Abe, Toshiyuki

Kyoto University Research Information Repository

TitleA cohomology group of a $\mathbb{Z}_2$-orbifold model ofthe symplectic fermionic vertex operator superalgebra(GroupTheory and Related Topics)Author(s)Abe, ToshiyukiCitation数理解析研究所講究録 (2007), 1564: 103-111Issue Date2007-07URL http://hdl.handle.net/2433/81139RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityA cohomology group of a $\mathbb{Z}_{2}$ -orbifold model ofthe symplectic fermionic vertex operator superalgebra1Toshiyuki AbeEhime university1 IntroductionIn this report we calculate a cohomological group of a model of an irrational$C_{2}$,-cofinite simple vertex operator algebra. The cohomological group is con-sidered by Miyamoto in a study on the category of modules for $C_{2}$,-cofinitevertex operator algebras, and this result is just a calculation of a concreteexample. In my talk, I introduced a homology of a certain functor. Butthe functor we considered is left exact. and hence the homology should beconsidered as a cohomology2. In this report we consider the cohomologicalgroup of the simple vertex operator algebra $SF^{+}$ which is one of examplesof irrational $C_{2}$-cofinite vertex operator algebra.2 PreliminariesWe do not state the definition of vertex operator algebras and its mod-ules. For them, plea,se refer to the literatures [LL], [MN] and [FHL]. Let$(t^{\gamma}.1^{r}(\cdot., \backslash l)$ . $1,$ $\omega$ ) be a simple vertex operator algebra over $\mathbb{C}$ , and $(M, Y(\cdot, x))$a weak $t^{r}$-module. We write $Y(a, x)= \sum_{n\in}\underline{7}a_{(n)}x^{-n-1}$ for $a\in V$ following[MN], where $a_{\langle n)}\in$ End $M$ . We also write $L_{n}$ for the n-th mode $\omega_{(n)}$ of the$1’\prime i?^{\backslash }asoro$ vector $\ ’$ . The vacuum vector 1 satisfies that for any $a\in V$ and$i\in \mathbb{Z}\geq 0\cdot a_{(i)}1=0$ and $a_{(-1)}1=a$ .A $vacuum- lik\epsilon$ vector $u\in\Lambda’I$ is a vector $u\in M$ satisfying $a_{(i)}u\cdot=0$ fora,ny $0\in t^{J}$’ and $i\in \mathbb{Z}\geq 0$ . We set Vac $(M)$ to be the set of all vacuum-likevectors in $\Lambda I$ . It is known that$\backslash ^{\tau}\prime a\cdot c(\Lambda I)=$ Ker $L_{-1}=\{u\in M|L_{-1}u=0\}$ .Actually. $L_{-1}=\omega_{(0)}$ shows that Vac $(M)\supset$ Ker $L_{-1}$ . On the other hand if$\frac{u\in KerL_{-l},then(-ik-1)a_{\{i)}v}{1Theori\dot{g}na1tit1eisAhonuo1}=\frac{1}{k!}L_{-1}^{k}a_{(i+k)}u..\cdot Sincea_{(j)}u=0forsufficient1yogygroupa^{-}\sim- orbifo1dmode1ofthesymp1ectic$fermionic vertex operator superalgebra.$\underline{)}After$ my talk. Professors Matsuo and Arakawa gave me this advice. I apologize thatI made audience confused a lot according to my knowledgeless.1564 2007 103-111 103large positive integer $j$ and $(^{-i-1}k)\neq 0$ for any $i,$ $k\in \mathbb{Z}\geq 0$ , we see that $a_{(i)}u=0$and that $u\in Vac(M)$ .We note that Vac $(M)$ is included in the $L_{0}$-eigenspace $AI_{0}$ of weight $0$because $L_{0}=\omega_{(1)}$ . Thus if $L_{0}$ does not have any eigenvector in $M$ , thenVac $(M)=0$ .Proposition 2.1. ([Li]) Let $u\in Vac(\Lambda l)$ , and suppose that $u\neq 0$ . Then theV-submodule $\langle u\rangle$ of $\Lambda l$ generated from $u$ is isomorphic to V. A linear map$T”arrow\langle u\rangle$ defined by a $|arrow a_{1-1)}u$ is a V-module isomorphism.Proof. Let $f$ : $Varrow\langle u\rangle$ be a linear map given by $f(a)=a_{(-1)}u$ It is knownthat $\langle u\rangle$ is spanned by vectors of the form $a_{(m)}u$ with $a\in V$ and $m\in \mathbb{Z}$ .Since $u\in t^{7}ac$ (A4), we see that $\langle u\rangle$ is in fact spanned by $a_{(-m)}u$ with $a\in T^{\gamma}$and $m\in \mathbb{Z}_{>0}$ . Thus $f$ is surjective. We also see that $\langle u\rangle=\{a_{(-1)}u|a\in V\}$because $(m-1)!a_{(-m)}u=(L_{-1}^{m-1}a)_{(-1)}u$ for $m\in \mathbb{Z}_{>0}$ .Now we see $that$$f(a_{(n)}b)=(a_{(n)}b)_{(-1)}u= \sum_{i=0}^{\infty}(\begin{array}{l}ni\end{array})(-1)^{i}(a_{(n-i)}b_{(-1+i)}u-(-1)^{n}b_{(n-1-i)}a_{(i)}u)$$=a_{(n)}b_{(-1)}u$$=a_{(n)}.f(b)$for $a,$ $b\in V$ and $n\in \mathbb{Z}$ . Therefore, $f$ is a$,$ $T^{J}r$-module homomorphism. Finallyker $f$ is a proper ideal of $V$ and hence ker $f=0$ because \dagger is simple. Thus$f$ is a V-module isomorphism. $\square$3 A cohomological group associated to $T/$Suppose that the adjoint module $V$ ha,s an injective resolution;$0arrow Varrow X^{0}arrow f_{0}$ . . . $arrow X^{n}arrow X^{n+1}f_{n}arrow^{f_{n+1}}$ . . . (exact).Then we have a cochain complex$0arrow Vac(X^{0})arrow ro$ Vac $(X^{1})arrow r_{1}$ . $arrow\backslash Iac(X^{n})arrow r_{n}$ Vac $(X^{n+1}) \frac{r_{n+1_{1}}}{\prime}$ ,where $r_{n}=f_{n}|_{P^{n}}\cdot$ . We denote the corresponding cohomological group by$H(V^{\gamma})=\oplus_{n=0}^{\infty}H^{n}(T^{r})$ ;$H^{n}(]^{!}’)=$ ker $r_{n}/{\rm Im} r_{n-1}$104for $n\in \mathbb{Z}\geq 0$ , where $r_{-1}=0$ . The cohomological group is independent of thechoice of injective resolutions.A vertex operator algebra $V$ is called $C_{2}$ -cofinite if the subspace $C_{2}(V)$spanned by vectors of the form $a_{(-2)}b$ with $a,$ $b\in t/’r$ has finite codimensionin $V$ . If $V$ is $C_{2}$-cofinite then we can show that any finitely generated weakV-module has a projective cover. Therefore, the contragredient module $V’$has a projective resolution. In particular, $V$ has an injective resolution.4 The vertex operator algebra $SF^{+}$Let $\mathfrak{h}$ be a finite dimensional vector space of dimension $2d$ with a $nonde-\wedge$generate skew-symmetric bilinear form $\langle\cdot, \rangle$ . Then the vector space $\mathfrak{h}=$$\mathfrak{h}\otimes \mathbb{C}[t^{\pm 1}]\oplus \mathbb{C}K$ has a Lie super-algebra structure as follows; the even partis $\mathbb{C}K$ and the odd part is $\mathfrak{h}\otimes \mathbb{C}[t^{\pm 1}]$ . and the super-commutation relationsare$\{\psi\otimes t^{m}, \phi\otimes t^{n}\}=\uparrow n\langle\psi, \phi\rangle\delta_{m,-n}I\iota’’$, $[K,\mathfrak{h}]\wedge=0$for $\phi,$ $\psi\in \mathfrak{h}$ and $m,$ $n\in \mathbb{Z}$ .Now we consider the super-algebra $\mathcal{A}$ $:=t^{\gamma}(\mathfrak{h})/\langle K-1\rangle\wedge$ , where $U(\mathfrak{h})\wedge$ is theuniversal enveloping algebra $of\mathfrak{h}\wedge$ and $\langle K-1\rangle$ is the two-sided ideal of $U(\mathfrak{h})\wedge$generated by $K-1$ . Let $I\geq 0$ be the left ideal of $A$ generated by $\psi\otimes t^{n}$ for all$\psi’\in \mathfrak{h}$ and $n\in \mathbb{Z}\geq 0$ . We then have a left A-module $\mathcal{A}/I_{>0}$ and denote it by$SF^{3}$ It is clear that $SF$ is isomorphic to the exterior $alge\overline{b}ra\Lambda(\mathfrak{h}\otimes t^{-1}\mathbb{C}[t^{-1}])$as vector spaces. We write $\psi_{(n)}$ for the left multiplication on $SF$ by $’\psi\otimes t^{n}$for $\psi\in \mathfrak{h}$ and $n\in \mathbb{Z}$ . Let 1 be the image of the unit of $\mathcal{A}$ in $SF$ . Then $SF$is spanned by vectors of the form$\psi_{(-n_{1})}^{1_{t}}\psi_{(-n_{2})}^{2}\cdots\psi_{(-n_{\Gamma})}^{r}1$ , $(\psi^{i}\in \mathfrak{h}, n_{i}\in \mathbb{Z}>0)$ .We define the vertex operator map $Y(\cdot, \approx)$ : $SFarrow(EndSF)[[z, z^{-1}]]$ by$Y(1, z)=id_{T}$ ,$Y( \psi_{t-1)}1, \approx)=\sum_{n\in \mathbb{Z}}\psi_{(n)})z^{-n-1}$,$Y(\psi_{(-n_{1})}^{1}’\psi_{(-n_{2})}^{2}\cdots l\psi_{(-n_{f})}^{f}1, z)$$=oO\partial^{(n_{1}-1)}Y(\psi_{t-1)}^{1}1, z)\cdots\partial^{(n_{f}-1)}Y(\psi_{t-1)}^{r}1, z)_{0}^{o}$ ,3The notation $SF$ comes from “Symplectic Fermion”.105for $’\psi’,$ $\psi^{i}\in \mathfrak{h},$ $n,,$ $n_{i}\in \mathbb{Z}_{>0}$ , where $\partial^{(k)}$ $:= \frac{1}{k!}\frac{d^{k}}{d\sim- k}$ for $k\in \mathbb{Z}\geq 0$ .Let $\{e^{i}, f^{i}\}_{i=1,\cdots,d}$ be a basis of $\mathfrak{h}$ satisfying$\langle e^{i}, e^{j}\rangle=\langle f^{i}, f^{j}\rangle=0$ and $\langle e^{i}, f^{j}\rangle=-\delta_{i,j}$for $1\leq i,j\leq d$. Then the Virasoro element $\omega$ is given by$\omega=\sum_{i=1}^{d}e_{(-1)}^{i}f_{(-1)}^{i}1$ .Finally we have a, vertex operator superalgebra $(SF, Y(\cdot, z), 1,\omega)$ of central$charge-2d$.The vertex operator superalgebra $SF$ has canonically an automorphism$\theta$ defined by$\theta(\cdot\psi_{(-n_{1})}^{1_{\text{ }}}\psi_{(-n_{2})}\cdots’\psi_{(-n_{t})})1)=(-1)^{r}’\psi_{(-n_{1})}^{1}\psi_{(-n_{2})}^{2}\cdots\cdot\psi_{(-n_{r})})^{f}1$for any $\psi_{i}\in \mathfrak{h},$ $n_{i}\in \mathbb{Z}_{>0}$ . The fixed point set $SF^{+}$ of $SF$ for $\theta$ is the evenpart of the vertex operator superalgebra $SF$ and the-l-eigenspac.e $SF^{-}$ isthe odd one. The even part $SF^{+}$ becomes a simple vertex operator a$,lgebra$of central $cbarge-2d$ . and $SF^{-}$ is an irreducible $SF^{+}$ -module.5 Projective and injective resolutions of $SF^{+}$It is known that $SF^{+}$ has four irreducible modules (see [A]). These are givenby $SF^{\pm}$ and irreducible components of the unique irreducible $\theta$-twisted SF-module. The lowest weights of $SF^{+}$ and $SF^{-}$ are $0$ and 1 respectively. Thoseof other two irreducible $SF^{+}$ -modules are $- \frac{d}{8}$ and $\frac{4-d}{8}$The two irreducible modules given as submodules of the irreducible $\theta-$twisted SF-module are projective and injective. This fa,ct is not so easy butcan be shown by using the structure of Zhu’s algebra of $SF^{+}$ studied in [A].On the other hand. $SF^{\pm}$ are not projective nor injective. Their projectivecovers can be constructed as follows.First we consider the SF-module $\overline{SF}=\mathcal{A}/I>0$ , where $I_{>0}$ is a left ideal of$\mathcal{A}$ genera,ted by $\psi t_{\vee^{\backslash }}^{-}:\cdot’$ ) $t^{n}$ with $\psi\in \mathfrak{h}$ and $n\underline{\in}\mathbb{Z}_{>0}$ . We see that $\overline{SF}$ is generatedfrom the vector $\wedge 1=1+I_{>0}$ and that $SF\cong\Lambda(\mathfrak{h}_{-}1_{\backslash }^{\backslash },-/I\mathbb{C}[t^{-1}])$ as vector spaces.We define the action of $\theta$ on $\hat{T}$ by$\theta(\prime d_{t-n_{1})}^{1},\psi_{\langle-n_{2})}^{2}\cdots’\psi_{(-n_{r})}^{f}1)=\wedge(-1)^{r},\psi_{(-n_{1})}^{1})’\psi_{\langle-n_{2})}^{2}\cdots\psi_{(-n_{r})}^{r}1\wedge$106$-\pm$for a,ny $\psi_{i}$) $\in \mathfrak{h},$ $n_{i}\in \mathbb{Z}\geq 0$ . We denote by $SF$ by $the\pm 1$ -eigenspace for $\theta$ . We$-\pm$ $-\pm$note tha,$t$ they are $SF^{+}$ -modules and $(SF )’\cong SF$ respectively. We usethe following conjecture.Conjecture. The $S$F-modules $\overline{SF}^{\pm}$ are projective and injective.$-\pm$Assuming this conjecture is true. we can find that $SF$ are projectivecovers of the $\llcorner\sigma\prime F^{+}$-module $SF^{\pm}$ respectively as follows. By construction, wehave a,n SF-module epimorphism $\phi_{0}$ : $\overline{SF}arrow SF$ defined by$\phi_{0}(\psi_{(-n_{1})}^{1}\psi_{(-n_{2})}^{2}. . . ’\psi_{(-n_{\Gamma})}r^{\wedge}1)=\psi_{(-n_{1})}^{1}\psi_{(-n_{2})}^{2}\cdots\psi_{(-n_{r})}^{r}1$for $\iota i_{i}\in \mathfrak{h},$ $n_{i}\in \mathbb{Z}\geq 0$ . By definition $\phi_{0}$ gives epimorphisms $\overline{SF}^{\pm}arrow SF^{\pm}$respectively. We set $M_{0}^{\prime^{r}}=$ ker $\phi_{0}$ . Then $W^{r_{0}}$ is an SF-submodule of $\hat{S}F$generated from $e_{(0)}^{i}1\wedge$ and $f_{(0)}^{i}1\wedge$ for 1 $\leq i\leq d$ . We also see that $J\psi_{0}^{r}=$$-+$ — $-\pm$$(Tt_{0}\cap SF)\oplus(\iota r_{0\cap SF}^{J}" )$ and the submodules $\mathfrak{j}fl_{0}^{7}/\cap SF$ are indecomposable.$-\pm$Hence $SF$ are projective covers of $SF^{\pm}$ respectively.We now state that $SF$ has the following projective resolution.Theorem 5.1. $Th,eSF^{+}$ -module $SF$ has a projective resolution. . . $arrow P^{n+1}arrow P^{n}arrow\cdotsarrow P^{0}arrow SFarrow 0$ ,with $P^{n}=SF$$-\oplus h(n+1)$; the direct sum of $h(n)$ -copies of $\overline{SF}$ .The number $h(n)$ is given as follows: Let$44=(\begin{array}{lllll}0 0 0 (^{2d})(^{2^{0}d})(_{2}^{2^{l}d})-1 0 0 \vdots 0 -1 0 \vdots\vdots \ddots \vdots \vdots 0 0 -l (_{2d-l}2d)\end{array})$be a $2d\cross 2d$-matrix, and set$\mathfrak{k}1(n)=A^{n-1}(\begin{array}{l}0|01\end{array})$ .107Then $h(n)$ is the $2d$-th component of $t$) $(r)$ . Hence$h(1)=1$ , $h(2)=2d$, $h(3)=d(2d+1)$ , $\cdot$ .. .In the case $d=1$ , we have $d(n)=n$ .Since $\overline{SF}’$ , the contragredient mcdule to $\overline{SF}$ , is isomorphic to $\overline{SF}$ , by thistheorem, we have an injective resolution$0arrow SFarrow P^{0}arrow P^{0}arrow\cdotsarrow P^{n}arrow\cdots$By studying the structure of $\overline{SF}$ in detail, we getTheorem 5.2. The irreducible $SF^{+}$ -modules $SF^{\pm}$ have injective resolutions$0arrow SF^{\pm}arrow P^{0,\pm}arrow\cdotsarrow P^{n,\pm}arrow P^{n+1,\pm}arrow\cdots$respectively, $wh.e\uparrow^{\backslash }\epsilon$$P^{n,\pm}=\{\begin{array}{ll}- (SF )^{\oplus h(n+1)} if n is even:(\downarrow\overline{SF} )^{\oplus h(n+1)} if n is odd.\end{array}$6 Cohomological group $H^{\bullet}(SF^{+})$By Theorem 5.2, we get tbe cochain complex$0arrow Vac(P^{0,+})arrow r0$ Vac $(P^{1.+})arrow^{1}f$. . . $arrow Vac(P^{n,+})arrow r_{n}$ Vac $(P^{n+1,+}) \frac{r_{n+1_{t}}}{r}$We note that Vac. $(\overline{SF}^{+})=\mathbb{C}e_{0}^{1}\cdots\epsilon_{(0)}^{d}f_{(0)}^{1}\cdots f_{(0)}^{d}i$ and Vac $(\overline{SF}^{-})=0$ . HenceVac $(P^{n,+})\cong\{\begin{array}{ll}\mathbb{C}^{h(n+1)} if n is even,0 if n is odd.\end{array}$for $n\geq 1$ . We ca,$n$ observe that${\rm Im} r_{n}=0$ for $n\in \mathbb{Z}\geq 0$ ,ker $r_{n}=\{\begin{array}{ll}Vac (P^{n,+}) if n is even,0 if_{7?}\cdot is odd.\end{array}$Therefore, we have108Theorem 6.1.$H^{i}(SF^{+})\cong \mathbb{C}^{h(i+1)}$ if $i$ is even,$H^{i}(SF^{+})=0$ if $i$ is odd.Remark 6.2. We can also define $H^{i}(SF^{-})$ . Then we have $H^{i}(SF^{-})\cong 0$ if$i$ is even and $H^{i}(SF^{-})\cong \mathbb{C}^{h(i+1)}$ if $i$ is odd.7 A projective resolution in the case $d=1$We explain the projective resolution of $l_{\vee}qF$ in the case $d=1$ . For simplicity,we set $\epsilon=e^{1}$ and $f=f^{1}$ . In this case, the submodule ker $\phi_{0}=T\phi_{0}^{7}$ isgenerated by $e_{(0)}i$ and $f_{(0)}\hat{1}$ , and the submodule generated from $e_{(0)}.f_{(0)}1\wedge$ isisomorphic to $SF$ because $e_{(0)}.f_{(0)}1\wedge$ is a vacuum-like vector. Therefore, wehave the following sequence of submodules;$0\subset SF\subset|V_{0}\subset\overline{SF}$ .One sees that $\overline{SF}/M_{0}^{r}\cong SF$ and $\nu v_{0}^{7}/SF\cong SF\oplus SF$ .Now we consider the SF-module epimorphism $\phi_{1}$ : $\overline{SF}\oplus\overline{SF}arrow T\phi_{0}^{r}$defined by$\phi_{1}(u1’.\iota 1)=u\epsilon_{(0)}1+t)f_{(0)}1\wedge\wedge\wedge\wedge$,where $u_{:}v\in\Lambda(\mathfrak{h}\otimes \mathbb{C}[t^{-1}])$ . Then we see that the kernel of $\phi_{1}$ , denotedby $f\phi_{1}^{7}$ , is the $SF$-submodule of $\overline{SF}^{\oplus 2}$ generated by the vectors $(e_{(0)}10)\wedge,$ ,$(f_{(0)}1, e_{(0)}1)$ and $(0.f_{(0)}1)\wedge$ .If we draw an extension of $X$ by $l’-$ as$X$$Y\downarrow$then we have the following pictures;$SF$$\swarrow$ $\searrow$$\overline{SF}=$ $SF$ $SF$$\searrow$ $\swarrow$$SF$109and$SF$\acute ’0 $=$ $\swarrow$ $\searrow$$SF$ $SF$We also see that$SF$ $SF$$\swarrow$ $\searrow$ $\swarrow$ $\searrow$$-\oplus 2$$SF$ $=SF$ $SF$ $SF$ $SF$ ,$\searrow$ $\swarrow$ $\searrow$ $\swarrow$$SF$ $SF$and$SF$ $SF$\mbox{\boldmath $\tau$}1 $=$ $\swarrow$ $\searrow$ $\swarrow$ $\searrow$$SF$ $SF$ $SF$By the same way, for $n\in \mathbb{Z}_{>0}$ , we consider a SF-module homomorphism$-\oplus n$ $-\oplus(n-1)$$\phi_{n-1}$ : $SF$ $arrow SF$ defined by$\phi_{n-1}(u^{1}1..u^{n}1)\wedge,.,\wedge$$=(u^{1}e_{(0)}1\wedge+u^{2}.f_{(0)}^{\wedge}1, u^{2^{\wedge}}\epsilon_{(0)}1+u^{3^{\wedge}}f_{(0)}1, \ldots, u^{n-1^{\wedge}}e_{(0)}1+u^{n}f_{(0)}1)\wedge$with $u^{1},$ $\ldots,$ $u^{n}\in\Lambda(\mathfrak{h}\otimes \mathbb{C}[t^{-1}])$ . Then we can show that${\rm Im}\phi_{n}=$ ker $\phi_{n-1}$for $7l\in \mathbb{Z}_{>0}$ and we have the exact sequence. . . $arrow^{\phi_{n+1}}\overline{SF}^{\oplus(n+1)}arrow\phi_{n}\overline{- S^{l}F}^{\oplus n}arrow^{\phi_{\mathfrak{n}-1}}$ . . . $arrow\phi_{1}\overline{SF}-\phi_{0}SFarrow 0$ .We recall the action of $\theta$ on $\overline{SF}$ . We extend the action of $\theta$ to tha,$t$ on$-\oplus n$$SF$ with diagonal action. Then it is easy to see that $\theta 0\phi_{n}0\theta=-\phi_{n}$for any $n\in \mathbb{Z}\geq 0$ . Therefore, the projective resolution above gives rise to twoprojective resolutions$-\epsilon^{\pm}$ $-\cdot\pm$. . . $arrow^{\phi_{n+1}}(SF^{n\cdot+1})^{\Phi(n+1)}arrow\phi_{n}(SF^{rightarrow n})^{\oplus n}arrow^{\phi_{n-1}}$ . . .. . . $arrow(\overline{SF}^{\mp})^{\oplus 2}arrow\phi_{1}\overline{\vee 9F}^{\pm}arrow,-\phi_{0}9F^{\pm}arrow 0$,110where $\vee^{\pm}cn$ is defined by$\epsilon_{n}^{\pm}=\{\begin{array}{ll}\mp if n is even\pm if n is odd.\end{array}$References[A] SF. Abe, A $\mathbb{Z}_{2}$-orbifold model of the symplectic fermionic vertexoperator superalgebra, Math. Z., online.[FHL] I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approachesto vertex operator algebras and modules, $\Lambda\cdot Iem$ . Amer. Math. Soc.104 (1993).[Li] H.-S. Li, Symmetric invariant bilinear forms on vertex operator al-gebra.$s_{J}$. J. Pure. and Appl. Algebra 96, Issue 3 (1994), 279-297.[LL] H.-S. Li and J. Lepowsky, Introduction to vertex operator algebmsand their representations, Prog. Math., Birkh\"auser, 2004.[MN] A. Matsuo and K. Nagatomo, Axioms for a Vertex Algebra andthe Locality of Quantum Fields, MSJ Memoirs 4, MathematicalSociety of Japan, (1999).111

A cohomology group of a $\mathbb{Z}_2$-orbifold model of the symplectic fermionic vertex operator superalgebra(Group Theory and Related Topics)

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A cohomology group of a $\mathbb{Z}_2$ -orbifold model of the symplectic fermionic vertex operator superalgebra(Group Theory and Related Topics)

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A cohomology group of a Z2\mathbb{Z}_2Z2​-orbifold model of the symplectic fermionic vertex operator superalgebra(Group Theory and Related Topics)

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A cohomology group of a $\mathbb{Z}_2$ -orbifold model of the symplectic fermionic vertex operator superalgebra(Group Theory and Related Topics)