REND. SEM. MAT. UNIVERS. POLITECN. TORINO

Vol. 42°, 1 (1984)

MirceaPuta

DIFFERENTIAL FORMS ON A COMPLEX FOLIATED

MANIFOLD AND GEOMETRIC QUANTIZATION

Summary. In this paper we-study some spectral and cohomological properties of differen­tial forms of type (0,q) on a complex foliated manifold in view their applications in geometric quantization.

1. Introduction.

The original motivation for the investigation of differential forms on a complex foliated manifold was provided by the cohomological correction of geometric quantization. It was suggested by B. Kostant [6] for the patological cases in which the representation spaces of geometric quantization are trivial.

In this paper we study some spectral and cohomological properties of differential forms of type (0,q) on a complex foliated manifold. The ma­terial is divided as follows. In Section 2 we briefly review the fundamentals of geometric quantization, including an example of R.J. Blattner and J.H. Rawnsley [4], which suggests the importance of the cohomological groups in geometric quantization. In Sections 3 and 4 we study some spectral properties of differential forms of type (0,g) on a complex foliated compact manifold and also present an G-invariant version of the Dolbeault-Kostant complex [5]. Finally, the last section discuss the L2-cohomology of a non-compact com­plex elliptic foliated manifold.

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Classificazione per soggetto: AMS (MOS) 1980-. 58 F 06

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2. Geometric Quantization.

Let (M, co) be a symplectic manifold; i.e. M is a real C^-manifold and co is a closed non-degenerate 2-form on M. To each f&C~My there corresponds a vectorfield fy G C°° TM defined by:

w(€/,.*?) = »?(/) ,

for all 17 G C°° 7VW, which is called the hamiltonian vectorfield generated by /. Then we can define the Poisson bracket {f,g} of / and g in C°°M by:

{/,*} = fy<g).

Under {...}, C°°M becomes a Lie algebra over JR. Let 7r: L -+M be a C°° hermitian complex line bundle over M with

hermitian structure <...>. We suppose that L is endowed with a c6nnection V such that (...) is preserved under paralel transport with respect to V. Such connections are in 1-1 correspondence with 1-forms a. on the comple­ment L * of the zero-section in L which are invariant under multiplication of L * by non-zero complex numbers, whose restriction to any fibre of L * is (l/iz)dz, and which satisfy:

"" V|S = is*(a)(£)- s ,

for all %ec°°T*M and s ec°°L, and also

i(a-a) = d\og\H\2 .

The curvature £2 of V is

r^«,r?) = [V|,V r ?]-VK > 7 ?] ,

for all J, 17 £ C°° TM. £1 is a 2-form on M and satisfies

n*£l = da .

In the sequel we shall make the fundamental assumption

1 £1 = - — co .

h

This requires that (2TT h) _ 1 co define an integral De Rham class on M. Prequantization associates to each f€C°°M a first order differential

operator 3(f) on L defined by

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We have

[6(/),.«(?)] = 6({ / ,g}) ,

so that 5 is a Lie algebra homomorphism. Quantization of the Lie algebra C°°M is accomplished through the

introduction of a polarization. A polarization of (M, co) is a C°° subbundle F of TM^ which is

(i) isotropic; (ii) maximal with respect (i); (iii) integrable.

Using F, we introduce two important spaces:

CFL = {sE C°°L | V{ 5 = 0, for all £ G C~F} ,

Cf M = {/G C°°M | [$ / f{] G C~F, for all % E C°°F} .

It is easily verified that CFM is a Lie subalgebra of C°°M and that 5(/)• leaves. C £ L stable when f€CFM. This action of C*Af on CQL is a one form of quantization. But what we really want is an action on a (pre-) Hilbert space. In general this requires the introduction of half-forms as in [2] and [3] (see also [10], [11]). However, in case when F is an elliptic foliation, i.e. F + F = 77WC, one can often proceed as follows: let JJL be the

1 F

2«-form (i/n\) cow, where n = y dim M Let s G C0 L, we define:

* l l o =

1/2

(s(x), s(x)) n

4 < L ) = {s€CJL | ||s.||<oo} ;

(^1^2)o = I ( j l ( 4 S 2 W ) ^

for 5 ! , s 2 G 4 ( L ) . In some cases, £F(L) # {0} and its completion is a Hilbert space under

(...)0. Then the action of CFM on £F(L) is called quantization. However in many cases, including the example below, £F(L) — {0}. In these cases B.Kostant [6] suggested the using of the cohomological groups as representation spaces. It

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was quickly verified by R.J. Blattner, J.H. Rawnsley, D.J. Simms and J. Sniatycki that one can quantise the one dimensional harmonic oscillator in Simms' polarization usign the first cohomology group as the representation space. This result was also extended by J.H. Rawnsley [9], to the case of a 72-dimensional harmonic oscillator in the Simms' polarization. Thus there are ample motivations for the study of differential forms on a complex foliated manifold.

The following example gives us a concret case in which the representa­tion space of geometric quantization is trivial.

EXAMPLE (Blattner-Rawnsley). Let M = 1R4 with linear coordinate func­tions pi,p2,ql>42 anc* let co = dpi Adq1 + dp2f\dq2 be the canonical symplectic form on IR4. Since co is an exact form M is quantizable and the prequantum bundle L^ is the trivial bundle Lw « IR4 x C. The her-mitian structure on Lu is given by

<(*,ci), (x,c2))= cxc2 .

We let Zt^pi + iq1, z2 = p2 + iq2. Then —— = —- (— i——r ) and

_i_ = j7_L + l- a\ . . r\ dZi 2^> dq J dzi 2 \dpi

7 = 2

Let F be the subbundle of TM^ spanned at each point by {£i,£2}- F ls

an elliptic polarization on M. If we identify any section sEC°°L with a C-valued function f on M according to

s(a?) = (*,/(*))

for xGMt we have for each fGC^M:

df b 3 / 9 \ 1 2 / 3 / « ( / ) j%i\bqi 3p;* 3p; dqjj 2ih y?i \f' dp, + ^ ty'

Also we can deduce that:

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3z;- bzj 4h '

3 8 1 V 3z;- 3zy 4h y '

Hence fEiC°°M will belong to CF L if and only if

- J <KlM*2l2) f-we *u

where w is holomorphic with respect to the complex structure on 1R4 de­termined by F. Thus £\(I) =• {0]•.

3. Differential forms of type (0,#) on a foliated manifold.

Let M be a smooth, orientable (n + m) dimensional, paracompact manifold and TM^ its complexified tangent bundle.

DEFINITION 3.1 ([5]). A complex foliation of M is a complex subbundle F C TMfg satisfying the following two conditions:

(i) F H F is of constant rank; (ii) F and F are integrable.

We shall suppose in all that follows that rank (F) = n. Choosing a direct summand F^ of F in TM^ with respect to some hermitian structure on 7MC we obtain

TMr =F®FL 1<E

DEFINITION 3.2 ([9]). A differential form of type (0,#) on M is a smooth section of the bundle A ̂ F*.

We shall denote by AF(M) [resp.jtfqF(M)] the space of differential

forms of type (0,g) [resp. the sheaf of germs of differential forms of type (0,4)] on M.

The exterior derivative along F, dF, is given by

dF : aeAF(M) —• dFaGAF+l(M),

where for any vector fields along F, Xx,..., X +l, we have:

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dFa{X„ ..., Xq+l) = S (-1)'+1 X,(ct(X„..., X,, ..., Xq+l)) + 1 = 1

+ 2 (X([Xj, X:], Xlt ... ,Xj-, ... ,Xy, ... , Xq + i) . i<j

The sheaf complex:

s4\{M) -^s4lF{M) - ^ ... - ^ W F ( M ) — * 0

is called the Dolbeault-Kostant complex [6]. It is a fine resolution of the sheaf ^p = ker(dF:j?^F -*s4F) and therefore \

W(Af,*F)*/i«(i45(M),rfF) .

EXAMPLES. . ' . ;

i) Let Af = IRxIR with the leaves {*}xIR. Then H°(M, %) = C°°JR and H\M, # F ) = 0.

ii) Let Af = 51 x 5 1 , where S1 is the unit circle, with the leaves {x} x S 1 . Then

H°(M, <£F) = HHM, <^F)••= (TS 1 .

Other properties of the-Dolbeault-Kostant complex can be founded in [5], [8], [9].

Hereafter we suppose that My is compact manifold and F is a complex, elliptic foliation on M, i.e. F + F = TM€: We choose a riemannian metric on M and extend it to a hermitian structure on TMC. Then the operator i F has an adjoint 5F with respect to the usual global inner product given by

•'AT

where a and 0 are (0,g) forms and "*" denotes the Hodge star operator on M. Therefore we can define the F-Laplacian AF on Aq

F(M) by setting

AF = dpbF -f dp dp .

It is an elliptic, self adjoint operator [6] and has an infinite sequence of eigenvalues (denoted by Sped (M, AF))

o = x S . F < x ? i J r < . . . — + < » , each eigenvalue being repeated as many times as its multiplicity indicates.

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DEFINITION 3.3. The F-dependent heat operator on AqF(M) is d/dt~Aq

py

where t is regarded as the time variable and AqF is the F-Laplacian on

AF(M).

PROPOSITION 3.1. Given a smooth qform <pEAF(M) there exist an F-de-pendent heat distribution \p(t,x), i.e. a g-form \p smoothly depending on t, such that

{ii~An * = 0 ' for t>0

lim \l>(t,x) == <pO0

Proof. The proof can be obtained using the general theory of elliptic opera­tors on a compact manifold [1]. The solution is given by

\p(t,x)= I epF(j,x,y)A* <p(y)dy ,

where eF(t,x,y) is the F-dependent heat kernel. It can be expressed in terms of eigenvalue and eigenforms of the F-Laplacian. Indeed, let {^F} |=1

be the orthonormal sequence of eigenforms so that $F is the eigenform cor­responding to eigenvalue X?F, for every f = 1,2,... (i.e. AF*pF = X?F<pF). Then the F-dependent kernel is given by

(i) 4 ( ^ ^ ) = | 1 ^ x ' ' F ^ W ( ^ ^ ( y ) J

where series converges uniformly with all derivatives on compact subsets of

q.e.d.

Now, using the above results we can define the following operators:

i) F-dependent harmonic projection, lfF\ yEAqF(M) -• lfF(</>) E Aq

F(Af),

ifpdp) = lim \j/(t,x)

ii) F-dependent Green operator, GqF:yE A%(M) -> G%[sp) G A%(M),

(2) ' GqF(<p)(x) = l dt J eq

F(t,x>y)A*(^-HqF^))(y)dy .

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With these in mind we can prove

PROPOSITION 3.2 (F-version of Hodge decomposition theorem). For each q\ 0 < q < n we have a unique orthogonal direct decomposition

AqF{M) = dp{A%'\M)) e 5F(A

qF

+1(M)) ®XqF(M) ,

where JK^ (M) is the kernel of A^.

Proof. By relation (2) we have

G f f °° t\q

(*)(*) = / dtl .S * i'F^F(x)^<pF(y)(ip-HqF^))(y)dy

C - -*x? ' = J dt X e i,FanpF(x), where a,- = (^,<^).

J i=l

Then the series X e ,,FanpF converges and moreover we have

II £ e-tX^Al4- <'~'X*'Fll4*||. X*

Ai,F ~—

-t\q-

Then the series 2 e t'FanftF converges uniformly with respect to t. This

justifies integration term by term. Therefore we have successively: G\

(tp)(x)=l 2 e t'FanplF(x)dt = '

Jo *l.F

1 \.J 2 anpF(x)j e l>Fdt= S W V , F ) ^ F W

Using the same argument as above we can conclude that the series 2 (#;Af F)<^F(^) converges uniformly. Then term by term differentiation

gives: A«G* (*)(*) = 2 fo/X? F ) Xf F 4(*.) = 2 A,- 4 (x) =

x? x* Ai,F *i,F

= ^O)-#£(^)O0,

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and therefore

ip = dp8pGpiv) + 8pdpGqp(v)+!%&). q.e.d.

Using the same expression (2) we can also prove:

PROPOSITION 3.3. Let \\ F be the smallest positive eigenvalue of AF on ^-forms. Then we have

X? f F<l/ | |G*| | .

PROPOSITION 3.4 (F-minimum principle). Let {$} be a complete ortho-normal base in A % (M) consisting of eigenf orms of the F-Laplacian. Then we have:

^ + i , F = inf { (A^ ,^ ) / | | ^ 2 | | | ^ e i4^ (M) , *>*(),

( ^ , ^ F ) = 0 , for all i = 1,2,...,»} .

4. Dolbeault-Kostant complex invariant under the action of a Lie group.

Let M be a smooth, in + m)-dimensional, paracompact manifold and G a compact Lie group which acts in a transitively way on Af.

DEFINITION 4.1. An G-invariant complex foliation of Af is a complex'sub-bundle FCTAfc, satisfying the following conditions:

(i) F n F is of constant rank, (ii) both F and F + F are integrable, (iii) F is invariant under the action of G.

We shall denote by AFG(M) [resp. $$%G (Af)L t n e space of G-invariant differential forms of type (0,g) [resp. the sheaf of G-invariant differential forms of type (0,g)L on Af. Let dF be also the restriction of dF at

DEFINITION 4.2. The sheaf complex

s4%G (M) -^s4FG (Af) - ^ ... - ^ j / S c (M) • 0

is called the Dolbeault-Kostant complex invariant under the action of G, or shorter the G -DK complex.

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PROPOSITION 4.1. For each q > 0, the sequence jG ,G ,G

0 —> 1R C~L^^°FG (M) -^s4FG (M) - ^ ... -^^nFG (M) —+ 0

is a fine resolution of IR, and therefore

W(MflR)^HHA<FG(M)id^).

Proof. For the proof is enough to observe that ker (df '-s4FG (M) -*s4FG (M)) = = IR, and then our result can be obtained via an G-invariant version of the F-Poincare lemma.

q.e.d.

REMARK. In a particular case when F = TMc we refined the G-invariant version of the Dolbeault isomorphism theorem. ' ,

Let @rFG (M) be the space of G-mvariant De Rham currents of type (0,q) on M [7], and dF the natural exterior derivative along F. Then the sheaf complex

0- dF L dF dF ft

%GW) ^ &FG(M) - * ... - > &FG(M) -+ 0

is called the dual complex_of the G-DK complex, invariant under the action of G.

As for as the differential forms we can prove

PROPOSITION 4.2. For each q > 0, the sequence

' ' —r i 9* dF I, dp dFi n,

0 —> ker d% ^ $FG (M) -*» ®FG (M) - » ... . -* 9FG (M) -» 0

is a fine resolution of ker dF and therefore

HHM, k e r ^ ) ^ H ^ ( ^ F G ( M ) , ^ ) .

In a particular case when M is a compact manifold and F is an elliptic G-invariant complex foliation on M we obtain an G-invariant version of Simms' theorem [12].

PROPOSITION 4.3. For each q > 0 we have the isomorphism

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5. L2-cohomology of a non-compact complex foliated manifold.

In this section we suppose that M is a non-compact (w + m)-dimension-al manifold and F is a complex elliptic foliation on M.

Let L2AF(M) be the space of L2-differential forms of type (0,q) on M. It is the completion of the space @q

F(M) of compactly supported C°° g-forms on M of type (0\q) with respect to the norm ||^||0 = (<^,^)0 = = (/ <£ A*•\p)V2. Thus L2Aq

F(M) is a Hilbert space with the inner product (...)0 given

O, i//)0 = / <^A* i//.

Then we have

PROPOSITION 5.1. The space L2AF(M) has the following orthogonal de­composition:

L2A% (M) = Tp^f^M) © IF@J+T(M) © 3C* (M)

where KqF{M) is the space of C°°-forms <£ of type (0,q) satisfying the

equation AFip = 0.

DEFINITION 5.1. The L2-cohomology of M is defined as follows

L2&qp{M)= {*£L2AF(M)\dFv = 0} ,

L2BqF(M) = dFL2AF~l(M)i

L2^(M,IR) = L 2 ^ ( M ) / ? 4 M .

We can prove now:

PROPOSITION 5.2. L2-F-harmonic forms, on M are dF-closed and 8F co--closed, i.e. Kq

F(M) = {# G L2AqF(M) | dF<p = 6 F ^ = 0}. Moreover, 5C£(M)

is isomorphic to L2lfF(My 1R).

Proof. We have:

whicn proves the first assertion. The second is a consequence of inclusions:

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dp@% \M)CL2BF(M)CL2BqF(M)CL2&F(M).

q.e.d.

REMARK. It is an open problem to decide if Proposition 5.2 remains true if we omit closures in the definition of L2-cohomology.

Acknowledgement. I wish to take this oportunity to express my sincere thanks to Dr. J.H. Rawnsley who kindly sent me reprints of your papers on geometric quantization. -x

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[5

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[7

R E F E R E N C E S

M.F. Atiyah, R. Bott, V.K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279-330.

R.J. Blattner, Quantization and representation theory, Proc. Sympos. Pure Math. Vol. 26, Amer. Math. Soc.Providence, (1974), 145-165.

R.J. Blattner, The meta-linear geometry of non-real polarizations, Lecture Notes in Math. Vol. 570,(1977), 11-46. y

R.J. Blattner, J.H. Rawnsley, Quantization of the action of U(k,l) on R2^l'\ preprint UCLA (1981).

H.R. Fischer, F.L. Williams, Complex foliated structure I. Cohomology of the Dol-beault-Kostant complexes, Trans, of the Amer. Math. Society Vol.252 (1979), 163-195.

B. Kostant, On the definition of quantization, Colloque Symplectique, Aix-en--Provence, (1974).

M. Puta, Sur les courants d'une variete differentielle invariants sous Faction d'un groupe de Lie, C.R. Acad. Sci. Paris, t. 281 (1975), 647-649.

M. Puta, Some spectral properties of the Kostant complex, Journ. of Math. Physics vol. 23, n° 10 (1982), 1749-1751.

[8

[9

[10] J.H. Rawnsley, On the pairing of polarizations, Comm. Math. Phys. 58 (1978), 1-8.

J.H. Rawnsley, On the cohomology groups of a polarization and diagonal quanti­zation, Trans, Amer. Math. Soc. 230 (1977), 235-255.

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[11] J.H. Rawnsley, Non-positive polarizations and half-forms. Lecture Notes in Math. Vol. 836, (1980), 145-153.

[12] D.J. Simms, Serre duality for polarized symplectic manifolds, Reports on Math. Phys. Vol. 12 (1977), 213-217.

PUTA MIRCEA - Seminarul de Geometrie-Topologie, University of Timisoara, 1900 Timi­soara, Romania.

Lavoro pervenuto in redazione il 12/VII/1982 ^