اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدWKB solutions of difference equations and reconstruction by the topological recursion
109
0
0.0
(
0
)
تأليف
Olivier Marchal
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $hbar$-difference equation: $Psi(x+hbar)=left(e^{hbarfrac{d}{dx}}right) Psi(x)=L(x;hbar)Psi(x)$ with $L(x;hbar)in GL_2( (mathbb{C}(x))[hbar])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $hbar$-differential systems to this setting. We apply our results to a specific $hbar$-difference system associated to the quantum curve of the Gromov-Witten invariants of $mathbb{P}^1$ for which we are able to prove that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve $y=cosh^{-1}frac{x}{2}$. Finally, identifying the large $x$ expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of $mathbb{P}^1$.
قيم البحث
اقرأ أيضاً
Starting from a $dtimes d$ rational Lax pair system of the form $hbar partial_x Psi= LPsi$ and $hbar partial_t Psi=RPsi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfie
s the topological type property. A consequence is that the formal $hbar$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painleve systems.
In this article, a novel description of the hypergeometric differential equation found from Gelfand-Kapranov-Zelevinskys system (referred to GKZ equation) for Giventals $J$-function in the Gromov-Witten theory will be proposed. The GKZ equation invol
ves a parameter $hbar$, and we will reconstruct it as the WKB expansion from the classical limit $hbarto 0$ via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant $mathbb{C}textbf{P}^{1}$ model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovins conjecture for this equivariant model.
Given a topological modular functor $mathcal{V}$ in the sense of Walker cite{Walker}, we construct vector bundles over $bar{mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction depends on a determ
ination of the logarithm of the eigenvalues of the Dehn twist and central element actions. We show that the intersection of the Chern class with the $psi$-classes in $bar{mathcal{M}}_{g,n}$ is computed by the topological recursion of cite{EOFg}, for a local spectral curve that we describe. In particular, we show how the Verlinde formula for the dimensions $D_{vec{lambda}}(mathbf{Sigma}_{g,n}) = dim mathcal{V}_{vec{lambda}}(mathbf{Sigma}_{g,n})$ is retrieved from the topological recursion. We analyze the consequences of our result on two examples: modular functors associated to a finite group $G$ (for which $D_{vec{lambda}}(mathbf{Sigma}_{g,n})$ enumerates certain $G$-principle bundles over a genus $g$ surface with $n$ boundary conditions specified by $vec{lambda}$), and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply-connected Lie group $G$ (for which $mathcal{V}_{vec{lambda}}(mathbf{Sigma}_{g,n})$ is the Verlinde bundle).
We formulate a notion of abstract loop equations, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian m
atrix models, and of the O(n) model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SU(N) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.
We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e. the
differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painleve 2 equation and the quantization of a degree three spectral curve with pole only at infinity.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
