Large N Elliptic Genus and AdS/CFT Correspondence

TL;DR

The paper investigates the AdS$_3$/CFT$_2$ correspondence for type IIB string theory on $AdS_3\times S^3\times K3$ by comparing the elliptic genus of the boundary theory with a bulk supergravity computation. To make a finite, meaningful comparison at large $N$, the author proposes an exclusion principle that truncates multi-particle states to those with total chiral-primary degree $\sum d_i\le N$. In the supergravity regime, the NS-sector elliptic genus matches the CFT NS elliptic genus for states with conformal weight $h\le (N+1)/4$, with the linear-in-$N$ part governed by coefficients $a(m,l)$ and $b(m,l)$ that agree between the two descriptions. This provides strong evidence for the proposed exclusion principle and the AdS/CFT duality in the regime where supergravity is valid, while clarifying limitations in accounting for black hole entropy and highlighting the need for genuine stringy states beyond supergravity for the full spectrum.

Abstract

According to one of Maldacena's dualities, type IIB string theory on AdS_3 X S^3 X K3 is equivalent to a certain N=(4,4) superconformal field theory. In this note we compute the elliptic genus of the boundary theory in the supergravity approximation. A finite quantity is obtained once we introduce a particular exclusion principle. In the regime where the supergravity approximation is reliable, we find exact agreement with the elliptic genus of a sigma model with target space K3^N/S_N.