Effective Field Theory Description of the Higher Dimensional Quantum Hall Liquid

We derive an effective topological field theory model of the four dimensional quantum Hall liquid state recently constructed by Zhang and Hu. Using a generalization of the flux attachment transformation, the effective field theory can be formulated as a U(1) Chern–Simons theory over the total configuration space CP₃, or as a SU(2) Chern–Simons theory over S⁴. The new quantum Hall liquid supports various types of topological excitations, including the 0-brane (particles), the 2-brane (membranes), and the 4-brane. There is a topological phase interaction among the membranes which generalizes the concept of fractional statistics.     

Large N 2D Yang-Mills Theory and Topological String Theory

We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A= 0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's “Ω^-1 points ”. We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach. Received: 3 March 1994 / Accepted: 2 February 1995     

Mutual Chern-Simons theory and its applications in condensed matter physics

In this paper, the mutual Chern-Simons (MCS) theory is introduced as a new kind of topological gauge theory in 2+1 dimensions. We use the MCS theory in gapped phase as an effective low energy theory to describe the Z ₂ topological order of the Kitaev-Wen model. Our results show that the MCS theory can catch the key properties for the Z ₂ topological order. On the other hand, we use the MCS theory as an effective model to deal with the doped Mott insulator. Based on the phase string theory, the t-J model reduces to a MCS theory for spinons and holons. The related physics in high T _c cuprates is discussed.      

Non-Equilibrium Thermodynamics and Topology of Currents

In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged quantities which describe such stochastic maps, in particular time-averaged currents and density functionals. We derive the large deviations principle for these quantities, as well as for global topological currents, and formulate variational, thermodynamic relations to establish large deviation properties of the topological currents. We illustrate the theory with a nontrivial example of a Heisenberg spin-chain with a topological driving of the Wess-Zumino type. The Cramér functional of the topological current is found explicitly in the instanton gas regime for the spin-chain model in the weak-noise limit. In the context of the Morse theory, we discuss a general reduction of continuous stochastic models with weak noise to effective Markov chains describing transitions between stable fixed points.     

Electrodynamics on matrix space: non-Abelian by coordinates

Faddeev and Niemi have proposed a decomposition of SU(N) Yang–Mills theory in terms of new variables, appropriate for describing the theory in the infrared limit. We extend this method to SO(2N) Yang–Mills theory. We find that the SO(2N) connection decomposes according to irreducible representations of SO(N). The low-energy limit of the decomposed theory is expected to describe soliton-like configurations with nontrivial topological numbers. How the method of decomposition generalizes for SO(2N+1) Yang–Mills theory is also discussed. Received: 22 November 2000 / Published online: 8 June 2001     

The Donaldson–Witten Function for Gauge Groups of Rank Larger Than One

We study correlation functions in topologically twisted , d=4 supersymmetric Yang–Mills theory for gauge groups of rank larger than one on compact four-manifolds X. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of b _2,+(X)>0 in terms of Seiberg–Witten invariants and the classical cohomology ring of X. For manifolds X of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over =1 vacua. We describe two applications of our expressions, one to superconformal field theory and one to large N expansions of SU(N) , d=4 supersymmetric Yang–Mills theory. Received: 30 March 1998 / Accepted: 17 April 1998     

Generalized two-mode harmonic oscillator model: squeezed number state solutions and nonadiabatic Berry’s phase

A generalized two-mode harmonic oscillator model is investigated within the framework of its general dynamical algebra so(3,2). Two types of eigenstates, formulated as extended su(1,1), su(2) squeezed number states are found respectively. The nonadiabatic Berrys phase for this system with the cranked time-dependent Hamiltonian is also given.Received: 16 January 2004, Published online: 10 August 2004PACS: 42.50.Dv Nonclassical states of the electromagnetic field, including entangled photon states; quantum state engineering and measurements - 03.65.Fd Algebraic methods - 03.65.Vf Phases: geometric; dynamic or topological     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Topological Geometrodynamics

The identification of spacetime as a 4-surface in the space H =M⁴×CP₂ (product of Minkowski space and complex projective space of complex dimension two) as means of obtaining Poincare invariant theory of gravitation was the triggering idea of topological geometrodynamics (TGD), which can be regarded as an attempt to unify basic interactions in terms of submanifold geometry instead of abstract manifold geometry as in case of General Relativity. One can however regard TGD also as a generalization of string model: instead of strings free particles are regarded as 3-surfaces. In this article I want to describe these two approaches and to show how they merge into a single coherent scheme provided macroscopic 3-space with matter is identified as a 3-surface containing particles as topological inhomogenities. Also the quantization program of TGD based on the idea that interacting field theory can be regarded as a classical, free field theory for Grassmann algebra valued Schrödinger amplitude in the space of all possible 3-surfaces of H, is described.     

Abelian model of gravitating magnetic monopole

A self-consistentU(1)-gauge model in gravitational field is investigated. The exact solutions of two types of corresponding field equations are obtained. These solutions can be interpreted as magnetic monopoles. The first solution is regular forr 0 and provides an everywhere regular geometry, the second one has a physical singularity. In order to guarantee the stability of the monopoles we introduce the notion of a gravitational topological charge using de Rham's cohomology theory. This topological charge describes the sizes and the inner structure of the monopole.     

On the multi trace superpotential and corresponding matrix model

We study supersymmetric U(N) gauge theory coupled to an adjoint scalar superfield with a cubic superpotential containing a multi trace term. We show that the field theory results can be reproduced from a matrix model whose potential is given in terms of a linearized potential obtained from the gauge theory superpotential by adding some auxiliary non-dynamical field. Once we get the effective action from this matrix model we could integrate out the auxiliary field getting the correct field theory results.Received: 2 May 2003, Published online: 18 December 2003     

SU(2)×SU(2) Electroweak Theory I: The B3 Field on the Physical Vacuum

Nonlinear optics confronts the U(1) theory of electrodynamics with the dilemma of the existence of nonlinear fields. The U(1) group is completely linear and Abelian and causes consideration of an SU(2) theory of electrodynamics. An SU(2) theory of electrodynamics, with a B ³ magnetic field, means that physics is forced to consider an SU(2) × SU(2) electroweak theory. It is then demonstrated that the B ³ field exists on the physical vacuum defined by the Higgs symmetry breaking of this extended electroweak theory.     

Generalized hypergeometric functions on the torus and the adjoint representation ofU q (sl 2)

We study the homology groups with coefficient in local systems arising in the free field representation of minimal models of conformal field theory on an elliptic curve with punctures. We define an action of the quantum enveloping algebraU _q (sl ₂) on a space of relative cycles, extending results obtained previously for the sphere. Absolute cycles are identified with singular vectors. In the case of one puncture, we find that the resulting topological representation is essentially the adjoint representation.     

Chern-Simons theory,coloured-oriented braids and link invariants

A method to obtain explicit and complete topological solution of SU(2) Chern-Simons theory onS ³ is developed. To this effect the necessary aspects of the theory of coloured-oriented braids and duality properties of conformal blocks for the correlators ofSU(2) _k Wess-Zumino conformal field theory are presented. A large class of representations of the generators of the groupoid of coloured-oriented braids are obtained. These provide a whole lot of new link invariants of which Jones polynomials are the simplest examples. These new invariants are explicity calculated as illustrations for knots up to eight crossings and twocomponent multicoloured links up to seven crossings.     

Two-dimensional topological gravity and equivariant cohomology

The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the derived functor of semi-relative cohomology, just as equivariant cohomology is the derived functor of basic cohomology. That homological algebra finds a place in the study of topological string theory should not surprise the reader, granted that topological string theory is the conformal field theorist's algebraic topology.In [7], we have shown that the cohomology of a topological conformal field theory carries the structure of a batalin-Vilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations {a ₁, ...,a _k },k2, satisfying quadratic relations generalizing the Jacobi rule. (The operad underlying the category of gravity algebras has been studied independently by Ginzburg-Kapranov [9].)The author is grateful to M. Bershadsky, E. Frenkel, M. Kapranov, G. Moore, R. Plesser and G. Zuckerman for the many ways in which they helped in the writing of this paper; also to the Department of Mathematics at Yale University for its hospitality while part of this paper was written.The author is partially supported by a fellowship of the Sloan Foundation and a research grant of the NSF.     

A geometric generalization of field theory to manifolds of arbitrary dimension

We introduce a generalization of the O(N) field theory to N-colored membranes of arbitrary inner dimension D. The O(N) model is obtained for , while leads to self-avoiding tethered membranes (as the O(N) model reduces to self-avoiding polymers). The model is studied perturbatively by a 1-loop renormalization group analysis, and exactly as .Freedom to choose the expansion point D, leads to precise estimates of critical exponents of the O(N) model. Insights gained from this generalization include a conjecture on the nature of droplets dominating the 3d-Ising model at criticality; and the fixed point governing the random bond Ising model. Received: 15 October 1998 / Accepted: 4 November 1998     

The Chiral Space of Local Operators in <Emphasis Type="Italic">SU</Emphasis>(2)-Invariant Thirring Model

The space of local operators in the SU(2) invariant Thirring model (SU(2) ITM) is studied by the form factor bootstrap method. By constructing sets of form factors explicitly we define a susbspace of operators which has the same character as the level one integrable highest weight representation of . This makes a correspondence between this subspace and the chiral space of local operators in the underlying conformal field theory, the su(2) Wess-Zumino-Witten model at level one.     

Khovanov-Rozansky Homology and Topological Strings

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks Mathematics Subject Classifications (2000): 57M25, 57M27, 18G60, 18E30, 14J32, 14N35, 81T30, 81T45 Dedicated to the memory of F.A. Berezin