RNS NS Background Sigma Model

Adopt the following local physical model. A worldsheet $\mathrm{\Sigma }$ is a finite disjoint union of compact connected Riemann surfaces, each equipped with a smooth positive-definite metric $h$ satisfying $h(Ju,Jv)=h(u,v)$ for its complex structure $J$, and a supplied Spin structure. Use exactly the real split $2\mathrm{\mid }2$ superspace defined by the earlier item def-rns-n11-worldsheet-superspace-data: its odd tangent bundle has real rank two and complex chiral decomposition $S_{+}\oplus S_{-}$, while the chiral coordinate functions ${\theta }^{\pm }$ are exchanged by the real structure. The curved data are the supplier's explicit constrained conformal-presentation geometry and its coordinate/Spin transforms: $E_M{}^A$ is the coframe, $\mathcal{E}=\mathrm{Ber}(E_A{}^M)$ is the inverse-frame Berezinian, $\mu ={\mathcal{E}}^{-1}$ is the coframe density, ${\mathrm{\nabla }}_{\pm }$ are its Euclidean curved odd derivatives, and $\mathcal{R}=4F_J$ is its real scalar curvature normalized relative to the real rotation generator. In particular ${\mathrm{\nabla }}_{\pm }$ are not the flat Lorentzian plus-sign generators $Q_{\pm }$. Write $d^2{\theta }_E$ for the supplier's real odd integral $I_E$, with $I_E({\theta }^{+}{\theta }^{-})=-i$, and use $d^{2}z=dxdy$ in an oriented conformal chart $z=x+iy$. The even superfield $X:\mathcal{W}\to M$ is real, so in the component convention below its coefficients obey the supplied reality condition. On each connected component, the body field is a smooth map $x:\mathrm{\Sigma }\to U$ into one target chart $U$ of a smooth ten-dimensional Lorentzian manifold $M$ with signature $(-,+,\dots ,+)$. The fields $G_{\mu \nu }$, $B_{\mu \nu }=-B_{\nu \mu }$, and $\mathrm{\Phi }$ are respectively a Lorentzian metric, a local two-form potential, and a real scalar on $U$; $H=dB$ is the resulting local three-form, with components $H_{\mu \nu \rho }=3{\mathrm{\partial }}_{[\mu }{B}_{\nu \rho ]}$. The target coordinates are $X^{\mu }$, $\mu =0,\dots ,9$, and the worldsheet coordinate indices are $a,b=1,2$. The metric $G$ is the string-frame metric by definition here: it is the metric entering the worldsheet kinetic term, with no Einstein-frame identification asserted.

Use $\mathrm{\hslash }=c=1$, dimensionless worldsheet and odd coordinates, and dimensionless metric components. The target coordinates $x^{\mu }$ have length dimension one, ${\alpha }^{\mathrm{\prime }}>0$ has length dimension two, $G$, $B$, $\mathrm{\Phi }$, and the RNS fermions ${\psi }_{\pm }^{\mu }$ are dimensionless, $H_{\mu \nu \rho }$ has inverse-length dimension, and the auxiliary component $F^{\mu }$ has inverse-length dimension. Fix the superfield normalization as $X^{\mu }=x^{\mu }+\sqrt{{\alpha }^{\mathrm{\prime }}}{\theta }^{+}{\psi }_{+}^{\mu }+\sqrt{{\alpha }^{\mathrm{\prime }}}{\theta }^{-}{\psi }_{-}^{\mu }+{\alpha }^{\mathrm{\prime }}{\theta }^{+}{\theta }^{-}{F}^{\mu }$, so the odd coefficients in the mathematical superspace definition are ${\mathrm{\Xi }}_{\pm }^{\mu }=\sqrt{{\alpha }^{\mathrm{\prime }}}{\psi }_{\pm }^{\mu }$ and the top coefficient is $A^{\mu }={\alpha }^{\mathrm{\prime }}{F}^{\mu }$. In conformal spin coordinates these symbols denote conformal-frame components: relative to the orthonormal generators ${\vartheta }^{\pm }=e^{-s}{\theta }^{\pm }$ they obey ${\psi }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f},\pm }=e^{-s}{\psi }_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h},\pm }$ and $F_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}=e^{-2s}{F}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}}$. The coefficients ${\psi }_{+}$ and ${\psi }_{-}$ are sections of $S_{+}^{\ast }\otimes x^{\ast }T{M}_{\mathbb{C}}$ and $S_{-}^{\ast }\otimes x^{\ast }T{M}_{\mathbb{C}}$, respectively, with odd component-algebra coefficients. Their signs label the multiplying odd coordinate; using the real spin metric to raise their spin index exchanges the two associated chiral summands. Their orthonormal components transform by $e^{\mp i\tau \mathrm{/}2}$ and conformal components by $a_{ij}^{-1}$ and ${\bar{a}}_{ij}^{-1}$, respectively, as in the supplier. This is a change of local bundle trivialization with the same dimensions, not an identification with the independently named Lorentzian spinors. Since $X$ is real in the specified Euclidean superspace, these coefficients satisfy $\overline{{\psi }_{+}}={\psi }_{-}$ and $\bar{F}=-F$ in this ordered-component convention. This is the Euclidean real-superfield condition; it does not identify the two chiral components with the independent Lorentzian Majorana-Weyl components of the flat RNS model. These assignments make the displayed actions dimensionless. For the even superfield $X:\mathcal{W}\to M$, adopt the local Euclidean superspace action $S_E^{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}=\frac{4i}{4\pi {\alpha }^{\mathrm{\prime }}}{\int }_{\mathcal{W}}{d}^{2}z{d}^2{\theta }_E{\mathcal{E}}^{-1}(G_{\mu \nu }(X)+B_{\mu \nu }(X)){\mathrm{\nabla }}_{+}{X}^{\mu }{\mathrm{\nabla }}_{-}{X}^{\nu },$ $S_E^{\mathrm{F}\mathrm{T}}=\frac{1}{4\pi }{\int }_{\mathcal{W}}{d}^{2}z{d}^2{\theta }_E{\mathcal{E}}^{-1}\mathrm{\Phi }(X)\mathcal{R}\mathrm{.}$ The multiplier $4i$ in the matter term is part of this explicit Euclidean formulation: with $D_{\pm }^2={\mathrm{\partial }}_z,{\mathrm{\partial }}_{\bar{z}}$ and $I_E({\theta }^{+}{\theta }^{-})=-i$, the supplier's tensor-density calculation gives $4i{I}_E(\mu (G+B){\mathrm{\nabla }}_{+}x{\mathrm{\nabla }}_{-}x)=\sqrt{h}{h}^{ab}{G}_{\mu \nu }{\mathrm{\partial }}_a{x}^{\mu }{\mathrm{\partial }}_b{x}^{\nu }+i{\epsilon }^{ab}{B}_{\mu \nu }{\mathrm{\partial }}_a{x}^{\mu }{\mathrm{\partial }}_b{x}^{\nu }$. This is an explicit public superspace-formulation normalization, not an asserted theorem of physical equivalence to a differently normalized uncontinued action. The bosonic truncation sets the independent matter components ${\psi }_{\pm }=F=0$ and the independent conformal-gauge gravitino/auxiliary components of $S$ to zero, while retaining the coframe's constraint-determined odd-coordinate terms. The proved weighted curvature-density identity holds for arbitrary $\mathrm{\Phi }(x)$ without discarding its derivatives by integration by parts. With these conventions the complete bosonic component action is $S_E^{\mathrm{b}\mathrm{o}\mathrm{s}}=\frac{1}{4\pi {\alpha }^{\mathrm{\prime }}}{\int }_{\mathrm{\Sigma }}{d}^2\sigma (\sqrt{h}{h}^{ab}{G}_{\mu \nu }(x){\mathrm{\partial }}_a{x}^{\mu }{\mathrm{\partial }}_b{x}^{\nu }+i{\epsilon }^{ab}{B}_{\mu \nu }(x){\mathrm{\partial }}_a{x}^{\mu }{\mathrm{\partial }}_b{x}^{\nu })+\frac{1}{4\pi }{\int }_{\mathrm{\Sigma }}{d}^2\sigma \sqrt{h}\mathrm{\Phi }(x)R^{(2)}(h),$ where ${\epsilon }^{ab}$ is the orientation-compatible antisymmetric density with ${\epsilon }^{12}=+1$, and $R^{(2)}=2K_h$ fixes the positive-curvature convention. The displayed superspace completion, its signs and normalization, and the component action are adopted together as a model postulate, not derived from the flat RNS action. This is local NS–NS data only; no RR background or localized source is present, and no global B-gerbe, boundary, or path-integral measure is asserted.

In $S_E^{\mathrm{F}\mathrm{T}}$, $F_J$ is the coefficient of $J_0$ in the mixed odd frame curvature, and $\mathcal{R}=4F_J=-16i{e}^{2S}{D}_{-}{D}_{+}S$ in the supplier's Euclidean conformal presentation. The integrand contains this real scalar superfunction, rather than Lie-algebra-valued curvature or a representation matrix. At the bosonic truncation $S=s$, $h=e^{-4s}\mathrm{\mid }dz{\mathrm{\mid }}^2$, the coframe retains its linear odd-coordinate terms and $\mu \mathcal{R}=16i{\theta }^{+}{\theta }^{-}{s}_{z\bar{z}}$. Real odd integration therefore gives the scalar-curvature density $\sqrt{h}{R}^{(2)}$ with $R^{(2)}=2K_h$, positive on the round sphere. This formulation asserts a model convention and local classical component reduction; it supplies no functional integration measure or general supergeometry classification.

Timo Weigand, §5.6.1 and Chapter 6.1

David Tong, §7.2.2

Hull, Papadopoulos, and Townsend, Introduction and §5

Knutt-Wehlau and Mann, §4 equations (4.2)–(4.5)

RNS N=(1,1) Worldsheet Superspace Data

RNS Flat Worldsheet Model

Flat Bosonic String Model

String Tension Alpha Prime