PhD Work

I did my PhD at Harish-Chandra Research Institute Allahabadfrom 2002 to 2008 under supervision of Prof. Rajesh Gopakumar. During my PhD I mainly worked on different aspects of the AdS/CFT correspondence. Here I briefly explain my thesis work.

• During my Ph.D. I worked on the finite temperature gauge theory and studying their ther- modynamics and phase structure. I had been trying to understand the phase diagram of the theory at week coupling and make connection with the geometric description of the dual supergravity theory. In [1] we have studied the thermal gauge theory partition function. At week coupling the gauge theory partition function can be written in terms of a unitary matrix model. We have obtained an exact expression for the partition function as a sum over representations of unitary group. This is an exact result for any finite N. Our analysis can also be generalized to write the partition function for gauge theory in presence of R charges.
  In large N limit, the exact answer is dominated by a saddle point. The dominant saddle points are basically different dominant representations of the unitary group. At large N we found that the answer shows some non-analytic behavior, as one varies the temperature, which is characteristic of a phase transition. That is in the large N limit, there is a dominant saddle point in the sum over representations and the nature of this saddle point exhibits non-analytic jumps as one varies the temperature.
  The solution in the large N limit shows three different saddle points. According to the AdS/CFT conjecture these three saddle points in weekly coupled gauge theory correspond to thermal AdS, small black hole and large black hole in the dual gravity description. For example, at low enough temperature the saddle point corresponds to global AdS dominates the partition function (confined phase of gauge theory) whereas the high temperature phase is dominated by the “big black hole” saddle point (deconfined phase in gauge theory). In fact we reproduced the complete Hawking-Page phase diagram of the dual gravity descriptions (strong coupling side), which was found earlier by usual eigenvalue analysis of the unitary matrix model.
  The most interesting and important feature of our analysis is that we found that the different saddle points of these matrix model can all be described in terms of free fermions. The representations of U(N) have an interpretation in terms of free fermions where the number of boxes in the Young tableaux behave like the momentum. In our analysis we find the momentum distribution function for free fermions for different saddle points. On the other hand, in the usual eigenvalue analysis of matrix model, one also finds the position distribution function for different saddle points since the eigen values are like positions of the fermions. Therefore we considered a phase space distribution for fermions which gives rise to these individual distribution functions. We found different phase space distributions of N non interacting free fermions for three different saddle points of dual description. These free fermionic phase space description of the bulk geometries (saddle points) are likely to be useful in trying to reconstruct the local bulk geometry from the gauge theory.

• We examine the Euclidean action approach, as well as that of Wald, to the entropy of black holes in asymptotically AdS spaces. From the point of view of holography these two approaches are somewhat complementary in spirit and it is not obvious why they should give the same answer in the presence of arbitrary higher derivative gravity corrections. For the case of the AdS5 Schwarzschild black hole, we explicitly study the leading correction to the Bekenstein-Hawking entropy in the presence of a variety of higher derivative corrections studied in the literature, including the Type IIB R4 term. We find a non-trivial agreement between the two approaches in every case. Finally, we give a general way of understanding the equivalence of these two approaches. Journal reference: Phys.Rev.D74:044007,2006 , ArXiv entry

• We study phase transition between electrically charged Ricci-flat black holes and AdS soliton spacetime of Horowitz and Myers in five dimensions. Boundary topology for both of them is S1×S1×R2. We consider Reissner-Nordstrom black hole and R-charged black holes and find that phase transition of these black holes to AdS soliton spacetime depends on the relative size of two boundary circles. We also perform the stability analysis for these black holes. In order to use the AdS/CFT correspondence, we work in the grand canonical ensemble. Journal reference: JHEP 0707:047,2007 , ArXiv entry

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