Talk on conformal field theory from conformal loop ensembles, and other research interests
presented for King's Lectureship position
home page of Dr. Benjamin Doyon

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My general research interest is in two-dimensional quantum field theory and its applications, but I'll only tell you about a part of my recent works, that has to do with conformal field theory and a very new probabilistic approach to it.

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The first thing I'd like to tell you about is my point of view on quantum field theory. The basic idea is that quantum field theory is in fact a theory for certain kinds of emergent behaviours. I will illustrate this with the classical Ising model on a lattice; let's say on an hexagonal lattice, like is shown here. The measure is simply an exponential of a sum of products of neighbouring spins. It is homogeneous, everywhere it looks the same except perhaps on the boundary sites where I simply fix the spins, and it is local, as it factorises into a product of factors associated to small regions -- here two neighbouring sites. In the picture, the edges in red are those where the energy is lowest, where the spins on both sides are opposite. Note that these edges necessarily form loops.

At small temperatures, the spins all tend to align; at large temperatures, they tend to be randomly distributed. And since the measure factorises in a local way as I said, there should be very little correlations between fluctuations of spins that are separated by a large distance. In fact, these correlations usually vanish exponentially. However, at the critical temperature that separates the large and small temperature regions, something special happens. Spins want to align, so they form large aligned regions, yet the average is still zero, so these large regions themselves fluctuate to an average moment of zero.  Hence, there will be correlations at large distances thanks to these fluctuating large regions. Also, any small local magnetic field will influence a large region of spins -- the system is very sensitive to external disturbances. It is critical. These large-distance correlations are emergent correlations, the result of many spins acting together but only through local interactions. Since they imply a large number of local interactions, a lot of the details of these interactions is lost, and the result is universal.

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Quantum field theory is a construction that describes these emergent correlations. Mathematically, it provides the asymptotics of expectations of products of local variables as the critical point is approached, and at the same time as the observation distance is made larger. Like is shown in this formula. Here, alpha is a parameter that tells us how fast we approach the critical point -- it is essentially an RG parameter, with alpha=0 being the UV fixed point. As compared to usual concepts of QFT, it is the process of approaching the critical point that is a regularisation-renormalisation procedure. What we are doing is looking at the lattice from further and further away, while approaching the critical point so that correlations are present at larger and larger distances. Correlations still vanish in this limit, albeit not exponentially, only with a power law. Correlation functions of QFT give the coefficients of these vanishing behaviours. These coefficients, when seen as functions of the points upon which they depend, here x and y, also contain in their functional form the exponent of the power-law vanishing, here 1/4. Hence, they contain all the information about how the correlations behave at large distances -- the emergent correlations. Here, the 1/4 exponent is characteristic of the spin variables in the Ising model, and the fact that the critical point beta-c is approached with a quantity proportional to the small parameter epsilon is also characteristic of the Ising model; there could be a non-trivial exponent involved here too.

Something similar happens for quantum systems, where we may have quantum critical points. There, it's not the temperature, but the quantum world that provides the fluctuations. A critical point is obtained by asjusting an internal parameter of the model, to a value that separates regions where the ground states are very different.

It may seem a priori that quantum field theory only describes a small, very particular aspect of statistical systems (or quantum systems). I have three things to recall in reply: first, the only situations where fluctuating many-body systems with local interactions show non-trivial large-distance correlations are near critical points, described by quantum field theory. Second, what is described is universal, so valid for many microscopic models. Finally, these are emergent behaviours, and are at the source of many of the important phenomena of theoretical physics -- for instance, the famous Kondo effect. Note also that armed with the viewpoint of emergent correlations and behaviours, one may wonder how far QFT ideas can reach: other complex systems? turbulence? macroeconomics?

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The basic ingredients of quantum field theory are local fields, in correspondence with local variables of the microscopic model, and correlation functions, describing the asymptotics of expectations of their products. But, after all, there is something more than just correlations at large distances. Indeed, there are large fluctuating objects leading to these correlations. These large objects should be seen as new objects emerging from the microscopic model, with their own, emerging probability measure. What are these objects? What is their measure theory? Clearly, understanding this would provide a better picture of the universal scaling limit than just looking at correlation functions. Then, once we have the emergent measure theory, how do we reproduce local fields and correlation functions from it? In the other direction, how do we prove that microscopic models lead to it in the scaling limit? These are the four questions that are at the source of some of my present research.

Some answers or partial answers are available, when we look at recent and not-so-recent developments in the context of these questions. In order to put things in context, I think it's nice to discuss the quantum case before going to the case of classical statistics. In many quantum situations, we know what the emergent objects are: the quantum particles corresponding to "modes" of oscillations, or the quasi-particles. These are the things we're interested in when taking relativistic quantum field theory as a theory for relativistic quantum particles. Then, their "measure theory" is given by the scattering matrix. From that, we may re-construct local field-operators on the Hilbert space, and correlation functions are quantum averages. In integrable quantum field theory, the factorisable scattering theory goes a long way in this direction -- this is the basis for my 2dQFT course. However, the last question is still unanswered: there is no known proof, for any non-trivial model, that these structures emerge from underlying quantum models.

In the case of thermal fluctuations, advances are much more recent, but very exciting. First, there is evidence that the correct emergent objects, in a large class of universal behaviours, are the domain walls, the red loops in the previous drawings. Indeed, at least at the critical point (alpha equals 0), they have a proper measure theory, recently developed by Sheffield and Werner, conformal loop ensembles. There is still little known about how to reproduce local fields and correlation functions from these fluctuating objects, but my recent work made encouraging progress. Finally, and importantly, there are proofs by Stan Smirnov, in special cases, that conformal loop ensemble indeed emerges from microscopic models at criticality. Note that if we succeed in reconstructing the quantum field theory structure from conformal loop ensembles, this would provide the first ever non-trivial quantum field theory construction from microscopic models. This is the more immediate goal of some of my research, which I may call "constructive CFT"

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So, let me now describe in more detail the work I did in this direction. I will tell you in turn about conformal loop ensemble, conformal field theory, and how I connected them. Conformal loop ensemble is a family of measures on sets of disjoint, self-avoiding simple loops lying in simply connected domains. Here is a drawing of a configuration. There are three axioms that define conformal loop ensemble, which I'll tell you just now; there is a unique, one-parameter family of solutions to these axioms (this parameter is essentially the central charge of the corresponding conformal field theory, between 0 and 1). The axioms are as follows.

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First, we require conformal invariance: the measure is invariant upon conformal transformations that preserve the domain, and the measures on different domains are obtained from each other by conformal transport. Note that it is not a very strong axiom: for a given domain, there are few conformal transformations that preserve it, so few constraints on the measure, and for other domains, it can be seen as only a definition. But as seen from the viewpoint of the scaling limit, it is hard to prove from statistical models.

Second, suppose we "fix" one of the loops, and look at what there is inside it. Then, the measure on these inside loops is exactly the CLE measure on the domain delimited by the fixed loop. This is very natural from the viewpoint of statistical models: a loop is a place where spins are opposite, so we know that all spins are the same along the loop inside it, hence it is like a "fixed-spin" boundary.

Third, there is something similar that we can say about the outside of the loop, although we have to be more careful because CLE is not defined on multiply connected domains. The proper way is to say that if we draw a line and look at the domain bounded by this line and by all loops that cross it, like in the drawing here, the measure for the loops in that domain is a CLE measure on that domain.

These are the three main axioms for conformal loop ensembles as developed very recently by Sheffield and Werner.

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Conformal field theory, on the other hand, is a widely developed subject. I'll tell you only about its most important structures. The first thing is, again, conformal invariance, or rather conformal covariance. It is expressed through local fields, and it is not as neat as conformal invariance of CLE. It says that there is a map in the space of local fields such that conformal transformations of the domain of definition and the positions of the fields is equivalent to this map. We usually assume that there are fields, called primary fields, that transform in the simplest way possible, as shown here. They are characterised by two real numbers, h and h-tilda, whose sum is the scaling dimension of the field, and whose difference is the spin. Again, this is not a very strong requirement by itself, and not immediately indicative of the supposed infinitely many conformal symmetries of CFT. But by basic principles of quantum field theory, in particular the Noether theorem, this implies that there exists a field, called the stress-energy tensor (rather, this is the holomorphic component of it, but I'll continue with the abuse of language), which has certain analytic properties. Essentially, its poles, at positions of other local fields, talk about how these other local fields transform. For instance, in the case of primary fields, this is what we obtain. That is, we know all of its singularity structure inside the domain D. This is the true statement of the infinite "symmetries" of CFT. By calculations in particular models, and by analysis of the algebra of its modes, we find that it transforms almost like a dimension-2, spin-2 primary field, except for a conformal anomaly, a term proportional to the Schwarz derivative. The proportionality constant is the central charge of the underlying Virasoro algebra of the modes. This anomaly tells us that the infinite conformal symmetry is "quantum mechanically broken". It is important to note that for any given domain, there aren't infinitely many symmetries, even broken. It's more when we look at many possible domains, and the true statement is in the analytic structure of the stress-energy tensor.

This structure, along with locality of the operator product expansions, and along with other internal symmetry currents when they are present, is at the basis of the algebraic formulation of conformal field theory -- vertex operator algebras and their modules. This is a very powerful formulation, with many non-trivial dimensions and correlation functions deducted from algebraic properties.

It's the stress-energy tensor, with these main properties, that I managed to construct in the context of CLE, where we don't have a priori the QFT principles, like Noether theorem, quantum mechanics, and the concept of symmetry generators.

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I told you about the singularity structure of the stress-energy tensor. But can we determine exactly the correlation function? If the correlation function is on the Riemann sphere, this is enough to fully determine the function of w, so indeed yes. If its on an open set, and we know the boundary conditions, then this is also enough. It turns out that on the upper half plane, there is a physical condition found by Cardy that tells us about these boundary conditions. Hence, by conformal transport, we can in principle determine the exact w dependence in any simply connected domain. Recently I found a way of writing this in complete generality, using a certain type of Hadamard derivatives.

Hadamard differentiability is something that generalises the concepts of differentiability in many-variable calculus to cases of infinite dimensionality. I studied the particular case where we look at derivatives in directions of small conformal transformations; the tangent space is a space of holomorphic functions on a domain A. In this case the derivative can be written as an integral along the boundary of A of the holomorphic function, times a function which can be seen as the Hadamard derivative (an element of the cotangent space). This function is not unique, but in certain situations, we can choose it to be holomorphic outside A -- I call it the holomorphic A-derivative.

Suppose now the space on which our function lives is a space of transformations that are conformal on the points z_i where the local fields are, and on the boundary of the domain D of our conformal field theory. Suppose we choose the domain A to be the Riemann sphere minus a neighbourhood of the point w where the stress-energy tensor is. Then let me define a "relative partition function", a ratio of partition functions on various domains, the domain D and involving some arbitrary domain B inside D. Then the full correlation function with the stress-energy tensor is a conformal derivative of a product of the relative partition function times the correlation function without the stress-energy tensor. This, it turns out, reproduces the correct singularity structure in w, and the correct boundary conditions; in particular, the correct one-point function of the stress-energy tensor. In fact, naturally, this should hold as well on multiply-connected domains.

It is this formula that is reproduced in conformal loop ensembles, and that allows me to identify the stress-energy tensor.

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Some of my results are as follows. I define a "renormalised" random variable that is essentially the indicator function of the event that no CLE loop intersect a given domain boundary -- that this domain is, in a way, separated from the rest, like in this picture. This event has zero measure, because in CLE there are infinitely many loops around every point almost surely. That's why I need to renormalise it, in a prescribed way which I will not talk about. From this renormalised random variable, I can construct the stress-energy tensor. It is obtained by choosing an ellipse for the domain boundary, by taking the second Fourier transform in the angle the ellipse makes with a fixed axis, and by taking the limit where the ellipse is very small, like in this formula. The fact that its an elliptical domain that's we choose related to the fact that the stress-energy tensor is L_{-2} on the identity, in operatorial language. I can also construct the relative partition function: just by naive arguments, it is a ratio of expectations of the renormalised random variable associated to the boundary of the domain D. Then, I prove basically two things: that we obtain the Ward identities and boundary conditions, in the form of the formula involving the conformal derivative; and that the stress-energy tensor transforms correctly, in particular with the Schwarz derivative. I also prove, under certain uniformity conditions, that any object that is "local" with respect to the loops and transforms like the stress-energy tensor, also gives rise to the correct Ward identies and boundary conditions -- it is the stress-energy tensor. This is a principle of universality -- it is useful to allow us to make a closer connection between CFT and underlying statistical models.

So, what I have found is that a local random variable in CLE can be interpreted as the stress-energy tensor; essentially, from this we can recover the QFT structure associated to local conformal invariance from CLE. Indeed, in a next step, I want to derive from this the full Virasoro vertex operator algebra. This construction also gives an interpretation of the central charge beyond the notion of quantum symmetries. Indeed, it arises because of the necessity of renormalising the random variable I told you about. Hence, it comes from the infinity of small loops -- it is essentially a density of small loops, with respect to scale. That it be related to the central extension of the Virasoro algebra can be seen by considering that a negative Virasoro mode "takes away" some space, that is put back with a positive Virasoro mode. The result is that some loops are lost, and this gives rise to the central extension.

Thank to the clear physical meaning of the stress-energy tensor in terms of particles, this construction also should provide a better connection between the quantum emergent objects, and the statistical emergent objects -- essentially, the curves are related to particle trajectories.

Also, it is this construction that led me to study conformal Hadamard derivatives in relation to CFT; another avenue that is to be explored is how much we can develop in this direction, independently of CLE.

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I'd like to finish by mentionning other areas that I have worked on with many collaborators. Recently I studied the entanglement entropy in extended quantum systems, both in the integrable context, and for general 1+1-dimensional quantum field theory. We obtained quite universal low-energy asymptotic results, and I showed that these results were valid for any 1+1-dimensional massive model. I also worked on quantum impurities in steady states out of equilibrium, where we studied electric currents in quantum dots and found general existence results and calculation techniques. I also worked on vertex operator algebras, developing twisted modules, as well as construction techniques which I think will be useful in relating CLE to vertex operator algebras. Then I studied correlation functions in integrable QFT in various situations: finite temperature, non-zero curvature. I focussed on the form factor techniques, extending the known concepts to these situations. Finally I'm also quite interested in the connection between correlation of twist fields in massive free-fermion models and classical integrability.