Math Department at MIT | Contacts: Pavel Etingof, Victor Kac

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Spring 2012 | Fridays 3:00 - 5:00pm at 2-136

quantum groups theory. One of its main ingredients is a connection on the configuration

spaces of points in the complex plane, called the universal KZ connection. It gives rise to

explicit relations between multiple zeta values, and explains the "torsor" structure of these

relations ; it also gives rise to a Lie algebra, which is the target of a morphism from a

motivic Lie algebra.

An elliptic analogue of the KZ connection, living over the moduli space

of elliptic curves with marked points, was later constructed (D. Calaque, P. Etingof

and the speaker). It allows for an analytic proof of the computation of the

prounipotent completion of elliptic pure braid groups (Bezrukavnikov). A theory

parallel to the theory of associators can then be developed. The analogues of

multiple zeta values are certain functions of the elliptic parameter, which

are shown to satisfy certain relations. The theory gives rise to a Lie algebra,

which seems to be related to motivic theory, and is shown to possess a semidirect

product structure. A new feature with respect to genus zero it the appearance of

iterated integrals of Eisenstein series: they are shown to satisfy some algebraic relations

with multiple zeta values. Another byproduct is the computation of the Zariski closure of

the genus one mapping class group in the automorphism groups of the prounipotent

completions of pure braid groups in genus one.

(Partially based on joint work with D. Calaque and P. Etingof.)

In the first hour, we will discuss the ring of G_m invariant functions on V_m, and its relationship to the ring of functions on (V//G)_m, where V//G is the categorical quotient. These invariant rings were previously studied in some special cases by D. Eck, E. Frenkel, and D. Eisenbud.

In the second hour,
we will discuss an application of these results to vertex algebras. Given a
vertex algebra V and a subalgebra A, a basic problem is to describe the
commutant Com(A,V) by giving generators, operator product expansions, and
normally ordered polynomial relations among the generators. We solve this
problem for a family of examples where V is a "free field" algebra and A is

an affine vertex algebra. This is a joint work with Bailin Song (University
of Science and Technology of China).

This is work in progress with Nora Ganter.

Let $\mathfrak{g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak{g}$-invariants in the basic representation of the affine Kac--Moody algebra $\hat{\mathfrak{g}}$ is known as a $\mathcal{W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal{F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal{F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal{F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the $\mathcal{W}$-algebra. (Joint work with T. Milanov.)