We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from this limit. Our analysis leads us to an equivalence between ordinary gauge fields and noncommutative gauge fields, which is realized by a change of variables that can be described explicitly. This change of variables is checked by comparing the ordinary Dirac-Born-Infeld theory with its noncommutative counterpart. We obtain a new perspective on noncommutative gauge theory on a torus, its T-duality, and Morita equivalence. We also discuss the D0/D4 system, the relation to M-theory in DLCQ, and a possible noncommutative version of the six-dimensional (2,0) theory.

https://iopscience.iop.org/article/10.1088/1126-6708/1999/09/032/pdf

String theory and noncommutative geometry

From the Publisher:
The old opposition of matter versus mind stubbornly persists in the way we study mind and brain. In treating cognition as problem solving, Andy Clark suggests, we may often abstract too far from the very body and world in which our brains evolved to guide us. Whereas the mental has been treated as a realm that is distinct from the body and the world, Clark forcefully attests that a key to understanding brains is to see them as controllers of embodied activity. From this paradigm shift he advances the construction of a cognitive science of the embodied mind.

/pdf/being-there-putting-brain-body-and-world-together-again-31tt4y51ml.pdf

Being There: Putting Brain, Body, and World Together Again

Modified Gravity and Cosmology

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

/pdf/deformation-quantization-of-poisson-manifolds-4tp4kn8ari.pdf

Deformation Quantization of Poisson Manifolds

This article reviews the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, on both the classical and the quantum level.

Noncommutative field theory

We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory.

https://iopscience.iop.org/article/10.1088/1126-6708/1998/02/003/pdf

Noncommutative Geometry and Matrix Theory: Compactification on Tori

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/pdf/non-commutative-differential-geometry-56zrwz9sv5.pdf

Non-commutative differential geometry

For purely mathematical reasons it is necessary to consider spaces which cannot be represented as point set sand where the coordinates describing the space do not commute.In other words,spaces which are described by algebras of coordinates which arenot commutative.If you conside rsuch spaces,then it is necessary to rethink most of the notions of classical geometry and redefine them. Motivated from pure mathematics it turns out that there are very striking parallels to what is done in quantum physics In the following lectures, I hope to discuss some of these parallels.

Non-Commutative Geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D−1, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.

Gravity coupled with matter and the foundation of non-commutative geometry

We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

http://cds.cern.ch/record/362017/files/9808042.pdf

Alain Connes

Papers

Noncommutative Geometry and Matrix Theory: Compactification on Tori

Non-commutative differential geometry

Non-Commutative Geometry

Gravity coupled with matter and the foundation of non-commutative geometry

Hopf Algebras, Renormalization and Noncommutative Geometry