2d Wall-Crossing, R-twisting, and a Supersymmetric Index

TL;DR

This work introduces a time-dependent, R-charge twisted ${\cal N}=2$ framework in 2d that defines an index $I_k$ invariant under deformations, enabling a direct link between UV Ramond ground-state R-charges and IR BPS soliton data. By formulating a Parisi–Sourlas-type SQM with periodic time, the authors derive a time-ordered, phase-sensitive soliton product picture that reproduces wall-crossing in the IR while remaining controlled at the conformal UV fixed point. The approach unifies UV and IR descriptions through a topological, time-dependent TFT, and extends naturally to 2d theories and beyond, offering a new, conceptually simple route to tt* results and potential higher-dimensional generalizations including KS wall-crossing in 4d.

Abstract

Starting from N=2 supersymmetric theories in 2 dimensions, we formulate a novel time-dependent supersymmetric quantum theory where the R-charge is twisted along the time. The invariance of the supersymmetric index under variations of the action for these theories leads to two predictions: In the IR limit it predicts how the degeneracy of BPS states change as we cross the walls of marginal stability. On the other hand, its equivalence with the UV limit relates this to the spectrum of the U(1) R-charges of the Ramond ground states at the conformal point. This leads to a conceptually simple derivation of results previously derived using tt* geometry, now based on time-dependent supersymmetric quantum mechanics.