Witten-Dijkgraaf-Verlinde-Verlinde Research Articles

Dubrovin–Frobenius manifolds associated with Bn and the constrained KP hierarchy

In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group Bn constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy.

Conjugate Frobenius Manifold and Inversion Symmetry

We give a conjugacy relation on certain type of Frobenius manifold structures using the theory of flat pencils of metrics. It leads to a geometric interpretation for the inversion symmetry of solutions to Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations.

WDVV equations: symbolic computations of Hamiltonian operators

We describe software for symbolic computations that we developed in order to find Hamiltonian operators for Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, and verify their compatibility. The computation involves nonlocal (integro-differential) operators, for which specific canonical forms and algorithms have been used.

WDVV-type relations for Welschinger invariants: Applications

We first recall Solomon’s relations for Welschinger invariants counting real curves in real symplectic fourfolds and the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-style relations for Welschinger invariants counting real curves in real symplectic sixfolds with some symmetry. We then explicitly demonstrate that, in some important cases (projective spaces with standard conjugations, real blowups of the projective plane, and two- and threefold products of the one-dimensional projective space with two involutions each), these relations provide complete recursions determining all Welschinger invariants from basic input. We include extensive tables of Welschinger invariants in low degrees obtained from these recursions with Mathematica. These invariants provide lower bounds for counts of real rational curves, including with curve insertions in smooth algebraic threefolds.

Explicit examples of Hurwitz Frobenius manifolds in genus one

We provide three new solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations by computing examples of prepotentials of Hurwitz Frobenius manifolds, their deformations, and real doubles. By applying the general formalism to two families of elliptic curves, we obtain solutions to the WDVV system depending on four and eight variables. We also correct mistakes in previously known solutions in dimensions four and six associated with the same families of curves.

Soliton surface associated with the equation of associativity for n=3 case with an metric

The Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, also called the associativity equations, is a system of nonlinear partial differential equations for one function, depending on a finite number of variables. The WDVV equations were introduced a few decades ago in the context of two-dimensional topological field theories. The task of giving the associativity equations a geometric interpretation has two complementary aspects. On one side, can write these equations in a form that does not depend on the choice of the coordinates. On the other side, one must demand that the geometrical structure should be capable to select a class of affinely related coordinates. The coordinate selection rule is important in the geometrization of the associativity equations. In this paper, we consider the soliton surface of the associativity equation. The equation of associativity originated from 2D topological field theory. 2D topological field theory represent the matter sector of topological string theory. These theories covariant before coupling to gravity due to the presence of a nilpotent symmetry and are therefore often referred to as cohomological field theories. The surface is constructed using Sym-Tafel formula, which is a connection between classical manifold geometry and soliton theory. The Sym-Tafel formula reconstructs a surface from knowledge of its fundamental forms, combines integrable nonlinearities, and allows the application of soliton theory methods to geometric problems. The soliton surfaces approach is necessary in the construction of so-called integrable geometries. Any class of soliton surfaces is integrable. Geometric objects associated with the surfaces of the solitons can usually be identified with the solutions to the strings. Thus in this work soliton surfaces for the associativity equation for n=3 case with an metric h11 not equal to 0 are considered, and first and second fundamental forms of soliton surfaces are found for this case. In addition, we study an area of surfaces for the associativity equation for  case n=3 with an metric h11 not equal to 0.

Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and N=4 mechanics

We propose a generalization of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation from ${\mathbb R}^n$ to an arbitrary Riemannian manifold. Its form is obtained by extending the relation of the WDVV equation with ${\cal N}{=}\,4$ supersymmetric $n$-dimensional mechanics from flat to curved space. The resulting `curved WDVV equation' is written in terms of a third-rank Codazzi tensor. For every flat-space WDVV solution subject to a simple constraint we provide a curved-space solution on any isotropic space, in terms of the rotationally invariant conformal factor of the metric.

On deformation and classification of ∨-systems

The ∨-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of ∨-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations of ∨-systems and use them to study the classification problem for ∨-systems in dimension three. We discuss also possible matroidal structures of ∨-systems in relation with projective geometry and give the catalogue of all known irreducible rank three ∨-systems.

Frobenius Structures on Double Hurwitz Spaces

We present a construction Frobenius structures of “dual type” on the moduli space of ramified coverings of with given ramification type over two points, generalizing Dubrovin's construction of [4, 5]. A complete hierarchy of hydrodynamic type is obtained from the corresponding deformed flat connection. This provides a suitable framework for the theory of weakly deformed soliton lattices of an enlarged class of integrable equations; as applications we consider the q-deformed Gelfand–Dickey hierarchy and the sine-Gordon equation, and explicitly compute the corresponding solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations.

The structure of the tautological ring in genus one

We prove Getzler’s claims about the cohomology of the moduli space of stable curves of genus one, that is, that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) relation and Getzler’s relation. In particular, the even cohomology ring is isomorphic to the tautological ring.

Enumerative and asymptotic analysis of a moduli space

We study exact and asymptotic enumerative aspects of the Hilbert series of the cohomology ring of the moduli space of stable pointed curves of genus zero. This manifold is related to the WDVV (Witten–Dijkgraaf–Verlinde–Verlinde) equations of string theory.

N = 4 mechanics, WDVV equations and polytopes

N = 4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten—Dijkgraaf—Verlinde—Verlinde (WDVV) equation for F. The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide An and B3 examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.

The WDVV symmetries in two-primary models

From the bi-Hamiltonian standpoint, we investigate symmetries of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations proposed by Dubrovin. These symmetries can be viewed as canonical Miura transformations between genus-zero bi-Hamiltonian systems of hydrodynamic type. In particular, we show that the moduli space of two-primary models under symmetries of the WDVV equations can be parameterized by the polytropic exponent h. We discuss the transformation properties of the free energy at the genus-one level.

A new class of solutions to the WDVV equation

The known prepotential solutions F to the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation are parametrized by a set { α } of covectors. This set may be taken to be indecomposable, since F { α } ⊕ ˙ { β } = F { α } + F { β } . We couple mutually orthogonal covector sets by adding so-called radial terms to the standard form of F. The resulting reducible covector set yields a new type of irreducible solution to the WDVV equation.

N=4 mechanics, WDVV equations and roots

N=4 superconformal multi-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial differential equations linear in U and generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. Putting U=0 yields a class of models (with zero central charge) which are encoded by the finite Coxeter root systems. We extend these WDVV solutions F in two ways: the A_n system is deformed n-parametrically to the edge set of a general orthocentric n-simplex, and the BCF-type systems form one-parameter families. A classification strategy is proposed. A nonzero central charge requires turning on U in a given F background, which we show is outside of reach of the standard root-system ansatz for indecomposable systems of more than three particles. In the three-body case, however, this ansatz can be generalized to establish a series of nontrivial models based on the dihedral groups I_2(p), which are permutation symmetric if 3 divides p. We explicitly present their full prepotentials.

The Pre-WDVV Ring of Physics and its Topology

We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex $\Delta_n$ is homotopy equivalent to a wedge of $(n-2)!$ spheres of dimension $n-4$. We also verify the Cohen-Macaulay property. Additionally, recurrences are given for the face enumeration of the complex and the Hilbert series of the associated pre-WDVV ring.

The Non-Perturbative = 2 SUSY Yang-Mills Theory from Semiclassical Absorption of Supergravity by Wrapped D Branes

The imaginary part of the two point functions of the superconformal anomalous currents are extracted from the cross-sections of semiclassical absorption of dilaton, RR-2 form and gravitino by the wrapped D5 branes. From the central terms of the two point functions anomalous Ward identity is established which relates the exact pre-potential of the ${\cal N}=2$ SUSY Yang-Mills theory with the vacuum expectation value of the anomaly multiplet. From the Ward identity, WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation can be derived which is solved for the exact pre-potential.

Integrable Structure Behind the WDVV Equations

An integrable structure behind the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations is identified with the reduction of the Riemann–Hilbert problem for the homogeneous loop group \(\widehat{GL}(N,\mathbb{C})\). The reduction requires the dressing matrices to be fixed points of an order-two loop group automorphism resulting in a subhierarchy of the \(\widehat{gL}(N,\mathbb{C})\) hierarchy containing only odd-symmetry flows. The model has Virasoro symmetry; imposing Virasoro constraints ensures the homogeneity property of the Darboux–Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations.

On associativity equations

We consider the associativity or Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations and discuss their solution class based on the existence of the residue formulas, which is most relevant for nonperturbative physics. We demonstrate that for this case, proving the associativity equations reduces to solving a system of linear algebraic equations. Particular examples of solutions related to Landau–Ginzburg topological theories, Seiberg–Witten theories, and the tau functions of semiclassical hierarchies are discussed in detail. We also discuss related questions including the covariance of associativity equations, their relation to dispersionless Hirota relations, and the auxiliary linear problem for the WDVV equations.

Generalized WDVV equations for F4 pure N=2 super-Yang–Mills theory

An associative algebra of holomorphic differential forms is constructed associated with pure N=2 super-Yang–Mills theory for the Lie algebra F 4. Existence and associativity of this algebra, combined with the general arguments in the work of Marshakov, Mironov and Morozov, proves that the prepotential of this theory satisfies the generalized WDVV (Witten–Dijkgraaf–Verlinde–Verlinde) system.