This lecture series will provide a mathematical introduction to supersymmetric field theories, with an emphasis on low dimensions. We will build intuition with examples from supersymmetric quantum mechanics (dimension 1), linking the physical theories to the geometry Dirac operators and the representation theory of Clifford algebras. Next we will give a brief impression of the general story, formulated in terms of representations of geometric bordism categories built from super Euclidean groups. The remainder of the lecture series will focus on 2-dimensional theories with chiral supersymmetry, the central players in Stolz and Teichner's elliptic objects. Emphasis will be placed on examples that generalize the mathematical structures captured by the physical theories in dimension 1.
In 2004, Stephan Stolz and Peter Teichner formulated a question in the context of their program of relating elliptic cohomology with moduli spaces of super-symmetric 2D QFTs: "Can one construct a 3-category that deloops the 2-category of von Neumann algebras?" A preliminary such 3-category was constructed in joint work with Chris Douglas and Arthur Bartels, but that 3-category is not very well behaved, as it is not (visibly) idempotent complete. In particular, it doesn't contain the 3-category of unitary fusion categories. In this lecture series, I will present the objects that, conjecturally, form a better behaved 3-category. These are called bicommutant categories, and are the higher categorical analogues of von Neumann algebras. Expected examples of bicommutant categories include: the category of bimodules over a von Neumann algebra, the category of measurable bundles of Hilbert spaces over some measure space, the category of unitary representations of a locally compact group, the category of G-graded Hilbert spaces for G a discrete group, possibly twisted by a 3-cocycle, the category of representation of a locally compact quantum group, the category of representations of a conformal net, and the category of unitary topological line defects in any unitary d = 2 QFT (or even d >= 3?)
The two pillars of QFT are unitarity and locality. Locality is axiomatized in terms of higher categories, and is familiar to functorial field theorists. Unitarity, familiar to functional analysts, is axiomatized in terms of "dagger structures": equivariance for reflections together with positivity data. I will explain what a unitary n-category is, and the fundamental roles played by PL(n) and stable tangential structures in the subject. In particular, I will explain the following unitary cobordism hypothesis joint with C. Krulewski, L. Müller, and L. Stehouwer: the stably-PL-framed bordism n-category is the free unitary symmetric monoidal n-category generated by a unitarily n-dualizable object. Time permitting, I will discuss how the theory of complete W*-categories as higher-categorical Hilbert spaces, as developed by A. Henriques, Nivedita, and D. Penneys fits into our higher-dagger picture.
to be announced