Integrable Spin Chains on the Conformal Moose

We consider N=1, D=4 superconformal U(N)^{pq} Yang-Mills theories dual to AdS_5xS^5/Z_pxZ_q orbifolds. We construct the dilatation operator of this superconformal gauge theory at one-loop planar level. We demonstrate that a specific sector of this dilatation operator can be thought of as the transfer matrix for a two-dimensional statistical mechanical system, related to an integrable SU(3) anti-ferromagnetic spin chain system, which in turn is equivalent to a 2+1-dimensional string theory where the spatial slices are discretized on a triangular lattice. This is an extension of the SO(6) spin chain picture of N=4 super Yang-Mills theory. We comment on the integrability of this N=1 gauge theory and hence the corresponding three-dimensional statistical mechanical system, its connection to three-dimensional lattice gauge theories, extensions to six-dimensional string theories, AdS/CFT type dualities and finally their construction via orbifolds and brane-box models. In the process we discover a new class of almost-BPS BMN type operators with large engineering dimensions but controllably small anomalous corrections.Comment: 53 pages, 14 eps figures; Added reference

From Doubled Chern-Simons-Maxwell Lattice Gauge Theory to Extensions of the Toric Code

We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group $\mathbb{R}$, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space $\mathbb{R}^2$ in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus $U(1)^2$ threaded by $k$ units of quantized "magnetic" flux, with $k$ being the level of the Chern-Simons theory. The holonomies of the torus $U(1)^2$ give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from $U(1)$ to $\mathbb{Z}(k)$. The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a $\mathbb{Z}(k)$-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle $\frac{2 \pi}{k}$. Non-Abelian $U(k)$ Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.Comment: 38 pages, 4 figure

Nonabelian Duality and Solvable Large N Lattice Systems

We introduce the basics of the nonabelian duality transformation of SU(N) or U(N) vector-field models defined on a lattice. The dual degrees of freedom are certain species of the integer-valued fields complemented by the symmetric groups' \otimes_{n} S(n) variables. While the former parametrize relevant irreducible representations, the latter play the role of the Lagrange multipliers facilitating the fusion rules involved. As an application, I construct a novel solvable family of SU(N) D-matrix systems graded by the rank 1\leq{k}\leq{(D-1)} of the manifest [U(N)]^{\oplus k} conjugation-symmetry. Their large N solvability is due to a hidden invariance (explicit in the dual formulation) which allows for a mapping onto the recently proposed eigenvalue-models \cite{Dub1} with the largest k=D symmetry. Extending \cite{Dub1}, we reconstruct a D-dimensional gauge theory with the large N free energy given (modulo the volume factor) by the free energy of a given proposed 1\leq{k}\leq{(D-1)} D-matrix system. It is emphasized that the developed formalism provides with the basis for higher-dimensional generalizations of the Gross-Taylor stringy representation of strongly coupled 2d gauge theories.Comment: TeX, 46 page

On the coset duals of extended higher spin theories

We study the holographic duality between the M x M matrix extension of Vasiliev higher spin theories on AdS3 and the large N limit of SU(N+M)/SU(N) x U(1) type cosets. We present a simplified proof for the agreement of the spectra and clarify the relation between this duality and the version in which the cosets are replaced by Kazama-Suzuki models of Grassmannian type.Comment: 27 pages, 1 tabl