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 Vertex Function in Witten's Formulation of String Field Theory

Abstract: Abstract Conformal mapping techniques are employed to obtain the oscillator representation of the three open string vertex in Witten's formulation of string field theory. It is argued that the physical state couplings are identical to those of the dual resonance model, and this is confirmed for the low lying states with our explicit construction.

Supersymmetric Gauge Theories from String Theory

Abstract: The subject of this thesis are various ways to construct four-dimensional quantum field theories from string theory. In a first part we study the generation of a supersymmetric Yang-Mills theory, coupled to an adjoint chiral superfield, from type IIB string theory on non-compact Calabi-Yau manifolds, with D-branes wrapping certain subcycles. Properties of the gauge theory are then mapped to the geometric structure of the Calabi-Yau space. In particular, the low energy effective superpotential, governing the vacuum structure of the gauge theory, can in principle be calculated from the open (topological) string theory. Unfortunately, in practice this is not feasible. Quite interestingly, however, it turns out that the low energy dynamics of the gauge theory is captured by the geometry of another non-compact Calabi-Yau manifold, which is related to the original Calabi-Yau by a geometric transition. Type IIB string theory on this second Calabi-Yau manifold, with additional background fluxes switched on, then generates a four-dimensional gauge theory, which is nothing but the low energy effective theory of the original gauge theory. As to derive the low energy effective superpotential one then only has to evaluate certain integrals on the second Calabi-Yau geometry. This can be done, at least perturbatively, and we find that the notoriously difficult task of studying the low energy dynamics of a non-Abelian gauge theory has been mapped to calculating integrals in a well-known geometry. It turns out, that these integrals are intimately related to quantities in holomorphic matrix models, and therefore the effective superpotential can be rewritten in terms of matrix model expressions. Even if the Calabi-Yau geometry is too complicated to evaluate the geometric integrals explicitly, one can then always use matrix model perturbation theory to calculate the effective superpotential. This intriguing picture has been worked out by a number of authors over the last years. The original results of this thesis comprise the precise form of the special geometry relations on local Calabi-Yau manifolds. We analyse in detail the cut-off dependence of these geometric integrals, as well as their relation to the matrix model free energy. In particular, on local Calabi-Yau manifolds we propose a pairing between forms and cycles, which removes all divergences apart from the logarithmic one. The detailed analysis of the holomorphic matrix model leads to a clarification of several points related to its saddle point expansion. In particular, we show that requiring the planar spectral density to be real leads to a restriction of the shape of Riemann surfaces, that appears in the planar limit of the matrix model. This in turns constrains the form of the contour along which the eigenvalues have to be integrated. All these results are used to exactly calculate the planar free energy of a matrix model with cubic potential. The second part of this work covers the generation of four-dimensional supersymmetric gauge theories, carrying several important characteristic features of the standard model, from compactifications of eleven-dimensional supergravity on $G_2$-manifolds. If the latter contain conical singularities, chiral fermions are present in the four-dimensional gauge theory, which potentially lead to anomalies. We show that, locally at each singularity, these anomalies are cancelled by the non-invariance of the classical action through a mechanism called ``anomaly inflow". Unfortunately, no explicit metric of a compact G_2-manifold is known. Here we construct families of metrics on compact weak G_2-manifolds, which contain two conical singularities. Weak G_2-manifolds have properties that are similar to the ones of proper G_2-manifolds, and hence the explicit examples might be useful to better understand the generic situation. Finally, we reconsider the relation between eleven-dimensional supergravity and the E_8\times E_8-heterotic string. This is done by carefully studying the anomalies that appear if the supergravity theory is formulated on a ten-manifold times the interval. Again we find that the anomalies cancel locally at the boundaries of the interval through anomaly inflow, provided one suitably modifies the classical action.

Framing the Di-Logarithm (over Z)

Abstract: Motivated by their role for integrality and integrability in topological string theory, we introduce the general mathematical notion of "s-functions" as integral linear combinations of poly-logarithms. 2-functions arise as disk amplitudes in Calabi-Yau D-brane backgrounds and form the simplest and most important special class. We describe s-functions in terms of the action of the Frobenius endomorphism on formal power series and use this description to characterize 2-functions in terms of algebraic K-theory of the completed power series ring. This characterization leads to a general proof of integrality of the framing transformation, via a certain orthogonality relation in K-theory. We comment on a variety of possible applications. We here consider only power series with rational coefficients; the general situation when the coefficients belong to an arbitrary algebraic number field is treated in a companion paper.