EMPG Blog

This week’s EMPJ talk was given by Samson Shatashvili, where he explained some relations between SYM theories and quantum integrable systems.  In particular, we learned how every 2d SYM theory can be identified with a quantum integrable system.

We began by looking at 4d SYM theories, and how these may be nicely reduced to two dimensional theories.   That is, we wanted to reduce theories whose space looked like , where we identified with time and assumed was a “nice” compact space.  For certain nice compact spaces, in particular

a “small manifold limit” could be taken, so that the effective 2d theory looks like .

It was these 2d SYM theories which we wished to identify with quantum integrable systems.

The reduced theory is , so it has the usual supersymmetry algebra with generators

This algebra can be twisted in two inequivalent ways:

and  which now satisfy the algebra

and

We focused in particular on the A twist(For each twist, one gets a commutative twisted chiral ring, which comes for free with any theory). We were interested in solving the equation .  We exploited the operator state correspondence, which says that if is a vacuum state, then so is , for some functions that live in the cohomology of .

Investigating the cohomology on a Riemann surface, we found

We then wrote out these operators in terms of an operator product expansion, which looked like

Since this defines the the ring structure, and also since we had , we discovered the vacua formed a representation of this ring.  And, it was this ring representation that we identified with a quantum integrable system.

Thanks very much to Samson for the preseminar!  It was a pleasure to have you.