On the Deformation Quantization Description of Matrix Compactifications

We study the Matrix theory description of M-theory compactified on $T^4$ and $T^5$. M-theory on $T^4$ is described by the six dimensional (2,0) fixed point field theory compactified on a five torus, $\widetilde T^5$. For M-theory on $T^5$ we suggest the existence of a new theory which is compactified on a $\widetilde T^5$. The IR description of this theory is given by the (2,0) theory with a compactified moduli space. This new theory appears to be a new kind of a non-critical...

We study compactifications of Matrix theory on twisted tori and non-commutative versions of them. As a first step, we review the construction of multidimensional twisted tori realized as nilmanifolds based on certain nilpotent Lie algebras. Subsequently, matrix compactifications on tori are revisited and the previously known results are supplemented with a background of a non-commutative torus with non-constant non-commutativity and an underlying non-associative structure on ...

We present a general framework for Matrix theory compactified on a quotient space R^n/G, with G a discrete group of Euclidean motions in R^n. The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We characterize the resulting noncommutative gauge theory in terms of the twisted group algebra of G associated with a projective regular representation. Also we show how to extend our treatments to incorporate orientifolds.

The foundations of matrix geometry are discussed, which provides the basis for recent progress on the effective geometry and gravity in Yang-Mills matrix models. Basic examples lead to a notion of embedded noncommutative spaces (branes) with emergent Riemannian geometry. This class of configurations turns out to be preserved under small deformations, and is therefore appropriate for matrix models. The relation with spectral geometry is discussed. A possible realization of suf...

The aim of this paper is to give a basic overview of Deformation Quantization (DQ) to physicists. A summary is given here of some of the key developments over the past thirty years in the context of physics, from quantum mechanics to quantum field theory. Also, we discuss some of the conceptual advantages of DQ and how DQ may be related to algebraic quantum field theory. Additionally, our previous results are summarized which includes the construction of the Fedosov star-prod...

We characterise the quantum group gauge symmetries underlying q-deformations of two-dimensional Yang-Mills theory by studying their relationships with the matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge theories, together with their refinements and supersymmetric extensions. We develop uniqueness results for quantum deformations and refinements of gauge theories in two dimensions, and describe several potential analytic and geometric realisation...

The aim of this thesis is to study the isopectral deformations from the point of view of Alain Connes' noncommutative geometry. This class of quantum spaces constituts a curved space generalisation of Moyal planes and noncommutative tori. First of all, we look at the construction of non-unital spectral triples, for which we propose modified axioms. We then check that Moyal planes fit into this axiomatic framework, and give the keypoints for the construction of non-unital spec...

The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordstr\"om, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particles physics, leading to methods that are useful for areas of physics, in which ...

We first review the historical developments, both in physics and in mathematics, that preceded (and in some sense provided the background of) deformation quantization. Then we describe the birth of the latter theory and its evolution in the past twenty years, insisting on the main conceptual developments and keeping here as much as possible on the physical side. For the physical part the accent is put on its relations to, and relevance for, "conventional" physics. For the mat...

We demonstrate the emergence of the U-duality group in compactification of Matrix theory on a 4-torus. The discussion involves non-trivial effects in strongly coupled 4+1 dimensional gauge theory, and highlights some interesting phenomena in the Matrix theory description of compactified M-theory.