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Generalized Ablowitz-Ladik hierarchy in topological string theory
Mathematical Physics · Submitted 27 Dec 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions becomes a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz-Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. | background 0.69 | 0.51 | 0.04 | asserted 0.62 |
| s1 | Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. | result 0.62 | 0.51 | 0.02 | asserted 0.99 |
| s2 | The generating function of these amplitudes with respect to the product of two independent Schur functions becomes a tau function of the 2D Toda hierarchy. | result 0.99 | 0.74 | 0.06 | asserted 1.00 |
| s3 | The associated Lax operators turn out to have a particular factorized form. | result 1.00 | 0.82 | 0.03 | asserted 1.00 |
| s4 | This factorized form of the Lax operators characterizes a generalization of the Ablowitz-Ladik hierarchy embedded in the 2D Toda hierarchy. | result 0.65 | 0.83 | 0.31 | asserted 1.00 |
| s5 | The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds. | interpretation 0.63 | 0.87 | 0.92 | asserted 1.00 |
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The paper
Original abstract
This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions becomes a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz-Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.
Human review
How the literature uses it
2 arXiv papers cite this in 7 sentences
Papers citing it · 4 sentences without a confident label
Builds on it 1 paper
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Recently, we extended our previous work [ref] on the integrable structure of a modified melting crystal model to open string amplitudes of topological string theory on a generalized conifold [ref] .
Orbifold melting crystal models and reductions of Toda hierarchy · Introduction · 0.96
Uses its method or tool 1 paper
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Our strategy is to construct a generating function of the topological vertex of Aganagic et al. (modified to depend on the parameters $w$ of the cubic Hodge integrals), and to show that it coincides with (an exponentiated version of) the generating function of Li et al. To this end, we make use of tools developed in a series of our work on the melting crystal models [ref] and topological string theory [ref] .
Three-partition Hodge integrals and the topological vertex · Introduction · 0.99
Compares with it 1 paper
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The structure of the generating operator ([ref]) is reminiscent of that of the generating operator for open string amplitudes on generalized conifolds of the bubble type [ref] .
Orbifold melting crystal models and reductions of Toda hierarchy · Converting $Z'_a,b(s,,)$ to a tau function · 0.95
What it cites · 34 sentences without a confident label
Builds on 4 papers
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This paper presents an extension of our previous work [ref] on the integrable structure of a modified “melting crystal model”.
Modified melting crystal model and Ablowitz-Ladik hierarchy · Introduction · 0.99
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Topological string theory on the $N$-th generalized conifold is related to the following generalization of ([ref]) [ref] : [equation] $Q_1,…,Q_2N-1$ are “Kähler parameters” assigned to the internal edges of the web diagram (see Fig. [ref]).
Non-commutative Donaldson-Thomas theory and vertex operators · Tau function in fermionic language · 0.86
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Topological string theory on the $N$-th generalized conifold is related to the following generalization of ([ref]) [ref] : [equation] $Q_1,…,Q_2N-1$ are “Kähler parameters” assigned to the internal edges of the web diagram (see Fig. [ref]).
Wall-crossing, free fermions and crystal melting · Tau function in fermionic language · 0.86
Uses the method or tool of 7 papers
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We deformed the modified melting crystal model by two sets of external potentials with coupling constants $= (t_1,t_2,…)$ and $= (t̅_1,t̅_2,…)$, and employed the method [ref] developed for the genuine melting crystal model [ref] to show that the deformed partition function is essentially a tau function $τ(s,,)$ of the 2D Toda hierarchy [ref] .
Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions · Introduction · 0.97
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We deformed the modified melting crystal model by two sets of external potentials with coupling constants $= (t_1,t_2,…)$ and $= (t̅_1,t̅_2,…)$, and employed the method [ref] developed for the genuine melting crystal model [ref] to show that the deformed partition function is essentially a tau function $τ(s,,)$ of the 2D Toda hierarchy [ref] .
Integrable structure of melting crystal model with external potentials · Introduction · 0.97
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We deformed the modified melting crystal model by two sets of external potentials with coupling constants $= (t_1,t_2,…)$ and $= (t̅_1,t̅_2,…)$, and employed the method [ref] developed for the genuine melting crystal model [ref] to show that the deformed partition function is essentially a tau function $τ(s,,)$ of the 2D Toda hierarchy [ref] .
Integrable Hierarchies and Dispersionless Limit · Introduction · 0.97
Compares with 1 paper
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In the latest paper [ref] , they proposed a new scheme of reductions of the Toda hierarchy, which is apparently different from ours, and applied the reduced hierarchy to local Gromov-Witten invariants of a class of orbifolds of the resolved conifold.
Rational reductions of the 2D-Toda hierarchy and mirror symmetry · Conclusion · 0.98
Cites as background 4 papers
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The relevance of the Ablowitz-Ladik hierarchy [ref] (equivalently, the relativistic Toda hierarchy [ref] ) to this model was conjectured by Brini from an order-by-order analysis of the genus expansion of the partition function [ref] .
The local Gromov-Witten theory of CP^1 and integrable hierarchies · Introduction · 0.82
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The relevance of the Ablowitz-Ladik hierarchy [ref] (equivalently, the relativistic Toda hierarchy [ref] ) to this model was conjectured by Brini from an order-by-order analysis of the genus expansion of the partition function [ref] .
Faces of Relativistic Toda Chain · Introduction · 0.82
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The relevance of the Ablowitz-Ladik hierarchy [ref] (equivalently, the relativistic Toda hierarchy [ref] ) to this model was conjectured by Brini from an order-by-order analysis of the genus expansion of the partition function [ref] .
A note on the integrable discretization of the nonlinear Schr\"odinger equation · Introduction · 0.82
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