Boundary value problems for the Lorentzian Dirac operator

TL;DR

This paper explores boundary conditions for the Lorentzian Dirac operator on globally hyperbolic spacetimes, examining how more general conditions affect the operator's index compared to classical elliptic cases.

Contribution

It investigates the replacement of Atiyah-Patodi-Singer boundary conditions with more general ones for the Lorentzian Dirac operator and analyzes the resulting index changes.

Findings

Boundary conditions influence the index of the Lorentzian Dirac operator.

Differences arise compared to the classical elliptic case.

General boundary conditions can alter the Fredholm property.

Abstract

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

Equations

$$M=\mathbb{R}\times \Sigma$$

$$\gamma (X)\gamma (Y)+\gamma (Y)\gamma (X)=-2(X,Y)$$

$$(\gamma (X)u,v)=(u,\gamma (X)v)$$

$$⟨{\gamma }_{\Sigma }(X)u,v⟩$$

$$=-(u,\gamma (\nu ){\gamma }_{\Sigma }(X)v)=-(\gamma (\nu )u,{\gamma }_{\Sigma }(X)v)=-⟨u,{\gamma }_{\Sigma }(X)v⟩.$$

$$D=\sum _{j=0}^n{\epsilon }_j\gamma (e_j){\nabla }_{e_j}$$

$$D=-\gamma (\nu )\left({\nabla }_{\nu }+\mathrm{i}{A}_{\Sigma }-\frac{n}{2}H\right)$$

$$(u,v{)}_{L^2}={\int }_{\Sigma }⟨u,v⟩\mathrm{dA}$$

$$\Vert u{\Vert }_{FE}^2=\Vert u{\Vert }_{L^2}^2+\Vert Du{\Vert }_{L^2}^2$$

$${\mathrm{res}}_{\Sigma }\oplus D:FE(M_0,V^R)$$

$${\mathrm{res}}_{\Sigma }:\{u\in FE(M_0,V^R)\mid Du=0\}\to L^2(\Sigma ,V^R)$$

$$Q:L^2({\Sigma }_0,V^R)\to L^2({\Sigma }_1,V^R)$$

$$\{u\in FE(M_0,V^R)\mid Du=0\}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{res}}_{\Sigma }\oplus D:C^{\infty }(M_0,V^R)\to C^{\infty }(\Sigma ;V^R)\oplus C^{\infty }(M_0,V^L)$$

$$\mathrm{ind}(B_0,B_1)=\dim (B_0\cap B_1)-\dim (H/(B_0+B_1))$$

$$\mathrm{ind}(B_0,B_1)=\mathrm{ind}(B_1,B_0).$$

$$\mathrm{ind}(B_0,B_1)=-\mathrm{ind}(B_0^{\perp },B_1^{\perp }).$$

$$\mathrm{ind}(B_0^{\prime },B_1)=\mathrm{ind}(B_0,B_1)+\dim (B_0^{\prime }/B_0).$$

$${\pi }_V:H\to V.$$

$${\pi }_{B_1^{\perp }}{|}_{B_0}:B_0\to B_1^{\perp }$$

$$r_j\oplus L:E\to H_j\oplus F$$

$$\ker (L)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(({\pi }_{B_0^{\perp }}\circ r_0)\oplus ({\pi }_{B_1^{\perp }}\circ r_1)){|}_{\ker L}:\ker L\to B_0^{\perp }\oplus B_1^{\perp }$$

$$\tilde{L}:H_0\to B_0^{\perp }\oplus B_1^{\perp },h_0\mapsto ({\pi }_{B_0^{\perp }}(h_0),{\pi }_{B_1^{\perp }}(Q(h_0)),$$

$$\tilde{L}=\left(\begin{array}{cc}{1}_{B_0^{\perp }}& 0\\ {\pi }_{B_1^{\perp }}Q{|}_{B_0^{\perp }}& {\pi }_{B_1^{\perp }}Q{|}_{B_0}\end{array}\right).$$

$$\mathrm{ind}\big{(}\widetilde{L}\big{)}=\mathrm{ind}\begin{pmatrix}1_{B_{0}^{\perp}}&0\\ \pi_{B_{1}^{\perp}}Q|_{B_{0}^{\perp}}&\pi_{B_{1}^{\perp}}Q|_{B_{0}}\end{pmatrix}=\mathrm{ind}\begin{pmatrix}1_{B_{0}^{\perp}}&0\\ 0&\pi_{B_{1}^{\perp}}Q|_{B_{0}}\end{pmatrix}=\mathrm{ind}\big{(}\pi_{B_{1}^{\perp}}Q|_{B_{0}}\big{)}.$$

$$P=({\pi }_{B_0^{\perp }}\circ r_0)\oplus ({\pi }_{B_1^{\perp }}\circ r_1):E\to B_0^{\perp }\oplus B_1^{\perp }.$$

$$({\pi }_{B_0^{\perp }}\circ {\mathrm{res}}_{{\Sigma }_0})\oplus ({\pi }_{B_1^{\perp }}\circ {\mathrm{res}}_{{\Sigma }_1})\oplus D:FE(M_0,V^R)\to B_0^{\perp }\oplus B_1^{\perp }\oplus L^2(M_0,V^L)$$

$$D:\ker ({\pi }_{B_0^{\perp }}\circ {\mathrm{res}}_{{\Sigma }_0})\cap \ker ({\pi }_{B_1^{\perp }}\circ {\mathrm{res}}_{{\Sigma }_1})\to L^2(M_0,V^L)$$

$$\mathrm{ind}(B_0,B_1)={\int }_{M_0}\hat{\mathrm{A}}(\nabla )\wedge \mathrm{ch}({\nabla }^E)+{\int }_{\partial M_0}\mathcal{T}-\frac{h(A_0)+h(A_1)+\eta (A_0)-\eta (A_1)}{2}.$$

$$\mathrm{ind}(B_0^{\perp },B_1^{\perp })=-{\int }_{M_0}\hat{\mathrm{A}}(\nabla )\wedge \mathrm{ch}({\nabla }^E)-{\int }_{\partial M_0}\mathcal{T}+\frac{h(A_0)+h(A_1)+\eta (A_0)-\eta (A_1)}{2}.$$

Full text

1. Introduction

The Atiyah-Singer index theorem [AS63] for elliptic operators on closed manifolds is one of the central mathematical discoveries of the 20th century. It contains famous classical results such as the Gauss-Bonnet theorem, the Riemann-Roch theorem or Hirzebruch’s signature theorem as special cases and has numerous applications in analysis, geometry, topology, and mathematical physics. For instance, it has been used in [L63] to obtain a topological obstruction to the existence of metrics with positive scalar curvature and a refinement of the index theorem was employed in [H74] to show that on many manifolds a change of metric in a neighborhood of a point will alter the dimension of the space of harmonic spinors. This contrasts with the space of harmonic forms whose dimensions are given topologically by the Betti numbers.

The index theorem for compact manifolds with boundary by Atiyah, Patodi, and Singer [APS75] requires the introduction of suitable nonlocal boundary conditions which are based on the spectral decomposition of the operator induced on the boundary. An exposition of the most general boundary conditions which one can impose in order to obtain a Fredholm operator can be found e.g. in [BB16].

While an analog of the Atiyah-Singer index theorem for Lorentzian manifolds is unknown and not to be expected, one for Lorentzian manifolds with spacelike boundary has been found recently [BS15]. More precisely, we consider the (twisted) Dirac operator on a spatially compact globally hyperbolic manifold. It is supposed to have boundary consisting of two disjoint smooth spacelike Cauchy hypersurfaces. The Dirac operator is now hyperbolic rather than elliptic but the operator induced on the boundary is still selfadjoint and elliptic so that Atiyah-Patodi-Singer boundary conditions still make sense. It was shown in [BS15] that under these boundary conditions the Dirac operator becomes Fredholm and its index is given formally by the same geometric expression as in the Riemannian case. As an application the chiral anomaly in algebraic quantum field theory on curved spacetimes was computed in [BS16].

In the present paper we investigate more general boundary conditions which turn the hyperbolic Dirac operator into a Fredholm operator. There are similarities and differences to the Riemannian case. It was already observed in [BS16] that the boundary conditions complementary to the APS boundary conditions, the anti-Atiyah-Patodi-Singer boundary conditions, also give rise to a Fredholm operator. This is false in the Riemannian case. On the other hand, we show that the conditions described in [BB12, BB16] for the Riemannian case work in the Lorentzian setting only under an additional assumption. One can think of these boundary conditions as graph deformations of the APS boundary conditions plus finite dimensional modifications. The finite dimensional modifications work just the same in the Lorentzian setting but the deformations need to be small, either in the sense that the linear maps whose graphs we are considering are compact or that they have sufficiently small norm. We show by example that these conditions cannot be dropped.

The paper is organized as follows. In the next section we summarize what we need to know about Dirac operators on Lorentzian manifolds. The most important fact is well-posedness of the Cauchy problem. In the third section we discuss some functional-analytic topics concerning Fredholm pairs. In the last section we combine everything and consider various examples of boundary conditions giving rise to Fredholm operators and compute their index.

2. The Dirac operator on Lorentzian manifolds

We collect a few standard facts on Dirac operators on Lorentzian manifolds. For a more detailed introduction to Lorentzian geometry see e.g. [BEE96, ON83], for Dirac operators on semi-Riemannian manifolds see [BGM05, B81].

2.1. Globally hyperbolic manifolds

Suppose that $M$ is an $(n+1)$-dimensional oriented time-oriented Lorentzian spin manifold with $n$ odd. We use the convention that the metric of $M$ has signature $(-+\cdots+)$.

A subset $\Sigma\subset M$ is called a Cauchy hypersurface if every inextensible timelike curve in $M$ meets $\Sigma$ exactly once. If $M$ possesses a Cauchy hypersurface then $M$ is called globally hyperbolic. All Cauchy hypersurfaces of $M$ are homeomorphic. We assume that $M$ is spatially compact, i.e. the Cauchy hypersurfaces of $M$ are compact.

If $\Sigma_{0},\Sigma_{1}\subset M$ are two disjoint smooth and spacelike Cauchy hypersurfaces with $\Sigma_{0}$ lying in the past of $\Sigma_{1}$ then $M$ can be written as

[TABLE]

such that $\Sigma_{0}=\{0\}\times\Sigma$, $\Sigma_{1}=\{1\}\times\Sigma$ and each $\Sigma_{t}=\{t\}\times\Sigma$ is a smooth spacelike Cauchy hypersurface. The metric of $M$ then takes the form $\langle\cdot,\cdot\rangle=-N^{2}\,dt^{2}+g_{t}$ where $N:M\to\mathbb{R}$ is a smooth positive function (the lapse function) and $g_{t}$ is a smooth $1$-parameter family of Riemannian metrics on $\Sigma$, see [BS06, Thm. 1.2] (and also [M16, Thm. 1]).

2.2. Spinors

Let $SM\to M$ be the complex spinor bundle on $M$ endowed with its invariantly defined indefinite inner product $(\cdot,\cdot)$. Denote Clifford multiplication with $X\in T_{p}M$ by $\gamma(X):S_{p}M\to S_{p}M$. It satisfies

[TABLE]

and

[TABLE]

for all $X,Y\in T_{p}M$, $u,v\in S_{p}M$ and $p\in M$.

Let $e_{0},e_{1},\ldots,e_{n}$ be a positively oriented Lorentz-orthonormal tangent frame. Then Clifford multiplication with the volume form $\Gamma=\mathrm{i}^{n(n+3)/2}\,\gamma(e_{0})\cdots\gamma(e_{n})$ satisfies $\Gamma^{2}=\mathrm{id}_{SM}$. This induces the eigenspace decomposition $SM=S^{R}M\oplus S^{L}M$ for the eigenvalues $\pm 1$ into right-handed and left-handed spinors. Since the dimension of $M$ is even, $\Gamma\gamma(X)=-\gamma(X)\Gamma$ for all $X\in TM$. In particular, $S^{R}M$ and $S^{L}M$ have equal rank and Clifford multiplication by tangent vectors reverses handedness.

Now let $\Sigma\subset M$ be a smooth spacelike hypersurface. Denote by $\nu$ be the past-directed timelike vector field on $M$ along $\Sigma$ with $\langle\nu,\nu\rangle\equiv-1$ which is perpendicular to $\Sigma$. The restriction of $S^{R}M$ or $S^{L}M$ to $\Sigma$ can be naturally identified with the spinor bundle of $\Sigma$, i.e. $S^{R}M|_{\Sigma}=S^{L}M|_{\Sigma}=S\Sigma$. The spinor bundle of $\Sigma$ carries a natural positive definite scalar product $\langle\cdot,\cdot\rangle$ induced by the Riemannian metric of $\Sigma$. The two inner products are related by $\langle\cdot,\cdot\rangle=(\gamma(\nu)\cdot,\cdot)$.

Clifford multiplication $\gamma_{\Sigma}(X)$ on $S\Sigma$ corresponds to $\mathrm{i}\gamma(\nu)\gamma(X)$ under this identification. Note that $\gamma(\nu)^{2}=\mathrm{id}$. Clifford multiplication on $\Sigma$ is skew-adjoint because

[TABLE]

2.3. The Dirac operator

Let $E\to M$ be a Hermitian vector bundle with a compatible connection $\nabla^{E}$. Sections of the vector bundles $V^{R}:=S^{R}M\otimes E$ and $V^{L}:=S^{L}M\otimes E$ are called right-handed (resp. left-handed) twisted spinors (or spinors with coefficients in $E$). The inner product $(\cdot,\cdot)$ on $S^{R}M$ and the scalar product on $E$ induce an (indefinite) inner product on $V^{R}$, again denoted by $(\cdot,\cdot)$. When restricted to a spacelike hypersurface the scalar product $\langle\cdot,\cdot\rangle$ on $S^{R}M$ and the one on $E$ induce a (positive definite) scalar product on $V^{R}$, again denoted by $\langle\cdot,\cdot\rangle$.

Let $D:C^{\infty}(M;V^{R})\to C^{\infty}(M;V^{L})$ be the Dirac operator acting on right-handed twisted spinors. In terms of a local Lorentz-orthonormal tangent frame $e_{0},e_{1},\ldots,e_{n}$ this operator is given by

[TABLE]

where $\varepsilon_{j}=\langle e_{j},e_{j}\rangle=\pm 1$ and $\nabla$ is the connection on $V^{R}$ induced by the Levi-Civita connection on $S^{R}M$ and $\nabla^{E}$. The Clifford multiplication $\gamma(X)$ of a tangent vector $X\in T_{p}M$ on a twisted spinor $\phi\otimes e\in V^{R}_{p}=S^{R}_{p}M\otimes E_{p}$ is to be understood as acting on the first factor, $\gamma(X)(\phi\otimes e)=(\gamma(X)\phi)\otimes e$. The Dirac operator is a hyperbolic linear differential operator of first order.

Along a smooth spacelike hypersurface $\Sigma\subset M$ with past-directed unit normal field $\nu$ the Dirac operator can be written as

[TABLE]

where $H$ is the mean curvature of $\Sigma$ with respect to $\nu$ and $A_{\Sigma}$ is the elliptic twisted Dirac operator of the Riemannian manifold $\Sigma$.

2.4. The Cauchy problem

Now we fix two smooth spacelike Cauchy hypersurfaces $\Sigma_{1}$ and $\Sigma_{0}$ in $M$. We assume that $\Sigma_{0}$ lies in the chronological past of $\Sigma_{1}$. Then we consider the region $M_{0}$ “in between” $\Sigma_{0}$ and $\Sigma_{1}$, more precisely, $M_{0}=J^{+}(\Sigma_{0})\cap J^{-}(\Sigma_{1})$ where $J^{-}$ and $J^{+}$ denote the causal past and future, respectively. Since $M$ is spatially compact, the region $M_{0}$ is a compact manifold with boundary, the boundary being the disjoint union of $\Sigma_{0}$ and $\Sigma_{1}$.

For any compact spacelike hypersurface $\Sigma\subset M$ we define the $L^{2}$-scalar product for $u,v\in C^{\infty}(\Sigma,V^{R})$ by

[TABLE]

where $\mathrm{dA}$ denotes the volume element of $\Sigma$ induced by its Riemannian metric. Recall that the inner product $\langle\cdot,\cdot\rangle$ on $V^{R}$ is positive definite. The completion of $C^{\infty}(\Sigma,V^{R})$ w.r.t. the $L^{2}$-norm will be denoted by $L^{2}(\Sigma,V^{R})$.

Similarly, using an auxiliary positive definite scalar product on $V^{L}$ we can define $L^{2}(M_{0},V^{L})$ where we integrate against the volume element induced by the Lorentzian metric on $M$. By compactness of $M_{0}$ different choices of auxiliary scalar products on $V^{L}$ will give rise to equivalent $L^{2}$-norms. Hence $L^{2}(M_{0},V^{L})$ is unanimously defined as a topological vector space.

Finally, we complete $C^{\infty}(M_{0},V^{R})$ w.r.t. the $L^{2}$-graph-norm of $D$

[TABLE]

and obtain the “finite-energy” space ${FE}(M_{0},V^{R})$.

Now the Dirac operator obviously extends to a bounded operator $D:{FE}(M_{0},V^{R})\to L^{2}(M_{0},V^{L})$. It can be checked that the restriction map $C^{\infty}(M_{0},V^{R})\to C^{\infty}(\Sigma,V^{R})$ extends uniquely to a bounded operator $\mathrm{res}_{\Sigma}:{FE}(M_{0},V^{R})\to L^{2}(\Sigma,V^{R})$ if $\Sigma$ is a spacelike Cauchy hypersurface.

In these function spaces the Cauchy problem is well posed:

Theorem 2.1**.**

Let $\Sigma\subset M_{0}$ be a smooth spacelike Cauchy hypersurface. Then the mapping

[TABLE]

is an isomorphism of Hilbert spaces.∎

In particular, we get well-posedness of the Cauchy problem for the homogeneous Dirac equation:

Corollary 2.2**.**

For any smooth spacelike Cauchy hypersurface $\Sigma\subset M_{0}$ the restriction mapping

[TABLE]

is an isomorphism of Hilbert spaces.∎

For details see [BS15].

2.5. The wave evolution

Applying Corollary 2.2 to $\Sigma=\Sigma_{0}$ and to $\Sigma=\Sigma_{1}$ we can define the wave evolution operator

[TABLE]

by the commutative diagram

[TABLE]

By construction, $Q$ is an isomorphism. One can check that $Q$ is actually unitary, that it restricts to isomorphisms $H^{s}(\Sigma_{0},V^{R})\to H^{s}(\Sigma_{1},V^{R})$ for all $s>0$ and that it extends to isomorphisms $H^{s}(\Sigma_{0},V^{R})\to H^{s}(\Sigma_{1},V^{R})$ for all $s<0$. Here $H^{s}$ denote the corresponding Sobolev spaces. As a consequence, $Q$ maps $C^{\infty}(\Sigma_{0},V^{R})$ to $C^{\infty}(\Sigma_{1},V^{R})$.

In fact, well-posedness of the Dirac equation also holds for smooth sections, i.e.,

[TABLE]

is an isomorphism of Fréchet spaces.

3. Fredholm pairs

In this section we collect a few functional-analytic facts which will be useful later.

Definition 3.1**.**

Let $H$ be a Hilbert space and let $B_{0},B_{1}\subset H$ be closed linear subspaces. Then $(B_{0},B_{1})$ is called a Fredholm pair if $B_{0}\cap B_{1}$ is finite dimensional and $B_{0}+B_{1}$ is closed and has finite codimension. The number

[TABLE]

is called the index of the pair $(B_{0},B_{1})$.

We list a few elementary properties of Fredholm pairs. For details see [K95, Ch. IV, § 4].

Remark 3.2**.**

1. 1.)

The pair $(B_{0},B_{1})$ is Fredholm if and only if $(B_{1},B_{0})$ is a Fredholm pair and in this case

[TABLE] 2. 2.)

The pair $(B_{0},B_{1})$ is Fredholm if and only if $(B_{0}^{\perp},B_{1}^{\perp})$ is a Fredholm pair and in this case

[TABLE] 3. 3.)

Let $B_{0}^{\prime}\subset H$ be a closed linear subspace with $B_{0}\subset B_{0}^{\prime}$ and $\dim(B_{0}^{\prime}/B_{0})<\infty$. Then $(B_{0},B_{1})$ is a Fredholm pair if and only if $(B_{0}^{\prime},B_{1})$ is a Fredholm pair and in this case

[TABLE]

The following lemma reformulates the concept of Fredholm pairs in terms of orthogonal projections. For a proof see e.g. [BW93, Lemma 24.3]. Here and henceforth, the orthogonal projection onto a closed subspace $V$ of a Hilbert space $H$ will be denoted by

[TABLE]

Lemma 3.3**.**

Let $B_{0},B_{1}\subset H$ be closed linear subspaces. Then $(B_{0},B_{1})$ is a Fredholm pair of index $k$ if and only if

[TABLE]

is a Fredholm operator of index $k$. In this case we have \ker\big{(}\pi_{B_{1}^{\perp}}|_{B_{0}}\big{)}=B_{0}\cap B_{1} and \mathrm{coker}\big{(}\pi_{B_{1}^{\perp}}|_{B_{0}}\big{)}\cong B_{0}^{\perp}\cap B_{1}^{\perp}.∎

Let $E$, $F$, $H_{0}$, and $H_{1}$ be Hilbert spaces and let $L:E\to F$, $r_{0}:E\to H_{0}$, and $r_{1}:E\to H_{1}$ be bounded linear maps. We assume that

[TABLE]

is an isomorphism for $j=0$ and for $j=1$. Then $r_{j}$ restricts to an isomorphism $\ker(L)\to H_{j}$. We define the isomorphism $Q:H_{0}\to H_{1}$ by the commutative diagram

[TABLE]

Proposition 3.4**.**

Assume that $r_{0}\oplus r_{1}:E\to H_{0}\oplus H_{1}$ is onto. Let $B_{j}\subset H_{j}$ be closed linear subspaces. Then the following are equivalent:

1. (i)

The pair $(B_{0},Q^{-1}B_{1})$ is Fredholm of index $k$; 2. (ii)

The pair $(QB_{0},B_{1})$ is Fredholm of index $k$; 3. (iii)

The operator $(\pi_{B_{0}^{\perp}}\circ r_{0})\oplus(\pi_{B_{1}^{\perp}}\circ r_{1})\oplus L:E\to B_{0}^{\perp}\oplus B_{1}^{\perp}\oplus F$ is Fredholm of index $k$; 4. (iv)

The restriction $L:\ker(\pi_{B_{0}^{\perp}}\circ r_{0})\cap\ker(\pi_{B_{1}^{\perp}}\circ r_{1})\to F$ is a Fredholm operator of index $k$.

Proof.

a) The equivalence of (i) and (ii) is clear because $Q:H_{0}\to H_{1}$ is an isomorphism.

b) Proposition A.1.(iv) in [BB12] with $P=L$ states that (iii) is equivalent to the operator

[TABLE]

being Fredholm of index $k$. Since $r_{1}|_{\ker L}=Q\circ r_{0}|_{\ker L}$ this is again equivalent to

[TABLE]

being Fredholm of index $k$. With respect to the splittings $\widetilde{L}:H_{0}=B_{0}^{\perp}\oplus B_{0}\to B_{0}^{\perp}\oplus B_{1}^{\perp}$ takes the operator matrix form

[TABLE]

Now $\widetilde{L}$ is Fredholm if and only if it is invertible modulo compact operators. This is equivalent to $\pi_{B_{1}^{\perp}}Q|_{B_{0}}$ being invertible modulo compact operators, i.e. being Fredholm. By deformation invariance of the index we have

[TABLE]

This shows that (iii) is equivalent to $\pi_{B_{1}^{\perp}}|_{QB_{0}}$ being a Fredholm operator of index $k$. The equivalence of (ii) and (iii) now follows with Lemma 3.3.

c) The equivalence of (iii) and (iv) is Proposition A.1.(iv) in [BB12] with

[TABLE]

Note that $P$ is onto because $r_{0}\oplus r_{1}:E\to H_{0}\oplus H_{1}$ is onto by assumption. ∎

4. Boundary value problems for the Dirac operator

We return to our twisted Dirac operator $D$ on a Lorentzian manifold as introduced in Section 2. Let $B_{0}\subset L^{2}(\Sigma_{0},V^{R})$ and $B_{1}\subset L^{2}(\Sigma_{1},V^{R})$ be closed subspaces. Denote by $Q:L^{2}(\Sigma_{0},V^{R})\to L^{2}(\Sigma_{1},V^{R})$ the wave evolution operator as defined in (3). Combining Theorem 2.1 and Proposition 3.4 we get

Theorem 4.1**.**

The following are equivalent:

1. (i)

The pair $(B_{0},Q^{-1}B_{1})$ is Fredholm of index $k$; 2. (ii)

The pair $(QB_{0},B_{1})$ is Fredholm of index $k$; 3. (iii)

The operator

[TABLE]

is Fredholm of index $k$; 4. (iv)

The restriction

[TABLE]

is a Fredholm operator of index $k$.∎

Definition 4.2**.**

If these conditions hold then we call $(B_{0},B_{1})$ a Dirac-Fredholm pair of index $k$.

Condition (iv) in Theorem 4.1 means that we consider the Dirac equation $Du=f$ subject to the boundary conditions $u|_{\Sigma_{0}}\in B_{0}$ and $u|_{\Sigma_{1}}\in B_{1}$. We now look at concrete examples.

4.1. (Anti) Atiyah-Patodi-Singer boundary conditions

For any subset $I\subset\mathbb{R}$ denote by $\chi_{I}:\mathbb{R}\to\mathbb{R}$ its characteristic function. Denote the Riemannian Dirac operators for the boundary parts by $A_{0}:=A_{\Sigma_{0}}$ and $A_{1}:=A_{\Sigma_{1}}$, compare (2). Since $A_{0}$ and $A_{1}$ are self-adjoint elliptic operators on closed Riemannian manifolds they have real and discrete spectrum. We consider the spectral projector $\chi_{I}(A_{0}):L^{2}(\Sigma_{0},V^{R})\to L^{2}(\Sigma_{0},V^{R})$ and similarly for $A_{1}$. This is the orthogonal projector onto the sum of the $A_{0}$-eigenspaces to all eigenvalues contained in $I$. Its range will be denoted by $L^{2}_{I}(A_{0}):=\chi_{I}(A_{0})(L^{2}(\Sigma_{0},V^{R}))\subset L^{2}(\Sigma_{0},V^{R})$ and similarly for $A_{1}$.

Now the Atiyah-Patodi-Singer boundary conditions correspond to the choice $B_{0}=L^{2}_{(-\infty,0)}(A_{0})$ and $B_{1}=L^{2}_{(0,\infty)}(A_{1})$. It was shown in [BS15, Thm. 7.1] that the Dirac operator subject to these boundary conditions is Fredholm. In other words, the pair $(B_{0},B_{1})=(L^{2}_{(-\infty,0)}(A_{0}),L^{2}_{(0,\infty)}(A_{1}))$ is Fredholm and its index is given by

[TABLE]

Here ${\widehat{\mathrm{A}}}(\nabla)$ is the ${\widehat{\mathrm{A}}}$-form built from the curvature of the Levi-Civita connection $\nabla$ on $M$ and $\mathrm{ch}(\nabla^{E})$ is the Chern character form of the curvature of $\nabla^{E}$. The form $\mathcal{T}$ is the corresponding transgression form. In particular, the boundary integral vanishes if the given metric and connections have product structure near the boundary. Moreover, $h(A)$ denotes the dimension of the kernel of an operator $A$ and $\eta(A)$ its $\eta$-invariant. See [BS15] for details.

By Remark 3.2.2 the complementary boundary conditions $(\mathrm{APS}_{0}(0)^{\perp},\mathrm{APS}_{1}(0)^{\perp})=(L^{2}_{[0,\infty)}(A_{0}),L^{2}_{(-\infty,0]}(A_{1}))$, the anti-Atiyah-Patodi-Singer boundary conditions, are also Fredholm and the index has the opposite sign,

[TABLE]

4.2. Generalized Atiyah-Patodi-Singer boundary conditions

For $a\in\mathbb{R}$ we define $\mathrm{APS}_{0}(a):=L^{2}_{(-\infty,a)}(A_{0})\subset L^{2}(\Sigma_{0},V^{R})$ and $\mathrm{APS}_{1}(a):=L^{2}_{(a,\infty)}(A_{1})\subset L^{2}(\Sigma_{1},V^{R})$. Since cutting the spectrum of the boundary operator at [math] is somewhat arbitrary we may want to fix $a_{0},a_{1}\in\mathbb{R}$ and consider the boundary conditions $(B_{0},B_{1})=(\mathrm{APS}_{0}(a_{0}),\mathrm{APS}_{1}(a_{1}))$. These boundary conditions are known as generalized Atiyah-Patodi-Singer boundary conditions. Since all eigenvalues of the Riemannian Dirac operators $A_{\Sigma_{i}}$ are of finite multiplicity, these boundary conditions differ from the Atiyah-Patodi-Singer boundary conditions only by finite dimensional spaces, more precisely,

[TABLE]

and similarly for $\mathrm{APS}_{1}$. Remark 3.2.3 implies that generalized Atiyah-Patodi-Singer boundary conditions also form a Dirac-Fredholm pair. Setting

[TABLE]

we get for their index

[TABLE]

In other words, the correction terms in the index formula are given by the total multiplicity of the eigenvalues of $A_{j}$ between [math] and $a_{j}$.

4.3. Boundary conditions in graph form

In the previous section we discussed certain finite dimensional modifications of the Atiyah-Patodi-Singer boundary conditions, which lead to corrections in the index formula. Now we introduce continuous deformations of Dirac-Fredholm pairs, leaving the index unchanged. Formally, the definition coincides with that of $D$-elliptic boundary conditions for the elliptic Dirac operator on Riemannian manifolds as introduced in [BB12, Def. 7.5 and Thm. 7.11].

Definition 4.3**.**

We call a pair $(B_{0},B_{1})$ of closed subspaces $B_{i}\subset L^{2}(\Sigma_{i},V^{R})$ boundary conditions in graph form if there are $L^{2}$-orthogonal decompositions

[TABLE]

such that

1. (i)

$W_{i}^{+},W_{i}^{-}$ are finite dimensional; 2. (ii)

$W_{i}^{-}\oplus V_{i}^{-}=L^{2}_{(-\infty,a_{i})}(A_{i})$ and $W_{i}^{+}\oplus V_{i}^{+}=L^{2}_{[a_{i},\infty)}(A_{i})$ for some $a_{i}\in\mathbb{R}$; 3. (iii)

There are bounded linear maps $g_{0}:V_{0}^{-}\to V_{0}^{+}$ and $g_{1}:V_{1}^{+}\to V_{1}^{-}$ such that

[TABLE]

where $\Gamma(g_{0/1}):=\{v+g_{0/1}v\hskip 2.0pt|\hskip 2.0ptv\in V_{0/1}^{\mp}\}$ denotes the graph of $g_{0/1}$.

Remark 4.4**.**

In the setting of Definition 4.3 we have

[TABLE]

where both decompositions are orthogonal. With respect to the splitting $V_{i}^{-}\oplus V_{i}^{+}$ the projections onto $\Gamma(g_{i})$ are given by

[TABLE]

see [BB12, Lemma 7.7 and Remark 7.8]. Here $g_{i}^{\dagger}:V_{i}^{\pm}\to V_{i}^{\mp}$ denotes the adjoint linear map.

The next Lemma shows that deforming Atiyah-Patodi-Singer boundary conditions for the Dirac operator to a graph preserves Fredholm property as well as the index.

Proposition 4.5**.**

Let $a_{0},a_{1}\in\mathbb{R}$ be given. Then there exists an $\varepsilon>0$ such that for any bounded linear maps $g_{i}:\mathrm{APS}_{i}(a_{i})\to\mathrm{APS}_{i}(a_{i})^{\perp}$ the pair $(\Gamma(g_{0}),\Gamma(g_{1}))$ is a Dirac-Fredholm pair of the same index as $(\mathrm{APS}_{0}(a_{0}),\mathrm{APS}_{1}(a_{1}))$ provided

1. (A)

$g_{0}$* or $g_{1}$ is compact or* 2. (B)

$\|g_{0}\|\cdot\|g_{1}\|<\varepsilon$.

Proof.

We put $\tilde{g}_{1}:=Q^{-1}\circ g_{1}\circ Q$. Now $(\Gamma(g_{0}),\Gamma(g_{1}))$ is a Dirac-Fredholm pair of index $k$ if and only if $(\Gamma(g_{0}),Q^{-1}\Gamma(g_{1}))=(\Gamma(g_{0}),\Gamma(\tilde{g}_{1}))$ is a Fredholm pair of index $k$. By Lemma 3.3 this is equivalent to

[TABLE]

being a Fredholm operator of index $-k$. Since the maps $(\mathrm{id}+g_{i}):\mathrm{APS}_{i}(a_{i})\to\Gamma(g_{i})$ and $(\mathrm{id}-g_{i}^{\dagger}):\mathrm{APS}_{i}(a_{i})^{\perp}\to\Gamma(-g_{i}^{\dagger})$ are isomorphisms, this is equivalent to

[TABLE]

being Fredholm of index $-k$. We write $B_{0}:=\mathrm{APS}_{0}(a_{0})$ and $B_{1}:=Q^{-1}\mathrm{APS}_{1}(a_{1})$. Using (6) we see that $L$ takes the form

[TABLE]

The operators $\alpha$ and $\delta$ are compact by [BS16, Lemma 2.6]. If $g_{0}$ or $g_{1}$ (and hence $g_{0}^{\dagger}$ or $\tilde{g}_{1}^{\dagger}$) is compact then $\gamma$ is compact as well. Since $\beta=\pi_{B_{0}}|_{B_{1}^{\perp}}$ is a Fredholm operator of index $\mathrm{ind}(B_{0}^{\perp},B_{1}^{\perp})=-\mathrm{ind}(B_{0},B_{1})$ the same is true for $L$.

If $\|g_{0}\|\cdot\|g_{1}\|<\varepsilon$ then

[TABLE]

is small so that $\beta-\gamma$ (and hence $L$) is again a Fredholm operator of the same index as $\beta$. ∎

Remark 4.6**.**

In Example 4.11 we will see that conditions (A) and (B) cannot be dropped. Without these assumptions boundary conditions in graph form do not give rise to a Fredholm operator in general.

Remark 4.7**.**

Proposition 4.5 can be applied in particular if $g_{0}=0$ or $g_{1}=0$. Thus in the setting of the lemma $(\Gamma(g_{0}),\mathrm{APS}_{1}(a_{1}))$ and $(\mathrm{APS}_{0}(a_{0}),\Gamma(g_{1}))$ are Dirac-Fredholm pairs of index $\mathrm{ind}(\mathrm{APS}_{0}(a_{0}),\mathrm{APS}_{1}(a_{1}))$.

If $g:B\to B^{\perp}$ is a bounded linear map and $\widetilde{B}\subset B$, then $\Gamma(g)/\Gamma(g|_{\widetilde{B}})\cong B/\widetilde{B}$. Combining Section 4.2 and Proposition 4.5 we get the following result.

Corollary 4.8**.**

Let $(B_{0},B_{1})$ be boundary conditions in graph form with $g_{0}$ or $g_{1}$ compact or $\|g_{0}\|\cdot\|g_{1}\|$ sufficiently small. Then $(B_{0},B_{1})$ is a Dirac-Fredholm pair and its index is given by

[TABLE]

4.4. Local boundary conditions

Suppose we have subbundles $E_{j}\subset V^{R}|_{\Sigma_{j}}$. Then the boundary condition $B_{j}=L^{2}(\Sigma_{j},E_{j})\subset L^{2}(\Sigma_{j},V^{R})$ is called a local boundary condition. The following Lemma relates local boundary conditions to graph deformations of generalized Atiyah-Patodi-Singer boundary conditions as discussed in Section 4.3.

Proposition 4.9**.**

Let $E_{j}\subset V^{R}|_{\Sigma_{j}}$ be a subbundle and $B_{j}=L^{2}(\Sigma_{j},E_{j})$. Then the following are equivalent:

1. (i)

There exists an $L^{2}$-orthogonal splitting

[TABLE]

and bounded linear maps $g_{j}:V_{j}^{\mp}\to V_{j}^{\pm}$ as in Definition 4.3 such that $B_{j}=W_{j}^{\pm}\oplus\Gamma(g_{j})$; 2. (ii)

For every $x\in\Sigma_{j}$ and $\xi\in T_{x}\Sigma_{j}$, $\xi\neq 0$, the projection $\pi_{(E_{j})_{x}}:(V^{R}|_{\Sigma_{j}})_{x}\to(E_{j})_{x}$ restricts to an isomorphism from the sum of eigenspaces to the negative (for $j=0$) or positive (for $j=1$) eigenvalues of $\mathrm{i}\sigma_{A_{j}}(\xi)$ onto $(E_{j})_{x}$.

Proof.

First note that the fiberwise projections $\pi_{(E_{j})_{x}}$ yield a projection map

[TABLE]

Then (ii) is equivalent to the operator

[TABLE]

being Fredholm for some (and then all) $a\in\mathbb{R}$, as stated in [BB12, Thm. 7.20].

First we show that (i) implies (ii). Since the sum of a Fredholm operator and a finite rank operator is again Fredholm, we can assume that $W_{j}^{-}=W_{j}^{+}=0$, $V_{j}^{-}=\mathrm{APS}_{j}(0)$ and $V_{j}^{+}=\mathrm{APS}_{j}(0)^{\perp}$. With respect to the splitting $L^{2}(\Sigma_{j},V^{R})=V_{j}^{-}\oplus V_{j}^{+}$ we then have by (6)

[TABLE]

and hence

[TABLE]

is an isomorphism.

Now we show that (ii) implies (i). We construct the decomposition in (i) and the map $g_{j}$ for $j=0$. The case $j=1$ is analogous with the labels $+$ and $-$ interchanged.

Assuming that $P-\pi_{\mathrm{APS}_{0}(0)^{\perp}}$ is Fredholm we set

[TABLE]

We then have $V_{0}^{+}\oplus W_{0}^{+}=L^{2}_{[0,\infty)}(\Sigma_{0},V^{R})$ and $V_{0}^{-}\oplus W_{0}^{-}=L^{2}_{(-\infty,0)}(\Sigma_{0},V^{R})$. Furthermore, for $w\in W_{0}^{+}$ we have

[TABLE]

so $W_{0}^{+}$ is contained in the kernel of $P_{0}-\pi_{\mathrm{APS}_{0}(0)^{\perp}}$ and is hence finite dimensional. For $w\in W_{0}^{-}$ we observe

[TABLE]

for all $x\in L^{2}(\Sigma_{0},V^{R})$. Thus $W_{0}^{-}$ is contained in the orthogonal complement of the range of $P_{0}-\pi_{\mathrm{APS}_{0}(0)^{\perp}}$ and is hence finite dimensional.

Setting $U_{0}:=(W_{0}^{+})^{\perp}\cap B_{0}$ we have $B_{0}=W_{0}^{+}\oplus U_{0}$ and $\pi_{\mathrm{APS}_{0}(0)}|_{U_{0}}:U_{0}\to V_{0}^{-}$ is an isomorphism. Now the composition

[TABLE]

is a bounded linear map with $U_{0}=\Gamma(g_{0})$. Then we have $B_{0}=W_{0}^{+}\oplus\Gamma(g_{0})$ which concludes the proof. ∎

Example 4.10**.**

Let $G$ be a smooth field of unitary involutions of $V^{R}$ along $\Sigma_{0}\cup\Sigma_{1}$ which anti-commutes with $A_{j}$, $j=0,1$. Then $G$ anti-commutes with all $\sigma_{A_{j}}(\xi)$, $\xi\in T^{*}\Sigma_{0}\cup T^{*}\Sigma_{1}$. We split $V^{R}|_{\Sigma_{j}}=E_{j}^{+}\oplus E_{j}^{-}$ into the sum of $(\pm 1)$-eigenspaces for $G$.

Let $\xi\in T_{x}\Sigma_{j}$, $\xi\neq 0$. W.l.o.g. we assume $|\xi|=1$. Then $(\mathrm{i}\sigma_{A_{j}}(\xi))^{2}=1$ and $\mathrm{i}\sigma_{A_{j}}(\xi)$ has only the eigenvalues $\pm 1$. Since $G$ and $\sigma_{A_{j}}(\xi)$ anti-commute $(E_{j}^{+})_{x}$, $(E_{j}^{-})_{x}$, and the $(\pm 1)$-eigenspaces for $\mathrm{i}\sigma_{A_{j}}(\xi)$ all have the same dimension, namely half the dimension of $(V^{R})_{x}$. The projections onto $E_{j}^{\pm}$ are given by $\pi_{(E_{j}^{\pm})_{x}}=\frac{1}{2}(1\pm G)$.

Now assume that $\mathrm{i}\sigma_{A_{j}}(\xi)u=-u$ and that $\pi_{(E_{j}^{\pm})_{x}}u=0$. Then

[TABLE]

and hence

[TABLE]

Thus both $\pi_{(E_{j}^{+})_{x}}$ and $\pi_{(E_{j}^{-})_{x}}$ are injective when restricted to the $(-1)$-eigenspace of $\mathrm{i}\sigma_{A_{j}}(\xi)$. For dimensional reasons these restrictions form an isomorphism onto $(E_{j}^{\pm})_{x}$. Thus Proposition 4.9 applies to both $E_{j}=E_{j}^{+}$ and $E_{j}=E_{j}^{-}$. Hence $(L^{2}(\Sigma_{0},E_{0}^{\pm}),L^{2}(\Sigma_{1},E_{1}^{\pm}))$ are boundary conditions in graph form. These boundary conditions are known as chirality conditions.

The description of the chirality boundary conditions in Definition 4.3 is as follows: $V_{0}^{-}=L^{2}_{(-\infty,0)}(A_{0})$, $W_{0}^{+}=\ker(A_{0})\cap L^{2}(\Sigma_{0},E_{0}^{\pm})$, $V_{0}^{+}=L^{2}_{(0,\infty)}(A_{0})$, $W_{0}^{-}=\ker(A_{0})\cap L^{2}(\Sigma_{0},E_{0}^{\mp})$, $g_{0}=\pm G$ and similarly for $\Sigma_{1}$.

In order to conclude from Lemma 4.5 that the chirality conditions yield a Fredholm operator one would need to verify condition (A) or (B). Condition (A) is not satisfied but for (B) this is not clear. So let us look at a special case.

Example 4.11**.**

Let $h$ be a fixed Riemannian metric on the closed spin manifold $\Sigma$ and equip $M=\mathbb{R}\times\Sigma$ with the “ultrastatic” metric $-dt^{2}+h$. Then (2) simplifies to

[TABLE]

where $A$ is the Dirac operator on $(\Sigma,h)$. If we now solve the equation $Du=0$ with initial condition $u(0,\cdot)=u_{0}$ where $u_{0}$ is an eigenspinor for $A$, $Au_{0}=\lambda u_{0}$, then the solution is given by $u(t,x)=e^{\mathrm{i}\lambda t}u_{0}(x)$. We choose $\Sigma_{j}=\{j\}\times\Sigma$. Then $Qu_{0}=e^{\mathrm{i}\lambda}u_{0}$.

Specializing even further, we put $\Sigma=S^{1}$ and choose $h$ such that $\Sigma$ has length $1$. The $1$-dimensional sphere has two spin structures. For the so-called trivial spin structure the Dirac operator $A$ has the eigenvalues $\lambda=2\pi k$ where $k\in\mathbb{Z}$. Thus $e^{\mathrm{i}\lambda}=1$ and hence $Q=\mathrm{id}$ where we have identified $L^{2}(\Sigma_{0},V^{R})=L^{2}(S^{1},SS^{1})=L^{2}(\Sigma_{1},V^{R})$ and $A_{0}=A=A_{1}$. Now choose

[TABLE]

Let $g_{0}:L^{2}_{(-\infty,0)}(A)\to L^{2}_{(0,\infty)}(A)$ be an isomorphism of Hilbert spaces and put $g_{1}=g_{0}^{-1}$. Then

[TABLE]

Now $(B_{0},Q^{-1}B_{1})=(B_{0},B_{1})=(B_{0},B_{0})$ is clearly not a Fredholm pair. This shows that in Proposition 4.5 conditions (A) and (B) cannot be dropped. Without these assumptions boundary conditions in graph form do not give rise to a Fredholm operator in general.

Example 4.12**.**

Finally, we want to point out that one also obtains Dirac-Fredholm pairs $(B_{0},B_{1})$ if $B_{0}$ is finite dimensional and $B_{1}$ has finite codimension or vice versa. According to Theorem 4.1 the Dirac operator with these boundary conditions is Fredholm with index $\dim(B_{0})-\mathrm{codim}(B_{1})$.