Chiral Differential Operators Research Articles

Cosets from equivariant 𝒲-algebras

The equivariant W \mathcal W -algebra of a simple Lie algebra g \mathfrak {g} is a BRST reduction of the algebra of chiral differential operators on the Lie group of g \mathfrak {g} . We construct a family of vertex algebras A [ g , κ , n ] A[\mathfrak {g}, \kappa , n] as subalgebras of the equivariant W \mathcal W -algebra of g \mathfrak {g} tensored with the integrable affine vertex algebra L n ( g ˇ ) L_n(\check {\mathfrak {g}}) of the Langlands dual Lie algebra g ˇ \check {\mathfrak {g}} at level n ∈ Z > 0 n\in \mathbb {Z}_{>0} . They are conformal extensions of the tensor product of an affine vertex algebra and the principal W \mathcal {W} -algebra whose levels satisfy a specific relation. When g \mathfrak {g} is of type A D E ADE , we identify A [ g , κ , 1 ] A[\mathfrak {g}, \kappa , 1] with the affine vertex algebra V κ − 1 ( g ) ⊗ L 1 ( g ) V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g}) , giving a new and efficient proof of the coset realization of the principal W \mathcal W -algebras of type A D E ADE .

Quantum coordinate ring in WZW model and affine vertex algebra extensions

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case.

Chiral differential operators on the upper half plane and modular forms

In this paper we study the vertex operator algebra Dch(H,Γ) constructed from the fixed points of the chiral differential operators on the upper half plane which is holomorphic at all the cusps, under the action of a congruence subgroup Γ. To this end, we introduce an SL(2,R)-invariant filtration labeled by partition pairs and study its successive quotient. We show that the successive quotient under the cuspidal condition is isomorphic to the space of modular forms. We also give a description of the structure of Dch(H,Γ) and compute its character.

Logarithmic modules for chiral differential operators of nilmanifolds

We describe explicitly the vertex algebra of (twisted) chiral differential operators on certain nilmanifolds and construct their logarithmic modules. This is achieved by generalizing the construction of vertex operators in terms of exponentiated scalar fields to Jacobi theta functions naturally appearing in these nilmanifolds. This construction furnishes non-trivial examples of logarithmic vertex algebra modules, a notion recently introduced by Bakalov.

AST419 - Chiral Differential Operators via Batalin-Vilkovisky Quantization

We show that the local observables of the curved beta gamma system encode the sheaf of chiral differential operators using the machinery of the book Factorization algebras in quantum field theory, by Kevin Costello and the second author, which combines renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry. We begin by constructing a quantization of the beta gamma system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of formal vector fields by the closed 2-forms on the disk. By results in the book listed above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal beta gamma vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs on the complex manifold.

Little strings, quasi-topological sigma model on loop group, and toroidal Lie algebras

We study the ground states and left-excited states of the Ak−1N=(2,0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP1 with target space the based loop group of SU(k). The ground states, described by L2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.

Chiral differential operators: Formal loop group actions and associated modules

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of 2-dimensional σ-models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a “formal loop group action” on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said “formal loop group actions” and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of “formal loop spaces”. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.

Localization of Affine W-Algebras

We introduce the notion of an asymptotic algebra of chiral differential operators. We then construct, via a chiral Hamiltonian reduction, one such algebra over a resolution of the intersection of the Slodowy slice with the nilpotent cone. We compute the space of global sections of this algebra thereby proving a localization theorem for affine W-algebras at the critical level.

W-algebras and chiral differential operators at the critical level

Let Acrit be the chiral algebra corresponding to the affine Kac–Moody algebra at the critical level gˆcrit. Let Zcrit be the center of Acrit. The commutative chiral algebra Zcrit admits a canonical deformation into a non-commutative chiral algebra Wℏ. In this paper we will express the resulting first order deformation via the chiral algebra Dcrit of chiral differential operators on G((t)) at the critical level.

Chiral algebras of (0, 2) models

We explore two-dimensional sigma models with (0,2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0,2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of K\ahler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the H\ohn-Stolz conjecture.

Chiral algebras of (0, 2) models

We explore two-dimensional sigma models with (0,2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0,2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of K\"ahler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the H\"ohn-Stolz conjecture.

A chiral Borel–Weil–Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of G-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld–Sokolov reduction as a derived functor.

Quasi-topological gauged sigma models, the geometric Langlands program, and knots

We construct and study a closed, two-dimensional, quasi-topological (0, 2) gauged sigma model with target space a smooth G-manifold, where G is any compact and connected Lie group. When the target space is a flag manifold of simple G, and the gauge group is a Cartan subgroup thereof, the perturbative model describes, purely physically, the recently formulated mathematical theory of “Twisted Chiral Differential Operators”. This paves the way, via a generalized T -duality, for a natural physical interpretation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups. In particular, the Hecke eigensheaves and Hecke operators can be described in terms of the correlation functions of certain operators that underlie the infinite-dimensional chiral algebra of the flag manifold model. Nevertheless, nonperturbative worldsheet twisted-instantons can, in some situations, trivialize the chiral algebra completely. This leads to a spontaneous breaking of supersymmetry whilst implying certain delicate conditions for the existence of BeilinsonDrinfeld D-modules. Via supersymmetric gauged quantum mechanics on loop space, these conditions can be understood to be intimately related to a conjecture by Hohn-Stolz [1] regarding the vanishing of the Witten genus on string manifolds with positive Ricci curvature. An interesting connection to Chern-Simons theory is also uncovered, whence we would be able to (i) relate general knot invariants of three-manifolds and Khovanov homology to “quantum” ramified D-modules and Lagrangian intersection Floer homology; (ii) furnish physical proofs of mathematical conjectures by Seidel-Smith [2] and Gaitsgory [3, 4] which relate knots to symplectic geometry and Langlands duality, respectively. ∗E-mail: tan@ias.edu ar X iv :1 11 1. 06 91 v1 [ he pth ] 2 N ov 2 01 1

Chiral differential operators on supermanifolds

The first part of this paper provides a new description of chiral differential operators (CDOs) in terms of global geometric quantities. The main result is a recipe to define all sheaves of CDOs on a smooth cs-manifold; its ingredients consist of an affine connection ∇ and an even 3-form that trivializes p1(∇). With ∇ fixed, two suitable 3-forms define isomorphic sheaves of CDOs if and only if their difference is exact. Moreover, conformal structures are in one-to-one correspondence with even 1-forms that trivialize c1(∇). Applying our work in the first part, we then construct what may be called “chiral Dolbeault complexes” of a complex manifold M, and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings (by fermion numbers). When M is compact, their cohomology computes (in various cases) the Witten genus, the two-variable elliptic genus and a spinc version of the Witten genus. This part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic CDOs and σ-models.

Supersymmetry and the formal loop space

For any algebraic super-manifold M we define the super-ind-scheme LM of formal loops and study the transgression map (Radon transform) on differential forms in this context. Applying this to the super-manifold M = S X , the spectrum of the de Rham complex of a manifold X, we obtain, in particular, that the transgression map for X is a quasi-isomorphism between the [ 2 , 3 ) -truncated de Rham complex of X and the additive part of the [ 1 , 2 ) -truncated de Rham complex of L X . The proof uses the super-manifold SS X and the action of the Lie super-algebra sl ( 1 | 2 ) on this manifold. This quasi-isomorphism result provides a crucial step in the classification of sheaves of chiral differential operators in terms of geometry of the formal loop space.

Algebras of twisted chiral differential operators and affine localization of $${\mathfrak {g}}$$ -modules

We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest” such modules are irreducible \({{\hat{{\mathfrak{g}}}}}\) -modules, and all irreducible \({{\mathfrak{g}}}\) -integrable \({{\hat{{\mathfrak{g}}}}}\) -modules at the critical level arise in this way.

Vertex Algebroids over Veronese Rings

We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.

Invariant chiral differential operators and the [formula omitted] algebra

Attached to a vector space V is a vertex algebra S ( V ) known as the β γ -system or algebra of chiral differential operators on V . It is analogous to the Weyl algebra D ( V ) , and is related to D ( V ) via the Zhu functor. If G is a connected Lie group with Lie algebra g , and V is a linear G -representation, there is an action of the corresponding affine algebra on S ( V ) . The invariant space S ( V ) g [ t ] is a commutant subalgebra of S ( V ) , and plays the role of the classical invariant ring D ( V ) G . When G is an abelian Lie group acting diagonally on V , we find a finite set of generators for S ( V ) g [ t ] , and show that S ( V ) g [ t ] is a simple vertex algebra and a member of a Howe pair. The Zamolodchikov W 3 algebra with c = − 2 plays a fundamental role in the structure of S ( V ) g [ t ] .

Formal loops III: Additive functions and the Radon transform

To any algebraic variety X and closed 2-form ω on X, we associate the “symplectic action functional” T ( ω ) which is a function on the formal loop space LX introduced by the authors earlier. The correspondence ω → T ( ω ) can be seen as a version of the Radon transform. We give a characterization of the functions of the form T ( ω ) in terms of factorizability (infinitesimal analog of additivity in holomorphic pairs of pants) as well as in terms of vertex operator algebras. These results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on X (whose lien is the sheaf of closed 2-forms) and the determinantal gerbe of the tangent bundle of LX (whose lien is the sheaf of invertible functions on LX). On the level of liens this relation associates to a closed 2-form ω the invertible function exp T ( ω ) .

The first Pontryagin class

We give a natural obstruction theoretic interpretation to the first Pontryagin class in terms of Courant algebroids. As an application we calculate the class of the stack of algebras of chiral differential operators. In particular, we establish the existence and uniqueness of the chiral de Rham complex.