Penrose limit, Spontaneous Symmetry Breaking and Holography in PP-Wave Background

TL;DR

This work proposes that the gauge theory dual to IIB string theory on a ten-dimensional pp-wave background effectively resides on a Euclidean ${\mathbb R}^4$, with light-cone time $x^+$ identified with the gauge theory renormalization group scale. It systematically derives how Penrose contraction reorganizes the bulk symmetry into a pair of Heisenberg algebras with an outer automorphism, and how the dual gauge theory, defined on ${\mathbb R}^4$ and organized by fixed R-charge $J$, reproduces the string spectrum through Hermite-transformed local operators and a Goldstone-like low-energy sector. The analysis connects dilaton and longitudinal graviton / four-form modes to specific BMN-type operators, and demonstrates how an infinite tower of string modes arises from phase-modulated gauge-theory operators, providing a concrete holographic dictionary in the pp-wave limit. The results illuminate how holography can operate in a Euclidean gauge theory setting and open avenues for generalizing BMN-type dualities to backgrounds with less supersymmetry and different symmetry-breaking patterns.

Abstract

We argue that the gauge theory dual to the Type IIB string theory in ten-dimensional pp-wave background can be thought to `live' on an {\it Euclidean} subspace spanning four of the eight transverse coordinates. We then show that light-cone time evolution of the string is identifiable as the RG flow of the gauge theory -- a relation facilitating `holography' of the pp-wave background. The `holography' reorganizes the dual gauge theory into theories defined over Hilbert subspaces of fixed R-charge. The reorganization breaks the SO(4,2)$\times$SO(6) symmetry to a maximal subgroup SO(4)$\times$ SO(4) spontaneously. We argue that the low-energy string modes may be regarded as Goldstone modes resulting from such symmetry breaking pattern.