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Title Cohomological quantization of local prequantum boundary field theory
Publication Date
Degree Level masters
University/Publisher Universiteit Utrecht
Abstract 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.
Subjects/Keywords quantum field theory; quantization; generalized cohomology; K-theory
Contributors Henriques, dr. A.G.
Language en
Rights info:eu-repo/semantics/OpenAccess
Country of Publication nl
Format text/plain
Record ID oai:dspace.library.uu.nl:1874/282756
Repository utrecht
Date Indexed 2016-09-27

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…give nontrivial approximations to the smooth theory, one sometimes loses important geometric information by passing 6 to homotopy types: for example, the generalized cohomology of the homotopy type of a quotient stack M//G does not always agree with…

…the G-equivariant cohomology of M , as one might expect. To get a more refined quantization procedure, we should look for a concrete generalized cohomology theory of smooth stacks which is more receptive to stacky structures and allows for twists by Bn…

…states. The K-theoretic quantization of a symplectic manifold gives only a first example of the quantization by pull-pushing in generalized cohomology. Various phenomena considered in the literature can be put in this abstract framework: • the string…

…stacks), where there is a general abstract theory of twisted cohomology spectra and pushforward maps, due to [ABG+ 08]. In principle, this general picture can be used to quantize not only the boundaries to two-dimensional theories, but also…

…the tensor (∞)-category of R-modules. Given a correspondence diagram like 1.2, we can pull in R-cohomology along the left map, and push along the right map to produce a single map in the category of R-modules. As in ordinary cohomology

…prequantum fields. As we will see in section 2.3, a similar kind of diagram describes a boundary theory to an ndimensional topological field theory: Fields∂ uuu w uuu ww w uuu u ww u u w u u u7 uu u {ww u u u uu u ∗ pp Fields u u u u u 3A pp u sss pp s s…

…pp ss p5 ysss Bn U (1) (1.3) Here the spaces involved are not the spaces of fields on a surface, but instead they are classifying spaces of fields. For example, for Chern-Simons theory, where the fields on a manifold Σ are (…

…method to such diagrams, we quantize the boundary theory: the result of the pull-push quantization can be interpreted as the partition function of the boundary theory (see section 5 for more details). Since the pushforward requires a choice of…