Mock theta functions and representation theory of affine Lie superalgebras and superconformal algebras

One of the beautiful properties in representation theory of ane Lie
algebras is the SL2pZq-invariance of the space of characters of integrable modules
discovered by Kac-Peterson in the early 1980's.
However, for ane Lie superalgebras, modular invariance had long been quite un-
clear except for only a few cases. Recently a remarkable breakthrough was brought
by Zwegers, who constructed a modular function from the supercharacter of the
ane slp2|1q-module of level 1 by adding non-holomorphic correction term, which
is called the "modication" procedure.
In my lecture, I will explain some recent progress on mock modular forms and
representation theory of ane Lie superalgebras, which has been obtained by joint
works with Victor Kac. The main subjects are concerned to the followings, on some
of which I will give detail explanations in my talk.
After quick review on the Zwegers method which was the starting point of our
work, we introduce the "basic" mock theta functions and explain their modication
to non-holomorphic modular forms. These basic mock theta functions play basically
important roles in our modication theory in general case.
Then, just corresponding to classical theta functions, we consider mock theta
functions in genaral setting, namely mock theta functions of higher level and higher
rank and higher atypicality. I will explain how to construct modication of these
functions and how to compute their modular transformation formulas explicitely.
Then we turn to integrable representations of ane Lie superalgebras with max-
imal atypicality and compute their characters and supercharacters by using the
Weyl-Kac type character formula. These are mock theta functions and the above
method works to give SL2pZq -invariant family of modied supercharacters for any
ane Lie superalgebra. In this way, we observe that modular properties hold also
for all ane Lie superalgebras. Furthermore, by considering admissible representa-
tions just like the case of ane Lie algebras, we obtain SL2pZq-invariant family of
modied (super)characters of admissible modules for ane Lie superalgebras.
Then we go to the quantum Hamiltonian reduction, namely representations of
W-algebras associated to ane Lie superalgebras. The theory of W-algebra is very
important, since many of superconformal algebras are constructed as W-algebras.
The modication of characters of ane Lie superalgebras naturally gives rise to the
modication of characters of W-algebras. So we get modular invariant family of
representations also for superconformal algebras.
As a consequence, vast numbers of SL2pZq-invariant families of modied mock
theta functions are in our hands obtained from representations of ane Lie superal-
gebras and superconformal algebras. It will be natural to expect that there should
exist "new" conformal eld theory corresponding to these new "mock modular se-
ries" representations of superconformal algebras.
References:
piq The rst important reference for the modication theory is
S. P. Zwegers : Mock theta functions, arXiv:0807.4834.