Gauge Theory in higher dimensions, II

By Simon Donaldson, E. Segal. 2009

297 citations recorded by the source corpus.

The main aim of the paper is to develop the "Floer theory" associated to Calabi-Yau 3-folds, exending the analogy of Thomas' "holomorphic Casson invariant". The treatment in the body of the paper is largely formal, assuming appropriate compactness properties of moduli spaces of $G_{2}$-instantons, but in the last section we make some remarks about these compactness isssues. Section 3 of the paper contains a general dscussion of deformations of the equations, for gauge field and submanifolds, associated to manifolds with exceptional holonomy.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/be8176bf1820c7cc62e616dde3cdcaf6c3877cb5

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds (2008, methodology): “Donaldson-Thomas theory is defined via integration over the moduli space of ideal sheaves [ 5 , 35] of X. An ideal sheaf is a torsion-free sheaf of rank 1 with trivial determinant.”
Quantum foam and topological strings (2003, background): “philosophy of virtual fundamental classes is applicable and yields a well-defined canonical zero-dimensional homology cycle [ 31 ][42].”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, background): “Donaldson and Thomas in [ 19 ],[68] addressed the cases b) and c). Analytical difficulties there are not completely resolved.”
Donaldson-Thomas type invariants via microlocal geometry (2009, background): “Introduction Donaldson-Thomas type invariants Donaldson-Thomas invariants [6] [19] are the virtual counts of stable sheaves (with fixed determinant) on Calabi-Yau threefolds.”
An introduction to potential theory in calibrated geometry (2007, background): “Calibrated manifolds have attracted particular attention in recent years due to their appearance in generalized Donaldson theories ([DT], [Ti]) and in modern versions of string theory in Physics ([GLW], [GP], [AFS], [Her], [MS], [G], [EM], [GW], [MC] for example).”
A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces (2017, background): “(iii) The notion of a G2-instanton was introduced by Donaldson–Thomas [11] and elaborated on in [12].”
Curve counting via stable pairs in the derived category (2007, background): “of qn for n ≤ 0 uniquely determine Θ(q), and the coeﬃcients of qn for 1 ≤ n ≤ div(β) determine the linear combination (3.17). 12In the Calabi-Yau case, the obstruction theory on P n(X,β) is self-dual [15, 4] and the virtual count Pn,β is the virtual Euler characteristic, up to sign. CURVE COUNTING V”