arXiv subjects

Saebyeok Jeong

Publications and source records attributed to Saebyeok Jeong.

18 recordsLinked to original sources

arXiv2026

Split Heun functions via blown-up surface defects

We study resonant solutions of the Heun equation and its confluent limits that arise in the Nekrasov-Shatashvili (NS) limit of four-dimensional $\mathcal{N}=2$ $\mathrm{SU}(2)$ gauge theories with fundamental hypermultiplets. At the resonant loci $2a/\hbar\in\mathbb{Z}$ in the Coulomb branch parameter $a$, the Floquet multipliers coalesce and the instanton expansions of the bulk and surface defect NS functions develop poles of increasing order. We derive blow-up equations involving exclusively NS functions and use them to resum these singular expansions. The resulting resummed bulk and surface defect NS functions reveal the analytic structure of the gauge-theoretic solutions near the resonant loci, including the branch structure of the accessory parameter and of the Floquet solutions that is obscured by the term-by-term instanton expansion. At resonance, the resummed accessory parameters and suitably normalized defect wavefunctions admit finite limits that describe periodic or antiperiodic solutions at the edges of spectral gaps and allow us to construct their logarithmic companions. We then identify distinct nested mass loci governing gap closure and semisimple resonant monodromy. On the larger locus the band-edge accessory parameters coalesce, while on the smaller locus two independent resonant (anti)periodic Floquet solutions survive. We develop the general resummation procedure for $N_f=(n_0,n_1)$ theories with $n_i\leq 2$ $(i=0,1)$, and demonstrate it explicitly for the $N_f=(1,1)$ theory.

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arXiv2026

Parallel surface defects, Hecke operators, and quantum Hitchin system

We examine two types of half-BPS surface defects $-$ regular monodromy surface defect and canonical surface defect $-$ in four-dimensional gauge theory with $\mathcal{N}=2$ supersymmetry and $Ω_{\varepsilon_1,\varepsilon_2}$-background. Mathematically, we investigate integrals over the moduli spaces of parabolic framed sheaves over $\mathbb{P}^2$. Using analytic methods of $\mathcal{N}=2$ theories, we demonstrate that the former gives a twisted $\mathcal{D}$-module on $\text{Bun}_{G_{\mathbb{C}}}$ while the latter acts as a Hecke operator. In the limit $\varepsilon_2 \to 0$, the cluster decomposition implies the Hecke eigensheaf property for the regular monodromy surface defect. The eigenvalues are given by the opers associated to the canonical surface defect. We derive, in our $\mathcal{N}=2$ gauge theoretical framework, that the twisted $\mathcal{D}$-modules assigned to the opers in the geometric Langlands correspondence represent the spectral equations for quantum Hitchin integrable system. A duality to topologically twisted four-dimensional $\mathcal{N}=4$ theory is discussed, in which the two surface defects are mapped to Dirichlet boundary and 't Hooft line defect. This is consistent with earlier works on the $\mathcal{N}=4$ theory approach to the geometric Langlands correspondence.

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arXiv2026

Refined Invariants and Quantum Curves from Supersymmetric Localization

We study an Aganagic-Vafa brane supported on a special Lagrangian submanifold $\mathcal{L}$ in a non-compact toric Calabi-Yau threefold $\mathcal{X}$. From the perspective of geometric engineering, the Aganagic-Vafa branes give rise to a special class of half-BPS codimension-two defects in 5d $\mathcal{N}=1$ supersymmetric field theories in the presence of $Ω$-background. We propose that the defect partition functions give generating functions of refined, non-negative, integral open BPS invariants of the pair $(\mathcal{X},\mathcal{L})$, across different Kähler moduli chambers they are expanded in. In the Nekrasov-Shatashvili limit, the partition function provides a partially resummed solution to a $q$-difference equation that quantizes the mirror curve of $\mathcal{X}$ in an unambiguous fashion, in a polarization determined by the discrete labels of the Aganagic-Vafa brane. We demonstrate our method at examples of $\mathbb{C}^3$, resolved conifold, resolved $A_1$-singularity, local $F_0$, and local $F_1$.

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arXiv2025

$Q$-operators, $q$-opers, and R-matrices in 5d $\mathcal{N}=1$ gauge theory

We study the quantization of the moduli space of multiplicative Higgs bundles through the lens of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories in $Ω$-background. We extend the 4d $\mathcal{N}=2$ gauge theoretical construction of key geometric and representation-theoretic structures, established in earlier works, to the five-dimensional uplift. We construct and analyze the $Q$-operators and $q$-opers associated with the canonical codimension-two defect: the $Q$-operators are defined via the insertion of the defect, while the $q$-opers arise as the $q$-difference chiral ring equations in its presence. The $q$-oper difference equations are further identified with the Baxter TQ equations for XXZ spin chains constructed from tensor products of bi-infinite evaluation modules over quantum affine algebras of type ${\mathfrak{gl}}(n)$. We define a $q$-difference module structure on the space of monodromy codimension-two defect partition functions and show that the eigenstates of the $Q$-operators, constructed from monodromy defects, simultaneously diagonalize the quantum Hamiltonians of the XXZ spin chain. A Fourier transformation exchanges the $Q$-operators associated with two XXZ spin chains bispectral dual to each other. Finally, we relate these constructions to the quantum cluster algebra arising from the BPS quiver of the 5d theory, and re-express the R-matrices in terms of the cluster variables.

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arXiv2025

Exact QFT duals of AdS black holes

We construct large $N$ saddle points of the matrix model for the $\mathcal{N}=4$ Yang-Mills index dual to the BPS black holes in $AdS_5\times S^5$, in two different setups. When the two complex chemical potentials for the angular momenta are collinear, we find linear eigenvalue distributions which solve the large $N$ saddle point equation. When the chemical potentials are not collinear, we find novel solutions given by areal eigenvalue distributions after slightly reformulating the saddle point problem. We also construct a class of multi-cut saddle points, showing that they sometimes admit nontrivial filling fractions. As a byproduct, we find that the Bethe ansatz equation emerges from our saddle point equation.

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arXiv2025

di-Langlands correspondence and extended observables

We explore the $\textit{difference Langlands correspondence}$ using the four dimensional ${\mathcal{N}}=2$ super-QCD. Surface defects and surface observables play the crucial role. As an application, we give the first construction of the full set of quantum integrals, i.e. commuting differential operators, such that the partition function of the so-called regular monodromy surface defect is their joint eigenvectors in an evaluation module over the Yangian $Y(\mathfrak{gl}(2))$, making it the wavefunction of a $N$-site $\mathfrak{gl}(2)$ spin chain with bi-infinite spin modules. We construct the $\mathbf{Q}$- and $\tilde{\mathbf{Q}}$-surface observables which are believed to be the $Q$-operators on the bi-infinite module over the Yangian $Y(\mathfrak{gl}(2))$, and compute their eigenvalues, the $Q$-functions, as vevs of the surface observables.

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arXiv2025

Bispectral duality and separation of variables from surface defect transition

We study two types of surface observables $-$ the $\mathbf{Q}$-observables and the $\mathbf{H}$-observables $-$ of the 4d $\mathcal{N}=2$ $A_1$-quiver $U(N)$ gauge theory obtained by coupling a 2d $\mathcal{N}=(2,2)$ gauged linear sigma model. We demonstrate that the transition between the two surface defects manifests as a Fourier transformation between the surface observables. Utilizing the results from our previous works, which establish that the $\mathbf{Q}$-observables and the $\mathbf{H}$-observables give rise, respectively, to the $Q$-operators on the evaluation module over the Yangian $Y(\mathfrak{gl}(2))$ and the Hecke operators on the twisted $\widehat{\mathfrak{sl}}(N)$-coinvariants, we derive an exact duality between the spectral problems of the $\mathfrak{gl}(2)$ XXX spin chain with $N$ sites and the $\mathfrak{sl}(N)$ Gaudin model with 4 sites, both of which are defined on bi-infinite modules. Moreover, we present a dual description of the monodromy surface defect as coupling a 2d $\mathcal{N}=(2,2)$ gauged linear sigma model. Employing this dual perspective, we demonstrate how the monodromy surface defect undergoes a transition to multiple $\mathbf{Q}$-observables or $\mathbf{H}$-observables, implemented through integral transformations between their surface observables. These transformations provide, respectively, $\hbar$-deformation and a higher-rank generalization of the KZ/BPZ correspondence. In the limit $\varepsilon_2\to 0$, they give rise to the quantum separation of variables for the $\mathfrak{gl}(2)$ XXX spin chain and the $\mathfrak{sl}(N)$ Gaudin model, respectively.

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arXiv2024

R-matrices and Miura operators in 5d Chern-Simons theory

We derive Miura operators for $W$- and $Y$-algebras from first principles as the expectation value of the intersection between a topological line defect and a holomorphic surface defect in 5-dimensional non-commutative $\mathfrak{gl}(1)$ Chern-Simons theory. The expectation value, viewed as the transition amplitude for states in the defect theories forming representations of the affine Yangian of $\mathfrak{gl}(1)$, satisfies the Yang-Baxter equation and is thus interpreted as an R-matrix. To achieve this, we identify the representations associated with the line and surface defects by calculating the operator product expansions (OPEs) of local operators on the defects, as conditions that anomalous Feynman diagrams cancel each other. We then evaluate the expectation value of the defect intersection using Feynman diagrams. When the line and surface defects are specified, we demonstrate that the expectation value precisely matches the Miura operators and their products.

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arXiv2024

The Yang-Mills duals of small AdS black holes

We study the large $N$ matrix model for the index of 4d $\mathcal{N}=4$ Yang-Mills theory and its truncations to understand the dual AdS$_5$ black holes. Numerical studies of the truncated models provide insights on the black hole physics, some of which we investigate analytically with the full Yang-Mills matrix model. In particular, we find many branches of saddle points which describe the known black hole solutions. We analytically construct the saddle points dual to the small black holes whose sizes are much smaller than the AdS radius. They include the asymptotically flat BMPV black holes embedded in large AdS with novel thermodynamic instabilities.

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arXiv2024

Miura operators as R-matrices from M-brane intersections

We propose that Miura operators are R-matrices of certain infinite-dimensional quantum algebras. We test our proposal by realizing Miura operators of $q$-deformed $W$- and $Y$-algebras in terms of R-matrices of the quantum toroidal algebra of $\mathfrak{gl}(1)$. Physically, the representations of this toroidal algebra arise from the algebra of local operators on M2-branes and M5-branes, in M-theory subject to an $\Omega$-background. We associate an R-matrix to each M2-M5 brane crossing, by studying its description as a gauge-invariant intersection of a topological line defect and a holomorphic surface defect in 5-dimensional non-commutative Chern-Simons theory. The Miura transformation is engineered using multiple M2-M5 intersections, relying crucially on the properties of the underlying R-matrices. We thereby identify each R-matrix with a Miura operator. In a dual Type IIB frame, the components of the Miura transformation are shown to coincide with the half-index of a 3d supersymmetric gauge theory on a Hanany-Witten system of D3-NS5 branes. As a further application, we demonstrate that $qq$-characters can be algebraically constructed from the Miura transformation.

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arXiv2024

BPZ equations for higher degenerate fields and nonperturbative Dyson-Schwinger equations

In the two-dimensional Liouville conformal field theory, correlation functions involving a degenerate field satisfy partial differential equations due to the decoupling of the null descendant field. On the other hand, the instanton partition function of a four-dimensional $\mathcal{N}=2$ supersymmetric theory in the $Ω$-background at a special point of the parameter space also satisfies a partial differential equation resulting from the constraints of the gauge field configurations. This partial differential equation can be proved using the nonperturbative Dyson-Schwinger equations. We show for the next-to-simplest case that the partial differential equations obtained from two different perspectives can be identified, thereby confirming an assertion of the BPS/CFT correspondence.

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arXiv2023

Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations

We propose an interesting BPS/CFT correspondence playground: the correlation function of two intersecting half-BPS surface defects in four-dimensional $\mathcal{N}=2$ supersymmetric $SU(N)$ gauge theory with $2N$ fundamental hypermultiplets. We show it satisfies a difference equation, the fractional quantum T-Q relation. Its Fourier transform is the $5$-point conformal block of the $\widehat{\mathfrak{sl}}_N$ current algebra with one of the vertex operators corresponding to the $N$-dimensional $\mathfrak{sl}_N$ representation, which we demonstrate with the help of the Knizhnik-Zamolodchikov equation. We also identify the correlator with a state of the $XXX_{\mathfrak{sl}_2}$ spin chain of $N$ Heisenberg-Weyl modules over $Y(\mathfrak{sl}_2)$. We discuss the associated quantum Lax operators, and connections to isomonodromic deformations.

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arXiv2021

Opers, surface defects, and Yang-Yang functional

We explore the non-perturbative Dyson-Schwinger equations obeyed by the partition functions of the $Ω$-deformed $\mathcal{N}=2, d=4$ supersymmetric linear quiver gauge theories in the presence of surface defects. We demonstrate that the partition functions of different types of defects (orbifold or vortex strings) are related by analytic continuation. We introduce Darboux coordinates on a patch of the moduli space of flat $SL(N)$-connections on a sphere with special punctures, which generalize the NRS coordinates defined in the $SL(2)$ case. Finally, we compare the generating function of the Lagrangian variety of opers in these Darboux coordinates with the effective twisted superpotential of the linear quiver theory in the two-dimensional $Ω$-background, thereby proving the NRS conjecture and its generalization to the $SL(3)$ case.

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arXiv2020

Riemann-Hilbert correspondence and blown up surface defects

The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional $\mathcal{N}=2$ gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in $SU(2)$ theory with $N_f=4$ fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with $5$ regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.

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arXiv2020

New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(ν_1,ν_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito's vertex representations of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We construct the vertex operator intertwining between these two types of representations. This object is identified with a $(ν_1,ν_2)$-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the $qq$-characters of the quiver gauge theories.

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arXiv2020

A note on chiral trace relations from qq-characters

We study chiral operators in four-dimensional N=2 supersymmetric gauge theories. We provide a general procedure to derive the exact relations among the vacuum expectation values of chiral operators in the Omega-background using the non-perturbative Dyson-Schwinger equations of qq-characters. We demonstrate our procedure using several examples. For pure SU(N) gauge theory and SU(N) gauge theory with 2N fundamental hypermultiplets, we verify the relations conjectured previously in the literature. We also briefly discuss the relations in SU(N) linear superconformal quiver gauge theories.

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arXiv2019

SCFT/VOA correspondence via $Ω$-deformation

We investigate an alternative approach to the correspondence of four-dimensional $\mathcal{N}=2$ superconformal theories and two-dimensional vertex operator algebras, in the framework of the $Ω$-deformation of supersymmetric gauge theories. The two-dimensional $Ω$-deformation of the holomorphic-topological theory on the product four-manifold is constructed at the level of supersymmetry variations and the action. The supersymmetric localization is performed to achieve a two-dimensional chiral CFT. The desired vertex operator algebra is recovered as the algebra of local operators of the resulting CFT. We also discuss the identification of the Schur index of the $\mathcal{N}=2$ superconformal theory and the vacuum character of the vertex operator algebra at the level of their path integral representations, using our $Ω$-deformation point of view on the correspondence.

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arXiv2018

Splitting of surface defect partition functions and integrable systems

We study Bethe/gauge correspondence at the special locus of Coulomb moduli where the integrable system exhibits the splitting of degenerate levels. For this investigation, we consider the four-dimensional pure $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theory, with a half-BPS surface defect constructed with the help of an orbifold or a degenerate gauge vertex. We show that the non-perturbative Dyson-Schwinger equations imply the Schrödinger-type and the Baxter-type differential equations satisfied by the respective surface defect partition functions. At the special locus of Coulomb moduli the surface defect partition function splits into parts. We recover the Bethe/gauge dictionary for each summand.

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