Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on the Partition Function of ABJM theory on S^3

Kazumi Okuyama

Abstract

We study the partition function Z of U(N)_k x U(N)_{-k} Chern-Simons matter theory (ABJM theory) on S^3 which is recently obtained by the localization method. We evaluate the eigenvalue integral in Z exactly for the N=2 case. We find that Z has a different dependence on k for even k and odd k. We comment on the possible implication of this result in the context of AdS/CFT correspondence.

A Note on the Partition Function of ABJM theory on S^3

Abstract

We study the partition function Z of U(N)_k x U(N)_{-k} Chern-Simons matter theory (ABJM theory) on S^3 which is recently obtained by the localization method. We evaluate the eigenvalue integral in Z exactly for the N=2 case. We find that Z has a different dependence on k for even k and odd k. We comment on the possible implication of this result in the context of AdS/CFT correspondence.

Paper Structure

This paper contains 13 sections, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the Partition Function of ABJM Theory on $S^3$
  3. Grand Partition Function of ABJM Theory
  4. Mirror Description of ABJM Theory
  5. Partition function of $U(2)_k\times U(2)_{-k}$ ABJM theory
  6. Discussions
  7. Computation of $A_{\ell,k}$
  8. Writing $A_{\ell,k}$ as the integral of $2(\ell-1)$ variables
  9. $A_{1,k}$ and $A_{2,k}$
  10. Evaluation of $Z_{2,k}$
  11. Odd $k$ Case
  12. Even $k$ Case
  13. Some Examples of $Z_{2,k}$ for Low $k$'s

Figures (1)

  • Figure 1: This is the contour $C=C_1+C_2+C_3+C_4$ used in Appendix B. $C_1$ and $C_3$ are the horizontal lines at ${\rm Im}z=0$ and ${\rm Im}z=1$, respectively. $C_2$ and $C_4$ are the vertical segments at $|{\rm Re}z|=\Lambda$, and we will take the limit $\Lambda\rightarrow\infty$ at the end of computation.