Green Schwarz Superstring Model

Adopt the flat ten-dimensional Type II Green–Schwarz superstring as a classical worldsheet model. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be an open Lorentzian worldsheet coordinate patch with dimensionless coordinates $(\tau ,\sigma )$; all displayed action densities are integrated over $\mathrm{\Omega }$. Let $M={\mathbb{R}}^{1,9}$ with $\eta =\mathrm{diag}(-,+,\dots ,+)$, and fix a finite-dimensional real Grassmann algebra $\mathrm{\Lambda }$. The bosonic field is a smooth even Grassmann-valued map $X:\mathrm{\Omega }\to {\mathrm{\Lambda }}_{\bar{0}}\otimes {\mathbb{R}}^{1,9}$; its body $X_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}}:\mathrm{\Omega }\to M$ is obtained by the augmentation $\mathrm{\Lambda }\to \mathbb{R}$ that kills all Grassmann generators. Extend $\eta$ and the gamma pairings coefficientwise over $\mathrm{\Lambda }$. This even field space includes the nilpotent variations ${\delta }_{\kappa }X$ displayed below. The fermionic fields are smooth maps ${\theta }^I:\mathrm{\Omega }\to {\mathrm{\Lambda }}_{\bar{1}}\otimes S_I$ for $I=1,2$, where each $S_I$ is a fixed real sixteen-dimensional Majorana–Weyl module of $\mathrm{Spin}(1,9)$; the two modules have opposite chiralities in Type IIA and equal chiralities in Type IIB. Grassmann-valued integrals are taken coefficientwise. The action below is the Lorentzian conformal-gauge expression; its local kappa symmetry is the symmetry of the worldsheet-metric formulation with the inherited constraints and metric compensation specified below.

Use worldsheet orientation $d\tau \wedge d\sigma$ (so ${ϵ}^{\tau \sigma }=+1$). The coordinates $X^m$ have length dimension one and the Grassmann-odd spinors have length dimension $1\mathrm{/}2$. Use symmetric ten-dimensional gamma matrices for each chirality, with ${\gamma }_m^{\alpha \beta }{\gamma }_{n\beta \gamma }+{\gamma }_n^{\alpha \beta }{\gamma }_{m\beta \gamma }=2{\eta }_{mn}{\delta }_{\gamma }^{\alpha },$ and the corresponding hatted-index identity for ${\theta }^2$. Set ${\mathrm{\partial }}_{\pm }={\mathrm{\partial }}_{\tau }\pm {\mathrm{\partial }}_{\sigma }$ and define the two supersymmetric pullbacks ${\mathrm{\Pi }}_{\pm }^m={\mathrm{\partial }}_{\pm }{X}^m+\frac{1}{2}({\theta }^1{\gamma }^m{\mathrm{\partial }}_{\pm }{\theta }^1+{\theta }^2{\gamma }^m{\mathrm{\partial }}_{\pm }{\theta }^2)\mathrm{.}$ Adopt ${\alpha }^{\mathrm{\prime }}>0$ with dimension $[{\alpha }^{\mathrm{\prime }}]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^2$, and set $T=1\mathrm{/}(2\pi {\alpha }^{\mathrm{\prime }})$. Adopt the conformal-gauge action $S_{GS}=T{\int }_{\mathrm{\Omega }}d\tau d\sigma [-\frac{1}{2}{\mathrm{\Pi }}_{+}^m{\mathrm{\Pi }}_{-m}-\frac{1}{2}{\mathrm{\partial }}_{\tau }{X}^m({\theta }^1{\gamma }_m{\mathrm{\partial }}_{\sigma }{\theta }^1-{\theta }^2{\gamma }_m{\mathrm{\partial }}_{\sigma }{\theta }^2)+\frac{1}{2}{\mathrm{\partial }}_{\sigma }{X}^m({\theta }^1{\gamma }_m{\mathrm{\partial }}_{\tau }{\theta }^1-{\theta }^2{\gamma }_m{\mathrm{\partial }}_{\tau }{\theta }^2)+\frac{1}{4}({\theta }^1{\gamma }_m{\mathrm{\partial }}_{\tau }{\theta }^1)({\theta }^2{\gamma }^m{\mathrm{\partial }}_{\sigma }{\theta }^2)-\frac{1}{4}({\theta }^1{\gamma }_m{\mathrm{\partial }}_{\sigma }{\theta }^1)({\theta }^2{\gamma }^m{\mathrm{\partial }}_{\tau }{\theta }^2)],$ The Wess–Zumino terms use the actual ${\mathrm{\partial }}_{\tau },{\mathrm{\partial }}_{\sigma }$ derivatives in Berkovits (3.36); ${\mathrm{\Pi }}_{-}$ and ${\mathrm{\Pi }}_{+}$ are respectively that source's $\mathrm{\Pi }$ and $\bar{\mathrm{\Pi }}$. A bilinear ${\theta }^I{\gamma }^m{\mathrm{\partial }}_a{\theta }^I$ uses the matching lower-index block.

Retain the two metric constraints ${\mathrm{\Pi }}_{-}^m{\mathrm{\Pi }}_{-m}=0,{\mathrm{\Pi }}_{+}^m{\mathrm{\Pi }}_{+m}=0.$ For smooth compactly supported Grassmann-odd parameter fields ${\kappa }^1:\mathrm{\Omega }\to {\mathrm{\Lambda }}_{\bar{1}}\otimes S_1^{\ast }$ and ${\kappa }^2:\mathrm{\Omega }\to {\mathrm{\Lambda }}_{\bar{1}}\otimes S_2^{\ast }$ of length dimension $-1\mathrm{/}2$, the adopted kappa transformations are ${\delta }_{\kappa }{\theta }^{1\alpha }={\mathrm{\Pi }}_{-}^m({\gamma }_m{)}^{\alpha \beta }{\kappa }_{\beta }^1,{\delta }_{\kappa }{\theta }^{2\widehat{\alpha }}={\mathrm{\Pi }}_{+}^m({\gamma }_m{)}^{\widehat{\alpha }\widehat{\beta }}{\kappa }_{\widehat{\beta }}^2,{\delta }_{\kappa }{X}^m=\frac{1}{2}({\delta }_{\kappa }{\theta }^1{\gamma }^m{\theta }^1+{\delta }_{\kappa }{\theta }^2{\gamma }^m{\theta }^2)\mathrm{.}$ The induced pullback variations, obtained from their definitions and Grassmann antisymmetry, are ${\delta }_{\kappa }{\mathrm{\Pi }}_{\pm }^m={\delta }_{\kappa }{\theta }^1{\gamma }^m{\mathrm{\partial }}_{\pm }{\theta }^1+{\delta }_{\kappa }{\theta }^2{\gamma }^m{\mathrm{\partial }}_{\pm }{\theta }^2\mathrm{.}$ On the metric-constraint surface they obey the chiral projection constraints ${\mathrm{\Pi }}_{-m}({\gamma }^m{)}_{\alpha \beta }{\delta }_{\kappa }{\theta }^{1\beta }=0,{\mathrm{\Pi }}_{+m}({\gamma }^m{)}_{\widehat{\alpha }\widehat{\beta }}{\delta }_{\kappa }{\theta }^{2\widehat{\beta }}=0,$ because the two successive gamma contractions equal ${\mathrm{\Pi }}_{-}^2$ and ${\mathrm{\Pi }}_{+}^2$, respectively. These are the spinorial constraints parameterized by (3.37)–(3.38) in the source.

The metric compensation is part of the adopted local symmetry. Denote the two zweibein components in the parent worldsheet action by $e,\widehat{e}$, normalized so that their first metric variation at conformal gauge couples as $T{\int }_{\mathrm{\Omega }}d\tau d\sigma (\delta e{\mathrm{\Pi }}_{-}^2+\delta \widehat{e}{\mathrm{\Pi }}_{+}^2)$. Their transformations in this gauge are ${\delta }_{\kappa }e={\kappa }_{\alpha }^1{\mathrm{\partial }}_{-}{\theta }^{1\alpha },{\delta }_{\kappa }\widehat{e}={\kappa }_{\widehat{\alpha }}^2{\mathrm{\partial }}_{+}{\theta }^{2\widehat{\alpha }}\mathrm{.}$ For this source comparison, write $S_L$ for Berkovits's auxiliary $L$-dependent constraint term, which is not a field term in the adopted fixed-metric action. With compactly supported kappa parameters, the source's equation (3.40) is its variation, ${\delta }_{\kappa }{S}_L=+T{\int }_{\mathrm{\Omega }}d\tau d\sigma [{\mathrm{\Pi }}_{-}^2({\kappa }_{\alpha }^1{\mathrm{\partial }}_{-}{\theta }^{1\alpha })+{\mathrm{\Pi }}_{+}^2({\kappa }_{\widehat{\alpha }}^2{\mathrm{\partial }}_{+}{\theta }^{2\widehat{\alpha }})]\mathrm{.}$ Berkovits states that this cancels the variation of the Green–Schwarz action, so the latter has the opposite variation, ${\delta }_{\kappa }{S}_{GS}=-T{\int }_{\mathrm{\Omega }}d\tau d\sigma [{\mathrm{\Pi }}_{-}^2({\kappa }_{\alpha }^1{\mathrm{\partial }}_{-}{\theta }^{1\alpha })+{\mathrm{\Pi }}_{+}^2({\kappa }_{\widehat{\alpha }}^2{\mathrm{\partial }}_{+}{\theta }^{2\widehat{\alpha }})]\mathrm{.}$ This is canceled by the displayed zweibein variation before imposing conformal gauge, and the net variation vanishes on the retained metric constraints. Thus the fixed-metric action is not asserted invariant off shell under an arbitrary local parameter while the metric variation is discarded. Parameters preserving the gauge must also preserve its zweibein gauge condition, or be accompanied by a compensating worldsheet gauge transformation. The auxiliary twistor-like variables in the source's (3.35) and (3.39) are not additional fields of the adopted GS action; they are the source's route to these classical transformations.

This postulate asserts these classical flat-space action, constraints and local gauge transformations. It does not assert a covariant quantum measure or a quantum-consistency result.

String Tension Alpha Prime

Nathan Berkovits, Origin of the Pure Spinor and Green-Schwarz Formalisms, arXiv:1503.03080v3, §3.3, (3.35)–(3.40)

Nathan Berkovits, ICTP Lectures on Covariant Quantization of the Superstring, arXiv:hep-th/0209059v1, §§3.1–3.2