Explicit construction of $N=2$ superconformal orbifolds

Abstract

Following Gepner’s approach, we propose a construction of models of tensor product orbifolds of the minimal models of two-dimensional field theory with $N=2$ superconformal symmetry. To build models that satisfy the requirements of modular invariance, we use a spectral flow transformation. We demonstrate this construction with a specific example and show that its application ensures the modular invariance of the partition function simultaneously with the mutual locality of the fields of the theory under consideration.

Appendix

Here, we derive the rules for modular transformations of characters in Eqs. (73) and (75) and shift relations (77) using the spectral flow and modular properties of the minimal model characters.

We first examine the modular properties of characters of an individual minimal model. Namely, for each pair $l$, $t$ labeling a representation of the $N=2$ superconformal minimal model and for each pair $n,p\in \frac{1}{2}\mathbb{Z}$, we introduce the character

$$ NS^+_{l,t+n,p}\equiv e^{-i 2\pi\epsilon}\operatorname{Tr}_{\mathcal{H}_{l,t+n}}\biggl\{\exp{\biggl[i 2\pi \biggl(L_0-\frac{c}{24}\biggr)\tau+i 2\pi J_0(z+p)\biggr]}\biggr\}.$$

(90)

In this expression, the trace is taken using the twisted representation of $\mathcal{H}_{l,t+n}$ with the insertion of the operator $\exp{[i 2\pi (z+p)J_0]}$. The space $\mathcal{H}_{l,t+n}$ consists of vectors of the space $\mathcal{H}_{l,t}$ on which the $n$-twisted $N=2$ Virasoro superalgebra is acting. Thus, for a half-integer $n$, we obtain the character of a representation in the R sector. It is also easy to see that for the NS-sector characters, half-integer $p$ generate insertions of the operator $(-1)^{F}$ up to a common phase factor arising from the charge of the corresponding highest-weight vector.

Using (33) and (38), we obtain

$$ NS^+_{l,t+n,p}\biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{cz^2}{6\tau}\biggr)= \exp{\biggl[i 2\pi\frac{c}{3}pn\biggr]} S^{l',t'}_{l,t}NS^+_{l',t'+p,-n}(\tau,z,\epsilon)$$

(91)

and

$$\begin{aligned} \, NS^+_{l,t+n,p}(\tau+1,z,\epsilon) ={}&\exp{\biggl[i 2\pi \biggl(\Delta_{l,t}-\frac{c}{24}\biggr)\biggr]}\times {} \nonumber \\ &\times \exp{\biggl[-i\pi\biggl(\frac{c}{3}n(n+1)+Q_{l,t}\biggr)\biggr]} NS^+_{l,t+n,p+n+1/2}(\tau,z,\epsilon). \end{aligned}$$

(92)

From definitions (90) and (13), we obtain the shift relations for integer $q$:

$$ NS^+_{l,t+n,p+q}=\exp{\biggl[i 2\pi q\biggl(\frac{c}{3}n+Q_{l,t}\biggr)\biggr]} NS^+_{l,t+n,p}, \qquad q\in \mathbb{Z}.$$

(93)

For the characters with integer $n,p$, we introduce the special notation

$$\begin{aligned} \, &NS^-_{l,t+n,p}\equiv e^{-i 2\pi\epsilon}\operatorname{Tr}_{\mathcal{H}_{l,t+n}}\biggl\{(-1)^{F}\exp{\biggl[i 2\pi \biggl(L_0-\frac{c}{24}\biggr)\tau+i 2\pi J_0(z+p)\biggr]}\biggr\}, \\ &R^+_{l,t+n,p}\equiv e^{-i 2\pi\epsilon}\operatorname{Tr}_{\mathcal{H}_{l,t+n+1/2}}\biggl\{\exp{\biggl[i 2\pi \biggl(L_0-\frac{c}{24}\biggr)\tau+i 2\pi J_0(z+p)\biggr]}\biggr\}, \\ &R^-_{l,t+n,p} \equiv e^{-i 2\pi\epsilon}\operatorname{Tr}_{\mathcal{H}_{l,t+n+1/2}}\biggl\{(-1)^{F}\exp{\biggl[i 2\pi \biggl(L_0-\frac{c}{24}\biggr)\tau+i 2\pi J_0(z+p)\biggr]}\biggr\}. \end{aligned}$$

(94)

Then we can write

$$\begin{aligned} \, &NS^-_{l,t+n,p}=\exp{\biggl[-i \pi \biggl(\frac{c}{3}n+Q_{l,t}\biggr)\biggr]}NS^+_{l,t+n,p+1/2}, \\ &R^\pm_{l,t+n,p}=NS^\pm_{l,t+n+1/2,p}, \end{aligned}$$

(95)

and obtain shift relations for the R-sector characters

$$ R^\pm_{l,t+n,p+q}=\exp{\biggl[i 2\pi q\biggl(\frac{c}{3}\biggl(n+\frac{1}{2}\biggr)+Q_{l,t}\biggr)\biggr]} R^\pm_{l,t+n,p}, \qquad q\in \mathbb{Z}.$$

(96)

Based on rules (91), (92) and relation (93), we can conclude that for integer $n$ and $p$, characters (90) and (94) transform as

$$\begin{aligned} \, & NS^+_{l,t+n,p}\biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{cz^2}{6\tau}\biggr)= \exp{\biggl[i 2\pi\frac{c}{3}pn\biggr]}S^{l',t'}_{l,t}NS^+_{l',t'+p,-n}(\tau,z,\epsilon), \\ & NS^-_{l,t+n,p}\biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{cz^2}{6\tau}\biggr)= \exp{\biggl[i 2\pi\frac{c}{3}np\biggr]}S^{l',t'+1/2}_{l,t}R^+_{l',t'+p,-n}(\tau,z,\epsilon), \\ & R^+_{l,t+n,p}\biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{cz^2}{6\tau}\biggr)= \exp{\biggl[i 2\pi\frac{c}{3}np\biggr]}S^{l',t'}_{l,t+1/2}NS^-_{l',t'+p,-n}(\tau,z,\epsilon), \\ & R^-_{l,t+n,p}\biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{cz^2}{6\tau}\biggr)= -i\exp{\biggl[i 2\pi\frac{c}{3}np\biggr]}S^{l',t'+1/2}_{l,t+1/2}R^-_{l',t'+p,-n}(\tau,z,\epsilon) \end{aligned}$$

(97)

and

$$\begin{aligned} \, NS^+_{l,t+n,p}(\tau+1,z,\epsilon)&= \exp{\biggl[i 2\pi\biggl(\Delta_{l,t}-\frac{c}{6}n^2-\frac{c}{24}\biggr)\biggr]}NS^-_{l,t+n,p+n}(\tau,z,\epsilon), \\ NS^-_{l,t+n,p}(\tau+1,z,\epsilon)&= \exp{\biggl[i 2\pi\biggl(\Delta_{l,t}-\frac{c}{6}n^2-\frac{c}{24}\biggr)\biggr]}NS^+_{l,t+n,p+n}(\tau,z,\epsilon), \\ R^+_{l,t+n,p}(\tau+1,z,\epsilon)&= \exp{\biggl[i 2\pi\biggl(\Delta_{l,t}-\frac{c}{6}n^2+\frac{1}{2}Q_{l,t}\biggr)\biggr]}R^+_{l,t+n,p+n}(\tau,z,\epsilon), \\ R^-_{l,t+n,p}(\tau+1,z,\epsilon)&= \exp{\biggl[i 2\pi\biggl(\Delta_{l,t}-\frac{c}{6}n^2+\frac{1}{2}Q_{l,t}\biggr)\biggr]}R^-_{l,t+n,p+n}(\tau,z,\epsilon). \end{aligned}$$

(98)

The modular properties and shift relations for the products of minimal-model characters in (71), arising in the construction of the partition function of the orbifold by group (65), now follow by a direct generalization of the corresponding expressions for the minimal model. The shift relations take the form

$$\begin{aligned} \, &NS^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},(p+s)\vec{\alpha}+(q+r)\vec{\beta}}=\exp{[i 2\pi s(\langle\vec{\alpha},n\vec{\alpha}+m\vec{\beta}\rangle+Q_{\vec{l},\vec{t}})]}\exp{[i 2\pi r(\langle\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle+Q^{\beta}_{\vec{l},\vec{t}})]}\times{} \\ &\hphantom{NS^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},(p+s)\vec{\alpha}+(q+r)\vec{\beta}}{}=} \times NS^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}}, \\ &R^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},(p+s)\vec{\alpha}+(q+r)\vec{\beta}} =\exp{[i 2\pi s(\langle\vec{\alpha},(n+1/2)\vec{\alpha}+m\vec{\beta}\rangle +Q_{\vec{l},\vec{t}})]} \times {} \\ &\hphantom{R^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},(p+s)\vec{\alpha}+(q+r)\vec{\beta}}={}}\times \exp{[i 2\pi r(\langle\vec{\beta},(n+1/2)\vec{\alpha}+m\vec{\beta}\rangle +Q^{\beta}_{\vec{l},\vec{t}})]} R^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}}, \end{aligned}$$

(99)

where

$$ \langle\vec{\mu},\vec{\nu}\rangle\equiv\sum_i\frac{c_i}{3}\mu_i\nu_i Q^{\beta}_{\vec{l},\vec{t}}\equiv\sum_i\beta_iQ_{l_i,t_i}.$$

(100)

The generalizations of (97) and (98) are given by the expressions

$$\begin{aligned} \, &NS^+_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}} \biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{c_{\mathrm{tot}}z^{2}}{6\tau}\biggr)={} \\ &=\exp{[i 2\pi\langle p\vec{\alpha}+q\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle]} S^{\vec{l}',\vec{t}'}_{\vec{l},\vec{t}}NS^+_{\vec{l}',\vec{t}'+p\vec{\alpha}+q\vec{\beta},-n\vec{\alpha}-m\vec{\beta}}(\tau,z,\epsilon), \\ &NS^-_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}} \biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{c_{\mathrm{tot}}z^{2}}{6\tau}\biggr)={} \\ &=\exp{[i 2\pi\langle p\vec{\alpha}+q\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle]} S^{\vec{l}',\vec{t}'+\vec{\alpha}/2}_{\vec{l},\vec{t}}R^+_{\vec{l}',\vec{t}'+p\vec{\alpha}+q\vec{\beta},-n\vec{\alpha}-m\vec{\beta}}(\tau,z,\epsilon), \\ &R^+_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}} \biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{c_{\mathrm{tot}}z^{2}}{6\tau}\biggr)={} \\ &=\exp{[i 2\pi\langle p\vec{\alpha}+q\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle]} S^{\vec{l}',\vec{t}'}_{\vec{l},\vec{t}+\vec{\alpha}/2}NS^-_{\vec{l}',\vec{t}'+p\vec{\alpha}+q\vec{\beta},-n\vec{\alpha}-m\vec{\beta}}(\tau,z,\epsilon), \\ &R^-_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}} \biggl(-\frac{1}{\tau},\frac{z}{\tau},\epsilon+\frac{c_{\mathrm{tot}}z^{2}}{6\tau}\biggr)={} \\ &=-i\exp{[i 2\pi\langle p\vec{\alpha}+q\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle]} S^{\vec{l}',\vec{t}'+\vec{\alpha}/2}_{\vec{l},\vec{t}+\vec{\alpha}/2} R^-_{\vec{l}',\vec{t}'+p\vec{\alpha}+q\vec{\beta},-n\vec{\alpha}-m\vec{\beta}}(\tau,z,\epsilon) \end{aligned}$$

(101)

and

$$\begin{aligned} \, &NS^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}}(\tau+1,z,\epsilon)= \\ &=\exp{\biggl[i 2\pi\biggl(\Delta_{\vec{l},\vec{t}}-\frac{c_{\mathrm{tot}}}{24}\biggr)\biggr]}\exp{[-i \pi|n\vec{\alpha}+m\vec{\beta}|^{2}]} NS^{\mp}_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},(p+n)\vec{\alpha}+(m+q)\vec{\beta}}(\tau,z,\epsilon), \\ &R^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},p\vec{\alpha}+q\vec{\beta}}(\tau+1,z,\epsilon)= \\ &=\exp{\biggl[i 2\pi\biggl(\Delta_{\vec{l},\vec{t}}+\frac{1}{2}Q_{\vec{l},\vec{t}}\biggr)\biggr]} \exp{[-i \pi|n\vec{\alpha}+m\vec{\beta}|^{2}]} R^\pm_{\vec{l},\vec{t}+n\vec{\alpha}+m\vec{\beta},(p+n)\vec{\alpha}+(m+q)\vec{\beta}}(\tau,z,\epsilon). \end{aligned}$$

(102)

In our case,

$$\begin{aligned} \, &\langle p\vec{\alpha}+q\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle =3pn+6(pm+qn)+\frac{102}{5}qm\Rightarrow {} \\ &\hphantom{\langle p\vec{\alpha}+q\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle {}}\Rightarrow \exp{[i 2\pi\langle p\vec{\alpha}+q\vec{\beta},n\vec{\alpha}+m\vec{\beta}\rangle]}=\exp{\biggl(i 4\pi\frac{qm}{5}\biggr)}, \\ &|n\vec{\alpha}+m\vec{\beta}|^{2}=3n^{2}+12nm+\frac{102}{5}m^{2}\Rightarrow {} \\ &\hphantom{|n\vec{\alpha}+m\vec{\beta}|^{2}{}}\Rightarrow \exp{[-i \pi|n\vec{\alpha}+m\vec{\beta}|^{2}]}=(-1)^{n}\exp{\biggl(-i 2\pi\frac{m^{2}}{5}\biggr)}. \end{aligned}$$

(103)