arXiv subjects

Yannick Mvondo-She

Publications and source records attributed to Yannick Mvondo-She.

13 recordsLinked to original sources

arXiv2026

Complexified Virasoro Flow and the Logarithmic Graviton at the Chiral Point

At the chiral point of topologically massive gravity, the massive graviton becomes degenerate with the left-moving graviton, leading to the appearance of a logarithmic mode and a corresponding rank-two Jordan structure. This logarithmic graviton plays a central role in the conjectured AdS$_3$/LCFT$_2$ correspondence, where it has been widely interpreted as the bulk counterpart of a logarithmic partner in the putative boundary logarithmic conformal field theory. In this work, we develop a geometric interpretation of this Jordan structure based on complexified Virasoro evolution. Starting from the logarithmic graviton of Grumiller and Johansson, we show that its logarithmic coefficient \[ y(\tau,\rho) = -i\tau-\ln\cosh\rho \] admits, near the AdS$_3$ boundary, the asymptotic form \[ y(\tau,\rho) = -s+\ln2+\mathcal O(e^{-2\rho}), \qquad s=\rho+i\tau. \] The same complex parameter naturally appears in the exponentiation of the Virasoro generator $L_0$ acting on a rank-two Jordan cell, \[ L_0=h\mathbf1+N, \qquad N^2=0, \] for which \[ e^{sL_0} = e^{sh}(1+sN). \] We show that the resulting Jordan evolution reproduces the characteristic logarithmic mixing of the logarithmic sector, while analytic continuation $s\rightarrow s+2\pi i$ generates the corresponding logarithmic monodromy. From this perspective, radial evolution, temporal evolution, Jordan mixing, and logarithmic monodromy may be viewed as different manifestations of a single complexified Virasoro flow. The analysis suggests a geometric interpretation of the indecomposable structures characteristic of logarithmic conformal field theory and offers a new perspective on the logarithmic graviton within the conjectural AdS$_3$/LCFT$_2$ framework.

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arXiv2026

Analytic Properties of the Jost Functions via the Poincar\'e-Picard Theorem

The analytic properties of the Jost functions are fundamental in quantum scattering theory and in the analytic continuation of the scattering matrix into the complex energy plane. In this work, the analyticity of the Jost functions is investigated from the perspective of parameter-dependent ordinary differential equations. Starting from the radial Schr\"odinger equation for a short-range central potential, a first-order differential system is derived for the coefficient functions associated with the Ricatti--Bessel and Ricatti--Neumann solutions. The multivalued dependence on the energy variable is shown to originate from the square-root relation between energy and momentum. By explicitly factorizing the momentum-dependent branching terms, the scattering problem is transformed into a differential system whose coefficients are single-valued analytic functions of the complex energy. Using the classical theory of analytic dependence of solutions of ordinary differential equations on parameters, it is shown that the transformed Jost functions are single-valued analytic functions of the energy variable for finite radial distance. The geometric interpretation of the factorization procedure is also discussed in terms of the topology of the associated Riemann surface.

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arXiv2026

From monodromy to $SL(2,\mathbb{R})$: reconstructing the logarithmic sector of chiral TMG from virasoro flow

We construct and analyze the logarithmic sector of chiral Topologically Massive Gravity (TMG) at the critical point $\mu \ell = 1$ from the perspective of Virasoro evolution and radial monodromy in $\mathrm{AdS}_3$. We show that the logarithmic graviton arises naturally as a generalized eigenstate of $L_0$, with its Jordan structure persisting uniformly across the full $SL(2,\mathbb{R})_L$ descendant tower generated by $L_{-1}$. A central result is that the logarithmic mixing of primary and descendant states can be equivalently interpreted as unipotent monodromy under analytic continuation of the radial coordinate $r \to e^{2\pi i} r$. This establishes a direct identification between the LCFT Jordan cell structure and a geometric monodromy operator acting in the bulk. We demonstrate that requiring monodromy-compatible Virasoro flow uniquely reconstructs the full indecomposable logarithmic module, including all descendant levels, and show explicit equivalence with the logarithmic graviton module previously obtained in the linearized analysis of chiral TMG. This provides a unified representation-theoretic and geometric characterization of logarithmic gravity in $\mathrm{AdS}_3$.

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arXiv2026

Virasoro Zero-Mode Flow in Critical Topologically Massive Gravity: Logarithmic Correlators and Partition Function Grading

We study the rank-two logarithmic sector of critical Topologically Massive Gravity through the Virasoro zero-mode action \[ U(s,\bar s)=e^{-sL_0-\bar s\bar L_0}. \] Writing $L_0=h\mathbf 1+\frac{1}{2}N$ and $\bar L_0=\bar h\mathbf 1+\frac{1}{2}N$, with $N^2=0$, separates the semisimple conformal grading from the nilpotent mixing of the logarithmic pair. We apply the Virasoro zero-mode action to logarithmic two-point functions and to the conformal grading of the logarithmic contribution to the one-loop partition function. For the correlators, the semisimple part generates the ordinary conformal power law, while the nilpotent mixing governed by the combination $s+\bar s$ produces the characteristic logarithmic dependence. For the critical-TMG weights $(h,\bar h)=(2,0)$, this gives a holomorphic power law together with logarithmic dependence on the full Euclidean separation. For the partition function, the relations $q=e^{-s}$ and $\bar q=e^{-\bar s}$ express its conventional multiplicative grading in terms of the additive parameters of the same Virasoro zero-mode action, providing an exact reparameterization of the established one-loop result. The two applications capture complementary information: logarithmic correlators resolve the off-diagonal Jordan mixing, whereas the ordinary graded trace organizes conformal weights, multiplicities, and descendant content without separately resolving the nilpotent matrix element. The Virasoro zero-mode action therefore provides a unified representation-theoretic description of these boundary structures in critical TMG.

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arXiv2026

Monodromy, Logarithmic Sectors, and Two-Point Functions in Critical Topologically Massive Gravity

We investigate the structure of logarithmic modes in critical topologically massive gravity (CTMG) at the chiral point $\mu \ell=1$ from the perspective of analytic continuation and monodromy. Starting from the degeneration of massive and left-moving graviton modes, we construct the logarithmic mode as a derivative in parameter space and show that it acquires a natural multivalued structure upon complexification of the radial coordinate. We demonstrate that this multivaluedness induces a nontrivial monodromy action on the space of linearized solutions, under which the left-moving and logarithmic modes form an indecomposable (Jordan block) representation. This monodromy is unipotent and provides a bulk realization of the logarithmic structure typically associated with logarithmic conformal field theories. We further show that the monodromy representation alone is sufficiently constraining to determine the characteristic logarithmic form and mixing structure of two-point functions, up to normalization, without assuming logarithmic conformal field theory data a priori. These results suggest a geometric interpretation in which logarithmic modes act as sources of branchlike behavior in the bulk, analogous to twist fields that generate monodromy. While this perspective is compatible with proposed connections to branched coverings, Hurwitz theory, and integrable hierarchies, establishing a precise correspondence is left for future work.

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arXiv2025

On palindromic numerators of bigraded symmetric orbifold Hilbert series and Kostka-Foulkes polynomials

From our work on partition functions in log gravity, we show that the palindromic numerators in two variables of bigraded symmetric orbifold Hilbert series take the form of sums of products of Kostka-Foulkes polynomials associated with a pair of partition $λ$ and $μ=(1^n)$. The log partition function also being a KP $τ$-function, our work gives a new description of Hall-Littlewood and Kostka-Foulkes polynomials as palindromic numerators of quotient expansions in the moduli space of formal power series solutions of the KP hierarchy. Using the structure and properties of the log partition function, we also show that the palindromic polynomials are eigenvalues of a differential operator arising from a recurrence relation and acting on the Hilbert series.

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arXiv2024

Urn models, Markov chains and random walks in cosmological topologically massive gravity at the critical point

We discuss a partition-valued stochastic process in the logarithmic sector of critical cosmological topologically massive gravity. By applying results obtained in our previous works, we first show that the logarithmic sector can be modelled as an urn scheme, with a conceptual view of the random process occurring in the theory as an evolutionary process whose dynamical state space is the urn content. The urn process is then identified as the celebrated Hoppe urn model. We next show a one-to-one correspondence between Hoppe's urn model and the genus-zero Feynman diagram expansion of the log sector in terms of rooted trees. In this context, the balls in the urn model are represented by nodes in the random tree model, and the "special" ball in this Pólya-like urn construction finds a nice interpretation as the root in the recursive tree model. Furthermore, a partition-valued Markov process in which a sequence of partitions whose distribution is given by Hurwitz numbers is shown to be encoded in the log partition function. Given the bijection between the set of partitions of $n$ and the conjugacy classes of the symmetric group $S_n$, it is shown that the structure of the Markov chain consisting of a sample space that is also the set of permutations of $n$ elements, leads to a further description of the Markov chain in terms of a random walk on the symmetric group. From this perspective, a probabilistic interpretation of the logarithmic sector of the theory as a two-dimensional gauge theory on the $S_n$ group manifold is given. We suggest that a possible holographic dual to cosmological topologically massive gravity at the critical point could be a logarithmic conformal field theory that takes into account non-equilibrium phenomena.

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arXiv2024

Fragmented perspective of self-organized criticality and disorder in log gravity

We use a statistical model to discuss nonequilibrium fragmentation phenomena taking place in the stochastic dynamics of the log sector in log gravity. From the canonical Gibbs model, a combinatorial analysis reveals an important aspect of the $n$-particle evolution previously shown to generate a collection of random partitions according to the Ewens distribution realized in a disconnected double Hurwitz number in genus zero. By treating each possible partition as a member of an ensemble of fragmentations, and ensemble averaging over all partitions with the Hurwitz number as a special case of the Gibbs distribution, a resulting distribution of cluster sizes appears to fall as a power of the size of the cluster. Dynamical systems that exhibit a distribution of sizes giving rise to a scale-invariant power-law behavior at a critical point possess an important property called self-organized criticality. As a corollary, the log sector of log gravity is a self-organized critical system at the critical point $μl =1$. A similarity between self-organized critical systems, spin glass models and the dynamics of the log sector which exhibits aging behavior reminiscent of glassy systems is pointed out by means of the Pòlya distribution, also known to classify various models of (randomly fragmented) disordered systems, and by presenting the cluster distribution in the log sector of log gravity as a distinguished member of this probability distribution. We bring arguments from a probabilistic perspective to discuss the disorder in log gravity, largely anticipated through the conjectured AdS$_3$/LCFT$_2$ correspondence.

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arXiv2024

Shannon information entropy, soliton clusters and Bose-Einstein condensation in log gravity

We give a probabilistic interpretation of the configurational partition function of the logarithmic sector of critical cosmological topologically massive gravity, in which the Hurwitz numbers considered in our previous works assume the role of probabilities in a distribution on cycles of permutations. In particular, it is shown that the permutations are distributed according to the Ewens sampling formula which plays a major role in the theory of partition structures and their applications to diffusive processes of fragmentation, and in random trees. This new probabilistic result together with the previously established evidence of solitons in the theory provide new insights on the instability originally observed in the theory. We argue that the unstable propagation of a seed soliton at single particle level induces the generation of fragments of defect soliton clusters with rooted tree configuration at multiparticle level, providing a disordered landscape. The Shannon information entropy of the probability distribution is then introduced as a measure of the evolution of the unstable soliton clusters generated. Finally, based on Feynman's path integral formalism on permutation symmetry in the $λ$-transition of liquid helium, we argue that the existence of permutation cycles in the configurational log partition function indicates the presence of Bose-Einstein condensates in log gravity.

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arXiv2024

From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity

We show that the partition function of the logarithmic sector of critical topologically massive gravity which represents a series expansion of composition of functions, can be expressed as a sum over rooted trees. Our work brings a connection between integrable hierarchies of mathematical physics, combinatorial Hopf algebras and rooted trees, by explaining how the $τ$-functions of the (potential) Burgers and KP integrable hierarchies appearing in the partition function of log gravity conceal the Hopf algebra of composition of functions, known as the Faà di Bruno algebra, of the same type as the celebrated Connes-Kreimer Hopf algebra of rooted trees and Feynman diagrams. In particular, the Hurwitz numbers appearing in the partition function arise as coefficients of isomorphism classes of rooted trees. A parallel is drawn between our findings and established results in the statistical physics literature concerning certain systems with quenched disorder on trees, associated to nonlinear partial differential equations admitting traveling wave solutions. This should be of particular interest in view of a further description of the disorder observed in log gravity.

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arXiv2024

Moduli space of logarithmic states in critical massive gravities

We take new algebraic and geometric perspectives on the combinatorial results recently obtained on the partition functions of critical massive gravities conjectured to be dual to Logarithmic CFTs throught the AdS$_3$/LCFT$_2$ correspondence. We show that the partition functions of logarithmic states can be expressed in terms of Schur polynomials. Subsequently, we show that the moduli space of the logarithmic states is the symmetric product $S^n \left( \mathbb{C}^2 \right)$. As the quotient of an affine space by the symmetric group, this orbifold space is shown to be described by Hilbert series that have palindromic numerators. The palindromic properties of the Hilbert series indicate that the orbifolds are Calabi-Yau, and allow for a new interpretation of the logarithmic state spaces in critical massive gravities as Calabi-Yau singular spaces.

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arXiv2022

Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity

We discuss integrable aspects of the logarithmic contribution of the partition function of cosmological critical topologically massive gravity. On one hand, written in terms of Bell polynomials which describe the statistics of set partitions, the partition function of the logarithmic fields is a generating function of the potential Burgers hierarchy. On the other hand, the polynomial variables are solutions of the Kadomtsev-Petviashvili equation, and the partition function is a KP $τ$ function, making more precise the solitonic nature of the logarithmic fields being counted. We show that the partition function is a generating function of Hurwitz numbers, and derive its expression. The fact that the partition function is the generating function of branched coverings gives insight on the orbifold target space. We show that the logarithmic field $ψ^{new}_{μν}$ can be regarded as a branch point field associated to the branch point $μl =1$.

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arXiv2019

On the combinatorics of partition functions in AdS3/LCFT2

Three-dimensional Topologically Massive Gravity at its critical point has been conjectured to be holographically dual to a Logarithmic CFT. However, many details of this correspondence are still lacking. In this work, we study the 1-loop partition function of Critical Cosmological Topologically Massive Gravity, previously derived by Gaberdiel, Grumiller and Vassilevich, and show that it can be usefully rewritten as a Bell polynomial expansion. We also show that there is a relationship between this Bell polynomial expansion and the Plethystic Exponential. Our reformulation allows us to match the TMG partition function to states on the CFT side, including the multi-particle states of t (the logarithmic partner of the CFT stress tensor) which had previously been elusive. We also discuss the appearance of a ladder action between the different multi-particle sectors in the partition function, which induces an interesting sl(2) structure on the n-particle components of the partition function.

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