A thesis that I once supervised:
Cohomological quantization of local prequantum boundary field theory
MSc thesis, Utrecht, August 2013
about the cohomological quantization of local prequantum field theory in the context of differential cohomology in a cohesive topos. See also at Type-semantics for quantization.
This thesis won the GQT Student prize 2012-2013.
We discuss how local prequantum field theories with boundaries can be described in terms n-fold correspondence diagrams in the (∞,1)-topos of smooth stacks equipped with higher circle bundles. This places us in a position where we can linearize the prequantum theory by mapping the higher circle groups into the groups of units of a ring spectrum, and then quantize the theory by a pull-push construction in the associated generalized cohomology theory.
We are particularly interested in the case of a 2d field theory, where the pull-push quantization takes values in the twisted K-theory of differentiable stacks. In such a way, we can produce quantum propagators along a cobordism and partition functions of boundary theories as maps between certain twisted cohomology spectra. Many quantization procedures found in the literature fit in this framework. For instance, propagators as maps between spectra have been considered in the context of string topology and in the realm of Chern-Simons theory, transgressed to two dimensions. Examples of partition functions of boundary theories are provided by the K-theoretic quantization of a Poisson manifold, seen as the boundary of its non-perturbative Poisson sigma model, together with the D-brane charges and “M-brane charges” that appear in string theory and M-theory.
The presentation talk is
Monday, August 26, 2013
at 14.00,
in BBL 071, Utrecht University.
An introduction to geometric quantization and detailed derivation of the Poisson manifold quantization as the boundary field theory for the 2d Poisson-Chern-Simons theory is in (Bongers, MSc 2014).
Related talk notes include
Motivic quantization of local prequantum field theory
at GAP XI, Pittsburgh, August 2013 (website)
An outlook in the general context of prequantum field theory is in section 6 of
More discussion of the formal nature of the cohomological quantization step is in
See also the $n$Lab entry
differential cohomology in a cohesive topos
Motivic quantization of local prequantum field theory
Cohomological quantization of local prequantum boundary field theory