The high-energy symmetry of string theory: construct the symmetry behind Gross's genus-independent $\alpha'\to\infty$ linear relations and determine the $\alpha'\to\infty$ S-matrix

In the zero-tension ($\alpha'\to\infty$) limit, string theory's infinite tower of massive states collapses onto zero mass, and the four-point amplitudes of the resulting massless states satisfy linear relations whose ratios are independent of the worldsheet genus. Discovered by D. J. Gross in 1988 via a saddle-point analysis, these relations are the hallmark of Ward identities of a conjectured infinite-parameter high-energy symmetry that Gross proposed as a potential defining symmetry of string theory. The tree-level derivation of the relations is now complete for the 26-dimensional open bosonic string (via decoupling of zero-norm states; closed-string and superstring analogues are partly proven), but the symmetry itself has never been constructed: its algebra, conserved charges, and action on the tower of massless states remain unknown; its validity beyond fixed-order perturbation theory is unestablished; and it is undecided whether the $\alpha'\to\infty$ S-matrix in more than two spacetime dimensions is trivial or nontrivial (the Coleman-Mandula dichotomy). The problem is to identify and construct this symmetry, establish its nonperturbative validity, determine the S-matrix it implies, and establish its scope for superstring/heterotic theories and multiparticle amplitudes.

Background

String theory's characteristic dimensionful parameter is the Regge slope $\alpha'$, related to the string tension by $T=1/(2\pi\alpha')$ and setting the mass scale of the string's oscillator excitations ($M^2\sim n/\alpha'$ at oscillator level $n$). Gross and Mende (1987-1988) analyzed string scattering at fixed scattering angle -- center-of-mass energy squared $s\to\infty$ with the ratio $t/s$ of the squared momentum transfer $t$ to $s$ held fixed -- and showed that the contribution of each worldsheet genus (the string analogue of loop order: each order corresponds to a worldsheet surface with a given number of handles/holes) is dominated by a saddle point of the worldsheet path integral and is exponentially suppressed, roughly as $\exp[-\alpha' s\, f(\theta)]$. In the resulting super-Planckian limit $\alpha'\to\infty$, every fixed-genus amplitude vanishes exponentially while the massive tower collapses onto zero mass: the $\alpha'=\infty$ string is a theory of infinitely many massless particles of arbitrarily high spin, including a spin-2 graviton.

Gross (1988) then derived, at each fixed genus, linear relations among the four-point amplitudes of these massless states: at each mass level the amplitudes are constrained to fixed ratios, with numbers that do not depend on the genus. Linear relations among amplitudes are the hallmark of Ward identities -- the constraints that a continuous symmetry imposes on a scattering matrix (S-matrix) -- so the relations suggest that the $\alpha'\to\infty$ theory possesses an infinite-parameter symmetry of an unfamiliar new kind, which Gross proposed might serve as a defining symmetry of string theory, one that would distinguish it from any field theory. Two facts frame the question. First, the Coleman-Mandula theorem states that under its hypotheses -- a nontrivial, unitary, analytic S-matrix in more than two spacetime dimensions with only finitely many particle types below any given mass -- the symmetry group can only be a direct product of spacetime (Poincaré) and internal symmetries, with no nontrivial mixing of the two. A symmetry acting across an infinite tower of massless higher-spin states evades the hypotheses, but the theorem then gives no guidance: either the $\alpha'\to\infty$ S-matrix is trivial, or genuinely new symmetry structures are possible. Second, zero-norm states: in the old covariant first-quantization of the bosonic string (a standard quantization in which physical states are built from oscillator modes and selected by the residual worldsheet conformal constraints), certain combinations of physical oscillator states have zero norm; they decouple from all physical amplitudes, and this decoupling generates an infinite system of Ward identities. Chan, Lee, Yang, and collaborators used exactly this mechanism to prove, at tree level (genus zero), the linear relations Gross had extracted from saddle points, and to correct the original saddle-point ratios.

The current status is summarized in the 2025 review by Lee and Yang. The tree-level proof of Gross's linear-relations conjecture is complete for the 26-dimensional open bosonic string (closed strings follow through the Kawai-Lewellen-Tye (KLT) relations, which express closed-string amplitudes as bilinear products of open-string ones; the superstring Neveu-Schwarz sector -- the half of the open superstring's bosonic spectrum with integer-moded worldsheet fermions -- is proven for in-plane polarizations and conjectured in general); the Ward identities are argued to hold at every genus, order by order; and in two-dimensional string theory the high-energy symmetry is explicitly known to be a $w_\infty$ algebra (an algebra of area-preserving-diffeomorphism-like generators) built from discrete zero-norm states. What remains unknown is the symmetry itself in the critical dimension: no algebra, no conserved charges, and no action on the state space have been constructed for the 26-dimensional (or 10-dimensional) theory; the nonperturbative status of the relations is unestablished, since the genus expansion diverges and validity at each fixed genus does not imply validity of the full $\alpha'\to\infty$ theory; and the structure of the $\alpha'\to\infty$ S-matrix -- trivial or nontrivial -- is undetermined. Candidate structures in the literature -- notably the exact $SL(K+3,\mathbb{C})$ symmetry of the tree-level string amplitudes (Lai, Lee and Yang) and the dynamical symmetries of high-energy amplitudes -- and the recent perturbative analyses of the high-energy expansion are conjectured to bear on this symmetry, but none is established as the spacetime symmetry whose Ward identities are the genus-independent relations, and none settles the nonperturbative questions.

Gross also stated a reconstruction program for the nontrivial case: solving the unitarity equations (the constraints that probability conservation, $S^\dagger S=1$, imposes on the amplitudes) together with the linear relations for the dilaton/tachyon amplitudes, with the accumulation of infinitely many massless modes as the obstruction.

Problem Statement

Identify and construct the conjectured infinite-parameter high-energy symmetry of string theory in the $\alpha'\to\infty$ limit -- the "string symmetry" that Gross proposed might be the defining symmetry of string theory -- and determine its consequences for the $\alpha'\to\infty$ S-matrix. Concretely, an answer must address, for the critical bosonic string (the 26-dimensional case in which the question was posed):

1. Construct the symmetry. Exhibit the algebraic structure and conserved charges whose Ward identities are the genus-independent linear relations satisfied by the four-point amplitudes of the massless states in the $\alpha'\to\infty$ limit (the relations established at tree level by zero-norm-state decoupling), and specify how these charges act on the infinite tower of massless states, including the tachyon (the negative-mass-squared ground state of the bosonic string), the dilaton (its massless scalar), and the graviton (the massless spin-2 state).

2. Establish nonperturbative validity. Determine whether the linear relations, derived at each fixed genus in a perturbation expansion that diverges in this limit, hold in the full nonperturbative $\alpha'\to\infty$ theory.

3. Determine the S-matrix structure. Decide whether the $\alpha'\to\infty$ S-matrix in more than two spacetime dimensions is trivial (all amplitudes vanish, as Coleman-Mandula-type reasoning applied to a theory with higher-spin conserved charges might force) or nontrivial; if nontrivial, reconstruct it.

4. Establish the symmetry's scope. Verify (or delimit) the corresponding relations and symmetry for superstring and heterotic theories and for multiparticle amplitudes.

Items 1-3 are one coherent objective: the symmetry is what explains the relations, and the S-matrix structure is what the symmetry implies. Item 4 is the source's named extension of the same question. Either such a consistent symmetry exists and its Ward identities carry the genus-independent relations, or no such symmetry exists and the relations fail nonperturbatively.

References

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2. David J. Gross and Paul F. Mende, "The high-energy behavior of string scattering amplitudes", Physics Letters B 197, 129-134 (1987). DOI: 10.1016/0370-2693(87)90355-8, https://doi.org/10.1016/0370-2693(87)90355-8
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