#### Alladi Ramakrishnan Hall

#### Consequences of Integrable Representations on Chern-Simons Theory

#### Arghya Chattopadhyay

##### IISER, Bhopal

*Partition function for Chern-Simons(CS) theory on any Seifert manifold M with a gauge group*

G and level K can be written as a sum over integrable representations of the corresponding affine

Lie algebra of the boundary Wess-Zumino-Witten Model. We consider the partition function

for U(N) Chern-Simons theory with level K written as a sum over Integrable representation(for

odd K) at large N, and show a natural manifestation of matrix integrals for CS theory studied

earlier.

For small coupling the dominant representations are characterised by Young diagrams with

number of boxes on the top-most row being less than the level K. On the other hand at some

critical value of the coupling, the dominant representations are always characterised by Young

diagrams with exact K number of boxes on the topmost row. The restriction over representations

dictate some constraints on the eigenvalue distribution of the matrix model as well. For CS theory on S2 × S1

this approach naturally describes the discreteness of the eigenvalues of the corresponding holonomy matrix which in turn translates into emergence of new phases. Our ongoing work deals with the effect of discreteness of the eigenvalues for the CS theory on S3.

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