ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-05 (gpt-6.1-sol)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Heterotic E8e8 And Spin32 Lattice Models

Physical scope

Two specified rank-16 lattice inputs in the ten-dimensional heterotic construction. The lattice properties are checked; the current-algebra and global gauge-group identifications are explicitly source-reported, not derived here.

Example

The specified rank-16 lattice inputs E8 direct-sum E8 and $D 16^{+}$ , defined in def-e8-and-d16plus-rank16-lattices, give the two standard ten-dimensional heterotic lattice constructions. The source reports the associated E8 times E8 and Spin(32)/Z2 current-algebra/global-group choices. This is a pair of explicit examples and makes no claim that no other rank-16 lattice or modular invariant exists.

Example references

Timo Weigand, Introduction to String Theory, Winter term 2011/12

Paul Ginsparg, Applied Conformal Field Theory, Les Houches lectures, arXiv:hep-th/9108028v1

David J. Gross, Jeffrey A. Harvey, Emil Martinec, and Ryan Rohm, Heterotic String, Physical Review Letters 54 (1985), 502–505, abstract.

Thomas Mohaupt, Selected Topics in String Theory, Liverpool PGR lecture notes (2022), §2.4, pp. 21–23.

Facts & Assumptions

Given: The ten-dimensional heterotic construction, the two specified rank-16 lattices, and the source-reported current-algebra/global-group associations.

[F1]

Heterotic String Worldsheet Construction pairs the left critical RNS sector with a right bosonic sector whose internal chiral momenta are indexed by a specified positive-definite even self-dual rank-16 lattice.

[F2]

E8 ⊕ E8 and D16+ rank-16 lattice models defines $E_{8} \oplus E_{8}$ and $D_{16}^{+}$ as the displayed rank-16 lattices.

[F3]

The E8 and D16+ models are positive even unimodular lattices proves that these two displayed lattices are positive-definite even unimodular rank-16 lattices.

[F4]

Even-lattice chiral boson sector properties: the rank-16 internal Heisenberg sector has central charge 16.

[A1]

Gross–Harvey–Martinec–Rohm report that the consistent ten-dimensional heterotic gauge-group choices include $E_{8} \times E_{8}$ and $Spin (32) / Z_{2}$ ; Mohaupt’s §2.4 discussion associates those groups with the $E_{8} \oplus E_{8}$ and $D_{16}^{+}$ lattice constructions, respectively. This example reports rather than reconstructs the current-algebra and global-form derivation.

Verification

1.1givenF1F2F3algebra

The product lattice $E_{8} \oplus E_{8}$ has rank $8 + 8 = 16$ by its definition [F2]; $D_{16}^{+}$ has rank 16 by its ambient space. By [F3], both satisfy the even self-dual positive lattice hypothesis in [F1], so each is an admissible specified input to the heterotic construction.

2.1F1F2A1step 1.1

Substituting $Λ = E_{8} \oplus E_{8}$ into [F1] gives the first source-reported construction, with current algebra/global group $E_{8} \times E_{8}$ by [A1]. Substituting $Λ = D_{16}^{+}$ gives the second, with source-reported global group $Spin (32) / Z_{2}$ . Since the example lists two inputs and makes no exhaustive classification claim, neither substitution depends on uniqueness of rank-16 lattices.

3.1F1F4step 2.1algebra∎

The construction uses the rank-16 chiral lattice sector, which contributes $c = 16$ by [F4]; adding the ten noncompact Heisenberg bosons gives $c_{R} = 26$ . This is consistent with the specified right-moving critical central charge, while the gauge-group labels remain source-reported data.

Depends on

Dependency tree · 2 levels

Sources