Broken (super) conformal Ward identities at finite temperature

TL;DR

This paper derives broken Ward identities for (super) conformal symmetries on thermal manifolds, linking thermal energy spectra with conformal spectra and reproducing known results like the generalized Cardy formula.

Contribution

It introduces and applies broken Ward identities for (super) conformal groups on thermal manifolds, providing new constraints and insights into the relationship between thermal and conformal spectra.

Findings

Reproduces known results including the generalized Cardy formula

Relates thermal energy spectrum with conformal spectrum

Provides systematic derivation of broken Ward identities

Abstract

When a (super) conformal field theory is placed on a non-trivial manifold, the (super) conformal symmetry is broken. However, it is still possible to derive broken Ward identities for these broken symmetries, which provide additional constraints on the theory. We derive and apply the broken Ward identities associated with the (super) conformal group on the thermal manifold $\mathcal{M}_\beta = S_\beta^1 \times \mathbb{R}^{d-1}$ and $\mathcal{M} = T^2 \times \mathbb{R}^{d-2}$. The novel constraints not only systematically reproduce known results, including an implicit formulation of the generalized Cardy formula, but also elegantly relate the thermal energy spectrum with the conformal spectrum.