# Non-orientable CFT A 2d CFT can be defined on an arbitrary Riemann surface. It is interesting when the Riemann surface is non-orientable. A Riemann surface is specified by the number of boundaries, the genus, and the number of crosscaps. A CFT on oriented surfaces with boundaries must satisfy six [[0602 Moore-Seiberg construction|Moore-Seiberg]] consistency relations. If the surface is non-oriented, there are three more: - crosscap constraint (two bulk operators on $\mathbb{RP}_2$) - Mobius constraint (one boundary operators on Mobius strip) - Klein bottle constraint (one bulk operator on Klein bottle) ## Refs - [[0602 Moore-Seiberg construction|sewing constraints]] - [[1993#Fioravanti, Pradisi, Sagnotti]]: gives all constraints in picture, but only the crosscap one explicitly - [[1995#Pradisi, Sagnitti, Stanev (Mar)]]: $SU(2)$, diagonal. - [[1995#Pradisi, Sagnotti, Stanev (Jun)]]: $SU(2)$, non-diagonal; comments on two ways of looking at the crosscap constraint (solve $X_i$ from $C_{ijk}$ or vice versa) - Mobius and Klein bottle constraints (implicitly) used in - [[1990#Bianchi, Sagnotti]] - [[1991#Bianchi, Sagnotti]] - [[1991#Bianchi, Pradisi, Sagnotti]] - [[1992#Bianchi, Pradisi, Sagnotti]] - reviews - [[2001#Stanev (Lectures)]]: gives all constraints explicitly for partition functions (no insertions) - book by Blumenhagen, Lust, and Theisen, Sec.6.7 - universal dynamics - [[2020#Tsiares]] - TQFT - [[2003#Fuchs, Runkel, Schweigert]] - holographic bulk - [[2015#Verlinde]]: local bulk states formulated in terms of crosscap states - [[2016#Maloney, Ross]]: argues to include singular bulk saddles - [[2016#Nakayama, Ooguri]]: weakly coupled gravity; finds that bulk causality and boundary bootstrap give incompatible superposition structure - [[2016#Lewkowycz, Turiaci, Verlinde]]: bulk locality, HKLL from crossing, soft theorems - [[2024#Wei (Z)]]: proposes to include dS branes in the bulk theory; new saddles (with end-of-the-world dS branes) for the non-orientable CFTs - examples - [[1994#Petkova, Zuber]]: $sl(2)$ minimal models; determine parity signs - [[2002#Hikida]]: Liouville theory; argues that the parity $\epsilon_i$ must be $+1$ from consistency of OPE; uses crosscap constraint to find that the solution is a linear combination of two functions, then use Mobius constraint to determine the combination