Perturbative partition function for a squashed S^5

TL;DR

This work computes the perturbative partition function for 6d ${\rm N}=(1,0)$ theories on ${S}^5\times{\mathbb R}$ by performing a twisted compactification along ${\mathbb R}$ and taking the small-radius limit to obtain a squashed ${S}^5$. The authors formulate the index via localization, derive the 1-loop determinants for vector and hypermultiplets, and express the final perturbative partition function in terms of the triple sine function ${S_3}$ with periods ${\bm\omega}=(\omega_1,\omega_2,\omega_3)$, where ${\omega_i}=1+i\phi_i$ encode the squashing. They treat special homogeneous squirtings (${\cal N}=1/4$ and ${\cal N}=3/4$) and discuss the role of the prepotential ${\cal F}$ in supplying the exponential classical action, proposing a general prescription for the classical factor in the absence of a known 5d action on general squashed ${S}^5$. The results provide a concrete and compact framework for testing AdS/CFT predictions on squashed backgrounds and pave the way for large-${N}$ analyses of deformed five-dimensional gauge theories.

Abstract

We compute the index of 6d N=(1,0) theories on S^5 x R containing vector and hypermultiplets. We only consider the perturbative sector without instantons. By compactifying R to S^1 with a twisted boundary condition and taking the small radius limit, we derive the perturbative partition function on a squashed S^5. The 1-loop partition function is represented in a simple form with the triple sine function.

Figures (1)

• Figure 1: The $(k,k)$ representation on the weight space. This example shows $(6,6)$.