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Vyacheslav "Slava" Rychkov (Вячеслав Рычков)

Permanent professor, Institut des Hautes Études Scientifiques

MHI-ENS Professor in High-Energy Physics, Ecole Normale Supérieure, Laboratoire de Physique

## What's new

29/6/2021 Navigator functions in the conformal bootstrap

With Marten Reehorst, David Simmons-Duffin, Benoit Sirois, Ning Su, and Balt van Rees, we recently proposed a new tool in the numerical conformal bootstrap which we named the "navigator function". The name comes from the fact that with the new tool one can "sail" in the sea of disallowed points to the bootstrap islands, following the gradient of the navigator function as a compass (while previously to find the islands one had to perform expensive scans). I believe this will allow extending the range of conformal bootstrap applications in quite amazing ways. To know more please see our paper and watch talks by Ning Su or Balt van Rees.

Since a couple of years, my PhD student Jiaxin Qiao and myself were collaborating with Petr Kravchuk on Lorentzian CFT. Our long paper has finally appeared last week. In it we show that Euclidean CFT correlation functions, constructed using the usual Euclidean rules such as convergent OPE, can be analytically continued to Minkowski space, obtaining tempered distributions satisfying Wightman axioms. This result is new: in the past people were happy to accept Wightman axioms as an extra assumption.

In our paper we also review a lot of old literature about the relation of Lorentzian and Euclidean quantum field theory, such as the Osterwalder-Schrader theorem. We also review modern Lorentzian CFT literature. Much of this literature is not on yet on the same rigorous footing as the Euclidean conformal bootstrap, and we point out the most pressing open problems.

2020 POSTDOC SEASON (completed, Junchen Rong will join IHES in Fall 2021) If you wish to come to Paris and work with me, please apply to the following postdoc positions:

The CFT/bootstrap group in Paris includes:

• Permanent researchers: myself, Balt van Rees (Ecole Polytechnique), Miguel Paulos (ENS), Eric Perlmutter (Saclay)

• Postdocs: Marten Reehorst (IHES), Emilio Trevisani, Edoardo Lauria (Polytechnique), Denis Karateev (ENS)

• PhD students: Jiaxin Qiao, Benoit Sirois (IHES), Xiang Zhao (Polytechnique), Zechuan Zheng (ENS)

Another project started in Rome 3 years ago came to an end. In January 2018 I went to a conference on Disordered Systems organized at La Sapienza University (Rome) by Giorgio Parisi and the Cracking the Glass Simons Collaboration. I was an outsider invited to give a review talk about the conformal bootstrap, but I did listen to the other talks. Many of them were about the Parisi-Sourlas supersymmetry and dimensional reduction, or rather why they are absent, in the Random Field Ising and other related models . I previously heard about this problem from my ENS colleague Édouard Brézin, so I got curious and decided to study the literature. Thanks are due to my other ENS colleague Nicolas Sourlas for mentioning a poorly-known paper by John Cardy. After a quick look, it became clear that this is interesting both from CFT and Renormalization Group perspectives, and that something could be done complementary to the existing treatments. I came back to IHES, talked to my postdocs Apratim Kaviraj and Emilio Trevisani, and we set to work.

The first paper devoted to supersymmetric CFT aspects appeared in December 2019, and the second paper, about the RG aspects, is finally out in September 2020. I already discussed the SUSY CFT paper below. The RG paper is much longer and more subtle. It develops a systematic theory of RG stability of the SUSY fixed point. Complexity stems from the fact that two out of three potentially destablizing perturbation classes (susy-writable, non-susy-writable, and susy-null) cannot even be written in the SUSY variables. Yet we managed to prove that two such perturbations, one non-susy-writable and one susy-null, may destablize SUSY below a critical dimension d ≈ 4.2. See the paper for full details.

Emilio Trevisani, myself and my then-PhD student Bernardo Zan discuss the problem of dimensional reduction and Parisi-Sourlas supersymmetry (January 2018, IHES)

26/9/2020 Gentle introduction to rigorous Renormalization Group: a worked fermionic example Conformal bootstrap and Renormalization Group are two complementary, and rival, approaches in the theory of critical phenomena. Now that the conformal bootstrap is well and running, I feel a bit sorry for the non-perturbative RG. Majority of high-energy physics researchers view its most popular current incarnation (Functional Renormalization Group) with suspect, as a bunch of unjustified approximations and truncations. My point of view is that Wilson was right, the strongly coupled fixed points exist in a true mathematical sense, and the problem is that we simply haven't found yet a numerically stable implementation of Wilson's vision. Where to start? Mathematicians proved many theorems using the renormalization group, so they must know how to control it rigorously. It would seem natural to turn to them for inspiration, but the mathematical RG papers look so daunting... See my talk at a Princeton workshop about the dream to understand better the rigorous RG and how to make it useful for practical calculations.

This dream started to come true in this paper, written in collaboration with two remarkable mathematical physicists, Alessandro Giuliani and Vieri Mastropietro. Alessandro and I met in 2017 at a conference at the Accademia dei Lincei (Rome). Once Alessandro told me that he can control rigorously Wilsonian RG for fermions, I got excited about the possibility to use his techniques to construct the simplest non-gaussian fermionic fixed point, in a model of symplectic fermions with a long-range kinetic term and a quartic interaction (a fermionic analogue of the bosonic long-range model I studied earlier with Connor Behan, Leonardo Rastelli and Bernardo Zan).

It was clear from the start to Alessandro and Vieri that this construction should be possible. I am the main culprit that it took 3 years to complete: I had to master all their tricks to the extent as to be able to review them for my hep-th colleagues (pedagogy was one of our goals) and I could not initially fathom the "Gallavotti-Niccolò" tree expansion used by the Italian school in the theory of fermionic fixed point. The final version of our paper implements instead a Banach contraction argument, arguably more "Wilsonian" in spirit (while the tree construction is in an appendix and in retrospect it does look simpler).

I am quite proud of this paper. We have a nice result, written in a self-contained way. It will certainly stimulate further work, e.g. on the general theory of fermionic fixed points, and on checking various common lore properties of RG fixed points (such as conformal invariance). It would be nice to have a similar pedagogical exposition for bosonic fixed points. Finally, talking about dreams, it would be nice to have a (computer-assisted) construction of the fixed point of the 3d Ising model. This would be similar to how my late IHES colleague Oscar Lanford constructed the fixed point for the Feigenbaum period doubling back in 1982. I have some ideas in this direction.

2/4/2020 Open Problems in Conformal Bootstrap website started. Please send your problems you want to be posted on this site.

1/2/2020 CARGESE SCHOOL. This summer I am co-organizing a theoretical physics summer school "Old and New Frontiers in Quantum Field Theory" in Cargèse, June 15-27 2020. The school will include many excellent speakers, both on the wise side and from the up-and-coming generations. The school is open to PhD students and young postdocs. Registration deadline is on March 31, 2020. UPDATE 6/2/2020: Application form was down, now repaired, please apply. Deadline extended till March 31. UPDATE: This school was eventually postponed to 2021 due to Covid.

1/2/2020 Conformal bootstrap and liquid Helium. I first learned about an 8-standard deviations discrepancy between the best experimental measurement of the critical exponent α in the superfluid transition of Helium-4, and its best theoretical determination, from this paper by Ettore Vicari. My dream since 2014 was that one day conformal bootstrap will tell who's right and who's wrong? Once Kos, Poland and Simmons-Duffin discovered the O(N) archipelago, the problem appeared within reach: just shrink the O(2) island until it excludes the Monte Carlo or the experiment. But somehow the O(2) island was not shrinking as fast as the Ising island. Finally, last December a 7-strong collaboration (Chester, Landry, Liu, Poland, Simmons-Duffin, Su, Vichi) cracked this problem. In the plot below, the vertical axis is directly related to the critical exponent α, the other axes being two other relevant operators of the O(2) model CFT. Experiment gives the brown region, earlier theory (Monte Carlo) the green parallelepiped. The tiny blue region is the conformal bootstrap, which confirms Monte Carlo and excludes the experiment. (At least) three key ingredients went into this work:

• Shrinking the island from all three directions, i.e. with 3 relevant operators (2 operators gave weaker results).

• A new clever scanning algorithm over the OPE coefficient ratios

• Major improvements in the SDPB code, increasing speed and parallelization, and reducing memory usage.

See their paper for the full details, and my commentary for another introduction to this beautiful result. Congratulations!

Last but not least, I am impressed by the work of Campostrini, Hasenbusch, Pelissetto, and Vicari which first pointed out this anomaly back in 2006 using Monte Carlo and High Temperature expansion, and stood by their result for a decade. I heard they had to face some serious criticisms, while some other theorists were happy to fiddle with their calculations until they perfectly agreed with the experiment.

27/1/2020 People and funding. CFT and bootstrap have been rapidly developing in recent years, with applications to such disparate subjects as statistical physics, quantum gravity, or supersymmetric quantum field theory. I'm very happy that Paris now hosts many permanent researchers working in this direction. Two more senior people are joining Paris CFT community in 2020. Balt van Rees moved from Durham University to become a professor of mathematical physics at the Ecole Polytechnique, the most prestigious Grande Ecole university in France. Eric Perlmutter is moving from Caltech to a permanent position at the famous Saclay Institute of Theoretical Physics (IPHT). Both these institutions are located in the south of Paris, a 10-minute drive from my own Institut des Hautes Etudes Scientfiiques (IHES), and I often go there for seminars.

This years postdoc hiring season also brought some good news. Yifei He is working on the applications of conformal bootstrap to the Potts model and percolation in 2d, as well as on S-matrix bootstrap, and she will stay in Paris on a postdoctoral fellowship from the Philippe Meyer Theoretical Physics institute at the Ecole Normale Superieure. Denis Karateev, a seasoned CFT warrior, will also join the Ecole Normale Superieure group. Marten Reehorst, graduating PhD student of Alessandro Vichi, will start a postdoc position at the IHES. With such a field of youngsters, the coming years will bring more Parisian CFT blasts!

In the meantime, the Simons bootstrap collaboration for the nonperturbative bootstrap has been extended for 3 more years, until Sep 2023.

27/1/2020 Parisi-Sourlas Supersymmetry. Since a couple of years, I have been collaborating with my two postdocs Apratim Kaviraj and Emilio Trevisani on the puzzle of the Random Field Ising model and Parisi-Sourlas supersymmetry. (Nicolas Sourlas is my colleague at ENS and we shop at the same Parisian market). This 40-year old story concerns the phase transition in the Ising model with magnetic impurities. By one kind of lore, this leads to a non-unitary supersymmetric CFT with scalar supercharges (Parisi-Sourlas supersymmetry). By further lore, Parisi- Sourlas SCFT in D dimensions is equivalent to a non-SUSY CFT in D-2 dimensions (dimensional reduction). It is known that this story is not fully correct, and in our project we are trying to understand why. Last December we put out a paper which puts on firmer ground the dimensional reduction part of the story, using non-perturbative CFT methods. We are now working on the second paper which aims to clarify the existence of the SUSY fixed point.

27/1/2020 Distributions in CFT. Petr Kravchuk, my PhD student Jiaxin Qiao, and myself have been working on deriving Lorentzian physics from Euclidean CFT axioms (see the 12/1/2020 post below). For example, we would like to show that CFT 4pt functions in Lorentzian signature satisfy Wightman axioms (in particular that they are tempered distributions). We have accumulated a lot of material and it's time to start publishing. Our first paper considers the simplest result: the distributional properties of CFT correlation functions in cross-ratio space on the boundary of the region of convergence. The next joint papers will consider CFT 4pt functions in flat Lorentzian space and on the Lorentzian cylinder. Jiaxin Qiao has also worked out an interesting classification of convergent OPE channels for CFT four-point functions, which he should publish soon.

12/1/2020 IS CATEGORY THEORY USEFUL FOR PHYSICS? "Of course!" - will say the selected wise. However, until recently I belonged to the great majority of people who believed that this subject was just "abstract nonsense". Things changed in a recent paper which I completed in collaboration with a Princeton PhD student Damon Binder. In this project we tried to clarify the pesky notion of O(N) symmetry when N is not an integer, and thus O(N) group does not exist. In theoretical physics, we love to perform analytic continuations in parameters which nominally have to be integers, and O(N) is one such example. Other example is the analytic continuation in the number of dimensions D. Even more excitingly, this continuation is not just a formal trick: models corresponding to the intermediate values of N do exist and are well-defined non-perturbatively (they are called loop models). Damon and mine's explanation for the non-integer O(N) symmetry is that it is a categorical symmetry, controlled not by a group or an algebra but by a Deligne category' - a symmetric tensor category defined by a famous Belgian mathematician Pierre Deligne (formerly at IHES, now at IAS) in 2004.

12/1/2020 "I remember telling someone that I wanted to learn about elementary particles by studying boiling water." (Alexander Polyakov) In perturbative QFT, we can go back and forth between Euclidean and Lorentzian Feynman integrals - this is called Wick rotation. Under certain conditions, this is also true non-perturbatively: correlation functions can be analytically continued from Euclidean to Lorentzian and back. Even if one is eventually interested in Euclidean physics, the existence of this continuation allows to explore typically "Lorentzian" constraints such as e.g. causality. Recently I gave four lectures about such analytic continuations in Conformal Field Theories. This is an ongoing project with IAS postdoc Petr Kravchuk and my PhD student Jiaxin Qiao, and the papers should appear soon.

5/2/2019 About a year ago I gave the 27th Occam lecture at Merton college, Oxford, entitled: "Reductionism vs bootstrap: are things big always made of things elementary". The video recording is here:

2018 POSTDOC SEASON - completed If you wish to come to Paris and work with me, you might have a chance of doing so by applying to the following postdoc positions. Please inform me by email if you believe your profile is close to my research interests. Notice that I split my time equally between the IHES and the ENS:

Please apply also at https://academicjobsonline.org/ajo/jobs/12194 for postdoc positions of the Simons Bootstrap Collaboration (6 positions available). I am not hiring this year through the Simons grant.

The CFT/bootstrap group in Paris consists of: Permanent researchers: myself and Miguel Paulos; Postdocs: Mikhail Isachenkov, Apratim Kaviraj, Emilio Trevisani; several PhD and Master students.

9/11/2018 In my talk at the annual Bootstrap Collaboration meeting (the Simons Foundation, New York), I talked about long-range interactions and structural phase transitions as possible bootstrap targets. The talk will be put online at some point, in the meantime here are the slides.

9/11/2018 I'd like to say a few words about a recent paper by Andy Stergiou and myself. Epsilon-expansion is a classic subject on which there was a lot in the 70s and 80s, and it's even now in use. Of course now there are alternative conformal bootstrap techniques, which are nonperturbative and sometimes give much more powerful results. But the role of the epsilon-expansion is still significant - it provides us with conjectural theories to shoot for with the bootstrap. Andy and I took a fresh look at the epsilon-expansion, trying to get results valid in full generality (without any symmetry assumptions). Only a handful of such results are known. We got a new such result, which limits any fixed point to lie in a known compact region of coupling space. We prove (lambda_ijkl)^2 <= (N/8)epsilon^2, where N is the number of scalar fields, and (lambda_ijkl)^2 is the square of the quartic coupling tensor at the fixed point. Writing this paper also gave us a chance to review a lot of old beautiful literature about the epsilon expansion.

31/10/2018 Long read: Walking, First-Order Phase Transitions, and Complex CFTs. part I & part II It's high time to report on this recent project I completed a couple of months ago with Victor Gorbenko and my PhD student Bernardo Zan. We linked two hitherto disconnected phenomena - walking in 4D gauge theories and weakly first-order phase transitions in lattice models such as the 2D Potts model. Kaplan, Lee, Son and Stephanov proposed in 2009 that QCD conformal window terminates because the Banks-Zaks fixed point collides with an as yet unidentified fixed point they called QCD*', after which the fixed points go to the complex plane'. I was not convinced about their reasoning and examples, either exotic (Efimov physics), or inappropriate (BKT phase transition), or holographic (raising doubts that everything is just a large N artifact). However, in the summer of 2016 I learned by chance that the 2D Potts model realizes the fixed point annihilation scenario (the example missed by Kaplan et al). So I got converted, and then joined forces with Victor and Bernardo to convert the rest of the world.

In part I we set up general theory of walking RG flows, and clarify numerous related confusions lingering in high energy physics due to the absence of reliable lattice data for many-flavor QCD. One such confusion is the light pseudodilaton, urban legend based on dodgy theoretical arguments and overly optimistic misinterpretations of the meager lattice data. We also introduced a novel concept of complex CFT', to clarify another confusion related to the RG fixed points at complex couplings. Some authors seem to think that these fixed points are unphysical e.g. referring to them as unstable'. We argued instead that these fixed points are bona fide nonperturbative CFTs with operator product expansion and other axioms, except that they are non-unitary in a rather radical way, with complex scaling dimensions (and the central charge). Thinking in terms of these CFTs we discovered a non-perturbative criterion for the existence of a walking RG flow - imaginary part of the scaling dimension of a certain operator has to be small.

In part II we treated in detail the 2D Potts model - a computable example where analytic continuation can be done thanks to the exact solution. While Q-state Potts models with Q=2,3 are well known (these are just unitary minimal models), here we needed to understand statistical physics literature on the Q-state Potts model with non-integer Q. That this model can be non-perturbatively defined may be a surprise in itself to a high-energy physicist, but it can, in terms of clusters and loops. It took us two years to decipher the stat-phys jargon, but the fruits we harvested were worth it. We were able to exhibit complex CFTs, compute characteristics of walking RG flow perturbing around them, and make predictions for lattice simulations of drifting critical exponents'. Weak first order phase transitions seem to appear in the theory of deconfined quantum criticality (quantum cond-mat), so our work was timely and will lead to further results.

Our second paper will appear in SciPost. Do you know about their peer-witnessed refereeing system?

13/7/2018 About 4 years ago, Filip Kos, David Poland and David Simmons-Duffin discovered the 3d Ising island by means of the multi-correlator conformal bootstrap analysis of 3d CFTs having Z2 symmetry. This result opened a new era in the conformal bootstrap, and in particular has been extended to O(N) models, producing the O(N) archipelago. I am happy to announce that a new bootstrap island has just been discovered by Junchen Rong and Ning Su, which corresponds to the 3d CFT describing a supersymmetric version of the Ising model. I heard that the same result has been obtained by David Poland and collaborators, whose paper should appear soon.

28/5/2018 With conformal bootstrap revival turning 10 years old, David Poland, Alessandro Vichi and myself wrote a long review article "The Conformal Bootstrap: Numerical Techniques and Applications" which will hopefully appear in the Reviews of Modern Physics but is already available online: https://arxiv.org/abs/1805.04405. Our main goal was to review the euclidean CFT, numerical techniques, and applications to the main CFT universality classes discussed in statistical, condensed matter and high energy physics. We could not do justice to analytical techniques and applications to SUSY theories which are under active development. This will have to wait for their own reviewers.

8/2/2018 Please apply at http://bootstrapcollaboration.com/bootstrap2018/school to the Bootstrap PhD school in Caltech (USA), July 8-14, 2018.

23/1/2018 I started teaching my master course at the ENS on "Topics in strongly coupled QFT". The course page is here.

31/12/2017 One of the subjects I've been focussing on recently is the Hamiltonian Truncation - a Hamiltonian approach to solving strongly coupled dynamics of quantum field theories, which takes inspiration of the Rayleigh-Ritz method in quantum mechanics. At present, most applications of this method have been in 1+1 dimensional field theories, but I am convinced that this method has a lot of unexplored potential and may one day lead to a way of solving even four-dimensional quantum field theories, such as QCD, which is radically different from the currently used lattice Monte Carlo techniques.

26/10/2017 THESE POSITIONS HAVE BEEN FILLED: POSTDOC OPENINGS, starting date Fall 2018

I have been recently appointed professor at the Institut des Hautes Études Scientifiques, a paradisiac home for theoretical physicists and mathematicians in Bures-sur-Yvette, a short metro ride to the hills south of Paris. I am also going to continue part-time affiliation with the Ecole Normale Supérieure in the center Paris.

If you are interested in working with me, please apply to these postdoc position openings at the IHES and the ENS:

There are also postdoc opening at other nodes of the Simons bootstrap collaboration:

11/8/2017 POSITION FILLED The Simons bootstrap collaboration is .

27/2/2017 Bootstrap in the news. Almost simultaneously, two popular articles about the conformal bootstrap appeared in New Scientist and Quanta. I like better the Quanta article, by Natalie Wolchover, which dives deeper into the ideas behind the bootstrap.

The connection with the early S-matrix bootstrap by Geoffrey Chew came out very well, and quotes by Sasha Polyakov, who has been a great inspiration all along, are great to see. Polyakov admits that he did not believe the first results coming out of the post-2008 bootstrap revival: "I thought originally that there is some mistake there,” This is very true - I remember his criticisms and incredulity when I came to Princeton to give a seminar at the IAS in 2008. But this was also a positive influence of sorts - to work harder and to convince my former PhD advisor was a unique challenge :)

Unfortunately, both articles do have some inaccuracies in the history of the bootstrap. In particular the contribution of the Italian group (Ferrara, Gatto, Grillo) is completely absent, although I did stress it in my interviews. The New Scientist piece, by Gabriel Popkin, did not mention my collaborator Riccardo Rattazzi by name; it is also not quite correct in describing the sequence of events leading to the application of bootstrap methods to the 3d Ising model.

Next time I meet Subir Sachdev, I have to ask him what he meant when he said that bootstrap "is like going from a Mercedes to a Roll-Royce". Is this a positive or a negative comment?

5/1/2017 POSITION FILLED I'm inviting applications for a PhD position at the Physics Department of Ecole Normale Supérieure (Paris), starting in the Fall of 2017, funded by the Simons Collaboration on Non-perturbative bootstrap. The application should be made via this AJO ad. For further details on the research project see here.

1/12/2016 POSITION FILLED I'm inviting applications for a postdoctoral position at the Physics Department of Ecole Normale Supérieure (Paris), starting in the Fall of 2017. The hired postdoctoral fellow will become a member of the Simons Collaboration on Non-perturbative bootstrap. The application should be made via the joint collaboration AJO ad.

4/12/2016 In my field (hep-th) PRLs have not been particularly important, compared say to cond-mat where it's been a feature for decades. This is changing now, as more and more high-energy physicists are trying to squeeze their work into the two-column, 4 page format. Presumably, visibility is the primary goal. It's questionable if this goal is achieved, and even if it is, then only at the expense of sacrificing the quality of the presentation. Inevitably details get omitted when writing a PRL, making the work harder to follow, reconstruct and reproduce. I spend a lot of time studying cond-mat/stat-phys literature and I think the PRL part of it is messed up and impenetrable. I believe that the hep-th literature is, in comparison, very well-documented and self-sufficient. Best people are taking time to write long papers and explain their ideas in full. We should cherish this tradition and keep it the same for the future generations. If somebody circulated a petition asking to never submit to PRL, I would be ready to sign it. Stop PRL-mania!

Disclaimer I did experiment with publishing in PRL in the past - sins of youth. Too bad one of my collaborators wants to submit our upcoming work there. Perhaps he will read this post and change his opinion? :)

5/11/2016 I'm inviting applications for a postdoctoral position at the Physics Department of Ecole Normale Supérieure (Paris), starting in the Fall of 2017. The hired postdoctoral fellow will become a member of the Simons Collaboration on Non-perturbative bootstrap. The application should be made via the joint collaboration AJO ad.

31/08/2016 The Simons Foundation announces its support of the Collaboration on Non-perturbative bootstrap.

16/03/2016 I slacked out on reporting the news, but today's paper 1603.04436 by Kos, Poland, Simmons-Duffin and Vichi is too exciting not to be mentioned. Using the multiple correlator bootstrap, they raise the precision on the 3d Ising model dimensions by another order of magnitude. See also this video for a bird's eye view of the whopping precision improvement achieved in the last two years using the multiple correlator bootstrap. Bravo Alessandro, David, David, and Filip!

1/09/2015 The long-range Ising model joins the list of theories known to be conformally invariant at their critical point. This paves the way to its future analysis by the conformal bootstrap techniques.

22/08/2015

Some months ago, I put a paper out, based on the results obtained under my guidance by Zhong Ming Tan, an outstanding student from the ENS master program, during his two month internship at CERN. I was invited to submit the paper to the newly created Fast Track Communication section of Journal of Physics A. The paper was published after an impressively fast refereeing process (although it included three referee reports). It was also featured in the Publisher's Pick section of the journal's website. In case you don't have access, I copy our featured comments below:

What prompted you to pursue this field of research?

We wanted to bridge a gap between two famous subjects: the epsilon-expansion first used to study the critical points of scalar field theories by Wilson and Fisher in 1972, and conformal symmetry whose presence in these critical points was first pointed out by A M Polyakov in 1970. The importance of conformal symmetry in two-dimensional critical phenomena is long understood, while its utility in dimensions larger than 2 has been recognized only recently. Last year, the world's best determinations of the critical exponents in the 3d Ising model have been obtained using conformal field theory. Much of these recent developments have been numerical. We wanted to find an analytically solvable example, and this naturally led us to study conformal invariance of the epsilon-expansion.

What is this latest paper all about?

One nice feature about the Wilson–Fisher critical points is that they are weakly coupled—their properties allow for a systematic expansion in epsilon, the deviation of the space dimensionality from 4 (hence the name of the method—the epsilon-expansion). In the traditional approach, the coefficients of these series are computed using perturbative field theory, by evaluating Feynman diagrams. In our paper we present an alternative way to compute the leading coefficients, which does not use the Feynman diagrams at all. Instead, our method exploits the consequences of the conformal symmetry of the critical point, and in particular one phenomenon which we call 'multiplet recombination'. It's a conformal field theoretic analogue of the Higgs mechanism familiar from particle physics. We were quite surprised to find out that it leads to such strong constraints on the structure of the epsilon-expansion series.

What do you plan to do next?

It would be very interesting to explore the consequences of 'multiplet recombination' for the conformal field theories other than the Wilson–Fisher fixed point, and to extend our calculations to higher-order terms of the epsilon-expansion series. The first of these tasks should be relatively straightforward, while the second will likely require significant new ideas.

27-28/07/2015 I gave 3 introductory lectures about the conformal bootstrap at a PhD school in DESY:

1. Introduction and the current status of the 3d Ising model project.

2. The structure of the conformal blocks and the convergence of the OPE.

3. Bootstrap in the Minkowski space and the large spin asymptotics of the CFT spectrum.

6/02/2015 I'd like to report a new dramatic success of the conformal bootstrap program, appeared in a paper of David Simmons-Duffin, building on his earlier work with Filip Kos and David Poland. He performed a joint analysis of three correlation functions <ssss>, <ssee>, <eeee> where s and e are the two primary scalars of the critical 3d Ising model. Conformal bootstrap proves, rigorously, that the dimensions of these operators must lie in a tiny sliver-like region in the shown plot. The previous best Monte-Carlo determinations of the same dimensions are the dashed rectangle. The best RG determination from the epsilon-expansion is two orders of magnitude less accurate. Notice how much more precise conformal bootstrap has become compared to those old techniques.

10/12/2014 I'm getting more and more interested in the development and application of Hamiltonian truncation methods in QFT. After last month's paper on TCSA in d>2, today Lorenzo Vitale from EPFL and myself put out a paper where we study the \phi^4 theory in 2 dimensions using the Fock-space Hamiltonian truncation. The method is so simple that it should be taught in introductory QFT classes. The most technical part is a renormalization procedure which allows to improve the accuracy of the method. The python code that we used to carry out the computations was so clean (thanks to Lorenzo) that we included it with the submission. Try it out for yourself!

On the left: the spectrum of the \phi^4 theory as a function of the quartic coupling. Notice the phase transition from the Z2-unbroken to the Z2-broken phase for g~3.

11/11/2014 It's been a busy period of life since my last post. I've been mostly busy working with Matthijs Hogervorst and Balt van Rees on generalizing Truncated Conformal Space Approach from d=2 to d>2 spacetime dimensions. Our first paper has been out in September, and after a short scuffle with the Phys.Rev.D editor and the referees who did not like our intentionally provocative title (were they afraid that we will soon put them out of business? they should! :)) will soon appear in print.

I've also recently attended an interesting meeting on the conformal bootstrap organized by the Princeton theory group. It was nice to see everyone so genuinely interested in what we are doing.

1/5/2014 Our 3d Ising work has been honored by an appreciating comment by one of the pioneers in the field of critical phenomena - Leo Kadanoff.

19/3/2014 Big day for the 3d Ising collaboration - our paper: "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents" has appeared on the arxiv. With this work, the bootstrap study of the critical 3d Ising model is entering the precision stage. We beat the precision world records for several previously considered exponents. We also determine precisely several OPE coefficients, about which nothing was known before. We also observe mysterious operator decouplings taking place, which may be a hint of the exact solvability of the theory.

Feb 2014 I'm attending the KITP program "New methods in non-perturbative quantum field theory", where the bootstrap is featured prominently. Overview talks on the subject were given by David Poland and myself. Leonardo Rastelli and Balt van Rees talked about their recent work on the bootstrap in SUSY theories. Finally, David Simmons-Duffin and myself gave talks about the ongoing project of solving the 3d Ising model at criticality (the second paper in the series is due soon). All the talks of the program are recorded.

Sep 2013 When you do dimensional regularization, is it just a trick or do fractional dimensions make mathematical sense? Can quantum field theory in fractional spacetime dimensions be non-perturbatively defined? In our recent paper we do exactly that, for conformal field theories. The idea is very simple: four point function in a conformal field theory is fixed up to a function g(u,v) of two conformal cross ratios. It's not necessary to think about what it means to have a vector with a fractional number of components, the number of cross-ratios - two - is always the same. In the paper we combined this idea with the conformal bootstrap techniques to study the famous Wilson-Fisher family of fixed points, which is supposed to interpolate between the free 4D scalar theory and critical 2D Ising model, by way of critical Ising model in D=3. Normally this family is studied by a perturbative method called the epsilon-expansion technique. Our method is in agreement, but is more precise, as the following plot shows. Red bands - predictions for the leading critical exponents using the epsilon-expansion. Black dots - our results.

6/7/2013 This week I was lecturing about "Phases of Quantum Field Theory" at the Italian Ph.D. school of theoretical physics, newly founded by my friends and collaborators Roberto Contino, Enrico Trincherini and Andrea Wulzer. My last lecture was about the Coleman-Mandula theorem, a surprisingly poorly documented result, given its fundamental status. The only detailed expositions of interest are the original 1967 paper of Coleman and Mandula, and an Appendix in volume 3 of Weinberg's QFT series, which follows the same logic spelling out some missing steps. Both are exceedingly formal and difficult to follow. Is the rigmarole really worth it? To quote from Coleman-Mandula: "Although [our proof] at times attains mathematical levels of obscurity, we make no claims for corresponding standards of rigor". In my opinion, the theorem can be explain physically and simply in a two-hour blackboard lecture, and that's what I did in Padova. I hope to write this up sooner or later.

24/6/2013 A talk about bootstrap at "Strings 2013" in Seoul, South Korea. Here is the link to the video and slides.

18/6/2013 I gave a review talk about the bootstrap program for CFTs at the "String-Math 2013" conference at Stony Brook to a mixed audience of theoretical physicists and mathematicians. Here are the links to the video and slides.

21-31/5/2013 "Back to the Bootstrap 3", the third workshop in the series, has just finished at CERN Theory Division. It was incredible fun and a chance to meet all friends, collaborators, and likeminded researchers working on strongly coupled quantum field theories.

18/5/2013 This year, new developments abound in the theory of d-dimensional conformal blocks. First Matthijs Hogervorst and I talked the magic rho-coordinate (see the post below). Then there was intriguing work by Liam Fitzpatrick, Jared Kaplan and David Poland about the large spacetime dimension limit. Finally, a few days ago Matthijs, Hugh Osborn and I put out a paper presenting, among other things, an extremely efficient algorithms for the numerical evaluation of conformal blocks.

5/3/2013 Conformal blocks are mysterious special functions. In today's paper with my student Matthijs Hogervorst we are trying to demystify them a bit. Conformal blocks are, first and foremost, sums of contributions of radial quantization states to a matrix element computing a four point function. This leads to power series representations with Gegenbauer polynomial coefficients. Convergence of these series can be optimized by a judicious choice of the radial quantization origin. We argue that the best choice is to insert the operators symmetrically. We analyze in detail the resulting "ρ-series" and show that it converges much more rapidly than for the commonly used variable z. Taking only the first term already approximates the block with a few % accuracy.

We believe the "ρ-series" will be very useful for the conformal bootstrap. Most excitingly, we use them to derive analytically some bootstrap bounds whose existence was previously found numerically.

13/12/2012 I finished a four-lecture PhD course at EPFL about Conformal Field Theory techniques in D>=3 dimensions. The lecture notes (65 pp) can be found here.

26/11/2012 I put out a paper with my student Axel Orgogozo about computing the S parameter in composite Higgs models.

16/11/2012 I gave a review talk in Padova about the current status and outlook of the bootstrap program. The slides can be found here.

13/11/2012 Today, my friends and collaborators Sheer El-Showk and Miguel Paulos put out an exciting paper: Bootstrapping Conformal Field Theories with the Extremal Functional Method. Let's see if I can summarize it here without going into much detail. Most of the work on the conformal bootstrap has so far been going along two directions:

• Carving out the space of CFTs, borrowing the name of a paper by David Poland, David Simmons-Duffin and Alessandro Vichi, and consists in deriving all sorts of a priori universal bounds on operator spectrum, OPE coefficients, central charges etc.

• Hunting for kinks, i.e. special points in these bounds. So far three such kinks have been identified: one in D=2, one in D=3, and one in the D=4 SUSY case (see Fig. 7 in the "Carving out..." paper). The positions of the D=2 and D=3 kinks correspond to the operator dimensions of the critical Ising model, while the interpretation of the SUSY kink is unclear.

Miguel and Sheer's paper adds a new direction to the list. They show that the theories saturating the bouns are so tightly constrained, that full spectrum can be recovered. In practice "full" means a couple dozen low-lying operators in the OPE of the lowest dimension scalar with itself, at which point the method runs out of steam due to a loss of numerical precision. They have demonstrated how this works for the 2D Ising model case, where it gives results in good agreement with the known exact solution, both for the operator dimensions and for the OPE coefficients.

17/10/2012 There was an interesting paper today related to my work, so I'd want to mention it here. Pedro Liendo, Leonardo Rastelli and Balt van Rees add an exciting new twist to the bootstrap program. They propose to study bootstrap in a half-space with conformally invariant boundary conditions, rather than in full space. The point if that the two point function in half-space is morally equivalent but somewhat simpler than the four point function in full space that one usually studies. Conformal blocks in half-space are also simpler and explicitly known in any number of spacetime dimensions. This allows the Stony Brook team to push through a number of analytic analyses which are currently too complicated to do in full space. E.g. they solve bootstrap to the leading order in 4-epsilon dimensions. They analyze the conformal block decomposition for the stress tensor two point function. They also derive quite a few bounds under the assumption that the boundary preserves positivity, which they conjecture to be true for the extraordinary (Z2-breaking) and special (Neumann) boundary condition.

3/09/2012 In QFT we are often resigned to dealing with perturbative series which are only asymptotic rather than convergent. The irreducible theoretical error is related to nonperturbative effects, which cannot be captured by summing Feynman diagrams. The situation is better in Conformal Field Theory, as I discuss in my last paper with Duccio Pappadopulo and Riccardo Rattazzi. There, Operator Product Expansion and conformal block decomposition provide expansions for correlation functions which do converge, and even very rapidly (exponentially fast). This technical result can be proven in complete generality. It is important for justifying the conformal bootstrap calculations.

29/03/2012 "Solving 3D Ising Model with Conformal Bootstrap", a paper I've been working on for the last six months with Sheer El-Showk, Miguel Paulos, David Poland, David Simmons-Duffin and Alessandro Vichi, appeared on the Arxiv.

This is the first paper of a project whose long-term goal is to solve the critical 3D Ising model using conformal field theory techniques. Below's one plot, showing how the 3D Ising lives at a corner point on the boundary of the parameter space allowed by conformal bootstrap. See the paper for the zoom of the dashed rectangle region, and for the plots showing impact of various assumptions about the gaps in the operator spectrum. While we have not solved the 3D Ising model yet, we have definitely cornered it!

21/02/2012 Today I begin a six-lecture PhD course on "Conformal field theory techniques in D>=3 dimensions". I will talk about the physical foundations of conformal symmetry, relation between scale and conformal invariance, constraints imposed by conformality on the dimensions of local operators and their correlation functions. I will also discuss conformal blocks and their uses in the conformal bootstrap program I am developing since 2008. I will finish by presenting some brand-new results about 3D Ising model obtained via this approach.

7/02/2012 Dispersion relations are fun tools in quantum field theory - it's amazing how much one can learn just from analyticity and unitarity of the S-matrix. In my today's paper with Adam Falkowski and Alfredo Urbano, I use this technique to derive a general result about the Higgs boson coupling to the W and Z bosons.

30/11/2011 Komargodski-Schwimmer proof of the a-theorem is definitely a highlight of the year. In October-November I gave a PhD course about it, and today I presented the proof to the interested colleagues in a journal club at the Institut Henri Poincaré.

16/11/2011 LHC has not found the Higgs boson yet, and perhaps it does not exist. If no Higgs then what? My today's paper (the first one with my PhD student Axel Orgogozo) is about strongly coupled, Higgsless, models of Electroweak Symmetry Breaking.

10/11/2011 In a small research manifesto based on two recent talks, I argue that one can compute 3D critical exponents using conformal field theory techniques. The proposed computation (which will still take some time to perform) is mathematically well defined, unlike standard methods based on the epsilon-expansion which manipulate divergent series.

25/10/2011 My friend and colleague Roberto Contino (Roma La Sapienza) will hold an invited professorhsip of Ecole Normale Supérieure for a month.

1/10/2011 Alfredo Urbano joins our lab as a postdoc. He has been hired thanks to my project "Radiation problem in Transplanckian scattering" supported by the UPMC program Emergence 2011.

30/09/2011 A new step forward for the bootstrap program in my today's paper "Spinning Conformal Blocks" with Miguel Costa (Porto), Joao Penedones (Perimeter) and David Poland (IAS). For the first time, we are able to compute conformal blocks for correlators of vector and tensor fields.

23/09/2011 David Poland (IAS), David Simmons-Duffin (Harvard) and my former student Alessandro Vichi (Berkeley) put out an interesting paper with a catchy title "Carving Out the Space of 4D CFTs". In this paper they develop to perfection the art of extracting bounds on operator dimensions in 4D CFTs (with or without extended symmetry, including SUSY) from conformal bootstrap. The field has come a long way from the first paper on this subject which I wrote back in 2008.

19/09/2011 I gave a talk at the "Hierarchies and Symmetries" workshop explaining why I think it should be possible to solve the 3D Ising model at criticality by conformal bootstrap methods which I have been developing over a number of years.

14/09/2011 First day of teaching

06/09/2011 I put on the arXiv the writeup of my plenary talk "EWSB Theory on the Eve of Higgs Boson Exclusion/Discovery" at EPS HEP 2011 conference in Grenoble this summer. What should we conclude from the fact that new physics has not been seen in the first inverse femtobarn of LHC data? Read the proceedings to learn my opinion...

30/08-04/09/2011 In the mountains (Pralognan-la-Vanoise).

26/08/2011 Attended a really interesting seminar by Juan Maldacena at CERN today. He is trying to extend the Coleman-Mandula theorem to interacting CFTs (i.e. to prove that there are no interacting CFTs with conserved higher spin currents). We had exactly the same urge with Riccardo Rattazzi earlier this year but we got only about 30% as far as Juan did.

24/08/2011 "Conformal field theories as building blocks of Nature". Colloquium at the CERN Theory division. Trying to argue, among other things, that we should move on from AdS/CFT to make progress...

18/07/2011 My new paper "Spinning Conformal Correlators" (with Miguel Costa, João Penedones and David Poland) appeared on the arXiv. If you ever worked in Conformal Field Theory, you must know that things are simple for scalars, but quickly become unwieldy for fields of nonzero spin. You must have also heard about the old idea of Dirac to use the auxiliary 6D space to make 4D conformal symmetry manifest. In this paper, we put this idea to good use in bringing the spinning correlators under control. Basically, our formalism makes computations for nonzero spin as easy as they are for scalars. This is neat, and will also be indispensable for the project of working out conformal blocks for correlators of nonzero spin (work in progress with the same collaborators).

14/07/2011 Today I gave a talk at KITP, Santa Barbara, about my work with Riccardo Rattazzi and Alessandro Vichi on constraining the Conformal Technicolor scenario, proposed back in 2004 by Markus Luty (who was in the audience) and Takemichi Okui.