Abelian Conformal Field theories and Determinant Bundles

Abstract

The present paper is the first in a series of papers, in which we shall construct modular functors and Topological Quantum Field Theories from the conformal field theory developed in [TUY]. The basic idea is that the covariant constant sections of the sheaf of vacua associated to a simple Lie algebra over Teichmüller space of an oriented pointed surface gives the vectorspace the modular functor associates to the oriented pointed surface. However the connection on the sheaf of vacua is only projectively flat, so we need to find a suitable line bundle with a connection, such that the tensor product of the two has a flat connection. We shall construct a line bundle with a connection on any family of pointed curves with formal coordinates. By computing the curvature of this line bundle, we conclude that we actually need a fractional power of this line bundle so as to obtain a flat connection after tensoring. In order to functorially extract this fractional power, we need to construct a preferred section of the line bundle. We shall construct the line bundle by the use of the so-called $bc$-ghost systems (Faddeev-Popov ghosts) first introduced in covariant quantization [FP]. We follow the ideas of [KNTY], but decribe it from the point of view of [TUY].

A couple of typos corrected

References in corpus (3)

• Geometric construction of modular functors from Conformal Field Theory
• Modular functors are determined by their genus zero data
• Asymptotics of the Hilbert-Smith norm of curve operators in TQFT

Cited by in corpus (24)

• Berezin-Toeplitz quantization for compact Kaehler manifolds. A Review of Results
• Geometric construction of modular functors from Conformal Field Theory
• Mapping Class Groups do not have Kazhdan's Property (T)
• The Verlinde formula for Higgs bundles
• The Witten-Reshetikhin-Turaev invariant for links in finite order mapping tori I
• Construction of the Witten-Reshetikhin-Turaev TQFT from conformal field theory
• Hitchin's Projectively Flat Connection, Toeplitz Operators and the Asymptotic Expansion of TQFT Curve Operators
• The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory
• Asymptotics of Toeplitz operators and applications in TQFT
• Hitchin's connection, Toeplitz operators and symmetry invariant deformation quantization
• Enumeration of chord diagrams on many intervals and their non-orientable analogs
• Convergence of Sewing Conformal Blocks
• Hitchin's Connection in Half-Form Quantization
• A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT
• Mapping class group invariant unitarity of the Hitchin connection over Teichmüller space
• A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications
• Asymptotic properties of the Hitchin-Witten connection
• On the unitary nature of abelian conformal blocks
• Coordinates for the Universal Moduli Space of Holomorphic Vector Bundles
• A mathematical formalism for the Kondo effect in WZW branes
• $\mathrm{Sp}(1)$-symmetric hyperkähler quantisation
• Asymptotic aspects of the Teichmüller TQFT
• Asymptotic formulas for curve operators in TQFT
• On the monodromy of the Hitchin connection