Giant gravitons, Harish-Chandra integrals, and BPS states in symplectic and orthogonal $$ \mathcal{N} $$ = 4 SYM

Abstract

We find generating functions for half BPS correlators in $$ \mathcal{N} $$ N = 4 SYM theories with gauge groups Sp(2N), SO(2N + 1), and SO(2N) by computing the norms of a class of BPS coherent states. These coherent states are built from operators involving Harish-Chandra integrals. Such operators have an interpretation as localized giant gravitons in the bulk of anti-de-Sitter space. This extends the analysis of [1] to Sp(2N), SO(2N + 1), and SO(2N) gauge theories. We show that we may use ordinary Schur functions as a basis for the sector of states with no cross-caps in these theories. This is consistent with the construction of these theories as orientifold projections of an SU(2N) theory. We make note of some relations between the symmetric functions that appear in the expansion of these coherent states and symplectic Schur functions. We also comment on some connections to Schubert calculus and Gromov-Witten invariants, which suggest that the Harish-Chandra integral may be extended to such problems.

Orbit averaging coherent states: holographic three-point functions of AdS giant gravitons

“…This would explain why certain angular momentum modes are allowed on the spin chain description, even though semiclassically they are forbidden by the boundary conditions. Our calculation in the gauge theory demonstrate the power of the methods introduced in [22,30,31] for computing correlators of fully-symmetric Schur polynomials. Our methods are in many respects more streamlined when compared to the approach introduced in [4] for dealing with symmetric Schur functions.…”

Generating functions for giant graviton bound states

“…Similar integrals have been constructed for the projectors onto representations labelled by a single column or row, with an arbitrary length [53,54]. Another development [55][56][57][58], which is closely related, has shown that many of the results from the combinatorial approach can be reproduced by writing suitable generating functions of coherent states as a matrix integral. In some cases these matrix integrals are related to the well known Harish-Chandra-Itzykson-Zuber integral and can be computed using localization.…”

Multi-matrix correlators and localization

“…More recently, there have been works showing that certain generating functions can be used to perform computations in the free-field theory limit [8][9][10][11][12]. This technique has been succesfully implemented in the computation of three-point correlators involving large operators made out of a single matrix field [13][14][15], as in the half-BPS sector of N = SYM [16,17], where the dual gravitational description is explicitly realized from the gauge theory.…”