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.
Notes for week 1
by Dominic
Notes for week 2 by Dominic
Notes for week 3
by Pierrick