Boundary conditions of the $q$-state Potts model on random planar maps: dual-theory loop functions, the $p=3/2$ boundary condition and the 'New' boundary state, and the $q\ge4$ regime

Kulanthaivelu and Wheater (2019) solved the boundary loop functions (disk partition functions) of the $q$-state Potts model coupled to random planar maps for $q<4$ with $p\le q$ allowed boundary spins, classified the allowed $(q,p)$ pairs, constructed the Kramers-Wannier dual matrix model for $q=3$, and conjectured that the unexpected $p=3/2$ boundary condition is the random-surface counterpart of the 'New' boundary condition of the flat-lattice three-state Potts conformal field theory. They left open four linked questions: explicitly solving the dual mixed Hermitian-complex matrix model and its non-resolvent mixed loop function $W_{\mathrm{mixed}}(z)$; computing cylinder (annulus) amplitudes beyond the planar limit, for example by topological recursion on their algebraic curves, to extract boundary operator spectra and test the $p=3/2$/'New' identification; interpreting the fractional $p=q/2$ boundary conditions; and characterizing the $q\ge4$ strong-coupling regime, where algebraicity breaks down, for possible new critical behaviour analogous to the exotic $O(n)$ critical points. The problem is therefore open as stated, with the dual-theory solution and the New-boundary-condition identification as its precise core.

Background

The $q$-state Potts model places a spin $\sigma\in\{1,\dots,q\}$ on every site of a lattice, with Boltzmann weight $e^{\beta}$ for equal neighbouring spins. Coupled to two-dimensional quantum gravity, the fixed lattice is replaced by a sum over random planar maps (graphs embedded in the sphere; here, triangulations), so that the geometry itself becomes dynamical, with Liouville quantum gravity coupled to the matter theory as continuum limit. Since Kazakov's exact solution of the Ising case, such spin systems on dynamical lattices have been exactly solvable by matrix models: the sum over triangulations carrying Potts spins is generated by a $q$-matrix integral over Hermitian $N\times N$ matrices, and the planar (large-$N$) limit is obtained by solving loop equations, the matrix-model Schwinger-Dyson equations; the Potts case was solved this way by Eynard and Bonnet.

The central objects are the one-boundary ('disk') partition functions, called loop functions or resolvents, $W(z)=\frac{1}{N}\langle\mathrm{Tr}\frac{1}{z-M}\rangle$, whose expansion in $1/z$ generates the weighted maps with a prescribed boundary. Kulanthaivelu and Wheater analysed boundary conditions in which only $p$ of the $q$ spin values are allowed, equally weighted, on a connected boundary. For $q<4$ the loop functions are finite-sheeted algebraic functions; these authors reproduced the long-known allowed values $q=2(1+\cos\frac{m\pi}{n})$ with $m,n$ coprime (also obtained combinatorially by Bernardi and Bousquet-Mélou), classified the allowed $p$ for each such $q$ into two distinct sequences, computed critical exponents in agreement with Eynard and Kristjansen's $O(n)$-model results, and found that $q=3$ carries an additional boundary condition with the fractional value $p=3/2$.

On the fixed ('flat') lattice, the critical $q=3$ Potts model is described by a conformal field theory (CFT), the $(6,5)$ minimal model with extended $\mathcal{W}_3$ symmetry, that is, a conformal symmetry algebra enlarged by a spin-three current in addition to the Virasoro generators. Its conformally invariant boundary conditions are classified by Cardy states, the CFT boundary states that preserve conformal invariance; Affleck, Oshikawa and Saleur showed there are eight for $q=3$, arising from fixed-spin (3), mixed (3) and free (1) conditions plus an eighth 'New' boundary condition whose microscopic realization needs negative Boltzmann weights ($w=1$ for equal and $w=-\frac{1}{2}$ for different adjacent boundary spins) and which lacks a simple spin interpretation; Fuchs and Schweigert confirmed the completeness of this classification. Kulanthaivelu and Wheater conjectured that their $p=3/2$ random-lattice boundary condition is, in the scaling limit, the random-surface counterpart of this New boundary condition.

They also extended Kramers-Wannier (high/low-temperature) duality, previously treated on random surfaces for the Ising case by Carroll, Ortiz and Taylor, to $q=3$: they constructed the dual matrix model $\hat{\mathcal{M}}_3$, with one Hermitian matrix $X$ and one complex matrix $U$, with action

$$\hat{\mathcal{S}}_3=\frac{1}{2}\,\mathrm{Tr}\big(X^2+2e^{\hat\beta}UU^\dagger\big)-\hat g\,X\big(UU^\dagger+U^\dagger U\big)-\frac{\hat g}{3}\big(U^3+U^{\dagger 3}+X^3\big),$$

proved that boundary weights map as on the flat lattice, and argued that the mixed boundary condition of the dual is the New boundary condition of the original theory. However, the loop function generating dual mixed boundaries,

$$W_{\mathrm{mixed}}(z)=\frac{1}{N}\left\langle\mathrm{Tr}\,\frac{1}{z-\big(X+U\frac{1}{z-X}U^\dagger\big)}\right\rangle,$$

is not a conventional resolvent, and the authors state that with the matrix-model technology available they do not know how to compute it.

Deciding continuum identifications of boundary conditions requires the two-boundary (cylinder, or annulus) amplitudes, whose scaling behaviour encodes the boundary operator spectrum, the set of scaling dimensions of the operators localized at the boundary. Beyond the planar limit such correlators are computed by topological recursion, the Eynard-Orantin procedure that recursively reconstructs genus-$g$ correlators from the planar spectral curve (the algebraic curve encoding the planar solution); it has been carried out for the $O(n)$ loop model on random lattices of all topologies by Borot and Eynard, and cylinder amplitudes were computed beyond the planar limit for minimal strings by Atkin and Wheater, but nothing analogous exists for the Potts model. Finally, for $q\ge4$ the finite-sheeted algebraic structure of the resolvents breaks down: at $q=4$ the $p=1$ solution involves elliptic functions (and $p$ is then unconstrained), while for $q>4$ the sheet structure does not terminate, excluding algebraic solutions. Because the $O(n)$ model on random lattices exhibits exotic strong-coupling critical points at $n>2$, where the central charge (the parameter counting the degrees of freedom of the continuum theory) exceeds one, found by Eynard, Zinn-Justin and Kristjansen, that have no flat-lattice counterpart, the analogous question for the Potts model at $q>4$ is open.

Problem Statement

Complete the boundary-condition solution of the $q$-state Potts model coupled to random planar maps initiated by Kulanthaivelu and Wheater, by resolving the following questions, all stated as open by those authors. They share one objective: a complete dictionary between the microscopic boundary conditions of the Potts model on random planar maps and the continuum conformal boundary states (Cardy states, including the New boundary condition) of the Potts model coupled to two-dimensional quantum gravity, together with the phase structure of the model in $q$. All four questions belong to the task and are required jointly: the boundary-condition solution is complete only when each is resolved, though each question is stated independently and can be attacked on its own.

1. Dual loop functions. Solve the Kramers-Wannier dual matrix model $\hat{\mathcal{M}}_3$ (one Hermitian matrix $X$, one complex matrix $U$, with the action $\hat{\mathcal{S}}_3$ quoted in the background): compute its planar loop functions explicitly, in particular the non-conventional mixed boundary loop function

$$W_{\mathrm{mixed}}(z)=\frac{1}{N}\left\langle\mathrm{Tr}\,\frac{1}{z-\big(X+U\frac{1}{z-X}U^\dagger\big)}\right\rangle,$$

which is not an ordinary resolvent, and determine their critical behaviour and scaling exponents, so that they can be compared directly with the loop functions of the original model.

2. Cylinder amplitudes and the New boundary condition. Compute cylinder (annulus) amplitudes, or equivalent higher-topology correlators, for the $q=3$ model on random lattices, which requires going beyond the planar limit, for example by applying topological recursion to the algebraic curves found in the source paper, and extract the boundary operator spectra from them. Use these to confirm or refute the conjecture that the $p=3/2$ algebraic boundary condition is, in the scaling limit, the random-lattice counterpart of the New boundary condition of the flat-lattice three-state Potts model. A refutation is admissible and consists of a computed universal (scaling) quantity, such as a boundary operator dimension or spectrum, that is incompatible with the New boundary condition's content in the $(6,5)$ conformal field theory.

3. Meaning of fractional $p$. Determine what the non-integer allowed boundary conditions, notably the $p=q/2$ family, correspond to physically, if anything: do they survive as local or non-local boundary conditions in the scaling limit, or are they artifacts of the random-lattice construction with no continuum counterpart?

4. The $q\ge4$ regime. Characterize the $q\ge4$ parameter space of the Potts model on random planar maps, where the finite-sheeted algebraicity of the resolvents breaks down (at $q=4$ the $p=1$ solution is elliptic and $p$ is unconstrained; for $q>4$ the sheet structure does not terminate, excluding algebraic solutions): determine whether new critical behaviour appears in this strong-coupling regime, analogous to the exotic critical points of the $O(n)$ model at $n>2$, and if so compute the corresponding critical points and exponents.

The four components are scientifically linked: solving the dual theory (1) supplies the quantities whose scaling must match the original model and feeds the comparison; the cylinder amplitudes (2) decide the continuum identification that motivates the duality analysis; and questions (3) and (4) complete the classification of boundary conditions and the phase structure that the same matrix-model framework must deliver.

References

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