A new kind of anomaly: on W-constraints for GKM

TL;DR

This paper investigates the origins of a unique W-constraint equation in the generalized Kontsevich model, revealing discrepancies with Ward identities and identifying a new anomaly that affects the model's formulation.

Contribution

It introduces the concept of a new anomaly in the W-constraints of GKM and analyzes the differences between the single equation and Ward identities, especially beyond the cubic case.

Findings

Discrepancy between W-constraints and Ward identities in GKM

The discrepancy reduces in the cubic model with odd times

Identification of a new anomaly affecting the interpretation of the single equation

Abstract

We look for the origins of the single equation, which is a peculiar combination of W-constrains, which provides the non-abelian W-representation for generalized Kontsevich model (GKM), i.e. is enough to fix the partition function unambiguously. Namely we compare it with the scalar projection of the matrix Ward identity. It turns out that, though similar, the two equations do not coincide, moreover, the latter one is non-polynomial in time-variables. This discrepancy disappears for the cubic model if partition function is reduced to depend on odd times (belong to KdV sub-hierarchy of KP), but in general such reduction is not enough. We consider the failure of such direct interpretation of the "single equation" as a new kind of anomaly, which should be explained and eliminated in the future analysis of GKM.