$N=1$ superconformal mode algebra in NS and Ramond sectors

Given: Work over $\mathbb{C}$. The modes $L_m$ ($m\in \mathbb{Z}$) and $C$ are even (parity $0$), and $G_r$ is odd (parity $1$); $C$ is central, meaning its bracket with every mode is zero. The bracket of two odd modes is denoted by braces; in any representation it acts by the operator anticommutator $ST+TS$. In the Neveu–Schwarz (NS) sector $r\in \mathbb{Z}+\frac{1}{2}$; in the Ramond (R) sector $r\in \mathbb{Z}$.

Here a complex Lie superalgebra means a complex vector space in the sense of Vector space over a field, graded as $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$, with a parity-preserving bilinear bracket satisfying $[A,B]=-(-1{)}^{\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }}[B,A]$ and, for homogeneous $A,B,D$, the graded Jacobi identity $(-1{)}^{\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }D\mathrm{\mid }}[A,[B,D]]+(-1{)}^{\mathrm{\mid }B\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }}[B,[D,A]]+(-1{)}^{\mathrm{\mid }D\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }}[D,[A,B]]=0$. A representation means a complex vector space $V=V_{\bar{0}}\oplus V_{\bar{1}}$ with this $\mathbb{Z}\mathrm{/}2$-grading and a parity-preserving linear map $\rho :\mathfrak{g}\to {\mathrm{End}}_{\mathbb{C}}(V)$ satisfying $\rho ([A,B])=[\rho (A),\rho (B){]}_{\mathrm{s}}$ for homogeneous $A,B$; here ${\mathrm{End}}_{\mathbb{C}}(V)$ is the set of complex-linear maps $V\to V$, with $[S,T{]}_{\mathrm{s}}=ST-(-1{)}^{\mathrm{\mid }S\mathrm{\mid }\mathrm{\mid }T\mathrm{\mid }}TS$. For the algebra defined below, it has central charge $c$ when $\rho (C)=c{\mathrm{id}}_V$.

An algebra $\ast$-structure is a parity-preserving conjugate-linear involution $A\mapsto A^{\ast }$ satisfying $[A,B{]}^{\ast }=[B^{\ast },A^{\ast }]$ for homogeneous elements. The convention specified below is the $\ast$-structure used here.

An algebraic $\ast$-representation here means such a representation on a nonzero graded complex vector space $\mathcal{H}={\mathcal{H}}_{\bar{0}}\oplus {\mathcal{H}}_{\bar{1}}$ with a positive-definite sesquilinear form $⟨\cdot ,\cdot ⟩$ satisfying $⟨v,u⟩=\overline{⟨u,v⟩}$ (conjugate-linear in the first variable and linear in the second), together with a graded dense linear domain $\mathcal{D}={\mathcal{D}}_{\bar{0}}\oplus {\mathcal{D}}_{\bar{1}}\subseteq \mathcal{H}$ for the norm $\mathrm{\parallel }u\mathrm{\parallel }=\sqrt{⟨u,u⟩}$, invariant under every mode. The adjoint condition is the formal identity $⟨\rho (A)u,v⟩=⟨u,\rho (A^{\ast })v⟩$ for every generator $A$ and $u,v\in \mathcal{D}$. It is an identity on this common domain; no operator closure or essential self-adjointness is included.

Virasoro algebra in the $L_m$ convention: The Virasoro bracket is $[L_m,L_n]=(m-n)L_{m+n}+\frac{m^3-m}{12}{\delta }_{m+n,0}C$, and its cubic central term satisfies the Lie-algebra Jacobi identity.

$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$: $\mathbb{C}$ is a field, so the defining brackets extend bilinearly over the stated coefficient field.

The $N=1$ super-Virasoro normalization used here is the standard one in Gaberdiel, §3.7.6 and §4.1, and Ginsparg, §5.4. It is distinct from an extended super-$W$ algebra.