Rigorous convergence of the zig-zag (Born-Oppenheimer) expansion underlying the Kachru-Witten computation of massless spectra of Landau-Ginzburg orbifolds

The massless spectrum of a heterotic string compactified at a Landau-Ginzburg orbifold point is computed, following Kachru and Witten (1993), as the cohomology of the right-moving supersymmetry charge $Q_+$ at zero worldsheet energy, evaluated by a formal zig-zag (Born-Oppenheimer) expansion: corrections $|\alpha_{-n}\rangle$ of increasingly negative $U$-charge are constructed inductively so that $Q_{+,R}|\alpha_{-n-1}\rangle=-Q_{+,L}|\alpha_{-n}\rangle$, and the resulting formal state $\sum_{n\geq 0}|\alpha_{-n}\rangle$ is asserted to converge on physical grounds - super-renormalizability of the two-dimensional theory and energy separation by $U$-charge - the authors noting that a rigorous proof of this assertion would be interesting. Three decades of follow-up literature has validated, generalized, reimplemented, and reformulated the construction (spectral-sequence and Koszul-complex reformulations, hybrid models, gauged-linear-sigma-model protection arguments, free-field confirmation of the quintic spectrum, public software), but no rigorous proof or refutation of the convergence of the expansion has been given. The open problem is to supply such a proof, or a rigorous counterexample, for general quasi-homogeneous Landau-Ginzburg orbifolds in all twisted sectors, in both the (2,2) and heterotic (0,2) cases.

Background

Heterotic string theory compactified on a Calabi-Yau manifold is a standard route from string theory to particle physics: the massless fields of the resulting four-dimensional theory include gauge-charged matter whose pattern is fixed by the compactification. In worldsheet terms one needs the massless spectrum, i.e. the states of the two-dimensional conformal field theory (CFT) describing string propagation that have zero worldsheet energy (for massless space-time particles the space-time part of the string drops out of the worldsheet Hamiltonian, so this is the zero-energy condition on the internal theory). Compactifications with (2,2) worldsheet supersymmetry - two right-moving and two left-moving supercharges on the two-dimensional worldsheet - correspond to Calabi-Yau compactifications with the tangent bundle as the gauge bundle (the 'standard embedding'); the general heterotic compactification has only (0,2) supersymmetry (right-moving supercharges only) and admits far more general gauge bundles, which makes its spectrum both more interesting phenomenologically and much harder to compute.

A Landau-Ginzburg (LG) model is a two-dimensional $N=2$ supersymmetric field theory of chiral superfields $X_1,\dots,X_N$ with a quasi-homogeneous superpotential $W$ - a polynomial homogeneous under a weighted rescaling of the fields, so that the theory flows in the infrared to a superconformal field theory and is super-renormalizable (only finitely many ultraviolet-divergent Feynman graphs). Witten argued that such super-renormalizable theories describe the tensor products of $N=2$ minimal models known as Gepner models, which represent special points in Calabi-Yau moduli spaces. Quotienting by a finite group $G$ of phase symmetries of $W$ produces an LG orbifold; the orbifold CFT splits into twisted sectors labeled by the elements of $G$, whose ground states and state structure differ sector by sector.

Kachru and Witten gave a technique for computing the complete massless spectrum of such LG orbifolds, including the heterotic (0,2) case, in which the left-moving fermion partners of the chiral fields are replaced by fermions coupled to a holomorphic vector bundle (the data that specify the heterotic gauge bundle). The tool is the right-moving supersymmetry charge $Q_+=Q_{+,R}+Q_{+,L}$ (often written $\overline{Q}_+$ in the (0,2) literature): by an argument analogous to Hodge theory for the Dolbeault operator, the states of zero right-moving worldsheet energy - in the right-moving Ramond sectors (right-moving fermions periodic) that they study, the kernel of the right-moving conformal Hamiltonian $L_{0,+}$ - are exactly the cohomology of $Q_+$, with the appropriate left-moving energy imposed as well. To compute this cohomology the superpotential is treated perturbatively ($W\to\epsilon W$). At zeroth order - the 'Born-Oppenheimer approximation', by analogy with the separation of fast and slow degrees of freedom in molecular physics - the cohomology of $Q_{+,R}$ in the free theory is an elementary finite algebraic object. Corrections are then built by induction on the $U$-charge: Kachru and Witten introduce a charge operator $U$ that assigns $+1$ to the right-moving fermion $\overline{\psi}_{+,i}$, $-1$ to $\psi_{+,i}$, and $0$ to every other field, so that $[U,Q_{+,R}]=Q_{+,R}$, $[U,Q_{+,L}]=0$, and the cohomology of $Q_{+,R}$ is concentrated at a single value of $U$. This charge is carried entirely by the fermions, so a state of $U$-charge $-n$ has energy at least that of a degenerate Fermi gas of $n$ fermions, which grows with $n$. The construction solves, at each order,

$$Q_{+,R}|\alpha_{-n-1}\rangle = -Q_{+,L}|\alpha_{-n}\rangle$$

(Kachru-Witten eq. (2.16); the inductive scheme is the standard 'zig-zag' argument for spectral sequences, attributed by the authors to Bott and Tu). This produces a formal state $|\alpha\rangle=\sum_{n\geq 0}|\alpha_{-n}\rangle$ with $Q_+|\alpha\rangle=0$: a would-be massless state of the interacting theory. Because $Q_{+,R}$ is built from a current of higher dimension (it carries an extra derivative), it dominates over $Q_{+,L}$ on the high-energy states of large negative $U$-charge, and Kachru and Witten argued that $|\alpha_{-n-1}\rangle$ can therefore be chosen much smaller in norm than $|\alpha_{-n}\rangle$, so that - together with super-renormalizability - the series should converge; they note explicitly, however, that a rigorous proof of this assertion would be interesting.

The output was striking: for the quintic hypersurface in projective four-space at its Fermat ('Gepner') point, the technique reproduced sector by sector the geometrically expected spectrum - $101$ families and one antifamily of $E_6$ matter (the ${\bf 27}$ and $\overline{\bf 27}$ representations), and gauge-neutral '$E_6$ singlet' fields reproducing the cohomology group $H^1(\mathbb{P}^4(5),\mathrm{End}(T))$ of endomorphism-valued one-forms of the tangent bundle ($224$ bundle moduli; $330$ singlets in all at the Fermat point, four of them extra states charged under an enhanced $U(1)^4$ gauge symmetry that are lifted away from that point). Precisely this agreement is subtle: worldsheet instantons (nonperturbative worldsheet configurations wrapping rational curves) can shift (0,2) spectra, and the authors stressed that the classical $H^1(\mathrm{End}(T))$ need not match the LG-point spectrum in general and that they did not know why it does for the quintic. The method nonetheless became a standard tool that has been generalized, coded, and repeatedly confirmed, while the convergence question that founds it - whether the formal perturbative state actually exists in the interacting quantum theory - has never been settled.

Problem Statement

Fix an $N=2$ Landau-Ginzburg orbifold in the sense of Kachru-Witten: chiral superfields $X_1,\dots,X_N$ with quasi-homogeneous superpotential $W$, a finite group $G$ of phase symmetries of $W$ (with, in the heterotic case, the accompanying left-moving fermion bundle data), and the right-moving supersymmetry charge $Q_+=Q_{+,R}+Q_{+,L}$ acting on the Hilbert space of each twisted sector. The Kachru-Witten construction builds, from each zeroth-order Born-Oppenheimer ground state $|\alpha_0\rangle$ representing a class in the cohomology of $Q_{+,R}$ (closed under $Q_{+,L}$ up to $Q_{+,R}$-exact states), a sequence of states $|\alpha_{-n}\rangle$ of $U$-charge $-n$ solving $Q_{+,R}|\alpha_{-n-1}\rangle=-Q_{+,L}|\alpha_{-n}\rangle$ order by order in perturbation theory in $W$, producing the formal zero-energy state $|\alpha\rangle=\sum_{n\geq 0}|\alpha_{-n}\rangle$ with $Q_+|\alpha\rangle=0$.

Prove rigorously, or refute rigorously, the convergence assertion of Kachru-Witten Sec. 2.1: that the perturbation series defining $|\alpha\rangle$ converges - in a sense to be made precise, for example as convergence in the Hilbert space of the interacting super-renormalizable theory, or as summability of the perturbation series for the energy of the would-be massless state - so that the $Q_+$-cohomology at zero $L_{0,+}$ eigenvalue (zero right-moving worldsheet energy) of the interacting theory is correctly computed, in every twisted sector, by this expansion.

The resolution is to be carried out within an explicitly stated mathematical framework - the regularization and Hilbert space of the super-renormalizable Landau-Ginzburg theory, the grading, and the precise notion of convergence or summability - so that the claim that the state with zero energy in the Born-Oppenheimer approximation has zero energy in the full interacting theory is a well-posed theorem. That claim is to hold for arbitrary quasi-homogeneous $W$ and orbifold groups $G$ in all twisted sectors, in both the (2,2) and (0,2) cases, or else the class of models and sectors for which convergence holds and the class for which it fails is to be characterized precisely. In case of refutation, the resolution is to exhibit a concrete model and prove divergence of the series or nonexistence of the asserted $Q_+$-closed zero-energy state, and to state precisely what this implies for the validity of the Kachru-Witten massless-spectrum computation.

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