The local Gromov–Witten invariants of configurations of rational curves

Karp, Dagan; Liu, Chiu-Chu Melissa; Mariño, Marcos

doi:10.2140/gt.2006.10.115

Cited by 18 publications

(28 citation statements)

References 28 publications

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“…We have encountered the same difficulty in an attempt to extend our results to more general tree-like web diagrams studied by Karp, Liu and Mariño [4]. Our attempt has been unsuccessful not only for open string amplitudes, but also for the closed string partition function.…”

Section: Calculation Of Open String Amplitudesmentioning

confidence: 86%

Open string amplitudes of closed topological vertex

Takasaki

Nakatsu

2015

J. Phys. A: Math. Theor.

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The closed topological vertex is the simplest "off-strip" case of non-compact toric Calabi-Yau threefolds with acyclic web diagrams. By the diagrammatic method of topological vertex, open string amplitudes of topological string theory therein can be obtained by gluing a single topological vertex to an "on-strip" subdiagram of the tree-like web diagram. If non-trivial partitions are assigned to just two parallel external lines of the web diagram, the amplitudes can be calculated with the aid of techniques borrowed from the melting crystal models. These amplitudes are thereby expressed as matrix elements, modified by simple prefactors, of an operator product on the Fock space of 2D charged free fermions. This fermionic expression can be used to derive q-difference equations for generating functions of special subsets of the amplitudes. These q-difference equations may be interpreted as the defining equation of a quantum mirror curve.

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Section: Calculation Of Open String Amplitudesmentioning

confidence: 86%

Open string amplitudes of closed topological vertex

Takasaki

Nakatsu

2015

J. Phys. A: Math. Theor.

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“…The toric Calabi-Yau threefold engineering T 2 theory is the resolved C 3 /(Z 2 × Z 2 ), whose toric diagram is shown in Figure 13. This multi-junction system has SU (2) Given the web diagram, the topological string partition function is simply given by [59][60][61]…”

Section: Warm Up: T 2 Theorymentioning

confidence: 99%

Topological strings and 5d T N partition functions

Hayashi

Kim

Nishinaka

2014

J. High Energ. Phys.

220

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Abstract:We evaluate the Nekrasov partition function of 5d gauge theories engineered by webs of 5-branes, using the refined topological vertex on the dual Calabi-Yau threefolds. The theories include certain non-Lagrangian theories such as the T N theory. The refined topological vertex computation generically contains contributions from decoupled M2-branes which are not charged under the 5d gauge symmetry engineered. We argue that, after eliminating them, the refined topological string partition function agrees with the 5d Nekrasov partition function. We explicitly check this for the T 3 theory as well as Sp(1) gauge theories with N f = 2, 3, 4 flavors. In particular, our method leads to a new expression of the Sp(1) Nekrasov partition functions without any contour integrals. We also develop prescriptions to calculate the partition functions of theories obtained by Higgsing the T N theory. We compute the partition function of the E 7 theory via this prescription, and find the E 7 global symmetry enhancement. We finally discuss a potential application of the refined topological vertex to non-toric web diagrams.

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“…For the r.h.s., we will use the fact that in positive degree all genus zero Gromov-Witten invariants can be computed by a combined use of the deformation invariance of GW invariants and the Aspinwall-Morrison formula [5,13,70]. The result is [13,47]…”

Section: Mirror Symmetrymentioning

confidence: 99%

Rational reductions of the 2D-Toda hierarchy and mirror symmetry

Brini¹,

Carlet²,

Романо³

et al. 2017

J. Eur. Math. Soc.

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We introduce and study a two-parameter family of symmetry reductions of the 2D Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1 + 1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz-Ladik hierarchy and E. Frenkel's q-deformed GelfandDickey hierarchy. We establish their characterization in terms of block Töplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the Takasaki-Takebe classical limit gives rise to a family of non-conformal Frobenius manifolds with flat identity. We use this to generalize the relation of the AblowitzLadik hierarchy to Gromov-Witten theory by proving an analogous mirror theorem for the general rational reduction: in particular, we show that the dual-type Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric Calabi-Yau threefolds obtained from minimal resolutions of the local orbifold line.

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The local Gromov–Witten invariants of configurations of rational curves

Cited by 18 publications

References 28 publications

Open string amplitudes of closed topological vertex

Open string amplitudes of closed topological vertex

Topological strings and 5d T N partition functions

Rational reductions of the 2D-Toda hierarchy and mirror symmetry

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