n s n Z 0 on complex-valued functions on the circle. Then sgn n + 1 ) n + 1 2) s

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1 . What is the eta invariant? The eta invariant was introce in the famos paper of Atiyah, Patoi, an Singer see [], in orer to proce an inex theorem for manifols with bonary. The eta invariant of a linear self-ajoint operator is roghly the ifference between the nmber of positive eigenvales an the nmber of negative eigenvales. One problem with this iea is that it oes not make sense in infinite imensions. However, we will have a way of reglarizing to make this qantity well-efine for ifferential operators; this is similar to the zeta-fnction reglarization of the eterminant of the Laplacian an methos se by physicists to reglarize qantities that are compte sing ivergent integrals. In fact, jst as the zeta fnction of elliptic operators is analogos to the Riemann zeta fnction, the eta fnction is analogos to Dirichlet L- fnctions. Assme that we know the eigenvales {λ}with mltiplicity of an essentially self-ajoint sally first orer ifferential operator D : C E C E on sections of a vector bnle E M, where M is a close Riemannian manifol. We efine the eta fnction η s to be η s λ sgn λ λ s, where we efine sgn. It trns ot that the eta fnction is holomorphic in s for large Re s, if D is elliptic; we will iscss this later. This is the zeta fnction if D has nonnegative eigenvales. The eta invariant is η, which means that we analytically contine to s. We see that this qantity is formally the nmber of positive eigenvales mins the nmber of negative eigenvales,. Note that there is no reason to expect that this nmber is efine ie η s is reglar at s ; it trns ot that it often is or that it is an integer often it is not. Remark: we can efine these for pseoifferential operators as well. Boring example: Let D, a ifferential operator acting i θ on complex-vale fnctions on the circle. In the notation above, M S, E M C. We remark that this is the most elementary example of a Dirac operator. We will now compte the eta invariant of this operator. Observe that the eigensections of this operator are the fnctions e inθ corresponing to eigenvale n Z, an in fact these eigenfnctions form an orthogonal basis of L S. Therefore, the eta

2 fnction is η s λ n Z sgn λ λ s sgn n n s n Z > n s n Z > n s. Note that the sm above converges absoltely for Res >, so the calclation is vali. Analytically contining the fnction η s to s, we obtain η. Ostensibly less boring bt jst as boring example: Let D i + θ on complex-vale fnctions on the circle. Then η s sgn λ λ s λ sgn n + n + n Z n + s n Z n Z s n + s, so again η. Finally, a non-boring example: Let D + c on complexvale fnctions on the circle, where c is a real constant. i θ Then η s λ sgn λ λ s n Z sgn n + c n + c s n> c n + c s n< c n + c s, which is nonzero in general. Bt now what o we o to obtain η, or to obtain a close-form expression for η s?. Families of Operators Proposition. Let Q be a C family of nonnegative self-ajoint operators with a complete system of eigenvales λ for which the eigenfnctions form a basis for L, sch that ζ s is efine an analytic at s an that there is a constant N > sch that ζ s T r Q s λ λ s converges absoltely for s > N, an so that zero eigenspace epens ifferentiably on an ths is of constant imension. Then the zeta fnction corresponing to Q satisfies ζ s s T r Q Q s.

3 Proof. For a simple proof, if one may assme that the eigenvales may be chosen to be ifferentiable in as in Rellich s Theorem, then one proves it like this: ζ s s λ s λ s T r QQ s for large s, an then by the ientity theorem, the analytic contination satisfies the same eqation. To prove it really sing the ieas of Seeley, note that ζ s πi T r λ s Q λ λ Γ for some contor Γ enclosing the real axis. For large Res, convergence is garantee by estimates in [4]. Next we ifferentiate to get Q λ Q λ I Q λ Q λ Q Q λ. 3 Then ζ s πi T r λ s Q λ Γ Q Q λ λ. If it happene that Q is of very large orer, Q λ is trace class an has a continos Schwarz kernel, an we may interchange trace with integral an commte operators within the trace. Then ζ s λ s T r Q Q λ λ πi Γ. πi T r Q λ s Q λ λ Γ πi T r Q sλ s Q λ λ Γ st r Q Q s. parts

4 4 Bt what if Q is not of large orer. Then if m >> we still have Q m satisfies the above, an ζ ms st r Q m Q ms m s j s j mst r T r T r Q j Q Q m j Q Q ms Q Q ms, Q ms m assming one can commte the operators aron in the trace, an ths the theorem is one. To careflly commte the operators in the trace, we see that if m is large enogh T r Q j Q Q m j Q ms m T r Q j Q Q ms j T r Q j Q Q ms/ Q ms/ j T r Q ms/ j Q j Q Q ms/ T r Q ms/ Q Q ms/ T r Q Q ms/ Q ms/ T r Q Q ms. Proposition. Let D any self-ajoint operator for which η s is efine an analytic at s, an that there is a constant B > sch that λ sgn λ + c λ + c s an λ λ + c s+ converge absoltely for s > B an c in a certain interval sch that c is not an eigenvale of D for all c in that interval. Then the eta fnction η c s corresponing to the operator D + c satisfies c η c s sζ D+c s + where ζ D+c is the zeta fnction corresponing to the nonnegative operator D + c, that is ζ D+c s µ> µ s,,

5 where the sm is over all eigenvales of the operator D + c. In particlar, if D is a first-orer elliptic essentially self-ajoint ifferential operator, then η c c is the resie of the simple pole of the meromorphic fnction ζ D+c s+ at s. Remark: It is known that secon-orer essentially self-ajoint elliptic ifferential operators on a manifol of imension n yiel zeta fnctions with at most simple poles, an they are locate at s n, s n, s n,... for n o an at s n, s n,..., s for n even. Frther, the resies at these poles are given by explicitly comptable integrals of locally-efine fnctions. Proof. We know that for each eigenvale λ of D, sgn λ + c oes not vary with c in the interval. Then 5 η c s λ c η c s λ sgn λ + c λ + c s/ sgn λ + c s λ + c s/ λ + c s λ s λ sgn λ + c λ + c s λ + c λ + c s s λ sζ D+c λ + c s+ s + Since both sies are analytic in s for large Res, the statement mst remain tre after analytic contination. Proposition 3. More general version of the last proposition For c in an open interval in R, let D c be a smooth family of self-ajoint operators for which η c s η Dc s is efine an analytic at s for all c, an that there is a constant B > sch that λ c sgn λ c λ c s an λ λc s+ converge absoltely for s > B an c in the interval, sch that im ker D c is constant in c. Then the eta fnction η c s satisfies c η c s st r Ḋ c Dc s+.

6 6 In particlar, if D c is a family of first-orer elliptic essentially selfajoint ifferential operator, then η c c is the resie of the simple pole of the meromorphic fnction T r Ḋ c Dc s+ at s. Proof. We know that for each eigenvale λ of D, sgn λ c oes not vary with c in the interval. By the work of Rellich, we may assme that λ c is ifferentiable in c. Then, for large Res, η c s λ c sgn λ c λ c s/ c η c s sgn λ c s λc s/ λ c c λ c λ c s λ c sgn λ c λ c s λ c c λ c s λ c s λ c s λ c λ c λ c λ c s λ c c λ c λ c λ c s c λ c λ c s c λ c s λ s+ c c λ c λ st r Ḋ c Dc s+ Since both sies are analytic in s, the statement mst be tre for all s. Again, this is st the wimpy version of the proof; one nees to se the resolvent for a rigoros proof that oes not reqire big hammers sch as the Rellich theorem. Remark 4. Until this moment we have always assme that the variation oes not change the qantity sgn λ c. However, we note that one may exactly accont for what happens to η c s as c varies in sch a way that an eigenvale goes throgh zero. That is, if λ is the offening eigenvale sch that λ c passes throgh zero when c c, we may instea consier the operator D c + εp, where P is the projection to the eigenspace corresponing to eigenvale λ c of D c. It trns ot that P can be written entirely in terms of powers of D c an is ths a classical pseoifferential operator as well. If ε is chosen to be sfficiently

7 7 small to eliminate the ifficlty at c. Frther, observe that η Dc+εP s η c s + sgn λ c + ε λ c + ε s sgn λ c λ c s, an pon analytic contination we see that η Dc+εP η c ±. Ths, with no assmptions on passing throgh eigenvales, η c mo is ifferentiable in c, an η c may be calclate precisely by etermining how many eigenvales pass zero. The same hols for the zeta fnction. 3. The Heat Kernel an Zeta Fnction Now we collect some facts abot the heat kernel an zeta fnctions. Let L be a m th orer, nonnegative elliptic classical pseo-ifferential operator on sections of a vector bnle E a close Riemannian manifol M of imension n whose principal symbol is the same as the m power of the Laplacian, ie m/. Then the Cachy problem for the heat eqation has a niqe soltion among soltions that grow less than e t in t: Problem: t + L x, t ; x, f x Soltion: x, t K t, x, y f y V y, M where K t, x, y Hom E y, E x is the heat kernel of L. The operator K satisfies the following asymptotic formla, for each k Z, as t : K t, x, y e x,y n/m 4πt /4t c x, y + c x, y t /m c k x, y t k/m + O t k+/m where x, y is the Riemannian istance from x to y, an each c j is smooth on M M, an c j x, y Hom E y, E x. In Eliean space, E is the trivial line bnle, x, y x y, c x, y, an c j x, y for each j >. Plgging in y x, we obtain, as t, c x, x + c K t, x, x x, x t /m + 4πt n/m... + c k x, x t k/m + O t k+/m. In the ifferential case, the coefficients c j x, y can be calclate by irectly plgging the asymptotic expansion into the ifferential eqation an solving for them. They epen only on the metric an symbol of the operator along the minimal geoesic connecting x an y. Note that in orer that K t, x, y satisfies the initial conition, it mst be tre,

8 8 that c x, x. Note that if e tl is the operator that maps f x to t, x, then T r e tl M T r K t, x, x V x. Uner the aitional assmption that L is an essentially self-ajoint classical pseo-ifferential operator, then we may choose an orthonormal basis of L E consisting of eigensections α k of L corresponing to eigenvales λ k conte with mltiplicity, an we have K t, x, y e tλ k α k x α k y, T r e tl T r K t, x, x V x e tλ k, T r e tl M 4πt n/m c + c t /m c k t k + O t k+ + im ker L an each sm absoltely an niformly converges at each t >. Here, c j M T r c j x, x V. Next, the zeta fnction of a nonnegative self-ajoint elliptic ifferential operator L is efine in analogy to the Riemann zeta fnction as ζ L s λ k λ s k. Note that in the case of the Laplacian L on complex-vale θ fnctions on the circle, which has eigenvales n corresponing to orthogonal eigenfnctions e ±inθ, we have ζ L s n> n s ζ R s, where ζ R s is the Riemann zeta fnction. Note that λ s Γ s t s e tλ t,

9 9 so we have that ζ L s λ k Γ s Γ s + Γ s Γ s + Γ s + Γ s λ s k Γ s t s M t s λ k e tλ k t T r K t, x, x V x im ker L t t s T r K t, x, x V x im ker L t M t s T r K t, x, x V x im ker L t M t s c + c 4πt n/m t /m c N t N/m t t s e tλ k im ker L c + c 4πt n/m t /m c N t N/m t t s e tλ k im ker L t 4π n/m Γ s 4π n/m Γ s N c j t s n m + j m t + φn s j N j c j s n m + j m + φ N s for large s, an this formla gives the meromorphic contination of the zeta fnction ζ L s with φ N s holomorphic for Res > n N. Observe that, as state earlier, in the ifferential case, ζ L s has at most simple poles, an they are locate at s n, s n, s n,... for n o an at s n, s n,..., s for n even note that has a simple zero at each nonpositive integer. The resie of the Γs pole at s n j is c j m m 4π n/m Γ n m m. j We remark that many pseoifferential operators have the same properties regaring the analytic contination. For instance, if A is a self-ajoint ifferential operator an p is any positive real nmber,

10 then A p A p/ is a pseoifferential operator, an ζ A p s λ ps λ ps/ ps ζa, λ λ an its analytic contination an poles can be obtaine from those of ζ A. In particlar, ζ A p ζa, an ζ A p p ζ A. In general, the asymptotic expansions of heat kernels corresponing to pseoifferential operators may have powers of t that increment by, an in aition logarithmic terms may appear. The logarithmic terms case the corresponing zeta fnctions to have poles of higher orer. Accoring Seeley s paper [4], for any classical pseoifferential operator A on a close manifol M, the the restriction of the kernel of A s to the iagonal in M M is meromorphic with poles only at s n k, k,,,... where m is the orer of the operator, n is the m imension of M, an the pole s k n, an its resie is given by m an explicit formla. The resies at s,,,... vanish, an the vale of the kernel at s is again given by an explicit formla. Explicitly, note that Γ s has a simple pole at s with resie. From this we see from the formla above that c n ζ L 4π, n/m an c n/m c n/m x, x is explicitly calclable from the metric an the local symbol of the operator, in the ifferential case. Now we may retrn to the Nonboring Example: Now we apply the Proposition to the operator D + c + c on the circle. By the i θ first proposition, we have that s + c η c s sζ D+c, so that η c c is times the resie of ζ D+c z at z. Bt note that D + c has the same principal symbol as the Laplacian, an ths its heat kernel satisfies T r e td+c π c x, x θ + t π c x, x θ+ 4πt... + t N π c N x, x θ + O t N+ π + tc t N c N + O t N+. 4πt

11 Then ζ D+c s π 4π / Γ s s N c j + 4π / Γ s s + holomorphic s, + j π so the resie is. Ths, near s πγ s + sζ D+c s s+, so c η c. Since when c, η c, we have that η c c c j for < c <. Note that the spectrm is invariant as c c +, so in fact η c c c moz, c R \ Z We have seen that η c, c Z. 4. Relationship between zeta an eta Accoring to Seeley s famos paper [], complex powers of pseoifferential operators are again pseoifferential. Ths, if A is a first orer self-ajoint elliptic pseo-ifferential operator, then B : 3 A + A, B : 3 A A are also elliptic an pseoifferential bt are nonnegative. Let ζ j s be the zeta fnction corresponing to B j for j,. Then if λ ranges over eigenvales of A, ζ s ζ s λ 3 λ + λ s λ 3 λ s λ s λ s + λ s λ s s λ> λ< λ> λ< s sgn λ λ s s η A s. λ λ s

12 Ths, η A s ζ s ζ s. s This gives the meromorphic contination of the eta fnction. If each ζ j has only simple poles an is reglar at s as it is for powers of self-ajoint elliptic ifferential operators, then η A s has only simple poles incling possibly at s. The resie of η A s at s is R A log ζ ζ. We nee to show that R A is in fact zero, an as a conseqence we will be able to ece that η A s is reglar at s. Note that by the formla for ζ L above, R A is an integral of a locally etermine qantity on the manifol. 5. Reglarity of η s at s The next step is to show that R A is constant on a family of operators A. In orer to allow for iscontinities proce by zero eigenvales, we can write η s η s + η s as a sm of two parts, η s corresponing to the eigenvales λ sch that λ < C an η s corresponing to eigenvales λ sch that λ > C, where C is interior to a spectral gap for A. The fnction η s is a finite sm of exponential fnction an is ths entire an also ifferentiable in. Ths, if we let η s enote η s moz, we see that η η. Ths, we may set assme change all eigenvales λ of A sch that λ < C to withot changing η ; this means that we may assme A is invertible for all. To prove R A is constant on sch a family of operators A, it sffices to show that for sch a family A, η. Let B A + A, which is elliptic an positive for small. Then by the propositions above, η A s st r A A s+ st r A B s, ζ B s s T r Ḃ B s s T r Ȧ B s so that these erivatives coincie at. By the above, ζ B s has a meromorphic contination which is reglar at s, so the same mst be tre for η A s. By the above relationship between ζ an η, an the meromorphic contination formla for ζ L s, we have with

13 ζ, ζ corresponing to 3 A + A, 3 A A that η A s is of the form η A s ζ s ζ s s R A s + k j n, a j s j m + φ k s,, where φ k s, is a smooth map into the space of holomorphic fnctions on Res > k. Bt then the resie of m η A s at s is then R A, which mst be zero by the latest calclations. Ths, R A is constant in. Ths R A is a homotopy invariant of A. By the comments at the beginning of this section, if η A s is the eta fnction rece molo Z, then η A s is holomorphic at s, an its vale there is given by an explicit integral formla constrcte ot of the complete symbols of A an Ȧ. 5.. Asie: a homotopy invariant for operators twiste by flat bnles. A conseqence of the above for flat bnles is as follows. Let α : π M U N be a nitary representation, an this efines a flat vector bnle M a V α over M with Hermitian metric. If A : C M, E C M, E is a ifferential operator acting on sections of E, then A extens natrally to A α : C M, E V α C M, E V α. Moreover, if A is self-ajoint, then A α is also selfajoint. Let η α s, A : η Aα s Nη A s. Since the operators A α an A N A... A N times are locally isomorphic, any invariant given by a local integral formla will coincie for the two operators. Ths, R A α R A N NR A, so that η α s, A is reglar at s. By the above, η α s, A is zero at s, so that η α s, A is a homotopy invariant of A. If A is instea pseoifferential, there is no niqe way of efining A α. However, sing a partition of nity, we can constrct an operator A α whose complete symbol is σ A α. We have shown Proposition 5. Notation as above η α, A is a finite homotopy invariant of A an takes vales in R Z. 5.. Back to the reglarity of the eta fnction. Rece eta invariant: If yo replace η by ξ η+h, where h is the imension of the nllspace, all of the reslts above apply. K-theory an self-ajoint symbols: Since R A is a homotopy invariant of A an with ajstment is a actally a stable homotopy 3

14 4 invariant of the symbol σ A, it sffices to check that R A for a sfficiently rich set of symbols that generate all K-theory classes. Yo can either se the Dirac operator or the bonary part of the signatre operator on o imensional manifols. Then, by invariance theory, the local integran mst be a Pontryagin-Chern form, an ths of even egree. Then, we have that R A on o-imensional manifols. Theorem 6. If M is an o-imensional manifol, an A is a selfajoint elliptic pseoifferential operator of positive orer on M, then η A s is holomorphic at s. The even-imensional case is mch trickier, an yo can see the proof in [3]. 6. Another meromorphic contination of the eta fnction If A is an self-ajoint elliptic classical pseoifferential operator of orer on a manifol of imension n, observe that η A s λ sgn λ λ s λ λ λ s λ λ λ s+ Γ t s+ λe tλ t s+ λ Γ t s+ T r Ae ta t s+ Now, it trns ot that T r Ae ta has an asymptotic expansion in powers of t, beginning with t n, an so the integral gives an analytic expression for η A s for s+ > n n, i.e. s > n. If A is a ifferential operator, then we have T r Ae t A N k c k A t k n + O t N n +, where as in the heat asymptotic expansion, if A is ifferential, c k A ck A, x V x, where c k A, x is a locally etermine qantity. Ths, the meromorphic contination of η A s for ifferential operators an classical pseoifferential operators is given by

15 5 η A s Γ s+ t s+ + Γ s+ + Γ s+ N Γ c s+ k A Γ s+ k k N k t s+ T r c k A t k n Ae ta t s+ T r Ae ta t t N k c k A t k n t t s+ k n t + holomorphic s N c k A s + k n + + holomorphic s This formla shows that the resie at s occrs when k n +, or k n n, or res s η A s c n A Γ. References [] M. F. Atiyah, V. K. Patoi, an I. M. Singer, Spectral asymmetry an Riemannian geometry. I, Math. Proc. Camb. Phil. Soc , [] N. Berline, E. Getzler, an M. Vergne, Heat Kernels an Dirac operators, Grnlehren er mathematischen Wissenschaften 98, Springer-Verlag, Berlin, 99. [3] P. Gilkey, Invariance Theory, the Heat Eqation, an the Atiyah-Singer Inex Theorem, Mathematics Lectre Series, Pblish or Perish Inc., Wilmington, Delaware, 984. [4] R. T. Seeley, Complex Powers of an elliptic operator, Proc. Symp. in Pre Math., Amer. Math. Soc., 967,

Math 273b: Calculus of Variations

Math 273b: Calculus of Variations Math 273b: Calcls of Variations Yacob Kreh Homework #3 [1] Consier the 1D length fnctional minimization problem min F 1 1 L, or min 1 + 2, for twice ifferentiable fnctions : [, 1] R with bonary conitions,