arXiv subjects

Chris Gerig

Publications and source records attributed to Chris Gerig.

6 recordsLinked to original sources

arXiv2019

Lagrangian torus invariants using ECH = SWF

We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes. They turn out not to be new invariants, instead they repackage the Gromov (and Seiberg-Witten) invariants of various torus surgeries. We then recover a result of Morgan-Mrowka-Szab\'o on product formulas for the Seiberg-Witten invariants along 3-tori.

math.SG↗

arXiv2019

$G_2$ Holonomy, Taubes' Construction of Seiberg-Witten Invariants and Superconducting Vortices

Using a reformulation of topological ${\cal N}=2$ QFT's in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a $G_2$ manifold constructed from the space of self-dual 2-forms over $X^4$, we show that superconducting vortices are mapped to M2 branes stretched between M5 branes. This setup provides a physical explanation of Taubes' construction of the Seiberg-Witten invariants when $X^4$ is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT's arising from ${\cal N}=2$ QFT's from all Gaiotto theories on arbitrary 4-manifolds.

hep-th↗

arXiv2019

No homotopy 4-sphere invariants using ECH = SWF

In relation to the 4-dimensional smooth Poincar\'e conjecture we construct a tentative invariant of homotopy 4-spheres using embedded contact homology (ECH) and Seiberg-Witten theory (SWF). But for good reason it is a constant value independent of the sphere, so this null-result demonstrates that one should not try to use the usual theories of ECH and SWF. On the other hand, a corollary is that there always exist pseudoholomorphic curves satisfying certain constraints in (punctured) 4-spheres.

math.SG↗

arXiv2018

Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms

This is the sequel to the author's previous paper which gives an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds. The main result of this paper asserts the following. Whenever the Seiberg-Witten invariants are defined over a closed minimal 4-manifold X, they are equivalent modulo 2 to "near-symplectic" Gromov invariants in the presence of certain self-dual harmonic 2-forms on X. A version for non-minimal 4-manifolds is also proved. A corollary to circle-valued Morse theory on 3-manifolds is also announced, recovering a result of Hutchings-Lee-Turaev about the 3-dimensional Seiberg-Witten invariants.

math.DG↗

arXiv2017

Taming the pseudoholomorphic beasts in $\mathbb{R}\times(S^1\times S^2)$

For a closed oriented smooth 4-manifold X with $b^2_+(X)>0$, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describes well-defined counts of pseudoholomorphic curves in the complement of the zero-set of such "near-symplectic" forms, and it is shown that they recover the Seiberg-Witten invariants (mod 2). This is an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds. The main result of this paper asserts the following. Given a suitable near-symplectic form w, a tubular neighborhood N of its zero-set, and a generic w-compatible almost complex structure J on X-N, there are well-defined counts of J-holomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to Reeb orbits on the ends. They can be packaged together to form "near-symplectic" Gromov invariants as a function of spin-c structures on X.

math.SG↗

arXiv2016

Generic transversality for unbranched covers of closed pseudoholomorphic curves

We prove that in closed almost complex manifolds of any dimension, generic perturbations of the almost complex structure suffice to achieve transversality for all unbranched multiple covers of simple pseudoholomorphic curves with deformation index zero. A corollary is that the Gromov-Witten invariants (without descendants) of symplectic 4-manifolds can always be computed as a signed and weighted count of honest J-holomorphic curves for generic tame J: in particular, each such invariant is an integer divided by a weighting factor that depends only on the divisibility of the corresponding homology class. The transversality proof is based on an analytic perturbation technique, originally due to Taubes.

math.SG↗