In Which We Heard About Entanglement in CFTs and the CHY Formalism

 

One of the goals of this workshop is to get graduate students up to speed on recent developments in “trending” topics of research. Usually, this is accomplished by visiting speakers, and often the format of this dissemination is a 1-hour talk. There’s a small problem with this format: an hour is often not nearly enough to go into the sort of detail required to get graduate students (especially those in the early stages of research) comfortable with new ideas. So we picked a reasonable, eminently divisible number of hours: twelve. 🙂

Vinay Malvimat (IIT-Kanpur) began the day with the first of his four lectures on entanglement entropy. After a quick introduction to entanglement entropy in quantum mechanical systems, Vinay turned to a discussion of entanglement in QFTs.

This was done using the half-spacetime Euclidean path integral (the path integral from to ) to define ground states of quantum field theories both free and interacting.

Vinay emphasized the importance of Renyi entropies,, which are defined as , specifically its geometrical interpretation viz. the n in Reyi secretly means you should make copies of the original field theory: one can glue 2n of these half-spacetime path integrals in fun ways into an n-fold cover of the original spacetime — with copy (of the full Euclidean spacetime) and copy glued along the interval. However, this abstract formulation is about as far as one can go in a generic d-dimensional QFT.

Specializing to the case of 2d CFTs, one is interested in computing the entanglement of an interval with the rest of the spatial line. Using the complex coordinatisation of , the n-fold cover becomes an -sheeted Riemann surface.

Of course, working on and n-sheeted Riemann surface is much harder than working on a 1-sheeted Riemann surface (also known by obscurantists as the complex plane). To do this, we make n copies of the fields, one for the field corresponding to each sheet, and give them decoupled dynamics. However, the new path integral on the complex plane must remember that the integral wasn’t just on n decoupled sheets but on n glued sheets. This is accomplished by “twist operators” at the ends of the intervals that impose the boundary conditions.

One point of confusion a lot of people faced was that of where these twist operators live. Imagine sitting at any point on the interval that is not either of the ends. In going in a loop around this point, we cross the interval twice, so there is no “twisting” — we go from sheet i to sheet i+1 and back again — as we return to the original field value. If we’re at the end of the interval, however, a loop will cause us to cross the interval once, so we go from sheet i to sheet i+1, thus necessitating the use of these twist operators. Hence, the twist operators live only on the ends of the intervals.

Any correlation function on the n-sheeted Riemann surface can be converted to a correlation function on the complex plan just by introducing twist operators — one for each end of the interval — and normalizing appropriately. So, we conclude that the partition function — a correlation function with no operators in it! — is proportional to a specific two-point function of the twist operators!

This is something we can compute straightforwardly: we simply compute the expectation value of the stress-energy tensor on the Riemann surface by translating the problem to a problem on the complex plane, and this allows us to fix the conformal weights of the twist operators. Lastly, as the Renyi entropy is straightforwardly computed and 2-point functions are determined solely by conformal invariance, we determine the single-interval entanglement entropy for an interval of length L to be

In the second session, Arnab Priya Saha (IMSc) spoke about CHY and the Weinberg soft theorem. The most logical starting point was to define the S-matrix and discuss its properties that follows from imposing locality and unitarity. He provided a good motivation for studying scattering amplitudes, by providing an overview of the early works of Park and Taylor on the MHV amplitude. It turns out the nice symmetric expressions that one obtains by calculating the MHV amplitude in the spinor-helicity formalism in fact hints towards hidden symmetries which might not be manifest. Also, for higher loops calculating a huge number of Feynman diagrams is indeed difficult. CHY formalism was introduced as a toolbox which maps the kinematic space (essentially a bunch of four-momenta dictated by the incoming and outgoing particles) to the moduli space of an n-punctured Riemann sphere. There was some confusion about how the position of these punctures are described on the Riemann sphere and how they’re related to the physical momenta. Once the paricipants were (partially?) satisfied with the definition, Arnab set up the scattering equations and wrote down a compact formula for tree-level scattering of massless particles.

A model that he chose as a testing ground for checking the predictions of the “CHY Formalism” was the scalar  theory but with two copies of color. He explicitly calculated the scattering amplitude for 3 and 4 particle scattering for this theory. A significant section which followed after this was spent in defining the reduced Pfaffian which finally helped in writing down the scattering amplitude for Yang-Mills Theory. He explicitly showed for Yang-Mills that the CHY Formalism agrees with the standard Feynman diagram result. He ended by showing that the soft limit of the amplitude calculated by the CHY formalism in fact reproduces Weinberg’s soft theorem. Due to time-constraint, he could not discuss gravity amplitudes in much details but simply mentioned the result that is obtained from this formalism.

A lot of our attendees requested a short overview of the AdS/CFT correspondence, and Pinaki Banerjee (IMSc) kindly agreed to review Maldacena’s original decoupling argument for us.

He began with some amusing comments on reasons to work on the AdS/CFT correspondence — by his estimates, Maldacena’s ’97 paper receives an average of over three citations a day! — before turning to a short discussion of different kinds of dualities in field theory and string theory.

To motivate Maldacena’s argument, Pinaki used the example of trying to understand electron-proton scattering from different points of view. One, that involves microscopic analyses of the various things that can happen — bremsstrahlung, vertex corrections, etc. — when electrons and protons exchange photons. The other point of view treats the field of a proton as being a classical background in which the electron moves.

Now make the replacement (proton goes to D-brane, electron goes to closed/open string) and run the same argument through. One finds that in the limit of \alpha’ going to zero, we have two descriptions of the low energy physics:

1. = 4  SYM with 10D SUGRA

2. Full IIB superstring theory in with 10D SUGRA

Now Maldacena does this totally cool amazing thing where he cancels 10D SUGRA. Yes, this is true. He was not joking. Seriously.

Anyway, that was the day.