Dynamical origin of the truncation that generates the ten-dimensional fermionic strings from the 26-dimensional bosonic string

All consistent ten-dimensional fermionic closed strings — type IIA, type IIB, both heterotic strings, and the non-supersymmetric type 0A/0B strings — can be realized inside the Hilbert space of the 26-dimensional bosonic closed string compactified on the $E_8\times SO(16)$ lattice: a universal, modular-invariant truncation of the bosonic spectrum converts bosonic lattice spinor states into space-time fermions, and, applied to the unique tadpole-free $SO(2^{13})$ unoriented open bosonic string, it reproduces from bosonic considerations alone the anomaly-cancelling $SO(32)$ Chan–Paton group of the type I superstring, the tadpole-cancelling Chan–Paton groups of the type 0 open descendants, and the tensions of their D-branes. The construction is, however, purely kinematic: the truncation is imposed at the level of perturbative conformal-field-theory spectra and one-loop amplitudes. The open problem is whether this truncation admits a dynamical origin — whether a non-perturbative mechanism of the bosonic string can be exhibited that selects and enacts the truncated subspace, so that space-time fermions, supersymmetry, and the observed anomaly and tadpole cancellations emerge dynamically from purely bosonic degrees of freedom — or whether a controlled argument rules such a mechanism out.

Background

The setting. The 26-dimensional bosonic closed string is the simplest string theory: its world-sheet degrees of freedom are 26 free bosonic coordinates $X^\mu$ (the dimension 26 is fixed by quantum conformal invariance of the world-sheet), and it contains no world-sheet fermions. To extract lower-dimensional physics one compactifies some of the 26 coordinates on a torus; the momenta of the compact zero modes then live on a lattice, and consistency of the one-loop torus amplitude — invariance under the modular group $SL(2,\mathbb{Z})$ of transformations of the world-sheet torus — requires this lattice to be even and self-dual (even meaning that every lattice vector has even squared length, self-dual meaning that the lattice coincides with its own dual). When the lattice is the weight lattice of a simply laced Lie group $G$ (a group all of whose roots have equal length), with left- and right-moving momenta in the same conjugacy class (a conjugacy class being a coset of the weight lattice with respect to the root lattice) — the authors call this the $E_N$ lattice of $G$ — the gauge symmetry is enhanced from abelian to $G_L\times G_R$.

Fermions out of bosons. In 1985 Casher, Englert, Nicolai and Taormina discovered that ten-dimensional superstrings already live inside this bosonic string: when the 26-dimensional closed bosonic string is compactified on the lattice of $E_8\times E_8$ (a sublattice of the more elegantly formulated $E_8\times SO(16)$ lattice, where $E_8$ is the exceptional Lie group of rank 8), a consistent truncation of its spectrum — removing the oscillator excitations of 12 of the 16 compactified bosons while keeping carefully selected zero modes, and flipping the signs of the contributions of the two spinor conjugacy classes of the lattice (the cosets whose momenta transform as spinor representations of the internal symmetry group) — produces exactly the spectrum of the type II superstrings (the two ten-dimensional supersymmetric closed strings, type IIA and type IIB, which differ in the chirality of their space-time supercharges), with space-time fermions and supersymmetry emerging from purely bosonic degrees of freedom. The fermions appear because lattice states in spinor representations of an internal $SO(8)$ can be tied to space-time rotations through a diagonal Lorentz algebra, so that a spatial rotation acts on them as a half-angle rotation. The mechanism is a stringy descendant of the field-theory phenomenon found by Jackiw–Rebbi and Hasenfratz–'t Hooft, in which monopole-charge composites of a purely bosonic gauge theory can behave as fermions; Freund had anticipated superstrings from 26 dimensions. Englert, Nicolai and Schellekens showed that the $E_8\times SO(16)$ formulation also produces the non-supersymmetric ten-dimensional strings: truncating both sectors gives the type 0A/0B closed strings (non-supersymmetric ten-dimensional strings that still contain space-time fermions), while truncating only one sector gives heterotic strings (closed strings whose left-moving sector is bosonic and whose right-moving sector is superstring-like), with all consistent ten-dimensional heterotic strings arising from the 24-dimensional even self-dual Euclidean lattices (the 'Niemeier' lattices, a finite classified set).

Open descendants. In 2001–2003 Englert, Houart and Taormina, with Chattaraputi, extended the truncation to open strings. An unoriented open string theory (obtained by quotienting by world-sheet parity, which introduces unoriented world-sheets and orientifold planes: fixed hypersurfaces of the parity action that carry negative tension) carries a Chan–Paton group: the gauge group generated by multiplicities at the open-string endpoints. Its one-loop vacuum amplitudes generically diverge through massless closed-string tadpoles (one-point couplings of the graviton–dilaton), and cancellation of these tadpoles fixes the Chan–Paton group: the uncompactified 26-dimensional bosonic open string is tadpole-free only for the group $SO(2^{13})$, i.e. $SO(8192)$. After compactification on the $E_8\times E_8$ sublattice the tadpole condition reduces the multiplicity from $2^{13}$ to $2^5$, so the group becomes $SO(32)$ — precisely the Chan–Paton group that cancels the ten-dimensional gauge and gravitational anomalies of the type I superstring (the unoriented open-plus-closed superstring); on the $E_8\times SO(16)$ lattice it yields instead the groups $[SO(32-n)\times SO(n)]^2$ that eliminate the tadpoles of the type 0 open descendants. The tensions of the D-branes (hypersurfaces on which open strings can end) of the fermionic theories are likewise predicted from bosonic data through a 'brane fusion' mechanism: at the enhanced-symmetry lattice point a single wrapped D25-brane carries the tension of $2^8/\sqrt{N}$ branes of a generic torus compactification of equal volume, where $N$ is the number of conjugacy classes of the lattice.

What is missing. Every one of these results is a perturbative conformal-field-theory and group-theory statement: partition functions, their modular transformations, and one-loop vacuum amplitudes. The truncation itself is imposed by hand as a projection onto a subspace of the bosonic Hilbert space; no dynamical process within the bosonic string is known that would select it. The authors are explicit that this is the open frontier: the underlying JHEP paper concludes that 'this host of results points towards a dynamical origin of truncation. But such dynamics cannot be handled in the context of the perturbative approach given here. To settle the issue, a manageable non-perturbative approach to the bosonic string is mandatory', and the review (ref. 1) adds that the group $E_8$ (and/or $SO(16)$) might play a key role in constructing some 'bosonic M-theory' — by analogy with M-theory, the hypothetical non-perturbative parent of the supersymmetric strings; Horowitz and Susskind had conjectured such a parent in the form of a 27-dimensional strong-coupling limit of the bosonic string. The unresolved target is the dynamical origin of the truncation.

Problem Statement

Does the truncation that extracts the ten-dimensional fermionic strings from the 26-dimensional bosonic string have a dynamical origin?

All consistent ten-dimensional fermionic closed strings — type IIA, type IIB, both heterotic strings, and the non-supersymmetric type 0A/0B strings — together with their D-branes and their tadpole- and anomaly-free open descendants, are obtained by a universal, modular-invariant truncation of the 26-dimensional bosonic string compactified on the $E_8\times SO(16)$ lattice: the oscillator excitations of 12 compactified bosons are removed, selected zero modes are retained, and the spinor-class lattice partition functions enter with flipped signs, as required by the spin-statistics theorem. In its established form this truncation is a kinematic, spectrum-level prescription, and the accompanying matching of the type I $SO(32)$ anomaly cancellation and the type 0 tadpole cancellations is a perturbative conformal-field-theory result.

The problem is to exhibit a dynamical mechanism — within bosonic string dynamics or a non-perturbative completion of the bosonic string — that selects and enacts this truncated subspace, so that the space-time fermions, the supersymmetry of the type II, type I and heterotic descendants, the retained zero modes, the spin-statistics sign flips, the $SO(32)$ Chan–Paton and anomaly structure of type I, and the tadpole cancellations of the type 0 open descendants follow as dynamical consequences from purely bosonic degrees of freedom. These listed consequences are joint requirements: the mechanism must enact the universal truncation — compactified on this lattice and applied to the unique tadpole-free $SO(2^{13})$ open bosonic string — so that the closed-string fermionic spectra and the open-descendant structure follow together; none of them suffices on its own. Alternatively, an acceptable resolution is a controlled no-go result showing that no mechanism within any non-perturbative completion of the bosonic string that reduces to it at weak coupling can enact the truncation.

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