Geometric Aspects of Supersymmetric Partition Functions
We explore the physical interpretations of concepts in (equivariant) K-theory using supersym- metric partition functions. The partition functions are first computed using supersymmetric localisation and then interpreted using suitable concepts in K-theory. We show that quantum field theories predict nontrivial results in K-theory. We focus on supersymmetric quantum field theories in one and three dimensions. Nevertheless, our approach and results can be generalized to other dimensions. We start by discussing the superconformal index of (4,4) superconformal quantum mechanics on hyperkahler cones. The index is computed by coupling the quantum mechanics to a killing vector field. The resulting object has a natural geometric interpretation in terms of the equivariant Euler characteristics of the algebraic differential forms on the resolved space. The superconformal symmetries of the quantum mechanics imply certain symmetries of the Euler characteristics. We also explain how holography predicts an asymptotic behaviour of the index. We then compute the topologically twisted based P 1 index of 3d N = 2 gauge theories. We explain the supersymmetric boundary conditions at the south pole of P 1 . We then prove the existence and uniqueness of the BPS locus compatible with our boundary conditions. A detailed computation of the one-loop determinant around the BPS locus is also provided. The resulting index is interpreted as the K-theoretic Euler characteristic of the corre- sponding based quasimap moduli space. We define the moduli space and construct its tangent-obstruction theory. We show that unmatched fermion modes lie in the tangent- obstruction spaces. We define the virtual structure sheaf using the tangent-obstruction theory and shows that its Euler characteristic is captured by the based P 1 index. In the final chapter we provide a contour integral representation of the based P 1 index by studying the 3d twisted index on a hemisphere. We illustrate the supersymmetric boundary conditions for the hemisphere geometry. We show that the contour integral representation naturally arises as the Coulomb branch localisation of the index. We mention why this presentation is important in the geometry literature.
Complex Geometry, Supersymmetric Quantum Field Theory
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