The Donaldson invariants for the elliptic surfaces
X with
p
g
(
X) > 0 were first studied by Donaldson (1990), and Friedman and Morgan (1994) using stable bundles. Kronheimer (1991) gave a ...topological calculation of the degree
d = 0 Donaldson invariant of the
K3 surface. Here we generalize Kronheimer's approach to calculate a degree
d =
g + 1 invariant for each elliptic surface
X with
p
g
(
X) =
g > 0. We shall construct a moduli space of flat connections
M
with virtual dimension 2
g
+ 2, derive that the degree
g + 1 Donaldson invariant satisfies
q
d
=
aQk
g − 1
+
bk
g + 1
, and compute the leading coefficient
a and how it changes under logarithmic transformations. The result agrees with Friedman and Morgan's, but our proofs do not use algebraic geometry.
We will also prove a relation over the anti-self-dual moduli space between the Pontryagin class of the base-point fibration and Donaldson's μ-class for certain smoothly embedded two-spheres with self-intersection −2.
We show that the SO(3) monopole cobordism formula from 15 implies that all closed, oriented, smooth four-manifolds with b1=0 and b+≥3 and odd with Seiberg–Witten simple type satisfy the ...superconformal simple type condition defined by Mariño, Moore, and Peradze 28,27. This implies the lower bound, conjectured by Fintushel and Stern 18, on the number of Seiberg–Witten basic classes in terms of topological data.
Let X be a smooth, closed, connected, orientable four-manifold with b1(X)=0 and b+(X)≥3 and odd. We show that if X has Seiberg–Witten simple type, then the SO(3)-monopole cobordism formula of 14 ...implies Witten's Conjecture relating the Donaldson and Seiberg–Witten invariants.
We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between ...N=2 supersymmetric gauge theories and two-dimensional conformal field theory.
Talk11http://salafrancesco.altervista.org/wugo2015/tanzini.pdf. presented by A.T. at the conference Interactions between Geometry and Physics — in honor of Ugo Bruzzo’s 60th birthday 17–22 August 2015, Guarujá, São Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).
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