Marathe K. Topics in Physical Mathematics
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438 Index
quantum, 21
quaternionic symplectic, 104
real general linear, 79
real symplectic, 103
special complex linear, 96
special orthogonal, 97
special real linear, 96
special unitary, 97, 102
sporadic, 23
unitary, 97, 102
unitary symplectic, 105
GRR theorem, 158
Haar measure, 96
Hamilton’s equations, 95, 241
Hamiltonian, 94, 241
harmonic oscillator, 385
Hauptmodul, 27
Hausdorff space, 37
Heegaard
genus, 340
splitting, 340
Heine–Borel theorem, 37
Heisenberg, 171, 177
Algebra, 12
Higgs
bundle, 255
generalized, 257
field, 255
generalized, 196, 257
mass, 266
mechanism, 174, 263
phenomenon, 263
higgsons, 271
Hilbert–Einstein action, 204
Hirzebruch
polynomial, 150
signature, 167
Hodge
decomposition
theorem, 59
Laplacian, 238
star operator, 89
Hodge–de Rham operator, 92
holonomy
bundle, 121
group, 120, 121
hom
ogeneous
space, 97
homologous, 398
homology module, 398
graded, 398
homotopy, 39
group
nth, 44
first, 40
stable, 50
Hopf
algebra, 20
quasitriangular, 20
Fibration, 48
fibration, 127, 239
complex, 139
quaternionic, 139
invariant, 69
horizontal
curve, 120, 128
distribution, 119
lift, 120, 128, 135
space, 135
subspace, 119
hyperbolic
manifold, 361
space, 98
Poincar´emodelof,98
immersion, 82
index, 16, 410, 413
analytic, 413
bundle, 215
topological, 413
infrared limit, 305
instanton, 249, 276
Belavin–Polyakov–Schwartz–Tyupkin,
277
BPST, 277
complex, 288
equation, 249, 277
gravitational, 203
number, 151, 152, 211
, 245, 249, 277, 281
i
n
tegral basis, 23
intermediate value property, 38
intersection form, 60
irreducible connections, 194
isotopy
class, 352
isotropy
group, 97
subgroup, 272
Jacobi
identity, 3
modular function, 27
Jones polynomial, 358
k-classifying, 138
K-group
complex, 162
quaternionic, 162
Index 439
real, 162
K-theory, 157, 158
algebraic, 157, 163
topological, 157, 163
K¨unneth formula, 58
Kac–Moody algebras, 10
Kaluza–Klein theory, 174
Kepler’s conjecture, 24
Kervaire invariant, 70
Killing form, 5, 112
KM basic classes, 302
knot, 43, 352
fundamental group of, 357
group, 43, 357
invariant, 43
signature, 357
type, 352
Laplace–Beltrami operator, 92
Laplacian, 412
operator, 92
Leech lattice, 24
Levi-Civita connection, 127, 133
Lie
algebra, 3
of a Lie group, 99
semi-simple, 5
simple, 5
bracket, 80
derivative, 85
group, 95
action of, 97
semi-simple, 5
simple, 5
subgroup, 96
link, 352
Lorentz
connection, 132
force, 242
manifold, 74
anti-dual, 134
complex, 74
differentiable, 74
differential, 74
di
m
ension of, 74
Lorentz, 89
orientable, 55, 84
oriented, 55, 84
pseudo-Riemannian, 89
real, 74
Riemannian, 89
self-dual, 134
symplectic, 93
topological, 74
with boundary, 76
mapping class group, 340
Maslov class, 155
Mathai–Quillen formalism, 213
matter fields, 107
Maurer-Cartan equations, 126
Maxwell
action, 237
connection, 237
field, 237
anti-dual, 239
Euclidean, 237
self-dual, 239
potential, 237
Maxwell’s equations, 172
metric
connection, 132
Hermitian, 111
Riemannian, 111
minimal
coupling, 199, 260
interaction, 199
mixed double complex, 401
mixing angle, 265, 266, 270
modular form, 27
module, 397
moduli space of instantons, 287
Monnshine Theorem, 32
monopole, 257
configuration space, 258
equation, 258
invariants, 305
number, 258
monster, 22
Lie algebra, 32
primes, 28
monstrous moonshine,
22
m
o
onshine module, 31
Morse
index, 314
inequalities, 314
series, 314
Morse Theory, 314
multiple complex, 401
natural transformation, 397
negative spinors, 307
non-dual, 251
non-Euclidean geometry, 137
observable, 210
operator
compact, 410
440 Index
differential, 406
elliptic, 408, 409
exterior differential, 86
Laplace, 412
pseudo-differential, 406, 409
trace class, 410
orbit space, 116
orientation, 55, 84
parallel
displacement, 120, 128
translation, 128
Parallel Postulate, 387
parity, 267
parity operator, 315
parity violation, 267
Parseval relation, 404
partition function, 210
Pauli spin matrices, 16
Perelman, 205
phase factor, 243
photon, 177
physical mathematics, xi–xiii, xv
Pin group, 415
Platonic solids, 388
Poincar´e
bundle, 214
conjecture, 41
generalized, 41
homology sphere, 62
lemma, 87
Poisson
bracket, 95
manifold, 95
structure, 95
Pontryagin class, 147
positive spinors, 307
principal bundle, 107, 113
product manifold, 78
property P, 311
pseudo-inner product, 16
pseudo-metric, 16, 88
QCD, 177
QED, 175
Q
E
D (quantum electrodynamics), 207
QFT, 208
quanta, 176
quantum
chromodynamics, 177
dimension, 9
electrodynamics, 175, 237
expectation, 210
field theory, 210
topological, 225
number, 266
topological, 47, 152
observable, 227, 365
topology, 21, 234, 371
quantum field theory (QFT), 208
quark, 176, 270
confinement, 176
mixing matrix, 270
quaternions, 103
quotient manifold, 81
Ramanujan, 344, 381, 389
rank, 56
regular
polyhedron, 388
projection, 352
value, 81
Ricci flow, 205
Ricci tensor, 132
trace-free part of, 133, 134
Riemann
curvature, 133
hypothesis, 382
Riemann–Roch theorem, 158
Riemann–Roch–Hirzebruch theorem, 158
Rochlin invariant, 66, 291
rotation group, 97
RR theorem, 158
RRH
formula, 160
theorem, 158
scalar curvature, 133
scale invariance, 173
Schwartz space, 404
s
ecti
onal curvature, 133
Seiberg–Witten
equations, 308
invariants, 305
Seifert surface, 357
semiclassical
approximation, 216
expectation, 216
short exact sequence, 399
signature, 16, 88
simple type, 302
KM, 302
SW, 309
simply connected, 40
smooth structure, 74
smoothability, 75
Sobolev
norm, 405
Index 441
space, 405
Spaces, 404
soldering form, 130
soliton, 257
space-time manifold, 132
sphere-packing problem, 24
spin
frame bundle, 115, 415
manifold, 115
structure, 115, 415
Spin
c
-structure, 307
spin manifold, 415
spinor, 118
bundle, 416
complex, 416
real, 416
bundles, 306
group, 415
representation, 307
spinors, 307
spontaneous symmetry breaking, 261
standard model, xiv, 178, 266, 268
gauge group of, 268
Stiefel
bundle, 239
manifold, 125, 153
Stiefel–Whitney number, 143
Stiefel-Whitney class, 141
Stiefel-Whitney classes, 139
Stokes’ theorem, 88
structure
constants, 3
equation, 124
group, 107
subbundle, 109
subcategory, 394
full, 394
submanifold, 75
open, 75
submersion, 81
superalgebra, 10
Lie, 10
supercharge operator, 315
supersymmetric theory, 315
supersymmetry
operator, 315
SW basic class, 309
SW equations, 308
SW-invariants, 305
symmetric
connection, 133
product, 395
symmetry group
global, 235
internal, 235
local, 183, 235
symplectic structure, 93
canonical, 93
tangent
bundl
e, 112
s
pace, 79
TCFT, 208
tensor
field, 83
G-invariant, 98
contravariant, 83
covariant, 83
skew symmetric, 83
symmetric, 83
of curvature type, 201
product, 395
graded, 9
space, 82
TFT, 208
Thom
class, 141
isomorphism, 141
space, 144
Thurston Geometrization Conjecture, 64
time reversal, 267
Todd class, 150
topological
charge, 152, 258
classical field theories, 208
field theories, 208
invariants, 147
quantum field theories, 208
torsion, 131
total differential, 401
TQFT, 208
trace class operator, 410
tubular neighborhood, 355
twisted product, 108
ultraviolate limit, 305
uncertainty principle, 171, 177
unknotting number, 356
vacuum
state, 261
vector, 31
Vassiliev Invariant, 367
vector bundle, 111
associated, 117
Hermitian, 111
Riemannian, 111
virtual, 159
442 Index
vector field, 80
fundamental, 116
Hamiltonian, 94
horizontal, 120
invariant, 98
left invariant, 98
locally Hamiltonian, 94
vertical, 116, 119
Verma module, 12
vertex
algebra, 10
operator, 32
vertical
bundle, 116
subspace, 119
Virasoro Algebra, 11
Virasoro vector, 31
virtual bundle, 215
VOA, 10
Ward correspondence, 281
wave–particle duality, 178
weak intermediate bosons, 267
Weil homomorphism, 144, 146
Wess–Zumino action, 224, 320
Weyl tensor, 134
anti-dual, 134
self-dual, 134
Whitney
duality, 142
sum rule, 151
Witten complex, 226, 314
Witten– Reshetikhin–Turaev invariants,
338
WRT invariants, 338, 342
Yang–Baxter equation, 21
Yang–Mills
action, 245
connection, 246
equations, 174, 246
field, 246
anti-dual, 249
self-dual, 249
functional, 245
theory, 174
Yang–Mills–Higgs
action, 255
equations, 257
Yukawa couplings, 259
zeta
f
unction
generalized, 217
of an operator, 219
regularization, 217