nLab Hitchin connection

Idea

For $G$ a compact Lie group and $\Sigma$ a surface, the Hitchin connection (Hitchin 90) is a projectively flat connection over the moduli space of Riemann surfaces $\mathcal{M}_\Sigma$ whose fiber over a complex structure on $\Sigma$ is a space of quantum states which quantization of Chern-Simons theory for gauge group $G$ assigns to the induced Kähler polarization on the moduli space of flat connection $Loc_G(\Sigma)$ (the phase space of Chern-Simons theory), hence whose fiber is the space of holomorphic sections of the prequantum line bundle of the CS theory with respect to the induced Kähler structure on $Loc_G(\Sigma)$ (induced, for abelian $G$, via the Weil complex structure and in the general nonabelian case via the Narasimhan-Seshadri theorem/Donaldson-Uhlenbeck-Yau theorem).

Therefore the existence and (projective) flatness of the Hitchin connection exhibits the relative independence of the geometric quantization of Chern-Simons theory from the choice of polarization.

The Hitchin connection is akin to the Knizhnik-Zamolodchikov connection, which is built by thinking holographic dually of the vector bundle of conformal blocks of the corresponding $G$-Wess-Zumino-Witten model 2d CFT on $\Sigma$.

A more axiomatic characterization of such projectively flat connections in terms of modular functors is due to (Segal 88, prop. 5.4). See (Segal 88, around (5.9)) for discussion of how to turn these projectively flat connections into genuine flat connections.

Reviews of the Hitchin connection include (Lauridsen 10, section 2).

Related concepts

• quantization of Chern-Simons theory

• AdS3-CFT2 and CS-WZW correspondence

• modular functor

References

The original article is

• Nigel Hitchin, Flat connections and geometric quantization, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)

(What is now called the Hitchin connection appears in theorem 3.6 there, its expression in local coordinates is around (3.15). That for abelian gauge group $U(1)$ the classical Riemann theta functions constitute the covariantly constant sections of the Hitchin connection in these coordinates is remark 4.12.)

More axiomatic/abstract discussion of these projectively flat connections in terms of modular functors is in

• Graeme Segal, section 5 of The definition of conformal field theory , preprint, 1988; also in Ulrike Tillmann (ed.) Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. (pdf)

More explicit expressions are discussed in

• Bert van Geemen, Aise Johan de Jong, On Hitchin’s Connection (arXiv:alg–geom/9701007)

A nice review and new concise account is in

• Eduard Looijenga, From WZW models to Modular Functors (arXiv:1009.2245)

• Johan Martens Jørgen Andersen, notes by Søren Jørgensen, section 1.2.1 and 4 of Topological quantum field theories and moduli spaces, 2011 (pdf)

Discussion for deformation quantization instead of geometric quantization is in

• Jørgen Andersen, Hitchin’s connection, Toeplitz operators and symmetry invariant deformation quantization (arXiv:math/0611126)

This also reproduces the original construciton in the context of Chern-Simons theory in

• Scott Axelrod, S. Della Pietra, Edward Witten, Geometric quantization of Chern-Simons gauge theory, Jour. Diff. Geom. 33 (1991), 787-902. (EUCLID)

More details (on metaplectic correction) and generalization to connections over more general manifolds are in

• Peter Scheinost, Martin Schottenloher, Metaplectic quantization of the moduli spaces of flat and parabolic bundles, J. reine angew. Mathematik, 466 (1996) (web)

Similar generalization away from moduli spaces of flat connections to general symplectic manifolds with Kähler structure on them is also in

• Lauridsen, Aspects of quantum mathematics – Hitchin connections and the AJ conjecture, PhD thesis Aarhus 2010 (pdf)

Relation between the Hitchin connection and bundles of conformal blocks is discussed in

• Yves LaszloHitchin’s and WZW connection are the same, J. Differential Geom. 49 (1998), no. 3, 547–576 (pdf)

For more on this see at quantization of 3d Chern-Simons theory.