Noncommutative Geometry and Physics 2005

Preface.

Contents.

Ideas from deformation quantization are applied to deform the expression of elements of an algebra. Extending these ideas to certain transcendental elements implies that in the Weyl algebra is naturally viewed as an indeterminate living in a discrete set or . This may yield a more mathematical understanding of Dirac's positron theory.

The main result of this paper is Theorem 2.12 which says that the quotient μ^-1({1})/U associated to a quasi-hamiltonian space (M, ω, μ: M → U) has a symplectic structure even when 1 is not a regular value of the momentum map μ. Namely, it is a disjoint union of symplectic manifolds of possibly different dimensions, which generalizes the result of Alekseev, Malkin and Meinrenken in [AMM98]. We illustrate this theorem with the example of representation spaces of surface groups. As an intermediary step, we give a new class of examples of quasi-hamiltonian spaces: the isotropy submanifold M_K whose points are the points of M with isotropy group K ⊂ U.

The notion of quasi-hamiltonian space was introduced by Alekseev, Malkin and Meinrenken in their paper [AMM98]. The main motivation for it was the existence, under some regularity assumptions, of a symplectic structure on the associated quasi-hamiltonian quotient. Throughout their paper, the analogy with usual hamiltonian spaces is often used as a guiding principle, replacing Lie-algebra-valued momentum maps with Lie-group-valued momentum maps. In the hamiltonian setting, when the usual regularity assumptions on the group action or the momentum map are dropped, Lerman and Sjamaar showed in [LS91] that the quotient associated to a hamiltonian space carries a stratified symplectic structure. In particular, this quotient space is a disjoint union of symplectic manifolds. In this paper, we prove an analogous result for quasi-hamiltonian quotients. More precisely, we show that for any quasi-hamiltonian space (M, ω, μ: M → U), the associated quotient M//U := μ^-1({1})/U is a disjoint union of symplectic manifolds (Theorem 2.12):

Smooth Deligne cohomology groups can be thought of as gauge transformation groups of "higher abelian grebes". For a compact oriented Riemannian manifold of dimension 4k + 1, the smooth Deligne cohomology has certain projective unitary representations. We account for a meaning of these representations in a context of a field theory.

The following sections are included:

• Introduction

• Review of elements of Poisson geometry

• Algebroids related with Poisson structures
  • Lie algebroids
  • Dirac structures

• Leibniz algebra and Leibniz algebroid
  • Abelian extension of Leibniz algebra
  • Super Leibniz algebra and its cochain complex

• References

In the present paper, we introduce spin groupoids associated with the degree two Stiefel-Whitney classes, and we also introduce spinor torsors equipped with certain connections compatible with an action of spin groupoid. Using the connection, we propose a Dirac-like operator acting on the space of smooth sections of the spinor torsor. We can show that its Dirac-Laplacian admits a heat kernel in a formal sense.

For a transitive Lie algebroid on a connected manifold M and its a representation on a vector bundle F, we study the localization map , where L_x is the adjoint algebra at x ∈ M. The main result in this paper is that: . Here p^1_* is the lift of to its counterpart over the universal covering space and is the F₀ = H⁰(L, F)-coefficient deRham cohomology. We apply these results to study associated vector bundles with respect to some principal fiber bundle and the structure of transitive Lie bialgebroids.

The following sections are included:

• Introduction

• System of second-order PDE's

• Compactification of the flat system

• Orbit of diffeomorphisms

• Construction of Diffeomorphisms

• System of third-order PDE's

• Compactification of the flat system for 3OPDE

• Orbit of contact diffeomorphisms and Contact Schwarzian Derivatives

• Construction of Contact Diffeomorphisms

• References

A Batalin-Vilkovisky formalism is a most general framework to construct consistent quantum field theories. Its mathematical structure is called a Batalin-Vilkovisky structure. First we explain the rather mathematical setting of a Batalin-Vilkovisky formalism. Next, we consider deformation theory of a Batalin-Vilkovisky structure. Especially, we consider deformation of topological sigma models in any dimension, which is closely related to deformation theories in mathematics, including deformation from commutative geometry to noncommutative geometry. We obtain a series of new nontrivial topological sigma models and we find these models have the Batalin-Vilkovisky structures based on a series of new algebroids.

The following sections are included:

• Introduction

• Beating Derrick's theorem
  • Solitons in d = 1 + 2 scalar field theory
  • Noncommutative deformation
  • Moyal star product
  • Fock-space realization

• d = 0 + 2 sigma model
  • U_⋆(1) sigma model in d = 0 + 2_θ
  • Grassmannian subsectors
  • BPS configurations

• d = 1 + 2 sigma model
  • d = 1 + 2 Yang-Mills-Higgs and Ward model
  • Commutative Ward solitons
  • Co-moving coordinates
  • Time-space versus space-space deformation
  • U_⋆(1) Ward solitons
  • Ward multi-solitons
  • Ward soliton scattering
  • U_⋆(n) Ward solitons

• Moduli space dynamics
  • Manton's paradigm
  • Ward model metric
  • Adiabatic two-soliton scattering

• Stability analysis
  • Fluctuation Hessian
  • Diagonal U_⋆(1) soliton: fluctuation spectrum
  • Single negative eigenvalue
  • Instability in unitary sigma model

• d = 1 + 1 sine-Gordon solitons
  • Reduction to d = (1 + 1)_θ: instantons
  • d = 1 + 1 sigma model metric
  • Moyal deformation in d = 1 + 1
  • Reduction to d = (1 + 1)_θ: solitons
  • Integrable noncommutative sine-Gordon model
  • Noncommutative sine-Gordon kinks
  • One-kink configuration
  • Tree-level scattering of elementary quanta

• References

In this talk I review the well known relation existing between extended supersymmetry and complex geometry in the non-linear sigma-models, and then briefly discuss some recent developments related to the introduction of the non-anti-commutativity in the context of the supersymmetric non-linear sigma-models formulated in extended superspace. This contribution is suitable for both physicists and mathematicians interesting in the interplay between geometry, supersymmetry and non(anti)commutativity.

The following sections are included:

• Introduction

• N.C. Cohomological field theory

• Universality of Partition Functions

• Z of Super U(1) N.C.Theory

• SUSY Gauge Theory on N.C.R⁴

• D-brane interpretation

• Deformed BRS Operators

• Solutions

• Localization Theorem

• Conclusion

• References

The following sections are included:

• Introduction

• Instanton Counting

• Superstring Perspective and Discrete Matrix Model

• Large N Limit

• Topological M-theory and Non-critical M-theory

• References

We generalize the differential algebra on superspace to non(anti)commutative superspace by defining the deformed wedge product. Then, we formulate a non(anti)commutative version of the super ADHM construction which gives deformed instantons in super Yang-Mills theory with U(n) gauge group.

The Hopf algebraic construction of the non(anti)commutative superspace is discussed. In that context, Noncommutative (super)space arises as a representation of the modified Hopf algebra by the Drinfel'd twist. This construction gives the same result which is given by using the ordinary Moyal product, but has some advantages in considering the symmetry of the noncommutative theory. We construct the twisted super-Poincaré algebra, and obtain the several non(anti)commutative superspaces.

In this paper, the differential operators realization of Lie superalgebra gl(2|2)⁽²⁾ is given with coherent state methods. According to the differential operators realization of the Lie superalgebra, the free field realization of the affine Kac-Moody superalgebra is given. The primary field of the superalgebra is also given. It is all know that the primary field corresponding to the highest weight representation of algebra. In our case, we obtain sixteen series[5], for the complexity of the expression and the limit of the space, we given a little more simple but more important primary fields. This corresponds to the highest weight representation of twisted Lie superalgebra psl(2|2)⁽²⁾.

Inspired by Prof. Lu's lecture note [1], we will give the global solutions of Yang-Mills equation on de-Sitter space-time with BHL metric, instead of the conformal metric.

An exact solution of the vacuum Einstein's field equations is presented in which there exists a congruence of null geodesics whose shear behaves like a travelling wave of the KdV equation. On the basis of this exact solution, the feasibility of solitonic information transmission by exploiting the nonlinearity intrinsic to the Einstein field equations is discussed.

This is an extended version of our previous paper [1] on difference discrete connection and curvature on cubic lattice. Some new results about properties of discrete topological number on bundles with Abelian gauge group are added.