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...ith the isometries of the background. The classical BRST invariance of the model implies that the background fields are constrained to satisfy the ten dimensional supergravity equations of motion. In =-=[12]-=- it was shown that the conformal invariance is preserved in the quantum regime at the oneloop level if the background is constrained by classical BRST invariance. The next logical step is to preserve ...

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...umber anomaly determines that the spinorial derivatives of the dilaton superfield Φ are proportional to the connection Ω. This relation is crucial to cancel the beta function in heterotic string case =-=[10]-=- and will be equally essential in our computation. Let us recall how this relation is obtained. Consider the coupling between ghost number currents and the connections in the action (2.2). Namely 1 2π...

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...mological approach. Since the pure spinor formulation describes a critical string, one expects that the conformal invariance is preserved on shell also at the quantum level and the β-functions vanish =-=[15]-=- [16]. However, as discussed in [7], the vanishing of the β-function is not sufficient to determine the corrections of the field equations. It turns out that the holomorphicity of 1the BRST current a...

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...f its nature, the pure spinor sigma model is a proper description for performing perturbative computations. Using this description it has been possible to compute the beta functions for the Heterotic =-=[6]-=- and Type II Superstring [7], showing that the classical BRST invariance implies in the conformal invariance3. Because in the PS formalism one can quantize in a Super-Poincare invariant manner, one co...

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...potency and holomorphicity imply that the torsion components can be gauge-fixed to the form Tαβ a = γaαβ, TAβ γ = 0, Tαa b = 2(γa b)α βΩβ . (2.21) In addition, the absence of chiral [6] and conformal =-=[7]-=- anomalies of the worldsheet action implies that Ωα is related to the dilaton superfield Φ by Ωα = 1 4 DαΦ. (2.22) 4 The BRST transformations of the minimal fields in (2.15) were determined in [8] to ...

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...d ∫ S = − d 2 σ √ −h( 1 2 hµν ∂µx m ∂νx n ηmn + ωµαP µν ∂νθ α + ˆωµˆα ˜ P µν ∂ν ˆ θ ˆα + + wµαP µν ∂νλ α + ˆwµˆα ˜ P µν ∂ν ˆ λ ˆα ) , 4 For discussion of pure spinor string in general background, see =-=[42, 43, 44, 45, 46, 47]-=-. (2.1) – 2 –where h µν is two dimensional world sheet metric, σ0 = τ ,σ1 = σ. Further, α = 1,... ,16 label Majorana-Weyl spinors and ˆα = 1,... ,16 label second Majorana-Weyl spinors and are 16 × 16...

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...the world-sheet curvature and Φ is the dilaton superfield. Although this term is not necessary for having a covariant action, it is required to have a quantum conformally invariant sigma-model action =-=[14]-=- [15]. 2.2 N = 1 Ten dimensional Supersymmetry We are interested in a background preserving N = 1 supersymmetry in four dimensions. The corresponding supersymmetric derivative algebra is a sub-algebra...

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...the world-sheet curvature and Φ is the dilaton superfield. Although this term is not necessary for having a covariant action, it is required to have a quantum conformally invariant sigma-model action =-=[14]-=- [15]. 2.2 Ten-dimensional, N = 1 Supersymmetry We are interested in a background preserving N = 1 supersymmetry in four dimensions. The corresponding supersymmetric derivative algebra is a sub-algebr...