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Logarithmic Frobenius structures and Coxeter discriminants
Abstract
We consider a class of solutions of the WDVV equation related to the special systems of covectors (called ∨-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of the corresponding hyperplanes. For the Coxeter arrangements the corresponding structures are shown to be almost dual in Dubrovin's sense to the Frobenius structures on the strata in the discriminants discussed by Strachan. For the classical Coxeter root systems this leads to the families of ∨-systems from the earlier work by Chalykh and Veselov. For the exceptional Coxeter root systems we give the complete list of the corresponding ∨-systems. We present also some new families of ∨-systems, which cannot be obtained in such a way from the Coxeter root systems.
... The class of ∨-systems is closed under the natural operations of taking subsystems [17] and under restriction of a system to the intersection of some of the hyperplanes α(x) = 0, where α ∈ A [16]. The class of ∨-systems contains multi-parameter deformations of the root systems A n and B n ( [9], see also [17] for more examples). ...
... In this section we consider the restriction operation for the trigonometric solutions of WDVV equations and show that this gives new solutions of WDVV equations. An analogous statement in the rational case was established in [16]. Let ...
... Remark 7.2. We note that for special values of the parameters configuration A is a restriction of a root system (cf [16] where the rational version of this configuration was considered). Thus if r = 0 and p = q = s then A reduces to the root system D 4 . ...
We consider a class of trigonometric solutions of Witten–Dijkgraaf–Verlinde–Verlinde equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric ∨-system, and we show that its subsystems are also trigonometric ∨-systems. Finally, while reviewing the root system case we determine a version of (generalised) Coxeter number for the exterior square of the reflection representation of a Weyl group.
... The class of ∨-systems is closed under the natural operations of taking subsystems [18] and under restriction of a system to the intersection of some of the hyperplanes α(x) = 0, where α ∈ A [17]. The class of ∨-systems contains multi-parameter deformations of the root systems A n and B n ( [9], see also [18] for more examples). ...
... In this section we consider the restriction operation for the trigonometric solutions of WDVV equations and show that this gives new solutions of WDVV equations. An analogous statement in the rational case was established in [17]. ...
... The proof of the next lemma is similar to the proof of [17,Lemma 1] in the rational case (see also [1]). ...
We consider a class of trigonometric solutions of WDVV equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric $\vee$-system and we show that their subsystems are also trigonometric $\vee$-systems. Finally, while reviewing the root system case we determine a version of (generalised) Coxeter number for the exterior square of the reflection representation of a Weyl group.
... It was shown in [14] that almost dual Frobenius multiplication has a natural restriction on discriminant strata D. However, the properties of Saito metric η on the discriminant did not seem to be investigated until the present work. The main object of the present paper is the Saito metric restricted on discriminant strata. ...
... In this section we revisit almost duality relation between Frobenius structures on discriminant strata. Multiplication * is well-defined on the strata and a version of such a duality was established in [14]. It was suggested earlier in [30] that discriminant strata are natural submanifolds so that multiplication • of tangential vectors is defined and belongs to the stratum. ...
... It was shown in [14] that the left-hand-side of equality (3.1) can be restricted to any stratum D with u and v being tangential vectors to D. More precisely, let Σ D be the union of the hyperplanes Π γ ∩ D in D, where γ ∈ R \ R D and consider the point x 0 ∈ D \ Σ D . Let u, v ∈ T x 0 D and consider extension of u and v to two local analytic vector fields u(x), v(x) ∈ T x V such that u(x 0 ) = u and v(x 0 ) = v. ...
Let $W$ be a finite Coxeter group and $V$ its reflection representation. The orbit space $\mathcal{M}_W= V/W$ has the remarkable Saito flat metric defined as a Lie derivative of the $W$-invariant bilinear form $g$. We find determinant of the Saito metric restricted to an arbitrary Coxeter discriminant stratum in $\mathcal{M}_W$. It is shown that this determinant is proportional to a product of linear factors in the flat coordinates of the form $g$ on the stratum. We also find multiplicities of these factors in terms of Coxeter geometry of the stratum. This result may be interpreted as a generalisation to discriminant strata of the Coxeter factorisation formula for the Jacobian of the group $W$. As another interpretation, we find determinant of the operator of multiplication by the Euler vector field in the natural Frobenius structure on the strata.
... Multiparameter deformations related to A n and B n root systems were constructed in [6]. The class of ∨-systems and the corresponding rational solutions (1.4) are closed under the natural operation of taking restrictions and subsystems [12], [13]. Trigonometric generalisation of solutions (1.4) also arise in theory of Frobenius manifolds. ...
... Also we note that outside W B we have a well-defined multiplication u(x) * v(x). Similarly to the rational case considered in [12] and trigonometric case with extra variable [1] the following lemma holds. ...
... We will need the following lemma (cf. [12], [1] ...
We give a family of solutions of Witten-Dijkgraaf-Verlinde-Verlinde equations in $n$-dimensional space. It is defined in terms of $BC_{n}$ root system and $n+2$ independent multiplicity parameters. We also apply these solutions to define some ${\mathcal N}=4$ supersymmetric mechanical systems.
... , and this induces the Legendre transformation between the two Frobenius manifolds, i.e. this change of primary differential induces a change of variable that maps (8) to (9). Thus the Frobenius manifold structures on C l+1 / W (k) (A l ) and M k,l+1−k are related by a Legendre transformation. ...
... By construction the 5-dimensional extended configuration is invariant under the Coxeter group F 4 . This configuration was obtained in [8] by restricting the E 8 system onto a certain discriminant, in their notation this is the solution (E 8 , A 3 1 ) . Whether a similar construction may be applied to the remaining exceptional cases is under investigation. ...
... This corresponds to the induced Frobenius structures on discriminant surfaces with a larger manifold [18]. That such induced structures on discriminant generate solutions to the WDVV equations of the form (1) was proved in [8]. ...
Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity are given in terms a configurations of vectors which satisfy certain algebraic conditions known as $\bigvee$-conditions. The simplest examples of such configuration are the root systems of finite Coxeter groups. In this paper conditions are derived which ensure that an extended configuration - a configuration in a space one-dimension higher -satisfy these $\bigvee$-conditions. Such a construction utilizes the notion of a small-orbit, as defined by Serganova. Symmetries of such resulting solutions to the WDVV-equations are studied; in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions and, for certain values of the free data, one obtains a transformation from extended $\bigvee$-systems to the trigonometric almost dual solutions corresponding to the classical extended affine Weyl groups.
... One further comment has to be made in the case when h ∨ U = 0 . We require here that the inverse metric used in the WDVV equations is the non-degenerate bilinear form ( , ) on h rather than one -possibly degenerate -constructed from the sum of derivatives of F as used in [14] . ...
... Type B pairs will occur when an extra set of Weyl invariant vectors is appended to the root system -see Section 6.2. Both these types of configuration appear in ∨-systems and deformed root systems [7,14,15,32]. ...
... Partial results have been obtained in [26], and these provide further examples of elliptic ∨-systems. In fact, interesting examples of ∨-system can be found by looking on the induced structures on disciminants [14,30] and clearly the same ideas could be applied here. ...
A functional ansatz is developed which gives certain elliptic solutions of the Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations. This ansatz is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, introduced by Veselov, of a ∨-system.Rational and trigonometric limits are studied together with examples of elliptic ∨-systems based on various Weyl groups. Jacobi group orbit spaces are studied: these carry the structure of a Frobenius manifold. The corresponding ‘almost dual’ structure is shown, in the AN and BN cases and conjecturally for an arbitrary Weyl group, to correspond to the elliptic solutions of the WDVV equations.Transformation properties, under the Jacobi group, of the elliptic trilogarithm are derived together with various functional identities which generalize the classical Frobenius–Stickelberger relations.
... The Lie-algebra root systems are only the tip of an iceberg of WDVV solutions. It has been shown [7,9] If we try the same idea on the A 3 system, we obtain the six edges of a regular tetrahedron and deform to encounter the fivedimensional moduli space of tetrahedral shapes (modulo scale). Again, the homogeneity condition (4.2) has a unique solution f α , but now the WDVV equation enforces the three conditions (Figure 3) 0, 0, 0 ...
... Yet, these properties are only necessary but not sufficient. Finally we remark that all our examples are part of a larger moduli space of 3 n = families of WDVV solutions [7,9]. ...
... Finding solutions to the WDVV equations is a nontrivial task, which was considered in numerous papers in various contexts (see, e.g. [2,3,4,5,6,7,8] and refs. to them). ...
... Having at hands the solution for the curved WDVV equations (2.11), (2.12), one may try to solve the equations for the prepotentials (2.13), (2.14) or (2.18), (2.19), respectively. It should be noted that even in the flat case, where a variety of the solutions to the WDVV equations is known [3,4,5,6,7], such a task is far from being completely solved. Nevertheless, many particular solutions are known for the standard supercharges [11,12,13] as well as for the case with spin variables [14]. ...
We present ${\cal N}{=}\,4$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds constructed within the Hamiltonian approach. The structure functions entering the supercharges and the Hamiltonian obey modified covariant constancy equations as well as modified Witten-Dijkgraaf-Verlinde-Verlinde equations specified by the presence of the manifold's curvature tensor. Solutions of original Witten-Dijkgraaf-Verlinde-Verlinde equations and related prepotentials defining ${\cal N}{=}\,4$ superconformal mechanics in flat space can be lifted to $so(n)$-invariant Riemannian manifolds. For the Hamiltonian this lift generates an additional potential term which, on spheres and (two-sheeted) hyperboloids, becomes a Higgs-oscillator potential. In particular, the sum of $n$ copies of one-dimensional conformal mechanics results in a specific superintegrable deformation of the Higgs oscillator.
... В работе [42] А. П. Веселов и О. А. Чалых нашли n-параметрические системы Веселова, деформирующие классические системы корней A n и B n . В работе [58] А. П. Веселов и М. В. Фейгин нашли естественную интерпретацию решений такого рода в терминах фробениусовых структур на дискриминантах групп Кокстера, возникающих в работе Строна, продолжив тем самым двойственность Дубровина на дискриминантные страты. Также были найдены новые ∨-системы. ...
... Также были найдены новые ∨-системы. В работах [58], [59] А. П. Веселов и М. В. Фейгин показали, что класс систем Веселова замкнут относительно естественных операций взятия подсистем и ограничений, а в недавнем препринте они исследовали подкласс гармонических ∨-систем и рассмотрели связи с теорией свободных по Саито конфигураций гиперплоскостей. Полное описание ∨-систем остается интересным открытым вопросом. ...
... By construction, the five-dimensional extended configuration is invariant under the Coxeter group F4. This configuration was obtained in Ref. 19 by restricting the E8 system onto a certain discriminant; in their notation, this is the solution (E 8 , A 3 1 ). ...
Rational solutions of the Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations of associativity are given in terms of configurations of vectors, which satisfy certain algebraic conditions known as ⋁-conditions [A. P. Veselov, Phys. Lett. A 261, 297–302 (1999)]. The simplest examples of such configurations are the root systems of finite Coxeter groups. In this paper, conditions are derived that ensure that an extended configuration—a configuration in a space one-dimension higher—satisfies these ⋁-conditions. Such a construction utilizes the notion of a small orbit, as defined in Serganova [Commun. Algebra, 24, 4281–4299 (1996)]. Symmetries of such resulting solutions to the WDVV equations are studied, in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions, and for certain values of the free data, one obtains a transformation from extended ⋁-systems to the trigonometric almost-dual solutions corresponding to the classical extended affine Weyl groups.
... The general solution to the WDVV equations is unknown, but various classes based on (deformed) Coxeter root systems have been found (see e.g. [18]). Secondly, another prepotential U is subject to a system of linear homogeneous differential equations of second order in a given F background. ...
We present a surprising redefinition of matrix fermions which brings the supercharges of the $\cal N$-extended supersymmetric $A_{n-1}$ Calogero model introduced in [1] to the standard form maximally cubic in the fermions. The complexity of the model is transferred to a non-canonical and nonlinear conjugation property of the fermions. Employing the new cubic supercharges, we apply a supersymmetric generalization of a "folding" procedure for $A_{2n-1}\oplus A_1$ to explicitly construct the supercharges and Hamiltonian for arbitrary even-$\cal N$ supersymmetric extensions of the $B_n$, $C_n$ and $D_n$ rational Calogero models. We demonstrate that all considered models possess a dynamical $osp({\cal N}|2)$ superconformal symmetry.
... One unexpected feature of the special case of twisted-Legendre fields (for almost-dual structures) is that they map rational solutions to the WDVV equations to trigonometric solutions, or more specifically, for those almost-dual solutions coming from classical extended-affine-Weyl Frobenius manifolds [6,13]. Since the restriction of Legendre fields to discriminant submanifolds are again Legendre, one should be able to derive similar results for the induced WDVV equations on discriminant submanifolds (for which the induced intersection form is flat), following the ideas in [8,14]. More generally, one needs to see if one can apply these ideas directly to rational/trigonometric -systems. ...
The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class - the so-called Legendre transformations - were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.
... The theory of Frobenius manifolds has multiple connections with other branches of mathematics, such as quantum cohomology, singularity theory, the theory of integrable systems, see, for example, [D1, D2, M]. The notion of a Frobenius manifold was introduced by B. Dubrovin in [D1], see also [D2,M,FV,St], where numerous variants of this notion are discussed. In all alterations, a Frobenius manifold is a manifold with a flat metric and a Frobenius algebra structure on tangent spaces at points of the manifold such that the structure constants of multiplication are given by third derivatives of a potential function on the manifold with respect to flat coordinates. ...
A Frobenius manifold is a manifold with a flat metric and a Frobenius algebra
structure on tangent spaces at points of the manifold such that the structure
constants of multiplication are given by third derivatives of a potential
function on the manifold with respect to flat coordinates.
In this paper we present a modification of that notion coming from the theory
of arrangements of hyperplanes. Namely, given natural numbers $n>k$, we have a
flat $n$-dimensional manifold and a vector space $V$ with a nondegenerate
symmetric bilinear form and an algebra structure on $V$, depending on points of
the manifold, such that the structure constants of multiplication are given by
$2k+1$-st derivatives of a potential function on the manifold with respect to
flat coordinates. We call such a structure a {\it Frobenius like structure}.
Such a structure arises when one has a family of arrangements of $n$ affine
hyperplanes in $\C^k$ depending on parameters so that the hyperplanes move
parallely to themselves when the parameters change. In that case a Frobenius
like structure arises on the base $\C^n$ of the family.
... Eine weitere Methode ist die Deformation von aus den Vektoren der Wurzelsystemen gebildeten Polytopen; die Wurzeln werden dabei als Kanten der Polytope aufgefasst [25,26,27,38]. Viele neue ∨-Systeme hat man über die ∨-Bedingungen aus bekannten ∨-Systemen durch Projektion auf bestimmte Unterräume gefunden [33,34]. Eine weitere Klasse liefern Deformationen "verallgemeinerter Wurzelsysteme" aus der Theorie der Lie-Superalgebren [34]. ...
... It is known that the ∨conditions are fulfilled for all root systems and for their special deformations discovered in the Calogero-Moser systems (see [3]). ∨-systems have been extensively studied by Veselov and his collaborators (see for instance [7], [8], [4]). Now we recall the notion of ∨-systems (see [19]). ...
In this paper, we extend purely non-local Hamiltonian formalism to a class of
Riemannian F-manifolds, without assumptions on the semisimplicity of the
product $\circ$ or on the flatness of the connection $\nabla$. In the flat case
we show that the recurrence relations for the principal hierarchy can be
re-interpreted using a local and purely non-local Hamiltonian operators and in
this case they split into two Lenard-Magri chains, one involving the even
terms, the other involving the odd terms. Furthermore, we give an elementary
proof that the Kohno property and the $\vee$-system condition are equivalent
under suitable conditions and we show how to associate a purely non-local
Hamiltonian structure to any $\vee$-system, including degenerate ones.
... were already considered in somewhat different, but closely related context in [28]. ...
The prepotentials for the quiver supersymmetric gauge theories are defined as
quasiclassical tau-functions, depending on two different sets of variables: the
parameters of the UV gauge theory or the bare compexified couplings, and the
vacuum condensates of the theory in IR. The bare couplings are introduced as
periods on the UV base curve, and the consistency of corresponding gradient
formulas for the tau-functions is proven using the Riemann bilinear relations.
It is shown, that dependence of generalised prepotentials for the quiver gauge
theories upon the bare couplings turns to coincide with the corresponding
formulas for the derivatives of tau-functions for the isomonodromic
deformations. Computations for the SU(2) quiver gauge theories with bi- and
tri-fundamental matter are performed explicitly and analysed in the context of
4d/2d correspondence.
... Numerous variations of the definition of the Frobenius structure see, for example, in [D1,D2,M,St,FV]. ...
We consider a family of generic weighted arrangements of hyperplanes and show
that the associated Gauss-Manin connection, the contravariant form on the space
of singular vectors, and the algebra of functions on the critical set of the
master function define a Frobenius like structure on the base of the family.
... The mathematical literature usually does not introduce a euclidean metric δ bc but defines an induced metric G bc = −x a F abc which is constant and nondegenerate. Hence, for any α =− 1 2 we can import all known WDVV solutions [80, 95, 21, 33, 34, 74, 75] up to constant coordinate transformations. The special case of D(2, 1; − 1 2 ) ≃ D(2, 1; 1) ≃ osp(4|2) only appears as a singular limit, where F can no longer be 'normalized' via (4.55) and the induced metric degenerates. ...
We survey the salient features and problems of conformal and superconformal
mechanics and portray some of its developments over the past decade. Both
classical and quantum issues of single- and multiparticle systems are covered.
... This same result holds in the weaker setting of semi-Hamiltonian systems where the metric is Egorov [18]. A class of such examples may be found by restricting flows to certain natural submanifolds of a Frobenius manifold [22, 11]. We end this section with an extended example which shows how the modularity properties of solutions to the Chazy equation results in, up to a reciprocal transformation, flows that are invariant under the modular transformations. ...
The space of Frobenius manifolds has a natural involutive symmetry on it: there exists a map I which sends a Frobenius manifold to another Frobenius manifold. Also, from a Frobenius manifold one may construct a so-called
almost dual Frobenius manifold which satisfies almost all of the axioms of a Frobenius manifold. The action of I on the almost dual manifolds is studied, and the action of I on objects such as periods, twisted periods, and flows is studied. A distinguished class of Frobenius manifolds sit at the
fixed point of this involutive symmetry, and this is made manifest in certain modular properties of the various structures.
In particular, up to a simple reciprocal transformation, for this class of modular Frobenius manifolds, the flows are invariant
under the action of I.
... We note that the z 2 log z-type terms have appeared in the WDVV-literature before (see, for example, [37, 56]) but one normally considers these as being derived as examples of dual Frobenius manifolds [27]. Their functional form suggests the type of term that may be present in a construction of deformed solutions for other Coxeter group orbit spaces. ...
This thesis is concerned with the relationship between integrable Hamiltonian partial differential equations and geometric structures on the manifold in which the dependent variables take their values. Chapters 1 and 2 are introductory chapters, and as such contain no original material. Chapter 1 covers some basic material from the theory of integrable systems, including the Hamiltonian formalism for PDE's, the concept of a bi-Hamiltonian system, and the dispersionless Lax equation. Chapter 2 is about Frobenius manifolds. It explains their relationship to the WDVV equations of topological quantum field theory, and how they form part of the theory of integrable systems via both the deformed Levi-Civita connection and a flat pencil of metrics. Chapter 3 is based on [J. T. Ferguson. Flat pencils of symplectic connections and Hamiltonian operators of degree 2. J. Geom. Phys., 58(4):468–486, 2008]. It is original, except for the background material in Section 3.1. In it we explain the (almost) symplectic geometry associated to Hamiltonian operators of degree 2, and use it to formulate the geometric conditions for two such operators to constitute a bi-Hamiltonian structure. In the case that these operators are associated to symplectic forms, these conditions are expressed as algebraic constraints on a multiplication of one-forms. We also express conditions for a Hamiltonian operator of degree two to be compatible with a hydrodynamic type Hamiltonian operator. Chapter 4 is based upon [J. T. Ferguson and I. A. B. Strachan. Logarithmic deformations of the rational superpotential/ Landau-Ginzburg construction of solutions of the WDVV equations. arXiv:math-ph/0605078, 2006], which was joint work with Ian Strachan. It is to appear in Communications in Mathematical Physics, and is original except for Section 4.1. It is concerned with the construction of new solutions to the WDVV equations which arise by analogy with the so-called waterbag reductions of the dispersionless KP hierarchy. Superpotentials of existing Frobenius manifolds are deformed by the addition of logarithmic terms, and this results in new WDVV solutions which deform existing ones, including a new class of polynomial solutions which deform solutions associated to the A_N Coxeter group. Chapter 5 follows on from Chapter 4, and considers in detail two integrable hierarchies which arise from the WDVV solutions studied there. It is particularly concerned with the Hamiltonian structures of these hierarchies. Appendix A attempts to incorporate some of the features of one of these hierarchies into a construction of a Frobenius structure from a bi-Hamiltonian structure.
... We note that the z 2 log z-type terms have appeared in the WDVV-literature before (see, for example, [10, 15]) but one normally considers these are being derived as examples of dual Frobenius manifolds [8]. Their functional form suggests the type of term that may be present in a construction of deformed solutions for other Coxeter group orbit spaces. ...
The superpotential in the Landau-Ginzburg construction of solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include logarithmic terms. This results in deformations - quadratic in the deformation parameters - of the normal prepotential solution of the WDVV equations. Such solution satisfy various pseudo-quasi-homogeneity conditions, on assigning a notional weight to the deformation parameters. This construction includes, as a special case, deformations which are polynomial in the flat coordinates, resulting in a new class of polynomial solutions of the WDVV equations.
... It would be interesting to obtain almost dual prepotentials for the Frobenius manifolds of the affine Weyl groups as well as for their discriminants (cf. rational case [7,12]). Comparison with a recent work on the elliptic solutions [16] might also be interesting. ...
We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system ($\vee$-system) and we determine all trigonometric $\vee$-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric $\vee$-system; this inverts a one-way implication observed by Veselov for the rational solutions.
We give a family of solutions of Witten–Dijkgraaf–Verlinde– Verlinde equations in n n -dimensional space. It is defined in terms of B C n BC_{n} root system and n + 2 n+2 independent multiplicity parameters. We also apply these solutions to define some N = 4 {\mathcal N}=4 supersymmetric mechanical systems.
We extend the relation between the Witten-Dijkgraaf-Verlinde-Verlinde equation and N = 4 supersymmetric mechanics to arbitrary curved spaces. The resulting curved WDVV equation is written in terms of the third rank Codazzi tensor. We provide the solutions of the curved WDVV equation for the so(n) symmetric conformally flat metrics. We also explicitly demonstrate how each solution of the flat WDVV equation can be lifted up to the curved WDVV solution on the conformally flat spaces.
We discuss the superconformal quantum mechanics arising from the M2-branes.
We begin with a comprehensive review on the superconformal quantum mechanics
and emphasize that conformal symmetry and supersymmetry in quantum mechanics
contain a number of exotic and enlightening properties which do not occur in
higher dimensional field theories. We see that superfield and superspace
formalism is available for $\mathcal{N}\le 8$ superconformal mechanical models.
We then discuss the M2-branes with a focus on the world-volume descriptions of
the multiple M2-branes which are superconformal three-dimensional Chern-Simons
matter theories. Finally we argue that the two topics are connected in
M-theoretical construction by considering the multiple M2-branes wrapped around
a compact Riemann surface and study the emerging IR quantum mechanics. We
establish that the resulting quantum mechanics realizes a set of novel
$\mathcal{N}\ge 8$ superconformal quantum mechanical models which have not been
reached so far. Also we discuss possible applications of the superconformal
quantum mechanics to mathematical physics.
It is shown that the description of certain class of representations of the
holonomy Lie algebra associated to hyperplane arrangement Delta is essentially
equivalent to the classification of V-systems associated to Delta. The flat
sections of the corresponding V-connection can be interpreted as vector fields,
which are both logarithmic and gradient. We conjecture that the hyperplane
arrangement of any V-system is free in Saito's sense and show this for a
special class of V-systems called harmonic, which includes all Coxeter systems.
In the irreducible Coxeter case the potentials of the corresponding gradient
vector fields turn out to be Saito flat coordinates, or their one-parameter
deformations.
The V-systems are special finite sets of covectors which appeared in the
theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations.
Several families of V-systems are known but their classification is an open
problem. We derive the relations describing the infinitesimal deformations of
V-systems and use them to study the classification problem for V-systems in
dimension 3. We discuss also possible matroidal structures of V-systems in
relation with projective geometry and give the catalogue of all known
irreducible rank 3 V-systems.
We define the Dunkl and Dunkl-Heckman operators in infinite number of
variables and use them to construct the quantum integrals of the
Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple
proof of integrability of the deformed quantum CMS systems related to classical
Lie superalgebras. We show how this naturally leads to a quantum version of the
Moser matrix, which in the deformed case was not known before.
We overcome the barrier of constructing \( \mathcal{N} = 4 \) superconformal models in one space dimension for more than three particles. The D (2, 1; α) superalgebra of our systems is realized on the coordinates and momenta of the particles, their superpartners and one complex pair of harmonic variables. The models are determined by two prepotentials, F and U, which must obey the WDVV and a Killing-type equation plus homogeneity conditions. We investigate permutation-symmetric solutions, with and without translation invariance. Models based on deformed A
n
and BCD
n
root systems are constructed for any value of α, and exceptional F
n
-type and super root systems admit solutions as well. Translation-invariant mechanics occurs for any number of particles at \( \alpha = - \frac{1}{2} \) (osp(4|2) invariance as a degenerate limit) and for four particles at arbitrary α (three series).
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 A
n and B
3 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.
We revisit the (untwisted) superfield approach to one-dimensional multi-particle systems with N=4 superconformal invariance. The requirement of a standard (flat) bosonic kinetic energy implies the existence of inertial (super-)coordinates, which is nontrivial beyond three particles. We formulate the corresponding integrability conditions, whose solution directly yields the superpotential, the two prepotentials and the bosonic potential. The structure equations for the two prepotentials, including the WDVV equation, follow automatically. The general solution for translation-invariant three-particle models is presented and illustrated with examples. For the four-particle case, we take advantage of known WDVV solutions to construct a D3 and a B3 model, thus overcoming a previously-found barrier regarding the bosonic potential. The general solution and classification remain a challenge.
We review the relation of N=4 superconformal multi-particle models on the
real line to the WDVV equation and an associated linear equation for two
prepotentials, F and U. The superspace treatment gives another variant of the
integrability problem, which we also reformulate as a search for closed flat
Yang-Mills connections. Three- and four-particle solutions are presented. The
covector ansatz turns the WDVV equation into an algebraic condition, for which
we give a formulation in terms of partial isometries. Three ideas for
classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and
matroids. Various examples and counterexamples are displayed.
Legendre transformations provide a natural symmetry on the space of solutions
to the WDVV equations, and more specifically, between different Frobenius
manifolds. In this paper a twisted Legendre transformation is constructed
between solutions which define the corresponding dual Frobenius manifolds. As
an application it is shown that certain trigonometric and rational solutions of
the WDVV equations are related by such a twisted Legendre transform.
From any given Frobenius manifold one may construct a so-called dual structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the WDVV equations of associativity. Jacobi group orbit spaces naturally carry the structures of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin.
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 U=0 one remains with the WDVV equation which suggests an ansatz for F in terms of a set of covectors to be found. One approach constructs such covectors from suitable polytopes, another method solves Veselov's \vee-conditions in terms of deformed Coxeter root systems. I relate the two schemes for the A_n example.
We consider ideals of polynomials vanishing on the W-orbits of the
intersections of mirrors of a finite reflection group W. We determine all such
ideals which are invariant under the action of the corresponding rational
Cherednik algebra hence form submodules in the polynomial module. We show that
a quantum integrable system can be defined for every such ideal for a real
reflection group W. This leads to known and new integrable systems of
Calogero-Moser type which we explicitly specify. In the case of classical
Coxeter groups we also obtain generalized Calogero-Moser systems with added
quadratic potential.
Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for which we get a complete orbifold or at least a Zariski open subset thereof, and we analyze these cases in some detail (e.g., we determine their orbifold fundamental group).
In this set-up, the principal results of Deligne-Mostow on the Lauricella hypergeometric differential equation and work of Barthel-Hirzebruch-Höfer on arrangements in a projective plane appear as special cases. Along the way we produce in a geometric manner all the pairs of complex reflection groups with isomorphic discriminants, thus providing a uniform approach to work of Orlik-Solomon.
We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg - Witten duality.
We present a complete proof that solutions of the WDVV equations in
Seiberg-Witten theory may be constructed from root systems. A generalization to
weight systems is proposed.
These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.
Differential-geometric structures on the space of orbits of a finite Coxeter group, determined by Groth\'endieck residues, are calculated. This gives a construction of a 2D topological field theory for an arbitrary Coxeter group.
This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.
CONTENTS
Introduction
§ 1. The manifold of non-regular orbits of the group
§ 2. and (reflection groups)
§ 3. and (singularity theory)
§ 4. The problem of avoiding an obstacle
§ 5. Projections of fronts and normal subgroups of the reflection groups , ,
§ 6. A summary of results on the discriminants of reflection groups
References
It is shown that the deformed Calogero–Moser–Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. A general structure of the ideals which are invariant under the action of the quantum CMS integrals is discussed in this context. The shifted super-Jack polynomials are introduced and combinatorial formulas for them and for super-Jack polynomials are given.
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A one-parameter deformation of Calogero–Moser quantum problem is introduced. It is shown that corresponding Schrödinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value. © 1998 American Institute of Physics.
The representation theory of symmetric Lie superalgebras and corresponding
spherical functions are studied in relation with the theory of the deformed
quantum Calogero-Moser systems. In the special case of symmetric pair
g=gl(n,2m), k=osp(n,2m) we establish a natural bijection between projective
covers of spherically typical irreducible g-modules and the finite dimensional
generalised eigenspaces of the algebra of Calogero-Moser integrals D_{n,m}
acting on the corresponding Laurent quasi-invariants A_{n,m}.
We present a new family of the locus configurations which is not related to ∨-systems thus giving the answer to one of the questions raised by one of the authors about the relation between the generalised quantum Calogero–Moser systems and special solutions of the generalised WDVV equations. As a by-product we have new examples of the hyperbolic equations satisfying the Huygens' principle in the narrow Hadamard's sense. Another result is new multiparameter families of ∨-systems which gives new solutions of the generalised WDVV equation.
The prepotential F(ai); defining the low-energy effective action of the SU(N) = 2 SUSY gluodynamics, satisfies an enlarged set of the WDVV-like equations FiFk−1Fj = FjFk−1Fi for any triple i, j, k = 1,…, N − 1, where matrix Fi is equal to . The same equations are actually true for generic topological theories. In contrast to the conventional formulation, when k is restricted to k = 0, in the proposed system there is no distinguished “first” time-variable, and the indices can be raised with the help of any “metric” ηmn(k) = (Fk)mn, not obligatory flat. All the equations (for all i, j, k) are true simultaneously. This result provides a new parallel between the Seiberg-Witten theory of low-energy gauge models in 4d and topological theories.
This work continues the study of F-manifolds (M,∘), first defined in [HeMa] (Int. Math. Res. Notices 6 (1999) 277–286, Preprint math.QA/9810132) and investigated in [He] (Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge University Press, Cambridge, 2002). The notion of a compatible flat structure ∇ is introduced, and it is shown that many constructions known for Frobenius manifolds do not in fact require invariant metrics and can be developed for all such triples (M,∘,∇). In particular, we extend and generalize recent Dubrovin's duality ([Du2], On almost duality for Frobenius manifolds, Preprint math.DG/0307374).
We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special value of parameter $\alpha = -2.$ As a corollary we present an explicit formula for the Hilbert-Poincar\`e series of this ideal and the generator of the minimal degree as a special Jack polynomial. A generalization to the case of the symmetric polynomials vanishing on the double shifted diagonals and the Macdonald polynomials specialized at $t^2 q = 1$ is also presented. We also give similar results for the interpolation Jack polynomials.
Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found (check-conditions). These conditions are satisfied for all root systems and their special deformations discovered in the theory of the Calogero-Moser systems by O.Chalykh, M.Feigin and the author. This leads to the new solutions for the generalized WDVV equations. Comment: 8 pages
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is considered. We describe the geometric conditions ($\vee$-conditions) on such a set which are necessary and sufficient for the corresponding function to satisfy the generalised WDVV equations. These conditions are satisfied for all Coxeter systems but there are also other examples discovered in the theory of the generalised Calogero-Moser systems. As a result some new solutions for the generalized WDVV equations are found.
In: Geometry, topology, and mathematical physics
- B Dubrovinonalmostdualityforfrobeniusmanifolds
- Math
B.DubrovinOnalmostdualityforFrobeniusmanifolds. math.DG/0307374. In: Geometry, topology, and mathematical physics, AMS Transl. Ser. 2, 212 (2004), 75–132,
Geometry of 2D topological field theories, in: Integrable Systems and Quantum Groups
- B Dubrovin
B. Dubrovin, Geometry of 2D topological field theories, in: Integrable Systems and Quantum Groups, Montecatini, Terme, 1993, in: Springer Lecture Notes in Math., vol. 1620, 1996, pp. 120–348.