Operators, correlators and free fermions for SO(N) and 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 5 ×RP 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 5 × S 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

Exact probes of orientifolds

“…N even states and for g = so(2n + 1), sp(n) the first N odd eigenstates [16,17,22,23]. The computations are straightforward, and reveal exact relations among various vevs.…”

“…Our aim is to explore some of these features at finite g s and α ′ /R 2 , taking advantage of the possibility of computing exactly the vev of certain Wilson loop operators for these field theories. While our focus is on non-local operators, the physics of local operators of these field theories at finite N has been explored in [16,17].…”

“…where the specific |Ψ N depends on the algebra g. For G = SO(N ), Sp(N ), these Slater determinants involving one-fermion wavefunctions of definite parity also appear in the study of certain local operators [16,17].…”

Anomalous dimensions of heavy operators from magnon energies

“…Further, significant progress was made in understanding the spectrum of anomalous dimensions of these operators in the studies [25,26,[29][30][31][32][33][34]. Extensions which consider orthogonal and symplectic gauge groups and other new ideas, have also been achieved [35][36][37][38][39][40].…”

FiniteNquiver gauge theory

“…As a simple example, consider a scalar field Z which is an N × N matrix transforming in the adjoint representation of U(N). A complete set of operators built using three fields is given by {Tr(Z It is a highly non-trivial problem to write a basis of local operators that is not over complete at finite N. This problem has been solved for multimatrix models with U(N) gauge group in [1,2,3,4,5,6,7,8,9] and for single matrix models with SO(N) or Sp(N) gauge groups in [10,11,12]. The result of these studies is a basis of local operators that also diagonalizes the free field two point function.…”