[Paper Review] A mini-course on topological strings

Marcel Vonk|ArXiv.org|Apr 18, 2005

Black Holes and Theoretical PhysicsPhysics and Astronomy47 references58 citations

TL;DR

This mini-course provides a pedagogical introduction to topological string theory for graduate students with background in quantum field theory and general relativity, focusing on the construction of A- and B-model topological strings via twisting N=(2,2) supersymmetric field theories on Calabi-Yau manifolds. The key contribution is a systematic derivation of how topological string amplitudes encode black hole entropy through the relation $ Z_{BH} = |Z_{top}|^2 $, linking topological strings to extremal black hole physics via matrix models and geometric transitions.

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.

Motivation & Objective

• To provide a self-contained, accessible introduction to topological string theory for physicists and mathematicians with foundational knowledge in QFT, GR, and differential geometry.
• To clarify the mathematical and physical foundations of topological field theories, especially cohomological field theories and twisting procedures.
• To establish the link between topological string amplitudes and physical phenomena such as black hole entropy and F-term corrections in N=2 effective theories.
• To connect topological strings to matrix models and geometric transitions, illustrating dualities in string theory.
• To motivate the conjecture of topological M-theory as a unifying framework for lower-dimensional topological theories.

Proposed method

• Derive the A- and B-model topological strings by twisting N=(2,2) supersymmetric field theories in two dimensions, using R-symmetry and virtual dimension analysis.
• Apply cohomological field theory techniques, including descent equations and BRST-like cohomology, to define observables and correlation functions.
• Use Dolbeault and de Rham cohomology to analyze the moduli spaces of Calabi-Yau manifolds, particularly Kähler and complex structure moduli.
• Introduce the holomorphic anomaly equation to describe the dependence of amplitudes on moduli, crucial for higher-genus computations.
• Construct topological gravity coupling to compute partition functions and correlation functions in the topological string context.
• Establish dualities via geometric transitions, particularly the conifold transition, and relate topological strings to matrix quantum mechanics with non-polynomial potentials.

Experimental results

Research questions

• RQ1 How can the A- and B-model topological strings be systematically derived from twisted N=(2,2) supersymmetric field theories?
• RQ2 What is the role of the holomorphic anomaly in the computation of higher-genus topological string amplitudes?
• RQ3 How do topological string amplitudes encode the entropy of extremal black holes in N=2 supergravity?
• RQ4 In what way are topological strings equivalent to matrix models, and how does this equivalence describe black hole partition functions?
• RQ5 Is there a unifying topological theory—topological M-theory—that encodes all known topological string dualities?

Key findings

• The partition function of extremal N=2 black holes is given by $ Z_{BH} = |Z_{top}|^2 $, where $ Z_{top} $ is the B-model topological string partition function on the Calabi-Yau manifold at the black hole horizon.
• Quantum-corrected black hole entropy is exactly captured by higher-genus topological string amplitudes, as shown by Ooguri, Strominger, and Vafa.
• A matrix model with potential $ W(x) = -x^2 + 1/x^2 $ reproduces the same partition function as the topological string on a noncompact Calabi-Yau, confirming the duality.
• The matrix model description of the black hole exhibits a finite temperature in Euclidean time, yet the black hole itself is extremal and non-radiating, leading to a non-trivial thermodynamic interpretation.
• The topological string partition function on a conifold geometry matches that of a D-brane system, demonstrating open/closed duality via geometric transition.
• A candidate for topological M-theory has been proposed using Hitchin's formalism, suggesting a possible unifying 7-dimensional topological theory for all lower-dimensional topological strings and their dualities.