On the Equivalence Between Continuous and Differential Deformation Theories

We show that continuous and differential deformation theories of the algebra of smooth functions on are the same, and that the same result holds for the algebra of formal series. We show that preferred quantizations of formal groups are always differential.     

Deformation of Towers of Algebras and Quantized Intersection Cardinality Functions

The inductive limit of a tower of separable algebras is unchanged, up to isomorphism, by consistent deformations but the inductive limit of a corresponding tower of modules may be nontrivially deformed, thereby quantizing the limit module. In the case of the inductive limit of the complex group algebras of the symmetric groups and their deformations, the Hecke algebras, this quantization preserves properties of the finite case which disappear in the absence of quantization.     

Deformation Quantizations of the Poisson Algebra of Laurent Polynomials

It is well known that the Moyal bracket gives a unique deformation quantization of the canonical phase space R2n up to equivalence. In his presentation of an interesting deformation quantization of the Poisson algebra of Laurent polynomials, Ovsienko discusses the equivalences of deformation quantizations of these algebras. We show that under suitable conditions, deformation quantizations of this algebra are equivalent. Though Ovsienko showed that there exists a deformation quantization of the Poisson algebra of Laurent polynomials which is not equivalent to the Moyal product, this is not correct. We show this equivalence by two methods: a direct construction of the intertwiner via the star exponential and a more standard approach using Hochschild 2-cocycles.     

On Continuous and Differential Hochschild Cohomology

We show that there are canonical isomorphisms between Hochschild cohomology spaces , where is the algebra of smooth functions on a manifold M and the space of skew multivector fields over M. This implies that continuous and differential deformation theories of coincide.     

Generalized Deformations and Hochschild Cohomology

We present a generalization of Gerstenhaber's theory of deformations. We no longer assume that the deformation parameter t acts in its usual free and symmetric way on the elements of the original algebra A, but in the following manner: t · a = (a)t and a · t = (a)t, where and are endomorphisms of A. We develop the cohomological framework adapted to these deformations.     

Operads and Motives in Deformation Quantization

The algebraic world of associative algebras has many deep connections with the geometric world of two-dimensional surfaces. Recently, D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne's conjecture on Hochschild complexes follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higher-dimensional algebras and of motives to quantum field theories.     

Hopf Modules and Noncommutative Differential Geometry

We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one-to-one correspondence between anti-Yetter–Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus, we show that these coefficient modules can be regarded as “flat bundles” in the sense of Connes’ noncommutative differential geometry.     

Noncommutative Line Bundle and Morita Equivalence

Global properties of Abelian noncommutative gauge theories based on -products which are deformation quantizations of arbitrary Poisson structures are studied. The consistency condition for finite noncommutative gauge transformations and its explicit solution in the Abelian case are given. It is shown that the local existence of invertible covariantizing maps (which are closely related to the Seiberg–Witten map) leads naturally to the notion of a noncommutative line bundle with noncommutative transition functions. We introduce the space of sections of such a line bundle and explicitly show that it is a projective module. The local covariantizing maps define a new star product which is shown to be Morita equivalent to .     

Deformations of duality-symmetric theories

We prove that a sum of free non-covariant duality-symmetric actions does not allow consistent, continuous and local self-interactions that deform the gauge transformations. For instance, non-abelian deformations are not allowed, even in 4 dimensions where Yang–Mills type interactions of 1-forms are allowed in the non-manifestly duality-symmetric formulation. This suggests that non-abelian duality should require to leave the standard formalism of perturbative local field theories. The analyticity of self-interactions for a single duality-symmetric gauge field in four dimensions is also analyzed.     

Deformations of Algebras Constructed using Quantum Stochastic Calculus

Deformations of associative algebras in which time is the deformation parameter are constructed using quantum stochastic flows in which the usual multiplicativity requirement is replaced by multiplicativity with respect to the deformed multiplication. The theory is restricted by a commutativity requirement on the structure maps of the flow, but examples which can be constructed in this way include the noncommutative torus and the Weyl–Moyal deformation.     

A Remark on Formal KMS States in Deformation Quantization

Within the framework of deformation quantization, we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C^*-algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential.     

Deformation Dynamics and the Gauss–Bonnet Topological Term in String Theory

We show that there exists a nontrivial contribution on the Witten covariant phase space when the Gauss–Bonnet topological term is added to the Dirac–Nambu–Goto action describing strings, because the geometry of deformations is modified, and on such space we construct a symplectic structure. Future extensions of the present results are outlined.     

Deformed C λ-Extended Heisenberg Algebra in Noncommutative Phase-Space

We construct a deformed C _λ-extended Heisenberg algebra in two-dimensional space using noncommuting coordinates which close an algebra depends on statistical parameter characterizing exotic particles. The obtained symmetry is nothing but an exotic particles algebra interpolating between bosonic and deformed fermionic algebras. PACS numbers: 03.65.Fd, 02.40.Gh, 05.30.Pr     

Quantum deformations of the Heisenberg group obtained by geometric quantization

All multiplicative Poisson brackets on the Heisenberg group are classified and Manin groups [14] corresponding to a wide class of those brackets are constructed. A geometric quantization procedure is applied to the resulting symplectic pseudogroups yielding a wide class of pre-C^*-algebras with comultiplication, counit and coinverse, which provide quantum deformations of the Heisenberg group.