# Mathematical Physics

## Supersymmetry on Curved Spaces

Over the last decades, there has been a very rich interplay between string theory/quantum field theory and mathematics, leading to several new and deep insights in both math and physics, involving topics such as topological field theory, mirror symmetry, Gromov-Witten invariants and so on. It is clear that string theory and quantum field theory both contains very rich and interesting mathematical structures, and those are what we try to study.

As a particular example, in recent years there has been a large interest in studying supersymmetric quantum field theories placed on curved spaces such as spheres, since it was realized[1] that these theories allows one to use a particular mathematical technique, called supersymmetric localization, to compute certain quantities of the theory exactly. In general, computing things exactly in quantum field theories is a very difficult task: most of the time all one can do is use various approximations, most commonly perturbation theory involving Feynman diagrams and so on. So having a class of examples where exact results can be computed can teach us interesting things about the structure of quantum field theories, allowing us to test things like different dualities, AdS/CFT, seeing if the theory has some hidden symmetries etc. It is also interesting to investigate what the theory “knows” about the curved space that we place it on, and seeing if it can be related to any interesting mathematical invariants.

In some more detail, to compute quantities in quantum field theory involves computing something called the path integral, which is an integral over an infinite dimensional space. Performing this integral is usually impossible, and mathematically it is very hard to even define what it means (in fact, it's one of the Millennium problems to do this for ordinary Yang-Mills theory). But it was observed in[1] that for particularly nice theories (i.e. supersymmetric), one can actually perform the integral exactly and get the answer in the form of a finite-dimensional integral. Or in other words, the theory localizes on a finite-dimensional subspace. This procedure has now been carried through for various theories in dimension 2,3,4 and 5, and many interesting results have been found. The work of this group has focused on theories in 5 dimensions (for example[2]-[4]), which are interesting in part by their relationship with the mysterious (2,0) theories in 6d, that are believed to hold a very special place among all quantum field theories.

Additional research themes are the study of rigid supersymmetry on curved spaces, instantons in gauge theories, topological field theories and geometry. Particular research fields pursued by the members of the group are: Localization of supersymmetric gauge theories, Rigid Supersymmetry on Curved Spaces, Instanton Partition Functions, Topological Field Theories, Topological String Theory, Matrix Models, Vertex Algebras, Supersymmetric Sigma Models, AKSZ/BV formalism.

## References

[1]  V. Pestun, “Localization of gauge theory on a four-sphere and supersymmetric Wilson loops”, arXiv:0712.2824
[2] J. Källen, M. Zabzine, J. Qiu, “The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere”, arXiv: 1206.6008
[3] J. Qiu, M. Zabzine, “5D Super Yang-Mills on Yp,q Sasaki-Einstein manifolds”, arXiv: 1307.3149
[4] J. Qiu, L. Tizzano, J. Winding, M. Zabzine, “Gluing Nekrasov Partition functions”, arXiv:1403.2945
[5] Marcos Marino, “Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories”, arXiv:1104.0783
[6] Guido Festuccia, Nathan Seiberg, “Rigid Supersymmetric Theories in Curved Superspace”, arXiv:1105.0689
[7] Yuji Tachikawa, “A review on instanton counting and W-algebras”, arXiv:1412.7121

### Reviews of Topological Field Theories

[8] Stefan Cordes, Gregory Moore, Sanjaye Ramgoolam, “Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories”, arXiv:hep-th/9411210
[9] Marcos Marino, “Chern-Simons Theory and Topological Strings”, arXiv:hep-th/0406005
[10] Hori K., Katz S., Klemm A., Pandharipande R., Thomas R., Vafa C., Vakil R., Zaslow E., “Mirror symmetry”, Clay Mathematics Monographs, Vol. 1, Amer. Math. Soc., Providence, RI, 2003.

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