The Spectrum of the Dirac Operator on the Odd Dimensional Complex Projective Space i]ZP2‘“`l
S. Seifarth, U. Semmelmann
SFB 288 Preprint No. 95
Diese Arbeit ist mit Untersttitzung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 288 entstanden und als Manuskript vervielfiitigt worden.
Berlin. Dezember 1993 OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON THE ODD DIMENSIONAL COMPLEX PROJECTIVE SPACE CP2"‘"l
S. SEIFARTH AND U. SEMMELMANN
ABSTRACT. We give explicitly the eigenvalues of the Dirac operator on the complex projective space CP"
0. INTRODUCTION In this paper we want to present a calculation of the eigenvalues of the Dirac operator on the complex projective space CP". This calculation has been done in [CFGKQ]. As their result differs from ours, we included two independent proofs of our result in a special case (for the subbundle S0). We prove the following theorem
Theorem 0.1. On the complex projective space of complex dimension n, endowed with the F·u.bini—Stady metric, the spectrum of the Dirac operator consists of the series 1,/AU, and ;l;/y,;_k,, where AU, =l2 + %l(3n —- 2k — 1)+ %(n — k)(·n. —- 1) k€{1,...,n} andl2mam{1,k—%} (01) n pu, =l2 + §l(3n — 2k +1)+ %(n — +1) k€{0,...,n—1}andl2max{O,k— Though the method of calculating the spectrum of the Dirac operator using the Casimir operator is well known, we wrote down everything as explicitly as possible to make the computations more transparent. Our paper is organized as follows. We begin with fixing the notations and parame trisations. ln the second part we recall geometric aspects like the spin structure and the spinor bundle. Then we define the Dirac operator, recall the connection with the Casimir operator and the Peter—Weyl theorem. ln section four we cite the branching rules from [CFG89] and [lT78l. Then we have the necessary material for the explicit calculation (section tive). OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON In the last section we included an independent proof of our assertion for the special case S0 and further remarks. 1. NOTATIONS We regard the Riemannian symmetric pair (cf. [Hel'78], ch.IV) (G, K) with G = S U (n + 1) a compact semi—simple Lie group and K : S(U(n) X U(1)) : {3( dc, n+1 (1.1) _ _ T = {diag(e"8‘, . .. ,e`p"*‘) |Hj E R, = 0}, 5:1 common to G and K, with (complexified) Lie algebra n+1 (1-2) *= {B ¤= d1¤g(U1 ,--- .%+1) I Bk € C, Z Bj = 0} j=1 Then 11+1 *' = {E 7161 I rj 6 C}, .i=1 where z—:;,(B) = Bk. The element rjsj we simply denote by (1-1,. . . , ·r,,+1) Because of the relation Bj = 0 we can set @,+1 : —— EQ2, Bj, i.e. the torus is in fact n—dimensional. Moreover, 11+1 n n Z Tieiipl Z" ZT5Bj ` 2:%+1% : Zi"} ‘ T¤+1 j=I j=1 j:1 j:]. By settting Z: Tj ·* 7},+1 OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP2""1 for j : 1,... ,n we get a parametrisation of the elements of t' as ·n.~vectors, which we will denote by [t1,... ,¢,,]. The inverse formula is t,+t forj= 1,... ,n T- __ J `_ t for j = ·n. + 1 with t = —T—£-3 2;}:, tj. ln the sequel we will use freely both parametrisations without further comments. The root system = (g, t) of t in g is generated by the simple roots (1.4) :={O¢;:6;—E;+1, i=l,...,TL+l}. By + we will denote the positive system, corresponding to this set of simple roots Z and by EG the corresponding halfsum of the positive roots. Then 6;; = -1,... ,——§). Since { is a maximal torus for E too, we can also define the root system K : = K(l?, t), generated by the simple roots (1.5) ::{oz,=s;—e,+1, i:1,... ,·n.}. The corresponding halfsum of the positive roots is 6K : ($,§— 1 ,... ,-%,0) A weight (1*1,... ,*5,+1) is dominant with respect to + if Tl 2 . .. 2 *5,+1, and integral, if Tg -—·rj is integral for all ·i, j. For the n—vecto1·s that means that [tl, . . . , tn] is dominant, if tl 2 . . . 2 tn 2 O and integral, if all ti are integers. 2. THE COMPLEX PROJECTIVE SPACE The complex projective space CP" can be realised as a compact symmetric space G/ K with G and K like in section 1. If we fix the point 1:0 = eK in G/K we may identify po with the tangent space in :1:0, pg : T:OG/K. We use the Riemannian metric induced from the negative of the Killing form B. Note that this is exactly (n + 1) times the Fubini—Study metric. Homogenous spin structures on symmetric spaces G/ K correspond to lifts 54 of the isotropy representation at K ——» SO(T,0G/K) 2 SO(2·n.), i.e. 54 is a homomorphism K —> Spin(2n) with cx : A 0 5, where A : S pin(2n) —> S O(2n) is the non-trivial twofold covering of S O(2·n.). OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP 2""1 The complex projective space CP" admits a spin structure if and only if n is odd. This spin structure is unique and homogenous. Let fi : C li f f2,, —> HomC(S, S) be an irreducible representation of the Cliilord algebra (unique up to equivalence). We will denote the restriction of fi to Spin(2n) as well by Then the spinor bundle S of CP" is given as the associated vector bundle 8 = G XR; S. Since the dimension of CP" is even, we have S = 8+ 6) S`, the decomposition into half—spinors. The representation ( ,5 0 54, S) of K = S(U X U( 1)) now decomposes into irreducible components. Lemma 2.1. [Par72], [CFG89]} The representation (fic Z2, S) of the group K = S(U(n) X U(1)) decomposes in (n + 1) irreducible components (2.1) S : QQ Sk k=O with the ha,U spinor modules S+ = @ Sk _ k;O mod.2 _ S : EB Sk. k£1 mod 2 Here Sk is the representation with highest weight uk:(&%il—k,..., #——k, %—k ,...,%—k) (vi.-Io) (n-k+x) Remark. It is easy to see that the highest weights Vk of the irreducible components Sk of S all are of the form 205;; — 5K for some element w in the Weyl group WG (cf. [Par72]). According to the decomposition (2.1) there is a decomposition of the spinor bundle in a sum of holomorphic subbundles (2.3) S : @98;,. k:O 3. THE Dimxc OPERATOR Our description of the action of the Dirac operator is similar to [CFG89]. We use the identification (3-1) U°°(G/K»$*) 2 (U°°(G)®$*)K» where ·· )K refers to the subspace of K —invariants with K acting on C'°°(G) by right translation and on sby sz:i 5*d: o Et. OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP2 '""1 Let {X,} be an orthonormal basis of po. Each X, determines a left invariant vector field r(X,) by infinitesimal right translation. In terms of the identification (3.1) the Dirac operator on 8* is given as (3.2) 0* Z Zr(X,) rg) p*(x,) ; (0~=·(o) ® s*)K Z» (0°¤(G) ® sr) It is easy to see that the Dirac operator preserves the K ·-invariants in C°°(G) ® 5*, and that it commutes whith the action of G. Sections in the spinor bundle may also be viewed as functions on G with values in se which are equivariant, i.e. functions f : G ——+ 5* with f(gk) = .s*(k"1)f(g) for all g G G and all k E K. The space of these functions we will denote by C'°°(G, 5*)K. Using a hermitian scalar product on the spinor module 5*, we can define a scalar product on C°°(G, se )K and consider the Hilbert space completion LI’(G, 5*) The Dirac operator extends to a self adjoint operator on .C2(G, 5*) Denote by G the unitary dual of G and let (lQ,7r,) for ry 6 G be a representative of the equivalence class ry. The L2 sections of the subbundle Sk are described by the Frobenius reciprocity and the Peter—Weyl theorem: (3.3) r’(s,,) Z Q; iq ® HOmK(v,, sk), ·v€G where ® is a direct, orthogonal sum and HomK(l@,5k) denotes the space of K invariant homomorphisms V, ——> Sk. All these summands are finite dimensional and invariant under the action of the Dirac operator D. The elements v ® A of KQ ® HomK(V,, Sk) define smooth sections ofthe bundle Sk by (11 ® A)(g) : Ao 1r.,(g’1)·v and are therefore in the domain of the Dirac operator D. The spectrum of D is completely determined by the spectra of the operators D., for *7 E G, where D, is defined as follows. The restriction of D to YQ ® HomK(l/Q, Sk) has the form 1 ® D.,. If we fix an ortho normal basis {X,} of p, D., can be written as (3.4) D,(A) : —— Z §(X,) 0 A 0 1r,,(X,). OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP 2""1 From (3.4) it follows that (3.5) D; A = Z F(X¢)F(X1) 0 A 0 ’H·(X1)”~,·(X¢) E A 0 7M·(X¢)*w·(Xs) + Z F(Xe)F(Xj) 0 A 0 M·(lXj» XJ) = (3.6) D3 A = · ZA 0 M·(X<)7M·(X¢) + ix B(lX¢»X1l»Zr)F(X¢lF(Xjl 0 A 0 ¤~,—(Z¤)· Since A E HomK(VY,, S), we can commute A and 1r,,(Z;), i.e. (3.7) A o vr.,,(Z;) = .s,(Z;) o A. Thus we get (3.8) Di A = — Z A ° "—»·(X¢)’H·(X¢) · ix B(lZ¤» X1]- X¢)F(Xi)P(Xj)0·(Z¢) 0 A Z A 0 1w·(X¢)¤~·(X¢)+ ix F(lZz»Xjl»X¢)F(X1)0—(Z4l 0 A Z A o 1r.,,(X;)1r.,.(Xi) — 2 Z s,(Z;) 0 s,(Z;) o A. Here we were using the fact that [E,p] C p, [Z,,Xj] : — Z B([Z,,Xj],X.)X; and the wellknown formula .s,(Z;) : —i E,5([Z;,Xj]),5(Xj). Before proceeding further we recall the definition of the Casimir operators SIG of g and DK of E QC = EX? + EZ,2 and QK : ZZ,2. a 1 Then D; A = —A o 1r.,,(QG) + A o 1r,,(QK) — 2.s.(QK) o A. Now we can again commute A and vr,,(QK) (A E HomK(lQ, We obtain D; A = —A o 1r,,(QG) — .s,(QK) o A. On the irreducible K module with highest weight p the operator —QK acts as mul tiplication by llu + 6KHz — II6KI OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP2""1 From the remark after lemma 2.1 follows that the Casimir operator acts on Sk as multiplication by lw6c=\\2 · II6z<|l" = |\6G|l2 · H6K||2 In the case G = SU(n +1) and K = S(U(n) x U(1)) we obtain s,(QK) = —§· id on all components Sk of S. Since (VH7) is an irreducible G representation of highest weight u., the operator OG acts as multiplication by HM + 6GHz · H6c=H2 Now it is possible to express the square of the Dirac operator in terms of the Casimir operator and to compute its action on HomK(V,, S) in terms of the highest weight Of (VV: 7r*v)· Proposition 3.1. The square of the Dirac operator D5 on HomK(l/Q, S) has the form (3-9) DZ = (HM + 6GHz · Hicllz + §)id» where u., is the highest weight of the representation (IQ, ir.,) and · the scalar product on t", induced by the Killing form,. 4. BRANCHINGS RULES By the lemma of Parthasaraty we have the decomposition of the spinor module as K representation (4.1) S : Q} Sk k=O with highest weights Vk : |·'%l—k,... , ”§i——k ,?—k,... ,%—k (¤-'=) To determine the spectrum of the Dirac operator on Sk, k = 0, . . . _, n, we have to find all representations 7 G G with HomK(VY,, Sk) ¢ {0}, i.e., all 7 E G such that the de— composition into irreducible components of VQ)? contains Sk. These representations can be found using the following branching rules: Proposition 4.1. [CFG89], [IT78]) The irreducible representation of highest weight p. : (m1,... ,m,,) of G : SU(n + 1) decomposes under the action of the subgroup K = S(U >< U in a direct sum of the irreducible representations of highest weight 2/ = (sl — 1,... ,s,, — l), where I : Ef‘:1(m; ~— s;), and where the si 's are integers such that mi2¤i2mz2·--2m¤2sn20OCR Output lIlIIl-IIVK THE SPECTRUM OF THE DIRAC OPERATOR ON The multiplicity of each of these representations is cnc. We are looking for p. = [ml, . .. ,m,l] 6 G such that ;,i[Y contains the representation with highest weight ul,. Uk = [sl — l,... ,s,, -1],1: Zf‘__:1(m; — sl), m12812m22322•··2mn2$n2O We obtain the system sl=...:s,,-;,: '*:,§l—k+l 3,...];+] = ... :3,,-] Z Thus we get ml:sl+:c, :z:?_0,x€N (4.2) m2==...=m,,-;,:'—L?,i—k+l m,,-;,+l : s,,-;,+l +e: = '—§——k+l+e, c G {0,1} m,l_.k+2 :...:m,l: '§l—k+l. l=:1:+z»:,therefore:1:-·=l—6 andhencel?_0,l2e. TTL; =‘- géi · k ·[···· E (4.3) m2=...=m,l-;,=:§—k+l mn-k+1 = 7‘—k+l+6 m,,-;,+2:...=m,,=’i?—k+l. From this system and the condition mj 2 0 (dominance) we get 1; max{k —’—?,e}. Now we transform ·n,——vectors into (n + 1)—vectors. Let p, := [ml, . . . ,m,l] be given by (4.3), then it corresponds to a (rz + 1)—vector ;2 : (#1,. .. ,,u,,+l) with pj = 0, where ;ij=mj—,uO forj=1,...,·n. /%+1 I ·‘#0 (4.4) 3:1 OCR Output We get OCR OutputTHE SPECTRUM OF THE DIRAC OPERATOR ON CP 2""1 10 Here we have computed the square of the length of the roots with respect to the scalar product induced from the trace form B0 : EF},-g;jB. Since · = - Hg,) we obtain Iii + 6c=||i; - H6cHi; = ;#5A“J‘(k)· Now we can use the lemma 3.1 to determine the spectrum of the square of the Dirac operator. We see that it consists of the values (5.2) }`*(k) + Q = qm(8l2 + 41(3n — 2k ZF 1)+ 4(n — k)(n ZF The spectrum of D2 with respect to the Fubini—Study metric is then (n + 1) times the last expression (5.2). So we have proved the theorem 0.1. 6. REMARKS (1) To compute the spectrum of D2 on the subbundle S0 we can proceed more directly using the Hopf fibration $2**+1 ——> CP". This is a principal S1 bundle and we can form the associated bundle 8,,, := S2'"'1 xpm C, where p,,,(z)v := z"‘·v denotes the S1 action on C (s 6 S1 and z G C). Now it is easy to see ([Kuw88]) that c1(S,,,) = m. If we use the isomorphism Sq 2 ICE (cf. [Hit74], [Kir86]), where IC is the canonical bundle of CP", we obtain c1(S0) = c1(}C$) = %c1(K) : %c1(CP") : +1). We remark that the connection on SO, induced by the Levi—Civita connection, is just the canonical Hermitian connection of the holomorphic line bundle S0. This follows from the fact that CP" is a Kahler manifold. Since the complex line bundles over CP" are classified by their first Chern class we get S0 2 5;;; . Since S0 is representable as an associated bundle, we can identify sections in S0 with S1 invariant functions on S2”+1 (6.1) I`(S,,) E o°°(s’*·+*,c) Under this identification the Bochner—Laplace operator V"V of SO corresponds to the difference of the Laplacian A on functions on S2"+1 and the Casimir operator of the representation (pm, C) for m = % ([BGV90], [Kuw88]). V°V = A —— 2 On the other hand we have for the square of the Dirac operator the Lichnerowic formula: OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP2'"'1 11 D2 = V°V + qi, where R denotes the scalar curvature. Now for the complex projective space, equipped with the Fubini—Study metric, the scalar curvature is R = n(n -1- 1). Then under the identification (6.1) we obtain D:n 2 nin.A [ %+2 _ )___ : -1 The spectrum of A on S2”"'1 is known. The eigenfunctions can be realised as the restrictions of homogeneous polynomials ([BGM7l]). Now it is easy to see which of these eigenfunctions are invariant under the S1 action p xg ((Kuw88]). From this we get the spectrum of A on functions in C °°(S2”+1, C)S1 as »\¤ = U + ¥)(l + @2*) We obtain the eigenvalues of D2 on sections in S0: #¤.¤=(’+¥)('+;’?*)·&i'$)i+@ - The above calculation is also possible for Sn 2 ICE. We get the same set of eigenvalues for D2 on 8,,. This can as well be seen from the relation »\;,,, = y,_¢'0, I2 % (cf. (6.6)). (2) The series AU, (resp. pu,) corresponds to the parameter e = 1 (resp. e = 0) in the branching rules. (3) The spectrum of the Dirac operator on the sections of the subbundle Sk is given by 1,/AU. and ;l;,/,u;_k. (4) The spectrum of the Dirac operator on CP] = S1 is given by ,1,,, Z (1+ 1)’ and ,),,0 Z P (5) The smallest possible eigenvalue is obtained on S g and S mi as M0.? = ‘@ wd has = "’¥Z· On Kahler manifolds M there is the following estimate for the smallest eigen value }\0(M) of the Dirac operator (cf. [Kir86]): (6-3) MMV 2 §%,,*-2R.), OCR Output THE SPECTRUM OF THE DIRAC OPERATOR ON CP ’""1 12 where R0 is the minimum on M of the scalar curvature. On the complex projective space we get (6.4) A0(c1>*·)2 g g2g¤g1n(n + 1) : Q1 Hence CP" realises equality in the inequality (6.4). (6) There is an other way to write the eigenvalues: m = (I+ '%i> REFERENCES [BGM71] M. Berger, P. Gauduchon, and E. Mazet. Le spectre d’une variété riemannienne. Lecture notes in Math., 194, 1971. [BGVQO] N. Berline, E. Getzler, and M. Vergne. Dirac Operator and Heat Kernel. Grundlehren der Math. Springer, 1990. [CFG89] M. Cahen, A. Frank, and S. Gutt. Spectrum of the Dirac operator on complex projective space P;q-1(C). Letters in Mathematical Physics, 18:165—176, 1989. [Hcl78] S. Helgason. Dijerential Geometry, Lie Groups and Symmetric Spaces. Academic Press, 1978. [HM4] N. Hitchin. Harmonic spinors. Adv. in Math., 14:1~55, 1974. [IT78] A. Ikeda and Y. Taniguchi. Spectra and eigenforms of the Laplacian on S" and P”( (S. Seifarth) INSTITUT FSR, ANGEWANDTE ANALYSIS UND Sroonssrrrx BERLIN, Mounsusra. 39, D—10117 BERLIN (U. Semmelmann) HUMEoLnr—UN1vERs1r.Kr zu BERLIN, INs·r1·rU·r FSR REINE MATHEMATIK (SFB 288), ZIEGELSTR. 13A, D—10099 BERLIN