arXiv subjects

Marcin R. Piatek

Publications and source records attributed to Marcin R. Piatek.

7 recordsLinked to original sources

arXiv2026

A perturbative Liouville prescription for the celestial three-gluon amplitude

We study the celestial three-gluon amplitude in a dilaton background through the Mellin-Liouville formulation proposed by Stieberger, Taylor and Zhu (STZ). The original map contains an ambiguity in the identification of Liouville and Mellin variables; we resolve it by requiring global conformal covariance and compatibility with the semiclassical expansion of Liouville theory. This uniquely fixes the operator normalization and the parameter dictionary, and leads to a controlled expansion in the Liouville coupling $b$. Starting from the full Liouville DOZZ three-point function, we derive the leading and first subleading terms in the $b^2$ expansion. The leading term reproduces the tree-level Yang-Mills amplitude in the small total momentum limit, as anticipated in the STZ proposal. The one-loop correction can be written in closed form using modified Bessel functions, and its soft limit exhibits a clear separation into geometric and logarithmic contributions. The resulting framework extends the STZ proposal to finite-$b$ corrections in a consistent and computable way.

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arXiv2026

Small-$b$ expansion of the DOZZ formula for light operators

We present a systematic small-$b$ expansion of the Liouville DOZZ three-point structure constant in the light-operator regime $α_i=bσ_i$ as $b\to0$. In this limit, the exact DOZZ function factorizes into a prefactor ${\cal P(b;σ_1,σ_2,σ_3)$ and a power series in $b^2$: \[ C(bσ_1,bσ_2,bσ_3)={\cal P}(b;σ_i)\Bigg[1+\sum_{n\ge1}b^{2n}\,Ω_n(σ_1,σ_2,σ_3)\Bigg]. \] Using Thorn's asymptotic expansion of the $Υ_b$-function we derive closed-form expressions for the leading coefficients $Ω_n(σ_i)$ and show that each $Ω_n$ is a symmetric polynomial in the variables $σ_i$. Our expansion provides explicit perturbative corrections to the semiclassical Liouville three-point function and therefore supplies a practical tool for applications in celestial holography, in particular, for generating loop-level corrections to the tree-level three-gluon scattering amplitude. Finally, we formulate a perturbative Liouville program for celestial amplitudes and outline directions for further development.

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arXiv2025

2D CFT and efficient Bethe ansatz for exactly solvable Richardson-Gaudin models

This work inaugurates a series of complementary studies on Richardson-Gaudin integrable models. We begin by reviewing the foundations of classical and quantum integrability, recalling the algebraic Bethe ansatz solution of the Richardson (reduced BCS) and Gaudin (central spin) models, and presenting a proof of their integrability based on the Knizhnik-Zamolodchikov equations and their generalizations to perturbed affine conformal blocks. Building on this foundation, we then describe an alternative CFT-based formulation. In this approach, the Bethe ansatz equations for these exactly solvable models are embedded within two-dimensional Virasoro CFT via irregular, degenerate conformal blocks. To probe new formulations within the Richardson-Gaudin class, we develop a high-performance numerical solver. The Bethe roots are encoded in the Baxter polynomial, with initial estimates obtained from a secular matrix eigenproblem and subsequently refined using a deflation-assisted hybrid Newton-Raphson/Laguerre algorithm. The solver proves effective in practical applications: when applied to picket-fence, harmonic oscillator, and hydrogen-like spectra, it accurately reproduces known rapidity trajectories and reveals consistent merging and branching patterns of arcs in the complex rapidity plane. We also explain how to generalize our computational approach to finite temperatures, allowing us to calculate temperature-dependent pairing energies and other thermodynamic observables directly within the discrete Richardson model. We propose an application of the solver to Gaudin-type Bethe equations, which emerge in the classical (large central charge) limit of Virasoro conformal blocks. We conclude by outlining future directions: direct minimization of the Yang-Yang function as an alternative root-finding strategy; revisiting time-dependent extensions; and ... .

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arXiv2023

Classical conformal blocks, Coulomb gas integrals, and quantum integrable models

In this paper, we recall Richardson's solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.

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arXiv2023

Self-dual solutions of a field theory model of two linked rings

In this work the connection established in [7, 8] between a model of two linked polymers rings with fixed Gaussian linking number forming a 4-plat and the statistical mechanics of non-relativistic anyon particles is explored. The excluded volume interactions have been switched off and only the interactions of entropic origin arising from the topological constraints are considered. An interpretation from the polymer point of view of the field equations that minimize the energy of the model in the limit in which one of the spatial dimensions of the 4-plat becomes very large is provided. It is shown that the self-dual contributions are responsible for the long-range interactions that are necessary for preserving the global topological properties of the system during the thermal fluctuations. The non self-dual part is also related to the topological constraints, and takes into account the local interactions acting on the monomers in order to prevent the breaking of the polymer lines. It turns out that the energy landscape of the two linked rings is quite complex. Assuming as a rough approximation that the monomer densities of half of the 4-plat are constant, at least two points of energy minimum are found. Classes of non-trivial self-dual solutions of the self-dual field equations are derived. ... .

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arXiv2017

Solvable spectral problems from 2d CFT and N=2 gauge theories

The so-called 2d/4d correspondences connect two-dimensional conformal field theory (2d CFT), N=2 supersymmetric gauge theories and quantum integrable systems. The latter in the simplest case of the SU(2) gauge group are nothing but the quantum-mechanical systems. In the present article we summarize our recent results and list open problems concerning an application of the aforementioned dualities in the studies of spectral problems for some Schrodinger operators with Mathieu-type periodic, periodic PT-symmetric and (Heun's) elliptic potentials.

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arXiv2014

A Topological Field Theory for the triple Milnor linking coefficient

The subject of this work is a three-dimensional topological field theory with a non-semisimple group of gauge symmetry with observables consisting in the holonomies of connections around three closed loops. The connections are a linear combination of gauge potentials with coefficients containing a set of one-dimensional scalar fields. It is checked that these observables are both metric independent and gauge invariant. The gauge invariance is achieved by requiring non-trivial gauge transformations in the scalar field sector. This topological field theory is solvable and has only a relevant amplitude which has been computed exactly. From this amplitude it is possible to isolate a topological invariant which is Milnor's triple linking invariant. The topological invariant obtained in this way is in the form of a sum of multiple contour integrals. The contours coincide with the trajectories of the three loops mentioned before. The introduction of the one-dimensional scalar field is necessary in order to reproduce correctly the particular path ordering of the integration over the contours which is present in the triple Milnor linking coefficient. This is the first example of a local topological gauge field theory that is solvable and can be associated to a topological invariant of the complexity of the triple Milnor linking coefficient.

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