The Penrose transform in the language of sheaves and D-modules

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The Penrose transform in the language of sheaves and D-modules

Andrea D’Agnolo (after a joint work with Pierre Schapira)

Abstract

The methods of D-module and sheaf theory are applied here to the study of integral transforms, such as the twistor correspondence of Penrose. This approach has the advantage of distinguishing the topological obstructions from the analytic ones. A section devoted to a short review of the classical paper [4] shows how this is effective.

AMS subject classification: 35C15, 32L25.

1 Historical introduction

Let TT be a complex four-dimensional vector space, and denote by IM the Grassmannian of two-planes in TT. Recall that IM is a compact complex manifold of dimension four, which is identified — according to Penrose — to a conformal compactification of the complexified Minkowski space. It follows in particular that the conformally invariant equations of physics, which are defined on the Minkowski space, nicely extend as differential operators acting on holomorphic vector bundles of IM. A distinguished family of such equations is given by the “massless field equations”, parameterized by a half- integer called “helicity”: these include the Dirac-Weyl neutrino equations, the linearized zero rest mass Einstein equations, and Maxwell’s wave equation. For simplicity, in these notes we will denote by h the massless field equation of helicity h (so that 0 is the classical wave equation).

The starting point of twistor theory was Penrose’s discovery that the holomorphic solutions φ ∈ Γ(U ; ker h) of the homogeneous massless field equations of positive helicity, on topologically “good” open subsets U ⊂ IM,

Appeared in: Bull. Soc. Roy. Sci. Li`ege 63 (1994), no. 3-4, 323–334, Algebraic Analysis Meeting (Li`ege, 1993).

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may be written as contour integrals φ =

Z

γ

ψ (1.1)

of free holomorphic functions ψ defined on corresponding open subsets bU of the projective three-space IP ^def= IP(TT). This is closely related to a similar integral formula that Bateman found in 1904 for the solutions of the wave equation. It was then realized that the freedom in the choice of ψ corresponds to the freedom in the choice of a representative for a ˇCech cohomology class of a line bundle on IP, and that the contour integral R

γ corresponds to the functor of direct image for sheaves.

In the classical paper [4], Eastwood, Penrose and Wells established the rig- orous mathematical setting for this integro-geometric construction in terms of sheaf cohomology. Let us explain in some details their construction.

Let IF^def= F12(TT) denote the flag manifold of type (1, 2) of TT. Recall that this is a compact complex manifold of dimension five, whose points are given by pairs (L1, L2) of linear subspaces L1 ⊂ L2 ⊂ TT of dimension one and two respectively. The so-called “twistor correspondence” is the double fibration

IF

g.&^f IP IM,

(1.2)

where the projections are given by f (L1, L2) = L2, g(L1, L2) = L1. The cohomological interpretation of formula (1.1) is as follows: bU ^def= g(f⁻¹(U )) ⊂ IP is the transform of U via (1.2); ψ is the pull back on IF of a representative of a ˇCech cohomology class in H¹( bU ; L) for a line bundle L of IP; and the contour integral corresponds to the first derived direct image functor R¹f∗. On the projective space IP the line bundles are classified by their first Chern class k. As usual, we denote by OIP(k) the −k-th tensor product H^⊗−k of the tautological line bundle H of IP (recall that H is the sub-bundle of the trivial bundle IP × TT, whose fiber over the point L1 of IP is the line L1 itself).

The holomorphic solutions φ ∈ ker(U ; h) of the homogeneous equation for

h are thus obtained by integrating along f the pull back by g of sections in H¹( bU ; OIP(k)). Here, the correspondence between the helicity h and the Chern class k is given by the formula:

h(k) = −(1 + k 2).

For instance, the solutions of the wave equation on U correspond to cohomology classes in H¹( bU ; OIP(−2)).

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As for the notations, in the same way as bU ⊂ IP, let us denote by bx, cW and bz the transform of x ∈ IM, W ⊂ IP and z ∈ IP respectively, via (1.2).

The above discussion is summarized in the following theorem.

Theorem 1.1. ([4]) Let U ⊂ IM be an open subset such that:

U ∩bz is contractible for every z ∈ bU . (1.3) Then the natural morphism:

P : H¹( bU ; O_IP(k)) −→ ker(U ; _h(k)) is an isomorphism.

(In [4], hypothesis (1.3) is actually weaker, since U ∩bz is only assumed connected and simply connected. Here, we restricted ourselves to the contractible case in order to simplify our discussion in the framework of derived categories.)

This result was the corner stone of Penrose’s “twistor programme” whose aim was to derive laws of physics from the rigidity of complex geometry.

An enormous literature on the subject is now available. From the point of view of the linear Penrose transform, the widest generalization of the above theorem is probably found in Baston and Eastwood’s book [1], where they treat the correspondence

G/Q .&

G/R G/P,

for G a complex semisimple Lie group, and P , R and Q = P ∩ R parabolic subgroups. In this framework, one gets isomorphisms between cohomology groups of homogeneous vector bundles on G/R and kernels and cokernels of invariant differential operators on G/P .

Our aim here is to present the new approach of [2] to the (linear) Penrose transform, in terms of (derived categories of) sheaves and D-modules.

The plan is the following.

In the next section, we will give a proof of Theorem 1.1 which, although simplified by the use of derived categories, is very close to the original proof of [4]. This will allow us to point out some problems that one encounters within the classical approach, thus providing a motivation for our work [2].

In section 3 we present our description of the Penrose transform, dis- cussing the extent to which it is possible to recover results such as Theo- rem 1.1 in this new framework.

We then describe the microlocal geometry underlying the Penrose transform. This will allow us to prove some interesting results which, in particular,

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clarify the computational problems usually encountered. As an application, we show how to easily recover the work of Wells [14] concerning the hyperfunction solutions of the massless field equations with positive helicity.

2 The case of positive helicity

Here, we will present a slight variation of the classical proof of Theorem 1.1 given in [4], for the case of positive helicity.

The twistor correspondence (1.2) satisfies the following properties:

(i) the maps f and g are smooth and proper,

(ii) the pair (g, f ) gives a closed embedding IF ,→ IP × TT.

In particular, for x ∈ IM the set f⁻¹(x) is a closed submanifold of IF, and f sends isomorphically f⁻¹(x) to the submanifold bx ' IP¹ of IP. Similarly, for z ∈ IP the set bz ' g⁻¹(z) ' IP² is a closed submanifold of X. It follows that hypothesis (1.3) of Theorem 1.1 asserts that for every z ∈ bU the fibers g⁻¹(z) are topologically trivial, and thus

H¹( bU ; OIP(k)) ' H¹(f⁻¹(U ); g⁻¹O_IP(k)). (2.1) Moreover,

H¹(f⁻¹(U ); g⁻¹O_IP(k)) ^def= H¹(RΓ(f⁻¹(U ); g⁻¹O_IP(k)))

' H⁰(RΓ(U ; Rf∗g⁻¹O_IP(k)[1])), (2.2) and so, in order to prove Theorem 1.1 in the case of positive helicity (i.e. for k < −2), we are left to analyze the complex Rf∗g⁻¹O_IP(k)[1].

To the smooth map g is associated the relative de Rham sequence:

0 −→ g⁻¹OIP −→ OIF dg

−→ Ω¹_g −→ Ω^d^g ²_g −→ 0, (2.3) where dg denotes the relative exterior differential, and Ωⁱ_g denotes the sheaf of relative i forms (i = 1, 2). Since the transition functions of the line bundle g^∗O_IP(k) are invariant along the fibers of g, we deduce from (2.3) the exact sequence:

0 −→ g⁻¹OIP(k) −→ OIF(k)−→ Ω^d^g ¹_g(k)−→ Ω^d^g ²_g(k) −→ 0,

where we wrote Ωⁱ_g(k) = Ωⁱ_g ⊗O_IF g^∗O_IP(k). In terms of derived categories, this gives an identification:

g⁻¹O_IP(k) ' [OIF(k) −→ Ω^d^g ¹_g(k) −→ Ω^d^g ²_g(k)]

| | |

0 1 2

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which we write for convenience sake:

g⁻¹OIP(k) ' [H0(k) −→ H^d^g 1(k + 1)−→ H^d^g 2(k + 2)]. (2.4) On a small neighborhood V of x ∈ U , the map f is identified to the first projection V × IP¹ −→ V . Roughly speaking, for i = 0, 1, 2, the “twist”

(k + i) attached to H_i is there to recall that the vector bundle H_i(k + i) behaves like O_IP¹(k + i) when restricted to the fibers of f , f⁻¹(x) ∼= IP¹.

In order to compute Rf∗g⁻¹OIP(k) using the isomorphism (2.4), recall that

H^j(IP¹; O_IP¹(k)) =





6= 0 if j = 1, k ≤ −2 6= 0 if j = 0, k ≥ 0

= 0 otherwise.

In our case of positive helicity, we have k < −2, and so R^jf∗H_i(k + i) = 0 for j 6= 1. The sheaves eHi = R¹f∗Hi(k +i) are coherent OIM-modules by Grauert theorem, and in fact they turn out to be locally free (i.e. holomorphic vector bundles). We thus get:

Rf∗g⁻¹OIP(k) ' [ eH0

R¹f∗dg

−→ He1

R¹f∗dg

−→ He2].

| | |

1 2 3

(2.5)

An explicit computation as in [4] shows that R¹f∗dg = h(k). Applying the functor H¹(RΓ(U ; ·)) to the above sequence we obtain the isomorphism:

H⁰(RΓ(U ; Rf∗g⁻¹O_IP(k))[1]) ' ker[Γ(U ; eH₀) −→ Γ(U; e^h(k) H₁)], which, together with (2.1), (2.2), gives the desired identification:

P : H¹( bU ; OIP(k))−→ ker(U;^∼ _h(k))

This method does not apply, as simply as above, to the case of non positive helicity. For instance, in the case h = 0 (which corresponds to the classical wave equation), when applying the functor Rf∗ to (2.4), we notice that the middle term on the right hand side vanishes, while the third term is concentrated in degree zero. The computation of Rf∗g⁻¹O_IP(−2) is not as straightforward as it was for (2.5), since the associated spectral sequence degenerate only at the second order. Similar technical difficulties are encountered for h < 0, when trying to apply the above reasoning.

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3 Sheaves and D-modules

We refer to [6] for an exposition of the theory of sheaves in the framework of derived categories: here we will just fix the notations.

Let X be a complex manifold of dimension dX, and denote by D^b(X) the derived category of the category of bounded complexes of sheaves of C-vector spaces. Let f : Y −→ X be a morphism of complex manifolds, and set dY /X = dY − d_X. We will consider the six classical functors in the derived category f⁻¹, Rf!, ⊗, and Rf∗, f^!, RHom. If A ⊂ X is a locally closed subset, we denote by CAthe constant sheaf on A. We denote by ωY /X

the relative dualizing complex defined by ωY /X = f^!C_X ' or_{Y /X}[d], where orY /X is the relative orientation sheaf on Y , and d = dim^RY − dim^RX.

The dual to F , denoted D_X⁰ F , is defined by D_X⁰ F = RHom(F, C_X). Fi- nally, we denote by D^b_R−c(X) the full triangulated subcategory of D^b(X) consisting of R-constructible objects (i.e. objects whose cohomology groups are R-constructible sheaves).

Here we will consider the correspondence:

Y

g.&^f

Z X,

(3.1)

where X, Y and Z are complex manifolds, and where we make the following hypotheses:

(H.1) f and g are smooth and proper,

(H.2) (g, f ) : Y ,→ Z × X is a closed embedding.

Definition 3.1. Let F ∈ Ob(D^b(X)) and H ∈ Ob(D^b(Z)). We set:

PH ^def= Rf∗g⁻¹(H), (3.2)

PFe ^def= Rg!(f⁻¹(F ) ⊗ ω_{Y /Z}), (3.3) and call them “Penrose transform” and “adjoint Penrose transform” respectively.

The name “adjoint Penrose transform” is justified by the following proposition, which is a straightforward consequence of classical adjunction formulas of sheaf theory such as the Poincar´e-Verdier duality formula.

Proposition 3.2. (i) Let F ∈ Ob(D^b(X)) and H ∈ Ob(D^b(Z)). Then Hom_Db(Z)( ePF, H)−→ Hom^∼ _Db(X)(F, PH).

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(ii) Let K ∈ Ob(D^b_R−c(X)), H ∈ Ob(D^b(Z)), and assume K has compact support. Then we have a commutative diagram whose rows are isomorphisms:

RaZ ∗(PK ⊗ H) −→^∼ RaX ∗(K ⊗ PH)

↓ ↓

Ra_{Z ∗}RHom(D_Z⁰ PK, H) −→ Ra^∼ _{X ∗}RHom(D⁰_XK, PH)

Let us recall some notions and notations on D-modules for which we refer to Schneiders’s lectures [11].

Denote by OX the sheaf of holomorphic functions on X, by ΩX the sheaf of holomorphic forms of maximal degree, and by DX the sheaf of rings of linear differential operators with holomorphic coefficients. Denote by D^b(DX) the derived category of the category of bounded complexes of left DX-modules.

We shall say that a coherent DX-module M is good, if for each compact subset K ⊂ X, there exists a coherent OX-module defined in a neighborhood of K, contained in M, and which generates M (see [8]). We denote by D^b_good(DX) the full triangulated subcategory of D^b(DX) consisting of objects M such that H^j(M) is coherent and good for every j ∈ Z. If M is a coherent DX-module (or an object of D^b(DX) whose cohomology groups are coherent) we denote by char(M) its characteristic variety, a closed complex analytic involutive set of the cotangent bundle T^∗X. Let f : Y −→ X be a morphism of manifolds. Denote by f_∗ and f⁻¹ the direct and inverse images in the sense of D-modules.

Let us now consider the double fibration (3.1).

Definition 3.3. For N ∈ Ob(D^b_good(DZ)) we set PN ^def= f_∗g⁻¹N , and call it the D-module Penrose transform of N .

As a corollary of the Cauchy-Kowalevski-Kashiwara theorem applied to g, and the direct image theorem applied to f (cf. e.g. [11]), we get the following result.

Proposition 3.4. Let N ∈ Ob(D_good^b (DZ)). Then the canonical morphism P RHom_D

Z(N , OZ) −→ RHom_D

X(PN , OX)[−d_{Y /X}] is an isomorphism.

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Let us set, as usual:

RHom_D

Z(·, ·) = RΓ(Z; RHom_D

Z(·, ·)).

Combining Propositions 3.2 and 3.4 we obtain the following result.

Theorem 3.5. (i) Let N ∈ Ob(D^b_good(DZ)), and let F ∈ Ob(D^b(X)).

Then there is a canonical isomorphism RHom_D

Z(N ⊗ ePF, O_Z)−→ RHom^∼ _D

X(PN ⊗ F, OX)[−dY /X].

(i) Let N ∈ Ob(D_good^b (DZ)), and let K ∈ Ob(D_R−c^b (X)) have compact support. Then there is a canonical diagram whose rows are isomorphisms

RHom_D_Z(N , PK ⊗ OZ) −→^∼ RHom_D_X(PN , K ⊗ OX)[−d_{Y /X}]

↓ ↓

RHom_D

Z(N ⊗ D_Z⁰ PK, OZ) −→ RHom^∼ _D_X(PN ⊗ D⁰_XK,OX)[−d_{Y /X}].

For x ∈ X, z ∈ Z and U ⊂ X, let bx = gf⁻¹(x), bz = fg⁻¹(z), bU = gf⁻¹(U ).

Corollary 3.6. Let N ∈ Ob(D^b_good(DZ)). Then (i) [Germ formula] For x ∈ X one has

RΓ(bx; RHomD_Z(N , O_Z)) ' RHom_D

X(PN , O_X)_x[−d_{Y /X}];

(ii) [Holomorphic solutions] Let U ⊂ X be an open subset such that U ∩bz is contractible for every z ∈ bU . (3.4) Then

RΓ( bU ; RHom_D

Z(N , OZ)) ' RΓ(U ; RHom_D

X(PN , OX))[−d_{Y /X}].

Proof. It is easy to check that PCx ' Cx_b for x ∈ X. On the other hand, hypothesis (H.2) gives an identification U ∩bz ' f⁻¹(U ) ∩ g⁻¹(z), and hence (3.4) implies that the fibers of g over f⁻¹(U ) are topologically trivial. One then has ePC_U ' C_U_b.

Let L be a holomorphic vector bundle on Z, L^∗ = Hom_O

Z(L, O_Z) its dual, and consider the locally free DZ-module:

DL^∗ = DZ⊗O_ZL^∗.

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One recovers L as the sheaf of holomorphic solutions to DL^∗: L ' RHom_D

Z(DL^∗, OZ).

In the framework of correspondence (1.2), applying Corollary 3.6 with L = O_IP(k), we get the isomorphisms:

RΓ(bx; OIP(k)) ' RHom_D_X(PDIP(−k), OX)x[1],

RΓ( bU ; OIP(k)) ' RΓ(U ; RHom_D_X(PDIP(−k), OX))[1],

where we set DIP(−k) = DOIP(−k). It is then possible to give the following interpretation to Theorem 1.1: the Penrose transform of the locally free D- module DIP(−k) is the D-module associated to the operator h.

4 Microlocal geometry

To the double fibration (3.1) are associated the correspondences of conormal bundles

T^∗Y ←− Y ×^t^f⁰

X T^∗X −→ T^f^π ^∗X, T^∗Y ←− Y ×^t^g⁰

Z T^∗Z −→ T^g^π ^∗Z.

Since the maps f and g are smooth, ^tf⁰ and ^tg⁰ are closed embeddings. Let Vf

def= ^tf⁰(Y ×

X ×T^∗X), Vg

def= ^tg⁰(Y ×

Z×T^∗Z).

It is then easy to check that the correspondence Vf ∩ V_g

gπ.&^f^π T^∗Z T^∗X is identified to the correspondence:

T_Y^∗(Z × X)

p^a₁.&^p² T^∗Z T^∗X,

(4.1)

where T_Y^∗(Z × X) denotes the cotangent bundle to Y ,→ Z × X, where p1, p2

are the first and second projection, and where p^a₁ = −p1. Since T_Y^∗(Z × X) is Lagrangian, it is well known that p1 is submersive if and only if p^a₂ is immersive. In the following we will assume:

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(H.3) the map p2 in (4.1) is a closed embedding (and so p^a₁ is submersive).

Let

V ^def= fπ(Vf ∩ V_g).

We will call V the characteristic set of the transform (3.1).

Theorem 4.1. Assume (H.1)–(H.3), and assume that X is connected. Then:

(i) H^j(PDL) is a holomorphic flat connection for j 6= 0,

(ii) char(H⁰(PDL)) = V , and H⁰(PDL) has simple characteristics along V ,

(iii) PDL is concentrated in degree zero if and only if there exists x ∈ X such that H^j(bx; L) = 0 for every j 6= dY − d_X.

Proof. Statements (i) and (ii) follow immediately if one notices that P “com- mutes to microlocalization”. By this we mean the following. Let EZ be the sheaf of finite order microdifferential operators on T^∗X (see [9], [5], or [10] for a detailed exposition). If M is a coherent DX-module, set EN = E_Z ⊗_˙π−1D_Z ˙π⁻¹N , where ˙π : ˙T^∗Z −→ Z is the cotangent bundle to Z with the zero-section removed. Then one proves that

EPN = P(EN ).

The conclusion follows if one recalls that a DX-module whose characteristic variety is contained in the zero section is a locally free O_X-module of finite rank endowed with a flat connection (cf. e.g. [11]).

The statement (iii) is easily proved using (i) and the germ formula.

In order to apply the last theorem to the twistor correspondence (1.2), recall that bx is identified with a projective one-space IP¹ linearly embedded in IP. It is then a straightforward computation to check that for x ∈ X one has

H⁰(bx; OZ(k)) =

0 for k < 0,

6= 0 and finite dimensional for k ≥ 0, H¹(bx; O^Z(k)) is infinite dimensional for any k, H^j(bx; O^Z(k)) = 0 for j 6= 0, 1 and for any k.

Corollary 4.2. The complex PDZ(−k) is concentrated in degree zero if and only if k < 0.

This explains why many authors just treat the case of positive helicity

“for simplicity’s sake”.

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5 Hyperfunction solutions

Let φ be a Hermitian form on TT of signature (+, +, −, −) and choose a basis for TT such that:

φ =

0 iI2

−iI₂ 0

. Consider the local chart:

C⁴ −→ IM

(x1, x2, x3, x4) 7→







x3− x₄ x1+ ix2

x1− ix2 x3+ x4

1 0

0 1







and notice that

(A^∗, I2) φ

A I2

= 0

if and only if A is Hermitian, i.e. if and only if it is the image of a point (x1, x2, x3, x4) in R⁴ ⊂ C⁴ via the above local chart. The manifold

M = {L2 ∈ IM; φ(v) = 0 ∀v ∈ L2}

is thus a compact, completely real submanifold of IM, which is identified to a conformal compactification of the real Minkowski space. We similarly define

F = {(L1, L2) ∈ IF; φ(v) = 0 ∀v ∈ L2}, P = {L1 ∈ IP; φ(v) = 0 ∀v ∈ L1},

and consider the embedding of diagrams F

e g.&^f^e

P M

,→

IF

g.&^f

IP IM

Recall after [14], that P is a real hypersurface of IP, and that eg : F −→ P is a sphere bundle (i.e. locally isomorphic to P × S¹ −→ P ).

We denote by

A_M = C_M ⊗ O_IM,

B_M = RHom(D_IM⁰ C_M, OIM),

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the sheaves of analytic functions and Sato hyperfunctions on M respectively.

Theorem 3.5 applied with K = C_M, N = D_IP(−k), gives:

RΓ(IP; PCM ⊗ O_IP(k)) −→ RHom^∼ _D

IM(PDIP(−k), AM)[−1]

↓ ↓

RHom(D⁰PC_M, OIP(k)) −→ RHom^∼ _D

IM(PDIP(−k), BM)[−1].

(5.1)

Since eg is locally isomorphic to P × S¹ −→ P , and P is simply connected, we find:

H^jPCM ∼=

C_P, for j = 0, 1, 0, otherwise.

Under the hypothesis k < 0, taking the zero-th cohomology in (5.1) we get the following diagram in which the horizontal arrows are isomorphisms

H¹(P ; OIP(k)) −→ Hom^∼ _D_IM(PDIP(−k), AM)

↓ ↓

H_P²(IP; OIP(k)) −→ Hom^∼ _D_IM(PDIP(−k), BM).

This is Theorem 6.1 of [14].

6 Acknowledgments

I wish to thank Jean-Pierre Schneiders for the fruitful discussions we had together during the preparation of this work.

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