Obstructions for twist star products

In this short note, we point out that not every star product is induced by a Drinfel’d twist by showing that not every Poisson structure is induced by a classical r-matrix. Examples include the higher genus symplectic Pretzel surfaces and the symplectic sphere \({\mathbb {S}}^2\).     

Projective Structure,Symplectic Connection and Quantization

Let X be a connected Riemann surface equipped with a projective structure . Let E be a holomorphic symplectic vector bundle over X equipped with a flat connection. There is a holomorphic symplectic structure on the total space of the pullback of E to the space of all nonzero holomorphic cotangent vectors on X. Using , this symplectic form is quantized. A moduli space of Higgs bundles on a compact Riemann surface has a natural holomorphic symplectic structure. Using , a quantization of this symplectic form over a Zariski open subset of the moduli space of Higgs bundles is constructed.     

Natural and Projectively Equivariant Quantizations by means of Cartan Connections

The existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of Thomas–Whitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of these connections. Our method yields the existence of a projectively equivariant quantization if and only if an -equivariant quantization exists in the flat situation in the sense of [18], thus solving one of the problems left open by M. Bordemann.Mathematics Subject Classification (2000). 53B05, 53B10, 53D50, 53C10     

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.     

Reciprocal Relativity of Noninertial Frames and the Quaplectic Group

The frame associated with a classical point particle is generally noninertial. The point particle may have a nonzero velocity and force with respect to an absolute inertial rest frame. In time–position–energy–momentum-space {t, q, p, e}, the group of transformations between these frames leaves invariant the symplectic metric and the classical line element ds² = d t². Special relativity transforms between inertial frames for which the rate of change of momentum is negligible and eliminates the absolute rest frame by making velocities relative but still requires the absolute inertial frame. The Lorentz group leaves invariant the symplectic metric and the line elements and . General relativity for particles under only the influence of gravity avoids the issue of noninertial frames as all particles follow geodesics and hence have locally inertial frames. For other forces, the question of the absolute inertial frame remains.) Born conjectured that the line element should be generalized to the pseudo-orthogonal metric . The group leaving this metrics and the symplectic metric invariant is the pseudo-unitary group of transformations between noninertial frames. We show that these transformations also eliminate the need for an absolute inertial frame by making forces relative and bounded by b and so embodies a relativity that is shape reciprocal in the sense of Born. The inhomogeneous version of this group is naturally the semidirect product of the pseudo-unitary group with the nonabelian Heisenberg group. This is the quaplectic group.     

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      

Toward $$\mathrm {U}(N|M)$$ knot invariant from ABJM theory

We study \(\mathrm {U}(N|M)\) character expectation value with the supermatrix Chern–Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half-BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives \(\mathrm {U}(N|M)\) character expectation values in terms of \(\mathrm {U}(1|1)\) averages for a particular type of character representations. This means that the \(\mathrm {U}(1|1)\) character expectation value is a building block for the \(\mathrm {U}(N|M)\) averages and also, by an appropriate limit, for the \(\mathrm {U}(N)\) invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern–Simons matrix model. We obtain the Rosso–Jones-type formula and the spectral curve for this case.     

Equivariant GW Theory of Stacky Curves

We extend Okounkov and Pandharipande’s work on the equivariant Gromov–Witten theory of ${\mathbb{P}^1}$ to a class of stacky curves ${\mathcal{X}}$ . Our main result uses virtual localization and the orbifold ELSV formula to express the tau function ${\tau_\mathcal{X}}$ as a vacuum expectation on a Fock space. As corollaries, we prove the decomposition conjecture for these ${\mathcal{X}}$ , and prove that ${\tau_\mathcal{X}}$ satisfies a version of the 2-Toda hierarchy. Coupled with degeneration techniques, the result should lead to treatment of general orbifold curves.     

Construction of a central extension of a Lie group from its class of symplectic cohomology

Bargmann’s group is a central extension of Galilei group motivated by quantum-theoretical considerations. Bargmann’s work suggests that one of the reasons of the failure of naïve attemps to construct actions on quantum wave functions has a cohomologic origin. It is this point, we develop in the context of Lie groups with symplectic actions. Studying the co-adjoint representation of a central extension of a group G$G$, we highlight the link between the extension cocycles and the symplectic cocycles of G$G$. Also, each extension coboundary corresponds to a symplectic coboundary. Finally, we emphasize the condition to be satisfied by the extension cocycle for the class of symplectic cohomology of the extension being null. The method is illustrated by application to Physics.     

An Infinite Genus Mapping Class Group and Stable Cohomology

We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion . L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.     

Gauge-theoretic invariants for topological insulators: a bridge between Berry,Wess–Zumino,and Fu–Kane–Mele

We establish a connection between two recently proposed approaches to the understanding of the geometric origin of the Fu–Kane–Mele invariant \(\mathrm {FKM}\in \mathbb {Z}_2\), arising in the context of two-dimensional time-reversal symmetric topological insulators. On the one hand, the \(\mathbb {Z}_2\) invariant can be formulated in terms of the Berry connection and the Berry curvature of the Bloch bundle of occupied states over the Brillouin torus. On the other, using techniques from the theory of bundle gerbes, it is possible to provide an expression for \(\mathrm {FKM}\) containing the square root of the Wess–Zumino amplitude for a certain U(N)-valued field over the Brillouin torus. We link the two formulas by showing directly the equality between the above-mentioned Wess–Zumino amplitude and the Berry phase, as well as between their square roots. An essential tool of independent interest is an equivariant version of the adjoint Polyakov–Wiegmann formula for fields \(\mathbb {T}^2 \rightarrow U(N)\), of which we provide a proof employing only basic homotopy theory and circumventing the language of bundle gerbes.     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

Two-dimensional topological gravity and equivariant cohomology

The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the derived functor of semi-relative cohomology, just as equivariant cohomology is the derived functor of basic cohomology. That homological algebra finds a place in the study of topological string theory should not surprise the reader, granted that topological string theory is the conformal field theorist's algebraic topology.In [7], we have shown that the cohomology of a topological conformal field theory carries the structure of a batalin-Vilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations {a ₁, ...,a _k },k2, satisfying quadratic relations generalizing the Jacobi rule. (The operad underlying the category of gravity algebras has been studied independently by Ginzburg-Kapranov [9].)The author is grateful to M. Bershadsky, E. Frenkel, M. Kapranov, G. Moore, R. Plesser and G. Zuckerman for the many ways in which they helped in the writing of this paper; also to the Department of Mathematics at Yale University for its hospitality while part of this paper was written.The author is partially supported by a fellowship of the Sloan Foundation and a research grant of the NSF.     

Geometric Quantization, Parallel Transport and the Fourier Transform

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle over the space of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.     

Courant–Dorfman Algebras and their Cohomology

We introduce a new type of algebra, the Courant–Dorfman algebra. These are to Courant algebroids what Lie–Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant–Dorfman algebra ${(\mathcal{R}, \mathcal{E})}$ we associate a differential graded algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ in a functorial way by means of explicit formulas. We describe two canonical filtrations on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ , and derive an analogue of the Cartan relations for derivations of ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; we classify central extensions of ${\mathcal{E}}$ in terms of ${H^2(\mathcal{E}, \mathcal{R})}$ and study the canonical cocycle ${\Theta \in \mathcal{C}^3(\mathcal{E}, \mathcal{R})}$ whose class ${[\Theta]}$ obstructs re-scalings of the Courant–Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; for Courant–Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.     

Equivariant Symbol Calculus for Differential Operators Acting on Forms

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces of differential operators transforming p-forms into functions, over . As an application, we classify the Vect(M)-equivariant maps from to over a smooth manifold M, recovering and improving earlier results of N.Poncin. This provides the complete answer to a question raised by P. Lecomte about the extension of a certain intrinsic homotopy operator.