Equivariant Quantization Research Articles

Quantum cohomology of [formula omitted] and the weighted hook walk on Young diagrams

Following the work of Okounkov and Pandharipande (2010) [OP1,OP2], Diaconescu [D], and the recent work of I. Ciocan-Fontanine et al. (in preparation) [CDKM] studying the equivariant quantum cohomology Q H ( C ⁎ ) 2 ⁎ ( Hilb n ) of the Hilbert scheme and the relative Donaldson–Thomas theory of P 1 × C 2 , we establish a connection between the J-function of the Hilbert scheme and a certain combinatorial identity in two variables. This identity is then generalized to a multivariate identity, which simultaneously generalizes the branching rule for the dimension of irreducible representations of the symmetric group in the staircase shape. We then establish this identity by a weighted generalization of the Greene–Nijenhuis–Wilf hook walk, which is of independent interest.

On [formula omitted]-equivariant quantizations

We investigate the concept of equivariant quantization over the superspace R p + q ∣ 2 r , with respect to the orthosymplectic algebra o s p ( p + 1 , q + 1 ∣ 2 r ) . Our methods and results vary upon the superdimension p + q − 2 r . When the superdimension is nonzero, we manage to obtain a result which is similar to the classical theorem of Duval, Lecomte and Ovsienko: we prove the existence and uniqueness of the equivariant quantization except in some resonant situations. To do so, we have to adapt their methods to take into account the fact that the Casimir operator of the orthosymplectic algebra on supersymmetric tensors is not always diagonalizable, when the superdimension is negative and even. When the superdimension is zero, the situation is always resonant, but we can show the existence of a one-parameter family of equivariant quantizations for symbols of degree at most two.

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.

A new integrable system on the sphere and conformally equivariant quantization

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi–Moser system, we disclose a novel integrable system on the sphere S n , namely the dual Moser system. The latter falls, along with the Jacobi–Moser and Neumann–Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann–Uhlenbeck and dual Moser systems is ensured by means of the conformally equivariant quantization procedure.

Quantum cohomology of the Springer resolution

Let G denote a complex, semi-simple, simply-connected group and B its associated flag variety. We identify the equivariant quantum differential equation for the cotangent bundle T ⁎ B with the affine Knizhnik–Zamolodchikov connection of Cherednik and Matsuo. This recovers Kimʼs description of quantum cohomology of B as a limiting case. A parallel result is proven for resolutions of the Slodowy slices. Extension to arbitrary symplectic resolutions is discussed.

Natural and Projectively Invariant Quantizations on Supermanifolds

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bordemann's method in order to solve it. The obtained quantization appears as the natural globalization of the $\mathfrak{pgl}({n+1|m})$-equivariant quantization on ${\mathbb{R}}^{n|m}$ constructed by P. Mathonet and F. Radoux in [arXiv:1003.3320]. Our quantization is also a prolongation to arbitrary degree symbols of the projectively invariant quantization constructed by J. George in [arXiv:0909.5419] for symbols of degree two.

Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.

Equivariant quantizations for AHS-structures

We construct an explicit scheme to associate to any potential symbol an operator acting between sections of natural bundles (associated to irreducible representations) for a so-called AHS-structure. Outside of a finite set of critical (or resonant) weights, this procedure gives rise to a quantization, which is intrinsic to this geometric structure. In particular, this provides projectively and conformally equivariant quantizations for arbitrary symbols on general (curved) projective and conformal structures.

Quantum cohomology of G/P and homology of affine Grassmannian

Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a quotient of the homology H_*(Gr_G) of the affine Grassmannian \Gr_G of G. As a consequence, all three-point genus zero Gromov-Witten invariants of $G/P$ are identified with homology Schubert structure constants of H_*(Gr_G), establishing the equivalence of the quantum and homology affine Schubert calculi. For the case G = B, we use the Mihalcea's equivariant quantum Chevalley formula for QH^*(G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology. Our main results extend to the torus-equivariant setting.

Second-Order Conformally Equivariant Quantization in Dimension 1|2

This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (su- per)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle S 1|2 equipped with the standard contact structure. The conformal Lie superalgebra K(2) of contact vector fields on S 1|2 contains the Lie su- peralgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.

Quantum cohomology of the Hilbert scheme of points in the plane

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ2. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.

Equivariant quantizations of symmetric algebras

Let g be a Lie bialgebra and let V be a finite-dimensional g -module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra U h ( g ) , resp. U q ( g ) . We investigate, in particular, such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of g and V. We classify these structure in the important special case, when g is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce a more general notion, co-Poisson module algebras and their quantizations, to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language.

On natural and conformally equivariant quantizations

The concept of conformally equivariant quantization was introduced by C. Duval, P. Lecomte and V. Ovsienko for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure. A natural generalization of this concept is to seek for a quantization procedure, over a manifold M, that depends on a pseudo-Riemannian metric, is natural and is invariant with respect to a conformal change of the metric. The existence of such a procedure was conjectured by P. Lecomte and proved by C. Duval and V. Ovsienko for symbols of degree at most 2 and by S. Loubon Djounga for symbols of degree 3. In two recent papers, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we shall show how the formalism developed in these works adapts in order to deal with the conformally equivariant quantization for symbols of degree at most 3. This will allow us to easily recover the results of C. Duval, V. Ovsienko and S. Loubon Djounga. We shall then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4.

Deformation Quantization on the Closure of Minimal Coadjoint Orbits

We consider a complex simple Lie algebra $${\mathfrak{g}}$$ , with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct $${\mathfrak{g}}$$ -invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra $${{\rm U}(\mathfrak{g})}$$ . This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.

Quantum cohomology of the Hilbert scheme of points on 𝒜_{𝓃}-resolutions

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A n A_{n} singularities. The operators encoding these invariants are expressed in terms of the action of the the affine Lie algebra g l ^ ( n + 1 ) \widehat {\mathfrak {gl}}(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of A n × P 1 A_{n}\times \mathbf {P}^1 . We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of types D D and E E .

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.

Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces

Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of \cite{cmp2} and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in \cite{seidel}. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\times$. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar \cite{Magyar}. It also uses Peterson's unpublished result \cite{Peterson} --- recently proved by Lam and Shimozono in \cite{Lam-Shi} --- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.

Quantum Integrability and Supersymmetric Vacua

This is an announcement of some of the results of a longer paper where the supersymmetric vacua of two dimensional N=2 susy gauge theories with matter are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The correspondence between the Heisenberg spin chain and the two dimensional U(N) theory with fundamental hypermultiplets is reviewed in detail. We demonstrate the isomorphism of the equivariant quantum cohomology of the cotangent bundle to the Grassmanian manifold Gr(N,L) and the ring of quantum integrals of motion of the length L SU(2) XXX spin chain, in the N-particle sector. This paper accompanies arXiv:0901.4744

Root systems and the quantum cohomology of ADE resolutions

We compute the ℂ∗-equivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity ℂ2∕G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov–Witten potential of [ℂ2∕G].

Szegö kernels, Toeplitz operators, and equivariant fixed point formulae

The aim of this paper is to apply algebro-geometric Szego kernels to the asymptotic study of a class of trace formulae in equivariant geometric quantization and algebraic geometry. Let (M,J) be a connected complex projective manifold, of complex dimension d, and let A be an ample line bundle on it. Let, in addition, G be a compact and connected g-dimensional Lie group acting holomorphically on M in such a way that the action can be linearized to A. For every k ∈ N, the spaces of global holomorphic sections H ( M,A ) are linear representations of G and therefore may be equivariantly decomposed over its irreducible representations. More precisely, let g be the Lie algebra of G, and let Λ ⊆ g be a set of weights parametrizing the family of all finite-dimensional irreducible representations of G. For every ∈ Λ, denote by V the corresponding G-module. Given ∈ Λ and a linear representation of G on a finite dimensional vector space W , we denote by W ⊆ W the -isotype of W , that is, the maximal invariant subspace of W equivariantly isomorphic to a direct sum of copies of V . For every k ∈ N, we then have equivariant direct sum decompositions