Elliptic genera and real Jacobi forms

Abstract

Abstract We construct real Jacobi forms with matrix index using path integrals. The path integral expressions represent elliptic genera of two-dimensional $ \mathcal{N} $ = (2, 2) supersymmetric theories. They arise in a family labeled by two integers N and k which determine the central charge of the infrared fixed point through the formula c = 3N (1 + 2N/k). We decompose the real Jacobi form into a mock modular form and a term arising from the continuous spectrum of the conformal field theory. For a given N and k we argue that the Jacobi form represents the elliptic genus of a theory defined on a 2N dimensional linear dilaton background with U(N) isometry, an asymptotic circle of radius $ \sqrt{{k\alpha \prime }} $ and linear dilaton slope $ N\sqrt{{{2 \left/ {k} \right.}}} $ . We also present formulas for the elliptic genera of their orbifolds.