topological quantum field theory from path integrals to higher categories bruce bartlett, phd...

Topological quantum field theory

from path integrals to higher categories

Bruce Bartlett, Phd student, Sheffield University January 2008, Stellenbosch

• A topological quantum field theory (TQFT) is a gadget which assigns invariants to geometric structures.

• They came out of physics in the late 1980's, but were soon defined formally by mathematicians.

• We will look at Chern-Simons theory, a famous example of a TQFT. Much remains to be understood.

• Amongst other things, this theory is about finding invariants of knots in three-dimensional space.

• Originally, Jones obtained these invariantsfrom studying von Neumann algebras. ButWitten explained them geometrically in terms ofa TQFT.

Outline

Fields Medallists 1982-2006

2006 Andrei OkounkovGrigori Perelman (declined)Terence TaoWendelin Warner

2002 Laruent LafforgueVladimir Voevodsky

1998 Richard BorcherdsTimothy GowersMaxim KontsevichCurtis McMullenSilver Plate : Andrew Wiles

1994 Jean BourgainPierre-Louis LionsJean-Cristophe YoccozEfim Zelmanov

1990 Vladimir DrinfeldVaughan JonesShigefumi MoriEdward Witten

1986 Simon DonaldsonGerd FaltingsMichael Freedman

1982 Alain ConnesWilliam ThurstonShing-Tung Yau

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1966 Michael AtiyahAlexander

GrothendieckPaul CohenStephen Smale









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Used ideas from inter alia quantum field theory to "bridge probability, representation theory and algebraic geometry".









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Proved the Geometrization conjecture for 3-manifolds by essentially employing,and making rigorous, the "renormalization group flow" from string theory.

Grisha Perelman.









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Proved the "moonshine conjecture" by rigourously establishing a quantum field theory with the "Monster group" as its symmetry group.









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His work is about geometric aspects of mathematical physics : quantization, knot theory, mirror symmetry, and topological quantum field theory.









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Quantum groups and their relation to knot theory.









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Used Von Neumann algebras and ideas from statistical mechanics to derive a new invariant of knots - the Jones polynomial.




1998 RichardTimothy GowersMaxim KontsevichCurtis McMullenSilver Plate : Andrew Wiles





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Used quantum field theory to explain the Jones polynomial in terms of a topological quantum field theory known as Chern-Simons theory. Geometrically one is "summing over all connections". Many other things.

In all this large and exciting field, which involves many of the leading physicists and mathematicians in the world, Edward Witten clearly stands out as the most influential and dominating figure. Michael Atiyah, Fields medal citation, 1990.









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Used gauge theory to find new invariants of smooth 4-dimensional manifolds. Corollary : there are infinitely many smooth structures on R4!

This is another example of a topological quantum field theory.









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Revolutionized this subject in 1994 with the Seiberg-Witten equations.









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Classification theorems for von Neumann algebras. The "father" of noncommutative geometry... which nowadays he uses to try and explain the Standard Model of particle physics.

Alain Connes at the British Mathematics Colloquium, Swansea, 2007.









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K-Theory and the Atiyah-Singer index theorem. Met Witten at Harvard in 1977... revolutionizing their future research. Formalized the notion of a topological quantum field theory.

John BaezMathematician

Urs SchreiberMathematicalPhysicist

David CorfieldPhilosopher

topological quantum field theory from path integrals to higher categories bruce bartlett, phd...

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