Heterotic String Worldsheet Construction

Fix a positive-definite even self-dual rank-16 lattice $\mathrm{\Lambda }\subset V$, with a lattice basis, in the preceding lattice and chiral-boson conventions. Work with closed oriented strings in the flat target $M={\mathbb{R}}^{1,9}$, with metric $\eta =\mathrm{diag}(-,+,\dots ,+)$, orientation $d{x}^0\wedge \cdots \wedge d{x}^9$, and units $\mathrm{\hslash }=c=1$, $[{\alpha }^{\mathrm{\prime }}]=L^2$. The ten-dimensional heterotic worldsheet construction pairs the left chiral part of the adopted free RNS model with a right bosonic sector of ten noncompact target coordinates and the internal chiral lattice state space ${\mathcal{V}}_{\mathrm{\Lambda }}$. The two noncompact chiral parts carry the same target momentum. Internal momenta $\lambda \in \mathrm{\Lambda }$ are dimensionless lattice labels, with weight $⟨\lambda ,\lambda ⟩\mathrm{/}2$; all oscillator state spaces here are algebraic, with finite excitations.

Choose the following left spin sectors and GSO projection, in the RNS spin-sector and GSO conventions. On the cylinder $(\tau ,\sigma )\in \mathbb{R}\times (\mathbb{R}\mathrm{/}2\pi \mathbb{Z})$ include both NS fermions, with ${\psi }_L^{\mu }(\tau ,\sigma +2\pi )=-{\psi }_L^{\mu }(\tau ,\sigma )$ and half-integer modes, and R fermions, with periodicity and integer modes. Their fermion Fock spaces are the exterior algebras of negative modes over the NS vacuum or a single complex irreducible ten-dimensional target Clifford module in R, as specified by the spin-sector definition. Let $F_L$ count nonzero fermionic oscillator excitations modulo two, with NS vacuum parity zero. On the Ramond ground module fix $\{{\mathrm{\Gamma }}^{\mu },{\mathrm{\Gamma }}^{\nu }\}=2{\eta }^{\mu \nu }I$ and ${\mathrm{\Gamma }}_{11}={\mathrm{\Gamma }}^0\cdots {\mathrm{\Gamma }}^9$. Retain precisely the images of $P_{NS}=\frac{1}{2}(I+(-1{)}^{F_L+1}),P_R=\frac{1}{2}(I+{\mathrm{\Gamma }}_{11}(-1{)}^{F_L})\mathrm{.}$ Thus the left sector is $N{S}^{+}\oplus R^{+}$, with ${ϵ}_L=+1$ in the GSO definition: the NS vacuum is removed and the retained Ramond ground states have ${\mathrm{\Gamma }}_{11}=+1$. Pair each retained left sector with the same right bosonic lattice sector. There is no right RNS spin sector or right RNS GSO sign. These specify sector data before physical-state constraints; they do not define a full spin-structure measure or prove modular invariance. The matter central charges are $c_L=15$ and $c_R=10+16=26$, before ghost contributions.

The inputs $E_8\oplus E_8$ and $D_{16}^{+}$ give the familiar source-reported connected gauge-group choices $E_8\times E_8$ and $G_{32}$, respectively. To fix the global-form notation, form the real Clifford algebra generated by an oriented orthonormal basis $e_1,\dots ,e_{32}$ with $e_i{e}_j+e_j{e}_i=-2{\delta }_{ij}$. Its subgroup of products of an even number of unit vectors is $\mathrm{Spin}(32)$. Put $\omega =e_1\cdots e_{32}$ and define $G_{32}=\mathrm{Spin}(32)\mathrm{/}\{1,\omega \}$. This is the semispin quotient denoted $\mathrm{Spin}(32)\mathrm{/}{\mathbb{Z}}_2$ here; the quotient by $\{1,-1\}$ instead gives $SO(32)$. The source-reported lattice-to-group association uses the half-spin convention in which $h_{16}=(1\mathrm{/}2,\dots ,1\mathrm{/}2)$ labels the half-spin representation on which $\omega$ acts as $+1$. No derivation of the current algebra or global gauge group, exhaustive classification of heterotic constructions, or empirical assertion is included.

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.

Brett McInnes, The Semispin Groups in String Theory, §II.

RNS Flat Worldsheet Model

RNS Spin Structures and Sectors

RNS GSO Projections

Positive-definite even unimodular lattice

E8 ⊕ E8 and D16+ rank-16 lattice models

The E8 and D16+ models are positive even unimodular lattices

See the lattice-sector definition and proof on this page. The right-moving internal state space is the rank-16 specialization of Even-lattice chiral boson sector; Even-lattice chiral boson sector properties proves its local vertex algebra, weight, and central-charge interfaces. The left matter central charge is the $D=10$ value from RNS Local Central-Charge Cancellation, and the right matter central charge is 10+16=26 by the rank calculation in the lattice-sector lemma. The associations of the two named lattices to the current algebras and global gauge groups are retained as source-reported construction data; this item does not claim to derive their global forms or to classify all modular invariants.