A Gutzwiller type trace formula for the magnetic Dirac operator
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Savale, Nikhil
For manifolds including metric-contact manifolds with non-resonant Reeb flow, we prove a Gutzwiller type trace formula for the associated magnetic Dirac operator involving contributions from Reeb orbits on the base. As an application, we prove a semiclassical limit formula for the eta invariant.
 For the chance to speak here, it's a great conference. I'm quite happy to give this talk. So this is the title of my talk. It's mostly going to be about spectral asymptotics. It's going to be semi-classical spectral asymptotics for the Dirac operators. So let me first begin by explaining what the problem is that I want to address. So we start with an odd dimensional manifold, the Riemannian manifold, which is oriented and further equipped with a spin structure. This already gives us a spin Dirac operator, but I want to talk about coupled Dirac operators. So we're going to be twisted by a line bundle, which further has a Hermitian metric, and a unitary connection, or a compatible connection, the metric. So that, and if you further give yourself a little one form, a real one form on your manifold, you have not just this one connection, but a family of connections. This base connection, a0 plus i times ra, where r is a real parameter. So you have a family of connections on your line bundle, and each one of those gives you a Dirac operator, coupled Dirac operator, acting on sections if this is the spin bundle, tends to be the line bundle. And what I'm interested in is the spectral asymptotics for this Dirac operator, and the semi-classical limits, as our question can. And by spectral asymptotics, I'll restrict myself for the sake of this talk to these two spectral invariants. I'm mostly interested in this spectral invariant, this is the eta invariant. And maybe I don't have to tell you very much what this is, because it already appeared yesterday, and everybody seemed to be very happy about what it was. Nobody complained in that talk, so I'll assume this. And so this, but formally speaking, this is the signature of the Dirac operator, the number of positive lines, the number of negative eigenvalues, and defined using regularization, because there's infinitely many of them. And then there's always the ambiguity of how you count 0 in the signature. So the related eta invariant, the related spectral invariant is the dimension of the kernel, the nullity of the Dirac operator. So this is what I'll look at. I'll look at the semi-classical limit of the eta invariant. And if you're a semi-classical analyst, you don't like r going to infinity. You're rather stated in terms of 1 over r, or 1 over r going to 0. And that's because this 1 over r has a physical significance as the Planck's constant semi-classical parameter. So this is the proper way of stating the same problem. So this is the problem I will press. It's related to another problem which is very well studied in quantization. This is a confidence on quantization. So there is an analog in geometric quantization, which is the following. So instead of a real, in the last slide, I had a real manifold. And it said you could start with a complex manifold, which has a Hermitian metric. And you could give yourself a Hermitian homomorphic line bundle on this complex manifold. And then you have tensor powers of this line bundle. And for each tensor power, we have a dobo Dirac operator acting on tensor powers. It's just the homomorphic dobo differential plus its adjoint. And in geometric quantization, what you consider are the spectral invariance of this dobo Dirac operator as this power p goes to n, v to these tensor powers. And the analogous invariance, the eta invariant of this Dirac operator is not so interesting, actually, it's 0. The spectrum is symmetric. But you could still consider the dimension of the kernel, which by Hartz theory is just cohomology of this tensor power for this line bundle. And instead of the eta invariant, there's still a nice spectral invariant. This is the analytic torsion or the homomorphic torsion, what we want to call it. This what this thing is, is there's a determinant on 0 q forms for each q. And it's the alternating product of the determinants on 0 q forms. So the determinant on Hart forms divided by determinant even forms. So you could consider these spectral invariance as p goes to infinity. And there's a lot of results that are very well known by now. The first one is just the physical homology. There is an asymptotic result. And this was, we'll explain yesterday in Martha Schultz talk. So again, I don't have to tell you, because you've got a whole talk on this yesterday. So this result of the MIE says that if you look at the q-cohomology and the p tensor power, it grows like p power n. And it's the complex dimension. And there's also the constant is also explicitly given. It's the integral of the curvature. And there's this minus sign, because this xq what appears in this, what the integration is over is the subset of the manifold, but the curvature is q negative eigenvalues. So this is a determinant. It's minus has the same sign as minus 1 q. It's positive. So that's the homomorphic and homophonic quantities of the MIE. This is just giving you the asymptotic of the dimension of the carmels. Now, to give the asymptotic of the torsion, you need to make an assumption. You need to assume that the line bundle is positive. Otherwise, it's not so easy otherwise. And assuming that the line bundle is positive, the asymptotic of the analytic torsion was given by Bisgut and Asero. So the result says that the logarithm of torsion has this asymptotic. This actually has two terms in it, but you can write them as 1. So there's a term which goes like p to the n, and there's a term which goes like p to the n times log p over here. But this is the leading term, and then there is a remainder. And this is actually not just an inequality. This is actually leading. So that's the asymptotic of torsion. What's related is the asymptotic. In this case, when the line bundle is positive, you have Kudar revanching and Riemann-Rach. You know exactly what the homology is. Contemplates in degrees 0, and then eventually it's given by the holomorphic Euler characteristic. And there is a further interesting refinement of the homology, which is given by the Baudmard-Cronor. This also appeared yesterday. This is the projection of smooth sections onto the holomorphic once. And the Baudmard-Cronor also has an expansion, which was first given by Katvan and Zeldic. And it starts at p power n in the same order. And this is the Baudmard-Cronor. It's a function of the Baudmard-Cronor. And another thing related to the Baudmard-Cronor is the totals quantization operator. So this is f as a function on your manifold. And you can compose f with the projection onto the homomorphic sections. And this gives a quantization procedure. This quantizes your function f. And this is a correct way of quantizing functions on at least on a Kela manifold. And this was first shown that the fact that this is a correct quantization procedure was shown by Baudmard-Mimerick in the tradition Meyer when they showed that this totals quantization has the right semi-classical limit. And I'm going to ask to verify these two properties here. So in particular, they have to show that the commutator of the totals quantizations is the totals quantization of the possible parameter at the leading order. So those are some of the results in the homomorphic setting. Any questions about these? OK, I think this is well known. So in our real case, we don't have a complex structure. So perhaps what's closer to our context is what's known as the almost Kela case, or the symphlectic case. And in this case, what we have is not a complex manifold, but just an almost complex structure on a symphlectic manifold. And it's almost Kela in the sense that if you contract the almost complex structure with the symphlectic form, you get a metric. So this is, in other words, the almost complex purpose compatible with the metric. So if you have almost Kela manifold like this, and if you have a prequantum line model, so the curvature of this connection on this line model is exactly the symphlectic form, then you can define, again, a renormal as Laplacian. So this is just this. The first term here is the Bach model Laplacian on tensor powers. And you can renormalize it by n times t minus n times b. And it's again the real dimension, p is just this power. And for this renormalized Laplacian, there is a spectral result, which was first given by Goodman in a rebate. What it says is that if you look at the spectrum of this renormalized Laplacian asymptotically, it splits up into two parts, one part which stays near 0, bounded away from 0, and then a second part which grows linearly. So that's the first result. The second result, then you might ask, how many eigenvalues are here close to 0? And there is an asymptotic result for this. So the number of eigenvalues which stay bounded away from 0, also grows like t to the n. Yeah? And so this was Goodman in a rebate. Furthermore, you can also ask for the distribution of the eigenvalues. You're not just the number. And there's a well-known distribution for how these eigenvalues are distributed asymptotically. So you can look at a function of the eigenvalues that appear here, and the way they're distributed is by this spectral density function. So f of this is maybe a little confusing way of writing it. This maybe I'll rewrite it here. So what appears on this left-hand side here is lambda is in this interval spec. And then you have this distribution lambda minus lambda t. This is the spectral distribution, and this is applied to this function f, this case function f. So this is the spectral distribution in this interval. And when you apply it to this function f, you get this formula, the asymptotic. This is an asymptotic formula. And then there's this spectral density function which appears in this formula. I should put it here. You have the first case, right? Sorry? In your formula, you have the second case. Yeah, they're kind of multiplicity eigenvalues. And so this is the formula for the spectral density function which was given later by a force we can remove in. So Kullin and Rewey showed that this should exist, and both the Rewey actually computed it in terms of the covariant derivative of the almost complex structure here. So in particular, if you have on a kilo manifold, this complex of this current, both we can Rewey. I'm not sure. I know you did it as well. Are we using a different method? Result is different. I'll let you sort it out. I'm not going to get it. You should take it here when it's a little. I know there is this 1 over 24. What's the constant rule? 1 over 24 was corrected by you, and I know this. But the snablages was correct. I know this. OK, so anyway, the point I was making is that if you're on a kilo manifold, the complex structure is parallel, and in this case, the spectrum is all concentrated at the origin of the snablages 0. All right. So, all right. And the reason I say that all these results together is because there's essentially, well, there are different proofs of these results. But there's at least one technique which works for all the results that has even the kilo case. I'm going to give this a collective case. And this is the rescaling a local index theory arguments. And this was well explained in the book of Maher and Marinus. Maher just spoke. So that's it. And it was also well explained by the Barstabh Uchul. The technique was well explained in Martha Stark yesterday. So that's what I wanted to say about the quantization case. Any questions? OK. OK. OK, so anyway, so now this is the quantization results. Now let's come back to the real case, the problem that I said in the beginning. So this is the coupled Dirac operator that I considered before. And now you could apply the same technique rescaling local index theory technique to this problem. And you get a result. You get a result using this technique. But so the result that you get is that the error invariant of this Dirac operator and the semi-classical limit is, goes like this is a little o. And this is half the dimension. So 2m plus 1 was the dimension of this r-dimensional manifold. So this is half the dimension. But as you see, unfortunately, in the previous results, you had a leading term. But in this case, we don't get a leading term. So you just get a pound using the same technique. So yeah, you're going to have to work a little more if you want to see a leading term here. And this problem actually was motivated by, was first considered by Taub's, this error invariant problem. And his result was that it was a little bigger than half the dimension in the result that he got. And this epsilon was between 0 and 1 half or something. And this problem was also considered by Sey, but his result is the same, but it's only in three dimensions under some additional hypotheses. And the reason Taub's considered this problem is because it came up in the proof of the Weinstein conjecture on contact manifolds. He proved the Weinstein conjecture in three dimensions of complete generality using the seven written equations. And along the way, he had to consider this asymptotic behavior. Which Weinstein conjecture is this? So this is the conjecture on existence of ribor bits on a contact manifold. Maybe I'll state it a little later tonight. But to contact manifolds. OK, so that's the first result we have for our error invariant, that's the problem. Now, this was our result. Now you can ask, well, is it sharp? Is this bound actually obtained in some example? So let's look at a very simple example on S1. On S1, the drop-up is very easy. It's just IDK theta plus r. And this back room of this, I hope everybody knows how to compute. You can compute the error invariant even. So an error invariant, maybe I might be off my back by a half or something here. But it's essentially the fractional part of r. So it's the fractional part of this parameter r, ribor r. And you can see that since it's a fractional part, it's going to be a little bit more complicated. Because the fractional part is going to be o of 1. And what our general result says is that it should be little o of r of the one half. So it's less than what you see. So our asymptotic result is not sharp in this case. OK, so that's S1. Already you see that it's not sharp. But you might say that this is too simple an example. So in general, you can look at S1 bundles. Or OK, so let's look at another example. This is a very general class of examples. If you want, you can look at S3, for example. And you can do the computation there. It's not so hard to do in S3, either. But I want to present a more general class of examples. So you could look at S1 bundles. So you could start with your complex manifold that appears in quantization. You could start with the positive line bundling, your complex manifold. And the real manifold to take would be the unit circle bundle of your complex manifold of your positive line bundle. And then there is a connection here, a shown connection that gives a splitting of the tangent space of the unit circle bundle up here. This is the vertical tangent space of the fibers. And then there's this horizontal loop to be identified with the whole back of the space in the base. And we need to choose a metric on this and a spin structure. A spin structure could be put back by a choice made on the base. For the metric, we choose the adiabatic family of metrics, which is great about study in the business of the rock opera, so you choose a family of metrics, which is respects the splitting. And it's something on this part that's pulled back from the base on this part. It's just equals to 1 on the generator. Yeah. And you combine them together using this adiabatic parameter. So this is one of them. For a family of metrics, it's put on this sort of bundle. We also need to choose an auxiliary line bundle since we want to twist the drug operator up here. We just choose the trivial one, and we choose that one form to be a deal to the generator of the S1 action. So with all these forces, we now have the well-defined coupled drug operator on this circle bundle. We could try to compute its spectrum to get a D& variant. So the spectrum, you can actually, well, this is not really computation, but you can relate it to the spectrum of the double board drug operator on the base. So the spectrum of the drug operator on the circle bundle, this is our coupled drug operator, it comes in two types. So the first type is coming from the zero eigenvalue on the base. So the zero eigenvalue of the double drug operator on the base. And it's co-homological because the zero eigenvalue of the Laplacian downstairs is co-homological. And this eigenvalue is very explicit, and it's actually just linear in R or correct R. So this is the type one eigenvalue. You know exactly what it is, how it looks like. And then there is eigenvalue of type 2, which is not linear in R, but it has this square root behavior in R. And it's coming from the positive eigenvalues of the Laplacian on the base. So it comes in two types. But so this is not really a computation. And this second type of eigenvalue, you see this mu, which is a spectrum of the Laplacian on the base. So it's not really a computation, per se. It's just a relation between spectrum of stream and gas layers. But this is enough to get a handle on the Eda invariant, just the fact that eigenvalues come in these two types. And in fact, for epsilon, if you choose epsilon reasonably small, then there is a whole theory of Eda better limits of the Eda invariant, which was started by this mu sheet. So it was started by Whitton, I guess, at some point. But a general result was given by this mu sheet, and then by Dyle, who generalized the previous score. And using this Eda invariant, let me if we can compute the Eda invariant for epsilon small enough. And you get an asymptotic result for the Eda invariant, in this case. This asymptotic actually goes down pretty far. It goes down to one, not just to a warning. But you see that this leading terms of this leading expression here is discontinuous in the coupling parameter r. I think you should decide also weight in sum. The circle under Eda form was computed by weight in sum. That's right. Yeah, that's right. Yeah, so in the computation of the Eda invariant, there's three pieces, there's a spectral flow part, there's an Eda-teleform. And there's a transgression form. And the Eda-teleform was computed by weight in sum. And what I did here was compute the spectral flow part. So the spectral flow part is not so easy to compute for general circle models, but for unit circle models of complex line model, you can compute it in this example. So the weeping sum computed part of this computation. OK, so what you see here is that the Eda invariant is discontinuous in this parameter. But because, again, you see that there's this greatest integer function that appears in the formula. Inverting integer function. And whenever r changes in a way that this crosses an integer, this Eda invariant, if you just look at this expression for a while, you see that it jumps by order r to the n. So this Eda invariant actually jumps. And so you see that in this example, the best you can do is prove O of r to the n. n is 2n plus 1 is, again, the dimension. But what we proved, the local index theory result gives you is O of little o of r to the n plus 1 half. So you're still off by a factor of 1 half. So between what you can prove and what you can see in an example. So this was what I was trying to do for a while. Trying to close this gap of 1 half. And so to prove a sharp result, we make further assumption. So the first assumption that I will make is that the one form is context. So one form is content. So what this means is that in range, it's never 0. So you have contact assumption. And then there is also a contact entomorphism that you can define by contraction of d of a with the metric. So there's all contact entomorphisms that's defined this way. It's anti-symmetric with respect to the metric. And the sharp result under these two assumptions, so you assume that if you assume that the one form is contact. And if you assume that the spectrum of this contact entomorphism, so this depends on each point, if you assume that the spectrum of this contact entomorphism is independent of the point on your manifold, then under these two assumptions, I can prove the sharp result. And these are two reasonable assumptions. Contact entomorphism is a very general class. And if you assume that your manifold is what's called a metric contact manifold, so the metric is coming from an almost complex structure on the contact half plane, then that's that satisfies the index. In that case, the script j is just j, the almost complex structure. And the spectrum of j is just plus or minus i. It's independent of the point on the manifold. So in that case, you have again this sharp result of this large set of examples. So that's the first result. And again, you see there's still no leading term here. So I wanted to get a leading term. And this leading term you can get if you make for the resumption. So if you further assume that the rate flow, so the rate vector field is defined using the contact form by these two relations, if you assume that there's so there's a flow for this rate vector field. And if you assume that the flow is non-resonant, so again, I have to put non-resonant on my slide. So what does non-resonant mean? So you look at a closed orbit. You look at a closed orbit, you have a point-query return map which is acting on a contact type of plane. And you look at the spectrum of the point-query return map. It comes in different types. So there's elliptic eigenvalues. There's hyperbolic ones. So the elliptic ones which lie under unit circle. There's hyperbolic ones which are real. And then there's lots of atomic ones. The elliptic ones come in pairs. Elliptic and hyperbolic ones, the lots of atomic ones come in multiples of four. And the non-resonant assumption is that these two sets of exponents, this flow k exponents, that appear here. So you have alpha j, u and alpha j, 0. These two sets are rationally related and independent. So there's no rational linear combination of these two, which is 0. So that's the non-resonant assumption. And under this assumption, you can get a leading tromb for the eta invariant. It's of order r of the n. And what appears in this leading tromb is, again, you see this in the form line, you see again, you see this in alpha j. This is similar to the both the Kiribay or Maureen Eskimo density function. And this non-resonant assumption, by the way, excludes the so-called one-counter example that is k-value. OK, so that's the result. Now I want to tell you what's involved in the remaining time. What's involved in the proof of these two results, is the short results. OK, so first of all, we're going to reduce the asymptotics of the eta invariant to trace asymptotics. So there's a well-molded formula for the eta invariant in terms of the heat trace, which is given by this muth and frit. So this is the formula. So this is an integral of the heat trace, d times e to the minus td squared from 0 to infinity. This integral, and this is an integral of time, all time. This time integral, you can separate into two parts, one for small time, one for large time. The large time part you can write as a single trace, and without an integral. Of a function, this is the sine error function here. And this function looks like the following. So the error function is rapidly decaying and everything. But the sine error function is also rapidly decaying and smooth except that the origin has a discontinuity. So this discontinuous functional trace, it doesn't have an expansion yet. So this trace of this sine error function is non-local. So this trace of this sine error function is non-local. And we have to work harder to get asymptotic of a discontinuous trace like this. But it's not so different from, if you're familiar with wild counting functions, you're looking at a wild counting function of an interval. You're looking at the trace of a characteristic function of that interval, which is also discontinuous. So then you can use a standard tabirian argument to get at this discontinuous functional trace using trace asymptotics. But this is a tabirian argument. This asymptotic is the A to N bearing of a trace formula, which is a physical trace formula. And this physical trace formula, what's important in this trace on the right-hand side here is this insertion of this data had this second function here. Without this data head drum, this is a function of d over root h. This trace asymptotics you can get at using local index to its local. But to get the sharp result, we need to consider a more refined trace with this data head inserted here with this combination. And what's important in the insertion of this data head is, get a check, rather. This is the inverse for your transform. It is that there's a full power of h in the denominator. Getting in with that full power of h in the denominator, this trace is not local. And but it still has an asymptotic. So there's an expansion in powers of h. And there's a local part in the asymptotics, but there's also a normal local part, which is coming from contributions of rib orbits. So this sum over gamma is sum over rib orbits of the contact form. And these constants that they're appearing in this formula are respectively the period of the rib orbit, the remanian length of the rib orbit, and the mass of index. So this is an analog of the usual with the trace formula. But what's new here is that the dynamic contributions in the usual with the trace formula there on the co-tangents on the Hamiltonian flow on the co-tangent space. But here they are on the base map. There's a dynamic system on the base map. This is somehow new here. And the even and varied asymptotics follows from this one. So I want to tell you a little bit about what's involved in the glistening trace formula. So as I said, the usual way to prove trace formulas is using FIOs and Hamilton. It involves Hamilton flow of the symbol. So if your operator is non-scaler, you have a non-scaler symbol, but you still have its scalar eigenvalues of the symbol. And you can look at Hamilton flow of the scalar eigenvalues. In this case, the trace formula, if the symbol is diagonalizable, smoothly diagonalizable, these are smoothly smooth functions. And you have a smooth flow for these. The trace formula will involve closed orbits of these Hamilton flows of these eigenvalues. But in our case, the symbol of the Dirac operator is clip of modification. And its eigenvalues are plus or minus absolute value of psi plus x times the co-vector on the co-tangents space, and a is the one form. And since this is absolute value, this is the square root singularity, this is not a smooth function of the co-tangent space. It's not smooth along this characteristic variety. And it's not diagonalizable, so the usual trace formula will not apply. But to prove this trace formula what you do is you try to look at the local model to begin with. So if you look at the local model, so this is the standard contact structure on R3 with some nice adapted metric, the Dirac operator looks like this. And these are the three-pillar-multimeter seas. It is a three-pillar-multimeter seas, and this is the coupling, which is coming from the contact flow. And then there's a free part, which corresponds to the ray vector field, and then there's a part which corresponds to a complex harmonic oscillator. And this complex harmonic oscillator has a well-understood spectrum. It has infinite dimension. It's conjugate to the real harmonic oscillator of one dimension. So these eigenspaces are actually infinite dimensional. They're called Landau levels. And in this case, and at least in the model case, you completely understand the spectrum. And what you try to do in general is try to make your general Dirac operator under your assumptions, try to make it look like the model one. So in general, it looks like this. This is the local expression for your Dirac operator. These are, again, the Clifford matrices. This is the coupling one form. And this is the tensor which diagonalizes the metric that appears in the local expression. And the idea is to try to conjugate. Conjugation does not change the spectrum. I try to conjugate it to make it look much like the Dirac, this model Dirac operator. So this is the model part. And then there will be an error. The error, you try to make it as best as you can. So you try to make it commute with the leading point. So this is a standard procedure for normal forms of operators. And then once you have this normal form, what you do is you break up the trace into these eigenspaces of this model operator, or the Landau levels. And then you study the trace on each level. That's the idea. Now, the normal form has an interesting, as an interesting procedure in this case. So the conjugation operator that appears, this conjugation of the Dirac operator that appears in the normal form, is conjugation by an FIO, the Fourier integral operator. And the Fourier integral operator that appears here is, as an interesting form, is now. OK, so it's conjugation by an FIO. This is E to the i t f of 2, while this f, while w stands for while quantization of a scalar symbol. OK, so this first part is the product of two operators. The first part is an FIO. And the second part, what's important here is that there's no h in the denominator in the second part. And it's called Clifford-Wall quantization. So Clifford-Wall quantization is quantization of not just a scalar symbol, but a scalar symbol which takes values in the exterior algebra piece of it. This scalar part is quantized by the while procedure. And the exterior algebra is quantized by the Clifford procedure, so the Clifford-Wall quantization of a symbol E, C. So this is something you want to quantize here. And this is just the scalar part is quantized by the while procedure. And then the Clifford part is quantized using Clifford quantization, in case of this or that. Clifford makes this. And once you quantize it this way, you can compute conjugations of such FIOs. And the reason for quantizing it this way is because what appears in the conjugation formulas in trying to produce a number of more are causal differentials of these symbols that you quantized. So this causal complex is just defined. The causal differentials are just defined on the exterior algebra, on the exterior algebra part it's either wedge or contraction, and on the symbol part it's either multiplication or differentiation. So these causal differentials appear in the conjugation formula. And then what gives you the normal formula is the Hodge theorem. It's a symphlectic Hodge theorem for a causal complex. So that's the rough idea of how the proof goes. And this normal form procedure generalizes a normal form in the book of every semi-flexible analysis. This is a new book. But he did it in three dimensions. He's a generalizing man. And so this gives the trace formula for small time. And then there's a large time trace formula, but that requires some more work. It requires a propagation of singularity's lemma, or the weak version of a propagation of singularity's lemma, which is not mine, but I found it to be a very good one. Which is not mine, but I found it in this book of the Marcy Andrews theorem. So that's roughly how the proof goes. I don't have so much time to give the details. So let me end by, I'd like to end with always some further questions. If anybody here wants to work on this, join me or do it without me. Then there's a lot of questions to be answered here. So first of all, you have this trace formula of the either invariant asymptotics. You could try to do the same under other assumptions. So instead of a contact form, if you choose some other form, there's a lot of models for one forms, and singularities of one forms on a man called. If you change your contact assumption, the assumption is not redundant by the way. It's fairly necessary. If you don't assume your manifold as contact, it's not clear what the dynamical system is that will appear in the trace formula. So if you change the assumption, the formula will change. The trace formula will almost surely change. So you could try to work it out. The other thing is we considered the data invariant. We could consider the determinant again, as in this book, the best row. We could consider the asymptotic numbers of the log determinant in this real case. And this is not so easy. Again, this move as rows. Example, you need positivity, which doesn't exist here. This assumption doesn't exist here in the real case. So you can look at asymptotics of the log determinant, and you would have to understand this case. Low line eigenvalues of the Dirac operator. There are people who work on this expansion of low line eigenvalues for magnetical fractions and so on. Then try to understand the propagator. So you could try to understand the wave equation of the Dirac operator. Our proof doesn't really go through this wave equation. It uses normal forms and stuff. So you could try to understand the propagator and propagation of singularities for this wave equation of the Dirac operator. And again, so that's another thing. And another thing you could try to do is try to consider, again, our technique is more refined than the local index theory technique. So you could try to apply our technique to the actual quantization problem. So actually, tens of powers. And the Dirac operator and tens of powers and stuff. Try to see what this local technique gives in that case. And finally, you could study the, so we consider the magnetical couple Dirac operator. You could also consider the magnetic operation. It's slightly different. You can try to prove a trace formula for the magnetical fashion, instead of the Dirac operator. And there's, so this is a semi-classical problem. You could consider the micro-ocal analog of this problem. And this is related to what's called the subroman in Laplacian, which I don't have time to explain. But that's a much bigger subject. Much bigger subject is subroman in geometry, but I don't have time to explain it. So that's all. Thank you. Thank you. Any questions? So you say one might try to apply those techniques to black-keyed and sense-contained space. Yeah. So the first thing to find out about analog is just a general question. It was only a general question, but you would have to figure out the right assumptions. What's the analog of contact assumption in the even dimensional case? I'm not so sure. But we could try to think about it. We could get something about black-keyed conjecture. At least some particles. I don't think this is a possibility. Even the usual good-solid transform involves Hamilton contribution from periodic Hamilton trajectories. And that doesn't really give you existence of a Hamilton, closed Hamilton trajectories. And there are examples where you don't have Hamilton trajectories, but you still have the trace model. So one more question. Let's have a speed. Let's have a given back equation. This cannot be properly mixtured.
 For the chance to speak here, it's a great conference.  I'm quite happy to give this talk.  So this is the title of my talk.  It's mostly going to be about spectral asymptotics.  It's going to be semi-classical spectral asymptotics  for the Dirac operators.  So let me first begin by explaining what the problem is  that I want to address.  So we start with an odd dimensional manifold,  the Riemannian manifold, which is oriented and further equipped  with a spin structure.  This already gives us a spin Dirac operator,  but I want to talk about coupled Dirac operators.  So we're going to be twisted by a line bundle, which  further has a Hermitian metric, and a unitary connection,  or a compatible connection, the metric.  So that, and if you further give yourself  a little one form, a real one form on your manifold,  you have not just this one connection,  but a family of connections.  This base connection, a0 plus i times ra,  where r is a real parameter.  So you have a family of connections on your line bundle,  and each one of those gives you a Dirac operator,  coupled Dirac operator, acting on sections  if this is the spin bundle, tends to be the line bundle.  And what I'm interested in is the spectral asymptotics  for this Dirac operator, and the semi-classical limits,  as our question can.  And by spectral asymptotics, I'll  restrict myself for the sake of this talk  to these two spectral invariants.  I'm mostly interested in this spectral invariant,  this is the eta invariant.  And maybe I don't have to tell you very much what this is,  because it already appeared yesterday,  and everybody seemed to be very happy about what it was.  Nobody complained in that talk, so I'll assume this.  And so this, but formally speaking,  this is the signature of the Dirac operator,  the number of positive lines, the number of negative eigenvalues,  and defined using regularization,  because there's infinitely many of them.  And then there's always the ambiguity  of how you count 0 in the signature.  So the related eta invariant, the related spectral invariant  is the dimension of the kernel, the nullity of the Dirac  operator.  So this is what I'll look at.  I'll look at the semi-classical limit of the eta invariant.  And if you're a semi-classical analyst,  you don't like r going to infinity.  You're rather stated in terms of 1 over r, or 1 over r going to 0.  And that's because this 1 over r has a physical significance  as the Planck's constant semi-classical parameter.  So this is the proper way of stating the same problem.  So this is the problem I will press.  It's related to another problem which is very well studied  in quantization.  This is a confidence on quantization.  So there is an analog in geometric quantization,  which is the following.  So instead of a real, in the last slide,  I had a real manifold.  And it said you could start with a complex manifold,  which has a Hermitian metric.  And you could give yourself a Hermitian homomorphic line  bundle on this complex manifold.  And then you have tensor powers of this line bundle.  And for each tensor power, we have a dobo Dirac operator  acting on tensor powers.  It's just the homomorphic dobo differential plus its adjoint.  And in geometric quantization, what  you consider are the spectral invariance  of this dobo Dirac operator as this power p goes to n,  v to these tensor powers.  And the analogous invariance, the eta invariant of this Dirac  operator is not so interesting, actually, it's 0.  The spectrum is symmetric.  But you could still consider the dimension of the kernel,  which by Hartz theory is just cohomology  of this tensor power for this line bundle.  And instead of the eta invariant,  there's still a nice spectral invariant.  This is the analytic torsion or the homomorphic torsion,  what we want to call it.  This what this thing is, is there's  a determinant on 0 q forms for each q.  And it's the alternating product of the determinants  on 0 q forms.  So the determinant on Hart forms divided by determinant  even forms.  So you could consider these spectral invariance  as p goes to infinity.  And there's a lot of results that are very well known by now.  The first one is just the physical homology.  There is an asymptotic result.  And this was, we'll explain yesterday  in Martha Schultz talk.  So again, I don't have to tell you,  because you've got a whole talk on this yesterday.  So this result of the MIE says that if you look  at the q-cohomology and the p tensor power,  it grows like p power n.  And it's the complex dimension.  And there's also the constant is also explicitly given.  It's the integral of the curvature.  And there's this minus sign, because this xq  what appears in this, what the integration is over  is the subset of the manifold, but the curvature  is q negative eigenvalues.  So this is a determinant.  It's minus has the same sign as minus 1 q.  It's positive.  So that's the homomorphic and homophonic quantities  of the MIE.  This is just giving you the asymptotic of the dimension  of the carmels.  Now, to give the asymptotic of the torsion,  you need to make an assumption.  You need to assume that the line bundle is positive.  Otherwise, it's not so easy otherwise.  And assuming that the line bundle is positive,  the asymptotic of the analytic torsion  was given by Bisgut and Asero.  So the result says that the logarithm of torsion  has this asymptotic.  This actually has two terms in it,  but you can write them as 1.  So there's a term which goes like p to the n,  and there's a term which goes like p to the n times log p  over here.  But this is the leading term, and then there is a remainder.  And this is actually not just an inequality.  This is actually leading.  So that's the asymptotic of torsion.  What's related is the asymptotic.  In this case, when the line bundle is positive,  you have Kudar revanching and Riemann-Rach.  You know exactly what the homology is.  Contemplates in degrees 0, and then eventually it's  given by the holomorphic Euler characteristic.  And there is a further interesting refinement  of the homology, which is given by the Baudmard-Cronor.  This also appeared yesterday.  This is the projection of smooth sections  onto the holomorphic once.  And the Baudmard-Cronor also has an expansion, which  was first given by Katvan and Zeldic.  And it starts at p power n in the same order.  And this is the Baudmard-Cronor.  It's a function of the Baudmard-Cronor.  And another thing related to the Baudmard-Cronor  is the totals quantization operator.  So this is f as a function on your manifold.  And you can compose f with the projection  onto the homomorphic sections.  And this gives a quantization procedure.  This quantizes your function f.  And this is a correct way of quantizing functions  on at least on a Kela manifold.  And this was first shown that the fact  that this is a correct quantization procedure was  shown by Baudmard-Mimerick in the tradition  Meyer when they showed that this totals quantization has  the right semi-classical limit.  And I'm going to ask to verify these two properties here.  So in particular, they have to show  that the commutator of the totals quantizations  is the totals quantization of the possible parameter  at the leading order.  So those are some of the results in the homomorphic setting.  Any questions about these?  OK, I think this is well known.  So in our real case, we don't have a complex structure.  So perhaps what's closer to our context  is what's known as the almost Kela case,  or the symphlectic case.  And in this case, what we have is not a complex manifold,  but just an almost complex structure  on a symphlectic manifold.  And it's almost Kela in the sense  that if you contract the almost complex structure  with the symphlectic form, you get a metric.  So this is, in other words, the almost complex purpose  compatible with the metric.  So if you have almost Kela manifold like this,  and if you have a prequantum line  model, so the curvature of this connection on this line  model is exactly the symphlectic form,  then you can define, again, a renormal as Laplacian.  So this is just this.  The first term here is the Bach model Laplacian  on tensor powers.  And you can renormalize it by n times t minus n times b.  And it's again the real dimension, p is just this power.  And for this renormalized Laplacian,  there is a spectral result, which  was first given by Goodman in a rebate.  What it says is that if you look at the spectrum  of this renormalized Laplacian asymptotically,  it splits up into two parts, one part which stays near 0,  bounded away from 0, and then a second part which grows linearly.  So that's the first result.  The second result, then you might ask,  how many eigenvalues are here close to 0?  And there is an asymptotic result for this.  So the number of eigenvalues which stay bounded away from 0,  also grows like t to the n.  Yeah?  And so this was Goodman in a rebate.  Furthermore, you can also ask for the distribution  of the eigenvalues.  You're not just the number.  And there's a well-known distribution  for how these eigenvalues are distributed asymptotically.  So you can look at a function of the eigenvalues that appear  here, and the way they're distributed  is by this spectral density function.  So f of this is maybe a little confusing way of writing it.  This maybe I'll rewrite it here.  So what appears on this left-hand side here  is lambda is in this interval spec.  And then you have this distribution lambda minus lambda  t.  This is the spectral distribution,  and this is applied to this function f, this case function  f.  So this is the spectral distribution in this interval.  And when you apply it to this function f,  you get this formula, the asymptotic.  This is an asymptotic formula.  And then there's this spectral density function  which appears in this formula.  I should put it here.  You have the first case, right?  Sorry?  In your formula, you have the second case.  Yeah, they're kind of multiplicity eigenvalues.  And so this is the formula for the spectral density function  which was given later by a force we can remove in.  So Kullin and Rewey showed that this should exist,  and both the Rewey actually computed it  in terms of the covariant derivative of the almost complex  structure here.  So in particular, if you have on a kilo manifold,  this complex of this current, both we can Rewey.  I'm not sure.  I know you did it as well.  Are we using a different method?  Result is different.  I'll let you sort it out.  I'm not going to get it.  You should take it here when it's a little.  I know there is this 1 over 24.  What's the constant rule?  1 over 24 was corrected by you, and I know this.  But the snablages was correct.  I know this.  OK, so anyway, the point I was making  is that if you're on a kilo manifold,  the complex structure is parallel,  and in this case, the spectrum is all  concentrated at the origin of the snablages 0.  All right.  So, all right.  And the reason I say that all these results together  is because there's essentially, well,  there are different proofs of these results.  But there's at least one technique  which works for all the results that has even the kilo case.  I'm going to give this a collective case.  And this is the rescaling a local index theory arguments.  And this was well explained in the book of Maher and Marinus.  Maher just spoke.  So that's it.  And it was also well explained by the Barstabh Uchul.  The technique was well explained in Martha Stark yesterday.  So that's what I wanted to say about the quantization case.  Any questions?  OK.  OK.  OK, so anyway, so now this is the quantization results.  Now let's come back to the real case, the problem that I  said in the beginning.  So this is the coupled Dirac operator  that I considered before.  And now you could apply the same technique rescaling  local index theory technique to this problem.  And you get a result.  You get a result using this technique.  But so the result that you get is that the error  invariant of this Dirac operator and the semi-classical limit  is, goes like this is a little o.  And this is half the dimension.  So 2m plus 1 was the dimension of this r-dimensional  manifold.  So this is half the dimension.  But as you see, unfortunately, in the previous results,  you had a leading term.  But in this case, we don't get a leading term.  So you just get a pound using the same technique.  So yeah, you're going to have to work a little more  if you want to see a leading term here.  And this problem actually was motivated  by, was first considered by Taub's, this error invariant  problem.  And his result was that it was a little bigger than half  the dimension in the result that he got.  And this epsilon was between 0 and 1 half or something.  And this problem was also considered by Sey, but his  result is the same, but it's only in three dimensions under  some additional hypotheses.  And the reason Taub's considered this problem is  because it came up in the proof of the Weinstein  conjecture on contact manifolds.  He proved the Weinstein conjecture in three dimensions  of complete generality using the seven written equations.  And along the way, he had to consider this asymptotic  behavior.  Which Weinstein conjecture is this?  So this is the conjecture on existence of ribor bits on a  contact manifold.  Maybe I'll state it a little later tonight.  But to contact manifolds.  OK, so that's the first result we have for our  error invariant, that's the problem.  Now, this was our result.  Now you can ask, well, is it sharp?  Is this bound actually obtained in some example?  So let's look at a very simple example on S1.  On S1, the drop-up is very easy.  It's just IDK theta plus r.  And this back room of this, I hope everybody knows how to  compute.  You can compute the error invariant even.  So an error invariant, maybe I might be off my back by a  half or something here.  But it's essentially the fractional part of r.  So it's the fractional part of this parameter r, ribor  r.  And you can see that since it's a fractional part, it's  going to be a little bit more complicated.  Because the fractional part is going to be o of 1.  And what our general result says is that it should be  little o of r of the one half.  So it's less than what you see.  So our asymptotic result is not sharp in this case.  OK, so that's S1.  Already you see that it's not sharp.  But you might say that this is too simple an example.  So in general, you can look at S1 bundles.  Or OK, so let's look at another example.  This is a very general class of examples.  If you want, you can look at S3, for example.  And you can do the computation there.  It's not so hard to do in S3, either.  But I want to present a more general class of examples.  So you could look at S1 bundles.  So you could start with your complex manifold that appears  in quantization.  You could start with the positive line  bundling, your complex manifold.  And the real manifold to take would be the unit circle  bundle of your complex manifold of your positive line bundle.  And then there is a connection here,  a shown connection that gives a splitting of the tangent space  of the unit circle bundle up here.  This is the vertical tangent space of the fibers.  And then there's this horizontal loop  to be identified with the whole back of the space in the base.  And we need to choose a metric on this and a spin structure.  A spin structure could be put back  by a choice made on the base.  For the metric, we choose the adiabatic family of metrics,  which is great about study in the business of the rock  opera, so you choose a family of metrics, which  is respects the splitting.  And it's something on this part that's  pulled back from the base on this part.  It's just equals to 1 on the generator.  Yeah.  And you combine them together using this adiabatic parameter.  So this is one of them.  For a family of metrics, it's put on this sort of bundle.  We also need to choose an auxiliary line bundle  since we want to twist the drug operator up here.  We just choose the trivial one, and we choose that one form  to be a deal to the generator of the S1 action.  So with all these forces, we now have  the well-defined coupled drug operator on this circle bundle.  We could try to compute its spectrum  to get a D& variant.  So the spectrum, you can actually,  well, this is not really computation,  but you can relate it to the spectrum of the double board  drug operator on the base.  So the spectrum of the drug operator on the circle bundle,  this is our coupled drug operator,  it comes in two types.  So the first type is coming from the zero eigenvalue  on the base.  So the zero eigenvalue of the double drug operator on the base.  And it's co-homological because the zero eigenvalue  of the Laplacian downstairs is co-homological.  And this eigenvalue is very explicit,  and it's actually just linear in R or correct R.  So this is the type one eigenvalue.  You know exactly what it is, how it looks like.  And then there is eigenvalue of type 2,  which is not linear in R, but it has this square root behavior  in R. And it's coming from the positive eigenvalues  of the Laplacian on the base.  So it comes in two types.  But so this is not really a computation.  And this second type of eigenvalue,  you see this mu, which is a spectrum of the Laplacian  on the base.  So it's not really a computation, per se.  It's just a relation between spectrum of stream and gas  layers.  But this is enough to get a handle on the Eda invariant,  just the fact that eigenvalues come in these two types.  And in fact, for epsilon, if you choose epsilon reasonably  small, then there is a whole theory  of Eda better limits of the Eda invariant, which  was started by this mu sheet.  So it was started by Whitton, I guess, at some point.  But a general result was given by this mu sheet,  and then by Dyle, who generalized the previous score.  And using this Eda invariant, let me  if we can compute the Eda invariant for epsilon  small enough.  And you get an asymptotic result for the Eda invariant,  in this case.  This asymptotic actually goes down pretty far.  It goes down to one, not just to a warning.  But you see that this leading terms of this leading expression  here is discontinuous in the coupling parameter r.  I think you should decide also weight in sum.  The circle under Eda form was computed by weight in sum.  That's right.  Yeah, that's right.  Yeah, so in the computation of the Eda invariant,  there's three pieces, there's a spectral flow part,  there's an Eda-teleform.  And there's a transgression form.  And the Eda-teleform was computed by weight in sum.  And what I did here was compute the spectral flow part.  So the spectral flow part is not so easy to compute  for general circle models, but for unit circle models  of complex line model, you can compute it in this example.  So the weeping sum computed part of this computation.  OK, so what you see here is that the Eda invariant  is discontinuous in this parameter.  But because, again, you see that there's  this greatest integer function that  appears in the formula.  Inverting integer function.  And whenever r changes in a way that this crosses an integer,  this Eda invariant, if you just look at this expression  for a while, you see that it jumps by order r to the n.  So this Eda invariant actually jumps.  And so you see that in this example,  the best you can do is prove O of r to the n.  n is 2n plus 1 is, again, the dimension.  But what we proved, the local index theory result gives you  is O of little o of r to the n plus 1 half.  So you're still off by a factor of 1 half.  So between what you can prove and what you can see in an example.  So this was what I was trying to do for a while.  Trying to close this gap of 1 half.  And so to prove a sharp result, we  make further assumption.  So the first assumption that I will make  is that the one form is context.  So one form is content.  So what this means is that in range, it's never 0.  So you have contact assumption.  And then there is also a contact entomorphism  that you can define by contraction of d of a  with the metric.  So there's all contact entomorphisms  that's defined this way.  It's anti-symmetric with respect to the metric.  And the sharp result under these two assumptions,  so you assume that if you assume that the one form is contact.  And if you assume that the spectrum of this contact  entomorphism, so this depends on each point,  if you assume that the spectrum of this contact  entomorphism is independent of the point on your manifold,  then under these two assumptions,  I can prove the sharp result.  And these are two reasonable assumptions.  Contact entomorphism is a very general class.  And if you assume that your manifold is what's  called a metric contact manifold,  so the metric is coming from an almost complex structure  on the contact half plane, then that's  that satisfies the index.  In that case, the script j is just j,  the almost complex structure.  And the spectrum of j is just plus or minus i.  It's independent of the point on the manifold.  So in that case, you have again this sharp result  of this large set of examples.  So that's the first result.  And again, you see there's still no leading term here.  So I wanted to get a leading term.  And this leading term you can get if you make  for the resumption.  So if you further assume that the rate flow, so the rate vector  field is defined using the contact form by these two  relations, if you assume that there's  so there's a flow for this rate vector field.  And if you assume that the flow is non-resonant,  so again, I have to put non-resonant on my slide.  So what does non-resonant mean?  So you look at a closed orbit.  You look at a closed orbit, you have a point-query return map  which is acting on a contact type of plane.  And you look at the spectrum of the point-query return map.  It comes in different types.  So there's elliptic eigenvalues.  There's hyperbolic ones.  So the elliptic ones which lie under unit circle.  There's hyperbolic ones which are real.  And then there's lots of atomic ones.  The elliptic ones come in pairs.  Elliptic and hyperbolic ones, the lots of atomic ones  come in multiples of four.  And the non-resonant assumption is  that these two sets of exponents, this flow k exponents,  that appear here.  So you have alpha j, u and alpha j, 0.  These two sets are rationally related  and independent.  So there's no rational linear combination of these two,  which is 0.  So that's the non-resonant assumption.  And under this assumption, you can get a leading tromb  for the eta invariant.  It's of order r of the n.  And what appears in this leading tromb is, again,  you see this in the form line, you see again,  you see this in alpha j.  This is similar to the both the Kiribay or Maureen Eskimo  density function.  And this non-resonant assumption, by the way,  excludes the so-called one-counter example that  is k-value.  OK, so that's the result.  Now I want to tell you what's involved in the remaining time.  What's involved in the proof of these two results,  is the short results.  OK, so first of all, we're going to reduce the asymptotics  of the eta invariant to trace asymptotics.  So there's a well-molded formula for the eta invariant  in terms of the heat trace, which is given by this muth  and frit.  So this is the formula.  So this is an integral of the heat trace, d times e to the  minus td squared from 0 to infinity.  This integral, and this is an integral of time, all time.  This time integral, you can separate into two parts, one  for small time, one for large time.  The large time part you can write as a single trace,  and without an integral.  Of a function, this is the sine error function here.  And this function looks like the following.  So the error function is rapidly decaying and everything.  But the sine error function is also rapidly decaying and  smooth except that the origin has a discontinuity.  So this discontinuous functional trace, it doesn't  have an expansion yet.  So this trace of this sine error function is non-local.  So this trace of this sine error function is non-local.  And we have to work harder to get asymptotic  of a discontinuous trace like this.  But it's not so different from, if you're familiar with  wild counting functions, you're looking at a wild counting  function of an interval.  You're looking at the trace of a characteristic function of  that interval, which is also discontinuous.  So then you can use a standard tabirian argument to get at  this discontinuous functional trace using trace asymptotics.  But this is a tabirian argument.  This asymptotic is the A to N bearing  of a trace formula, which is a physical trace formula.  And this physical trace formula, what's important in this  trace on the right-hand side here is this insertion of this  data had this second function here.  Without this data head drum, this is a function of d over  root h.  This trace asymptotics you can get at using  local index to its local.  But to get the sharp result, we need to consider a more  refined trace with this data head inserted here with this  combination.  And what's important in the insertion of this data head  is, get a check, rather.  This is the inverse for your transform.  It is that there's a full power of h in the denominator.  Getting in with that full power of h in the denominator, this  trace is not local.  And but it still has an asymptotic.  So there's an expansion in powers of h.  And there's a local part in the asymptotics, but there's also  a normal local part, which is coming from contributions of  rib orbits.  So this sum over gamma is sum over rib orbits of the  contact form.  And these constants that they're appearing in this formula  are respectively the period of the rib orbit, the remanian  length of the rib orbit, and the mass of index.  So this is an analog of the usual with the trace formula.  But what's new here is that the dynamic contributions in the  usual with the trace formula there on the co-tangents on  the Hamiltonian flow on the co-tangent space.  But here they are on the base map.  There's a dynamic system on the base map.  This is somehow new here.  And the even and varied asymptotics follows from this  one.  So I want to tell you a little bit about what's involved in  the glistening trace formula.  So as I said, the usual way to prove trace formulas is  using FIOs and Hamilton.  It involves Hamilton flow of the symbol.  So if your operator is non-scaler, you have a  non-scaler symbol, but you still have its scalar eigenvalues  of the symbol.  And you can look at Hamilton flow of the scalar eigenvalues.  In this case, the trace formula, if the symbol is  diagonalizable, smoothly diagonalizable, these are  smoothly smooth functions.  And you have a smooth flow for these.  The trace formula will involve closed orbits of these  Hamilton flows of these eigenvalues.  But in our case, the symbol of the Dirac operator is  clip of modification.  And its eigenvalues are plus or minus absolute value of  psi plus x times the co-vector on the co-tangents  space, and a is the one form.  And since this is absolute value, this is the square root  singularity, this is not a smooth function of the  co-tangent space.  It's not smooth along this characteristic variety.  And it's not diagonalizable, so the usual  trace formula will not apply.  But to prove this trace formula what you do is you try to  look at the local model to begin with.  So if you look at the local model, so this is the standard  contact structure on R3 with some nice adapted metric, the  Dirac operator looks like this.  And these are the three-pillar-multimeter  seas.  It is a three-pillar-multimeter seas, and this is  the coupling, which is coming from the contact flow.  And then there's a free part, which corresponds to the  ray vector field, and then there's a part which  corresponds to a complex harmonic oscillator.  And this complex harmonic oscillator has a well-understood  spectrum.  It has infinite dimension.  It's conjugate to the real harmonic oscillator of one  dimension.  So these eigenspaces are actually infinite dimensional.  They're called Landau levels.  And in this case, and at least in the model case, you  completely understand the spectrum.  And what you try to do in general is try to make your  general Dirac operator under your assumptions, try to make  it look like the model one.  So in general, it looks like this.  This is the local expression for your Dirac operator.  These are, again, the Clifford matrices.  This is the coupling one form.  And this is the tensor which diagonalizes the metric that  appears in the local expression.  And the idea is to try to conjugate.  Conjugation does not change the spectrum.  I try to conjugate it to make it look much like the Dirac,  this model Dirac operator.  So this is the model part.  And then there will be an error.  The error, you try to make it as best as you can.  So you try to make it commute with the leading point.  So this is a standard procedure for normal forms of  operators.  And then once you have this normal form, what you do is  you break up the trace into these eigenspaces of this  model operator, or the Landau levels.  And then you study the trace on each level.  That's the idea.  Now, the normal form has an interesting, as an interesting  procedure in this case.  So the conjugation operator that appears, this conjugation of  the Dirac operator that appears in the normal form, is  conjugation by an FIO, the Fourier integral operator.  And the Fourier integral operator that appears here is,  as an interesting form, is now.  OK, so it's conjugation by an FIO.  This is E to the i t f of 2, while this f,  while w stands for while quantization of a scalar  symbol.  OK, so this first part is the product of two operators.  The first part is an FIO.  And the second part, what's important here is that there's  no h in the denominator in the second part.  And it's called Clifford-Wall quantization.  So Clifford-Wall quantization is quantization of not just a  scalar symbol, but a scalar symbol which takes values in  the exterior algebra piece of it.  This scalar part is quantized by the while procedure.  And the exterior algebra is quantized by the Clifford  procedure, so the Clifford-Wall quantization of a symbol  E, C. So this is something you want to quantize here.  And this is just the scalar part is quantized by the  while procedure.  And then the Clifford part is quantized using Clifford  quantization, in case of this or that.  Clifford makes this.  And once you quantize it this way, you can compute  conjugations of such FIOs.  And the reason for quantizing it this way is because what  appears in the conjugation formulas in trying to produce  a number of more are causal differentials of these symbols  that you quantized.  So this causal complex is just defined.  The causal differentials are just defined on the exterior  algebra, on the exterior algebra part it's either wedge or  contraction, and on the symbol part it's either  multiplication or differentiation.  So these causal differentials appear in the  conjugation formula.  And then what gives you the normal formula is the  Hodge theorem.  It's a symphlectic Hodge theorem for a causal complex.  So that's the rough idea of how the proof goes.  And this normal form procedure generalizes a normal form in  the book of every semi-flexible analysis.  This is a new book.  But he did it in three dimensions.  He's a generalizing man.  And so this gives the trace formula for small time.  And then there's a large time trace formula, but that  requires some more work.  It requires a propagation of singularity's lemma, or the  weak version of a propagation of singularity's lemma, which  is not mine, but I found it to be a very good one.  Which is not mine, but I found it in this book of  the Marcy Andrews theorem.  So that's roughly how the proof goes.  I don't have so much time to give the details.  So let me end by, I'd like to end with always some further  questions.  If anybody here wants to work on this, join me or do it  without me.  Then there's a lot of questions to be answered here.  So first of all, you have this trace formula of the  either invariant asymptotics.  You could try to do the same under other assumptions.  So instead of a contact form, if you choose some other form,  there's a lot of models for one forms, and singularities of  one forms on a man called.  If you change your contact assumption, the assumption is  not redundant by the way.  It's fairly necessary.  If you don't assume your manifold as contact, it's not  clear what the dynamical system is that will appear in the  trace formula.  So if you change the assumption, the formula will  change.  The trace formula will almost surely change.  So you could try to work it out.  The other thing is we considered the data invariant.  We could consider the determinant again, as in this  book, the best row.  We could consider the asymptotic numbers of the log determinant  in this real case.  And this is not so easy.  Again, this move as rows.  Example, you need positivity, which doesn't exist here.  This assumption doesn't exist here in the real case.  So you can look at asymptotics of the log determinant, and you  would have to understand this case.  Low line eigenvalues of the Dirac operator.  There are people who work on this expansion of low line  eigenvalues for magnetical fractions and so on.  Then try to understand the propagator.  So you could try to understand the wave equation of the  Dirac operator.  Our proof doesn't really go through this wave equation.  It uses normal forms and stuff.  So you could try to understand the propagator and  propagation of singularities for this wave equation of the  Dirac operator.  And again, so that's another thing.  And another thing you could try to do is try to consider,  again, our technique is more refined than the local index  theory technique.  So you could try to apply our technique to the actual  quantization problem.  So actually, tens of powers.  And the Dirac operator and tens of powers and stuff.  Try to see what this local technique gives in that case.  And finally, you could study the, so we consider the  magnetical couple Dirac operator.  You could also consider the magnetic operation.  It's slightly different.  You can try to prove a trace formula for the  magnetical fashion, instead of the Dirac operator.  And there's, so this is a semi-classical problem.  You could consider the micro-ocal analog of this problem.  And this is related to what's called the subroman in Laplacian,  which I don't have time to explain.  But that's a much bigger subject.  Much bigger subject is subroman in geometry, but I don't  have time to explain it.  So that's all.  Thank you.  Thank you.  Any questions?  So you say one might try to apply those techniques to  black-keyed and sense-contained space.  Yeah.  So the first thing to find out about analog is just a general  question.  It was only a general question, but you would have to  figure out the right assumptions.  What's the analog of contact assumption in the even  dimensional case?  I'm not so sure.  But we could try to think about it.  We could get something about black-keyed conjecture.  At least some particles.  I don't think this is a possibility.  Even the usual good-solid transform involves Hamilton  contribution from periodic Hamilton trajectories.  And that doesn't really give you existence of a Hamilton,  closed Hamilton trajectories.  And there are examples where you don't have Hamilton  trajectories, but you still have the trace model.  So one more question.  Let's have a speed.  Let's have a given back equation.  This cannot be properly mixtured.
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