A mini-course on topological strings

TL;DR

The notes provide a compact, self-contained introduction to topological string theory by deriving the $A$- and $B$-models from twisted $N=(2,2)$ supersymmetric theories and coupling them to gravity to form topological strings. They lay out the essential mathematical background (Calabi–Yau geometry, moduli, and cohomology) and the two central twists that lead to A- and B-models, which compute Gromov–Witten invariants and Dolbeault cohomology data, respectively, via holomorphic and anti-holomorphic structures. The document then surveys a wide range of applications, including relations to matrix models (DV correspondence), geometric transitions, black hole entropy, and holographic dualities, highlighting the central role of holomorphic anomalies and the genus expansion in extracting physical information. Overall, the notes articulate how topological strings provide tractable probes of stringy geometry with deep connections to four-dimensional physics, enumerative geometry, and nonperturbative phenomena. The framework offers actionable tools for computing exact quantities in CY compactifications and for exploring open/closed dualities and higher-genus corrections in a controlled setting.

Abstract

These are the lecture notes for a short course in topological string theory that I gave at Uppsala University in the fall of 2004. The notes are aimed at PhD students who have studied quantum field theory and general relativity, and who have some general knowledge of ordinary string theory. The main purpose of the course is to cover the basics: after a review of the necessary mathematical tools, a thorough discussion of the construction of the A- and B-model topological strings from twisted N=(2,2) supersymmetric field theories is given. The notes end with a brief discussion on some selected applications.

Figures (17)

• Figure 1: A discontinuous function, with three open intervals $A, B$, and $C$ and their inverse images. Note that the inverse image $f^{-1}(B)$ is not an open interval: the end point marked by the dot is included in the interval.
• Figure 2: The manifold $\mathcal{M}$ is covered by patches $U_{(a)}$, parameterized by coordinates $x_{(a)} \in \mathbb R^n$. To go from one coordinate to another, one uses the transition functions $\phi_{(ba)}$.
• Figure 3: (a) The sphere is an orientable manifold without handles or boundaries. (b) An orientable manifold with one boundary and one handle. (c) The Möbius strip: an unorientable manifold with one boundary and no handles.
• Figure 4: The 1-dimensional submanifolds $S^{(1)}$ and $S^{(2)}$ represent the same homology class, since their difference is the boundary of $U$.
• Figure 5: A simple example of a vector bundle $E$: the cylinder. The circle $\mathcal{M} = S^1$ is the base space (the other two circles are only drawn to visualize the geometry), and the fibers $W$ are lines. $S$ is a section of this vector bundle.