Supersymmetric zeta functions and determinants

TL;DR

This work introduces supersymmetric zeta functions and determinants as analytic spectral tools that extend beyond supersymmetric indices. By expressing these zeta functions as Mellin transforms of (plethystic) single-particle data, the authors derive Cardy-like and Casimir-energy relations unified across 2d, 4d, and 6d theories, including numerous explicit 2d and 4d examples and AD theories. The central claim is that residues and special values of the supersymmetric zeta functions universally encode anomaly coefficients and central charges, enabling extraction of UV data from BPS spectra even in non-Lagrangian settings. The framework provides a coherent bridge between spectral regularization, modular/elliptic structures, and holographic or EFT interpretations, with broad potential for studying universal aspects of SUSY field theories. Overall, the paper offers a comprehensive toolkit for connecting BPS spectra, zeta-regularization, and anomaly data via a suite of new spectral functions.

Abstract

We define supersymmetric zeta functions and supersymmetric determinants, which can reveal spectral properties complementary to those captured by the supersymmetric indices. They play a crucial role in analyzing the Cardy-like behaviors of the supersymmetric indices and the supersymmetric Casimir energies associated with the supersymmetric partition functions. We investigate a variety of examples of the supersymmetric zeta functions and determinants for two-, four-, and six-dimensional supersymmetric field theories.

Figures (5)

• Figure 1: The vacuum exponent $\mathfrak{D}_{\textrm{NS}}^{\textrm{2d $(0,2)$ chiral}_r}(0)$.
• Figure 2: The vacuum exponent $\mathfrak{D}_{\textrm{NS}}^{\textrm{2d $(0,2)$ Fermi}_r}(0)$.
• Figure 3: The vacuum exponent $\mathfrak{D}_{\textrm{NS}}^{\textrm{2d $(2,2)$ chiral}_r}(0)$.
• Figure 4: The vacuum exponent $\mathfrak{D}^{\textrm{4d $\mathcal{N}=1$ chiral}_r}(0;1,1)$.
• Figure 5: The special value $\zeta'(-1,a)$.