Anomaly matching across dimensions and supersymmetric Cardy formulae

Abstract

’t Hooft anomalies are known to induce specific contributions to the effective action at finite temperature. We present a general method to directly calculate such contributions from the anomaly polynomial of a given theory, including a term which involves a U(1) connection for the thermal circle isometry. Based on this observation, we show that the asymptotic behavior of the superconformal index of 4d$$ \mathcal{N} $$ N = 1 theories on the “second sheet” can be calculated by integrating the anomaly polynomial on a particular background. The integration is then performed by an equivariant method to reproduce known results. Our method only depends on the anomaly polynomial and therefore the result is applicable to theories without known Lagrangian formulation. We also present a new formula that relates the behavior of 6d$$ \mathcal{N} $$ N = (1, 0) SCFTs on the second sheet to the anomaly polynomial.

Higher-derivative corrections to flavoured BPS black hole thermodynamics and holography

“…2 One way to prove (1.5) is therefore to extend the three-dimensional effective field theory approach of [28,29] to the flavoured case: the twisted supersymmetric reduction on a small Euclidean time circle discussed there can also be performed in the presence of background vector multiplets coupling to flavour currents; this leads to additional supersymmetric Chern-Simons contact terms in three dimensions [44,45]. Here we choose a different route and present a quicker, though more formal, way to reach the same result, which extends the equivariant integration of the anomaly polynomial presented in [30] (see also [46]) to the flavoured case. Equivariant integration of anomaly polynomials is a technique that has already proven effective for different scopes, such as obtaining the anomaly polynomial of lower-dimensional theories [47,48], or reproducing the supersymmetric Casimir energy [49].…”

“…The sign choice in (1.6) corresponds to two equivalent saddles. We will give a derivation of (1.5) via equivariant integration of the anomaly polynomial, following the approach of [30]. The expression is valid at finite N and should apply both to Lagrangian and non-Lagrangian theories, not necessarily holographic.…”

Continuous generalized symmetries in three dimensions

“…We would now like to demonstrate that the symmetry structure (2.40) can also be understood in terms of the results presented in this work. The new point of view, following [61], is that we first perform a dimensional reduction of the action (2.38). We integrate on a circle with local coordinate τ , and further restrict C (1) to be along that circle, to obtain…”

“…A and compute its effective action upon integrating over the whole tower of fermions. This is done in a related model in [61,62]. We can then gauge the shift symmetry θ ∼ θ + 2π.…”