Topological Landau-Ginzburg Theory Research Articles

On B-type Open–Closed Landau–Ginzburg Theories Defined on Calabi–Yau Stein Manifolds

We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair $(X,W)$, where $X$ is a non-compact Calabi-Yau manifold and $W$ has compact critical set. When $X$ is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to $X$. In particular, we show that the D-brane category is described by projective matrix factorizations defined over the ring of holomorphic functions of $X$. We also discuss simplifications of the analytic models which arise when $X$ is holomorphically parallelizable and illustrate these analytic models in a few classes of examples.

Differential Models for B-Type Open–Closed Topological Landau–Ginzburg Theories

We propose a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is any non-compact Calabi-Yau manifold and $W$ is any holomorphic complex-valued function defined on $X$ whose critical set is compact. The models are constructed at cochain level using smooth data, including the twisted Dolbeault algebra of polyvector valued forms and a twisted Dolbeault category of holomorphic factorizations of $W$. We give explicit proposals for cochain level versions of the bulk and boundary traces and for the bulk-boundary and boundary-bulk maps of the Landau-Ginzburg theory. We prove that most of the axioms of an open-closed topological field theory are satisfied on cohomology and conjecture that the remaining axioms are also satisfied.

Charge and spin fractionalization in strongly correlated topological insulators

We construct an effective topological Landau–Ginzburg theory that describes general SU(2) incompressible quantum liquids of strongly correlated particles in two spatial dimensions. This theory characterizes the fractionalization of quasiparticle quantum numbers and statistics in relation to the topological ground-state symmetries, and generalizes the Chern–Simons, BF (‘background field’) and hierarchical effective gauge theories to an arbitrary representation of the SU(2) symmetry group. We mainly focus on fractional topological insulators with time-reversal symmetry, which are treated as SU(2) generalizations of the quantum Hall effect.

Landau-Ginzburg realization of open string TFT

We investigate B-type topological Landau-Ginzburg theory with one variable, with D2-brane boundary conditions. We find that the allowed brane configurations are determined in terms of the possible factorizations of the superpotential, and compute the corresponding open string chiral rings. These are characterized by bosonic and fermionic generators that satisfy certain relations. Moreover we show that the disk correlators, being continuous functions of deformation parameters, satisfy the topological sewing constraints, thereby proving consistency of the theory. In addition we show that the open string LG model is, in its content, equivalent to a certain triangulated category introduced by Kontsevich, and thus may be viewed as a concrete physical realization of it.

ADIABATIC EFFECTIVE ACTION FOR VORTICES IN NEUTRAL AND CHARGED SUPERFLUIDS

Adiabatic effective action for vortices in neutral and charged superfluids at zero temperature are calculated using the topological Landau-Ginzburg theory recently proposed by Hatsuda, Yahikozawa, Ao and Thouless, and vortex dynamics are examined. The Berry phase term arising in the effective action naturally yields the Magnus force in both neutral and charged superfluids. It is shown that in neutral superfluid there is only one degree of freedom, namely the center of vorticities, and the vortex energy is proportional to the sum of all vorticities so that it is finite only for the vanishing total vorticity of the system. On the other hand the effective mass and the vortex energy for a vortex in charged superfluids are defined individually as expected. The effects of the vortex core on these quantities are also estimated. The possible depinning scenario which is governed by the Magnus force and the inertial mass is also discussed.

ON CONNECTION BETWEEN TOPOLOGICAL LANDAU-GINZBURG GRAVITY AND INTEGRABLE SYSTEMS

We study flows on the space of topological Landau-Ginzburg theories coupled to topological gravity. We argue that flows corresponding to gravitational descendants change the target space from a complex plane to a punctured complex plane and lead to the motion of punctures. It is shown that the evolution of the topological theory due to these flows is given by the dispersionless limit of KP hierarchy. We argue that the generating function of correlators in such theories is equal to the logarithm of the tau function of the generalized Kontsevich model.

Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy

Based on the dispersionless KP (dKP) theory, we give a comprehensive study of the topological Landau-Ginzburg (LG) theory characterized by a rational potential. Writing the dKP hierarchy in a general form, we find that the hierarchy naturally includes the dispersionless (continuous) limit of Toda hierarchy and its generalizations having finite number of primaries. Several flat solutions of the topological LG theory are obtained in this formulation, and are identified with those discussed by Dubrovin. We explicitly construct gravitational descendants for all the primary fields. Giving a residue formula for the 3-point functions of the fields, we show that these 3-point functions satisfy the topological recursion relation. The string equation is obtained as the generalized hodograph solutions of the dKP hierarchy, which show that all the gravitational effects to the constitutive equations (2-point functions) can be renormalized into the coupling constants in the small phase space.

Integrable N = 2 Landau-Ginzburg theories from quotients of fusion rings

The discovery of integrable N = 2 supersymmetric Landau-Ginzburg theories whose chiral rings are fusion rings suggests a close connection between fusion rings, the related Landau-Ginzburg superpotentials, and N = 2 quantum integrability. We examine this connection by finding the natural SO( N) K analogue of the construction that produced the superpotentials with Sp( N) K and SU( N) K fusion rings as chiral rings. The chiral rings of the new superpotentials are not directly the fusion rings of any conformal field theory, although they are natural quotients of the tensor subring of the SO( N) K fusion ring. The new superpotentials yield solvable (twisted N = 2) topological field theories. We obtain the integer-valued correlation functions as sums of SO( N) K Verlinde dimensions by expressing the correlators as fusion residues. The SO(2 n + 1) 2 k + 1 and SO(2 k + 1) 2 n + 1 related topological Landau-Ginzburg theories are isomorphic, despite being defined via quite different superpotentials.

TOPOLOGICAL FIELD THEORIES AND THE PERIOD INTEGRALS

We discuss topological Landau-Ginzburg theories coupled to two-dimensional topological gravity. We point out that the basic recursion relations for correlation functions of the two-dimensional gravity have exactly the same form as the Gauss-Manin differential equations for the period integrals of superpotentials. Thus the one-point functions on the sphere of the Landau-Ginzburg theories are given exactly by the period integrals. We discuss various examples, A-D-E minimal models and the c=3 topological theories.

TOPOLOGICAL LANDAU-GINZBURG MATTER FROM SP(N)K FUSION RINGS

We find and analyze the Landau-Ginzburg potentials whose critical points determine chiral rings which are exactly the fusion rings of SP (N)K WZW models. The quasihomogeneous part of the potential associated with SP (N)K is the same as the quasihomogeneous part of that associated with SU (N+1)K, showing that these potentials are different perturbations of the same Grassmannian potential. Twisted N=2 topological Landau-Ginzburg theories are derived from these superpotentials. The correlation functions, which are just the SP (N)K Verlinde dimensions, are expressed as fusion residues. We note that the SP (N)K and SP (K)N topological Landau-Ginzburg theories are identical, and that while the SU (N)K and SU (K)N topological Landau-Ginzburg models are not, they are simply related.

Topological phase of N = 2 superconformal field theory and topological Landau-Ginzburg field theory

We construct a topological invariant version of two-dimensional Landau-Ginzburg field theory which turns out to describe the topological phase of N = 2 superconformal field theory. The theory provides us with an alternative approach to two-dimensional topological field theory.