Research report: Discrete Laplacian structure and kernel reduction of the gap equation in odd-dimensional large-N vector models at imaginary chemical potential

A theoretical study of large vector models has found that the thermal gap equations in odd spacetime dimensions are linked by a hidden finite-difference structure. Working mainly with the large-N Gross–Neveu model at finite temperature and imaginary chemical potential, the researchers show that a single centred difference in the square of the thermal mass connects the kernel in dimension d to the kernel in dimension d−2. Repeating this elementary step produces the higher-order, discrete-Laplacian relations among dimensions 3, 5, 7, 9, 11, 13, 15 and beyond.

The result is an analytic description of the theory rather than a new numerical solution of its gap equation. The stated aim is to expose the mathematical organization hidden inside the saddle-point condition, not to claim a new phase transition or a complete solution of the theory. The construction is kinematic: it follows from the form of the thermal integral and the polynomial generated by the integration measure, rather than from a new interaction or a model-specific dynamical effect.

This distinction is important because the same structure appears in both fermionic and bosonic vector models. At the centre of the analysis is the homogeneous thermal mass σ. After the fermionic fields are integrated out in the Gross–Neveu model, the large-N effective action contains a vacuum contribution and a thermal contribution.

Changing the momentum variable to the energy E turns the thermal term into an integral weighted by the polynomial (E²−σ²)ⁿ, where n=(d−3)/2, multiplied by a thermal occupation factor containing Fermi–Dirac distributions at chemical potentials ±iα. Expanding the polynomial produces alternating binomial coefficients. These coefficients coincide with the weights in centred finite-difference stencils: the three-point pattern 1, −2, 1 occurs in dimension 7, the five-point pattern 1, −4, 6, −4, 1 in dimension 11, and the seven-point pattern 1, −6, 15, −20, 15, −6, 1 in dimension 15.

The complementary odd dimensions carry the remaining stencils. Dimension 5 produces the two-point weights 1 and −1, dimension 9 produces 1, −3, 3, −1, and dimension 13 produces 1, −5, 10, −10, 5, −1. Because the same polynomial appears in both branches, the apparent split between dimensions 4k+3 and 4k+1 is not a separate hierarchy.

A centred first difference in x=σ² lowers the polynomial degree by one, thereby relating every odd dimension d to d−2. Iterations continue until the ladder reaches dimension 3, whose kernel has degree zero and is therefore terminal. In the continuum limit, this relation becomes a differential identity involving the thermal kernels in adjacent odd dimensions.

For the distinguished sequence d=4k+3, the iterated construction yields a compact relation between dimension d and the higher dimension d′=2d−3. A centred finite difference of order 2k acting on the thermal kernel in d′ reproduces the kernel in d, up to normalization and higher-order terms in the spacing. The study gives a three-point relation from dimension 9 to 5 and a five-point relation from dimension 19 to 11.

The moving lower integration limit does not obstruct the identification because the polynomial vanishes at E=√x, making the boundary terms vanish for the relevant kernels. The relations become exact in the continuum limit as the finite spacing tends to zero and the difference operator becomes a derivative. They do not depend on α: setting the imaginary chemical potential to zero leaves the dimensional recursion unchanged.

The same hierarchy appears beyond the fermionic Gross–Neveu model. In the large-N CPᴺ⁻¹ model, the bosonic thermal distribution replaces the fermionic one, but radial momentum integration generates the identical polynomial kernel and therefore the same finite-difference stencils. The large-N bosonic O(N) model is expected to behave similarly for the same kinematic reason.

This is not an equivalence of the complete theories or a claim that their dynamics coincide; rather, the dimensional descent is inherited from a common measure and kernel structure, independently of particle statistics. The authors therefore frame the structure as a universal feature of the relevant large-N vector-model thermal sector, while leaving deeper dynamical questions open. At critical coupling, the analysis can also be reduced to special single-valued polylogarithmic expressions, including Bloch–Wigner–Ramakrishnan combinations.

After lower-order terms cancel, the gap condition in dimension d is represented by the vanishing of a remainder function R_(d−2), which the authors interpret as the thermal free-energy remainder of a dimensionally reduced theory. Numerical roots for dimensions 7, 11 and 15 give βσ values of 2.17755, 3.55044 and 4.93425, respectively. These are solutions of the reduced transcendental conditions considered in the study, rather than experimental measurements or universal constants.

The associated critical cutoff scales are selected by the full saddle-point condition, which also relates the ultraviolet regulator to the dynamically generated infrared mass. The reduced free-energy condition admits a thermodynamic interpretation. Since a finite-temperature free energy has the form F=U−TS, the selected thermal mass marks a balance between the energy cost of massive excitations and the entropic gain from populating thermal modes.

In the hyperbolic-geometric reading proposed by the authors, the ultraviolet cutoff truncates a modular cusp and leaves a finite polylogarithmic sector. They also report a monotonically decreasing sequence of cutoff ratios—1.50000, 1.27078, 1.19067 and further values for dimensions d=4k+3—which appears numerically to approach unity as the dimension grows. The geometric interpretation remains conjectural, and a rigorous connection to modular invariants and hyperbolic structures is left for future work.

A second reduction identifies the three-dimensional thermal kernel with the free-energy density of a one-dimensional system containing two twisted fermionic harmonic-oscillator modes. Its partition function contains factors 1+exp[−β(m−iα)] and 1+exp[−β(m+iα)], whose logarithms reproduce the dimension-3 integral. Higher-dimensional kernels are then generated by an integral transform that inserts the appropriate power of (E²−σ²).

The authors caution that this is a generating representation for the thermal sector, not a full equivalence between the Gross–Neveu field theory and a one-dimensional quantum system. The exact inverse of the centred finite-difference ladder is also left unresolved because it depends on boundary conditions or a corresponding discrete Green’s function. Overall, the study offers a structural view of finite-temperature large-N vector theories.

The vacuum sector has a complementary organization: after a ultraviolet cutoff is introduced, its divergent terms form a truncated asymptotic expansion with coefficients related to the binomial expansion of (1+x)⁻¹ᐟ². The thermal sector, by contrast, has a finite polynomial hierarchy expressible through discrete Laplacians. The authors distinguish their contribution from a wholly new algebraic identity: the underlying dimensional recursion is a differential relation in the continuum, and the finite-difference formulation reveals why the recurring binomial coefficients arise.

Open questions include deriving the ladder from a deeper symmetry, testing it in Gross–Neveu–Yukawa and gauge theories, and establishing whether its resemblance to thermal operator-product-expansion inversion formulas reflects a deeper common structure.

Subject of Research: Discrete finite-difference relations in thermal gap equations of odd-dimensional large-N vector models at imaginary chemical potential.

Article Title: Discrete Laplacian structure and kernel reduction of the gap equation in odd-dimensional large-N vector models at imaginary chemical potential

Article References: Filothodoros, E. G. (2026). Discrete Laplacian structure and kernel reduction of the gap equation in odd-dimensional large-N vector models at imaginary chemical potential. The European Physical Journal C, 86(10), Article 1199. https://doi.org/10.1140/epjc/s10052-026-16430-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16430-4

Keywords: large-N vector models, Gross–Neveu model, imaginary chemical potential, thermal gap equation, discrete Laplacian, dimensional reduction, finite differences, single-valued polylogarithms, twisted fermionic oscillators, CPᴺ⁻¹ model, Discrete, Laplacian