Chern-Simons theory on Seifert manifold and matrix model

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 boundary Wess-Zumino-Witten (WZW) 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 × 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. * arghya@iiserb.ac.in † suvankar@iiserb.ac.in ‡ neetuj@iiserb.ac.in arXiv:1902.07538v2 [hep-th]

New phase in Chern-Simons theory on lens space

“…Therefore the gapped phase is equivalent to the dominant representation obtained in [5][6][7][8]. However, the eigenvalue density of the gapped phase saturates the upper bound at λ = 1/π log cosh π ≡ λ * and seizes to exists beyond λ * [9]. In this paper we discover that there exists another phase (we call this phase cap-gap phase) for λ > λ * .…”

“…Hence this phase is equivalent to the dominant phase obtained in [5][6][7][8]. However, due to the constraint (11) on ρ(θ) the eigenvalue density saturates the upper bound at λ = 1/π log cosh(π/p) ≡ λ * [9]. Therefore the gapped phase is not valid anymore for λ > λ * .…”

“…However if we use an analytic continuation in λ [9], we would also get a real saddle point equation for YD distribution u(h) in h plane with a coth kernel similar to what considered in [7]. Solution of this equation renders a YD distribution which never crosses the maximum value 1 for p = 1.…”

“…But the dominant representations fail to be integrable for all values of λ. It was first pointed out in [9]. In this paper we address this issue in detail.…”

“…The unitary matrix model (9) was studied in [9,17]. It was observed that the system has a gapped phase in the large k, N limit and the eigenvalue distribution is given by,…”