ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 7, Pages 8–16 (April 2, 2001) S 1079-6762(01)00089-0

RELATIVE ZETA DETERMINANTS AND THE GEOMETRY OF THE DETERMINANT LINE BUNDLE

SIMON SCOTT

(Communicated by Michael Taylor)

Abstract. The spectral ζ-function regularized geometry of the determinant line bundle for a family of first-order elliptic operators over a closed manifold encodes a subtle relation between the local family’s index theorem and fundamental non-local spectral invariants. A great deal of interest has been directed towards a generalization of this theory to families of elliptic boundary value problems. We give here precise formulas for the relative zeta metric and curvature in terms of Fredholm determinants and traces of operators over the boundary. This has consequences for anomalies over manifolds with boundary.

0. Introduction

In this paper we study the zeta-function regularized geometry of the determinant line bundle for a family of first-order elliptic boundary value problems (EBVPs). By an EBVP we mean an elliptic differential operator over a compact manifold with boundary, endowed with a global (injectively elliptic) boundary condition. In [8] it was shown that the determinant line bundle for such a family has a natural Hermitian metric defined by the Fredholm determinant of a canonically associated operator over the boundary. The resulting canonical metric has an analytical status which is essentially opposite to that of the usual zeta function metric [1, 7]. That the two metrics, and their associated connections, are nevertheless precisely related derives from a certain ‘relativity principle for determinants’, which asserts that for preferred classes of unbounded operators, ratios of ζ-determinants can be written canonically in terms of Fredholm determinants (Theorem 1). This is interesting because the (usual) ζ-determinant, on the other hand, does not define an extension of the Fredholm determinant (operators with Fredholm determinants do not have ζ-determinants)^{1}; indeed there is no extension of the Fredholm determinant to a (multiplicative) determinant on general classes of elliptic pseudodifferential operators (ψdos).

This fact leads to the construction of ‘higher’ differential geometries for families of ψdos, via regularized traces and determinants, such as that defined by the spectral ζ-function. Our results indicate that the ‘relativity principle’ governs a precise relation between relative higher differential geometries and the (usual) canonical differential geometry associated to the standard trace. Applied to families of EBVPs

\begin{matrix}c \\ ◯ \end{matrix}2001 American Mathematical Society

Received by the editors December 15, 1999, and, in revised form, September 15, 2000.

2000 Mathematics Subject Classification. Primary 58G20, 58G26, 11S45; Secondary 81T50.

1We consider here infinite-dimensional Hilbert spaces.

this means that geometric anomalies, associated with the break down of symmetry invariance in higher traces, are located on the boundary. Details of the constructions here, which build on ideas in [2, 3, 4, 5, 8, 10], will be presented in [9].

1. Relative zeta determinants

Let A_{1}, A_{2} be invertible closed operators on a Hilbert space H with a common spectral cut R_{θ} = {re^{iθ} : r ≥ 0}. This means that there exists an ε > 0 such that the resolvents (A_{i} − λ)^{−1} are holomorphic in the sector

Λ_{θ,ε} = {z ∈ ℂ\{0} | | arg(z) − θ| < ε}

and such that the operator norms ∥(A_{i} − λ)^{−1}∥ are O(|λ|^{−1}) as λ → ∞ in Λ_{θ,ε}. For ℜ(s) > 0 one has the complex power operators

A_{i}^{−s} = \frac{i}{2π} ∫_{Γ} λ_{θ}^{−s} (A_{i} − λ)^{−1} dλ,

where λ_{θ}^{−s} = |λ|^{−s}e^{−is arg(λ)}, θ − 2π ≤ arg(λ) ≤ θ, is the branch of λ^{−s} defined by the spectral cut R_{θ}, and Γ is the contour traversing R_{θ} from ∞ to a small circle around the origin, clockwise around the circle, and then back to ∞ along R_{θ}.

If we assume that the A_{i}^{−s} are trace class in some half-plane ℜ(s) > r > 0, one then has spectral zeta functions

ζ_{θ}(A_{i}, s) = Tr A_{i}^{−s} = ∑_{μ∈sp(A_{i})} μ^{−s}, ℜ(s) > r.

We further assume that ∂_{λ}^{m}(A_{i} −λ)^{−1} are trace class for m+1 > r with asymptotic expansions as λ → ∞ along the ray R_{θ}

Tr (∂_{λ}^{m}(A_{i} − λ)^{−1}) ∼ ∑_{j=0}^{∞} ∑_{k=0}^{n_{j}} a_{j,k}^{(i)}(−λ)^{−(α_{j}+m)} log^{k}(−λ) (1.1)

with 0 < α_{j} + m ↗ +∞. From Seeley’s work (see also [4]) it is well known that the expansion (1.1) defines a meromorphic continuation of ζ_{θ}(A_{i}, s) to all of ℂ, the term with coefficient a_{j,k}^{(i)} corresponding to a pole of Γ(s)ζ_{θ}(A_{i}, s) at s = α_{j} − 1 of

order k + 1. In particular, if a_{J,k}^{(1)} = a_{J,k}^{(2)} = 0, where α_{J} = 1 with k ≥ 1, then there is no pole at s = 0 and one has the ζ-determinants

det_{ζ,θ}A_{1} = e^{−ζ′_{θ}(A_{1},0)}, det_{ζ,θ}A_{2} = e^{−ζ′_{θ}(A_{2},0)},

where ζ′_{θ} = d/ds(ζ_{θ}). In this case we refer to each of A_{1}, A_{2} as ζ-admissible. We refer to (A_{1}, A_{2}) as ζ-comparable if

a_{j,k}^{(1)} = a_{j,k}^{(2)} for j < J, (1.2)

and

a_{J,k}^{(1)} = a_{J,k}^{(2)} for k ≥ 1,

and if the relative resolvent (A_{1} − λ)^{−1} − (A_{2} − λ)^{−1} is trace class such that

Tr ((A_{1} − λ)^{−1} − (A_{2} − λ)^{−1}) = −\frac{∂}{∂λ} log det_{F} S_{λ}.

Here the ‘scattering’ operator S_{λ} = S_{λ}(A_{1}, A_{2}) is an operator of the form Id + W_{λ} on a Hilbert space H′ ⊆ H with W_{λ} of trace class, so that S_{λ} has a Fredholm determinant det_{F} S_{λ} := 1 + ∑_{k≥1} Tr (⋀^{k} W_{λ}). The relative spectral ζ-function

ζ_{θ}(A_{1}, A_{2}, s) = Tr (A_{1}^{−s} − A_{2}^{−s}) = \frac{i}{2π} ∫_{Γ} λ_{θ}^{−s} Tr ((A_{1} − λ)^{−1} − (A_{2} − λ)^{−1}) dλ

is then well defined and holomorphic in s for ℜ(s) > 0, and equal to ζ_{θ}(A_{1}, s) − ζ_{θ}(A_{2}, s). Using (1.2), we see that as λ → ∞ in Λ_{θ,ε}, there is an asymptotic expansion

Tr ((A_{1} − λ)^{−1} − (A_{2} − λ)^{−1}) ∼ ∑_{j=J+1}^{∞} ∑_{k=0}^{n_{j}} c_{j,k}(−λ)^{−α_{j}} log^{k}(−λ) + \frac{c_{J,0}}{λ} .

The meromorphic extension of ζ_{θ}(A_{1}, A_{2}, s) to ℂ is therefore regular at zero and we can define the relative ζ-determinant det_{ζ,θ}(A_{1}, A_{2}) = e^{−ζ′_{θ}(A_{1},A_{2},0)} .

The regularized limit LIM_{λ→∞}^{θ} of a function with an asymptotic expansion

f(λ) ∼ ∑_{j=0}^{∞} ∑_{k=0}^{m_{j}} b_{j,k}(−λ)^{−β_{j}} log^{k}(−λ)

as λ → ∞ in Λ_{θ,ε}, where β_{j} ↗ +∞ and β_{N} = 0, is defined to be the constant term in the expansion b_{N,0}. We have (with S := S_{0}):

Theorem 1. For ζ-comparable operators A_{1}, A_{2}

(1.3) det_{ζ,θ}(A_{1}, A_{2}) = det_{F} S . e^{−LIM_{λ→∞}^{θ} log det_{F} S_{λ}} .

If A_{1}, A_{2} are ζ-admissible, det_{ζ,θ}(A_{1}, A_{2}) = det_{ζ,θ}A_{1}/det_{ζ,θ}A_{2} .

Proof. Since λ^{−s} log det_{F} S_{λ} → 0 at the ends of Γ for ℜ(s) > 0, we can integrate by parts in

ζ_{θ}(A_{1}, A_{2}, s) = −\frac{i}{2π} ∫_{Γ} λ_{θ}^{−s} ∂_{λ} log det_{F} S_{λ} dλ .

Taking the s derivative we end up with ζ′_{θ}(A_{1}, A_{2}, s) = d/ds_{|s=0}(sg(s)), where

g(s) = −\frac{i}{2π} ∫_{Γ} λ_{θ}^{−s} (log det_{F} S_{λ})/λ dλ

has a simple pole at s = 0. From the Laurent expansion of (log det_{F} S_{λ})/λ around 0, and its asymptotic expansion as λ → ∞ in Λ_{θ,ε}, we can use the methods of [4] to obtain the full pole structure of ζ′_{θ}(A_{1}, A_{2}, s) on ℂ. Evaluation at s = 0 then yields (1.3). The final statement follows from ζ_{θ}(A_{1}, A_{2}, s) = ζ_{θ}(A_{1}, s) − ζ_{θ}(A_{2}, s).

In the case ζ_{θ}(A_{1}, A_{2}, 0) = 0 the regularized LIM can be replaced by the usual lim, and ζ_{θ}(A_{1}, A_{2}, s) exists at 0 without continuation. In particular, this applies to determinant class operators, that is, for A_{1}, A_{2} with Fredholm determinants. The ζ_{θ}(A_{i}, s) are then undefined for all s, but A_{1}, A_{2} are always ζ-comparable and from (1.3)

(1.4) det_{ζ,θ}(A_{1}, A_{2}) = det_{F} (A_{2}^{−1}A_{1}) = \frac{det_{F} A_{1}}{det_{F} A_{2}} .

Thus for any A of determinant class, det_{ζ,θ}(A, Id) = det_{F} (A). This is independent of θ, equivalent to the fact that Γ can be closed at ∞ and replaced by a bounded contour.

1.1. Elliptic boundary value problems. Let X be a compact connected Riemannian manifold with boundary ∂X = Y and let E^{1}, E^{2} be Hermitian vector bundles over X. Let A : C^{∞}(X, E^{1}) → C^{∞}(X, E^{2}) be a first-order elliptic operator of Dirac type. By this we mean that there is a collar neighborhood U = [0, 1) × Y of the boundary in which A has the form

A_{|U} = σ (\frac{∂}{∂u} + B + R) , (1.5)

where B : C^{∞}(Y, E_{|Y}^{1} ) → C^{∞}(Y, E_{|Y}^{1} ) is a first-order selfadjoint elliptic operator over the closed manifold Y, R is an operator of order 0, and σ : E_{|U}^{1} → E_{|U}^{2} a unitary isomorphism constant in u.

For each real s > 1/2, restriction to the boundary defines a continuous operator γ : H^{s}(X, E^{1}) → H^{s−1/2}(Y, E_{|Y}^{1} ) on the Sobolev completions, and we have the Cauchy data space H(A, s) = γKer(A, s), where

Ker(A, s) = {ψ ∈ H^{s}(X, E^{1}) | Aψ = 0}.

Because of the Unique Continuation Property, γ : Ker(A, s) → H(A, s) is a bijection while the Poisson operator K_{A} : H^{s−1/2}(Y, E_{|Y}^{1} ) ⟶ Ker(A, s) ⊂ H^{s}(X, E^{1}) of A defines a left inverse to γ.

The classical pseudodifferential operator (ψdo) P (A) := γK_{A} of order 0 is a projection on H^{s−1/2}(Y, E_{|Y}^{1} ) with range H(A, s), called the Calderón projection. Associated to P (A) we have the pseudodifferential Grassmannian Gr_{−1}(A) parameterizing (orthogonal) projections P on H_{Y} = L^{2}(Y, E_{|Y}^{1} ) such that P − P (A) is a ψdo of order −1. The smooth Grassmannian Gr_{−∞}(A) is the infinite-dimensional dense submanifold of Gr_{−1}(A) of those P such that P − P (A) is a smoothing operator. Each P ∈ Gr_{−1}(A) defines an EBVP

A_{P} = A : dom (A_{P}) ⟶ L^{2}(X, E^{2}),

with dom (A_{P}) = {ψ ∈ H^{1}(X, E^{1}) | P γψ = 0}. As a Fredholm operator, A_{P} is modeled by the boundary operator S_{A}(P) = P ∘P (A) : H(A) → W = range (P), in so far as the Poisson operator effects canonical isomorphisms KerA_{P} ≅ KerS_{A}(P), CokerA_{P} ≅ CokerS_{A}(P), leading to the relative-index formulas

ind (A_{P_{1}}) − ind (A_{P_{2}}) = ind (P_{2}, P_{1}), ind (A_{P}) = ind (S_{A}(P)),

where (P_{i}, P_{j}) := P_{j} ∘ P_{i} : range (P_{i}) → range (P_{j}). Moreover, if A_{P} is invertible, then so is S_{A}(P) and we can then define the Poisson operator of A_{P} by

K_{A}(P) := K_{A}S_{A}(P)^{−1}P : H^{s−1/2}(Y, E_{|Y}^{1} ) ⟶ H^{s}(X, E^{1}).

From the identities A_{P}^{−1}A = I −K_{A}(P)γ and A_{P} A_{P}^{−1} = Id_{L^{2}} we obtain the relativeinverse formula

A_{P_{1}}^{−1} = A_{P_{1}}^{−1}A_{P_{2}}A_{P_{2}}^{−1} = (A_{P_{1}}^{−1}A)A_{P_{2}}^{−1} = A_{P_{2}}^{−1} − K_{A}(P_{1})γA_{P_{2}}^{−1}. (1.6)

For references to details of these facts we refer the reader to [3, 4, 10].

Proposition 1.1. Let A_{z} be a 1-parameter family of Dirac-type operators depending smoothly on a complex parameter z. Let Ȧ_{z} = \frac{d}{dz} A_{z} and let P_{1}, P_{2} ∈ Gr_{−1}(A) such that P_{1} − P_{2} has a smooth kernel and such that A_{z,P_{i}} are invertible for each z. Then Ȧ_{z}(A_{z,P_{1}}^{−1} − A_{z,P_{2}}^{−1}) is a trace class operator on L^{2}(X, E_{2}) with

(1.7) Tr (Ȧ_{z}(A_{z,P_{1}}^{−1} − A_{z,P_{2}}^{−1})) = \frac{d}{dz} log det_{F} (\frac{S_{z}(P_{1})}{P_{1}S_{z}(P_{2})}) .

[For simplicity we assume here that (P_{2}, P_{1}) is invertible. For the general case see [9].]

Proof. We compute that

\frac{d}{dz} A_{z}.K_{z}(P_{1}) = −A_{z,P_{1}} \frac{d}{dz} K_{z}(P_{1}), (1.8)

P_{1}γA_{z,P_{2}}^{−1}A_{z,P_{1}} = −K_{z}(P_{1})^{−1}K_{z}(P_{2})P_{2}γ = −S_{z}(P_{1})S_{z}(P_{2})^{−1}P_{2}γ, (1.9)

P_{2} \frac{d}{dz} γK_{z}(P_{1}) = \frac{d}{dz} (K_{z}(P_{2})^{−1}K_{z}(P_{1})P_{1}) = \frac{d}{dz} (S_{z}(P_{2})^{−1} S_{z}(P_{1})P_{1}) , (1.10)

where S_{z} := S_{A_{z}}. Using (1.6) and (1.8) we obtain

Tr (Ȧ_{z}(A_{z,P_{1}}^{−1} − A_{z,P_{2}}^{−1})) = Tr (P_{1}γA_{z,P_{2}}^{−1}A_{z,P_{1}} \frac{d}{dz} K_{z}(P_{1})),

while (1.9), (1.10) reduce this to the right side of (1.7).

We assume now that E^{1} = E^{2} and A is a first-order elliptic operator of Dirac type. Let P_{1}, P_{2} ∈ Gr_{−1}(A) with P_{1} −P_{2} smoothing. Assume further that A_{P_{1}}, A_{P_{2}} are invertible, ζ-admissible, with spectral cut R_{θ}. Then by setting A_{λ} = A − λ for λ ∈ Γ, an application of Theorem 1 yields the following formula:

Theorem 2. With the above assumptions (putting S_{A} = S)

(1.11) \frac{det_{ζ,θ}(A_{P_{1}})}{det_{ζ,θ}(A_{P_{2}})} = det_{F} (\frac{S(P_{1})}{P_{1}S(P_{2})}) · e^{−LIM_{λ→∞}^{θ} log det_{F} ((P_{1}S_{λ}(P_{2}))^{−1}S_{λ}(P_{1}))} .

Further, the regularized limit is independent of the operator A, and depends only on the boundary conditions P_{1}, P_{2}.

The final statement follows from the fact that the left side of (1.7) is the logarithmic derivative of the left side of (1.11) ([2], Prop 1.3).

As an example, applied to selfadjoint boundary problems for a Dirac operator over an odd-dimensional spin manifold, equation (1.11) yields the relative determinant formula of [10] as a special case.

2. Geometry of the determinant line bundle

2.1. Families of EBVPs. Let π : Z ⟶ B be a smooth Riemannian fibration of manifolds with fibre X_{b} diffeomorphic to a compact manifold X with boundary Y, and let E^{i} → Z, i = 0, 1, be vertical Clifford bundles with compatible connection. Then we have a family of Dirac operators 𝔻 = {D_{b} | b ∈ B} : F^{0} ⟶ F^{1}, with F^{i} the infinite-dimensional Fréchet bundle on B with fibre C^{∞}(X_{b}, E_{b}^{i}), where E_{b}^{i} = E_{|X_{b}}^{i}. (Here 𝔻 can be a family of total or ‘chiral’ Dirac operators.)

The corresponding structures are inherited on the boundary fibration ∂π : ∂Z ⟶ B of closed manifolds with fibre Y_{b} = ∂X_{b}. We assume a product geometry in a collar neighborhood U of ∂Z so that in U_{b} = [0, 1) × ∂X_{b} one has (D_{b})_{|U_{b}} = σ_{b}(∂/∂u + D_{Y_{b}}), with 𝔻_{∂Z} = {D_{Y_{b}} | b ∈ B} : F_{∂Z} → F_{∂Z} the family of selfadjoint boundary Dirac operators defined by ∂π. Thus we have a smooth fibration of Grassmannians Gr(𝔻) → B with fibre Gr_{−∞}(D_{b}). A Grassmann section for 𝔻 is defined to be a smooth section ℙ = {P_{b} ∈ Gr_{−∞}(D_{b}) | b ∈ B} of Gr(𝔻), any such section differing from the Calderón section P (𝔻) = {P (D_{b}) | b ∈ B} by a smooth family of smoothing operators.

The primary role of a Grassmann section is to define a smooth family of EBVPs (𝔻, ℙ) parameterizing the operators D_{P_{b}} = D_{b} : dom (D_{P_{b}}) ⟶ L^{2}(X_{b}, E_{b}^{1}), and hence an associated determinant line bundle^{2} DET (𝔻, ℙ) with determinant section b ↦ det(D_{P_{b}}). On the other hand, a Grassmann section defines a smooth infinite-rank Hermitian subbundle W → B of F_{∂Z} with fibre W_{b} = range (P_{b}) ⊂ L^{2}(Y_{b}, E_{b|Y}^{1} ). For each pair of Grassmann sections ℙ_{1}, ℙ_{2} we therefore have a smooth family of boundary Fredholm operators

(ℙ_{2}, ℙ_{1}) = {P_{1,b} ∘ P_{2,b} : W_{b}^{2} → W_{b}^{1}} ∈ C^{∞}(B, Hom(W^{2}, W^{1})),

with determinant line bundle DET (ℙ_{2}, ℙ_{1}), and from [8] there is a canonical isomorphism

(2.1) DET (𝔻, ℙ_{1}) ≅ DET (𝔻, ℙ_{2}) ⊗ DET (ℙ_{2}, ℙ_{1}),

where 𝕊(ℙ_{i}) := (P (𝔻), ℙ_{i}). This means that the abstract relative determinant is a canonical section of the relative determinant line bundle DET (ℙ_{2}, ℙ_{1}). By choosing ℙ_{2} = P (𝔻), ℙ_{1} = ℙ this becomes a section of

DET (𝕊(ℙ)) ≅ DET (𝔻, ℙ). (2.2)

2.2. The ζ and C metrics. The bundle isomorphism (2.2) means that DET (𝔻, ℙ) inherits an Hermitian metric from DET (𝕊(ℙ)). This is the canonical metric ∥ · ∥_{C}, given over the open subset Ω ⊂ B where the operators D_{P} are invertible by

∥det D_{P} ∥_{C}^{2} := det_{F} (S(P)^{∗}S(P)) = det_{F} (P (D) · P · P (D)).

Here Δ_{P} = D^{∗}D : dom (Δ_{P}) → L^{2}(X, E^{0}) is the Dirac Laplacian with domain

dom (Δ_{P}) = {ψ ∈ H^{2}(X, E^{0}) : P γψ = 0, P^{⋆}γDψ = 0},

where P^{⋆} = σ(I − P)σ^{−1} is the adjoint boundary condition for D^{∗} (thus (D_{P})^{∗} = D_{P^{⋆}}^{∗}).

On the other hand, from recent work of Grubb [3] we know that ζ(Δ_{P} , s) is regular at s = 0 for P ∈ Gr_{−∞}(D). The resulting Quillen metric on DET (𝔻, ℙ) is defined over Ω by ∥detD_{P} ∥_{ζ}^{2} = det_{ζ}Δ_{P} .

Theorem 3. Let ℙ_{1}, ℙ_{2} be Grassmann sections for 𝔻 and let D_{P_{1}} ∈ (𝔻, ℙ_{1}), D_{P_{2}} ∈ (𝔻, ℙ_{2}) be invertible at b ∈ B. Then

(2.3) \frac{∥det(D_{P_{1}})∥_{ζ}}{∥det(D_{P_{2}})∥_{ζ}} = \frac{∥det(D_{P_{1}})∥_{C}}{∥det(D_{P_{2}})∥_{C}} .

That is,

(2.4) \frac{det_{ζ}(Δ_{P_{1}})}{det_{ζ}(Δ_{P_{2}})} = \frac{det_{F} (S(P_{1})^{∗}S(P_{1}))}{det_{F} (S(P_{2})^{∗}S(P_{2}))}.

2For a smooth family of Fredholm operators 𝔸 = {A_{b} | b ∈ B} parameterized by B, the complex lines Det (A_{b}) = ⋀^{max} Ker(A_{b})^{∗} ⊗⋀^{max} Coker(A_{b}) fit together to define the determinant line bundle DET (𝔸) → B endowed with a canonical section b ↦ det(A_{b}) (see [1, 7, 8]).

Or, from (1.4),

(2.5) det_{ζ}(Δ_{P_{1}}, Δ_{P_{2}}) = det_{ζ}(S(P_{1})^{∗}S(P_{1}) , S(P_{2})^{∗}S(P_{2})).

Equivalently, since S(P (D)) = Id,

(2.6) det_{ζ}(Δ_{P}) = det_{ζ}(Δ_{P (D)}) · det_{F} (S(P)^{∗}S(P)) .

Remark 2.1. In fact, this holds for P_{1} − P_{2} differing just by a ψdo of order less than − dim(X). Equations (2.3)–(2.5) say that the relative ζ-metric and relative C-metric on DET (𝔻, ℙ_{1}) ⊗ DET (𝔻, ℙ_{2})^{∗} coincide. Similarly, (2.6) corresponds to the isomorphism (2.2).

Proof. We study Δ_{P} by embedding in a fully elliptic first-order system. Associated to Δ we have the first-order elliptic operator acting on sections of E^{0} ⊕ E^{1}

Δ̂ = (\begin{matrix}0 & D^{∗} \\ D & −I \end{matrix} ) : H^{1}(X, E^{0} ⊕ E^{1}) → L^{2}(X, E^{0} ⊕ E^{1}) .

In the collar U_{b} ≅ [0, 1) × Y, Δ̂ has the form Δ̂_{|U} = σ̂ (\frac{∂}{∂u} + B̂ + R̂) , where

σ̂ = (\begin{matrix}0 & σ^{−1} \\ σ & 0 \end{matrix}) , B̂ = (\begin{matrix}B & 0 \\ 0 & −σBσ^{−1} \end{matrix} ) , R̂ = (\begin{matrix}0 & −σ^{−1} \\ 0 & 0 \end{matrix}) ,

satisfying the relations σ^{2} = −I, σ^{∗} = −σ̂, σ̂B̂ + B̂σ̂ = 0, σ̂R̂ + R̂σ̂ = −I. Hence Δ̂ is of Dirac type with Calderón projection P (Δ̂) on L^{2}(X, E_{|Y}^{0} ⊕ E_{|Y}^{1} ) (see §2),

and for each P̃ ∈ Gr_{−1}(Δ̂) we have a first-order EBVP

Δ̂P̃ = Δ̂ : dom (Δ̂P̃) → L^{2}(X, E^{0} ⊕ E^{1}).

We recover the resolvent (Δ_{P} − λ)^{−1} via the canonical embeddings

Gr_{−1}(D) ⟶ Gr_{−1}(Δ̂), P ⟼ P̂ := P ⊕ P^{⋆},
î: H^{2}(X, E^{0}) ⟶ H^{2}(X, E^{0}) ⊕ H^{1}(X, E^{1}), ψ ⟼ (ψ, Dψ) .

More precisely, setting Δ̂_{λ} = ( \begin{matrix}−λ & D^{∗} \\ D & −I \end{matrix} ), î restricts to an isomorphism Ker(Δ − λ) ≅ Ker(Δ̂_{λ}) and to an inclusion î : dom (Δ_{P}) → dom (Δ̂_{λ,P̂}) into the domain of the first-order selfadjoint (note that σ̂(I − P̂)σ^{−1} = P̂) local-elliptic boundary problem Δ̂_{λ,P̂}, and one has

Δ̂_{λ,P̂}^{−1} = (\begin{matrix}(Δ_{P} − λ)^{−1} & D_{P^{⋆}}^{∗}(Δ̃_{P^{⋆}} − λ)^{−1} \\ D_{P} (Δ_{P} − λ)^{−1} & λ(Δ̃_{P^{⋆}} − λ)^{−1} \end{matrix}) , (2.7)

where Δ̃_{P^{⋆}} = D_{P} D_{P^{⋆}}^{∗}. The relative resolvent is trace class for P_{1}, P_{2} ∈ Gr_{−∞}(D), and we use (2.7) and (1.7) to compute that

Tr ((Δ_{P_{1}} − λ)^{−1} − (Δ_{P_{2}} − λ)^{−1}) = −\frac{∂}{∂λ} log det_{F} (\frac{S_{λ}(P̂_{1})}{P̂_{1}S_{λ}(P̂_{2})}) ,

where S_{λ}(P̂) := SΔ̂_{λ}(P̂). This determines the scattering operator for (Δ_{P_{1}}, Δ_{P_{2}}). Combined with results from [3] we find that Δ_{P_{1}}, Δ_{P_{2}} are ζ-comparable. Applying Theorem 1 with θ = π we obtain

(2.8) \frac{det_{ζ}(Δ_{P_{1}})}{det_{ζ}(Δ_{P_{2}})} = det_{F} (\frac{S(P̂_{1})}{P̂_{1}S(P̂_{2})}) · e^{−LIM_{λ→+∞} log det_{F} ((P̂_{1}S_{−λ}(P̂_{2}))^{−1}S_{−λ}(P̂_{1}))}.

On the other hand, for a 1-parameter family Δ^{r} = D_{r}^{∗}D_{r} the identity (1.7) leads to

\frac{d}{dr} log \frac{det_{ζ}(Δ_{P_{1}}^{r})}{det_{ζ}(Δ_{P_{2}}^{r})} = \frac{d}{dr} log \frac{det_{C}(Δ_{P_{1}}^{r})}{det_{C}(Δ_{P_{2}}^{r})} .

Combined with (2.8) we find that

\frac{det_{ζ}(Δ_{P_{1}})}{det_{ζ}(Δ_{P_{2}})} = \frac{det_{C}(Δ_{P_{1}})}{det_{C}(Δ_{P_{2}})} · α(P_{1}, P_{2}),

where α(P_{1}, P_{2}) depends only on P_{1}, P_{2}. Finally, by studying the ‘vertical’ boundary data variation

\frac{d}{dr} log det_{ζ}(Δ_{P_{r}}) = Tr (S(P̂_{r})^{−1} \frac{d}{dr} S(P̂_{r})) − LIM_{λ→+∞}Tr (S_{−λ}(P̂_{r})^{−1} \frac{d}{dr} S_{−λ}(P̂_{r})) ,

the homogeneous (boundary symmetry group) structure of the Grassmannian forces α(P_{1}, P_{2}) = 1. (Note: It is not generally true that α = 1 for other classes of boundary conditions.)

2.3. The ζ- and C-connection and curvature. We briefly describe the extension of these methods to the regularized connection forms. To define a connection on DET (𝔻, ℙ) we make a choice of splitting T Z = T (Z/B) ⊕ T^{H}Z. This induces a natural connection ∇^{Z} on F^{i} (see [1]). Similarly, we obtain a connection ∇^{∂Z} on F_{∂Z}, and we assume that ∇_{|U}^{Z} = γ^{∗}∇^{∂Z}. Given Grassmann sections ℙ_{i}, i = 1, 2, the associated subbundles W^{i} of F_{∂Z} inherit a Hermitian metric with compatible connection ∇^{i} = ℙ_{i} · ∇^{∂Z} · ℙ_{i} , and hence induce a connection ∇^{1,2} on Hom(W^{1}, W^{2}). The C-connection on DET (𝔻, ℙ) is defined over Ω by ∇^{C} det(D_{P})/ det(D_{P}) = Tr (S(P)^{−1}∇^{1,2}S(P)), with ℙ_{1} = P (𝔻) and ℙ_{2} = ℙ. (See [8] for details.)

On the other hand, with a suitable boundary modification defined using ℙ·∇^{∂Z}·ℙ, ∇^{Z} descends to a connection ∇^{ℙ} on the subbundle F_{ℙ} with fibre dom (D_{P_{b}}) at b ∈ B. For ℜ(s) >> 0, let ω_{P} (s) = −Tr (Δ_{P}^{−s}D_{P} ∇^{ℙ}D_{P}^{−1}). This has a continuation to ℂ with a simple pole at s = 0. Following [1], the ζ-connection form on DET (𝔻, ℙ) over Ω is defined by (d/ds)_{|s=0}(sω(s)). The C- and ζ-connections are compatible with their metrics and we obtain the following result:

Theorem 4. Let ℙ_{1}, ℙ_{2} be choices of Grassmann sections. Let R_{C}^{ℙ_{i}}, R_{ζ}^{ℙ_{i}} be the curvature 2-forms of the canonical and zeta connection on DET (𝔻, ℙ_{i}). Then one has

R_{ζ}^{ℙ_{1}} − R_{ζ}^{ℙ_{2}} = R_{C}^{ℙ_{1}} − R_{C}^{ℙ_{2}}. (2.9)

Equivalently,

R_{ζ}^{ℙ} = R_{ζ}^{P (𝔻)} + R_{C}^{ℙ}. (2.10)

Equation (2.9) holds in Ω^{2}(B), since the endomorphism bundle of a complex line bundle is canonically trivial. The second identity (2.10), which says that the ζ curvature consists of an interior (cohomologically trivial) part plus a boundary correction term, follows from (2.9) because ∇^{C} on DET (𝔻, P (𝔻)) is the trivial connection.

As an example, consider the ‘universal’ family (𝔻, ℙ) = {D_{P} | P ∈ Gr_{−∞}(D)} of EBVPs parameterized by Gr_{−∞}(D) relative to a fixed operator D. Let R_{ζ} be the ζ curvature of the corresponding determinant bundle. Then DET (𝕊(ℙ)) is the determinant bundle of [6], used to construct loop group representations, ∇^{Y} is just the trivial connection, R_{ζ}^{P (𝔻)} = 0, and ∇^{C} the connection used in [6], and we obtain:

Corollary 2.2.

R_{ζ} = iω_{Gr},

where ω_{Gr} is the Kähler form on the Grassmannian.

References

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[8] Scott, S. G., ‘Splitting the curvature of the determinant line bundle’, Proc. Amer. Math. Soc. 128 (2000), 2763–2775. MR 2000m:58055

[9] Scott, S. G., In preparation.

[10] Scott, S.G., and Wojciechowski, K.P., ‘The ζ-determinant and Quillen’s determinant for a Dirac operator on a manifold with boundary’, Geom. Funct. Anal. 10 (2000), 1202–1236. CMP 2001:05

Department of Mathematics, King’s College, London WC2R 2LS, U.K.

E-mail address: sscott@mth.kcl.ac.uk