Exact correlators of giant gravitons from dual $N = 4$ SYM theory

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.

“…(Similar results had been obtained in a complex matrix model [3].) The 1-d free fermion system can be mapped to an integer quantum Hall (IQH) droplet in two dimensions.…”

Section: Jhep06(2006)060supporting

confidence: 53%

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

Dai¹,

Wang²,

Wu³

2006

“…In general, some of the best understood aspects of this duality naturally arise through the use of supersymmetry. A particularly striking example of this was realized in a remarkable paper by Lin, Lunin and Maldacena (LLM) [1], which constructed explicit regular 1/2 BPS states in IIB supergravity and demonstrated their relation to the free fermion picture of the corresponding 1/2 BPS sector of the N = 4 super-Yang Mills theory [2,3]. Based on the correspondence with chiral primaries satisfying ∆ = J, LLM examined all regular 1/2 BPS states with SO(4) × SO(4) isometry in IIB supergravity with only the metric and self-dual five-form turned on.…”

Section: Introductionmentioning

confidence: 99%

“…The four-dimensional metric, two scalars H and G, and the 2-form field strength F µν are related to their ten-dimensional counterparts according to [1,4,5] ds 2 10 = g µν dx µ dx ν + e H (e G dΩ 2 3 + e −G d Ω 2 3 ), F (5) = (1 + * 10 )F (2) ∧ Ω 3 .…”

Section: Introductionmentioning

confidence: 99%

Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity

Chen¹,

Cremonini²,

Lin³

et al. 2007

This paper focuses on supergravity duals of BPS states in N = 4 super YangMills. In order to describe these duals, we begin with a sequence of breathing mode reductions of IIB supergravity: first on S 3 , then S 3 × S 1 , and finally on S 3 × S 1 × CP 1 . We then follow with a complete supersymmetry analysis, yielding 1/8, 1/4 and 1/2 BPS configurations, respectively (where in the last step we take the Hopf fibration of S 3 ). The 1/8 BPS geometries, which have an S 3 isometry and are time-fibered over a six-dimensional base, are determined by solving a non-linear equation for the Kähler metric on the base. Similarly, the 1/4 BPS configurations have an S 3 × S 1 isometry and a four-dimensional base, whose Kähler metric obeys another non-linear, Monge-Ampère type equation. Despite the non-linearity of the problem, we develop a universal bubbling AdS description of these geometries by focusing on the boundary conditions which ensure their regularity. In the 1/8 BPS case, we find that the S 3 cycle shrinks to zero size on a five-dimensional locus inside the six-dimensional base. Enforcing regularity of the full solution requires that the interior of a smooth, generally disconnected five-dimensional surface be removed from the base. The AdS 5 × S 5 ground state corresponds to excising the interior of an S 5 , while the 1/8 BPS excitations correspond to deformations (including topology change) of the S 5 and/or the excision of additional droplets from the base. In the case of 1/4 BPS configurations, by enforcing regularity conditions, we identify three-dimensional surfaces inside the four-dimensional base which separate the regions where the S 3 shrinks to zero size from those where the S 1 shrinks. We discuss a large class of examples to show the emergence of a universal bubbling AdS picture for all 1/2, 1/4 and 1/8 BPS geometries.

“…This is in analogy with large representation chiral primary operators of N = 4 supersymmetric Yang-Mills theory which are interpreted as giant gravitons [11,12,13]. In the latter case, there is a beautiful picture of "bubbling geometries" which relates geometrical data of the space-time on the string theory side of AdS/CFT to the representations of SU(N) which are used to characterize chiral primary operators [14].…”

Section: Introductionmentioning

confidence: 99%

“…At present we do not understand the origin of this selection rule, but we do observe that, as a result of it, the Wilson loop operator acts as a probe of this aspect of the structure of a graviton. A single hook giant graviton is a single spherical 3-brane extended in either the AdS 5 or S 5 direction [13].…”

Section: Introductionmentioning

confidence: 99%

Wilson loops in Script N = 4 SYM and fermion droplets

Okuyama¹,

Semenoff²

2006

126

The matrix models which are conjectured to compute the circle Wilson loop and its correlator with chiral primary operators are mapped onto normal matrix models. A fermion droplet picture analogous to the well-known one for chiral primary operators is shown to emerge in the large N limit. Several examples are computed. We find an interesting selection rule for the correlator of a single trace Wilson loop with a chiral primary operator. It can be non-zero only if the chiral primary is in a representation with a single hook. We show that the expectation value of the Wilson loop in a large representation labelled by a Young diagram with a single row has a first order phase transition between a regime where it is identical to a large column representation and a regime where it is a large wrapping number single trace Wilson loop.