On the topological and crosscap entropies in non-oriented RCFTs

Abstract

We establish a relation between the boundary and the topological entropies for the conformal minimal models in some of the simplest models of the unitary A–A series. We show that in these models the boundary entropy is a difference of topological entropies. Furthermore, we define the crosscap entropy as the analog to the boundary entropy in non-oriented theories. The crosscap entropy is defined as the logarithm of the degeneracy of the ground state due to the presence of crosscaps, and it can be expressed in terms of the crosscap coefficients. This crosscap entropy has not an explicit relation to the topological entropy as the boundary entropy. However, we propose a new quantity, $\hat{S} = \text {ln}P_{0i}$ defined in terms of the modular transformation P between the open and closed channel of the Möbius partition function. With this quantity, the crosscap entropy can be regarded as a difference of entropies, very similar to the boundary entropy. We also compute the left-right entanglement entropy (LREE) for crosscap states, and we express it in terms of $\hat{S}$. An explicit example of the LREE of the crosscap state in Wess–Zumino–Witten is carried out.

Appendix

The matrix S for minimal models in the A–A series is given in [32, 51]. The matrix P can be constructed by means of its definition in terms of the matrix S and T, although an simple expression for it has been given in [33]. For the Ising model, the crosscap coefficients are [24]

$$\begin{aligned} \Gamma _{0 0} = \sqrt{\frac{2 + \sqrt{2}}{2}} , \Gamma _{0 \varepsilon } = \sqrt{\frac{2-\sqrt{2}}{2}} , \Gamma _{0 \sigma }= 0 . \end{aligned}$$

(A.1)

In this case, there is only a crosscap corresponding to the parity symmetry ${{\mathscr {P}}}_0$

$$\begin{aligned} |C_{{\mathscr {P}}_0}\rangle = \Gamma _{0 0} \vert C_0 \rangle \!\rangle + \Gamma _{0 \varepsilon } \vert C_\varepsilon \rangle \!\rangle , \end{aligned}$$

(A.2)

From (4.7), the LREE is

$$\begin{aligned} S_{|C_{{\mathscr {P}}_0}\rangle } =\frac{\pi \ell }{12 \varepsilon } - \frac{2+\sqrt{2}}{4} \text {ln} \Big (\frac{2 + \sqrt{2}}{2}\Big ) -\frac{2-\sqrt{2}}{2} \text {ln} \Big (\frac{2 - \sqrt{2}}{2}\Big ) \end{aligned}$$

(A.3)

For the Tricritical model, the primary fields the crosscap coefficients are

$$\begin{aligned} \Gamma _{00}&= \frac{ \sqrt{2 + \sqrt{2}}\, s_1}{\sqrt{s_2}} ,&\Gamma _{0 \epsilon }&= \frac{ \sqrt{2-\sqrt{2}} \,s_2}{\sqrt{s_1}} ,\nonumber \\ \Gamma _{0 \epsilon ^{\prime }}&= \frac{ \sqrt{2+\sqrt{2}} \,s_2}{\sqrt{s_1}} ,&\Gamma _{0 \epsilon ^{\prime \prime } }&= \frac{ \sqrt{2-\sqrt{2}} \,s_1}{\sqrt{s_2}} ,\nonumber \\ \Gamma _{0 \sigma }&=0 ,&\Gamma _{0 \sigma ^{\prime }}&=0 . \end{aligned}$$

(A.4)

These results agree with the quotients between them found using the sewing constraints in [41].

The crosscap state is

$$\begin{aligned} |C_{{\mathscr {P}}_0}\rangle = \Gamma _{00} \vert C_0 \rangle \!\rangle + \Gamma _{0 \epsilon } \vert C_\varepsilon \rangle \!\rangle + \Gamma _{0 \epsilon ^{\prime }}\vert C_{\varepsilon ^{\prime }} \rangle \!\rangle + \Gamma _{0 \epsilon ^{\prime \prime }}\vert C_{\varepsilon ^{\prime \prime }} \rangle \!\rangle , \end{aligned}$$

(A.5)

The corresponding LREE is given by

$$\begin{aligned} S_{|C_{{\mathscr {P}}_0}\rangle }= & {} \frac{7\pi \ell }{60 \,\varepsilon } - 4 \Big [s_2^2\, \text{ ln }\Big (\frac{s_2^2}{s_1}\Big ) - s_1^2\,\text{ ln }\Big (\frac{s_1^2}{s_2}\Big )\Big ]\nonumber \\&+\, (s_1^2+s_2^2) \Big [(2-\sqrt{2})\text{ ln }(2-\sqrt{2}) + (2+\sqrt{2})\text{ ln }(2+\sqrt{2}) \Big ] . \end{aligned}$$

(A.6)

For the Tetracritical model, we label the different conformal fields with $i = 0, \ldots 9$ it the order given for the conformal weights in 3.1. The crosscap coefficients are:

$$\begin{aligned} \Gamma _{0 0}&= \frac{\sqrt{3}+1}{3^{1/4}}\frac{s_1}{\sqrt{s_2}} ,&\Gamma _{0 1}&= \frac{\sqrt{3}-1}{3^{1/4}}\frac{s_2}{\sqrt{s_1}} ,&\Gamma _{0 3}&= \frac{\sqrt{3}+1}{3^{1/4}}\frac{s_2}{\sqrt{s_1}} , \nonumber \\ \Gamma _{0 5}&= \frac{\sqrt{2}}{3^{1/4}}\frac{s_2}{\sqrt{s_1}} ,&\Gamma _{0 6}&= \frac{\sqrt{3}-1}{3^{1/4}}\frac{s_1}{\sqrt{s_2}}&\Gamma _{0 8}&= \frac{\sqrt{2}}{3^{1/4}}\frac{s_1}{\sqrt{s_2}} ,\nonumber \\ \end{aligned}$$

(A.7)

and $\Gamma _{0 2}= \Gamma _{0 4} = \Gamma _{0 7}= \Gamma _{0 9}=0$.

The crosscap state and its corresponding LREE can be obtained also for the Tetracritical model. This can be performed in a straightforward way and will not be carried out here.

The matrix P for the $SU(2)_k$ WZW model for k even is [48], then

$$\begin{aligned} P_{jl}&= \frac{2}{\sqrt{k+2}} \text {sin} \frac{\pi (2j+1)(2l+1)}{2(k+2)} , \quad \quad j+l \in {\mathbb {Z}} \end{aligned}$$

(A.8)

$$\begin{aligned} {{{\hat{\mathcal{D}}}}}&= \frac{1}{P_{00}}= \frac{\sqrt{k+2}}{2} \text {csc}\Big (\frac{\pi }{2(k+2)}\Big ) ,&{\mathcal{D}}&= \frac{1}{S_{00}}= \sqrt{\frac{k}{k+2}} \text {csc}\Big (\frac{\pi }{k+2}\Big ) \nonumber \\ {{\hat{d}}_j}&= \frac{P_{0j}}{P_{00}}= \text {csc}\Big (\frac{\pi }{2(k+2)}\Big ) \text {sin}\Big ( \frac{(2 j +1)\pi }{2(k+2)} \Big ) \end{aligned}$$

(A.9)