Pure Spinor Type Ii Curved Background Sigma Model

Adopt the following minimal Type II superspace sigma-model data in the exact component order and connection conventions of Bedoya–Chandía. This specifies the local classical action and background constraints; it does not assert a quantum measure, a restored quantum master equation, or the vanishing of one-loop beta functions.

Let $\mathrm{\Sigma }$ be an oriented smooth two-dimensional worldsheet with local complex coordinate $z$. The target $\mathcal{M}$ is a smooth $10\mathrm{\mid }32$-dimensional supermanifold with local even and odd coordinates $Z^M=(x^m,{\theta }^{\mu },{\tilde{\theta }}^{\bar{\mu }})$, and the classical configuration is a smooth supermap $Z:\mathrm{\Sigma }\to \mathcal{M}$, coefficientwise in a fixed finite-dimensional real Grassmann algebra. All displayed equations are local on this chart; background superfields and the invertible supervielbein $E_M{}^A$ are smooth. A frame index $A$ is $a$, $\alpha$, or $\bar{\alpha }$, with $a=0,\dots ,9$ and each spinor index ranging from $1$ to $16$. Barred spinor indices label the second spin bundle, not complex conjugation: it has opposite chirality in IIA and equal chirality in IIB. Put ${\mathrm{\Pi }}^A=\mathrm{\partial }{Z}^M{E}_M{}^A$, ${\bar{\mathrm{\Pi }}}^A=\bar{\mathrm{\partial }}{Z}^M{E}_M{}^A$; these are the pullback components of the supervielbein. The Grassmann-odd first-order worldsheet fields are $d_{\alpha },{\tilde{d}}_{\bar{\alpha }}$. Bosonic ghost fields $({\lambda }^{\alpha },w_{\alpha })$ and $({\tilde{\lambda }}^{\bar{\alpha }},{\tilde{w}}_{\bar{\alpha }})$ obey $\lambda {\gamma }^a\lambda =\tilde{\lambda }{\gamma }^a\tilde{\lambda }=0$ and the conjugate gauge shifts $\delta w_{\alpha }={\mathrm{\Omega }}_a({\gamma }^a\lambda {)}_{\alpha }$ and $\delta {\tilde{w}}_{\bar{\alpha }}={\tilde{\mathrm{\Omega }}}_a({\gamma }^a\tilde{\lambda }{)}_{\bar{\alpha }}$ for arbitrary local parameters, with conventions as in Postulate references. Write $K^{\alpha }{}_{\beta }={\lambda }^{\alpha }{w}_{\beta }$ and ${\tilde{K}}^{\bar{\alpha }}{}_{\bar{\beta }}={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{w}}_{\bar{\beta }}$, always in this order.

The action, source equations (2.2)–(2.3), is $\begin{array}{rl}{\displaystyle S=\frac{1}{2\pi {\alpha }^{\mathrm{\prime }}}\int d^{2}z[}& {\displaystyle \frac{1}{2}{\mathrm{\Pi }}^a{\bar{\mathrm{\Pi }}}^b{\eta }_{ab}+\frac{1}{2}{\mathrm{\Pi }}^A{\bar{\mathrm{\Pi }}}^B{B}_{BA}+d_{\alpha }{\bar{\mathrm{\Pi }}}^{\alpha }+{\tilde{d}}_{\bar{\alpha }}{\mathrm{\Pi }}^{\bar{\alpha }}}\\ {\displaystyle }& {\displaystyle +K^{\alpha }{}_{\beta }{\bar{\mathrm{\Omega }}}_{\alpha }{}^{\beta }+{\tilde{K}}^{\bar{\alpha }}{}_{\bar{\beta }}{\tilde{\mathrm{\Omega }}}_{\bar{\alpha }}{}^{\bar{\beta }}+d_{\alpha }{\tilde{d}}_{\bar{\beta }}{P}^{\alpha \bar{\beta }}}\\ {\displaystyle }& {\displaystyle +K^{\alpha }{}_{\beta }{\tilde{d}}_{\bar{\gamma }}{C}_{\alpha }{}^{\beta \bar{\gamma }}+{\tilde{K}}^{\bar{\alpha }}{}_{\bar{\beta }}{d}_{\gamma }{\tilde{C}}_{\bar{\alpha }}{}^{\bar{\beta }\gamma }+K^{\alpha }{}_{\beta }{\tilde{K}}^{\bar{\alpha }}{}_{\bar{\beta }}{S}_{\alpha \bar{\alpha }}{}^{\beta \bar{\beta }}]+S_{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}+S_{FT},}\\ {\displaystyle S_{FT}}& {\displaystyle =\frac{1}{2\pi }\int d^{2}zr\mathrm{\Phi }(Z)\mathrm{.}}\end{array}$ Here $B$ is the graded superspace two-form, $P$ the RR bispinor superfield, and $r$ the worldsheet scalar-curvature density in the source convention. The worldsheet connection pullbacks are ${\bar{\mathrm{\Omega }}}_{\alpha }{}^{\beta }={\bar{\mathrm{\Pi }}}^A{\mathrm{\Omega }}_{A\alpha }{}^{\beta }$ and ${\tilde{\mathrm{\Omega }}}_{\bar{\alpha }}{}^{\bar{\beta }}={\mathrm{\Pi }}^A{\tilde{\mathrm{\Omega }}}_{A\bar{\alpha }}{}^{\bar{\beta }}$. $S_{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ is the free minimal constrained first-order ghost action: on independent local pure-spinor coordinates ${\gamma }^i,{\tilde{\gamma }}^i$ and their conjugates ${\beta }_i,{\tilde{\beta }}_i$ its action is $S_{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}=(2\pi {\alpha }^{\mathrm{\prime }}{)}^{-1}\int d^{2}z({\beta }_i\bar{\mathrm{\partial }}{\gamma }^i+{\tilde{\beta }}_i\mathrm{\partial }{\tilde{\gamma }}^i)$, with the conjugates normalized by the source canonical bracket. The sign/rescaling of these conjugates is part of this source convention; the action is not obtained by silently identifying the momentum normalization with another reference's flat action. There is no nonminimal quartet here.

The independent left and right structure connections include a scale part and a Lorentz part. To state all gauge-compatible couplings, put $G_{ab}={\gamma }^{ab}$ on the first spin bundle and ${\tilde{G}}_{ab}={\gamma }^{ab}$ on the second. The source decomposition (2.4) is $\begin{array}{rlrl}{\displaystyle {\mathrm{\Omega }}_{A\alpha }{}^{\beta }}& {\displaystyle ={\mathrm{\Omega }}_A{\delta }_{\alpha }{}^{\beta }+\frac{1}{4}{\mathrm{\Omega }}_{Aab}(G^{ab}{)}_{\alpha }{}^{\beta },}& {\displaystyle {\tilde{\mathrm{\Omega }}}_{A\bar{\alpha }}{}^{\bar{\beta }}}& {\displaystyle ={\tilde{\mathrm{\Omega }}}_A{\delta }_{\bar{\alpha }}{}^{\bar{\beta }}+\frac{1}{4}{\tilde{\mathrm{\Omega }}}_{Aab}({\tilde{G}}^{ab}{)}_{\bar{\alpha }}{}^{\bar{\beta }},}\\ {\displaystyle C_{\alpha }{}^{\beta \bar{\gamma }}}& {\displaystyle =C^{\bar{\gamma }}{\delta }_{\alpha }{}^{\beta }+\frac{1}{4}{C}_{ab}{}^{\bar{\gamma }}(G^{ab}{)}_{\alpha }{}^{\beta },}& {\displaystyle {\tilde{C}}_{\bar{\alpha }}{}^{\bar{\beta }\gamma }}& {\displaystyle ={\tilde{C}}^{\gamma }{\delta }_{\bar{\alpha }}{}^{\bar{\beta }}+\frac{1}{4}{\tilde{C}}_{ab}{}^{\gamma }({\tilde{G}}^{ab}{)}_{\bar{\alpha }}{}^{\bar{\beta }},}\\ {\displaystyle S_{\alpha \bar{\alpha }}{}^{\beta \bar{\beta }}}& {\displaystyle =S{\delta }_{\alpha }{}^{\beta }{\delta }_{\bar{\alpha }}{}^{\bar{\beta }}+\frac{1}{4}{S}_{ab}(G^{ab}{)}_{\alpha }{}^{\beta }{\delta }_{\bar{\alpha }}{}^{\bar{\beta }}+\frac{1}{4}{\tilde{S}}_{ab}{\delta }_{\alpha }{}^{\beta }({\tilde{G}}^{ab}{)}_{\bar{\alpha }}{}^{\bar{\beta }}}\\ {\displaystyle }& {\displaystyle +\frac{1}{16}{S}_{abcd}(G^{ab}{)}_{\alpha }{}^{\beta }({\tilde{G}}^{cd}{)}_{\bar{\alpha }}{}^{\bar{\beta }}\mathrm{.}}\end{array}$ No scale component is set to zero merely because the Lorentz connection is present.

The independent weight-$(1,1)$ operator families used in the source's one-loop expansion are the products of one entry from each list $({\mathrm{\Pi }}^a,{\mathrm{\Pi }}^{\alpha },d_{\alpha },K^{\alpha }{}_{\beta }),({\bar{\mathrm{\Pi }}}^a,{\bar{\mathrm{\Pi }}}^{\bar{\alpha }},{\tilde{d}}_{\bar{\alpha }},{\tilde{K}}^{\bar{\alpha }}{}_{\bar{\beta }})\mathrm{.}$ Independence is in the source's pure-spinor-gauge quotient and classical equations-of-motion basis, not in an unconstrained sixteen-pair ghost theory. The other spinor pullbacks are eliminated by the actual equations ${\bar{\mathrm{\Pi }}}^{\alpha }+{\tilde{d}}_{\bar{\beta }}{P}^{\alpha \bar{\beta }}+{\tilde{K}}^{\bar{\beta }}{}_{\bar{\gamma }}{\tilde{C}}_{\bar{\beta }}{}^{\bar{\gamma }\alpha }=0,{\mathrm{\Pi }}^{\bar{\alpha }}-d_{\beta }{P}^{\beta \bar{\alpha }}+K^{\beta }{}_{\gamma }{C}_{\beta }{}^{\gamma \bar{\alpha }}=0.$ This specifies the coupling slots referred to after source (4.6); no finite one-loop coefficient or vanishing theorem is included in this postulate.

The background tensors are the supervielbein torsions $T,\tilde{T}$, the two connection curvatures $R,\tilde{R}$, and $H=dB$. Covariant differentiation uses the left or right connection on each corresponding spinor index. The source classical background premises are nilpotent, anticommuting charges $Q=\oint {\lambda }^{\alpha }{d}_{\alpha }$ and $\tilde{Q}=\oint {\tilde{\lambda }}^{\bar{\alpha }}{\tilde{d}}_{\bar{\alpha }}$ and conserved currents $\bar{\mathrm{\partial }}(\lambda d)=\mathrm{\partial }(\tilde{\lambda }\tilde{d})=0$ on the classical equations of motion. The nilpotency and anticommutation conditions (2.5)–(2.7), for all pure spinors, are

${\lambda }^{\alpha }{\lambda }^{\beta }{H}_{\alpha \beta A}={\lambda }^{\alpha }{\lambda }^{\beta }{\lambda }^{\gamma }{R}_{\alpha \beta \gamma }{}^{\delta }={\lambda }^{\alpha }{\lambda }^{\beta }{\tilde{R}}_{\alpha \beta \bar{\gamma }}{}^{\bar{\delta }}=0,$

${\lambda }^{\alpha }{\lambda }^{\beta }{T}_{\alpha \beta }{}^a={\lambda }^{\alpha }{\lambda }^{\beta }{T}_{\alpha \beta }{}^{\gamma }={\lambda }^{\alpha }{\lambda }^{\beta }{T}_{\alpha \beta }{}^{\bar{\gamma }}=0,$

${\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{H}_{\bar{\alpha }\bar{\beta }A}={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{R}_{\bar{\alpha }\bar{\beta }\gamma }{}^{\delta }={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{\tilde{\lambda }}^{\bar{\gamma }}{\tilde{R}}_{\bar{\alpha }\bar{\beta }\bar{\gamma }}{}^{\bar{\delta }}=0,$

${\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{T}_{\bar{\alpha }\bar{\beta }}{}^a={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{T}_{\bar{\alpha }\bar{\beta }}{}^{\gamma }={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{\tilde{T}}_{\bar{\alpha }\bar{\beta }}{}^{\bar{\gamma }}=0.$

$H_{\alpha \bar{\beta }A}=T_{\alpha \bar{\beta }}{}^a=T_{\alpha \bar{\beta }}{}^{\gamma }=T_{\alpha \bar{\beta }}{}^{\bar{\gamma }}={\lambda }^{\alpha }{\lambda }^{\beta }{R}_{\bar{\gamma }\alpha \beta }{}^{\delta }={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{\tilde{R}}_{\gamma \bar{\alpha }\bar{\beta }}{}^{\bar{\delta }}=0.$

The holomorphicity conditions (2.15) are the following vanishing expressions:

$T_{\alpha (ab)}=H_{\alpha ab}=T_{\alpha \beta a}-H_{\alpha \beta a}=T_{a\alpha }{}^{\beta }=T_{a\alpha }{}^{\bar{\beta }}+P^{\gamma \bar{\beta }}{T}_{\gamma \alpha a}={\lambda }^{\alpha }{\lambda }^{\beta }{R}_{a\alpha \beta }{}^{\gamma }=0,$

${\tilde{R}}_{a\alpha \bar{\beta }}{}^{\bar{\gamma }}-{\tilde{C}}_{\bar{\beta }}{}^{\bar{\gamma }\delta }{T}_{\delta \alpha a}=T_{\gamma \alpha }{}^{\bar{\beta }}+\frac{1}{2}{P}^{\delta \bar{\beta }}{H}_{\delta \gamma \alpha }={\tilde{R}}_{\delta \alpha \bar{\beta }}{}^{\bar{\gamma }}+\frac{1}{2}{\tilde{C}}_{\bar{\beta }}{}^{\bar{\gamma }\rho }{H}_{\rho \delta \alpha }=P^{\delta \bar{\gamma }}{T}_{\delta \alpha }{}^{\beta }-{\mathrm{\nabla }}_{\alpha }{P}^{\beta \bar{\gamma }}-C_{\alpha }{}^{\beta \bar{\gamma }}=0,$

${\mathrm{\nabla }}_{\alpha }{\tilde{C}}_{\bar{\beta }}{}^{\bar{\gamma }\delta }+{\tilde{C}}_{\bar{\beta }}{}^{\bar{\gamma }\rho }{T}_{\rho \alpha }{}^{\delta }+P^{\delta \bar{\rho }}{\tilde{R}}_{\bar{\rho }\alpha \bar{\beta }}{}^{\bar{\gamma }}-S_{\alpha \bar{\beta }}{}^{\delta \bar{\gamma }}={\lambda }^{\alpha }{\lambda }^{\beta }({\mathrm{\nabla }}_{\alpha }{C}_{\beta }{}^{\gamma \bar{\delta }}-P^{\rho \bar{\delta }}{R}_{\rho \alpha \beta }{}^{\gamma })=0,$

${\lambda }^{\alpha }{\lambda }^{\beta }({\mathrm{\nabla }}_{\alpha }{S}_{\beta \bar{\delta }}{}^{\gamma \bar{\rho }}+C_{\beta }{}^{\gamma \bar{\sigma }}{\tilde{R}}_{\bar{\sigma }\alpha \bar{\delta }}{}^{\bar{\rho }}+{\tilde{C}}_{\bar{\delta }}{}^{\bar{\rho }\sigma }{R}_{\sigma \alpha \beta }{}^{\gamma })=0,$

The antiholomorphicity conditions (2.16) are:

$T_{\bar{\alpha }(ab)}=H_{\bar{\alpha }ab}=T_{\bar{\alpha }\bar{\beta }a}+H_{\bar{\alpha }\bar{\beta }a}=T_{a\bar{\alpha }}{}^{\bar{\beta }}=T_{a\bar{\alpha }}{}^{\beta }-P^{\beta \bar{\gamma }}{T}_{\bar{\gamma }\bar{\alpha }a}={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}{\tilde{R}}_{a\bar{\alpha }\bar{\beta }}{}^{\bar{\gamma }}=0,$

$R_{a\bar{\alpha }\beta }{}^{\gamma }-C_{\beta }{}^{\gamma \bar{\delta }}{T}_{\bar{\delta }\bar{\alpha }a}=T_{\bar{\gamma }\bar{\alpha }}{}^{\beta }+\frac{1}{2}{P}^{\beta \bar{\delta }}{H}_{\bar{\gamma }\bar{\delta }\bar{\alpha }}=R_{\bar{\delta }\bar{\alpha }\beta }{}^{\gamma }+\frac{1}{2}{C}_{\beta }{}^{\gamma \bar{\rho }}{H}_{\bar{\delta }\bar{\rho }\bar{\alpha }}=P^{\beta \bar{\delta }}{T}_{\bar{\delta }\bar{\alpha }}{}^{\bar{\gamma }}-{\mathrm{\nabla }}_{\bar{\alpha }}{P}^{\beta \bar{\gamma }}+{\tilde{C}}_{\bar{\alpha }}{}^{\bar{\gamma }\beta }=0,$

${\mathrm{\nabla }}_{\bar{\alpha }}{C}_{\beta }{}^{\gamma \bar{\delta }}+C_{\beta }{}^{\gamma \bar{\rho }}{T}_{\bar{\rho }\bar{\alpha }}{}^{\bar{\delta }}-P^{\rho \bar{\delta }}{R}_{\rho \bar{\alpha }\beta }{}^{\gamma }-S_{\beta \bar{\alpha }}{}^{\gamma \bar{\delta }}={\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}({\mathrm{\nabla }}_{\bar{\alpha }}{\tilde{C}}_{\bar{\beta }}{}^{\bar{\gamma }\delta }+P^{\delta \bar{\rho }}{\tilde{R}}_{\bar{\rho }\bar{\alpha }\bar{\beta }}{}^{\bar{\gamma }})=0,$

${\tilde{\lambda }}^{\bar{\alpha }}{\tilde{\lambda }}^{\bar{\beta }}({\mathrm{\nabla }}_{\bar{\alpha }}{S}_{\delta \bar{\beta }}{}^{\rho \bar{\gamma }}+C_{\delta }{}^{\rho \bar{\sigma }}{\tilde{R}}_{\bar{\sigma }\bar{\alpha }\bar{\beta }}{}^{\bar{\gamma }}+{\tilde{C}}_{\bar{\beta }}{}^{\bar{\gamma }\sigma }{R}_{\sigma \bar{\alpha }\delta }{}^{\rho })=0.$

All products, free indices and signs in these equations are in the source order. Symmetrization is graded: interchanging homogeneous superindices $A,B$ in an antisymmetrization contributes $-(-1{)}^{\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }}$, with $\mathrm{\mid }a\mathrm{\mid }=0$ and $\mathrm{\mid }\alpha \mathrm{\mid }=\mathrm{\mid }\bar{\alpha }\mathrm{\mid }=1$.

The torsion, three-form and curvature Bianchi identities adopted in the same convention are ${\mathrm{\nabla }}_{[A}{T}_{BC]}{}^D+T_{[AB}{}^E{T}_{EC]}{}^D-R_{[ABC]}{}^D=0,$ ${\mathrm{\nabla }}_{[A}{H}_{BCD]}+\frac{3}{2}{T}_{[AB}{}^E{H}_{ECD]}=0,{\mathrm{\nabla }}_{[A}{R}_{BC]D}{}^E+T_{[AB}{}^F{R}_{FC]D}{}^E=0.$ Use $\tilde{R}$ and the corresponding connection for an output right-spinor index; with vector output either stated connection gives its own torsion Bianchi identity. The $H$ identities for the two connections express the same $dH=0$ condition. These are background identities, not a quantum Ward-identity restoration result.

Finally adopt the source ghost-number/dilaton compatibility equation (2.21), ${\mathrm{\nabla }}_{\alpha }\mathrm{\Phi }=4{\mathrm{\Omega }}_{\alpha },{\mathrm{\nabla }}_{\bar{\alpha }}\mathrm{\Phi }=4{\tilde{\mathrm{\Omega }}}_{\bar{\alpha }}\mathrm{.}$ This is an explicit source background premise. It is not deduced here from classical torsion Bianchi identities alone, nor promoted to a construction of a regulator or measure.

Pure Spinor Superstring Model

Oscar A. Bedoya and Osvaldo Chandía, One-loop Conformal Invariance of the Type II Pure Spinor Superstring in a Curved Background, arXiv:hep-th/0609161v2, §§2–2.5, equations (2.2)–(2.7), (2.11), (2.15)–(2.21), and the operator basis described following (4.6).