Differential Topology and Quantum Field Theory

Front Cover
Elsevier, 1991 - Mathematics - 386 pages
The remarkable developments in differential topology and how these recent advances have been applied as a primary research tool in quantum field theory are presented here in a style reflecting the genuinely two-sided interaction between mathematical physics and applied mathematics. The author, following his previous work (Nash/Sen: Differential Topology for Physicists, Academic Press, 1983), covers elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory. The explanatory approach serves to illuminate and clarify these theories for graduate students and research workers entering the field for the first time.

  • Treats differential geometry, differential topology, and quantum field theory
  • Includes elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory
  • Tackles problems of quantum field theory using differential topology as a tool
 

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Contents

CHAPTER
1
homotopy
3
Homotopy groups
9
Cohomology and homology groups
12
Fibre bundles and fibrations
14
Differentiable structures for manifolds
19
CHAPTER II
27
Ellipticity and hypoellipticity
32
The Teichmiiller space 7P
148
The dimension of the moduli space
157
Weierstrass gaps and Weierstrass points
166
Group extensions
178
Representations
185
CHAPTER VII
192
Critical submanifolds
199
Supersymmetric quantum mechanics and Morse theory
207

Ellipticity and vector bundles
36
Pseudodifferential operators
41
Pseudodifferential operators and Sobolev spaces
47
CHAPTER III
56
Sheaf cohomology
59
Ktheory
65
Bott periodicity
75
Some characteristic classes
78
Fredholm operators and KX
88
CHAPTER IV
89
Some examples
97
Twisted complexes
116
The index theorem for families of operators
119
The index for real families
121
Index theory and fixed points
123
Index theory for manifolds with boundary
127
CHAPTER V
137
Riemann surfaces and divisors
139
Serre duality line bundles and Kahler manifolds
144
CHAPTER VIII
216
Secondary characteristic classes
223
Monopoles and symmetries of instantons
242
Monopole moduli and monopole scattering
250
Critical point theory and gauge theories
256
The space of metrics
262
CHAPTER X
269
Gravitational anomalies
279
Anomalies from a Hamiltonian perspective
291
CHAPTER XI
301
Relation to the Virasoro algebra
307
Operator products fusion rules and axiomatics
313
Donaldsons polynomial invariants
332
Knots and knot invariants
339
ChernSimons theory and the Jones polynomial
350
References
361
Index
375
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