Calculating Superconformal Index of 𝓝 = 4 Super-Yang-Mills Theory for Finite N

Born from superstring theory, the AdS/CFT correspondence is a proposed relationship which claims that a strongly-coupled four-dimensional gauge theory is equivalent to a gravitational theory in five-dimensional AdS spacetime. Put simply, the AdS/CFT correspondence maps five-dimensional physics onto four-dimensional physics, allowing the analysis of a gauge theory at strong coupling, a difficult task, with a weakly coupled theory in curved spacetime, i.e., the AdS spacetime.

Thus far, AdS/CFT has been studied intensively in the large-N limit (N is an integer representing the size of the gauge group SU (N)), where the gauge theory is strongly coupled. In this study, I investigated AdS/CFT when N is finite and the gravitational theory is not weakly coupled.

It has been shown that even in this scenario, when quantum gravitational effects are important, some quantities related to topology and supersymmetry can still be calculated on the AdS side. In a similar vein, I calculated the superconformal index (SCI) of the N = 4 super-Yang-Mills theory (SYM), the maximally supersymmetric gauge theory.

I considered the duality between 4D 𝓝 = 4 SYM and the string theory in AdS₅ x S⁵. In the large-N limit, the supergravity contribution (through the plethystic exponential and single-particle index) on the AdS side was used to obtain the SCI.

When N is finite, both the quantum gravitational effect and the contribution of D3-branes become significant. Thus, the finite-N corrections for some quantities can be reproduced as the contribution of D-3 branes. I performed such a calculation to obtain the SCI with finite N on the AdS side.

I then compared the result to the known SCI calculated on the gauge theory side, and I confirmed that the finite-N corrections for higher-order terms could be correctly reproduced by the inclusion of wrapping D3-brane contributions.

It was recently discovered that it is possible to calculate the entropy of a black hole from the SCI. This method could, thus, provide a new approach to calculation of black hole entropy.