High-temperature expansion of supersymmetric partition functions
JHE
High-temperature expansion of supersymmetric
Arash Arabi Ardehali 0 1 2 4
James T. Liu 0 1 2 4
Phillip Szepietowski 0 1 2 3
CFT Correspondence 0 1 2
/N Expansion 0 1 2
0 Open Access , c The Authors
1 Utrecht University , 3508 TD Utrecht , The Netherlands
2 The University of Michigan , Ann Arbor, MI 48109-1040 , U.S.A
3 Institute for Theoretical Physics & Spinoza Institute
4 Michigan Center for Theoretical Physics, Randall Laboratory of Physics
Di Pietro and Komargodski have recently demonstrated a four-dimensional counterpart of Cardy's formula, which gives the leading high-temperature (β → 0) behavior of supersymmetric partition functions ZSUSY(β). Focusing on superconformal theories, we elaborate on the subleading contributions to their formula when applied to free chiral and U(1) vector multiplets. In particular, we see that the high-temperature expansion of ln ZSUSY(β) terminates at order β0. We also demonstrate how their formula must be modified when applied to SU(N ) toric quiver gauge theories in the planar (N → ∞) limit. Our method for regularizing the one-loop determinants of chiral and vector multiplets helps to clarify the relation between the 4d N = 1 superconformal index and its corresponding supersymmetric partition function obtained by path-integration.
Supersymmetric gauge theory; Anomalies in Field and String Theories; AdS-
1 Introduction 2 3 4
Cardy’s formula for 2d CFTs
Subleading corrections to the Di Pietro-Komargodski formula
Free chiral multiplet
Free U(1) vector multiplet
The N = 4 theory as an example
A Di Pietro-Komargodski formula for large-N toric quivers
A SCFTs with squashed three-sphere as spatial manifold
A.1 Free chiral multiplet
A.2 Free U(1) vector multiplet
A.3 Toric quivers in the planar limit
B Single-trace index and the central charges
B.1 Finite-N theories
B.2 Ak SQCD fixed points in the Veneziano limit
B.3 Toric quivers in the planar limit
Some time ago, Cardy famously employed modular invariance to obtain the
hightemperature behavior of conformal field theory (CFT) partition functions in two
dimen
This result has since been exploited in a variety of contexts, including the
statistical physics of black holes [2–4]. Cardy’s formula gives the leading order divergence
Here, cL is the left-handed CFT central charge, and we have focused on the holomorphic
sector for simplicity. Note that the term “Cardy formula” is often applied to the expression,
derived from the above relation, for the micro-canonical entropy of a 2d CFT at high
energies. However, in the present work, by “Cardy formula” we always refer to the above
canonical version for the asymptotic high-temperature expansion of ln Z.
Similar formulae had been long sought in higher dimensions without much success,
partly because Cardy’s main tool, modular invariance, has no known higher-dimensional
counterpart. Recently, Di Pietro and Komargodski have combined ideas from
supersymmetry and hydrodynamics [5, 6] to obtain the high-temperature behavior of supersymmetric
(SUSY) partition functions in four and six dimensions [7]. Here we expand on their result
in the context of four-dimensional superconformal field theories (SCFTs).
By SUSY partition function we mean the one computed with periodic boundary
conthe partition function is represented as a weighted sum over the states, and makes it
independent of exactly marginal couplings [8]. Therefore, one might anticipate that the
partition function displays universal high-temperature behavior depending only on the 4d
central charges. This was realized by Di Pietro and Komargodski, who demonstrated the
The generalization to squashed 3-sphere (and therefore non-equal fugacities in the index)
will be discussed in appendix A.
In the following, we will refer to ZSUSY obtained by path-integration as the “SUSY
partition function”, and to I as the “index”. The relation between these two quantities
where c and a are the central charges of the 4d SCFT, and where the spatial manifold is
taken to be the round S3. (In the main text we focus on the round S3, while relegating
the case of the squashed sphere to appendix A.)
The formula (1.2) can be thought of as the leading order result in a high-temperature
expansion. In this paper we explore the subleading corrections to eq. (1.2) and provide
To explore the subleading behavior of SUSY partition functions, it proves helpful to
understand the relation between their path-integral representation and their representation
as a weighted sum. The latter is called the superconformal index [9, 10], and may be defined
with two fugacities as
I(p, q) = Tr h(−1)F e−βˆ(Δ−2j2− 23 r)pj1+j2+ 12 rq−j1+j2+ 21 ri .
r = 0 contribute to the index, so
I(β) = I(e−β, e−β) = Tr h(−1)F e−β(Δ− 21 r)i .
I(β) = eβEsusy ZSUSY(β) = e 27
the prefactor. However, as mentioned in [7, 14], and as highlighted below, an alternative
regularization of the computations in [12–14] would eliminate that extra factor.
The fact that the path-integral and the (...truncated)