Introduction .................................................... 1
Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector
bundles ........................................................ 11
1.1 The Clifford algebra ................................... 11
1.2 The standard Hodge theory .............................. 12
1.3 The Levi-Civita superconnection ........................ 14
1.4 Superconnections and Poincaré duality .................. 15
1.5 A group action ......................................... 16
1.6 The Lefschetz formula .................................. 16
1.7 The Riemann-Roch-Grothendieck theorem .................. 17
1.8 The elliptic analytic torsion forms .................... 19
1.9 The Chern analytic torsion forms ....................... 21
1.10 Analytic torsion forms and Poincaré duality ............ 22
1.11 The secondary classes for two metrics .................. 22
1.12 Determinant bundle and Ray-Singer metric ............... 23
Chapter 2. The hypoelliptic Laplacian on the cotangent
bundle ......................................................... 25
2.1 A deformation of Hodge theory .......................... 25
2.2 The hypoelliptic Weitzenböck formulas .................. 29
2.3 Hypoelliptic Laplacian and standard Laplacian .......... 30
2.4 A deformation of Hodge theory in families .............. 33
2.5 Weitzenböck formulas for the curvature ................. 35
2.6 FΜΦb,±н-bωII, EΜ,2Φb,±н-bωII and Levi-Civita
superconnection ........................................ 40
2.7 The superconnection АΜΦ,н-ωII and Poincaré duality ...... 40
2.8 A 2-parameter rescaling ................................ 41
2.9 A group action ......................................... 43
Chapter 3. Hodge theory, the hypoelliptic Laplacian and its
heat kernel .................................................... 44
3.1 The cohomology of T*X and the Thom isomorphism ......... 44
3.2 The Hodge theory of the hypoelliptic Laplacian ......... 45
3.3 The heat kernel for U2Φb,±нc ............................ 50
3.4 Uniform convergence of the heat kernel as b→0 .......... 53
3.5 The spectrum of U´2Φb,±н as b→0 ......................... 55
3.6 The Hodge condition .................................... 58
3.7 The hypoelliptic curvature ............................. 60
Chapter 4. Hypoelliptic Laplacians and odd Chern forms ......... 62
4.1 The Berezin integral ................................... 63
4.2 The even Chern forms ................................... 64
4.3 The odd Chern forms and a 1-form on R*2 ................. 65
4.4 The limit as t→0 of the forms ub,t, vb,t, wb,t ........... 68
4.5 A fundamental identity ................................. 68
4.6 A rescaling along the fibers of T*X .................... 69
4.7 Localization of the problem ............................ 70
4.8 Replacing T*X by TxX ⊕ T*xX and the rescaling of
Clifford variables on T*X .............................. 76
4.9 The limit as t→0 of the rescaled operator .............. 80
4.10 The limit of the rescaled heat kernel .................. 82
4.11 Evaluation of the heat kernel for Δv/4+a∇p ............. 87
4.12 An evaluation of certain supertraces ................... 91
4.13 A proof of Theorems 4.2.1 and 4.4.1 .................... 92
Chapter 5. The limit as t→0 and b→0 of the superconnection
forms .......................................................... 98
5.1 The definition of the limit forms ...................... 98
5.2 The convergence results ............................... 101
5.3 A contour integral .................................... 102
5.4 A proof of Theorem 5.3.1 .............................. 104
5.5 A proof of Theorem 5.3.2 .............................. 104
5.6 A proof of the first equations in (5.2.1) and
(5.2.2) ............................................... 109
Chapter 6. Hypoelliptic torsion and the hypoelliptic
Ray-Singer metrics ............................................ 113
6.1 The hypoelliptic torsion forms ........................ 113
6.2 Hypoelliptic torsion forms and Poincare duality ....... 115
6.3 A generalized Ray-Singer metric on the determinant
of the cohomology ..................................... 116
6.4 Truncation of the spectrum and Ray-Singer metrics ..... 120
6.5 A smooth generalized metric on the determinant
bundle ................................................ 122
6.6 The equivariant determinant ........................... 123
6.7 A variation formula ................................... 125
6.8 A simple identity ..................................... 126
6.9 The projected connections ............................. 126
6.10 A proof of Theorem 6.7.2 .............................. 127
Chapter 7. The hypoelliptic torsion forms of a vector
bundle ........................................................ 131
7.1 The function τ (c,η,x) ................................ 131
7.2 Hypoelliptic curvature for a vector bundle ............ 133
7.3 Translation invariance of the curvature ............... 134
7.4 An automorphism of E .................................. 135
7.5 The von Neumann supertrace of exp(-LEc) ................ 136
7.6 A probabilistic expression for Q´c .................... 138
7.7 Finite dimensional supertraces and infinite
determinants .......................................... 139
7.8 The evaluation of the form Trs [g exp(-LEc)] ........... 148
7.9 Some extra computations ............................... 152
7.10 The Mellin transform of certain Fourier series ........ 155
7.11 The hypoelliptic torsion forms for vector bundles ..... 160
Chapter 8. Hypoelliptic and elliptic torsions: a comparison
formula ....................................................... 162
8.1 On some secondary Chern classes ....................... 162
8.2 The main result ....................................... 163
8.3 A contour integral .................................... 164
8.4 Four intermediate results ............................. 165
8.5 The asymptotics of the I0k ............................ 166
8.6 Matching the divergences .............................. 169
8.7 A proof of Theorem 8.2.1 .............................. 170
Chapter 9. A comparison formula for the Ray-Singer metrics .... 171
Chapter 10. The harmonic forms for b→0 and the formal Hodge
theorem ....................................................... 173
10.1 A proof of Theorem 8.4.2 .............................. 173
10.2 The kernel of A2Φнc as a formal power series .......... 175
10.3 A proof of the formal Hodge Theorem ................... 178
10.4 Taylor expansion of harmonic forms near b=0 ........... 180
Chapter 11. A proof of equation (8.4.6) ....................... 182
11.1 The limit of the rescaled operator as t→0 ............. 182
11.2 The limit of the supertrace as t→0 .................... 187
11.3 A proof of equation (8.4.6) ........................... 189
Chapter 12. A proof of equation (8.4.8) ....................... 190
12.1 Uniform rescalings and trivializations ................ 190
12.2 A proof of (8.4.8) .................................... 192
Chapter 13. A proof of equation (8.4.7) ....................... 194
13.1 The estimate in the range t≥bβ ........................ 194
13.2 Localization of the estimate near π-1Xg ................ 196
13.3 A uniform rescaling on the creation annihilation
operators ............................................. 198
13.4 The limit as t→0 of the rescaled operator ............. 200
13.5 Replacing X by TXX .................................... 202
13.6 A proof of (13.2.11) .................................. 205
13.7 A proof of Theorem 13.6.2 ............................. 206
Chapter 14. The integration by parts formula .................. 214
14.1 The case of Brownian motion ........................... 215
14.2 The hypoelliptic diffusion ............................ 217
14.3 Estimates on the heat kernel .......................... 219
14.4 The gradient of the heat kernel ....................... 220
Chapter 15. The hypoelliptic estimates ........................ 224
15.1 The operator U´2Φb,±н ................................. 224
15.2 A Littlewood-Paley decomposition ...................... 226
15.3 Projectivization of T*X and Sobolev spaces ............ 227
15.4 The hypoelliptic estimates ............................ 229
15.5 The resolvent on the real line ........................ 238
15.6 The resolvent on С .................................... 240
15.7 Trace class properties of the resolvent ............... 243
Chapter 16. Harmonic oscillator and the J0 function ........... 247
16.1 Fock spaces and the Bargman transform ................. 247
16.2 The operator В(ξ) ..................................... 249
16.3 The spectrum of В(iξ) ................................. 251
16.4 The function J0(у,λ) .................................. 253
16.5 The resolvent of В(iξ)+P .............................. 261
Chapter 17. The limit of U´2Φb,±н as b→0 ....................... 264
17.1 Preliminaries in linear algebra ....................... 268
17.2 A matrix expression for the resolvent ................. 268
17.3 The semiclassical Poisson bracket ..................... 270
17.4 The semiclassical Sobolev spaces ...................... 271
17.5 Uniform hypoelliptic estimates for Ph ................. 272
17.6 The operator P0h and its resolvent Sh,λ for λ∈R ........ 277
17.7 The resolvent Sh,λ for λ∈С ............................ 281
17.8 A trivialization over X and the symbols Sd,kρ,σ,c ....... 283
17.9 The symbol Q0h(x,ξ) - λ and its inverse е0,h,λ(х,ξ) ..... 289
17.10 The parametrix for Sh,λ ............................... 306
17.11 A localization property for E0,E1 ..................... 307
17.12 The operator P±Sh,λ .................................... 308
17.13 A proof of equation (17.12.9) ......................... 309
17.14 An extension of the parametrix to λ∈V ................ 318
17.15 Pseudodifferential estimates for P±Sh,λi± .............. 319
17.16 The operator Θh,λ ...................................... 323
17.17 The operator Th,λ ...................................... 326
17.18 The operator (J1/J0)(hDX/√2,λ) ......................... 329
17.19 The operator Uh,λ ...................................... 331
17.20 Estimates on the resolvent of Th,h2λ ................... 337
17.21 The asymptotics of (Lc - λ)-1 .......................... 340
17.22 A localization property ............................... 348
Bibliography .................................................. 353
Subject Index ................................................. 359
Index of Notation ............................................. 361
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