Revisiting the Ramond sector of the $\mathcal{N}\!=\!1$ superconformal minimal models

TL;DR

This paper investigates the Ramond sector of  superconformal minimal models, revealing a breakdown of standard assumptions and proposing new methods to compute characters and fusion rules, thus advancing understanding of their boundary conditions and modular properties.

Contribution

It identifies a fundamental issue in the Ramond sector representation theory and introduces a novel approach to compute characters and fusion rules for the even series of models.

Findings

Established the existence of linear-dependence relations in the Ramond sector

Derived a new character formula for the irreducible c/24 module

Uncovered new conformally invariant boundary conditions

Abstract

Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight $c/24$ in the even series of $\mathcal{N}\!=\!1$ superconformal minimal models with central charge $c$. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible $c/24$ module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.

Equations

$$c=\frac{3}{2}\left(1-\frac{8}{m(m+2)}\right),m=2,3,4,\dots .$$

$$\left[L_m,L_n\right]$$

$$\left\{G_r,G_s\right\}$$

$$\left[L_m,G_r\right]$$

$$\text{det}{V}_0^{+}$$

$$\text{det}{V}_{\ell >0}^{\pm }$$

$$V_{\star }:G_0|h_{\star }^{+}⟩=|\text{mbox}null⟩⟶{\tilde{V}}_{\star }:G_0|{\tilde{h}}_{\star }^{+}⟩=0.$$

$$|{\tilde{\mathbf{\chi }}}_{\star }⟩={\hat{\mathcal{L}}}_0\left[f_1^{(0)},\dots ,f_{\frac{1}{2}{P}_{\text{R}}(\Delta {\ell }_0)}^{(0)}\right]|{\tilde{h}}_{\star }^{+}⟩,$$

$${\hat{\mathcal{L}}}_k$$

$$:=f_1^{(k)}{L}_{-1}^{\Delta {\ell }_k}+\cdots +f_{\frac{1}{2}{P}_{\text{R}}(\Delta {\ell }_k)}^{(k)}{G}_{-1}{G}_{1-\Delta {\ell }_k}$$

$$+g_1^{(k)}{G}_{-1}{L}_{-1}^{\Delta {\ell }_k-1}{G}_0+\cdots +g_{\frac{1}{2}{P}_{\text{R}}(\Delta {\ell }_k)}^{(k)}{G}_{-\Delta {\ell }_k}{G}_0,$$

$${\hat{\mathcal{L}}}_1|{\tilde{\mathbf{\chi }}}_{\star }⟩=0.$$

$${\hat{\mathcal{L}}}_k{\hat{\mathcal{L}}}_{k-1}|{\ell }_{k-2}⟩=0,k⩾2,$$

$${\chi }_{\text{R}}({\mathcal{M}}_{\frac{m}{2},\frac{m}{2}+1}^{\pm })$$

$$\displaystyle\hskip 48.36967pt-q^{\frac{1}{2}m(m+2)(n+\frac{1}{2})^{2}}\Bigr{)}\pm\frac{1}{2}.$$

$$Z^{+}$$

$$+|{\chi }_{\text{R}}^{c/24}(q){|}^2+2\sum _{(r,s)\in {\Delta }_{\text{R}}}^{\prime }|{\chi }_{\text{R}}({\mathcal{M}}_{r,s}^{+}){|}^2,$$

$$L_0|w_{\star }^{\pm }⟩=\frac{c}{24}|w_{\star }^{\pm }⟩,⟨w_{\star }^{-}|w_{\star }^{-}⟩=1,|w_{\star }^{+}⟩=G_0|w_{\star }^{-}⟩=|\text{mbox}null⟩,$$

$$|h_{\star ,\gamma }^{\text{R}}⟩⟩$$

$$|w_{\star ,\gamma }^{\text{R}}⟩⟩$$

$$\big{|}\textstyle\frac{c}{24}^{\textrm{R}}_{\pm}\big{\rangle}\!\big{\rangle}\coloneqq|h^{\textrm{R}}_{\star,\pm}\rangle\!\rangle\pm|w^{\textrm{R}}_{\star,\pm}\rangle\!\rangle.$$

$$Z_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{\text{open}}(q)$$

$$\displaystyle+\frac{1}{2}\underset{\{\textrm{R}\}}{\mathrm{Tr}}\big{[}e^{-RH_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{\text{open}}}\big{]}+\frac{1}{2}\underset{\{\widetilde{\textrm{R}}\}}{\mathrm{Tr}}\big{[}(-1)^{F}e^{-RH_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{\text{open}}}\big{]},$$

$$Z_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{\text{closed}}(\tilde{q})=⟨⟨\Theta {\alpha }_{\gamma }\Vert e^{-L{H}^{\text{closed}}}\Vert {\beta }_{{\gamma }^{\prime }}⟩⟩,$$

$$Z$$

$$+\frac{1}{2}\left(m_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{h_{\star }^{+}}+m_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{w_{\star }^{-}}\right){\chi }_{\text{R}}^{c/24}(q)+\sum _{i\in {\Delta }_{\text{R}}}^{\prime }{m}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^i{\chi }_{\text{R}}^i(q)$$

$$+\frac{1}{2}\left({\tilde{m}}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{h_{\star }^{+}}-{\tilde{m}}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{w_{\star }^{-}}\right)$$

$$=\frac{1}{2}\sum _{i\in {\Delta }_{\text{NS}}}\left(n_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^i{\chi }_{\text{NS}}^i(q)+{\tilde{n}}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^i{\chi }_{\widetilde{\text{NS}}}^i(q)\right)$$

$$+\sum _{i\in {\Delta }_{\text{R}}}{m}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^i{\chi }_{\text{R}}^i(q)+\frac{1}{2}{\tilde{m}}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{c/24},$$

$$\Vert {\alpha }_{\gamma }⟩⟩=\sum _{j\in {\Delta }_{\text{NS}}}{B}_{{\alpha }_{\gamma }}^{j_{\gamma }^{\text{NS}}}|j_{\gamma }^{\text{NS}}⟩⟩+\sum _{j\in {\Delta }_{\text{R}}}{B}_{{\alpha }_{\gamma }}^{j_{\gamma }^{\text{R}}}|j_{\gamma }^{\text{R}}⟩⟩,$$

$$\displaystyle\big{\langle}\!\big{\langle}\Theta\textstyle\frac{c}{24}_{\pm}^{\textrm{R}}\big{|}e^{-LH^{\textrm{closed}}}\big{|}\textstyle\frac{c}{24}_{\mp}^{\textrm{R}}\big{\rangle}\!\big{\rangle}=2.$$

$$\sum _{i\in {\Delta }_{\text{NS}}}{n}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^i{S}_{ij}^{[\text{NS},\text{NS}]}=2({\delta }_{\gamma +}{\delta }_{{\gamma }^{\prime }+}+{\delta }_{\gamma -}{\delta }_{{\gamma }^{\prime }-})B_{{\alpha }_{\gamma }}^{j_{\gamma }^{\text{NS}}}{B}_{{\beta }_{{\gamma }^{\prime }}}^{j_{{\gamma }^{\prime }}^{\text{NS}}}$$

$$\sum _{i\in {\Delta }_{\text{R}}}{m}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^i\frac{1}{{\lambda }_i}{S}_{ij}^{[\text{R},\widetilde{\text{NS}}]}=({\delta }_{\gamma +}{\delta }_{{\gamma }^{\prime }-}+{\delta }_{\gamma -}{\delta }_{{\gamma }^{\prime }+})B_{{\alpha }_{\gamma }}^{j_{\gamma }^{\text{NS}}}{B}_{{\beta }_{{\gamma }^{\prime }}}^{j_{{\gamma }^{\prime }}^{\text{NS}}}$$

$$\sum _{i\in {\Delta }_{\text{NS}}}{\tilde{n}}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^i{S}_{ij}^{[\widetilde{\text{NS}},\text{R}]}{\lambda }_j=4({\delta }_{\gamma +}{\delta }_{{\gamma }^{\prime }+}+{\delta }_{\gamma -}{\delta }_{{\gamma }^{\prime }-})B_{{\alpha }_{\gamma }}^{j_{\gamma }^{\text{R}}}{B}_{{\beta }_{{\gamma }^{\prime }}}^{j_{{\gamma }^{\prime }}^{\text{R}}}$$

$${\tilde{m}}_{{\alpha }_{\gamma }{\beta }_{{\gamma }^{\prime }}}^{c/24}=4({\delta }_{\gamma +}{\delta }_{{\gamma }^{\prime }-}+{\delta }_{\gamma -}{\delta }_{{\gamma }^{\prime }+})B_{{\alpha }_{\gamma }}^{{\frac{c}{24}}_{\gamma }^{\text{R}}}{B}_{{\beta }_{{\gamma }^{\prime }}}^{{\frac{c}{24}}_{{\gamma }^{\prime }}^{\text{R}}}$$

$$\Vert k_{+}^{\text{NS}}⟩⟩$$

$$\Vert l_{+}^{\widetilde{\text{NS}}}⟩⟩$$

$$\Vert d_{-}^{\text{R}}⟩⟩=\sum _{j\in {\Delta }_{\text{NS}}}\frac{\sqrt{2}}{{\lambda }_d}\frac{S_{dj}^{[\text{R},\widetilde{\text{NS}}]}}{\sqrt{S_{0j}^{[\text{NS},\text{NS}]}}}|j_{-}^{\text{NS}}⟩⟩.$$

$$\displaystyle{\Big{\|}}{\textstyle\frac{c}{24}}^{\widetilde{{\textrm{R}}}}_{{-}}{\Big{\rangle}\!\!\Big{\rangle}}=\frac{1}{2\sqrt{S^{[\widetilde{\textrm{NS}},\textrm{R}]}_{0,c/24}}}\big{|}\textstyle\frac{c}{24}^{\textrm{R}}_{-}\big{\rangle}\!\big{\rangle}.$$

Full text

*Introduction.—*Two-dimensional (2D) conformal field theories (CFTs) Belavin et al. (1984) play a central role in physics, with applications ranging from string theory and gauge-gravity duality to critical phenomena and strongly correlated electron systems. The presence of additional symmetries besides conformal symmetry can lead to CFTs with a particularly rich mathematical structure. For instance, a minimal generalization of the conformal Virasoro algebra consistent with supersymmetry is the $\mathcal{N}\!=\!1$ super-Virasoro algebra Ramond (1971); Neveu and Schwarz (1971a); *neveu1971b, given in Eqs. (2)–(4) below. The $\mathcal{N}\!=\!1$ superconformal minimal models (SMMs) Friedan et al. (1984, 1985); Qiu (1986); Goddard et al. (1986) are an infinite, discrete series of superconformal field theories (SCFTs) corresponding to unitary, irreducible representations of this algebra, with values $c\!<\!3/2$ of the central charge given by

[TABLE]

As opposed to early proposals for the experimental realization of SMMs at classical multicritical points (e.g., the tricritical Ising model Friedan et al. (1985)), which requires much fine tuning, advances in condensed matter physics in the past ten years or so suggest several of the models (1) may more promisingly be realized as bona fide quantum critical points or even stable quantum critical phases in an increasingly diverse array of platforms, ranging from anyonic spin chains Feiguin et al. (2007); Gils et al. (2009) to boundaries of topological superconductors Grover et al. (2014); Li et al. (2017) and lattice models of interacting Majorana fermions Rahmani et al. (2015a); *rahmani2015; O’Brien and Fendley (2018); Sannomiya and Katsura (2019).

A first step towards the realization of SMMs in nature by these means is their unambiguous identification in numerical experiments. The entanglement properties of critical (1+1)D quantum many-body systems, to which powerful entanglement-based numerical methods such as the density-matrix renormalization group (DMRG) White (1992) give direct access, can probe various universal quantities in the underlying CFT and are particularly promising in this regard. For instance, in Ref. Grover et al. (2014) the central charge $c\!=\!7/10$ of the $m\!=\!3$ SMM in the tricritical Ising universality class was determined from a DMRG calculation of the ground-state entanglement entropy Holzhey et al. (1994). Besides the single value of the central charge, a fuller characterization of the underlying CFT may in principle be achieved by a study of the low-lying entanglement spectrum, which was argued to match the set of scaling dimensions in the boundary CFT Läuchli (2013); Ohmori and Tachikawa (2015); Cardy and Tonni (2016) for a particular choice of boundary conditions on the entangling surface. This choice of entanglement boundary conditions in turn singles out one among a set of allowed conformally invariant boundary conditions Cardy (1984), i.e., a pair of boundary (Cardy) states Cardy (1989). By determining numerically how different entanglement boundary conditions affect degeneracies in the low-lying entanglement spectrum, one may in principle directly probe the entire set of fusion rules of the underlying CFT Cardy (1989); Verlinde (1988).

Perhaps surprisingly, the fundamental question of boundary CFT, i.e., the construction of Cardy states — which additionally yields, via Ref. Cardy (1989), the bulk fusion rules — has not been satisfactorily settled so far for the even-$m$ series of SMMs. A problem first posed by Ishibashi in his seminal 1989 paper Ishibashi (1989), the construction of Cardy states for the SMMs was completed in the odd-$m$ case by Nepomechie Nepomechie (2001). Subtleties in the even-$m$ case, due to the presence of the Ramond-sector highest weight (HW) $c/24$, were noticed by Apikyan and Sahakyan Apikyan and Sahakyan (1998); *apikyan1999, but the complete set of Cardy states was not obtained. Here we show that the standard assumption of representation theory — the nested Verma submodule structure of null states — fails for the $c/24$ module, requiring an alternate approach to the construction of an irreducible module and the computation of its character. We propose as resolution an infinite hierarchy of linear-dependence relations among null states in an auxiliary module whose character is identical to, but easier to compute than, that of the original module. Based on this newly derived character we argue that modular invariance of the torus partition function Cardy (1986) requires the Ramond-sector ground-state manifold to contain states of both fermion parities. We subsequently construct Ishibashi states Ishibashi (1989) and solve the Cardy equations Cardy (1989), finding an extra Cardy state, Eq. (30), beyond those discussed in the literature. Inverting the Cardy equations, we correspondingly find two extra fusion rules, Eqs. (31)–(32) and (37), absent from previous discussions. To our knowledge, this is the first time the full set of Cardy states and fusion rules have been derived for the even-$m$ series of SMMs.

Failure of standard representation theory for the $c/24$ module.—The $\mathcal{N}\!=\!1$ super-Virasoro algebra is given by

[TABLE]

where $L_{m}$ ($G_{r}$) are the bosonic (fermionic) Laurent modes of the energy-momentum tensor (supercurrent); $m\!\in\!\mathbb{Z}$ while $r\!\in\!\mathbb{Z}\!+\!\frac{1}{2}$ in the Neveu–Schwarz (NS) sector and $r\!\in\!\mathbb{Z}$ in the Ramond (R) sector. We focus on the holomorphic part of the algebra; identical results are obtained for the antiholomorphic part. The central result in representation theory is the Kac determinant Friedan et al. (1985), an expression for the determinant of the Gram matrix in the degenerate subspace at level $\ell$ in a Verma module $V^{+}$ ($V^{-}$) with even (odd) fermion parity and HW $h$,

[TABLE]

where $h_{r,s}(c)$ is a prescribed function of $c$ and the integers $r,s$ which determines the finite set of allowed HWs in the SMMs. We focus on the R sector, where $r\!-\!s$ is odd and the maximal level degeneracy $P_{\textrm{R}}(\ell)$ is obtained from the generating function $\sum_{\ell=0}^{\infty}q^{\ell}P_{\textrm{R}}(\ell)=\prod_{n=1}^{\infty}(1+q^{n})/(1-q^{n})\eqqcolon 1/\varphi_{\textrm{R}}(q)$.

Unitarity and irreducibility require that negative-norm states should be absent from the Verma module and null (i.e., zero-norm) states should be systematically removed. A large class of linearly-independent null states consists of states obtained from the repeated action of lowering operators $L_{-m}$, $G_{-r}$ ($m>0,r\geqslant 0$) onto a singular vector $|\bm{\chi}\rangle$, that is, a descendant state satisfying the HW condition: $L_{n}|\bm{\chi}\rangle=0\ \mbox{for}\ n>0$ and $G_{r}|\bm{\chi}\rangle=0\ \mbox{for}\ r>0$. Such states form a null Verma submodule Di Francesco et al. (1997). A singular vector is itself necessarily a null state, but the converse is not generally true, as will be seen shortly. For all Virasoro and most super-Virasoro HWs, the first null state dictated by the Kac determinant (at level $\ell=rs/2$) happens to also be a singular vector; the same Kac determinant can thus be used again to find another singular vector in the resulting null Verma submodule, and such a procedure repeats, leading to a nested structure of null Verma submodules which comprises all null states Roc . Based on this nested structure, Kiritsis derived character formulas for NS and R irreducible Verma modules with HWs not equal to $c/24$ Kiritsis (1988). The $c/24$ HW only appears in the R sector of the even-$m$ series of the SMMs, and we focus on those for the rest of the paper.

The crucial concurrence between first null state and singular vector is however not generically warranted by the Kac determinant, and breaks down for the module with HW $h_{\star}\!\coloneqq\!h_{\frac{m}{2},\frac{m}{2}+1}(c)\!=\!c/24$. As for other R modules the HW states come in degenerate pairs $|h_{\star}^{\pm}\rangle$ of opposite fermion parity where $|h_{\star}^{-}\rangle\!=\!G_{0}|h_{\star}^{+}\rangle$, but contrary to other R modules the odd-parity state $|h_{\star}^{-}\rangle$ is annihilated by $G_{0}$ since $G_{0}^{2}\!=\!L_{0}\!-\!\frac{c}{24}$ (and is thus also null). As a result the linear system corresponding to the singular vector condition is in general overdetermined. For example, when $m\!=\!2$, one has $\langle h^{+}_{\star}|L_{1}L_{-1}|h^{+}_{\star}\rangle\!=\!0$ but $G_{1}L_{-1}|h^{+}_{\star}\rangle\!=\!\frac{3}{2}|h^{-}_{\star}\rangle\!\neq\!0$, i.e., the first null state $L_{-1}|h^{+}_{\star}\rangle$ built from the even-parity HW state $|h^{+}_{\star}\rangle$ is not singular. Analogously for $m\!=\!4$, one can prove that no linear combinations of level-$3$ descendants of $|h^{+}_{\star}\rangle$ are singular.

*Character of the $c/24$ module.—*To resolve this issue we first propose the study of an auxiliary module $\widetilde{V}_{\star}$, defined by contrast with the original module $V_{\star}\!\coloneqq\!V(c,h_{\star})$ as

[TABLE]

The entire class of null states built solely on $|h^{-}_{\star}\rangle$ has been effectively subsumed into a single representative zero vector so that $|\widetilde{h}^{-}_{\star}\rangle$ does not appear and the Verma module is halved, which greatly simplifies the construction of irreducible HW representations. The Kac determinant for $\widetilde{V}_{\star}$ remains formally unchanged: \textrm{det}\,\widetilde{V}^{+}_{\star,0}\!=\!1,\ \textrm{det}\,\widetilde{V}^{-}_{\star,0}\!=\!0,\ \textrm{det}\,\widetilde{V}^{\pm}_{\star,\ell>0}\!=\!\prod_{r,s\geqslant 1}\!\left(h\!-\!h_{r,s}(c))^{P_{\textrm{R}}\left(\ell-\frac{rs}{2}\right)}. Thus irreducible modules constructed from $V_{\star}$ and $\widetilde{V}_{\star}$, albeit fundamentally different, necessarily possess the same character.

We first investigate the structure of the reducible auxiliary modules $\widetilde{V}_{\star}^{\pm}$ built entirely upon the HW state $|\widetilde{h}^{+}_{\star}\rangle$. According to the Friedan–Qiu–Shenker (FQS) prescription Friedan et al. (1985), a set of linearly independent vectors spanning the level-$\ell$ degenerate subspace is given by $G_{-m_{1}}G_{-m_{2}}\cdots L_{-n_{1}}L_{-n_{2}}\cdots|\widetilde{h}^{+}_{\star}\rangle$, where $0<m_{1}<m_{2}<\cdots$, $0<n_{1}\leqslant n_{2}\leqslant\cdots$, and $\sum_{i}m_{i}+\sum_{i}n_{i}\!=\!\ell$. Imposing $G_{0}|\widetilde{h}^{+}_{\star}\rangle=0$ in Eq. (7) leads to two major differences in the structure of $V_{\star}^{\pm}$ and $\widetilde{V}_{\star}^{\pm}$:

1. (i)

In contrast to $V^{\pm}_{\star}$, singular vectors $|\bm{\widetilde{\chi}}_{\star}\rangle$ are restored in $\widetilde{V}^{\pm}_{\star}$ and first appear at levels dictated by the Kac determinant. However, generically $|\bm{\widetilde{\chi}}_{\star}\rangle$ is not annihilated by $G_{0}$ although $|\widetilde{h}^{+}_{\star}\rangle$ is. 2. (ii)

At a given level and for a fixed fermion parity, the set of null descendant states built upon the two degenerate singular vectors $|\bm{\widetilde{\chi}}_{\star}\rangle$ and $G_{0}|\bm{\widetilde{\chi}}_{\star}\rangle$ by the FQS prescription is in general linearly dependent; thus null states in $\widetilde{V}_{\star}^{\pm}$ do not form Verma submodules.

As will now be argued, the linear-dependence relations among null states evoked in (ii) are organized into an infinite hierarchy that plays a role analogous to that of the nested embedding of null Verma submodules for $h\!\neq\!c/24$ HWs, and allows us to compute the irreducible character of the original $V_{\star}^{\pm}$ modules.

The trivial or zeroth echelon in this hierarchy corresponds simply to the first singular vector $|\bm{\widetilde{\chi}}_{\star}\rangle$ of a given fermion parity, which appears at level $\ell_{0}\!\coloneqq\!\frac{m}{4}(\frac{m}{2}+1)$ according to the Kac determinant and can thus be written as a linear combination of level-$\ell_{0}$ descendants of $|\widetilde{h}_{\star}^{+}\rangle$,

[TABLE]

where $f^{(0)}_{1},\ldots,f^{(0)}_{\frac{1}{2}\!P_{\textrm{R}}(\Delta\ell_{0})}$ are the coefficients of this linear combination, and we define the $k$th-echelon ($k\!\geqslant\!0$) generalized lowering operator,

[TABLE]

with $\Delta\ell_{k}\!\coloneqq\!\ell_{k}-\ell_{k-1}$ and $\ell_{k}\!\coloneqq\!(1+k)^{2}\ell_{0}$. Eq. (9) is the most general fermion-parity-preserving operator that raises the level of a state by $\Delta\ell_{k}$, and $\ell_{k}$ for $k\!\geqslant\!1$ is the level at which higher-level singular vectors appear according to the Kac determinant. The first $\frac{1}{2}P_{\textrm{R}}(\Delta\ell_{k})$ terms in Eq. (9) involve bosonic generators while the remaining terms involve fermionic generators times the zero mode operator $G_{0}$. From Eq. (7) the latter trivially vanish when acting on $|\widetilde{h}_{\star}^{+}\rangle$ and are thus excluded from Eq. (8).

The first echelon in the hierarchy corresponds to the linear dependence of null states built upon $|\bm{\widetilde{\chi}}_{\star}\rangle$ and $G_{0}|\bm{\widetilde{\chi}}_{\star}\rangle$, which first appears at level $\ell_{1}$ and can be expressed as

[TABLE]

The set of $f^{(1)}_{i}$ and $g^{(1)}_{i}$ coefficients implicit in Eq. (10) is uniquely determined by the $f^{(0)}_{i}$ up to an overall multiplicative constant. The second echelon expresses the fact that linear dependence relations generated from (10) become themselves linearly dependent at higher levels. In general, the $k$th echelon consists of linear-dependence relations among the linear-dependence relations of the ($k-1$)th-echelon, and can be summarized compactly as

[TABLE]

where the coefficients $f_{i}^{(k)},g_{i}^{(k)}$, again omitted for simplicity, are uniquely determined from the $f_{i}^{(k-1)},g_{i}^{(k-1)}$ up to an overall multiplicative constant, and $|\ell_{k-2}\rangle$ denotes an arbitrary state at level $\ell_{k-2}$. In Ref. Sup we substantiate this hierarchy conjecture with explicit calculations for $m\!=\!2$; a calculation for $m\!=\!4$ with a computer algebra system yields analogous results.

The hierarchy of linear-dependence relations embodied in Eqs. (10)–(11) can now be used to calculate the character of the original irreducible $c/24$ module $\mathcal{M}^{\pm}_{\frac{m}{2},\frac{m}{2}+1}$: the number of linearly independent null states to be removed from $\widetilde{V}_{\star}^{\pm}$ at each level is reduced by one whenever a linear-dependence relation occurs. Implementing this procedure recursively, one obtains Sup :

[TABLE]

Modular invariance of the torus partition function.—Since the full SCFT is nonlocal, we now restrict ourselves to the Gliozzi–Scherk–Olive (GSO)-projected spin model Friedan et al. (1985) with even fermion parity. On a torus, the required invariance of the partition function under modular transformations of the torus constrains which HW representations appear in the theory and how often Cardy (1986). A modular invariant contribution to the partition function for even $m$ was found to be Matsuo and Yahikozawa (1986); Cappelli (1987); Kastor (1987)

[TABLE]

where $\Delta_{\textrm{NS}}$ ($\Delta_{\textrm{R}}$) denotes the set of independent HWs in the NS (R) sector, $\chi_{\widetilde{\textrm{NS}}}$ corresponds to the NS character twisted by the insertion of the fermion parity operator, and the primed sum over $\Delta_{\textrm{R}}$ means that the $c/24$ HW $(r\!=\!\frac{m}{2},s\!=\!\frac{m}{2}\!+\!1)$ is excluded. The asymmetry in Eq. (13) between the latter and other R HWs comes from the fact that the $c/24$ HW occupies the self-symmetric point of the Kac table Matsuo and Yahikozawa (1986); Cappelli (1987); Kastor (1987). The function \chi_{\textrm{R}}^{c/24}(q)\!\coloneqq\!\varphi_{\textrm{R}}(q)^{-1}\!\sum_{n\in\mathbb{Z}}\bigl{(}q^{\frac{1}{2}m(m+2)n^{2}}-q^{\frac{1}{2}m(m+2)(n+\frac{1}{2})^{2}}\bigr{)}, whose form is highly constrained by modular invariance, should be the character of the $c/24$ module but disagrees with the result (12) of an explicit evaluation in representation theory. To resolve this paradox we introduce an additional pair of R HW ground states $|w^{\pm}_{\star}\rangle$, obeying

[TABLE]

and giving rise to a second irreducible $c/24$ module built on the non-null HW state $|w_{\star}^{-}\rangle$. The level degeneracies of the modules built on $|h_{\star}^{+}\rangle$ and $|w_{\star}^{-}\rangle$ only differ at level zero, since the parity of the non-null HW state is opposite for both, and the sum of their characters inferred from Eq. (12) yields precisely $\chi_{\textrm{R}}^{c/24}(q)$. We thus interpret Eq. (13) as an off-diagonal modular invariant involving the product of holomorphic and antiholomorphic characters for two distinct $c/24$ modules, in sharp contrast to the diagonal form for $m$ odd. The introduction of an additional R ground state also resolves the arbitrariness in the original definition $|h^{-}_{\star}\rangle\!=\!G_{0}|h^{+}_{\star}\rangle$ Friedan et al. (1985), which could have equally been chosen as $|h^{+}_{\star}\rangle\!=\!G_{0}|h^{-}_{\star}\rangle$.

Boundary SCFT and a new Cardy state.—We now explore the consequences of this symmetrization of the R ground-state manifold on the boundary SCFT. Superconformally invariant boundary states, the Cardy states, can be expanded on a basis of Ishibashi states $|h_{\gamma}\rangle\!\rangle$, $\gamma=\pm 1$, which are constructed for each irreducible module and obey the gluing conditions $(L_{n}\!-\!\overline{L}_{-n})|h_{\gamma}\rangle\!\rangle\!=\!0$ and $(G_{r}+i\gamma\overline{G}_{-r})|h_{\gamma}\rangle\!\rangle\!=\!0$ Ishibashi (1989). For the $c/24$ HW, we now have two sets of Ishibashi states,

[TABLE]

where $U_{\gamma}$ is an antiunitary operator that commutes with the holomorphic fermion parity operator $(-1)^{F}$ and obeys $U_{\gamma}L_{n}U_{\gamma}^{-1}=L_{n}$, $U_{\gamma}G_{r}U_{\gamma}^{-1}=-i\gamma G_{r}(-1)^{F}$. The sums run over a complete set of states in each module, with ${\sf q,q^{\prime}}$ a set of quantum numbers sufficient to label each state (holomorphic fermion parity, level, and other quantum numbers). An equally valid basis, which facilitates the construction of the Cardy states, is given by the bonding and antibonding combinations of Eqs. (15) and (16),

[TABLE]

Cardy states Cardy (1989) $\|\alpha_{\gamma}\rangle\!\rangle,\|\beta_{\gamma^{\prime}}\rangle\!\rangle$ are defined by the property that the partition function $Z_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}$ of the (GSO-projected) SCFT on a cylinder of length $L$ and circumference $R$ can be evaluated in either the open-string or closed-string pictures. In the open-string picture, one has periodic time evolution along the $R$ direction according to a Hamiltonian $H_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{\text{open}}=\frac{\pi}{L}\left(L_{0}-\frac{c}{24}\right)_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}$ with boundary conditions $\alpha_{\gamma},\beta_{\gamma^{\prime}}$ along the $L$ direction,

[TABLE]

where traces are over holomorphic states only and all four spin structures on the cylinder are considered separately. In the closed-string picture, one has a transition amplitude between Cardy states with finite time evolution along the $L$ direction,

[TABLE]

with Hamiltonian $H^{\textrm{closed}}=\frac{2\pi}{R}\left(L_{0}+\overline{L}_{0}-\frac{c}{12}\right)$, and $\Theta$ is the antiunitary CPT operator. The parameters $q=e^{-\pi R/L}=e^{2\pi i\tau}$ and $\widetilde{q}=e^{-4\pi L/R}=e^{2\pi i\widetilde{\tau}}$ are related by a modular $S$ transformation $\widetilde{\tau}=-1/\tau$. The open-string partition function is evaluated as

[TABLE]

where the multiplicities $n^{i}_{\alpha_{\gamma}\beta_{\gamma^{\prime}}},\widetilde{n}^{i}_{\alpha_{\gamma}\beta_{\gamma^{\prime}}},m^{i}_{\alpha_{\gamma}\beta_{\gamma^{\prime}}},\widetilde{m}^{i}_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}\!\in\!\mathbb{Z}$ denote how many times the irreducible HW module $i$ appears in the spectrum of the SCFT with boundary conditions $\alpha_{\gamma},\beta_{\gamma^{\prime}}$ for a given choice of spin structure. In the last equality we define $m_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{h^{+}_{\star}}\!+\!m_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{w^{-}_{\star}}\!\eqqcolon\!2m_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{c/24}$ and $\widetilde{m}_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{h^{+}_{\star}}\!-\!\widetilde{m}_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{w^{-}_{\star}}\!\eqqcolon\!\widetilde{m}_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{c/24}$, establishing later that $m_{\alpha_{\gamma}\beta_{\gamma^{\prime}}}^{c/24}$ is integer. Note that the two independent $c/24$ modules in general contribute a nontrivial constant from the twisted sum in the R sector, which is at the origin of the new Cardy state to be discussed shortly. Conversely, expanding the Cardy states on the basis of Ishibashi states as

[TABLE]

and similarly for $\|\beta_{\gamma^{\prime}}\rangle\!\rangle$, where $B_{\alpha_{\gamma}}^{j^{\textrm{NS,R}}_{\gamma}}\!\coloneqq\!\langle\!\langle j_{\gamma}^{\textrm{NS,R}}\|\alpha_{\gamma}\rangle\!\rangle$ are the expansion coefficients, the closed-string partition function (19) can be calculated and expressed in terms of these coefficients and the characters $\chi_{\textrm{NS}}^{j}(\widetilde{q})$, $\chi_{\widetilde{\textrm{NS}}}^{j}(\widetilde{q})$, $\chi_{\textrm{R}}^{j}(\widetilde{q})$ Sup . The aforementioned constant term in Eq. (Revisiting the Ramond sector of the $\mathcal{N}\!=\!1$ superconformal minimal models) is matched by a corresponding term in Eq. (19), which arises from

[TABLE]

Equating Eqs. (Revisiting the Ramond sector of the $\mathcal{N}\!=\!1$ superconformal minimal models) and (19), and using the transformation properties of the characters under a modular $S$ transformation Matsuo and Yahikozawa (1986); Cappelli (1987); Kastor (1987), we find a set of four Cardy equations Sup ,

[TABLE]

where $S$ is the modular $S$-matrix Verlinde (1988), $j$ runs over HWs in the appropriate sector, and we define $\lambda_{c/24}=1$, $\lambda_{j\neq c/24}=\sqrt{2}$.

Adopting the method in Ref. Cardy (1989), the Cardy states can be obtained as solutions to the above equations. For HWs $k,l\in\Delta_{\textrm{NS}}$, we obtain

[TABLE]

while for HWs $d\!\in\!\Delta_{\textrm{R}}$, we obtain

[TABLE]

Finally, for even $m$ the existence of the self-symmetric $c/24$ HW yields an additional Cardy state,

[TABLE]

Fusion rules.—Identifying the multiplicities appearing in the open-string partition function (Revisiting the Ramond sector of the $\mathcal{N}\!=\!1$ superconformal minimal models) with the fusion coefficients of the SCFT Cardy (1989), one obtains the complete set of fusion rules for the even-$m$ series of SMMs,

[TABLE]

as well as the corresponding Verlinde formula,

[TABLE]

We have checked explicitly that all fusion coefficients are integers.

*Conclusion.—*In summary, we have shown that the standard representation-theoretic approach to the determination of irreducible characters in CFT fails for the R-sector $c/24$ HW in the even-$m$ series of $\mathcal{N}\!=\!1$ SMMs, and have conjectured an infinite hierarchy of linear-dependence relations that allows us to compute the character of this module. Modular invariance on the torus can be restored with this character provided that the R ground-state manifold is augmented by a non-null state with odd fermion parity, which in turn yields additional bulk fusion channels and superconformally invariant boundary states.

We gratefully acknowledge discussions with T. Creutzig as well as financial support from NSERC grant #RGPIN-2014-4608, the CRC Program, CIFAR, and the University of Alberta.