Conformal Field Theory and ApplicationsIPAM Fall 2001September 10, 2001 – December 14, 2001ScheduleWeekly SeminarsParticipantsWorkshops
Organizing CommitteeEric D’Hoker (UCLA) Program:The program will take place from September 10 to December 14, 2001 at the Institute for Pure and Applied Mathematics (IPAM), a new NSF funded international mathematical sciences research institute located on the UCLA campus. The program is open to the entire international Pure and Applied Mathematics and Physics communities. For application information, please see information at end. At the beginning and throughout the entire period of the program, from September -December there will be a number of tutorials and courses in order to facilitate interdisciplinary interactions. The course material will be targeted to the workshops. The courses will be at least twice a week throughout the program. In addition there will be special lectures and other events. Three 5-day conference/workshop meetings will be organized on various cutting edge fundamentals and applications of conformal field theory in mathematics and in physics. A 5 day final conference will be held at Lake Arrowhead Conference Center, a mountain resort facility owned by UCLA. Please see further down for detailed program schedule. Scientific Content of the ProgramMathematics has recently seen extensive interaction with physics, especially with the ideas of quantum field theory (QFT), string theory, conformal field theory (CFT) and Seiberg-Witten theory. Physicists have discovered many truly amazing mathematical relationships. While there is a thriving collaboration between string theorists and mathematicians, there has been somewhat less activity between conformal field theorists and mathematicians. The purpose of the proposed semester program in CFT is to encourage this interaction. Quantum field theory provides a highly successful formulation of the physics of elementary particles and of statistical mechanical systems. Methods of quantum field theory may be applied to problems of topological invariants of knots and instanton moduli spaces as well and have led to a wealth of new results in pure mathematics. This despite the fact that a complete mathematically rigorous construction of general quantum field theories is still lacking. Conformal field theories are quantum field theories that are invariant under conformal – and in particular scaling – symmetry. Their physical importance derives from the fact that they describe crucial phenomena of statistical mechanical systems, i.e., their behavior near a second order phase transition. While the microscopic dynamics may vary considerably from one system to another, the scaling behavior near a second order phase transition falls into universality classes, each of which is described by a conformal field theory. Conformal field theories in two dimensions are exceptionally strongly constrained since the conformal algebra (or Virasoro algebra) has an infinite number of generators. The classification of two-dimensional conformal field theories is thereby reduced to problems in the representation theory of infinite-dimensional Virasoro, Kac-Moody and vertex operator algebras. Their purely algebraic formulation elevates two-dimensional conformal field theories to some of the rare examples where quantum field theories may be defined and constructed in a mathematically rigorous fashion. An infinite number of so-called minimal models have been solved exactly and the universality classes of a large number of physical systems (such as spin, vertex and clock models) have been identified with specific minimal models. However, other important classes of conformal field theory, such as the Liouville theory remain only partially solved. Dynamics in the vicinity of second order phase transitions may be accurately described with the help of conformal theories perturbed by the addition of operators that break the conformal symmetry. For some conformal systems some of these perturbations preserve the integrability properties and many familiar integrable systems may be described in terms of integrable deformations of conformal field theories. The perturbative expansion of string theory in powers of the coupling constant may be formulated in terms of conformal field theories on families of two-dimensional compact Riemann surfaces, whose genus is the order of the expansion. Virasoro and vertex operator algebras were discovered in string theory and the development of both subjects continues to hand in hand. The recent Anti-de Sitter/Conformal Field Theory (AdS/CFT) conjecture by Maldacena proposes a precise correspondence between superstring theory (and supergravity) on the AdSxS and supersymmetric conformal invariant Yang-Mills theory in three, four and six dimensions. Though still at the level of a conjecture, the correspondence has already generated a wealth of new results on conformal field theories in dimensions higher than two. Two outstanding mathematical outgrowths of CFT are the Verlinde formulas and the theory of vertex operator algebras. The Verlinde formulas gave new and completely unexpected information on the moduli of stable vector bundles on a compact Riemann surface. The theory of vertex operator algebras developed by Borcherds was essential in his work, most notable in his solution of the monstrous moonshine conjecture. Vertex operator algebras came directly from the physicists’ theories and have developed into a thriving mathematical discipline. Several mathematicians have been very active in developing connections between quantum groups, integrable systems and vertex operator algebras from a completely mathematical point of view. The goal of this program is to bring together leading investigators and junior researchers in the many different areas of mathematical and physical research on conformal field theories and their applications. The role of the tutorial sessions right at the start of the program is to introduce researchers quickly to the physical and mathematical problems of current interest. The role of the three conference/workshop sessions is to have focused efforts on the subjects of applications of conformal field theories to physical systems in the first and on their purely mathematical aspects in the second Finally, in the third, we plan to bring mathematicians and physicists who have worked primarily on two-dimensional conformal field theories with string theorists who are aiming at understanding supersymmetric conformal field theories in dimensions higher than 2. Long-term participantsThe following Mathematicians and Physicists are planning to participate for extended periods of at least two weeks, at least tentatively. Several other senior figures will also be involved.
Weekly Seminars· Conformal Theory for Beginners Organizer: David Gieseker Tuesdays and Thursdays September 13, 2001 – December 6, 2001 Time: 1:30 - 3:00 Location: IPAM Building, Room 1200
· Seminar in Geometry and Physics Organizer: Kefeng Liu, UCLA Tuesdays and Thursdays September 25, 2001 – December 6, 2001 Time: 11:00 - 12:30 Location: IPAM Building, Room 1180
Speaker: Stan Osher, UCLA Mathematics Dept. Tuesdays September 25, 2001 – December 24, 2001 Time: 2:00 - 3:30 Location: IPAM Building, Room 1180
Program Schedule:The program will take place from September 10 to December 14, 2001 at the Institute for Pure and Applied Mathematics (IPAM), a new NSF funded international mathematical sciences research institute located on the UCLA campus. Tutorials/Learning Seminars:At the beginning and throughout the entire period of the program, from September 10-December 10 there will be a number of tutorials and courses in order to facilitate interdisciplinary interactions. The course material will be targeted to the workshops. The courses will be at least twice a week throughout the program. In addition there will be special lectures and other events. Course Leaders:
David Gieseker
(UCLA ) Possible topics include: Elementary conformal field theory, affine Kac-Moody algebra, Vertex Operator Algebra.
Culminating Workshop at Lake Arrowhead, California.New Directions in Conformal Field TheoryDecember 9-14, 2001 High Dimensional Conformal Field Theory and String Theory: This workshop will include resident participants at the junior and senior levels and will explore new directions and approaches of research in these fields. At this final event of the program the focus is on interactions between the junior and senior participants and developing future directions for collaborations. The conference will be held at Lake Arrowhead Conference Center, a mountain resort facility owned by UCLA ideally designed to such purposes. Senior Participants
David Gieseker Arrowhead Speakers:Mina Aganagic (Harvard) Contact Us:Institute for Pure and Applied
Mathematics (IPAM)
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