Equivalence of star products on a symplectic manifold; an introduction to Deligne''s ech cohomology classes

These notes grew out of the Quantisation Seminar 1997–1998 on Deligne's paper [P. Deligne, Déformations de l'algèbre des fonctions d'une variété symplectique: Comparison entre Fedosov et De Wilde, Lecomte, Selecta Math. (New Series) 1 (1995) 667–697] and the lecture of the first author in the Workshop on Quantisation and Momentum Maps at the University of Warwich in December 1997.We recall the definitions of the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manfold and show the properties of the relations between these classes by elementary methods based on ech cohomology.     

Notes on discrete torsion in orientifolds

In this paper we discuss discrete torsion in orientifolds. In particular, we apply the physical understanding of discrete torsion worked out several years ago, as group actions on B$B$ fields, to the case of orientifolds, and recover some old results of Braun and Stefanski concerning group cohomology and twisted equivariant K theory. We also derive new results including phase factors for nonorientable worldsheets and analogues for C$C$ fields.     

Para-Hopf Algebroids and their Cyclic Cohomology

We introduce the concept of para-Hopf algebroid and define their cyclic cohomology in the spirit of Connes–Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, including the Connes–Moscovici algebra , are para-Hopf algebroids     

Some Remarks on the Cohomology of Krichever–Novikov Algebras

We calculate the continuous cohomology of the Lie algebra of meromorphic vector fields on a compact Riemann surface from the cohomology of the holomorphic vector fields on the open Riemann surface pointed in the poles. This cohomology has been given by Kawazumi. Our result shows the Feigin–Novikov conjecture.     

Effects of pressure on flame propagation in a premixture containing fine fuel droplets

Combustion experiments on fuel droplet–vapor–air mixtures have been performed with a rapid expansion apparatus which generates monodispersed droplet clouds with narrow diameter distribution using the condensation method. The effects of fine fuel droplets on flame propagation were investigated for ethanol droplet–vapor–air mixtures at various pressures from 0.2 to 1.0 MPa. A stagnant fuel droplet–vapor–air mixture, generated in a rapid expansion chamber, was ignited at the center of the chamber using an ignition wire. Spherical flame propagation under constant-pressure conditions was observed with a high-speed video camera and flame speed was measured. Total equivalence ratio, and the ratio of liquid fuel mass to total fuel mass, was varied from 0.6 to 1.4 and from zero to 56%, respectively. The mean droplet diameter of fuel droplet–vapor–air mixtures was set at 8.5 and 11 μm. It was found that the flame speed of droplet–vapor–air mixtures less than 0.9 in the total equivalence ratio exceeds that of premixed gases of the same total equivalence ratio at all pressures. The flame speed of fuel droplet–vapor–air mixtures decreases as the pressure increases in all total equivalence ratios. At large ratios of liquid fuel mass to total fuel mass, the normalized flame speed (the flame speed of droplet–vapor–air mixtures divided by the flame speed of the premixed gas with the same total equivalence ratio), increases with the increase in pressure for fuel-lean mixtures, and it decreases for fuel-rich mixtures. The outcome is reversed at small ratios of liquid fuel mass to total fuel mass; the normalized flame speed decreases with the increase in pressure for fuel-lean mixtures, and increases for fuel-rich mixtures. The results suggest that the increase in pressure promotes droplet evaporation in the preheat zone.     

Unitons of Harmonic Maps from R2 to U(p,q)

We calculate a second cohomology class which determines a deformation quantization up to equivalence for a deformation quantization with separation of variables on a Kähler manifold, following P. Deligne.     

Integrality theorems in the theory of topological strings

We give a simplified derivation of the expression of instanton numbers and of mirror map in terms of Frobenius map on p-adic cohomology and use this expression to prove integrality theorems. Modifying this proof we verify that the Aganagic–Vafa formulas for the number of holomorphic disks can be expressed in terms of Frobenius map on p-adic relative cohomology; this expression permits us to prove integrality of this number.     

K-Theory, Reality, and Orientifolds

We use equivariant K-theory to classify charges of new (possibly non-supersymmetric) states localized on various orientifolds in Type II string theory. We also comment on the stringy construction of new D-branes and demonstrate the discrete electric-magnetic duality in Type I brane systems with p+q=7, as proposed by Witten.     

BRST Cohomology of the Chapline–Manton Model

We completely compute the local BRST cohomology H(s|d) of the combined Yang–Mills 2-form system coupled through the Yang–Mills Chern–Simons term (Chapline–Manton model). We consider the case of a simple gauge group and explicitly include in the analysis the sources for the BRST variations of the fields (antifields). We show that there is an antifield independent representative in each cohomological class of H(s|d) at ghost number 0 or 1. Accordingly, any counterterm may be assumed to preserve the gauge symmetries. Similarly, there is no new candidate anomaly beside those already considered in the literature, even when one takes the antifields into account. We then characterize explicitly all the nontrivial solutions of the Wess–Zumino consistency conditions. In particular, we provide a cohomological interpretation of the Green–Schwarz anomaly cancellation mechanism.     

On the computation of the Lichnerowicz–Jacobi cohomology

Lichnerowicz–Jacobi cohomology of Jacobi manifolds is reviewed. The use of the associated Lie algebroid allows to prove that the Lichnerowicz–Jacobi cohomology is invariant under conformal changes of the Jacobi structure. We also compute the Lichnerowicz–Jacobi cohomology for a large variety of examples.     

Morita Invariant Properties of Twisted Poisson Manifolds

In this letter, we study some invariants of integrable twisted Poisson manifolds under Morita equivalence. We show that two twisted symplectic manifolds are Morita equivalent if and only if their fundamental groups are isomorphic. Moreover, the first cohomology group of twisted Poisson manifolds is invariant under Morita equivalence. Finally, we prove that if two twisted Poisson manifolds are Morita equivalent, then their representation categories are equivalent.      

Anatomy of a gauge theory

We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson–Schwinger equations, the quantum equations of motion. These equations emerge from Hopf- and Lie algebra theory and free quantum field theory only. In the course of our analysis, we exhibit an intimate relation between the Slavnov–Taylor identities for the couplings and the existence of Hopf sub-algebras defined on the sum of all graphs at a given loop order, surpassing the need to work on single diagrams.     

Cohomology and associated deformations for not necessarily co-associative bialgebras

In this Letter, a cohomology and an associated theory of deformations for (not necessarily co-associative) bialgebras are studied. The cohomology was introduced in a previous paper (Lett. Math. Phys. 25, 75–84 (1992)). This theory has several advantages, especially in calculating cohomology spaces and in its adaptability to deformations of quasi-co-associative (qca) bialgebras and even quasi-triangular qca bialgebras.     

Slow-roll inflation in non-geometric flux compactification

By implementing a genetic algorithm we search for stable vacua in Type IIB non-geometric flux compactification on an isotropic torus with orientifold 3-planes. We find that the number of stable dS and AdS vacua are of the same order. Moreover we find that in all dS vacua the multi-field slow-roll inflationary conditions are fulfilled. Specifically we observe that inflation is driven by the axio-dilaton and the Kähler moduli. We also comment on the existence of one stable dS vacuum in the presence of exotic orientifolds.     

Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map in the Cartan model of equivariant cohomology. As a byproduct, we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.     

Projective unitary representations of smooth Deligne cohomology groups

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle at level 2. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations being finite is determined by the cohomology of the manifold.     

BRST cohomology of the super-Virasoro Algebras

We study the superextension of the semi-infinite cohomology theory of the Virasoro Algebra. In particular, we examine the BRST complex with coefficients in the Fock Space of the RNS superstring. We prove a theorem of vanishing cohomology, and establish the unitary equivalence between a positive definite transversal space, a physical subspace and the zeroth cohomology group. The cohomology of a subcomplex is identified as the covariant equivalent of the well-known GSO subspace. An exceptional case to the vanishing theorem is discussed.Supported by NSF Grant DMS-8703581     

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds

We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painlevé VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations.     

Classification of Equivariant Star Products on Symplectic Manifolds

In this note, we classify invariant star products with quantum momentum maps on symplectic manifolds by means of an equivariant characteristic class taking values in the equivariant cohomology. We establish a bijection between the equivalence classes and the formal series in the second equivariant cohomology, thereby giving a refined classification which takes into account the quantum momentum map as well.