RNS Spin Structures and Sectors

Let $D\in {\mathbb{Z}}_{>0}$, let $M={\mathbb{R}}^{1,D-1}$ carry the Minkowski metric $\eta =\mathrm{diag}(-,+,\dots ,+)$ from the adopted RNS model, and let $X:\mathrm{\Sigma }\to M$ be a $C^2$ target map. For the closed Euclidean conformal-field formulation, take $\mathrm{\Sigma }$ to be a connected smooth oriented two-manifold without boundary with a smooth Riemannian metric $h$; its conformal structure makes it a Riemann surface. Write $K_{\mathrm{\Sigma }}=T^{\ast (1,0)}\mathrm{\Sigma }$ for its holomorphic cotangent line bundle. In the chiral-line description, a spin structure is a holomorphic line bundle $S\to \mathrm{\Sigma }$ together with an isomorphism $S^{\otimes 2}\cong K_{\mathrm{\Sigma }}$. For one common geometric nonchiral spin structure set $S_L=S$ and $S_R=\bar{S}$, so $S_L^{\otimes 2}\cong K_{\mathrm{\Sigma }}$ and $S_R^{\otimes 2}\cong \overline{K_{\mathrm{\Sigma }}}$. A factorized chiral state-space description may instead supply these two square-root line bundles independently before projection; that independent choice is additional sector data, not one common geometric spin structure. If spin data are recorded as a frame-bundle lift, it means a principal $\mathrm{Spin}(2)$-bundle $P$ and a $\mathrm{Spin}(2)$-equivariant two-fold map $\tilde{\lambda }:P\to P_{SO}(T\mathrm{\Sigma },h)$ covering the identity and satisfying $\tilde{\lambda }(pg)=\tilde{\lambda }(p){\lambda }_2(g)$ for the covering homomorphism ${\lambda }_2:\mathrm{Spin}(2)\to SO(2)$, where $P_{SO}(T\mathrm{\Sigma },h)$ is the principal bundle of positively oriented $h$-orthonormal tangent frames; the associated spinor bundle is $P{\times }_{\tau }W$ for a specified real Clifford module $W$ and representation $\tau$. No existence criterion or equivalence theorem between these two presentations is used here. These Euclidean conformal-field data are part of the chosen quantum-sector formulation of the Lorentzian RNS model; this definition does not prove a Euclidean path integral or equivalence of Lorentzian and Euclidean observables.

The Grassmann-odd classical chiral matter fields are sections of $S_L\otimes X^{\ast }T{M}_{\mathbb{C}}$ and $S_R\otimes X^{\ast }T{M}_{\mathbb{C}}$, respectively, where $T{M}_{\mathbb{C}}=TM{\otimes }_{\mathbb{R}}\mathbb{C}$; in the Lorentzian local field domain of the adopted postulate their components are $C^1$. The modes below are algebraic state-space data and assert no operator-domain completion. The Lorentzian target metric remains $\eta$; no positive target inner product is imported from the auxiliary Euclidean worldsheet description. On the spatial circle use $\tau \in \mathbb{R}$ and $S^1=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$; NS means ${\psi }^{\mu }(\tau ,\sigma +2\pi )=-{\psi }^{\mu }(\tau ,\sigma )$ and has modes $r\in \mathbb{Z}+\frac{1}{2}$; R means periodicity ${\psi }^{\mu }(\tau ,\sigma +2\pi )={\psi }^{\mu }(\tau ,\sigma )$ and has modes $n\in \mathbb{Z}$.

For one chirality, let $M_{\mathbb{C}}=M{\otimes }_{\mathbb{R}}\mathbb{C}$. The NS algebraic Fock space is the exterior algebra generated by the negative half-integer modes acting on a vacuum $\mathrm{\mid }0{⟩}_{NS}$; every positive mode annihilates that vacuum. In R, choose a nonzero complex vector space $V_R$ with operators ${\mathrm{\Gamma }}^{\mu }$ satisfying ${\mathrm{\Gamma }}^{\mu }{\mathrm{\Gamma }}^{\nu }+{\mathrm{\Gamma }}^{\nu }{\mathrm{\Gamma }}^{\mu }=2{\eta }^{\mu \nu }{I}_{V_R}$, and set ${\psi }_0^{\mu }={\mathrm{\Gamma }}^{\mu }\mathrm{/}\sqrt{2}$. The algebraic R Fock space is the exterior algebra generated by the negative nonzero integer modes over $V_R$; every positive integer mode annihilates $V_R$. Thus both sectors obey $\{{\psi }_r^{\mu },{\psi }_s^{\nu }\}={\eta }^{\mu \nu }{\delta }_{r+s,0}I$, with the integer zero modes acting in the explicitly supplied target module. No positive inner product on these covariant modules is asserted. The closed factorized sectors are NS-NS, NS-R, R-NS, and R-R, with independent chiral labels before projection; a geometric nonchiral spin structure supplies the common left/right geometry described above.

For an open NN strip use the $C^1$ Lorentzian components ${\psi }_{+}^{\mu },{\psi }_{-}^{\mu }$ of the adopted Majorana field on $(\tau ,\sigma )\in \mathbb{R}\times [0,\pi ]$, with signs ${ϵ}_0,{ϵ}_{\pi }\in \{+1,-1\}$ in the endpoint conditions ${\psi }_{+}^{\mu }(\tau ,0)={ϵ}_0{\psi }_{-}^{\mu }(\tau ,0)$ and ${\psi }_{+}^{\mu }(\tau ,\pi )={ϵ}_{\pi }{\psi }_{-}^{\mu }(\tau ,\pi )$. Doubling across the two endpoints gives monodromy ${ϵ}_0{ϵ}_{\pi }$ around the doubled spatial circle; hence the doubled modes are NS when this product is $-1$ and R when it is $+1$. Mixed ND/DN target directions are outside this convention.

RNS Flat Worldsheet Model

Given Spin and Spin-c structures and their spinor bundles

Free two-dimensional fermion CFT with spin sectors

Timo Weigand, Introduction to String Theory, Chapter 6.3

Paul Ginsparg, Applied Conformal Field Theory, Chapter 7

Oliver Schlotterer, Scattering amplitudes in open superstring theory, §§2.2.1–2.2.4

Spencer Dembner, Notes on Conformal Geometry, Uniformization, and Ricci Flow, §2a, Lemma 2.9 and Theorem 2.10

The free-fermion supplier is used for the Euclidean square-root spin-structure and NS/R mode conventions; its positive Fock inner product, Euclidean target metric, and common nonchiral spin sum are not imported. The Lorentzian target Clifford module and its $\eta$-CAR are specified above. Ginsparg, Chapter 7, treats a single nonchiral spin structure and is used only for the NS/R and spin-structure conventions. No claim is made about mixed ND/DN target directions or a global spin-structure measure.