Abstract
We consider a class of gauged linear sigma models (GLSMs) in two dimensions that flow to non-compact (2, 2) superconformal field theories in the infra-red, a prototype of which is the SL(2, ℝ)/U(1) (cigar) coset. We compute the elliptic genus of the GLSMs as a path-integral on the torus using supersymmetric localization. We find that the result is a Jacobi-like form that is non-holomorphic in the modular parameter τ of the torus, with mock modular behavior. This agrees with a previously-computed expression in the cigar coset. We show that the lack of holomorphicity of the elliptic genus arises from the contributions of a compact boson carrying momentum and winding excitations. This boson has an axionic shift symmetry and plays the role of a compensator field that is needed to cancel the chiral anomaly in the rest of the theory.
Highlights
The key feature of a function f (τ ) of this type is that it transforms like a holomorphic modular form of weight k, but it suffers from a holomorphic anomaly: (4πτ2)k ∂τ f (τ ) = −2πi g(τ ), (1.2)
We evaluate the elliptic genus of the class of gauged linear sigma models (GLSMs) introduced in [10, 19] using supersymmetric localization, by adapting the method developed in [20] for the compact models
The elliptic genus of GLSMs for compact theories has been computed in the 1990s following [18], using the fact that one can compute it in the free-field limit
Summary
We have identified a simple physical source of the holomorphic anomaly in such models, namely as arising from the chiral anomaly of a two-dimensional field theory This seems to be an example of the general idea that introducing compensators for anomalous symmetries destroys some other nice property of the theory. We adapt this method to the non-compact models of interest to us, derive their elliptic genera, and show how the non-holomorphic contributions can be understood as arising from the contributions of the compensator multiplet P. While this paper was being prepared for publication, the author received communication of [22] which contains overlapping results
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
