Large N Field Theory and AdS Tachyons

In non-supersymmetric orbifolds of N =4 super Yang-Mills, conformal invariance is broken by the logarithmic running of double-trace operators -- a leading effect at large N. A tachyonic instability in AdS_5 has been proposed as the bulk dual of double-trace running. In this paper we make this correspondence more precise. By standard field theory methods, we show that the double-trace beta function is quadratic in the coupling, to all orders in planar perturbation theory. Tuning the double-trace coupling to its (complex) fixed point, we find conformal dimensions of the form 2 + i b, as formally expected for operators dual to bulk scalars that violate the stability bound. We also show that conformal invariance is broken in perturbation theory if and only if dynamical symmetry breaking occurs. Our analysis is applicable to a general large N field theory with vanishing single-trace beta functions.Comment: 26 pages, 6 figures. v3: small changes, version published on JHEP

JJ-trace identities and invariant theory

We generalize the notion of trace identity to $J$-trace. Our main result is that all $J$-traces of $M_{n,n}$ are consequence of those of degree $\frac12n(n + 3)$. This also gives an indirect description of the queer trace identities of $M_n(E)$

Orientifold daughter of N=4 SYM and double-trace running

We study the orientifold daughter of N=4 super Yang-Mills as a candidate non-supersymmetric large N conformal field theory. In a theory with vanishing single-trace beta functions that contains scalars in the adjoint representation, conformal invariance might still be broken by renormalization of double-trace terms to leading order at large N. In this note we perform a diagrammatic analysis and argue that the orientifold daughter does not suffer from double-trace running. This implies an exact large N equivalence between this theory and a subsector of N=4 SYM.Comment: 12 page

Predictions for PP-wave string amplitudes from perturbative SYM

The role of general two-impurity multi-trace operators in the BMN correspondence is explored. Surprisingly, the anomalous dimensions of all two-impurity multi-trace BMN operators to order g_2^2\lambda' are completely determined in terms of single-trace anomalous dimensions. This is due to suppression of connected field theory diagrams in the BMN limit and this fact has important implications for some string theory processes on the PP-wave background. We also make gauge theory predictions for the matrix elements of the light-cone string field theory Hamiltonian in the two string-two string and one string-three string sectors.Comment: 46 pages, 12 figures. V3:typos correcte

Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of $N$ D-brane probes for both $N \to \infty$ and finite $N$. The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``Plethystic Exponential'' provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.Comment: 59+1 pages, 7 Figure

On the trace identity in a model with broken symmetry

Considering the simple chiral fermion meson model when the chiral symmetry is explicitly broken, we show the validity of a trace identity -- to all orders of perturbation theory -- playing the role of a Callan-Symanzik equation and which allows us to identify directly the breaking of dilatations with the trace of the energy-momentum tensor. More precisely, by coupling the quantum field theory considered to a classical curved space background, represented by the non-propagating external vielbein field, we can express the conservation of the energy-momentum tensor through the Ward identity which characterizes the invariance of the theory under the diffeomorphisms. Our ``Callan-Symanzik equation'' then is the anomalous Ward identity for the trace of the energy-momentum tensor, the so-called ``trace identity''.Comment: 11 pages, Revtex file, final version to appear in Phys.Rev.

On higher holonomy invariants in higher gauge theory II

This is the second of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. We provide a definition of trace over a crossed module such to yield surface knot invariants upon application to 2-holonomies. We show further that the properties of the trace are best described using the theory quandle crossed modules.Comment: Latex, 34 pages, no figure