Quantum Hamiltonian Reduction Research Articles

Bernstein inequality and holonomic modules

In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, and spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows one to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.

The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions

We show that the Morita equivalences $\mathrm{Cliff}(4) \simeq {\mathbb H}$, $\mathrm{Cliff}(7) \simeq \mathrm{Cliff}(-1)$, and $\mathrm{Cliff}(8) \simeq {\mathbb R}$ arise from quantizing the Hamiltonian reductions ${\mathbb R}^{0|4} // \mathrm{Spin}(3)$, ${\mathbb R}^{0|7} // G_2$, and ${\mathbb R}^{0|8} // \mathrm{Spin}(7)$, respectively.

Finite dimensional Hopf actions on algebraic quantizations

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory, due to A. Perucca.

Compatibility of t-structures for quantum symplectic resolutions

Let W be a smooth complex quasiprojective variety with the action of a connected reductive group G. Adapting the stratification approach of Teleman to a microlocal context, we prove a vanishing theorem for the functor of G-invariant sections---i.e., of quantum Hamiltonian reduction---for G-equivariant twisted D-modules on W. As a consequence, when W is affine we establish an effective combinatorial criterion for exactness of the global sections functors of microlocalization theory. When combined with our earlier derived equivalence results, this gives precise criteria for "microlocalization of representation categories."

Representations of affine superalgebras and mock theta functions II

We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra sℓˆ2|1 can be modified, using Zwegers' real analytic corrections, to form an SL2(Z)-invariant family of functions. Using a variation of Zwegers' correction, we obtain a similar result for ospˆ3|2. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the N=2 (resp. N=3) superconformal algebras with central charge c=3(1−2m+2M), where m∈Z≥0, M∈Z≥2, gcd⁡(2m+2,M)=1 if m>0 (resp. c=−32m+1M, where m∈Z≥0, M∈Z≥2gcd⁡(4m+2,M)=1), whose modified supercharacters form an SL2(Z)-invariant family of functions.

The Shatashvili-Vafa G 2 superconformal algebra as a quantum Hamiltonian reduction of D(2, 1; α)

We obtain the superconformal algebra associated to a sigma model with target a manifold with G2 holonomy, i.e., the Shatashvili-Vafa G2 algebra as a quantum Hamiltonian reduction of the exceptional Lie superalgebra D(2, 1; α) for α = 1. We produce the complete family of W-algebras SW (3/2, 3/2, 2) (extensions of the N = 1 superconformal algebra by two primary supercurrents of conformal weight 3/2 and 2 respectively) as a quantum Hamiltonian reduction of D(2, 1; α). As a corollary we find a free field realization of the Shatashvili-Vafa G2 algebra, and an explicit description of the screening operators.

BRST COHOMOLOGIES FOR SYMPLECTIC REFLECTION ALGEBRAS AND QUANTIZATIONS OF HYPERTORIC VARIETIES

We study algebras constructed by quantum Hamiltonian reduction associated with symplectic quotients of symplectic vector spaces, including deformed preprojective algebras, symplectic reection algebras (rational Cherednik algebras), and quantization of hypertoric varieties introduced by Musson and Van den Bergh in [MVdB]. We determine BRST cohomologies associated with these quantum Hamiltonian reductions. To compute these BRST cohomologies, we make use of the method of deformation quantization (DQ-algebras) and F-action studied by Kashiwara and Rouquier in [KR], and Gordon and Losev in [GL].

Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry

We investigate whether there are unitary families of W-algebras with spin one fields in the natural example of the Feigin-Semikhatov W^(2)_n-algebra. This algebra is conjecturally a quantum Hamiltonian reduction corresponding to a non-principal nilpotent element. We conjecture that this algebra admits a unitary real form for even n. Our main result is that this conjecture is consistent with the known part of the operator product algebra, and especially it is true for n=2 and n=4. Moreover, we find certain ranges of allowed levels where a positive definite inner product is possible. We also find a unitary conformal field theory for every even n at the special level k+n=(n+1)/(n-1). At these points, the W^(2)_n-algebra is nothing but a compactified free boson. This family of W-algebras admits an 't Hooft limit that is similar to the original minimal model 't Hooft limit. Further, in the case of n=4, we reproduce the algebra from the higher spin gravity point of view. In general, gravity computations allow us to reproduce some leading coefficients of the operator product.

REPRESENTATIONS OF AFFINE SUPERALGEBRAS AND MOCK THETA FUNCTIONS

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $\hat{s\ell}_{2|1}$ (resp. $\hat{ps\ell}_{2|2}$) can be modified, using Zwegers' real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-\frac{2m+2}{M})$, where $m \in \mathbb{Z}_{\geq 0}$, $M\in \mathbb{Z}_{\geq 2}$, $\gcd(2m+2,M)=1$ if $m>0$ (resp. $6(\frac{m}{M}-1)$, where $m \in \mathbb{Z}_{\geq 1}, M\in \mathbb{Z}_{\geq 2}$, $\gcd(2m,M)=1$ if $m>1$), whose modified characters and supercharacters form a modular invariant family.

Coset Constructions of Logarithmic (1, p) Models

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p = 2, 3, . . .) of central charge c 1, p =1 − 6(p − 1)2/p. This family includes the theories corresponding to the singlet algebras $${\mathcal{M}(p)}$$ and the triplet algebras $${\mathcal{W}(p)}$$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The $${W^{(2)}_n}$$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of $${\widehat{\mathfrak{sl}}(n)_k}$$ , generalising the Bershadsky–Polyakov algebra $${W^{(2)}_3}$$ . Inspired by work of Adamović for p = 3, vertex algebras $${\mathcal{B}_p}$$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra $${\mathcal{B}_p}$$ is a quotient of $${W^{(2)}_{p-1}}$$ at level −(p − 1)2/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra $${\mathcal{W}(p)}$$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra $${\mathcal{M}(p)}$$ is similarly realised inside $${\mathcal{B}_p}$$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p = 2 and 3.

Quantization of Drinfeld Zastava in type $A$

Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra \hat{sl}_n . We introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on Z in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra produces a quantization Y of the coordinate ring of Z . The same quantization was obtained in the finite (as opposed to the affine) case generically in [4]. We prove that, for generic values of quantization parameters, Y is a quotient of the affine Borel Yangian.

Isomorphisms of quantizations via quantization of resolutions

In this paper, we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations of Kleinian singularities obtained by three different constructions are isomorphic to each other. The constructions are via symplectic reflection algebras, quantum Hamiltonian reduction, and W-algebras. Next, we prove that parabolic W-algebras in type A are isomorphic to quantum Hamiltonian reductions associated to quivers of type A. Finally, we show that the symplectic reflection algebras for wreath-products of the symmetric group and a Kleinian group are isomorphic to certain quantum Hamiltonian reductions. Our results involving W-algebras are new, while for those dealing with symplectic reflection algebras we just find new proofs. A key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate Poisson varieties.

Good gradings of basic Lie superalgebras

We classify good ℤ-gradings of basic Lie superalgebras over an algebraically closed field \(\mathbb{F}\) of characteristic zero. Good ℤ-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W-algebras. We also describe the centralizer of a nilpotent even element and of an \(\mathfrak{s}\mathfrak{l}_2\)-triple in \(\mathfrak{g}\mathfrak{l}\left( {\left. m \right|n} \right)\) and \(\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {\left. m \right|2n} \right)\).

Almost-commuting variety, D-modules, and Cherednik algebras

We study a scheme M closely related to the set of pairs of n by n-matrices with rank 1 commutator. We show that M is a reduced complete intersection with n+1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Nil in M. We introduce a category, C, of D-modules whose characteristic variety is contained in Nil. Simple objects of that category are analogous to Lusztig's character sheaves. We construct a functor of Quantum Hamiltonian reduction from category C to the category O for type A rational Cherednik algebra.

The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant's Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand–Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand–Tsetlin modules using Gelfand–Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand–Tsetlin modules for finite W-algebras.

Differential operators and Cherednik algebras

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction [GS1]; the other involving quantum hamiltonian reduction of an algebra of differential operators [GG]. In this paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on $${\fancyscript {D}}$$ -modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result [GS1, Theorem 1.4] without recourse to Haiman’s deep results on the n! theorem [Ha1]. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG].

A Lie-Theoretic Construction of Spherical Symplectic Reflection Algebras

We propose a construction of the spherical subalgebra of a symplectic reection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semisimple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra.

Twisted spin Sutherland models from quantum Hamiltonian reduction

Recent general results on Hamiltonian reductions under polar group actions are applied to study some reductions of the free particle governed by the Laplace–Beltrami operator of a compact, connected, simple Lie group. The reduced systems associated with arbitrary finite-dimensional irreducible representations of the group by using the symmetry induced by twisted conjugations are described in detail. These systems generically yield integrable Sutherland-type many-body models with spin, which are called twisted spin Sutherland models if the underlying twisted conjugations are built on non-trivial Dynkin diagram automorphisms. The spectra of these models can be calculated, in principle, by solving certain Clebsch–Gordan problems, and the result is presented for the models associated with the symmetric tensorial powers of the defining representation of SU(N).

On the self-adjointness of certain reduced laplace-beltrami operators

The self-adjointness of the reduced Hamiltonian operators arising from the Laplace-Beltrami operator of a complete Riemannian manifold through quantum Hamiltonian reduction based on a compact isometry group is studied. A simple sufficient condition is provided that guarantees the inheritance of essential self-adjointness onto a certain class of restricted operators and allows us to conclude the self-adjointness of the reduced Laplace-Beltrami operators in a concise way. As a consequence, the self-adjointness of spin Calogero-Sutherland type reductions of ‘free’ Hamiltonians under polar actions of compact Lie groups follows immediately.

Hamiltonian reductions of free particles under polar actions of compact Lie groups

Classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds are investigated. The reduced systems are described under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifolds meeting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of spin Calogero-Sutherland type.