- Published 1999

Using an adiabatic collapse trick we determine, by two different methods, the eta invariants of many Dirac type operators on circle bundles over Riemann surfaces. These results, coupled with a delicate spectral flow computat ion, are then used to determine the virtual dimensions of moduli spaces of finite energy Seiberg-Witten monopoles on 4-manifolds bounding such circle bundles. TABLE OF CONTENTS Introduction 62 1. The eta invariant of a first order elliptic operator 66 §1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . 66 §1.2. The Atiyah-Patodi-Singer theorem . . . . . . . . . . . . . 67 § 1.3. Variational formulm . . . . . . . . . . . . . . . . . . . . 70 2. E t a i nva r i an t s o f D i r ac o p e r a t o r s 77 §2.1. The differential geometric background . . . . . . . . . . . . 77 §2.2. The eta invariant of the spin Dirac operator . . . . . . . . . 80 §2.3. The eta invariant of the adiabatic Dirac operator . . . . . . . 82 §2.4. The eta invariant of the coupled adiabatic Dirac operator 83 3. F in i t e e n e r g y S e i b e r g W i t t e n m o n o p o l e s 88 §3.1. The 4-dimensional Seiberg-Witten equations . . . . . . . . . 89 Received June 17, 1998 61 62 L.I. NICOLAESCU Isr. J. Math. §3.2. The 3-dimensional Seiberg-Witten equations . . . . . . . . . 92 §3.3. Spectral flows and perturbat ion theory . . . . . . . . . . . . 97 §3.4. Virtual dimensions . . . . . . . . . . . . . . . . . . . . 103 A p p e n d i x A. Proo f of the first transgression formula 112 A p p e n d i x B. Proo f of the second transgression formula 114 A p p e n d i x C. Elementary computat ion of the eta invariants 115 A p p e n d i x D. Technical identit ies 118 References 121 Introduct ion The eta invariant was introduced in mathematics in the celebrated papers lAPS1-3] as a correction term in an index formula for a non-local, elliptic boundary value problem and since then it has been subjected to a lot of scrutiny because of its appearance in many branches of mathematics. Contrary to the index density of an elliptic operator, the eta invariant is a nonlocal object and this explains why it is so much harder to compute. Most concrete computat ions rely on special topologic or geometric features. For example, one can use the Atiyah-Patodi-Singer theorem to compute the eta invariant of the signature operator because in this case the eta invariant is a combination of a topological term (the signature of a 4k-dimensional manifold with boundary) and a local contribution (the integral of the L-genus). For Sl-bundles over Riemann surfaces, this approach was successfully carried out in [Ko] (see also [O] for similar results in the more general case of Seifert manifolds). For the Dirac operator associated to a spin structure such an approach is not possible because the index of the Atiyah-Patodi-Singer problem is notoriously dependent upon the metric. However, if all the manifolds involved have positive scalar curvature then a Lichnerowicz type argument allows the computat ion of the index and thus, in this case, the computation of the eta invariant is a local problem. The first goal of this paper is to compute the eta invariant of some Dirac operators on the total space of a nontrivial circle bundle N over a Riemann surface E of genus > 1. The second goal is to use the eta invariant information to determine the virtual dimensions of the moduli spaces of finite energy solutions of the Seiberg-Wit ten equations on a 4-manifold bounding a disjoint union of circle bundles over Riemann surfaces. As in [N], we will work with product-like metrics on N such that the fibers are very short. Such metrics have negative scalar curvatures and thus are beyond Vol. 114, 1999 ETA INVARIANTS OF DIRAC OPERATORS 63 the reach of the Lichnerowicz vanishing approach. Instead, using the results of Bismut Cheeger [BC] and Dai [Dai] we will compute the eta invariant for the usual Dirac operator using its known adiabatic limit (i.e. its limiting value as the geometry of N changes so that the fibers become shorter and shorter). To recover the eta invariant (at least for short fibers) one can use known variational tbrrnulae and some very precise information about the very small eigenvalues (in the sense of [Dai]) of the Dirac operators determined by metrics with very short fibers. It turns out that the variational formulae in this case involve no spectral flow contribution. Once this computat ion is performed we embark on a related problem. More precisely, we will determine the eta invariant of a very special scalar perturbat ion of the Dirac operator. These perturbed Dirac operators (we called them adiabatic Dirac operators) arose in IN] where we studied the adiabatic limits of the Seiberg Wit ten equations on circle bundles (see also [MOY]). We again use a variational approach. This time, however, there is a spectral flow contribution which requires some "spectral care". An adiabatic approach was also used in [SS] to compute the eta invariant of Dirac operators on circle bundles over Riemann surfaces of genus > 2. There are two main differences. The first difference comes from the spin structure considered in [SS] which extends to the disk bundle bounding our circle bundle. We perform our computations on Dirac operators associated to spin c structures pulled back from the base of our fibration and these, as explained in [KS], have notable topological properties. For example, the pullback of a spin structure from the base does not extend to a spin-structure on the bounding disk bundle, though it extends as a spin<structure. This explains why the adiabatic limit in [SS] is different from ours and shows that the eta invariants can distinguish spin structures!!! The second difference is in the manner in which the adiabatic limit is computed. In [SS], using the representation theory of PSL2(IR) the authors determine explicitly the adiabatically important part of the spectrum which allows them to determine the adiabatic limit of eta itself. We achieve this in two ways. The first method uses the results of Bismut, Cheeger and Dai. In Appendix C we present a second method, which works for the adiabatic Dirac operators. Their whole eta functions can be computed directly and "elementarily", and can be elegantly described in terms of Riemann's zeta function and some topological invariants. This argument extends easily to the more general case of Seifert manifolds. We present this extension in a separate paper [N1] to isolate the very complex eombi64 L.I. NICOLAESCU Isr. J. Math. natorics, generated by the singular fibers, from the analytical arguments, which work without any modification in the general case. The eta invariant is an essential ingredient in the computation of the virtual dimension of the moduli space of finite energy solutions of the Seiberg-Witten equations on a 4-manifold with boundary a disjoint union of Sl-bundles over Riemann surfaces. For closed 4-manifolds the virtual dimension of the moduli space of solutions of the Seiberg-Witten equations corresponding to the spin c structure a is given by d(a) = ~(c l (a) 2 (2e + 3~)) where cl (a) denotes the Chern class of the line bundle determined by the spin c structure, while e respectively T are the Euler characteristic and resp. the signature of the 4-manifold. In the non-closed case the above formula is no longer true. There is a correction term determined by the asymptotic value of a finite-energy solution. We compute this correction term via the Atiyah-Patodi-Singer and the Seiberg-Witten analogues of the results in [MMR] describing the structure of the finite energy moduli space. There is an additional difficulty one has to overcome. The operators describing the deformation complex of this moduli space are based not just on the adiabatic operator alone. They depend on a very explicit (though complicated) perturbation of the direct sum (Dirac operator ® odd signature operator). The final determination of the virtual dimension relies on an excision trick which requires a spectral flow computation. Some of the eigenvalues changing sign do not do this transversally and detecting them is a very delicate perturbat ion theoretic problem. The theoretical basis of our approach is described in [FL] and [KK] which deal with similar degeneration problems in the case of the odd signature operators twisted by flat connections. We obtain explicit formula~ for the virtual dimensions for any 4-manifold bounding disjoint unions of circle bundles. We briefly describe one instance when the asymptot ic limit of a finite energy solution is irreducible. The total space N of a degree e # 0 Sl-bundle over a Riemann surface ~ of genus g can be equipped with a spin structure obtained by pullback from a fixed spin structure on ~. The sp/n c structures can be identified with second degree integral cohomology classes a E H2(N) ~Z 2g @ Zle I. The three dimensional Seiberg-Wit ten equations have solutions only if a is a torsion class a -k mod ]e I. Set Rk = {n e Z; l < ] n [ < g 1 , n = k mode}. Vol. 114, 1999 ETA INVARIANTS OF DIRAC OPERATORS 65 In [N] we have shown that the space of irreducible solutions of a certain perturbation of the Seiberg-Witten equations on N is smooth, and its components are bijectively parametrized by Rk. Fix a spin c structure 6 on N extending the spin ~ structure k on N. Suppose N is a four manifold with boundary 0 N = N and C := (¢, A) is a finite energy solution of the Seiberg-Witten equations on N ~ N U IR+ × N. If the asymptotic limit of (~ is an irreducible solution on N lying in the connected component labelled by n c Rk, then the expected dimension of a neighborhood of C in its moduli space is 1(--~--~2 /N FA AFA-(2e (N)+ 3sign(1Q)) ) 4 ~ ( s i g n ( g ) 1 ) + n + ~ ( 2 9 1 ) ~ ( g s i g n ( g ) ) . We tested our results in special case of "tunnelings". These are finite energy solutions of the Seiberg-Witten equations on an infinite cylinder ~ x N. Our results are in perfect agreement with the computations in [MOY] obtained by entirely different methods. There are similarities between our paper and [MOY], but there are also many important differences. The paper [MOY] is interested in finite energy solutions of the Seiberg Wit ten equations only on cylinders R x M where M is a Seifert fibration. The techniques used there are algebraic-geometric in nature and allow them to obtain detailed information about the nature of solutions, leading eventually to virtual dimension formula~. In this paper (and its sequel [N1]) we are interested in finite energy solutions on any 4-manifold with cylindrical ends of the form R+ x M where M is again a Seifert manifold. This is outside the realm of algebraic geometry so we use entirely different methods, differential-geometric in nature. We obtain virtual dimension formulm in this general situation and, additionally, detailed information about the eta invariants of many Dirac operators. As shown in [N1] and IN2], these eta invariants contain a remarkable amount of topological information. On the other hand, some informations about tunnelings obtained in [MOY] are not accessible by our techniques. This paper is divided into three sections and four appendices. The first section is essentially a brief survey of known facts concerning the eta invariant: definition, the At iyah-Patodi-Singer theorem, variational formulee and the spectral flow. We included these facts as a service to the reader, to eliminate any ambiguity concerning the various sign conventions. There does not seem to be general agreement on these conventions and, additionally, we used some "folklore" results for which we could not indicate satisfactory references. 66 L.I. NICOLAESCU Isr. J. Math. The second section contains the main steps in the computat ion of the eta invariants discussed above. We begin by describing the geometric background and the various Dirac operators. Then using variational formulae for the eta invariant and the adiabatic results of Bismut-Cheeger-Dai we compute in the second part the eta invariant of the Dirac operator on a circle bundle with very short fibers (Theorem 2.4). In the third part, we compute the eta invariant of the adiabatic Dirac operator a per turbat ion of the Dirac operator which arose in IN]. This is achieved in Theorem 2.6 via a variational formula and a spectral flow computation. The computat ions of certain transgression terms involved in the variational formuse are deferred to appendices. An alternative method of computation is described in Appendix C. The last par t of this section is devoted to extending the previous computat ions to the Dirac operators coupled with flat line bundles. We use essentially the same variational strategy. However, new phenomena arise during the computat ion of some spectral flow contributions. The third section is devoted to applications to Seiberg-Witten theory. The first two subsections describe the 3and 4-dimensional Seiberg-Witten equations and some basic facts about them established in [MOY] and IN]. The third subsection is entirely devoted to the computat ion of a spectral flow. This is a very delicate job since one has to worry about eigenvalues changing sign in a nontransversal manner. In the last subsection we compute virtual dimensions of finite energy Seiberg Wit ten moduli spaces on 4-manifolds founding circle bundles over Riemann surfaces and we conclude by comparing our answers in the special case of tunnelings to those in [MOY]. ACKNOWLEDGEMENT: While working on the eta invariants I benefited very much from conversations with X. Dai, J. Lott, and M. Ouyang. I want to express here my gratitude. 1. T h e e t a i n v a r i a n t o f a f i rs t o r d e r e l l ip t ic o p e r a t o r §1.1. DEFINITION. The elliptic selfadjoint operators on closed compact manifolds behave in many respects as common finite dimensional symmetric matrices. The eta invariant extends the notion of signature from finite dimensional matrices to elliptic operators. We will denote the trace of an infinite dimensional operator (when it Vol. 114, 1999 ETA INVARIANTS OF DIRAC OPERATORS 67 exists) by "Tr" while "tr" is reserved for finite dimensional operators. We have the following result. PROPOSITION 1.1: (a) Consider a dosed, compact, oriented R iemann manifold (N ,g ) o f dimension d, E --+ N a hermitian vector bundle and A: C ° ° ( E ) --+ C ~ ( E ) a first-order selfadjoint dl ipt ic operator. Then (1.1) ~)A(S) -F( s + ~ t (s-1) /2Tr(Ae-tA'~)dt = E dimV~ d i m V _ ~ ~-) A s A>0 (V~ = ker(A A)) is well defined for all 9~e s >> 0 and extends to a meromorphic function on C. Its poles are all simple and can be located only at s = ( d + 1 n ) /2, n = O, 1 ,2 , . . . . (b) I f d is odd, then the residue of ~IA (s) at s = 0 is zero so that s = 0 is a regular point. For a proof of this proposition we refer to [APS3]. When d is odd we define the eta invariant of A by zl(A) := qA(0). R e m a r k 1.2: (a) From the definition it follows directly that 7/(-A) = -z~(A) and 7/(AA) = zl(A), VA > O. (b) In [BF] it is shown that if A is an operator of Dirac type then one can define its eta invariant directly by setting s = 0 in (1.1). In other words, in this case -2/? , ( A ) = t-1/2Tr ( Ae -tA~ )dt. In the sequel, we will reserve the letter D to denote Dirae type operators. §1.2. THE ATIYAH-PATODI SINGER THEOREM. The importance of the eta invariant in mathematics is due mainly to its appearance in the formula for the index of an elliptic boundary value problem first considered by Atiyah Patodi-Singer in [APS1]. Suppose that (M, g) is a compact, (d+ 1)-dimensional, oriented Riemann manifold with boundary N = OM. We assume d is odd and that the metric g is a product on a tubular neighborhood ( -1 , 0] × N of the boundary, i.e. g = du 2 +go, where go is a metric on N (see Fig. 1). We orient N such that the outer normal followed by the orientation of N gives the orientation of M. (This is precisely the orientation that makes the Stokes' formula come out right.) 68 L . I . N I C O L A E S C U Isr. J. Ma th .

@inproceedings{Nicolaescu1999EtaIO,
title={Eta Invariants of Dirac Operators on Circle Bundles over Riemann Surfaces and Virtual Dimensions of Finite Energy Seiberg-witten Moduli Spaces},
author={Liviu I. Nicolaescu},
year={1999}
}