Aspects Of Topological String Theory

This dissertation mainly studies the aspects of topological string theory,including topological stirngs and spectral theory,the relationship with TBA-like equations,integrability and Krlov complexity.In the chapter 2 and chapter 3,we introduce the relevant knowledge background of topological field theory,and explain that topological string theory is a N=(2,2)topological field theory coupling the worldsheet gravity.And we give some specific introduction of the local toric Calabi-Yau 3-manifold,which is closely related to the spectrum problems,we call it topological strings/spectrum problems correspondence(TS/ST correspondence).The spectrum problems can be solved by a perturbation method-the all-orders WKB method.However,the WKB coefficients diverge double-factorially,so we need to consider the non-perturbative effects,and after considering its contribution,we can deduce an exact quantization condition for the spectrum.In the chapter 4,we study the relationship between the quantum periods and TBAlike difference equations.TBA equation is a nonlinear thermodynamic Bethe ansatz equation and is an effective method for solving the integrable systems,especially the spin chain.The relation can be clearly seen from two aspects:First,from the perspective of physics,the topological string theory is a integrable system,and the toric Calabi-Yau geometries we consider correspond to the 5d supersymmetric gauge theories,e.g.the 5d pure SU(2)gauge theory is engineered by the local P1×P1 geometry.In the 4d limit,we can get some integrable systems,such as the Toda chain,and they can be solved by the TBA equations.Second,from the perspective of mathematics,the integral kernel of TBA-like equation has the form of 1/cosh(x),so solving the spectral equation of the integral kernel is equivalent to solving the TBA-like equation,and has been confirmed in the local P1 × P1 model.We consider the cases of the Om,1 operators,and give two methods to derive the TBA-like equations.The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.In chapter 5,we use a novel method-the bootstrap method to continue the study of spectrum problems of the topological strings.This method is called the bootstrap,and needs not to solve the Schrodinger equation,but just uses the positive constrain.We find that for the topological strings,the bootstrap level is bounded by π/h,due to the constrains from the asymptotic behavior of the wave functions,which is different from the nonrelativistic quantum mechanical systems.We use a two-indexed operators with both x and p,which turns out to improve greatly the efficiency of the bootstrap procedure,and we can achieve quite high numerical precisions.And we have also verified some non-relativistic quantum mechanical systems,and this method is also applicable.Finally we discuss the integrability and Krylov complexity.The integrability of topological strings is guaranteed by its exact quantization condition,but for the general models,it is not a simple matter to determine whether they are integrable.Krylov complexity is a possible method,which can be calculated using Lanczos coefficients.This method replaces the trajectory of classical particles with the growth of operators,because the concept of trajectory becomes ill-defined after considering quantum effects.If the complexity increases exponentially,the system is generally quantum chaotic,otherwise it is integrable.We study this approach and obtain an upper bound of the slope of the Lanczos coefficients for the chaotic systems,which is proportional to temperature.In addition,we find that for the topological string theory,it does meet expectations,but for the non-relativistic model,their behavior is similar to a chaotic system,which shows Krylov complexity is not a good way to judge whether it is integrable or not.