Symmetries and Invariant Differential Pairings

The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings, explains how they naturally arise, and formulates various associat...

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Datum:	2007
1. Verfasser:	Eastwood, M.G.
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spelling	nasplib_isofts_kiev_ua-123456789-1472232025-02-09T14:15:54Z Symmetries and Invariant Differential Pairings Eastwood, M.G. The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings, explains how they naturally arise, and formulates various associated problems. This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). It is a pleasure to acknowledge many extremely useful conversations with Vladim´ır Souˇcek and Jens Kroeske. Support from the Australian Research Council is also gratefully acknowledged. 2007 Article Symmetries and Invariant Differential Pairings / M.G. Eastwood // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 19 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53A30; 58J70; 53A20 https://nasplib.isofts.kiev.ua/handle/123456789/147223 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings, explains how they naturally arise, and formulates various associated problems.
format	Article
author	Eastwood, M.G.
spellingShingle	Eastwood, M.G. Symmetries and Invariant Differential Pairings Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Eastwood, M.G.
author_sort	Eastwood, M.G.
title	Symmetries and Invariant Differential Pairings
title_short	Symmetries and Invariant Differential Pairings
title_full	Symmetries and Invariant Differential Pairings
title_fullStr	Symmetries and Invariant Differential Pairings
title_full_unstemmed	Symmetries and Invariant Differential Pairings
title_sort	symmetries and invariant differential pairings
publisher	Інститут математики НАН України
publishDate	2007
url	https://nasplib.isofts.kiev.ua/handle/123456789/147223
citation_txt	Symmetries and Invariant Differential Pairings / M.G. Eastwood // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 19 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT eastwoodmg symmetriesandinvariantdifferentialpairings
first_indexed	2025-11-26T17:52:35Z
last_indexed	2025-11-26T17:52:35Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 113, 10 pages Symmetries and Invariant Differential Pairings? Michael G. EASTWOOD Department of Mathematics, University of Adelaide, SA 5005, Australia E-mail: meastwoo@member.ams.org Received November 14, 2007; Published online November 23, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/113/ Abstract. The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings, explains how they naturally arise, and formulates various associated problems. Key words: conformal invariance; differential pairing; symmetry 2000 Mathematics Subject Classification: 53A30; 58J70; 53A20 1 Introduction The Lie derivative is an extremely familiar operation in differential geometry. Given a smooth vector field V and tensor field φ, the Lie derivative LV φ is certainly an intrinsic or invariant construction. In practise, we can take this to mean that the result is independent of any local coördinate formula that may be used. Alternatively, we may employ an arbitrary torsion-free affine connection ∇ to write explicit formulae such as φbc LV7−→ V a∇aφbc + (∇bV a)φac + (∇cV a)φba, (1) checking that the result is independent of the connection used. In this formula, the indices are ‘abstract indices’ in the sense of Penrose [18]. It follows that the expression is coördinate-free whilst its freedom from choice of connection follows immediately from the formulae ∇̂aV c = ∇aV c + Γab cV b and ∇̂aφbc = ∇aφbc − Γab dφdc − Γac dφbd, (2) capturing all possible torsion-free connections via the choice of tensor Γab c = Γba c. For further details on this point of view, see [18]. Writing the Lie derivative as LV φ suggests that we are regarding it as a linear differential operator acting on the tensor field φ for a fixed choice of vector field V . However, it is clear from expressions such as (1) that we may equally well fix φ and regard the result as a linear differential operator acting on vector fields V . Sometimes, this is the natural viewpoint. For example, if φab is symmetric then we may rewrite (1) as (LV φ)bc = φbc∇aV c + φac∇bV c + (∇cφab)V c and now if φab is a metric and we choose ∇a to be its Levi-Civita connection, then LV φ = 0 ⇐⇒ ∇aVb +∇bVa = 0, ?This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The full collection is available at http://www.emis.de/journals/SIGMA/symmetry2007.html mailto:meastwoo@member.ams.org http://www.emis.de/journals/SIGMA/2007/113/ http://www.emis.de/journals/SIGMA/symmetry2007.html 2 M.G. Eastwood which is the usual way of viewing the Killing equation on a Riemannian manifold. Of course, the best way of viewing LV φ is as a bilinear differential operator or as a differential pairing. It is invariantly defined on any manifold and is an example of the subject matter of this article. One can easily imagine other expressions along the lines of the right hand side of (1) that turn out to be independent of choice of connection. Restricting the choice of connection leads to a greater range of expressions that might be invariant. In the next two sections we shall find examples that are invariant when the connections are deemed to be Levi-Civita connections for metrics taken from a fixed conformal class. 2 First order symmetries Suppose L : E → F is a linear differential operator between vector bundles E and F . A linear differential operator D : E → E is said to be a symmetry of L if and only if LD = δ L for some linear differential operator δ : F → F. When L is the Laplacian on R3 (and the bundles E and F are trivial), its first order symmetries are well-known [17]. Specifically, they comprise an 11-dimensional vector space spanned by the following f 7→ f, f 7→ ∂f ∂x , f 7→ ∂f ∂y , f 7→ ∂f ∂z , f 7→ x ∂f ∂x + y ∂f ∂y + z ∂f ∂z , f 7→ x ∂f ∂y − y ∂f ∂x , f 7→ (x2 − y2 − z2) ∂f ∂x + 2xy ∂f ∂y + 2xz ∂f ∂z + xf, f 7→ y ∂f ∂z − z ∂f ∂y , f 7→ (y2 − z2 − x2) ∂f ∂y + 2yz ∂f ∂z + 2yx ∂f ∂x + yf, f 7→ z ∂f ∂x − x ∂f ∂z , f 7→ (z2 − x2 − y2) ∂f ∂z + 2zx ∂f ∂x + 2zy ∂f ∂y + zf, (3) where x, y, z are the usual Euclidean coördinates on R3. Following [10], it is convenient to rewrite the general first order symmetry as f 7→ V a∇af + 1 6(∇aV a)f + Cf, where V a is an arbitrary vector field of the form V a = −sa −ma bx b + λxa + rbx bxa − 1 2xbx bra (4) and C is an arbitrary constant. In fact, vector fields of the form (4) on Rn are precisely the conformal Killing fields, i.e. the solutions of the equation ∇aVb +∇bVa = 2 ngab∇cVc, (5) where gab is the usual Euclidean metric on Rn. More generally, on Rn the first order symmetries of the Laplacian may be written as f 7→ DV f + Cf , where DV f ≡ V a∇af + n−2 2n (∇aV a)f (6) for an arbitrary conformal Killing field V a. This expression certainly resembles the Lie derivative and we now make this precise as follows. If φbc···de is an n-form, then 0 = (n+ 1)(∇[aV a)φbc···de] = (∇aV a)φbc···de − [ (∇bV a)φac···de + · · ·+ (∇eV a)φbc···da ] , Symmetries and Invariant Differential Pairings 3 where we are following [18] in employing square brackets to denote the result of skewing over the indices they enclose and also noting that a skew tensor with n+ 1 indices on Rn necessarily vanishes. Hence, the Lie derivative on n-forms simplifies: LV φbc···de = V a∇aφbc···de + (∇bV a)φac···de + · · ·+ (∇eV a)φbc···da = V a∇aφbc···de + (∇aV a)φbc···de. Thus, if h is a section of the bundle Λn of n-forms, then LV h = V a∇ah+ (∇aV a)h. Therefore, the expression (6) coincides with the Lie derivative if f is interpreted as a section of the line bundle (Λn)(n−2)/2n (supposing that the manifold is orientable so that such fractional powers are allowed). In particular, we deduce that [DV ,DW ] = D[V,W ], as is also readily verified by direct computation. Also we conclude that, with this interpretation, the differential pairing DV f between V a and f is invariant in the sense discussed for Lie derivative in Section 1. 3 Conformal geometry An alternative viewpoint on these matters is as follows. If ∇a is the Levi-Civita connection for a metric gab, then the Levi-Civita connection ∇̂a for the conformally related metric ĝab = Ω2gab is given on 1-forms by ∇̂aφb = ∇aφb −Υaφb −Υbφa + Υcφcgab, (7) where Υa = ∇a log Ω and Υc = gcdΥd. Thus, associated with a conformal class of metrics we have a restricted supply of torsion-free connections related by formulae such as (2) where Γab c = Υaδb c + Υbδa c − gabΥc (8) for closed 1-forms Υa. It is convenient introduce a line bundle L on a general conformal manifold as follows. A choice of metric gab in the conformal class trivialises L, allowing us to view a section of L as a function f . We decree, however, that viewing the same section by means of the trivialisation due to ĝab = Ω2gab gives us the function f̂ = Ωf . More generally, sections of the bundle Lw transform by f̂ = Ωwf and are called conformal densities of conformal weight w. Since ∇af̂ = ∇a(Ωwf) = Ωw(∇af + wΥaf) we may view Lw as equipped with a family of connections related by ∇̂af = ∇af + wΥaf. (9) On an oriented conformal n-manifold, the line bundle Λn is also trivialised by a choice of metric. Specifically, we may use the inverse metric gab to raise indices and then normalise the volume form εbc···de so that εbc···deεbc···de = n!. Since ĝab = Ω−2gab it follows that ε̂bc···de = Ωnεbc···de and hence that we may identify Λn = L−n. It is easily confirmed that (7) implies ∇̂aφbc···de = ∇aφbc···de − nΥaφbc···de for n-forms φbc···de, which is consistent with (9) for w = −n, as it should be. Combining (7) and (9) gives us a formula for the change of connection on Λ1 ⊗ Lw, namely ∇̂aφb = ∇aφb + (w − 1)Υaφb −Υbφa + Υcφcgab. 4 M.G. Eastwood Let us now consider how the Laplacian ∆ = ∇a∇a acting on densities of weight w is affected by a conformal rescaling of the metric. We compute: ∇̂a∇̂bf = ∇a∇̂bf + (w − 1)Υa∇̂bf −Υb∇̂af + (Υc∇̂cf)gab and so ∆̂f = ĝab∇̂a∇̂bf = Ω−2gab ( ∇a∇̂bf + (w − 1)Υa∇̂bf −Υb∇̂af + (Υc∇̂cf)gab ) = Ω−2 ( ∇a∇̂af + (n+ w − 2)Υa∇̂af ) = Ω−2 ( ∇a(∇af + wΥaf) + (n+ w − 2)Υa(∇af + wΥaf) ) . Here, we are using gab to raise indices on the right hand side. Regrouping as ∆̂f = Ω−2 ( ∆f + (n+ 2w − 2)Υa∇af + w(∇aΥa + (n+ w − 2)ΥaΥa)f ) , we see that when w = 1− n/2 there are no first order derivatives in f and ∆̂f = Ω−2 ( ∆f − (n/2− 1)(∇aΥa + (n/2− 1)ΥaΥa)f ) . (10) On the other hand, the Riemann curvature tensor transforms by R̂abcd = Ω2(Rabcd − Ξacgbd + Ξbcgad − Ξbdgac + Ξadgbc) where Ξab ≡ ∇aΥb −ΥaΥb + 1 2ΥcΥcgab. It follows immediately that the scalar curvature R = gacgcdRabcd transforms by R̂ = Ω−2 ( R− 2(n− 1)(∇aΥa + (n/2− 1)ΥaΥa) ) . (11) From (10) and (11) we conclude that ∆̂f − n−2 4(n−1)R̂f = Ω−2 ( ∆f − n−2 4(n−1)Rf ) . If we also absorb the factor of Ω−2 into a change of conformal weight, then we conclude that Y ≡ ∆− n−2 4(n−1)R is a conformally invariant differential operator Y : L1−n/2 → L−1−n/2. This is the conformal Laplacian or Yamabe operator. The peculiar multiple of the scalar curvature that must be added to the Laplacian to achieve conformal invariance is often referred to a curvature correction term. Now we are in a position to compare with the first order symmetries of the Laplacian on Rn found in Section 2. Of course, the curvature correction terms are absent for the flat metric on Rn. Therefore, there is no difference between the Laplacian and the Yamabe operator provided we restrict our attention to conformal rescalings that take the flat metric to another flat metric. But flat-to-flat rescalings are precisely what are provided by the conformal Killing fields (5). Specifically, the conformal Killing fields are those vector fields whose flows preserve the conformal structure on Rn and, moreover, the flat-to-flat conformal factors generated in this way are general [2]. The conformal invariance of the Yamabe operator on a general conformal manifold therefore implies the invariance of the Laplacian on Rn under Lie derivative LV for conformal Killing fields V . Here, as we have just seen, LV should be interpreted as acting on the bundles L1−n/2 and L−1−n/2. At the end of Section 2 we found that the symmetry DV given by (6) may be regarded as the Lie derivative on (Λn)(n−2)/2n. These viewpoints now coincide because Λn = L−n =⇒ (Λn)(n−2)/2n = L1−n/2. The same reasoning applies more generally. The geometric interpretation of conformal Killing fields combines with the conformal invariance of the Yamabe operator and we have proved: Symmetries and Invariant Differential Pairings 5 Theorem 1. Suppose that V a is a conformal Killing field on a Riemannian manifold, i.e. a solution of the equation (5). Then DV given by (6) is a symmetry of the Yamabe operator. More precisely,( ∆− n−2 4(n−1)R ) DV = δV ( ∆− n−2 4(n−1)R ) , where DV = V a∇a + n−2 2n (∇aV a) and δV = V a∇a + n+2 2n (∇aV a). In fact, it follows from proof of [10, Theorem 1] that these are the only symmetries of the Yamabe operator. We have already observed that the differential pairing DV f given by (6) is invariant in the very strong sense of being intrinsically defined on any manifold. Conformal invariance is a weaker statement but one that is easily verified directly. From (2) and (8), ∇̂aV b = ∇aV b + ΥaV b −ΥbVa + ΥcV cδa b =⇒ ∇̂aV a = ∇aV a + nΥaV a whilst for f a density of weight w we have (9). It follows immediately that V a∇af − w n (∇aV a)f (12) is a conformally invariant differential pairing and for w = 1− n/2 this agrees with (6). 4 Higher order symmetries There are some second order symmetries of the Laplacian on Rn that should be regarded as trivial. If D = h∆ for some smooth function h, then ∆D = δ∆ simply by taking δf = ∆(hf). All symmetries preserve harmonic functions but D = h∆ does this by dint of annihilating them. Ignoring these trivial symmetries, the second order symmetries of the Laplacian in R3 were found by Boyer, Kalnins, and Miller [3]. They add an extra 35 dimensions to the 11-dimensional space (3). In [10] all higher symmetries were found in all dimensions and, ignoring the trivial ones, there is an additional finite-dimensional space of second order ones. More precisely, there are second order symmetries DV f = V ab∇a∇bf + n n+2(∇aV ab)∇bf + (n−2)n 4(n+1)(n+2)(∇a∇bV ab)f (13) for any trace-free symmetric tensor V ab satisfying ∇aVbc +∇bVca +∇cVab = 2 n+2 ( gab∇dVcd + gbc∇dVad + gca∇dVbd ) . (14) Such V ab are called conformal Killing tensors (of valence 2) and on Rn form a vector space of dimension (n−1)(n+2)(n+3)(n+4) 12 . As detailed in [10], this is the ‘additional space’ of second order symmetries referred to above. As with first order symmetries (6), the pairing between valence 2 symmetric trace-free tensors V ab and densities f of weight 1−n/2 given by the right hand side of (13) is conformally invariant under flat-to-flat conformal rescalings. Unlike the first order case, however, the pair- ing (13) is not invariant under general conformal rescalings of a general Riemannian manifold 6 M.G. Eastwood but, like the Laplacian, becomes so with the addition of suitable curvature correction terms. Specifically, for V ab any trace-free symmetric contravariant tensor and f of weight w V ab∇a∇bf − 2(w−1) n+2 (∇aV ab)∇bf + w(w−1) (n+1)(n+2)(∇a∇bV ab)f + w(n+w) (n+1)(n−2)RabV abf, (15) where Rab is the Ricci tensor, is a conformally invariant differential pairing on an arbitrary Riemannian manifold. When w = 1− n/2, we obtain DV f ≡ V ab∇a∇bf + n n+2(∇aV ab)∇bf + (n−2)n 4(n+1)(n+2)(∇a∇bV ab)f − n+2 4(n+1)RabV abf as a conformally invariant analogue of the flat symmetry operator (13). Whether this operator actually provides a symmetry of the Yamabe operator in general, however, is unclear. By analogy with Theorem 1, one might hope that if V ab were a conformal Killing tensor (14), then it would follow that( ∆− n−2 4(n−1)R ) DV = δV ( ∆− n−2 4(n−1)R ) (16) for DV as above and δV f ≡ V ab∇a∇bf + n+4 n+2(∇aV ab)∇bf + n+4 4(n+1)(∇a∇bV ab)f − n+2 4(n+1)RabV abf. This is currently unknown. The problem is that there does not seem to be any useful geometric interpretation of V ab being a conformal Killing tensor in parallel to V a being a conformal Killing field. In principle, the validity or otherwise of (16) should boil down to a calculation once the differential consequences of (14) are determined. Furthermore, the differential consequences of (14) can all be found by prolonging the system, as described in [5] for example. Nevertheless, it will surely be a difficult calculation. 5 Further examples of invariant pairings The simple conformally invariant pairings we have found so far, namely (12) and (15), may be extended by allowing V a and V ab to have a general conformal weight as follows. It is convenient to set Φab ≡ 1 n− 2 ( Rab − 1 nRgab ) . Proposition 1. Suppose that V a has conformal weight v and f has conformal weight w. Then (v + n)V a∇af − w(∇aV a)f (17) is conformally invariant. Suppose V ab is trace-free symmetric with conformal weight v and f has conformal weight w. Then (n+ v + 2)(n+ v + 1)V ab∇a∇bf − 2(w − 1)(n+ v + 1)(∇aV ab)∇bf + w(w − 1)(∇a∇bV ab)f + w(n+ v + w)(n+ v + 2)ΦabV abf (18) is conformally invariant. Proof. These are easily verified using the formulae for conformal rescaling developed in Sec- tion 3. � Symmetries and Invariant Differential Pairings 7 Notice that there are several special cases occurring for particular values of v and w. The form of (17) suggests setting w = 0 or v = −n and, when we do, we find (v + n)V a∇af is invariant ∀V a =⇒ f 7→ ∇af is invariant and −w(∇aV a)f is invariant ∀ f =⇒ V a 7→ ∇aV a is invariant. These are familiar invariant linear differential operators. The first one is the exterior derivative d : Λ0 → Λ1 and the second may be identified as d : Λn−1 → Λn. More interesting are the consequences of setting w = 1 and v = −n − 1 in (18). We obtain conformally invariant linear differential operators f 7→ ∇a∇bf − 1 n(∆f)gab + Φabf and V ab 7→ ∇a∇bV ab + ΦabV ab, respectively. The theory of conformally invariant linear differential operators is well understood [1, 4, 12, 14] and includes these two as simple examples. Other special values of the weights v and w show up as zeroes of the coefficients in (18). We have already seen that for w = 0 there is an invariant differential operator f 7→ ∇af . Similarly, for v = −n − 2 the operator V ab 7→ ∇aV ab is conformally invariant. It is also interesting to note that, conversely, we can build the invariant pairing (18) from these various linear invariant operators. Specifically, if V ab has weight v and f has weight w 6= 0 then, at least where f does not vanish, f−(n+v+1)/wV ab has weight − n− 1 and f−(n+v+2)/wV ab has weight − n− 2. Therefore, ∇a∇b(f−(n+v+1)/wV ab) + Φabf −(n+v+1)/wV ab and ∇b(f−(n+v+2)/wV ab) are invariant. If the second of these is multiplied by f2/w we obtain a vector field of weight −n and conclude that ∇a ( f2/w∇b(f−(n+v+2)/wV ab) ) is also invariant. It follows, therefore, that the combination w(n+ v + 2)(n+ v + w)f (n+v+1+w)/w ( ∇a∇b(f−(n+v+1)/wV ab) + Φabf −(n+v+1)/wV ab ) − w(n+ v + 1)(n+ v + 1 + w)f (n+v+w)/w ( ∇a ( f2/w∇b(f−(n+v+2)/wV ab) )) is invariant. A short calculation reveals that this expression agrees with (18) and, although we supposed w 6= 0 and also that f 6= 0 in carrying out this derivation, the final conclusion is valid regardless. These tricks reveal some sort of relationship between invariant differential pairings and invariant linear differential operators, at least when one of the quantities to be paired is a conformal density. These tricks are unavailable more generally but there are, nevertheless, many conformally in- variant differential pairings. For example, if V ab is symmetric trace-free with conformal weight v and φa has conformal weight w, then (n+ v + 2)V ab∇aφb − (w − 2)(∇aV ab)φb. is invariant. Operators such as this one and (17) are referred to as first order , not only because there are no higher derivatives involved than first order but also because there are no cross terms 8 M.G. Eastwood of the form ∇V ./ ∇φ where ./ denotes some algebraic pairing (we shall come back to this point soon). The theory of first order invariant differential pairings on a conformal manifold, or more generally on a manifold with AHS structure, is reasonably well understood thanks to recent work of Kroeske [16]. Following the approach of Fegan [14] in the conformal case and Čap, Slovák, and Souček [7] more generally, Kroeske classifies the first order invariant pairings provided that the weights of the tensors involved are so that invariant linear differential operators are excluded. This corresponds well to the relationship that seemed to be emerging in our examples. For higher order pairings, however, Kroeske [16] finds additional unexpected pairings and is able to present a satisfactory theory only in the flat projective setting. 6 Some precise formulations When E and F are smooth vector bundles on some smooth manifold, it is well-known that linear differential operators D : E → F of order ≤ k are in 1–1 correspondence with vector bundle homomorphisms JkE → F where JkE is the kth jet bundle of E. Indeed, it is usual to abuse notation and also write D : JkE → F , sometimes adopting this viewpoint as the definition of a linear differential operator. There are canonical short exact sequences of vector bundles 0 → ⊙kΛ1 ⊗ E → JkE → Jk−1E → 0, (19) where ⊙kΛ1 is the bundle of symmetric covariant tensors of valence k and the composition⊙kΛ1 ⊗ E → JkE D−→ F is called the symbol of D. For further details, see [19] for example. Differential pairings may be formulated similarly. Suppose another vector bundle V is given and we wish to define what is a differential pairing E × F → V . The jet exact sequences (19) may be written as JkE = E + Λ1 ⊗ E + ⊙2Λ1 ⊗ E + ⊙3Λ1 ⊗ E + · · · + ⊙kΛ1 ⊗ E, meaning that JkE is filtered with successive quotients as shown. It follows that JkE ⊗ JkF = E ⊗ F + Λ1 ⊗ E ⊗ F ⊕ E ⊗ Λ1 ⊗ F + ⊙2Λ1 ⊗ E ⊗ F ⊕ Λ1 ⊗ E ⊗ Λ1 ⊗ F ⊕ E ⊗ ⊙2Λ1 ⊗ F + · · · and, following [16], we define the bi-jet bundle Jk(E,F ) as the quotient of JkE ⊗ JkF corre- sponding to the first k + 1 columns of this composition series. There are bi-jet exact sequences starting with 0 → Λ1 ⊗ E ⊗ F ⊕ E ⊗ Λ1 ⊗ F → J1(E,F ) → E ⊗ F → 0 and 0 → ⊙2Λ1 ⊗ E ⊗ F ⊕ Λ1 ⊗ E ⊗ Λ1 ⊗ F ⊕ E ⊗ ⊙2Λ1 ⊗ F → J2(E,F ) → J1(E,F ) → 0. Symmetries and Invariant Differential Pairings 9 A differential pairing E × F → V of order k may now defined as a homomorphism of vector bundles Jk(E,F ) → V . Notice that it is the total order of the operator that is constrained to be less than of equal to k. For first order pairings, for example, we are excluding terms of the form ∇ψ ./ ∇φ (written in the presence of chosen connections). For second order pairings we exclude ∇2ψ ./ ∇φ, ∇ψ ./ ∇2φ, and ∇2ψ ./ ∇2φ. The symbol of a pairing is defined as the composition k⊕ j=0 ⊙jΛ1 ⊗ E ⊗ ⊙k−jΛ1 ⊗ F −→ Jk(E,F ) −→ V. Several other basic notions for linear differential operators immediately carry over to bilinear differential pairings. Suppose, for example, that we consider homogeneous vector bundles on a homogeneous space G/P . If E and F are induced from P -modules E and F, then Jk(E,F ) will also be induced from a P -module, say Jk(E,F). If V is induced from V, a G-invariant differential pairing will correspond to a P -module homomorphism Jk(E,F) → V. In particular, whenG = SO(n+1, 1) andG/P is the conformal sphere (as detailed in [9] for example) then these P -module homomorphisms correspond to flat-to-flat invariant differential pairings. This gives an approach to classification as adopted in [16]. Nevertheless, it remains a difficult problem. 7 Other sources of invariant pairings Firstly, there are other conformally invariant differential operators whose symmetries give rise to invariant differential pairings. The square of the Laplacian is treated in [11] and the Dirac operator in [13]. There is also another source of pairings derived from the structure of the symmetry algebra as follows. Let us consider the symmetries of the Laplacian as in Sections 2 and 4. It is clear that the composition of symmetries is again a symmetry. In particular, if we compose two first order symmetries (6) on Rn then we find [10] DVDW f = DV }W f + 1 2D[V,W ]f − n−2 4n(n+1)〈V,W 〉f + 1 nV aWa∆f where DV }W is given by (13) and (V }W )ab = 1 2V aW b + 1 2V bW a − 1 ng abV cWc, [V,W ]a = V b∇bV a −W b∇bV a, 〈V,W 〉 = (n+ 2)(∇bV a)(∇aW b)− n+2 n (∇aV a)(∇bW b)− n+2 n V a∇a∇bW b − n+2 n W a∇a∇bV b + ∆(VaW a). The last of these is constant when V a and W a are Killing fields on Rn and in this case coincides with their inner product under the Killing form in so(n+1, 1). Remarkably, there is a curvature corrected version (n+ 2)(∇bV a)(∇aW b)− n+2 n (∇aV a)(∇bW b)− n+2 n V a∇a∇bW b − n+2 n W a∇a∇bV b + ∆(VaW a)− 2(n+2) (n−2) RabV aW b + 2n (n−1)(n−2)RVaW a that provides an invariant differential pairing on arbitrary vector fields. This approach to constructing invariant pairings is sometimes called quantisation: the symbol of an operator is specified, one attempts to build an invariant operator with this symbol, and then one composes these operators. Further examples are given by Duval and Ovsienko [8] in the conformal case and Fox [15] in the projective case. Finally, there is a general construction of invariant pairings for very particular weights called cup products by Calderbank and Diemer [6]. This construction applies in any parabolic geometry but further discussion is beyond the scope of this article. 10 M.G. Eastwood Acknowledgements It is a pleasure to acknowledge many extremely useful conversations with Vladimı́r Souček and Jens Kroeske. Support from the Australian Research Council is also gratefully acknowledged. References [1] 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. [2] Baird P., Wood J.C., Harmonic morphisms between Riemannian manifolds, Oxford University Press, 2003. [3] Boyer C.P., Kalnins E.G., Miller W. Jr., Symmetry and separation of variables for the Helmholtz and Laplace equations, Nagoya Math. J. 60 (1976), 35–80. [4] Branson T.P., Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293–345. [5] Branson T.P., Čap A., Eastwood M.G., Gover A.R., Prolongations of geometric overdetermined systems, Internat. J. Math. 17 (2006), 641–664, math.DG/0402100. [6] Calderbank D.M.J., Diemer T., Differential invariants and curved Bernstein–Gelfand–Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67–103, math.DG/0001158. [7] Čap A., Slovák J., Souček V., Invariant operators on manifolds with almost Hermitian symmetric structures. III. Standard operators, Differential Geom. Appl. 12 (2000), 51–84, math.DG/9812023. [8] Duval C., Ovsienko V., Conformally equivariant quantum Hamiltonians, Selecta Math. 7 (2001), 291–230. [9] Eastwood M.G., Notes on conformal differential geometry, Rendi. Circ. Mat. Palermo Suppl. 43 (1996), 57–76. [10] Eastwood M.G., Higher symmetries of the Laplacian, Ann. Math. 161 (2005), 1645–1665, hep-th/0206233. [11] Eastwood M.G., Leistner T., Higher symmetries of the square of the Laplacian, in Symmetries and Overde- termined Systems of Partial Differential Equations, IMA Volumes, no. 144, Springer Verlag, 2007, 319–338, math.DG/0610610. [12] Eastwood M.G., Rice J.W., Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys. 109 (1987), 207–228, Erratum, Comm. Math. Phys. 144 (1992), 213. [13] Eastwood M.G., Somberg P., Souček V., Symmetries of the Dirac operator, in preparation. [14] Fegan H.D., Conformally invariant first order differential operators, Quart. J. Math. 27 (1976), 371–378. [15] Fox D.J.F., Projectively invariant star products, Int. Math. Res. Not. 2005 (2005), no. 9, 461–510, math.DG/0504596. [16] Kroeske J., Invariant differential pairings, Acta Math. Univ. Comenian., to appear, math.DG/0703866. [17] Miller W. Jr., Symmetry and separation of variables, Addison-Wesley, 1977. [18] Penrose R., Rindler W., Spinors and space-time, Vol. 1, Cambridge University Press, 1984. [19] Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179–239. http://arxiv.org/abs/math.DG/0402100 http://arxiv.org/abs/math.DG/0001158 http://arxiv.org/abs/math.DG/9812023 http://arxiv.org/abs/hep-th/0206233 http://arxiv.org/abs/math.DG/0610610 http://arxiv.org/abs/math.DG/0504596 http://arxiv.org/abs/math.DG/0703866 1 Introduction 2 First order symmetries 3 Conformal geometry 4 Higher order symmetries 5 Further examples of invariant pairings 6 Some precise formulations 7 Other sources of invariant pairings References

Symmetries and Invariant Differential Pairings

Institution

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