Towards Noncommutative Topological Quantum Field Theory: Tangential Hodge-Witten cohomology

2nd International Conference on Mathematical Modeling in Physical Sciences 2013

物理学;数学

Zois, I.P.^1

PPCR, R and D Department, Kapodistriou Street, 3, Agia Paraskevi, GR-15342, Athens, Greece^1

关键词: Black-hole entropy;    Generalisation;    Moduli space;    Morse the-ory;    Non-commutative;    Non-commutative geometry;    Quantum field theory;    Quantum gravity;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/490/1/012235/pdf
DOI  :  10.1088/1742-6596/490/1/012235

来源: IOP

【 摘 要 】

Some years ago we initiated a program to define Noncommutative Topological Quantum Field Theory (see [1]). The motivation came both from physics and mathematics: On the one hand, as far as physics is concerned, following the well-known holography principle of 't Hooft (which in turn appears essentially as a generalisation of the Hawking formula for black hole entropy), quantum gravity should be a topological quantum field theory. On the other hand as far as mathematics is concerned, the motivation came from the idea to replace the moduli space of flat connections with the Gabai moduli space of codim-1 taut foliations for 3 dim manifolds. In most cases the later is finite and much better behaved and one might use it to define some version of Donaldson-Floer homology which, hopefully, would be easier to compute. The use of foliations brings noncommutative geometry techniques immediately into the game. The basic tools are two: Cyclic cohomology of the corresponding foliation C*-algebra and the so called "tangential cohomology" of the foliation. A necessary step towards this goal is to develop some sort of Hodge theory both for cyclic (and Hochschild) cohomology and for tangential cohomology. Here we present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian.