 Well, thanks for the invitation. It's certainly a pleasure to have the opportunity to speak at this DuMorifest. And there have been many interesting talks and I have to also add to the occasion. So way back in 1977, in the early days of Bacchal Thermodynamics, Gibbent and Hawking first studied the Schwarzschild solution in Euclidean signature to study its thermodynamics. And then they considered rotating Bacchal, the CARE solution. The CARE solution was different from Schwarzschild because when they continued to imaginary time, the CARE solution becomes complex. So they called it a quasi-Euclidean metric. But they showed they could get sensible results from this complex metric. By computing the action of the quasi-Euclidean solution, they got sensible results with the thermodynamics of rotating Bacchal. So that was at least one case in which it appeared appropriate to consider complex solutions of the Einstein equations. Since then, there have been numerous other reasons. For example, one interesting application of complex spacetime metrics was by Lukow and Sorkin in 1995. They were interested in topology change and they considered Lorenz signature more physical so they wanted to study topology change in Lorenz signature. So here in my picture, time runs vertically and a closed universe is trying to break into two. You could certainly have a smooth Euclidean signature metric on that spacetime, but you can't have a smooth Lorenz signature spacetime roughly because the time would have the stagnation point at this saddle point here. But they considered a metric which was in Lorenz signature away from the critical points and they slightly regularized it by letting the metric become complex near the critical point. And they showed they could get what looked like sensible results. They had a good motivation for which direction in the complex plane the metric should be perturbed near this point. And if they perturbed it that way, several things work nicely. But here's another motivation for complex metrics. It has to do with the Hartle-Hawking wave function of the universe. So in D dimensions, well that Y be a D minus one amount of all I should have written that with a metric called GD minus one. The Hartle-Hawking wave function is supposed to be a function of D minus one metrics on a fixed manifold Y. You're supposed to compute it by summing overall manifold M of the dimension D whose boundary is Y. And for each M you do a path integral of all metrics on M whose boundary value is GD minus one. So here's the picture. Y and the metric on it are given. You complete the picture by picking an M and extending the metric of Y over M. And then you do a gravitational path integral over all such metrics. And that's supposed to give the Hartle-Hawking wave function. Well, this is similar to what you do for an ordinary quantum field rather than gravity with one important difference. If you were discussing a scalar field on Y you would pick the boundary values on Y. You'd pick a particular M. You wouldn't try summing over M's and you'd pick a metric on M. But for an ordinary quantum field, for a particular M you'd integrate overall values on M fixing the boundary values on Y. And that would give you a wave function for an ordinary quantum field. For gravity instead, you don't know what M should use. You sum over all M's and integrate over all metrics on M. Now, for usual quantum fields Euclidean path integrals are used to find ground states and Hartle-Hawking wanted something like that. So they wanted to find a state by a Euclidean path integral on M. So M here was supposed to be Euclidean and the metric was supposed to have Euclidean signature. Now, when you try to do this a few strange things happen immediately. First of all, for normal quantum fields the action is bounded below. That's not true for gravity but let's suppose for a second it was. If the Einstein-Hilbert action were bounded below then the asymptotic behavior of the wave function in a semi-classical limit which here means large volume. I mean, semi-classical means that the metric G D minus one is such that Y has large volume. The action is large for large volume and this asymptotic behavior with the exponential of minus the greatest lower bound on the action of the metric for any metric that satisfies the boundary condition. In other words, in a semi-classical region if the action were bounded below the integral would be dominated by the metric that achieves the greatest lower bound if it can be achieved and otherwise by metrics that come close. So if the greatest lower bound is positive we'd get exponential decay of the wave function for large volumes. That's what happens in ordinary quantum field theory. If you have an ordinary field phi with a positive definite action the corresponding wave function decays exponential you're actually more than exponential it decays like a Gaussian for precisely this reason. So it decays like the exponential of minus the lowest greatest lower bound on the action allowed by the boundary condition and that grows quadratically with phi. Well, this would be a very bad answer for gravity because we don't wanna predict that it's exponentially unlikely to observe a large universe. What saves us is actually something which sounds problematical. The Einstein-Hilbert action in Euclidean signature is unbounded below. So there's no lower bound on the action of the metric that satisfies the boundary question. So we don't predict that we can't observe a large universe since there's no lower bound on the action what does the gravitational path integral mean? Well, Gibbons Hawking and Perry suggested that it's a middle dimensional contour integral in a space of complex valued metrics. But what's the contour? Well, there's no good answer to that question. To this day, the only concrete idea is to think of the contour of the canteen language as the sum of left-shift symbols associated at the critical points. Semi-classically that just means that you evaluate the path integral by summing over classical solutions with no clear guidance on what solutions to pick and for each solution, you evaluate its contribution perturbatively, making a weak rotation in field space so that you can do the perturbation theory. Now, to try to actually calculate the hard-locking wave function, in examples, you need to pick some classical solutions. And here, again, you run into an obstacle, what looks like an obstacle, but perhaps it's really a benefit. In simple cases, although I'm not sure if this is done rigorously, there appear to be no real Euclidean signature solutions that obey the boundary conditions. For instance, let's take the cosmological constant to be positive and let's take the most obvious Y, which is a sphere. I'm not sure why, but I'm hearing some background noise. I'm not sure I can do anything at my end to suppress it. Anyway, we take the D minus one manifold to be a sphere with a round metric of large radius. Large means large compared to the length set by the cosmological constant. Then it's believed that there is no real Euclidean classical solution that satisfies the boundary condition. So it's believed that there's no M that would be an unsigned manifold with a cosmological constant whose boundary is this large sphere. I'm not aware that that's known, but perhaps it is known rigorously. However, if we broaden our horizons a little and allow complex solutions of the Einstein equations, then there is one that gives an answer which at least at some level of detail is a sensible answer. In more detail than today's lecture, there are certainly unresolved puzzles, but at some level it makes sense. So how are we going to find a complex solution of this problem of finding an Einstein metric whose boundaries around sphere with large radius? Well, we'll start in Euclidean signature with the metric of a d-sphere, where rho, which is the radius, is essentially this one over square root of g times lambda. So here's the metric of a round d-sphere. Well, assuming that theta is a real variable and this is the metric one around d minus one sphere, this would be the metric one around sphere. So usually theta would run on the real axis from minus pi over two to pi over two. But there are a lot of things we can make by considering a curve theta of u in the complex theta plane where u is a real variable. So I've drawn the complex theta plane showing some zeros of the function cosine theta. The reason the zeros are important is that if we want to make a manifold with boundary, sorry, if you want, well, okay. If the curve theta of u is going to have an endpoint, the endpoint has to be one of the zeros or else we'll get a manifold with boundary. So an important option is for the curve to end on one of these zeros. And that way we can make a manifold without boundary. So first let's discuss some real solutions that we can make with curves in the theta plane. So first we could go on the real axis from minus pi over two to pi over two. And that's simply the Euclidean sphere. But there's something else we can do that makes a real metric. We can go on the imaginary axis from minus infinity to plus infinity. And that again gives us a real metric but now it has Lorenz signature. So the metric is this one and it's actually the metric of what's known as the sitter space. And these are both considered physically sensible in the appropriate context. Now here's another solution that's considered important and this is the Haudelhawking wave function. Here we start at pi over two and there's an endpoint there but it doesn't represent a boundary because that's a zero of the cosine. We go along the real axis to the origin and then we go on the imaginary axis to plus infinity. So the real axis part gives us a hemisphere and the imaginary axis part gives us half of the sitter space. One has Euclidean signature, one has Lorenz signature. But they meet nicely at theta equals zero where they both describe a sphere of this radius row. So this is a solution that describes creation from nothing of a closed universe that then expands exponentially fast. You sort with nothing at theta equals pi over two by the time you get to zero, you have a sphere of radius row which was one over the square root of G lambda and then you proceed along the imaginary axis and the sphere grows. So now if you, we don't have to go all the way up to infinity. If you take this problem that we had here that had no real solution, which was to find a solution where the boundary is a big sphere, that solution has a problem. Oh, sorry, that problem has a solution where I just pick a point on the imaginary axis and I stopped there. So I started with nothing. I ended up with a big sphere. So because this function cos u goes to infinity, if you give me the radius you want, I just pick u so that row cos u was whatever you asked for and stopping it at that value of u, that value of imaginary part of theta, we get a complex metric that solved the problem. Now, you might feel funny about this piecewise smooth metric, but you can smooth it out and also solve the problem with a smooth complex metric, which is almost Euclidean at the beginning and almost Lorentian in the future. And if the endpoint on the imaginary axis was chosen properly, this is a smooth complex metric that solved the problem of finding an Einstein metric. This boundary is a sphere of some given large radius. So as I said, the last two pictures solved the problem of finding a classical solution with cosmological constant, whose boundaries around sphere of large radius. The only trick is that the solution is complex. Because it is complex, the action, which for such a metric, which with the Einstein theory with the cosmological constant is a multiple of the volume, it's also complex. And it has the general form. It has a real part that comes from this part of the contour. The action is the same for either contour by essentially Cauchy's theorem. The action has a real part that comes from this part and an imaginary part that comes from this part. The real part is actually negative and it's minus half what's called the decider entropy. And then there's an imaginary part. So the semi-classical answer is the exponential of half the decider entropy minus i times something that I wrote i of rho, but it's just an i of u. Well, i of the radius that we have in the final state, which is an i of Cauchy, really. Roe was the radius at the endpoint, not the radius at zero. So as I've written here, rho was the radius of the sphere we're producing from nothing. So this answer is considered physically sensible, more or less. The real part means that because there are a lot of states in decider space because the decider entropy is big, this has a large probability somehow because there are a lot of microstates. And the oscillatory behavior means that you're producing something real. It's not just a tunneling process. It's tunneled into a real solution in learning signature. So this is an example of a complex solution that seems to give a sensible answer. A more elaborate example than the two I mentioned at the beginning that involved the rotating blockhole and also the topology change in learning signature. But once we allow complex metrics for opening Pandora's box, we can do lots of other things that won't give physically sensible results. For example, here's a complex metric on the d-sphere that obeys Einstein's equations. It has an action different from the standard value of a round sphere. So I just took a trajectory that went between two different zeros of the cosine. And if they're not adjacent, we get a non-standard complex metric that we probably don't want in the gravitational path integral. Another thing we don't want in the gravitational path integral is a closed loop in the theta plane. It gives a complex classical solution whose topology is a circle times SD minus one. It's action is zero. And here's a complex solution which if we allow it gives a contribution to creating a universe from nothing whose negative part is more negative, sorry, whose real part is more negative than the in half of the given talking entropy, which probably means an unphysical result. Then we can make it even worse by starting at five pi over two or seven pi over two and so on. The farther to the right we start, the worst is what we get. So in short, we need a principle that would help us select what complex solutions we consider sensible. So in a recent paper, conservative and Siegel made a suggestion for distinguish class of allowable complex metrics with a property that ordinary quantum field theory makes sense when coupled to allowable metrics. Their motivation wasn't quantum gravity at all. They were aiming to develop an alternative to some of the standard axioms of quantum field theory. The new axiom set would assert that quantum field theory can be consistently coupled to allowable metrics. But it's natural to consider that criterion in this context. And in fact, many authors such as Halibut and Hortel in the 80s have, there are matter fields in nature coupled to quantum gravity. So if we pick a solution of our problem, the classical solution with a complex metric, if we wanna calculate quantum corrections, it had better be true that whatever quantum fields there are in nature can be consistently coupled to the metric we assumed. So it's been long recognized that part of this problem is that, well, the classical metrics you can consider should have the property that quantum matter fields, at least the one in the real world, maybe all quantum matter fields, if the real world is based on a sufficiently generic theory should be such that quantum field theories can be coupled to it. So if there's a good class of complex metrics for quantum field theory, it's natural to think that this might be the class who should work using quantum gravity. So the main observation in my talk is that the good complex metrics that seem to have given useful results. In the examples I've described and also some other examples we won't have time for, seem to be allowable in the sense of conceivage and seagull and the obvious bad ones, including the ones I've mentioned and others that can be constructed similarly don't seem to be allowable. I should add before I go on that in one of the papers I've already mentioned by Luco and Sorkin, well, they were mostly in two dimensions, but they had a class of good complex metrics, which although not developed so systematically was rather similar to the notion of conceivage and seagull. The idea of conceivage and seagull was to require that the theory of a P form field A should make sense for every P from zero up to D minus one were in D dimensions. So P form field can't be above degree D minus one. So if F is its curvature and Q is P plus one, the usual action is this expression and the requirement they impose for a metric G to be allowable is that the real part of the action should be positive for every real non zero F and all Q. What we state this condition for Q equals zero, although my explanation doesn't quite explain why because Q was P plus one. The intuitive idea is that positivity of the real part of the action implies that the path integral of an anti-symmetric tensor field coupled to the metric G make sense. In other words, in the theory of an anti-symmetric tensor field with the integrating of rule A, the exponential of minus IQ. So if IQ has a positive real part, that's going to make that interval convergent. Now you might, you might be a little skeptical of whether anti-symmetric tensor fields are so important. Why not symmetric tensors, for instance? Well, according to a result by Weinberg in May that was formulated with some help by Sidney Coleman, the only massless fields that have local energy momentum tensors and therefore can potentially be defined in curves space-time are the anti-symmetric tensors. And for P equals zero and one, the anti-symmetric tensors have non-linear versions, non-linear signal models and non-linear engaged fields. But at least in the realm of classical field theory, these are the only classical field theories that you can hope to couple in a curve space-time. So by including anti-symmetric tensor fields of all ranks, you're essentially including all quantum field theories that can be derived from underlying classical field theories. I've stated this for massless fields, but you don't get anything essentially different if you add massive fields. So it's well motivated to consider this class of theories. It's not guaranteed. No statement about all quantum field theories is guaranteed, but at least the statement applies to quantum field theories that can be associated to underlying classical field theories. Consavage and Seagull give rather explicit description of the allowable metrics. This is a point-wise criterion. Well, sorry, I should go back here. This expression has a positive real part for every F, but that's a point-wise expression. It means that this quadratic form on the space of Q forms is point-wise positive. So allowability is a point-wise condition that this quadratic form made from the metric tensor has a positive real part for all Q, and therefore we're going to get a point-wise condition. So we can speak of G being allowable at a given point. It means that at that point, the quadratic forms were positive. So at a point where G is allowable, there's a real basis of the tangent space where the metric is diagonal with complex coefficients lambda I, because this is a complex metric. But allowability amounts to the condition that the sum of the arguments of lambda I is strictly less than pi. And well, this condition is useful for various things. For example, it shows that the space of allowable metrics is contractible onto the space of Euclidean metrics because some of the arguments being less than pi means that none of the lambdas is negative. And therefore you can rotate them to the positive axis by the shortest path in the complex plane in a unique way. And that would contract the space of allowable metrics onto the space of Euclidean metrics. Another nice fact is the following. Well, first of all, if G is allowable, then the volume of M has positive real parts since that's the Q equals zero case of allowability. But conservative and single show that if M has an allowable metric, then the induced metric on any sub-mountable N of M is also allowable. And therefore, the real part of the volume of M is also positive, which we can take as an indication that perturbative string theory makes sense on such an M. So that M, the allowability with the condition that let's not just quantum fields, but quantum fields and strings be well-defined. Now, the condition for allowability shows that a Lorenz signature metric isn't quite allowed. But it's on the border of the space of allowable metrics. It can be perturbed in either of two ways to make it allowable. We give a positive or negative imaginary coefficient to the dt squared term. Here, epsilon corresponds to the Feynman I epsilon. And well, I'll explain in a moment what the two choices mean. The two choices differ by the sign of the square root of the determinant of G. So allowability means that the square root of the determinant of G is not imaginary. And we pick the branch of the square root that has a positive real part. That's the definition of a part of the definition of that's the Q equals the real case of allowability. But when we approach Lorenz signature, the square root of the determinant of G is becoming negative. So the square root is becoming imaginary. But it approaches the positive or negative imaginary axis depending on the sign of the I epsilon term. So since the action is proportional to the square root of the determinant of G, the overall sign of Lorenz signature action is going to depend on which way we approach this limit. In standard physics, the sign of Lorenz signature action actually involves a convention. In classical physics, the square root of minus 1 didn't really appear. When you go to quantum physics, Feynman said to take the integrant of the Feynman pathogen to be the exponential of I times the action. But it could have taken it to be the exponential of minus I times the action. So we get these two options depending on the sign of the regulator here. They're both really used in physics, for example, in the Schringer-Kellertische approach to thermal physics. The usual path integral propagates the ket, and the opposite path integral propagates the bra, which is which really depends on the convention. Either way, the epsilon in this formalism is playing a similar role to the primary epsilon. The Lorenz signature evolution is always accompanied by a little bit of Euclidean evolution as a regulator. Now, clearly, we can't go continuously from epsilon positive to epsilon negative because the sign of the action is jumping. That helps in understanding what's wrong with some of the oddball metrics that I described before. So the oddball metrics that I described before and various other ones that I won't tell you about today violate this condition. So, for example, for the metric I've drawn, well, one constraint is that an allowable metric can't cross the lines where the real part of theta is n plus half times pi because on those lines, cosine squared theta is negative. You can end at one of those lines, but you can't cross it. Crossing one of those lines is like crossing from positive to negative epsilon. So for example, this trajectory is ruled out because it crosses the line where real part of theta is pi over two and the same applies to this one. And similarly, the real part of theta can't have a maximum or minimum on the trajectory which rules out closed loops in the theta plane and so on. Now, there are other examples I would have wanted to talk about given more time. One of the more interesting is why the quasi Euclidean metrics of Gibbons and Hawking, which I mentioned at the start, turned out to be allowable whenever the black hole such that one would expect quantum corrections to be well-defined. So quantum corrections to a rotating black hole are not necessarily well-defined because for example, in an asymptotically Newtowski spacetime, a particle at large distances from the black hole has a very large angular momentum compared to its mass and that causes the thermal ensemble with rotation included to be unstable. So the quasi Euclidean metric is actually not allowable in the asymptotically flat case. But in the asymptotically anti-dual case, if you put a constraint on the angular momentum so that the ensemble expected to be well-haved, then you find that the quasi Euclidean metric is allowable. So this is a rather interesting example but I thought there wouldn't really be time for it today. In short, in today's lecture, I'll explain a little of why it's motivated to consider complex solutions of Einstein's equations in the context of the quantum gravity, why a good class of allowed metrics is needed and why the class identified by Konsevich and Siegel and in embryo previously by Luko and Sorkin has at least some of the right properties. There's a lot missing. The biggest thing which is missing is that to understand the so-called Euclidean path into more gravity, who has much more than a knowledge of what's a good class of complex metrics, ideally you want an integration cycle which is extremely far from being understood. Thank you today and let me wish, congratulate again, T-Bowd or more on this occasion. Thank you very much, questions. So I have a question is that when you were continuing this Lorenzian metric from this Lorenzian signature to the complex, you're replacing complex metric. So when this allowable metric, is there any canonical way to determine which of this Lorenzian metric can be extended to this allowable metric with physically I'm not saying, is there any canonical way? All complex, all Lorenzian signature metrics can be perturbed to make them allowable. Roughly speaking by just adding plus or minus or epsilon to the DT squared term. So Lorenzian signature metrics aren't quite allowable. They need the I epsilon to make them allowable and they can be made allowable in two ways, which as I've explained, are related to each of the IS and each of the minus IS. And Feynman, it's not always formulated this way but the Feynman I epsilon is somewhat similar to making this regularization. There is another question. So you told us if I understood correctly that the asymptotically flat Euclidean given token metric is not allowable for all the gap solution. So what is the physical interpretation of this? Well, let me see if I can manage to write on this, although I'm not well equipped to do that unfortunately. The given, so the Schwarzschild black hole is related to this ensemble e to the minus beta h. But then when you add angular momentum, you're considering an ensemble which has a chemical potential for the angular momentum. So you're not just looking at the ensemble e to the minus beta h, you have an extra term proportional to the angular momentum. And if it's possible for a particle to have such a large j compared to h that a mega j is which comes with a minus sign is bigger than h, then it's going to make a large contribution to this ensemble. And if there's an arbitrarily large contribution then this ensemble isn't going to be well defined. And that's actually the situation for Kearra black hole in asymptomatically five schools. So for Kearra black hole, this ensemble, you can't expect it to make sense because particles far away from the black hole make very, very big contributions to this partition function. So if you try to calculate, so if you try to use the, sorry, I meant the trace, not the explanation, but try to use the quasi Euclidean metric to calculate this trace. We're trying to calculate a trace that isn't well defined and that shows up in the fact that the quasi Euclidean metric isn't allowable. However, if you go to estimate the anti-dissiduous points and also if you put an upper bound on a magnet. So, okay. In the early days of the ADS CFT correspondent talking and others pointed out that this ensemble is better defined in anti-dissiduous space than in asymptomatically flat space. If you go to a situation where you, there's background noise. I don't know how to control. If you go to a situation where the ensemble can be expected to be well-behaved, that turns out to be equivalent to the quasi Euclidean metric being allowable. You find that it just barely works if the condition, the sum of the arg lambdas is bounded by pi, strictly less than pi, but it saturates that near the horizon or somewhere. Maybe it's near the boundary. We'll take one question from the online community, Nikita Nekrasov, and then afterwards one more. Nikita. Yes. Thank you. I actually had two questions, but maybe I'll ask a simple one. Instinct perturbation theory, we probably want the metric on the wall sheet to be nearly Lorentzian along the long tubes corresponding to the propagators and be nearly Euclidean near the vertices. Is it a complex allowable metric which smoothly supports? Yes, that's basically the type of metric described by Sorkin and Lukow in 1995. They weren't motivated by string theory. They don't even mention what they're doing is relevant to string theory. So what Nikita's saying is that if you want to do string perturbation theory in Lorentz signature with real-time propagation of real strings, well, if I can maybe I should just draw it. I had it on an early slide. So we had topology change. As Nikita's saying, sorry, I'm not set up to do this well, unfortunately. OK, imagine I drew the rest of the picture. You're trying to draw it vertically in this incomplete drawn picture near the saddle point where there is stagnation. We don't have a good Lorentz signature metric. Then Sorkin and Lukow perturbed the metric with a small imaginary part and showed a number of nice things happens, which is hard to summarize now in the available time. But one nice thing that happens is that the integral that's supposed to give the Euler characteristic, square root of the Gauss-Penangial, gets a positive imaginary part, which is appropriate because the saddle point contributes minus 1 to the Euler characteristic. Anyway, the answer to your question is yes. So if you want to do string perturbation theory with everything being in real time, you should allow this small imaginary bit regularization near the interaction points. You can also include the Charm and I epsilon away from it. But this is a framework for nearly Lorentz signature metrics for real time string perturbation theory. That's, I think, what Sorkin and Lukow could have said, although they didn't because they weren't interested in string theory. Yeah, but in string field theory, we have more complicated vertices. For example, if we do all closed string field theory here, like SWEBUCK, so we have more complicated junctions, so we should require much more E.P. geometric. OK, I see what you mean. Right. I've only answered four cubic vertices, like in late-con string theory. My guess is it will be OK, but I haven't thought about higher order vertices. And just very quick. Yes. In the u-plane, in the theta plane, the trajectories which you have with hyperbolicization, if I vary the trajectory, it doesn't change the action keeping the bound endpoint. Yeah, the action only depends on the homology class of the cycle. Right, but are these solutions, but the solutions are different? Are they equivalent, or are they actually different solutions? And somehow, you need to take into account the number of trajectories in the nothing to do. Well, it's not extremely clear what is the equivalent relation on trajectories you want. Clearly, two homotopic trajectories should be considered equivalent. But it's not totally clear if they're homologous, but not homotopic, whether they should be considered equivalent. But even that requires some kind of complexification of the geomorphisms, right? The geomorphism group doesn't have a complexification, although some authors have written as if it did. So you can't divide by complex between opposites. You need to formulate things in a different way, which isn't well known. But homology is like the first attempt to replace. I think it's clear that homotopic trajectory should be considered equivalent. OK, thank you. That's the only thing which is completely clear. So there's one more question from the audience here. Yeah, my question is. I'm giving one example. These two are homotopic. So we surely will consider those equivalent. Yes, go ahead. You said talk just that the option should be extended as an homomorphic function of the metric. Well, if you're going to try to do this, it's hard to imagine what else you can do than to treat. When I said that the homotopic trajectory should be considered equivalent, it's built in that the action is homomorphic. And then you would use Koshy's theorem to show that the action is the same for homotopic trajectories. The reason for the last question, in part, was that that argument would actually prove that homologous trajectories have the same action. And it's not clear to me whether perturbation theory is equivalent for homologous trajectories. I think that's a very interesting question, actually. I suspect it isn't, but I'm not 100% sure. But I think to play this game at all, you have to interpret the action. I didn't say it explicitly, but when we complexified the metric, we also analyzed and continued the action and everything else in sight. OK, is there another question in the audience? I don't see any. So thank you very much again, Ethel.