Monogenic Functions in Conformal Geometry

Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural ex...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2007
Main Authors:	Eastwood, M., Ryan, J.
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Language:	English
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Cite this:	Monogenic Functions in Conformal Geometry / M. Eastwood, J. Ryan // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ.

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spelling	Eastwood, M. Ryan, J. 2019-02-13T19:31:40Z 2019-02-13T19:31:40Z 2007 Monogenic Functions in Conformal Geometry / M. Eastwood, J. Ryan // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53A30; 58J70; 15A66 https://nasplib.isofts.kiev.ua/handle/123456789/147228 Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition. This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson. It is a pleasure to acknowledgment useful conversations with Vladim´ır Souˇcek. He is certainly one person for whom the ‘well-known’ material in this article is actually known. Michael Eastwood is a Professorial Fellow of the Australian Research Council. This research was begun during a visit by John Ryan to the University of Adelaide in 2005, which was also supported by the Australian Research Council. This support is gratefully acknowledged. John Ryan also thanks the University of Adelaide for hospitality during his visit. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Monogenic Functions in Conformal Geometry Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Monogenic Functions in Conformal Geometry
spellingShingle	Monogenic Functions in Conformal Geometry Eastwood, M. Ryan, J.
title_short	Monogenic Functions in Conformal Geometry
title_full	Monogenic Functions in Conformal Geometry
title_fullStr	Monogenic Functions in Conformal Geometry
title_full_unstemmed	Monogenic Functions in Conformal Geometry
title_sort	monogenic functions in conformal geometry
author	Eastwood, M. Ryan, J.
author_facet	Eastwood, M. Ryan, J.
publishDate	2007
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/147228
citation_txt	Monogenic Functions in Conformal Geometry / M. Eastwood, J. Ryan // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ.
work_keys_str_mv	AT eastwoodm monogenicfunctionsinconformalgeometry AT ryanj monogenicfunctionsinconformalgeometry
first_indexed	2025-11-24T11:38:35Z
last_indexed	2025-11-24T11:38:35Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 084, 14 pages Monogenic Functions in Conformal Geometry? Michael EASTWOOD † and John RYAN ‡ † Department of Mathematics, University of Adelaide, SA 5005, Australia E-mail: meastwoo@member.ams.org ‡ Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA E-mail: jryan@uark.edu Received August 29, 2007; Published online August 30, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/084/ Abstract. Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition. Key words: Clifford analysis; monogenic functions; Dirac operator; conformal invariance 2000 Mathematics Subject Classification: 53A30; 58J70; 15A66 1 Introduction This article is dedicated to the memory of Tom Branson, a true gentleman, scholar, and friend. A significant and recurring theme in his work was the interaction between the following topics Lie Groups Homogeneous Spaces Global Symmetry Representation Theory ↔ Differential Geometry Local Structure Local Symmetry PDEs Items in the left hand box are concerned with geometry and global analysis on a homogeneous space G/P where G is a Lie group with subgroup P . Items in the right hand box are concerned with local differential geometry ‘modelled’ on the homogeneous space G/P . Without trying to explain what this might mean in general, one of the best known examples is to have conformal differential geometry in dimension n ≥ 3 in the right hand box and the sphere Sn, viewed as a homogeneous space for SO(n +1, 1), in the left hand box. The round sphere provides a better ‘flat model’ of conformal geometry than does Euclidean space because all the conformal Killing fields integrate to genuine conformal motions (and there is a conformal embedding Rn ↪→ Sn given by inverse stereographic projection). Tom Branson was one of the first (e.g. [4]) systemati- cally to exploit the interplay between ‘curved’ conformal differential geometry and its flat model. He was also keen to promote the interaction between conformal geometry and the field of Clifford analysis, where monogenic functions were defined and are now studied. Some disclaimers are in order. The results in this article are, in some sense, already known. More precisely, there are a few mathematicians, including David Calderbank and Vladimı́r Souček, who are well aware of results in this vein. There is no doubt that Tom Branson would have had his own distinctive viewpoint. Our aim, therefore, is to give what we believe to be a particularly efficient formulation and method of proof with minimal prerequisites. Motivation and consequences will be given as we go along. ?This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson. The full collection is available at http://www.emis.de/journals/SIGMA/MGC2007.html mailto:meastwoo@member.ams.org mailto:jryan@uark.edu http://www.emis.de/journals/SIGMA/2007/084/ http://www.emis.de/journals/SIGMA/MGC2007.html 2 M. Eastwood and J. Ryan Clifford analysis on Euclidean space Rn is usually developed by considering smooth functions with values in the Clifford algebra of Rn. In this context one may equally well consider smooth differential forms on Rn, i.e. smooth functions with values in Λ• = ⊕n p=0 Λp where Λp is the bundle of p-forms. In fact, there is a canonical identification between Λ•Rn and the Clifford algebra. Under this identification, the so-called ‘Dirac operator’ D : Λ• → Λ• is precisely Dω = dω − d∗ω, where d is exterior derivative and d∗ is its adjoint. Viewed like this, D enjoys an evident extension to any Riemannian manifold where it is called the Hodge-de Rham operator. There is, however, an equally natural but different extension of D to any Riemannian manifold with spin structure. This alternative extension turns out to be conformally invariant and hence to be preferred as the starting point for Clifford analysis on manifolds. In this article, we shall present this alternative extension in as simple a manner as possible. In particular, we shall not need Clifford algebras for the basic exposition, only to compare the resulting Dirac operator with the classical one defined on the usual spin bundles in § 4 (and, even then, only implicitly). The simplest exposition does not even require any representation theory beyond the basics. We conclude this introduction by establishing some notation and conventions. We shall use Penrose’s ‘abstract index notation’ [17] for tensors on a manifold and also for representations of the orthogonal group SO(n). Thus, we shall write ωa to denote a smooth 1-form, and Xa to de- note a vector field. Square brackets denote skewing over the indices they enclose – if ωab = ω[ab], then ωab denotes a 2-form. Similarly, we shall use round brackets to denote symmetrisation: φ(ab) = 1 2(φab + φba). Repeated indices denote the natural pairing of vectors and co-vectors: X ω = Xaωa. In local coördinates this is the Einstein summation convention but the ab- stract index convention is itself coördinate free. Following standard practise, on a Riemannian manifold we shall write gab for the metric and gab for its inverse: gabg bc = δa c where δa c is the Kronecker delta or identity matrix. We shall often ‘raise and lower indices’ without comment – if Xa is a vector field then Xa ≡ gabX b is the corresponding 1-form. Sometimes we shall write 〈u, v〉 instead of uava. We need to set some conventions for the exterior and interior product on differential forms – if ωbc···d has p indices, then (v ∧ ω)abc···d ≡ v[aωbc···d] and (v ω)c···d ≡ pvbωbc···d. (1) These conventions have the convenient feature that u ∧ (v ω) + v (u ∧ ω) = 〈u, v〉ω. (2) For further details see [18]. 2 Conformally invariant first order operators We begin with three simple examples. If ωa is a 1-form on a Riemannian manifold M and we rescale the metric according to ĝab = Ω2gab, then it is easily verified that ∇̂aωb = ∇aωb −Υaωb −Υbωa + Υcωcgab where Υa = ∇aΩ / Ω and ∇a is the metric connection for gab whilst ∇̂a is the metric connection for ĝab. It follows that ∇̂[aωb] = ∇[aωb]. Monogenic Functions in Conformal Geometry 3 This is just the conformal invariance of the exterior derivative d : Λ1 → Λ2. More instructive examples are obtained by insisting that when we rescale the metric so that ĝab = Ω2gab we also rescale the 1-form ω̂a = Ωwωa. In this case ∇̂aω̂b = ∇̂a(Ωwωb) = (∇aΩw)ωb + Ωw∇̂aωb = ΩwwΥaωb + Ωw ( ∇aωb −Υaωb −Υbωa + Υcωcgab ) = Ωw ( ∇aωb + (w − 1)Υaωb −Υbωa + Υcωcgab ) . Notice that the right hand side rescales by the same power of Ω as we decreed for our 1-form ωa. Thus, if we say that ωa is conformally weighted of weight w instead of writing ω̂a = Ωwωa, then we have shown that ∇̂aωb = ∇aωb + (w − 1)Υaωb −Υbωa + Υcωcgab (3) for ωa of conformal weight w. More precisely, we define a conformal manifold to be a smooth manifold equipped with an equivalence class of Riemannian metrics under the equivalence rela- tion gab 7→ ĝab = Ω2gab for any smooth function Ω. On such a manifold we define a line bundle L as trivialised by any choice of metric – its sections are identified as ordinary functions f . For a conformally related metric ĝab = Ω2gab, however, the same section gives a different func- tion f̂ = Ωf . Let us write Λ0[w] instead of Lw and refer to its sections as conformal densities of weight w. Similarly, Λp[w] = Λp ⊗ Lw is the bundle of conformally weighted p-forms of weight w. For any metric gab in the conformal class these bundles acquire natural connections, namely by trivialising L and employing the metric connection. Equation (3) now has a precise interpretation – it says how the connection ∇ : Λ1[w] → Λ1 ⊗ Λ1[w] changes under a conformal rescaling of the metric. The metric gab itself acquires a tautological interpretation as a confor- mally invariant section of ⊙2Λ1[2]. Similarly, its inverse gab has conformal weight −2 and we may raise and lower indices at the expense of changing the weight – if ωa has weight w, then ωa has weight w − 2. If w = 2, then (3) reads ∇̂aωb = ∇aωb + Υaωb −Υbωa + Υcωcgab. It follows that ∇̂(aωb) − 1 n∇̂ cωcgab = ∇(aωb) − 1 n∇ cΥcgab. Thus, we have a conformally invariant operator Λ1[2] → ⊙2 ◦Λ 1[2], where ⊙2 ◦Λ 1 denote the trace-free symmetric covariant 2-tensors. Taking the trace of (3) yields ∇̂aωa = ∇aωa + (n + w − 2)Υaωa. We obtain, as our third example, a conformally invariant differential operator Λ1[−(n− 2)] → Λ0[−n]. Our fundamental theorem constructs first order conformally invariant operators from repre- sentations of SO(n) or Spin(n). Let us denote by W, the defining representation of SO(n) on Rn and by gab ∈ W⊗W or by 〈 , 〉 the standard Euclidean metric, preserved by SO(n). We shall 4 M. Eastwood and J. Ryan identify Λ2W with the Lie algebra so(n) of SO(n) or Spin(n). Explicitly, let us take the action of Xab ∈ Λ2W on wa ∈ W to be given by Xa bwb. Also let wa 7→ −2ga[bwc] define ι : W → W⊗ Λ2W. A representation E of SO(n) gives rise to an irreducible tensor bundle E on any oriented Rieman- nian n-manifold. Specifically, we shall regard E as induced from the orthonormal co-frame bun- dle so that W gives rise to Λ1, the co-tangent bundle. Similarly, if E is a Spin(n)-representation, then we find induced bundles E on any oriented Riemannian spin n-manifold. On a conformal manifold it is convenient, following Calderbank [7], to induce bundles from the orthonormal frames of Λ1[1]. The reason is that 〈ω, µ〉 = gabωaµb is conformally invariant for ωa and µa of conformal weight 1 and so these frames are well-defined. A further reason is that, although there is no conformally invariant connection on Λ1[1], the conformal change (3) is especially simple: ∇̂aωb = ∇aωb −Υbωa + Υcωcgab = ∇aωb − Γab cωc where Γabc = −2ga[bΥc]. (4) We shall denote by E the bundle induced on an oriented conformal manifold from a representa- tion E of SO(n). From the defining representation W of SO(n) we obtain the bundle Λ1[1]. Theorem 1. Suppose that E and F are representations of SO(n) and that π : W⊗ E → F is a homomorphism of SO(n)-modules. Let ρ : Λ2W⊗ E → E denote the action of so(n) on E. Suppose that the composition W⊗ E ι⊗Id−−−→ W⊗ Λ2W⊗ E Id⊗ρ−−−→ W⊗ E π−→ F is equal to wπ : W ⊗ E → F for some constant w. Then there is a conformally invariant first order linear differential operator D : E[w] → F [w − 1] whose symbol is induced by π. If, instead, E and F are representations of Spin(n), then we obtain the corresponding conclusion on any Riemannian spin manifold. Proof. Recall the formula (4) for a change of connection on Λ1[1]. Therefore, for sections φ of the associated bundle E, ∇̂φ = ∇φ− Γφ where Γφ is the image of Υ⊗ φ under the composition Λ1 ⊗ E ι⊗Id−−−→ Λ1 ⊗ Λ2 ⊗ E Id⊗ρ−−−→ Λ1 ⊗ E. For sections of E[w], therefore, ∇̂φ = ∇φ + wΥ⊗ φ− Γφ. Hence, for D = π∇ we have D̂φ = Dφ + wπ(Υ⊗ φ)− πΓφ = Dφ, as required. � Monogenic Functions in Conformal Geometry 5 The three conformally invariant operators given at the beginning of this section are easily obtained from Theorem 1. Specifically, the irreducible decomposition W⊗W = Λ2W⊕ ⊙2 ◦W⊕ R is given explicitly by ξaωb = ξ[aωb] + ( ξ(aωb) − 1 nξcωcgab ) + 1 nξcωcgab from which we obtain three possible symbols with E = W so that E = Λ1[1]. The composition W⊗ E ι⊗Id−−−→ W⊗ Λ2W⊗ E Id⊗ρ−−−→ W⊗ E is ξaωb 7→ −2ga[bξc]ωd 7→ ξbωa − gabξ cωc ≡ σab and to apply Theorem 1 we must see whether πσ is a multiple of π(ξ ⊗ ω). We find ξaωb π7−→ ξ[aωb] gives σ[ab] = −ξ[aωb], ξaωb π7−→ ξ(aωb) − 1 nξcωcgab gives σ(ab) − 1 nσc cgab = ( ξ(aωb) − 1 nξcωcgab ) , ξaωb π7−→ ξaωa gives σa a = −(n− 1)ξaωa. We obtain our three basic conformally invariant operators F w F E[w] F [w − 1] Λ2W −1 Λ2[2] Λ1 Λ2⊙2 ◦W 1 ⊙2 ◦Λ 1[2] Λ1[2] ⊙2 ◦Λ 1[2] R −(n− 1) Λ0 Λ1[−(n− 2)] Λ0[−n] in accordance with Theorem 1. In stating Theorem 1 we were imprecise concerning whether E and F should be real or complex representations of SO(n) or Spin(n). In fact, there are two versions – if the representations are real (as in our three examples), then the corresponding induced bundles are real but if the representations are complex (as is more usual in representation theory) then the bundles are complex. The hypotheses of Theorem 1 are automatically satisfied if E is an irreducible complex rep- resentation of SO(n) or Spin(n). In this case the decomposition of W ⊗ E into irreducibles is multiplicity-free. Thus, if π : W ⊗ E → F is projection onto any of the irreducibles, then any other so(n)-invariant homomorphism W ⊗ E → E must be a multiple thereof. In this case, it is just a matter of identifying the constant w in order to classify the first order conformally invariant differential operators between irreducible tensor or spinor bundles. This is precisely what Fegan does in [10], using Casimir operators to compute w. In [8], Čap, Slovák, and Souček extended Fegan’s method to higher order operators. 3 Monogenic functions and Dirac operators The construction and conformal invariance of monogenic functions also comes from Theorem 1. In distinction to the discussion at the end of the previous section, the representation E will be real and reducible. As a vector space, let E = Λ•W where, as usual, W is the (real) defining representation of SO(n). Define a homomorphism W⊗ E ε−→ E by v ⊗ e 7→ v.e ≡ v ∧ e− v e. (5) 6 M. Eastwood and J. Ryan There are two natural ways in which E is an so(n)-module. The more obvious corresponds to the SO(n)-action defined by that on W. Evidently, ε is a homomorphism of SO(n)-modules. The geometric consequence of this is a differential operator from E to itself defined on any Riemannian manifold as the composition E ∇−→ Λ1 ⊗ E ε−→ E, where∇ is induced by the metric connection on the co-frame bundle. This is the Hodge–de Rham operator. For this choice of ρ : Λ2W⊗ E → E, however, W⊗ E ι⊗Id−−−→ W⊗ Λ2W⊗ E Id⊗ρ−−−→ W⊗ E ε−→ E is not a multiple of ε : W⊗E → E. The Hodge–de Rham operator is not conformally invariant. An alternative but equally natural way in which E is an so(n)-module is given by so(n)⊗ E = Λ2W⊗ E 3 v ∧ w ⊗ e σ7−→ −1 8(v.w.e− w.v.e) ∈ E. (6) The constant −1/8 ensures that this is, indeed, a representation of so(n). To verify this and other properties of this construction, the following lemmata are useful. Lemma 1. For all u, v ∈ W and e ∈ E we have u.v.e + v.u.e = −2〈u, v〉e. (7) Proof. A simple computation from (2). � Lemma 2. For all u, v, w ∈ W and e ∈ E we have u.v.w.e− v.w.u.e− u.w.v.e + w.v.u.e = −4〈u, v〉w.e + 4〈u, w〉v.e. (8) Proof. From (7) we find u.v.w.e− v.w.u.e = u.v.w.e + v.u.w.e− v.u.w.e− v.w.u.e = −2〈u, v〉w.e + 2〈u, w〉v.e. The other terms in (8) are dealt with similarly. � Proposition 1. The action (6) makes E into a representation of so(n). With this structure ε : W⊗ E → E is a homomorphism of so(n)-modules. Proof. On simple vectors, the Lie bracket on so(n) = Λ2W is given by [t ∧ u, v ∧ w] = 1 2 ( 〈u, v〉t ∧ w − 〈u, w〉t ∧ v − 〈t, v〉u ∧ w + 〈t, w〉u ∧ v ) . The assertions are straightforward calculations on simple vectors using (8). � Lemma 3. If we define W⊗W⊗ E → E by v ⊗ w ⊗ e 7→ v.w.e, then gab ⊗ e 7→ −ne. Proof. If u1, u2, . . . , un is an orthonormal basis of W, then gab = n∑ i,j=1 ui⊗uj . The result follows immediately from (7). � Proposition 2. The composition W⊗ E ι⊗Id−−−→ W⊗ Λ2W⊗ E Id⊗σ−−−→ W⊗ E ε−→ E (9) is equal to −n−1 2 ε : W⊗ E → E. Monogenic Functions in Conformal Geometry 7 Proof. Let us perform this computation using abstract indices. For this purpose, if we write ε : W⊗ E → E as wa ⊗ eα 7→ waε a α βeβ, then Lemma 3 says that εa α βεaβ γ = −nδα γ . (10) Let us write σ : Λ2W⊗ E → E as Xab ⊗ eα 7→ Xabσ ab α βeβ, where σab α γ = −1 8(εa α βεb β γ − εb α βεa β γ). Then (9) becomes waeα 7→ εa α β(−2ga[bwc]σ bc β γeγ) = −2εa α βwcσa c β γeγ and so we are required to show that −2εa α βσa c β γ = −n−1 2 εc α γ i.e. 4εa α βσa c β γ = (n− 1)εc α γ . Using (10), we compute: 4εa α βσa c β γ = −1 2εa α β(εaβ ηεc η γ − εc β ηεaη γ) = n 2 εc α γ + 1 2εa α βεc β ηεaη γ . To continue, let us write (8) with abstract indices: εa α βεb β ηεc η γ − εb α βεc β ηεa η γ − εa α βεc β ηεb η γ + εc α βεb β ηεa η γ = −4gabεc α γ + 4gacεb α γ . Tracing over a and b and using (10) gives −nεc α γ − εa α βεc β ηεaη γ − εa α βεc β ηεaη γ − nεa α γ = −4nεc α γ + 4εc α γ . If follows that εa α βεc β ηεaη γ = (n− 2)εc α γ and hence that 4εa α βσa c β γ = n 2 εc α γ + n−2 2 εc α γ = (n− 1)εc α γ , as required. � Theorem 2. Let E denote that representation of Spin(n) corresponding to (6). Let E denote the corresponding bundle on a conformal spin manifold induced from the orthonormal frames of Λ1[1]. Then there is a conformally invariant first order linear differential operator E[−n−1 2 ] D−→ E[−n+1 2 ] whose symbol is induced by ε : W⊗ E → E. Proof. Proposition 1 ensures that the statement of this theorem makes sense and now Propo- sition 2 ensures that the criterion of Theorem 1 is satisfied. � Remark 1. This operator D extends the Dirac operator from Clifford analysis on Rn to a general spin manifold. Elements of its kernel are referred to as ‘monogenic functions’. Remark 2. Almost no representation theory is needed here – we have only used that Spin(n) is the simply-connected connected Lie group whose Lie algebra is so(n). In fact, we shall see in the next section that spin is essential – the representation (6) of so(n) does not arise from an action of SO(n). The proof of Theorem 2 is purely computational and the only ingredients in this computation are (8) and (10) both of which follow easily from Lemma 1. In fact, the particular constant in Lemma 3 (leading to (10)) is unimportant. We would obtain the same result (with the same conformal weight) from gab ⊗ e 7→ κe for any non-zero κ. 8 M. Eastwood and J. Ryan 4 Clifford algebras and the classical Dirac operator The Dirac operator from Clifford analysis is usually introduced via the Clifford algebra C`(W), defined as the tensor algebra ⊗• W modulo the two-sided ideal generated by v ⊗ w + w ⊗ v + 2〈v, w〉 ∀ v, w ∈ W. Writing the multiplication in C`(W) as juxtaposition, there is a canonical identification E ≡ Λ•W '−−→ C`(W) by v ∧ w 7→ 1 2(vw − wv). (11) The conclusion (7) of Lemma 1 is precisely that ε : W ⊗ E → E extends to a representation C`(W), namely to a homomorphism of algebras C`(W) → End(E). Equivalently, if we transport the Clifford algebra structure from C`(W) to Λ•W using (11) then (5) gives the Clifford product between W and Λ•W and so the action of C`(W) on E induced by ε becomes the action of C`(W) on itself by left multiplication. For further details see [13]. A standard rationale for introducing the Clifford algebra is in providing a concrete realisation of Spin(n), namely as a subgroup of the group of invertible elements in C`(Rn). Then it is clear that E under (11) is a real representation of Spin(n) – simply restrict the action of C`(Rn) to Spin(n). The basic spin representations of so(n) are complex representations. Hence, in order to make contact with E as an so(n)-module given by (6), it is necessary to complexify. Let us suppose that n is even. Then we may write CW as the direct sum of two totally null subspaces, using the complexified metric as a dual pairing between them. Specifically, let us take CW = U⊕ U∗ 3 α + β so that ‖α + β‖2 = 2α β. (12) Let use abstract indices to write elements of U as αa and elements of U∗ as βa, without being alarmed that the index a now runs only over half the range that it did in previous sections. Forms now decompose according to ‘type’. Specifically, CE = Λ•CW and ΛrCW = ⊕ p+q=r ΛpU⊗ ΛqU∗. Splitting the formulae (1) according to this decomposition, we find for ΛpU⊗ ΛqU∗ 3 ω ! ωbc · · · d︸︷︷︸ p q︷︸︸︷ fg · · ·h, { (α + β) ∧ ω ! α[aωbc···d] fg···h + (−1)pβ[eωbc···d fg···h], (α + β) ω ! pβbωbc···d fg···h + (−1)pqαfωbc···d fg···h. Consequently, as a complexification of the action of W on E = Λ•W given by (5) we obtain an action of CW on Λ•U⊗ Λ•U∗ given by (α + β).ω ! α[aωbc···d] fg···h + (−1)pβ[eωbc···d fg···h] − pβbωbc···d fg···h − (−1)pqαfωbc···d fg···h. In accordance with (7) and (12), or as may be verified by direct computation, this action has the property that (α + β).(α + β).ω = −2αaβ aω. There is, however, another action with this property. Specifically, it is easily verified that (α + β):ω ! √ 2 ( α[aωbc···d] fg···h − pβbωbc···d fg···h ) =⇒ (α + β):(α + β):ω = −2αaβ aω. Monogenic Functions in Conformal Geometry 9 Proposition 3. There is an automorphism of Λ•U⊗Λ•U∗ that converts (α+β).ω into (α+β):ω for all α + β ∈ CW. Proof. More precisely, we want to find Φ : Λ•U ⊗ Λ•U∗ → Λ•U ⊗ Λ•U∗, an invertible linear transformation, so that (α + β):Φ(ω) = Φ((α + β).ω) ∀ (α + β) ∈ U⊕ U∗ = CW and ω ∈ Λ•U⊗ Λ•U∗. (13) Let us take Φ(1) ≡ 1 + δb f + δ[b [fδc] g] + · · ·+ δ[b [fδc g · · · δd] h] + · · · . Then, in order for (13) to hold, we should have α:Φ(1) = Φ(α.1) = Φ(α) whence Φ(αb) = √ 2 ( αb + α[bδc] f + α[bδc fδd] g + · · · ) . Similarly, we should have β:Φ(1) = Φ(β.1) = Φ(β), which forces Φ(βf ) = − √ 2 ( βf + 2β[fδb g] + 3β[fδb gδc h] + · · · ) . Now, however, there is something to check because α.β + β.α = −2αaβ a and so for consistency it must be that α:Φ(β) + β:Φ(α) = −2αaβ aΦ(1). (14) This is readily verified as follows αb:Φ(βf ) = −2 ( αbβ f + 2α[bβ [fδc] g] + · · · ) , βf :Φ(αb) = −2 ( βaαa + 2βaα[aδb] f + 3βaα[aδb fδc] g + · · · ) = −2 ( βaαa + βaαaδb f − βaαbδa f + βaαaδ[b fδc] g − 2α[bβ [fδc] g] + · · · ) =⇒ αb:Φ(βf ) + βf :Φ(αb) = −2βaαa ( 1 + δb f + δ[b fδc] g + · · · ) = −2αaβ aΦ(1). So far, we know Φ on C⊕ CW. But now we may use the desired property Φ(v.w) = v:Φ(w) for v, w ∈ CW of Φ to extend its definition to C⊕CW⊕Λ2CW: it is only necessary for consistency to check (14) and, similarly, that α:Φ(α) = 0 and β:Φ(β) = 0. Next one uses the desired property (13) to extend the definition of Φ to ⊕3 r=0 ΛrCW and so on by induction. The details are left to the reader. � Remark 3. For those readers who know the theory of Clifford algebras, we have in Λ•CW and Λ•U⊗Λ•U∗ = End(U) two different realisations of the Clifford algebra C`(CW). The mapping Φ constructed in the proof of Proposition 3 is the unique unital isomorphism between them. The geometric consequences of Proposition 3 are as follows. Recall that the Dirac operator D in the sense of Clifford analysis was built in § 3 from (5) and (6) both of which are defined in 10 M. Eastwood and J. Ryan terms of v.e for v ∈ W and e ∈ E ≡ Λ•W. We have just seen that, in case n is even, we can write CE = Λ•CW = Λ•(U⊕ U∗) = Λ•U⊗ Λ•U∗. Therefore, Proposition 3 may be viewed as providing an automorphism Φ of CE so that Φ(v.e) = v:Φ(e) ∀ v ∈ CW and e ∈ CE. In particular, this is true for v ∈ W ↪→ CW. Though it is essential to complexify W and also to choose a splitting CW = U⊕ U∗ in order to define Φ as a complex linear automorphism of CE and though it is also necessary in order to write down the formula (α + β):ω ! √ 2 ( α[aωbc···d] fg···h − pβbωbc···d fg···h ) for { α + β ∈ U⊕ U∗ = CW, ω ∈ Λ•U⊗ Λ•U∗ = CE, (15) it is not necessary for v to be complex in order that v:ω be perfectly well-defined. The key point to observe about (15) is that, when viewed on Λ•U⊗Λ•U∗, the action ω 7→ v:ω is entirely on Λ•U with Λ•U∗ as a passenger. It follows that the complexification of the representation (6) of so(n) is isomorphic to S⊗ Λ•U∗ where S is Λ•U regarded as an so(n)-module according to so(n)⊗ S = Λ2W⊗ Λ•U 3 v ∧ w ⊗ ω 7→ −1 8(v:w:ω − w:v:ω) ∈ S. This action manifestly preserves the splitting Λ•U = ΛevenU ⊕ ΛoddU. It is the standard spin representation � @ ...• • • • • 0 0 0 1 0 ⊕ � @ ...• • • • • 0 0 0 0 1 of so(n). In any case, the geometric import of these observations is that there is a complex vector bundle S defined on an arbitrary spin manifold such that the differential operator D : E → E defined in §3, when acting on complex-valued sections, becomes C⊗R E ∼= S ⊗ CN 6D⊗Id−−−−→ S ⊗ CN ∼= C⊗R E. Here CN denotes the trivial bundle of rank N = 2n/2 obtained as the induced bundle from the trivial representation of Spin(n) on U∗. In other words, the Dirac operator D from Clifford analysis may be viewed simply as 2n/2-copies of an operator 6D : S → S. The operator 6D is the classical Dirac operator. There is, however, an awkward proviso to this conclusion, namely that D and 6D should be acting on complex-valued sections. The theory of real spin-bundles is quite complicated [13]. These particular complications are simply avoided in Clifford analysis by using the real operator D instead. Similar conclusions hold in the odd-dimensional case: CE = Λ•CW ∼= ...• • • •〉0 0 0 1⊗ CN =⇒ C⊗R E ∼= S ⊗ CN and D = 6D ⊗ Id, where 6D is the classical Dirac operator in odd dimensions and N = 2(n+1)/2. In fact, all complications are avoided by using the operator D. This ‘one-size-fits-all’ approach avoids complex bundles and, at the same time, there is no need to treat even-dimensional and odd-dimensional manifolds differently. The integral formulae for monogenic functions developed in [3], for example, are obtained in a uniform dimension-free manner. The main point of this article is to provide a simple formulation and näıve computational proof of Theorem 2 and its main consequence, the conformal invariance of monogenic functions. But the conformal invariance of the classical Dirac operator is another consequence. In arbitrary dimensions, this invariance was first shown by Kosmann-Schwarzbach [14]. Monogenic Functions in Conformal Geometry 11 5 Rarita–Schwinger operators and their generalisations Having defined and established the conformal invariance of the Dirac operator in the context of Clifford analysis, it is reasonably straightforward to do the same thing for the classical Rarita– Schwinger operator [5] and its symmetric analogues [6]. Throughout this section E is to be regarded as an so(n)-module under (6). For the Rarita–Schwinger operator, define a representation F of so(n) via the exact sequence 0 → F → W⊗ E ε−→ E → 0, (16) bearing in mind that ε is a homomorphism of so(n)-modules in accordance with Proposition 1. Lemma 3 provides a canonical splitting of this sequence. Specifically, using abstract indices as in the proof of Proposition 2, we define Π : W⊗ E → F by Taα Π7−→ Taα + 1 nεaα βεc β γTcγ . Finally, to define the Rarita–Schwinger operator via Theorem 1, we consider the so(n)-module homomorphism θ : W⊗ F → F given as the composition W⊗ F ↪→ W⊗W⊗ E ε̃−→ W⊗ E Π−→ F where ε̃(w ⊗ v ⊗ e) = v ⊗ w.e or, using abstract indices, Tabα ε̃7−→ εb α βTbaβ. Let us write τ for the representation of so(n) = Λ2W on F. By construction, it is the restriction to F of the action Λ2W⊗W⊗ E 3 X ⊗ w ⊗ e 7→ Xw ⊗ e + w ⊗ σ(X ⊗ e) ∈ W⊗ E (17) on W⊗ E. Proposition 4. The composition W⊗ F ι⊗Id−−−→ W⊗ Λ2W⊗ F Id⊗τ−−−→ W⊗ F θ−→ F is equal to −n−1 2 θ : W⊗ F → F. Proof. In full, we may expand this composition as W⊗ F ↪→ W⊗W⊗ E ι⊗Id⊗Id−−−−−→ W⊗ Λ2W⊗W⊗ E Id⊗τ−−−→ W⊗W⊗ E︸︷︷︸ ? ε̃−→ W⊗ E Π−→ F and, according to (17), the homomorphism ? is the sum of two parts, namely W⊗W⊗ E ι⊗Id⊗Id−−−−−→ W⊗ Λ2W⊗W⊗ E −→ W⊗W⊗ E (18) and W⊗ W ⊗ E ι⊗Id⊗Id−−−−−→ W⊗ Λ2W⊗ W ⊗ E −→ W⊗ W ⊗ E, (19) in which the boxed vector spaces are passengers. The first one is Tabα 7→ −2ga[bTc]dα 7→ Tbaα − gabT c cα 12 M. Eastwood and J. Ryan and composing with ε̃ gives Tabα 7→ εb α β(Tabβ − gabT c cβ) = −εaα βT d dβ, when acting on W⊗ F. Evidently, this is in the kernel of Π and so (18) makes no contribution to the overall composition. On the other hand (19) can be continued to W⊗ W ⊗ E ι⊗Id⊗Id−−−−−→ W⊗ Λ2W⊗ W ⊗ E −→ W⊗ W ⊗ E ε̃−→ W ⊗ E, which has already been computed in Proposition 2. It is −n−1 2 ε̃. Therefore, composing with Π gives −n−1 2 θ, as advertised. � Theorem 3. Let F denote the bundle defined on a conformal spin manifold corresponding to the representation F of so(n). Then there is a conformally invariant first order linear differential operator F [−n−1 2 ] D−→ F [−n+1 2 ] whose symbol is induced by θ : W⊗ F → F. Proof. Proposition 4 ensures that the criterion of Theorem 1 is satisfied. � The operators in this theorem are the Rarita–Schwinger operators in the context of Clifford analysis. The symmetric analogues of [6] are obtained by a similar construction starting with the exact sequence 0 → Fj → ⊙jW⊗ E Id⊗ε−−−→ ⊙j−1W⊗ E → 0, generalising (16). Details are left to the reader. 6 Flat structures and comparison of notations The operators that we have constructed in previous sections are conformally invariant in the ‘curved’ setting. Explicitly, this means that for each operator there is a universal formula in terms of a chosen Riemannian metric and its Levi-Civita connection so that, for an arbitrary metric, using the same formula with any metric in the same conformal class gives the same operator. This is a very strong notion of invariance. There are several weaker notions one of which is to restrict attention to conformally flat metrics, asking only for a local formula in terms of a flat metric from the conformal class and only that the result be invariant under arbitrary flat-to-flat conformal rescalings. Evidently, an invariant operator in the fully curved sense gives rise to an invariant operator in the conformally flat sense as just defined. That curved invariance is strictly stronger, however, was demonstrated by Graham [12] who showed that the operator ∆3 : Λ0[1] → Λ0[−5] on R4, where ∆ is the Laplacian for the standard flat metric on R4 does not arise from a curved invariant operator although it is easily verified to be invariant under flat-to-flat conformal rescalings (see also [11]). In any case, the Dirac operator is conformally invariant in the curved case and therefore in- variant under flat-to-flat conformal rescalings. Such rescalings are scarce. In fact, the only way that they can arise is by so-called ‘Möbius transformations’, i.e. the transformations obtained from the action of SO(n + 1, 1) on the n-sphere and viewed in Rn by stereographic projection. Hence, monogenic functions locally defined on Rn are invariant under Möbius transformations. Monogenic Functions in Conformal Geometry 13 This is a well-known phenomenon from Clifford analysis usually attributed to the invariance of the generalised Cauchy integral formula [3] on Rn under Möbius transformations. Mono- genic functions on conformally flat spin manifolds such as cylinders and tori provide a natural extension [15, 16] of automorphic forms from the realm of complex analysis. The n-sphere is a spin manifold homogeneous under its conformal motions, finitely covered by the group G = SO(n + 1, 1) (as indicated below and fully explained, for example, in [9]). It follows that the Dirac operator is G-invariant on Sn (some authors write G-equivariant). This is yet another sense of ‘conformal invariance’ and a notion that is purely algebraic – the question of classi- fying the G-invariant operators on Sn acting between irreducible homogeneous vector bundles is a (solved) question in representation theory. It is this sense of conformal invariance that is in effect throughout the following discussion. Finally, for completeness, we present the conformal invariance of the Dirac operator on the flat model in terms of the Dynkin diagram notation of [1]. The conformal sphere Sn is a homogeneous space for the Lie group SO(n + 1, 1). For convenience we shall take this to mean the connected component of the group of (n + 2)× (n + 2) matrices preserving the quadratic form 0 0 1 0 Id 0 1 0 0  where Id denotes the n × n identity matrix. This group acts on Sn as its conformal motions and, for a suitable choice of basepoint, Sn = SO(n+1, 1)/P where P is the subgroup consisting of matrices of the form λ · · 0 M · 0 0 λ−1  , for { λ > 0 M ∈ SO(n). The irreducible real homogeneous bundles on Sn are thus parameterised by the representations λ · · 0 M · 0 0 λ−1  7→ λ−wρ(M) for w ∈ R and ρ an irreducible real representation of SO(n). It is convenient to specify such a representation of P by recording its highest (or lowest) weight as a linear combination of the fundamental weights for so(n + 1, 1). With the conventions of [1] we find, for example, that Λ0[w] =× • • • •〉 w 1 0 0 0 Λ1 =× • • • •〉 −2 1 0 0 0 Λ2 =× • • • •〉 −3 0 1 0 0 Λ1[w] = × • • • •〉 w−2 1 0 0 0 on S9 and Λ0[w] = � @ × • • • • • w 1 0 0 0 0 Λ1 = � @ × • • • • • −2 1 0 0 0 0 Λ2 = � @ × • • • • • −3 0 1 0 0 0 Λ1[w] = � @ × • • • • • w−2 1 0 0 0 0 on S10. On odd-dimensional spheres Sn the Dirac operator acts as ...× • • • •〉 −n/2 0 0 0 1 −→ ...× • • • •〉 −n/2−1 0 0 0 1 . On even-dimensional spheres, however, there are two irreducible Dirac operators: � @ ...× • • • • • −n/2 0 0 0 1 0 −→ � @ ...× • • • • • −n/2−1 0 0 0 0 1 and � @ ...× • • • • • −n/2 0 0 0 0 1 −→ � @ ...× • • • • • −n/2−1 0 0 0 1 0 . In odd dimensions the generalised Rarita–Schwinger operators of [6] act as ...× • • • • •〉 −n/2−j j 0 0 0 1 −→ ...× • • • • •〉 −n/2−j−1 j 0 0 0 1 . 14 M. Eastwood and J. Ryan Acknowledgements It is a pleasure to acknowledgment useful conversations with Vladimı́r Souček. He is certainly one person for whom the ‘well-known’ material in this article is actually known. Michael Eastwood is a Professorial Fellow of the Australian Research Council. This research was begun during a visit by John Ryan to the University of Adelaide in 2005, which was also supported by the Australian Research Council. This support is gratefully acknowledged. John Ryan also thanks the University of Adelaide for hospitality during his visit. References [1] Baston R.J., Eastwood M.G., The Penrose transform: its interaction with representation theory, Oxford University Press, 1989. [2] Baston R.J., Eastwood M.G., Invariant operators, in Twistors in Mathematics and Physics, London Math. Soc. Lecture Note Ser., Vol. 156, Cambridge University Press, 1990, 129–163. [3] Brackx F., Delanghe R., Sommen F., Clifford analysis, Research Notes in Math., Vol. 76, Pitman, 1982. [4] Branson T.P., Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293–345. [5] Bureš J., The Rarita–Schwinger operator and spherical monogenic forms, Complex Variables Theory Appl. 43 (2000), 77–108. [6] Bureš J., Sommen F., Souček V., Van Lancker P., Symmetric analogues of Rarita–Schwinger equations, Ann. Global Anal. Geom. 21 (2002), 215–240. [7] Calderbank D.M.J., Dirac operators and Clifford analysis on manifolds with boundary, Preprint no. 53, Department of Mathematics and Computer Science, University of Southern Denmark, 1997, available at http://bib.mathematics.dk/preprint.php?lang=en&id=IMADA-PP-1997-53. [8] Čap A., Slovák J., Souček V., Invariant operators on manifolds with almost Hermitian symmetric structures, III. Standard operators, Diff. Geom. Appl. 12 (2000), 51–84, math.DG/9812023. [9] Eastwood M.G., Graham C.R., Invariants of conformal densities, Duke Math. J. 63 (1991), 633–671. [10] Fegan H.D., Conformally invariant first order differential operators, Quart. J. Math. Oxford (2) 27 (1976), 371–378. [11] Gover A.R., Hirachi K, Conformally invariant powers of the Laplacian – a complete nonexistence theorem, J. Amer. Math. Soc. 17 (2004), 389–405, math.DG/0304082. [12] Graham C.R., Conformally invariant powers of the Laplacian II: nonexistence, J. London Math. Soc. (2) 46 (1992), 566–576. [13] Harvey F.R., Spinors and calibrations, Academic Press, 1990. [14] Kosmann-Schwarzbach Y., Propriétés des dérivations de l’algèbre des tenseurs-spineurs, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A355–A358. [15] Kraußhar R.S., Ryan J., Clifford and harmonic analysis on cylinders and tori, Rev. Mat. Iberoamericana 21 (2005), 87–110. [16] Kraußhar R.S., Ryan J., Some conformally flat spin manifolds, Dirac operators and automorphic forms, J. Math. Anal. Appl. 325 (2007), 359–376, math.AP/0212086. [17] Penrose R., Rindler W., Spinors and space-time, Vol. 1, Cambridge University Press, 1984. [18] Woodhouse N.M.J., Geometric quantization, Oxford University Press, 1980. http://bib.mathematics.dk/preprint.php?lang=en&id=IMADA-PP-1997-53 http://arxiv.org/abs/math.DG/9812023 http://arxiv.org/abs/math.DG/0304082 http://arxiv.org/abs/math.AP/0212086 1 Introduction 2 Conformally invariant first order operators 3 Monogenic functions and Dirac operators 4 Clifford algebras and the classical Dirac operator 5 Rarita-Schwinger operators and their generalisations 6 Flat structures and comparison of notations References

Monogenic Functions in Conformal Geometry

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