BPS coherent states and localization

Abstract

We introduce coherent states averaged over a gauge group action to study correlators of half BPS states in $$ \mathcal{N} $$ N = 4 SYM theory. The overlaps of these averaged coherent states are a generating function of correlators and can be written in terms of the Harish-Chandra-Itzykzon-Zuber (HCIZ) integral. We show that this formula immediately leads to a computation of the normalization of two point functions in terms of characters obtained originally in the work of Corley, Jevicki and Ramgoolam. We also find various generalizations for An−1 quivers that follow directly from other solvable integrals over unitary groups. All of these can be computed using localization methods. When we promote the parameters of the generating function to collective coordinates, there is a dominant saddle that controls the effective action of these coherent states in the regime where they describe single AdS giant gravitons. We also discuss how to add open strings to this formulation. These will produce calculations that rely on correlators of matrix components of unitaries in the ensemble that is determined by the HCIZ integral to determine anomalous dimensions. We also discuss how sphere giants arise from Grassman integrals, how one gets a dominant saddle and how open strings are added in that case. The fact that there is a dominant saddle helps to understand how a 1/N expansion arises for open strings. We generalize the coherent state idea to study 1/4 and 1/8 BPS states as more general integrals over unitary groups.

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.…”

“…Consequently, a promising starting point is an integral representations for the relevant operators. This philosophy was recently employed, in a very similar setting, in [55]. In this article we have obtained these integral representations, for operators dual to a boundstate of two sphere giant gravitons or two AdS giant gravitons.…”

Giant gravitons in twisted holography

“…An insertion of a determinant of this form corresponds to a Giant Graviton brane wrapping a 1-dimensional complex curve, which approaches the boundary of AdS 3 at a point z along the line b = au − m + O(a −1 ), where ad − bc = 1 are the coordinates of SL(2, C). 10 When considering insertions of multiple determinants D(m i ; u i ; z i ) there might be many brane configurations with the same asymptotics controlled by parameters m i , u i , z i .…”

“…A method of computing correlations functions of determinant operators in the large N limit was presented in [43]. Following their prescription 11 (also implemented in [10,22,62]), we fermionize the determinants:…”

“…With this choice of conventions, the ribbon diagrams of genus g contribute at order N 2−2g in the 't Hooft expansion. 10 The boundary behavior of a D1-brane dual to a determinant can be derived by adding a probe D1'-brane transverse to the stack of N D1-branes and finding its image in the backreacted geometry (see [17]). 11 Notice that the tree level computations of [43] give an exact answer in the chiral algebra (when considering BRST closed operators) since the 't Hooft coupling dependence drops out in this subsector.…”