Operators, Correlators and Free Fermions for SO(N) and Sp(N)

TL;DR

This paper develops a basis for local operators in free SO(N) gauge theory, deriving correlation functions and product rules, and explores their string theory duals, connecting to free fermions and generalizing to Sp(N).

Contribution

It introduces an exact operator basis for free SO(N) gauge theory, derives correlation functions, and relates the results to string theory duals and free fermion descriptions, extending to Sp(N).

Findings

Derived an exact formula for multi-trace operator correlations.

Established a product rule involving Littlewood-Richardson numbers.

Connected SO(N) basis to free fermions and generalized to Sp(N).

Abstract

Using the recently constructed basis for local operators in free SO(N) gauge theory we derive an exact formula for the correlation functions of multi trace operators. This formula is used to obtain a simpler form and a simple product rule for the operators in the SO(N) basis. The coefficients of the product rule are the Littlewood-Richardson numbers which determine the corresponding product rule in free U(N) gauge theory. SO(N) gauge theory is dual to a non-oriented string theory on the AdS_5xRP^5 geometry. To explore the physics of this string theory we consider the limit of the gauge theory that, for the U(N) gauge theory, is dual to the pp-wave limit of AdS_5xS^5. Non-planar unoriented ribbon diagrams do not survive this limit. We give arguments that the number of operators in our basis matches counting using the exact free field partition function of free SO(N) gauge theory. We connect the basis we have constructed to free fermions, which has a natural interpretation in terms of a class of 1/2-BPS bubbling geometries, which arise as orientifolds of type IIB string theory. Finally, we obtain a complete generalization of these results to Sp(N) gauge theory by proving that the finite N physics of SO(N) and Sp(N) gauge theory are related by exchanging symmetrizations and antisymmetrizations and replacing N by -N.