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 abelianization 47-50, 51 Alexander polynomial 6, 7, 11

Bohm-Aharonov effect 4-5, 7 boundary components 42-4 braids 8-9

bundles, holomorphic moduli spaces of 26-9 stability of 27, 29, 48

Chern-Simons Lagrangian 52-5,

62, 63, 64, 66, 71 curves, degeneration of 50-1

Donaldson-Floer theory 1

electromagnetism 2

Feynman integrals 3-4, 14, 67 Feynman path-integral approach 52-65

and Hamiltonian formulation 63-5

stationary-phase approximations 55-63

see also Chern-Simons Lagrangian field theory, conformal 46-7, 51 flatness, projective 45-51 framings 17, 63

gauge theories 2-5 on a circle 42-3

gauge transformations 37, 53, 57 geometry, projective 30

Hecke algebra 9, 10 Heisenberg commutation relations

18, 19 Higgs bundles 48, 50 Hilbert spaces, quantum 34, 35 Hitchin's moduli space 50 homotopy invariance 13

integer lattice 19

invariant theory, geometric 30-1

Jacobians 20, 21

Jones polynomials 7-11, 46, 67-8, 70

Kahler metrics 32, 33

Kelvin's theory of vortex atoms 5-6

knot, definition of 5

Lie groups, disconnected 72 line-bundles 31, 33, 48, 49 holomorphic 18, 20, 28, 34, 38 Quillen 38-9, 65 links

definition of 5 oriented 8, 9, 15 loop groups 43, 47, 65, 69

manifolds, symplectic 17, 31, 66 homogeneous 36, 46 quantization of 33-5moment maps 31, 36, 37, 40 momentum, angular 31 'multiplicity spaces' 42, 44

Narasimhan-Seshadri theorem 27, 39, 40

orbifolds 72 orbits

co-adjoint 35-6, 43 volumes of 58

phase ambiguity 65 points irreducible 24

marked 40-2, 44, 46, 50, 70 semi-stable 31 stable 31

poles 42, 50, 51, 60

quantization canonical 17-19 of quotient torus 20-2 of symplectic manifolds 33-5 quantum field theories 1,11 topological 12-23; axioms for 12- 17

quotients 30-1 complex 32 Mumford 33, 34

symplectic 31-3, 35-6, 45, 64, 67; infinite-dimensional case of 37- 44

Ray-Singer torsion 61 Reidemeister torsion 61, 71

representations 24-6, 42, 44, 47, 48, 50

Riemann surfaces with boundary 42-4, 47 connections on 37-40

skein relations 8, 10, 67-9 spaces, symplectic 37

quantization of 40, 42 stability 27, 29, 48 stationary-phase approximations

55-63, 67, 71 surgery methods 69-70, 71

Tait's conjectures 6, 7 Teichmiiller space 19, 28, 45, 50, 65 theta-divisor 20 «-functions 19-23, 51, 70 torus

Hilbert space of 69-70 quotient 20-2

vacuum vectors 23, 66-7 vector bundles, holomorphic 20 Verlinde algebra 51, 70 'Virasora algebra' 47

Weyl group 48, 50 Wilson line 54 Witten axioms 70 Witten functor 17 Witten invariants 68, 69, 71

Yang-Baxter equations 10 Yang-Mills theory 1, 3 Young diagrams 10