The Chern-Gauss-Bonnet Theorem via supersymmetric Euclidean field theories

We present a new proof for the Chern-Gauss-Bonnet theorem. We represent the Euler class integral as the partition function for zero-dimensional field theory with on-shell supersymmetry. We rewrite the supersymmetric partition function as a BV integral and deform the Lagrangian submanifold. The new Lagrangian submanifold localizes the BV integral to the critical points of the Morse function.

We discuss two dimensional N-extended supersymmetry in Euclidean signature and its R-symmetry. For N=2, the R-symmetry is SO(2)\times SO(1,1), so that only an A-twist is possible. To formulate a B-twist, or to construct Euclidean N=2 models with H-flux so that the target geometry is generalised Kahler, it is necessary to work with a complexification of the sigma models. These issues are related to the obstructions to the existence of non-trivial twisted chiral superfields in ...

This survey discusses our results and conjectures concerning supersymmetric field theories and their relationship to cohomology theories. A careful definition of supersymmetric Euclidean field theories is given, refining Segal's axioms for conformal field theories. We state and give an outline of the proof of various results relating field theories to cohomology theories.

Supersymmetric quantum mechanical models are computed by the Path integral approach. In the $\beta\rightarrow0$ limit, the integrals localize to the zero modes. This allows us to perform the index computations exactly because of supersymmetric localization, and we will show how the geometry of target space enters the physics of sigma models resulting in the relationship between the supersymmetric model and the geometry of the target space in the form of topological invariants...

Utilizing (4,0) superfields, we discuss aspects of supersymmetric sigma models and the ADHM construction of instantons a' la Witten.

We investigate the target space geometry of supersymmetric sigma models in two dimensions with Euclidean signature, and the conditions for N=2 supersymmetry. For a real action, the geometry for the N=2 model is not the generalized Kahler geometry that arises for Lorentzian signature, but is an interesting modification of this which is not a complex geometry.

We construct a topological Chern-Simons sigma model on a Riemannian three-manifold M with gauge group G whose hyperkahler target space X is equipped with a G-action. Via a perturbative computation of its partition function, we obtain new topological invariants of M that define new weight systems which are characterized by both Lie algebra structure and hyperkahler geometry. In canonically quantizing the sigma model, we find that the partition function on certain M can be expr...

The present paper gives calculations in detail to prove several special cases of Atiyah-Singer theorem through supersymmetric $\sigma$-models. Some technical tricks are employed to calculate the determinants of fluctuation operators of the path integrals. An intuitive and geometric argument is applied to overcome the complicated calculation on spin fields twisted by gauge fields.

These notes explore some aspects of formal derived geometry related to classical field theory. One goal is to explain how many important classical field theories in physics -- such as supersymmetric gauge theories and supersymmetric sigma-models -- can be described very cleanly using derived geometry. In particular, I describe a mathematically natural construction of Kapustin-Witten's P^1 of twisted supersymmetric gauge theories.

We find a new class of (2,0)-supersymmetric two-dimensional sigma models with torsion and target spaces almost complex manifolds extending similar results for models with (2,2) supersymmetry. These models are invariant under a new symmetry which is generated by a Noether charge of Lorentz weight one and it is associated to the Nijenhuis tensor of the almost complex structure of the sigma model target manifold. We compute the Poisson bracket algebra of charges of the above (2,...