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Determinants of Elliptic Pseudo-Differential Operators 3 DETERMINANTS OF ELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS MAXIM KONTSEVICH AND SIMEON VISHIK March 1994 Abstract. Determinants of invertible pseudo-differential operators (PDOs) close to positive self-adjoint ones are defined through the zeta-function regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A) det(B)) con- sidered as a function on pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural class of PDOs on odd-dimensional manifolds generalizing the class of elliptic dif- ferential operators, the multiplicative anomaly is identically 1. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a quadratic non-linearity hidden in the defi- nition of determinants of PDOs through zeta-functions. The natural explanation of this non-linearity follows from complex-analytic prop- erties of a new trace functional TR on PDOs of non-integer orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain arXiv:hep-th/9404046v1 7 Apr 1994 several new results. In particular, we describe a structure of derivatives of zeta- functions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zeta-regularized determinants to gen- eral elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition. Contents 1. Introduction 2 1.1. Second variations of zeta-regularized determinants 2. Determinants and zeta-functions for elliptic PDOs. Multiplicative anomaly 12 1 2 MAXIM KONTSEVICH AND SIMEON VISHIK 3. Canonical trace and canonical trace density for PDOs of noninteger orders. Derivatives of zeta-functions at zero 24 3.1. Derivatives of zeta-functions at zero as homogeneous polynomials on the space of logarithms for elliptic operators 4. Multiplicative property for determinants on odd-dimensional man- ifolds 45 4.1. Dirac operators 4.2. Determinants of products of odd class elliptic operators 4.3. Determinants of multiplication operators 4.4. Absolute value determinants 4.5. A holomorphic on the space of PDOs determinant and its monodromy 5. Lie algebra of logarithmic symbols and its central extension 70 6. Determinant Lie groups and determinant bundles over spaces of elliptic symbols. Canonical determinants 77 6.1. Topological properties of determinant Lie groups as C×-bundles over elliptic symbols 6.2. Determinant bundles over spaces of elliptic operators and of elliptic symbols. 6.3. Odd class operators and the canonical determinant 6.4. Coherent systems of determinant cocycles on the group of elliptic symbols 6.5. Multiplicative anomaly cocycle for Lie algebras 6.6. Canonical trace and determinant Lie algebra 7. Generalized spectral asymmetry and a global structure of deter- minant Lie groups 119 7.1. PDO-projectors and a relative index 8. Determinants of general elliptic operators 130 8.1. Connections on determinant bundles given by logarithmic symbols. Another determinant for general elliptic PDOs 8.2. The determinant defined by a logarithmic symbol as an extension of the zeta-regularized determinant 8.3. Determinants near the domain where logarithms of symbols do not exist References 155 1. Introduction Determinants of finite-dimensional matrices A, B ∈ Mn(C) possess a multiplicative property: det(AB) = det(A) det(B). (1.1) DETERMINANTS OF ELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS 3 An invertible linear operator in a finite-dimensional linear space has different types of generalizations to infinite-dimensional case. One type is pseudo-differential invertible elliptic operators A: Γ(E) → Γ(E), acting in the spaces of smooth sections Γ(E) of finite rank smooth vector bundles E over closed smooth manifolds. Another type is invertible operators of the form Id +K where K is a trace class operator, acting in a separable Hilbert space H. For operators A, B of the form Id+K the equality (1.1) is valid. However, for a general elliptic PDO this equality cannot be valid. It is not trivial even to define any determinant for such an elliptic operator. Note that there are no difficulties in defining of the Fredholm determinant detFr(Id+K). One of these definitions is 2 m detFr(Id+K):= 1+Tr K + Tr ∧ K + . + Tr(∧ K)+ . (1.2) The series on the right is absolutely convergent. For a finite-dimensional linear opera- tor A its determinant is equal to the finite sum on the right in (1.2) with K := A−Id. Properties of the linear operators of the form Id+K (and of their Fredholm determi- nants) are analogous to the properties of finite-dimensional linear operators (and of their determinants). In some cases an elliptic PDO A has a well-defined zeta-regularized determinant detζ(A) = exp −∂/∂sζA(s) , s=0 where ζA(s) is a zeta-function of A. Such zeta-regularized determinants were invented by D.B. Ray and I.M. Singer in their papers [Ra], [RS1]. They were used in these papers to define the analytic torsion metric on the determinant line of the cohomol- ogy of the de Rham complex. This construction was generalized by D.B. Ray and I.M. Singer in [RS2] to the analytic torsion metric on the determinant line of the ∂¯-complex on a K¨ahler manifold. However there was no definition of a determinant for a general elliptic PDO until now. The zeta-function ζA(s) is defined in the case when the order d(A) is real and ∗ nonzero and when the principal symbol ad(x, ξ) for all x ∈ M, ξ ∈ Tx M, ξ =6 0, has no eigenvalues λ in some conical neighborhood U of a ray L from the origin in the spectral plane U ⊂ C ∋ λ. But even if zeta-functions are defined for elliptic PDOs A, B, and AB (so in particular, d(A), d(B), d(A)+ d(B) are nonzero) and if the principal symbols of these three operators possess cuts of the spectral plane, then in general det(AB) =6 det(A) det(B). It is natural to investigate algebraic properties of a function F (A, B) := det(AB) (det(A) det(B)). (1.3) . 4 MAXIM KONTSEVICH AND SIMEON VISHIK This function is defined for some pairs (A, B) of elliptic PDOs. For instance, F (A, B) is defined for PDOs A, B of positive orders sufficiently close to self-adjoint positive PDOs (with respect to a smooth positive density g on M and to a Hermitian structure h on a vector bundle E, A and B act on Γ(E)).1 (In this case, zeta-functions of A, B and of AB can be defined with the help of a cut of the spectral plane close to R−. Indeed, for self-adjoint positive A and B the operator AB is conjugate to A1/2BA1/2 and the latter operator is self-adjoint and positive.) Properties of the function F (A, B), (1.3), are connected with the following remark (due to E. Witten). Let A be an invertible elliptic DO of a positive order possessing some cuts of the spectral plane. Then under two infinitesimal deformations for the coefficients of A in neighborhoods U1 and U2 on M on a positive distance one from another (i.e., U¯1 ∩ U¯2 = ∅) we have −1 −1 δ1δ2 log detζ (A)= − Tr δ1A · A δ2A · A . (1.4) This equality is proved in Section 1.1. Here, δjA are deformations of a DO A in Uj without changing of its order. The operator on the right is smoothing (i.e., its Schwarz kernel is C∞ on M × M). Hence it is a trace class operator and its trace is well-defined. Note that the expression on the right in (1.4) is independent of a cut of the spectral plane in the definition of the zeta-regularized determinant on the left in (1.4). It follows from (1.4) that log detζ (A) is canonically defined up to an additional local functional on the coefficients of A. Indeeed, for two definitions, log detζ(A) and ′ log detζ (A), for a given A, we have ′ δ1δ2 log detζ (A) − log detζ (A) =0. (1.5) The equality δ1δ2F (A) = 0 for deformations δjA in Uj, U¯1 ∩ U¯2 = ∅, is the character- istic property of local functionals. It follows from (1.4) that f(A, B) := log det(AB) − log det(A) − log det(B) (1.6) is a local (on the coefficients of invertible DOs A and B) functional, if these zeta- regularized determinants are defined. Namely, if δjA and δjB are infinitesimal vari- ations of A and of B in Uj, j =1, 2, U¯1 ∩ U¯2 = ∅, then δ1δ2f(A, B)=0. (1.7) 1The explicit formula for F (A, B) in the case of positive definite commuting elliptic differential operators A and B of positive orders was obtained by M. Wodzicki [Kas]. For positive definite elliptic PDOs A and B of positive orders a formula for F (A, B) was obtained in [Fr]. However it was obtained in another form than it is written and used in the present paper. The authors are very indebted to L. Friedlander for his information about the multiplicative anomaly formula obtained in [Fr]. DETERMINANTS OF ELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS 5 This equality is deduced from (1.4) in Section 1.1.
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