Projectively unitary Hermitian structure for the Axelrod-Della Pietra-Witten/Hitchin connection on nonabelian theta functions over Teichmüller space: general compact simple gauge groups and the exact unitary density $U$

Geometric quantization of Chern-Simons gauge theory with compact simple gauge group $G$ on a closed surface of genus $g \geq 2$ produces the bundle of nonabelian theta functions over Teichmüller space, carrying a projectively flat connection constructed by Axelrod, Della Pietra and Witten (1991) and, independently, by Hitchin (1990). Axelrod, Della Pietra and Witten could establish the unitarity of this connection — the existence of a Hermitian structure it preserves up to a scalar — only in genus 1, where the correcting density relative to the $L^2$ inner product is exactly $U = H^{1/2}$ with $H = \det{}'\Delta$ the determinant of the Laplacian; in genus $\geq 2$ they showed only that $U = H^{1/2}$ works to leading order in $1/k$, leaving the unitary structure, the exact form of $U$, and the marked-point case explicitly open. Later work settled the connection side (projective flatness, identification with the WZW/Knizhnik-Zamolodchikov connection, parabolic extensions to marked points, comparison with the Reshetikhin-Turaev TQFT) and important unitarity cases: genus 0 for all simple Lie algebras (Belkale 2012) and $SU(2)$ in genus $\geq 2$ with density $\exp(-F_\sigma + O(1/k))$ (Andersen 2012). The open core: construct the projectively unitary Hermitian structure for general compact simple $G$ in genus $\geq 2$, with or without marked points, and determine the density $U$ exactly — the latter open even for $SU(2)$.

Background

Chern-Simons gauge theory is a topological quantum field theory in three dimensions whose classical action, for a connection $A$ on a principal $G$-bundle over a 3-manifold $X$, is $\frac{k}{4\pi}\int_X \mathrm{Tr}(A \wedge dA + \tfrac{2}{3} A \wedge A \wedge A)$; the integer $k$ is the level. Witten (1989) proposed that canonical quantization of this theory on $\Sigma \times \mathbb{R}$, for a closed oriented surface $\Sigma$, produces the Jones polynomial of knots and projective unitary representations of the mapping class group of $\Sigma$ (the group of isotopy classes of orientation-preserving diffeomorphisms). Making this precise is the problem of geometrically quantizing the theory.

Canonical quantization starts from a finite-dimensional phase space: the symplectic quotient of the affine space of connections on $\Sigma$ by the gauge group, i.e. the moduli space $\mathcal{M}$ of flat $G$-connections on $\Sigma$. Choosing a complex structure $\sigma$ on $\Sigma$ identifies $\mathcal{M}$ (by Narasimhan-Seshadri type results) with a moduli space of semistable holomorphic $G$-bundles, giving $\mathcal{M}$ a Kähler structure. Kähler (geometric) quantization then takes as the quantum Hilbert space the space of holomorphic sections of the $k$-th tensor power of the prequantum line bundle over $\mathcal{M}$ — here the determinant line bundle, whose sections are called nonabelian theta functions (for a torus and rank one this reduces to classical theta functions on the Jacobian). As $\sigma$ ranges over Teichmüller space $\mathcal{T}$ (the space of complex structures on $\Sigma$ modulo diffeomorphisms isotopic to the identity), these spaces form a holomorphic vector bundle $H^{(k)} \to \mathcal{T}$, the bundle of nonabelian theta functions; Axelrod, Della Pietra and Witten work with its metaplectic (half-form) corrected version, obtained by tensoring with half-forms to remove a spurious dependence on the choice of polarization.

Symmetry of the underlying three-dimensional theory requires a projectively flat connection on this bundle — a connection whose curvature is a scalar 2-form — so that parallel transport identifies the fibers projectively and yields projective representations of the mapping class group. Hitchin (1990) constructed such a connection (the Hitchin connection) for genus $g \geq 2$, any simple simply-connected $G$, and no marked points, by a heat-equation argument using the algebra of holomorphic symmetric tensors on $\mathcal{M}$; Axelrod, Della Pietra and Witten (1991) constructed one independently by geometric quantization with the half-form correction. Both constructions exhibit the level shift: the projective curvature and monodromy involve not $k$ but $k + h$, where $h$ is the dual Coxeter number of $G$ (a Lie-theoretic constant, e.g. $h = n$ for $G = SU(n)$); Axelrod, Della Pietra and Witten note that the deeper meaning of this step is somewhat mysterious. On the conformal-field-theory side, the sheaf of vacua of the Wess-Zumino-Witten model (Tsuchiya-Ueno-Yamada) carries a projectively flat connection on conformal blocks — the Knizhnik-Zamolodchikov (KZ) connection in genus zero — and Laszlo (1998) proved that the Hitchin and WZW connections coincide (genus $> 2$, no marked points).

Unitarity. The $L^2$ inner product on holomorphic sections induced by the symplectic volume of $\mathcal{M}$ is not preserved by the connection. The unitarity problem is to find a Hermitian structure on the quantum bundle that the connection preserves up to a scalar function on $\mathcal{T}$ (projective unitarity), together with a concrete description of it. Formally one seeks the structure as an integral over the moduli space, $\int_{\mathcal{M}} \langle s_1, s_2 \rangle\, U\, \omega^m/m!$, a pushdown or direct-image construction (Lempert and Szőke developed the general framework of direct images of fields of Hilbert spaces), where $\omega$ is the symplectic form, $2m = \dim \mathcal{M}$, and the density $U$ is an unknown positive function on $\mathcal{M} \times \mathcal{T}$. In genus 1, $U = H^{1/2}$ exactly, where $H = \det{}'\Delta$ is the determinant of the Laplacian with zero modes removed — first derived from the path integral by Elitzur, Moore, Schwimmer and Seiberg (1989) and rigorously verified by Axelrod, Della Pietra and Witten. In genus $\geq 2$, $U$ depends on $k$ and no closed form is known; Axelrod, Della Pietra and Witten verified that $U = H^{1/2}$ satisfies the local unitarity condition only to leading order in $1/k$. Two neighboring pictures calibrate expectations: in a real polarization of $\mathcal{M}$, quantization concentrates on the Bohr-Sommerfeld fibers — the fibers of the real polarization on which the prequantum line bundle restricts trivially (Jeffrey-Weitsman), giving a combinatorial model related to the skein-theoretic TQFT of Blanchet-Habegger-Masbaum-Vogel; and the Reshetikhin-Turaev theory (of which the BHMV construction is the $SU(2)$ instance) is unitary by quantum-group construction, with Andersen and Ueno (2015) giving a geometric construction of that TQFT from conformal field theory for $U_q(\mathrm{sl}_N)$. Neither route produces the concrete geometric Hermitian structure and density on the moduli-space quantization that the Chern-Simons problem asks for. Surfaces with marked points correspond to parabolic bundles (bundles with flags at the marked points, encoding the holonomy around the punctures) and are the case relevant to knot and braid representations.

Problem Statement

Let $G$ be a compact, connected, simply-connected simple Lie group with dual Coxeter number $h$ (a Lie-theoretic constant of $G$, e.g. $h = n$ for $G = SU(n)$); let $\Sigma_g$ be a closed oriented surface of genus $g \geq 2$; let $\mathcal{T}_g$ be its Teichmüller space (the space of complex structures on $\Sigma_g$ modulo diffeomorphisms isotopic to the identity); and let $\widetilde{H}^{(k)} \to \mathcal{T}_g$ be the bundle of nonabelian theta functions at level $k$, with $k$ a positive integer, with its half-form (metaplectic) correction, whose fiber over a complex structure $\sigma$ is the half-form corrected space of holomorphic sections of the $k$-th power of the determinant/prequantum line bundle over the moduli space $\mathcal{M}_\sigma$ of semistable $G$-bundles identified with flat connections via $\sigma$. By the work of Axelrod, Della Pietra and Witten and of Hitchin, extended to the metaplectic setting by Andersen, Gammelgaard and Lauridsen, $\widetilde{H}^{(k)}$ carries a projectively flat connection $\nabla$ (the ADW/Hitchin connection; projectively flat means its curvature is a scalar 2-form, so that parallel transport identifies the fibers projectively), and this connection is now known to agree with the WZW connection on conformal blocks where both are defined (Laszlo: genus $> 2$, no marked points; Andersen-Henke: genus 0 with at least three marked points).

(1) Unitary structure. Construct a Hermitian structure on $\widetilde{H}^{(k)}$ relative to which $\nabla$ is projectively unitary — i.e. $\nabla$ preserves the Hermitian form up to a scalar function on $\mathcal{T}_g$ — for every compact, connected, simply-connected simple $G$, every genus $g \geq 2$, and every positive integer level $k$; establish the uniqueness of this structure up to scalar and its projective invariance under the mapping class group of $\Sigma_g$ (the group of isotopy classes of orientation-preserving diffeomorphisms of $\Sigma_g$). The same question for surfaces with marked points (parabolic moduli, where the connection now exists by the work of Biswas, Mukhopadhyay and Wentworth and of Ouaras) is an equally open extension.

(2) Identification of the density. Relative to the $L^2$ inner product defined by the symplectic volume form $\omega^m/m!$ on the moduli space (where $\omega$ is the symplectic form of $\mathcal{M}_\sigma$ and $2m = \dim \mathcal{M}_\sigma$), determine the density $U^{(k)}_\sigma$ relating the unitary structure of (1) to the $L^2$ structure — equivalently, carry out the pushdown construction of $U$ described by Axelrod, Della Pietra and Witten — exactly, not merely to leading order in $1/k$.

The structure of (1) and the density of (2) must specialize to the known cases: genus 1, where $U = H^{1/2}$ with $H = \det{}'\Delta$ the determinant of the Laplacian on the torus with zero modes removed; genus 0, where Belkale's projectively unitary metrics on conformal blocks apply; and $G = SU(2)$, $g \geq 2$, where Andersen's Hermitian structure with density $\exp(-F_\sigma + O(1/k))$ applies, $F_\sigma$ denoting the Ricci potential of the moduli space (a local potential for its Ricci form). The target is the general compact simple case in genus $\geq 2$, with or without marked points, together with the exact density $U$; the $SU(2)$ case is an instance of it.

References

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