Exactly marginal ground-ring deformations of the two-dimensional (c=1) string

The two-dimensional (c=1) bosonic string, a free matter field coupled to Liouville gravity, possesses a 'ground ring': the commutative ring of physical states of spin zero and ghost number zero in the BRST cohomology. At the self-dual (SU(2)) compactification radius this ring is the ring of polynomial functions on a three-dimensional quadric cone $Q$ defined by $a_1a_2-a_3a_4=0$ . Witten (1991) conjectured that the discrete-state moduli of the theory deform this relation to $a_1a_2-a_3a_4=\phi(a_i)$ , computed the second-order obstruction (the quadratic beta function), and posed the problem of determining which $\phi$ correspond to exactly marginal (integrably marginal) perturbations to all orders in perturbation theory, together with the classification of the associated lifted symmetry algebra and a geometric interpretation of the deformed theory. The literature shows that the conjecture has been verified only for the special case $\phi=\mu$ (the cosmological constant, by Barbon 1992); that the moduli space was enlarged by Witten and Zwiebach (1992), who reformulated the second-order obstruction as $\beta=F\wedge F$ and developed the symmetry structure as a homotopy Lie algebra; that the ground-ring geometry was extended to the A-D-E radii (Ghoshal, Jatkar and Mukhi 1993) and identified with the deformed conifold of topological string theory (Ghoshal and Vafa 1995); and that the matrix-model side classifies all Fermi-surface deformations as Toda-lattice flows (Alexandrov, Kazakov and Kostov 2002-2003). The all-orders exact-marginality classification in ground-ring terms, the geometric interpretation of the deformed moduli algebra, the completion of the symmetry-lifting classification, and the conjectural three-dimensional formulation all remain without a complete answer in the literature.

Background

The system. The $c=1$ string is bosonic string theory with a two-dimensional target space: a free 'matter' field $X$ (time, possibly compactified on a circle of radius $R$ ) is coupled to the Liouville field $\phi$ , which plays the role of the spatial direction and carries a linearly growing dilaton so that the total central charge is 26. It is the simplest exactly solvable string theory: the double-scaled limit (the limit in which the depth of the oscillator potential and the matrix size $N$ grow together so that the string coupling stays finite) of the quantum mechanics of an $N\times N$ Hermitian matrix in an inverted harmonic-oscillator potential — a system of free fermions whose filled Fermi sea occupies single-particle energies below $-\mu$ ( $\mu>0$ is the cosmological constant) — is believed to define the theory nonperturbatively. In this matrix model, the eigenvalue phase space with coordinates (eigenvalue $q$ , momentum $p$ ) and the profile of the Fermi surface play the role of a classical phase space and of the background geometry.

Discrete states and the ground ring. Physical states of a string theory are classes in the BRST cohomology (the cohomology of the gauge-fixing ghost charge after worldsheet gauge fixing), graded by ghost number. In the $c=1$ string, besides the continuum of tachyon momenta, there are 'discrete states': physical operators existing only at special values of the momentum. When $X$ is compactified at the self-dual radius (the radius at which the matter conformal field theory acquires an $SU(2)\times SU(2)$ current-algebra symmetry), the discrete states organize into $SU(2)$ multiplets, and their couplings are governed by the algebra $W_{1+\infty}$ of polynomial area-preserving vector fields on a plane (an infinite-dimensional higher-spin symmetry algebra). The 'ground ring' is the commutative ring formed by the ghost-number-zero, spin- $(0,0)$ BRST classes under the operator product expansion (the multiplication of local operators in a conformal field theory). At the self-dual radius it is generated by four operators $a_1,a_2,a_3,a_4$ with the single relation

a_1a_2-a_3a_4=0,

that is, it is the ring of polynomial functions on a three-dimensional quadric cone $Q\subset\mathbb{C}^4$ (the surface swept out by complex lines through the origin over $\mathbb{P}^1\times\mathbb{P}^1$ , where $\mathbb{P}^1$ is the Riemann sphere; its small resolution is the conifold of Calabi-Yau geometry).

Witten's geometric picture. Witten (1991) showed that this algebraic structure organizes both the symmetries of the $c=1$ string and its relation to the matrix model. Writing $a_3=\rho e^{\psi}$ and $a_4=\rho e^{-\psi}$ , the uncompactified theory consists of the states invariant under the vector field $K=\partial/\partial\psi$ (which measures the difference of left- and right-moving momenta). The symmetries of the uncompactified theory are the volume-preserving diffeomorphisms of $Q$ that commute with $K$ ; they project to area-preserving diffeomorphisms of the $a_1$ - $a_2$ plane that preserve the locus $a_1a_2=0$ . Under the dictionary, $(a_1,a_2)$ are the light-cone coordinates $p\pm q$ of the matrix-model eigenvalue phase space, the locus $a_1a_2=0$ is the Fermi surface, $a_1a_2$ is the inverted-oscillator Hamiltonian, and $\psi$ is matrix-model time. The infinitesimal moduli of the compactified theory at the self-dual point are the spin- $(1,1)$ discrete-state operators $Z^+=W^+_{s,n}\overline{W}^+_{s,n'}$ , where $W^+_{s,n}$ and $\overline{W}^+_{s,n'}$ are left- and right-moving higher-spin currents (the generators of $W_{1+\infty}$ ); these operators transform like polynomial functions on $Q$ .

The deformation program and what was left open. Witten conjectured that turning on the $Z^+$ moduli deforms the ground-ring relation to $a_1a_2-a_3a_4=\phi(a_i)$ for a general polynomial $\phi$ , the cosmological constant corresponding to $\phi=\mu$ ; the deformed ground ring would then be the coordinate ring of a deformed hypersurface, and for deformations of the special form $a_3a_4=h(a_1,a_2)$ the 'branch locus' $h=0$ in the $a_1$ - $a_2$ plane would be the perturbed Fermi surface of a matrix model with one-body Hamiltonian $h$ . A marginal operator (a spin- $(1,1)$ operator whose addition to the worldsheet action preserves conformal invariance at linear order) is exactly marginal (integrably marginal) only if conformal invariance survives to all orders in perturbation theory; the obstruction first appears in the beta function. Witten computed the second-order obstruction: with $Z=\sum_i a_i\,\partial/\partial a_i$ the scaling operator, $\eta$ the flat metric in which $a_1a_2-a_3a_4=\eta_{ij}a^ia^j/2$ , and $\Delta=\eta^{ij}\partial_i\partial_j$ the associated Laplacian,

\beta=\frac{1}{2}\left(\eta^{ij}(\partial_i Z\phi)(\partial_j Z\phi)-(Z^2\phi)\,\Delta\phi\right),

which vanishes, for example, for $\phi=\sum_n\epsilon_n(a_1a_2)^n$ (arbitrary deformations of the Fermi-surface profile) and, plausibly by a criterion of Chaudhuri and Schwartz, for perturbations supported on abelian subalgebras of the current algebra. Obstructions at third and higher order were not determined, and the global validity of the geometric interpretation remained conjectural. Witten also identified an explicit gap in the symmetry analysis: an area-preserving vector field $f\,\partial/\partial a_1+g\,\partial/\partial a_2$ of the plane lifts to a polynomial $K$ -invariant volume-preserving vector field on $Q$ only if $f$ is divisible by $a_1$ and $g$ by $a_2$ (necessary conditions only), so the full classification of the lifted symmetry algebra and its relation to the conserved quantities of the matrix model was left open. Finally, he speculated that the theory might admit a three-dimensional formulation with volume form $\Theta=d\psi\,da_1\,da_2$ and a dynamical two-form $\omega$ obeying $d\omega=0$ , which would still have to be reconciled with the rich discrete-state structure of the compactified theory near the $SU(2)$ point.

Problem Statement

Complete the geometric characterization, via the ground ring, of the moduli space of exactly marginal deformations of the $c=1$ string, as posed by Witten (1991) and refined by Witten and Zwiebach (1992). All four parts below serve this single objective: being able to decide, for any proposed discrete-state deformation, whether it is exactly marginal, and to identify the geometry and symmetries of the resulting background.

Ground-ring deformation ansatz. Prove or disprove, beyond first order in the deformation and beyond the subspace of functions on the cone $Q$ , that turning on the discrete-state moduli deforms the ground ring of the self-dual-radius theory from $\mathbb{C}[a_1,a_2,a_3,a_4]/(a_1a_2-a_3a_4)$ to $\mathbb{C}[a_1,a_2,a_3,a_4]/(a_1a_2-a_3a_4-\phi(a_i))$ , taking into account that the moduli space is larger than the functions on $Q$ (Witten-Zwiebach two-forms $F=d_Ld_R\phi$ , with $d_L$ and $d_R$ the exterior derivatives acting on the left- and right-chiral ground rings, and the distribution $\sigma$ supported at the apex of $Q$ ).
All-orders obstruction and classification. Determine the obstruction conditions to every order in perturbation theory — the higher-order analogues of the second-order beta function $\beta=\frac{1}{2}(\eta^{ij}(\partial_iZ\phi)(\partial_jZ\phi)-(Z^2\phi)\Delta\phi)$ , where $Z=\sum_i a_i\,\partial/\partial a_i$ is the scaling operator, $\eta$ the flat metric defined by $a_1a_2-a_3a_4=\eta_{ij}a^ia^j/2$ , and $\Delta=\eta^{ij}\partial_i\partial_j$ the associated Laplacian — and thereby classify exactly which deformations $\phi$ , together with the extra moduli, are exactly marginal, i.e. define conformally consistent string backgrounds at finite coupling; determine the corresponding nonperturbative integrability conditions.
Symmetry lifting. Classify precisely which polynomial area-preserving vector fields of the $a_1$ - $a_2$ plane lift to $K$ -invariant (with $K=\partial/\partial\psi$ the vector field measuring the difference of left- and right-moving momenta) volume-preserving polynomial vector fields on the deformed hypersurface $a_1a_2-a_3a_4=\phi$ (the source gives only the necessary divisibility conditions), and identify the full lifted symmetry algebra — its $W_{1+\infty}$ structure and its homotopy Lie structure, the latter being the tower of multilinear brackets of all orders obeying generalized Jacobi identities that the BRST construction produces — together with its exact match to the conserved quantities of the corresponding matrix model.
Geometric interpretation. Give the deformed moduli algebra and its perturbed current algebra a geometric and space-time interpretation, reconciling the discrete-state structure of the compactified theories (at the self-dual radius and at the other isolated enhanced-symmetry, A-D-E, radii) with the decompactified matrix-model picture.

The four parts are jointly required for a solution: a partial result — such as an obstruction established at finite order for a single $\phi$ , or a classification restricted to a subspace — settles only the corresponding sub-question and does not by itself solve the problem.

The problem is solved when the moduli space of exactly marginal deformations is characterized completely enough that a proposed $\phi$ (with its extra moduli) can be decided to be exactly marginal or obstructed, and the symmetry algebra and target-space geometry of the resulting background are identified.

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