C CENTRE R DE RECHERCHES Le Bulletin M MATHÉMATIQUES Printemps/Spring 2015 — Volume 21, No 1 — Le Centre de recherches mathématiques A conversation with Marco Bertola (Concordia), Robert Bran- type IIb string theory on AdS 5, and N = 4 Super Yang–Mills denberger (McGill), John Harnad (Concordia) and Johannes theory. Walcher (McGill) on December 8, 2014. Integrability, which is our third theme, has also been play- ing a role in mathematical investigations of string theory Bulletin: Can you elaborate on the theme of the semester? for a long time: its role in the AdS/CFT correspondence J. Walcher [JW]: Our semester has a 3-pronged theme: it’s emerged around 2004, and has been playing an increasing called AdS/CFT, holography and integrability, and I’ll start role in the quantitative development of the correspondence. trying to explain what AdS/CFT is. AdS/CFT stands for That is what our semester is about: it’s the intersection of the Anti–de Sitter/Conformal Field Theory correspondence. It AdS/CFT correspondence as a holographic duality, and inte- was the last of the major dualities discovered in the wake of grable methods as far as they are relevant to the AdS/CFT the second super string revolution of the mid-1990s, and it correspondence. plays a rather special role in the web of dualities. First of all it’s not a duality between two different string theories or two I will let Robert take it from here. different field theories, but rather it involves an equivalence RB: I will start from the physicist’s point of view, and in par- between, on the one side, a theory of quantum gravity, string ticular from the point of view of someone interested in gravity theory, and on the other side a more standard, although very and cosmology. In gravity and cosmology we had for a 100 interesting, quantum field theory. And the dictionary between years basic problems: our theories predict singularities, and these two has the feature of being holographic. although singularities may be interesting from a mathematical Bulletin: Can you explain what holographic means? point of view, they are unphysical. No physical apparatus will ever measure an infinity—the physical apparatus will break R. Brandenberger [RB]: Holographic means that there is one down. And the reason that our theories predict singularities theory, in d dimensions, which is equivalent to a different the- is that they are not correct. The AdS/CFT correspondence ory which lives on the boundary. It’s like in optics, it’s the provides a way to solve these problems. For example, it pro- same use of the word as in optics. vides a means to understand the physical resolution of the JW: The ideas that later developed into AdS/CFT go back singularity at the center of a black hole that is predicted by to the early 1970s, shortly after the birth of string theory. Einstein’s theory. This question is going to be explored in one The first proposals that gravity should be holographic are of the workshops, the one which Alex Maloney and Patrick due to ’t Hooft, Polyakov, and Susskind, and the Bekenstein– Hayden are coordinating, the one on gravity, in September Hawking entropy of black holes was another important clue. [crm.math.ca/2015/Gravity15/]. But the big breakthrough came in 1998, or late 1997, when I’m a cosmologist and cosmologists have this huge problem Juan Maldacena, who is one of the organizers of the the- of the Big Bang singularity, so again we know that the the- matic semester, proposed this exact correspondence between ory breaks down, and we hope that through the AdS/CFT crm.math.ca correspondence we can have a model for what happened in 19th century to the 1970s, when it became a clearly devel- the very early universe, what replaces the Big Bang. That’s oped theory. The fantastic new things, which occurred in the going to be the focal point of the workshop on cosmology, the last 10 or 15 years, were absolutely unexpected connections first workshop in July [crm.math.ca/2015/Extension15/]. between integrable systems and other areas of physics and of There’s the hint that one can take the gravitation theory, mathematics. I always think of integrability as a subject that which is singular, and through the AdS/CFT correspondence, has nine lives, because it keeps running out of steam and then which is this holographic correspondence, map it into a non- finding a new reason to come back into the center of atten- gravitational theory, on the boundary. Then, since nongravi- tion. I’m going to just list some of these things (not in order tational theories are under much better mathematical control, of importance), and everything on my list is covered in this we can hope that we can evolve the theory on the boundary. thematic program. If we have the picture in mind that we are evolving towards One thing that is close to my heart is the topic of tau func- the Big Crunch singularity, and we go to the boundary, then tions, which is more of a mathematical application, but closely we hope that we can let time continue beyond where it would related to the theory of Riemann surfaces, essential to string stop in the gravitational theory. There are already indications theory. It turned out that tau functions also play a completely that this would work for a very simple model of cosmology different role, and are extremely important, in a mathemati- where there is just a homogeneous and isotropic universe, no cal sense, as generating functions for invariants in enumera- fluctuations, no people, nothing. And the goal of this cos- tive geometry and graph theory, which are also very central mology workshop is to extend it to a more realistic situation to the string theory application. Tau functions are generat- where we include fluctuations, where we include the possibil- ing functions not just for the enumerative geometry related ity of having galaxies in this picture. This is really going to to Riemann surfaces, but also for the invariants such as Jones be a workshop in the sense that this has never been achieved and HOMFLY polynomials appearing in knot theory. It was before and we are getting the people who are working on this known for a long time that there’s a close connection between topic to come together. That is a really novel feature of this topological field theory, Chern–Simons theory, and knot in- program. variants, but it turns out that there is also a relation to inte- There are also applications of the AdS/CFT correspondence grable systems through the tau function. That’s a fascinat- in different areas of physics. For example, a big challenge ing aspect that will hopefully also be developed during this is to understand gauge theories beyond perturbation theory, program. because in some gauge theories the coupling constant is ac- Another connection that emerged recently was a very surpris- tually large and we cannot trust existing techniques. So now ing connection between scattering amplitudes in supersym- we can go backwards using the AdS/CFT correspondence, we metric Yang–Mills theory and soliton theory. Two aspects of can map this complicated gauge theory into a higher dimen- it entered, one was explicit on-shell calculations of scattering sional gravitational theory, and it works out that if the gauge amplitudes which turned out to have integral representations theory is complicated, the gravity theory is going to be in the involving the integration of differential forms with singular- more easily tractable regime. This has applications both to ities over Grassmannians. This gave exactly the singularity condensed matter physics and to relativistic heavy ion col- structure that arises. It is characterized by what we call the lisions, and that’s the theme of the workshop that Keshav moduli space of solitons. Solitons are essential elementary in- Dasgupta and Charles Gale and Sangyong Jeon are coordi- gredients in integrable systems. It turns out that the param- nating [crm.math.ca/2015/Applications15/]. eters that characterize solitons are exactly what we integrate J. Harnad [JH]: I want to mainly talk about the aspects of over when we compute these on-shell scattering amplitudes— the semester that connect up to the notion of completely in- a totally unexpected connection. At present, this connec- tegrable systems. The point is that the thematic program tion is still not understood, but I have great hopes for this is trying to build a bridge between very new developments, program, where we’re bringing together people from the dif- which are everything related to AdS/CFT holography, and ferent specialties, those specialized in integrable systems and work that really dates back to the mid or early 19th century, those specialized in scattering amplitudes and other aspects of which is on integrable systems. A big breakthrough in inte- quantum field theory, to try to cross-fertilize the two subjects. grable systems occurred in the 1960s, when it was realized There is also a connection through a very intensely studied that there were in fact real genuine physical systems, which new area of mathematics, which is known as cluster alge- are integrable in the sense that they have the maximum num- bras. The topological cell structure of positroid cells in Grass- ber of conservation laws, but are actually infinite-dimensional: mannians may appear to be a totally unrelated mathemati- they have an infinite number of degrees of freedom. cal subject. On the contrary, it turns out that the struc- At the same time, the fact that this is closely connected to ture of positroid cells, which exactly characterizes, on the one quantum mechanics became apparent and there were many hand, nonsingular solitons, and on the other hand, the domain quantum integrable systems that were studied simultaneously of integration which represents the scattering amplitudes, under the general heading of “quantum inverse scattering are very closely related.