One loop partition function for Topologically Massive Higher Spin Gravity

TL;DR

This work computes the one-loop partition function for Topologically Massive Higher Spin Gravity (TMHSG) in AdS$_3$, first for spin $3$ and then generalized to arbitrary spin. By decomposing fluctuations into transverse-traceless, trace, and longitudinal components and carefully accounting for ghost determinants, the authors show that the resulting partition function contains nonholomorphic (left-right) contributions that do not factorize holomorphically at the chiral point $ll o 1$. The nonholomorphic structure matches expectations from a high-spin extension of logarithmic CFTs rather than a chiral CFT, and the general spin analysis reveals systematic nonholomorphic contributions associated with trace and longitudinal modes. These results strengthen the conjecture that TMHSG is dual to a high-spin LCFT and motivate further study of the corresponding ${\u200B ext{W}}_N$-like algebra in the quantum regime.

Abstract

We calculate the one loop partition function for topologically massive higher spin gravity (TMHSG) for arbitrary spin by taking the spin-3 TMHSG action constructed in arXiv:1107.0915 and subsequently generalising it for an arbitrary spin. We find that the final result can be put into a product form which cannot be holomorphically factorized giving strong evidence that the topologically massive higher spin gravity is dual to a high spin extension of logarithmic CFT rather than a chiral one.