Landau-Ginzburg Superpotentials Research Articles

Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau–Ginzburg superpotentials

For the root systems of type Bl,Cl and Dl, we generalize the result of [7] by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in [7]. It also depends on certain additional data. We also construct Landau–Ginzburg superpotentials for these Frobenius manifold structures.

A new approach for computing the geometry of the moduli spaces for a Calabi–Yau manifold

It is known that moduli spaces of Calabi–Yau (CY) manifolds are special Kähler manifolds. This structure determines the corresponding low-energy effective theory which arises in superstring compactifications on CY manifolds. In the case where a CY manifold is given as a hypersurface in the weighted projective space, we propose a new procedure for computing the Kähler potential of the moduli space. Our method is based on the fact that the moduli space of CY manifolds is a marginal subspace of the Frobenius manifold which arises on the deformation space of the corresponding Landau–Ginzburg superpotential.

Seidel’s morphism of toric $4$-manifolds

Following McDuff and Tolman's work on toric manifolds [McDT06], we focus on 4-dimensional NEF toric manifolds and we show that even though Seidel's elements consist of infinitely many contributions, they can be expressed by closed formulas. From these formulas, we then deduce the expression of the quantum homology ring of these manifolds as well as their Landau-Ginzburg superpotential. We also give explicit formulas for the Seidel elements in some non-NEF cases. These results are closely related to recent work by Fukaya, Oh, Ohta, and Ono [FOOO11], Gonz\'alez and Iritani [GI11], and Chan, Lau, Leung, and Tseng [CLLT12]. The main difference is that in the 4-dimensional case the methods we use are more elementary: they do not rely on open Gromov-Witten invariants nor mirror maps. We only use the definition of Seidel's elements and specific closed Gromov-Witten invariants which we compute via localization. So, unlike Alice, the computations contained in this paper are not particularly pretty but they do stay on their side of the mirror. This makes the resulting formulas directly readable from the moment polytope.

Doran-Harder-Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves

We prove the Doran-Harder-Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi-Yau manifold $X$ degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi-Yau manifold of $X$ can be constructed by gluing the two mirror Landau-Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau-Ginzburg superpotentials.

Lagrangian Floer theory on compact toric manifolds, I

The present authors introduced the notion of \emph{weakly unobstructed} Lagrangian submanifolds and constructed their \emph{potential function} $\mathfrak{PO}$ purely in terms of $A$-model data in [FOOO2]. In this paper, we carry out explicit calculations involving $\mathfrak{PO}$ on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential. Combining this study with the results from [FOOO2], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to Entov-Polterovich's symplectic quasi-states.

Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds

In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2\pi} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value $T^{2\pi} = e^{-1}$) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology. We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the {\it adapted Floer cohomology}. We then prove that the adapted Floer cohomology of any non-singular torus fiber of Fano toric manifolds is well-defined, invariant under the Hamiltonian isotopy and isomorphic to the Bott-Morse Floer cohomology of the fiber.

Formula omitted] RG flows, product groups, and a-maximization

We explore new IR phenomena and dualities, arising for product groups, in the context of N = 1 supersymmetric gauge theories. The RG running of the multiple couplings can radically affect each other. For example, an otherwise IR interacting coupling can be driven to be instead IR free by an arbitrarily small, but nonzero, initial value of another coupling. Or an otherwise IR free coupling can be driven to be instead IR interacting by an arbitrarily small non-zero initial value of another coupling. We explore these and other phenomena in N = 1 examples, where exact results can be obtained using a-maximization. We also explore the various possible dual gauge theories, e.g., by dualizing one gauge group with the other treated as a weakly gauged flavor symmetry, along with previously proposed duals for the theories deformed by A k -type Landau–Ginzburg superpotentials. We note that this latter duality, and all similar duality examples, always have nonempty superconformal windows within which both the electric and dual A k superpotentials are relevant.

D-branes from matrix factorizations

B-type D-branes can be obtained from matrix factorizations of the Landau–Ginzburg superpotential. We here review this promising approach to learning about the spacetime superpotential of Calabi–Yau compactifications. We discuss the grading of the D-branes, and present applications in two examples: the two-dimensional torus, and the quintic. To cite this article: K. Hori, J. Walcher, C. R. Physique 5 (2004).

Two-dimensional string theory, topological field theories and the deformed matrix model

In this paper the c = 1 string theory is studied from the point of view of topological field theories. A change in the prescription is proposed, which reproduces the results of the 1/ x 2 deformed matrix model. It is proposed that the deformed matrix model is related to a D-series Landau-Ginzburg superpotential.

On the integrability of N = 2 supersymmetric massive theories

In this paper we propose a criterion to establish the integrability of N = 2 supersymmetric massive theories. The basic data required are the vacua and the spectrum of Bogomolnyi solitons, which can be neatly encoded in a graph (nodes=vacua and links=Bogomolnyi solitons). Integrability is then equivalent to the existence of solutions of a generalized Yang-Baxter equation which is built up from the graph (graph-Yang-Baxter equation). We solve this equation for two general types of graphs: circular and daisy, proving, in particular, the integrability of the following Landau-Ginzburg superpotentials: A n ( t 1), A n ( t 2), D n ( τ), E 6( t 7) and E 8( t 16). For circular graphs the solution are intertwiners of the affine Hopf algebra U q( A 1 (1)) , while for daisy graphs the solution corresponds to a SUSY generalization of the Boltzmann weights of the chiral Potts model in the trigonometric regime. A chiral-Potts-like solution is conjectured for the more tricky case D n ( t 2). The scattering theory of circular models, for instance A n ( t 1) or D n ( τ), is Toda-like. The physical spectrum of daisy models, as A n ( t 2), E 6( t 7) or E 8( t 16), is given by confined states of radial solitons. The scattering theory of the confined states is again Toda-like. Bootstrap factors for the confined solitons are given by fusing the SUSY chiral Potts S-matrices of the elementary constituents; i.e the radial solitons of the daisy graph.

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.

On the complete solution of a perturbed topological conformal field theory at c=3

The problem of determination of the flat coordinates for a model of a topological conformal field theory corresponding to the 'twisted' version of an N=2 superconformal Landau-Ginzburg field theory with central charge c=3 is analysed. The model is characterized by a Landau-Ginzburg superpotential of the form: W= 1/4 x4+ 1/4 y4. All possible relevant and marginal perturbations with their corresponding couplings (expressed as functions of the dimensionless flat-coordinate) are added to the above superpotential to give the perturbed topological field theory model. It is seen that the couplings can be completely determined (and hence also the dependence of the perturbations on the flat coordinate) by imposing the conditions of flatness on the space of couplings of the perturbed theory.

String compactifications and N=2 superconformal coset constructions

Abstract We consider the class of string compactifications obtained through the tensoring of the N =2 superconformal theories recently constructed by Kazama and Suzuki. A classification of all c =9 and c =6 models obtained from N =2 theories based on hermitian symmetric spaces is given. The number of E 6 generations and antigenerations for c =9 models based on SU( m +1)/SU( m )×U(1) is also provided. A number of these models correspond to deformations of others previously obtained from tensoring of minimal N =2 theories as can be deduced from the corresponding Landau-Ginzburg superpotential. Many others are genuine new compactifications. It is also shown how more general modular invariant projections lead to new models both in the minimal and Kazama-Suzuki constructions. As an example, we obtain the standard Z 3 -orbifold as a particular projection of the ( k =1) 9 Gepner model.