物理的数学与未来
Physical Mathematics and the Future
by Gregory W. Moore
英文

Abstract: These are some thoughts meant to accompany one of the summary talks at
Strings2014, Princeton, June 27, 2014. This is a snapshot of a personal and perhaps
heterodox view of the relation of Physics and Mathematics, together with some guesses
about some of the directions forward in the field of Physical Mathematics. At least, this is
my view as of July 21, 2014.


Contents
1. Predicting the future 2
2. What is Physical Mathematics? 3
3. String-Math vs. Strings Meetings 7
4. Geometry, Symmetry, and Physics 8
4.1 Duality Symmetries and BPS states 9
4.1.1 BPS states 9
4.1.2 Topological string theory 10
4.1.3 3-manifolds, 3d mirror symmetry, and symplectic duality 11
4.1.4 Knot theory 12
4.1.5 Special holonomy in six, seven, and eight dimensions 13
4.2 Hyperk¨ahler and Quaternionic-K¨ahler geometry 14
4.2.1 Construction of explicit hyperk¨ahler metrics 14
4.2.2 Construction of explicit quaternion-K¨ahler metrics 14
4.2.3 Monopole moduli spaces 15
4.2.4 Line and surface defects and the geometry of (quantum) Hitchin
moduli spaces 15
4.3 Four-manifolds 16
4.4 Geometric Representation Theory 17
4.5 Generalizing Geometry 20
4.6 Topology 21
4.6.1 Anomaly cancellation 21
4.6.2 K,M,S 22
4.6.3 Elliptic cohomology 23
4.6.4 Topological phases of matter 23
4.7 Geometry and Representation Theory of Some Infinite Dimensional Groups 24
4.8 Supergeometry 25
5. Defining Quantum Field Theory 26
5.1 Theories without actions 27
5.2 Theories with many actions 28
5.3 The trouble with scale dependence 28
5.4 S matrix magic 28
5.5 Oddball theories 29
5.6 Geometrization of field theoretic phenomena 30
5.7 Locality, locality, locality 31
5.8 Theories without defects are defective 31
5.9 So, what should we do about this? 32
6. Exploring Quantum Field Theory 33
6.1 Classification questions 33
6.2 Geometry on the space of field theories 34
7. String theory compactification 35
8. String Field Theory 38
9. Noncommutativity and spacetime 39
10. Exceptional Structures 40
11. Should we count in number theory? 41
12. Keep true to the dreams of thy youth: M-theory 43
13. Money and jobs 44
14. An uneasy marriage 45