Erler (2022) proved, via the Bergman-Zwiebach dilaton theorem (the statement that closed string field theory admits a distinguished field variation that shifts the string coupling), that the classical action of closed string field theory (SFT) vanishes on any on-shell solution with a compact target space. The determination of this value in physically interesting backgrounds, and the very formulation of closed SFT there, remains open: on noncompact backgrounds such as the orbifolds C/Zn\mathbb{C}/\mathbb{Z}_n, where twisted tachyon condensation changes the asymptotic structure of spacetime, the on-shell action can receive boundary contributions, and closed SFT must be supplemented by boundary terms and tadpole vertices whose construction in the full interacting theory is unknown. The problem asks for this completion, for the resulting on-shell action value in tachyon-condensing noncompact backgrounds, and for its match to the first-quantized vacuum sphere amplitude, with the existence and on-shell action of the Yang-Zwiebach closed string tachyon vacuum as a closely linked subquestion.

Background

Closed string field theory is the second-quantized formulation of closed strings in which the dynamical variable is a full string field Φ\Phi (a state of ghost number 2 in the closed string Hilbert space, subject to level matching — the equality of left- and right-moving oscillator excitation levels), with an action built from a nilpotent coderivation L(κ)=Q+κL2+κ2L3+…L(\kappa)=Q+\kappa L_2+\kappa^2 L_3+\dots on the symmetrized tensor algebra of string states (a coderivation is a map on this algebra that acts as a derivation on symmetrized products of states), where QQ is the BRST operator (the generator of the worldsheet residual gauge symmetry), the LnL_n are graded-symmetric multi-string products satisfying L∞L_\infty relations (a hierarchy of homotopy-Lie identities guaranteeing gauge invariance), κ\kappa is the closed string coupling, and the bilinear ω\omega is the BPZ symplectic form (the pairing of left- and right-moving string states defined by two-point functions on the sphere, after Belavin, Polyakov and Zamolodchikov). The dilaton is the massless scalar field of string theory whose expectation value fixes κ∼e⟨ϕ⟩\kappa\sim e^{\langle\phi\rangle}, so the closed SFT action carries an overall factor 1/κ21/\kappa^2.

The Bergman-Zwiebach dilaton theorem states that closed SFT admits a distinguished field variation, generated at leading order by the ghost-dilaton state, that shifts the coupling κ→κ(1+ϵ)\kappa\to\kappa(1+\epsilon). Erler (2022) recast this variation at the coderivation level and, after a symplectic-form-preserving rescaling of the string field, produced a variation δu\delta_u that changes the overall normalization of the action, δuSSFT=SSFT\delta_u S_{\mathrm{SFT}}=S_{\mathrm{SFT}}. Since the action is stationary on a classical solution π1LeΦ=0\pi_1 L e^{\Phi}=0 (the closed string equations of motion), the action must vanish on-shell: SSFT=0S_{\mathrm{SFT}}=0. This mirrors the low-energy effective action, Seff=κ−2∫dDx e−2ϕLS_{\mathrm{eff}}=\kappa^{-2}\int d^D x\, e^{-2\phi}\mathcal{L}, which also vanishes on-shell because a constant dilaton shift rescales it — provided the target space is compact, so that the constant shift is an allowed variation. On a noncompact target space the action need only be stationary under variations that vanish at infinity; a constant dilaton shift is not of this type, and on-shell boundary contributions can survive, the stringy analogue of the Gibbons-Hawking-York term (the boundary term κ−2∫∂MdD−1xh K\kappa^{-2}\int_{\partial M} d^{D-1}x\sqrt{h}\,K that must be added to the gravitational action for a well-posed variational principle with Dirichlet boundary data).

Two physically central applications make the value of the on-shell action pressing. First, closed string tachyon condensation: twisted tachyons (tachyonic string modes localized at the fixed point of the orbifold) on the orbifolds C/Zn\mathbb{C}/\mathbb{Z}_n (the complex plane modulo a rotation by 2π/n2\pi/n, i.e. a cone whose total angle at the apex is 2π/n2\pi/n and whose deficit angle is 2π(1−1/n)2\pi(1-1/n)) are expected to condense and reduce the conical deficit angle, and the on-shell action of the condensed configuration computes the corresponding shift of vacuum energy; Okawa and Zwiebach's level-truncation computation (truncating the string field to finite mass level) gave a nonzero value in rough agreement with a conjecture of Dabholkar (that the vacuum energy shift equals the deficit-angle change of the effective spacetime), but the exact value is unknown and Dabholkar's conjecture is believed to fail once massless states are included. Second, bulk tachyon condensation: Yang and Zwiebach found level-truncation evidence for a nonperturbative 'tachyon vacuum' of bosonic closed SFT, whose on-shell action is expected to vanish on compact target spaces, but whose existence has never been proven. Relatedly, the on-shell action should equal the first-quantized vacuum sphere amplitude (the tree-level closed string partition function on the sphere), whose definition is obstructed by a cancellation between the infinite volume of the conformal Killing group (the group of worldsheet sphere reparametrizations not fixed by the punctures) and the infinite volume of noncompact target space; it has been extracted only in special limits such as the semiclassical regime of Liouville theory (the conformal field theory describing a fluctuating worldsheet metric).

Problem Statement

Determine the value of the classical on-shell action of closed string field theory in physically interesting backgrounds, and complete the formulation of closed SFT required to pose this question. Specifically: (1) construct the boundary terms and tadpole vertices that must supplement the closed SFT action on backgrounds where the action is nonzero — determining what constant or tadpole terms the completion in fact requires, in particular whether it entails a field-independent constant piece S0S_0 (nonzero already when the string field vanishes) in addition to the usual homogeneous action ShomS_{\mathrm{hom}}, and what condition consistency of the dilaton shift imposes on them, including whether the inhomogeneous variation δuShom=S0+Shom\delta_u S_{\mathrm{hom}}=S_0+S_{\mathrm{hom}} can be realized within the ordinary closed SFT action — and establish the relation of these terms to the Gibbons-Hawking-York boundary terms of the low-energy effective action; (2) with this completed formulation, compute the on-shell action for twisted tachyon condensation on noncompact orbifolds C/Zn\mathbb{C}/\mathbb{Z}_n, deciding whether it vanishes or receives nonzero boundary contributions; (3) establish how this value is matched by the first-quantized vacuum sphere amplitude, which requires a well-defined procedure canceling the conformal Killing group volume against the target-space volume. These three deliverables — (1), (2) and (3) — constitute the resolution and are all required; they share one objective, the value and definability of the closed SFT on-shell action beyond compact target spaces. Closely linked to this objective, but not required for a resolution, is the subquestion of whether the Yang-Zwiebach tachyon vacuum exists as an exact solution of bosonic closed SFT and has vanishing on-shell action, together with the extension of the coderivation-level dilaton variation to closed superstring field theories as the corresponding formal superstring question.

References

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