scholarly journals Loop Equations for Gromov-Witten Invariant of P1

2019 ◽  
Author(s):  
Gaëtan Borot ◽  
◽  
Paul Norbury ◽  
◽  
◽  
...  
Keyword(s):  
Witten Invariant

ON ROBERTS’ PROOF OF THE TURAEV-WALKER THEOREM

1996 ◽  
Vol 05 (04) ◽  
pp. 427-439 ◽  
Author(s):  
RICCARDO BENEDETTI ◽  
CARLO PETRONIO
Keyword(s):  
Euler Characteristic ◽  
A Priori ◽  
Algebraic Approach ◽  
Fusion Rule ◽  
Point Of View ◽  
Witten Invariant ◽  
Manifolds With Boundary ◽  
3 Dimensional ◽  
Skein Theory ◽  
Natural Way

In this paper we discuss the beautiful idea of Justin Roberts [7] (see also [8]) to re-obtain the Turaev-Viro invariants [11] via skein theory, and re-prove elementarily the Turaev-Walker theorem [9], [10], [13]. We do this by exploiting the presentation of 3-manifolds introduced in [1], [4]. Our presentation supports in a very natural way a formal implementation of Roberts’ idea. More specifically, what we show is how to explicitly extract from an o-graph (the object by which we represent a manifold, see below), one of the framed links in S3 which Roberts uses in the construction of his invariant, and a planar diagrammatic representation of such a link. This implies that the proofs of invariance and equality with the Turaev-Viro invariant can be carried out in a completely “algebraic” way, in terms of a planar diagrammatic calculus which does not require any interpretation of 3-dimensional figures. In particular, when proving the “term-by-term” equality of the expansion of the Roberts invariant with the state sum which gives the Turaev-Viro invariant, we simultaneously apply several times the “fusion rule” (which is formally defined, strictly speaking, only in diagrammatic terms), showing that the “braiding and twisting” which a priori may exist on tetrahedra is globally dispensable. In our point of view the success of this formal “algebraic” approach witnesses a certain efficiency of our presentation of 3-manifolds via o-graphs. In this work we will widely use recoupling theory which was very clearly exposed in [2], and therefore we will avoid recalling notations. Actually, for the purpose of stating and proving our results we will need to slightly extend the class of trivalent ribbon diagrams on which the bracket can be computed. We also address the reader to the references quoted in [2], in particular for the fundamental contributions of Lickorish to this area. In our approach it is more natural to consider invariants of compact 3-manifolds with non-empty boundary. The case of closed 3-manifolds is included by introducing a correction factor corresponding to boundary spheres, as explained in §2. Our main result is actually an extension to manifolds with boundary of the Turaev-Walker theorem: we show that the Turaev-Viro invariant of such a manifold coincides (up to a factor which depends on the Euler characteristic) with the Reshetikhin-Turaev-Witten invariant of the manifold mirrored in its boundary.

scholarly journals ARNOLD CONJECTURE AND GROMOV–WITTEN INVARIANT

Topology ◽  
1999 ◽  
Vol 38 (5) ◽  
pp. 933-1048 ◽  
Author(s):  
Kenji Fukaya ◽  
Kaoru Ono
Keyword(s):  
Witten Invariant ◽  
Arnold Conjecture

Seiberg-Witten invariants, the topological degree and wall crossing formula

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Maciej Starostka
Keyword(s):  
Simple Proof ◽  
Topological Degree ◽  
Witten Invariant ◽  
Homotopy Invariance ◽  
Finite Dimensional ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractFollowing S. Bauer and M. Furuta we investigate finite dimensional approximations of a monopole map in the case b 1 = 0. We define a certain topological degree which is exactly equal to the Seiberg-Witten invariant. Using homotopy invariance of the topological degree a simple proof of the wall crossing formula is derived.

scholarly journals Some refined higher type adjunction inequalities on 4-manifolds

2015 ◽  
Vol 26 (05) ◽  
pp. 1550038
Author(s):  
Chanyoung Sung
Keyword(s):  
Witten Invariant

We further sharpen higher type adjunction inequalities of Ozsváth and Szabó on a 4-manifold M with a non-zero Seiberg–Witten invariant for a Spin c structure 𝔰, when an embedded surface Σ ⊂ M satisfies [Σ] ⋅ [Σ] ≥ 0 and

Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number, I: Non-vanishing theorem

2015 ◽  
Vol 26 (06) ◽  
pp. 1541004 ◽  
Author(s):  
Masashi Ishida ◽  
Hirofumi Sasahira
Keyword(s):  
Betti Number ◽  
Witten Invariant ◽  
Vanishing Theorem ◽  
Connected Sums ◽  
Four Manifolds ◽  
Stable Cohomotopy ◽  
Invent Math

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg–Witten invariant [S. Bauer and M. Furuta, Stable cohomotopy refinement of Seiberg–Witten invariants: I, Invent. Math.155 (2004) 1–19; S. Bauer, Stable cohomotopy refinement of Seiberg–Witten invariants: II, Invent. Math.155 (2004) 21–40.] of connected sums of 4-manifolds with positive first Betti number.

scholarly journals ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

2005 ◽  
Vol 92 (1) ◽  
pp. 99-138 ◽  
Author(s):  
J. FERNÁNDEZ DE BOBADILLA ◽  
I. LUENGO-VELASCO ◽  
A. MELLE-HERNÁNDEZ ◽  
A. NÉMETHI
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curve ◽  
Topological Type ◽  
Homology Sphere ◽  
Plane Curves ◽  
Curve Singularity ◽  
Witten Invariant ◽  
Rational Homology ◽  
Rational Homology Sphere

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

scholarly journals SOME TOPOLOGICAL ASPECTS OF 4-FOLD SYMMETRIC QUANDLE INVARIANTS OF 3-MANIFOLDS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250064 ◽  
Author(s):  
ERI HATAKENAKA ◽  
TAKEFUMI NOSAKA
Keyword(s):  
Double Covering ◽  
Witten Invariant ◽  
Chern Simons ◽  
Homotopy Invariants

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf–Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S3 constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S3 branched along the link. We also reconstruct the Chern–Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].

scholarly journals SOME COMPUTATIONS WITH SEIBERG–WITTEN INVARIANT ACTIONS

2001 ◽  
Vol 16 (04n06) ◽  
pp. 227-233 ◽  
Author(s):  
LORENZO CORNALBA
Keyword(s):  
Effective Action ◽  
Low Energy ◽  
Two Dimensional ◽  
Witten Invariant

We show, with a two-dimensional example, that the low energy effective action which describes the physics of a single D-brane is compatible with T-duality whenever the corresponding U (N) non-Abelian action is form-invariant under the noncommutative Seiberg–Witten transformations.

Gromov–Witten type invariants on circle bundles over symplectic manifolds

2016 ◽  
Vol 13 (09) ◽  
pp. 1650114
Author(s):  
Yong Seung Cho ◽  
Young Do Chai
Keyword(s):  
Moduli Space ◽  
Contact Structure ◽  
Quantum Cohomology ◽  
Symplectic Manifolds ◽  
Base Space ◽  
Total Space ◽  
Witten Invariant ◽  
Circle Bundles ◽  
Type Invariant ◽  
The One

We consider circle bundles over symplectic manifolds to study Gromov–Witten type invariants. We investigate the moduli space of pseudo-coholomorphic maps, Gromov–Witten type invariant, the quantum type cohomology of the total space which has a natural contact structure. We then compare Gromov–Witten invariant, and quantum cohomology of the base space with the one of the total space, and derive some relations between them.

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