Lecture 2

The Index and Determinant of the Dirac Operator

This chapter is something of a digression, but is put in for two reasons. First, the patching-together properties of the index of the Dirac operator provide a good example of the formal structure of topological field theory, and will serve to motivate some of the abstract-looking category theory which we shall come to in the next chapter. Secondly, the construction of the determinant gives us our first example of a more-or-less “realistic” —not topological— field theory. It will be the basic tool when we turn to conformal field theory in Chapter Four.

An n-dimensional topological field theory gives us a number F (X) for every closed oriented n-dimensional manifold X. Among the invariants of manifolds which arise in this way are the signature of a 4k-manifold, i.e. the signature of the quadratic form on the middle-dimensional homology of X given by the intersection pairing, and the Â-genus, which is defined as the index of the Dirac operator on an even-dimensional spin manifold. I shall concentrate here on the Â-genus Â(X), but the signature sign(X) is a closely related invariant, and I shall return to it in §2.?. (Both Â(X) and sign(X) are additive, rather than multiplicative, for disjoint unions, so to fit them directly into the framework of Chapter 1 we should consider e^{Â(X)} and e^{sign(X)}.

Let us recall the most basic facts about Dirac operators and Fredholm operators.

2.1 The Dirac operator

Dirac defined his operator in Euclidean space ℝ^{n} as

D = ∑ γ_{i} \frac{∂}{∂x_{i}} ,

where γ_{1}, ⋯, γ_{n} are N × N skew-hermitian matrices satisfying γ_{i}^{2} = −1 and γ_{i}γ_{j} = −γ_{j}γ_{i} when i ≠ j. His idea was to find a first-order operator D whose square was the Laplacian:

D^{2} = − ∑ (\frac{∂}{∂x_{i}} )^{2} .

To give the matrices γ_{i} is the same as to give for each vector ξ ∈ ℝ^{n} a matrix c_{ξ} depending linearly on ξ such that c_{ξ}^{2} = −||ξ||^{2}, for if ξ = (ξ_{i}) we can define c_{ξ} = ∑ ξ_{i}γ_{i}. In other words, we are giving a matrix representation of the Clifford algebra C(ℝ^{n}).

To make sense of this on a general Riemannian manifold M we must give for each x ∈ M a complex vector space Δ_{x} with an inner product, called the space of spinors at x, and for each cotangent vector ξ at x a skew transformation

c_{ξ} : Δ_{x} → Δ_{x}

such that c_{ξ}^{2} = −||ξ||^{2}. The spaces Δ_{x} must fit together to form a vector bundle Δ on M. If each space Δ_{x} is an irreducible representation of the Clifford algebra C(T_{x}^{∗}) then Δ is called a spin bundle. A choice of such a bundle is traditionally called a spin^{c}-structure on M. To define the Dirac operator we also need to choose a connection on Δ which is compatible with the Levi-Civita connection of M. A connection in Δ is a rule which enables us to differentiate any spinor field s —i.e. a section s of Δ— along any tangent vector field ξ to M. I shall write the derivative ∇_{ξ}s. Compatibility with the Levi-Civita connection means that

∇_{ξ}(c_{η}s) = c_{∇_{ξ}η}s + c_{η}∇_{ξ}s.

When we have a spin bundle with a connection, we define the Dirac operator D_{M} as the operator given locally by

D_{M} = ∑ c_{ξ_{i}}∇_{ξ_{i}} (2.1.1)

where {ξ_{i}} is a set of tangent vector fields which form an orthonormal basis of the tangent space at each point. (Of course D_{M} is independent of the choice of the ξ_{i}). Solutions s of the equation D_{M}s = 0 are called harmonic spinor fields.

We must distinguish two cases. If M is even-dimensional —say of dimension n = 2k— then the spin bundle Δ is of dimension 2^{k}, and it automatically splits as a sum

Δ = Δ^{even} ⊕ Δ^{odd},

where Δ^{even} and Δ^{odd} are the (±1)-eigenspaces of the operator

ω = i^{\frac{1}{2} n(n+1)}c_{ξ_{1}}c_{ξ_{2}} ... c_{ξ_{n}},

which has square 1. The operator ω depends on the orientation of M, and changes sign if it is reversed. Reversing the orientation therefore interchanges Δ^{even} and Δ^{odd}. The Dirac operator takes sections of Δ^{even} to sections of Δ^{odd}, and vice versa. We shall write it

D_{M} = D_{M}^{even} ⊕ D_{M}^{odd},

where D_{M}^{even} : Γ(Δ^{even}) → Γ(Δ^{odd}) is the adjoint of D_{M}^{odd} : Γ(Δ^{odd}) → Γ(Δ^{even}). When one speaks of the index of the Dirac operator, one always means the index of D_{M}^{even}, as the self-adjoint operator D_{M} has index zero.

If M is of odd dimension n = 2k + 1, on the other hand, then the spin bundle Δ has dimension 2^{k}, and does not split into even and odd parts. In this case the operator ω acts as the scalar ±1 on each Δ_{x}, and, by replacing c_{ξ} by −c_{ξ} if necessary, we can assume ω acts as +1. Reversing the orientation of M therefore changes D_{M} to −D_{M}.

If M is a manifold with a boundary ∂M, then at each point x ∈ ∂M we have the map c_{v(x)} : Δ_{x} → Δ_{x} such that c_{v(x)}^{2} = −1, where v(x) is the unit inward normal vector to ∂M at x. If M is even-dimensional, these maps define an isomorphism

Δ_{M}^{even}|∂M ≅ Δ_{M}^{odd}|∂M,

and either of these bundles can be identified with Δ_{∂M}. If M is odd-dimensional, then the (±i)-eigenspaces of c_{v(x)} for x ∈ ∂M split Δ_{M}|∂M as the sum of two bundles which can be identified with Δ_{∂M}^{even} and Δ_{∂M}^{odd}.

2.2 Fredholm operators

If E and F are topological vector spaces, a Fredholm operator T : E → F is a continuous linear map which has an inverse, or “parametrix”, modulo operators of finite rank, i.e. for which there is a continuous P : F → E such that P ∘ T and T ∘ P differ from the identity by finite rank operators. If T is Fredholm it is easy to see that the kernel and cokernel

ker(T) = { ξ ϵ E : T ξ = 0}
coker(T) = F/T (E)

are finite dimensional, and that the image T (E) is a closed subspace of F. The converse is true, too, if E and F are Fréchet spaces. If T : E → F is Fredholm, its index χ(T) is defined by

χ(T) = dim(ker(T)) − dim(coker(T)).

The important property of the index is that in many situations it does not change when T is deformed continuously. For example, if E and F are Banach spaces, and the space Fred(E; F) of Fredholm operators is given the norm topology, then T_{0} and T_{1} belong to the same connected component of Fred(E; F) if and only if they have the same index. But, quite generally, if {T_{t}} is a family of Fredholm operators, then χ(T_{t}) is a continuous function of t providing one can find a family {P_{t}} of parametrices such that the operators P_{t} ∘ T_{t} − 1 and T_{t} ∘ P_{t} − 1 are compact, and vary continuously with t in the uniform topology. This is always the case if {T_{t}} is a family of elliptic differential operators on a compact manifold, and T_{t} depends smoothly on t.

2.3 Localizing the index

Like the signature, the Â-genus of a manifold appears to be of an altogether global nature, and when a closed manifold X is a union X = X_{1} ∪ X_{2} of two manifolds with boundary whose intersection is their common boundary Y, there seems at first no reason why Â(X) should be a sum of contributions from X_{1} and X_{2}. The Atiyah-Singer index theorem, however, tells us that Â(X) is in some sense a sum of local contributions, for it asserts

Â(X) = ∫_{X} α_{X}
= ∫_{X_{1}} α_{X} + ∫_{X_{2}} α_{X}
(2.3.1)  = Â(X_{1}) + Â(X_{2}),

say, where α_{X} is a differential form constructed locally from the geometry of X.

The formula (2.3.1) splits Â(X) into contributions from X_{1} and X_{2}; but the contributions are not integers, and —more importantly— they depend on the Riemannian structure of X_{1} and X_{2}, while Â(X) does not. Nevertheless, the image of Â(X_{1}) or Â(X_{2}) in ℝ/ℤ depends only on the structure of X in an arbitrarily small neighbourhood of the interface Y, for if either X_{1} or X_{2} is replaced by another manifold which is indistinguishable in the neighbourhood of Y then Â(X) can only change by an integer. Let us write Z_{Y} for the set of real numbers congruent to Â(X_{1}) modulo ℤ : it is a set with a free transitive action of the additive group ℤ, i.e. an “affine space” for ℤ, or ℤ-torsor. (The word “torsor” seems rebarbative, but at least it is short).

That there should be a ℤ-torsor Z_{Y}, depending only on Y, in which X_{1} and X_{2} define elements Â(X_{1}) and −Â(X_{2}) whose “difference” is Â(X), is quite easy to see directly, without using the index theorem. Let us write K, K_{1} and K_{2} for the harmonic spinor fields on X, X_{1}, and X_{2}. Because the Dirac operator is of first order and elliptic, an element s ∈ K is the same thing as a pair s_{1} ∈ K_{1}, s_{2} ∈ K_{2} such that s_{1}|Y = s_{2}|Y. Furthermore, s_{1} and s_{2} are completely determined by their boundary values s_{1}|Y and s_{2}|Y, and so one can regard K_{1} and K_{2} as subspaces of the space Γ_{Y} of all smooth spinor fields on Y. Thus we have K = K_{1} ∩ K_{2}. In fact rather more is true.

Proposition 2.3.2 (i) The subspaces K_{1} and K_{2} of Γ_{Y} are closed, and there is an exact sequence

(2.3.3) 0 → ker(D_{X}) → K_{1} ⊕ K_{2} → Γ_{Y} → coker(D_{X}) → 0

where the middle map is the sum of the inclusions.

(ii) There is an orthogonal decomposition Γ_{Y} = K_{2} ⨁ K_{2}^{⊥}, and hence an exact sequence

(2.3.4) 0 → ker(D_{X}) → K_{1} → K_{2}^{⊥} → coker(D_{X}) → 0

where the middle map is the orthogonal projection.

Before giving the proof let us notice how the result relates to the factorization of the index of D_{X}. If the spaces K_{1} and K_{2}^{⊥} were finite-dimensional, (2.3.4) would tell us that χ(D_{X}) was the difference between their dimensions. Of course they are infinite-dimensional, but we shall see that they belong to a special class of closed subspaces of Γ_{Y} which are sufficiently close to each other for any two of them to have a well-defined relative dimension, say dim(K_{1} : K_{2}^{⊥}). Thus (2.3.4) implies

χ(D_{X}) = dim(K_{1} : K_{2}^{⊥}).

The subspaces of Γ_{Y} in question form its restricted Grassmannian Gr_{Y}. This is a space whose set of connected components π_{0}Gr_{Y} forms a ℤ-torsor Z_{Y}, the components being distinguished by their relative dimension.

2.4 Polarized vector spaces and the restricted Grassmannian

The concept of a polarized topological vector space will play a prominent role throughout these lectures. The vector spaces we shall consider will always be assumed to be locally convex and complete.

A polarization of a vector space E is a class of allowable decompositions E = E^{+} ⊕ E^{−} which are fairly close to each other. The meaning of “fairly close” is somewhat elastic, depending on our precise purposes. At the least, we want to permit any finite dimensional changes to E^{+} and E^{−}, but for the present a very loose definition will suffice. To state it, it is useful to identify decompositions E = E^{+}⊕E^{−} with the corresponding operators J : E → E such that J|E^{±} = ±1.

Definition 2.4.1 A coarse polarization of a vector space E is a class J of operators J : E → E such that (i) J^{2} = 1 modulo compact operators

(ii) any two operators in J differ by a compact operator, and

(iii) J does not contain ±1.

Example 2.4.2 If E is the space of smooth functions on the circle S^{1} then we have a decomposition E = E^{+} ⊕ E^{−}, where E^{+} is spanned by the functions e^{inθ} for n < 0, and E^{−} by e^{inθ} for n ⩾ 0. The polarization so defined does not depend on the parametrization of the circle, for the operators J corresponding to two different choices differ by an integral operator with a smooth kernel. (See [PS] page 91). We could also transfer any finite number of the functions e^{inθ} from E^{+} to E^{−}, or vice versa, without changing the polarization.

The functions e^{inθ} are characterized as the eigenfunctions of the operator i\frac{d}{dθ} on S^{1}, which in fact is the Dirac operator on S^{1}. For any odd-dimensional compact Riemannian spin manifold Y, the space Γ_{Y} of smooth spinor fields on Y is correspondingly polarized by Γ_{Y} = Γ_{Y}^{+} ⊕ Γ_{Y}^{−} where Γ_{Y}^{+} is spanned by the eigenfunctions of the Dirac operator with eigenvalues ⩾ 0, and Γ_{Y}^{−} by those with eigenvalues < 0.

When we have a polarized vector space E we can define its restricted Grassmannian Gr(E) as the set of all subspaces which can occur as the “negative energy” part E^{−} in one of the allowable decompositions E = E^{+} ⊕ E^{−}. If

E = E^{+}^{∼} ⊕ E^{∼}^{−} is another allowable decomposition then the projection of Ẽ^{−} on to E^{−} along E^{+} is automatically Fredholm, and its index is called the relative dimension dim (Ẽ^{−} : E^{−}.) The set Gr(E) is naturally an infinite dimensional manifold, for any W ∈ Gr(E) which is near E^{−} is the graph of a compact operator E^{−} → E^{+}, and so a neighbourhood of E^{−} can be identified with the vector space Hom_{cpt}(E^{−}; E^{+}). It is easy to see that two points of Gr(E) are in the same connected component if and only if their relative dimension is zero, and so π_{0}Gr(E) is a ℤ-torsor as desired. The component of Gr(E) to which a subspace E^{−} belongs will be called its virtual dimension, and written simply dim(E^{−}).

2.5 The polarization of spinors on the boundary

My aim in this section is to show that there is a polarization of the spinor fields Γ_{Y} on a compact manifold Y such that whenever Y is the boundary of X_{1} the boundary values of harmonic spinor fields on X_{1} form a closed subspace K_{X_{1}} belonging to the restricted Grassmannian Gr(Γ_{Y}). If we only want a coarse polarization this is quite easy. We have already remarked that the self-adjoint Dirac operator on Y splits Γ_{Y} as Γ_{Y}^{+} ⊕Γ_{Y}^{−} according to the sign of the eigenvalues, and it is not too hard to show that the projections K_{X_{1}} → Γ_{Y}^{+} and K_{X_{1}} → Γ_{Y}^{−} are Fredholm and compact respectively. For our future purposes, however, we need a more precise result.

Let us examine the construction of the projection operator Γ_{Y} → K_{X_{1}}, which is called the Calderon projector. We must use a little of the technology of pseudodifferential operators. The main points are that a pseudo-differential operator is determined up to a smoothing operator by its symbol, and that the symbol of the Calderon projector can be calculated explicitly from the symbol of the Dirac operator D_{X_{1}} by a local formula. Thus only the jets of the symbol of D_{X_{1}} at points of Y are relevant. The following argument is taken from Hörmander [].

There is no loss of generality in taking X_{1} to be part of a closed manifold X = X_{1} ∪ X_{2}. Let P be a parametrix for D_{X} on X, i.e. an inverse modulo smoothing operators. We can choose it so that D_{X} ∘P is exactly the identity on distributions with support in X_{1}. (Pseudo-differential operators extend automatically to act on distributions.) Let s_{1} be a harmonic spinor field on X_{1}, and extend it by zero to X. Then D_{X}s_{1} is a δ-function distribution along Y. To be precise, we can write s_{1} = χs̃_{1}, where χ is the characteristic function of X_{1}, and s̃_{1} is a smooth extension of s_{1} to X. We have

D_{X}s_{1} = D_{X}(χs̃_{1})
= χD_{X} s̃_{1} + c_{dχ}s̃_{1}
= c_{dχ}s̃_{1},

where c_{dχ} denotes Clifford multiplication by the distributional 1-form dχ. Now c_{dχ} is supported on Y: it can be written γ_{Y} δ_{Y}, where δ_{Y} is the δ-function along Y and γ_{Y} is the unit conormal vector field to Y. So

D_{X}s_{1} = (γ_{Y} · (s_{1}|Y))δ_{Y} .

Let us define an operator C : Γ_{Y} → Γ_{X_{1}} by

C(s) = {P ((γ_{Y} s) · δ_{Y})}|X_{1}.

This has its image in K_{X_{1}}. (One needs to check that C(s), which is automatically smooth in the interior of X_{1}, extends smoothly to the boundary. For this, see [].) Furthermore, if s is the boundary value of s_{1} ∈ K_{X_{1}} then

C(s) = P Ds_{1} = s_{1} + Ss_{1},

where S is a smoothing operator on X.

Thus C differs from the identity on K_{X_{1}} ⊂ Γ_{Y} only by a smoothing operator, and we can easily correct it by adding a smoothing operator to make it exactly the identity on K_{X_{1}}. The resulting operator can be regarded in two ways:

(i) an operator C : Γ_{Y} → Γ_{X_{1}}, in which guise it is the integral formula mentioned earlier which expresses a harmonic spinor field in terms of its boundary values, or (ii) or as a projection operator C : Γ_{Y} → Γ_{Y}.

Hörmander shows that C : Γ_{Y} → Γ_{Y} is a pseudo-differential operator of order zero, and he gives a formula ([]) for its symbol in terms of that of P.

To prove that K_{X_{1}} is close to Γ_{Y}^{−} we must compare the projection C with the orthogonal projection on to Γ_{Y}^{−}. The latter, however, is just the Calderon projection C_{0} corresponding to regarding Y as the boundary of Y × ℝ_{+} with the product metric. For we can take as a parametrix for D_{Y ×ℝ} the Feynman propagator

P_{0} : C_{cpt}^{∞}(ℝ; Γ_{Y}) → S(ℝ; Γ_{Y})

defined by

(P_{0}f)(t) = ∑_{λ<0} ∫_{ℝ_{+}} e^{sλ}f_{λ}(t + s) ds + ∑_{λ>0} ∫_{ℝ_{−}} e^{sλ}f_{λ}(t + s) ds,

where f = ∑ f_{λ} is the decomposition of an element f ∈ Γ_{Y} into eigenfunctions of D_{Y}, and S denotes the rapidly decreasing smooth functions. The calculus of pseudo-differential operators, however, gives us a rival parametrix

P̃_{0} : C_{cpt}^{∞}(ℝ; Γ_{Y}) → C_{cpt}^{∞}(ℝ; Γ_{Y}).

It is easy to see that P_{0} and P̃_{0} must differ by a smoothing operator.

Thus, finally, we need to know how the Calderon projection C depends on the metric of the manifold X. If we are content to assume that the metric of X agrees with that of Y × ℝ to infinite order along Y then there is no more to say than that the symbol of C can be calculated locally from that of D_{X}. If the metric is not a product, but we are interested only in a coarse polarization, we need only check that the leading term of the symbol of C depends only on the metric of Y, for a pseudo-differential operator of order −1 is compact. But the fundamental result is

Theorem 2.5.1 The polarization of Γ_{Y} defined by the Calderon projection for a manifold X_{1} with ∂X_{1} = Y depends on the first [n/2] normal derivatives of the metric of Y, where n = dim(Y).

In other words, if dim(X_{1}) = 2 then the polarization is independent of X_{1}, while if dim(X_{1}) = 4 it depends on both the metric and the second fundamental form of Y. I shall return to the importance of this for quantum field theory in section 2.

Proof of 2.5.1 We first calculate the symbol of the parametrix P of D_{X}. We can work in local coordinates (x_{0}, ⋯, x_{n}; ξ_{0}, ⋯, ξ_{n}) for T^{∗}X, so that the symbol of D_{X} is the matrix-valued function γ_{ξ} = ∑ γ_{i}(x)ξ_{i}. Then P is of order −1, and its symbol is

p_{−1}(x; ξ) + p_{−2}(x; ξ) + ⋯

where p_{−k} is homogeneous in ξ of degree −k : it is not defined when ξ = 0. Clearly p_{−1} = γ_{ξ}^{−1}, and we find by recurrence that

p_{−k} = (−1)^{k−1} γ_{ξ}^{−1} D γ_{ξ}^{−1} D γ_{ξ}^{−1} ⋯ D γ_{ξ}^{−1},

where there are k factors γ_{ξ}^{−1}, and D denotes the differential operator

D = ∑ γ_{i}(x)\frac{∂}{∂x_{i}} .

Because γ_{ξ}^{2} = −||ξ||^{2} we see that p_{k}(x; ξ) is, for each x, a matrix-valued polynomial in ξ, divided by a power of ||ξ||^{2}. (Notice that ||ξ||^{2} depends on x as well as ξ.)

To calculate the symbol of C we choose the coordinates so that (x_{0}; ξ_{0}) are normal to Y, which is defined by x_{0} = 0. Then Hörmander’s formula ([]) for the symbol

c_{0}(y; η) + c_{−1}(y; η) + ⋯,

where y = (x_{1}, ⋯, x_{n}) and η = (ξ_{1}, ⋯, ξ_{n}), amounts to saying that c_{−k}(y; η) is obtained from p_{−k−1}(0, y; ξ_{0}, η) simply by taking the residue of the latter, regarded as a matrix-valued function of ξ_{0} for fixed (y; η), at its unique pole in the upper half-plane, i.e. at ξ_{0} = i||η||. Thus c_{−k}, like p_{−k−1}, involves k normal derivatives of the metric along Y.

On an n-dimensional manifold a pseudo-differential operator is Hilbert-Schimdt if its order is strictly less than −\frac{1}{2}n, so we need to retain all terms of order ⩾ −\frac{1}{2}n to define the polarization, i.e. we need [\frac{1}{2}n] normal derivatives of the metric.

2.6 Subdividing the boundary: the appearance of categories

To express the localizability of the index of the Dirac operator on even-dimensional closed manifolds we were led to associate algebraic objects Z_{Y} to closed manifolds Y of one lower dimension. If Y in turn is the union of manifolds Y_{1}, Y_{2} which intersect in their common boundary Σ, we may ask whether Z_{Y} can be constructed from objects Z_{Y_{1}} and Z_{Y_{2}} associated to the pieces. This is indeed the case. But just as, when X = X_{1} ∪_{Y} X_{2}, the contributions of X_{1} and X_{2} to the index of D_{X} were not themselves integers, but were elements of the ℤ-torsor Z_{Y} associated to Y, so the objects Z_{Y_{1}} and Z_{Y_{2}} will not be ℤ-torsors, but instead will be objects of a new category Z_{Σ} associated to Σ. The category Z_{Σ} is a groupoid, i.e. all of its morphisms are isomorphisms. The group of automorphisms of each object of Z_{Σ} is ℤ, and the morphisms from any object to any other form a ℤ-torsor. In particular Z_{Y} is the set of all morphisms from Z_{Y_{1}} to Z_{Y_{2}}.

Before explaining this any further we should take some thought of the slipperiness of the slope we have stepped upon. It would be easy to ask what happens if Σ = Σ_{1} ∪ Σ_{2}, and to slither into a wilderness of 2-categories. Personally, I think it is worth going as far as categories. The justification for that must be mainly aesthetic, but there is one objective fact that is relevant, arising from Morse theory.

By choosing a generic Morse function f : X → ℝ on a closed manifold X we can slice X up as a union

X = X_{0} ∪ X_{1} ∪ .... ∪ X_{m},

where each slice X_{k} = f^{−1}([t_{k}, t_{k+1}]) contains only one critical point of f, and is a cobordism Y_{k} ⇝ Y_{k+1} between two smooth level sets of f. We cannot assume that the slices X_{k} have any simple standard form, though X_{k} differs from the cylinder Y_{k} × [0, 1] by attaching a single “handle”. But by cutting the manifolds Y_{k} into two we can describe the situation much more explicitly. We write

Y_{k} = Yk^{′} ∪ (S^{p−1} × D^{q}),

where p is the index of the handle to be attached, and p + q = n. The two parts in this splitting intersect in S^{p−1} × S^{q−1}. Then

Y_{k+1} = Yk^{′} ∪ (D^{p} × S^{q−1}),

and the cobordism from Y_{k} to Y_{k+1} is simply the union of the trivial cobordism Yk^{′} × [0, 1] with the standard cobordism D^{p} × D^{q} from S^{p−1} × D^{q} to D^{p} × S^{q−1}. Thus in the end everything is reduced to understanding the category Z_{S^{p−1}×S^{q−1}}, the objects Z_{S^{p−1}×D^{q}} and Z_{D^{p}×S^{q−1}} which belong to it, and the morphism

Z_{S^{p−1}×D^{q}} → Z_{D^{p}×S^{q−1}}

defined by the standard cobordism. In principle, at least, this is all very explicit.

We shall return to the preceding considerations in §?. Meanwhile, let us describe the category Z_{Σ} and its properties. The best known example of a groupoid is the fundamental groupoid π_{1}(B) of a path-connected space B. This is the category whose objects are the points of B, and whose morphisms from b_{0} to b_{1} are the homotopy classes of path in B from b_{0} to b_{1}. Thus the group of automorphisms of b_{0} is the fundamental group π_{1}(B, b_{0}), and the morphisms from b_{0} to b_{1} form a torsor for this group. Up to equivalence of categories, one can also say that π_{1}(B) is the category whose objects are the universal covering spaces B̃ of B, and whose morphisms B̃_{0} → B̃_{1} are the covering maps, i.e. the maps which cover the identity map of B. (The usual construction of the universal covering space of B as the space of homotopy classes of paths in B with a chosen starting-point defines a functor from the first definition of π_{1}(B) to the second.)

The object Z_{Y} associated to an odd-dimensional manifold Y was defined as π_{0}(Gr_{Y}), where Gr_{Y} was a Grassmannian of subspaces of the space Γ_{Y} of spinor fields on Y. The groupoid Z_{Σ} is the fundamental groupoid of an analogous Grassmannian J_{Σ} formed by a certain class of subspaces of Γ_{Σ}. The space J_{Σ} is connected, and its fundamental group is ℤ, so the sets of morphisms in the category π_{1}(J_{Σ}) are ℤ-torsors, as we want. Both points of view on the fundamental groupoid are relevant. If Y is a manifold with boundary Σ then the boundary values of harmonic spinor fields on Y form a space belonging to J_{Σ}, and hence define an object of π_{1}(J_{Σ}) = Z_{Σ}. But a point of J_{Σ} can also, as we shall see, be regarded as a self-adjoint boundary condition for the Dirac operator on Y. So each point σ of J_{Σ} defines a polarisation of Γ_{Y}, and hence a restricted Grassmannian Gr_{Y,σ}. As σ varies the sets Z_{Y,σ} = π_{0}(Gr_{Y,σ}) from a covering space Z_{Y} of J_{Σ}, and hence an object of the category π_{1}(J_{Σ}).

To define J_{Σ} we begin with the formula which expresses the self-adjointness of the Dirac operator D_{Y} on an arbitrary manifold Y

− ⟨D_{Y} φ_{1}, φ_{2}⟩ + ⟨φ_{1}, D_{Y} φ_{2}⟩ = div⟨φ_{1}, γφ_{2}⟩. (2.6.1)

Here ψ_{1} and ψ_{2} are spinor fields, and ⟨ , ⟩ denotes their pointwise inner product. The expression ⟨ψ_{1}, γψ_{2}⟩ denotes the 1-form (or vector field) whose components with respect to a local orthonormal framing ξ_{i} of the tangent bundle are, in the notation of (2.1.1), the functions ⟨ψ_{1}, c_{ξ_{i}}ψ_{2}⟩. Integrating (2.6.1) over Y with ∂Y = Σ gives

(2.6.2)  − ∫ ⟨D_{Y} ψ_{1}, ψ_{2}⟩dy + ∫_{Y} ⟨ψ_{1}, D_{Y} ψ_{2}⟩dy = ∫_{Σ}⟨ψ_{1}, γψ_{2}⟩dσ,
= B(ψ_{1}, ψ_{2}),

say, where dy and dσ are the Riemannian volume elements. The right-hand side of (2.6.2) is a hermitian form on the space Γ_{Σ} of spinor fields. If Y is odd-dimensional, then Clifford multiplication by the unit normal vector to the boundary Σ splits the spin bundle Δ_{Σ} as Δ_{Σ}^{0} ⊕ Δ_{Σ}^{1}, and so we have

⟨ψ_{i}, γψ_{2}⟩ = ⟨ψ_{1}^{0}, ψ_{2}^{0}⟩ − ⟨ψ_{1}^{1}, ψ_{2}^{1}⟩

on Σ.

2.7 Determinants

For the remainder of this lecture we turn from the index to the determinant of the Dirac operator. The next two subsections are concerned with the algebraic and analytic properties of infinite dimensional determinants, and not with the Dirac operator directly. We shall make use of this material in later lectures.

If E is a topological vector space, an operator T : E → E has a determinant in the most straightforward sense if it is of the form T = 1 + A, where A is of trace class. For then A has a sequence of eigenvalues {λ_{k}} —with multiplicities— such that ∑ |λ_{k}| < ∞, and we can define

det(1 + A) = Π(1 + λ_{k}). (2.7.1)

We shall say that such an operator T is of determinant class.

Even in finite dimensions the determinant of an operator T : E → F is not a number. If dim(E) = m and dim(F) = n we define the determinant line of T as the one-dimensional space

Det(T) = (∧^{m}E)^{∗} ⊗ (∧^{n}F) = Hom(∧^{m}E; ∧^{n}F),

and then we define the determinant det(T) as the obvious element of the line Det(T) if m = n, and as 0 if m ≠ n. The essential properties of Det(T) and det(T) are

(i) det(T) ≠ 0 ⇔ T is invertible;

(ii) Det(T_{2} ∘ T_{1}) ≅ Det(T_{2}) ⊗ Det(T_{1}),

and, in terms of this isomorphism,

det(T_{2} ∘ T_{1}) ↔ det(T_{2}) ⊗ det(T_{1});

(iii) if

0 →  E_{1} → E_{2} → E_{3} → 0
↓ T_{1} ↓ T_{2} ↓ T_{3}
0 → F_{1} → F_{2} → F_{3} → 0

is a commutative diagram with exact rows, then

Det(T_{2}) ≅ Det(T_{1}) ⊗ Det(T_{3})

canonically, and

det(T_{2}) ↔ det(T_{3}) ⊗ det(T_{3}).

Quillen was the first to point out that in this second, slightly more abstract, sense the determinant can be defined just as easily for an arbitrary Fredholm operator T : E → F, and has the same three properties. I shall give the definition first in the case when T has index 0. Then a point of the line Det(T) is defined as an equivalence class of pairs (S, λ), where λ ∈ ℂ and S : E → F is an isomorphism such that S − T is of trace class, and

(S_{1}, λ_{1}) ∼ (S_{2}, λ_{2})

if and if only if

λ_{2} = det(S_{2}^{−1} ∘ S_{1})λ_{1},

where det(S_{2}^{−1}∘S_{1}) is defined in the straightforward way (2.7.1). The determinant det(T) is defined as the point of the line Det(T) represented by (T, 1) if T is invertible, and as 0 otherwise.

The first thing to notice about this definition is that if the kernel and cokernel of T have dimension n then

(2.7.2) Det(T) ≅ (∧^{n}ker(T))^{∗} ⊗ ∧^{n}coker(T)

canonically. For if α_{1}, ..., α_{n} is a basis of ker(T)^{∗} and v_{1}, ..., v_{n} ∈ F is a basis of coker(T) we can map the element

(α_{1} ∧ ... ∧ α_{n}) ⊗ (v_{1} ∧ ... ∧ v_{n})

of the right-hand side of (2.7.2) to the class of (T^{∼}, 1) in Det(T), where

T^{∼} = T + ∑ v_{i} ⊗ α^{∼}_{i}

and α_{i}^{∼} ∈ E^{∗} is an extension of α_{i}. Having made this observation it is clear that there is a unique way to extend the definition of the Det(T) to Fredholm operators of any index so that the following two properties are preserved.

(a) Det(T) has the usual meaning if E and F are finite dimensional, and

(b) Det(T_{1} ⊗ T_{2}) ≅ Det(T_{1}) ⊗ Det(T_{2}).

Of course det(T) is defined as 0 if the index of T is not zero.

It is completely elementary to check that the definitions of Det and det for Fredholm operators have the basic properties (i), (ii), (iii) above. One might wonder, however, what has been achieved, for to give a line with a distinguished vector conveys no information except whether the vector is non-zero. In fact the real interest of the construction appears only when we have a family of Fredholm operators {T_{x}}_{x∈X}. Then, under exactly the same very general circumstances which ensure that the index of T_{x} is locally constant, we find that {Det(T_{x})}_{x∈X} is a complex line-bundle on X, and {det(T_{x})} is a continuous section. Once again, the proof of this presents no difficulty.

A feature of the definition of the line Det(T) is that it depends on the operator T only modulo the addition of trace-class operators — and hence does not depend on T at all in finite dimensions. A situation where this can be exploited is the following.

The determinant bundle and the extension of GL_{res}

Whenever we have a polarized topological vector space E we can define not only the restricted Grassmannian Gr(E) as in §2.4, but also the restricted general linear group GL_{res}(E), which consists of all isomorphisms E → E which preserve the class of allowed splittings E = E′ ⊕ E′′. Clearly GL_{res}(E) acts on GR(E).

If W_{0} and W_{1} both belong to Gr(E), then we have a preferred class of Fredholm operators T : W_{0} → W_{1} singled out by the polarization : we compose the inclusion W_{0} → E with any allowable projection E → W_{1}. For any such T the line Det(T) depends only on W_{0} and W_{1} (and, of course, on the polarization). I shall denote it by Det(W_{0} : W_{1}). Clearly we have

(2.7.3) Det(W_{0} : W_{1}) ⊗ Det(W_{1} : W_{2}) ≅ Det(W_{0} : W_{2})

and

Det(W_{0} : W_{1}) ≅ Det(gW_{0} : gW_{1})

for any g in GL_{res}(E). Putting these two facts together we see that for g ∈ GL_{res}(E) the line

L_{g} = Det(W : gW)

is independent of the choice of W ∈ Gr(E), and satisfies

L_{f} ⊗ L_{g} ≅ L_{fg}.

This permits us to define the fundamental central extension GL_{res}^{∼}(E) of GL_{res}(E) as the group of all pairs (g, λ) with g ∈ GL_{res}(E) and λ a non-zero element of L_{g}. It is an extension by ℂ^{×}:

1 → ℂ^{×} → GL_{res}^{sim}(E) → GL_{res}(E) → 1.

2.8 The ζ-function determinant

If T : E → E is a self-adjoint Fredholm operator in a vector space with an inner product then the complex line Det(T) is “real” i.e. there is an operation of complex conjugation in Det(T) which picks out a real line inside it. The element det(T) belongs to this real line.

If T is a self-adjoint first order elliptic differential operator, such as the Dirac operator, we can say a great deal more. Then T has a discrete spectrum {λ_{k}}_{k∈ℤ} lying on the real axis, with λ_{k} → ±∞ as k → ±∞, and (assuming for the moment that no λ_{k} is zero) we can define the ζ-function

ζ_{T} (s) = ∑ λ_{k}^{−s}.

Here λ_{k}^{−s} is defined as |λ_{k}|^{−s}e^{−iπs} if λ_{k} is negative. The ζ-function is initially defined as a holomorphic function of s in a half-plane Re(s) > a where the series converges, but it is known that it can be analytically continued to a meromorphic function in the entire complex plane, and that ζ_{T} is regular at s = 0. Motivated by the formula

ζ′_{T} (s) = − ∑(logλ_{k})λ_{k}^{−s},

we can now define the ζ-function determinant det_{ζ}(T) as the complex number

det_{ζ}(T) = e^{−ζ′_{T} (0)}.

We assumed here that 0 was not an eigenvalue of T. In fact what we have really found is an isomorphism

det_{ζ} : Det(T) → ℂ, (2.8.1)

for we have

Proposition 2.8.2 If T is a Dirac operator, and S is an invertible operator such that S − 1 is of trace class, then det_{ζ}(S) is defined, and

det_{ζ}(P S) = det(P)det_{ζ}(S)

if P ∈ 1+ (trace class).

The isomorphism (2.8.1) has no reason to respect the real structure of the line Det(T), so det_{ζ} gives us a real line contained in ℂ, even when T is not invertible and det(T) = 0. This real line can be written e^{iπη(T)/2}ℝ, where η(T) ∈ ℝ/2ℤ is called the η-invariant of T. The Quillen determinant det(T) ∈ Det(T) lies in the real sub-line, so η(T) is essentially the phase of det_{ζ}(T).

The fact that the phase of the determinant can be defined even when the determinant vanishes is interesting topologically. It is well known [AS] that the space of self-adjoint Fredholm operators in Hilbert space has the homotopy type of the infinite unitary group U_{∞} = ⋃ U_{n}. If they are given an appropriate topology the same is true of the unbounded self-adjoint operators we are considering here. As the fundamental group π_{1}(U_{∞}) is ℤ, there is a continuous map, defined up to homotopy, from self-adjoint Fredholm operators to the circle ℝ/ℤ which induces an isomorphism of π_{1}. On the subspace of Dirac operators the η-invariant is a definite choice of this map.

There is yet another way to look at the η-invariant. For each way of splitting the spectrum of T into two subsets Λ_{+} and Λ_{−} such that almost all the positive eigenvalues belong to Λ_{+} and almost all the negative ones to Λ_{−} we can define a holomorphic function of s for Re(s) ≫ 0 by

η(T; s) = ∑_{λ∈Λ_{+}} |λ|^{−s} − ∑_{λ∈Λ_{−}} |Λ|^{−s}.

It is known that this function can be continued analytically to s = 0, where its value is the real number η(T; 0). It jumps by ±2 when an eigenvalue is reassigned to the other half of the spectrum, so η(T; 0) ∈ ℝ/2ℤ is independent of the chosen splitting, and is precisely the number η(T) defined above, as we shall verify in a moment. But we can say more, for the choice of a splitting of the spectrum picks out for us the subspace of E spanned by the eigenfunctions with eigenvalues in Λ_{−}, and this is a point of the restricted Grassmannian Gr(E), and defines a point of the ℤ-torsor π_{0}Gr(E). In other words, the η-invariant can be regarded as an embedding

\frac{1}{2}η : π_{0}Gr(E) → ℝ.

2.9 Fock spaces

To understand how the determinant of the Dirac operator behaves when manifolds are sewn together we need the concept of a Fock space. It is one of the main ideas in quantum field theory.

If E is a polarized topological vector space the Fock space F(E) is a “renormalized” version of the exterior algebra ∧(E), where the renormalization is defined in terms of the polarization. There is a range of possible definitions. The rough definition used by physicists — due to Dirac — makes sense when the polarization of E is defined by a self-adjoint operator D : E → E, and one has an orthonormal basis {e_{k}}_{k∈ℤ} of E consisting of eigenvectors of D whose eigenvalues λ_{k} tend to ±∞ as k → ±∞. Then F(E) has an orthonormal basis given by the formal expressions

e_{k} = e_{k_{0}} ∧ e_{k_{1}} ∧ e_{k_{2}} ∧ ⋯,

where the sequence k = {k_{0}, k_{1}, k_{2}, ⋯ } satisfies k_{0} > k_{1} > k_{2} > ⋯, and differs from {0, −1, −2, −3, ⋯ } only by including a finite number of positive integers and omitting a finite number of negative ones. The defect of this as a definition is that it seems to depend on the operator D. A good feature, however, is that it makes clear why — even when D is given — it is only the projective space of rays in F(E), and not the vector space itself, which can be defined canonically. For if we choose another basis {ẽ_{k}} of eigenvectors of D — say ẽ_{k} = u_{k}e_{k}, where |u_{k}| = 1 — then

Ω̃ = ẽ_{0} ∧ ẽ_{−1} ∧ ẽ_{−2} ∧ ⋯

should certainly define the same ray as

Ω = e_{0} ∧ e_{−1} ∧ e_{−2} ∧ ⋯,

but there is no way of fixing a scalar κ such that Ω̃ = κΩ. On the other hand, once the single number κ is prescribed there is no further indeterminacy, in the sense that we must have ẽ_{k} = κu_{k}e_{k}, where

u_{k} = ∏_{r⩾0}u_{k_{r}}/∏_{r⩾0}u_{r}

is a well-defined number.

To give a mathematically more satisfactory definition of F(E) we observe that we should be able to multiply elements of F(E) by vectors in E, and so F(E) should be a module for the exterior algebra ∧(E). There should also be an adjoint action of the exterior algebra ∧(E^{∗}) by the inner product: if α ∈ E^{∗} then

αe_{k} = ∑(−1)^{i}⟨α, e_{k_{i}}⟩e_{k}^{(i)},

where e_{k}^{(i)} is e_{k} with its i^{th} factor omitted. The action of an element of E on F(E) is traditionally called a creation operator, and that of an element of E^{∗} an annihilation operator. The two actions fit together to form an action of the Clifford algebra C(E ⊕ E^{∗}), where E ⊕ E^{∗}, has its natural hyperbolic quadratic form, i.e. if ξ ∈ E and α ∈ E^{∗} then

αξ + ξα = ⟨α, ξ⟩ (2.9.1)

in C(E ⊕ E^{∗}). From this point of view, the Fock space F(E) is characterized as an irreducible C(E ⊕ E^{∗})-module which, for each splitting E = E^{+} ⊕ E^{−} allowed by the polarization, contains a vector Ω_{E^{−}} which is annihilated by both E^{−} ⊂ E and (E^{−})^{∘} ⊂ E^{∗}. A finite-dimensional Clifford algebra has a unique irreducible representation, up to isomorphism, but in infinite dimensions the irreducible representations of C(E ⊕ E^{∗}) are parametrized by the polarizations of E ⊕ E^{∗}. For a given choice of E^{−} ∈ Gr(E) we have a definite Fock space

F_{E^{−}}(E) = ∧((E^{−})^{∗}) ⊗ ∧(E/E^{−}) (2.9.2)

with a definite vacuum vector Ω_{E^{−}}, but for different allowable choices E_{1}^{−}, E_{2}^{−} the isomorphism

F_{E_{1}^{−}}(E) → F_{E_{2}^{−}}(E) (2.9.3)

is canonical only up to a scalar: Schur’s lemma replaces the physicists’ renormalization constant.

The description just given was vague about the topology of F(E). If E is a Hilbert space then one can clearly construct F_{E^{−}}(E) as a Hilbert space, prescribing that the creation and annihilation operators are each others’ adjoints. But it is worth mentioning a more general and abstract approach. We want F_{E^{−}}(E) to contain a ray L_{W} for each W ∈ Gr(E) : roughly,

L_{W} = ℂw_{0} ∧ w_{1} ∧ w_{2} ∧ ⋯,

where {w_{i}} is a basis of W. Now we have seen that for a given E^{−} ∈ Gr(E) there is a holomorphic line bundle Det_{E^{−}} on Gr(E) whose fibre at W is Det(E^{−} : W). We can characterize F_{E^{−}}(E) by saying that it is a topological vector space with a holomorphic map

Det_{E^{−}} → F_{E^{−}}(E) (2.9.4) which is linear on each fibre of Det_{E^{−}}, and that it is universal among topological vector spaces with such maps. Then the vacuum vector Ω_{E^{−}} is the image of 1 ∈ Det(E^{−} : E^{−}) in F_{E^{−}}(E), and the isomorphism (2.9.3) arises from a canonical isomorphism

F_{E_{2}^{−}}(E) ≅ F_{E_{1}^{−}} ⊗ Det(E_{1}^{−} : E_{2}^{−}) (2.9.5)

(cf. (2.7.3)).

The existence of a vector space with the universal property (2.9.4) is clear: we take the dual of the space of holomorphic sections of the line bundle Det_{(E^{−})}^{∗}. Of course we shall not get a Hilbert space, but a pre-Hilbert structure in E induces one in F_{E^{−}}(E). (See [PS] Chapter 10).

One feature of the Fock space which is clear from any of the definitions is that F(E) is naturally graded by the ℤ-torsor π_{0}Gr(E) : the degree of the vacuum vector Ω_{E^{−}} is the virtual dimension of E^{−}.

Finally, we need to know that reversing the polarization of the space E^{−}, i.e. changing J to −J, or interchanging E^{+} and E^{−}, essentially changes the Fock space F(E) to its dual. To be precise, if Ẽ denotes E with the reversed polarization, and we choose E^{−} ∈ Gr(E) and E^{+} in Gr(Ẽ), not necessarily complementary, then there is a canonical pairing

F_{E^{+}}(Ẽ) × F_{E^{−}}(E) → L_{E^{+},E^{−}}, (2.9.6)

where L_{E^{+},E^{−}} is the determinant line of the Fredholm operator E^{+} ⊕ E^{−} → E defined by adding the inclusions. Restricted to the rays

Det(E^{+} : W̃) × Det(E^{−} : W)

the pairing (2.9.6) is

(S̃, S) ↦ Det(S̃ + S : E^{+} ⊕ E^{−} → E).

2.10 Patching the determinant

We shall now return to the Dirac operator on a closed even-dimensional manifold X which is a union of two pieces X = X_{1} ⨿_{Y} X_{2}. We saw how the index of D_{X} can be calculated from contributions associated to X_{1} and X_{2}, and now we should like to do the same for the determinant of D_{X}. There are two aspects to this. If we forget for a moment that the determinant of D_{X} is not quite a number, then we expect the operator D_{X_{1}} on a manifold with boundary to have a determinant only when we equip it with a boundary condition. An appropriate boundary condition is defined by a point W of the restricted Grassmannian Gr\overline{Y}.

A boundary condition for D_{X_{2}} corresponds to a point of the opposite Grassmanian Gr_{Y}. Thus, roughly speaking, both det(D_{X_{1}}) and det(D_{X_{2}}) are functions on Gr_{Y}. We are aiming for a formula of the type

(2.10.1) det(D_{X}) = ⟨det(D_{X_{1}}), det(D_{X_{2}})⟩,

expressing the result as an L^{2} inner product, i.e. some kind of integral over the infinite-dimensional space of all boundary conditions. This fits in well with the point of view of quantum field theory, where det(D_{X}) is regarded as a pathintegral over the space of all spinor fields on X, but it cannot be interpreted too literally, for the necessary integration theory is quite out of reach. Instead, we shall simply prescribe the Fock space F_{Y} formed from the space Γ_{Y} of spinor fields on Y as our candidate for the Hilbert space of functions on Gr_{Y}, and we shall define elements of F_{Y} so that (2.10.1) is true.

Before doing so, however, we must return to the second aspect of the problem, namely the fact that det(D_{X}) is not a number but actually an element of the abstractly defined line Det(D_{X}). This fits into the formalism very attractively. We saw that the object canonically associated to Y is not a Hilbert space F_{Y} but a projective space ℙF_{Y}. Whereas a closed manifold X gives us a line Det(D_{X}), the corresponding object for a manifold X_{1} with boundary Y is a vector space F_{X_{1}} together with an isomorphism

ℙ(F_{X_{1}}) ≅ ℙF_{Y} .

This makes good sense, for if ℙ is a complex projective space then the category of vector spaces V with isomorphisms ℙ(V) ≅ ℙ is equivalent to the category of complex lines. Indeed if V is one such space then any other is of the form V ⊗ L for some line L, for an isomorphism ℙ(V_{0}) ≅ ℙ(V_{1}) gives an isomorphism V_{0} ⊗ L ≅ V_{1}, where L is the line of homomorphisms V_{0} → V_{1} which induce the given map of projective spaces.

In the present situation we define F_{X_{1}} as the Fock space F_{K_{1}}(Γ_{Y}) formed from Γ_{Y} and the space K_{1} of boundary values of harmonic spinor fields on X_{1}. For the other half, we form F_{X_{2}} similarly from Γ̃_{Y} = ΓȲ. The projective spaces ℙF_{Y} and ℙFȲ are dual, and so F_{X_{2}} is in duality with F_{X_{1}} ⊗ L for some line L which can be denoted by

⟨F_{X_{1}} , F_{X_{2}}⟩.

Proposition 2.10.2 We have

(a) ⟨F_{X_{1}} , , F_{X_{2}}⟩ = Det(D_{X}), and

(b) ⟨det(D_{X_{1}}) , det(D_{X_{2}})⟩ = det(D_{X}),

where det(X_{i}) denotes the vacuum vector in F_{X_{i}}.

Proof. According to (2.9.6) the line ⟨F_{X_{1}} , F_{X_{2}}⟩ and the point ⟨det(D_{X_{1}}), det(D_{X_{2}})⟩ in it can be identified with the pointed determinant line of

K_{1} ⊕ K_{2} → Γ_{Y} .

But the diagram in the proof of (2.3.2) showed that this Fredholm operator was equivalent, in a sense which preserves the determinant as well as the index, to the Dirac operator

D_{X} : Γ_{X}^{even} → Γ_{X}^{odd}.

2.11 Patching the ζ-function determinant

We now turn to the ζ-function determinant of the self-adjoint Dirac operator on an odd-dimensional manifold. Essentially the same discussion applies in even dimensions to the determinant of the self-adjoint operator D_{X} = D_{X}^{even} ⊕D_{X}^{odd}, in contrast to the “chiral” operator D_{X}^{even} which was treated in the previous section. We consider a closed manifold X = X_{1} ⨿_{Y} X_{2}, as usual, but now dim(Y) is even. We shall obtain a formula

(2.11.1) det_{ζ}(D_{X}) = ⟨det_{ζ}(D_{X_{1}}), det_{ζ}(D_{X_{2}})⟩,

where the determinants on the right are elements of dual Hilbert spaces H_{Y} , HȲ associated to Y, which we think of as consisting of functions of the boundary data for D_{X_{1}} and D_{X_{2}}.

We saw in §2.6 that appropriate boundary data in this case are maximal isotropic subspaces of Γ_{Y} belonging to a certain polarization-class, or, equivalently, certain unitary isomorphisms u : Γ_{Y}^{even} → Γ_{Y}^{odd} which form a space U_{Y} which is a principal homogeneous space for the group of unitary transformations of Γ_{Y}^{even} of determinant class. The Hilbert space H_{Y} is once again a kind of Fock space: it can be regarded as

∧^{middle}(Γ_{Y}) ≅ Hom(∧(Γ_{Y}^{even}) ; ∧(Γ_{Y}^{odd})).

But the important thing is that it is a definite vector space, and not just a projective space. The most concrete definition is to say that H_{Y} contains a unit vector ε_{u} for each u ∈ U_{Y}, and is obtained from the formal algebraic span of these vectors by completing with respect to the inner product defined by

⟨ε_{u_{1}}, ε_{u_{2}}⟩ = det\frac{1}{2}(1 + u_{1}^{−1}u_{2}). (2.11.2)

To see that this does indeed define a positive inner product it is enough, by continuity, to consider the same formula applied to the unitary group U_{n}. But then (2.11.2) is simply the inner product induced by the natural embedding of U_{n} in End(∧ℂ^{n}).

For a connected manifold X_{1} with non-empty boundary Y we can now define

det_{ζ}(X_{1}) = ε_{u} ∈ H_{Y} ,

where u ∈ U_{Y} represents the isotropic subspace of Γ_{Y} consisting of the boundary values of harmonic spinor fields on X_{1}. A self-adjoint boundary condition for D_{X_{1}} is an element β ∈ UȲ = U_{Y}^{−1}, and we have

Proposition 2.11.3 (i) The ζ-function determinant of D_{X_{1}} with boundary condition β is

det_{ζ}(D_{X_{1}}, β) = ⟨det_{ζ}(D_{X_{1}}) , ε_{β}⟩
= det\frac{1}{2}(1 + βu).

(ii) For the closed manifold X = X_{1} ∪ X_{2} we have

det_{ζ}(D_{X}) = ⟨det_{ζ}(D_{X_{1}}) , det_{ζ}(D_{X_{2}})⟩.

46  Lecture 2  July 27, 1999

Bibliography

[AS] AS