QM Research Seminar: The TQFT Factorization Axiom and Bohr-Sommerfeld Quantization

Speaker: William Elbæk Mistegård (University of Southern Denmark)

Abstract:

In his seminal work on quantum Chern-Simons field theory and the Jones polynomial, Witten envisioned a 2+1 dimensional TQFT. Further, he presented two procedures for constructing the underlying modular functor, namely by conformal field theory (CFT) and by geometric Kähler quantization of the moduli spaces of flat principal bundles on two-manifolds. Subsequently, Reshetikhin and Turaev constructed a TQFT using surgery of three-manifolds and representation theory of quantum groups, and this is now known as the WRT-TQFT. The CFT approach was initially developed by Tsuchiya, Ueno and Yamada, and then further refined to a full modular functor by Andersen and Ueno, who also established that this is equivalent to the WRT-TQFT. The geometric Kähler quantization approach was initialized by Axelrod, Witten and Pietra, and Hitchin. The quantization approach results in a projective representation of the mapping class group of a two-manifold, which is known due to Laszlo to be projectively equivalent to the representation of the WRT-TQFT. However, the full TQFT is not yet described from the point of view of quantization. In particular, the so-called TQFT factorization axiom has not been proved using only geometric quantization with respect to Kähler polarizations. However, Andersen have understood factorization from the point of view of geometric quantization with respect to reducible non-negative polarizations.

Jeffrey and Weitsman introduced a different quantization procedure. Given a trinion decomposition of a closed two-manifold S, one can obtain a system of functions on the moduli space of flat bundles on the two-manifold S. This is given by taking the trace of the holonomy representations of the boundary circles of the trinions. This results in a fibration of the moduli space, the fibres of this map being Lagrangian tori. The vertical distribution of this fibration is a polarization in the sense of geometric quantization. Jeffrey and Weitsman considered the so-called Bohr-Sommerfeld quantization of this system (i.e. this polarization), and proved that it results in a finite dimensional Hilbert space, which we denote by V(S). Further, they showed that V(S) has dimension equal to the dimension of the Hilbert space of the WRT-TQFT (this dimension is the famous Verlinde formula).

In this talk, we present aspects of an ongoing project on quantization and TQFT, which is complementary to the factorization construction of Andersen. We consider the two-manifold S' obtained by cutting along a circle of the trinion decomposition. This two-manifold have two boundaries and an induced trinion decomposition. Following Jeffrey and Weitsman, we consider for each element j in the gauge group, the associated Lagrangian torus fibration of the moduli space of flat bundles on S' with boundary holonomy conjugate to j. We consider the Bohr-Sommerfeld quantization of this system, which we denote by V(S',j). We show that in accordance with the TQFT factorization axiom, the Hilbert space of the closed surface, denoted above by V(S), splits as a direct sum indexed by elements of the WRT-TQFT label set, the summand indexed by j being equal to V(S',j). Further, we show that this direct sum decomposition is conjugate to factorization isomorphism of the WRT-TQFT. Thus these results gives a geometric illumination of the factorization axiom of the WRT-TQFT by means of the Bohr-Sommerfeld quantization of Chern-Simons theory.