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Localization of N=4 Superconformal Field Theory on S^1 x S^3 and Index

Satoshi Nawata

TL;DR

This work provides a geometric interpretation of the ${N}=4$ superconformal index by realizing it as the partition function of the theory on a Scherk–Schwarz twisted ${S^1\times S^3}$ background. Utilizing a δ_ε-exact deformation guided by a suitable conformal Killing spinor, the authors localize the path integral to flat gauge connections on ${S^1\times S^3}$ and compute the one-loop determinants around these loci, producing a matrix model that matches the letter-counting description of the index. The localization locus is identified with the moduli space of flat connections, which on ${S^1}\times S^3$ is the quotient $T/W$ of the maximal torus by the Weyl group, and the final matrix integral takes the form ${\cal I}(t,y,v,w)=\int_G [dU] \exp\{\sum_m \frac{1}{m} f(t^m,y^m,v^m,w^m) \mathrm{Tr}(U^\dagger)^m \mathrm{Tr} U^m\}$, with the single-particle index $f$ equal to the PSU$(1,2|3)$ character. This establishes a direct link between the superconformal index, TQFT localization, and a computable matrix model, and points to natural generalizations to other SCFTs and dimensional reductions.

Abstract

We provide the geometrical meaning of the ${\cal N}=4$ superconformal index. With this interpretation, the ${\cal N}=4$ superconformal index can be realized as the partition function on a Scherk-Schwarz deformed background. We apply the localization method in TQFT to compute the deformed partition function since the deformed action can be written as a $δ_ε$-exact form. The critical points of the deformed action turn out to be the space of flat connections which are, in fact, zero modes of the gauge field. The one-loop evaluation over the space of flat connections reduces to the matrix integral by which the ${\cal N}=4$ superconformal index is expressed.

Localization of N=4 Superconformal Field Theory on S^1 x S^3 and Index

TL;DR

This work provides a geometric interpretation of the superconformal index by realizing it as the partition function of the theory on a Scherk–Schwarz twisted background. Utilizing a δ_ε-exact deformation guided by a suitable conformal Killing spinor, the authors localize the path integral to flat gauge connections on and compute the one-loop determinants around these loci, producing a matrix model that matches the letter-counting description of the index. The localization locus is identified with the moduli space of flat connections, which on is the quotient of the maximal torus by the Weyl group, and the final matrix integral takes the form , with the single-particle index equal to the PSU character. This establishes a direct link between the superconformal index, TQFT localization, and a computable matrix model, and points to natural generalizations to other SCFTs and dimensional reductions.

Abstract

We provide the geometrical meaning of the superconformal index. With this interpretation, the superconformal index can be realized as the partition function on a Scherk-Schwarz deformed background. We apply the localization method in TQFT to compute the deformed partition function since the deformed action can be written as a -exact form. The critical points of the deformed action turn out to be the space of flat connections which are, in fact, zero modes of the gauge field. The one-loop evaluation over the space of flat connections reduces to the matrix integral by which the superconformal index is expressed.

Paper Structure

This paper contains 15 sections, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. ${\cal N}=4$ SCFT on $\mathbb{R} \times S^3$
  3. ${\cal N}=4$ Index
  4. Action of ${\cal N}=4$ SCFT on $\mathbb{R} \times S^3$ and Noether Charges
  5. Functional Integral Interpretation of ${\cal N}=4$ Index
  6. Scherk-Schwarz Deformed Action
  7. Off-shell Formulation
  8. Localization
  9. $\delta_\epsilon$-Exact Term and Critical Points
  10. One-Loop Evaluations
  11. Conclusions and Future Directions
  12. Superconformal Algebra
  13. Notations for Field Theory on $\mathbb{R} \times S^3$
  14. Superconformal Transformations by $Q$ and $S$
  15. Self-dual and Anti-self-dual Field Strength in Euclidean Signature

Figures (2)

  • Figure 1: Schematic figures of the Scherk-Schwarz deformed background. Left: The 3-sphere at the top is identified with the one at the bottom by the rotation $e^{2\beta J_3}$. Here the time translation is vertical, which is common in physics literatures. The curve $\gamma_0$ depicts the integral curve of the vector field $\partial/\partial\tilde{x}^0$ which corresponds to the time direction of the space-time $M$. Right: the right 2-plane $\mathbb{C}$ is identified with the right one by rotating $e^{-\beta \tilde{R}_k}$. Here the time direction $S^1$ can be seen as the base manifold of the fibre bundle with fibre a 2-plane $\mathbb{C}$, which is common in mathematics literatures.
  • Figure 2: A schematic figure which explains the geometric meaning of a Wilson line.