Supersymmetric gauge theory and matrix models

Theory of random matrices plays a crucial role both in mathematics and physics. The field was initiated almost a century ago by statisticians and introduced in physics in the 50s-60s by Wigner and Dyson. The theory has diverse application in different branches in physics and mathematics. We focus on a subset of random matrix theory : unitary matrix models (UMMs). Partition functions of different super-symmetric gauge theories, in particular Chern-Simons theories on certain manifolds boil down to UMMs. Different topological string theories reduce to UMM. These models also have applications in a broad class of condensed matter systems. Therefore, UMM is a useful tool to solve varieties of physical systems. Our main goal is to use the versatility of UMM to explore a dual quantum mechanical description of string theory (quantum gravity).

Interesting directions

  • In we found some identifications between dominant representations of U(N) (or Young diagrams), which are basically saddlepoints of SYM partition function and dif- ferent bulk geometries or saddle points (thermal AdS, small black holes and big black holes). But these identifications hold of course at saddle points, where as the full bulk partition function is basically sum over all geometries and the SYM partition function can also be cast as a sum over all possible Young diagrams. Therefor it would be interesting to reconstruct an exact mapping between the local bulk geometries and Young diagrams in non-supersymmetric case .

  • It would be nice to find a more general phase space functional (as a functional of phase space density) from which one can reconstruct different functionals i.e. functional of momentum distributions and functional of position distributions on appropriately integrating out.

References

[1] S. Dutta and R. Gopakumar, “Free Fermions and Thermal AdS/CFT,” JHEP 0803, 011 (2008) [arXiv:0711.0133 [hep-th]].

[2] P. Dutta, S.Dutta, “Phase Space Distribution for Two-Gap Solution in Unitary Matrix Model,” JHEP 1604, 104 (2016).

[3] P. Dutta, S.Dutta, “Phase Space Distribution of Riemann Zeros,” J.Math.Phys 58 053504 2017.

[4] A. Chattopadhyay, P. Dutta, S.Dutta, “Emergent Phase Space Description of Unitary Matrix Model,” JHEP 1711 186 (2017).