(All papers and books listed here are based on work partially supported by the NSF and the Alfred P. Sloan foundation).
Lectures on mathematical aspects of (twisted) supersymmetric gauge theories.arXiv:1401.26762 This is another mostly expository paper based on lectures I gave at the Les Houches winter school on mathematical physics in 2012. It is related to the papers arXiv:1308.0370, arXiv:1303.2632 and arXiv:1111.4234.
Integrable lattice models from four-dimensional field theoriesarXiv.1308:0370. This mostly expository paper explains the results of arXiv:1303.2632 (below) from a point of view that I hope is more accessible. Lots of pictures are included. More precisely, this paper gives a general construction of an integrable lattice model (and a solution of the Yang-Baxter equation with spectral parameter) from a four-dimensional field theory which is a mixture of topological and holomorphic. Spin-chain models arise in this way from a twisted, deformed version of N=1 gauge theory.
Supersymmetric gauge theory and the Yangian. arXiv:1303.2632. This paper develops a new connection between supersymmetric gauge theories and the Yangian. I show that a twisted, deformed version of the pure N=1 supersymmetric gauge theory is controlled by the Yangian, in the same way that Chern-Simons theory is controlled by the quantum group. This result is used to give an exact calculation, in perturbation theory, of the expectation value of a certain net of n+m Wilson operators in the deformed N=1 gauge theory. This expectation value coincides with the partition function of a spin-chain integrable lattice model on an n-by-m doubly-periodic lattice.
Notes on supersymmetric and holomorphic field theories in dimensions $2$ and $4$. Dedicated to Dennis Sullivan on the occassion of his 70th birthday. This is now arXiv:1111.4234.
Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model. Joint with Si Li, arXiv:1201.4501.
Factorization algebras in quantum field theory, joint with Owen Gwilliam. This book still requires some work. A draft is available here as a pdf file. Last updated: August 28, 2012. Submitted to Cambridge University Press.
Abstract: I describe how the Witten genus of a complex manifold $X$ can be seen from a rigorous analysis of a certain two-dimensional quantum field theory of maps from a surface to $X$.
Renormalization and effective field theory. This is my book on the foundations of perturbative quantum field theory, from a point of view which emphasizes the Wilsonian low-energy effective field theories and the Batalin-Vilkvovisky formalism. The AMS has kindly allowed me to make a PDF of this book available on this website. The AMS also has a website for the book.
I also have some slides from talks I've been giving about this material.
Renormalization and the Batalin-Vilkovisky formalism arXiv:0706.1533
This paper contains an early version of some of the ideas in the longer paper above. The aim of this paper was to get, as quickly
as possible, to the construction of a non-trivial theory (perturbative
Chern-Simons theory). Unfortunately, though, this paper doesn't
do a good job of explaining the Wilsonian philosophy underlying
everything, so I wouldn't really recommend reading it.