Abstract

Chern-Simons (CS) theories with rank $N$ and level $k$ on Seifert manifold are discussed. The partition functions of such theories can be written as a function of modular transformation matrices summed over different integrable representations of affine Lie algebra $u(N{)}_{k}$ associated with the boundary Wess-Zumino-Witten model. Using properties of modular transform matrices we express the partition functions of these theories as a unitary matrix model. We show that the eigenvalues of unitary matrices are discrete and proportional to hook lengths of the corresponding integrable Young diagram. As a result, in the large $N$ limit, the eigenvalue density develops an upper cap. We consider CS theory on ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ coupled with fundamental matters and express the partition functions in terms of modular transformation matrices. Solving this model at large $N$ we find the dominant integrable representations and show how large $N$ representations are related to each other by transposition of Young diagrams as a result of level rank duality. Next we consider $U(N)$ CS theory on ${S}^{3}$ and observed that in Seifert framing the dominant representation is no longer an integrable representation after a critical value of 't Hooft coupling. We also show that CS on ${S}^{3}$ admits multiple (two-gap phase) large $N$ phases with the same free energy.

Highlights

  • Study of topological objects in physics is an extremely interesting subject

  • A field theoretic realization of knots and links was discovered by Witten in his groundbreaking work in 1989 [4]. In that paper he showed that physical observables (Wilson loops) in Chern-Simons (CS) gauge theory in three dimensions are related to knot polynomials in the same dimensions and opened up a plethora of new possibilities for both mathematicians and physicists

  • The first goal of this paper is to show that using the form of modular transform matrices S and T for given representations, the partition function can be written as a unitary matrix model

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Summary

INTRODUCTION

Study of topological objects in physics is an extremely interesting subject. One of the earliest examples of topological objects is the Dirac monopole [1]. Witten showed that the quantum version of Chern-Simons theory preserves topological invariance but at the expense of a choice of “framing.” Correlation functions of Wilson loop operators along different knots depend on the linking number between the knots involved in the computation [4,5]. In our previous work [15] we observed that the discreteness in eigenvalue distribution imposes a constraint on the dominant representations of SUðNÞ: The maximum number of columns must be less than k, which is nothing but the integrability condition This observation motivated us to look at the relation between CS theories on different manifolds directly starting from its relation with the current algebra of the corresponding WZW theory. V and discuss how the dominant representations found in the current paper are different than what we considered in our previous works [14,15]

CHERN-SIMONS PARTITION FUNCTION ON THE SEIFERT MANIFOLD
Connection with the WZW model
Gross-Witten-Wadia potential—a toy model
Large N representations
Level-Rank duality and transposition of diagrams
CHERN-SIMONS THEORY ON S3
Two-gap phase in Chern-Simons theory on S3
CONCLUSION
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