Supersymmetry and Fredholm Modules Over Quantized Spaces (original) (raw)

Supersymmetric Quantum Theory and Non-Commutative Geometry

Communications in Mathematical Physics, 1999

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing noncommutative Riemannian, symplectic, complex-Hermitian and (Hyper-) Kähler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.

Almost-Kähler deformation quantization

Letters in Mathematical Physics, 2001

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kähler manifold which respects the almost-Kähler structure to construct a Fedosov-type deformation quantization on this manifold. 1 2 A.V. KARABEGOV AND M. SCHLICHENMAIER parameterized by the elements of the affine vector space (1/iν)[ω] + H 2 (M, C)[[ν]] via the mapping * → cl( * ).

Supersymmetry in noncommutative superspaces

Journal of High Energy Physics, 2003

Non commutative superspaces can be introduced as the Moyal-Weyl quantization of a Poisson bracket for classical superfields. Different deformations are studied corresponding to constant background fields in string theory. Supersymmetric and non supersymmetric deformations can be defined, depending on the differential operators used to define the Poisson bracket. Some examples of deformed, 4 dimensional lagrangians are given. For extended superspace (N > 1), some new deformations can be defined, with no analogue in the N = 1 case.

Quantization of Kähler manifolds. III

Letters in Mathematical Physics, 1994

We use Berezin's dequantization procedure to dene a formal -product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal -product is convergent on a dense subalgebra of the algebra of smooth functions.

On Deformation Theory and Quantization

2008

Deformation theory requires solving Maurer-Cartan equation (MCE) associated to an DGLA (L-infinity algebra). The universal solution of [HS] is obtained iteratively, as a fixed point of a contraction, analogous to the Picard method. The role of the Kuranishi functor in this construction is emphasized. The parallel with Lie theory suggests that deformation theory is a higher “dimensional” version. The deformation determined by the solution of the Maurer-Cartan equation associated to a contraction, splits the epimorphism, leading to a “doubling and gluing” interpretation. The *-operator associated to a contraction is introduced, and the connection with Hodge structures and generalized complex structures ( dd∗ -lemma) is established. The relations with bialgebra deformation quantization on one hand and ConnesKreimer renormalization on the other, are suggested.

Quantization of K�ahler manifolds II

1993

We use Berezin's dequantization procedure to dene a formal -product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal -product is convergent on a dense subalgebra of the algebra of smooth functions.

Generalized Kähler Manifolds and Off-shell Supersymmetry

Communications in Mathematical Physics, 2006

We solve the long standing problem of finding an off-shell supersymmetric formulation for a general N = (2, 2) nonlinear two dimensional sigma model. Geometrically the problem is equivalent to proving the existence of special coordinates; these correspond to particular superfields that allow for a superspace description. We construct and explain the geometric significance of the generalized Kähler potential for any generalized Kähler manifold; this potential is the superspace Lagrangian.

Q A ] 1 7 Ju l 2 00 5 Irreducible highest-weight modules and equivariant quantization

2005

The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers [2, 7, 8]. Roughly speaking, a deformation quantization of a Poisson manifold (P, { , }) is a formal associative product on (Fun P)[[]] given by f 1 ⋆ f 2 = f 1 f 2 + c(f 1 , f 2) + O(2) for any f 1 , f 2 ∈ Fun P , where the skew-symmetric part of c is equal to { , }, and the coefficients of the series for f 1 ⋆ f 2 should be given by bi-differential operators. The fact that any Poisson manifold can be quantized in this sense was proved by Kontzevich in [15]. However, finding exact formulas for specific cases of Pois-son brackets is an interesting separate problem. There are several well-known examples of such explicit formulas. One of the first was the Moyal product quan-tizing the standard symplectic structure on R 2n. Another one is the standard quantization of the Kirilov-Kostant-Souriau bracket on the dual space g * to a Lie algebra g (see [10]...