective action in terms of a scalar density that does
not integrate to zero. An example of a type-B anomaly is the coefficient $c$
multiplying the Weyl tensor squared part of the Weyl anomaly in 4D CFTs.
Another example, is the coefficient of two-point functions of operators with
integer scaling dimension $\Delta=\frac{D}{2}+n$, where $D$ is the spacetime
dimension and $n\in\mathbb{Z}_{\geq 0}$. As we review in Sec. 2.2, one way to
discover this anomaly is by noting the presence of a logarithmic divergence in
the two-point function in momentum space.

In a certain sense, type-A anomalies are more ‘rigid’ quantities compared to
type-B anomalies. For instance, it has been argued [2] that type-A anomalies
should match in phases of a CFT with unbroken and spontaneously-broken
conformal invariance. Instead, a similar argument for type-B anomalies is much
harder to make, and the general expectation is that type-B anomalies do not
match across different phases of a CFT with or without spontaneous breaking of
the conformal symmetry. Indeed, we will present some concrete examples in this
paper where the matching does not work.222The robustness of type-A anomalies
compared to that of type-B anomalies is also manifest on their dependence on
exactly-marginal couplings. For example, the Wess–Zumino consistency
conditions can be used to show that the $a$ anomaly is independent of exactly-
marginal couplings, but in general the $c$ anomaly may have a nontrivial
dependence. We refer the reader to [3] (below Eq. (6)) for a discussion on
this point. The vast majority of the type-B anomalies that we will consider in
this paper also depend non-trivially on exactly-marginal couplings.

The main goal of this paper is to examine non-perturbative properties of
type-B anomalies for certain integer-dimension operators (the so-called
Coulomb-branch operators) on the Higgs branch of 4D ${\cal N}=2$ CFTs. In
contrast to the general expectation, we argue that there are cases in this
context, where these anomalies match on the Higgs branch and prove this
matching using a formal, non-perturbative argument. To our knowledge, there is
no precedence of such anomaly matching in the literature.

More specifically, we focus on 4D ${\cal N}=2$ superconformal field theories
(SCFTs) with non-trivial superconformal manifolds, namely continuous families
of ${\cal N}=2$ SCFTs related by exactly-marginal deformations. In the
conformal phase, such theories typically possess a rich spectrum of
$\frac{1}{2}$-BPS Coulomb-branch operators (CBOs), whose scaling dimension is
integral and protected by supersymmetry. Hence, there are natural type-B
anomalies associated with the two-point functions of these operators. These
two-point functions involve a chiral and an anti-chiral CBO, ${\cal O}_{I}$
and $\bar{\cal O}_{J}$ respectively. We call the corresponding type-B anomaly
$G_{I\bar{J}}^{\rm CFT}$.

Additionally, each SCFT on the ${\cal N}=2$ superconformal manifold possesses
a moduli space of Higgs-branch vacua where the conformal symmetry is
spontaneously broken.3334D ${\cal N}=2$ SCFTs also possess moduli spaces of
Coulomb-branch vacua parametrised by vacuum expectation values of Coulomb-
branch operators. We do not consider such vacua in this paper. In the non-
conformal theory of the Higgs branch we isolate a specific contact term in the
three-point function $\langle T^{\mu}_{~{}\mu}(x_{1}){\cal
O}_{I}(x_{2})\bar{\cal O}_{J}(x_{3})\rangle$ that leads to a Weyl anomaly. We
define its dimensionless coefficient, that we call
$G_{I\bar{J}}^{{\mathbb{H}}}$, as the value of the type-B anomaly of interest.
This coefficient captures a corresponding dilaton coupling in the effective
action on the Higgs branch. More details about the definition of
$G_{I\bar{J}}^{{\mathbb{H}}}$ appear in Sec. 2.2 and Sec. 3.

Both $G_{I\bar{J}}^{\rm CFT}$ in the unbroken phase and
$G_{I\bar{J}}^{{\mathbb{H}}}$ in the broken phase are, in general, non-trivial
functions of the exactly-marginal couplings that parametrise the
superconformal manifold. We will argue that whenever there is at least one
point on the superconformal manifold where these quantities match, the
matching will necessarily extend to a finite region of couplings around this
point. In other words, matching at one point, e.g. at weak coupling,
guarantees that $G_{I\bar{J}}^{\rm CFT}$ and $G_{I\bar{J}}^{{\mathbb{H}}}$ are
the same functions of the exactly-marginal couplings in a finite region of the
superconformal manifold. All the relevant concepts underlying the quantities
$G_{I\bar{J}}^{\rm CFT}$ and $G_{I\bar{J}}^{{\mathbb{H}}}$ are reviewed and
defined in Secs. 2 and 3.

The argument that we present in this paper is not a general argument for
type-B anomaly matching. It relies heavily on the presence of ${\cal N}=2$
supersymmetry and the fact that we examine type-B anomalies associated with
CBOs on Higgs-branch moduli spaces. The latter is crucial for two reasons.
First, the Higgs-branch moduli space is parametrised by the vacuum expectation
values (vevs) of a set of $\frac{1}{2}$-BPS operators, called Higgs-branch
operators (HBOs).444In standard gauge-theory examples of 4D ${\cal N}=2$
SCFTs, like the 4D ${\cal N}=2$ superconformal QCD, the CBOs are Casimirs of
adjoint scalar fields and the HBOs are gauge-invariant mesonic operators. Ref.
[4] argued that the states corresponding to HBOs have vanishing Berry phases
under parallel transport on the superconformal manifold,555Equivalently, [4]
argued that the holomorphic vector bundle of Higgs-branch superconformal
primaries is equipped with a flat connection. which is a fact that we will
use. Second, the argument relies crucially on a superconformal Ward identity
whose integrated form receives exactly the same contributions in the unbroken
and broken phases. We find that this occurs when the moduli space respects the
$U(1)_{r}$ symmetry. This is the case for the Higgs-branch moduli spaces, but
not for Coulomb-branch moduli spaces.

We now list the three ingredients that constitute our formal argument in more
detail:

  * (1)

We argue that derivatives of the correlation functions on the Higgs branch
with respect to the exactly-marginal couplings can be defined using the same
connection on the superconformal manifold that appears in the unbroken CFT
phase in the context of conformal perturbation theory. This argument appears
in Sec. 4.

  * (2)

We use a Ward identity for the superconformal currents to show that the three-
point function $\langle T_{\mu\nu}(x_{1}){\cal O}_{I}(x_{2})\bar{\cal
O}_{J}(x_{3})\rangle$ of the energy-momentum tensor with two CBOs is
covariantly constant on the superconformal manifold. The argument works in the
same manner in the unbroken and broken phases because the dilatino that
couples to the superconformal current on the Higgs branch cannot exhibit a
massless pole on the external momentum for the correlation functions that we
consider. As we have already mentioned, it is important for this feature that
the vacuum respects the $U(1)_{r}$ symmetry. The Ward identities of interest
are presented in Sec. 5.

  * (3)

Having shown that both $G_{I\bar{J}}^{\rm CFT}$ and
$G_{I\bar{J}}^{{\mathbb{H}}}$ are covariantly-constant quantities, it suffices
to show that they match at one point of the superconformal manifold. Then, the
vanishing covariant derivatives guarantee that the matching holds in a finite
range of exactly-marginal couplings. We summarise this argument in Sec. 6.

In Sec. 7 we present several concrete examples (in 4D ${\cal N}=2$
superconformal QCD, the large-node limit of a 4D ${\cal N}=2$ circular quiver
and 4D ${\cal N}=4$ SYM theory), where the tree-level matching can be
established by direct computation and the non-perturbative matching follows
from our general arguments. It is worth noting that a special case of type-B
anomaly matching in the 4D ${\cal N}=4$ SYM theory can also be argued
independently using supersymmetry and the matching of chiral anomalies,
forming a partial, independent check of our claim. In contrast, the case of
the 4D ${\cal N}=2$ circular quiver with a finite number of nodes is an
example where the anomaly does not match already at tree level. This is a
manifestation of the more general expectation that type-B anomalies do not
match across moduli spaces.

When all the assumptions are met, the proposed anomaly matching has immediate
implications for the non-perturbative structure of ${\cal N}=2$ SCFTs. The
two-point function coefficients of CBOs in ${\cal N}=2$ SCFTs (in the
conformal phase) are, as we noted, non-trivial functions of the exactly-
marginal coupling constants. Nevertheless, it has been recently shown [5, 6,
7, 8, 9] that they can be obtained directly from the $S^{4}$ partition
function of the theory and that they obey an integrable set of differential
equations with respect to the exactly-marginal couplings, called $tt^{*}$
equations. In Lagrangian theories the $S^{4}$ partition function can be
determined exactly using supersymmetric localisation techniques [10]. As a
result, the proposed anomaly matching implies that there is a corresponding
set of data for the non-conformal theory on the Higgs branch, which is also
determined by the same functions of the exactly-marginal couplings. In Sec. 8
we outline the supersymmetric-localisation data that determine the anomalies
of interest in the conformal phase of a 4D ${\cal N}=2$ circular-quiver
theory.

The anomaly matching of CBO data on the Higgs branch of 4D ${\cal N}=2$ SCFTs
has an additional interesting application in the context of dimensional
deconstruction. Some time ago it was proposed that a certain limit on the
Higgs branch of a 4D ${\cal N}=2$ circular quiver CFT—the same quiver that we
analyse in Secs. 7.2 and 8—defines the 6D (2,0) theory of type $A$ on a
2-torus [11]. In this construction, the exactly-marginal coupling of the 4D
theory controls the complex structure of the torus and the 4D CBOs map to the
Kaluza-Klein (KK) modes of local $\frac{1}{2}$-BPS operators of the 6D theory.
The 4D type-B anomaly matching that we present in this paper then has a
striking implication: there exist data of these local 6D $\frac{1}{2}$-BPS
operators that depend non-trivially on the complex structure of the torus, and
yet can be computed non-perturbatively from the 4D perspective by doing a
supersymmetric-localisation computation in the CFT phase of the 4D quiver. So
far, most of the structure of the non-Lagrangian 6D (2,0) theory has remained
a mystery, lying beyond the reach of existing techniques. Important exceptions
are recent results on partition functions [12, 13, 14, 15, 16, 17], chiral-
algebra structures [18] or the superconformal-bootstrap programme [19]. The
results implied by the anomaly matching in this paper provide the first
example of exact, non-supersymmetric data of local operators in the 6D (2,0)
theory. We present the details of this relationship in Sec. 9. A collection of
useful facts as well as our conventions are relegated to a set of appendices
at the end of the manuscript.

##  2 Review of Useful Concepts

###  2.1 4D ${\cal N}=2$ Superconformal Manifolds

The ${\cal N}=2$ superconformal field theories in four spacetime dimensions
are often members of continuous families of theories connected by exactly-
marginal deformations that preserve the ${\cal N}=2$ supersymmetry. The
continuous space of such theories is called a superconformal manifold and is
parametrised by the corresponding exactly-marginal couplings
${\bm{\lambda}}=\\{\lambda^{i}\\}$. The index $i$ labels different directions
on the tangent space of the superconformal manifold.

The R-symmetry group of a 4D ${\cal N}=2$ SCFT is $SU(2)_{R}\times U(1)_{r}$.
The theory possesses two types of ${\frac{1}{2}}$-BPS superconformal-primary
operators: Coulomb-branch operators (CBOs), which are charged under the
$U(1)_{r}$ symmetry but are neutral under the $SU(2)_{R}$, and Higgs-branch
operators (HBOs), which are charged under the $SU(2)_{R}$ symmetry but are
neutral under the $U(1)_{r}$. A generic ${\cal N}=2$ theory also exhibits
moduli spaces of vacua, where (a subset of) the above operators obtain a non-
vanishing vev. We will return to the moduli spaces in the next subsection. For
the moment we focus exclusively on the vacuum that preserves the full ${\cal
N}=2$ superconformal algebra of the theory.

The CBOs can preserve either the left- or right-chiral part of the ${\cal
N}=2$ Poincaré supersymmetry. We will denote the first set as ${\cal O}_{I}$
and the second set as $\bar{\cal O}_{J}$ (one can think of one set as the
complex conjugate of the other). By default, the scaling dimension $\Delta$ of
a (anti-)chiral primary CBO is related to its $U(1)_{r}$ charge $r$ through
the relation $\Delta=|r|$ and is naturally an integer greater or equal to 2
that does not depend (generically) on the value of exactly-marginal couplings.
In fact, CBOs of scaling dimension 2 are special, because a certain
supersymmetric descendant of such operators has scaling dimension 4 and
defines exactly-marginal deformations of the theory that preserve the ${\cal
N}=2$ supersymmetry. Therefore, for generic 4D ${\cal N}=2$ SCFTs with a non-
empty set of $\Delta=2$ CBOs there is naturally a non-vanishing superconformal
manifold.666Clearly, there are examples where this is not the case, e.g. in
Argyres-Douglas theories.

Each of these superconformal primaries forms a chiral (or anti-chiral) ring
under OPE multiplication. The data of the chiral ring are encoded in the two-
and three-point functions

$\displaystyle\langle{\cal O}_{I}(x)\bar{\cal O}_{J}(y)\rangle$
$\displaystyle=$ $\displaystyle\frac{G_{I\bar{J}}}{|x-y|^{2\Delta}}~{},~{}~{}$
(2.1) $\displaystyle\langle{\cal O}_{I}(x){\cal O}_{J}(y)\bar{\cal
O}_{K}(z)\rangle$ $\displaystyle=$
$\displaystyle\frac{C_{IJ\bar{K}}}{|x-y|^{\Delta_{I}+\Delta_{J}-\Delta_{K}}|x-z|^{\Delta_{I}+\Delta_{K}-\Delta_{J}}|y-z|^{\Delta_{J}+\Delta_{K}-\Delta_{I}}}~{}.$
(2.2)

In (2.1) $\Delta$ is the common scaling dimension of the operators, while in
(2.2) $\Delta_{I}$ etc. denotes the scaling dimension of each operator. The
three-point function coefficients $C_{IJ\bar{K}}$ are related to the OPE
coefficients $C_{IJ}^{L}$ in the CBO chiral ring via the relation
$C_{IJ\bar{K}}=C_{IJ}^{L}G_{L\bar{K}}$.

Clearly, as one varies the exactly-marginal couplings on a superconformal
manifold one has the freedom to choose an arbitrary (possibly coupling-
constant-dependent) normalisation of the operators. One of the possible
normalisations, which was discussed in [8], is to arrange for the OPE
coefficients $C_{IJ}^{K}$ to be either one or zero. This is a special case of
the so-called ‘holomorphic gauge’. In this case, all the non-trivial
information about the structure of the CBO chiral ring is contained in the
two-point function coefficients $G_{I\bar{J}}$, which will soon play a
protagonistic role in their interpretation as type-B anomalies. These
coefficients are, in general, non-trivial functions of the exactly-marginal
coupling constants ${\bm{\lambda}}$, the computation of which requires
powerful non-perturbative tools. In recent years this has been possible using
supersymmetric localisation on $S^{4}$. For further details on normalisation
conventions and related issues we refer the reader to [8].

The coupling-constant dependent choice of the operators on the superconformal
manifold raises various practical questions. How does one sensibly relate data
for different values of ${\bm{\lambda}}$, and how does one write equations
that behave covariantly under coupling-constant dependent redefinitions of the
operators? A related question is the following: When varying the exactly-
marginal couplings to relate correlation functions at nearby points on the
superconformal manifold, one is instructed to compute correlation functions
with integrated insertions of the exactly-marginal operators. For example, the
naive derivative of an $n$-point function $\langle
O_{1}(x_{1})O_{2}(x_{2})\cdots O_{n}(x_{n})\rangle$ at separated points
$x_{1},\ldots,x_{n}$ with respect to the exactly-marginal coupling
$\lambda^{i}$ is

$\left\langle\int d^{4}x\,\Phi_{i}(x)O_{1}(x_{1})O_{2}(x_{2})\cdots
O_{n}(x_{n})\right\rangle\;,$ (2.3)

where $\Phi_{i}$ is the corresponding exactly-marginal operator. This quantity
is not well defined. It exhibits ultraviolet (UV) divergences when the
integrated insertion $\Phi_{i}$ collides with any of the other fixed
insertions $O_{\ell}(x_{\ell})$ and needs to be regularised.

One approach to address this problem is to define the regulated version of
(2.3) as a modified (covariant) derivative with respect to the coupling
$\lambda^{i}$

$\nabla_{i}\langle O_{1}(x_{1})O_{2}(x_{2})\cdots
O_{n}(x_{n})\rangle\equiv\left\langle\int
d^{4}x\,\Phi_{i}(x)O_{1}(x_{1})O_{2}(x_{2})\cdots
O_{n}(x_{n})\right\rangle_{\text{regulated}}\;.$ (2.4)

For example, one can regulate by cutting balls of finite size around the
operator insertions and removing divergent terms in the limit of vanishing
cutoff, as was done in [20, 21, 22, 23]. Different subtraction schemes define
different types of connections on the conformal manifold, some of which may
not have desirable features. In this paper we follow [22, 24] and adopt a
natural scheme where one subtracts the terms that are divergent in the spatial
cutoff and keeps the finite remainder. In [25] it was shown that this
prescription reproduces the Berry connection in radial quantisation.

Alternatively, one can keep the definition

$\partial_{i}\langle O_{1}(x_{1})O_{2}(x_{2})\cdots
O_{n}(x_{n})\rangle=\left\langle\int
d^{4}x\,\Phi_{i}(x)O_{1}(x_{1})O_{2}(x_{2})\cdots O_{n}(x_{n})\right\rangle$
(2.5)

intact and proceed to cancel divergences by modifying the OPEs with suitable
contact terms. Some of these contact terms express a coupling-constant
dependent operator mixing. A natural notion of connection on the conformal
manifold arises in this manner from contact terms in OPEs that involve the
exactly-marginal operators. This approach was advocated in [26].

These observations endow the conformal manifold of SCFTs with an intrinsic
geometric structure. For CBOs in a 4D ${\cal N}=2$ superconformal manifold
${\cal M}$ this structure can be summarised in the following manner [24, 8].
The CBOs are sections of a holomorphic vector bundle on ${\cal M}$ and the
two-point function coefficients $G_{I\bar{J}}$ (which are functions of the
exactly-marginal couplings ${\bm{\lambda}}$) can be viewed as components of a
Hermitian metric on these bundles. Crucially, these bundles possess a
connection $\nabla$ that defines parallel transport. Physically, parallel
transport on these bundles gives the relation between CBOs in SCFTs at
different values of the exactly-marginal couplings. $\nabla$ is compatible
with the metric $G$, namely the covariant derivative with respect to
$\lambda^{k}$ vanishes

$\nabla_{k}G_{I\bar{J}}=0~{}\Leftrightarrow~{}\partial_{k}G_{I\bar{J}}-(A_{k})^{L}_{I}G_{L\bar{J}}-(A_{k})^{\bar{L}}_{\bar{J}}G_{I\bar{L}}=0~{}.$
(2.6)

In these expressions $(A_{k})_{M}^{L}$, $(A_{k})_{\bar{M}}^{\bar{L}}$ are the
components of the connection on the bundle of CBOs. The indices $L$, $\bar{L}$
run over the set of chiral and anti-chiral CBOs.

Eq. (2.6) can be viewed as an automatic consequence of the definition of the
connection.777For general CBOs in 4D ${\cal N}=2$ SCFTs (2.6) can be proven
using the regularisation prescription mentioned below Eq. (2.4) and by
employing the superconformal Ward identities of Sec. 5. This argument is
reviewed in Sec. 5.1. It expresses the coupling-constant dependence of the
two-point functions in the scheme that defines the connection, and in that
respect it is a trivial relation. However, it may be useful to appreciate that
if one were to be given the connection through some independent means, then
the coupling-constant dependence would follow from this equation. The role and
usefulness of the $tt^{*}$ equations is related to this statement [24, 8]. The
$tt^{*}$ equations provide the curvature of the connection on the chiral ring
of CBOs in terms of the two- and three-point function coefficients in the
chiral ring. As we have already mentioned, in a normalisation scheme where the
OPE coefficients in the chiral ring are trivialised, all the information is
encoded in the two-point function coefficients. Then, the $tt^{*}$ equations
(combined with (2.6)) become an integrable set of differential equations for
the coupling-constant dependence of the two-point functions. A prescription
based on supersymmetric localisation on $S^{4}$ provides a non-trivial
solution to these equations [5, 6, 7, 8, 9].

So far we have focussed on the geometric structure of the conformal manifold
in the unbroken, conformal phase of the theory. We will explore the role of
the connection in the broken phase of the Higgs branch in Sec. 4.

###  2.2 Two-point Functions of CBOs as Type-B Anomalies

When an even-$D$-dimensional CFT possesses operators with scaling dimension
$\Delta=\frac{D}{2}+n$, $n\in\mathbb{Z}_{\geq 0}$, there are corresponding
type-B Weyl anomalies associated with the two-point function coefficients of
these operators. Since the CBOs in 4D ${\cal N}=2$ SCFTs are naturally
operators of this type we will focus the remaining discussion on them.

As we noted in Eq. (2.1), the two-point function of two CBOs in position space
is a perfectly well-defined correlation function at separated points. However,
the Fourier transform of this function is not well defined and a type-B Weyl
anomaly arises in momentum space as a logarithmic contribution to the two-
point function

$\langle{\cal O}_{I}(p)\bar{\cal
O}_{J}(-p)\rangle\simeq(-1)^{n+1}\frac{\pi^{2}G_{I\bar{J}}}{2^{2n}\Gamma(n+1)\Gamma(n+\frac{D}{2})}p^{2n}\log\left(\frac{p^{2}}{\mu^{2}}\right)~{},$
(2.7)

where $\simeq$ means that this is correct up to non-logarithmic scheme-
dependent terms that we drop. This is clearly proportional to the two-point
function coefficients $G_{I\bar{J}}$. The logarithm has introduced a scale
$\mu$ and the conformal anomaly

$\mu\frac{d}{d\mu}\langle{\cal O}_{I}(p)\bar{\cal
O}_{J}(-p)\rangle=(-1)^{n}\frac{\pi^{2}G_{I\bar{J}}}{2^{2n-1}\Gamma(n+1)\Gamma(n+\frac{D}{2})}p^{2n}$
(2.8)

translates in position space to a specific contact term

$\mu\frac{d}{d\mu}\langle{\cal O}_{I}(x)\bar{\cal O}_{J}(0)\rangle\propto
G_{I\bar{J}}\Box^{n}\delta(x)~{}.$ (2.9)

This contact term reflects an inconsistency between the Ward identities for
diffeomorphism invariance and the Ward identities for Weyl invariance. The
clash between these Ward identities in the conformal phase is reviewed in Sec.
3.1.

Alternatively, one may consider the generating functional $W$ of correlation
functions for CBOs. Adding spacetime-dependent sources to the action for the
operators ${\cal O}_{I}$, $\bar{\cal O}_{J}$

$\displaystyle\delta S=\int d^{4}x\,\left(t^{I}(x){\cal
O}_{I}(x)+\bar{t}^{J}(x)\bar{\cal O}_{J}(x)\right)\;,$ (2.10)

for a theory on a curved manifold with metric $g_{\mu\nu}$, $W$ becomes a
functional of $g_{\mu\nu}$ and $t^{I}$, $\bar{t}^{J}$. A (generalised)888The
term ‘generalised’ reflects an interesting modification of the standard
geometric part of the Weyl transformation,
$\delta_{\sigma}=2\sigma(x)g_{\mu\nu}(x)\frac{\delta}{\delta g_{\mu\nu}(x)}$,
when the operators ${\cal O}_{I}$, $\bar{\cal O}_{J}$ are irrelevant. As
explained in [27], the modification requires the inclusion of a metric
$\beta$-function in the definition of $\delta_{\sigma}$. Otherwise, the Osborn
equation (2.11) does not make sense. We refer the reader to [27] for further
details on this point. Weyl transformation expresses the type-B anomaly ${\cal
A}$ via Osborn’s equation [28]

$\delta_{\sigma}W(\\{t,\bar{t}\\},g_{\mu\nu})=\int
d^{4}x\,\sqrt{g}\,\sigma(x){\cal
A}\left(\\{t,\bar{t}\\},g_{\mu\nu}\right)~{},$ (2.11)

where $\sigma$ is the infinitesimal parameter of the Weyl transformation and

${\cal A}=G_{I\bar{J}}t^{I}\Delta_{c}\bar{t}^{J}$ (2.12)

is (up to an overall numerical constant) a differential operator of the form

$\Delta_{c}=\Box^{n}+{\rm curvature~{}terms}\;,$ (2.13)

with $G_{I\bar{J}}$ the two-point function coefficients listed above. In this
form, it is clear that the type-B anomalies $G_{I\bar{J}}$ reflect a specific
piece in the three-point function $\langle T_{\mu\nu}(x_{1}){\cal
O}_{I}(x_{2})\bar{\cal O}_{J}(x_{3})\rangle$.

As an example, let us examine the case of two CBOs with scaling dimension
$\Delta=4$. In momentum space the three-point function $\langle
T_{\mu\nu}(p_{1}){\cal O}_{I}(p_{2})\bar{\cal O}_{J}(p_{3})\rangle$ contains
the term (see e.g. [29], Eq. (3.58))

$\langle T_{\mu\nu}(p_{1}){\cal O}_{I}(p_{2})\bar{\cal
O}_{J}(p_{3})\rangle=\ldots+\frac{1}{3}\frac{\pi^{2}G_{I\bar{J}}}{16\Gamma(3)\Gamma(4)}p_{2}^{2}p_{3}^{2}\left(\eta_{\mu\nu}-\frac{(p_{1})_{\mu}(p_{1})_{\nu}}{p_{1}^{2}}\right)~{}.$
(2.14)

For the corresponding three-point function of the trace of the energy-momentum
tensor $T\equiv T^{\mu}_{~{}\mu}$

$\langle T(p_{1}){\cal O}_{I}(p_{2})\bar{\cal
O}_{J}(p_{3})\rangle=\ldots+\frac{\pi^{2}G_{I\bar{J}}}{16\Gamma(3)\Gamma(4)}p_{2}^{2}p_{3}^{2}~{},$
(2.15)

which isolates the $G_{I\bar{J}}t^{I}\Box^{2}\bar{t}^{J}$ contact term in
(2.11)-(2.13).

More generally, the anomalous term for operators ${\cal O}_{I}$, $\bar{\cal
O}_{J}$ with common scaling dimension $\Delta=2+n$ is

$\langle T(p_{1}){\cal O}_{I}(p_{2})\bar{\cal
O}_{J}(p_{3})\rangle=\ldots+(-1)^{n}\frac{\pi^{2}G_{I\bar{J}}}{2^{2n}\Gamma(n+1)\Gamma(n+2)}p_{2}^{n}p_{3}^{n}~{}.$
(2.16)

For a CBO that has $\Delta=2$ we get, in particular,

$\langle T(p_{1}){\cal O}_{I}(p_{2})\bar{\cal
O}_{J}(p_{3})\rangle=\ldots+\pi^{2}G_{I\bar{J}}~{}.$ (2.17)

We will examine this case in more detail via several examples in Sec. 7.

Let us summarise the key points of the preceding discussion. In a 4D ${\cal
N}=2$ SCFT there is a chiral ring of CBOs with integer scaling dimensions that
are independent of the exactly-marginal couplings on the superconformal
manifold. In the vacuum of unbroken conformal invariance the two-point
function coefficients $G_{I\bar{J}}$ of these operators play two related
roles:

  * $(a)$

They define a metric on the holomorphic bundles of CBOs.

  * $(b)$

They express a type-B Weyl anomaly that captures a particular part in three-
point functions of the CBOs with the energy-momentum tensor. For three-point
functions involving the trace of the energy-momentum tensor the Weyl anomaly
appears in position space as a contact term.

###  2.3 Type-B anomalies on the Higgs Branch

The above statements concerned properties of the theory exclusively in a
vacuum with unbroken conformal invariance. Typically, at each point
${\bm{\lambda}}$ of a superconformal manifold an SCFT possesses a non-trivial
moduli space of vacua labelled by the non-vanishing vevs of a set of
operators. 4D ${\cal N}=2$ SCFTs have different types of moduli spaces of
vacua: Higgs-branch, Coulomb-branch and mixed Coulomb-Higgs moduli spaces. In
each of these spaces the non-vanishing vev spontaneously breaks the conformal
invariance as well as part of the R-symmetry of the theory. By definition, the
Higgs-branch moduli spaces are characterised by non-vanishing vevs of the
Higgs-branch operators. Hence, in these vacua the $SU(2)_{R}$ is spontaneously
broken but the $U(1)_{r}$ is preserved. In contrast, the Coulomb-branch moduli
spaces are characterised by non-vanishing vevs of Coulomb-branch operators.
Accordingly, in these vacua the $U(1)_{r}$ is spontaneously broken but the
$SU(2)_{R}$ is preserved.

In what follows we will focus on the properties of Higgs-branch vacua. We
assume that over each point ${\bm{\lambda}}$ of a 4D ${\cal N}=2$
superconformal manifold ${\cal M}$ there is a corresponding Higgs-branch
moduli space, which will be denoted as ${\mathbb{H}}_{\bm{\lambda}}$ (the
letter ${\mathbb{H}}$ stands for Higgs). We will isolate a certain class of
data on ${\mathbb{H}}_{\bm{\lambda}}$ and ask how they vary as we change the
exactly-marginal couplings ${\bm{\lambda}}$ and move across the superconformal
manifold.

The data we are interested in are the type-B Weyl anomalies of CBOs. Clearly,
in vacua with spontaneous breaking of conformal invariance, several effects
take over and modify the behaviour of the theory across different scales, from
the ultraviolet (UV) to the infrared (IR). For example, the full dependence of
the two- and three-point functions of primary operators on the spacetime
coordinates is now more complicated and involves non-trivially the
characteristic scale of the vacuum. Moreover, typical excitations become
massive and in the extreme IR composite operators can be lifted from the
spectrum. For instance, from a Lagrangian perspective several elementary
fields become massive with a mass set by the characteristic scale of the
vacuum and the composite operators of these fields are lifted at ultra-low
energies. This can also happen for certain exactly-marginal operators. In that
case, the extreme-IR effective theory will be independent of those exactly-
marginal couplings. On a related note, in weakly-coupled corners of the Higgs
branch it is expected that the extreme-IR effective theory can be expressed in
terms of abelian vector multiplets and neutral hypermultiplets. This theory
obeys non-renormalisation theorems that make it independent of the exactly-
marginal couplings [30].

The data of interest in this paper are associated with the full
renormalisation-group (RG) flow, and will typically involve heavy operators.
Most of these data cannot be defined independently in the extreme-IR theory
and exhibit a non-trivial dependence on the exactly-marginal couplings. It
should be noted, however, that we will also encounter examples where some of
the data of interest involve massless fields. One of these cases are type-B
anomalies associated with untwisted operators in the circular-quiver theory of
Sec. 7. In that instance, the quantities of interest have a direct
interpretation in the extreme-IR effective theory.

One of the crucial features of the RG running induced by the vacuum, is the
fact that it leaves the form of the Ward identities intact. In the unbroken
vacuum we noted that type-B anomalies express a clash between the Ward
identities of diffeomorphism and Weyl transformations. In Sec. 3 we review the
argument that shows the relation of these anomalies to two-point functions in
the unbroken phase and extend it to vacua with spontaneous breaking of
conformal invariance. In the process, we demonstrate that the anomaly survives
the breaking but is no-longer directly connected to the two-point functions.
This effect is a result of the modified analytic structure that correlation
functions exhibit in the broken vacuum. In this context, our primary goal in
the rest of the paper will be to investigate how CBO type-B anomalies behave
on the Higgs branch of 4D ${\cal N}=2$ SCFTs as a function of the exactly-
marginal couplings.

Since there is no direct connection between these Weyl anomalies and the two-
point functions on the Higgs branch, it is most appropriate to define the
former by analysing directly the anomalous part of the three-point function
between the CBOs and the trace of the energy-momentum tensor, $\langle
T(x_{1}){\cal O}_{I}(x_{2})\bar{\cal O}_{J}(x_{3})\rangle$. In momentum space,
the anomaly can be read off in a special kinematic regime, which is summarised
by the formula

$G^{{\mathbb{H}}}_{I\bar{J}}=(-1)^{n}\frac{2^{2n}\Gamma{(n+1)\Gamma{(n+2)}}}{\pi^{2}(n!)^{2}}\lim_{p_{1}\to
0}\lim_{p_{2},p_{3}\to
0}\left[\frac{d}{dp_{2}^{n}}\frac{d}{dp_{3}^{n}}\left(\langle T(p_{1}){\cal
O}_{I}(p_{2})\bar{\cal O}_{J}(p_{3})\rangle\right)\right]\;.$ (2.18)

In this expression the three-point function is evaluated on a Higgs-branch
vacuum of ${\mathbb{H}}_{\bm{\lambda}}$, $T=T^{\mu}_{~{}\mu}$ is the trace of
the energy-momentum tensor and ${\cal O}_{I}$, $\bar{\cal O}_{J}$ are chiral
and anti-chiral CBOs. The quantities $G^{\mathbb{H}}_{I\bar{J}}$ are
dimensionless functions of the exactly-marginal couplings ${\bm{\lambda}}$. A
priori, these functions can be different from the two-point function
coefficients $G_{I\bar{J}}^{\rm CFT}$ that appeared in the unbroken phase, Eq.
(2.1). In the following sections we will argue, however, that under certain
assumptions these quantities can in fact be identical, namely
$G_{I\bar{J}}^{\rm CFT}=G^{\mathbb{H}}_{I\bar{J}}$.

This anomaly-matching statement is far from automatic. In [2] it was argued
that type-A anomalies match on moduli spaces, but the expectation is that this
does not generically happen with type-B anomalies. In special cases where
supersymmetry relates a type-A anomaly to a type-B anomaly—and both are
related to chiral anomalies—one anticipates the matching to work also for the
corresponding type-B anomalies. One such example arises in ${\cal N}=4$ SYM
theory where the type-B anomaly for $\Delta=2$, ${\frac{1}{2}}$-BPS operators
is believed to match on the Coulomb branch on these grounds. In Sec. 7 we will
independently confirm this expectation when the anomaly in question is viewed
as a type-B anomaly on the Higgs branch of an ${\cal N}=2$ SCFT with an
adjoint hypermultiplet.

##  3 Ward Identities for Diffeomorphism and Weyl Transformations

We begin the detailed discussion of type-B anomalies by exhibiting how these
anomalies arise from a clash between the Ward identities of diffeomorphism and
Weyl transformations. We focus on the case of scalar operators with integer
scaling dimension in four-dimensional CFTs. In this section there are no
assumptions of supersymmetry. For pedagogical reasons, we first review the
standard argument that exhibits the anomaly in the unbroken phase, closely
following the notation of [2]. We then discuss how the argument is modified in
phases with spontaneously-broken conformal symmetry.

Consider, for concreteness, two scalar operators $O(x)$, $\bar{O}(x)$ with
common scaling dimension $\Delta=2+n$, $n\in\mathbb{Z}_{\geq 0}$. These
operators could be a chiral and an anti-chiral CBO in a 4D ${\cal N}=2$ SCFT,
but for the purposes of the present argument we do not need to make this
restriction. Let $J$ and $\bar{J}$ be sources of these operators in the
action. Then, the generating functional for the correlation functions of $O$,
$\bar{O}$ and the energy-momentum tensor $T^{\mu\nu}$ is

$W(g,J,\bar{J})=\int d^{4}xd^{4}y\,\Gamma^{(2)}(x,y)J(x)\bar{J}(y)+\int
d^{4}xd^{4}yd^{4}z\,\Gamma^{(3)}_{\mu\nu}h^{\mu\nu}(x)J(y)\bar{J}(z)+\ldots\;,$
(3.1)

where $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ is a background-metric
perturbation and the ellipsis denotes other contributions to the generating
functional.

In momentum space, the Ward identities for diffeomorphism and Weyl
transformations are respectively [2]

$q^{\mu}\Gamma_{\mu\nu}^{(3)}(q,k_{1},k_{2})=\frac{1}{2}(k_{1})_{\nu}\Gamma^{(2)}(k_{1}^{2})+\frac{1}{2}(k_{2})_{\nu}\Gamma^{(2)}(k_{2}^{2})~{},$
(3.2)
$\eta^{\mu\nu}\Gamma_{\mu\nu}^{(3)}(q,k_{1},k_{2})=\frac{\Delta}{2}\left(\Gamma^{(2)}(k_{1}^{2})+\Gamma^{(2)}(k_{2}^{2})\right)~{}.$
(3.3)

These identities are valid both in the broken and unbroken phases of the CFT.
The conservation of momentum yields $q=k_{1}+k_{2}$. In addition, we set

$r:=k_{1}-k_{2}~{}.$ (3.4)

Then, on general grounds the kinematical expansion of $\Gamma^{(3)}_{\mu\nu}$
takes the form

$\Gamma^{(3)}_{\mu\nu}=\bar{A}\eta_{\mu\nu}+Bq_{\mu}q_{\nu}+C(q_{\mu}r_{\nu}+q_{\nu}r_{\mu})+Dr_{\mu}r_{\nu}~{}.$
(3.5)

The factors $\bar{A},B,C,D$ depend on the Lorentz invariants
$q^{2},k_{1}^{2},k_{2}^{2}$. In the broken phase they can also depend on the
symmetry-breaking scale $v$, but we will keep this dependence implicit. Let us
also define the combination

$A:=\bar{A}-\frac{1}{4}\left(\Gamma^{(2)}(k_{1}^{2})+\Gamma^{(2)}(k_{2}^{2})\right)~{}.$
(3.6)

Inserting (3.5) into (3.2) we obtain

$q_{\nu}\left(A+Bq^{2}+Cq\cdot r\right)+r_{\nu}\left(Cq^{2}+Dq\cdot
r-\frac{1}{4}\Gamma^{(2)}(k_{1}^{2})+\frac{1}{4}\Gamma^{(2)}(k_{2}^{2})\right)=0~{}.$
(3.7)

Setting the independent coefficients of $q_{\nu}$ and $r_{\nu}$ separately to
zero yields

$A+q^{2}B+q\cdot rC=0~{},$ (3.8) $q^{2}C+q\cdot
rD-\frac{1}{4}\Gamma^{(2)}(k_{1}^{2})+\frac{1}{4}\Gamma^{(2)}(k_{2}^{2})=0~{}.$
(3.9)

Finally, inserting (3.5) into (3.3) (and using the fact that $\Delta=2+n$) we
obtain

$4A+q^{2}B+2q\cdot
rC+r^{2}D=\frac{n}{2}\left(\Gamma^{(2)}(k_{1}^{2})+\Gamma^{(2)}(k_{2}^{2})\right)~{}.$
(3.10)

###  3.1 Unbroken Phase

At this point, let us focus more specifically on the case of unbroken
conformal invariance. In this phase there are no singularities at $q^{2}=0$,
so following [2] one can set $q^{2}=0$ and restrict to the special kinematic
regime where $k_{1}^{2}=k_{2}^{2}=k^{2}$. As a consequence of the last
restriction

$q\cdot r=k_{1}^{2}-k_{2}^{2}=0~{}.$ (3.11)

In this regime, Eqs. (3.8) and (3.9) (associated with diffeomorphism
transformations) give respectively

$\displaystyle A(k^{2})$ $\displaystyle=$ $\displaystyle 0~{},$ (3.12)
$\displaystyle k^{2}D(k^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{4}k^{2}\frac{\partial\Gamma^{(2)}}{\partial k^{2}}$
(3.13)

whereas Eq. (3.10) (associated with Weyl transformations) yields

$4A(k^{2})+4k^{2}D(k^{2})=n\Gamma^{(2)}(k^{2})~{}.$ (3.14)

Inserting the precise form of the two-point function (see Eq. (2.7))

$\Gamma^{(2)}(k^{2})=Gk^{2n}\log\left(\frac{k^{2}}{\mu^{2}}\right)~{},$ (3.15)

with a constant $G$ proportional to the two-point function coefficient, we
find that (3.14) becomes

$4A=-Gk^{2n}~{}.$ (3.16)

This result is in direct contradiction with Eq. (3.12). This contradiction is
the explicit manifestation of the type-B anomaly in the conformal phase.

###  3.2 Broken Phase

In the broken phase the analytic structure of the correlation functions is
different [2]. Since there is a pole in $q^{2}$ coming from the dilaton
propagator, one cannot directly set $q^{2}=0$, but one can instead take
$q^{2}\neq 0$ and consider the limit $q^{2}\to 0$. We will continue to discuss
the kinematic regime $k_{1}^{2}=k_{2}^{2}=k^{2}$ with an additional limit
$k^{2}\to 0$ taken at the end of the computation. It is also useful to notice
that

$r^{2}=k_{1}^{2}+k_{2}^{2}-2k_{1}\cdot
k_{2}=2(k_{1}^{2}+k_{2}^{2})-q^{2}\xrightarrow[q^{2}\to
0]{}2(k_{1}^{2}+k_{2}^{2})=4k^{2}~{}.$ (3.17)

We observe that the $B$ term on the RHS of (3.5) is the term that carries the
linear coupling of the energy-momentum tensor to the dilaton. The dilaton is
the massless Goldstone boson associated with the spontaneous breaking of the
conformal symmetry. There are several ways to view the linear coupling of the
dilaton to the energy-momentum tensor: $(a)$ One way is by shifting
classically the value of the elementary fields around the condensed vacuum
(with vev $v$), which produces an expression for the energy-momentum tensor
with a linear term in the dilaton, or $(b)$ Equivalently, one can argue that
the effective action for the dilaton in the broken vacuum contains terms of
the form

$\displaystyle S_{eff}(\sigma)=\int
d^{4}x\Big{(}-\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma+\frac{1}{v}\sigma(x)T^{\mu}\,_{\mu}(x)+O(v^{-3})\Big{)}\;,$
(3.18)

where $T^{\mu\nu}$ is the original expression of the energy-momentum tensor in
the unbroken phase. This action leads to the equation of motion999In this form
the energy-momentum tensor is no-longer traceless. However, in terms of the
full vev-dependent energy-momentum tensor of the first viewpoint $(a)$, the
tracelessness condition is unaltered. For a recent discussion and review of
these standard points, see for example [31].

$T^{\mu}\,_{\mu}(x)=-v\,\Box\sigma(x)\,.$ (3.19)

As a result, as we send $q^{2}\to 0$ the $B$ term on the RHS of (3.5) will
exhibit a $1/q^{2}$ pole that cancels the $q^{2}$ in front of $B$. The
resulting expression contains a term proportional to $k^{2n}$ in the low-
momentum expansion

$\lim_{q^{2}\to 0}q^{2}B\sim\widetilde{G}k^{2n}$ (3.20)

for a non-zero value $\widetilde{G}$. As we will show in several examples
later on, this constant captures the coupling of the dilaton to the operators
$O$, $\bar{O}$. Consequently, for the contributions (3.20), Eqs. (3.8) and
(3.10) become respectively

$A+\widetilde{G}k^{2n}=0~{},$ (3.21)
$4A+\widetilde{G}k^{2n}=-\bigg{[}4k^{2}D-n\Gamma^{(2)}(k^{2})\bigg{]}_{k^{2n}}~{}.$
(3.22)

The expression $\big{[}4k^{2}D-n\Gamma^{(2)}(k^{2})\big{]}_{k^{2n}}$ is
shorthand notation for the $k^{2n}$ contribution in the combination
$4k^{2}D-n\Gamma^{(2)}(k^{2})$ when considering the small-$k^{2}$ expansion.
Using (3.21) to express $A$ in terms of $\widetilde{G}k^{2n}$ (3.22) can be
recast into the form

$\widetilde{G}k^{2n}=\frac{1}{3}\bigg{[}4k^{2}D-n\Gamma^{(2)}(k^{2})\bigg{]}_{k^{2n}}~{}.$
(3.23)

Employing the expression (3.13), which is still valid, we obtain

$\widetilde{G}k^{2n}=\frac{1}{3}\bigg{[}k^{2}\frac{\partial\Gamma^{(2)}}{\partial
k^{2}}-n\Gamma^{(2)}(k^{2})\bigg{]}_{k^{2n}}~{}.$ (3.24)

The RHS of this equation, however, vanishes because the generic $k^{2n}$
contribution to $\Gamma^{(2)}(k^{2})$,

$\Gamma^{(2)}(k^{2})\sim ak^{2n}$ (3.25)

for some constant $a$, cancels out. Therefore Eq. (3.24) implies that
$\tilde{G}$ vanishes, which leads to an inconsistency.

From this general argument, the following important conclusions can be
reached. The type-B anomaly persists in the broken phase but its coefficient
(expressed by the constant $\widetilde{G}$ in the above equations) is not
related directly to the structure of the two-point function. However, it can
be read off from the structure of the three-point function $\langle
T^{\mu}_{~{}\mu}(q){\cal O}(k_{1})\bar{\cal O}(k_{2})\rangle$ in the
appropriate kinematic regime.

##  4 Coupling-constant Dependence on Moduli Spaces

Having established the specifics of the anomaly of interest, we would next
like to understand better how it may depend on exactly-marginal couplings. In
Sec. 2.1 we observed that a proper comparison of CFT correlation functions
along a conformal manifold requires the introduction of a covariant
derivative. The associated connection on the bundle of operators captures
details of the regularisation and operator mixing that occur in the UV when
two operators collide. In what follows, we want to discuss the proper
treatment of similar effects in correlation functions defined on a vacuum with
spontaneously-broken conformal symmetry. Once again, we concentrate on Higgs-
branch vacua in 4D ${\cal N}=2$ SCFTs. Since we want to distil information
about type-B anomalies of CBOs, we will tailor the discussion to three-point
functions of the energy-momentum tensor with two CBOs

$C_{TI\bar{J}}(x_{1},x_{2},x_{3})\equiv\langle T_{\mu\nu}(x_{1}){\cal
O}_{I}(x_{2})\bar{\cal O}_{J}(x_{3})\rangle~{},$ (4.1)

expressed here in position space for insertions at separated spacetime points.
We will also focus on correlation functions that are evaluated in the
spontaneously-broken phase.

Besides its spacetime dependence, the quantity (4.1) depends non-trivially on
the exactly-marginal couplings and the vev characterising the vacuum state. To
analyse the dependence on the exactly-marginal couplings one may consider
derivatives of $C_{TI\bar{J}}$ with respect to these couplings. Repeating the
logic of Sec. 2.1 we observe that the precise, covariant definition of these
derivatives requires a connection $\nabla$ on the conformal manifold

$\nabla_{\lambda^{i}}C_{TI\bar{J}}=\left\langle\int
d^{4}x\,\Phi_{i}(x)\,T(x_{1}){\cal O}_{I}(x_{2})\bar{\cal
O}_{J}(x_{3})\right\rangle_{\text{regulated}}~{}.$ (4.2)

We have denoted by $\Phi_{i}$ the exactly-marginal operator that corresponds
to the coupling $\lambda^{i}$ and by the subscript ‘regulated’ that potential
UV divergences in the integrated four-point function have been regularised.
Since these divergences are a UV effect they can be regularised with the same
prescription used in Sec. 2.1 for the unbroken phase. In this manner, we
recover a connection $\nabla$ on the moduli space, which is independent of the
details of the vacuum state.

In Eq. (4.2) we used the fact that we can vary the exactly-marginal couplings
while keeping the vacuum state constant. As the choice of a vacuum amounts to
the choice of a superselection sector for QFTs in more than two spacetime
dimensions, this assumption may seem trivial. However, since the precise
details of a theory on a non-trivial vacuum are characterised by non-vanishing
vevs, one also needs to examine if these vevs can be chosen in a coupling-
constant independent fashion.101010In a Lagrangian formulation, the
correlation functions on the moduli space can be evaluated by defining the vev
of elementary fields, shifting the fields in the action around their new
vacuum, and subsequently performing the path integral. In the new path
integral the asymptotic behaviour of the fields at infinity is that of the
trivial vacuum but the action contains extra interactions that depend on the
vevs. It is therefore important, when we take derivatives with respect to the
exactly-marginal couplings, to know if the extra interactions involve
additional coupling-constant dependence.

On the Higgs branch one has non-vanishing vevs for Higgs-branch operators
$O_{H}$

$\langle v|O_{H}|v\rangle=v_{O}~{}.$ (4.3)

Therefore, it is important to analyse to what extent coupling-constant-
independent vevs $v_{O}$ can be consistently chosen. We propose that such a
choice can be made on the Higgs branch because the connection on the bundle of
Higgs-branch superconformal-primary operators is flat [4] and therefore one
can trivially choose the vacuum so that $\partial_{\lambda^{i}}v_{O}=0$. Under
these conditions Eq. (4.2) is correct. It should be appreciated that we had to
use a special feature of the Higgs-branch moduli space to reach this
conclusion.

In summary, the same connection $\nabla$ can be used to parallel transport
data both in the CFT phase (where the Higgs vev vanishes) and on Higgs-branch
moduli spaces (where the Higgs vev does not vanish).

##  5 Superconformal Ward Identities

The main ingredient of this section is a specific set of superconformal Ward
identities. We consider these identities first in the unbroken phase of the
CFT, where they are well known (see e.g. [32, 24]), and subsequently in the
broken phase of the Higgs branch, where to our knowledge they are less
explored. In this context, they can also be viewed as a particular class of
superconformal soft theorems along the lines of [31].

###  5.1 Unbroken Phase

Recall that a 4D ${\cal N}=2$ SCFT possesses a left-chiral
supercurrent111111It possesses also a right-chiral supercurrent. Since the
right-chiral version of the Ward identities in this section is completely
analogous to the left-chiral version, we will not repeat it explicitly.
$G^{{\cal I}\mu}_{\alpha}$, where ${\cal I}=1,2$ is an $SU(2)_{R}$ R-symmetry
index, $\mu=0,1,2,3$ a spacetime index and $\alpha=\pm$ a Weyl spinor
index.121212Our notation for the 4D ${\cal N}=2$ superconformal algebra can be
found in App. Appendix A. 4D $\mathcal{N}=2$ Superconformal Algebra
Conventions. We mostly follow the conventions of [33, 34]. $G^{{\cal
I}\mu}_{\alpha}$ is a conformal-primary operator with scaling dimension
$\Delta=\frac{7}{2}$. The currents

$j^{{\cal I}\mu}=\xi^{\alpha}(x)G^{{\cal I}\mu}_{\alpha}(x)$ (5.1)

are classically conserved for the conformal Killing spinors (on
${\mathbb{R}}^{4}$)

$\xi^{\alpha}(x)=\nu^{\alpha}+x^{\dot{\alpha}\alpha}\mu_{\dot{\alpha}}~{},$
(5.2)

where $\nu^{\alpha}$ and $\mu_{\dot{\alpha}}$ are arbitrary constant spinors.
Setting $\nu^{\alpha}\neq 0$, $\mu_{\dot{\alpha}}=0$ gives the left-chiral
Poincaré supercharges $Q^{\cal I}_{\alpha}$ while $\nu^{\alpha}=0$,
$\mu_{\dot{\alpha}}\neq 0$ gives the right-chiral superconformal charges
$\bar{S}^{{\cal I}\dot{\alpha}}$.

Now let us consider the correlation function of $n$ operators $O_{\ell}$,
$\ell=1,2,\ldots,n$

$\langle O_{1}(x_{1})\cdots O_{n}(x_{n})\rangle~{}.$ (5.3)

A standard derivation yields the Ward identity gives (see [24] for further
details)

$\displaystyle{\partial}_{\mu}\langle j^{{\cal I}\mu}(x)O_{1}(x_{1})\cdots
O_{n}(x_{n})\rangle$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{n}\delta(x-x_{\ell})\bigg{[}\xi^{\alpha}(x_{\ell})\langle
O_{1}(x_{1})\cdots[Q^{\cal I}_{\alpha},O_{\ell}](x_{\ell})\cdots
O_{n}(x_{n})\rangle$ (5.4) $\displaystyle-$
$\displaystyle\sigma^{\mu}_{\alpha\dot{\beta}}\left({\partial}_{\mu}\xi^{\alpha}\right)(x_{\ell})\langle
O_{1}(x_{1})\cdots[\bar{S}^{{\cal I}\dot{\beta}},O_{\ell}](x_{\ell})\cdots
O_{n}(x_{n})\rangle\bigg{]}\,.$

In the unbroken phase integrating over $x$ gives zero on the LHS of this
equation. This is because we are integrating a total derivative and there is
no boundary contribution from infinity for the $(n+1)$-point function $\langle
j^{{\cal I}\mu}(x)O_{1}(x_{1})\cdots O_{n}(x_{n})\rangle$. As a result, (5.4)
becomes

$\displaystyle\sum_{\ell=1}^{n}\bigg{[}\xi^{\alpha}(x_{\ell})\langle
O_{1}(x_{1})\cdots[Q^{\cal I}_{\alpha},O_{\ell}](x_{\ell})\cdots
O_{n}(x_{n})\rangle$
$\displaystyle-\sigma^{\mu}_{\alpha\dot{\beta}}\left({\partial}_{\mu}\xi^{\alpha}\right)(x_{\ell})\langle
O_{1}(x_{1})\cdots[\bar{S}^{{\cal I}\dot{\beta}},O_{\ell}](x_{\ell})\cdots
O_{n}(x_{n})\rangle\bigg{]}=0~{}.$ (5.5)

Taking

$\xi^{\alpha}(x)=(x-x_{0})^{\dot{\alpha}\alpha}\mu_{\dot{\alpha}}$ (5.6)

for arbitrary $x_{0}$, we obtain the superconformal Ward identity

$\sum_{\ell=1}^{n}\langle
O_{1}(x_{1})\cdots[(x_{\ell}-x_{0})^{\dot{\alpha}\alpha}Q^{\cal
I}_{\alpha}-\bar{S}^{{\cal I}\dot{\alpha}},O_{\ell}](x_{\ell})\cdots
O_{n}(x_{n})\rangle=0~{}.$ (5.7)

The right-chiral version of this Ward identity is

$\sum_{\ell=1}^{n}\langle
O_{1}(x_{1})\cdots[(x_{\ell}-x_{0})^{\dot{\alpha}\alpha}\bar{Q}_{{\cal
I}\dot{\alpha}}+S^{\alpha}_{\cal I},O_{\ell}](x_{\ell})\cdots
O_{n}(x_{n})\rangle=0~{}.$ (5.8)

In applications of these identities one needs to keep in mind the following
key point. When the operators $O_{\ell}$ are superconformal primaries (in
particular, when they are annihilated by the $\bar{S}$ supercharges) (5.7)
becomes

$\sum_{\ell=1}^{n}\langle
O_{1}(x_{1})\cdots[(x_{\ell}-x_{0})^{\dot{\alpha}\alpha}Q^{\cal
I}_{\alpha},O_{\ell}](x_{\ell})\cdots O_{n}(x_{n})\rangle=0~{}.$ (5.9)

Choosing $x_{0}$ to be one of the $x_{\ell}$, the corresponding operator
insertion can be ‘hidden’ from the Ward identity. Analogous statements apply
to the right-chiral version (5.8).

As a first example of a concrete application, consider the two-point function
$\langle{\cal O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle$, where as above
${\cal O}_{I}$ is a chiral-primary CBO and $\bar{\cal O}_{J}$ an anti-chiral-
primary CBO. By default, both operators are superconformal primaries and obey
the shortening conditions

$[\bar{Q}_{{\cal I}\dot{\alpha}},{\cal O}_{I}]=0~{},~{}~{}[\bar{S}^{{\cal
I}\dot{\alpha}},{\cal O}_{I}]=0~{},~{}~{}[S^{\alpha}_{\cal I},{\cal
O}_{I}]=0~{},$ (5.10) $[Q^{\cal I}_{\alpha},\bar{\cal
O}_{I}]=0~{},~{}~{}[\bar{S}^{{\cal I}\dot{\alpha}},\bar{\cal
O}_{I}]=0~{},~{}~{}[S^{\alpha}_{\cal I},\bar{\cal O}_{I}]=0~{}.$ (5.11)

We recall that the ${\cal N}=2$-preserving, exactly-marginal deformations of a
4D ${\cal N}=2$ SCFT can be written in the form

$\delta S=\lambda^{k}\int d^{4}x\,Q^{4}\cdot\phi_{k}(x)+\bar{\lambda}^{k}\int
d^{4}x\,\bar{Q}^{4}\cdot\bar{\phi}_{k}(x)~{},$ (5.12)

where $\phi_{k}$, $\bar{\phi}_{k}$ are chiral and anti-chiral CBOs with
scaling dimension two and the notation $Q^{4}\cdot\phi$ refers to the nested
(anti)commutator

$\varepsilon_{{\cal I}{\cal J}}\varepsilon_{{\cal K}{\cal
L}}\varepsilon^{\alpha\beta}\varepsilon^{\gamma\delta}\\{Q^{\cal
I}_{\alpha},[Q^{\cal K}_{\beta},\\{Q^{\cal J}_{\gamma},[Q^{\cal
L}_{\delta},\phi]\\}]\\}~{}.$ (5.13)

Analogous expressions apply to the complex-conjugate (anti)commutator.
Suppressing the spinor indices, but keeping the $SU(2)_{R}$ indices explicit,
it will also be convenient to use a notation of the form

$Q^{4}\cdot\phi\sim Q^{1}\cdot Q^{1}\cdot Q^{2}\cdot Q^{2}\cdot\phi$ (5.14)

For our first observation, let us single out one of supercharges and write the
exactly-marginal deformations in the following way

$\delta S=\lambda^{k}\int d^{4}x\,\\{Q^{1},{\cal
Y}_{k}\\}(x)+\bar{\lambda}^{k}\int d^{4}x\,\\{\bar{Q}_{1},\bar{\cal
Y}_{k}\\}(x)~{},$ (5.15)

where the index $k$ runs over the exactly-marginal couplings and ${\cal
Y}_{k}$ (resp. $\bar{\cal Y}_{k}$) are obtained by stripping off one of the
supercharges in the exactly-marginal operator (here we chose $Q^{1}$ and
$\bar{Q}_{1}$). Then, by definition

$\nabla_{k}\langle{\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=\left\langle\int d^{4}x\,\\{Q^{1},{\cal Y}_{k}\\}(x){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\right\rangle~{},$ (5.16)
$\nabla_{\bar{k}}\langle{\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=\left\langle\int d^{4}x\,\\{\bar{Q}_{1},\bar{\cal
Y}_{k}\\}(x){\cal O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\right\rangle~{}.$
(5.17)

Applying (5.7) to the 3-point function $\langle{\cal Y}_{k}(x){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle$ with $x_{0}=x_{1}$ and the right-
chiral version of (5.7) to $\langle\bar{\cal Y}_{k}(x){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle$ with $x_{0}=x_{2}$ we obtain that
the RHS of both (5.16), (5.17) vanish. To summarise, in the conformal phase of
an ${\cal N}=2$ SCFT we obtain

$\nabla_{k}\langle{\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=0~{},~{}~{}\nabla_{\bar{k}}\langle{\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle=0~{},$ (5.18)

a known statement that appeared, e.g., in [32, 24].

As a second example, which is more closely related to our discussion, consider
the three-point function $\langle T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle$ of the energy-momentum tensor $T_{\mu\nu}$ with two
CBOs. As we review in Appendix Appendix A. 4D $\mathcal{N}=2$ Superconformal
Algebra Conventions, the energy-momentum tensor is a conformal primary in the
short superconformal multiplet $\hat{\cal C}_{(0,0)}$. The bottom component in
this multiplet is a scalar superconformal-primary field ${\cal T}$ with
dimension $\Delta=2$ and vanishing $SU(2)_{R}$ and $U(1)_{r}$ R-charges, that
obeys the shortening conditions

$(Q^{\cal I})^{2}\cdot{\cal T}=0~{},~{}~{}(\bar{Q}_{\cal I})^{2}\cdot{\cal
T}=0~{},~{}~{}{\cal I}=1,2~{}.$ (5.19)

The energy-momentum tensor is obtained by the successive application of one
$Q^{1}$, one $Q^{2}$, one $\bar{Q}_{1}$ and one $\bar{Q}_{2}$ supercharge on
the superconformal primary ${\cal T}$. Different orders of the application of
these supercharges gives operators $T_{\mu\nu}$ that differ by total
derivative terms. A particular combination of these different possibilities
provides the conformal primary $T_{\mu\nu}$. Equivalently, $T_{\mu\nu}$ can be
written with any ordering of supercharges by adding the appropriate
descendants. For example, one can schematically write $T_{\mu\nu}$
(suppressing spacetime indices, spinor indices and sigma-matrices on the RHS)
as

$T_{\mu\nu}=Q^{1}Q^{2}\bar{Q}_{1}\bar{Q}_{2}{\cal
T}+c_{1}Q^{1}\bar{Q}_{1}\partial{\cal T}+c_{2}Q^{2}\bar{Q}_{2}\partial{\cal
T}+c_{3}\partial^{2}{\cal T}\;,$ (5.20)

with suitable numerical constants $c_{1},c_{2},c_{3}$.

Armed with these properties of the energy-momentum tensor, we proceed to study
the covariant derivatives

$\nabla_{k}\langle T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=\left\langle\int
d^{4}x\,Q^{4}\cdot\phi_{k}(x)T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\right\rangle~{},$ (5.21) $\nabla_{\bar{k}}\langle
T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=\left\langle\int
d^{4}x\,\bar{Q}^{4}\cdot\bar{\phi}_{k}(x)T_{\mu\nu}(y){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\right\rangle~{}.$ (5.22)

For concreteness, let us focus on the first of these correlation functions
(similar arguments apply to the second, complex-conjugate version). Repeating
the steps from the preceding computation of the two-point function, one of the
$Q^{1}$ supercharges from the exactly-marginal deformation can be singled out
and moved around inside the correlation function using the superconformal Ward
identities (5.7) as applied to the four-point function

$\langle Q^{1}(Q^{2})^{2}\cdot\phi_{k}(x)T_{\mu\nu}(y){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle$ (5.23)

with $x_{0}=x_{1}$.131313In (5.23) and the expressions that follow we will
mostly suppress the spinor indices to avoid cluttering the notation. There is
obviously no contribution to this Ward identity from the action of the
superconformal charges on the CBOs ${\cal O}_{I}$ and $\bar{\cal O}_{J}$, but
there are two potential contributions of the form

${\cal I}_{1}=\langle Q^{1}(Q^{2})^{2}\cdot\phi_{k}(x)\,Q^{1}\cdot
T_{\mu\nu}(y)\,{\cal O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle~{},$ (5.24)
${\cal I}_{2}=\langle Q^{1}(Q^{2})^{2}\cdot\phi_{k}(x)\,\bar{S}^{1}\cdot
T_{\mu\nu}(y)\,{\cal O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle$ (5.25)

from the action of the supercurrent on the energy-momentum tensor. Explicit
calculations, which we will present in a moment, show that these potential
contributions are actually both zero. For the reader who is not interested in
the details, this fact establishes that

$\nabla_{k}\langle T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=0~{}.$ (5.26)

Similarly, employing the complex-conjugate version of the superconformal Ward
identities (5.9) one obtains

$\nabla_{\bar{k}}\langle T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=0~{}.$ (5.27)

Both results are consistent with the fact that all tensor structures in the
three-point function $\langle T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle$ (in the unbroken phase) are proportional to the
coefficients of the two-point functions $\langle{\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle$; see e.g. [29]. On the one hand, we established in
(5.18) that the latter are covariantly constant. On the other, we have just
shown that the full three-point function $\langle T_{\mu\nu}(y){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle$ is covariantly constant
(including the part of the type-B anomaly) without using any knowledge about
its structure and its relation to the two-point function of the CBOs.

### Explicit Calculation of ${\cal I}_{1}={\cal I}_{2}=0$

Here we present the full evaluation of (5.24), (5.25), starting with ${\cal
I}_{1}$. This involves the commutator $Q^{1}\cdot T_{\mu\nu}$:

  * •

We notice that the first two terms on the RHS of (5.20) cannot contribute in
this commutator. Indeed, if we have an operator $\Lambda$ with the property
$(Q^{1})^{2}\cdot\Lambda=0$, then

$\displaystyle Q^{1}\cdot Q^{1}\cdot\bar{Q}_{1}\cdot\Lambda$ $\displaystyle=$
$\displaystyle
Q^{1}\cdot\\{Q^{1},\bar{Q}_{1}\\}\cdot\Lambda-Q^{1}\cdot\bar{Q}_{1}\cdot
Q^{1}\cdot\Lambda$ (5.28) $\displaystyle=$ $\displaystyle
Q^{1}\cdot\partial\Lambda-\\{Q^{1},\bar{Q}_{1}\\}\cdot
Q^{1}\cdot\Lambda+\bar{Q}^{1}\cdot Q^{1}\cdot Q^{1}\cdot\Lambda$
$\displaystyle=$ $\displaystyle
Q^{1}\cdot\partial\Lambda-Q^{1}\cdot\partial\Lambda=0~{}.$ (5.29)

For the first term in $Q^{1}\cdot T_{\mu\nu}$ we apply this identity to
$\Lambda=Q^{2}\cdot\bar{Q}_{2}\cdot{\cal T}$ while for the second term we
apply it to $\Lambda=\partial{\cal T}$. In both cases, the identity
$(Q^{1})^{2}\cdot\Lambda=0$ follows from the shortening condition (5.19).

  * •

For the last two terms we can use the superconformal Ward identity (5.7) once
again to move around the second $Q^{1}$ coming from the exactly-marginal
deformation inside ${\cal I}_{1}$. As before, there is no contribution from
the action of the superconformal charges on the CBOs. There is also no
contribution from $\bar{S}^{1}$ on the last two terms of $Q^{1}\cdot
T_{\mu\nu}$ because the former commutes with all supercharges to act directly
on ${\cal T}$, which is a primary. Additionally, there is no contribution from
the action of the second $Q^{1}$ on the last two terms of $Q^{1}\cdot
T_{\mu\nu}$, because it involves the action $(Q^{1})^{2}\cdot{\cal T}$ that
vanishes through (5.19).

The second potential contribution ${\cal I}_{2}$ involves the commutator
$\bar{S}^{1}\cdot T_{\mu\nu}$. This is slightly more involved, but one can
again argue using the same superconformal Ward identity (5.7) that it
vanishes. Because of (5.20) $\bar{S}^{1}\cdot T_{\mu\nu}$ involves four terms,
which we will examine in detail:

  * •

Since $\bar{S}^{1}\cdot{\cal T}=0$, the last term $c_{3}\partial^{2}{\cal T}$
in (5.20) does not contribute.

  * •

The third term in (5.20) gives

$\bar{S}^{1}\cdot Q^{2}\cdot\bar{Q}_{2}\cdot\partial{\cal
T}=-Q^{2}\cdot\bar{S}^{1}\cdot\bar{Q}_{2}\cdot\partial{\cal
T}=Q^{2}\cdot\\{\bar{S}^{1},\bar{Q}_{2}\\}\cdot\partial{\cal T}~{},$ (5.30)

where in the last equality we used that $\bar{S}^{1}\cdot{\cal T}=0$. Using
the superconformal algebra relation (see Appendix Appendix A. 4D
$\mathcal{N}=2$ Superconformal Algebra Conventions for the explicit form with
all spinor indices reinstated)

$\\{\bar{S}^{1},\bar{Q}_{2}\\}=-R^{1}_{~{}2}\;,$ (5.31)

and the fact that the primary ${\cal T}$ is also the highest weight of the
$SU(2)_{R}$ representation $R^{1}_{~{}2}\cdot{\cal T}=0$, we find
$\bar{S}^{1}\cdot Q^{2}\cdot\bar{Q}_{2}\cdot\partial{\cal T}=0$. Here
$R{{}^{1}}_{2}=R_{+}$ is the raising operator of $SU(2)_{R}$ [33].

  * •

The second term in (5.20) yields the contribution

$\bar{S}^{1}\cdot Q^{1}\cdot\bar{Q}_{1}\cdot\partial{\cal
T}=-Q^{1}\cdot\bar{S}^{1}\cdot\bar{Q}_{1}\cdot\partial{\cal
T}=Q^{1}\cdot\\{\bar{S}^{1},\bar{Q}_{1}\\}\cdot\partial{\cal T}~{}.$ (5.32)

From the superconformal algebra we have that

$\\{\bar{S}^{1},\bar{Q}_{1}\\}=\bar{M}+\frac{i}{2}D-R^{1}_{~{}1}~{},$ (5.33)

where $\bar{M}$ are Lorentz generators and $D$ the dilatation generator. Both
$\bar{M}$ and $R^{1}_{~{}1}$ annihilate the superconformal primary ${\cal T}$,
since it is a scalar with zero $SU(2)_{R}$ and $U(1)_{r}$ quantum numbers and
here $R{{}^{1}}_{1}=R+{\frac{1}{2}}r$. Moreover, $[D,{\cal T}]=2i{\cal T}$.
Taking into account the presence of the derivative in ${\partial}{\cal T}$ we
end up with a potential contribution that is proportional to the correlator

$\left\langle Q^{1}(Q^{2})^{2}\cdot\phi_{k}(x)\,Q^{1}\partial{\cal T}(y){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\right\rangle~{}.$ (5.34)

Using the superconformal Ward identity (5.7) to move $Q^{1}$ around we find
that this correlation function vanishes.

  * •

The final potential contribution comes from the first term in (5.20)

$\bar{S}^{1}\cdot Q^{1}\cdot Q^{2}\cdot\bar{Q}_{2}\cdot\bar{Q}_{1}\cdot{\cal
T}=Q^{1}\cdot Q^{2}\cdot\bar{S}^{1}\cdot\bar{Q}_{2}\cdot\bar{Q}_{1}\cdot{\cal
T}~{}.$ (5.35)

We notice that

$\displaystyle\bar{S}^{1}\cdot\bar{Q}_{2}\cdot\bar{Q}_{1}\cdot{\cal T}$
$\displaystyle=$
$\displaystyle\\{\bar{S}^{1},\bar{Q}_{2}\\}\cdot\bar{Q}_{1}\cdot{\cal
T}-\bar{Q}_{2}\cdot\\{\bar{S}^{1},\bar{Q}_{1}\\}\cdot{\cal T}$ (5.36)
$\displaystyle=$ $\displaystyle-R^{1}_{~{}2}\cdot\bar{Q}_{1}\cdot{\cal
T}-\bar{Q}_{2}\cdot\left(\bar{M}+\frac{i}{2}D-R^{1}_{~{}1}\right)\cdot{\cal
T}$ $\displaystyle=$ $\displaystyle-[R^{1}_{~{}2},\bar{Q}_{1}]\cdot{\cal
T}+\bar{Q}_{2}\cdot{\cal T}$ $\displaystyle=$
$\displaystyle\bar{Q}_{2}\cdot{\cal T}+\bar{Q}_{2}\cdot{\cal
T}=2\bar{Q}_{2}\cdot{\cal T}~{}.$

Hence, the contribution to $\bar{S}^{1}\cdot T_{\mu\nu}$ is proportional to
$Q^{1}\cdot Q^{2}\cdot\bar{Q}_{2}\cdot{\cal T}$, which leads to the
correlation function

$\left\langle Q^{1}(Q^{2})^{2}\cdot\phi_{k}(x)\,Q^{1}\cdot
Q^{2}\cdot\bar{Q}_{2}\cdot{\cal T}(y)\,{\cal O}_{I}(x_{1})\,\bar{\cal
O}_{J}(x_{2})\right\rangle~{}.$ (5.37)

Using the superconformal Ward identity (5.7) once again, we move the second
$Q^{1}$ around. There is no contribution from the resulting action of $Q^{1}$
or $\bar{S}^{1}$ on the CBOs. From the action of $Q^{1}$ on the $Q^{1}\cdot
Q^{2}\cdot\bar{Q}_{2}\cdot{\cal T}$ factor there is no contribution, because
both $Q^{1}$s move past $Q^{2}$ and $\bar{Q}_{2}$ and $(Q^{1})^{2}\cdot{\cal
T}=0$. From the action of $\bar{S}^{1}$ on $Q^{1}\cdot
Q^{2}\cdot\bar{Q}_{2}\cdot{\cal T}(y)$ we also obtain zero because

$\displaystyle\bar{S}^{1}\cdot Q^{1}\cdot Q^{2}\cdot\bar{Q}_{2}\cdot{\cal T}$
$\displaystyle=$ $\displaystyle Q^{1}\cdot
Q^{2}\cdot\bar{S}^{1}\cdot\bar{Q}_{2}\cdot{\cal T}=Q^{1}\cdot
Q^{2}\cdot\\{\bar{S}^{1},\bar{Q}_{2}\\}\cdot{\cal T}$ (5.38) $\displaystyle=$
$\displaystyle-Q^{1}\cdot Q^{2}\cdot R^{1}_{~{}2}\cdot{\cal T}=0~{}.$

###  5.2 Broken Phase

We are now in position to explore similar properties in the broken phase on
the Higgs branch. In general, Ward identities are operator equations and
retain the same form independently of the vacuum. Nevertheless, when a Ward
identity is applied for a current whose corresponding symmetry is broken on
the vacuum, some care needs to be taken with the asymptotic behaviour of the
fields at infinity. The latter phenomenon is typically responsible for extra
contributions that are absent in the unbroken phase.

More specifically, consider the superconformal Ward identity (5.4), which is
of central importance to our discussion. In this form the Ward identity
continues to hold on the Higgs branch, despite the fact that the Higgs vevs
break the superconformal symmetry. As has been already noted, in order to
derive (5.1) one needs to integrate $x$ over the whole of ${\mathbb{R}}^{4}$.
The integral of the RHS of (5.4), which involves contact terms, is the same in
the broken and unbroken phases, but the LHS can be different if there is a
boundary term for the $(n+1)$-point function $\langle
j^{I\mu}(x)O_{1}(x_{1})\cdots O_{n}(x_{n})\rangle$ at $|x|\to\infty$. In the
unbroken phase all correlation functions run sufficiently fast to zero at
infinity and a boundary term is absent. In the broken phase the supermultiplet
that contains the dilaton couples linearly to the supermultiplet that contains
the energy-momentum tensor and supercurrents–c.f. Eq. (A.5)—to create a
massless pole in the analytic structure of the $(n+1)$-point function in
momentum space. It is precisely this pole that can generate potential
contributions otherwise absent in the unbroken phase. For the case at hand,
the following observations can be made.

In momentum space the integrated LHS of (5.4) corresponds to taking a low-
momentum limit. In this limit the above massless pole dominates the correlator
and yields the contribution

$\lim_{p\to 0}p_{\mu}\langle v|j^{I\mu}(p)O_{1}(q_{1})\cdots
O_{n}(q_{n})|v\rangle\simeq\lim_{p\to 0}p_{\mu}\langle
v|j^{I\mu}(p)|\chi^{\alpha}(p)\rangle\,\frac{p_{\alpha\dot{\alpha}}}{p^{2}}\,\langle\bar{\chi}^{\dot{\alpha}}(p)|O_{1}(q_{1})\cdots
O_{n}(q_{n})|v\rangle~{},$ (5.39)

where $\chi^{\alpha}$ is the Weyl dilatino that couples to the supercurrent
and propagates as an intermediate massless state. We are using bra-ket
notation, with the appropriate time-ordering prescriptions left implicit,
while $|v\rangle$ is a Higgs-branch vacuum state. In momentum space Eq. (5.1)
reads141414In this argument we focus again on the left-chiral supercurrents.
Analogous statements can be made in an obvious manner for the right-chiral
parts of the supercurrents.

$j^{{\cal
I}\mu}(p)=\left(\nu^{\alpha}-i\mu_{\dot{\alpha}}\frac{{\partial}}{{\partial}p_{\alpha\dot{\alpha}}}\right)G^{{\cal
I}\mu}_{\alpha}(p)~{}.$ (5.40)

Also, let us denote the linear coupling

$\langle v|G_{\beta}^{{\cal I}\mu}(p)|\chi^{\alpha}\rangle=f^{\cal
I}p^{\mu}\delta^{\alpha}_{\beta}\;,$ (5.41)

with a non-vanishing coefficient $f^{\cal I}$ that is left unspecified, and
the last factor in (5.39)

${\cal
F}^{\dot{\alpha}}\equiv\langle\bar{\chi}^{\dot{\alpha}}|O_{1}(q_{1})\cdots
O_{n}(q_{n})|v\rangle~{}.$ (5.42)

Then (5.39) becomes

$\displaystyle\lim_{p\to 0}p_{\mu}\langle v|j^{{\cal
I}\mu}(p)O_{1}(q_{1})\cdots O_{n}(q_{n})|v\rangle\simeq\lim_{p\to
0}\left(p_{\alpha\dot{\alpha}}\nu^{\alpha}{\cal F}^{\dot{\alpha}}f^{\cal
I}-i\mu_{\dot{\alpha}}{\cal F}^{\dot{\alpha}}f^{\cal
I}\right)=-i\mu_{\dot{\alpha}}{\cal F}^{\dot{\alpha}}f^{\cal I}~{}.$ (5.43)

The first term on the RHS, which is proportional to the constant part
$\nu^{\alpha}$ of the Killing spinor $\xi^{\alpha}$, represents the
contribution of the Poincaré supercharges. This is proportional to the
momentum $p$ and vanishes in the IR limit. As expected, there is no boundary
contribution from this term on the LHS of (5.4) in accordance with the fact
that the Poincaré supersymmetries are not broken on the moduli space. The
second term on the RHS of (5.43) represents the non-vanishing contributions of
the superconformal charges, whose corresponding symmetry is broken by the
vacuum. This term expresses potentially non-vanishing effects of the symmetry
breaking in the superconformal Ward identities of interest.

So far the arguments about the structure of the superconformal Ward identities
(5.4) in the broken phase have been quite general. The key ingredient has been
the generic linear coupling of the dilatino to the supercurrent in a
superconformal theory without relying on any special properties of the moduli
space or the $n$ operator insertions $O_{\ell}$ ($\ell=1,2,\ldots,n$).
However, now we would like to specialise to the case of the four-point
functions $\langle Q^{4}\cdot\phi_{k}(z)T_{\mu\nu}(y){\cal
O}_{i}(x_{1})\bar{\cal O}_{j}(x_{2})\rangle$ and
$\langle\bar{Q}^{4}\cdot\bar{\phi}_{k}(z)T(y){\cal O}_{i}(x_{1})\bar{\cal
O}_{j}(x_{2})\rangle$ that appeared in the second example of Sec. 5.1, with
the goal of exploring whether (5.26), (5.27) hold in the broken phase.

Let us repeat the steps that we performed in the unbroken phase. When one
applies the Ward identities (5.4) (and their complex conjugate) to the
superconformal current $j^{1\mu}$ ($\bar{j}_{1}^{\mu}$) in order to move
around a $Q^{1}$ (respectively, a $\bar{Q}_{1}$) from the exactly-marginal
interaction, a new boundary term can arise involving five-point functions of
the form

${\partial}_{\mu}\langle j^{1\mu}(x){\cal Y}_{k}(z)T(y){\cal
O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle~{},~{}~{}{\partial}_{\mu}\langle\bar{j}_{1}^{\mu}(x)\bar{\cal
Y}_{k}(z)T(y){\cal O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle~{}.$ (5.44)

As we argued in (5.43) (and implied for its complex conjugate version) in
momentum space this boundary term (and its conjugate version) will be
proportional to the following amplitudes

$\langle\bar{\chi}^{\dot{\alpha}}(p)|{\cal Y}_{k}(q_{1})T(q_{2}){\cal
O}_{i}(q_{3})\bar{\cal
O}_{j}(q_{4})|v\rangle~{},~{}~{}\langle\chi_{\alpha}(p)|\bar{\cal
Y}_{k}(q_{1})T(q_{2}){\cal O}_{i}(q_{3})\bar{\cal O}_{j}(q_{4})|v\rangle~{}.$
(5.45)

We notice that the $U(1)_{r}$ symmetry is not preserved in any of these two
amplitudes.151515As detailed in App. Appendix A. 4D $\mathcal{N}=2$
Superconformal Algebra Conventions, the energy-momentum tensor $T$, as well as
the combination ${\cal O}_{i}\bar{\cal O}_{j}$ are both $U(1)_{r}$ neutral.
The supercurrent $j^{\mu}$ and the dilatino have both charge $+{\frac{1}{2}}$,
while $\mathcal{Y}_{k}$ has charge $-{\frac{1}{2}}$, with opposite charge
assignments for their conjugates. In our notation $\langle\bar{\chi}|$ has
charge $-{\frac{1}{2}}$. Therefore, as the $U(1)_{r}$ is preserved on the
Higgs branch, they both have to vanish and the potential boundary-term
contributions to the superconformal Ward identity (5.4) cannot arise in this
particular case. We would like to emphasise that this argument crucially
relies on the absence of an amplitude with the intermediate propagation of a
dilatino without the simultaneous breaking of the $U(1)_{r}$ symmetry.
Consequently, this line of reasoning would not go through for the same
correlation functions on the Coulomb branch, where the $U(1)_{r}$ symmetry is
broken by the corresponding vevs.

To conclude, we have shown that there is no modification of the integrated
Ward identities (5.1) for the correlation functions of interest on the Higgs
branch. Hence, one can retrace the steps of the second example of Sec. 5.1 to
argue that the relations

$\nabla_{k}\langle T_{\mu\nu}(y){\cal O}_{I}(x_{1})\bar{\cal
O}_{J}(x_{2})\rangle=0~{},~{}~{}\nabla_{\bar{k}}\langle T_{\mu\nu}(y){\cal
O}_{I}(x_{1})\bar{\cal O}_{J}(x_{2})\rangle=0$ (5.46)

hold both in the unbroken phase, as well as in the broken phase along the
Higgs branch. This is a powerful statement with important implications that
will be highlighted in the following sections. The analogous identities for
three-point functions with the trace of the energy-momentum tensor (that are
involved in the type-B anomaly) are a special case of (5.46).

##  6 Type-B Anomaly Matching on the Higgs Branch

In Sec. 2.3 we denoted the value of the type-B anomaly at an arbitrary point
${\bm{\lambda}}$ of the conformal manifold ${\cal M}$ as
$G_{I\bar{J}}^{\mathbb{H}}({\bm{\lambda}})$ in the broken phase of the Higgs
branch ${\mathbb{H}}_{\bm{\lambda}}$ and as $G_{I\bar{J}}^{\rm
CFT}({\bm{\lambda}})$ in the unbroken conformal phase.

The identities (5.46) now imply that the type-B anomalies
$G_{I\bar{J}}^{\mathbb{H}}$ on the Higgs branch are covariantly constant
across the superconformal manifold ${\cal M}$, namely

$\nabla_{a}G_{I\bar{J}}^{\mathbb{H}}=0~{}.$ (6.1)

In this equation the index $a$ can be any holomorphic or anti-holomorphic
index on the tangent space of the superconformal manifold ${\cal M}$ and
expresses a covariant derivative with respect to any exactly-marginal coupling
of the theory. Note that in Sec. 5.1 we also showed that

$\nabla_{a}G_{I\bar{J}}^{\rm CFT}=0$ (6.2)

are obeyed in the conformal phase (see Eqs. (5.18)). A non-perturbative formal
proof of type-B anomaly matching between the broken and unbroken phases, in a
range of values of the exactly-marginal couplings, can now be obtained in the
following manner.

Let us assume that the anomalies match at a special point ${\bm{\lambda}}^{*}$
of the conformal manifold, i.e. that

$G_{I\bar{J}}^{\mathbb{H}}({\bm{\lambda}}^{*})=G_{I\bar{J}}^{\rm
CFT}({\bm{\lambda}}^{*})~{}.$ (6.3)

In a moment we will present examples of Lagrangian theories where this
equality can be established at weak coupling with a tree-level computation. In
those cases ${\bm{\lambda}}^{*}$ is a weak-coupling point of the conformal
manifold. We stress that (6.3) is an assumption in the present argument. It
does not hold in all 4D ${\cal N}=2$ SCFTs.

Then, the combination of the general identities (6.1), (6.2) with the
assumption (6.3) yields

$\nabla_{a}\left(G_{I\bar{J}}^{\mathbb{H}}({\bm{\lambda}})-G_{I\bar{J}}^{\rm
CFT}({\bm{\lambda}})\right)=0~{},~{}~{}G_{I\bar{J}}^{\mathbb{H}}({\bm{\lambda}}^{*})-G_{I\bar{J}}^{\rm
CFT}({\bm{\lambda}}^{*})=0~{}.$ (6.4)

The first equality implies that an arbitrary number of covariant derivatives
on $G_{I\bar{J}}^{\mathbb{H}}({\bm{\lambda}})-G_{I\bar{J}}^{\rm
CFT}({\bm{\lambda}})$ vanish at ${\bm{\lambda}}={\bm{\lambda}}^{*}$.
Therefore, combined with the second equality we infer that an arbitrary number
of ordinary derivatives of
$G_{I\bar{J}}^{\mathbb{H}}({\bm{\lambda}})-G_{I\bar{J}}^{\rm
CFT}({\bm{\lambda}})$ also vanishes at ${\bm{\lambda}}={\bm{\lambda}}^{*}$,
namely

${\partial}_{a_{1}}\cdots{\partial}_{a_{n}}\left(G_{I\bar{J}}^{\mathbb{H}}-G_{I\bar{J}}^{\rm
CFT}\right)|_{{\bm{\lambda}}={\bm{\lambda}}^{*}}=0$ (6.5)

for arbitrary integer $n$.161616Alternatively, one can show that the system of
first-order differential equations in (6.4) admits only the trivial solution.
As a result, we conclude that the quantities
$G_{I\bar{J}}^{\mathbb{H}}({\bm{\lambda}})-G_{I\bar{J}}^{\rm
CFT}({\bm{\lambda}})$ vanish at all points ${\bm{\lambda}}$ (at least in a
finite neighbourhood of ${\bm{\lambda}}^{*}$) establishing the type-B anomaly
matching on the Higgs branch non-perturbatively, for finite values of the
exactly-marginal couplings. This argument works independently in each sector
of CBOs with the same scaling dimension.

The above statement can be practically very powerful: It allows for the
evaluation of the quantities $G_{I\bar{J}}$ at generic points of the Higgs
branch by means of a corresponding CFT computation in the unbroken phase at
the origin of the moduli space. The latter can be performed by taking suitable
derivatives of the $S^{4}$ partition function with respect to the exactly-
marginal couplings [5, 6, 7, 8, 9]. In theories with a known Lagrangian
formulation the $S^{4}$ partition function can be further reduced, using
supersymmetric localisation methods [10], to a finite-dimensional integral.

In Sec. 9 we will employ these considerations to a specific 4D ${\cal N}=2$
circular quiver that deconstructs the 6D ${\cal N}=(2,0)$ theory on a torus,
to obtain certain predictions for a new class of exactly-computable data in
the 6D $(2,0)$ theory.

##  7 Examples at Tree-Level

We now present three examples of Lagrangian theories where type-B anomalies on
the Higgs branch are evaluated with a simple tree-level computation at weak
coupling.171717Here we will perform the computation in components. It could
also be performed conveniently in ${\cal N}=1$ superspace language using
superspace Feynman diagrams, which could be useful in explicit higher-loop
computations. We hope to return to an explicit higher-loop demonstration of
the proposed type-B anomaly matching in future work. The first example
concerns ${\cal N}=2$ superconformal QCD (SCQCD). This is a simple case where
the anomaly matching can be established at tree level and therefore holds
nonperturbatively according to our general argument. The second example
involves a certain 4D ${\cal N}=2$ superconformal circular quiver. For quivers
with a finite number of nodes $N$, we find a mismatch already at tree level.
However, this mismatch is suppressed by $\frac{1}{N}$ and therefore the
anomalies match in the large-$N$ limit. The latter plays a crucial role in the
deconstruction of 6D ${\cal N}=(2,0)$ theory on $\mathbb{T}^{2}$ that will be
discussed in Sec. 9. The final, third example revisits (and extends) the issue
of type-B anomaly matching in 4D ${\cal N}=4$ SYM theory. This provides a
check of the Ward identities and theorem proposed in Secs. 5, 6.

###  7.1 ${\cal N}=2$ SCQCD

$k$$\tilde{Q}_{2}$$Q_{2}$$Q_{1}$$\tilde{Q}_{1}$$k$$k$ Fig. 1: The quiver
diagram of ${\cal N}=2$ superconformal QCD.

The field content of ${\cal N}=2$ SCQCD theory is captured by the quiver
diagram in Fig. 1. The circular node depicts the gauge group of ${\cal N}=2$
SYM theory, which we take here to be $SU(k)$. The ${\cal N}=2$ SYM theory is
coupled to $2k$ hypermultiplets. In the quiver of Fig. 1 we have chosen to
depict explicitly only a $U(k)\times U(k)$ part of the full $U(2k)$ flavour
symmetry group using two square nodes. The links reflect pairs of fundamental
${\cal N}=1$ chiral multiplets that build up two sets of ${\cal N}=2$
hypermultiplets.

It is convenient to denote the hypermultiplets as $(q_{\cal
I})^{a}_{\mathfrak{i}}$, where $a=1,\ldots,k$ is a colour index,
${\mathfrak{i}}=1,\ldots,2k$ is a flavour index and ${\cal I}=1,2$ is an
$SU(2)_{R}$ index. In terms of the ${\cal N}=1$ chiral fundamentals

$q_{1}=\bigg{\\{}{Q_{1}\atop
Q_{2}}{\mathfrak{i}=1,\ldots,k\atop~{}~{}~{}~{}\mathfrak{i}=k+1,\ldots,2k}$
(7.1)
$q_{2}=\bigg{\\{}{\overline{\tilde{Q}}_{1}\atop\overline{\tilde{Q}}_{2}}{\mathfrak{i}=1,\ldots,k\atop~{}~{}~{}~{}\mathfrak{i}=k+1,\ldots,2k}~{}.$
(7.2)

The ${\cal N}=2$ SCQCD theory has a multi-dimensional Higgs-branch moduli
space [30]. In this subsection, we consider the specific direction where

$\langle(q_{1})^{a}_{\mathfrak{i}}\rangle=v\delta_{\mathfrak{i}}^{a}~{},~{}~{}\langle(q_{2})^{a}_{\mathfrak{i}}\rangle=0$
(7.3)

using the same letter $q_{\cal I}$ for the bottom scalar components of the
corresponding supermultiplets.181818We hope it will be clear from the context
when we will be referring to a component versus a full superfield.

We will be working in components following the conventions of [35, 34]. The
Lagrangian of the theory can be written as ${\cal L}={\cal L}_{V}+{\cal
L}_{H}$, where $(V)$ denotes the vector-multiplet part and $(H)$ the
hypermultiplet part. We remind the reader that besides the gauge field, whose
field strength is $F_{\mu\nu}$, the ${\cal N}=2$ vector multiplet possesses a
complex scalar field $\varphi$ and Weyl fermions $\lambda^{\cal I}$ all in the
adjoint representation of the gauge group. The hypermultiplets contain, beside
the complex scalar fields $q_{\cal I}$, corresponding fermions
$\psi,\tilde{\psi}$. For reference, in this notation,

$\displaystyle{\cal L}_{V}$ $\displaystyle=$
$\displaystyle-{\rm{Tr}}\bigg{[}\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+i\bar{\lambda}_{\cal
I}\bar{\sigma}^{\mu}D_{\mu}\lambda^{\cal
I}+D^{\mu}\varphi\overline{D_{\mu}\varphi}$ (7.4)
$\displaystyle+ig\sqrt{2}\left(\epsilon_{{\cal I}{\cal J}}\lambda^{\cal
I}\lambda^{\cal J}\bar{\varphi}-\epsilon^{{\cal I}{\cal J}}\bar{\lambda}_{\cal
I}\bar{\lambda}_{\cal
J}\varphi\right)+\frac{g^{2}}{2}[\varphi,\bar{\varphi}]^{2}\bigg{]}~{},$
$\displaystyle{\cal L}_{H}$ $\displaystyle=$
$\displaystyle-\bigg{[}D^{\mu}\bar{q}^{\cal I}D_{\mu}q_{\cal
I}+i\bar{\psi}\bar{\sigma}^{\mu}D_{\mu}\psi+i\tilde{\psi}\bar{\sigma}^{\mu}D_{\mu}\bar{\tilde{\psi}}+i\sqrt{2}g\left(\varepsilon^{{\cal
I}{\cal J}}\bar{\psi}\bar{\lambda}_{\cal I}q_{\cal J}-\varepsilon_{{\cal
I}{\cal J}}\bar{q}^{\cal I}\lambda^{\cal J}\psi\right)$ (7.5)
$\displaystyle+g\tilde{\psi}\lambda^{\cal I}q_{\cal I}-g\bar{q}^{\cal
I}\bar{\lambda}_{\cal
I}\bar{\tilde{\psi}}+g\tilde{\psi}\varphi\psi-g\bar{\psi}\bar{\varphi}\bar{\tilde{\psi}}+g^{2}\bar{q}_{\cal
I}(\bar{\varphi}\varphi+\varphi\bar{\varphi})q^{\cal I}+g^{2}V(q)\bigg{]}~{}.$

Here $V(q)$ is a quartic expression in the $q$s, the detailed form for which
will not be important; for the explicit form of $V(q)$ we refer the reader to
[34]. The $D_{\mu}$ are standard gauge-covariant derivatives.

To analyse the Higgs branch we implement the following shift on the
hypermultiplets

$(q_{1})^{a}_{\mathfrak{i}}\to
v\delta^{a}_{\mathfrak{i}}+(q_{1})^{a}_{\mathfrak{i}}~{},~{}~{}(q_{2})^{a}_{\mathfrak{i}}\to(q_{2})^{a}_{\mathfrak{i}}~{}.$
(7.6)

After the shift, the following terms in (7.5) are especially important for the
purposes of our computation

$\mathcal{L}_{H}\supset-g^{2}v{\rm{Tr}}\left[(\bar{\varphi}\varphi+\varphi\bar{\varphi})(Q_{1}+\bar{Q}_{1})\right]-2g^{2}v^{2}{\rm{Tr}}\left[\varphi\bar{\varphi}\right]~{}.$
(7.7)

The last term shows that the adjoint complex scalar field $\varphi$ has mass
squared $m^{2}=2g^{2}v^{2}$.

A crucial aspect of the spontaneous breaking of conformal invariance on the
moduli space is the associated massless Goldstone boson, the dilaton. In the
present example, the dilaton takes the form

$\sigma:={\rm{Tr}}\left[Q_{1}+\bar{Q}_{1}\right]~{}.$ (7.8)

As we noted previously, the dilaton has a linear coupling to the energy-
momentum tensor $T_{\mu\nu}$ (see e.g. [2]) of the form

$\begin{split}\leavevmode\hbox to206.36pt{\vbox
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}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} } \pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{162.18063pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{122.43692pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\sim-\frac{1}{6}v(q_{\mu}q_{\nu}-\eta_{\mu\nu}q^{2})\sigma~{}.$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{11.38092pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.69432pt}{11.95694pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T_{\mu\nu}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{11.38092pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.96947pt}{-15.40694pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$q$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{93.89418pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{91.03712pt}{12.07361pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@moveto{53.21504pt}{0.0pt}\pgfsys@curveto{53.21504pt}{1.10458pt}{52.31963pt}{2.0pt}{51.21504pt}{2.0pt}\pgfsys@curveto{50.11046pt}{2.0pt}{49.21504pt}{1.10458pt}{49.21504pt}{0.0pt}\pgfsys@curveto{49.21504pt}{-1.10458pt}{50.11046pt}{-2.0pt}{51.21504pt}{-2.0pt}\pgfsys@curveto{52.31963pt}{-2.0pt}{53.21504pt}{-1.10458pt}{53.21504pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21504pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{split}$ (7.9)

This coupling can be seen most easily in the classical expression of
$T_{\mu\nu}$ (after the shift (7.6)) as a term

$-\frac{1}{6}v\left({\partial}_{\mu}{\partial}_{\nu}-\eta_{\mu\nu}\Box\right)\sigma~{},$
(7.10)

which is linear in the vev $v$.

The dilaton also has a cubic coupling to the massive adjoint scalar fields
$\varphi$, which follows immediately from the first trace on the RHS of (7.7)

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}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}\pgfsys@moveto{85.35828pt}{31.2982pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}\pgfsys@moveto{85.35828pt}{-31.2982pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\bar{\varphi}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{128.03741pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{110.92847pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\sim-\frac{2}{k}g^{2}v~{}.$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{split}$ (7.11)

The $1\over k$ factor follows from isolating the trace part of
$Q_{1}+\bar{Q}_{1}$ in (7.7).

Our main goal here will be to evaluate, at leading order in the Yang-Mills
coupling $g$, the three-point function $\langle T(q){\cal O}_{2}(p)\bar{\cal
O}_{2}(\bar{p})\rangle$ in the vanishing-momentum limit, where

${\cal O}_{2}={\rm{Tr}}[\varphi^{2}]$ (7.12)

is the single Coulomb-branch operator of this theory with scaling dimension
$\Delta=2$.

The computation involves the Feynman diagram of Fig. 2,191919This is in
complete analogy with the corresponding case of the ${\cal N}=4$ SYM theory
analysed in Ref. [2]. which receives the following contributions:

$T$$q$$\sigma$$q$$\ell$$q-\ell$$p$$\bar{p}$${\rm{Tr}}[\varphi^{2}]$${\rm{Tr}}[\bar{\varphi}^{2}]$$\ell-p$
Fig. 2: The Feynman diagram that determines the leading contribution to
$G_{2\bar{2}}^{\mathbb{H}}$ in the 4D ${\cal N}=2$ SCQCD theory.

  * •

The $T$-$\sigma$ coupling (7.9) gives the factor
$-\frac{1}{6}q^{2}v(-3)=\frac{vq^{2}}{2}$.

  * •

The dilaton propagator gives the factor $\frac{2ik}{q^{2}}$, which cancels the
above $q^{2}$.

  * •

There is a factor $(-i)(-\frac{2}{k}g^{2}v)$ from the vertex
$\sigma\varphi\bar{\varphi}$ (7.11).

  * •

There is a one-loop momentum integration that yields the factor

$I=\int\frac{d^{4}\ell}{(2\pi)^{4}}\frac{i}{\ell^{2}-m^{2}}\frac{i}{(q-\ell)^{2}-m^{2}}\frac{i}{(\ell-p)^{2}-m^{2}}\xrightarrow[p,q\to
0]{}-\frac{1}{(4\pi)^{2}}\frac{1}{2m^{2}}~{},$ (7.13)

where at the RHS of this expression we evaluated the limit of the integral at
vanishing momentum, $p,q\to 0$.

  * •

The contraction of gauge indices in the triangle give the extra factor
$4(k^{2}-1)$. From the point of view of the index contractions it is twice as
if one is evaluating the tree-level contribution to the two-point function
$\langle{\rm{Tr}}[\varphi^{2}](p){\rm{Tr}}[\bar{\varphi}^{2}](-p)\rangle$,
which involves the Feynman diagram

$\begin{split}\leavevmode\hbox to238.8pt{\vbox
to40.63pt{\pgfpicture\makeatletter\hbox{\hskip 41.5417pt\lower-20.31683pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }
{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.20001pt}\pgfsys@invoke{
}{}\pgfsys@moveto{113.81104pt}{0.0pt}\pgfsys@lineto{156.49017pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{113.81104pt}{0.0pt}\pgfsys@lineto{156.49017pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{-25.6073pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-38.2087pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\rm{Tr}}[\varphi^{2}]$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{182.09747pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{170.2669pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{${\rm{Tr}}[\bar{\varphi}^{2}]$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{0.0pt}\pgfsys@curveto{78.24509pt}{19.91684pt}{78.24509pt}{19.91684pt}{113.81104pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{0.0pt}\pgfsys@curveto{78.24509pt}{-19.91684pt}{78.24509pt}{-19.91684pt}{113.81104pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{split}$ (7.14)

Collecting all these factors we obtain the following result

$G^{\mathbb{H}}_{2\bar{2}}=\frac{1}{\pi^{2}}\lim_{q,p\to 0}\langle T(q){\cal
O}_{2}(p)\bar{\cal O}_{2}(\bar{p})\rangle=\frac{2(k^{2}-1)}{(2\pi)^{4}}\;,$
(7.15)

at leading order in the gauge coupling $g$.

This result can be compared against the coefficient of the two-point function
$\langle{\rm{Tr}}[\varphi^{2}]{\rm{Tr}}[\bar{\varphi}^{2}]\rangle$ in the CFT
phase. Specifically, for the CFT in position space one has (see e.g. [36])

$\langle{\rm{Tr}}[\varphi^{2}](x){\rm{Tr}}[\bar{\varphi}^{2}(0)]\rangle=\frac{2(k^{2}-1)}{(2\pi)^{4}}\frac{1}{|x|^{4}}~{}.$
(7.16)

In momentum space, this result takes the form

$\langle{\rm{Tr}}[\varphi^{2}](p){\rm{Tr}}[\bar{\varphi}^{2}(-p)]\rangle=-\frac{2(k^{2}-1)}{(4\pi)^{2}}\log(p^{2})~{}$
(7.17)

and from these equations we read off $G^{\rm
CFT}_{2\bar{2}}=\frac{2(k^{2}-1)}{(2\pi)^{4}}$. As a result, in this case we
have verified explicitly the tree-level matching of type-B anomalies

$G^{\text{CFT}}_{2\bar{2}}=G^{\mathbb{H}}_{2\bar{2}}~{}.$ (7.18)

We expect that analogous steps produce a similar matching of type-B anomalies
for all the CBOs in this theory. These CBOs are freely generated as multi-
trace products of the single-trace Casimirs ${\rm{Tr}}[\varphi^{l}]$. We will
return to this aspect in future work.

###  7.2 ${\cal N}=2$ Circular Quivers

$k$$k$$\cdots\cdots$$k$$\cdots\cdots$$k$$k$$Q^{(\alpha)}$$\tilde{Q}^{(\alpha)}$$Q^{(\alpha-1)}$$\tilde{Q}^{(\alpha-1)}$
Fig. 3: The circular ${\cal N}=2$ superconformal quiver. Each of the $N$
nodes denotes an $SU(k)$ ${\cal N}=2$ vector multiplet. The links are ${\cal
N}=1$ bifundamental chiral multiplets. The dashed perpendicular lines at the
left and right ends are identified.

Our second example involves the circular superconformal quiver of Fig. 3.
Every two consecutive nodes in this quiver are linked by a pair of ${\cal
N}=1$ chiral bifundamental fields $Q^{(\alpha)}$, $\tilde{Q}^{(\alpha)}$ that
make up an ${\cal N}=2$ bifundamental hypermultiplet, while there are also
adjoint chiral superfields $\Phi^{(\alpha)}$ that are part of the ${\cal N}=2$
vector multiplet. By a slight abuse of notation we will continue denoting the
bottom components of $Q^{(\alpha)}$, $\widetilde{Q}^{(\alpha)}$ with the same
symbol, while $\Phi^{(\alpha)}$ will have bottom component
$\varphi^{(\alpha)}$. The node label $\alpha$ takes the $N$ discrete values
$-\lfloor N/2\rfloor+1,-\lfloor N/2\rfloor+2,\ldots,\lfloor N/2\rfloor$. The
gauge group at each node is $SU(k)$ and we focus on the case where all the
gauge couplings $g^{(\alpha)}$ are equal, $g^{(\alpha)}\equiv g$. This
particular point on the space of couplings is usually referred to as the
‘orbifold point’, since this version of the theory can be obtained as an
${\cal N}=2$ preserving $\mathbb{Z}_{N}$ orbifold of ${\cal N}=4$ SYM theory
with gauge group $SU(kN)$.

For our purposes it is convenient to introduce the following notation. We set
${\mathfrak{q}}=e^{2\pi i/N}$ and perform the discrete Fourier
transformations:

$\displaystyle
Q^{(\alpha)}=\frac{1}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{\alpha\beta}\widehat{Q}^{(\beta)}~{},$
(7.19)
$\displaystyle\widetilde{Q}^{(\alpha)}=\frac{1}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{\alpha\beta}\widehat{\widetilde{Q}}^{(\beta)}~{},$
(7.20)
$\displaystyle\varphi^{(\alpha)}=\frac{1}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{\alpha\beta}\hat{\varphi}^{(\beta)}~{},$
(7.21)

where $\beta$ is summed over all the quiver nodes. For the Hermitian
conjugates we set

$\bar{Q}^{(\alpha)}=\frac{1}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{-\alpha\beta}\widehat{\bar{Q}}^{(\beta)}$
(7.22)

and similarly for the remaining scalars. For the real and imaginary parts of
the Fourier coefficients $\widehat{Q}$ we introduce the notation

$\displaystyle\widehat{Q}^{(\alpha)}=\frac{1}{\sqrt{2}}\frac{1}{g}\left(X^{(\alpha)}+iY^{(\alpha)}\right)~{},$
(7.23)
$\displaystyle\widehat{\bar{Q}}^{(\alpha)}=\frac{1}{\sqrt{2}}\frac{1}{g}\left(X^{(-\alpha)}-iY^{(-\alpha)}\right)~{}.$
(7.24)

This theory has a rich moduli space of vacua. Here, we will only be interested
in a very specific direction along the Higgs branch, where in terms of the
elementary fields we consider vevs

$\langle
Q^{(\alpha)}\rangle=\frac{v}{\sqrt{2}}{\,\hbox{1\kern-2.27621ptl}}_{k\times
k}~{},~{}~{}\langle\widetilde{Q}^{(\alpha)}\rangle=0~{},$ (7.25)

with ${\,\hbox{1\kern-2.27621ptl}}_{k\times k}$ denoting the $k\times k$
identity matrix, in analogy with the vev (7.3) of Sec. 7.1. In this case, the
dilaton $\sigma$ is proportional to the trace part of $X^{(0)}$. Specifically

$\sigma=\frac{\sqrt{2N}}{g}{\rm{Tr}}\left[X^{(0)}\right]~{}.$ (7.26)

Let us highlight some of the pertinent features of the classical action of
this quiver; we use the conventions of [37]. ${\cal N}=2$ supersymmetry
requires at each node $\alpha$ the ${\cal N}=1$ superpotential

$W^{(\alpha)}=-i\sqrt{2}g{\rm{Tr}}\left[\widetilde{Q}^{(\alpha)}\Phi^{(\alpha)}Q^{(\alpha)}-Q^{(\alpha)}\Phi^{(\alpha+1)}\widetilde{Q}^{(\alpha)}\right]~{}.$
(7.27)

The corresponding scalar-potential $V$ receives two contributions
$V=V_{F}+V_{D}$. The $F$-term contributions are

$V_{F}=\sum_{\alpha}{\rm{Tr}}\left[\bar{F}_{Q^{(\alpha)}}F_{Q^{(\alpha)}}+\bar{F}_{\widetilde{Q}^{(\alpha)}}F_{\widetilde{Q}^{(\alpha)}}+\bar{F}_{\varphi^{(\alpha)}}F_{\varphi^{(\alpha)}}\right]~{},$
(7.28)

with

$\displaystyle
F_{Q^{(\alpha)}}=-i\sqrt{2}g\left(\widetilde{Q}^{(\alpha)}\varphi^{(\alpha)}-\varphi^{(\alpha+1)}\widetilde{Q}^{(\alpha)}\right)~{},$
(7.29) $\displaystyle
F_{\widetilde{Q}^{(\alpha)}}=-i\sqrt{2}g\left(\varphi^{(\alpha)}Q^{(\alpha)}-Q^{(\alpha)}\varphi^{(\alpha+1)}\right)~{},$
(7.30) $\displaystyle
F_{\varphi^{(\alpha)}}=-i\sqrt{2}g\left(Q^{(\alpha)}\widetilde{Q}^{(\alpha)}-\widetilde{Q}^{(\alpha-1)}Q^{(\alpha-1)}\right)\;,$
(7.31)

while the D-term contributions are

$V_{D}=\frac{g^{2}}{2}\sum_{\alpha}D^{(\alpha)A}D^{(\alpha)A}~{},$ (7.32)

where

$\displaystyle D^{(\alpha)A}$ $\displaystyle=$
$\displaystyle{\rm{Tr}}\bigg{[}T^{A}\bigg{(}\left[\varphi^{(\alpha)},\bar{\varphi}^{(\alpha)}\right]+Q^{(\alpha)}\bar{Q}^{(\alpha)}-\widetilde{\bar{Q}}^{(\alpha)}\widetilde{Q}^{(\alpha)}$
(7.33)
$\displaystyle~{}~{}~{}~{}~{}\qquad\qquad-\bar{Q}^{(\alpha-1)}Q^{(\alpha-1)}+\widetilde{Q}^{(\alpha-1)}\widetilde{\bar{Q}}^{(\alpha-1)}\bigg{)}\bigg{]}~{}.$

In this expression $A$ is an $SU(k)$ Lie-algebra index and $T^{A}$ the
generators of the Lie algebra with normalisation
${\rm{Tr}}[T^{A}T^{B}]=\delta^{AB}$. In (7.33) we assume canonical kinetic
terms for all the vector and chiral multiplets.

When expanding around the vevs (7.25), one obtains expressions of the
following form from the D-terms

$D^{(\alpha)A}=D_{0}^{(\alpha)A}+\frac{v}{g\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{\alpha\beta}\left(1-{\mathfrak{q}}^{-\beta}\right){\rm{Tr}}\left[T^{A}X^{(\beta)}\right]~{},$
(7.34)

where $D_{0}^{(\alpha)A}$ is 0${}^{\text{th}}$ order in the $v$ expansion.
Feeding this expansion into $V_{D}$ gives

$\displaystyle V_{D}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\sum_{\alpha}D_{0}^{(\alpha)A}D_{0}^{(\alpha)A}+\frac{vg}{\sqrt{N}}\sum_{\alpha,\beta}{\mathfrak{q}}^{\alpha\beta}\left(1-{\mathfrak{q}}^{\beta}\right)D_{0}^{(\alpha)A}{\rm{Tr}}\left[T^{A}X^{(\beta)}\right]$
(7.35)
$\displaystyle+\frac{v^{2}}{2}\sum_{\alpha}\left(1-{\mathfrak{q}}^{\alpha}\right)\left(1-{\mathfrak{q}}^{-\alpha}\right){\rm{Tr}}\left[T^{A}X^{(\alpha)}\right]{\rm{Tr}}\left[T^{A}X^{(-\alpha)}\right]~{}.$

We can further recast this expression by using the identity

${\rm{Tr}}\left[T^{A}X\right]\left[T^{A}Y\right]={\rm{Tr}}\left[XY\right]-\frac{1}{k}{\rm{Tr}}\left[X\right]{\rm{Tr}}\left[Y\right]~{},$
(7.36)

but we will not write explicitly the resulting expressions. Instead, we will
refer the interested reader to [37] for related computations. The observation
that is needed here is that the field $X^{(0)}$ does not appear in a cubic
scalar coupling and does not receive a mass, in agreement with the fact that
it is directly related to the dilaton, (7.26).

The F-term part of the potential, $V_{F}$, provides more interesting
contributions. We note the following useful expansions

$F_{\widetilde{Q}^{(\alpha)}}=F_{0\widetilde{Q}^{(\alpha)}}-\frac{ivg}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{\alpha\beta}\left(1-{\mathfrak{q}}^{\beta}\right)\hat{\varphi}^{(\beta)}~{},$
(7.37)
$\bar{F}_{\widetilde{Q}^{(\alpha)}}=\bar{F}_{0\widetilde{Q}^{(\alpha)}}+\frac{ivg}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{\alpha\beta}\left(1-{\mathfrak{q}}^{\beta}\right)\hat{\bar{\varphi}}^{(-\beta)}~{},$
(7.38)

where $F_{0\widetilde{Q}^{(\alpha)}}$, $\bar{F}_{0\widetilde{Q}^{(\alpha)}}$
are 0${}^{\text{th}}$ order expressions in the $v$ expansion. Then,

$\displaystyle
V_{F}\supset\sum_{\alpha}{\rm{Tr}}\left[\bar{F}_{0\widetilde{Q}^{(\alpha)}}F_{0\widetilde{Q}^{(\alpha)}}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{vg}{\sqrt{N}}\sum_{\alpha,\alpha^{\prime}}\left(1-{\mathfrak{q}}^{\alpha}\right){\rm{Tr}}\left[\left(X^{(-\alpha-\alpha^{\prime})}-iY^{(-\alpha-\alpha^{\prime})}\right)\left(\hat{\bar{\varphi}}^{(-\alpha^{\prime})}\hat{\varphi}^{(\alpha)}-{\mathfrak{q}}^{\alpha^{\prime}}\hat{\varphi}^{(\alpha)}\hat{\bar{\varphi}}^{(-\alpha^{\prime})}\right)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{vg}{\sqrt{N}}\sum_{\alpha,\alpha^{\prime}}\left(1-{\mathfrak{q}}^{\alpha}\right){\rm{Tr}}\left[\left(X^{(-\alpha-\alpha^{\prime})}+iY^{(-\alpha-\alpha^{\prime})}\right)\left(\hat{\bar{\varphi}}^{(-\alpha)}\hat{\varphi}^{(\alpha^{\prime})}-{\mathfrak{q}}^{\alpha^{\prime}}\hat{\varphi}^{(\alpha^{\prime})}\hat{\bar{\varphi}}^{(-\alpha)}\right)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+v^{2}g^{2}\sum_{\alpha}\left(1-{\mathfrak{q}}^{\alpha}\right)\left(1-{\mathfrak{q}}^{-\alpha}\right){\rm{Tr}}\left[\hat{\bar{\varphi}}^{(\alpha)}\hat{\varphi}^{(\alpha)}\right]~{}.$
(7.39)

From this formula we read off the mass

$m_{\alpha}^{2}=2v^{2}g^{2}\left(1-{\mathfrak{q}}^{\alpha}\right)\left(1-{\mathfrak{q}}^{-\alpha}\right)$
(7.40)

for the modes $\hat{\varphi}^{(\alpha)}$. In addition, we read off a cubic
coupling between the dilaton and the modes $\hat{\varphi}^{(\alpha)}$

$V_{F}\supset-\frac{\sqrt{2}vg^{2}}{kN}\sum_{\alpha}\left(1-{\mathfrak{q}}^{\alpha}\right)\left(1-{\mathfrak{q}}^{-\alpha}\right)\sigma\,{\rm{Tr}}\left[\hat{\varphi}^{(\alpha)}\hat{\bar{\varphi}}^{(\alpha)}\right]~{}.$
(7.41)

This information is all we need to perform the computation of the three-point
function $\langle T{\cal O}\bar{\cal O}\rangle$ at tree-level for general CBOs
${\cal O}$. The vector multiplet at each quiver node contains an adjoint
complex scalar field $\varphi^{(\alpha)}$ from which we can build the Casimir

${\cal O}_{\ell}^{(\alpha)}={\rm{Tr}}[(\varphi^{(\alpha)})^{\ell}]~{}.$ (7.42)

This is a CBO with scaling dimension $\Delta=\ell$.202020We can also consider
multi-trace products of the Casimir, which are also CBOs, but we will not do
this explicitly in this paper. The single-trace operators are generators of
the Coulomb-branch chiral ring. For this theory it is customary to define the
operators $\widehat{\cal O}_{\ell}^{(\alpha)}$ via the discrete Fourier
transform

${\cal
O}_{\ell}^{(\alpha)}=\frac{1}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{\alpha\beta}\widehat{\cal
O}_{\ell}^{(\beta)}~{}.$ (7.43)

The operator $\widehat{\cal O}_{\ell}^{(0)}$, which has vanishing discrete-
Fourier momentum, is called an ‘untwisted’ CBO operator of the quiver theory.
All the other operators are ‘twisted’ operators.

For calculations in the broken phase it is useful to express $\widehat{\cal
O}_{\ell}^{(\alpha)}$ as a sum over traces of the discrete-Fourier-transformed
elementary fields $\hat{\varphi}^{(\alpha)}$ (see Eq. (7.21))

$\widehat{\cal
O}^{(\alpha)}_{\ell}=\frac{1}{\sqrt{N}}\sum_{\beta}{\mathfrak{q}}^{-\alpha\beta}{\rm{Tr}}\left[\left(\varphi^{(\beta)}\right)^{\ell}\right]=\frac{1}{N^{\frac{\ell-1}{2}}}\sum_{\alpha_{1},\ldots,\alpha_{\ell-1}}{\rm{Tr}}\left[\left(\prod_{n=1}^{\ell-1}\hat{\varphi}^{(\alpha_{n})}\right)\hat{\varphi}^{(\alpha-\sum_{m=1}^{\ell-1}\alpha_{m})}\right]~{}.$
(7.44)

As an illustration of the above structure, let us focus henceforth on
$\Delta=2$ CBOs, which can be cast in the following form

$\widehat{\cal
O}^{(\alpha)}_{2}=\frac{1}{\sqrt{N}}\sum_{\alpha^{\prime}}{\rm{Tr}}\left[\hat{\varphi}^{(\alpha^{\prime})}\hat{\varphi}^{(\alpha-\alpha^{\prime})}\right]~{}.$
(7.45)

Our purpose is to evaluate the tree-level contribution to the type-B anomaly
in $\langle T\widehat{\cal O}_{2}^{(\alpha)}\overline{\widehat{\cal
O}}_{2}^{(\alpha)}\rangle$ in the broken phase. As in the previous subsection,
this involves the computation of a Feynman diagram similar to that of Fig. 2.
We will call the tree-level result $G^{(\alpha){\mathbb{H}}}_{2\bar{2}}$ when
it refers to the operator $\widehat{\cal O}^{(\alpha)}_{2}$.

First, let us consider separately the case where the external CBO is
untwisted, i.e. the case $\alpha=0$ in (7.45). We obtain

$\displaystyle G^{(0){\mathbb{H}}}_{2\bar{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\left(\frac{v}{2\sqrt{2}}q^{2}\right)\left(\frac{2Nki}{q^{2}}\right)\left(-i\frac{-\sqrt{2}vg^{2}}{kN}\right)$
(7.46) $\displaystyle\times\sum_{\alpha^{\prime}\neq
0}\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)\frac{1}{N}\frac{-1}{(4\pi)^{2}}\frac{1}{2v^{2}g^{2}}\frac{8(k^{2}-1)}{2\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)}$
$\displaystyle=$
$\displaystyle\frac{2(k^{2}-1)}{(2\pi)^{4}}\left(1-\frac{1}{N}\right)~{}.$

The result in the conformal phase is $G^{(0){\rm
CFT}}_{2\bar{2}}=\frac{2(k^{2}-1)}{(2\pi)^{4}}$. We observe a mismatch, which
is suppressed in the large-$N$ limit.

For the twisted sector CBOs we can compute the anomaly in a similar fashion.
In this case, the calculation requires the value of the integral

$I=\int\frac{d^{4}\ell}{(2\pi)^{4}}\frac{i}{\ell^{2}-m_{1}^{2}}\frac{i}{\ell^{2}-m_{1}^{2}}\frac{i}{\ell^{2}-m_{2}^{2}}~{}.$
(7.47)

For $m_{1}=m_{2}=m$, which occurs in ${\cal I}^{(0)}$, we obtain

$I=-\frac{1}{(4\pi)^{2}}\frac{1}{2m^{2}}~{}.$ (7.48)

For $m_{1}=m\neq 0$ and $m_{2}=0$ we obtain

$I=-\frac{1}{(4\pi)^{2}}\frac{1}{m^{2}}~{}.$ (7.49)

More generally, when both $m_{1},m_{2}\neq 0$ we obtain

$I=-\frac{1}{(4\pi)^{2}}\frac{m_{1}^{2}-m_{2}^{2}-m_{2}^{2}\log\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)}{(m_{1}^{2}-m_{2}^{2})^{2}}~{}.$
(7.50)

Collecting all the contributions to the diagram for $\alpha\neq 0$ we find

$G^{(\alpha){\mathbb{H}}}_{2\bar{2}}=-\frac{1}{(2\pi)^{4}}\frac{2(k^{2}-1)}{N}\left[1+\sum_{\alpha^{\prime}\neq\,0,\alpha}\mathfrak{L}(\alpha^{\prime};\alpha)\right]~{},$
(7.51)

where

$\displaystyle\begin{split}&\mathfrak{L}(\alpha^{\prime};\alpha)=\left[\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)-\left(1-{\mathfrak{q}}^{-\alpha+\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{\alpha-\alpha^{\prime}}\right)\right]^{-2}\\\
&\Bigg{\\{}\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)\Bigg{[}\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)-\left(1-{\mathfrak{q}}^{-\alpha+\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{\alpha-\alpha^{\prime}}\right)\\\
&\hskip
113.81102pt-\left(1-{\mathfrak{q}}^{-\alpha+\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{\alpha-\alpha^{\prime}}\right)\log\frac{\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)}{\left(1-{\mathfrak{q}}^{-\alpha+\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{\alpha-\alpha^{\prime}}\right)}\Bigg{]}\\\
&+\left(1-{\mathfrak{q}}^{-\alpha+\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{\alpha-\alpha^{\prime}}\right)\Bigg{[}\left(1-{\mathfrak{q}}^{-\alpha+\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{\alpha-\alpha^{\prime}}\right)-\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)\\\
&\hskip
128.0374pt+\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)\log\frac{\left(1-{\mathfrak{q}}^{\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{-\alpha^{\prime}}\right)}{\left(1-{\mathfrak{q}}^{-\alpha+\alpha^{\prime}}\right)\left(1-{\mathfrak{q}}^{\alpha-\alpha^{\prime}}\right)}\Bigg{]}\Bigg{\\}}\\\
&\hskip 28.45274pt=1~{}.\end{split}$ (7.52)

The term 1 inside the parenthesis of the RHS of (7.51) comes from the case
where one of the propagators inside the triangle is massless. There is no case
where only massless fields, or two massless fields run inside the loop. The
remaining contributions, which are captured by the sum over the quantity
$\mathfrak{L}$, involve only massive propagators. For all $\alpha^{\prime}\neq
0,\alpha$ we obtain $\mathfrak{L}(\alpha^{\prime};\alpha)=1$. As a result we
find that

$G^{(\alpha){\mathbb{H}}}_{2\bar{2}}=\frac{2(k^{2}-1)}{(2\pi)^{4}}\left(1-\frac{1}{N}\right)$
(7.53)

independent of $\alpha$.212121The case $\alpha=0$ is special. In that case the
anomaly receives an extra contribution from the IR chiral ring. For an updated
discussion see [38].

In comparison, the tree-level two-point function in the conformal phase

$\langle\widehat{\cal O}^{(\alpha)}_{2}~{}\overline{\widehat{\cal
O}}^{(\alpha)}_{2}\rangle=\frac{1}{N}\sum_{\alpha^{\prime},\alpha^{\prime\prime}}\left\langle{\rm{Tr}}\left[\hat{\varphi}^{(\alpha^{\prime})}\hat{\varphi}^{(\alpha-\alpha^{\prime})}\right]~{}{\rm{Tr}}\left[\hat{\bar{\varphi}}^{(\alpha^{\prime\prime})}\hat{\bar{\varphi}}^{(\alpha-\alpha^{\prime\prime})}\right]\right\rangle$
(7.54)

yields

$G^{(\alpha){\rm CFT}}_{2\bar{2}}=\frac{2(k^{2}-1)}{(2\pi)^{4}}~{},$ (7.55)

which is also independent of the discrete momentum $\alpha$, but does not
agree exactly with (7.53) on the Higgs branch, except at leading order in the
large-$N$ limit. Consequently, this is an example of a 4D ${\cal N}=2$ SCFT
where the type-B anomalies do not match across the Higgs branch by failing to
do so already at tree-level. However, since the mismatch is suppressed in the
large-$N$ limit, there will be a matching of this class of type-B anomalies at
leading order in a $\frac{1}{N}$ expansion. This fact will be very useful in
Sec. 9.

We expect that there is a corresponding matching of type-B anomalies in the
large-$N$ limit for all the single-trace CBOs $\widehat{\cal
O}^{(\alpha)}_{\ell}$ and their multi-trace versions. We aim to return to this
point in the future.

###  7.3 ${\cal N}=4$ SYM Theory

The case of the ${\cal N}=4$ SYM theory can be treated as a special case of an
${\cal N}=2$ theory. The ${\cal N}=4$ SYM theory has an enhanced
$SO(6)_{R}\supset SU(2)_{R}\times U(1)_{r}$ R-symmetry and six real adjoint
scalar fields $\varphi_{1},\ldots,\varphi_{6}$ charged under it. The ${\cal
N}=2$ SCA can be embedded inside the ${\cal N}=4$ algebra so that the ${\cal
N}=2$ $SU(2)_{R}$ symmetry rotates the $\varphi_{1},\ldots,\varphi_{4}$
fields, while the ${\cal N}=2$ $U(1)_{r}$ the $\varphi_{5},\varphi_{6}$. In
the context of this embedding, the ${\cal N}=4$ Coulomb branch can be split
into an ${\cal N}=2$ Higgs-branch direction and an ${\cal N}=2$ Coulomb-branch
direction. Correspondingly, the Higgs branch is characterised by vevs of the
scalars $\varphi_{1},\ldots,\varphi_{4}$, while the Coulomb branch by vevs of
the scalars $\varphi_{5},\varphi_{6}$. To repeat the computations of the
previous subsections we focus on ${\frac{1}{2}}$-BPS chiral-primary operators
that are single- (or multi-) trace Casimirs of the scalar fields
$\varphi_{5},\varphi_{6}$. These fields are ${\cal N}=2$ CBOs under the above
superconformal-algebra embedding. The type-B anomalies of these operators on
the Higgs branch can then be computed at tree-level as in Secs. 7.1, 7.2, to
establish the tree-level matching of type-B anomalies on the ${\cal N}=4$
Coulomb branch.

This computation was performed in [2] for ${\rm{Tr}}[\varphi_{5}\varphi_{6}]$
type-B anomalies in the ${\cal N}=4$ $SU(2)$ theory, where the Coulomb-branch
direction was taken to be associated with $\varphi_{1}$ vevs. The authors of
[2] verified at tree level that the corresponding type-B anomalies of the
broken and unbroken phases match. Our arguments in this paper are improving
the analysis of [2] in the following directions:

  * (a)

The formal proof of Sec. 6 can be used to argue that the above matching holds
non-perturbatively at finite Yang-Mills coupling. This conclusion is
consistent with expectations, based on the relationship between type-B and
chiral anomalies for this high degree of supersymmetry. Therefore, the ${\cal
N}=4$ example could be viewed as a mild check of the general claims of Secs.
5, 6.

  * (b)

One can use our approach to extend the tree-level matching to more general
${\frac{1}{2}}$-BPS operators of the ${\cal N}=4$ SYM theory with $\Delta>2$.
In that case the arguments of Secs. 5, 6 would imply the non-perturbative
matching for type-B anomalies that are not related to chiral anomalies.

##  8 The Type-B Anomaly from SUSY Localisation

We now turn to the explicit calculation of the type-B anomalies for the
Lagrangian ${\cal N}=2$ theories that we discussed in Sec. 7 using
supersymmetric localisation.

###  8.1 The Partition Function on $S^{4}_{\epsilon_{1},\epsilon_{2}}$

The partition function of Lagrangian ${\cal N}=2$ $SU(k)$ SCFTs ${\cal
Z}_{S^{4}_{\epsilon_{1},\epsilon_{2}}}(\tau,\bar{\tau},m;\epsilon_{1},\epsilon_{2})$
on the ellipsoid $S^{4}_{\epsilon_{1},\epsilon_{2}}$ is explicitly known from
the localisation calculation of [10, 39]:

$\displaystyle{\cal
Z}_{S^{4}_{\epsilon_{1},\epsilon_{2}}}(\tau,\bar{\tau},m;\epsilon_{1},\epsilon_{2})=\int[da]\left|Z_{\rm
4D}(\tau,a,m;\epsilon_{1},\epsilon_{2})\right|^{2}\;.$ (8.1)

Here we use the notation $\tau=\\{\tau^{(\alpha)}\\}$ for the set of marginal
gauge couplings, with the index $\alpha$ counting the number of gauge groups,
$a=\\{a^{(\alpha)}_{b}\\}$ for the set of Coulomb-branch parameters, with the
index $b=1,...,k$ labelling the Coulomb-branch parameters within a given
colour group. $m$ denotes the set of all masses. In the case of SCQCD we have
only one gauge group with one marginal coupling $\tau$, $k$ traceless Coulomb-
branch parameters $a=\\{a_{b}\\}$ and $N_{f}=2k$ fundamental masses
$m=\\{m^{{\mathfrak{i}}}\\}$ with ${\mathfrak{i}}=1,\ldots,2k$ a flavour
index. In the case of the circular ${\cal N}=2$ superconformal quiver with $N$
$SU(k)$ nodes, $m=m_{\rm bif}=\\{m_{\rm bif}^{(\alpha)}\\}$ is the set of $N$
bifundamental-hypermultiplet masses.

The holomorphic half of the integrand of the partition function $Z_{\rm
4D}(\tau,a,m;\epsilon_{1},\epsilon_{2})$ is the IR Coulomb-branch partition
function on $\mathbb{R}^{4}_{\epsilon_{1},\epsilon_{2}}$, which factorises as

$\displaystyle Z_{\rm 4D}$
$\displaystyle=Z_{\text{4D,cl}}Z_{\text{4D,1-loop}}Z_{\text{4D,inst}}\;.$
(8.2)

The one-loop part

$\displaystyle Z_{\text{4D,1-loop}}(a,m_{\rm
bif};\epsilon_{1},\epsilon_{2})=\prod_{\alpha}Z_{\text{1-loop}}^{\text{vec}}(a;\epsilon_{1},\epsilon_{2})Z_{\text{1-loop}}^{\text{bif}}(a,m_{\rm
bif}\;;\epsilon_{1},\epsilon_{2})\;,$ (8.3)

contains vector and hypermultiplet contributions,

$\displaystyle Z^{\text{bif}}_{\text{1-loop}}(a,m_{\rm
bif};\epsilon_{1},\epsilon_{2})$
$\displaystyle=\prod_{b,c=1}^{k}\Gamma_{2}\left(a_{b}^{(\alpha)}-a_{c}^{(\alpha+1)}-m_{\rm
bif}^{(\alpha)}+\frac{\epsilon_{+}}{2}\big{|}\epsilon_{1},\epsilon_{2}\right)\;,$
(8.4) $\displaystyle
Z^{\text{vec}}_{\text{1-loop}}(a;\epsilon_{1},\epsilon_{2})$
$\displaystyle=\prod_{b,c=1}^{k}\Gamma_{2}\left(a_{b}^{(\alpha)}-a_{c}^{(\alpha)}\big{|}\epsilon_{1},\epsilon_{2}\right)^{-1}\;,$
(8.5)

where $\epsilon_{+}=\epsilon_{1}+\epsilon_{2}$ and
$\Gamma_{2}(x|\epsilon_{1},\epsilon_{2})$ is the Barnes double-Gamma function.
Here we only give the contribution of a bifundamental hypermultiplet, from
which the fundamental is easily obtained. Similarly, the instanton part is
given by [40]

$\displaystyle Z_{\text{4D,inst}}(\tau,a,m_{\rm
bif};\epsilon_{1},\epsilon_{2})=\sum_{\nu}\prod_{\alpha}\mathbf{q}_{(\alpha)}^{\sum_{b=1}^{k}|\nu^{(\alpha)}_{b}|}Z_{\text{inst}}^{\text{vec}}(a;\nu)Z_{\text{inst}}^{\text{bif}}(a,m_{\rm
bif};\nu)$ (8.6)

where $\mathbf{q}_{(\alpha)}=e^{2\pi i\tau^{(\alpha)}}$. A Young diagram
$\nu^{(\alpha)}_{b}$ appears for each of the Coulomb moduli
$a^{(\alpha)}_{b}$, collectively denoted by $\nu=\\{\nu_{b}^{(\alpha)}\\}$,
while the instanton number is given in terms of the total number of boxes of
the Young diagram $|\nu_{b}^{(\alpha)}|$. The vector and hypermultiplet
instanton contributions respectively read

$\displaystyle Z_{\text{inst}}^{\text{bif}}(\tau,a,m_{\rm bif};\nu)$
$\displaystyle=\prod_{b,c=1}^{k}{\rm
N}_{\nu_{c}^{(\alpha+1)}\nu_{b}^{(\alpha)}}\left(a_{b}^{(\alpha)}-a^{(\alpha+1)}_{c}-m_{\rm
bif}^{(\alpha)}-\frac{\epsilon_{+}}{2}\right)\;,$ (8.7) $\displaystyle
Z_{\text{inst}}^{\text{vec}}(a,m_{\rm bif};\nu)$
$\displaystyle=\prod_{b,c=1}^{k}{\rm
N}_{\nu_{c}^{(\alpha)}\nu_{b}^{(\alpha)}}\left(a^{(\alpha)}_{b}-a^{(\alpha)}_{c}\right)^{-1}\;,$
(8.8)

where the functions $\textrm{N}_{\lambda\mu}(a)$ involved above are defined as

$\displaystyle\textrm{N}_{\lambda\mu}(a)=\prod_{(i,j)\in\lambda}\left[a+\epsilon_{1}(\lambda_{i}-j+1)+\epsilon_{2}(i-\mu_{j}^{t})\right]\prod_{(i,j)\in\mu}\left[a+\epsilon_{1}(j-\mu_{i})+\epsilon_{2}(\lambda_{j}^{t}-i+1)\right]\;.$
(8.9)

Specifically, for the circular ${\cal N}=2$ superconformal quiver, the
Coulomb-branch partition function (8.2) takes the form [41]

$\displaystyle Z_{\text{4D,cl}}=\exp\left[-\frac{2\pi
i}{\epsilon_{1}\epsilon_{2}}\sum_{\alpha=-\lfloor\frac{N}{2}\rfloor+1}^{\lfloor\frac{N}{2}\rfloor}\tau^{(\alpha)}\sum_{b=1}^{k}\left(a_{b}^{(\alpha)}\right)^{2}\right]\;,$
(8.10) $\displaystyle
Z_{\text{4D,1-loop}}=\prod_{i,j=1}^{\infty}\prod_{\alpha=\lfloor-\frac{N}{2}\rfloor+1}^{\lfloor\frac{N}{2}\rfloor}\prod_{1\leq
b\leq c\leq k}\frac{\left[a^{(\alpha)}_{b}-a_{c}^{(\alpha+1)}-m_{\rm
bif}^{(\alpha)}+\epsilon_{1}(j-1/2)-\epsilon_{2}(i-1/2)\right]}{\left[a^{(\alpha+1)}_{b}-a^{(\alpha+1)}_{c}+\epsilon_{1}i-\epsilon_{2}(j-1)\right]}$
(8.11) $\displaystyle\qquad\qquad\times\prod_{1\leq b<c\leq
k}\frac{\left[a_{b}^{(\alpha+1)}-a^{(\alpha)}_{c}+m_{\rm