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.

Fluctuating topological invariants in string theory as an Abelian gauge theory

Abstract: It is shown that the topological invariants associated with the two-dimensional world-surface in string theory have nontrivial fluctuations around their nonexistent classical dynamics. Additionally it is proved that the underlying geometrical structure in a covariant phase space formulation for such topological string actions mimics entirely that of an Abelian gauge theory

On the Algebraic Structure of the Holomorphic Anomaly for N=2 Topological Strings

Abstract: The special geometry ($(t,{\bar t})$-equations) for twisted $N=2$ strings are derived as consistency conditions of a new contact term algebra. The dilaton appears in the contact terms of topological and antitopological operators. The holomorphic anomaly, which can be interpreted as measuring the background dependence, is obtained from the contact algebra relations.

Deformations of special geometry: in search of the topological string

Abstract: The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus $g\leq3$ contributions are constructed explicitly for a general class of effective actions associated with a special-K\"ahler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identification of a topological string free energy from an effective action is primarily based on conceptual arguments and does not involve any of its more specific properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that non-holomorphic deformations of special geometry can be dealt with consistently.

Refined open topological strings revisited

Abstract: In this work we verify consistency of refined topological string theory from several perspectives. First, we advance the method of computing refined open amplitudes by means of geometric transitions. Based on such computations we show that refined open BPS invariants are non-negative integers for a large class of toric Calabi-Yau threefolds: an infinite class of strip geometries, closed topological vertex geometry, and some threefolds with compact four-cycles. Furthermore, for an infinite class of toric geometries without compact four-cycles we show that refined open string amplitudes take form of quiver generating series. This generalizes the relation to quivers found earlier in the unrefined case, implies that refined open BPS states are made of a finite number of elementary BPS states, and asserts that all refined open BPS invariants associated to a given brane are non-negative integers in consequence of their relation to (integer and non-negative) motivic Donaldson-Thomas invariants. Non-negativity of motivic Donaldson-Thomas invariants of a symmetric quiver is therefore crucial in the context of refined open topological strings. Furthermore, reinterpreting these results in terms of webs of five-branes, we analyze Hanany-Witten transitions in novel configurations involving lagrangian branes.

Effective actions and topological strings : Off-shell mirror symmetry and mock modularity of multiple M5-branes

Abstract: This thesis addresses two different topics within the field of string theory. In the first part it is shown how Hodge-theoretic methods in conjunction with open string mirror symmetry can be used to compute non-perturbative effective superpotential couplings for type II/F-theory compactifications with D-branes and fluxes on compact Calabi-Yau manifolds. This is achieved by studying the flat structure of operators which derives from the open/closed B-model geometry. We analyze the variation of mixed Hodge structure of the relative cohomology induced by a family of divisors, which is wrapped by a D7-brane. This leads to a Picard-Fuchs system of differential operators, which can be used to compute the moduli dependence of the superpotential couplings as well as the mirror maps at various points in the open/closed deformation space. These techniques are used to obtain predictions for genuine A-model Ooguri-Vafa invariants of special Lagrangian submanifolds in compact Calabi-Yau geometries and real enumerative invariants of on-shell domain wall tensions. By an open/closed duality the system of differential equations can also be obtained from a gauged linear σ-model, which describes a non-compact Calabi-Yau four-fold compactification without branes. This is used in the examples of multi-parameter models to study the various phases of the combined open/closed deformation space. It is furthermore shown how the brane geometry can be related to a F-theory compactification on a compact Calabi-Yau four-fold, where the Hodge-theoretic techniques can be used to compute the G-flux induced GukovVafa-Witten potential. The dual F-theory picture also allows to conjecture the form of the Kahler potential on the full open/closed deformation space. In the second part we analyze the background dependence of theories which derive from multiple wrapped M5-branes. Using the Kontsevich-Soibelman wall-crossing formula and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes which are wrapped around a rigid divisor inside a compact Calabi-Yau manifold. The non-holomorphicity of the modified elliptic genus in this situation is traced back to the contribution of non-trivial BPS bound states of M5-branes. As a byproduct a result from pure mathematics obtained by Gottsche is re-derived in physical terms, which concerns the structure of the moduli spaces of stable sheaves on certain complex surfaces. The anomaly equation fulfilled by the modified elliptic genus is shown to be consistent with the holomorphic anomaly equations observed in the context of N = 4 topological VafaWitten theory on P2 and theories of E-strings obtained from wrapping M5-branes on del Pezzo surfaces. In selected examples it is argued how the holomorphic anomaly equation supplemented with appropriate boundary conditions can be used to explicitly compute the BPS degeneracies for certain charges. This work is based on the original publications [1, 2, 3, 4].