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Type Ii Worldsheet GSO Spin Anomaly

Physical scope

Critical ten-dimensional Type II closed pure NS–NS worldsheet, no orientifold, with fermions coupled to the target tangent bundle and canceled local anomaly polynomial. The smooth ten-dimensional criterion uses the source-reported Type II anomaly evaluation; the orbifold clause is restricted to the source's light-cone product target R^{1,1}×[Y/G] with an eight-dimensional transverse target Y.

Statement

Assume full AC. Let $Σ$ be a closed oriented smooth two-dimensional worldsheet with the separate left/right spin structures used by the chiral GSO gauging, and let $ϕ : Σ \to X$ be a smooth map to a Hausdorff second-countable smooth ten-dimensional target $X$ . Choose an auxiliary smooth positive-definite metric on $X$ solely to define the orthonormal frame bundle of $T X$ . The worldsheet fermions are sections of the corresponding chiral worldsheet spinor bundles tensored with $ϕ^{*} T X$ . In a critical Type II $N = (1, 1)$ NS–NS sigma-model background with no orientifold, once the perturbative local anomaly polynomial cancels between the two chiralities, the source-reported global GSO anomaly vanishes exactly when $w_{1} (T X) = w_{2} (T X) = 0$ , equivalently when $X$ is orientable and the oriented orthonormal frame bundle of $T X$ admits a lift to $Spin (10)$ .

For the source's light-cone orbifold presentation, take the ten-dimensional target to be $R^{1, 1} \times [\hat{Y} / G]$ , where $\hat{Y}$ is an eight-dimensional smooth transverse target and the discrete group $G$ acts trivially on $R^{1, 1}$ . Under the quotient and Borel admissibility data below, the source-reported anomaly criterion is $w_{1} (E_{G}) = w_{2} (E_{G}) = 0$ for the equivariant transverse tangent bundle $E_{G}$ ; these are the first two equivariant Stiefel–Whitney classes, not the classes of an unspecified coarse quotient. The theorem does not cover RR backgrounds, orientifolds, or general ten-dimensional orbifold targets outside this light-cone product case.

Statement references

Matilda Delgado, Lorenz Eberhardt, and Marija Tomašević, What makes spacetime spin in string theory?, arXiv:2606.18380v2

Masashi Kawahira, Shotaro Kawanago, Shota Saito, and Hiroki Wada, Anomalies and discrete torsion in Type II and Type I worldsheet theories, arXiv:2608.30774v1

Facts & Assumptions

Given: A smooth ten-dimensional target $X$ with an auxiliary smooth positive-definite metric, an oriented closed Type II NS–NS worldsheet with tangent fermions in $ϕ^{*} T X$ , canceled local anomaly polynomial, and the source-reported global anomaly input [A1]. In the orbifold clause the input also includes the quotient presentation and admissible Borel bundle specified in [F5].

[F1]

RNS Spin Structures and Sectors fixes separate left/right spin structures and the NS/R sectors for the two chiral worldsheet fermion systems.

[F2]

RNS GSO Projections fixes the flat-sector chiral GSO projector conventions and IIA/IIB relative chirality choices. Its stated scope does not prove curved-target gauging, modular invariance, or anomaly cancellation; the existence and anomaly of the gauged sigma-model in this item belong to the Given data and [A1].

[F3]

The first Stiefel–Whitney class classifies orientability: for the stated smooth-manifold scope, $w_{1} (T X) = 0$ exactly when $X$ is orientable.

[F4]

Spin and Spin-c obstruction and torsor classification: for an oriented real bundle, a Spin lift exists exactly when $w_{2} = 0$ , with full AC and the stated smooth-manifold base hypotheses; this mathematical criterion is proved using the Spin and determinant-reduction suppliers.

[F5]

Fix the orbifold presentation rather than only its coarse space: $G$ is a specified discrete group, $\hat{Y}$ a Hausdorff second-countable smooth eight-dimensional manifold, and $ρ : G \times \hat{Y} \to \hat{Y}$ a specified smooth proper action with finite stabilizers, preserving the stated NS–NS sigma-model background. The tangent action is $g \cdot (y, v) = (ρ_{g} (y), d ρ_{g} ∣_{y} (v)) .$ Choose a contractible free right $G$ -space $E G$ for which $B_{G} = E G \times_{G} \hat{Y}$ is a paracompact Hausdorff CGWH space of CW homotopy type and $E_{G} = E G \times_{G} T \hat{Y} ⟶ B_{G}$ is a numerable real rank-eight bundle. These admissibility conditions are explicit data of this clause, not a consequence of the smooth-manifold Spin supplier. The quotient relation is $(e g, y, v) \sim (e, ρ_{g} (y), d ρ_{g} (v))$ . Put $H_{G}^{i} (\hat{Y}; F_{2}) = H^{i} (B_{G}; F_{2})$ and define $w_{i}^{G} (T \hat{Y}) = w_{i} (E_{G})$ by Stiefel–Whitney classes from the projective-bundle relation. This defines the classes under that supplier's exact AC, base and numerability hypotheses. The bundle is not $T (B_{G})$ ; no tangent bundle of the infinite-dimensional Borel base is used. No freeness of the action on $\hat{Y}$ is required. The group $G$ is the target orbifold group, distinct from the chiral $Z_{2}$ gauged in [F2].

[A1]

Kawahira–Kawanago–Saito–Wada, §3.1, evaluates the global Type II anomaly for the smooth ten-dimensional target and identifies its mixed anomaly character by $(w_{1} (T X), w_{2} (T X))$ ; under the stated cancellation of the local anomaly polynomial it vanishes exactly when both classes vanish. Delgado–Eberhardt–Tomašević, §§4.1 and 4.3, gives the complementary light-cone transverse-bundle calculation and, for the orbifold data of [F5], reports that the anomaly vanishes exactly when $w_{1} (E_{G}) = w_{2} (E_{G}) = 0$ . The latter source's covariant discussion is not used to claim a general ten-dimensional orbifold theorem. Both are source-reported physical anomaly inputs, not consequences of the Borel construction or local deductions in this item. Their bordism-group computation, reduced-eta calculation, and physical anomaly-line identification are not locally supplied by the current GTS carriers.

[A2]

Full AC is assumed as defined in The Axiom of Choice, as required by [F3]–[F5].

Proof

1.1givenF1F2A1

By the Given data and [A1], the anomaly under consideration is the residual global anomaly of the separate chiral GSO gauging after the local polynomial has canceled; [F1]–[F2] fix sector conventions only. The restrictions to pure NS–NS backgrounds and no orientifold remove the RR and unoriented sectors from this claim. Thus [A1] applies to the pullback bundle $T X$ and says cancellation is equivalent to $w_{1} (T X) = w_{2} (T X) = 0$ .

2.1F3F4A2step 1.1algebra

By [F3], the first equality is equivalent to orientability. Once oriented, the auxiliary metric makes $T X$ an oriented Euclidean bundle of rank ten over the Hausdorff second-countable smooth manifold $X$ . The hypotheses of [F4] now hold, so $w_{2} (T X) = 0$ is equivalent to a lift of this orthonormal frame bundle to $Spin (10)$ . Hence the smooth-target condition is exactly that $X$ is orientable and admits the stated Spin structure.

3.1F5A1∎

For the specified light-cone orbifold presentation, [F5] defines the rank-eight equivariant transverse tangent bundle $E_{G}$ and its classes in $H^{*} (B_{G}; F_{2})$ . Substituting these classes into the matching orbifold input in [A1] gives exactly the stated criterion. This is a conditional use of [A1], not a deduction of the physical anomaly from characteristic-class definitions. In particular [F4] is not applied to $B_{G}$ , which need not be a finite-dimensional smooth manifold, and no general equivariant Spin-lift theorem is inferred here.

Remarks

The anomaly equivalence in the first and third steps remains conditional on the exact source-reported bordism/eta class [A1]. The ordinary smooth-target Spin criterion is supplied by the proved GTS classification theorem; it does not compute the physical anomaly. The Pontryagin–Thom supplier is for normal Spin bordism over an independent $B G$ , and the Dai–Freed supplier is for complex Spin Dirac determinant lines; neither computes this mixed Spin $\times Z_{2}$ physical anomaly. The Borel bundle and its characteristic classes are defined explicitly in [F5]; a theorem identifying equivariant Spin lifts with a classifying-space lifting problem, and the source-specific physical orbifold anomaly evaluation, are not locally supplied. The two bordism/anomaly-line results are therefore not declared dependencies of this argument. The flat-sector conventions in [F2] specify the meaning of the chiral projection only; they do not prove a curved-target gauging or the anomaly evaluation [A1].

Type Ii Worldsheet GSO Spin Anomaly

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources