Cohomological quantization of local prequantum boundary field theory
Summary
We discuss how local prequantum field theories with boundaries can be described in terms of n-fold correspondence diagrams in the infinity-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. In such a way, we can produce quantum propagators along cobordisms and partition functions of boundary theories as maps between certain twisted cohomology spectra. We are particularly interested in the case of 2d boundary field theories, where the pull-push quantization takes values in the twisted K-theory of differentiable stacks.
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 partitions functions of boundary theories are provided by the D-brane charges appearing in string theory and the K-theoretic quantization of symplectic manifolds. Here we extend the latter example to produce a K-theoretic quantization of Poisson manifolds, viewed as boundaries of the non-perturbative Poisson sigma-model. This involves geometric quantization of symplectic groupoids as well as the K-theoretic formulation of Kirillov’s orbit method. At the end we give an outlook on the 2d string sigma-model on the boundary of the membrane, quantized over tmf-cohomology with partition function the Witten genus.