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 invariants from studying von Neumann algebras. But Witten explained them geometrically in terms of a TQFT. Outline

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Used ideas from inter alia quantum field theory to "bridge probability, representation theory and algebraic geometry".

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Proved the Geometrization conjecture for 3-manifolds by essentially employing, and making rigorous, the "renormalization group flow" from string theory.

Grisha Perelman.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Proved the "moonshine conjecture" by rigourously establishing a quantum field theory with the "Monster group" as its symmetry group.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale His work is about geometric aspects of mathematical physics : quantization, knot theory, mirror symmetry, and topological quantum field theory.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Quantum groups and their relation to knot theory.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Used Von Neumann algebras and ideas from statistical mechanics to derive a new invariant of knots - the Jones polynomial.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 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.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Used gauge theory to find new invariants of smooth 4-dimensional manifolds. Corollary : there are infinitely many smooth structures on R 4 ! This is another example of a topological quantum field theory.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Revolutionized this subject in 1994 with the Seiberg-Witten equations.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 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.

Fields Medallists Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986Simon Donaldson Gerd Faltings Michael Freedman 1982Alain Connes William Thurston Shing-Tung Yau Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale K-Theory and the Atiyah-Singer index theorem. Met Witten at Harvard in revolutionizing their future research. Formalized the notion of a topological quantum field theory.

John Baez Mathematician Urs Schreiber Mathematical Physicist David Corfield Philosopher