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Here is my question, below that some motivation:

For $G$ a compact abelian Lie group and $\Sigma$ a surface, with $M_G = \mathrm{Loc}_G(\Sigma)$ denoting the space of flat $G$-connections on $\Sigma$ (the "Jacobian", itself a torus)... is there a natural map

  • from the moduli space of 3-framings of $\Sigma$ (i.e. of trivializations of $T \Sigma \oplus \underline{\mathbb{R}}$);

  • to the moduli space of stable $\mathrm{Spin}^c$-structures on $M_G$

?

Here "a natural map" may be too vague (though say if some interesting map springs to mind), but to make it more specific I first need to say the following.

Suppose we are handed a line bundle $\theta \in \mathcal{O}(\mathrm{Loc}_G(\Sigma))$, thought of as the prequantum line bundle of $G$-Chern-Simons theory, then traditional geometric quantization constructs a projectively flat vector bundle on the Riemann moduli space $\mathcal{M}_\Sigma$ -- the Hitchin connection -- whose fiber over a complex structure of $\Sigma$ is the space of holomorphic sections of $\theta$ with respect to a complex structure on $\mathrm{Loc}_G(\Sigma)$ that is naturally induced from that of $\Sigma$.

In the perspective of $\mathrm{Spin}^c$-quantization this construction is understood as forming in $\mathrm{KU}$-theory the push-forward of $\theta$ to the point, with respect to the $\mathrm{KU}$-orientation given by the complex structure regarded as a $\mathrm{Spin}^c$-structure.

These traditonal constructions are time-honored, but, when one gets to the bottom of it, are somewhat ad-hoc. Something is clearly right about them, but the main reason why we choose complex structures on $\Sigma$ to induce complex structures on $\mathrm{Loc}_G(\Sigma)$ to finally perform Kähler quantization is that it happens to work.

On the other hand, the cobordism hypothesis gives a systematic way to approach this: regarding the "Chern-Simons 3-bundle" $\mathbf{B}G \to \mathbf{B}^3 U(1)$ (the map on moduli stacks refining the cup product) as a fully dualizable object in $\mathrm{Corr}_3(\mathrm{Sh}_\infty(\mathrm{Mfd})_{/\mathbf{B}^3 U(1)})$ defines a "local prequantum field theory" $\mathrm{Bord}_3^{\mathrm{fr}}\to \mathrm{Corr}_3(\mathrm{Sh}_\infty(\mathrm{Mfd})_{/\mathbf{B}^3 U(1)})$ which sends a closed surface $\Sigma$ to the transgression bundle $\theta$.

To turn this into an actual QFT with values in (3-fold) $\mathrm{KU}$-modules at least in codimension 1 (that's all I will consider here) we are to naturally produce stable $\mathrm{Spin}^c$-structures on $\mathrm{Loc}_G(\Sigma)$. The cobordism hypothesis says that the "geometric moduli" on $\Sigma$ that we may and have to use for this are not a priori complex structures on $\Sigma$, but are 3-framings on $\Sigma$: it asks us to construct a flat $\mathrm{KU}$-module bundle on the moduli space of 3-framings of $\Sigma$.

Therefore finally the more concrete version of my question:

what might be a natural map from the moduli of 3-framings on $\Sigma$ to that of stable $\mathrm{Spin}^c$-structures on $\mathrm{Loc}_G(\Sigma)$ such that for given $\theta$ the induced flat bundle on 3-framing moduli is a pullback of the Hitchin connection along a map from 3-framing moduli to complex moduli (or something close)?

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  • $\begingroup$ Chris Schommer-Pries has kindly pointed out that there should be a positive answer to the second of my questions: the projectively flat Hitchin connection on the Riemann moduli space canonically lifts to a genuinely flat connection on the 3-framing moduli space. This follows with a) Segal's deprojectivization, 2) the fact that "Atiyah 2-framings" provide "level-12 riggings" and 3) the observation (which Chris kindly highlighted) that there is a canonical functor from the 1-type of 3-framings to that of "Atiyah 2-framings" (see ncatlab.org/nlab/show/modular+functor#TopologicalLift). $\endgroup$ – Urs Schreiber Sep 15 '14 at 9:13

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