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Quantum field theory, supersymmetry, and enumerative geometry. (English) Zbl 1122.81010

IAS/Park City Mathematics Series 11. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3431-2/hbk). viii, 285 p. (2006).
In the introduction, the editors make the point that the challenge to mathematicians is not so much to rigorize and axiomatize what physicists are doing, but to develop a framework for which it fits into mathematics and allows further discoveries to be made. The statement was first used for quantum field theory in the first graduate school series and now, in volume 11, it is just as important.
Contents: William Fulton, Enumerative geometry (5–29); Aaron Bertram, Computing Gromov-Witten invariants with algebraic geometry (31–60); Daniel S. Freed, Classical field theory and supersymmetry (61–161); John W. Morgan, Introduction to supermanifolds (163–181); Clifford V. Johnson, Notes on introductory general relativity (183–285).
The first set of lectures (by William Fulton), is in enumerative geometry and whilst this does not connect overtly with physics, it is an interesting lecture, especially the work on Schubert calculus and the diagrams that enable one to understand the structures more easily. This is followed by a set of lectures on computing invariants with algebraic geometry (by Aaron Bertram). These are again very formal but interspersed in these lectures are statements indicating some of the physical interest.
The next set of lectures (by David S. Freed) is concerned with the classical field theories and supersymmetry. He points out that the lectures are versions of a course he gave which starts with some basic formal structures of mechanics and ends with sufficient background about supersymmetric field theory. He describes this as an impossible task, but he does this very well. Starting with basic notation for classical mechanics in terms of geometrical terms (so for example the position of a particle \(x(t)\) is a function from time to space) he continues through differential geometry, Hamiltonian and Lagrangian mechanics, electromagnetic theory and Minkowski space-time. In the next few lectures the concepts are extended to Lagrangian field theory, gauge theory and particle theory, quantization and supersymmetry. It is probably fair to say that this set of lectures connects more closely with physics.
Following this there is a short, but well written, lecture on supermanifolds (by John Mogan) and then lectures on “Notes on Introducing General Relativity” (by Clifford Johnson). The latter is an excellent set of lectures that uses much of the sophisticated mathematics introduced, expressed in terms more closely aligned to conventional physics. This is particularly evident in the discussion on the basis of string theory. But this set of lectures is more than an example of mathematical structures.
There is a very interesting and entertaining section on 4 thought experiments in special relativity. (Including Harry Parker after a game of Quidscratch!) All of it is laid out very logically and is very readable.
The topics on general relativity are comprehensive and include a discussion of black holes and rotating black holes (using the Kerr metric). The latter can be used to get an analogous “First Law of Thermodynamics”. As the author points out this is really saying that gravity has some underlying structure that has yet to be fully understood. Thus the mathematical framework is being developed that allows further discoveries and, as alluded to earlier, this was the aim of all of the lectures.

MSC:

81-06 Proceedings, conferences, collections, etc. pertaining to quantum theory
00B25 Proceedings of conferences of miscellaneous specific interest
83-06 Proceedings, conferences, collections, etc. pertaining to relativity and gravitational theory
81Qxx General mathematical topics and methods in quantum theory
81Txx Quantum field theory; related classical field theories
81Sxx General quantum mechanics and problems of quantization
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
70Sxx Classical field theories
81T60 Supersymmetric field theories in quantum mechanics
70H03 Lagrange’s equations
70H05 Hamilton’s equations
78A25 Electromagnetic theory (general)
83A05 Special relativity
14M30 Supervarieties
58A50 Supermanifolds and graded manifolds
83C57 Black holes
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