Vafa-Witten
invariants are
enumerative invariants of compact oriented 4-manifolds
conjectured by the
String Theorists Vafa and Witten in 1994. Notionally they should
have a gauge
theory definition - something like Euler characteristics of
moduli space of
instantons - but as the moduli spaces are potentially
noncompact, this does not
lead to a rigorous, deformation-invariant definition. For
compact Kähler
surfaces
(better, projective surfaces), a rigorous definition is
possible, due
to Tanaka-Thomas and Thomas, roughly as invariants counting
torsion-free
coherent sheaves with Higgs fields.
Vafa and Witten
proposed that generating
functions of Vafa-Witten invariants should have modular
properties: they should
be modular forms (or possibly mock modular for surfaces with
geometric genus pg
= 0, when there is a 'modular anomaly'). Modular forms are
special functions
coming from Number Theory. Their appearance in Vafa-Witten
generating functions
is surprising, and currently very mysterious geometrically.
For surfaces X
with pg
= 0, the situation is different. Vafa-Witten invariants depend
on the Kähler
class
of the surface, which defines a stability condition. String
Theorists say
the generating function of Vafa-Witten invariants should be
modified to an
'almost holomorphic completion', which is non-holomorphic but
exactly modular.
When Vafa-Witten
invariants of projective
surfaces have been calculated (often modulo some conjectures),
the modular
properties have been confirmed.
Provisional
programme:
Week 1: Sergey
Alexandrov
(String Theory guest speaker): short discussion of Vafa-Witten
invariants in String Theory.
Weeks 1-2: Dominic:
What
are Vafa-Witten invariants? Sketch of gauge theory ‘definition’,
Tanaka-Thomas 1,2 and Thomas. Try to get to modularity statement,
at least for
SU(2).
Weeks 2-3:
Pierrick: Overview
of the Vafa-Witten
paper, quantum field theory origin of the Vafa-Witten equations,
physics
motivation: S-duality of maximally supersymmetric Yang-Mills
theory,
Vafa-Witten equations from topological twist. If time: relation
with
exceptional holonomy.
Week 4: Andrew
Graham
(guest speaker from Number Theory): introduction to (quasi)
modular forms, and
their relation to special values of L-functions.
Week 5: Hulya:
Explicit
calculations
and conjectures for generating series of Vafa-Witten invariants
of
surfaces with pg > 0, in rank
2 and rank 3.
Contributions of instanton and monopole branches to Vafa-Witten
invariants, and
discussion on modularity (in physics terms S-duality) exchanging
these two
contributions.
Weeks 6-8:
TBA.