Chiral Equivariant Cohomology of a Point: A First Look

The chiral equivariant cohomology contains and generalizes the classical equivariant cohomology of a manifold M with an action of a compact Lie group G. For any simple G, there exist compact manifolds with the same classical equivariant cohomology, which can be distinguished by this invariant. When M is a point, this cohomology is an interesting conformal vertex algebra whose structure is still mysterious. In this paper, we scratch the surface of this object in the case G = SU(2).     

Topology of generalized complex quotients

Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold M$M$. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well.     

A Finite Analog of the AGT Relation I: Finite <Emphasis Type="Italic">W</Emphasis>-Algebras and Quasimaps’ Spaces

Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on \mathbbP²{\mathbb{P}^2} . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties.     

Crossed Module Actions on Continuous Trace C*-Algebras

We lift an action of a torus \({\mathbb{T}^n}\) on the spectrum of a continuous trace algebra to an action of a certain crossed module of Lie groups that is an extension of \({\mathbb{R}^n}\). We compute equivariant Brauer and Picard groups for this crossed module and describe the obstruction to the existence of an action of \({\mathbb{R}^n}\) in our framework.     

Torus Fibrations and Localization of Index III

This paper is the third of the series concerning the localization of the index of Dirac-type operators. In our previous papers we gave a formulation of index of Dirac-type operators on open manifolds under some geometric setting, whose typical example was given by the structure of a torus fiber bundle on the ends of the open manifolds. We introduce two equivariant versions of the localization. As an application, we give a proof of Guillemin-Sternberg’s quantization conjecture in the case of torus action.     

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds.     

Topology and Flux of T-Dual Manifolds with Circle Actions

We present an explicit formula for the topology and H-flux of the T-dual of a general type II, compactification, significantly generalizing earlier results. Our results apply to T-dualities with respect to any circle action on spacetime X. As before, T-duality exchanges type IIA and type IIB string theories. A new consequence is that the T-dual spacetime is a singular space when the fixed point set ${X^\mathbb{T}}$ is non-empty; the singularities correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges of type II string theories on the singular dual are classified by twisted equivariant cohomology groups. We also discuss the K-theory approach.     

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA ₁ ⁽¹⁾ introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.     

Torus Equivariant Spectral Triples for Odd-Dimensional Quantum Spheres Coming from <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Extensions

The torus group (S ¹)^ℓ+1 has a canonical action on the odd-dimensional sphere S _q ^2ℓ+1. We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial K-homology class thus generalizing our earlier results for SU _q (2). We also relate the triple we construct with the C ^*-extension      

Interwining Operators for Vertex Representations of Toroidal Lie Algebras

The construction of the vertex representation of the toroidal Lie algebra ^[n] depends on the way of labelling the points in the dual ⁿ of the torus Tⁿ. Thus there is a built-in symmetry of the vertex representation with respect to the symmetry of Zⁿ. In conjunction with this, the energy operator L₀ gives rise to intertwining operators which reflect the symmetry of the vertex representation with respect to S¹ action on .     

Twisted K-Theory and Finite-Dimensional Approximation

We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators.     

Vertex operator representations of the quantum affine algebra U q(B r (1) )

We construct the level one vertex operator representations of the q-deformation U _q(B _r ⁽¹⁾ ) of the affine Kac-Moody algebra B _r ⁽¹⁾ . Beside the q-deformed vertex operators introduced by Frenkel and Jing, this construction involves a q-deformation of free fermionic fields.     

Topological Open Strings on Orbifolds

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3} orbifold point of local \mathbb P²{{\mathbb P}^2}, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov–Witten invariants of \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3}, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.     

Infinite dimensional groups and Riemann surface field theories

We show how to obtain positive energy representations of the groupG of smooth maps from a union of circles toU(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case whereN=1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity ofG which have the form of a Fock representation of an infinite dimensional Heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs. We give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum (or cyclic) vector. (This gives a geometric construction of correlation functions which had been obtained earlier using tau functions.) By choosing particular functions inG with non-zero winding numbers on the boundary we obtain analogues of vertex operators described by Segal in the genus zero case. These special elements ofG (which have a simple interpretation in terms of function theory on theRiemann surface) approximate fermion (or Clifford algebra) operators. They enable a rigorous derivation of a form of boson-fermion correspondence in the sense that we construct generators of a Clifford algebra from the unitaries representing these elements ofG.     

Ergodic Properties of the Quantum Geodesic Flow on Tori

We study ergodic averages for a class of pseudodifferential operators on the flatN-dimensional torus with respect to the Schrödinger evolution. The later can be consider a quantization of the geodesic flow on . We prove that, up to semi-classically negligible corrections, such ergodic averages are translationally invariant operators.Mathematics Subject Classifications (2000) 58J50, 58J40, 81S10.     

A Vertex Formalism for Local Ruled Surfaces

We develop a vertex formalism for topological string amplitudes on ruled surfaces with an arbitrary number of reducible fibers embedded in a Calabi-Yau threefold. Our construction is based on large N duality and localization with respect to a degenerate torus action. We also discuss potential generalizations of our formalism to a broader class of Calabi-Yau threefolds using the same underlying principles.     

Exts and the AGT Relations

We prove the connection between the Nekrasov partition function of \({\mathcal{N}=2}\) super-symmetric U(2) gauge theory with adjoint matter and conformal blocks for the Virasoro algebra, as predicted by the Alday–Gaiotto–Tachikawa relations. Mathematically, this is achieved by relating the Carlsson–Okounkov Ext vector bundle on the moduli space of rank 2 sheaves with Liouville vertex operators. Our approach is geometric in nature, and uses a new method for intersection-theoretic computations of the Ext operator.     

PV Cohomology of the Pinwheel Tilings, their Integer Group of Coinvariants and Gap-Labeling

In this paper, we first remind how we can see the “hull” of the pinwheel tiling as an inverse limit of simplicial complexes (Anderson and Putnam in Ergod Th Dynam Sys 18:509–537, 1998) and we then adapt the PV cohomology introduced in Savinien and Bellissard (Ergod Th Dynam Sys 29:997–1031, 2009) to define it for pinwheel tilings. We then prove that this cohomology is isomorphic to the integer Čech cohomology of the quotient of the hull by S ¹ which let us prove that the top integer Čech cohomology of the hull is in fact the integer group of coinvariants of the canonical transversal Ξ of the hull. The gap-labeling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap-labeling, showing that m^t ( C(X,\mathbb Z) ) = \frac1264\mathbb Z [ \frac15]{\mu^t \left( C(\Xi,\mathbb {Z}) \right) = \frac{1}{264}\mathbb {Z} \left [ \frac{1}{5}\right ]}.     

Dirac Operators on Quantum Projective Spaces

We construct a family of self-adjoint operators D _N , ${N\in{\mathbb Z}}We construct a family of self-adjoint operators D _N , N ? \mathbb Z{N\in{\mathbb Z}} , which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space \mathbb CP^l_q{{\mathbb C}{\rm P}^{\ell}_q} , for any ℓ ≥ 2 and 0 < q < 1. They provide 0⁺-dimensional equivariant even spectral triples. If ℓ is odd and N=\frac12(l+1){N=\frac{1}{2}(\ell+1)} , the spectral triple is real with KO-dimension 2ℓ mod 8.