On the Eigenvalues of the Twisted Dirac Operator ͒ Marcos Jardim and Rafael F

JOURNAL OF MATHEMATICAL PHYSICS 50, 063513 ͑2009͒

On the eigenvalues of the twisted Dirac operator ͒ Marcos Jardim and Rafael F. Leãoa Department of Mathematics, IMECC—UNICAMP, Caixa Postal 6065, Campinas, 13083-970 Sao Paulo, Brazil ͑Received 3 June 2008; accepted 22 April 2009; published online 24 June 2009͒

Given a compact Riemannian spin manifold whose untwisted Dirac operator has trivial kernel, we ﬁnd a family of connections ٌAt for t෈͓0,1͔ on a trivial vector bundle of rank no larger than dim M +1, such that the ﬁrst eigenvalue of the twisted Dirac operator D is nonzero for t1 and vanishes for t=1. However, if one At restricts the class of twisting connections considered, then nonzero lower bounds do exist. We illustrate this fact by establishing a nonzero lower bound for the Dirac operator twisted by Hermitian–Einstein connections over Riemann surfaces. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3133944͔

I. INTRODUCTION Given a compact Spin manifold ͑M ,g͒, one can consider the spinor bundle S and the associated Dirac operator D. There is a vast literature about this operator, in particular, concerning the behavior of its spectrum, see Ref. 4. One of the most studied problems is to ﬁnd lower bounds for the eigenvalues of D. The most well-known result was obtained by Friedrich4 and can be stated as follows: if ͑M ,g͒ is a compact Riemannian spin manifold, with positive scalar curvature R, then the eigenvalues of the associated Dirac operator satisfy the inequality

1 n ␭2 Ն R , 4 n −1 0

where R0 is the minimum of the scalar curvature and n is the dimension of M. If the equality is satisﬁed, then the scalar curvature R is constant and M is an Einstein manifold. In addition, if further geometric structures in ͑M ,g͒ are imposed, then the above lower bound can be improved. For example, when ͑M ,g͒ is a compact Kähler manifold, then Kirchberg7 proved that the eigenvalues of the Dirac operator satisfy the inequality

1 k +1 R if k = dim M is odd 4 k 0 C ␭2 Ն Ά 1 k · R if k = dim M is even. 4 k −1 0 C The influence on the eigenvalues of the Dirac operator of other geometric structures on M is found in Refs. 10 and 11; see also the survey in Ref. 5, Kirchberg also considered estimates in terms of other curvature tensors, see Ref. 8 and the references therein. However, this is not the only Dirac operator one can define on a spin manifold ͑M ,g͒. For .instance, consider a Hermitian vector bundle with a compatible connection ͑E,ٌA͒ over ͑M ,g͒ Aٌ Using the connection on E, one can define the twisted Dirac operator DA on S E. This operator is extremely important in classical field theory, for it describes particles such as electrons and neutrinos coupled to external gauge fields. Very little is known about the behavior of the Aٌ eigenvalues of DA in terms of the coupling connection .

͒ a Electronic mail: [email protected].

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More precisely, ﬁx the base manifold and geometry ͑M ,g͒ and a Hermitian vector bundle E→M. Let A be the afﬁne space of compatible connections on E and consider the functional

␭:A→R+,

␭ ٌ͑A͉͉͒ ۋ Aٌ 1 that associates with each connection ٌA ෈A the absolute value of the first nonzero eigenvalue of the associated Dirac operator DA. In analogy with the untwisted case, it would be interesting to determine whether this functional possesses a nonzero lower bound, and whether it is bounded above. In Ref. 14 Vafa and Witten proved that the functional ␭ does admit a universal upper bound which depends only on the geometry of the compact base manifold ͑M ,g͒ but not on the twisting bundle E; see also Ref. 2 for a clear geometrical proof. On the other hand, lower bounds are known only for very special cases, see, for instance, Ref. 12, and rely on strong restrictions both .on the base manifold ͑M ,g͒ and especially on the connection ٌA Despite these particular results, the problem of how the bounds for eigenvalues of the twisted Dirac operator, and consequently the spectrum, changes with the twisting connection is not well understood. The first natural question is whether the functional ␭ has a nonzero lower bound. In this article, we construct a general example showing that the first eigenvalue of the twisted Dirac operator depends strongly on the twisting connection. More precisely, Let ͑M ,g͒ be a compact Riemannian spin manifold such that the free, untwisted Dirac operator has trivial kernel. Then we find a trivial bundle Cគ N →M, where either N=dim M if M is even dimensional or N dim M +1 if M is odd dimensional, and a family of connections ٌAt on Cគ N for t෈͓0,1͔, such that= the first eigenvalue of the twisted Dirac operator D is nonzero for t1 and vanishes for t=1. In At other words, a Vafa–Witten type of result for an universal lower bound for the first eigenvalue of the twisted Dirac operator is impossible; lower bounds for the first nonzero eigenvalue necessarily depend on the topology and the geometry of the twisting bundle. However, if one restricts the class of twisting connections considered, then nonzero lower bounds do exist. We illustrate this fact by establishing a nonzero lower bound for the Dirac operator twisted by Hermitian–Einstein connections over Riemann surfaces. More generally, we believe that interesting lower bounds for the twisted Dirac operator can be obtained whenever the twisting connection satisfies some classical field equation, such as the anti-self-duality Yang–Mills equation.

II. ARBITRARILY SMALL EIGENVALUES Let ͑M ,g͒ be a compact Riemannian spin manifold such that the untwisted Dirac operator has trivial kernel. Two important cases that satisfy this condition are when ͑M ,g͒ has positive scalar curvature, as discussed in Sec. I, or when ͑M ,g͒ is a torus equipped with a nontrivial spin structure. In this section, we will construct a one-parameter family of connections ٌAt for t෈͓0,1͔ on a trivial bundle Cគ N =CN ϫM for which the first eigenvalue of the associated twisted Dirac operator D is nonzero and arbitrarily small. The main idea is to show that the trivial bundle គ N admits two At C connections, ٌ0 and ٌ1, such that the Dirac operator twisted by ٌ0 has trivial kernel while the .Dirac operator twisted by ٌ1 has nontrivial kernel The difficult part of our construction is to find the right trivial bundle Cគ N and the connection .This will be done in two separate cases .1ٌ

A. The even dimensional case Let us ﬁrst assume that the manifold ͑M ,g͒ is even dimensional, with dim M =2n. Let ,E→M be a Hermitian vector bundle over M and ٌA be a compatible connection on E. In this case the spinor bundle splits as follows:

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S = S+ S−, ͑1͒ + − which implies that the twisted Dirac operator also splits DA =DA DA, where

+ + → − DA:S E S E,

− − → + ͑ ͒ DA:S E S E. 2 + The index for the operator DA can be topologically calculated with the following formula:

͑ +͒ ͑ ͒n͵ ͑ ͒ ∧ ˆ ͑ ͒ ͑ ͒ ind DA = −1 ch E A M , 3 M

where Aˆ is the so-called Aˆ -genus of M, a characteristic class that can be written in terms of the Pontrjagin classes, and ch͑E͒ is the Chern character of the bundle E. This expression, together with the analytical index

͑ +͒ + − ͑ ͒ ind DA = dim kerDA − dim kerDA 4 and the fact that both ways to calculate the index are equal by the Atiyah–Singer index theorem, allows us to conclude that we can ﬁnd a Hermitian vector bundle E with connection ٌA such that + DA has nontrivial kernel. Indeed, if the index of DA is not zero, then DA has a nontrivial kernel. + Conversely, if the Dirac operator has trivial kernel, then the index of DA must vanish; in particular, the Aˆ -genus of M vanishes by our assumption of the untwisted Dirac operator having trivial kernel. Now take E→M to be the pullback of the generating bundle H over S2n by a map f :M →Sn of degree 1; more precisely, H→S2n is a complex vector bundle of rank n, such that ͑ ͒ Յ Յ ͑ ͒ 2n͑ 2n͒ ck H =0 for 1 k n−1 and cn H is a generator of H S . Then the only nonvanishing Chern ͑ ͒ class of E is the top Chern class cn E , which can be represented, in de Rham cohomology, by a generator of H2n͑M͒. Now given an arbitrary connection ٌA on E→M, it follows from the index ͑ ͒ ͑ +͒ formula 3 that ind DA =1, hence DA must have nontrivial kernel. Now let HЈ→S2n be a bundle complimentary to the bundle H→S2n constructed above; notice HЈ, so thatءthat HЈ also has rank n. Take EЈ= f

E EЈ Ӎ Cគ 2n. ͑5͒ This is the desired trivial bundle. ˜ .Choosing any connection ٌB in EЈ, set ٌA =ٌA ٌB, which is a connection on E EЈӍCគ 2n Next, consider the twisted Dirac operator D˜A =DA DB; the fact that DA has nontrivial kernel in E implies that D˜A also does.

B. The odd dimensional case Now let us assume that dim M =2n−1, so that dim͑M ϫS1͒=2n. Following Atiyah,2 let 2n−1→ ͑ ͒ ␲ ͑ ͑ ͒͒Ӎ F:S U n be a generator of 2n−1 U n Z. Consider the following one-parameter family of connections on Cគ n →S2n−1:

−1 ,␶ = dគ + ␶F dFٌ joining the trivial connection dគ on Cគ n to its gauge transform under the map F. Alternatively, this may be regarded as a connection on the rank n bundle H→S2n−1ϫS1 constructed as follows. Let I=͓0,1͔ and consider the following equivalence relation on Cn ϫS2n−1ϫI:

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͑ ␶ ͒ϳ͑ ␶ ͒ ͑ ͒ ␶ ␶ v,x, 0 w,y, 1 if x = y, w = F x v, 0 =0, 1 =1. It is easy to see that H=͑Cn ϫS2n−1ϫI͒/ϳ is a rank n vector bundle over S2n−1ϫS1. Furthermore, the one-parameter family of connections ٌ␶ can also be regarded as a connection on the trivial .bundle over S2n−1ϫI, which in turn descends to a connection ٌA on H→S2n−1ϫS1 Now let f :M →S2n−1 be a degree 1 map, and let E→M ϫS1 be the pullback of H under the → ϫ ٌ គ n product map f 1S1. We will also denote by ␶ the pullback to C M of the one parameter family of connections on Cគ n →S2n−1 and by ٌA the pullback to E→M ϫS1 of the connection on H→S2n−1ϫS1 constructed above. ٌ Associated with the family of connections t over M is the one-parameter family of Dirac operators Dt given by ␶ D␶ = D0 + X, គ n គ n where D0 is the Dirac operator on S C twisted by the trivial connection on C , and X is the algebraic operator on S Cគ n locally given by

2n−1 F˜ץ ͚ e · ˜F−1 , ץ i i=1 xi where ˜F:M →U͑n͒ is the composition Fؠ f. By construction, we have D␶ coincides with the ϫ͕␶͖ ٌA restriction to M of the Dirac operator DA twisted by the connection deﬁned on the bundle 1 E→M ϫS as above. By Ref. 3, the spectral ﬂow of the family D␶, which measures the number of eigenvalues that go from positive to negative as the value of ␶ goes from 0 to 1, is equal to the

index of DA, which, according to Ref. 2, is equal to 1. ء .has a zero eigenvalue ءIt follows that, for some value ␶ of the parameter ␶, the operator D␶ 1− ء A -dគ +␶ F dF and proceed as in the last paragraph of Sec. II A to obtain a connec= ءWe set ٌ =ٌ␶ A គ 2n˜ٌ tion on the trivial bundle C over M, such that the D˜A has a nontrivial kernel.

C. Completing the construction The construction of the connection ٌ0 is trivial. Note that sections of the trivial bundle S Cគ N can be written in the form ␺ 1 ␺ ␺ = ΂ 2 ΃, ͑6͒ ] ␺ N គ គ N which implies that, for the trivial connection d on C , the associated twisted Dirac operator D0 can be written as

␺ D 1 D␺ ␺ 2 ͑ ͒ D0 = ΂ ΃, 7 ] ␺ D N where D is the free, untwisted Dirac operator associated with ͑M ,g͒. In particular, this means that the only effect of the coupling with the trivial connection is to change the multiplicity of the eigenvalues, leaving the spectrum unchanged. Since we assumed that D has trivial kernel, it follows easily that D0 must also have trivial kernel. Summing up, if ͑M ,g͒ is a Riemannian spin manifold whose untwisted Dirac operator has trivial kernel, then one can choose a connection, the trivial one, on the trivial vector bundle Cគ N such that the twisted Dirac operator also has trivial kernel.

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Now we have in Cគ N the two connections with the desired properties: one is the trivial con- ˜ nection dគ and the other is ٌA =ٌA ٌB. The Dirac operator associated with the first has trivial kernel, while the one associated with the latter has nontrivial kernel. Since the space of connections over a fixed vector bundle is an affine space modeled in the space of 1-forms over M with values in the endomorphism bundle, we can write

˜ A = dគ + ␣, ͑8ٌ͒ where ␣ is a 1-form over M with values in the endomorphism bundle. This can be used to deﬁne the family of connections,

At = dគ + t␣, ͑9ٌ͒ and the family of twisted Dirac operators D in the obvious way. Clearly, D =D and At A0 0 D =D˜, and by construction, D does not have kernel while D does. Since the functional ␭ A1 A A0 A1 deﬁned in ͑1͒ is continuous ͑see Ref. 2͒, we conclude that ␭͑D ͒ is nonzero for t1 and vanishes At for t=1, as desired.

III. UNIFORM BOUND FOR RIEMANN SURFACES In this section we show that for suitable conditions on the base manifold and on the twisting Aٌ connection , it is possible to find lower bounds for the first nonzero eigenvalue of DA. First of all, note that the topological index formula ͑3͒ implies that if E is a vector bundle with ͑ ͒ c1 E 0 over a Riemann surface, then for any connection A on E the twisted Dirac operator DA has nonvanishing index, which implies that for it has nontrivial kernel. Because of this fact, lower bounds only make sense for non-null eigenvalues. Riemann surfaces are naturally Kähler manifolds, with the Kähler form given by the volume form ␻=i␰∧¯␰. We can use this to make the connection ٌA satisfies a compatibility condition -known as the Hermitian–Einstein condition. A connection ٌA is called a Hermitian–Einstein con nection if ␻ ͑ ͒ FA =−icI, 10 A ␻ٌ where FA denotes the curvature 2-form of and is the contraction by the Kähler form. For background on the importance of the Hermitian–Einstein condition, see Ref. 9. Another important feature of a Riemann surface M is that the spinor bundle can be explicitly described in terms of forms ͑see Ref. 13͒; more precisely, it is well known that the spinor bundle Ӎ∧͑0,0͒ ∧͑0,1͒ associated with the complex structure is SC , so the usual spinor bundle is

Ӎ −1/2 Ӎ͑∧͑0,0͒ ∧͑0,1͒͒ −1/2 ͑ ͒ S SC KM KM , 11

where KM is the canonical bundle of M. Furthermore, the complex Dirac operator D coincides D 1/2 with a twisted real Dirac S, where S is the connection on KM induced by the Chern connection on M. Now let E→M be a holomorphic vector bundle of negative degree on M that admits a Hermitian–Einstein connection ٌA. In Ref. 6 the authors have shown that the eigenvalues of the twisted complex Dirac operator satisfy the following lower bound:

4␲ deg͑E͒ ␭2 Ն − . ͑12͒ rk͑E͒vol͑M͒ This lower bound for the eigenvalues of the twisted complex Dirac operator can be translated into a lower bound for the eigenvalues of the twisted real Dirac operator in the following manner. First, ˜ Aٌ ͒ ͑͒ ͑ ͒ ͑ ͒ ͑ −1/2 consider L=KM E, so that deg L =deg E −rk E 1−g , where g is the genus of M. Let be S ٌAٌ the tensor connection I+I . Then the twisted complex Dirac operator D˜A coincides with D ͑ ͒ the twisted real Dirac operator A. Thus applying the lower bound 12 to the bundle L, we obtain

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4␲͑1−g͒ 4␲ deg͑E͒ ␭2 Ն − , ͑13͒ vol͑M͒ rk͑E͒vol͑M͒ which is a lower bound for the eigenvalues of the twisted real Dirac operator on M. Of course, this lower bound is only signiﬁcant provided deg͑E͒Ͻrk͑E͒͑1−g͒. The Gauss–Bonnet formula can be used to rewrite ͑13͒ in the following manner:

R deg͑E͒ ␭2 Ն 0 ͩ1− ͪ if g 1, 2 ͑1−g͒rk͑E͒

4␲ deg͑E͒ ␭2 Ն − if g =1. rk͑E͒vol͑M͒ Comparing with the results in Ref. 1 ͑see Theorems 5.3, 5.10, and 5.22 there͒, we conclude that the above estimates are actually attained in the case where M has constant scalar curvature, E→M is a line bundle, and the connection on E has constant curvature.

IV. CONCLUSION We have constructed a general example showing that the first eigenvalue of the twisted Dirac operator on a vector bundle over an arbitrary compact Riemannian spin manifold ͑M ,g͒ whose untwisted Dirac operator has trivial kernel depends strongly on the twisting connection. This was done by providing a one-parameter family of connections ٌAt on a trivial vector bundle over M of appropriate rank such that the first eigenvalue of the twisted Dirac operator D is nonzero for t At 1 and vanishes for t=1. Therefore, one cannot expect absolute lower bounds for the nonzero eigenvalues of a Dirac operator twisted by arbitrary connections. However, if one imposes conditions on the class of twisting connections, e.g., the connection satisfies some classical field equation such as the anti-self-duality Yang–Mills equation or the Hermitian–Einstein condition, then lower bounds do exist. We illustrate this fact by establishing a sharp, nonzero lower bound for the complex Dirac operator twisted by Hermitian–Einstein connections over Riemann surfaces.

ACKNOWLEDGMENTS M.J. is partially supported by the CNPQ Grant No. 305464/2007-8 and the FAPESP Grant No. 2005/04558-0. R.F.L.’s research was supported by a CNPQ doctoral grant.

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