Topological string theory

About: Topological string theory is a research topic. Over the lifetime, 1206 publications have been published within this topic receiving 54758 citations.

Resurgence in topological string theory

Abstract: The study of physical theories in the nonperturbative regime is an interesting but difficult problem. In high energy theoretical physics the use of dualities inspired by the original AdS/CFT correspondence has become the main technique for calculating nonperturbative data. String theories, and in particular, topological string theory, lack nonperturbative definitions. The perturbative free energy, as series in the string coupling constant, is asymptotic, with zero radius of convergence. This is a general feature of many physical systems and it is the main concern of the mathematical theory of resurgence. One of the main results of this theory describes, in a quantitative way, the relation between the perturbative and nonperturbative information of a system. Encoded in the asymptotic growth of the series coefficients of perturbation theory is the information necessary to reconstruct nonperturbative sectors. All these sectors can be put together in a formal object called the transseries, whose different coefficients are related to each other by resurgence relations. The resurgent approach has been applied succesfully to problems in mathematics, on differential and difference equations, and in physics, on quantum mechanics and even quantum field theory. It is currently a very active area of research merging the efforts of both physicists and mathematicians. This thesis performs a resurgent analysis of the perturbative topological string theory. Using the holomorphic anomaly equations it is possible to compute coefficients of the perturbative free energy to very high order and analyze their asymptotic growth. In agreement with resurgence, it is found that nonperturbative sectors coming from a transseries control this growth. It is shown that this transseries can be computed as a solution of a natural extension of the holomorphic anomaly equations. The first half of this thesis is concerned with the main properties of the theory of resurgence and with the computation of the perturbative topological string free energy. These results are then applied to a concrete topological string example. A careful study of the asymptotic growth of the perturbative free energies is performed and various resurgence relations are uncovered. These relations involve elements of the transseries describing the full nonperturbative free energy. General properties of the transseries satisfying the holomorphic anomaly equations are described, including the role of the instanton actions, the presence of holomorphic ambiguities and the possibility of resonance. The numerical results are found to match, to high precision, the elements of the computed transseries. The asymptotic nature of the higher instanton sectors is also studied and a complicated net of resurgence relations is found.

Topological string entanglement

Abstract: We investigate how topological entanglement of Chern-Simons theory is captured in a string theoretic realization. Our explorations are motivated by a desire to understand how quantum entanglement of low energy open string degrees of freedom is encoded in string theory (beyond the oft discussed classical gravity limit). Concretely, we realize the Chern-Simons theory as the worldvolume dynamics of topological D-branes in the topological A-model string theory on a Calabi-Yau target. Via the open/closed topological string duality one can map this theory onto a pure closed topological A-model string on a different target space, one which is related to the original Calabi-Yau geometry by a geometric/conifold transition. We demonstrate how to uplift the replica construction of Chern-Simons theory directly onto the closed string and show that it provides a meaningful definition of reduced density matrices in topological string theory. Furthermore, we argue that the replica construction commutes with the geometric transition, thereby providing an explicit closed string dual for computing reduced states, and Renyi and von Neumann entropies thereof. While most of our analysis is carried out for Chern-Simons on S^3, the emergent picture is rather general. Specifically, we argue that quantum entanglement on the open string side is mapped onto quantum entanglement on the closed string side and briefly comment on the implications of our result for physical holographic theories where entanglement has been argued to be crucial ingredient for the emergence of classical geometry.

Open-closed duality and double scaling

Abstract: Nonperturbative terms in the free energy of Chern-Simons gauge theory play a key role in its duality to the closed topological string. We show that these terms are reproduced by performing a double scaling limit near the point where the perturbation expansion diverges. This leads to a derivation of closed string theory from this large-N gauge theory along the lines of noncritical string theories. We comment on the possible relevance of this observation to the derivation of superpotentials of asymptotically free gauge theories and its relation to infrared renormalons.

A Note on q-Deformed Two-Dimensional Yang-Mills and Open Topological Strings

Abstract: In this note we make a test of the open topological string version of the OSV conjecture in the toric Calahi–Yau manifold X = O(–3) → P2 with background D4-hranes wrapped on Lagrangian suhmanifolds. The D-brane partition function reduces to an expectation value of some inserted operators of a q-deformed Yang–Mills theory living on a chain of P1 's in the base P2 of X. At large N this partition function can be written as a sum over squares of chiral blocks, which are related to the open topological string amplitudes in the local P2 geometry with branes at both the outer and inner edges of the toric diagram. This is in agreement with the conjecture.