The η-form and a generalized Maslov index

Ulrich Bunke^{∗}and H. Koch^{†} October 8, 1997

Contents

1 Introduction 1 2 Definition of η(L_{0}, L_{1}) 2 3 The Maslov cocycle 4 4 Computation of τ(L_{0}, L_{1}, L_{2}) 16

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:bunke@cfgauss.uni-math.gwdg.de

†Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, GERMANY, E-mail:koch@IWR.Uni-Heidelberg.De

1 Introduction

In this note we consider the η-form of a family of Dirac operators D(b), b ∈ B, on the interval [0, 1] over a base space B. The η-form was introduced by Bismut-Cheeger [3] as the boundary contribution to the local index theorem for families of Dirac operators. It also appeared in the study of adiabatic limits of η-invariants [2]. In our case the operator D(b) depends on b ∈ B only through the boundary conditions. If B is a point, then the η-form reduces to the usual η-invariant of D which was explicitly calculated by Lesch-Wojciechowski [9]. In [5] we found a relation between the η-invariant and the Maslov index. The Maslov index was first introduced in Wall [12] as a measure of the nonadditivity under gluing of the signature of manifolds with boundary. This non-additivity was generalized to arbitrary Dirac operators in [5].

In the present note we relate the η-form with a family version of the Maslov index. The family version of the Maslov index conjecturally plays the same role in the non-additivity of the family index of families of Dirac operators on manifolds with cylindrical ends (or APS-boundary conditions) as the usual Maslov index does for the usual index.

Let V be a finite-dimensional Hilbert space equipped with a hermitean symplectic structure Ω (this just means that ıΩ is a non-degenerate hermitean form of index (l, l), dim_{C}(V) = 2l). If {L_{0}(b), L_{1}(b)}_{b∈B} is a smooth family of pairs of transverse Lagrangian subspaces of V, then we define the η-form η(L_{0}, L_{1}) ∈ C^{∞}(B, Λ^{ev}T^{∗}B). Our main result is

Theorem 1.1 If {L_{0}(b), L_{1}(b), L_{2}(b)}_{b∈B} is a smooth family of triples of pairwise transverse Lagrangian subspaces of V, then

(1) d(η(L_{0}, L_{1}) + η(L_{1}, L_{2}) + η(L_{2}, L_{0})) = 0,

(2) and if we define the cohomology class τ(L_{0}, L_{1}, L_{2}) by

τ(L_{0}, L_{1}, L_{2}) := [η(L_{0}, L_{1}) + η(L_{1}, L_{2}) + η(L_{2}, L_{0})] ∈ H^{even}(B, R) ,

2 DEFINITION OF η(L_{0}, L_{1}) 2

then τ(L_{0}, L_{1}, L_{2}) = ch(L_{0}^{+}) − ch(L_{0}^{−}), where L_{0} = L_{0}^{+} ⊕ L_{0}^{−} is the splitting of the bundle of Lagrangian subspaces L_{0} ⊂ B × V into the positive and negative eigenspaces of the quadratic form Q(x_{0}) := Ω(x_{1}, x_{2}), where x_{i} ∈ L_{i}, x_{0} = x_{1} + x_{2}.

The proof of the theorem is based on a local index theorem for families of Dirac operators on manifolds with cylindrical ends and boundaries with local boundary condition. Instead of saying how the existing proofs Bismut-Cheeger [3], Melrose-Piazza [11] should be modified in order to include the additional boundaries we prefer to work out again the essential arguments. Our approach is modelled on the b-calculus proof of [11], but is more direct and might be of independent interest.

2 Definition of η(L_{0}, L_{1})

Our definition of the η-form is modelled on the case of even-dimensional base space of [2]. Since the total spaces of the fibrations considered in [2] are odd-dimensional, this corresponds to the case of odd-dimensional fibres.

Let V be a finite-dimensional complex Hilbert space with scalar product (., .). Let I ∈ End(V) be a complex structure, i.e. I^{2} = −1, I^{∗} = −I. We assume that tr I = 0. Then Ω(x, y) := (Ix, y) is a hermitean symplectic structure on V. We consider the formally selfadjoint differential operator D := I \frac{d}{dt} acting on C^{∞}([0, 1], V).

A complex subspace L ⊂ V is called Lagrangian, if L ⊥ IL and L⊕IL = V. We want to consider D as an unbounded operator on the Hilbert space L^{2}([0, 1], V). In order to define selfadjoint extensions D of D we choose two Lagrangian subspaces L_{i} ⊂ V, i = 0, 1. Then we define the domain dom(D) of D by

dom(D) := {f ∈ C^{∞}([0, 1], V) | f(i) ∈ L_{i}, i = 0, 1} .

Then D is essentially selfadjoint, and we denote its unique selfadjoint extension by D, too. We have ker(D) ≅ L_{0} ∩ L_{1}. In particular D is invertible iff L_{0} ∩ L_{1} = 0.

We now turn to families. Let B be some manifold. We consider a pair of smooth families of Lagrangian subspaces B ∋ b ↦ L_{i}(b), i = 0, 1, and we assume that L_{0}(b) ∩ L_{1}(b) = 0, ∀b ∈ B. We obtain a corresponding family {D(b)}_{b∈B} of invertible operators. We want to apply the superconnection formalism in order to define the η-form of that family. Since this formalism involves derivatives of the family with respect to b we prefer to work with an unitarily equivalent family {D̃(b)}_{b∈B} which has the advantage that its domain is independent of b ∈ B.

The η-form is a local object with respect to the base space. In order to define it we only consider a germ of the family near a point b_{0} ∈ B. Let U_{I}(V) denote the group of unitary operators on V which commute with I. The group U_{I}(V) acts transitively on the space Λ of all Lagrangian subspaces of V. Thus we can find germs of smooth families of unitaries b ↦ U_{i}(b) ∈ U_{I}(V), i = 0, 1, with U_{i}(b_{0}) = 1 and U_{i}(b)L_{i}(b) = L_{i}(b_{0}). We can define germs of smooth families A_{i}(b) := log(U_{i}(b)) of anti-hermitean matrices using the standard branch of the logarithm. Let χ_{i} ∈ C^{∞}([0, 1]) be cut-off functions with χ_{0}(t) = 1 for t < 1/5, χ_{0}(t) = 0 for t > 2/5, χ_{1}(t) = 1 for t > 4/5, and χ_{1}(t) = 0 for t < 3/5. We set W (t, b) := exp(χ_{0}(t)A_{0}(b) + χ_{1}(t)A_{1}(b)). Then b ↦ W (., b) can be considered as a germ of a family of unitary multiplication operators on L^{2}([0, 1], V). We set

D̃(b) := W (., b)DW^{∗}(., b)
= D − χ′_{0}IA_{0}(b) − χ′_{1}IA_{1}(b) ,

where ”′” denotes the derivative with respect to t. We define the selfadjoint extension of D̃(b) using the Lagrangian subspaces L_{0}(b_{0}), L_{1}(b_{0}). Then D̃(b) is unitarily equivalent to D(b) and its domain is independent of b.

We now turn to the definition of the η-form of the family {D̃(b)}_{b∈B} following [2], Thm. 2.43. Let C_{1} denote the graded algebra over C generated by σ satisfying σ^{2} = 1, σ^{∗} = σ, and deg(σ) = 1. Let H denote the germ at b_{0} of the trivial Hilbert space bundle with fibre L^{2}([0, 1], V) ⊗ C_{1} over B. We define the superconnection A_{s}, s > 0, on H associated to D̃ by

A_{s} = ∇ − d(χ_{0}A_{0} + χ_{1}A_{1}) + \sqrt{s}σD̃ ,

where d differentiates along B. Here ∇ is the canonical connection of H and ∇−d(χ_{0}A_{0} + χ_{1}A_{1}) = W ∇W^{∗}. For Re(u) > 1 we can define the holomorphic family of germs of smooth, even differential forms

η(u) := \frac{1}{2\sqrt{π}} ∫_{0}^{∞} tr_{σ}^{even}(σD̃e^{−A_{s}^{2}})s^{u−1/2}ds .  (1)

Here tr_{σ}^{even}(…) stands for the even form part of tr(σ …). As usual the asymptotic expansion of the heat kernels for small times implies that η(u) has a meromorphic continuation with respect to u to all of C having at most first order poles. Following [7] we define

Definition 2.1

η(D̃) := P.F. η(0) ,

where ”P.F.” stands for the finite part of the Laurent expansion of η(u) at u = 0. As defined above the form η(D̃) may depend on the choices made for the definition of D̃. But the following lemma justifies the notation η(L_{0}, L_{1}) := η(D̃).

Lemma 2.2 η(D̃) does not depend on the choices of the families U_{i} and the cut-off functions χ_{i}.

Proof. Let Û_{i}, χ̂_{i} be another choice and define Ŵ as above. Let ˆ D̃ denote the corresponding family of operators. We set V = W Ŵ^{∗}. Then ˆ D̃ = V^{∗}D̃V. If Â_{s} is the superconnection associated to ˆ D̃, then Â_{s} = V^{∗}A_{s}V. It now follows from the cyclicity of the trace that η̂(u) = η(u), where η̂(u) corresponds to Â_{s}. □

3 The Maslov cocycle

Now we turn to the generalized Maslov index. Let {L_{i}(b)}_{b∈B}, i = 0, 1, 2, be smooth families of Lagrangian subspaces of V such that L_{i}(b) ∩ L_{j}(b) = {0}, ∀b ∈ B, i ≠ j. Let ch : K^{0}(B) → H^{ev}(B, R) be the Chern character. In the present section we prove Proposition 3.1 The form η(L_{0}, L_{1}) + η(L_{1}, L_{2}) + η(L_{2}, L_{0}) is closed. Moreover

τ(L_{0}, L_{1}, L_{2}) := [η(L_{0}, L_{1}) + η(L_{1}, L_{2}) + η(L_{2}, L_{0})] ∈ ch(K^{0}(B)) ⊂ H^{even}(B, R) .

\underline{Remark:} The zero component τ(L_{0}, L_{1}, L_{2})^{0} ∈ Z is the Maslov index (in its hermitean symplectic generalization) of the triple (L_{0}, L_{1}, L_{2}) (see [5]). For an exposition of the usual Maslov index we refer to [10]). In Proposition 4.1 below we explicitly compute the class τ(L_{0}, L_{1}, L_{2}) ∈ H^{even}(B, R) in terms of the hermitean symplectic geometry of the family {L_{0}(b), L_{1}(b), L_{2}(b)}_{b∈B}.

Proof. The idea of the proof is to formulate an index problem for a family of Dirac operators ∂/^{+} such that the form η(L_{0}, L_{1}) + η(L_{1}, L_{2}) + η(L_{2}, L_{0}) represents the Chern character of the index bundle of ∂/^{+}.

We consider a compact oriented surface M_{c} with boundary and corners which is diffeomorphic to a rectangular convex 12-gon in the hyperbolic plane. We label the boundary pices ∂_{i}M_{c}, i = 0, …, 11, according to their cyclic order.

We choose a Riemannian metric on M_{c} such that ∂_{i}M_{c} are isometric to the interval [0, 1], and such that the boundary pieces intersect in twelve rectangular corners. We assume that the metric is product in a neighbourhood of the interior of the pieces ∂_{i}M, and that neighbourhoods of the corners are isometric to a neighbourhood of the vertex of the euclidean quadrant R^{+} × R^{+}.

Let M be the oriented, non-compact Riemann surface with six boundary components ∂_{i}M, i = 0, …, 5, isomorphic to R which is obtained by gluing infinite cylinders [0, ∞) × [0, 1] along the boundary pieces of M_{c} with odd label. The boundary components of M are again labelled according to their cyclic order.

We consider the spinor bundle S = S^{+} ⊕ S^{−} of M and fix a finite-dimensional Hilbert space W of dimension dim(V)/2. We consider the graded vector bundle E = S ⊗ (W ⊕ W^{op}). By ∂/ we denote the corresponding twisted Dirac operator. Let ∂/^{±} be the parts mapping sections of E^{±} to those of E^{∓}. We formulate an index problem for ∂/^{+} by putting boundary conditions at ∂_{j}M depending on L_{i}(b), i = 0, 1, 2.

The metric of M is flat near infinity and the boundaries (the flat region). We claim that the holonomy of the parallel transport in S along ∂M_{c} is trivial, where we consider M_{c} as a submanifold of M. Note that M is topologically a disc. Thus we can choose a trivialization of the tangent bundle T M. Measured with respect to the trivialization the parallel transport in T M along ∂M_{c} gives a rotation by −4π. The trivialization of T M induces one of the Spin(2)-principal bundle of M. The parallel transport along ∂M_{c} in this bundle corresponds to a rotation by −2π in the structure group. This implies the claim. We fix some point in ∂M_{c} and identify the fibres of the bundle S near infinity and ∂M with the fibre over this point using the parallel transport inside the flat region. We denote this fibre by S, too. Analogously we denote the fibre of E over this point by E.

Let (s, t) be oriented euclidean orthonormal coordinates near a point in the flat region. Then we have ∂/ = σ_{s}∂_{s} + σ_{t}∂_{t}, where σ_{s}, σ_{t} ∈ Hom^{odd}(E, E) depend on the choice of coordinates. Again we consider the components σ_{s}^{±}, σ_{t}^{±} ∈ Hom^{odd}(E^{±}, E^{∓}). The operator I := σ_{s}^{−}σ_{t}^{+} ∈ Aut(E^{+}) is invariantly defined. It satisfies I^{∗} = −1, I^{2} = −1, trI = 0, and it defines a hermitean symplectic structure on E^{+}. We fix an isometry V ≅ E^{+} which is compatible with the complex structures I on V and E^{+}.

Now we introduce the family of boundary conditions defining the family {∂/^{+}(b)}_{b∈B}. We let ∂/^{+}(b) be the differential operator ∂/^{+} mapping

{ϕ ∈ C_{c}^{∞}(M, E^{+}) | ϕ(x) ∈ L_{i}(b)∀x ∈ ∂_{i}M or x ∈ ∂_{i+3}M}

to L^{2}(M, E^{−}).

First we show that ∂/^{+}(b) gives rise to a smooth family of Fredholm operators such that the index bundle is well-defined. Then we apply the superconnection formalism in order to compute the Chern character of the index bundle.

First we conjugate the family {∂/^{+}(b)}_{b∈B} to a family {˜∂/^{+}(b)}_{b∈B} with constant domain. This will be done again on the level of germs at a point b_{0} ∈ B. In the remainder of the present section we replace B by a sufficiently small neighbourhood of b_{0}. We define a germ of a family of smooth U(V)-valued functions W^{+}(b, m), m ∈ M, such that near ∂_{i}M, ∂_{i+3}M, i = 0, 1, 2,

W^{+}(b, (s, t)) = exp(χ_{0}(s)A_{i}(b)) ,  (2)

where A_{i}, χ_{0} were defined above. Here (s, t) are orthonormal euclidean coordinates, s being normal to the boundary given by s = 0. W^{+} is determined by (2) near ∂M and we continue W^{+} to the interior of M by the constant 1 ∈ U(V). Similarly we set W^{−} = −σ_{s}^{+}W^{+}σ_{s}^{−} near ∂M and continue W^{−} to the interior of M by 1 ∈ U(E^{−}).

Let ˜∂/^{+} := W^{−}∂/(W^{+})^{∗}. Then ˜∂/^{+}(b) is a germ of a family of (now b-dependent) Dirac operators with domain (now independent of b) given as above by the Lagrangian subspaces L_{i}(b_{0}) at ∂_{i}M, ∂_{i+3}M, i = 0, 1, 2.

Recall that M has 6 cylindrical ends isomorphic to [0, ∞) × [0, 1]. We can consider [0, ∞) × [0, 1] as a subset of the cylinder R × [0, 1]. For i = 0, …, 5 the family ˜∂/^{+}(b) induces families of translation invariant operators ˜∂/_{i}^{+}(b) on the infinite cylinder R × [0, 1] together with boundary conditions. Let ˜∂/_{i}^{−}(b) be the formal adjoint (with the adjoint boundary condition) of ˜∂/_{i}^{+}(b) and let

˜∂/_{i}(b) := ( \begin{matrix}0 & ˜∂/_{i}^{−}(b) \\ ˜∂/_{i}^{+}(b) & 0 \end{matrix} ) .

Then ˜∂/_{i}(b) acts on sections of a bundle which we also denote by E.

Lemma 3.2 (1): The symmetric operator ˜∂/_{i}(b) defined on the space of smooth sections with compact support satisfying the boundary conditions is essentially selfadjoint. (2): It has a bounded inverse ˜∂/_{i}(b)^{−1}.

Proof. To prove the Lemma it is better to consider the original non gauge-transformed operator. We fix the point b ∈ B and consider without loss of generality the case i = 0.

Let (t, s) ∈ R × [0, 1] be the coordinates of the cylinder. Then we put ∂/ := σ_{t}∂_{t} + σ_{s}∂_{s} acting on C^{∞}(R × [0, 1], E). We let dom(∂/) be the space of all f ∈ C_{c}^{∞}(R × [0, 1], E) such that f(t, k) ∈ (L_{k}(b) + σ_{t}^{+}L_{k}(b)), k = 0, 1. Then ∂/ becomes an symmetric operator on L^{2}(R × [0, 1], E) which is unitarily equivalent with ˜∂/_{0}(b). We show that ∂/ is essentially selfadjoint.

Let D := σ_{t}^{−}σ_{s}^{+}∂_{s} be defined on {f ∈ C^{∞}([0, 1], E^{+})|f(k) ∈ L_{k}(b), k = 0, 1}. Then D is essentially selfadjoint on L^{2}([0, 1], E^{+}). Let σ(D) be the spectrum of D with multiplicity. Moreover let ϕ_{λ}, λ ∈ σ(D), be an orthonormal base of L^{2}([0, 1], E^{+}) of eigenvectors of D. Then we let H ⊂ L^{2}([0, 1], E) be the subspace spanned by ϕ_{λ}, σ_{t}^{+}ϕ_{λ}. We can write

L^{2}(R × [0, 1], E) = ⨁_{λ∈σ(D)} H_{λ} ⊗ L^{2}(E) = ⨁_{λ∈σ(D)} H_{λ} .

Note that E = E^{+} ⊕ E^{−} induces a splitting H_{λ} = H_{λ}^{+} ⊕ H_{λ}^{−}. We define

∂/_{λ} := σ_{t}∂_{t} + λ ( \begin{matrix}0 & σ_{t}^{−} \\ −σ_{t}^{+} & 0 \end{matrix} ) .

Let ˇ∂/ := ⊕_{λ∈σ(D)}∂/_{λ} be defined on

dom(ˇ∂/) := {f ∈ C_{c}^{∞}(R × [0, 1], E) | f = ∑_{λ∈σ(D)}^{finite} f_{λ}, f_{λ} ∈ H_{λ}} .

Then ˇ∂/ is symmetric and ˇ∂/ ⊂ ∂/. If we show that ˇ∂/ is essentially selfadjoint, then we are done since ¯ ˇ∂/ ⊂ ¯∂/ ⊂ ∂/^{∗} ⊂ ˇ∂/^{∗} implies ¯∂/ = ∂/^{∗}, where ¯∂/ denotes the closure of ∂/. Now ∂/_{λ} is essentially selfadjoint on the domain dom(∂/_{λ}) := C_{c}^{∞}(R, H_{λ}). Assume that f ∈ dom(ˇ∂/^{∗}). Then |⟨f, ˇ∂/ψ⟩| ≤ C(f)∥ψ∥, ∀ψ ∈ dom(ˇ∂/). We write f = ∑_{λ∈σ(D)} f_{λ}. Then we can conclude that |⟨f_{λ}, ∂/_{λ}u⟩| ≤ C(f_{λ})∥u∥, ∀u ∈ dom(∂/_{λ}). Thus f_{λ} ∈ dom(∂/_{λ}^{∗}) = dom(¯∂/_{λ}). Thus any finite sum of f_{λ} is in dom(¯ˇ∂/). We have lim_{N→∞} ∑_{λ∈σ(D),|λ|≤N} f_{λ} = f, and lim_{N→∞} ¯ ˇ∂/ ∑_{λ∈σ(D),|λ|≤N} f_{λ} = ˇ∂/^{∗}f exists. Hence f ∈ dom(¯ˇ∂/). This proves that ˇ∂/ is essentially selfadjoint and thus (1).

We now show (2). Note that ∂/_{λ}^{2} ≥ λ^{2}. Since L_{0}(b)∩L_{1}(b) = {0} we have inf_{λ∈σ(D)} λ^{2} = c > 0. It follows that ∂/_{λ}^{−1} is bounded and ∥∂/_{λ}^{−1}∥ < c^{−1}. Thus ∂/^{−1} = ⊕_{λ∈σ(D)}∂/_{λ}^{−1} exists and is bounded by c^{−1}. □

The distribution kernel of ˜∂/_{i}(b)^{−1} gives parametrices for ˜∂/(b) at infinity, in the interior, and also near the boundary of M. By patching we build a global parametrix Q(b) such that the smoothing remainders R(b) := ˜∂/(b)Q(b) − 1, R_{1}(b) := Q(b)˜∂/(b) − 1 have compact support.

Lemma 3.3 (1) The operator ˜∂/(b) is essentially selfadjoint.

(2) The domain H of ˜∂/(b) is independent of b.

(3) {˜∂/^{+}(b)} viewed as a family of bounded operators from H^{+} to L^{2}(M, E^{−}) is a smooth family of Fredholm operators.

Proof. We start with (1). Let h ∈ L^{2}(M, E) be arbitrary. We claim that Q(b)h ∈ dom(¯˜∂/(b)). We approximate h in L^{2}(M, E) by a sequence h_{α} ∈ C_{c}^{∞}(M, E). Then Q(b)h_{α} ∈ C^{∞}(M, E) satisfies the boundary conditions. Let … ⊂ K_{N} ⊂ K_{N+1} ⊂ … ⊂ M be an exaustion of M by compact subsets. Let {χ_{N}}_{N∈N} be a sequence of smooth functions such that supp(χ_{N}) ⊂ K_{N+1}, χ_{|K_{N}} = 1, and such that sup_{m∈M} ∥dχ_{N}(m)∥ → 0 as N → ∞. We have h_{α} = ˜∂/(b)Q(b)h_{α} − R(b)h_{α}. Note that χ_{N}Q(b)h_{α} ∈ dom(˜∂/). The existence of

lim_{α} lim_{N→∞} ˜∂/_{i}(b)χ_{N}Q(b)h_{α} = lim_{α} lim_{N→∞}[χ_{N}(R(b) + 1)h_{α} + grad(χ_{N})Q(b)h_{α}]
= (R(b) + 1)h

implies the claim.

Let now f ∈ dom(˜∂/(b)^{∗}). We must show that f belongs to the domain of ¯ ˜∂/(b). We write f = Q(b)˜∂/(b)^{∗}f − R_{1}(b)f. By the above Q˜∂/_{i}(b)^{∗}f belongs to the domain of ¯ ˜∂/(b). Moreover R_{1}(b)f is smooth and has compact support. Since ˜∂/(b)^{∗}f ∈ L^{2}(M, E) the trace f_{|∂M} ∈ L_{loc}^{2}(∂M, E) is well-defined. It follows from

|⟨f, ˜∂/(b)ψ⟩| ≤ C(f)∥ψ∥, ∀ψ ∈ dom(˜∂/(b))

that f_{|∂M} satisfies the same boundary conditions used to define ˜∂/(b). We conclude that R_{1}(b)f ∈ dom(˜∂/(b)) and hence f ∈ ¯ ˜∂/(b). This finishes the proof of (1).

Since

˜∂/(b) = ˜∂/(b_{0}) + V (b) ,  (3)

where V (b) is a smooth family of bounded operators the domain H of ˜∂/(b) does not depend on b ∈ B. This proves (2).

For (3) we employ the spectral comparison theorem for manifolds which coincide at infinity stating that σ_{ess}(˜∂/(b)) = ∪_{i}σ_{ess}(˜∂/_{i}(b)), hence inf σ_{ess}(˜∂/(b)) > 0 for all b ∈ B. From (3) it follows by perturbation theory that the family {˜∂/^{+}(b)} is a smooth family of Fredholm operators when it is viewed as a family of bounded operators from H^{+} to L^{2}(M, E^{−}). □

We now apply the superconnection formalism in order to obtain a formula for the Chern character ch(index(˜∂/^{+})) of the index bundle of {˜∂/^{+}(b)}_{b∈B}.

As a first step we make ˜∂/^{+}(b) surjective following [1], Ch.9.5. Let N ∈ N be given. Then we consider the manifold M′ := M ∪ ∗. We extend the bundle E^{+} to a bundle (E′)^{+} over M′ such that the fibre over the point ∗ is C^{N}. If we are given a map ψ : C^{N} → C_{c}^{∞}(M, E^{−}), then we define the operator ˜∂/_{ψ}^{+}(b) : H^{+} ⊕ C^{N} → L^{2}(M, E^{−}) by ˜∂/_{ψ}^{+}(b)(f ⊕ u) := ˜∂/^{+}(b)f + ψ(u).

Assume that B is compact. Then as in [1], Lemma 9.30, there exists an integer N and a linear map ψ : C^{N} → C_{c}^{∞}(M, E^{−}) such that ˜∂/_{ψ}^{+}(b) is surjective for all b ∈ B. In this case we have a bundle ker(˜∂/_{ψ}^{+}) := {ker(˜∂/_{ψ}^{+}(b))}_{b∈B}. The index bundle index(˜∂/^{+}) ∈ K^{0}(B) is represented by ker(˜∂/_{ψ}^{+}) − B × C^{N}. We fix ψ as above, and in addition we can assume that supp(ψ(u)) ∈ M_{c}, ∀u ∈ C^{N}.

In the remainder of the present section we replace M by M′, E^{+} by (E′)^{+}, and ˜∂/^{+} by ˜∂/_{ϵψ}^{+} such that ˜∂/_{ψ}^{+} is surjective.

First we take ϵ = 1. Let ∇ denote the trivial connection on the bundle B × L^{2}(M, E). We set W := W^{+} ⊕ W^{−}. Let ˜∇ = W ∇W^{∗}. Then we define the superconnection

B_{t} = ˜∇ + \sqrt{t}˜∂/ .

The curvature B_{t}^{2} has the form

B_{t}^{2} = t˜∂/^{2} + \sqrt{t}R ,

where R is a one-form with values in the odd endomorphisms of E. The heat operator e^{−B_{t}^{2}} can be constructed using the Volterra series [1], Prop.9.46. Let P_{t}(x, y) denote the smooth integral kernel of e^{−B_{t}^{2}}. In order to simplify the notation we omit the smooth dependence on b ∈ B.

Note that M is a manifold with a cylindrical end N × [0, ∞), where N is isometric to the disjoint union of six copies of the unit interval. Let (n, r) denote corresponding coordinates. Though e^{−B_{t}^{2}} is not of trace class we define

Tr′_{s}e^{−B_{t}^{2}} := ∫_{M_{c}} tr_{s}P_{t}(x, x)dx + lim_{u→∞} ∫_{0}^{u} ∫_{N} tr_{s}P_{t}((n, r), (n, r))dn dr .

We first argue that this limit exists. We claim that for some C < ∞, c > 0,

|tr_{s}P_{t}((n, r), (n, r))| < Ce^{−cr^{2}} .

The constants C, c can be choosen uniformly for t varying in compact subsets of (0, ∞). Consider the infinite cylinder Z := N × R. Let E^{Z} be the bundle on Z induced by E. Let {˜∂/^{Z}(b)}_{b∈B} denote the family of translation invariant operators on E^{Z} induced by {˜∂/(b)}_{b∈B}. The domain H^{Z} of ˜∂/(b)^{Z} is again independent of b ∈ B. We then obtain a translation invariant superconnection B_{t}^{Z} on the bundle B × L^{2}(Z, E^{Z}). Let P_{t}^{Z}((n, r), (m, s)) denote the corresponding heat kernel. The Clifford multiplication by ıσ_{r} is unitary, odd, and commutes with (B_{t}^{Z})^{2}. Thus

tr_{s}P_{t}^{Z}((n, r), (n, r)) = 0, ∀(n, r) ∈ Z .

A standard finite-propagation speed estimate [6] gives for r > 0

|P_{t}^{Z}((n, r), (n, r)) − P_{t}((n, r), (n, r))| < Ce^{−cr^{2}} ,

and this proves the claim.

Let D̃^{Z} be the family of operators on N ×V induced by ˜∂/^{Z}. Then D̃^{Z} can be identified with the direct sum of two copies of the direct sum of three copies of D̃ with boundary conditions given by the families of pairs (L_{0}, L_{1}), (L_{1}, L_{2}), (L_{2}, L_{0}). By A_{t}^{Z} we denote the superconnection corresponding to D̃^{Z}.

Set γ := σ_{r} ∈ End(E). It follows from Duhamel’s formula that the integral kernel of e^{−B_{t}^{2}} depends smoothly on t. The comparison with the cylinder Z shows that Tr′_{s}e^{−B_{t}^{2}} can be differentiated with respect to t, and that one can commute Tr′_{s} and d/dt.

Lemma 3.4

\frac{d}{dt}Tr′_{s}e^{−B_{t}^{2}} = −\frac{1}{\sqrt{4πt}}Tr_{s}γD̃e^{−(A_{t}^{Z})^{2}} − \frac{1}{2\sqrt{t}}dTr′_{s}˜∂/e^{−B_{t}^{2}}

Proof. First we claim that

\frac{d}{dt}Tr′_{s}e^{−B_{t}^{2}} = −Tr′_{s}[B_{t}, \frac{dB_{t}}{dt} e^{B_{t}^{2}}] .

Let ρ_{u} denote the characteristic function of M_{c} ∪ N × [0, u]. Using Duhamel’s formula we get

\frac{d}{dt}Tr′_{s}e^{−B_{t}^{2}} = −Tr′_{s} ∫_{0}^{1} e^{−sB_{t}^{2}} \frac{dB_{t}^{2}}{dt} e^{−(1−s)B_{t}^{2}}ds
= − lim_{u→∞} ∫_{0}^{1} Tr_{s}ρ_{u}e^{−sB_{t}^{2}} \frac{dB_{t}^{2}}{dt} e^{−(1−s)B_{t}^{2}}ds
= − lim_{u→∞} ∫_{0}^{1} Tr_{s}[\frac{dB_{t}}{dt} , B_{t}]e^{−sB_{t}^{2}}ρ_{u}e^{−(1−s)B_{t}^{2}}ds
= − lim_{u→∞} lim_{v→∞} ∫_{0}^{1} Tr_{s}ρ_{v}[\frac{dB_{t}}{dt} , B_{t}]e^{−sB_{t}^{2}}ρ_{u}e^{−(1−s)B_{t}^{2}}ds
= − lim_{v→∞} lim_{u→∞} ∫_{0}^{1} Tr_{s}ρ_{v}[\frac{dB_{t}}{dt} , B_{t}]e^{−sB_{t}^{2}}ρ_{u}e^{−(1−s)B_{t}^{2}}ds
= − lim_{v→∞} ∫_{0}^{1} Tr_{s}ρ_{v}[\frac{dB_{t}}{dt} , B_{t}]e^{−B_{t}^{2}}ds
= −Tr′_{s}[B_{t}, \frac{dB_{t}}{dt} e^{−B_{t}^{2}}] .

In order to justify that lim_{v→∞} and lim_{u→∞} can be interchanged one can again use the comparison with the infinite cylinder Z. We use

\frac{dB_{t}}{dt} = \frac{1}{2\sqrt{t}} ˜∂/

in order to write

−Tr′_{s}[B_{t}, \frac{dB_{t}}{dt} e^{B_{t}^{2}}] = −\frac{1}{2\sqrt{t}}Tr′_{s}[ ˜∇, ˜∂/e^{−B_{t}^{2}}] − \frac{1}{2}Tr′_{s}[˜∂/, ˜∂/e^{−B_{t}^{2}}]  (4)
= −\frac{1}{2\sqrt{t}}dTr′_{s}˜∂/e^{−B_{t}^{2}} − \frac{1}{2}Tr′_{s}[˜∂/, ˜∂/e^{−B_{t}^{2}}] .  (5)

Before explaining the transition from (4) to (5) we consider the second term of (5). Let z denote the Z_{2} grading operator. By integration by parts we obtain

−\frac{1}{2}Tr′_{s}[˜∂/, ˜∂/e^{−B_{t}^{2}}]

= −\frac{1}{2}Tr′z˜∂/^{2}e^{−B_{t}^{2}} − \frac{1}{2}Tr′z˜∂/e^{−B_{t}^{2}} ˜∂/
= −\frac{1}{2} lim_{u→∞} ∫_{0}^{u} ∫_{N} trz(˜∂/^{2}e^{−B_{t}})((n, u), (n, u))dndu − \frac{1}{2} ∫_{M_{c}} trz(˜∂/^{2}e^{−B_{t}^{2}})(m, m)dm

−\frac{1}{2} lim_{u→∞} ∫_{0}^{u} ∫_{N} trz(˜∂/e^{−B_{t}} ˜∂/)((n, u), (n, u))dndu − \frac{1}{2} ∫_{M_{c}} trz(˜∂/e^{−B_{t}^{2}} ˜∂/)(m, m)dm

= −\frac{1}{2} lim_{u→∞} ∫_{N} tr(γz˜∂/e^{−B_{t}})((n, u), (n, u))
= \frac{1}{2} lim_{u→∞} ∫_{N} tr_{s}(γ˜∂/e^{−B_{t}})((n, u), (n, u)) .

In order to evaluate this limit we can replace the kernel P_{t} by P_{t}^{Z}. We use the Volterra series [1], Prop.9.46, in order to compute P_{t}^{Z}. Note that on Z

e^{−t(˜∂ /^{Z})^{2}}((n, r), (m, s)) = e^{−t(D̃^{Z})^{2}}(n, m)\frac{e^{−(r−s)^{2}/4t}}{\sqrt{4πt}} ,  (6)

and

P_{t}^{Z} = e^{−t(˜∂ /^{Z})^{2}} + ∑_{k=1}^{∞}(−1)^{k}t^{k/2} ∫_{Δ^{k}} e^{−tσ_{0}(˜∂ /^{Z})^{2}}R^{Z} … R^{Z}e^{−tσ_{k}(˜∂ /^{Z})^{2}}dσ ,

where Δ_{k} denotes the standard k-simplex. Inserting (6) we obtain

P_{t}^{Z}((n, r), (m, s))

= \frac{e^{−(r−s)^{2}/4t}}{\sqrt{4πt}} (e^{−t(D̃^{Z})^{2}} + ∑_{k=1}^{∞}(−1)^{k}t^{k/2} ∫_{Δ^{k}} e^{−tσ_{0}(D̃^{Z})^{2}}R^{Z} … R^{Z}e^{−tσ_{k}(D̃^{Z})^{2}}dσ) (n, m) .

We now apply

γ˜∂/^{Z} = −\frac{∂}{∂r} + γD̃^{Z}

and evaluate the result at r = s. We then obtain

\frac{1}{2} lim_{u→∞} ∫_{N} tr_{s}γ(˜∂/e^{−B_{t}^{2}})(n, u)dn = \frac{1}{2} lim_{u→∞} ∫_{N} tr_{s}γ(˜∂/^{Z}e^{−(B_{t}^{Z})^{2}})(n, u)dn = \frac{1}{4\sqrt{πt}}Tr_{s}γD̃^{Z}e^{−(A_{t}^{Z})^{2}} .

Now we consider the first term of (5). By a similar computation as above one can show that on the cylinder Z

tr_{s}[ ˜∇^{Z}, ˜∂/^{Z}e^{−(B_{t}^{Z})^{2}}]((n, r), (n, r)) ≡ 0 .

Thus on M this quantity vanishes rapidly as r → ∞. We can take Tr′_{s} and

−\frac{1}{2\sqrt{t}}Tr′_{s}[ ˜∇, ˜∂/e^{−B_{t}^{2}}] = −\frac{1}{2\sqrt{t}}dTr′_{s}˜∂/e^{−B_{t}^{2}}

is an exact form. This finishes the proof of the lemma. □

Let ∇^{0} denote the induced connection on the bundle ker(˜∂/^{+}), and let ch(∇^{0}) := tr_{s}e^{−(∇^{0})^{2}} be the corresponding Chern form.

Lemma 3.5 Let |.| be any continuous seminorm on the space of smooth forms on B. For t → ∞ we have

|Tr′_{s}e^{−B_{t}^{2}} − ch(∇^{0})| = O(t^{−1/2})
|\frac{1}{2\sqrt{t}}Tr′_{s}˜∂/e^{−B_{t}^{2}}| = O(t^{−3/2}) .

Proof. The proof is the same as that of [1], and Theorem 9.19, Corollary 9.22, Theorem 9.23. Since M is non-compact we better replace K(E) defined in [1], p.279, by smooth families of finite-dimensional operators. □

We conclude that

∫_{s}^{∞} \frac{1}{4\sqrt{πt}}Tr_{s}γD̃^{Z}e^{−(A_{t}^{Z})^{2}}dt =: η̂(s)

exists and that

ch(∇^{0}) = Tr′_{s}e^{−B_{s}^{2}} + η̂(s) + dα(s) ,

where

α(s) := − ∫_{s}^{∞} \frac{1}{2\sqrt{t}}Tr′_{s}˜∂/e^{−B_{t}^{2}}dt .  (7)

Recall that we have included a parameter ϵ in our definition of B_{s}, and we will write B_{s}(ϵ) for a moment. Since \frac{d}{dϵ}B_{s}(ϵ) is a finite-dimensional operator, [1], Theorem 9.25 applies, and we have

\frac{d}{dϵ}Tr′_{s}e^{−B_{s}(ϵ)^{2}} = −dTr_{s} \frac{d}{dϵ}B_{s}(ϵ)e^{−B_{s}(ϵ)^{2}} .

If we put

β(s) := − ∫_{0}^{1} Tr_{s} \frac{d}{dϵ}B_{s}(ϵ)e^{−B_{s}(ϵ)^{2}}dϵ ,

then

Tr′_{s}e^{−B_{s}(1)^{2}} − Tr′_{s}e^{−B_{s}(0)^{2}} = dβ(s) .

We thus have

ch(∇^{0}) = Tr′_{s}e^{−B_{s}(0)^{2}} + η̂(s) + dα(s) + dβ(s) .  (8)

We now want to take the limit s → 0. Since \frac{d}{dϵ}B_{s}(ϵ) is a finite-dimensional, the limit β(0) := lim_{s→0} β(s) exists in C^{∞}(B, Λ^{∗}T^{∗}B).

Lemma 3.6 Let |.| be any continuous seminorm on the space of smooth forms on B. For t → 0 we have

|Tr′_{s}e^{−B_{t}(0)^{2}} − N| = o(1)  (9)
|\frac{1}{2\sqrt{t}}Tr′_{s}˜∂/e^{−B_{t}(0)^{2}}| = O(t^{−1/2}) .  (10)

Proof.We first show (9). We employ finite propagation speed estimates and the comparison with the cylinder in order to show that on the end of M

|tr_{s}P_{t}(n, r)| < Ce^{−r^{2}/t} .

4 COMPUTATION OF τ(L_{0}, L_{1}, L_{2}) 16

The local index theorem [1], Ch.10, gives |tr_{s}P_{t}(x)| = o(1) for x ∈ M since S is twisted with a bundle of the form W ⊕W^{op}. The contribution of the point {∗} is just tr_{s}P_{t}(∗) = N. This together yields (9).

Now we consider (10). On the cylinder Z we have tr_{s}˜∂/^{Z}e^{−(B_{t}^{Z})^{2}}(n, r) ≡ 0. We conclude that on N × [0, ∞) ⊂ M we have |tr_{s}˜∂/e^{−B_{t}^{2}}(n, r)| < Ce^{−r^{2}/t}. Moreover, for x in a small neighbourhood of the boundary of M_{c} we have |tr_{s}˜∂/e^{−B_{t}^{2}}(x)| < Ce^{−c/t}. If x is in the interior of M_{c} we can employ the method of [1], Ch. 10.5, in order to show that

|\frac{1}{2\sqrt{t}}tr_{s}˜∂/e^{−B_{t}^{2}}(x)| = O(t^{−1/2}) .

The estimate (10) follows. □

Now we can take the limit s → 0 in (7). From (8) we obtain

ch(∇^{0}) − N = η̂(0) + dα(0) + dβ(0) .

If η(u, L_{i}, L_{j}) denotes the form (1) which is defined using the boundary condition given by family of pairs (L_{i}, L_{j}), then η̂(0) = lim_{u→0}(η(u, L_{0}, L_{1}) + η(u, L_{1}, L_{2}) + η(u, L_{2}, L_{0})) (in particular this shows that the combination η(u, L_{0}, L_{1}) + η(u, L_{1}, L_{2}) + η(u, L_{2}, L_{0}) is regular at u = 0). We conclude that

τ(L_{0}, L_{1}, L_{2}) = [ch(∇^{0}) − N] = ch(index(˜∂/^{+})) .

This proves Proposition 3.1. □

4 Computation of τ(L_{0}, L_{1}, L_{2})

Let L_{0}, L_{1}, L_{2} ⊂ V be pairwise transverse Lagrangian subspaces. Then V = L_{1} ⊕ L_{2} and we can write x_{0} = x_{1} + x_{2}, x_{i} ∈ L_{i}, i = 0, 1, 2. We define a hermitean quadratic form Q on L_{0} by

Q(x_{0}) := (Ix_{1}, x_{2}) ,

where (I, .,) is the symplectic form on V associated to I and the Hilbert space structure of V. It is easy to see that Q is nondegenerate. Thus we can split L_{0} = L_{0}^{+} ⊕ L_{0}^{−} into the positive and negative eigenspace of Q. Returning now to the family case we obtain a decomposition L_{0} = L_{0}^{+} ⊕ L_{0}^{−} of the bundle of Lagrangian subspaces L_{0} ⊂ B × V which is induced by the two other subbundles L_{1}, L_{2}.

Proposition 4.1 We have

τ(L_{0}, L_{1}, L_{2}) = ch(L_{0}^{+}) − ch(L_{0}^{−}) ∈ H^{∗}(B, R) .

Proof. The proof of the proposition consists of two steps.

1. Using the K-theoretic relative index theorem [4] we reduce to an index problem for a family of Dirac operators on the disc. The parameter dependence of this family is again built in through the boundary conditions.

2. We then consider the ”universal” family of such operators which is parametrized by a space which is homotopy equivalent to the space of all triples of pairwise transverse Lagrangian subspaces of V. It suffices to verify the assertion of the proposition in this special case.

First we want to compactify M by cutting off the cylindrical ends and gluing in half discs. The resulting manifold M̂ is then topologically a disc. Let ˆ∂/^{+} be the corresponding Dirac operator. We want to find a family of boundary conditions parametrized by B such that ch(index(∂/^{+})) = ch(index(ˆ∂/^{+})).

Let Y ⊂ R^{2} denote the subset

Y := {(s, t) ∈ R^{2} | s ≥ 0, t ∈ [−1/2, 1/2] or s ≤ 0, t^{2} + s^{2} ≤ 1/4} .

Then Y is a Riemannian surface with C^{1}-boundary and one cylindrical end. Let S_{Y} be the spinor bundle of Y. Let E_{Y} := S_{Y} ⊗ (W ⊕ W^{op}) and ∂/_{Y} be the Dirac operator on E_{Y}. We trivialize S_{Y} and E_{Y} using the flat Levi Civita connection and denote the typical fibre of E_{Y} by E. Then

∂/_{Y}^{±} = σ_{s}^{±} \frac{∂}{∂s} + σ_{t}^{±} \frac{∂}{∂t} .

The space V := E^{+} is a symplectic vector space with symplectic structure induced by I = σ_{s}^{−}σ_{t}^{+}.

Let now L_{0}, L_{1} be transversal Lagrangian subspaces of V. We want to construct a family of Lagrangian subspaces L(s, t) = L(s, t)(L_{0}, L_{1}) which is parametrized by (s, t) ∈ ∂Y, and which depends smoothly on L_{0}, L_{1}, such that L(s, t) = L_{0} for s ≥ 0, t = 1/2 and L(s, t) = L_{1} for s ≥ 1, t = −1/2. If B ∋ b → (L_{0}(b), L_{1}(b)) is a smooth family of pairs of transverse Lagrangian subspaces, then we require that the index of the Dirac operator ∂/_{Y}^{+} subject to the family of boundary conditions L(., .)(L_{0}(b), L_{1}(b)) is trivial.

We first set

L̂(s, t)(L_{0}) := \{ \begin{matrix}L_{0} & (s, t) ∈ ∂Y, t ≥ 0 \\ IL_{0} & (s, t) ∈ ∂Y, s ≥ 0, t = −1/2 \\ σ_{s}^{−}n(s, t)L_{0} & (s, t) ∈ ∂Y, s ≤ 0, t ≤ 0 \end{matrix} ,

where n(s, t) := 2(sσ_{s}^{+}+tσ_{t}^{+}) is the Clifford multiplication by the normal vector. It is easy to see that L̂(s, t)(L_{0}) is C^{1} with respect to (s, t). If B ∋ b → L_{0}(b) is a smooth family of Lagrangian subspaces, then we consider the family {ˆ∂/_{Y}^{+}(b)}_{b∈B} given by ∂/_{Y}^{+} subject to the boundary conditions given by L̂(., .)(L_{0}(b)). Since L_{0} and IL_{0} are transverse, we see as in Section 3 that {ˆ∂/_{Y}^{+}(b)}_{b∈B} gives rise to a family of Fredholm operators.

Lemma 4.2 In K^{0}(B) we have 2 index(ˆ∂/_{Y}^{+}) = 0.

Proof. We claim that ˆ∂/_{Y}^{+} is equivalent with its adjoint (ˆ∂/_{Y}^{+})^{∗}. Thus 2 index(ˆ∂/_{Y}^{+}) = index(ˆ∂/_{Y}^{+}) − index(ˆ∂/_{Y}^{+})^{∗} = 0.

We now show the claim. First we describe the adjoint (ˆ∂/^{+})^{∗}. Note that E^{−} is a hermitean symplectic vector space with symplectic structure induced by σ_{s}^{+}σ_{t}^{−}. Then the adjoint of ˆ∂/_{Y}^{+} is the operator ˆ∂/_{Y}^{−} subject to the boundary conditions given by the family of Lagrangian subspaces of E^{−}

B ∋ b → {(s, t) ∈ ∂Y ↦ τ(s, t)L̂(s, t)(L_{0}(b))} ,

where τ(s, t) is the Clifford multiplication by the tangent vector at (s, t) ∈ ∂Y. If ψ^{+}(s, t) is a section of E_{Y}^{+}, then we set (U^{+}ψ^{+})(s, t) := σ_{t}^{+}ψ^{+}(s, −t). Then U^{+} is unitary, and U^{+}ψ^{+} is a section of E_{Y}^{−}. It is easy to check that the equality of differential operators

U^{+}ˆ∂/_{Y}^{−}U^{+} = ˆ∂/_{Y}^{+}

is compatible with the boundary conditions. This shows the claim. □

Let Ω(x, y) = (Ix, y) be the (hermitean) symplectic form on V. Let Λ denote the manifold of all Lagrangian subspaces of V. For L ∈ Λ let L_{L} denote the subset of all Lagrangian subspaces L′ ∈ Λ which are transverse to L. The following discussion is parallel to that in [8] p. 117/118. Let P_{L′} denote the projection from V to L′ with kernel L. It is easy to check that Ω(P_{L′}x, y) + Ω(x, P_{L′}y) = Ω(x, y). We define the hermitean quadratic form

Q_{L′}(x, y) := Ω(P_{L′}x, y) − \frac{1}{2}Ω(x, y) .  (11)

Indeed

Q_{L′}(x, y) = Ω(P_{L′}x, y) − \frac{1}{2}Ω(x, y)
= −Ω(x, P_{L′}y) + \frac{1}{2}Ω(x, y)
= \overline{Ω(P_{L′}y, x) − \frac{1}{2}Ω(y, x)}
= \overline{Q_{L′}(y, x)} .

We have

Q_{L′}(x, y) = −\frac{1}{2}Ω(x, y), ∀x ∈ L, y ∈ V .  (12)

Any hermitean quadratic form Q satisfying (12) determines a Lagrangian subspace L′ such that Q = Q_{L′}. In fact let P′ be determined by Q and (11), then L′ is just the 1-eigenspace of P′. Thus we can identify L_{L} with the space of hermitean quadratic forms satisfying (12). In particular, L_{L} is an affine space where the affine structure only depends on L.

We now can construct the desired family L(s, t)(L_{0}, L_{1}). Note that IL_{0}, L_{1} ∈ L_{L_{0}}, and there is a natural affine path L(r) = L(r)(L_{0}, L_{1}) with L(0) = IL_{0}, L(1) = L_{1}. We choose a smooth cut-off function χ ∈ C^{∞}([0, 1]) with χ(t) ∈ [0, 1], χ(t) = 0 near t = 0 and χ(t) = 1 near t = 1. We set L(s, t) = L̂(s, t) for all (s, t) ∈ ∂Y except for t = −1/2, where we set L(s, t) := L(χ(s))(L_{0}, L_{1}) for s ∈ [0, 1] and L(s, t) := L_{1} for s ≥ 1. Then L(s, t) is C^{1} with respect to (s, t) and depends smoothly on L_{0}, L_{1}. Let {∂/_{Y}^{+}(b)}_{b∈B} denote the family of Dirac operators given by ∂/_{Y}^{+} subject to the boundary conditions defined by {L(., .)(L_{0}(b), L_{1}(b))}_{b∈B}. Then ˆ∂/_{Y}^{+} and ∂/_{Y}^{+} are homotopic families and thus 2 index(∂/_{Y}^{+}) = 0 in K^{0}(B). The upshot of the construction above is that we associated to a pair L_{0}, L_{1} of transversal Lagrangian subspaces a canonical path γ(L_{0}, L_{1}) from L_{0} to L_{1} which is parametrized by ∂Y. The path γ(L_{0}, L_{1}) depends smoothly on the pair (L_{0}, L_{1}) and has (in a certain sense that will become clear below) the minimal winding number.

Now we can cut-off the six cylindrical ends of M and glue in the pieces Y_{i} = {(s, t) ∈ Y |s ≥ 1}, i = 0, …, 5. The resulting manifold M̂ is topologically a two-dimensional disc.

To be more precise let Z_{i} = [0, ∞) × [−1/2, 1/2], i = 0, …, 5, denote the cylindrical ends of M. Then we cut at {1} × [−1/2, 1/2]. We identify (s, t) ∈ Z_{i} with (1 − s, t) ∈ Y_{i}, s ∈ (0, 1). Moreover, we use σ_{s} in order to glue the bundles. Then ∂/^{+} glues with ∂/_{Y}^{−}. Assume that on the component [0, ∞) × {−1/2} of ∂Z_{i} we have the boundary condition given by L_{i} (resp. L_{i−3}) and on [0, ∞)×{1/2} we have the one given by L_{i+1} (resp. L_{i−2}), where L_{3} = L_{0}. Then on the boundary part of M̂ which comes from Y_{i} we choose the path σ_{s}τγ(L_{i}, L_{i+1}) (resp. σ_{s}τγ(L_{i−3}, L_{i−2})), where τ again denotes the Clifford multiplication with the unit vector tangent to the boundary. This path indeed connects L_{i} with L_{i+1}.

Thus we have constructed a closed path γ̂(L_{0}, L_{1}, L_{2}) of Lagrangian subspaces of V which is parametrized by ∂M̂, and which depends smoothly on the triple (L_{0}, L_{1}, L_{2}). We use this path in order to define the boundary condition for the W ⊕ W^{op}-twisted Dirac operator ˆ∂/^{+} on M̂. Recall that we identify V with the fibres of the bundle E_{∂M̂}^{+} using the parallel transport along ∂M̂.

It follows from the K-theoretic relative index theorem [4] that index(∂/^{+}) = index(ˆ∂/^{+}) ∈ K^{0}(B)[1/2]. Indeed, the relative index theorem states that

index(∂/^{+}) + ∑_{i=0}^{5} index(∂/_{Y_{i}}^{+}) = index(ˆ∂/^{+}) + ∑_{i=0}^{5} index(∂/_{Z_{i}}^{+}) ,

where Z_{i} = [−1/2, 1/2] × R and ∂/_{Z_{i}}^{+} is the W ⊕ W^{op}-twisted Dirac operator subject to the boundary conditions given by L_{i} at {−1/2} × R, L_{i+1} at {−1/2} × R (resp. L_{i−3} at {−1/2} × R, L_{i−2} at {−1/2} × R). But index(∂/_{Z_{i}}^{+}) = 0 in K^{0}(B)[1/2] for symmetry reasons and index(∂/_{Y_{i}}^{+}) = 0 by Lemma 4.2.

To be precise, the relative index theorem in [4] is stated for manifolds without boundary. But argument given there carries over to the present case without any essential modification.

Deforming the metric of M̂ to the standard metric of the two disc we do not change the index. Below we will assume that M̂ is isometric to the two disc. The parallel transport in E^{+} along ∂M̂ with respect to the globally flat metric gives an identification of V with the fibres of E^{+} near ∂M̂ which is topologically different from the one used above. This fact has to be taken into account below. We have now finished the first part of the proof of the proposition.

We start with the second part. Let Λ^{3} be the space of triples (L_{0}, L_{1}, L_{2}) of pairwise transverse Lagrangian subspaces of V. Let Sp(V) denote the group of symplectic automorphisms of V. Note that iΩ is a non-degenerate hermitean form of signature (l, l), where l = dim_{C}(V)/2. Thus Sp(V) ≅ U(l, l). The group Sp(V) acts on Λ^{3}. We claim that Λ^{3} is the disjoint union of orbits of Sp(V).

First it is easy to see that Sp(V) acts transitively on the space Λ. Let L_{0} ∈ Λ. Then any L_{1} ∈ L_{L_{0}} can be written as {Bx + x | x ∈ IL_{0}} for some B ∈ End(IL_{0}, L_{0}). The condition that L_{1} is Lagrangian translates to Ω(Bx, y) + Ω(x, By) = 0 for all x, y ∈ IL_{0}. This is equivalent to (BI)^{∗} = BI, where ∗ is defined with respect to the hermitean metric of V. Thus we can parametrize L_{L_{0}} by the symmetric endomorphisms of L_{0}. Writing V = L_{0} ⊕ IL_{0} it is easy to check that

A := ( \begin{matrix}1 & B \\ 0 & 1 \end{matrix} ) ∈ Sp(V) ,

AL_{0} = L_{0} and AIL_{0} = L_{1}. Thus Sp(V) acts transitively on the set Λ^{2} of pairs (L_{0}, L_{1}) of transverse Lagrangian subspaces.

Let G denote the stabilizer of the pair (L_{0}, IL_{0}). Let j : Gl(L_{0}) → Gl(IL_{0}) denote the unique isomorphism such that

Gl(L_{0}) ∋ A ↦ ( \begin{matrix}A & 0 \\ 0 & j(A) \end{matrix} ) ∈ G .  (13)

Then j(A) = −I(A^{−1})^{∗}I. If L_{2} ∈ Λ is transverse to L_{0} and IL_{0}, then we write

L_{2} = ( \begin{matrix}1 & B \\ 0 & 1 \end{matrix} ) IL_{0}

for some invertible B as above. If g ∈ G is represented by A ∈ Gl(L_{0}) according to (13), then

gL_{2} = ( \begin{matrix}1 & −ABIA^{∗}I \\ 0 & 1 \end{matrix} ) IL_{0} .

The action of G on L_{L_{0}} is hence given by action of Gl(L_{0}) on the symmetric endomorphisms of L_{0} by conjugation. Thus the signature of the symmetric BI is the only invariant of the orbit of G generated by L_{2} inside the space of Lagrangian subspaces which are transverse to L_{0} and IL_{0}. We conclude that Λ^{3} is the disjoint union of orbits of Sp(V) of points (L_{0}, IL_{0}, L_{2}), which are distinguished by the signature of a matrix BI defined by L_{2}.

We now consider the orbit generated by a triple (L_{0}, IL_{0}, L_{2}) ∈ Λ^{3}. The stabilizer U of (L_{0}, IL_{0}, L_{2}) can be identified with the subgroup of Gl(L_{0}) fixing the hermitean form on L_{0} defined by BI. Let K ⊂ U denote a maximal compact subgroup. We can choose K such that it fixes the metric (., .), hence K is a subgroup of the unitary group of V. But then it fixes I, too. Using the explicit formulas given above one checks that γ̂(L_{0}, IL_{0}, L_{2}) = kγ̂(L_{0}, IL_{0}, L_{2}) for all k ∈ K.

We now globally trivialize S, E using the parallel transport given by the globally flat metric of M̂. Along the boundary ∂M̂ the old and the new trivialization of S are related by a twist of −2π in the structure group of S. Note that S^{±} are the ±ı eigenspaces of the Clifford multiplication by the volume form of M̂. The image of the path γ̂(L_{0}, L_{1}, L_{2}) in the new trivialization can be obtained (up to homotopy) by γ(L_{0}, L_{1}, L_{2})(z) := z^{−ıI} γ̂(L_{0}, L_{1}, L_{2})(z), z ∈ S^{1} = ∂M̂. We see that γ(L_{0}, L_{1}, L_{2}) is K-invariant, too.

Let (x, y) be oriented, flat orthonormal coordinates on M̂ and write ˆ∂/ = σ_{x}∂_{x}+σ_{y}∂_{y}. If we let K act on E^{−} by, say, K ∋ k ↦ −σ_{x}^{+}kσ_{x}^{−} ∈ End(E^{−}), then ˆ∂/^{+} is K-equivariant. We now consider the family of Dirac operators parametrized by Sp(V), given by ˆ∂/^{+} subject to the boundary conditions Sp(V) ∋ g ↦ γ(gL_{0}, gIL_{0}, gL_{2}). This family is K-equivariant and we go over to the quotient family parametrized by Sp(V)/K which we denote by ˇ∂/^{+}.

Let X := index(ˆ∂/^{+}) ∈ R(K) be the K-equivariant index of ˆ∂/^{+} subject to the boundary conditions given by γ(L_{0}, IL_{0}, L_{2}), where R(K) denotes the representation ring of K. Then index(ˇ∂/^{+}) = [Sp(V) ×_{K} X] ∈ K^{0}(Sp(V)/K). The following Lemma implies the proposition for the family ˇ∂/^{+}.

Lemma 4.3 Let (n, m) be the signature of the quadratic form defined by BI on L_{0}. Then X = C^{n} − C^{m}, where C^{n}, C^{m} are the m- and n-dimensional standard representations of the corresponding factors of K ≅ U(n) × U(m).

Proof. Let T ⊂ K be a maximal torus. If Y ∈ R(K), then Y_{T} denotes the restriction to T. It is sufficient to show that X_{T} = C_{T}^{n} − C_{T}^{m}.

We first consider the case that W ≅ C. Then ˆ∂/^{+} can be expressed in terms of complex geometry. Indeed we have

ˆ∂/^{+} = ( \begin{matrix}−∂ & 0 \\ 0 & ¯∂ \end{matrix} ) .

Writing ¯∂ = ∂_{x} + ı∂_{y}, ∂ = ∂_{x} − ı∂_{y} we obtain

σ_{x}^{+} = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , σ_{y}^{+} = ( \begin{matrix}ı & 0 \\ 0 & ı \end{matrix} )

and hence

I = ( \begin{matrix}ı & 0 \\ 0 & −ı \end{matrix} ) .

A one-dimensional subspace C(a, b) ⊂ V, a, b ∈ C, is Lagrangian iff |a| = |b|. We parametrize the Lagrangian subspaces of V by S^{1} associating to ϕ ∈ S^{1} the space C(ϕ, 1).

The space of pairwise transverse triples Λ^{3} consists of two components which are distinguished by the cyclic order of the parameters ϕ_{i} ∈ S^{1} of L_{i}. Let Λ_{1}^{3} be the component with order ϕ_{0} < ϕ_{1} < ϕ_{2} and Λ_{−1}^{3} be the component with order ϕ_{0} < ϕ_{2} < ϕ_{1}. The T - equivariant index of ˆ∂/^{+} only depends on the homotopy class of the path γ(L_{0}, L_{1}, L_{2}).

We fix an identification π_{1}(S^{1}) = Z, γ ↦ [γ], such that the path mapped to 1 has positive orientation. We leave to the reader to compute [γ(L_{0}, L_{1}, L_{2})] ∈ Z for (L_{0}, L_{1}, L_{2}) ∈ Λ_{±1}^{3}. The result is [γ(L_{0}, L_{1}, L_{2})] = 0 on Λ_{1}^{3} and [γ(L_{0}, L_{1}, L_{2})] = 2 on Λ_{−1}^{3}.

Lemma 4.4 Let W ≅ C, and let the boundary condition of ˆ∂/^{+} be given by a closed path γ of Lagrangian subspaces. Then index( ˆ∂/^{+}) = −[γ] + 1.

Proof. Let n ∈ Z be represented by the path γ_{n}(ϕ) = C(ϕ^{n}, 1), ϕ ∈ S^{1}. The kernel of ˆ∂/^{+} with boundary condition given by γ_{n} can be identified with the space of pairs (f, g) of functions on M̂, where ¯∂g = 0, ∂f = 0 and z^{n}g(z) = f(z) at S^{1}. This space is non-trivial for n ≤ 0, and it is spanned by (z̄^{−n}, 1), (z̄^{−n−1}, z), …, (1, z^{−n}). One can check that (ˆ∂/^{+})^{∗} is given by

(∂/^{+})^{∗} = ( \begin{matrix}¯∂ & 0 \\ 0 & −∂ \end{matrix} ) ,

and that the boundary condition is given by the path γ_{n+2}. The kernel of (ˆ∂/^{+})^{∗} can be identified with the space of pairs (f, g) of functions on M̂ with ¯∂f = 0, ∂g = 0, and z^{n−2}g(z) = f(z). The kernel of (ˆ∂/^{+})^{∗} is non-trivial for n ≥ 2, and it is spanned by by (z^{n−2}, 1), (z^{n−3}, z̄), …, (1, z̄^{n−2}). It follows that index( ˆ∂/^{+}) = −n + 1. □

We now finish the proof of Lemma 4.3 in the case W = C. We must show that the signature of the quadratic form given by BI is (1, 0) on Λ_{1}^{3} and (0, 1) on Λ_{−1}^{3}. Let l_{0} = (1, 1), l_{1} = (−1, 1), and l_{2} = (−i, 1) generate the Lagrangian subspaces L_{0}, IL_{0}, L_{2}. Then (L_{0}, IL_{0}, L_{2}) ∈ Λ_{1}^{3}. We have l_{2} ∼ l_{0} + \frac{1+ı}{1−ı}l_{1} and Il_{0} = −il_{1}. It follows that BI = 1. The other case is similar.

In order to complete the proof of Lemma 4.3 in the general case one reduces to the special case W = C by considering the direct sums. □

We now finish the proof of the proposition. Let {Λ_{i}^{3}} denote the set of components of Λ^{3} and choose x_{i} ∈ Λ_{i}^{3} for all i. Let U_{i} be the stabilizer of x_{i} in Sp(V). Then ⋃_{i} Sp(V)/U_{i} ≅ Λ^{3} parametrizes the universal family of boundary conditions for ˆ∂/^{+} given by a family of path’ B ∋ b ↦ γ(L_{0}(b), L_{1}(b), L_{2}(b)). Indeed, any such family can be pulled back from the universal one using the canonical map B → ⋃_{i} Sp(V)/U_{i}. Since the fibres of π_{i} : Sp(V)/K → Sp(V)/U_{i} are symmetric spaces of non-compact type, π_{i} is a homotopy equivalence. Since the proposition is proved for ˇ∂/^{+} (for each component separately) it is also true for the universal family, and hence in general. □

\underline{Remark :} We sketch another proof of Proposition 4.1 which avoids the use of the universal family and the equivariant index.

We first consider a model case where V_{model} := C^{2} and I := diag(ı, −ı). Let l_{i}, i = 0, 1, 2, be the Lagrangian subspaces of V_{model} parametrized by ϕ_{0} := 0, ϕ_{1} := ±arg(\frac{1−i}{1+i}), ϕ_{2} := −ϕ_{1}, i.e., l_{i} = {(e^{ıϕ_{i}}x, x) ∈ C^{2} | x ∈ C}. Let ∂/_{±}^{+} be the Dirac operator on M with boundary conditions given by the triple l_{0}, l_{1}, l_{2}. We have index(∂/_{±}^{+}) = ±1. This can be proved in the same way as Proposition 4.1, but the proof simplifies due to the facts that dim(V_{model}) = 2, and that B is a point.

We now turn to the general case. Let V be any finite dimensional Hilbert space with hermitean symplectic structure, and let {L_{0}, L_{1}, L_{2}}_{b∈B} be a family of pairwise transverse Lagrangian subspaces of V. Then we can write L_{i} = {x + A_{i}Ix | x ∈ IL_{0}}, i = 1, 2, where A_{i} are smooth symmetric bundle endomorphisms of the subbundle L_{0} ⊂ B × V such that A_{1} − A_{2} is invertible.

We show that this family of triples is homotopic to a family in some standard form. Consider the family A(t)_{i} := \frac{1}{2}((1 − t)(A_{i} − A_{3−i}) + tA_{i}), t ∈ [0, 1]. Then A(t)_{1} − A(t)_{2} = A_{1} − A_{2} is invertible for all t. Moreover, A(1)_{i} = A_{i} and A(0)_{1} = A_{1} − A_{2} = −A(0)_{2}. Thus up to homotopy we can assume that A_{1} = −A_{2} = A, where A is invertible. There is a further index bundle preserving homotopy of A to A/|A|. Thus we can assume that A^{2} = 1.

Let ∂/^{+} be the family of operators on M defined by the family {L_{0}, L_{1}, L_{2}}_{b∈B} associated with A. Let L_{0}^{±} be the ±1-eigenspaces of A. We will define isomorphisms Φ : ker(∂/_{+}^{+}) ⊗ L_{0}^{+} ⊕ ker(∂/_{−}^{+}) ⊗ L_{0}^{−} ≅ ker(∂/^{+}), Ψ : coker(∂/_{+}^{+}) ⊗ L_{0}^{+} ⊕ coker(∂/_{−}^{+}) ⊗ L_{0}^{−} ≅ coker(∂/^{+}).

First we fix a basis vector v_{model} ∈ l_{0}. Then any v ∈ L_{0}^{±} defines an unique symplectic embedding v_{∗} : V_{model} ↪ V such that v_{∗}(v_{model}) = v. If v, w ∈ L_{0}^{±} and μ ∈ C, then we have (μv + w)_{∗} = μv_{∗} + w_{∗}. One can check that v_{∗}(ker(∂/_{±}^{+})) ⊂ ker(∂/^{+}), v_{∗}(coker(∂/_{±}^{+})) ⊂ coker(∂/^{+}). We define Φ(f ⊗v⊕f′⊗v′) := v_{∗}(f)+v′_{∗}(f′), Ψ(g⊗w⊕g′⊗w′) := w_{∗}(g)+w′_{∗}(g′). In follows that

index(∂/^{+}) = index(∂/_{+}^{+})[L_{0}^{+}] + index(∂/_{−}^{+})[L_{0}^{−}] = [L_{0}^{+}] − [L_{0}^{−}] ∈ K^{0}(B) .

This finishes our sketch of an alternative proof of Proposition 4.1.

\underline{Example :} For the purpose of illustration let us consider an example. Let V := C^{4} equipped with some complex structure I such that L_{0} = C^{2} ⊂ C^{4} is Lagrangian. Let B := P^{2}C. If T → B denotes the tautological bundle of B, then we have an orthogonal splitting of the trivial bundle B × L_{0} as T ⊕ T^{⊥}. For b ∈ B let Q_{b} be the quadratic form on L_{0} given by the matrix diag(1, −1) with respect to the splitting L_{0} = T_{b} ⊕ T_{b}^{⊥}. The family of quadratic forms {Q_{b}}_{b∈B} induces a family of Lagrangian subspaces {L_{2}(b)}_{b∈B} such that L_{2}(b) is transverse to L_{0}, IL_{0} for all b ∈ B. Thus τ(L_{0}, IL_{0}, L_{2}) is defined, and we have

τ(L_{0}, IL_{0}, L_{2}) = ch(T) − ch(T^{∗}) = 2c_{1}(T) .

This class is non-trivial.

\underline{Remark :} It would be desirable to have an explicit formula for the η-form generalizing the result of [9].

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