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.

The omega deformation from string and M-theory

Abstract: We present a string theory construction of Omega-deformed four-dimensional gauge theories with generic values of ϵ1 and ϵ2. Our solution gives an explicit description of the geometry in the core of Nekrasov and Witten’s realization of the instanton partition function, far from the asymptotic region of their background. This construction lifts naturally to M-theory and corresponds to an M5-brane wrapped on a Riemann surface with a selfdual flux. Via a 9-11 flip, we finally reinterpret the Omega deformation in terms of noncommutative geometry. Our solution generates all modified couplings of the Ω-deformed gauge theory, and also yields a geometric origin for the quantum spectral curve of the associated quantum integrable system.

On gauge theory and topological string in Nekrasov-Shatashvili limit

Abstract: We study the Nekrasov-Shatashvili limit of the $ \mathcal{N} $ = 2 supersymmetric gauge theory and topological string theory on certain local toric Calabi-Yau manifolds. In this limit one of the two deformation parameters ϵ1,2 of the Ω background is set to zero and we study the perturbative expansion of the topological amplitudes around the remaining parameter. We derive differential equations from Seiberg-Witten curves and mirror geometries, which determine the higher genus topological amplitudes up to a constant. We show that the higher genus formulae previously obtained from holomorphic anomaly equations and boundary conditions satisfy these differential equations. We also provide a derivation of the holomorphic anomaly equations in the Nekrasov-Shatashvili limit from these differential equations.

Resumming the string perturbation series

Abstract: We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local $$ {\mathrm{\mathbb{P}}}^1\times {\mathrm{\mathbb{P}}}^1 $$ , which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.

Exact solutions to quantum spectral curves by topological string theory

Abstract: We generalize the conjectured connection between quantum spectral problems and topological strings to many local almost del Pezzo surfaces with arbitrary mass parameters. The conjecture uses perturbative information of the topological string in the unrefined and the Nekrasov-Shatashvili limit to solve non-perturbatively the quantum spectral problem. We consider the quantum spectral curves for the local almost del Pezzo surfaces of $$ {\mathbb{F}}_2,{\mathbb{F}}_1,{\mathrm{\mathcal{B}}}_2 $$ and a mass deformation of the E 8 del Pezzo corresponding to different deformations of the three-term operators O1,1, O1,2 and O2,3. To check the conjecture, we compare the predictions for the spectrum of these operators with numerical results for the eigenvalues. We also compute the first few fermionic spectral traces from the conjectural spectral determinant, and we compare them to analytic and numerical results in spectral theory. In all these comparisons, we find that the conjecture is fully validated with high numerical precision. For local $$ {\mathbb{F}}_2 $$ we expand the spectral determinant around the orbifold point and find intriguing relations for Jacobi theta functions. We also give an explicit map between the geometries of $$ {\mathbb{F}}_0 $$ and $$ {\mathbb{F}}_2 $$ as well as a systematic way to derive the operators O m,n from toric geometries.

Exact duality symmetries in CFT and string theory

Abstract: The duality symmetries of WZW and coset models are discussed. The exact underlying symmetry responsible for semiclassical duality is identified with the symmetry under affine Weyl transformations. This identification unifies the treatment of duality symmetries and shows that in the compact and unitary case they are exact symmetries of string theory to all orders in α' and in the string coupling constant. Non-compact WZW models and cosets are also discussed. A toy models is analyzed suggesting that duality can survive as an exact symmetry.