Transcript Slide 1
Topological quantum field theory from path integrals to higher categories Bruce Bartlett, Phd student, Sheffield University January 2008, Stellenbosch Outline • 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. Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 2002 Laruent Lafforgue Vladimir Voevodsky 1986 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 2002 Laruent Lafforgue Vladimir Voevodsky 1986 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Used ideas from inter alia quantum field theory to "bridge probability, 1990 Vladimir Drinfeld representation theory Vaughan Jonesand algebraic geometry". Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Proved the Geometrization conjecture for 3-manifolds byDrinfeld essentially employing, 1990 Vladimir and making rigorous, the Vaughan Jones "renormalization Shigefumigroup Moriflow" from string theory. Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Grisha Perelman. Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 1990 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1986 Simon Donaldson Proved the "moonshine conjecture" Gerd Faltings by rigourously establishing a Freedman quantumMichael field theory with the "Monster group" as its symmetry 1982 group. Alain Connes William Thurston Shing-Tung Yau 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 1990 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1986 Donaldson His workSimon is about geometric aspects Gerd Faltings of mathematical physics : Michael Freedman quantization, knot theory, mirror symmetry, and topological quantum 1982 Alain Connes field theory. William Thurston Shing-Tung Yau 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 Quantum groups and their relation 2006 Andrei Okounkov to knot theory. Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 Used Von Neumann algebras and 2006 Andrei Okounkov ideas from statistical mechanics to Grigori Perelman (declined) derive a new invariant of knots - the Terence Tao Jones polynomial. Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 Used quantum field theory to 2006 Andrei explain the JonesOkounkov polynomial in Perelman (declined) terms of aGrigori topological quantum field Terence Tao theory known as Chern-Simons Wendelin Warner theory. Geometrically one is "summing over all connections". 2002 Laruent Many other things.Lafforgue Vladimir Voevodsky 1998 1994 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman Richard Timothy Gowers In all this large and exciting whichConnes involves many of the 1982 field,Alain Maxim Kontsevich leading physicists and mathematicians in Thurston the world, Edward William Curtis McMullen Witten clearly stands out as the most influential Shing-Tung Yauand dominating Silver Platefigure. : AndrewMichael Wiles Atiyah, Fields medal citation, 1990. ... Jean Bourgain Pierre-Louis Lions 1966 Michael Atiyah Jean-Cristophe Yoccoz Alexander Grothendieck Efim Zelmanov Paul Cohen Stephen Smale Fields Medallists 1982-2006 Used gauge 2006 Andrei theory Okounkov to find new invariantsGrigori of smooth Perelman 4-dimensional (declined) manifolds. Terence Corollary Tao : there are infinitely Wendelin many smooth Warner structures on R4! 2002 Laruent Lafforgue This is another Vladimir example Voevodsky of a topological quantum field theory. 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner Revolutionized this subject in 1994 2002 Laruent Lafforgue with the Seiberg-Witten equations. Vladimir Voevodsky 1998 1994 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner Classification theorems for von 2002 Laruent Lafforgue Neumann algebras. The "father" of Vladimir Voevodsky noncommutative geometry... which nowadays he uses to try and explain 1998 Richard Borcherds the Standard Model of particle Timothy Gowers physics. Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Alain Connes at the British Mathematics Colloquium, Swansea, 2007. Fields Medallists 1982-2006 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 2002 Laruent Lafforgue Vladimir Voevodsky 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau 1998 Richard Borcherds Timothy Gowers K-TheoryMaxim and theKontsevich Atiyah-Singer Curtis McMullen index theorem. Met Witten at Silver Plate : Andrew Wiles Harvard in 1977... revolutionizing their future research. Formalized the 1994 Bourgainquantum notion of Jean a topological Pierre-Louis Lions field theory. Jean-Cristophe Yoccoz Efim Zelmanov ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale John Baez Mathematician Urs Schreiber Mathematical Physicist David Corfield Philosopher