# Boundary CFT A boundary CFT is a CFT that lives on a manifold with a boundary. It is not as simple as it sounds because a conformal transformation will generally relocate the boundary. Studying the necessary boundary conditions and the new data that specify the theory is nontrivial. ## Data When there are no boundaries, 2d CFTs are completely specified by the left and right [[0033 Central charge|central charges]] $(c_L,c_R)$ and the set of data $\{h_i, \bar{h}_i, C_{i j k}\}$, where $h_i$ and $\bar{h}_i$ are the conformal weights and $C_{ijk}$ are the [[0030 Operator product expansion|OPE]] coefficients. In the presence of a boundary, only the diagonal Virasoro subalgebra is preserved, which sets $c_L=c_R$. Now, the data needed to specify the theory become$\{h_i,\bar{h}_i,h_I;C_{ijk}, D_{i}^{(a)I}, B_{IJ}^{(abc)K}\},$where the $C$, $D$ and $B$ are called bulk-to-bulk, bulk-to-boundary, and boundary-to-boundary structure constants, respectively. We have used capital letters to label boundary operators. Notice there is only one conformal weight for a boundary operator because there is only one [[0032 Virasoro algebra|Virasoro algebra]] on the boundary. ## Pair of pants and more In addition to pairs of pants, the building blocks of a BCFT correlation function also include two more types. They are pictured below: ![[NumasawaTsiares2022_fig1.png]] where dotted lines represent a boundary operator insertion, and an orange circle represents a bulk operator insertion. The three pictures correspond to the three types of structure constants, respectively. ## Explicit models - the doubling trick - [[1984#Cardy]] - [[0096 Rational CFT|RCFT]] - [[1989#Cardy]]: fusion rules, Verlinde formula - [[1991#Cardy, Lewellen]] - [[1998#Runkel]]: A-series - [[1999#Runkel]]: D-series - [[1999#Behrend, Pearce, Petkova, Zuber]] - Ishibashi states - [[1989#Ishibashi]] - [[0562 Liouville theory|Liouville]] - [[2001#Hosomichi]] - [[2001#Ponsot, Teschner]] - [[2003#Ponsot]]: agrees with Hosomichi - universal dynamics - [[2021#Kusuki]] - [[2022#Numasawa, Tsiares]] - sewing constraints - [[1991#Cardy, Lewellen]] - [[1992#Lewellen]]: complete list of sewing conditions - [[1996#Pradisi, Sagnotti, Stanev]]: claims to complete and provide corrections to the work of Lewellen (but see below) - [[1999#Behrend, Pearce, Petkova, Zuber]]: says that the results of Pradisi et al are the same as Lewellen; also comments that a simpler expression than Lewellen can be stated - [[2006#Fjelstad, Fuchs, Runkel, Schweigert]]: for identical boundary conditions, all correlators are determined by one-, two,- and three-point correlators on the disk - useful: [[1999#Runkel]] appendix A contains a detailed derivation - global symmetry breaking and entanglement asymmetry - [[2024#Kusuki, Murciano, Ooguri, Pal]] - applications - [[0301 Entanglement entropy|entanglement entropy]] - [[2014#Ohmori, Tachikawa]]: original proposal - [[2025#Roy, Lukyanov, Saleur]]: numerical test ## Related - [[0181 AdS-BCFT]] \[Acknowledgement: I thank Zixia Wei for explaining many things about BCFT.\]