Topological Strings and Loop Equations

1991 ◽  
pp. 53-76 ◽  
Author(s):  
Robbert Dijkgraaf ◽  
Herman Verlinde
Keyword(s):  
Topological Strings

scholarly journals Intertwining operator and integrable hierarchies from topological strings

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jean-Emile Bourgine
Keyword(s):  
Topological Strings ◽  
Vertex Operator ◽  
Integrable Hierarchies ◽  
Kp Hierarchy ◽  
Tau Function ◽  
Time Deformation ◽  
Toroidal Algebra ◽  
Crystal Model ◽  
Melting Crystal ◽  
Definition Of

Abstract In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

scholarly journals Topological Strings and (Almost) Modular Forms

2007 ◽  
Vol 277 (3) ◽  
pp. 771-819 ◽  
Author(s):  
Mina Aganagic ◽  
Vincent Bouchard ◽  
Albrecht Klemm
Keyword(s):  
Topological Strings ◽  
Modular Forms

scholarly journals Topological Strings from Quantum Mechanics

2016 ◽  
Vol 17 (11) ◽  
pp. 3177-3235 ◽  
Author(s):  
Alba Grassi ◽  
Yasuyuki Hatsuda ◽  
Marcos Mariño
Keyword(s):  
Quantum Mechanics ◽  
Topological Strings

scholarly journals Phase transitions inq-deformed 2D Yang-Mills theory and topological strings

2006 ◽  
Vol 73 (2) ◽  
Author(s):  
Xerxes Arsiwalla ◽  
Rutger Boels ◽  
Marcos Mariño ◽  
Annamaria Sinkovics
Keyword(s):  
Phase Transitions ◽  
Topological Strings ◽  
Yang Mills

String theory

2018 ◽  
Author(s):  
Ivan Kostov
Keyword(s):  
String Theory ◽  
Matrix Models ◽  
Topological Strings ◽  
Mirror Symmetry ◽  
Topological String ◽  
Enumerative Geometry ◽  
Topological String Theory ◽  
Superstring Theory ◽  
String Dualities ◽  
Supersymmetric Gauge

This article discusses the link between matrix models and string theory, giving emphasis on topological string theory and the Dijkgraaf–Vafa correspondence, along with applications of this correspondence and its generalizations to supersymmetric gauge theory, enumerative geometry, and mirror symmetry. The article first provides an overview of strings and matrices, noting that the correspondence between matrix models and string theory makes it possible to solve both non-critical strings and topological strings. It then describes some basic aspects of topological strings on Calabi-Yau manifolds and states the Dijkgraaf–Vafa correspondence, focusing on how it is connected to string dualities and how it can be used to compute superpotentials in certain supersymmetric gauge theories. In addition, it shows how the correspondence extends to toric manifolds and leads to a matrix model approach to enumerative geometry. Finally, it reviews matrix quantum mechanics and its applications in superstring theory.

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