Thermal one point functions in large $N$ vector models and holography

Abstract

This thesis develops analytic methods to evaluate thermal one point functions in holography and a class of conformal field theories(CFT) described by large $N$ vector models.

For planar charged black holes in $AdS_{d+1}$ we find a suitable large $d$ limit where thermal one point functions can be evaluated exactly. We show that for operators with large scaling dimensions, these 1-point functions probe the geometry interior to the horizon, including the geometry inside the inner horizon.

We then study one point functions of higher spin currents in large $N$ vector models on $S^1 \times R^{d−1}$, with $d$ being odd, and demonstrate they exhibit universal features. We show that these 1-point functions at the non-trivial fixed point of the theory tend to its Stefan-Boltzmann value at large spin. At large $d$, 1-point functions at the non-trivial fixed point decay exponentially compared to its Stefan-Boltzmann value.

Considering the critical $O(N)$ model on $S^1 \times S2$, we evaluate the finite size corrections to the partition function, stress tensor and the two point function. We find these for the first time despite extensive study of this model in the literature. Applying the OPE inversion formula we show that the finite size corrections to higher spin currents at the non-trivial fixed point of the theory tend to that of the free theory at large spin.