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Satoshi Nawata

Publications and source records attributed to Satoshi Nawata.

At least 19 recordsLinked to original sources
arXiv2026

Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity

We study the representation theory of the vertex operator algebras (VOAs) associated with rank-one 4d $\mathcal{N} = 3$ superconformal field theories. For the $\mathbb{Z}_3$ S-fold theory, whose VOA $\mathcal{W}_{\mathbb{Z}_3}$ has central charge $c_{\mathrm{2d}} = -15$, we use the $\mathcal{N} = 1$ Lagrangian description to obtain the unflavored Schur index in terms of Dedekind eta functions, while Wilson-loop indices yield the unflavored non-vacuum characters. These characters all solve a modular linear differential equation (MLDE) whose solution space also contains a logarithmic character. Combining flavored MLDEs from null states with Zhu's associative algebra and a free-field realization, we study four highest-weight modules of $\mathcal{W}_{\mathbb{Z}_3}$ and their flavored characters in closed form. A parallel analysis applies to the $\mathcal{N} = 3$ theories obtained by gauging a discrete $\mathbb{Z}_n$ flavor subgroup of $\mathcal{N} = 4$ $U(1)$ and $SU(2)$ super-Yang--Mills, for which we also obtain closed-form Schur indices and a new free-field realization of the VOA of the $\mathbb{Z}_4$ quotient.

hep-th↗
arXiv2025

Central Charges and Vacuum Moduli of 2d $\mathcal{N}=(0,4)$ Theories from Class $\mathcal{S}$

We investigate 2d $\mathcal{N}=(0,4)$ supersymmetric theories obtained from a topologically-twisted reduction of 4d $\mathcal{N}=2$ class $\mathcal{S}$ theories on a Riemann surface. This study addresses subtle aspects of central charges, unbroken gauge groups, and emergent superconformal R-symmetries of these theories. Focusing on infrared vacuum structures, we propose conjectural formulas for the central charges. For theories with the gauge group $SU(2)$, we use a Lagrangian description to analyze the vacuum moduli spaces. In particular, we examine two distinct branches -- the special Higgs branch and the twisted Higgs branch -- by computing their Hilbert series, and find agreement with the proposed central charge formulas.

hep-th↗
arXiv2025

An introduction to 2d conformal field theory

These lecture notes provide a comprehensive introduction to 2d conformal field theory, covering foundational topics such as free fields, minimal models, Zamolodchikov's $c$-theorem, Wess-Zumino-Witten models, modular invariant partition functions, and the connections to entanglement entropy. The material presented is based on lectures at the Southeast University Summer School and the course at Fudan University.

hep-th↗
arXiv2025

2d dualities from 4d

We find new $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,2)$ dualities through the twisted compactifications of 4d supersymmetric theories on $S^2$. Our findings include dualities for both $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,2)$ non-Abelian gauge theories, as well as $\mathcal{N}=(0,2)$ Gauge/Landau-Ginzburg duality.

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arXiv2025

Branes and DAHA Representations

Using brane quantization, we study the representation theory of the spherical double affine Hecke algebra of type $A_1$ in terms of the topological A-model on the moduli space of flat SL(2,C)-connections on a once-punctured torus. In particular, we provide an explicit match between finite-dimensional representations and A-branes with compact support; one consequence is the discovery of new finite-dimensional indecomposable representations. We proceed to embed the A-model story in an M-theory brane construction, closely related to the one used in the 3d/3d correspondence; as a result, we identify modular tensor categories behind particular finite-dimensional representations with PSL(2,Z) action. Using a further connection to the fivebrane system for the class S construction, we go on to study the relationship of Coulomb branch geometry and algebras of line operators in 4d N=2* theories to the double affine Hecke algebra.

hep-th↗
arXiv2024

Branes and Representations of DAHA $C^\vee C_1$: affine braid group action on category

We study the representation theory of the spherical double affine Hecke algebra (DAHA) of $C^\vee C_1$, using brane quantization. By showing a one-to-one correspondence between Lagrangian $A$-branes with compact support and finite-dimensional representations of the spherical DAHA, we provide evidence of derived equivalence between the $A$-brane category of $\mathrm{SL}(2,\mathbb{C})$-character variety of a four-punctured sphere and the representation category of DAHA of $C^\vee C_1$. The $D_4$ root system plays an essential role in understanding both the geometry and representation theory. In particular, this $A$-model approach reveals the action of an affine braid group of type $D_4$ on the category. As a by-product, our geometric investigation offers detailed information about the low-energy effective dynamics of the SU(2) $N_f=4$ Seiberg-Witten theory.

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arXiv2024

Freezing and BPS jumping

We report a novel BPS jumping phenomenon of 5d $\mathcal{N}=1$ supersymmetric gauge theories whose brane configuration is equipped with an O$7$-plane. The study of the relation between O$7{}^+$-plane and O$7{}^-$-plane reveals that such BPS jumps take place when the Higgsing is triggered near the O7-plane upon a particular parameter tuning of the theories. We propose two types of gauge theories whose BPS spectra jump. One is the SU($2N+8$) gauge theory with a symmetric hypermultiplet converted to the SU($2N$) gauge theory with an antisymmetric hypermultiplet. The other is pure SO($2N+8$) gauge theory jumping to pure Sp($N$) gauge theory. We explicitly confirm our proposal through the (un)refined instanton partition functions. Furthermore, we discuss feasible generalizations involving an O$p$-plane for supersymmetric gauge theories of eight supercharges in four and three dimensions ($p=6, 5$ respectively).

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arXiv2024

Class $\mathcal{S}$ on $S^2$

We study 2d $\mathcal{N}=(0,2)$ and $\mathcal{N}=(0,4)$ theories derived from compactifying class $\mathcal{S}$ theories on $S^2$ with a topological twist. We present concise expressions for the elliptic genera of both classes of theories, revealing the TQFT structure on Riemann surfaces $C_{g,n}$. Furthermore, our study highlights the relationship between the left-moving sector of the (0,2) theory and the chiral algebra of the 4d $\mathcal{N}=2$ theory. Notably, we propose that the (0,2) elliptic genus of a theory of this class can be expressed as a linear combination of characters of the corresponding chiral algebra.

hep-th↗
arXiv2024

Quantum toroidal algebras and solvable structures in gauge/string theory

This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using $\mathcal{W}$-algebras as our starting point, we elucidate the interconnection of affine Yangians, quantum toroidal algebras, and double affine Hecke algebras. Our exploration delves into the representation theory of the quantum toroidal algebra of $\mathfrak{gl}_1$ in full detail, highlighting its connections to partitions, $\mathcal{W}$-algebras, Macdonald functions, and the notion of intertwiners. Further, we also discuss integrable models constructed on Fock spaces and associated $\mathcal{R}$-matrices, both for the affine Yangian and the quantum toroidal algebra of $\mathfrak{gl}_1$. The article then demonstrates how quantum toroidal algebras serve as a unifying algebraic framework that bridges different areas in physics. Notably, we cover topological string theory and supersymmetric gauge theories with eight supercharges, incorporating the AGT duality. Drawing upon the representation theory of the quantum toroidal algebra of $\mathfrak{gl}_1$, we provide a rather detailed review of its role in the algebraic formulations of topological vertex and $qq$-characters. Additionally, we briefly touch upon the corner vertex operator algebras and quiver quantum toroidal algebras.

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arXiv2023

ABCD of qq-characters

The qq-characters are powerful tools to reveal symmetries and integrabilities of Seiberg-Witten theories. The goal of this paper is to provide analytic expressions of qq-characters based on Young diagrams in 5d $\mathcal{N} = 1$ pure Yang-Mills theories with BCD-type gauge groups, by focusing on the unrefined limit. Using these expressions, we investigate the relationships among qq-characters of classical gauge groups. For SO(n) gauge groups, we construct a quantum-toroidal-like algebra via the Ward-identity approach, which allows us to derive the qq-characters.

hep-th↗
arXiv2023

3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry

The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d $\mathcal{N}=4$ theories. In this paper, starting with known mirror pairs of 3d $\mathcal{N}=4$ quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0-form, 1-form, and 2-group) and the mirror maps between them.

hep-th↗
arXiv2023

Instantons on Young diagrams with matters

We present the unrefined instanton partition functions of various 5d gauge theories with matter beyond the fundamental representation as sums over Young diagrams. By using these explicit expressions, we verify a range of identities among the instanton partition functions predicted by Higgsing procedures of fivebrane web diagrams and representation theory.

hep-th↗
arXiv2023

Instanton counting and O-vertex

We present closed-form expressions of unrefined instanton partition functions for gauge groups of type $BCD$ as sums over Young diagrams. For $\mathrm{SO}(n)$ gauge groups, we provide a fivebrane web picture of our formula based on the vertex-operator formalism of the topological vertex with a new type called O-vertex for an O5-plane.

hep-th↗
arXiv2022

Magnetic quivers and line defects -- On a duality between 3d N=4 unitary and orthosymplectic quivers

Supersymmetric Sp(k) quantum chromodynamics with 8 supercharges in space-time dimensions 3 to 6 can be realised by two different Type II brane configurations in the presence of orientifolds. Consequently, two types of magnetic quivers describe the Higgs branch of the Sp(k) SQCD theory. This is a salient example of a general phenomenon: a given hyper-Kahler Higgs branch may admit several magnetic quiver constructions. It is then natural to wonder if these different magnetic quivers, which are described by 3d N=4 theories, are dual theories. In this work, the unitary and orthosymplectic magnetic quiver theories are subjected to a variety of tests, providing evidence that they are IR dual to each other. For this, sphere partition function and supersymmetric indices are compared. Also, we study half BPS line defects and find interesting regularities from the viewpoints of exact results, brane configurations, and one-form symmetry.

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arXiv2021

On knots, complements, and 6j-symbols

This paper investigates the relation between colored HOMFLY-PT and Kauffman homology, $\text{SO}(N)$ quantum $6j$-symbols and $(a,t)$-deformed $F_K$. First, we present a simple rule of grading change which allows us to obtain the $[r]$-colored quadruply-graded Kauffman homology from the $[r^2]$-colored quadruply-graded HOMFLY-PT homology for thin knots. This rule stems from the isomorphism of the representations $(\mathfrak{so}_6,[r]) \cong (\mathfrak{sl}_4,[r^2])$. Also, we find the relationship among $A$-polynomials of SO and SU-type coming from a differential on Kauffman homology. Second, we put forward a closed-form expression of $\text{SO}(N)(N\geq 4)$ quantum $6j$-symbols for symmetric representations, and calculate the corresponding $\text{SO}(N)$ fusion matrices for the cases when representations $R = [1],[2]$. Third, we conjecture closed-form expressions of $(a,t)$-deformed $F_K$ for the complements of double twist knots with positive braids. Using the conjectural expressions, we derive $t$-deformed ADO polynomials.

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arXiv2020

Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams

We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any knot.

math.GT↗
arXiv2020

Refined large N duality for knots

We formulate large $N$ duality of $\mathrm{U}(N)$ refined Chern-Simons theory with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the $Ω$-background. This form enables us to relate refined Chern-Simons invariants of a torus knot/link in $S^3$ to refined BPS invariants in the resolved conifold. Assuming that the extra $\mathrm{U}(1)$ global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2-M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be interpreted as a positivity conjecture of refined Chern-Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

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