Correlation functions and representation bases in free N=4 Super Yang-Mills

We study exact correlation functions of N=4 SYM at zero coupling. It has been known that it is convenient to label local gauge invariant operators by irreducible representations of symmetric groups/Brauer algebras. We first review the construction of representation bases from the viewpoint of the enhanced symmetry structure of the free theory. We present a basis of multi-matrix models using elements of Brauer algebras, generalising our previous construction for two matrices. We will compute multi-point functions of the basis with the exact N-dependence. In particular we study three-point functions of a class of BPS operators, and we find that they are given by a branching rule of the Brauer algebra. The three-point functions take a factorised form if representations on the operators satisfy a relation.Comment: 28 pages, typos correcte

From Gauss Graphs to Giants

We identify the operators in ${\cal N}=4$ super Yang-Mills theory that correspond to ${1\over 8}$-BPS giant gravitons in AdS$_5\times$S$^5$. Our evidence for the identification comes from (1) counting these operators and showing agreement with independent counts of the number of giant graviton states, and (2) by demonstrating a correspondence between correlation functions of the super Yang-Mills operators and overlaps of the giant graviton wave functions.Comment: 16 pages; v2: matches published versio

CFT4 as SO(4,2)-invariant TFT2

We show that correlators of local operators in four dimensional free scalar field theory can be expressed in terms of amplitudes in a two dimensional topological field theory (TFT2). We describe the state space of the TFT2, which has $SO(4,2)$ as a global symmetry, and includes both positive and negative energy representations. Invariant amplitudes in the TFT2 correspond to surfaces interpolating from multiple circles to the vacuum. They are constructed from SO(4,2) invariant linear maps from the tensor product of the state spaces to complex numbers. When appropriate states labeled by 4D-spacetime coordinates are inserted at the circles, the TFT2 amplitudes become correlators of the four-dimensional CFT4. The TFT2 structure includes an associative algebra, related to crossing in the 4D-CFT, with a non-degenerate pairing related to the CFT inner product in the CFT4. In the free-field case, the TFT2/CFT4 correspondence can largely be understood as realization of free quantum field theory as a categorified form of classical invariant theory for appropriate SO(4,2) representations. We discuss the prospects of going beyond free fields in this framework.Comment: 54 pages, 7 figures; version 2: Published version - extended discussion of CFT4/TFT2 in terms of emergent space-time; refs added; typos correcte

Nonplanar integrability at two loops

In this article we compute the action of the two loop dilatation operator on restricted Schur polynomials that belong to the su(2) sector, in the displaced corners approximation. In this non-planar large N limit, operators that diagonalize the one loop dilatation operator are not corrected at two loops. The resulting spectrum of anomalous dimensions is related to a set of decoupled harmonic oscillators, indicating integrability in this sector of the theory at two loops. The anomalous dimensions are a non-trivial function of the 't Hooft coupling, with a spectrum that is continuous and starting at zero at large N, but discrete at finite N.Comment: version to appear in JHE