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Tomoki Nakanishi

Publications and source records attributed to Tomoki Nakanishi.

At least 19 recordsLinked to original sources
arXiv2026

Relation between generalized and ordinary cluster algebras

Recently, Ramos and Whiting showed that any generalized cluster algebra of geometric type is isomorphic to a quotient of a subalgebra of a certain cluster algebra. Based on their idea and method, we show that the same property holds for any generalized cluster algebra with $y$-variables in an arbitrary semifield. We also present the relations between the $C$-matrices, the $G$-matrices, and the $F$-polynomials of a generalized cluster pattern and those of the corresponding composite cluster pattern.

math.RT↗
arXiv2025

Cluster Algebras and Dilogarithm Identities

This is a reasonably self-contained exposition of the fascinating interplay between cluster algebras and the dilogarithm in the past two decades. The dilogarithm has a long and rich history since it was studied by Euler. The most intriguing property of the function is that it satisfies various functional relations, which we call dilogarithm identities (DIs). In the 1990s, various DIs were conjectured in the study of integrable models, but most of them were left unsolved. On the other hand, cluster algebras are a class of commutative algebras introduced by Fomin and Zelevinsky around 2000. In this text, we explain how the above DIs are proved using the techniques and results of cluster algebras. Also, we employ the DI associated with each period in a cluster pattern of cluster algebra as the leitmotif and present several proofs, variations, and generalizations of them with various methods and techniques. The quantum DIs are also treated from a unified point of view compared to the classical ones.

math.RA↗
arXiv2024

$S^1$ reduction of 4D $\mathcal{N}=4$ Schur index and 3D $\mathcal{N}=8$ mass-deformed partition function

We study the compactification of 4D $\mathcal{N}=4$ SYM on $S^1$ from the viewpoint of the superconformal index. In the cases that the gauge group of the 4D SYM is $U(N)$ and $Usp(2N)$, the resulting 3D theory is believed to be the ABJM theory with the Chern-Simons level $k=1$ and $k=2$, respectively. This suggests that the small $S^1$ limit of the superconformal index of these 4D $\mathcal{N}=4$ SYMs is identical to the sphere partition function of the ABJM theories. Using a recently observed relation between the 4D and 3D R-charges for theories with twelve or more supercharges, we explicitly confirm this identity in the Schur limit of the 4D index. Our result provides a direct quantitative check of the relation between 4D $\mathcal{N}=4$ SYMs and 3D $\mathcal{N}=8$ ABJM theories.

hep-th↗
arXiv2024

Local and global patterns of rank 3 $G$-fans of totally-infinite type

We focus on the $G$-fans associated with cluster patterns whose initial exchange matrices are of infinite type. We study the asymptotic behavior of the $g$-vectors around the initial $G$-cone under the alternating mutations for two indices of infinite type. In the rank 3 case, we classify them into several patterns. As an application, the incompleteness of the $G$-fans of infinite type is proved. We observed that the local pattern of a rank 3 $G$-fan of totally-infinite type classified by the above types correlates with its global pattern. Following the classification of the local patterns (together with the Markov constant), we present several prototypical examples of the global patterns of the rank 3 $G$-fans of totally-infinite type, many of which are new in the literature.

math.CO↗
arXiv2024

Pentagon relation in quantum cluster scattering diagrams

We formulate the pentagon relation for quantum dilogarithm elements in the structure group of a quantum cluster scattering diagram (QCSD). As an application, we show the nonpositivity of a certain class of nonskew-symmetric QCSDs. Also, we explicitly present various consistency relations for QCSDs of rank 2 completely or up to some degree, many of which are new in the literature.

math.CO↗
arXiv2024

Liouville Irregular States of Half-Integer Ranks

We conjecture a set of differential equations that characterizes the Liouville irregular states of half-integer ranks, which extends the generalized AGT correspondence to all the $(A_1,A_\text{even})$ and $(A_1,D_\text{odd})$ types Argyres-Douglas theories. For lower half-integer ranks, our conjecture is verified by deriving it as a suitable limit of a similar set of differential equations for integer ranks. This limit is interpreted as the 2D counterpart of a 4D RG-flow from $(A_1,D_{2n})$ to $(A_1,D_{2n-1})$. For rank $3/2$, we solve the conjectured differential equations and find a power series expression for the irregular state $|I^{(3/2)}\rangle$. For rank $5/2$, our conjecture is consistent with the differential equations recently discovered by H. Poghosyan and R. Poghossian.

hep-th↗
arXiv2023

Dilogarithm identities in cluster scattering diagrams

We extend the notion of $y$-variables (coefficients) in cluster algebras to cluster scattering diagrams. Accordingly, we extend the dilogarithm identity associated with a period in a cluster pattern to the one associated with a loop in a cluster scattering diagram. We show that these identities are constructed from and reduced to a trivial one by applying the pentagon identity possibly infinitely many times.

math.CO↗
arXiv2023

Two Formulas for $F$-Polynomials

We discuss a product formula for $F$-polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock-Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of $F$-polynomials in a given seed that depends only on the $\mathbf{c}$-vectors and $\mathbf{g}$-vectors along a finite sequence of mutations from the initial seed to the given seed.

math.CO↗
arXiv2023

$S^1$ Reduction of 4D $\mathcal{N}=3$ SCFTs and Squashing Independence of ABJM Theories

We study the compactification of 4D $\mathcal{N}=3$ superconformal field theories (SCFTs) on $S^1$, focusing on the relation between the 4D superconformal index and 3D partition function on the squashed sphere $S^3_b$. Since the center $\mathfrak{u}(1)$ of the $\mathfrak{u}(3)$ R-symmetry of the 4D theory can mix with an $\mathcal{N}=6$ abelian flavor symmetry in three dimensions, the precise 4D/3D relation for the global symmetry is not obvious. Focusing on the case in which the 3D theory is the ABJM theory, we demonstrate that the above R-symmetry mixing can be precisely identified by considering the Schur limit (and/or its $\mathcal{N}=3$ cousin) of the 4D index. As a result, we generalize to the ABJM theories recent discussions on the connection between supersymmetry enhancement of the 4D index and squashing independence of the $S^3_b$ partition function.

hep-th↗
arXiv2023

Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams

We review some important results by Gross, Hacking, Keel, and Kontsevich on cluster algebra theory, namely, the column sign-coherence of $C$-matrices and the Laurent positivity, both of which were conjectured by Fomin and Zelevinsky. We digest and reconstruct the proofs of these conjectures by Gross et al. still based on their scattering diagram method, however, without relying on toric geometry. At the same time, we also give a detailed account of the correspondence between the notions of cluster patterns and scattering diagrams. Most of the results in this text are found in or translated from the known results in the literature. However, the approach, the construction of logic and proofs, and the overall presentation are new. Also, as an application of the results and the techniques in the text, we show that there is a one-to-one correspondence between $g$-vectors and cluster variables in cluster patterns with arbitrary coefficients.

math.CO↗
arXiv2023

Cluster Algebras and Scattering Diagrams, Part III. Cluster Scattering Diagrams

This is a self-contained exposition of several fundamental properties of cluster scattering diagrams introduced and studied by Gross, Hacking, Keel, and Kontsevich. In particular, detailed proofs are presented for the construction, the mutation invariance, and the positivity of theta functions of cluster scattering diagrams. Throughout the text we highlight the fundamental roles of the dilogarithm elements and the pentagon relation in cluster scattering diagrams.

math.CO↗
arXiv2022

Duality Cascades and Affine Weyl Groups

Brane configurations in a circle allow subsequent applications of the Hanany-Witten transitions, which are known as duality cascades. By studying the process of duality cascades corresponding to quantum curves with symmetries of Weyl groups, we find a hidden structure of affine Weyl groups. Namely, the fundamental domain of duality cascades consisting of all the final destinations is characterized by the affine Weyl chamber and the duality cascades are realized as translations of the affine Weyl group, where the overall rank in the brane configuration associates to the grading operator of the affine algebra. The structure of the affine Weyl group guarantees the finiteness of the processes and the uniqueness of the endpoint of the duality cascades. In addition to the original duality cascades, we can generalize to the cases with Fayet-Iliopoulos parameters. There we can utilize the Weyl group to analyze the fundamental domain similarly and find that the fundamental domain continues to be the affine Weyl chamber. We further interpret the Weyl group we impose as a "half" of the Hanany-Witten transition.

hep-th↗
arXiv2021

Brane Transitions from Exceptional Groups

It is a well-known result by Hanany and Witten that, when two five-branes move across each other, D3-branes stretching between them are generated. Later the same brane configurations played a crucial role in understanding the worldvolume theory of multiple M2-branes. Recently the partition function of multiple M2-branes was transformed to the Fredholm determinant for quantum algebraic curves, where the characteristic 3/2 power law of degrees of freedom is reproduced and the determinant enjoys a large symmetry given by exceptional Weyl groups. The large exceptional Weyl group reproduces the Hanany-Witten brane transitions and, besides, contains brane transitions unknown previously. Aiming at understanding the new brane transitions better, we generalize our previous study on the D5 quantum curve to the E7 case, which requires delicate handling of degeneracies. By combining the results of these two cases, we propose a "local" rule for the brane transitions.

hep-th↗
arXiv2019

Synchronicity phenomenon in cluster patterns

It has been known that several objects such as cluster variables, coefficients, seeds, and $Y$-seeds in different cluster patterns with common exchange matrices share the same periodicity under mutations. We call it synchronicity phenomenon in cluster patterns. In this expository note we explain the mechanism of synchronicity based on several fundamental results on cluster algebra theory such as separation formulas, sign-coherence, Laurent positivity, duality, and detropicalization obtained by several authors. We also show that all synchronicity properties studied in this paper are naturally extended to cluster patterns of generalized cluster algebras, up to the Laurent positivity conjecture.

math.RA↗
arXiv2019

Asymptotic sign coherence conjecture

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and formulate a conjecture describing their asymptotic behavior. This conjecture, which is called the asymptotic sign coherence conjecture, states that for any infinite sequence of matrix mutations that satisfies certain natural conditions, the corresponding c-vectors eventually become sign coherent. We prove this conjecture for rank 2 cluster algebras of infinite type and for a particular sequence of mutations in a cluster algebra associated with the Markov quiver.

math.CO↗
arXiv2017

Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

We introduce and study a Hamiltonian formalism of mutations in cluster algebras using canonical variables, where the Hamiltonian is given by the Euler dilogarithm. The corresponding Lagrangian, restricted to a certain subspace of the phase space, coincides with the Rogers dilogarithm. As an application, we show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from the Hamiltonian/Lagrangian point of view.

math.RA↗