BPSR-balls in Script N = 4 SYM onR×S³, quantum Hall analogy and AdS/CFT holography

Abstract

In this paper, we propose a new approach to study the BPS dynamics in N = 4 supersymmetric U(N ) Yang-Mills theory on R × S 3 , in order to better understand the emergence of gravity in the gauge theory. Our approach is based on supersymmetric, space-filling Q-balls with R-charge, which we call R-balls. The usual collective coordinate method for non-topological scalar solitons is applied to quantize the half and quarter BPS R-balls. In each case, a different quantization method is also applied to confirm the results from the collective coordinate quantization. For finite N , the half BPS R-balls with a U(1) R-charge have a moduli space which, upon quantization, results in the states of a quantum Hall droplet with filling factor ν = 1. These states are known to correspond to the "sources" in the Lin-Lunin-Maldacena geometries in IIB supergravity. For large N , we find a new class of quarter BPS R-balls with a non-commutativity parameter. Quantization on the moduli space of such R-balls gives rise to a non-commutative ChernSimons matrix mechanics, which is known to describe a fractional quantum Hall system. In view of AdS/CFT holography, this demonstrates a profound connection of emergent quantum gravity with non-commutative geometry, of which the quantum Hall effect is a special case.

Large charges on the Wilson loop in $$ \mathcal{N} $$ = 4 SYM. Part II. Quantum fluctuations, OPE, and spectral curve

“…In this regime, the role of the large charge effective field theory is played by a probe string action in AdS 5 × S 5 , which becomes classical in the large J limit. As we demonstrated in the previous paper [41], the leading large charge answer for the correlation functions can be computed by evaluating light vertex operators on a nontrivial classical string solution with large angular momentum, which was constructed in [8,49,50]. In special kinematics, we can compare the results from holography with exact results from supersymmetric localization [14] and verify the agreement of the two approaches.…”

“…The string tension is 2g, so the two terms are the same order in the large charge limit. We reviewed the classical string dual to the Wilson loop with Z J and ZJ in [41], which was discussed previously in [8,49,50]. We computed the action and the vertex operators dual to Z and Z on the classical solution, which determined the leading large charge behavior of the two-point and higher-point functions in (2.9) and (2.11).…”

“…This configuration in global coordinates has manifest translational symmetry along τ . The classical string dual to this operator was discussed in [8,49,50] and reviewed in [41]. To…”

“…Next, we determine W xx to zeroth order in c 2 . 49 We need to find g R xx (r; k) from (4.10), supplemented with the boundary condition g R xx (r m ; k) = 0. To zeroth order, the differential equation becomes…”

“…) 49 We study Wyy to linear order and Wxx only to zeroth order because the zeroth order ODE in (C.2) is nicer than (C.15), and makes the perturbative analysis simpler for Wyy than Wxx.…”