’t Hooft anomalies and the holomorphy of supersymmetric partition functions

Abstract

We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, G F , for 2d N = (0, 2) and 4d N = 1 supersymmetric quantum field theories. In any diffeomorphism-invariant scheme and in the presence of G F 't Hooft anomalies, the supersymmetric Ward identities imply that the partition function has a non-holomorphic dependence on the flavor parameters. We show this explicitly for the 2d torus partition function, Z T 2 , and for a large class of 4d partition functions on half-BPS four-manifolds, Z M 4in particular, for M 4 = S 3 × S 1 and M 4 = Σ g × T 2 . We propose a new expression for Z M d−1 ×S 1 , which differs from earlier holomorphic results by the introduction of a non-holomorphic "Casimir" pre-factor. The latter is fixed by studying the "high temperature" limit of the partition function. Our proposal agrees with the supersymmetric Ward identities, and with explicit calculations of the absolute value of the partition function using a gauge-invariant zeta-function regularization.

AdS black holes, holography and localization

“…where the quantity E c:e ; called supersymmetric Casimir energy (Assel et al 2014(Assel et al , 2015Lorenzen and Martelli 2015;Benetti Genolini et al 2017;Martelli and Sparks 2016;Closset et al 2019), is due to the regularization of the one-loop determinants and it can be interpreted as the vacuum expectation value of the Hamiltonian. The superconformal index can be computed explicitly through localization or, being invariant under continuous deformations, just by enumerating the gauge invariant states annihilated by Q and Q y in the weakly coupled UV theory.…”

The BPS limit of rotating AdS black hole thermodynamics

“…In fact, in [43,44] E 0 has been interpreted as the consequence of a supercurrent anomaly, which is physical in nature. It may be possible to see this as a mixed anomaly and thus shift it away by adding local counterterms that restore supersymmetry at the expense of breaking part of the diffeomorphisms, along the lines of [45][46][47]. 10 The solution admits an extremal limit.…”

Comments on anomalies in supersymmetric theories*

“…In this section we mostly review well known facts. They are discussed in detail in [8,9] and more recently in [10]. Consider a supersymmetric field theory with global ("flavour") symmetries.…”