Noncommutative two-dimensional gauge theories

We elaborate on the dynamics of noncommutative two-dimensional gauge field theories. We consider U(N) gauge theories with fermions in either the fundamental or the adjoint representation. Noncommutativity leads to a rather non-trivial dependence on theta (the noncommutativity parameter) and to a rich dynamics. In particular the mass spectrum of the noncommutative U(1) theory with adjoint matter is similar to that of ordinary (commutative) two-dimensional large-NSU(N) gauge theory with adjoint matter. The noncommutative version of the ?t Hooft model receives a non-trivial contribution to the vacuum polarization starting from three-loops order. As a result the mass spectrum of the noncommutative theory is expected to be different from that of the commutative theory.     

Obtaining non-Abelian field theories via the Faddeev-Jackiw symplectic formalism

In this Letter we construct non-Abelian field theories employing the Faddeev-Jackiw symplectic formalism. The original Abelian fields were modified in order to introduce the non-Abelian algebra. We construct the SU(2) and SU(2)⊗U(1) Yang-Mills theories having as starting point the U(1) Maxwell electromagnetic theory.     

AdS superalgebras with brane charges

We consider the inclusion of brane charges in AdS₅ superalgebras that contain the maximal central extension of the super-Poincaré algebra on ∂AdS₅. For theories with N supersymmetries on the boundary, the maximal extension is OSp(1/8N,R), which contains the group Sp(8N,R)⊃U(2N,2N)⊃SU(2,2)×U(N) as extension of the conformal group. An “intermediate” extension to U(2N,2N/1) is also discussed, as well as the inclusion of brane charges in AdS₇ and AdS₄ superalgebras. BPS conditions in the presence of brane charges are studied in some details.     

Noncommutative linear sigma models

We examine noncommutative linear sigma models with U(N) global symmetry groups at the one-loop quantum level, and contrast the results with our previous study of the noncommutative O(N) linear sigma models where we have shown that Nambu–Goldstone symmetry realization is inconsistent with continuum renormalization. Specifically we find no violation of Goldstone's theorem at one-loop for the U(N) models with the quartic term ordering consistent with possible noncommutative gauging of the model. The difference is due to terms involving noncommutative commutator interactions, which vanish in the commutative limit. We also examine the U(2), and O(4) linear sigma models with matter in the adjoint representation, and find that the former is consistent with Goldstone's theorem at one-loop if we include only trace invariants consistent with possible noncommutative gauging of the model, while the latter exhibits violations of Goldstone's theorem of the kind seen in the fundamental of O(N) for N>2.     

A scaling limit with many noncommutativity parameters

We derive the worldsheet propagator for an open string with different magnetic fields at the two ends, and use it to compute two distinct noncommutativity parameters, one at each end of the string. The usual scaling limit that leads to noncommutative Yang–Mills can be generalized to a scaling limit in which both noncommutativity parameters enter. This corresponds to expanding a theory with U(N) Chan–Paton factors around a background U(1)^N gauge field with different magnetic fields in each U(1).     

Elliptic Hypergeometry of Supersymmetric Dualities II. Orthogonal Groups, Knots, and Vortices

We consider Seiberg electric-magnetic dualities for 4d ${\mathcal{N} = 1}$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N + 1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of knot theory, generalized AGT duality for (3 + 3)d theories, and a 2d vortex partition function are described.     

Projective spacetime

It is suggested that the world is locally projectively flat rather than Euclidean. From this postulate it is shown that an (N+1)-particle system has the global geometry of the symmetric spaceSO(4,N+1)/SO(4)×SO(N+1). A complex representation also exists, with structureSU(2,N+1)/S[U(2)×U(N+1)]. Several aspects of these geometrics are developed. Physical states are taken to be eigenfunctions of the Laplace-Beltrami operators. The theory may provide a rational basis for comprehending the groupsSO(4, 2),SU(2)×U(1),SU(3), etc., of current interest.     

Unitary group decomposition of Hamiltonian operators. I. Structure and sizes in the Hartree-Fock basis

The extent to which the Hartree-Fock (HF) procedure converts a two-body interaction into an effective one-body operator has been studied. For this purpose, the nuclear Hamiltonian is classified according to its tensor properties under the unitary group U(N) and its direct-sum subgroup U(m) + U(N ? m) generated by the Hartree-Fock procedure. Here N is the total number of HF single particle states of which m are occupied. The sizes of different tensor parts in m-particle spaces have been determined for both U(N) and U(m) + U(N ? m). In terms of these sizes a ratio is defined which provides a measure of the conversion efficiency of HF and a global measure of the goodness of HF s.p. basis. This ratio is evaluated for N = Z even-even nuclei in 0d-1s shell and 0f-1p shell using realistic Hamiltonian operators.     

Chern–Simons theory with vector fermion matter

We study three-dimensional conformal field theories described by U(N) Chern?CSimons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger?CDyson equation in light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ?² as a function of the ??t?Hooft coupling ??=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |??|=1; the conformal theory does not exist for |??|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three-point functions up to two loops. We also discuss a light-cone Hamiltonian formulation of this theory where a W _?? algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.     

N = ∞ phase transition in a class of exactly soluble model lattice gauge theories

An exactly soluble class of model U(N) lattice gauge theories is considered. The ground state is discussed as a separable N-fermion problem solved by mathieu functions. Some exact correlation functions are presented. The N = ∞ limit exhibits a third order phase transition demarcating the strong and weak phases at (g²N)^?1 ≈ 0.55.     

Holographic description of large N gauge theory

Based on the earlier work [S.-S. Lee, Nucl. Rev. B 832 (2010) 567], we derive a holographic dual for the D-dimensional U(N) lattice gauge theory from a first principle construction. The resulting theory is a lattice field theory of closed loops, dubbed as lattice loop field theory which is defined on a (D+1)-dimensional space. The lattice loop field theory is well defined non-perturbatively, and it becomes weakly coupled and local in the large N limit with a large ?t Hooft coupling.     

Massive hyper-Kähler sigma models and BPS domain walls

With the non-Abelian hyper-Kähler quotient by U(M) and SU(M) gauge groups, we give the massive hyper-Kähler sigma models that are not toric in the N=1 superfield formalism. The U(M) quotient gives N!/[M!(N-M)!] (N is the number of flavors) discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua. We derive a BPS domain-wall solution in the case of N = 2 and M = 1 in the U(M) quotient model.     

Large-N reduction in spin systems and Griffiths singularities

We apply the quenched Eguchi-Kawai reduction procedure to N-component spin models. We first recover the equivalence of the O(N) symmetric Heisenberg model with the spherical model at large N, and we extend it to the case where a quenched random external field is present. When the random field has a gaussian distribution, we show that Griffiths singularities disappear as N → ∞.     

Conservation laws in the quantum Hall Liouvillian theory and its generalizations

It is known that the localization length scaling of noninteracting electrons near the quantum Hall plateau transition can be described in a theory of the bosonic density operators, with no reference to the underlying fermions. The resulting “Liouvillian” theory has a U(1|1) global supersymmetry as well as a hierarchy of geometric conservation laws related to the noncommutative geometry of the lowest Landau level (LLL). Approximations to the Liouvillian theory contain quite different physics from standard approximations to the underlying fermionic theory. Mean-field and large-N generalizations of the Liouvillian are shown to describe problems of noninteracting bosons that enlarge the U(1|1) supersymmetry to U(1|1)×SO(N) or U(1|1)×SU(N).These noninteracting bosonic problems are studied numerically for 2⩽N⩽8 by Monte Carlo simulation and compared to the original N=1 Liouvillian theory. The N>1 generalizations preserve the first two of the hierarchy of geometric conservation laws, leading to logarithmic corrections at order 1/N to the diffusive large-N limit, but do not preserve the remaining conservation laws. The emergence of nontrivial scaling at the plateau transition, in the Liouvillian approach, is shown to depend sensitively on the unusual geometry of Landau levels.     

On the low-energy effective action of N = 2 supersymmetric Yang-Mills theory

We investigate the perturbative part of Seiberg's low-energy effective action of N = 2 supersymmetric Yang-Mills theory in Wess-Zumino gauge in the conventional effective field theory technique. Using the method of constant field approximation and restricting the effective action with at most two derivatives and not more than four-fermion couplings, we show some features of the low-energy effective action given by Seiberg based on U(1)_R anomaly and non-perturbative β-function arguments.     

Guided random walks for solving Hamiltonian lattice gauge theories

Motivated by developments for many-particle quantum systems, a Monte Carlo method for solving Hamiltonian lattice gauge theories without fermions is presented in which a stochastic random walk is guided by a trial wave function. To the extent that a substantial portion of the local structure of the theory can be incorporated in the trial function, the method offers significant advantages relative to existing techniques. The method is applicable to the study of SU(N) lattice gauge theories, and its utility is demonstrated by solving the compact U(1) gauge theory in three spatial dimensions.     

Large N classical equations and their quantum significance

We show that the large N limits of a wide variety of vector models may be obtained by studying the classical equations of motion. In particular, we derive a constraint which allows us to choose solutions of the classical field equations which directly give the correlation functions of N → ∞ quantum system. Models studied here include quantum mechanics on a sphere, two-dimensional linear and nonlinear O(N) field theories and the CP^N model.     

Duality and quantum-algebra symmetry of the AN−1(1) open spin chain with diagonal boundary fields

We show that the transfer matrix of the A_N−1⁽¹⁾ open spin chair with diagonal boundary fields has the symmetry U_q(SU(l)) × U_q(SU(N−l)) × U(1), as well as a “duality” symmetry which maps l ↔ N − l. We exploit these symmetries to compute exact boundary S-matrices in the regime with q real.     

SO(N) supergravity and unification of all fundamental forces

From the group theoretical arguments, we find that among allSO(N) supergravitiesN=10 is the minimal supersymmetry group which unifies all fundamental forces of weak, electromagnetic, strong and gravitational interactions. The (super)symmetry is broken throughSO(10)→SU(3)?SU(2)?U(1)→SU(3)?U(1). All observed particles of the low energy physics (three generations of quarks and leptons, γ,Z, W ^± and gluons) and graviton can be minimally accomodated with the correctSU(3)?SU(2)?U(1) quantum numbers. Some characteristic predictions, which can be checked in the coming high energy experiments, are briefly discussed.     

Non-Fermi liquid theory of quantum Hall effects

Within the Grassmannian U(2N)/U(N) × U(N) nonlinear σ-model representation of localization, one can study the low-energy dynamics of both a free and interacting electron gas. We study the crossover between these two fundamentally different physical problems. We show how the topological arguments for the exact quantization of the Hall conductance are extended to include the Coulomb interaction problem. We discuss dynamical scaling and make contact with the theory of variable range hopping.