GEOMETRIC QUANTIZATION AND NO-GO THEOREMS

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1 POISSON GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 51 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2000 GEOMETRIC QUANTIZATION AND NO-GO THEOREMS VIKTOR L. GINZB URG and RI CHARD MONTGOMERY Department o Mathematics, University o Caliornia at Santa Cruz Santa Cruz, CA 95064, U.S.A. ginzburg@math.ucsc.edu, rmont@cats.ucsc.edu Dedicated to the memory o Stanis law Zakrzewski Abstract. A geometric quantization o a Kähler maniold, viewed as a symplectic maniold, depends on the complex structure compatible with the symplectic orm. The quantizations orm a vector bundle over the space o such complex structures. Having a canonical quantization would amount to inding a natural (projectively) lat connection on this vector bundle. We prove that or a broad class o maniolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist. This is a consequence o a no-go theorem claiming that the entire Lie algebra o smooth unctions on a compact symplectic maniold cannot be quantized, i.e., it has no essentially nontrivial inite-dimensional representations. 1. Introduction. The quantization o a classical mechanical system is, in its most ambitious orm, a representation R o some subalgebra A o the Lie algebra o smooth unctions by sel-adjoint operators on a Hilbert space Q. The Lie algebra structure on the space o unctions is given by the Poisson bracket and the representation is usually assumed to satisy some extra conditions which we will discuss later. It is generally accepted, however, that such a quantization does not exist when the algebra A is too large. (See, e.g., [Atk, Av1, Av2], and also [GGT, GGG] or a detailed discussion. We will return to this subject later.) In other words, the quantization problem in the strict orm stated above has no solution. Results claiming that there are no such quantizations are oten reerred to as no-go theorems. Thus, one oten tries either to just construct the Hilbert space Q, without quantizing the unctions, or to only ind the algebra o operators representing A without a Hilbert space on which they would act. The latter program, which can successully be carried 2000 Mathematics Subject Classiication: Primary 53D50; Secondary 37J15. The work is partially supported by the NSF and by the aculty research unds granted by the University o Caliornia, Santa Cruz. The paper is in inal orm and no version o it will be published elsewhere. [69]

2 70 V. L. GINZBURG AND R. MONTGOMERY out on symplectic maniolds, is called deormation quantization (see [We] or a review) and we are not concerned with it here. The ormer question, addressed by geometric quantization (see, e.g., [Wo]), is the subject o the present paper. One o the main problems with geometric quantization, arising already or nice symplectic maniolds such as S 2, is that the construction o the geometric quantization space inevitably involves an extra structure (polarization). This leads to the question o whether the quantization spaces constructed or dierent polarizations can be naturally identiied. (Under rather weak additional hypotheses the spaces are isomorphic.) In this paper we show that the answer to this question is negative or a broad class o maniolds including S 2. The problem o geometric quantization has no solution either! Beore we recall what geometric quantization is and outline our proo, let us return to the no-go theorems. The irst such theorem is a classical result due to Groenewold and Van Hove stating that the algebra o polynomials on R 2n has no representation that would restrict to the Schrödinger representation o the Heisenberg algebra, i.e., the algebra o linear unctions. (The Schrödinger representation is the unique unitary representation o the Heisenberg group; see, e.g., [LV] or more details and urther reerences.) This result lies at the oundation o the general principle that a suiciently large algebra o unctions A cannot be quantized. (See [Atk, Av1, Av2, Gr, GGH, GGT, GGG], and also Section 3 or more details.) The sel-adjoint representations o A are required to satisy certain extra conditions to warrant the title quantizations. Although there is no consensus on what the conditions are, their main goal is to ensure that the representation is small. For instance, in the majority o examples, the conditions include that the representation o the constant unit unction is consti, where const 0. (This is the case with the Groenewold Van Hove theorem.) Such conditions exclude representations like the one arising rom the natural action o the group o symplectomorphisms on the space o L 2 -unctions. When the symplectic maniold M in question is compact (and connected), its quantization is usually assumed to be inite dimensional with the dimension equal to the Riemann Roch number RR(M). A suiciently large Lie algebra A o unctions on M has no essentially non-trivial inite dimensional representations, i.e., each such representation actors through a representation o R = A/{A, A}. This rather well-known act alone is suicient to conclude that under some natural hypotheses about the maniold, M cannot be quantized in a canonical way. In other words, the geometric quantization spaces obtained or dierent polarizations cannot be naturally identiied. (See Section 3). We now return to the question o naturally identiying various quantization spaces. Our approach is inspired by recent results on quantization o moduli spaces o lat connections. (See, e.g., [ADPW, Ati, Hi] and reerences therein.) Given an integral compact symplectic maniold (M, ω), we consider the space J o all complex structures compatible with ω (i.e., complex polarizations). Then, or every J J, the quantization Q J (M, k) is deined to be the space o J-holomorphic sections o the pre-quantum line bundle L k. We take k suiciently large to ensure that a vanishing theorem applies, so that dim Q J (M, k) = RR(M, kω). (By deinition, L is a line bundle with a connection whose curvature is ω. The pair, and J, gives rise to the structure o a holomorphic line bundle on L, and so on L k.)

3 GEOMETRIC QUANTIZATION AND NO-GO THEOREMS 71 Fix k, and consider the collection {Q J (M, k)} j J as a vector bundle E over J. Here we ignore the act that the lower bound on k necessary or the vanishing theorem may depend on J. (This leaves open the interesting question: Is there a universal J-independent bound?) An identiication o quantizations (or their projectivizations) is the same as a (projectively) lat connection on E. The identiication is natural i it is equivariant with respect to the group o symplectomorphisms Ham. Strictly speaking this group does not act on E, but it has a central extension Cont 0 which acts. The Lie algebra o Cont 0 is the algebra A = C (M) with respect to the Poisson bracket {, }. (The group Cont 0 is a subgroup o the group o contactomorphisms o the unit circle bundle associated with L.) I it existed, a (projectively) lat Cont 0 -invariant connection would give rise to a projective representation R o A on the iber o E. Since this iber is inite dimensional, the representation R must actor through A/{A, A} = R as we pointed out above. On the other hand, such a representation R cannot exist i or some J 0 J, the Kähler maniold (M, ω, J 0 ) has a continuous group G o Hamiltonian symmetries. For R would restrict to a non-trivial representation o the Lie algebra o G on Q J0 (M, k). This contradicts the act that R actors through A/{A, A}. Hence, a Cont 0 -invariant (projectively) lat connection does not exist or a broad class o maniolds M including homogeneous spaces and, in particular, S 2. The details are given in Section 2. O course, it may well happen that J is empty. In this case, instead o working with holomorphic sections o L k, one considers the index o the Spin C -Dirac operator D or o the rolled-up operator, [Du]. The index is a virtual space, which still has the right dimension RR(M, kω). For and D there are again vanishing theorems (see [GU] and [BU]), ensuring that the index is a genuine vector space Q J (M, k). This space is equal to H 0 (M, O(L k )) when the maniold is Kähler and k is large enough. Both o the operators depend on a certain extra structure on M, e.g., an almost complex structure or. These extra structures orm a space serving, similarly to J, as the base o the index vector bundle E, and the above argument applies word-or-word. (This can be viewed as an answer to the question asked in [Fr].) Acknowledgments. The authors are grateul to Joseph Bernstein, Alexander Givental, Victor Guillemin, Leonid Polterovich, and Jean-Claude Sikorav or useul discussions. The irst author would like to thank the Tel Aviv University or its hospitality during the period when the work on this manuscript was started. 2. Natural lat connections on the vector bundle o quantizations. Let M be a compact Kähler maniold with symplectic orm ω, which is assumed throughout this section to represent an integral cohomology class. As usual in geometric quantization, ix a Hermitian line bundle L over M with c 1 (L) = [ω] (the prequantization line bundle) and a Hermitian connection on L whose curvature is ω. Consider the space J o all complex structures J on M which are compatible with ω in the sense that ω(, J ) is a Riemannian metric on M. For every J J, the connection on L gives rise to the structure o a holomorphic line bundle on L. Then, given a suiciently large k, the vanishing theorem applies to the line bundle L k or a ixed J J. In other words, H q (M, O(L k )) = 0 when q > 0 and k k 0, where k 0 depends on J. Thus, we can take the space o J-holomorphic

4 72 V. L. GINZBURG AND R. MONTGOMERY sections H 0 (M, O(L k )), k k 0, o L k as the quantization o M. Denote it by Q J (M, k) or just Q J (M) when k is ixed or irrelevant. Let J 0 be a C 1 -small neighborhood o a ixed complex structure J 0 J. It is not diicult to see that one can take the same k 0 or all J J 0. Note that sometimes the same is true or the entire space J. For example, this is the case when dim R M = 2. Fixing k k 0, we obtain a vector bundle E over J 0 whose iber over J is Q J (M, k). Let Ham be the group o Hamiltonian symplectomorphisms o M. The elements o Ham are symplectomorphisms which can be given as time-one lows o time-dependent Hamiltonians. It is clear that Ham acts (locally) on J 0. To lit this action to E, consider the group Cont o dieomorphisms o the unit circle bundle U o L which preserve the connection orm θ. Clearly, θ is a contact orm on U. Thus, Cont consists o those contact transormations which preserve the contact orm θ itsel (not just the contact ield), and which, as a consequence, are also bundle automorphisms. Let Cont 0 be the identity connected component in Cont, i.e., the elements o Cont 0 are isotopic to id in Cont. Every element o Cont 0 naturally covers a symplectomorphism o M, which belongs to Ham. The projection Cont 0 Ham is surjective, and it makes Cont 0 into a one-dimensional central extension o Ham by U(1). The Lie algebra o Cont 0 is just C (M). Since Cont 0 acts on L, and so on L k, it also acts (locally) on E and the latter action is a lit o the Ham-action on J 0. A connection on E is said to be natural i it is invariant under the Cont 0 -action. Now we are in a position to state our main observation, which will be proved in the next section: Theorem 1. Assume that the stabilizer G o J 0 in Ham has positive dimension and that the ininitesimal representation o G on Q J0 (M) is non-trivial. Then there is no natural (projectively) lat connection on E. When M is two-dimensional, the theorem applies to M = S 2 only, showing that the geometric quantizations o S 2 or dierent complex structures cannot be identiied. Note that there are many (projectively) lat connections on E, or J and J 0 are contractible, and many natural connections on E, but there is no connection which is simultaneously lat and natural. Remark As mentioned above, Theorem 1 extends word-or-word to compact symplectic, not necessarily Kähler, maniolds. In this case, J is the space o almostcomplex structures compatible with the symplectic structure and J 0 is a neighborhood o a given structure J 0 in J. The quantization bundle E over J 0 is deined using the vanishing theorems or either the Spin C -Dirac operator D or the rolled-up operator (see [GU, BU]). Note also that in this case J is a contractible Fréchet maniold. 2. What makes this theorem somewhat surprising is a recent collection o constructions o projectively lat connections related to topological quantum ield theory. Axelrod Della Pietra Witten [ADPW], and ollowing them Atiyah [Ati] and Hitchin [Hi], constructed quantizations Q J o the moduli space M Σ o lat vector bundles over a Riemann surace Σ. Here the additional polarization data is a complex structure on Σ. Their connections are natural with respect to transormations o M Σ induced by those o Σ, and not with

5 GEOMETRIC QUANTIZATION AND NO-GO THEOREMS 73 respect to all o Cont 0 (M Σ ). Note also that our Theorem 1 seems to contradict what is said in [Ati], page Hodge theory or a compact maniold X associates the vector space H p g o g- harmonic p-orms on X to each Riemannian metric g on X. This space is canonically isomorphic to the p-th real cohomology o X. Consequently, Hodge theory deines a lat connection on the vector bundle H p M over the space M o Riemannian metrics on X. This connection is Di (X) invariant. As a result, we have an induced representation o Di (X) on each H p g. O course, this representation is trivial on the identity component Di 0 (X) o X. Consequently, this induces the usual representation o the mapping class group Di (X)/Di 0 (X) on cohomology. 4. When the local action o Ham on J 0 is ree, it induces a projectively lat connection along the orbit o Ham. This connection is natural but does not seem to be o any interest or quantization. 3. No-go Theorems. Theorem 1 is an easy consequence o the general no-go theorems discussed in this section. Let (M, ω) be a connected symplectic maniold. Now ω is not assumed to be integral and M need not be compact. Let A = Cc (M) be the Lie algebra o smooth compactly supported unctions on M with respect to the Poisson bracket. Denote by A 0 the commutant A 0 = {A, A} o A. In act, A 0 is just the algebra o unctions with zero mean and, thereore, A 0 is a maximal ideal o codimension one. Theorem 2. The commutant A 0 is the only ideal o inite codimension in the Lie algebra A. This theorem has a long history. For a compact maniold, it is due to Avez, [Av2], who proposed a very interesting proo relying on the properties o the symplectic Laplacian. An algebraic version o Theorem 2, which applies to a broad class o Poisson algebras, has been obtained by Atkin [Atk]. This class includes the algebra o compactly supported unctions and the algebra o (real) analytic unctions when (M, ω) is (real) analytic. Furthermore, it appears that the reasoning and the key results o [Atk] (see Theorem 6.9 and Section 9) apply to the Poisson algebra o polynomial unctions on a coadjoint orbit or a compact semisimple Lie algebra, which would give a generalization o the no-go theorem o [GGH]. A simple direct proo o Theorem 2 can be obtained by adapting the methods o [Om] (Chapter X), which, in turn, go back to Shanks and Pursell [SP]. Remark 2. Theorem 2 is just a relection o the general act that the algebra A, like many ininite-dimensional algebras o vector ields, is in a certain sense simple. This assertion should not be taken literally A has many ideals o ininite codimension (unctions supported within a given set) but the Lie group o A is already simple in the algebraic sense [Ba]. (For more details see [Av1, Av2, ADL, Om, Atk], and reerences therein.) In many o the papers quoted above, in varying generality, the ollowing description o maximal ideals in A is given. For any x M, let I x be the ideal o A ormed by unctions vanishing at x together with all their partial derivatives. It is well known and easy to see that I x is a maximal ideal. In other words, the Lie algebra o ormal power series with

6 74 V. L. GINZBURG AND R. MONTGOMERY Poisson bracket is simple. These and A 0 are the only maximal ideals in A, i.e., every maximal ideal is either A 0 or I x or some x. Corollary 3. Any nontrivial inite-dimensional representation o A actors through a representation o A/A 0 = R. Thus, i a quantization o A is to be understood as just a inite-dimensional representation, we conclude that there are no non-trivial quantizations. It is also worth noticing that the corollary still holds or representations R in a Hilbert space by bounded operators, provided that when M is compact R(1) is a scalar operator [Av2]. Now we are in a position to prove Theorem 1 by reducing it to the no-go theorem (Theorem 2). Proo. Arguing by contradiction, assume that there is a natural projectively lat connection on E. This connection will be thought o as a lat connection on the projectivization bundle P E o E. Our goal is to construct, using this connection, a representation o A = C (M), the Lie algebra o Cont 0, on the iber P Q = P Q J0 (M) whose existence would contradict Theorem 2. For A, denote by φ t the (local) low on E generated by in time t and by φt the (local) low on J 0 induced by the Hamiltonian low o on M in time t. (In act, φ t is induced by the contact low o on the unit circle bundle.) Let Π(J 1, J 2 ) be the parallel transport rom the iber o P E over J 1 to the iber over J 2. Since the connection on P E is lat, this operator is well deined. Finally, deine a linear homomorphism R(): P Q P Q as R()(v) = d dt Π( φ t ) (J 0 ), J 0 φt (v), t=0 where v P Q. In other words, v is moved to the iber over φ t (J 0) using the group action and then transported back to P Q by means o the connection. We claim that R is a (projective) representation o A in Q, i.e., R({, g}) = [R(), R(g)] in the Lie algebra o the group o projective transormations o Q. To see this, recall that φ τ 2 {,g} = φ τ φ τ τ g φ g + O(τ 3 ). Furthermore, Π(φ τ 2 {,g} (J 0), J 0 ) is equal, up to O(τ 3 ), to the parallel transport rom the iber over φ τ φτ gφ τ φ τ g (J 0 ) to P Q. Thus, 1 R({, g}) = lim τ 0 τ 2 Π( φ τ φ τ gφ τ ) φ τ g (J 0 ), J 0 φτ φτ φ τ g g. Let us now ocus on [R(), R(g)]. By deinition, 1 [R(), R(g)] = lim τ 0 τ 2 (commutator), where commutator = {(Π(φ τ (J 0 ), J 0 ) φ τ )(Π(φ τ g(j 0 ), J 0 ) φ τ g) (Π(φ τ (J 0 ), J 0 ) φ τ ) 1 (Π(φ τ g(j 0 ), J 0 ) φ τ g) 1 }.

7 GEOMETRIC QUANTIZATION AND NO-GO THEOREMS 75 To calculate the commutator, we use the assumption that the connection is natural, i.e., Cont 0 -invariant. Explicitly, this assumption means that Π(J 1, J 2 ) φ t h = φ t hπ(φ t hj 1, φ t hj 2 ) or any h A and t R. Observing also that Π(J 1, J 2 ) 1 = Π(J 2, J 1 ), we transorm the commutator on the right hand side o the expression or [R(), R(g)] as ollows: commutator = Π ( φ τ ) (J 0 ), J 0 φτ Π ( φ τ ) g(j 0 ), J 0 φτ g τ φ Π( J 0, φ τ (J 0 ) ) φ τ g Π ( J 0, φ τ g(j 0 ) ) = Π ( φ τ ) ( (J 0 ), J 0 Π φ τ φ τ g(j 0 ), φ τ (J 0 ) ) φτ φτ g τ φ Π( J 0, φ τ (J 0 ) ) φ τ g Π ( J 0, φ τ g(j 0 ) ) = Π ( φ τ ) ( (J 0 ), J 0 Π φ τ φ τ g(j 0 ), φ τ (J 0 ) ) Π ( φ τ φ τ gφ τ (J 0), φ τ φ τ g(j 0 ) ) φτ φτ φ τ g g Π ( J 0, φ τ g(j 0 ) ) = Π ( φ τ ) ( (J 0 ), J 0 Π φ τ φ τ g(j 0 ), φ τ (J 0 ) ) Π ( φ τ φ τ gφ τ (J 0), φ τ φ τ g(j 0 ) ) Π ( φ τ φ τ gφ τ φ τ φ τ τ g φ = Π ( φ τ φ τ gφ τ g φ τ g (J 0 ), φ τ φ τ gφ τ ) φ τ g (J 0 ), J 0 φτ φτ φ τ g (J 0) ) g. Comparing this with the ormula or R({, g}), we see that R is indeed a representation. 4. Concluding remarks. One natural connection on E seems to be o a particular interest. For the sake o simplicity, we describe it or the case when M is a Kähler maniold and, thus, J 0 is the space o complex structures compatible with a ixed symplectic orm. Let s be a section o E and J(t) a path in J 0. Observe that every iber E J is a linear subspace in the linear space C (M; L) o smooth sections o the prequantization line bundle L over M. We set J(0) s(0) = P s (0), where s (0) C (M; L) is the derivative o s(j(t)) with respect to t at t = 0 and P is the orthogonal projection to E J(0), the space o holomorphic sections o L or J(0). It is easy to check that is indeed a connection. (A similar connection can be deined or the vector bundle o quantizations in the almost complex case.) The ollowing two questions on the properties o appear interesting already or M = S 2 : Is there an explicit expression or the curvature o? The curvature o evaluated on the vectors / t 1 and / t 2 tangent to a two-parameter amily J(t 1, t 2 ) is equal, as is easy to see, to [ P/ t 1, P/ t 2 ] where P = P (t 1, t 2 ) is the orthogonal projection to E J(t1,t 2). (This holds only when M is Kähler.) By an explicit expression we mean a ormula which can be used, or example, to see directly that the curvature is nonzero. From a dierent perspective Theorem 1 shows that the vector bundle

8 76 V. L. GINZBURG AND R. MONTGOMERY E J 0 is not Cont 0 -equivariantly trivial. Then an explicit expression or the curvature may yield some inormation on the Cont 0 -equivariant Chern classes o E. To state the second question, inspired to some extend by the results o [Gu], consider the curvature or E with iber Q J (M, k) over J as a unction o k. Is it true that the curvature o goes to zero as k? Reerences [Ati] [Atk] [Av1] [Av2] [ADL] [ADPW] [Ba] [BU] [Du] [Fr] [GGG] [GGH] [GGT] [Gr] [Gu] [GU] [Hi] [LV] M. F. Atiyah, Geometry and physics o knots, Cambridge University Press, Cambridge, C. J. Atkin, A note on the algebra o Poisson brackets, Math. Proc. Cambridge Philos. Soc. 96 (1984), A. Avez, Représentation de l algèbre de Lie des symplectomorphismes par des opérateurs bornés, C. R. Acad. Sci. Paris Sér. A 279 (1974), A. Avez, Remarques sur les automorphismes ininitésimaux des variétés symplectiques compactes, Rend. Sem. Mat. Univ. Politec. Torino 33 ( ), A. Avez, A. Diaz-Miranda and A. Lichnerowicz, Sur l algèbre des automorphismes ininitésimaux d une variété symplectique, J. Dierential Geom. 9 (1974), S. Axelrod, S. Della Pietra and E. Witten, Geometric quantization o Chern Simons gauge theories, J. Dierential Geom. 33 (1991), A. Banyaga, Sur la structure du groupe des diéomorphismes qui préservent une orme symplectique, Comment. Math. Helv. 53 (1978), D. Borthwick and A. Uribe, Almost complex structures and geometric quantization, Math. Res. Lett. 3 (1996), J. J. Duistermaat, The heat kernel Leschetz ixed point ormula or the Spin C Dirac operator, Birkhäuser, Boston, D. S. Freed, Review o The heat kernel Leschetz ixed point ormula or the Spin C Dirac operator by J. J. Duistermaat, Bull. Amer. Math. Soc. 34 (1997), M. J. Gotay, J. Grabowski and H. B. Grundling, An obstruction to quantizing compact symplectic maniolds, Proc. Amer. Math. Soc. 128 (2000), M. J. Gotay, H. B. Grundling and A. Hurst, A Groenewold Van Hove theorem or S 2, Trans. Amer. Math. Soc. 348 (1996), M. J. Gotay, H. B. Grundling and G. M. Tuyman, Obstruction results in quantization theory, J. Nonlinear Sci. 6 (1996), J. Grabowski, Isomorphisms and ideals o the Lie algebras o vector ields, Invent. Math. 50 (1978), V. Guillemin, Star products on compact pre-quantizable symplectic maniolds, Lett. Math. Phys. 35 (1995), V. Guillemin and A. Uribe, The Laplace operator on the nth tensor power o a line bundle: eigenvalues which are uniormly bounded in n, Asymptotic Anal. 1 (1988), N. J. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), G. Lion and M. Vergne, The Weil representation, Maslov index and theta series, Birkhäuser, Boston, 1980.

9 GEOMETRIC QUANTIZATION AND NO-GO THEOREMS 77 [Om] H. Omori, Ininite dimensional Lie transormation groups, Lect. Notes in Math., no. 427, Springer Verlag, New York, [SP] M. E. Shanks and L. E. Pursel, The Lie algebra o smooth maniolds, Proc. Amer. Math. Soc. 5 (1954), [We] A. Weinstein, Deormation quantization, Séminaire Bourbaki, Vol. 1993/94. Astérisque No. 227 (1995), Exp. No. 789, 5, [Wo] N. M. J. Woodhouse, Geometric quantization, Oxord University Press, New York, 1992.