Polynomial modular Frobenius manifolds

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class–so-called modular Frobenius manifolds–lie at the fixed points of this symmetry. In this paper a classification of semi-simple modular Frobenius manifolds which are polynomial in all but one of the variables is begun, and completed for three and four dimensional manifolds. The resulting examples may also be obtained from higher dimensional manifolds by a process of folding. The relationship of these results with orbifold quantum cohomology is also discussed.     

Flat 3-webs via semi-simple Frobenius 3-manifolds

We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are hexagonal and admit at least one infinitesimal symmetry at each singular point. Singularities of the web are also discussed.     

A Remark on Deformations of Hurwitz Frobenius Manifolds

In this note, we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux–Egoroff system. As an application, we explain how Shramchenko’s deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental–van de Leur twisted loop group action on the space of semi-simple Frobenius manifolds.     

Towards Lax Formulation of Integrable Hierarchies of Topological Type

To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in Buryak et al. (J Differ Geom 92:153–185, 2012), Buryak et al. (J Geom Phys 62:1639–1651, 2012) to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action.     

Geometry of Skyrmions

A Skyrmion may be regarded as a topologically non-trivial map from one Riemannian manifold to another, minimizing a particular energy functional. We discuss the geometrical interpretation of this energy functional and give examples of Skyrmions on various manifolds. We show how the existence of conformal transformations can cause a Skyrmion on a 3-sphere to become unstable, and how this may be related to chiral symmetry breaking.     

Proof of a conjecture on the genus two free energy associated to the An singularity

In a recent paper Dubrovin et al. (1998), it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so called genus two $G$-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two $G$-functions vanish. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.     

Quantum Algebra as Deformation Symmetry

The deformation maps as well as the general algebraic maps among algebras with three generators are systematically investigated in terms of symplectic geometry and geometric quantization on 2-D manifolds, from which the explicit Hamiltonian of Heisenberg model with SU_q(2) symmetry and arbitrary spin values are given. The deformation symmetries in differential dynamical systems and the q-deformed transformations of SO(3) group in usual R³ are also discussed.     

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

The Darboux–Egoroff system of PDEs with any number $n\ge 3$ of independent variables plays an essential role in the problems of describing $n$-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux–Egoroff system and describe some symmetries generated by these operators.The constructed recursion operators are not pseudodifferential, but are Bäcklund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux–Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev.In the structure of the obtained third and fifth order symmetries of the Darboux–Egoroff system, one finds the third and fifth order flows of an $(n-1)$-component vector modified KdV hierarchy.The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental–van de Leur twisted loop group action on the space of semisimple Frobenius manifolds.We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.     

A first approximation for quantization of singular spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.     

Genus one contribution to free energy in Hermitian two-matrix model

We compute the genus one correction to free energy of Hermitian two-matrix model in large N limit in terms of theta-functions associated to the spectral curve. We discuss the relationship of this expression to the isomonodromic tau-function, the Bergmann tau-function on Hurwitz spaces, the G-function of Frobenius manifolds and the determinant of Laplacian in a singular metric over the spectral curve.     

General relativistic gravitation as the theory of broken symmetry of intransitive groups of transformations

General relativistic gravitational theories are constructed from suitable intransitive continuous groups of transformations. A minimal invariant variety forms the unperturbed universe. The formalism of the group is generalized to have the symmetry of its action on this manifold broken by gauge potentials. The theory is expressed in these potentials and it is shown how the present symmetry breaking is related to a general metric. The physical interpretation of the formalism is outlined.Work supported by the Department of Energy and presented at the Meeting in Honor of the Retirement of Prof. A. Taub, August 10, 1978.     

Hypercommutative Operad as a Homotopy Quotient of BV

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.     

Frobenius manifolds near the discriminant and relations in the tautological ring

We give a criterion for extending a generically semisimple (not necessarily conformal) Frobenius manifold locally near a smooth point of the discriminant to a cohomological field theory. As an application, we show that a large set of tautological relations related to the Givental–Teleman classification for any generically semisimple cohomological field theories follow from Pixton’s generalized Faber–Zagier relations.     

Coupling of matter to supergravity from the rheonomic symmetry point of view and the origin of the auxiliary fields

We write a first-order action for the Wess-Zumino supermultiplet regarding it as a 0-form on the graded Poincaré supergroup manifold. The standard supersymmetry transformations are reproduced by the rheonomic symmetry mechanism. When this action is coupled to the geometrical action of supergravity on the group manifold, the same rheonomic symmetry mechanism automatically generates the so-called minimal set of auxiliary fields.     

A Note on Invariant Temporal Functions

The purpose of this article is to present a result on the existence of Cauchy temporal functions invariant by the action of a compact group of conformal transformations in arbitrary globally hyperbolic manifolds. Moreover, the previous results about the existence of Cauchy temporal functions with additional properties on arbitrary globally hyperbolic manifolds are unified in a very general theorem. To make the article more accessible for non-experts, and in the lack of an appropriate single reference for the Lorentzian geometry background of the result, the latter is provided in an introductory section.     

Rigid rotor as a toy model for Hodge theory

We apply the superfield approach to the toy model of a rigid rotor and show the existence of the nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations, under which, the kinetic term and the action remain invariant. Furthermore, we also derive the off-shell nilpotent and absolutely anticommuting (anti-) co-BRST symmetry transformations, under which, the gauge-fixing term and the Lagrangian remain invariant. The anticommutator of the above nilpotent symmetry transformations leads to the derivation of a bosonic symmetry transformation, under which, the ghost terms and the action remain invariant. Together, the above transformations (and their corresponding generators) respect an algebra that turns out to be a physical realization of the algebra obeyed by the de Rham cohomological operators of differential geometry. Thus, our present model is a toy model for the Hodge theory.     

On the genus two free energies for semisimple Frobenius manifolds

We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as the sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called ??genus two G-function.?? Conjecturally, the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities, as well as for ?¹-orbifolds with positive Euler characteristics. We explain the reasons for the conjecture and prove it in particular cases.     

Lattice spinor gravity

We propose a regularized lattice model for quantum gravity purely formulated in terms of fermions. The lattice action exhibits local Lorentz symmetry, and the continuum limit is invariant under general coordinate transformations. The metric arises as a composite field. Our lattice model involves no signature for space and time, describing simultaneously a Minkowski or euclidean theory. It is invariant both under Lorentz transformations and euclidean rotations. The difference between space and time arises from expectation values of composite fields. Our formulation includes local gauge symmetries beyond the generalized Lorentz symmetry. The lattice construction can be employed for formulating models with local gauge symmetries purely in terms of fermions.     

The action of transformations of the point group of a molecule on its instantaneous nuclear configuration

The action of the transformations of a point group of a rigid molecule in a specified electronic state on the instantaneous nuclear configuration of the molecule is considered. This action is shown to be equivalent to permutations of identical nuclei in the effective potential of interaction of the nuclei in this state, which is invariant with respect to the transformations of the point group. Therefore, the point group characterizing the electronic state should be used as the rigorous symmetry group of the total electronic-vibrational-rotational motion.