U(1) spin Chern-Simons theory and Arf invariants in two dimensions

Abstract

The level-k U(1) Chern-Simons theory is a spin topological quantum field theory for k odd. Its dynamics is captured by the 2d CFT of a compact boson with a certain radius. Recently it was recognized that a dependence on the 2d spin structure can be given to the CFT by modifying it using the so-called Arf invariant. We demonstrate that one can reorganize the torus partition function of the modified CFT into a finite sum involving a finite number of conformal blocks. This allows us to reproduce the modular matrices of the spin theory. We use the modular matrices to calculate the partition function of the spin Chern-Simons theory on the lens space $L(a,\pm 1)$, and demonstrate the expected dependence on the 3d spin structure.

17 pages; version 2: comments and references added, typos corrected, journal version

References in corpus (5)

• Generalized Global Symmetries
• Fermionic minimal models
• Classification of abelian spin Chern-Simons theories
• Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions
• Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories

Cited by in corpus (7)

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• A T-Duality of Non-Supersymmetric Heterotic Strings and an implication for Topological Modular Forms
• Spin Structures and AdS${}_4$ Holography
• Emergent Supersymmetry on the Edges
• Topological phase, spin Chern-Simons theory and level rank duality on lens space