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A positive two-dimensional representation of the $N = 1$ Ramond zero-mode algebra

Facts & Assumptions

Given: Let $V = C^{1 ∣ 1}$ have even basis vector $e_{0}$ and odd basis vector $e_{1}$ . Write $h$ for the $L_{0}$ eigenvalue. Give $V$ the Hermitian inner product for which $(e_{0}, e_{1})$ is orthonormal, and let the parity operator be $P = diag (1, - 1)$ .

[F1]

$N = 1$ superconformal mode algebra in NS and Ramond sectors: In the Ramond sector the defining odd bracket is ${G_{r}, G_{s}} = 2 L_{r + s} + \frac{r^{2} - 1 / 4}{3} δ_{r + s, 0} C$ .

[F2]

$N = 1$ superconformal mode algebra in NS and Ramond sectors: The adjoint convention is $L_{m}^{*} = L_{- m}$ , $G_{r}^{*} = G_{- r}$ and $C^{*} = C$ .

[F3]

$C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ : $C$ is a field, so the specified matrices act on the complex vector space $V$ .

Example

Consider only the Ramond zero-mode subalgebra generated by $L_{0}, G_{0}, C$ . Define $ρ (C) = \frac{3}{2} I_{V}, ρ (L_{0}) = \frac{17}{16} I_{V}, ρ (G_{0}) = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}) .$ This is an example of a positive finite-dimensional representation of that zero-mode subalgebra. It is not asserted to extend to a representation of all $N = 1$ modes.

Proof

1.1givenF1F3algebra

Since $ρ (L_{0})$ is scalar, $[ρ (L_{0}), ρ (G_{0})] = 0$ . Also $ρ (G_{0})^{2} = I_{V}$ , so ${ρ (G_{0}), ρ (G_{0})} = 2 I_{V}$ . The Ramond relation in [F1] at $r = s = 0$ has right-hand side $2 ρ (L_{0}) - ρ (C) / 12 = \frac{17}{8} I_{V} - \frac{1}{8} I_{V} = 2 I_{V}$ , which agrees. The image of $C$ is scalar and hence central. Thus all brackets of this zero-mode subalgebra are satisfied.

2.1givenF2step 1.1algebra∎

The matrices $ρ (L_{0}), ρ (C), ρ (G_{0})$ are self-adjoint, matching [F2]. The parity operator commutes with $ρ (L_{0})$ and $ρ (C)$ and anticommutes with $ρ (G_{0})$ , so the even/odd grading is respected. The standard Hermitian form is positive definite because $∥ a e_{0} + b e_{1} ∥^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} > 0$ for every nonzero vector. This verifies positivity for this finite-dimensional zero-mode representation only; it is not a unitarity or module-classification result.

A positive two-dimensional representation of the $N = 1$ Ramond zero-mode algebra

Facts & Assumptions

Example

Proof

Depends on

Dependency tree · 2 levels

Sources

A positive two-dimensional representation of the N=1N=1 Ramond zero-mode algebra

Facts & Assumptions

Example

Proof

Depends on

Dependency tree · 2 levels

Sources

A positive two-dimensional representation of the $N = 1$ Ramond zero-mode algebra