In 2013 Fabbri, Fioravanti, Piscaglia and Tateo derived the thermodynamic Bethe ansatz (TBA) governing the low-energy limit of the type IIA superstring on AdS4 x CP3 and discovered an infinite family of finite Y-systems indexed by a rank N and a truncation level p, of a novel 'crossed' type in which two massive nodes enter only through mixed products of shifted functions. Supported by numerical agreement down to 10^-15, they conjectured a stationary Rogers dilogarithm sum rule for the ultraviolet central-charge coefficient c(N,p)(0)c^{(0)}_{(N,p)}, a periodicity of the Y-functions in imaginary rapidity, and a non-stationary dilogarithm sum rule valid for arbitrary initial data, and asked whether these crossed systems are mutation-equivalent to known cluster algebras. arXiv:1308.1861 poses these statements as conjectures, which remain open. The problem is to establish these exact statements rigorously; through the coset connection, their resolution would fix the exact ultraviolet data of the perturbed (CPN−1)p×U(1)(\mathbb{CP}^{N-1})_p \times U(1) conformal field theories interpolating to the SU(N) x U(1) sigma model.

Background

In the Alday-Maldacena decoupling limit (the low-energy limit of gauge theories with large spin charge, after Luis F. Alday and Juan Maldacena), the type IIA superstring on the product space AdS4 x CP3 is described at low energy by a two-dimensional relativistic sigma model with global SU(N) x U(1) symmetry — the case N = 4 corresponding to the AdS4 x CP3 string — coupled to a massless Dirac fermion with a Thirring self-interaction. Basso and Rej conjectured this model to be integrable and proposed its asymptotic Bethe ansatz and spinon S-matrix (the two-particle scattering matrix of the fundamental massive excitations). The thermodynamic Bethe ansatz (TBA), introduced by Alexander B. Zamolodchikov for scaling 1+1-dimensional quantum field theories, is the standard non-perturbative tool: it determines the exact finite-volume ground-state energy E(R) on a circle of circumference R from infinitely many coupled non-linear integral equations for pseudo-energies ϵA(θ)\epsilon_A(\theta), one per particle species or auxiliary node, as functions of the rapidity θ\theta; their exponentials YA=eϵAY_A = e^{\epsilon_A} obey functional 'Y-system' relations of schematic form YA(θ+iπ/2)YA(θ−iπ/2)=∏B(1+YB)mABY_A(\theta+i\pi/2)Y_A(\theta-i\pi/2) = \prod_B (1+Y_B)^{m_{AB}}, which usually tie the nodes A to the vertices of a Dynkin diagram.

Fabbri, Fioravanti, Piscaglia and Tateo (FFPT) derived the ground-state TBA of this sigma model and found a Y-system of a new 'crossed' type: the two massive (particle) nodes enter through the mixed products Y0(θ+iπ/2)Yˉ0(θ−iπ/2)Y_0(\theta+i\pi/2)\bar Y_0(\theta-i\pi/2) rather than through the square of a single function, with the magnon (momentum-carrier) nodes Y(i,j)Y_{(i,j)} arranged on an AN−1×Ap−1A_{N-1} \times A_{p-1} lattice and sourced by the two massive nodes. Truncating the magnon column at level p and letting the rank N be arbitrary produces an infinite family of finite Y-systems indexed by (N, p), conjecturally the TBA of the massive perturbations of the coset conformal field theories (CPN−1)p×U(1)(\mathbb{CP}^{N-1})_p \times U(1) — conformal models constructed as gauge cosets, here SU(N)p/(SU(N−1)p×U(1))SU(N)_p/(SU(N-1)_p \times U(1)) times a free boson, perturbed by their least relevant operator — the limit p→∞p \to \infty being the original sigma model, whose ultraviolet Casimir coefficient c(N,∞)=2N−1c_{(N,\infty)} = 2N-1 (equal to 7 for N = 4) confirms the degree-of-freedom count proposed by Basso and Rej. At N = 2 the system reduces to the Dp+1D_{p+1}-related TBA identified with points of the fractional sine-Gordon model studied by Denis Bernard and Andre Leclair.

The Rogers dilogarithm L(x)=Li2(x)+12log⁡x log⁡(1−x)=−∫0xdt log⁡(1−t)t+12log⁡x log⁡(1−x)L(x) = \mathrm{Li}_2(x) + \frac{1}{2}\log x\,\log(1-x) = -\int_0^x dt\, \frac{\log(1-t)}{t} + \frac{1}{2}\log x\,\log(1-x) is the function through which Y-systems encode ultraviolet data: at a rapidity-independent (stationary) solution yA{y_A} of the Y-system, the sum of L(yA/(1+yA))L(y_A/(1+y_A)) over all nodes equals the effective central charge of the ultraviolet conformal field theory times π2/6\pi^2/6 in the standard ADE cases. For the (N, p) family FFPT found the stationary constants in closed trigonometric form only for N≤4N \le 4; for general (N, p) they computed them numerically to 10−1510^{-15} (N < 5, p < 20, degrading to 10−1010^{-10} at N = 4, p = 61) and conjectured the stationary sum rule c(N,p)(0)=p(1+pN−p)/(p+N−1)c^{(0)}_{(N,p)} = p(1+pN-p)/(p+N-1). In this family the same stationary sum defines the coefficient c(N,p)(0)c^{(0)}_{(N,p)}, from which the actual effective central charge is obtained as c(N,p)=c(N,p)(0)−c(N,p)(∞)c_{(N,p)} = c^{(0)}_{(N,p)} - c^{(\infty)}_{(N,p)}, the subtraction c(N,p)(∞)=p(N−1)(p−1)/(p+N)c^{(\infty)}_{(N,p)} = p(N-1)(p-1)/(p+N) — the analogous dilogarithm sum over the stationary solutions zA{z_A} with Y0=Yˉ0=0Y_0 = \bar Y_0 = 0 — being analytically known (source equations (4.3), (4.13)). The same numerics revealed the periodicity YA(θ+iπP(N,p))=YA(θ)Y_A(\theta+i\pi P_{(N,p)}) = Y_A(\theta) with P(N,p)=2(p+N−1)/NP_{(N,p)} = 2(p+N-1)/N for solutions generated from arbitrary initial conditions, from which the conformal dimension Δ(N,p)=1−N/(2(p+N−1))\Delta_{(N,p)} = 1 - N/(2(p+N-1)) of the perturbing operator was read off — consistent with the field identified by Paul Fendley in his study of the CPN−1\mathbb{CP}^{N-1} sigma model as a perturbed conformal field theory — and FFPT stated a non-stationary Rogers dilogarithm sum rule over a full period for arbitrary initial data.

For standard (Dynkin-related, 'ADE-type') Y-systems, periodicity and dilogarithm identities of exactly this kind are theorems: they follow from cluster-algebra structures, in which a quiver (an oriented graph) evolves under elementary 'mutations' at its vertices and the Y-functions are cluster variables, as surveyed by Atsuo Kuniba, Tomoki Nakanishi and Junji Suzuki and developed by Tomoki Nakanishi and Roberto Tateo for sine-Gordon-type systems. FFPT therefore explicitly posed the question whether their crossed Y-systems are genuinely new objects or lead to quivers mutation-equivalent to the known ABCD-related cases. The paper itself stresses that its results were rigorously derived only at (N,p)=(4,∞)(N,p) = (4,\infty), the general family being supported by numerics and structural analogy. The crossed structure had a precedent in the mirror TBA of the AdS4/CFT3 correspondence worked out by Diego Bombardelli, Davide Fioravanti and Roberto Tateo and in the AdS4 x CP3 quasi-classical-string Y-system of Nikolay Gromov and Fedor Levkovich-Maslyuk, but had never been classified.

Problem Statement

The (N,p) crossed Y-systems of arXiv:1308.1861 consist of functions Y0(θ)Y_0(\theta), Yˉ0(θ)\bar Y_0(\theta) (the massive nodes) and Y(i,j)(θ)Y_{(i,j)}(\theta) with 1≤i≤N−11 \le i \le N-1, 1≤j≤p−11 \le j \le p-1 (the magnon wings), of a rapidity variable θ\theta, obeying finite coupled functional equations in which the two massive nodes enter only through the crossed products Y0(θ+iπ/2)Yˉ0(θ−iπ/2)Y_0(\theta+i\pi/2)\bar Y_0(\theta-i\pi/2) and Y0(θ−iπ/2)Yˉ0(θ+iπ/2)Y_0(\theta-i\pi/2)\bar Y_0(\theta+i\pi/2) while the magnon nodes obey the usual AN−1×Ap−1A_{N-1} \times A_{p-1}-type relations (i.e. they are tied through the Cartan matrices of types AN−1A_{N-1} and Ap−1A_{p-1}), with shifts ±iπ/N\pm i\pi/N, sourced by the massive pair (source equations (3.6)-(3.7)), for integers N≥2N \ge 2 and p≥1p \ge 1. The open problem is to establish rigorously, for every rank N≥2N \ge 2 and truncation level p≥1p \ge 1, the three exact statements that the source could only verify numerically and called its 'main mathematical conjectures', together with the structural classification question it posed:

  1. Periodicity: any solution generated by iterating the recursion from arbitrary initial data satisfies YA(θ+iπP(N,p))=YA(θ)Y_A(\theta+i\pi P_{(N,p)}) = Y_A(\theta) with P(N,p)=2(p+N−1)/NP_{(N,p)} = 2(p+N-1)/N (source equation (4.17)).

  2. Stationary dilogarithm identity: the rapidity-independent (stationary) solution yA{y_A}, defined as the solution of the source's algebraic system (4.6), satisfies ∑AL(yA1+yA)=π26 c(N,p)(0)\sum_A L\left(\frac{y_A}{1+y_A}\right) = \frac{\pi^2}{6}\, c^{(0)}_{(N,p)} with c(N,p)(0)=p(1+pN−p)p+N−1c^{(0)}_{(N,p)} = \frac{p(1+pN-p)}{p+N-1} (source equation (4.11)). Closed trigonometric forms of the stationary constants are known only for N≤4N \le 4.

  3. Non-stationary sum rule: for arbitrary initial conditions, the iterated solution sampled along the imaginary direction, YA(n)=YA(θ+iπn/N)Y_A^{(n)} = Y_A(\theta+i\pi n/N), satisfies the source's equation (5.1): ∑n=12(N+p−1)[L(Yˉ0(n)1+Yˉ0(n))+L(Y0(n)1+Y0(n))+∑i=1N−1∑j=1p−1L(Y(i,j)(n)1+Y(i,j)(n))]=2p(1+pN−p)π26\sum_{n=1}^{2(N+p-1)}\left[L\left(\frac{\bar Y_0^{(n)}}{1+\bar Y_0^{(n)}}\right)+L\left(\frac{Y_0^{(n)}}{1+Y_0^{(n)}}\right)+\sum_{i=1}^{N-1}\sum_{j=1}^{p-1}L\left(\frac{Y_{(i,j)}^{(n)}}{1+Y_{(i,j)}^{(n)}}\right)\right] = \frac{2p(1+pN-p)\pi^2}{6}, where LL is the Rogers dilogarithm.

  4. Cluster-algebra status: determine, for each rank N≥2N \ge 2 and truncation level p≥1p \ge 1, whether the crossed (N,p) Y-systems lead to quivers that are mutation-equivalent to known ABCD-related cases or are genuinely new objects; a case-dependent classification is a complete resolution of this item.

Items 2 and 3 are not independent: the stationary solution is admissible constant initial data for item 3, and item 3 evaluated there reduces to item 2, so a proof of item 3 discharges item 2.

In the physics interpretation, resolving items 1-3 fixes the exact ultraviolet data of the perturbed coset conformal field theories (CPN−1)p×U(1)(\mathbb{CP}^{N-1})_p \times U(1): the periodicity yields the conformal dimension Δ(N,p)=1−N2(p+N−1)\Delta_{(N,p)} = 1-\frac{N}{2(p+N-1)} of the perturbing operator and the sum rules yield the effective central charges.

References

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