SciPost Phys. 11, 036 (2021)

Proving the 6d Cardy formulaand matching global gravitational anomalies
Chi-Ming Chang1,2? , Martin Fluder3,4,5† , Ying-Hsuan Lin5,6‡ and Yifan Wang3,7,8◦
1 Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
2 Center for Quantum Mathematics and Physics (QMAP),
University of California, Davis, CA 95616, USA
3 Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
4 Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa 277-8583, Japan
5 Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125, USA
6 Hsinchu County Environmental Protection Bureau, Hsinchu, Taiwan
7 Center of Mathematical Sciences and Applications,
Harvard University, Cambridge, MA 02138, USA
8 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA? wychang@ucdavis.edu, † mfluder@princeton.edu,‡ yhlin@caltech.edu, ◦ yifanw@princeton.edu

Abstract
A Cardy formula for 6d superconformal field theories (SCFTs) conjectured by Di Pietro
and Komargodski in [1] governs the universal behavior of the supersymmetric partition
function on Sβ1 × S5 in the limit of small β and fixed squashing of the S5 . For a general
6d SCFT, we study its 5d effective action, which is dominated by the supersymmetriccompletions of perturbatively gauge-invariant Chern-Simons terms in the small β limit.
Explicitly evaluating these supersymmetric completions gives the precise squashing dependence in the Cardy formula. For SCFTs with a pure Higgs branch (also known asvery Higgsable SCFTs), we determine the Chern-Simons levels by explicitly going onto
the Higgs branch and integrating out the Kaluza-Klein modes of the 6d fields on Sβ1 . We
then discuss tensor branch flows, where an apparent mismatch between the formulain [1] and the free field answer requires an additional contribution from BPS strings.
This “missing contribution” is further sharpened by the relation between the fractionalpart of the Chern-Simons levels and the (mixed) global gravitational anomalies of the
6d SCFT. We also comment on the Cardy formula for 4d N = 2 SCFTs in relation to Higgs
branch and Coulomb branch flows.
Copyright Chang et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

1

Received 30-04-2021
Accepted 22-07-2021
Check forupdatesPublished 23-08-2021
doi:10.21468/SciPostPhys.11.2.036

SciPost Phys. 11, 036 (2021)

Contents
1 Introduction
1.1 Review of the 4d Cardy formula
1.2 The 4d Cardy formula from moduli space flows
1.3 6d Cardy formula
1.4 Sketch of the proof

2
3
6
7
9

2 Supersymmetric Chern-Simons terms on the squashed S 5 background
2.1 Squashed S 5 background from reduction of squashed S 1 × S 5 background
2.2 Off-shell 5d supergravity multiplets
2.3 Squashed S 5 backgrounds in off-shell supergravity
2.4 Supersymmetric Euclidean Chern-Simons terms and their evaluation on thesquashed S 5
2.4.1 FFF term
2.4.2 Ff Ff F term
2.4.3 FWW term
2.4.4 FRR term
2.5 Summary

11
11
12
15

3 Deriving the Chern-Simons levels
3.1 Integrating out free field KK modes
3.2 On the Higgs branch
3.3 On the tensor branch

22
23
24
26

4 Chern-Simons levels and global anomalies

29

5 Conclusion

30

A Conventions for characteristic classes

33

B Central extensions of supersymmetry algebras and (extended) BPS objects for
4d N = 2 and 6d N = (1, 0) theories
B.1 4d N = 2
B.2 6d N = (1, 0)

33
33
34

C Supersymmetric S 1 × S 5 background for the index and modified index

35

D Geometric invariance and the equivalence between gauges

37

References

38

1

17
18
19
19
20
21

Introduction

Universal features provide key checks of dualities and reliable handles on otherwise stronglyinteracting non-perturbative phenomena. It is long understood that the high temperature/energy limit of quantum field theories exhibits universality. In particular, the Cardy formulafor 2d conformal field theories (CFTs) [2] relates the torus partition function, which counts

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SciPost Phys. 11, 036 (2021)

operators (with refinement), to the Weyl anomaly coefficient (or Virasoro central charge) c in
the high temperature limit1
log ZS 1 ×S 1 =β

π2 c+ O(β 0 , log β) .
3β

(1.1)

Here, β is the circumference of one of the circles of the torus, while the other is of unit radius.
The O(log β) is due to the potential “noncompactness” of the CFT. For a holographic theory, thisformula maps to the universality of the Bekenstein-Hawking entropy of BTZ black holes [3].
In [1], Di Pietro and Komargodski introduced analogs of the Cardy formula for 4d and6d superconformal field theories (SCFTs), establishing universal relations between perturbative anomalies and the Cardy limit (high temperature limit) of the superconformal index.2
These higher-dimensional Cardy formulae involve a sum of terms that depend on the background geometry and gauge fields, whose overall coefficients are determined by the perturbative anomaly coefficients. Supersymmetry plays a key role in their higher dimensional Cardy
formulae, particularly because supersymmetric partition functions are (expected to be) geometric invariants – roughly speaking, quantities that depend only on a subset of the bosonicbackground data [9–13]. Let us elaborate this point. Unlike the partition function of a 2d
CFT on an arbitrary Riemann surface, which only depends on the complex structure (in theabsence of chemical potentials), the metric dependence for the partition function of higher
dimensional CFTs is much more complicated in general. However, the supersymmetric partition function ZMd for a SCFT on a spacetime manifold Md is (believed to be) sensitive to amuch smaller subset of the metric data; in the case of even dimensions, this subset comprises
the topology and the complex structure moduli of Md .3 Taking Md to be S 1 × Md−1 , thecomplex structure dependence of ZMd translates into the dependence on the “transversely
holomorphic foliation” structure of Md−1 . In fact, the sole dependence of ZS 1 ×Md−1 on thetransversely holomorphic foliation structure of Md−1 has been proven for 3d N = 2 theories
in [11, 12] and conjectured for 5d N = 1 theories in [14].4
The 4d Cardy formula was proven in [1] for Lagrangian theories continuously connected
to free theories via renormalization group (RG) flows triggered by marginal or relevant deformations (moduli flows were not considered, which we remedy in this paper). They evaluatedthe 3d geometric invariants on squashed sphere backgrounds (the evaluation on lens space
was later done in [15]), and determined the coefficients from the Kaluza-Klein (KK) reductionof the free theory. The 6d Cardy formula was first conjectured in [1], and further evidence forthe conjecture from the superconformal indices of free theories was found in [1, 16, 17].
In the first part of this paper, we derive the universal background dependence of 6d SCFTs
on squashed S 5 in the Cardy limit, by studying the 5d effective action from the reduction on
S 1 . Throughout, we make use of various effective actions, whose relations are summarized
in Figure 1. A key ingredient in the proof of the 6d Cardy formula is the classification of
the 5d supersymmetric Chern-Simons terms in [18] by the present authors, and their explicit
forms found earlier in [19–24]. We evaluate these 5d supersymmetric Chern-Simons termson the most general supersymmetric squashed S 5 background, and produce the conjectured
expression of [1]. Along the way, we find further evidence for these supersymmetric ChernSimons terms to be geometric invariants.
β

The convention for the torus moduli is τ = i 2π .
There are other universal limits of the partition function of CFTs in d > 2. The governing formulae are also
called Cardy formulae in the literature. See e.g. [4–8].
3
This is proven for 4d N = 1 SCFTs in [11, 12], and it would be interesting to pursue a similar argument in 6dby classifying Q-exact background couplings.
4
In [11, 12], it was shown that supersymmetry further demands such a background dependence to be holomorphic. Recently, in [13], it was pointed out that this statement is actually (slightly) scheme dependent. Nonetheless,since the different regularization schemes are related by local counter-terms, which are of order O(β) [13], they
will not affect the singular terms in the Cardy limit.
1

2

3

SciPost Phys. 11, 036 (2021)

6d/4d SCFT
on vacuum moduli space
6d/4d Wilsonian effective action
mmodul i < Λ, E < Λ

integrate out massive

reduce on S 1
integrate out massive
5d/3d Wilsonian effective action
mmodul i < Λ < β −1 , E < Λ

6d/4d effective action
E < mmoduli
reduce on S 1
integrate out all

integrate out massless

5d/3d effective action

Figure 1: Diagram depicting the relations among different effective actions. We denote by Λ the cutoff of the effective action, and mmoduli is the mass scale associatedwith the moduli scalar vacuum expectation value. The field contents are as follows:
the Wilsonian effective action contains light dynamical fields and background fields;
the 6d/4d effective action contains massless dynamical fields and background fields;
the 5d/3d effective action contains only background fields. On the one hand, the ↓→direction is in principle correct but difficult to carry out (unless the effective theory
is weakly coupled). On the other hand, except on the pure Higgs branch, where theGreen-Schwarz/Wess-Zumino type terms relevant for (mixed) gravitational anomalies are absent in the 6d/4d effective action, it is not understood how to perform thedashed arrow on the right. This is the source of the puzzle of Section 3.3.
In the second part, we determine the Chern-Simons levels that appear in the Cardy formulafor a specific SCFT. This is considerably harder in 6d than in 4d, because 6d SCFTs do nothave marginal or relevant supersymmetry preserving operator deformations, and there are
no known interacting 6d SCFTs with weakly-coupled (UV) Lagrangian descriptions [25, 26].
Fortunately, we can consider moduli space flows. Indeed, for theories with a pure Higgs branch(i.e. very Higgsable theories in the language of [27]), where the effective theory far away fromthe origin is particularly simple and described by free hypermultiplets, we can determine these
Chern-Simons levels exactly. This gives a proof of the 6d Cardy formula for very Higgsable
SCFTs, which include in particular rank-N E-string theories [28, 29] and (G, G)-type minimal
conformal matter theories [30].
A similar procedure applied to 4d extends the validity of the 4d proof of [1] to a larger
class of theories that includes various non-Lagrangian Argyres-Douglas type theories [31, 32].
We also comment on moduli space flows on the tensor and mixed branches in 6d, and on the
Coulomb and mixed branches in 4d.

1.1

Review of the 4d Cardy formula

For any N ≥ 1 SCFT in 4d, we define the supersymmetric Sβ1 × S 3 partition function (orsuperconformal index) ZS 1 ×S 3 by [33–36],5
β



P2
†
ZS 1 ×S 3 = Tr H (−1) F e−β̂{Q,Q } e−β i=1 ωi ( ji +R) ,β

5For simplicity, we neglect the fugacities for possible flavor symmetries of the system in this subsection.

4

(1.2)

SciPost Phys. 11, 036 (2021)

where Tr H denotes the trace over the Hilbert space H on S 3 in radial quantization, and j1 , j2
and R are the Cartan generators of the Lorentz and U(1)R -symmetry groups.6 The supercharge
Q and its conjugate Q† generate an su(1|1) subalgebra of the full N = 1 superconformal algebra. The combinations ji + R commute with the su(1|1) and pair with the chemical potentials(squashing parameters) ωi that refine and regularize the sum. Due to supersymmetry, only
states annihilated by Q and Q† contribute to the partition function, and consequently the β̂dependence drops out. In the Cardy limit β → 0, this supersymmetric partition function has
the expansion (we fix the radius of S 3 to be r3 ≡ 1)
log ZS 1 ×S 3 =βπ2 ω1 + ω2
κ + O(β 0 , log β) .
6β ω1 ω2

(1.3)

The coefficient κ is related to the pertuburtive mixed gravitational-R-symmetry anomaly, whichappears in the anomaly polynomial 6-form as
I6 3k
FR ∧ tr (R ∧ R) ,48(2π)3(1.4)

with R the Riemann curvature 2-form and FR the field strength of the background U(1)R gauge
field VR . The relation between κ and the anomaly coefficient k is κ = −k.7 By supersymmetry,κ and k are in turn related to the 4d conformal anomalies asκ = −k = 16(c − a) .

(1.5)

This provides a universal relationship between perturbative anomalies and the spectrumof protected BPS operators, and has been explicitly checked in examples by localization computations [1, 15, 43, 44]. The combination of (1.3) and (1.5) is dubbed the 4d Cardy formula,due to its similarity to the 2d Cardy formula (1.1).8
In [1], Di Pietro and Komargodski proved the 4d Cardy formula by considering the 4d
SCFT compactified on Sβ1 , which leads to a 3d effective action for a set of background fields
that include the 3d metric, the background graviphoton gauge field A, and the R-symmetry
background gauge field VR . To preserve supersymmetry, the fermionic degrees of freedom
are chosen to have periodic boundary condition along the Sβ1 . Thus, the 3d effective action
contains non-local terms due to integrating out certain massless modes. However, since thesemodes are uncharged under the Kaluza-Klein U(1)KK symmetry, such non-local terms are subThe superconformal index differs from the supersymmetric partition function by a Casimir factor [37–40],
which vanishes in the Cardy limit β → 0. The two are often used interchangeably in this paper, since we are only
concerned with the singular terms. Furthermore, one can replace the three-sphere more generally with Seifert
manifolds [9, 41, 42].
7
See footnote 9 for a caveat.
8
In 4d N = 2 SCFTs, an analogous Cardy formula was derived for the Schur index [44] (see also [45])6log ZSSchur
1 ×S 3 =βπ2 κ+ O(β 0 , log β) ,
2β

(1.6)

which also naturally arises in the Cardy limit of the associated 2d chiral algebra [46]. The relative factor of 32
compared to (1.3) in the case ωi = 1 is due to the ratio between the U(1)R backgrounds that are turned on in the
Schur limit for N = 2 SCFTs versus the one for a generic N = 1 SCFT, i.e.
Schur
VU(1)
=
R

3
VU(1)R .
2(1.7)

Note that the Schur limit also involves turning on the SU(2)R background gauge field along Sβ1 in the N = 2 SCFT,
but this does not affect the Cardy limit.

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SciPost Phys. 11, 036 (2021)

leading O(β 0 , log β) in the β → 0 limit.9 The leading term in the small β expansion of theeffective action is a Chern-Simons term of order O(β −1 ), Z
iκiW = − log Z =− VR ∧ dA + SUSY completion + O(β 0 , log β) ,
(1.8)
24π
where the graviphoton A is of order O(β −1 ).10 Evaluating this supersymmetric effective action on the squashed three-sphere background, one produces the Cardy formula (1.3). Whatremains is to establish the relation (1.5) between κ and the anomaly coefficient k.
Let us first provide a qualitative explanation of the relation (1.5) before proving it. Inκcanonical normalization, the Chern-Simons term in (1.8) has level 12, which is in general
fractional. Consequently, under a large background U(1)KK gauge transformation, the partition function will pick up a phase. This is nothing but a manifestation of the global (mixed)
gravitational anomaly of the 4d CFT [48–50].
Proceeding with the proof, the coefficient κ first must be invariant under both marginal andrelevant deformations. Otherwise, by promoting the coupling constants of the deformations
to background fields, the Chern-Simons term becomes gauge non-invariant under small background gauge transformations (which contradicts the absence of such perturbative anomalies
in 4d). In particular, this argument forbids not only the continuous dependence of κ on the
coupling constants, but also potential jumps of κ, since this would lead to gauge non-invariant
domain wall configurations (for the coupling) after promoting the coupling constants to background fields.11
Now, let us assume that our 4d SCFT is connected to a weakly coupled theory by either
marginal or irrelevant operators that preserve U(1)R .12 This happens, for example, when the
SCFT is located on the conformal manifold of an N = 1 conformal gauge theory or as theIR fixed point of an asymptotically-free gauge theory.13 Given such a free-field point, one
can explicitly KK-reduce the free fields along the Sβ1 . The coefficient κ receives contributions
from the massive KK-modes via 1-loop diagrams. Summing over such contributions under
the appropriate regularization, one finds the relation (1.5) between the coefficient κ and themixed gravitational-R anomaly coefficient k, computed at the free point. Since both κ and k
are invariant under RG flows triggered by deformations in the action, the relation (1.5) holds
at the superconformal fixed point as well.

1.2

The 4d Cardy formula from moduli space flows

The argument in the previous section can be extended to SCFTs connected to free theories
by moduli space flows. Given a 4d SCFT with a vacuum moduli space, generic Sβ1 reduc-tions would lift these flat directions. However, for supersymmetric (Scherk-Schwarz type) Sβ1
reductions which we consider here, the moduli space is (partly) preserved (and sometimes
9

Strictly speaking, this is only true if the effective action of the 3d massless modes has a minimum at the origin
upon turning on general chemical potentials. In particular, there are counter-examples when c − a < 0 in whichcase the 1/β term in the Cardy limit gets shifted by a non-universal piece due to the existence of nontrivial minimaof the potential for the holonomies around S 1 [15, 44, 47]. Here, for simplicity we restrict to theories where thisphenomena does not occur.
10
Compared with the convention in [1], our graviphoton field Ahere = 2π
β athere .
11
A priori, such jumps in κ could happen if there are extra massless degrees of freedom charged under U(1)KK
at special loci of the coupling space. Since we are assuming β1 is the largest scale, this is not possible in the present
context.
12
As explained in [25], there are no relevant deformations of an N = 1 SCFT without breaking the N = 1
supersymmetry and U(1)R symmetry. Moreover, marginal deformations can only become exactly marginal or
marginally irrelevant as explained in [51].
13
In the latter case, one needs to perform a-maximization to identify the superconformal U(1)R symmetry froma combination of the UV U(1)R symmetry and flavor symmetries [52].

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SciPost Phys. 11, 036 (2021)

enhanced) in the 3d theory. In particular, for 4d N = 2 SCFTs, the Higgs branch moduli space
is preserved and the Coulomb branch moduli space is enhanced (by line operators wrapping
the S 1 ) [53]. However, for 4d N = 1 SCFTs described by IR fixed points of asymptotically freegauge theories, it is known that upon S 1 reduction parts of the Coulomb branch are lifted due
to Affleck-Harvey-Witten type monopole-instantons [54]. In such cases, we use φ to denote thedirections on the 4d moduli space that are not lifted upon S 1 reduction. Since the 4d moduli
φ is unambiguously identified as a subspace of the 3d moduli, we will use the same notation. The 3d Wilsonian effective action, which does not involve integrating out the (massless)scalars φ parametrizing the moduli space (and their superpartners), takes the form of (1.8),
but with κ promoted to a function κ(φ) of the moduli fields.14 However, any nontrivial dependence of κ(φ) on φ is again in violation of background small U(1)KK gauge invariance, so
the coefficient κ(φ) must be constant in φ.15
From the above argument, one would hope to extract κ from a weakly coupled description
on the 4d moduli space which may involve massive particles and (extended) solitons coupled
to the moduli fields.16 Since we are interested in the supersymmetric partition function, weonly expect BPS states to contribute [55–57]. Consequently, κ is simply determined by thespectrum of BPS states and their U(1)R charges (in the full 4d theory on S 1 × S 3 but prior to
integrating out massive fields).
For a 4d N = 2 SCFT with a pure Higgs branch of quaternionic dimension dH , which
may not have a UV Lagrangian description, the low energy description is simply given bydH massless hypermultiplets. There are no other BPS particles on the Higgs branch due tothe absence of a scalar central charge, but there can be BPS strings (see Appendix B.1 for a
discussion) [58–62]. Recall that the U(1)R symmetry of the N = 1 subalgebra is related to theSU(2)R × U(1) r symmetry of the N = 2 algebra by
RN =1 =

4
1
I3 + rN =2 ,
3
3

(1.9)

where I3 is the Cartan element of the SU(2)R . Upon supersymmetric S 1 reduction, the 3d
Higgs branch is identical to the 4d Higgs branch [63]. Far on the Higgs branch, the effective
theory is described by dH free hypermultiplets. Thus, by a free field computation as in [1],one finds that17
2dH
1(1.10)
κ = − Tr (U(1) r ) =.
3
3
On the other hand, from anomaly matching [64],dH = 24(c − a) ,(1.11)

which confirms (1.5).
One may worry about potential contributions from the BPS vortex strings [59–62]. For Lagrangian theories, by looking at the Higgs branch localization formulae for the index [65, 66],
we explicitly see that they do not give additional contributions in the Cardy limit. Combinedwith the above agreement, we expect this to hold in general.
The effective action contains the usual kinetic term of φ plus higher derivative interactions which are of orderO(β 0 , log β)).
15
We remark that this argument is not affected by the presence of the path-integral measure for φ, since perturbative anomalies, which could potentially absorb such gauge non-invariant pieces are absent in 3d.
16Since an explicit mass term (with a U(1)R invariant mass parameter) pairs chiral and anti-chiral fermions
with the same U(1)R charge, integrating out such massive particles cannot contribute to the Chern-Simons level κ.
Consequently, κ can only receive contributions from massive particles when the U(1)R symmetry is spontaneously
broken, so that the mass parameter could carry nonzero U(1)R charge.
17
The N = 2 hypermultiplet contains two left-handed Weyl fermions with (I3 , rN =2 ) = (0, −1).
14

7

SciPost Phys. 11, 036 (2021)

The above argument extends the validity of the 4d Cardy formula to N = 2 SCFTs witha pure Higgs branch, on which there is no additional contribution to κ beyond the massless
hypermultiplets. In particular, one can explicitly check that it is obeyed by a large class ofArgyres-Douglas type theories (for instance) using the relation to 2d chiral algebras [31, 45,46].18On the contrary, the Coulomb branch is known to host a zoo of BPS particles which undergo complicated decays and recombinations as one explores the moduli space (from one
chamber to another), characterized by the wall-crossing phenomena. However, as we haveargued, κ(φ) must remain constant. In other words, one can pick any chamber and computeκ from the stable BPS spectrum within the given chamber. If the theory has a weakly coupledchamber, such as in Lagrangian theories where the stable BPS particles are W-bosons and massive hypermultiplets, it is easy to see that (1.5) holds.19 However, a complication arises when
there is no weakly coupled chamber. This happens for instance on the Coulomb branch of 4d
N = 2 Argyres-Douglas theories, which has mutually non-local BPS monopoles and dyons,and consequently the effective theory is never weakly coupled [31, 32]. In such cases, the interactions between the BPS particles cannot be neglected in the Cardy limit and will contributeto κ.
One may attempt to determine κ using a different procedure by first integrating out themassive BPS states in 4d which induces various higher derivative terms for the moduli fields,
and then studying the S 1 reduction of the effective action that only involves the massless moduli (and the supersymmetric partners). For the supersymmetric partition function, only F-type
higher derivative terms contribute (D-terms are necessarily Q-exact). This includes the supersymmetric Wess-Zumino term [67–69] (i.e. it contains the Wess-Zumino term for the Weyla-anomaly used in [70, 71] to prove the 4d a-theorem), which is essential for matching the
perturbative mixed anomaly of the CFT on the Coulomb branch involving the spontaneously
broken U(1) r symmetry [72]. When the Coulomb branch is one dimensional, it was arguedin [73] that this is the only F-term that can contribute on S 1 × S 3 .20 However, it is not clear
to us how the Wess-Zumino terms (and its supersymmetric completion) or other F-terms cancontribute to the level κ. We leave this for future investigation.21

1.3

6d Cardy formulaNow, let us consider a 6d N = (1, 0) SCFT in radial quantization. The states in the Hilbert
space are labeled by the energy E, the spins j1 , j2 , j3 of the SO(2)3 ⊂ SO(6) rotation of the S 5 ,and the R-charge R of the SU(2)R R-symmetry. The theory has eight Poincaré Q-supercharges
Q j1 , j2 , j3 ,R , and eight superconformal cousins. In principle, the full spectrum of this theory is
encoded in its S 1 ×S 5 partition function with anti-periodic boundary condition for the fermionsalong the S 1 . However, such a quantity is generally difficult to compute because the fermionicboundary condition breaks supersymmetry.
A more manageable quantity is the superconformal index, which is related by a Casimir
factor to the S 1 ×S 5 partition function with periodic boundary conditions for the fermions, andwith suitable chemical potentials turned on [40,74]. It is also a counting function, but only forstates saturating a BPS bound. To define it, we first pick a particular supercharge Q ≡ Q −−−+ ,See footnote 8 for an explanation of a relative factor of 32 where comparing the Cardy formulae that appearedin [44, 46] with that in [1].
19Note that the massive N = 2 vector multiplet contains two left-handed Weyl fermions with
(I3 , rN =2 ) = (±1/2, 1), respectively, so each W-boson contributes − 23 to κ.
20
In [73], it was proven that for one dimensional Coulomb branches, the only F-terms are proportional to the
Weyl tensor and its derivatives which all vanish on S 1 × S 3 .
21
A logical possibility is that (a) integrating out the massive matter in 4d and (b) compactifying on S 1 do notcommute, and thus the order of first (a) then (b) misses subtle contributions to κ.
18

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SciPost Phys. 11, 036 (2021)

whose anticommutator reads{Q, Q† } = E − j1 − j2 − j3 − 4R ≥ 0 .

(1.12)

Given this choice, the states annihilated by the supercharge Q and therefore saturating the BPS
bound (1.12) can be counted (with signs) by the superconformal index [74–76] (see also [77]for a review),


P I I
P3
†
ZS 1 ×S 5 = Tr H (−1) F e−β̂{Q,Q }−β I µf Hf −β i=1 ωi ( ji +R) .
(1.13)
βHere, the trace is taken over the Hilbert space H of states on S 5 , and β is the circumferenceof the S 1 . Furthermore, we have introduced (complex) chemical potentials {ωi }3i=1 (withRe ωi > 0) for the isometries, and µfI for the flavor symmetries. Finally, HfI are the generatorsof the Cartan subalgebra of the Lie algebra of the flavor symmetry Gf .22
Di Pietro and Komargodski [1] conjectured a Cardy formula for the Cardy limit β → 0 of
the superconformal index, stating that the singular terms in this limit take the form
log ZS 1 ×S 5 = −
β


 
 
2
2
2
κ1 2π 3 (ω1 + ω2 + ω3 )(κ2 − 3κ3 /2) 2ππ+
ω1 ω2 ω3 360 β
72
β
G





µ2f κf f 2π
(ω1 + ω2 + ω3 )2 κ3 2π+++ O(β 0 , log β) ,
48
β
24
β

(1.14)

with the constants κi completely fixed by the perturbative anomalies as follows.23 We write
the 8-form anomaly polynomial as
I8 =


1 
αc2 (SU(2)R )2 + β c2 p1 + γp12 + δp2 + µGf p1 c2 (Gf ) ,4!

(1.15)

where the explicit formulae for the Chern classes ci and Pontryagin classes pi in terms of the
field strengths and curvature are summarized in Appendix A. Here, Gf is the flavor symmetryof the 6d SCFT. Then, the relations areκ1 = −40γ − 10δ ,3
κ2 − κ3 = 16γ − 2δ ,2

κ3 = −2β ,Gκf f = −48µGf .

(1.16)

N = (2, 0) theory check
Before pursuing a proof, let us explicitly check the 6d Cardy formula in N = (2, 0) theories
where we have a closed form expression for the superconformal index in the limit ω1 = ω2 =
ω3 = 1 (unsquashed) and µf = 1, computed by localization in [74, 76]. The result for thetheory of type-g, where g is a simply laced simple Lie algebra, isg
ZS 1 ×S 5
β

=e

β ∨
6 hg |g|

β2π

 r2g Y
4π2  r1 − e−β(α·ρg ) η e− β g ,(1.17)gα∈∆+gwhere h∨g is the dual Coxeter number, rg is the rank, |g| is the dimension, ∆+ is the set of
positive roots, and ρg is the Weyl vector. Notice that the result for g = E is only conjectural, as
The normalization of HfI is chosen to be Tr (HfI HfJ ) = δ I J .
As in the 4d/3d case, we assume that the 5d effective action of the massless modes has a minimum at the origin
as β → 0 (see footnote 9). We checked this explicitly when the 6d index has a known matrix model description,i.e. for N = (2, 0) theories and E-string theories.
22
23

9

SciPost Phys. 11, 036 (2021)

the instanton contributions are unknown. In the Cardy limit, the last factor of (1.17) becomes
the dominant contribution, and we findg
Sβ ×S 5log Z 1

=

rg π2
6β

+ O(β 0 , log β) .

(1.18)

To compare, the Cardy formula (1.14) with the Chern-Simons levels κi given in Table 2 dictatesthat the Sβ1 × S 5 partition function in the Cardy limit islog ZS 1 ×S 5 =β

rg π2
24βω1 ω2 ω3(2ω1 ω2 + 2ω2 ω3 + 2ω3 ω1 + µ2f − ω21 − ω22 − ω23 ) + O(β 0 , log β) ,(1.19)
which with ω1 = ω2 = ω3 = 1 (unsquashed) and µf = 1 matches with the localization result (1.18).
The β → 0 limit of the type-g, N = (2, 0) superconformal index commutes with the “chiral algebra limit” of the 6d theory in which its superconformal index reduces to the vacuum
character of a 2d Wg algebra [78]. This implies that the Cardy formula of the corresponding
2d VOA coincides with the 6d (supersymmetric unsquashed) Cardy formula (1.18), and themodular properties of the characters lead to a high/low-temperature relation in the 6d parent
between the Casimir energy and the Cardy limit. See [44, 46] for analogous statements in the
4d case.

1.4

Sketch of the proof

We presently outline our proof of the 6d Cardy formula, given by (1.14) and (1.16), for theories
with (at least) a pure Higgs branch. The first step is to analyze the most general 5d effective
action of 6d SCFTs compactified on Sβ1 . As argued in [1] and in Section 3, the 5d effective
action has an expansion in the small radius β → 0 limit asGκf f G 
κ2 − 32 κ3κ1κ3iiW = − log Z =
I1 +
I2 −
I3 −
I f + O(β 0 , log β) ,8π2 360
144
24
24 4

(1.20)

which contains four types of supersymmetric Chern-Simons terms
ZA ∧ dA ∧ dA + SUSY completion ,

I1 ≡
Z
I2 ≡
Z
I3 ≡
G
I4 f

Z
≡

A ∧ tr (R ∧ R) + SUSY completion ,(1.21)A ∧ Tr (FR ∧ FR ) + SUSY completion ,A ∧ Tr (FGf ∧ FGf ) + SUSY completion .

Here, A is the U(1)KK graviphoton (which in the β → 0 limit scales as β −1 ), R denotes the
Riemann curvature 2-form of the 5d background metric hi j ,

2
β
2
i
(1.22)
ds6 = dτ +Ai dx
+ hi j dx i dx j ,2π
FR is the field strength of the SU(2)R background gauge field, and lastly FGf is the field strengthof background flavor symmetry gauge fields.24 The key realization is that only these supersymmetric Chern-Simons terms contribute to singular terms in the β → 0 limit.
24

Recall that on the supersymmetric background, the SU(2)R gauge field takes value in the Cartan of SU(2)R(see Appendix C).

10

SciPost Phys. 11, 036 (2021)

The squashing dependence of the 6d Cardy formula (1.14) is recovered by evaluating theeffective action (1.20) on the rigid supersymmetric background of a squashed S 5 , with thesquashing parameters ωi = 1 + ai of the 5d metric (see Section 2.1). We stress that thecontributions from the additional supersymmetric pieces in (1.21) are absolutely crucial to our
result; without them, the result is not a geometric invariant depending only on the squashingparameters.
The second part of the 6d Cardy formula is the relation (1.16) between the Chern-Simonslevels κi in (1.14) and the perturbative anomaly coefficients. To derive this relation, we firstargue that the Chern-Simons levels are constant on the vacuum moduli space, similar to ourargument in Section 1.1 for the 4d case. Consider the 5d Wilsonian effective action that does
not involve integrating out the (massless) moduli scalars φ (and the superpartners). It takes
the form (1.20) but with the Chern-Simons levels κi promoted to functions κi (φ) of the moduli
fields. Note that the scalars descend to moduli fields in the 5d theory, since these flat directions
are not lifted under supersymmetric Sβ1 reduction. Any nontrivial dependence of κ(φ) on φ isin violation of background small gauge invariance, so the coefficients κi (φ) must be constanton the entire vacuum moduli space.
For 6d SCFTs with a pure Higgs branch, on which we just have several massless hypermultiplets and no other BPS particles (see Appendix B.2 for a discussion), the above argument
allows us to determine the Chern-Simons levels by simply reducing free fields. This procedure
proves the relation (1.16) for theories possessing a pure Higgs branch, with the understandingthat γ = − 74 δ, which is inherently true for such theories.
To derive the relation (1.16) on the tensor branch, we compute the one-loop contributions
from the free fields to the Chern-Simons levels κ, and find that there must be additional contributions. It is known that the 6d tensor branch supports BPS strings which may contributeto κ upon reduction on S 1 . Indeed, this is evident from the conjectured localization formulae
for the S 1 × S 5 partition function of 6d SCFTs [74, 76, 77, 79] (see also Section 3.3). However,
it is not clear how to systematically include contributions from such states to κ. Instead, if
we first integrate out the massive states on the 6d tensor branch, we obtain a tower of higher
derivative interactions in the effective action. In particular, one of the leading F-terms is givenby (the supersymmetric completion of) the Green-Schwarz term. Then, one needs to study
the contribution from such terms to κ upon reduction on S 1 ; we will not pursue this in thepresent paper.
To provide another perspective on the would-be contributions from BPS strings to κ, we
consider the relation between the Chern-Simons levels in the 5d effective action and the global
anomalies of the 6d SCFT, and explain that the extra contributions are essential for the global
anomaly matching. The global gravitational anomalies of the 6d SCFT can receive contributions from the Green-Schwarz term in the tensor branch effective action [80–86] (just as for
perturbative anomalies), therefore we expect that such terms are responsible for additional
contributions to the 5d Chern-Simons levels upon compactification on S 1 .
The remainder of this paper is organized as follows. In Section 2, we prove the squashingdependence of the 6d Cardy formula, by solving for appropriate 5d backgrounds with rigid supersymmetry, and evaluating the supersymmetric Chern-Simons terms on such backgrounds.
In Section 3, we fix the Chern-Simons levels in terms of the perturbative anomaly coefficients,
by combining non-renormalization arguments and free field computations on the pure Higgs
branch. An analogous computation on the pure tensor branch is then performed which yields
a naive mismatch with (1.14) but the “missing” contributions must come from the BPS strings.
In Section 4, we remark on the relation between the Chern-Simons levels and global gravitational anomalies, which further highlights the “missing” contributions on the tensor branch.
Section 5 closes with a summary and comments on future directions. Finally, in the four appendices we provide more details on various aspects discussed in the main text. In particular,

11

SciPost Phys. 11, 036 (2021)

in Appendix C we detail the 6d supersymmetric background on S 1 × S 5 , including its “modified/complexified version” and how it relates to our 5d backgrounds.

Supersymmetric Chern-Simons terms on the squashed S 5 background2

In this section, we study the 5d supersymmetric Chern-Simons terms (1.21) and their supersymmetric completions arising from the dimensional reduction of the 6d theory, and evaluate
them on the squashed S 5 background. These Chern-Simons terms correspond to the higherderivative terms in 5d Poincaré supergravity classified in [18].
We begin with the S 1 × S 5 geometric background and reduce to the 5d background that
also includes a graviphoton field, and further embed this bosonic background into supergravity to obtain the full 5d supersymmetric background. We review various ingredients, i.e. various (matter) multiplets and a version of Poincaré supergravity that arises from a particular
choice of gauge-fixing of the (conformal) standard Weyl multiplet together with a vector anda linear multiplet.25 Then, following the general formalism of [36], we provide the rigid supersymmetric background for a generically squashed S 5 . Finally, we detail the corresponding
higher-derivative terms (supersymmetric Chern-Simons terms) and their evaluation on the
supersymmetric squashed S 5 background.
Readers who are not interested in the technicalities of 5d supersymmetric solutions forvarious multiplets may skip this section and consult the results summarized in Section 2.5.

2.1

Squashed S 5 background from reduction of squashed S 1 × S 5 backgroundWe begin by specifying the appropriate S 1 × S 5 background of the 6d theory. This can be done
most straightforwardly by employing a conformal transformation from flat R6 to R × S 5 , andthen compactifying the non-compact direction with twisted boundary conditions for various
fields induced by the chemical potentials in (1.13). The introduction of such chemical potentials reduces the amount of preserved supercharges. We absorb the twists for the isometriesof the S 5 into the geometry (for the sake of setting some background fields to zero), and fix
the (deformed) metric of S 1 × S 5 to be
dsS21 ×S 5

=

r52

3 
X

d yi2 + yi2

i=1


2 
iai
dφi +
dτ
+ dτ2 ,
r5

(2.1)

where τ ∼ τ+β is the β-periodic S 1 coordinate, and { yi , φi }i are polar coordinates, satisfying
y12 + y22 + y32 = 1 as well as φi ∼ φi + 2π. Finally, r5 is the radius of the five-sphere, and thechemical potentials ωi are related to the metric deformations via
ωi = 1 + ai ,(2.2)

the round case being ωi = 1. The twisting parameters ai can in general be complex numbers. The full 6d supersymmetric background and the background for the “modified” index
are detailed in Appendix C.
As we are interested in the compactification to five dimensions, we can conveniently rewrite
the 6d metric as follows
dsS21 ×S 5 = κ̃−2 (dτ + ir5 Y )2 + ds52 ,
(2.3)
25

Note that this is a different choice of Poincaré supergravity than the one employed in [87].

12SciPost Phys. 11, 036 (2021)

with the (squashed) S 5 metric ds52 ,ds52 =

3Xd yi2 + yi2 dφi2 + κ̃−2 Y 2 ,i=1

Y = κ̃2

3
X

ai yi2 dφi ,

i=1
3
X

κ̃−2 = 1 −(2.4)

y 2j a2j .

j=1

When the twisting parameters ai ∈ R, they correspond to the squashing parameters of thesquashed S 5 (ai = 0 corresponds to the round S 5 limit). A comparison of the metrics (1.22)
and (2.3) (up to a conformal transformation) shows that the graviphoton field A in the squashed
S 5 background is
2πi.
A = mKK r5 Y ,
mKK =
(2.5)
β
In the following, we set r5 = 1.

2.2

Off-shell 5d supergravity multiplets

The 5d higher-derivative supersymmetric Chern-Simons terms can be written in terms of various 5d (off-shell) supergravity multiplets. On the one hand, we work with the standard Weyl
multiplet coupled to vector multiplets [19, 20]; on the other hand, we work with the 5d offshell Poincaré supergravity. The former is enough if the Chern-Simons term preserves conformal invariance, which in Poincaré supergravity is explicitly broken. The Poincaré supergravity
is obtained by gauge-fixing the (conformal) standard Weyl multiplet [19, 20, 88, 89]. In thefollowing, we pick a particular gauge-fixing condition which naturally reduces the su(2)R symmetry of the standard Weyl multiplet to its u(1)R truncation, which is convenient because there
is a general description of 5d rigid supersymmetric backgrounds of (the u(1)R truncation of)
the standard Weyl multiplet [14, 90].
We review the various multiplets of interest, before presenting the solution on the squashedS 5 in the next section.
Standard Weyl multiplet
The (full) 5d standard Weyl multiplet consists of the following matter contentSW = gµν , D , Vµi j , vµν , bµ , ψµi , χ i ,

(2.6)

ij

given by the metric gµν , a dilaton D, an su(2)R gauge field Vµ , a two-form field vµν , a gauge
field bµ of the Weyl symmetry, and two su(2) Majorana fermions ψµi and χ i . To obtain a rigid
supersymmetric background [36], we have to set the fermions to zero and thus find (at least)
one non-vanishing spinors such that the supersymmetry variation of the fermions vanish. In
particular, for a generic transformation
δ = "¯i Q i + η̄i Si ,(2.7)

where Q i and Si are the supercharges and their conformal cousins, and " i , ηi are the respective
parameters specifying the transformation, the supersymmetry variation of the fermions of thestandard Weyl multiplet reads
1 νρ
v γµνρ " i − γµ ηi ,2
δχ i = " i D − 2γρ γµν " i ∇µ vνρ + γµν Fµν i j (V )" − 2γµ " i εµνρσλ v νρ v σλ + 4γµν vµν ηi .

δψµi = Dµ " i +

13

(2.8)

SciPost Phys. 11, 036 (2021)

Here, F (V ) is the field strength of the su(2)R symmetry gauge field V . The covariant derivative
is given by
1
1
(2.9)
Dµ " i = ∂µ " i + bµ " i + ωµ ab γab " i − Vµ i j " j .
2
4
We recall that the doublet indices i, j ∈ {1, 2} are raised and lowered with the totally antisymmetric tensor εi j using NW-SE and SW-NE conventions, i.e.,ζi = εi j ζ j ,ζi = εi j ζ j ,(2.10)

with ε12 = −ε12 = 1. Notice that bµ corresponds to the gauge field for Weyl transformations (and to obtain non-conformal supergravity we have to gauge-fix it). Furthermore, in thepresent context we shall work with a u(1)-truncated version of the standard Weyl multiplet –i.e. the gauge field (Vµ )i j only has components along the Cartan generator of su(2)R – in whichcase the supersymmetry conditions (2.8) can be recast in terms of certain simple geometricconstraints on the 5d background [14, 90].
Vector multiplet
The 5d supergravity vector multiplet contains the fields

V = Wµ , M , Ωαi , Y i j ,(2.11)

where Wµ is the gauge field (the background gauge field for the 5d flavor symmetry), M ascalar (the “scalar mass parameter”), Ωαi the gaugino, and Y i j a triplet of auxiliary scalars. In
order to obtain a rigid supersymmetric background, we set the fermions to zero, and thus theirvariation [20, 91]
1
1/ "i + Y i j " j − M η jδΩi = − γµν Fµν (W )" i − DM
42

(2.12)

has to vanish. Here, Fµν (W ) denotes the field strength of the vector multiplet gauge field W .
This version of the vector multiplet is naturally coupled to the standard Weyl multiplet SW ,with the same supersymmetry parameters, " i and ηi .
Linear multiplet
As we shall see, in order to describe Poincaré supergravity, we have to add a vector compensator
multiplet as well as another either linear or hypermultiplet compensator. We opt to go withthe former option, and thus introduce the 5d linear multiplet L here. It contains the fields

L = L i j , ϕαi , E µ , N ,(2.13)
where N and L i j are scalars, E µ is a divergence-less vector, i.e.
∇µ Eµ = 0 ,(2.14)and ϕαi are their fermionic partners. Again, to preserve rigid supersymmetry, we set thefermions to zero and require their variation [20, 91]
11/ L i j " j + γµ " i Eµ + " i N + 2γµν vµν " j L i j − 6L i j η j
δϕ i = − D22
to vanish.

14(2.15)

SciPost Phys. 11, 036 (2021)

Poincaré supergravity
Now, to obtain 5d Poincaré supergravity, we add compensators (denoted by hatted symbols)
to the standard Weyl multiplet SW and gauge-fix the conformal symmetry [20, 23, 92]. In
the following, we shall pick the compensators to be given by a vector multiplet V̂ and a linear
multiplet L̂.26 To gauge-fix the superconformal transformations and the dilatation one may
impose the following “standard gauge” conditions27“Standard gauge”:

M̂ = m ,bν = 0 ,L̂ i j =i
L̂(σ3 )i j ,2Ω̂i = 0 .

(2.16)

Here, m is (generally chosen to be) an arbitrary constant of mass dimension one, and bν isthe gauge field for dilatations. The first constraint fixes dilatations (up to a constant), the second constraint fixes special conformal transformations, the third reduces the su(2)R symmetry
down to u(1)R , and the last one fixes the S-supersymmetry.
There is another way to gauge-fix the Weyl and the superconformal symmetries given by
the “KK gauge” condition,27“KK gauge”:L̂ i j =i
L̂(σ3 )i j ,2L̂ = 1 ,bν = 0 ,ϕ̂ i = 0 .

(2.17)As before, the first constraint in (2.17) breaks su(2)R to u(1)R , the second and third fixes
dilatations and special conformal transformations, respectively, and the last one fixes the Ssupersymmetry transformation. Note that while L̂ i j in the compensator linear multiplet iscompletely fixed by the gauge condition, the scalar M̂ in the compensator vector multiplet isunfixed; We are free to choose its value to be non-constant and proportional to the warping
factor κ̃ defined in (2.4),
M̂ = MKK ,where

MKK = −mK K κ̃

(2.18)

is the (fixed) “KK-mass”.28
The compensator vector multiplets V̂ in these two gauges have different physical meanings.
In the standard gauge, V̂ should be interpreted as a background vector multiplet for a Cartan
component of the flavor symmetry, whose mass parameter is a constant. In the KK gauge,V̂ should be interpreted as the background vector multiplet for the U(1)KK symmetry, whosemass parameter is position-dependent and determined by the warping parameter κ̃.
The two gauge-fixing conditions, (2.16) and (2.17), are related to one-another by a Weyltransformation (prior to gauge-fixing it); we refer to Appendix D for more details. In fact, we
explicitly find that evaluating the FRR terms for either gauge-fixing (2.16) or (2.17) leads tothe same answer in the case of (generic) squashed S 5 . This gives more credence to the fact thatthe answers should be “geometric invariants”, i.e. only dependent on some general geometricstructure (in our case believed to be the transversely holomorphic foliation) and not on theparticular choice of background fields.
The fields of the standard Weyl multiplet SW together with the compensator vector V̂and linear multiplet L̂, gauge-fixed using the conditions (2.16) make up the full (off-shell)
Poincaré supergravity multipletiPstd = gµν , D , Vµi j , vµν , Ŵµ , Ŷ i j , Ê µ , L̂ , N̂ , ψµα, χαi , ϕ̂αi .
(2.19)
26

Alternatively, one can add vector multiplets and a hypermultiplet as compensators; this was considered in [21,

88].
27

The naming of the two gauge-fixing conditions is for the convenience of referencing in this paper, and is notstandard in the literature. There are various other ways to fix the conformal symmetry and get different versionsof Poincaré supergravity, none of which is distinguished (see e.g. [92]).
28
Since the 6d geometry (2.4) is warped, the KK-mass is not constant but rather proportional to the warping
factor κ̃ defined in (2.3).

15

SciPost Phys. 11, 036 (2021)

In the case of the “KK gauge” (2.17), we get the following (independent) component fields in
the Poincaré multipletiPKK = gµν , D , Vµi j , vµν , M̂ , Ŵµ , Ŷ i j , Ê µ , N̂ , ψµα, χαi , Ω̂αi .
(2.20)In the rest of this paper, we will work in the KK gauge if the compensator gauge field Ŵ
is identified with the graviphoton gauge field A. This involves the supersymmetric completionof the A ∧ tr (R ∧ R) term, which contains the graviphoton and is written in terms of Poincaré
supergravity. On the other hand, when we evaluate the supersymmetric completion of the
A ∧ Tr (FGf ∧ FGf ) term, we have to include some other (non-compensator) background vector
multiplets, in which the gauge fields AGf of the flavor symmetry reside.
To get a supersymmetric background of Poincaré supergravity, we take the rigid limit, andthus set the fermionic fields to zero. Hence, we are required to find non-trivial solutions for
the Killing spinors " i to the vanishing fermionic supersymmetry transformations (2.8), (2.12)
and (2.15), whilst imposing the gauge-fixing conditions of (2.16) or (2.17).

2.3

Squashed S 5 backgrounds in off-shell supergravity

To evaluate the 5d supersymmetric Chern-Simons terms on the squashed S 5 background (2.4),
we consider the rigid limit of the various supergravity multiplets discussed in the previous
section. We first present the solution for the standard Weyl multiplet, SW , and then proceedwith solving for the flavor vector multiplets, the KK vector multiplets and finally the full (KKgauge fixed) Poincaré supergravity.
Solution for the standard Weyl multiplet
The solution for the bosonic background fields (V i j , v, D) of the rigid standard Weyl multiplet
are detailed in [18], and we simply state the result29
1
dY ,
4κ̃

 3
X
2
κ̃2 ,
D =2
ai2 + 2atot − atotv =(2.21)

i=1

Vij = −

ii(atot − 1)Y + d(φ1 + φ2 + φ3 ) σ3 j ,2

P3
where atot = i=1 ai . This solution for the background fields in the standard Weyl multiplet
satisfies the supersymmetry equations (2.8) with the following (conformal) Killling spinors


−i





Æ

"

1



κ̃β̃  i = p ,2 2  1 

−i



Æ

"

2



κ̃β̃  −i = p 
,2 2  −1 

−1
−1
 


 a 1
i
4
1
1
2
2
ab
"1 ,
(1 − atot )κ̃ −+ ∂a log κ̃β̃ γ + vab γ
η =
6
6
3
κ̃β̃




 a 1
i
4
1
2
2
2
ab
η = −
(1 − atot )κ̃ −+ ∂a log κ̃β̃ γ + vab γ
"2 .
6
6
3
κ̃β̃

(2.22)

29
There is some residual freedom in our choice of solution, which we have fixed in order to obtain a (somewhat)
simple-looking answer.

16

SciPost Phys. 11, 036 (2021)

We have chosen the frame
1

e1 =



q


( y22 − 1)d y1 − y1 y2 d y2 ,1 − y22
a3 − a1y1 y3
2(dφ1 − dφ3 ) +
X ,
e =q
β̃
1 − y221
d y2 ,
e3 = q
1 − y22y2
1 + a24e =q−dφ2 +X ,β̃
1 − y2y3(2.23)

2

e5 =

1

1
X + Y,
κ̃κ̃β̃with the definitions

X =

3
X

yi2 dφi ,

Y

= κ̃2

3
X

i=1

ai yi2 dφi ,β̃ = 1 +Xai yi2 ,(2.24)ii=1

and the 5d gamma matrices are given by
γ1 = σ3 ⊗ 12×2 ,γ2 = σ2 ⊗ 12×2 ,γ5 = −σ2 ⊗ σ2 ,γ5 = −σ2 ⊗ σ1 ,γ3 = −σ2 ⊗ σ3 ,(2.25)

with σi the standard Pauli matrices.
Flavor multiplet solution
Next we present the solution for nV flavor (background) vector multiplets VfI , I = 1, . . . , nV ,coupled to the standard Weyl multiplet SW . As mentioned above, we fix the scalar MfI to
be the constant mass mfI of the I th vector multiplet. Then, we find the following solution to
the vector multiplet supersymmetry condition (2.12) (of course this is also a solution to thestandard gauge-fixing of Poincaré supergravity given in (2.16))30
Flavor solution VfI :MfI(WfI )µ dx µ
I i(Yf ) j

= µfI ,
=

µfI

=

iµfI

1 − β̃ κ̃

1 − atotβ̃1(κ̃β̃)2X+

1
κ̃2 β̃


(2.26)Y ,(atot − 1)κ̃
2
3
+
+
−
2
β̃ κ̃ (κ̃β̃)2



σ3

i

j.

KK multiplet solution
For this solution, we would like a connection between the KK gauge field WKK in the vector
multiplet and the graviphoton in the 6d metric (2.3). This will lead to a different solution
for the vector multiplet than the one in (2.26), and in particular, the mass MKK will not be
constant. Of course this is expected, since there is a warping factor in the 6d metric. Although
the two solutions are distinct, we spell out a way to relate them in Appendix D.
30
Notice, that for our purposes it is crucial that the solution presented here is continuously connected to theround S 5 solution, presented in [18]. In particular, this means that WfI vanishes in the round limit, ai → 0 – there
is another general class of “topologically non-trivial” solutions, which is related to 5d instanton backgrounds.

17

SciPost Phys. 11, 036 (2021)

Then, the “KK solution” VKK is given by
KK solution VKK := −mKK κ̃ ,MKK
(WKK )µ dxµ

i(YKK ) j= mKK Y ,imKK 2κ̃ (1 − atot ) (σ3 )i j ,=2(2.27)

where we remark that indeed WKK precisely agrees with the graviphoton gauge field in (2.3),
and we can further pick the constant mKK = 2πiβ to fully match to 6d.
Poincaré solution in the “KK gauge”
Finally, in order to evaluate the last supersymmetric Chern-Simons term we require a solution
to Poicaré supergravity given by imposing the KK gauge-fixing conditions (2.17). This is because the relevant Chern-Simons term, A∧tr (R∧R), contains the U(1)K K gauge field, A, rather
than a flavor gauge field. Then, the corresponding solutions are given by the standard Weyl
solution (2.21) together with the KK solution (2.27) – which is now a compensator vector
multiplet within the Poincaré multiplet, i.e. V̂ = VKK – and the following compensator linear
multipletL̂ i j =Êµ dxµ=i(σ3 )i j ,2y12 y32 4
1 − y22
+

N̂ =



y22

β̃


1 − y224κ̃β̃(a1 − a3 ) + (a12 + a32 )κ̃2

(a1 − a3 )X − dφ1 + dφ3β̃(1 + a2 )4
2
2(1 + a2 ) + (a2 − 1)κ̃ − 3
X − dφ2 ,
β̃
β̃



(2.28)+ (atot − 1)κ̃ .

This solution satisfies the supersymmetry conditions (2.8), (2.12) and (2.15) together with the
KK gauge fixing conditions (2.17), and thus constitutes a solution to KK gauge-fixed Poincaré
supergravity.

2.4

Supersymmetric Euclidean Chern-Simons terms and their evaluation on thesquashed S 5With the squashed S 5 solutions for the various supergravity multiplets in hand, we can proceed with evaluating the explicit 5d supersymmetric Chern-Simons terms corresponding tothe (minimal) supersymmetric completion of the pieces in the 5d effective theory, arising in
the Cardy limit of the 6d superconformal index. We first introduce their explicit expressions
for Riemannian manifolds, which requires a careful Wick rotation from the expressions in theliterature, and then provide their evaluation on the rigid supersymmetric background.31
There are four supersymmetric Chern-Simons terms: FFF, Ff Ff F, FWW, and FRR, whose
linear combinations correspond to I1 , . . . , I4 in (1.21).32 The first two fall into the same classin the classification of [18], while FWW and FRR constitute a subset of the remaining threeclasses. The reason we can omit the F-Ric2 term is that it can be related to a linear combinationof FWW and FRR by a field redefinition.
31

In the following, we shall present the Euclidean version of the relevant supersymmetric Chern-Simons terms;
this is somewhat subtle and to the knowledge of the authors has not been discussed in the literature before.
32
Here, we are using the notation of [18], and label the Chern-Simons terms by the featured fields: F and Ff for
the KK and flavor-gauge fields/field strengths, W for the Weyl tensor, and R for the Ricci scalar.

18

SciPost Phys. 11, 036 (2021)

The FFF and Ff Ff F terms can be treated uniformly by first assuming a generic set of vector
multiplets labeled by I, J, K ∈ {0, 1, . . . , nV − 1} with structure constant c I J K . The (Euclidean)
supersymmetry completion of the A ∧ dA ∧ dA term in the rigid limit can be written as follows [19, 20] (see also [89])33Z
13
S1 (V I ) =c I J K W I ∧ F J (W ) ∧ F K (W ) − M I F J (W ) ∧ ∗F K (W )
2
2
M5

3
(2.29)+ M I dM J ∧ ∗dM K − 3M I M J 2F K (W ) + M K v ∧ ∗v
2




1 J K RIJ
K ij
+ M 3(Y )i j (Y ) + M M− D vol5 ,42
where we denoted by F I (W ) the two-form field strength of the gauge field W I , and the volume
five-form is explicitly given by34
vol5 =pg d y1 ∧ d y2 ∧ dφ1 ∧ dφ2 ∧ dφ3 .

(2.30)

2.4.1 FFF term
We start by considering the first supersymmetric Chern-Simons term, given by the supersymmetry completion of the 5d Chern-Simons action
A ∧ dA ∧ dA,(2.31)

where (as before) A is the U(1)KK graviphoton. The minimal supersymmetric extension is
simply given by a U(1) vector multiplet VKK coupled to the standard Weyl multiplet SW .
Therefore, there is no need to be working in Poincaré supergravity, and hence, we are notrequired to impose any gauge-fixing conditions.
Here, we are solely dealing with the gauge fields arising from U(1)KK -photons, and thus
we may set I = J = K = 0 and c000 = 1 in (2.29). We may then plug in our solutions into thesupersymmetric action (2.29), i.e. we take V ≡ VKK , with corresponding solutions in (2.21)
and (2.27).35 We readily observe that up to some total derivative terms, the integrand reduces
to the 5d contact volume, i.e.
Z
Z
m3KK
S1 (VKK , VKK , VKK ) =η ∧ dη ∧ dη +d ∗ (· · · ) ,(2.32)
2
MM
5

5

where η is the contact 1-form on the squashed S 5 , given byη = X + β̃ Y ,(2.33)

where X and Y are given in (2.24). This is true for a more general family of backgrounds.
Then, a simple application of the Duistermaat-Heckman fixed-point formula [93] shows that
S1 (VKK , VKK , VKK ) =m3KK (2π)3.
2 ω1 ω2 ω3(2.34)

33
Here, we use the standard conventions for the 5d Hodge star operator, i.e. for p-forms α and β we haveα ∧ ∗β = p!1 αµ1 ...µp β µ1 ...µp ∗ 1.
34
This action also gives rise to the third supersymmetric Chern-Simons term S3 by replacing two instances of thevector multiplet by composite expressions in terms of the linear multiplet, i.e. we have S1 (V̂ , V , V ), where by V wedenote the vector multiplet expressed in terms of a linear compensator multiplet; See below.
35
One could also simply evaluate the standard Poincaré supergravity action, which includes a compensator linear
multiplet piece. This would not correspond to the minimally extended version of the corresponding Chern-Simons
term, and it would only contribute to the subleading pieces O(β 0 ) in the β-expansion.

19

SciPost Phys. 11, 036 (2021)

2.4.2 Ff Ff F term
We now turn towards the supersymmetric completion of the term
A ∧ Tr (FGf ∧ FGf ) ,(2.35)

where we recall that FGf is the field strength for a background flavor multiplet. For simplicity,
we focus on the background
FGf = FfI HfI = dWIf HfI ,(2.36)
where HfI are the generators of the Cartan subalgebra of the Lie algebra of the flavor symmetry
Gf with the normalization Tr (HfI HfJ ) = δ I J .
We start again with the action in equation (2.29), where we now pick two of the vector
multiplets to be pure flavor multiplets,i j 
i
(2.37),
VfI = WfI µ , MfI , ΩfI α , YfI
while the third one is given by the KK vector multiplet, VKK . The resulting action, S1 (VfI , VfJ , VKK )with c I J K = δ(I J δK)0 , is then given by
Z




1
2WfI ∧ FKK + WKK ∧ FfI ∧ FfJ
6
M5



1
1
− 2MfI FKK + MKK FfI ) ∧ ∗FfJ + 2MfI dMKK + MKK dMfI ) ∧ ∗dMfJ
2 
2

J
J
I
− 2Mf 2MKK Ff + Mf FKK ∧ ∗v − 3MfI MfJ MKK v ∧ ∗v





1 I J
R
J
I
J
+ (Yf )i j 2Mf (YKK )i j + MKK (Yf )i j vol5 − Mf Mf MKK D −vol5 ,
42(2.38)
and we plug in the standard Weyl solution (2.21) together with one instance of the KK solution (2.27) and two instances of the flavor solution (2.26). By the same arguments as before,
we end up withm(2π)3 2(2.39)S1 (VfI , VfJ , VKK ) = KKµ ,2 ω1 ω2 ω3 f
P
where µ2f = I (µfI )2 .
S1 VfI , VfJ , VKK



=

δI J

2.4.3 FWW term
Let us now turn to the remaining two supersymmetric Chern-Simons terms. They are givenby the supersymmetry completion of the terms A ∧ tr (R ∧ R) and A ∧ Tr (FR ∧ FR ), where (asbefore) A is the U(1)KK -photon and FR = F (V ) the background su(2)R field strength. These
terms decompose into (a linear combination of) two supersymmetric higher-derivative terms,
which are themselves the supersymmetric completion of the Weyl- and Ricci-squared higherderivative actions in 5d. We shall first focus on the flavor-Weyl2 (FWW) term, and below
discuss the flavor-Ricci2 (FRR) one.
First, we turn to the supersymmetric completion of the Weyl-squared higher-derivative
term. This term gives the supersymmetric completions of a combination of the following
Chern-Simons terms
A ∧ tr (R ∧ R) and A ∧ Tr (FR ∧ FR ) .
(2.40)As in the case of the FFF supersymmetric Chern-Simons term, the FWW term can be written
purely in terms of the vector multiplet coupled to the standard Weyl multiplet. It was first20

SciPost Phys. 11, 036 (2021)

written down in [21] (see also [89]), and its Euclidean (Wick rotated) version in the rigid
limit is given as follows36
S2 (V I ) =

Z
cI
M5

p



 


1 2
1 2
D 4 2
2 2
g M
C − v
+
D +
− v vµν F I (W )µν
8
12
6 9


1 I
4
I ρσ
ρσ
µν 1
M v + F (W )− (Y I )i j F (V )µν i j v µν
+ Cµνρσ v
3231 µνρσλ
3
I
τδ
ij
−
ε
(W )µ
Cνρτδ Cσλ − F (V )νρ F (V )σλ i j
12
4


σλ
ρ
σ λτ
I
µν 2 ρτv ∇τ v + v τ ∇ v− εµνρσλ F (W )3
1
− M I F (V )µν i j F (V )µν i j + 4∇ν vµρ ∇µ v νρ − 8vµν ∇ν ∇ρ v µρ3
1 I
+ M 16Rνρ vµν v µ ρ + 12∇µ vνρ ∇µ v νρ − 2Rv 2
9


1 I
1
+ F (W )µνvµν v 2 + 4vµρ v ρλ vνλ + 4M I v µν vνρ v ρσ vσµ
3
3

2 I
µν ρσ
λτ
+ M εµνρσλ v v ∇τ v
,
3
I



where we have defined

C 2 = C µνρσ Cµνρσ ,(2.41)(2.42)

with the Weyl tensor given byCµνρσ = Rµνρσ − 31 (gµρ Rνσ − gνρ Rµσ − gµσ Rνρ + gνσ Rµρ )1+ 12(gµρ gνσ − gµσ gνρ )R .

As before, we use the solution for the KK vector multiplet VKK (2.27) together with thestandard Weyl solution (2.21), and a straightforward yet tedious computation yields
S2 (VKK ) ==(a12 + a22 + a32 ) − (a1 a2 + a1 a3 + a2 a3 )3ω1 ω2 ω3ω21 + ω22 + ω23
2ω1 ω2 ω3(2π)3 mKK(ω1 + ω2 + ω3 )2−(2π)3 mKK ,6ω1 ω2 ω3(2.43)

where we have set c I = δ I,0 indicating that we are dealing with a single Abelian U(1)KK gauge
field.
2.4.4

FRR term

Finally, the FRR supersymmetric Chern-Simons term coincides precisely with the supersymmetric completion of the pieceA ∧ Tr (FR ∧ FR ) .
(2.44)
It can be written in terms of Poincaré supergravity, which we introduced as a gauge-fixing of thestandard Weyl multiplet above. Notice that this term explicitly breaks conformal invariance.
The full Lagrangian was first constructed in [89] (in Lorentzian signature), by taking thevector multiplet action (2.29), and then writing part of the vector multiplet fields as composite
362Here, we mostly follow the notation in [89], but with Dhere = 16Dthere + 128
3 Tthere , vhere = 4Tthere , as well as

I
I
Mhere
= −ρthere.

21

SciPost Phys. 11, 036 (2021)

expressions in terms of the gauge-fixed linear multiplet. Therefore, we require two steps to get
the Lagrangian; we write down the vector multiplet action and then replace the underlined
fields by their composite expressions, which we provide further below. The Ricci scalar squared
term then arises from the Y i j Y i j term. By choosing c I,0,0 = c I and all other c I J K to zero in thevector multiplet action, we obtain the following (Euclidean) Ricci scalar squared action in therigid limit
Z

1
1
p
S3 (V̂ I ) =c I g M̂ I Y i j Y i j − 2ρY i j (Ŷ I )i j + M̂ I ρ 2 R − M̂ I F µν F µν
8
4
M
5

1
1
+ ρ F µν F̂ I (Ŵ )µν − M̂ I ∂µ ρ∂ µ ρ − M̂ I ρ∂ µ ∂µ ρ
2
2

1 I 2
− M̂ ρ D + 6v 2 − ρ 2 F̂ I (Ŵ )µν v µν + 2 M̂ I ρ F µν v µν
41
I µ νρ σλ+ εµνρσλ (Ŵ ) F F,
8(2.45)

where the composite expressions (in the rigid limit) are given byρ =N̂( L̂ i j L̂1i j) 2,2

1F µν = 2∇[µ (( L̂ i j L̂ i j )− 2 Êν] ) +Y ij =

3( L̂ i j L̂

1

i j)

1
2

F (V )µν i j L̂ i j −

i j)

3
2

L̂ kl D[µ L̂ kp Dν] L̂ l p ,2

V i Dµ L jk
1 µ k
8( L̂ i j L̂ i j )( L̂ i j L̂ i j )( L̂ i j L̂ i j ) 2
1
1
Da L̂ k(i D a L̂ j)m L̂ km −N̂ 2 L̂ i j
+
3
3
i
j
i
j
2
2
( L̂ L̂ i j )4( L̂ L̂ i j )

1
1
1
+
D − 2v 2 L̂ i j −Êµ Ê µ L̂ i j +Êµ L̂ k(i Dµ L̂ j) k ,
3
1
3
i
j
i
j
i
j
4( L̂ L̂ i j ) 2
4( L̂ L̂ i j ) 2( L̂ L̂ i j ) 2
1
2

L̂ i j R −1
2∂ µ Dµ L̂ i j −2( L̂ i j L̂with v 2 defined as(2.46)

v 2 = v µν vµν ,(2.47)

Dµ L̂ i j = ∂µ L̂ i j + 2Vµ (i j L̂ j)k ,(2.48)

the covariant derivative Dµ L i j given by

and the symmetrization defined as as X (i Y j) = 21 (X i Y j + X j Yi ), and similarly for the antisymmetrization.
We can now plug in our explicit supersymmetric solutions for KK gauge-fixed Poincaré
supergravity, i.e. equations (2.8), (2.12) and (2.15), with gauge fixing conditions (2.17).
Again, a tedious analysis shows that it integrates toS3 (V̂ ) = −(ω1 + ω2 + ω3 )2(2π)3 mKK ,ω1 ω2 ω2

(2.49)

where we have set c I = δ I,0 as we are to dealing with a single Abelian U(1)KK -photon.

2.5

Summary

Let us now conclude this technical section with a summary of our results. By carefully evaluating the supersymmetric completion of the four Chern-Simons terms appearing in the 5d
22

SciPost Phys. 11, 036 (2021)

effective action, and imposing the following relation between the KK-masses mKK to the S 1
circumference β
2πi
mKK =
,
(2.50)
βwe have shown that
I1 ≡ 2S1 (VKK , VKK , VKK ) =Z

A ∧ dA ∧ dA + SUSY completion

M5

(2π)3
=
ω1 ω2 ω3Z


1
I2 ≡ −4 S2 (VKK ) − S3 (V̂ )
6

=

2πi
β

3

,

A ∧ tr (R ∧ R) + SUSY completionM5= − 2(2π)3Z

1I3 ≡ − S3 (V̂ ) =2

I4 ≡ 2S1 VKK , VfI , VfJ







(ω21 + ω22 + ω23 ) 2πi,
ω1 ω2 ω3β(2.51)A ∧ Tr (FR ∧ FR ) + SUSY completion

M5(2π)3 (ω1 + ω2 + ω3 )2=2
ω1 ω2 ω3
Z

=



2πi
β



,

A ∧ Tr (FGf ∧ FGf ) + SUSY completionM5(2π)3 2 2πiµ.
=ω1 ω2 ω3 fβ

3Deriving the Chern-Simons levels

Having shown that the 5d supersymmetric Chern-Simons terms produce the form of the Cardy
formula, it remains to determine the Chern-Simons levels κi in (1.14). Let us revisit the originof these terms from the 6d perspective: they come from integrating out massive Kaluza-Klein(KK) modes, and are accompanied by non-local pieces arising from the massless degrees of
freedom. The latter are uncharged under the U(1)KK symmetry, so their contributions are oforder O(β 0 ) or O(log β). Since we are interested in the parts of the 5d effective action that are
singular as β → 0, we only need to take into account the massive KK contributions, and doing
so allows us to determine the Chern-Simons levels. We emphasize that such 5d Chern-Simons
terms are unaffected by the local counter-terms in 6d.
The Chern-Simons terms that arise from integrating out massive KK modes can be computed explicitly for various 6d free fields. For interacting SCFTs, our strategy is to connect themto free theories by renormalization group flows on the vacuum moduli space. To see that thisproduces the correct answer at the origin (i.e. the interacting superconformal point), we recall
the argument given in Section 1.4 that in the 5d Wilsonian effective action, any dependenceκ(φ) on the moduli fields violates background gauge invariance.
For Higgs branch flows, the free field computation allows us to determine the precise relation between the Chern-Simons levels and perturbative anomaly coefficients. This completes
the proof of the 6d Cardy formula for general SCFTs with a Higgs branch. We also carry out thecomputation for tensor branch flows, and discover that additional contributions from the BPS
strings wrapping the S 1 , are essential, though to compute them we need further information
about such objects.

23

SciPost Phys. 11, 036 (2021)

3.1

Integrating out free field KK modes

The 5d Chern-Simons terms arise from integrating out KK modes from the S 1 reduction of 6dtensor, vector and hypermultiplets. It is well-known that only chiral fermions ψ∓ and (anti)self-dual 2-forms B can generate such Chern-Simons terms. The Chern-Simons terms of thefirst two types in (1.21) are given in [94]. The result is one-loop exact due to the usual nonrenormalization argument. For a conjugate pair of KK modes of U(1)KK charge ±n, we get thefollowing contributions
1
1
3
Iψn =
n
I
+
nI2 ,
1
−
48π2
384π2
1
1
(3.1)
Iψn = −
n3 I1 −
nI2 ,
2
+
48π
384π2
4
8
IBn = −
n3 I1 +
nI2 ,
2
48π
384π2
where the various Chern-Simons terms are given in (1.21).
For 2m fermions transforming in the 2m representation of US p(2m), a similar computation
gives
1
ne
I (US p(2m)) ,Ĩψn = ±
2
∓
32π
Z
(3.2)

e
I (US p(2m)) =A ∧ Tr FUSp(2m) ∧ FUSp(2m) + SUSY completion .
iA tensor multiplet contains the fields (B, ψ−), while a vector multiplet only contains theifermions ψ+ , where i ∈ {1, 2} is the fundamental index for SU(2)R . On the other hand, nH hyi
permultiplets contain 2nH fermions ψ−, which are uncharged under SU(2)R and transformed
in 2nH representation of US p(2nH ).
Putting these contributions together and using zeta function regularization to perform thesum over KK modes,
∞
∞
X
X
1
1
(n3 ) =,(n) = − ,
(3.3)
120
12
n=1
n=1

we obtain

2−4 1
2+8 1
2 1
I1 −
I2 −
I3 ,
2
2
48π 120384π 1232π2 12
−2 1
−2 1
−2 1
IV =
I1 −
I2 −
I3 ,
2
2
48π 120384π 1232π2 122
12
1
2 1 USp(2nH )
n
IHH =
I1 −
I2 −
I
,
2
2
48π 120384π 1232π2 12 4IT =US p(2nH )where I3 and I4

(3.4)

are
USp(2nH )I3 = eI (SU(2)R ∼= US p(2)) ,I4= e
I (US p(2nH )) .

(3.5)

Thus, we conclude thatn T I T + nV IV + nH IH(nH − nV − n T ) 1(nH − nV + 5n T ) 1(−nV + n T ) 1
1 1 USp(2nH )
I1 −
I2 −
I3 −
I
,
2
2
2
24π
120
192π
12
16π
24
16π2 24 4(3.6)
and comparing with the expression of (1.14) for free fields rewritten asn T + nV − nH
nH + 5n T − nVn T − nVi
1 USp(2nH )
iW =
I1 +
I2 +
I3 +
I
8π2
360
288
24
24 4
(3.7)
+ O(β 0 , log β) .
=24

SciPost Phys. 11, 036 (2021)

3.2

On the Higgs branch

On the Higgs branch, the free-field analysis above is sufficient to fully capture the 6d Cardy
formula. The reason is as follows. From the perspective of the 6d effective action (see Figure 1), the free-field analysis can potentially miss contributions from extended BPS objects.
In Appendix B.2, we classify such objects by studying the central extensions of the supersymmetry algebra, and find that the only extended BPS objects allowed on the Higgs branch arecodimension-two vortex branes. However, such objects wrapped on S 1 have infinite energy(unlike the BPS strings on the tensor branch), and thus cannot contribute to the Chern-Simonslevels in the 5d effective action. We now proceed with explicitly deriving the relation between
the Chern-Simons levels κi in the 5d effective action (1.20) and the (perturbative) anomaly
coefficients in the anomaly polynomial (1.15).
Let us assume that the 6d SCFT has a non-Abelian flavor symmetry group Gf and a Higgs
branch, whose infrared limit consists of free vector and hypermultiplets. At an arbitrary pointon the Higgs branch, the global symmetry group SU(2)R × Gf is broken toSU(2)R × Gf → SU(2) D × G1 × · · · × Gn ,(3.8)

where SU(2) D is the diagonal subgroup of SU(2)R and SU(2)X , with SU(2)X ×G1 ×· · ·×Gn ⊂ Gf .37
The infrared theory consists of m half-hypermultiples in the doublet of SU(2) D , and free halfhypermultiplets in a pseudo-real representation r1 ⊕ r2 ⊕ ·P
· · ⊕ rn of G1 × · · · × Gn . Let us denotenthe dimension of the representation ri by dri , and dtot = i=1 dri .
By using the perturbative anomaly matching [64] and global anomaly matching on the
Higgs branch, we prove the formula (1.16) for the class of theories considered here. Let us
start with the anomaly polynomial of the UV SCFT, which contains the terms(8)

IU V ⊃

1 
δp2 + γp12 + β c2 (SU(2)R )p1 + 24µc2 (Gf )p1 .
4!

(3.9)

Under the general symmetry breaking pattern (3.8), we have the following relations involving
the second Chern classes:
c2 (Gf ) = c2 (SU(2) D ) +

n
X
i=1

ai c2 (Gi ) ,(3.10)

c2 (SU(2)R ) = c2 (SU(2) D ) ,where ai are some coefficients that could be determined by the details of the symmetry breaking pattern (3.8). Thus, the UV anomaly polynomial (3.9) admits a rewriting in terms of theresidual symmetry group as

 n
 
X
1
(8)
2
IU V ⊃
δp2 + γp1 + (β + 24µ)c2 (SU(2) D )p1 + 24µ
ai c2 (Gi ) p1 .
(3.11)
4!
i=1
On the Higgs branch, the anomaly polynomial of the IR effective theory receives contributions
from the hypermultiplets and contains the terms
§




dtot
dtot
1
1
7
(8)
IHB =
−
m+
p2 +
m+
p12
4!
60
2
240
2
ª
n
(3.12)m
1 X Gi+ c2 (SU(2) D )p1 +T (ri )c2 (Gi )p1 ,
2
2 i=1
37

We have assumed that Gi are simple Lie groups. Our argument can straightforwardly be generalized to thecase with U(1) factors appearing on the right-hand side of (3.8).

25

SciPost Phys. 11, 036 (2021)

where T G (r) is defined bytr r (FG2 ) = T G (r)Tr (FG2 ) .

(3.13)We have also used2
Tr (Fusp(dtot

2) = tr fund (Fusp
(d
)

tot

X

) =
)

tr ri (FG2 ) =X

i

T Gi (ri )Tr (FG2 ) .
ii(3.14)i

By perturbative anomaly matching [64], we conclude that




dtot
dtot
1
7
δ = −
m+
,
γ =
m+
,
60
2
240
2
m
β + 24µ =,48µai = T Gi (ri ) .
2(3.15)

Notice that for theories with a Higgs branch,7
γ = − δ.
4

(3.16)

Now, let us turn to the corresponding effective action on the Higgs branch. We start withthe general expression
Z
Z
 


iκ1
2
i
3
SU(2)RGiWU V =
A
∧
A
∧
dA
∧
dA
−
κ
−
κ
p
+
4κ
c
+
4κ
c
2
3
1
3 2
f 2 ,
2880π2 M
96 M
3
2
5
5
(3.17)
which under the Higgs branch symmetry breaking pattern (3.8) reduces to the following expressionZ
iκ1iWU V =A ∧ dA ∧ dA
2880π2 M
5
 
 n

(3.18)
Z

X
2
3
i
A∧
κ2 − κ3 p1 + 4(κ3 + κ f )c2 (SU(2) D ) + 4κ fai c2 (Gi )
.
−
96 M
3
2
i=1
5

On the other hand, plugging in the field content on the Higgs branch to the effective action (3.7), we findZdm + 2tot
A ∧ dA ∧ dA
WHB = − i2880π2 M
5(3.19)
 

Z

n
X
dtot
i
1
Gi
m+
p1 − 4mc2 (SU(2) D ) − 4
T (ri )c2 (Gi ) .
−
A∧
96 M
3
2
i=1
5

Thus, comparing (3.18) with (3.19), we find the relations
31
κ2 − κ3 = (2m + dtot ) ,24
κ f ai = −T Gi (ri ) .

1
κ1 = − (2m + dtot ) ,2
κ3 + κ f = −m ,(3.20)

Hence, from (3.15) and (3.20) we derive the universal formulae for SCFTs with a Higgs
branch (with the relation between γ and δ (3.16)):
κ1 = 60δ ,3
κ2 − κ3 = −30δ,2κ3 = −2β ,κ f = −48µ .
26(3.21)

SciPost Phys. 11, 036 (2021)

This is consistent with the general formulae conjectured in [1], namely
κ1 = −40γ − 10δ ,3
κ2 − κ3 = 16γ − 2δ,2κ3 = −2β ,(3.22)

κ f = −48µ .

3.3

On the tensor branch

To extend the analysis to theories without a Higgs branch, a natural step is to derive the coefficients κi and κf on the tensor branch. In this section, we compute the “naive” Chern-Simonslevels κi and κf from the one-loop free field contributions on the tensor branch of several theories, and show that additional contributions must be present. In particular, in theories that also
have a Higgs branch, we find a “mismatch” between the two results.38 Although in 6d there
are no massive particles, there are massive (tensionful) BPS strings – including “M-strings” in
N = (2, 0) theories and “E-strings” in the E-string theory – that can potentially contribute to
the Chern-Simons levels. A key difference between the Higgs branch and the tensor or mixed
branch is that BPS strings are absent in the former, but are in general present in the latter.
In the M-theory picture, such BPS strings are M2-branes stretched between M5-M5 or M5-M9
branes, and are massive only when there are separations between the M5/M9 branes (i.e. on
the tensor or mixed branch).
Using supersymmetric localization, the superconformal index can be computed by a contour integral on the complexified tensor branch, and one can explicitly see that the BPS strings
crucially contribute to the Cardy limit of the superconformal index. The integrand is a productof three copies of the supersymmetric partition function on R4 × T 2 [74, 76, 79], which canfurther be written as a product of the partition function of the free tensor multiplets and a sum
over the elliptic genera of multiple BPS strings winding around the T 2 [95, 96]. The Cardy(β → 0) limit of the superconformal index reduces to the Cardy limit of the elliptic genera,
which is controlled by the central charge (1.1) of the theory living on the BPS strings. Therefore, a potential way to compute the Chern-Simons levels on the tensor branch is to relatethem to the 2d central charges of the BPS strings using the above relations. We leave this for
future investigation.
Below, we elaborate on the one-loop free field contributions and highlight the differencecompared to (1.14) which should come from BPS strings.
In the fourth columns of Table 2 and 3, we give the values of the Chern-Simons levels
κi ’s and κf inferred from the free field content of various theories on their tensor branches.
The values of the κi ’s can be straightforwardly computed using the numbers of the free multiplets given in Table 1. In the following, we give more details on the computation of theChern-Simons level κf , which only receives contributions from the hypermultiplets. On thetensor branch, the hypermultiplets transform under the US p(2nH ) symmetry. By the one-loop
computation in Section 3.1, we findUSp(2nH )κf38= −1 .

(3.23)A quick way to see that additional contributions must be present is the following. Many 6d SCFTs have a onedimensional tensor branch, on which the massless free field content is the same, but have very different Cardy limits
according to (1.14). It is plausible that the difference comes from the Green-Schwarz type terms, which appear
upon integrating out the appropriate BPS objects appearing on the tensor branch (see Appendix B.2). Likewise,for 4d SCFTs that have a complex one-dimensional Coulomb branch, the contributions from the supersymmetricWess-Zumino terms – again upon integrating out the heavy BPS states – may be crucial for recovering the Cardy
formula at the fixed point.

27

SciPost Phys. 11, 036 (2021)

Table 1: Field content on the tensor branch of various theories.
TheorynH

nVnT

E-string

N −1

0

N

type-g (2, 0)

rg

0

rg

2

k N

(k − 1)(N − 1)

N −1

TN ,ΓD2(2k − 8)Nk(2k − 1)(N − 1) + (k − 4)(2k − 7)N

2N − 1TN ,ΓE

086N − 78

4N − 1TN ,ΓE

16N

160N − 133

6N − 1TN ,ΓE

16N

334N − 248

12N − 1TN ,ΓA

k−1
k

6
7
8

2

Table 2: Anomaly coefficients κ1,2,3 for an assortment of 6d SCFTs I. We exhibit boththe values computed at the SCFT point using the formula (1.16), and the naive valuesby simply adding free fields on the tensor branch.

Free theory

E-string

type-g (2, 0)

Coefficient

SCFT

Naive tensor branch

Difference

κ1−nH + nV + n T
1
2 (nH − nV + 5n T )−nH + nV + n T
1
2 (nH − nV + 5n T )0

κ3
US p(2nH )κfnV − n T

nV − n T

0

−1

−1

0

κ1

−30N + 1

1−30N

κ2 − 32 κ3

15N − 123N − 1212N

κ3N (6N + 5)−N

6N (N + 1)

E
κf 8

−12N

0

−12N

κ1

0

0

0

κ2 − 32 κ3

3rg

3rg

0

κ3
SU(2)
κf

−rg

−rg

0

−rg

−rg

0

κ2 − 32 κ30

In general a subgroup H of the UV flavor symmetry group G acts nontrivially on the hypermultiplets, and it is embedded into the US p(2nH ) symmetry. The κfG can be computed by theformulaeκfH = IH,→G κfG ,(3.24)
USp(2nH )κfH = IH,→US p(2nH ) κf
= −IG,→USp(2nH ) ,where IH,→G and IH,→US p(2nH ) are the embedding indices of the embedding of the UV flavor
subgroup H into the UV flavor group or the US p(2nH ) symmetry or nH free hypermultiplets.
For the E-string, TN ,E6 , TN ,E7 , and TN ,E8 theories, the UV flavor group G does not act on
the hypermultiplets, i.e. the subgroup H is trivial. Hence, the Chern-Simons level κf for thosetheories on the tensor branches is zero.
For the type-g, N = (2, 0) theories viewed as N = (1, 0) theories, the UV SU(2) symmetryis embedded as the diagonal subgroup of US p(2) rg ⊂ US p(2rg ). The embedding index isISU(2),→US p(2)rg = rg ,

and

28

IUSp(2)rg ,→USp(2rg ) = 1 .

(3.25)

SciPost Phys. 11, 036 (2021)

Table 3: Anomaly coefficients κ1,2,3 for an assortment of 6d SCFTs II. We exhibit
both the values computed at the SCFT point using the formula (1.16), and the naive
values by simply adding free fields on the tensor branch.

TN ,ΓA

Coefficient

SCFT

Naive tensor branch

Difference

κ1

−k2

−k2

0


k + 6N − 6k2 − 2 (N − 1)

0

−2k

−2k

0

κ1

−2k2 + k − 1−2k2 + k + 30N − 1−30N

κ2 − 23 κ3

k − 2k + 3N − 52
k2 (4N −2)−4kN+k−10N +1

k − 2k − 9N − 52
k2 (4N −2)−16kN+k+26N +1

12N
12(k − 3)N4 − 4k16 − 4k

−12

κ1

−79

90N − 79−90N

κ2 − 23 κ3

3N + 73
2−33N + 73
2

36N

κ3
(E6 ) L(E )κf
or κf 6 R

166N − 77

82N − 77

84N

−24

0

−24

κ1

−134

150N − 134−150N

κ2 − 23 κ3

3N + 64−57N + 64

60Nκ3(E7 ) L(E )κf
or κf 7 R

382N − 132

154N − 132

228N

−36

0

−36

κ1

−249

330N − 249−330N

κ2 − 23 κ3

3N + 2432−129N + 243
2

132N

κ3
(E8 ) L(E )κf
or κf 8 R

1078N − 247

322N − 247

756N

−60

0

−60

κ2 − 23 κ3

k−1

1
2

k + 6N − 6k2 − 2 (N − 1)

κ3
SU(k)R
SU(k) Lκfor κf

TN ,ΓDk

TN ,ΓETN ,ΓE

6

7

8

2

2

κ3
SO(2k)R
SO(2k) Lκfor κf

TN ,ΓE

1
2



2

0

2

We find the Chern-Simons level
SU(2)

κf

= −rg ,

(3.26)

for the type-g N = (2, 0) theories on the tensor branch.
For TN ,ΓA , there are k2 N hypermultiplets on the tensor branch. Only k2 of them are
k−1

charged under the UV flavor symmetry SU(k) L . In the IR limit, the k2 free hypermultipletstransform under US p(2k2 ). The UV SU(k) L symmetry embeds into the IR US p(2k2 ) symmetry
as SU(k) ,→ SU(k)×SU(k) ,→ SU(k2 ) ,→ US p(2k2 ). We find the following embedding indicesISU(k),→SU(k)×SU(k) = k ,ISU(k)×SU(k),→SU(k2 ) = 1 ,SU(k) L

Therefore, the Chern-Simons levels κf

SU(k) Lκf

SU(k)Rand κfSU(k)R= κf

ISU(k2 )→USp(2k2 ) = 2 .

(3.27)

on the tensor branch are= −2k .

(3.28)For TN ,ΓD , there are 2k(2k − 8)N hypermultiplets on the tensor branch, but only k(2k − 8)kof them are charged under the UV flavor symmetry SO(2k) L . In the IR limit, the k(2k − 8) free
hypermultiplets transform under US p(2k(2k − 8)). The UV SO(2k) L symmetry embeds into
29

SciPost Phys. 11, 036 (2021)

the IR US p(2k(2k − 8)) symmetry as SO(2k) ,→ SO(2k) × US p(2k − 8) ,→ US p(2k(2k − 8)).
We find the embedding indicesISO(2),→SO(2k)×US p(2k−8) = 2k − 8 ,ISO(2k)×USp(2k−8),→USp(2k(2k−8)) = 2 .
SO(2k)R

SO(2k) L

Therefore, the Chern-Simons levels κf

SO(2k) Lκf4and κfSO(2k)R= κf(3.29)

on the tensor branch are= 16 − 4k .

(3.30)

Chern-Simons levels and global anomalies

There is a direct relation between the Chern-Simons levels κi in the 5d effective action andthe phases from global anomalies in the 6d theory [48, 49, 97]. This connection immediately
implies that the Chern-Simons levels κi can only jump by an appropriate quantized amountalong renormalization group flows into the Higgs or tensor branch moduli spaces. This supports the argument of Section 1.4 that κi are in fact constant on the entire vacuum moduli
space.
We put the theory on a six manifold M6 (1.22) that is an S 1 fibration over a five manifold
M5 , and study the large diffeomorphisms of the coordinate τ of the S 1 fiber. The coordinate
τ has periodicity β, i.e. τ ∼ τ + β. The boundary condition for the fermionic degrees of
freedom along the S 1 fiber is chosen to be periodic to preserve supersymmetry. Due to thisboundary condition, the fermionic degrees of freedom on M5 have integer charges under the
background graviphoton,Z
1dA ∈ Z ,
(4.1)
2π Σ
2

where Σ2 is a two-cycle in M5 . Therefore, the five manifold M5 must be a spin manifold.39
In the low energy limit, the tensor or Higgs branch effective theories always contain freefermions. For the partition function on M6 to be nonzero, we need to choose a backgroundsuch that there is no fermionic zero mode. Let us now assume that M5 = S 1x × M4 , with the5radius of S 1x being R5 , i.e.
5x 5 ∼ x 5 + 2πR5 .

(4.3)

The boundary condition of the fermionic degrees of freedom on S 1x is chosen to be periodic.
5
The fermion zero modes of the IR free fermions can be lifted by turning on a nontrivial flat
connection of either the R-symmetry or flavor symmetry background gauge fields along the x 5
direction. This is equivalent to trivial background gauge fields but with nontrivial boundary
conditions for the charged degrees of freedom.
Consider a background diffeomorphism as follows
τ → τ+

nβ 5
x ,
2πR5

(4.4)

39
If the fermionic degrees of freedom have antiperiodic boundary conditions along the S 1 , then after the dimensional reduction the fermionic degrees of freedom on M5 would have half-integer charges under the background
U(1)KK graviphoton gauge field A given in the metric (1.22). The five manifold M5 could be a more general spincmanifold. More precisely, the fermionic degrees of freedom on M5 have integer charges under a spinc connection
Ac = 21 A, which satisfies
Z
Z
1
1
dAc =
w2 mod Z ,
(4.2)
2π Σ
2 Σ
2

2

where w2 is the second Stiefel-Whitney class of M5 .

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SciPost Phys. 11, 036 (2021)

where n ∈ Z to preserve the boundary conditions for the fermionic degrees of freedom along
the S 1x . The background diffeomorphism corresponds to the background large gauge trans5
formation of the graviphoton A,n
A → A + dx 5 .
(4.5)
R5In general, for theories with (mixed) gravitational anomalies, the partition function is notinvariant under such a background diffeomorphism, and transforms by a phase,Z[A + δA] = e−iπη Z[A] .

(4.6)

The 5d effective action completely captures this anomalous diffeomorphism. Under thelarge gauge transformation (4.5), the effective action transforms as
Z


κ1
π
3
π
π Gf
δW = ndA ∧ dA −(κ2 − κ3 )p1 −κ3 c2 (SU(2)R ) −κf c2 (Gf ) .
72
2
12
12
M4 480π(4.7)
The above integral satisfies quantization conditions given by various index theorems on thespin manifold M4
Z1
m1 ≡dA ∧ dA ∈ Z ,2(2π)2 M
4Z
1m2 ≡p1 ∈ 2Z ,
24 M
4
Z
(4.8)
c2 (SU(2)R ) ∈ Z ,

m3 ≡
M4

Z
mf ≡

c2 (Gf ) ∈ Z .

M4

We find that the anomalous phase is given by
η =

nm1
2nm2
3
n
G
κ1 +
(κ2 − κ3 ) −(m3 κ3 + mf κf f ) mod 2 .
60
3
2
12

(4.9)

From the fifth columns of Table 2 and 3, we find explicitly the mismatches between the
global anomaly η of the UV SCFT (computed using the formula (1.16)) and the IR effective
theory on the tensor branch. However, anomaly matching on the tensor branch is more difficult
due to the (possible) subtle contributions of Green-Schwarz type terms which are generated
from integrating out massive BPS strings. Recent progress on understanding the contributionof the Green-Schwarz type terms to global anomalies has been made in [80–86]. For theories
that also have a Higgs branch (where the formula (1.16) was proven), anomaly matching
between the two branches then predicts what the Green-Schwarz type contributions must be.

5Conclusion

We have proven the universality of the Cardy limit of the superconformal index for 6d SCFTs
with a pure Higgs branch, embodied in a precise formula (1.14) conjectured by Di Pietro
and Komargodski in [1] for the singular terms, with coefficients related to the perturbative
anomalies via (1.16).

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SciPost Phys. 11, 036 (2021)

Summary of proofi) Compactifying the 6d SCFT on Sβ1 , in the Cardy limit (β → 0), one obtains the 5deffective action (1.20) that contains four supersymmetric Chern-Simons terms I j , forj = 1, . . . 4, with explicit expressions given in (1.21).
ii) We evaluate {I j }4j=1 on the supersymmetric squashed S 5 background, and determine
the explicit dependence on the squashing parameters ωi . The dependence is in agreement with the proposal (1.14). The pieces that supersymmetrically complete the ChernSimons terms contribute nontrivially; in fact, their contributions are crucial for the final
answer to be a geometric invariant, i.e. dependent only on the transverse holomorphic
structure.
iii) To prove the relation (1.16) between the Chern-Simons levels κ j and the coefficientsα, β, γ and δ in the 8-form anomaly polynomial (1.15), the following are the steps.
a) First, we use background infinitesimal gauge invariance to argue that the ChernSimons levels κ j are invariant under RG flows of the 6d SCFT on its moduli space.
b) The IR effective theory on the Higgs branch is a theory of hypermultiplets (free atlarge moduli), whose field content can be determined explicitly from the generic(global) symmetry breaking pattern along the RG flow.
c) We explicitly KK-reduce the free hypermultiplets along the Sβ1 , and determine theChern-Simons levels κ j by integrating out massive KK modes at one-loop. The
result can be expressed in terms of the perturbative anomaly coefficients α, β, γand δ as in (3.22).
d) Since κi , α, β, γ and δ are all invariant under the pure Higgs branch flow, therelation (3.22) holds at the UV superconformal fixed point as well.
A puzzle on the tensor branchiv) By looking at various examples, we remarked that the invariance of the Chern-Simonslevels along the tensor branch flow requires extra contributions to the κi ’s, in addition
to the one-loop contributions from the free fields. We leave a more in-depth study ofthese extra contributions for future work [98].40
Global anomaly matchingv) We relate the Chern-Simons couplings κ j to global gravitational anomalies. By explicitly
studying the variation of the effective action under large diffeomorphisms, we derive a
relation between the anomalous factor η in (4.6) and the Chern-Simons levels κ j .
Future prospects
An obvious avenue for exploration is to study more general 5d manifolds with non-trivial
topology. This is especially interesting in view of the relation between the 5d Chern-Simonscoefficients and global anomalies, the latter being sensitive to the topology of the space.
40
We note that the full 6d anomaly polynomial for a large class of theories are not fully determined as of thewriting of this paper.

32

SciPost Phys. 11, 036 (2021)

The 2d Cardy formula is intimately related to modular properties of the torus partition
function. In fact, modularity properties for N = (2, 0) [74, 76, 79] have also been understood.
It is an intriguing question to ask whether the Cardy formula for N = (1, 0) theories also hasa modular origin.
An alternative abstract way of determining the Chern-Simons levels for the 5d effective
action in the Cardy limit was given in [99, 100]. Their argument involves putting the CFT on
conical geometries, and assuming that the partition function is well-defined. In particular, theanswer should not depend on the different resolutions of the singularity. It would be interesting
to extend their analysis to the supersymmetric setting, in which case the supersymmetric partition function is known to be insensitive to the resolutions of conical singularities [101–103],
which may put their arguments on a more rigorous footing.
Lastly, we comment on the relation between the Cardy formula proven in this paper, andrecent work relating the “Cardy limit” of superconformal indices in 4d and 6d to black holes
entropies [5–8]. There is an ambiguity in nomenclature here. Whereas the Cardy limit considered in this paper fixes the squashing parameters, the limit considered in [5–7] has the(complexified) squashing parameters scale with β. The fugacities ωi in [5–8] are related to
our notation by(here)

βω1(there)= ω1,(here)

βω2(there)= ω2,(here)

βω3(there)= 2πi + ω3

,

(5.1)

(there)and the “Cardy limit” defined in [5–7] is the small ωilimit. The terms in the 5d effective
action that dominate in this new limit are different from our limit – additional terms thatare non-invariant under background perturbative gauge transformations must be included.41
Since this new limit appears important for black hole entropy-matching, it would be interesting
to prove the universal relation between this new limit and the anomalies, by similar arguments
as in the present work applied to a different 5d squashed sphere background. As a matter offact, we show in Appendix C, that the squashed S 5 background for the “modified 6d index” canbe obtained by a simple shift of ω3 . Nonetheless, it remains to understand the supersymmetriccompletion of the additional pieces appearing in the corresponding effective action.42

Acknowledgments
We are grateful to Lorenzo Di Pietro for extremely helpful correspondences, and are deeply
indebted to Yuji Tachikawa for patiently explaining the various subtleties of global anomaly
matching in 6d. We also thank Zohar Komargodski, Kantaro Ohmori, and Amos Yarom for
insightful discussions. Finally, we thank Lorenzo Di Pietro, Zohar Komargodski, and Yuji
Tachikawa for discerning comments on the draft. CC is supported in part by the U.S. Department of Energy grant DE-SC0009999. The work of MF is supported by the SNS fellowship P400P2-180740, the Princeton physics department, the JSPS Grant-In-Aid for Scientific
Research Wakate(A) 17H04837, the WPI Initiative, MEXT, Japan at IPMU, the University ofTokyo, the David and Ellen Lee Postdoctoral Scholarship. YL is supported by the Sherman
Fairchild Foundation, and both MF and YL by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632. YW is supported
41

The present authors find no clear relation between the physical 5d effective action and the “5d effective action”
in [8]. In particular, the effective action in [8] is not supersymmetric.
42
The effective action in the limit of the modified index (see [8]) contains singular gauge non-invariant pieces aswell as gauge-invariant ones. The supersymmetric completions of the gauge-invariant terms have been consideredin the present paper, and to study the limit of the modified index we simply need to evaluate them on the modified
5d background (see Appendix C). However, the supersymmetric completions of the gauge non-invariant terms are(to the knowledge of the authors) not known to date.

33

SciPost Phys. 11, 036 (2021)

in part by the US NSF under Grant No. PHY-1620059 and by the Simons Foundation Grant
No. 488653. YL and YW thank the Bootstrap 2019 conference, and CC, MF and YW thankthe Aspen Center for Physics, which is supported by National Science Foundation grant PHY1607611, for hospitality during the finishing stages of this paper.

A

Conventions for characteristic classes

We denote by p1 and p2 the first and second Pontryagin classes, and c1 (U(1)) and c2 (G) the
first and second Chern class of the groups U(1) and G, respectively. They can be written interms of the corresponding curvatures as follows:1
trR2 ,
2(2π)2


1
1 4 1
2 2
p2 =
trR
+
(trR
)
,
−
(2π)4
4
8
1
c1 (U(1)) =FU(1) ,2π
1
c2 (G) =TrFG2 .
2(2π)2
p1 = −(A.1)

Here, the trace tr is over the vector representation, and the trace Tr is defined as
Tr(·) ≡

1
tradj (·) ,
2h∨

(A.2)

where h∨ is the dual Coxeter number.

B

Central extensions of supersymmetry algebras and (extended)
BPS objects for 4d N = 2 and 6d N = (1, 0) theories

In this Appendix, we provide a brief classification of BPS objects appearing in Higgs/Coulomb/
mixed branch flows for 4d N = 2 theories and tensor/Higgs/mixed branch flows for 6d
N = (1, 0) theories.

B.1

4d N = 2

The BPS states of 4d N = 2 theories can be classified by looking at the central extensions of
the supersymmetry algebra,
j

µν

j

µν (i j)Z̄ ,α̇β̇ µν(i j){Q iα , Q β } = εαβ εi j Z + σαβ Zµν,{Q̄ iα̇ , Q̄ } = εα̇β̇ εi j Z̄ + σ̄
β̇
j

µ

β̇

αβ̇

{Q iα , Q̄ } = σ(B.1)(εi j (Pµ + Zµ ) + Zµ(i j) ) .
(i j)

(i j)

Here, Z is the usual central charge for BPS particles, whereas Zµ , Zµ and Zµν are the branecharges for BPS strings and domain-walls, respectively.43 In particular, the brane charges are
43
Equivalently, one can study non-conformal modifications of the conformal supercurrent multiplets which contain additional brane currents, or non-conformal extensions of the conformal supergravity by introducing compensator multiplets (for reviews see [104–106]).

34

SciPost Phys. 11, 036 (2021)

related to the (higher-form) brane currents for a p-form symmetry byZ(p)Zµ(p)µ ...µ ≡ T (p)1 2

dd−1 x J0µ µ ...µ ,
1 2

p

p

(B.2)

where the integral is over the spatial directions, and T (p) is the brane tension with mass dimension p + 1 (since the current J (p) has dimension d − p − 1). In terms of coupling to backgroundnon-conformal supergravity, the brane tension T (p) is given by the scalar vev of certain compensator multiplets. As argued in [105, 107], Zµ should vanish for 4d N = 2 theories thatcome from deforming SCFTs, because they are expected to have Ferrara-Zumino (FZ) multiplets (with respect to any N = 1 subalgebra) [108].
As for the other brane charges that appear in the central extension of the 4d N = 2 supersymmetry algebra, the central charge Z for BPS particles is given by the complex scalars
in the vector multiplets, whereas the tensions for the BPS strings and domain walls also depend on the real SU(2)R triplet scalars in the linear multiplets. For effective theories on themoduli space of an SCFT, these tensions are determined by the vevs of vector multiplet andhypermultiplet scalars. In particular, the tension for a (unit-charge) BPS string is given byqq̃, where q, q̃ denote the chiral scalars in a hypermultiplet (this follows from the composite
expression of a linear multiplet in terms of hypermultiplets). As for the BPS domain wall, the3-form current is proportional to the Hodge dual of dφ where φ is a complex scalar in a vector
multiplet, and the tension for a BPS domain is given by qq̃(φ(+∞) − φ(−∞)), where the q, q̃
are hypermultiplet scalars charged under the vector multiplet.44
Consequently, the possible massive stable BPS states are:(i) BPS particles on the Coulomb branch,(ii) BPS strings on the Higgs branch,(iii) BPS particles, strings and domain-walls on the mixed branches.

B.2

6d N = (1, 0)

The BPS states of 6d N = (1, 0) theories can be classified by looking at the central extensionsof the supersymmetry algebra,
µ

µνρ

(i j)
.
{Q iα , Q β } = γαβ εi j (Pµ + Zµ ) + γαβ Zµνρ
j

(B.3)

Here, Zµ is the string charge and Zµνρ the 4-brane charge, and due to supersymmetry, thecorresponding brane tensions are respectively given by the real scalar in the tensor multiplet
and the real SU(2)R -triplet scalars in the linear multiplet (which contains a 4-form gauge fieldthat couples to the brane). By a similar reasoning as for 4d N = 2, the possible massive BPS
states are(i) BPS strings on the tensor branch,(ii) BPS codimension-two vortex branes on the Higgs branch, and(iii) BPS strings and vortex branes on the mixed branches.
Upon compactification on S 1 (and before compactifying the 5d theory on S 5 ), the (wrapped)
vortex branes have infinite energy and thus do not contribute to the Chern-Simons levels in
the 5d effective action. Thus, in conclusion, there are no massive BPS states contributing to
the 5d effective action for Higgs branch flows.
Note that in N = 1 notation, this is just the usual statement that domain wall tensions are determined by thedifference between the values of the superpotential at adjacent vacua.
44

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SciPost Phys. 11, 036 (2021)

C

Supersymmetric S 1 × S 5 background for the index and modified
index

This appendix discusses the supersymmetric background for 6d theories on S 1 × S 5 twisted bychemical potentials. The notation used in this section is independent of the main text.
We start with the 6d metric in equation (2.1), where we set r5 = 1, and pick the following
frame
e1 = dτ ,e2= dθ ,(C.1)
3
j+3
e = sin θ dψ ,e
= y j (dφ j + ia j dτ) , j = 1, 2, 3 ,where we have introduced the spherical coordinates
y1 = sin θ cos ψ ,y2 = sin θ sin ψ ,y3 = cos θ ,(C.2)

with ψ ∈ [0, 2π) and θ ∈ [0, π).
In order to preserve some supersymmetry, we couple the theory to 6d background (offshell) conformal supergravity. More precisely, we couple to the “6d Weyl 1 multiplet”, also
known as the “6d standard Weyl multiplet” [109–111],45 which contains the bosonic fields
given by the metric, gµν , the gauge field for Weyl transformations bµ , an su(2)R symmetry−
gauge field Vµ i j , an antisymmetric tensor Tabc
and finally a real scalar D as well as fermioniciifields, ψµ , χ su(2)R Majorana-Weyl spinors of positive, negative chirality, respectively.
We are interested in the rigid limit [36], and thus set the fermionic fields to zero. Furthermore, to preserve some SUSY, we have to find non-trivial Killing spinors " i and their conformal
cousins, ηi , such that the variations of the fermionic fields vanish. The first constraint, δχ i = 0,
is automatically solved by setting−
Tabc= D = 0.

(C.3)

With this, the remaining supersymmetry condition is then given by1δψµi = ∂µ " i + ωµ ab Γab " i + Vµ i j " j + Γµ ηi = 0 .
4

(C.4)

In the following, we pick the 6d Gamma matrices as follows
Γi = iσ2 ⊗ γi(i = 1, . . . , 5) ,

Γ6 = σ1 ⊗ 14×4 ,(C.5)

where we recall that by γi are the 5d Gamma matrices, defined in equation (2.25). We may
solve (C.4) by turning on a background u(1)R ⊂ su(2)R gauge-field,


V0
i
µ
dτ (σ3 )i j .
Vµ j dx = 1 +(C.6)
2
We explicitly find the following solutions for the 6d Killing spinors " i and their conformal
45This is to make a distinction from another 6d N = (1, 0) Weyl multiplet – “6d Weyl 2 multiplet” or “dilaton
Weyl multiplet” – which has the same gauge but different matter fields [109, 110].

36

SciPost Phys. 11, 036 (2021)

cousins ηi :




04×1


θ ψ 
 i sin 2 ei 2 


p
τ
i
j+1
θ −i ψ2  ," j (τ) = 2e− 2 (1+atot +(−1) (2+V0 ))+ 2 φtot 
−
sin
e
2



θ ψ 
 i cos 2 ei 2 ψ− cos θ2 e−i 2ψ− sin θ2 ei 2



(C.7)

ψ 

 −i sin θ2 e−i 2 


τ
i
j+1
1
θ iψ

η j (τ) = p e− 2 (1+atot +(−1) (2+V0 ))+ 2 φtot  cos 2 e 2  ,2ψ  i cos θ2 e−i 2 

04×1
P3
P3
where φtot = i=1 φi and similarly atot = i=1 ai , and with 04×1 a column vector with four
zeroes as entries.
So far we have effectively been dealing with R × S 5 . Now, we compactify the τ-direction,and require the Killing spinors to satisfy the following consistency condition as we go aroundthe τ-circle" j (τ + β) ≡ eπin " j (τ) ,n ∈ {0, 1} ,(C.8)
where n = 0 gives us back the usual index as defined in (1.13), while n = 1 gives us the
background for the modified index [5, 112].46 Therefore, we arrive at the conditions
V0 + ω1 + ω2 + ω3 =

2πin
,
β

(C.9)

where ωi = 1 + ai . The analogous constraint in 4d was found in [112] by considering the 4dbackground S 1 × S 3 with modified periodicity for the fermions along the S 1 .
To go from the usual index (i.e. n = 0 in (C.9)) considered throughout the present paper
to the modified index (i.e. n = 1 in (C.9)), we can simply shift (e.g.) ω3 , i.e.
ω3 −→ ω3 +

2πi
.
β

(C.10)

Applying this shift to the evaluation of the supersymmetric completions of the higher-derivative
terms (2.51), we find the following answers for the modified background:iω23
ω3 (2π)4i (2π)5
3
1
= −
+
+
(2π)
+
O
β
,
ω1 ω2 β 2
ω1 ω2 β
ω1 ω2

3
2
2

2ω3 (2π)4 2i(2π) ω1 + ω2
2i (2π)5
mod
I2,
=
+
−
+ O β1 ,n=12
ω1 ω2 βω1 ω2 β
ω1 ω2
mod
I1,
n=1

iπ2 (2π)3 π2 (2ω1 + 2ω2 + ω3 ) (2π)2−
ω1 ω2 β 2
ω1 ω2
β

3
2
2

2iπ ω1 + 2ω2 ω1 + ω2+
+ O β1 ,
ω1 ω2

mod
I3,
n=1 = −

modI4,
n=1 = −

i(2π)3 (mfI )2
ω1 ω2(C.11)

+ O β1 .

See [57], where a similar modification has originally been considered in the N = 2 superconformal Schurindex.
46

37

SciPost Phys. 11, 036 (2021)

We remark that this is the answer (together with the constraint (C.9)) in the strict β → 0
limit. In fact, the constraint is crucial to render the result in (C.11) invariant under the exchange of ωi . Finally, in order to move to the “new Cardy limit” [5], we need to further rescaleβωi → ωi ,(C.12)and then take the ωi → 0 limit. However, as mentioned in the main text, the backgroundgauge-invariant supersymmetric Chern-Simons terms treated in this paper are not the complete set of terms contributing to that limit.

D

Geometric invariance and the equivalence between gauges

In the main text, we introduced two different solutions for background vector multiplets coupled to the standard Weyl multiplet: one was the “flavor solution” (2.26), and the other was
the “KK solution” (2.27), where the gauge field is naturally identified with the 6d graviphoton. Accordingly, we introduced two distinct gauge-fixed version of Poincaré supergravity: the“standard gauge-fixed” (2.16) and the “KK gauge-fixed” (2.17) one. As one can explicitly confirm, the various types of solutions (in different combinations) lead to the exact same results
for the evaluation of the supersymmetric Chern-Simons terms, giving more credence to the
expectation that they are geometric invariants, i.e. only dependent on some simple geometricproperties, believed to be the transversally holomorphic foliation inherent in the u(1)R truncated rigid supersymmetric backgrounds [14]. In this appendix, we provide a relation between
the various solutions.
Let us first focus on the vector multiplet. A vector multiplet coupled to the standard Weyl
multiplet is conformal. The (bosonic) fields in the standard Weyl and vector multiplet transform under Weyl rescaling as follows: (for completeness, we added the Weyl transformationof the linear multiplet)

SW :

g

→ Λ2 g ,

Vij →

Vij ,

D

→ Λ−2 D ,V:

M→ ΛM ,Wµ →Wµ ,(Y )i j→ Λ2 (Y )i j ,

L:

i

3 i

Eµ →

3

L j

→ Λ L j,Λ Eµ ,v → Λ2 v ,
(D.1)

4

→ Λ N.

N

From these expressions, it is clear that starting with the KK solution, we can fix the scalar M
to be a constant by Λ = µf M −1 , where m is the constant flavor mass. However, this wouldchange the metric, which is not what we want. Instead, we may connect the flavor solution to
the KK solution by Weyl transforming the standard Weyl solution (2.21) according to the firstline in (D.1) with the explicit Weyl factor
Λ = 2κ̃ +

1
β̃ κ̃

.

(D.2)

Then, one explicitly finds
dWKK = d [w(Wf )] +4β̃(e1 ∧ e2 + e3 ∧ e4 ) ,(D.3)

where by w(Wf ) we mean the Weyl transform of the ingredients in the standard Weyl multiplet
defining Ŵ , i.e. in the constant-M solution, Wf = −µf (1 − (β̃ κ̃)−1 )e5 , which Weyl-transforms
as follows47
Ŵ → w(Ŵ ) = (1 − (Λβ̃ κ̃)−1 )(Λe5 ) .
(D.4)
47The reason for this curious Weyl transformation is that the ingredients in the (general) flavor solutions are
actually bilinears in the Killing spinors, i.e. W = (1 − 1/S)K1 , where K1 ∼ ε† γ(1) ε and S ∼ ε† ε (see [14] for moredetails), which transform as in (3.8) of [14] under Weyl transformations.

38

SciPost Phys. 11, 036 (2021)

Consequently, in the supersymmetry condition (2.12), part of the explicit Λ in (D.2) will get
absorbed by Y and the κ̃ piece will be absorbed by M , making it non-constant. Thus, we movefrom the flavor solution with constant M to a solution where M is non-constant. Upon a (now)full Weyl transformation acting on the Standard Weyl as well as the vector multiplet back byΛ =12κ̃ + 1

,

(D.5)

β̃ κ̃

we end up with the same metric but a different solution given by the KK solution.
By the same logic, one can relate the two gauge fixing conditions. These arguments thensuggest that the resulting superconformal Chern-Simons terms, i.e. the FFF, Ff Ff F and FWW
terms, should be the same when evaluated on various solutions (which we explicitly checked),as they are by construction Weyl invariant. However, the reason for the FRR term, whose Weyl
invariance is broken, to remain unchanged (explicitly checked) must be explained by the factthat it is a geometric invariant, i.e. solely depends on the transversely holomorphic structureof the solution.

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