Classification of All 1/2 BPS Solutions of the Tiny Graviton Matrix Theory

TL;DR

Using the half BPS solutions, this paper shows how the tiny graviton Matrix theory and the mass deformed D = 3, = 8 superconformal field theories are related to each other.

Abstract

The tiny graviton matrix theory [hep-th/0406214] is proposed to describe DLCQ of type IIB string theory on the maximally supersymmetric plane-wave or AdS_5xS^5 background. In this paper we provide further evidence in support of the tiny graviton conjecture by focusing on the zero energy, half BPS configurations of this matrix theory and classify all of them. These vacua are generically of the form of various three sphere giant gravitons. We clarify the connection between our solutions and the half BPS configuration in N=4 SYM theory and their gravity duals. Moreover, using our half BPS solutions, we show how the tiny graviton Matrix theory and the mass deformed D=3, N=8 superconformal field theories are related to each other.

Figures (12)

• Figure 1: A fuzzy three sphere $S^3_f$ is obtained from $S^4_f$ by cutting it in a narrow strip close to its equator.
• Figure 2: $X^i$ corresponding to $m$ concenteric giants and the ${\cal L}_5$ corresponding to the $k^{th}$ block. The size of the $k^{th}$ three sphere is then $R^2_k\propto J_k$ ( cf.\ref{['S3f-radius']}) and hence $\sum_{k=1}^m R^2_k= R^2$.
• Figure 3: $X$'s corresponding to the most generic 1/2 BPS solutions of TGMT.
• Figure 4: Some Young Tableaux with $J$ boxes. (a) A totally anti-symmetric representation. This corresponds to a single giant graviton of radius $\sqrt{J}$ grown in $S^5$or$J$ tiny gravitons residing in $AdS_5$. (b) A totally symmetric representation which corresponds to a single giant graviton in $AdS_5$or$J$ tiny gravitons in $S^5$. (c) A generic Young tableau of $l$ rows and $J$ boxes. If we view the tableaux from above (focusing on columns) we see giants grown in $S^5$, and if we view it from the left side (focusing on rows) we see giants in $AdS_5$. In a $U(N)$ Young tableau number of rows cannot exceed $N$, a realization of the stringy exclusion principle MST. From the viewpoint of giants grown in $AdS$, however, this is the number of concentric 3-branes, and not their size, which cannot exceed $N$. The fact that each Young tableau has two interpretations in terms of giants in $S^5$ or $AdS_5$ is a manifestation of particle-hole duality in the two dimensional fermion picture discussed in LLM.
• Figure 5: A Young tableau of $J$ boxes has three interpretations. i) A representation of $U(N)$, ii) Schur Polynomials and representation of permutation group ${\cal S}_J$ and iii) Partition of $J$ into non-negative integers. The first two interpretations were noticed in LLMBerensteindeMelloKoch as discussed there, they lead to an equivalent description of all 1/2 BPS configurations of ${\cal N}=4$$U(N)$ SYM in terms of two dimensional fermion system. The third one, however, is the one, is relevant to the 1/2 BPS solutions of the TGMT.