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1 Traces and Determinants of Pseudodifferential Operators Simon Scott King's College London OXFORD UNIVERSITY PRESS

2 CONTENTS INTRODUCTION 1 1 Traces Definition and uniqueness of a trace Traces and DGAs Example: bounded operators Matrix algebras Example: commutative algebras Example: non-commutative torus Example: de Rham currents Example: smoothing operators Example: pseudodifferential operators Characters and (algebraic) ff-theory Cyclic homology K0{A) Chern character on K0(A) Example: topological even K-theory Superconnections: Chern-Weil on KQ(M) Trace ideals in B(iJ) B( f) has no trace if H is infinite-dimensional, and a uniqtie trace if H is finite-dimensional Traces on commutator ideals The ideal F(iJ) of finite-rank operators has a unique trace On the ideal C(H) of compact operators there is no trace Schatten ideals Cp and the classical trace Ci: trace class operators Tr on smoothing operators on Rn Tr on Q Exotic traces Holomorphic functional calculus on a Banach algebra Functional integrals Spectral cut and Agmon angle Traces on functional integrals Functional integrals over infinite contours Functional integrals on B(H) and trace ideals Products of functional integrals 62

3 viii Contents Complex powers, logarithms, and heat operators Integer powers Logarithm and complex power functions Logarithm and complex power operators Dependence on the spectral cut Spectral zeta functions on Banach algebras Heat operators Functional calculus on the de Rham algebra Quillen-Chern-Weil theory Traces on i/>dos Symbol algebras Classical symbols Pseudodifferential operators (i/>dos) Classical Vdos Unique trace on * Commutator structure of \, Commutator structure of Unique quasi-trace on $c\z" Meromorphic extension of TR to Gaugings of i/'dos and meromorphic continuation Trace formulae for the Laurent coefficients Example: zeta functions Complex power gaugings Spectral zeta functions Higher Laurent expansion Zeta quasi-traces Dependence on the choice of principal angle Resolvent trace expansion Heat trace invariants Traces on subalgebras of ipdos Classical I? trace TR on even-even (odd-class) operators Leading symbol traces Suspended traces Traces on families of smoothing operators Traces on geometric families of V'dos Notes Determinants Logarithmic representations of semigroups Global logarithms Logs pull-back 156

4 ix Local logarithms Determinant structures Counting log-determinant structures Determinant structures and DGAs Logarithms on categories Example: Fredholm index Predholm operators Index as a determinant structure The restricted general linear group Glres,} (H) Families index A logarithm on the category CatFred Index bundle Universal odd logarithm Universal odd log Example: GlIea(H) Logs and KX{A) Example: log-holonomy Logs on Gloo(^l) Odd Chern character on Ki(A) K-i{A) and topological odd if-theory Logarithm structures on a Banach algebra Suspended log Example: classical logarithm The Fredholm-determinant structure detf and the first odd Chern form Spectral zeta function and heat trace formulae detf and Gli(I + Ci(J?)) Constructions of the Fredholm determinant Combinatorial spectral invariant Character of the fundamental representation Canonical trivialization of the determinant line Determinant line of a Fredholm operator Graded Fredholm determinant Gaussian and Fermionic integrals Determinants of self-adjoint operators Exotic determinants de Rham determinants Classical local logarithm on Q(M, End (E)) Odd log on C (M, G) Trace structures on logarithmic ^dos Log-classical <0dos Log-polyhomogeneous symbols Log-classical ipdos Logarithm operator 216

5 X Contents Trace structure of *Z>N Higher residue traces on *Z'N Commutator structure of *C'N TRon ^C^-N{M,E) Meromorphic extension of TR to *z-n Zeta functions of log-classical i/dos Traces on proper subalgebras of *Z'N Traces on families of log-classical ipdos Determinant structures on classical ^dos Residue determinant Residue determinant and C(A, 0) Residue determinant and the index A 'better' residue determinant structure Leading symbol determinants The 'order' determinant structure on A uniqueness result Zeta quasi-determinants Variation of the C-determinant Multiplicative anomaly and Polyakov formula Further log-determinant structures on i/>dos ipdo determinant structures on trace ideals TR determinants on even-even operators Index determinant and odd Chern forms Suspended determinants and the eta invariant Pole structure of i](d, z) and 77aus Determinants on geometric families of ^dos Notes Computations, transition formulae, and the local index formula Computations using classical zeta functions Dirac-type operators on the circle Relative zeta determinants and formal quotients Special values of the Riemann zeta function Laplacians on higher dimensional spheres Laplacian on the torus Computations on compact quotients Pushing down Schwartz kernels Extension to flat T-bundles Traces and regularized traces on compact quotients Spectral-geometric trace formula Selberg-Arthur trace formula A regularized spectral-geometric trace formula 294

6 xi Spectral-geometric trace formulae on tori: Poisson summation Spectral-geometric trace formulae on S Variational computation of det^d on S Spectral-geometric heat trace on tori Laplacian determinant from the heat trace on S Determinant of the Laplacian coupled to a flat bundle on a two-torus Regularized det d% on an elliptic curve Computations on higher genus Riemann surfaces A Selberg heat trace formula Relative Laplacian (-determinant on a surface Conformal variation of the Laplacian determinant Relative holomorphic torsion on negatively curved manifolds Computations using the complex powers Complex powers and transition formulae Basic properties Functional integral computation on S Resolvent trace and zeta trace Mellin transform formulae Resolvent trace and zeta trace transition formulae Heat trace and zeta trace Resolvent trace to heat trace to zeta trace General heat-zeta trace transition formulae Relative determinant formulae Relative heat kernel regularization Multiplicativity property Contour integral (-determinants on S Residue determinant computations Relative formulae for C(A 0) Determinant property on/ + f<q Relative Laplacians Local Atiyah-Singer index formula Residue determinant and the index Riemann-Roch-Hirzebruch formula Local Riemann-Roch formula on a surface An elementary proof of the local Atiyah-Singer index formula Statement of the formula Two traces 375

7 Resolvent symbols Computing the index density Reformulation for heat coefficients Notes Pseudodifferential operator trace formulae Homogeneous distributions Local distributions Fourier transform Extendibility of homogeneous distributions Example: dimension Gauged distributions Extensions via a gauging Regularized limit method Pull-back property of extensions Log-homogeneous distributions Log-homogeneous functions Extending log-homogeneous functions Distributions on manifolds Densities and distributions Push-forward map Localizations Schwartz kernel theorem SKT push-forward The unique trace on smoothing operators Pseudodifferential operators Oscillatory integrals Regularization of oscillatory integrals Vdos on manifolds Kernel singularity structure Stationary phase approximation expansion Gaugings of V>dos: holomorphic parameters Holomorphic families of symbols Holomorphic families of V'dos TRon#c\z"'N TRon*c^z" Traciality properties of TR TR and finite-part integrals TR on ^n,^ TRmer on *z" Laurent expansion of the gauged trace density Global densities defined on M by the expansion Full Laurent expansion Trace and quasi-trace densities 513

8 xiii The pole (the residue trace) Formulae for sphere integrals Existence and traciality of res Uniqueness of res The constant term (a quasi-trace) Global existence of the density Trace defect formulae Traciality on even-even operators TRmer on *Z"-N Construction of the quasi-trace density Densities and (quasi-)traces A higher residue trace and various quasi-traces on TRquasi #Z»'N Primitives in *Z"'N Complex power gauging Weakly parameter-dependent?/>dos Trace formulae for the basic gauging on WCN,Z"'" Formulae for the zeta determinant Trace formulae for zeta-regularized logs Determinant defect/anomaly formulae Residue determinant Commutators of logarithmic i/>dos are classical Residue density on logarithmic i/jdos res([5, T]) 0 for S, T = f^ res(log A) is independent of Residue determinant and ((A, 0) res log AB = res log A + res log B log is a logarithm map Notes Geometric families of ipdos operators and determinant line bundles Families of y>dos associated to fibrations Vertical i/>dos Classical vertical i/)dos of non-constant order Vertical V'dos on fibred manifolds Traces and quasi-traces on vertical V'dos Form-valued vertical i/'dos Vertical ellipticity and traces Asymptotic expansion of the resolvent trace form Zeta forms and zeta-determinant forms Determinant structures on vertical tpdos Vertical pseudodifferential logarithms Review of facts about connections on 7r* (E) 618

9 5.2.3 A -form local family index theorem A -Chern form local family Example: counting holomorphic index theorem 627 sections Determinant line bundles for families of V'dos The determinant line bundle Line bundle structure Determinant bundle functor Regularized determinant functions metric and connection C metric ( connection Example: conformal anomaly and the Polyakov formula Example: Quillen's computation of determinants of Cauchy-Riemann operators over a Riemann surface Parametrix formulae and Notes 662 REFERENCES 664 INDEX 673

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