Young Researchers School 2024 Maynooth: Lectures on CFT, BCFT and DCFT

TL;DR

This work provides a structured, self-contained tour of two-dimensional conformal field theory, starting from the conformal group and its Witt/Virasoro algebras to the construction of primary/descendant fields, OPEs, and fusion. It then extends to boundary and defect extensions—BCFT and DCFT—through tools like the folding trick, Ishibashi and Cardy states, and the Petkova-Zuber and Verlinde formalisms. A central thread is the role of modular invariance and the S- and T-transformations in constraining spectra and boundary/defect data, with Ising model examples illustrating the construction. The notes culminate in a coherent framework for boundary states, bulk-boundary couplings, and topological defects, highlighting their relevance to condensed matter, string theory, and beyond.

Abstract

These notes were presented at the Young Researchers School (YRS) in Maynooth in April 2024 and provide an introduction to Conformal Field Theory CFT, Boundary Conformal Field Theory (BCFT) and Defect Conformal Field Theory (DCFT). This class is mostly self-contained and includes exercises with solutions. The first part of these notes is concerned with the basics of CFT, and was taught by the author during the pre-school for the YRS 2024. Here the aim is to convey the notion of conformal families, their fusion and the construction of partition functions. The second part of these notes is dedicated to boundaries and defects in CFT and was presented by the author at the main school. As far as boundaries are concerned, emphasis is placed on boundary operators and their state spaces, as well as the boundary state formalism with the Cardy constraint. Topological defects are discussed in analogy, i.e. defect state spaces and the relevant consistency constraint are derived. Verlinde lines are constructed as their simplest solution and their properties are inspected.

Figures (12)

• Figure 1: The Exponential map \ref{['ExpMap']} maps a cylinder onto the plane. Constant time slices on the cylinder are mapped into concentric circles on the plane and time flows radially outward.
• Figure 2: Two vectors $\alpha_{1,2}$ (red) span the green lattice. Their fundamental domain is shaded in green. Two other vectors (purple), $\beta_{1,2}$ span the same lattice. The modular parameters of $\alpha_{1,2}$ and $\beta_{1,2}$ are related by $\tau\to\tau+\tau_1$, which is a modular transformation.
• Figure 3: The folding trick. The semicircle traced by the antiholomorphic sector is reflected at the boundary into the lower half-plane. The gluing condition \ref{['glueT']} prescribes the analytic continuation justifying this move.
• Figure 4: By virtue of the folding trick, a bulk field $\phi_{i\bar{\imath}}$ can be viewed as two individual chiral fields $\phi_i$ in the upper half-plane and $\phi_{\bar{\imath}}$ in the lower half-plane. The latter becomes a mirror charge. As shown in the third panel, when approaching the boundary, the two chiral fields can be expanded in terms of operators confined to the boundary, as in \ref{['bulkBdyOPE']}.
• Figure 5: Left: A strip with two distinct boundary conditions $\alpha$ and $\beta$. Right: On the plane we see that this setup is induced by a BCCO $\psi^{\alpha\beta}$ situated on the real line.