Second Cohomology Space of the Lie Superalgebra of contact vector fields $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on $S^{1|2}$

TL;DR

This paper computes the second cohomology space of the Lie superalgebra of contact vector fields on a (1,2)-dimensional superspace, providing explicit cocycles that span this space, advancing understanding of its structure.

Contribution

It explicitly determines the second cohomology space of $rak{K}(2)$ with coefficients in weighted densities, including explicit cocycles, which was previously unknown.

Findings

Explicit cocycles spanning the second cohomology space are provided.

The structure of the second cohomology space for $rak{K}(2)$ is characterized.

The results contribute to the understanding of deformations and extensions of the superalgebra.

Abstract

We investigate the second cohomology space of the Lie superalgebra $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on the (1,2)-dimentional real superspace. We explicitly give cocycles spanning this cohomology space.

Equations

$$L_{X\frac{d}{dx}}^{\lambda }(f)=X{f}^{\prime }+\lambda X^{\prime }f,$$

$${\mathfrak{F}}_{\lambda }^2=\{F{\alpha }_2^{\lambda }:F\in {\mathcal{C}}^{\infty }(S^{1|2})\},$$

$${\mathfrak{L}}_{v_F}^{\lambda }(\Phi )=(v_F(G)+\lambda F^{\prime }G){\alpha }_2^{\lambda }$$

$${\alpha }_n=dx+\sum _{i=1}^2{\theta }_{i}d{\theta }_i.$$

$$\{F,G\}=F{G}^{\prime }-F^{\prime }G-\frac{1}{2}(-1{)}^{|F|}\sum _{i=1}^2{\overline{\eta }}_i(F)\cdot {\overline{\eta }}_i(G),$$

$$\mathrm{Vect}(S^{1|2})=\left\{F_0{\partial }_x+\sum _{i=1}^2{F}_i{\partial }_i\mid F_i\in C^{\infty }(S^{1|2}),i=1,2\right\},$$

$$\mathcal{K}(2)=\left\{X\in \mathrm{Vect}(S^{1|2})|there\; existsF\in C^{\infty }(S^{1|2})such\; that{L}_X({\alpha }_2)=F{\alpha }_2\right\}.$$

$$X_F=F{\partial }_x-\frac{1}{2}\sum _{i=1}^2(-1{)}^{|F|}{\overline{\eta }}_i(F){\overline{\eta }}_i,\text{where F∈C∞(S1∣2).}$$

$$[X_F,X_G]=X_{\{F,G\}}.$$

$${\mathfrak{L}}_{X_F}^{\lambda }=X_F+\lambda F^{\prime },\lambda \in \mathbb{K}.$$

$$[{\mathfrak{L}}_{X_F}^{\lambda },{\mathfrak{L}}_{X_G}^{\lambda }]={\mathfrak{L}}_{X_{\{F,G\}}}^{\lambda }.$$

$${\mathfrak{F}}_{\lambda }^2=\left\{F{\alpha }_2^{\lambda }\mid F\in C^{\infty }(S^{1|2})\right\}.$$

$$\mathcal{K}(1{)}_i=\{X_F\in \mathcal{K}(2)|{\partial }_{i}F=0,i=1,2\}.$$

$${\mathfrak{F}}_{\lambda }^2\simeq {\mathfrak{F}}_{\lambda }\oplus \Pi ({\mathfrak{F}}_{\lambda +\frac{1}{2}}).$$

$$C^k(\mathfrak{g},V):=\mathrm{Hom}({\wedge }^k\mathfrak{g};V).$$

$$H^k(\mathfrak{g},V)=Z^k(\mathfrak{g},V)/B^k(\mathfrak{g},V).$$

$$\delta ({\rm Y})(x,y):=(-1{)}^{|x||{\rm Y}|}x.{\rm Y}(y)-(-1{)}^{|y|(|x|+|{\rm Y}|)}y.{\rm Y}(x)-{\rm Y}([x,y]),$$

$$\begin{array}[]{lllll}\delta(\Omega)(x,y,z):=&(-1)^{|\Omega||x|}x.\Omega(y,z)-(-1)^{|y|(|\Omega|+|x|)}y.\Omega(x,z)+(-1)^{|z|(|\Omega|+|x|+|y|)}z.\Omega(x,y)\\[5.0pt] &-\Omega([x,y],z)+(-1)^{|y||z|}\Omega([x,z],y)-(-1)^{|x|(|y|+|z|)}\Omega([y,z],x),\end{array}$$

$$\mathrm{Hom}(\mathfrak{g},V{)}_b={\oplus }_{a\in {\mathbb{Z}}_2}\mathrm{Hom}({\mathfrak{g}}_a,V_{a+b});b\in {\mathbb{Z}}_2.$$

$$\mathcal{K}(1)_{i}=\Big{\{}X_{F}~{}|~{}~{}F=f_{0}+f_{i}\theta_{i},\hbox{ where }f_{0},~{}f_{i}\in C^{\infty}(S^{1})\Big{\}}.$$

$$\hbox{H}_{diff}^{2}(\mathcal{K}(2),\mathfrak{F}_{\lambda}^{2})\simeq\left\{\begin{array}[]{ll}\mathbb{K}^{3}&\makebox{ if }~{}\lambda=0,\\ \mathbb{K}^{2}&\makebox{ if }~{}\lambda=1,\\ \mathbb{K}&\makebox{ if }~{}\lambda=3,\\ 0&\makebox{ otherwise }.\end{array}\right.$$

$$\begin{array}[]{ll}\bullet\;\displaystyle\Upsilon^{0}_{1}(X_{F},X_{G})=\displaystyle(FG^{\prime}-F^{\prime}G)\\ -\Big{(}\frac{1}{4}+\frac{3}{4}(-1)^{|F||G|}\Big{)}(-1)^{|F|(|G|+1)}\Big{(}\overline{\eta}_{1}(F)\overline{\eta}_{1}(G)+\overline{\eta}_{2}(F)\overline{\eta}_{2}(G)\Big{)}\\ +\frac{1}{2}\Big{(}(-1)^{|F|}\overline{\eta}_{1}\overline{\eta}_{2}(F)\overline{\eta}_{2}(G)-\overline{\eta}_{2}(F)\overline{\eta}_{1}{\overline{\eta}}_{2}(G)\Big{)}\theta_{1}\\ +\frac{1}{2}\Big{(}\overline{\eta}_{1}(F)\overline{\eta}_{1}\overline{\eta}_{2}(G)-(-1)^{|F|}\overline{\eta}_{1}\overline{\eta}_{2}(F)\overline{\eta}_{1}(G)\Big{)}\theta_{2},\\ \\[5.0pt] \bullet\;\displaystyle\Upsilon^{0}_{2}(X_{F},X_{G})=\displaystyle\Big{(}\overline{\eta}_{1}\overline{\eta}_{2}(F)G-F\overline{\eta}_{1}\overline{\eta}_{2}(G)\Big{)}\\ +\frac{1}{2}\Big{(}(-1)^{|G|}\overline{\eta}_{1}(F)\overline{\eta}_{1}\overline{\eta}_{2}(G)-\overline{\eta}_{1}\overline{\eta}_{2}(F)\overline{\eta}_{1}(G)\Big{)}\theta_{1}\\ +\frac{1}{2}\Big{(}(-1)^{|G|}\overline{\eta}_{2}(F)\overline{\eta}_{1}\overline{\eta}_{2}(G)-\overline{\eta}_{1}\overline{\eta}_{2}(F)\overline{\eta}_{2}(G)\Big{)}\theta_{2}\\ +2\Big{(}FG^{\prime\prime}-F^{\prime\prime}G\Big{)}\theta_{1}\theta_{2}\\ \\[5.0pt] \bullet\;\displaystyle\Upsilon^{0}_{3}(X_{F},X_{G})=\displaystyle\Big{(}F^{\prime}\overline{\eta}_{1}\overline{\eta}_{2}(G)-\overline{\eta}_{1}\overline{\eta}_{2}(F)G^{\prime}\Big{)}.&\hbox{ if }\lambda=0,\\ \\[5.0pt] \bullet\;\displaystyle\Upsilon^{1}_{0}(X_{F},X_{G})=\displaystyle\Big{(}(-1)^{|F|}F^{\prime}\overline{\eta}_{1}\overline{\eta}_{2}(G^{\prime})-(-1)^{|G|}\overline{\eta}_{1}\overline{\eta}_{2}(F^{\prime})G^{\prime}\Big{)}\alpha_{2}\\ \\[5.0pt] \bullet\;\displaystyle\Upsilon^{1}_{1}(X_{F},X_{G})=\displaystyle(-1)^{(|F|+|G|)}\Big{(}\overline{\eta}_{1}\overline{\eta}_{2}(F)\overline{\eta}_{1}\overline{\eta}_{2}(G^{\prime})-\overline{\eta}_{1}\overline{\eta}_{2}(F^{\prime})\overline{\eta}_{1}\overline{\eta}_{2}(G)\Big{)}\alpha_{2}\\ &\hbox{ if }\lambda=1,\\ \\[5.0pt] \bullet\;\displaystyle\Upsilon^{3}_{0}(X_{F},X_{G})=\displaystyle\Big{[}(-1)^{|F|+1}\Big{(}\overline{\eta}_{1}(F^{\prime\prime})\overline{\eta}_{1}(G^{\prime\prime})+\overline{\eta}_{2}(F^{\prime\prime})\overline{\eta}_{2}(G^{\prime\prime})\Big{)}\\ +2\Big{(}\overline{\eta}_{1}\overline{\eta}_{2}(F^{\prime\prime})\overline{\eta}_{1}\overline{\eta}_{2}(G^{\prime})-\overline{\eta}_{1}\overline{\eta}_{2}(F^{\prime})\overline{\eta}_{1}\overline{\eta}_{2}(G^{\prime\prime})\Big{)}\\ +6\Big{(}\overline{\eta}_{1}(F^{\prime\prime})\overline{\eta}_{2}(G^{(3)})-\overline{\eta}_{2}(F^{(3)})\overline{\eta}_{1}(G^{\prime\prime})\Big{)}\Big{]}\alpha_{2}^{3}&\hbox{ if }\lambda=3.\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\end{array}$$

$$H_{diff}^2(\mathcal{K}(1),{\mathfrak{F}}_{\lambda }^2)=H_{diff}^2(\mathcal{K}(1),{\mathfrak{F}}_{\lambda }^1)\oplus H_{diff}^2(\mathcal{K}(1),\Pi ({\mathfrak{F}}_{\lambda +\frac{1}{2}}^1)).$$

$$H^{2}(\mathcal{K}(1)_{i},\mathfrak{F}_{\lambda}^{2})_{0}=\left\{\begin{array}[]{ll}\mathbb{K}^{3}&\makebox{ if }~{}\lambda=0,\\ \mathbb{K}^{2}&\makebox{ if }~{}\lambda=1,\\ \mathbb{K}&\makebox{ if }~{}\lambda=3,5,\\ 0&\makebox{ otherwise }.\end{array}\right.$$

$$\begin{array}[]{ll}\bullet\;\displaystyle\Omega_{0,1}^{i}(X_{F},X_{F})=\displaystyle(FG^{\prime}-F^{\prime}G)-\Big{(}\frac{1}{4}+\frac{3}{4}(-1)^{|F||G|}\Big{)}\overline{\eta}_{i}(F)\eta_{i}(G),\\ \\[5.0pt] \bullet\;\displaystyle\Omega_{0,2}^{i}(X_{F},X_{G})=\displaystyle(-1)^{|F|+|G|}\Big{(}F^{\prime}\eta_{i}(G^{\prime})-\eta_{i}(F^{\prime})G^{\prime}\Big{)}\theta_{3-i},\\ \\[5.0pt] \bullet\;\displaystyle\Omega_{0,3}^{i}(X_{F},X_{G})=\Big{(}\frac{1}{2}+\frac{1}{4}(-1)^{|F||G|}\Big{)}(-1)^{|F|+|G|}\Big{(}F\eta_{i}(G^{\prime})-\eta_{i}(F^{\prime})G\Big{)}\theta_{3-i},\\ \\[5.0pt] \bullet\;\displaystyle\Omega_{1,1}^{i}(X_{F},X_{G})=\Big{(}\overline{\eta}_{i}(F^{\prime\prime})G-(-1)^{|F|}F\overline{\eta}_{i}(G^{\prime\prime})\Big{)}\theta_{3-i}-\frac{1}{2}\Big{(}\eta_{i}(F)\eta_{i}(G^{\prime\prime})+\eta_{i}(F^{\prime\prime})\eta_{i}(G)\Big{)}\theta_{i}\theta_{3-i}\alpha_{2},\\ \\[5.0pt] \bullet\;\displaystyle\Omega_{1,2}^{i}(X_{F},X_{G})=\Big{(}F^{\prime}\overline{\eta}_{i}(G^{\prime\prime})-\overline{\eta}_{i}(F^{\prime\prime})G^{\prime}\Big{)}\theta_{3-i}\alpha_{2},\\ \\[5.0pt] \bullet\;\displaystyle\Omega_{3}^{i}(X_{F},X_{G})=\eta_{i}(F^{\prime\prime})\overline{\eta}_{i}(G^{\prime\prime})\alpha_{2}^{3},\\ \\[5.0pt] \bullet\;\displaystyle\Omega_{5}^{i}(X_{F},X_{G})=\Big{(}(F^{(3)}G^{(4)}-(F^{(4)}G^{(3)})+\frac{3}{2}(\eta_{i}(F^{(4)})\eta_{i}(G^{(2)})\\ -(-1)^{|F||G|}\eta_{i}(F^{(2)})\eta_{i}(G^{(4)})-4\eta_{i}(F^{(3)})\eta_{i}(G^{(3)}))\Big{)}\alpha_{2}^{5}.\end{array}$$

$$\hbox{H}^{2}(\mathcal{K}(1)_{i},\mathfrak{F}_{\lambda}^{2})_{1}=\left\{\begin{array}[]{ll}\mathbb{K}^{2}&\makebox{ if }~{}\lambda=\frac{1}{2},\frac{3}{2}\\ \mathbb{K}&\makebox{ if }~{}\lambda=-\frac{1}{2},\frac{5}{2},\frac{9}{2}\\ 0&\makebox{ otherwise }.\end{array}\right.$$

$$\begin{array}[]{llll}\bullet\;\displaystyle\Omega_{-\frac{1}{2}}^{i}(X_{F},X_{G})=\displaystyle\Big{[}(FG^{\prime}-F^{\prime}G)\theta_{3-i}-\Big{(}\frac{1}{4}+\frac{3}{4}(-1)^{|F||G|}\Big{)}\overline{\eta}_{i}(F)\eta_{i}(G)\theta_{3-i}\Big{]}\alpha_{2}^{-\frac{1}{2}}~{}~{}~{}~{}~{}~{}~{}\hbox{ if }\lambda=-\frac{1}{2},\\ \\[2.0pt] \bullet\;\left\{\begin{array}[]{ll}\displaystyle\Omega_{\frac{1}{2}}^{i}(X_{F},X_{G})=\displaystyle\Big{[}(-1)^{|F|+|G|}\Big{(}F^{\prime}\eta_{i}(G^{\prime})-\eta_{i}(F^{\prime})G^{\prime}\Big{)}\Big{]}\alpha_{2}^{\frac{1}{2}},&\\[5.0pt] \widetilde{\Omega}_{\frac{1}{2}}^{i}(X_{F},X_{G})=\Big{[}\Big{(}\frac{1}{2}+\frac{1}{4}\big{(}1+(-1)^{|F||G|}\big{)}\Big{)}(-1)^{|F|+|G|}\Big{(}F\eta_{i}(G^{\prime})-\eta_{i}(F^{\prime})G\Big{)}\Big{]}\alpha_{2}^{\frac{1}{2}},~{}~{}~{}~{}\hbox{ if }\lambda=\frac{1}{2},\end{array}\right.\\ \\[2.0pt] \bullet\;\left\{\begin{array}[]{ll}\displaystyle\Omega_{\frac{3}{2}}^{i}(X_{F},X_{G})=\displaystyle\Big{(}\overline{\eta}_{i}(F^{\prime\prime})G-(-1)^{|F|}F\overline{\eta}_{i}(G^{\prime\prime})\Big{)}-\frac{1}{2}\Big{(}\eta_{i}(F)\eta_{i}(G^{\prime\prime})+\eta_{i}(F^{\prime\prime})\eta_{i}(G)\Big{)}\theta_{i}\alpha_{2}^{\frac{3}{2}},&\\[5.0pt] \widetilde{\Omega}_{\frac{3}{2}}^{i}(X_{F},X_{G})=\Big{(}F^{\prime}\overline{\eta}_{i}(G^{\prime\prime})-\overline{\eta}_{i}(F^{\prime\prime})G^{\prime}\Big{)}\alpha_{2}^{\frac{3}{2}},~{}\hbox{ if }\lambda=\frac{3}{2},\\ \end{array}\right.\\[2.0pt] \bullet\;\displaystyle\Omega_{\frac{5}{2}}^{i}(X_{F},X_{G})=\displaystyle\eta_{i}(F^{\prime\prime})\overline{\eta}_{i}(G^{\prime\prime})\theta_{i}\alpha_{2}^{\frac{5}{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{ if }\lambda=\frac{5}{2},\\ \\[2.0pt] \bullet\;\displaystyle\Omega_{\frac{9}{2}}^{i}(X_{F},X_{G})=\displaystyle\Big{(}(F^{(3)}G^{(4)}-(F^{(4)}G^{(3)})+\frac{3}{2}(\eta_{i}(F^{(4)})\eta_{i}(G^{(2)})\\ -(-1)^{|F||G|}\eta_{i}(F^{(2)})\eta_{i}(G^{(4)})-4\eta_{i}(F^{(3)})\eta_{i}(G^{(3)}))\Big{)}\theta_{i}\alpha_{2}^{\frac{9}{2}}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{ if }\lambda=\frac{9}{2},\\ \end{array}$$

$$\hbox{H}^{2}(\mathcal{K}(1)_{i},\Im_{\lambda}^{1,i})=\left\{\begin{array}[]{ll}\mathbb{K}^{2}&\makebox{ if }~{}\lambda=\frac{1}{2},\;\frac{3}{2}\\ \mathbb{K}&\makebox{ if }~{}\lambda=0,\;3,\;5\\ 0&\makebox{ otherwise }.\end{array}\right.$$

$$\begin{array}[]{llll}\bullet\;\displaystyle\Omega_{0}^{i}(X_{F},X_{G})=\displaystyle(FG^{\prime}-F^{\prime}G)-\big{(}\frac{1}{4}+\frac{3}{4}(-1)^{|F||G|}\big{)}\overline{\eta}_{i}(F)\eta_{i}(G)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{ if }\lambda=0,\\ \\[2.0pt] \bullet\;\left\{\begin{array}[]{ll}\displaystyle\Omega_{\frac{1}{2}}^{i}(X_{F},X_{G})=\displaystyle\Big{[}(-1)^{|F|+|G|}\Big{(}F^{\prime}\eta_{i}(G^{\prime})-\eta_{i}(F^{\prime})G^{\prime}\Big{)}\Big{]}\alpha_{1,i}^{\frac{1}{2}},&\\[5.0pt] \widetilde{\Omega}_{\frac{1}{2}}^{i}(X_{F},X_{G})=\Big{[}\Big{(}\frac{1}{2}+\frac{1}{4}\big{(}1+(-1)^{|F||G|}\big{)}\Big{)}(-1)^{|F|+|G|}\Big{(}F\eta_{i}(G^{\prime})-\eta_{i}(F^{\prime})G\Big{)}\Big{]}\alpha_{1,i}^{\frac{1}{2}},~{}\hbox{ if }\lambda=\frac{1}{2},\end{array}\right.\\[2.0pt] \bullet\;\left\{\begin{array}[]{ll}\displaystyle\Omega_{\frac{3}{2}}^{i}(X_{F},X_{G})=\displaystyle\Big{(}\overline{\eta}_{i}(F^{\prime\prime})G-(-1)^{p(F)}F\overline{\eta}_{i}(G^{\prime\prime})\Big{)}-\frac{1}{2}\Big{(}\eta(F)\eta(G^{\prime\prime})+\eta(F^{\prime\prime})\eta(G)\Big{)}\theta_{i}\alpha^{\frac{3}{2}}_{1,i},&\\[5.0pt] \widetilde{\Omega}_{\frac{3}{2}}^{i}(X_{F},X_{G})=\Big{(}F^{\prime}\overline{\eta}_{i}(G^{\prime\prime})-\overline{\eta}_{i}(F^{\prime\prime})G^{\prime}\Big{)}\alpha_{1,i}^{\frac{3}{2}},~{}\hbox{ if }\lambda=\frac{3}{2},\\ \end{array}\right.\\ \\[2.0pt] \bullet\;\displaystyle\Omega_{3}(X_{F},X_{G})=\eta_{i}(F^{\prime\prime})\overline{\eta}_{i}(G^{\prime\prime})\alpha_{1,i}^{3}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbox{ if }\lambda=3\\ \\[2.0pt] \bullet\;\displaystyle\Omega_{5}(X_{F},X_{G})=\Big{(}(F^{(3)}G^{(4)}-(F^{(4)}G^{(3)})+\frac{3}{2}(\eta_{i}(F^{(4)})\eta_{i}(G^{(2)})\\ -(-1)^{|F||G|}\eta_{i}(F^{(2)})\eta_{i}(G^{(4)})-4\eta_{i}(F^{(3)})\eta_{i}(G^{(3)}))\Big{)}\alpha_{1,i}^{5}.\end{array}$$

$$\begin{array}[]{lcll}\Phi:&{\mathfrak{F}}_{\lambda}^{2}&\longrightarrow&\Im_{\lambda}^{1,i}\oplus\Pi(\Im_{\lambda+\frac{1}{2}}^{1,i})\\ &F\alpha_{2}^{\lambda}&\longmapsto&\Big{(}(1-\theta_{3-i}\partial_{{\theta}_{3-i}})(F)\alpha_{1,i}^{\lambda},~{}\Pi\big{(}(-1)^{|F|+1}\partial_{\theta_{3-i}}(F)\alpha_{1,i}^{\lambda+\frac{1}{2}}\big{)}\Big{)},\end{array}$$

Full text

1 Introduction

Let $\mathfrak{vect}(1)$ be the Lie algebra of vector fields on $S^{1}$. Consider the 1-parameter deformation of the $\mathfrak{vect}(1)$-action on $\mathcal{C}^{\infty}(S^{1})$:

[TABLE]

where $X,\;f\in\mathcal{C}^{\infty}(S^{1})$ and $X^{\prime}:=\frac{dX}{dx}$. This deformation shows that on the level of Lie algebras (and similary below, for Lie superalgebras) it is natural to choose $\mathbb{C}$ as the ground fields.

Denote by $\mathcal{F}_{\lambda}$ the $\mathfrak{vect}(1)$-module structure on $\mathcal{C}^{\infty}(S^{1})$ denoted by $L^{\lambda}$ for a fixed $\lambda$. Geometrically, $\mathcal{F}_{\lambda}=\{fdx^{\lambda}/f\in\mathcal{C}^{\infty}(S^{1})\}$ is the space of weighted densities of weighted $\lambda\in\mathbb{K}$. The space $\mathcal{F}_{\lambda}$ coincides with the space of vector fields, fuctions and differential 1-form for $\lambda=-1,\;0$ and 1, respectively.

Now, we consider the superspace $S^{1|n}$ equipped with its standard contact structure 1-form $\alpha_{n}=dx+\displaystyle\sum_{i=1}^{n}\theta_{i}d\theta_{i}$, and introduce the superspace $\mathfrak{F}^{n}_{\lambda}$ of $\lambda$-densities on the supespace $S^{1|n}$. Let $\mathcal{K}(n)$ be the Lie superalgebra of contact vector fields, $\mathfrak{F}^{n}_{\lambda}$ in naturally a $\mathcal{K}(n)$-module.

The study of $H^{2}(Vect(S^{1}),\mathcal{F}_{\lambda})$ is given in [4], and the super corresponding case $H^{2}(\mathcal{K}(1),\mathfrak{F}^{1}_{\lambda})$ is given by B. Agrebaoui, I. Basdouri and M. Boujelbene in [1] . In this paper, we compute the second cohomology space of the $\mathcal{K}(2)$-module of tensor densities:

[TABLE]

where the action of $\mathcal{K}(2)$ on $\mathfrak{F}^{2}_{\lambda}$ is given by the Lie derivatives:

[TABLE]

where $F,\;G\in\mathcal{C}^{\infty}(S^{1|2}),\;\Phi=G\alpha^{\lambda}_{2}$ and $F^{\prime}=\partial_{x}F$. This space classify central extensions of $\mathcal{K}(2)$ by tensor densities $\mathfrak{F}^{2}_{\lambda}$.

2 Geometry of the superspace $S^{1|2}$

2.1 The Lie superalgebra of contact vector fields on $S^{1|2}$

Let $S^{1\mid 2}$ be the superspace with local coordinates $(x;\;\theta_{1},\theta_{2}),$ where $\theta=(\theta_{1},\theta_{2})$ are the odd variables. Any contact structure on $S^{1\mid 2}$ can be given by the following $1$-form:

[TABLE]

On the space $C^{\infty}(S^{1|2})$, we consider the contact bracket

[TABLE]

where the superscript ’ stands for $\frac{\partial}{\partial x},$ $\overline{\eta}_{i}=\frac{\partial}{\partial{\theta_{i}}}-\theta_{i}\frac{\partial}{\partial x}$ and $|F|$ is the parity of $F$. Note that the derivations $\overline{\eta}_{i}$ are the generators of $2$-extended supersymmetry and generate the kernel of the form (2.1) as a module over the ring of smooth functions. Let $\mathrm{Vect}(S^{1|2})$ be the superspace of vector fields on $S^{1|2}$:

[TABLE]

and consider the superspace $\mathcal{K}(2)$ of contact vector fields on $S^{1|2}$. That is, $\mathcal{K}(2)$ is the superspace of vector fields on $S^{1|2}$ preserving the distribution singled out by the $1$-form $\alpha_{2}$:

[TABLE]

where $L_{X}$ is the Lie derivative along the vector field $X.$

The Lie superalgebra $\mathcal{K}(2)$ is spanned by the fields of the form:

[TABLE]

The bracket in $\mathcal{K}(2)$ can be written as:

[TABLE]

For every contact vector field $X_{F}$, one define a one-parameter family of first-order differential operators on $C^{\infty}(S^{1|2})$:

[TABLE]

We easily check that

[TABLE]

We thus obtain a one-parameter family of ${\mathcal{K}}(2)$-modules on $C^{\infty}(S^{1|2})$ that we denote $\mathfrak{F}^{2}_{\lambda}$, the space of all weighted densities on $C^{\infty}(S^{1|2})$ of weight $\lambda$ with respect to $\alpha_{2}$:

[TABLE]

In particular, we have $\mathfrak{F}_{\lambda}^{0}=\mathcal{F}_{\lambda}$. Obviously the adjoint ${\mathcal{K}}(2)$-module is isomorphic to the space of weighted densities on $C^{\infty}(S^{1|2})$ of weight $-1.$ The case $n=0$ corresponds to the classical setting: ${\mathcal{K}}(0)=\mathrm{Vect}(S^{1})=\{F\partial_{x}|F\in C^{\infty}(S^{1})\}$. Note that, the Lie superalgebra $\mathcal{K}(1)$ is isomorphic to

[TABLE]

Therefore, the spaces of weighted densities $\mathfrak{F}_{\lambda}^{n}$ are also $\mathcal{K}(n-1)$-modules. In [3, 5] it was proved that, as $\mathcal{K}(1)$-module, we have

[TABLE]

where $\Pi$ is the change of parity operator.

2.2 Lie superalgebra cohomology

Let $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ be a Lie superalgebra acting on a super vector space $V=V_{0}\oplus V_{1}$. The space of $k$-cochaines with values in $V$ is the $\mathfrak{g}$-module

[TABLE]

The coboundary operator $\delta_{k}:C^{k}(\mathfrak{g},V)\longrightarrow C^{k+1}(\mathfrak{g},V)$ is a $\mathfrak{g}$-map satisfying $\delta_{k}\circ\delta_{k-1}=0.$ The kernel of $\delta_{k},$ denoted $\hbox{Z}^{k}(\mathfrak{g},V),$ is the space of $k$-cocycles, among them, the elements in the range of $\delta_{k-1}$ are called $k$-coboundries. We denote $\hbox{B}^{k}(\mathfrak{g},V)$ the space of $k$-coboundries.

By definition, the $k^{th}$ cohomology space is the quotient space

[TABLE]

We will only need the formula of $\delta_{n}$ (which will be simply denoted $\delta$) in degrees 0, 1 and 2: for $v\in C^{0}(\mathfrak{g},V)=V,\,\delta v(x):=(-1)^{|x||v|}x.v,$ for $\Upsilon\in C^{1}(\mathfrak{g},V),$

[TABLE]

and for $\Omega\in C^{2}(\mathfrak{g},V),$

[TABLE]

where $x,y,z\in\mathfrak{g}.$

$\mathrm{Hom}(\mathfrak{g},\,V)$ is $\mathbb{Z}_{2}$-graded via

[TABLE]

$\Box$ In this paper, we study the differential cohomology spaces $H^{2}(\mathcal{K}(2),\mathfrak{F}^{2}_{\lambda})$. That is, we consider only cochains $(X_{F},X_{G})\mapsto\Omega(F_{1},F_{2})\alpha_{2}^{\lambda}$ where $\Omega$ is a differential operator.

3 The space $H^{2}(\mathcal{K}(2),\mathfrak{F}^{2}_{\lambda})$

Recall that the Lie superalgebra $\mathcal{K}(2)$ has two subsuperalgebras $\mathcal{K}(1)_{i}\hbox{ for }~{}i=1,~{}2$ isomorphic to $\mathcal{K}(1)$ defined by

[TABLE]

For $i=1,~{}2,$ let $\Im_{\lambda}^{1,i}$ be the $\mathcal{K}(1)_{i}$-module of tensor densities of degree $\lambda$ on $S_{i}^{1\mid 1},$ where $S_{i}^{1\mid 1}$ is the superline with local coordinates $(x,\theta_{i}).$

The main result in this paper is the following:

Theorem 3.1**.**

[TABLE]

The non trivial spaces $\hbox{H}_{diff}^{2}(\mathcal{K}(2),\mathfrak{F}_{\lambda}^{2})$ are spanned by the following 2-cocycles:

[TABLE]

3.1 Relationship between $H^{2}_{diff}(\mathcal{K}(2),\mathfrak{F}^{2}_{\lambda})$ and $H^{2}_{diff}(\mathcal{K}(1),\mathfrak{F}^{2}_{\lambda})$

According to (2.6), we can see that the second cohomology space $H^{2}_{diff}(\mathcal{K}(1),\mathfrak{F}^{2}_{\lambda})$ is closely related to the space $H^{2}_{diff}(\mathcal{K}(1),\mathfrak{F}^{1}_{\lambda})$:

[TABLE]

Thus, we can easely deduce the following result:

Proposition 3.2**.**

1) The cohomology space

[TABLE]

The respective nontrivial $2$-cocycles are

[TABLE]

*where $X_{F},~{}X_{G}\in\mathcal{K}_{i},~{}~{}i=1,~{}2.$

2. The cohomology space*

[TABLE]

The respective nontrivial $2$-cocycles are

[TABLE]

where $X_{F},~{}X_{G}\in\mathcal{K}(1)_{i},~{}~{}i=1,~{}2.$

To prove proposition 3.2, we need the following result (see [1] ).

Proposition 3.3**.**

[1*]**

The cohomology space*

[TABLE]

The respective nontrivial $2$-cocycles are

[TABLE]

where $X_{F},~{}X_{G}\in\mathcal{K}(1)_{i}~{}\hbox{and}~{}\alpha_{1,i}=dx+\theta_{i}d\theta_{i},~{}i=1,~{}2.$

Proof of Proposition 3.2: Let $F\alpha_{2}^{\lambda}=(f_{0}+f_{1}\theta_{1}+f_{2}\theta_{2}+f_{12}\theta_{1}\theta_{2})\alpha_{2}^{\lambda}\in{\mathfrak{F}}_{\lambda}^{2}$. The map

[TABLE]

where $i=1,2$ and $\Pi$ stands for the parity change map, provides us with an isomorphism of $\mathcal{K}(1)_{i}$-modules. In fact, we easily check that

[TABLE]

This map induces the following isomorphism between cohomology spaces:

[TABLE]

We deduce from this isomorphism and Proposition 3.3, the $2$-cocycles (3.14–3.15). $\Box$

Recall that the adjoint $\mathcal{K}(n)$-module, is isomorphic to $\mathfrak{F}^{n}_{-1}.$ Thus, the $\mathcal{K}(1)_{i}$-isomorphism (2.6) yields the following $\mathcal{K}(1)_{i}$-isomorphism:

[TABLE]

where $\mathcal{H}^{i}$ is isomorphic to $\mathfrak{F}^{1,i}_{-\frac{1}{2}}$. More precisely, any element $X_{F}\in\mathcal{K}(2)$ is decomposed into $X_{F}=X_{F_{1}}+X_{F_{2}\theta_{3-i}}$, where $\partial_{\theta_{3-i}}F_{1}=\partial_{\theta_{3-i}}F_{2}=0$, and then $X_{F_{1}}\in\mathcal{K}(1)$ and $X_{F_{2}\theta_{2}}$ is identified to $\Pi(F_{2}\alpha_{1,i}^{-\frac{1}{2}})\in\Pi(\mathcal{H}^{i})$ and it will be denoted $X_{\overline{F}_{2}}$. Moreover, we can easily that:

[TABLE]

The following lemma gives the general form of each 2-cocycle.

Lemma 3.4**.**

Let $\Omega\in Z^{2}_{diff}(\mathcal{K}(2),\mathfrak{F}^{2}_{\lambda})$. Up to a coboundary, the map $\Omega$ are given by

[TABLE]

Proof. Evry differential operator $\Omega$ can be expressed in the form

$\Omega(X_{F},X_{G})=\displaystyle\sum_{i,j,k,l\geq 0}a_{ij}^{kl}\overline{\eta}_{1}^{i}\overline{\eta}_{2}^{j}(F)\overline{\eta}_{1}^{k}\overline{\eta}_{2}^{l}(G)\alpha^{\lambda},\;where\;\frac{\partial}{\partial_{x}}a_{ij}^{kl}=0$, where the coefficients $a_{ij}^{kl}$ are arbitrary functions. Using the 2-cocycle equation, we show that $\frac{\partial}{\partial_{x}}a_{ij}^{kl}=0$. That is, the coefficients $a_{ij}^{kl}$ are not depending on the variable $x$, but they are depending on $\theta_{i},\;i=1,2$ and on the parity of $F$ and $G$. The dependence on the parity of $F$ and $G$ becomes from the fact that $\Omega$ is skew-symmetric:

[TABLE]

indeed, $\overline{\eta}_{i}$ is an odd operator. The main result in this subsection is the following:

Proposition 3.5**.**

There exist, up to a scalar factor and a coboundary, only three non trivial 2-cocycles $\Omega_{0},\;\Omega_{1}$ and $\widetilde{\Omega}_{1}$ from $\mathcal{K}(2)$ to $\mathfrak{F}^{2}_{\lambda}$, given by

[TABLE]

*such that any nonzero linear combination is a nontrivial 2-cocycle and their restrictions to $\mathcal{K}(1)_{1}$ or $\mathcal{K}(1)_{2}$ are coboundaries.

Proof.

Let $\Upsilon$ a 2-cocycle of $\mathcal{K}(2)$ vanishing on $\mathcal{K}(1)_{i},\;i=1,2$ (for example $\mathcal{K}(1)_{1}$).

Assume that $\Upsilon/\mathcal{K}(1)_{1}$ is a coboundary, that is, there exist $\Phi\in\mathfrak{F}^{2}_{\lambda}$ such that

[TABLE]

By replacing $\Upsilon$ by $\Upsilon-\delta(\Phi)$, we can suppose that $\Upsilon$ vanishes on $\mathcal{K}(1)_{1}.$ But in this case according to (3.18), the 2-cocycle relations read

[TABLE]

where $X_{F_{1}},X_{G_{1}}\in\mathcal{K}(1)_{1}$ and $H,\;H_{1},\;H_{2},\;H_{3}\in\mathcal{H}^{1}$. According to (3.20) and lemma (3.4), we deduce the expression of $\Upsilon$. To be more precise, we get

[TABLE]

$\bullet$ For $\lambda=0.$

The 2-cocycle \Omega_{0}(X_{F},X_{G})=\Big{(}F^{\prime}\overline{\eta}_{1}\overline{\eta}_{2}(G)-\overline{\eta}_{1}\overline{\eta}_{2}(F)G^{\prime}\Big{)} is nontrivial. Indeed, suppose that there exist a map $b$ from $\mathcal{K}(2)$ into $\mathfrak{F}^{2}_{\lambda}$ having the general form

[TABLE]

where $a_{ij}\in\mathcal{C}^{\infty}(S^{1|2})$, such that $\Omega_{0}=\delta(b)$. By a direct computation, we find that the term $(f_{12}g_{0}^{\prime}-f_{0}^{\prime}g_{12})$ exist in the expression of $\Omega_{0}$ but not in the $\delta(b)$, wich gives $\Omega_{0}$ is nontrivial 2-cocycle.

$\bullet$ For $\lambda=1$.

For the same reason that for $\lambda=0$, we show that $\Omega_{1}$ and $\widetilde{\Omega}_{1}$ are nontrivial 2-cocycles. On the other hand $\Omega_{1}$ and $\widetilde{\Omega}_{1}$ are not cohomologous, otherwise there exist a map $b$ in the general form given by (3.21) such that, $\alpha\Omega_{1}+\beta\widetilde{\Omega}_{1}=\delta(b)$, where $(\alpha,\;\beta)\in\mathbb{R}^{2}\backslash\{(0,0)\}$, but the term $\alpha(f_{12}g_{0}^{\prime}-f_{0}^{\prime}g_{12})+\beta(f_{12}^{\prime}g_{12}-f_{12}g_{12}^{\prime})$ exist in the expression of $\alpha\Omega_{1}+\beta\widetilde{\Omega}_{1}$ but not in the $\delta(b)$, which gives that $\alpha(f_{12}g_{0}^{\prime}-f_{0}^{\prime}g_{12})+\beta(f_{12}^{\prime}g_{12}-f_{12}g_{12}^{\prime})=0$, i.e.f $\alpha=\beta=0$, and then $\Omega_{1}$ and $\widetilde{\Omega}_{1}$ are not cohomologous. ∎

3.2 Proof of Theorem 3.1

Proof.

Consider a 2-cocycle $\Upsilon\in Z^{2}_{diff}(\mathcal{K}(2),\mathfrak{F}^{2}_{\lambda})$. If $\Upsilon/_{\mathcal{K}(1)^{\otimes 2}}$ is trivial then the 2-cocycle $\Upsilon$ is completely described by proposition 3.5. Thus, assume that $\Upsilon/_{\mathcal{K}(1)^{\otimes 2}}$ is non trivial. Of course, by considering proposition 3.2, we deduce that nontrivial spaces $H^{2}_{diff}(\mathcal{K}(2),\mathfrak{F}^{2}_{\lambda})$ only can appear if $\lambda\in\{-\frac{1}{2},0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3,\frac{9}{2},5\}.$

The $\mathcal{K}(1)_{i}$-isomorphism:

[TABLE]

Together with proposition 3.2 describe, up to a coboundary and up to a scalar factor, the restriction of any 2-cocycle $\Upsilon$ to $\mathcal{K}(1)_{1}$ and to $\mathcal{K}(1)_{2}$. As before, we consider separately the even and the odd cases. Even cohomology spaces only can appear if $\lambda\in\{0,1,2,3,5\}$ and odd cohomology spaces only can appear if $\lambda\in\{-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{9}{2}\}$. In each case, after having the restriction of $\Upsilon$ to both $\mathcal{K}(1)_{1}$ and $\mathcal{K}(1)_{2}$, we complete the expression obtained by the corresponding other terms having the same parity of the last expression then we apply the 2-cocycles conditions we get:

[TABLE]

where $\Upsilon^{0}_{1},\;\Upsilon^{0}_{2}$ and $\Upsilon^{3}_{0}$ are given by (3.12).

Now, assume that there exist a map $b$ in the general form given by (3.21) such that $\alpha\Upsilon^{0}_{1}(X_{F},X_{G})+\beta\Upsilon^{0}_{2}(X_{F},X_{G})=\delta(b)(X_{F},X_{G}).$

The term $(f_{12}g_{0}-f_{0}g_{12})$ exist in the expression of $\alpha\Upsilon^{0}_{1}+\beta\Upsilon^{0}_{2}$ but not in the $\delta(b)$, wich gives that $\alpha\Upsilon^{0}_{1}+\beta\Upsilon^{0}_{2}\neq\delta(b),\;\forall(\alpha,\beta)\in\mathbb{R}^{2}\backslash\{(0,0)\}$. So $\Upsilon^{0}_{1}$ and $\Upsilon^{0}_{2}$ are not cohomologous. This completes the proof.

∎

Acknowledgments

I would like to thank Professors Mabrouk Ben Ammar and Boujemaa Agreboui for helpful suggestions and remarks .