Differential Topology and Quantum Field TheoryThe 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 twosided 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 pseudodifferential operators, AtiyahSinger 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.

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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 
361  
375  
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Common terms and phrases
action anomaly Atiyah base point boundary calculation chapter characteristic classes Chern class ChernSimons cohomology class cohomology group compact complex manifold compute conformal field theory conformal invariance connection consider construct coordinates correlation functions corresponding critical points curvature defined definition denote diffeomorphisms differential operator dim ker dimension Dirac operator eigenvalues element elliptic operator equation example Fermions fibre finite dimensional formula gauge transformations genus given gives global group G Hilbert space holomorphic homeomorphic homology homotopy groups index theorem infinite instanton integral isomorphism Jones polynomial Ktheory knots KU(X leading symbol Lie algebra line bundle loop metric moduli space monopoles Morse theory Mt(f nontrivial notation obtain pforms partition function polynomial Pontrjagin classes primary field pseudodifferential operator quantum field theory quotient representation result Riemann surface sheaf sheaf cohomology smooth tensor topological trivial vanishes vector bundles Virasoro algebra Witten write YangMills zero