In the late 1960s, Symanzik introduced the idea that certain Euclidean field theories could be represented as gases of random walks and loops, interacting according to their local times. This idea, developed further by Dynkin, Brydges, Fröhlich, Spencer, and many others, has led to a broad class of such identities, now commonly referred to as `isomorphism theorems'. One useful way to view these results is as a mechanism for transferring tools between the two settings: probabilistic techniques can be applied to field theoretic questions, while field theoretic ideas can be exploited in the study of random walks. In the classical setting, these theorems relate the local times of Markovian random walks to the squares of Gaussian free fields. In this talk, I will focus on the supersymmetric hyperbolic sigma model and its reinforced random walk representation in terms of the vertex reinforced jump process (VRJP), a non-Markovian random walk whose jump rates are reinforced by the accumulated local times of the walker. I will further discuss how the `mixture representation' of the VRJP can be understood through the geometry the hyperbolic sigma model (in particular, by its foliation by flat Euclidean leaves), and explain how supersymmetry provides a conceptual explanation the VRJP 'magic formula'. Finally, I will describe a non-reversible analogue of the VRJP, its connection to a new `$\Z_2$-equivariant' generalisation of the hyperbolic sigma model.
A complex algebra equipped with a conjugate-linear involution which can be faithfully represented as a norm-closed algebra of bounded operators on a Hilbert space is called a C-algebra. Examples include the algebra of continuous functions on a locally compact Hausdorff space vanishing at infinity. This is indeed the unique class of commutative C-algebras up to isomorphism. All relations between locally compact Hausdorff spaces can be translated as relation between their algebra of functions, and thus the theory of C*-algebras can be considered a generalization of the theory of locally compact spaces. It offers a framework in which to study algebras of observables in quantum field theory. In this talk, I will discuss aspects of a recent construction of 2-dimensional topological quantum field theories (TQFTs) from certain inclusions of C-algebras, which we call discrete. I will explain how this notion is an axiomatization of the fixed point subalgebra of a compact group action on a C-algebra. Starting from such an inclusion, the value of the resulting TQFT on a disk is characterized by an associated C*-algebra, called the symmetric enveloping algebra; a concrete realization of an abstract object that have appeared in the algebraic approach to conformal field theories, in the theory of quantum groups and of subfactors. The values of the TQFT on other surfaces give extensions of the symmetric enveloping algebra which come equipped with actions of the mapping class groups.
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