arXiv subjects

Nathan Haouzi

Publications and source records attributed to Nathan Haouzi.

13 recordsLinked to original sources

arXiv2026

Meromorphic amplitudes from 3-dimensional supersymmetry

We establish a new connection between supersymmetric theories and scattering amplitudes. We show that the Coon amplitude coincides with the 3d $\mathcal{N}=2$ half-index of the XYZ model with nontrivial boundary conditions. Our 3d theory, intrinsically defined in the UV, flows to a sigma model in the IR whose partition function is the Veneziano amplitude. Crossing symmetry is realized as a consequence of 3d $\mathcal{N}=2$ mirror symmetry between XYZ and SQED. We use this correspondence to construct a meromorphic modification of the Coon amplitude by promoting the long-standing dressing factor $\mathfrak{q}^{ST}$ responsible for a branch cut to an elliptic completion thereof. This illustrates that one does not have to give up single-valuedness to achieve positivity at the physical poles.

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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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arXiv2023

A new realization of quantum algebras in gauge theory and Ramification in the Langlands program

We realize the fundamental representations of quantum algebras via the supersymmetric Higgs mechanism in gauge theories with 8 supercharges on an $Ω$-background. We test our proposal for quantum affine algebras, by probing the Higgs phase of a 5d quiver gauge theory on a circle. We show that our construction implies the existence of tame ramification in the Aganagic-Frenkel-Okounkov formulation of the geometric Langlands program, a correspondence which identifies $q$-conformal blocks of the quantum affine algebra with those of a Langlands dual deformed ${\cal W}$-algebra. The new feature of ramified blocks is their definition in terms of Drinfeld polynomials for a set of quantum affine weights. In enumerative geometry, the blocks are vertex functions counting quasimaps to quiver varieties describing moduli spaces of vortices. Physically, the vortices admit a description as a 3d ${\cal N}=2$ quiver gauge theory on the Higgs branch of the 5d gauge theory, uniquely determined from the Drinfeld polynomial data; the blocks are supersymmetric indices for the vortex theory supported on a 3-manifold with distinguished BPS boundary conditions. The top-down explanation of our results is found in the 6d $(2,0)$ little string theory, where tame ramification is provided by certain D-branes. When the string mass is taken to be large, we make contact with various physical aspects of the point particle superconformal limit: the Gukov-Witten description of ramification via monodromy defects in 4d Super Yang-Mills (and their S-duality), the Nekrasov-Tsymbaliuk solution to the Knizhnik-Zamolodchikov equations, and the classification of massive deformations of tamely ramified Hitchin systems. In a companion paper, we will show that our construction implies a solution to the local Alday-Gaiotto-Tachikawa conjecture.

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arXiv2020

Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N} = 4$ Gauge Theories

We analyze symmetries corresponding to separated topological sectors of 3d ${\cal} N=4$ gauge theories with Higgs vacua, compactified on a circle. The symmetries are encoded in Schwinger-Dyson identities satisfied by correlation functions of a certain gauge-invariant operator, the "vortex character." Such a character observable is realized as the vortex partition function of the 3d gauge theory, in the presence of a 1/2-BPS line defect. The character enjoys a double refinement, interpreted as a deformation of the usual characters of finite-dimensional representations of quantum affine algebras. We derive and interpret the Schwinger-Dyson identities for the 3d theory from various physical perspectives: in the 3d gauge theory itself, in a 1d gauged quantum mechanics, in 2d $q$-Toda theory, and in 6d little string theory. We establish the dictionary between all approaches. Lastly, we comment on the transformation properties of the vortex character under the action of three-dimensional Seiberg duality.

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arXiv2020

On the Quantization of Seiberg-Witten Geometry

We propose a double quantization of four-dimensional ${\cal N}=2$ Seiberg-Witten geometry, for all classical gauge groups and a wide variety of matter content. This can be understood as a set of certain non-perturbative Schwinger-Dyson identities, following the program initiated by Nekrasov [arXiv:1512.05388]. The construction relies on the computation of the instanton partition function of the gauge theory on the so-called $Ω$-background on $\mathbb{R}^4$, in the presence of half-BPS codimension 4 defects. The two quantization parameters are identified as the two parameters of this background. The Seiberg-Witten curve of each theory is recovered in the flat space limit. Whenever possible, we motivate our construction from type IIA string theory.

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arXiv2020

Quantum Geometry and $θ$-Angle in Five-Dimensional Super Yang-Mills

Five-dimensional $Sp(N)$ supersymmetric Yang-Mills admits a $\mathbb{Z}_2$ version of a theta angle $θ$. In this note, we derive a double quantization of the Seiberg-Witten geometry of $\mathcal{N}=1$ $Sp(1)$ gauge theory at $θ=π$, on the manifold $S^1\times\mathbb{R}^4$. Crucially, $\mathbb{R}^4$ is placed on the $Ω$-background, which provides the two parameters to quantize the geometry. Physically, we are counting instantons in the presence of a 1/2-BPS fundamental Wilson loop, both of which are wrapping $S^1$. Mathematically, this amounts to proving the regularity of a $qq$-character for the spin-1/2 representation of the quantum affine algebra $U_q(\widehat{A_1})$, with a certain twist due to the $θ$-angle. We motivate these results from two distinct string theory pictures. First, in a $(p,q)$-web setup in type IIB, where the loop is characterized by a D3 brane. Second, in a type I' string setup, where the loop is characterized by a D4 brane subject to an orientifold projection. We comment on the generalizations to the higher rank case $Sp(N)$ when $N>1$, and the $SU(N)$ theory at Chern-Simons level $κ$ when $N>2$.

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arXiv2019

Supersymmetric Wilson Loops, Instantons, and Deformed ${\cal W}$-Algebras

Let $\mathfrak{g}$ be a simple Lie algebra. We study 1/2-BPS Wilson loops of supersymmetric 5d $\mathfrak{g}$-type quiver gauge theories on a circle, in a non-trivial instanton background. The Wilson loops are codimension 4 defects of the gauge theory, and their interaction with self-dual instantons is captured by a modified 1d ADHM quantum mechanics. We compute the partition function as its Witten index. This index is a "$qq$-character" of a finite-dimensional irreducible representation of the quantum affine algebra $U_q(\hat{\mathfrak{g}})$. Using gauge/vortex duality, we can understand the 5d physics in 3d gauge theory terms. Namely, we reinterpret the 5d theory with vortex flux from the point of view of the vortices themselves. This vortex perspective has an advantage: it has yet another dual description in terms of deformed $\mathfrak{g}$-type Toda Theory on a cylinder, in free field formalism. We show that the gauge theory partition function is equal to a chiral correlator of the deformed Toda Theory, with stress tensor and higher spin operator insertions. We derive all the above results from type IIB string theory, compactified on a resolved $ADE$ singularity times a cylinder with punctures, with various branes wrapping the blown-up 2-cycles.

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arXiv2017

The ABCDEFG of Little Strings

Starting from type IIB string theory on an $ADE$ singularity, the $(2,0)$ little string arises when one takes the string coupling $g_s$ to 0. In this setup, we give a unified description of the codimension-two defects of the little string, labeled by a simple Lie algebra ${\mathfrak{g}}$. Geometrically, these are D5 branes wrapping 2-cycles of the singularity, subject to a certain folding operation when the algebra is non simply-laced. Equivalently, the defects are specified by a certain set of weights of $^L {\mathfrak{g}}$, the Langlands dual of ${\mathfrak{g}}$. As a first application, we show that the instanton partition function of the ${\mathfrak{g}}$-type quiver gauge theory on the defect is equal to a 3-point conformal block of the ${\mathfrak{g}}$-type deformed Toda theory in the Coulomb gas formalism. As a second application, we argue that in the $(2,0)$ CFT limit, the Coulomb branch of the defects flows to a nilpotent orbit of ${\mathfrak{g}}$.

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arXiv2016

Little String Defects and Bala-Carter Theory

We give a physical realization of the Bala-Carter labels that classify nilpotent orbits of semi-simple Lie algebras, for the case $\mathfrak{g}=A,D,E$. We start from type IIB string theory compactified on an $ADE$ singularity and study the six-dimensional (2,0) $\mathfrak{g}$-type little string on a Riemann surface with punctures. The defects are introduced as D-branes wrapping the 2-cycles of the singularity. At low energies, the little string becomes the (2,0) conformal field theory of type $\mathfrak{g}$. As an application, we derive the full list of $E_n$ little string defects, and their Bala-Carter label in the CFT limit. Furthermore, we investigate new relations between the quiver gauge theory describing the D-brane defects at low energies, and the weighted Dynkin diagrams of $\mathfrak{g}$. We also give a physical version of the dimension formula of a nilpotent orbit based on its weighted Dynkin diagram.

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arXiv2016

Little String Origin of Surface Defects

We derive the codimension-two defects of 4d $\mathcal{N} = 4$ Super Yang-Mills (SYM) theory from the (2, 0) little string. The origin of the little string is type IIB theory compactified on an ADE singularity. The defects are D-branes wrapping the 2-cycles of the singularity. We use this construction to make contact with the description of SYM defects due to Gukov and Witten [arXiv:hep-th/0612073]. Furthermore, we derive from a geometric perspective the complete nilpotent orbit classification of codimension-two defects, and the connection to ADE-type Toda CFT. The only data needed to specify the defects is a set of weights of the algebra obeying certain constraints, which we give explicitly. We highlight the differences between the defect classification in the little string theory and its (2, 0) CFT limit.

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arXiv2015

ADE Little String Theory on a Riemann Surface (and Triality)

We initiate the study of (2,0) little string theory of ADE type using its definition in terms of IIB string compactified on an ADE singularity. As one application, we derive a 5d ADE quiver gauge theory that describes the little string compactified on a sphere with three full punctures, at low energies. As a second application, we show the partition function of this theory equals the 3-point conformal block of ADE Toda CFT, q-deformed. To establish this, we generalize the A_n triality of \cite{AHS} to all ADE Lie algebras; IIB string perspective is crucial for this as well.

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arXiv2014

$A_n$-Triality

A_n-type AGT correspondence anticipates that conformal blocks of A_n Toda CFT are related to partition functions of a family of 4d N=2 SCFTs. We use gauge/vortex duality to both give a precise form of the correspondence, and to prove it. Gauge/vortex duality relates the 4d theories and the 2d theories living on its vortices. Partition functions of the 2d theories on vortices provide Coulomb-gas representation of A_n Toda conformal blocks with discrete internal momenta. This gives a triality of relations between the gauge theory, its vortices and the Toda CFT. We prove that A_n triality holds for conformal blocks of A_n Toda on a sphere with all full punctures. The lift to one higher dimensional theories, compactified on a circle of arbitrary radius, and q-deformation of the Toda CFT, play a key role.

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arXiv2013

Gauge/Liouville Triality

Conformal blocks of Liouville theory have a Coulomb-gas representation as Dotsenko-Fateev (DF) integrals over the positions of screening charges. For q-deformed Liouville, the conformal blocks on a sphere with an arbitrary number of punctures are manifestly the same, when written in DF representation, as the partition functions of a class of 3d U(N) gauge theories with N=2 supersymmetry, in the Omega-background. Coupling the 3d gauge theory to a flavor in fundamental representation corresponds to inserting a Liouville vertex operator; the two real mass parameters determine the momentum and position of the puncture. The DF integrals can be computed by residues. The result is the instanton sum of a five dimensional N=1 gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.

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