Abstract Recently it was shown that mirror duals of 3d and 4d theories with four supercharges can be described by generalized quiver theories, constructed using strongly coupled SCFTs as elementary building blocks that replace and improve standard bifundamentals. In this work we study and extend the family of such improved bifundamentals and discuss the network of star-triangle dualities they satisfy. We provide a field theoretic proof of the star-triangle dualities, which only assumes the basic Seiberg dualities, using the sequential deconfinement technique.

Abstract Aimed at complex geometers and representation theorists, this survey explores higher dimensional analogs of the rich interplay between Riemann surfaces, Virasoro and Kac‐Moody Lie algebras, and conformal blocks. We introduce a panoply of examples from physics — field theories that are holomorphic in nature, such as holomorphic Chern‐Simons theory — and interpret them as (derived) moduli spaces in complex geometry; no comfort with physics is presumed. We then describe frameworks for quantizing such moduli spaces, offering a systematic generalization of vertex algebras and conformal blocks via factorization algebras, and we explain how holomorphic field theories generate examples of these higher algebraic structures. We finish by describing how the conjecture of Seiberg duality predicts a surprising relationship between holomorphic gauge theories on algebraic surfaces and how it suggests analogs of the Hori–Tong dualities already studied by algebraic geometers.

Abstract This work proves that the Seiberg Duality Conjecture holds for star-shaped quivers: the Gromov–Witten theories of mutation-related varieties are equivalent. In particular, it is known that there are only finitely many quivers that are mutation equivalent to a $D$-type quiver. We prove that the Seiberg Duality Conjecture holds for all quivers that are mutation equivalent to a $D_{3}$-type quiver, and find the change of Kähler variables.

We show that near the edges of the conformal window of supersymmetric SU(Nc) QCD, perturbed by Anomaly Mediated Supersymmetry Breaking (AMSB), chiral symmetry can be broken depending on the initial conditions of the RG flow. We do so by perturbatively expanding around Banks-Zaks fixed points and taking advantage of Seiberg duality. Interpolating between the edges of the conformal window, we predict that non-supersymmetric QCD breaks chiral symmetry up to Nf ≤ 3Nc − 1, while we cannot say anything definitive for Nf ≥ 3Nc at this moment.

We introduce a generative AI model to obtain type IIB brane configurations that realize toric phases of a family of 4D N=1 supersymmetric gauge theories. These 4D N=1 quiver gauge theories are world volume theories of a D3-brane probing a toric Calabi-Yau 3-fold. The type IIB brane configurations are given by the coamoeba projection of the mirror curve associated with the toric Calabi-Yau 3-fold. The shape of the mirror curve and its coamoeba projection, as well as the corresponding type IIB brane configuration and the toric phase of the 4d N=1 theory, all depend on the complex structure moduli parametrizing the mirror curve. We train a generative AI model, a conditional variational autoencoder (CVAE), that takes a choice of complex structure moduli as input and generates the corresponding coamoeba. This enables us not only to obtain a high-resolution representation of the entire phase space for a family of 4D N=1 theories corresponding to the same toric Calabi-Yau 3-fold, but also to continuously track the movements of the mirror curve and the branes wrapping the curve in the corresponding type IIB brane configurations during phase transitions associated with Seiberg duality.

We analyze symmetries corresponding to separated topological sectors of 3d 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 3-dimensional Seiberg duality.

In this paper, we compute the NSVZ beta functions for [Formula: see text] four-dimensional quiver theories arising from D-brane probes on singularities, complete with anomalous dimensions, for a large set of phases in the corresponding duality tree. While these beta functions are zero for D-brane probes, they are nonzero in the presence of fractional branes. As a result there is a nontrivial RG behavior. We apply this running of gauge couplings to some toric singularities such as the cones over Hirzebruch and del Pezzo surfaces. We observe the emergence in string theory, of “Duality Walls,” a finite energy scale at which the number of degrees of freedom becomes infinite and beyond which Seiberg duality does not proceed. We also identify certain quiver symmetries as T-duality-like actions in the dual holographic theory.

We consider 5d KK dualities, that is multiple 5d gauge theories with the same 6d infinite coupling limit. We provide a prescription to associate 4dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 quivers to the 5d dual quivers, such that the 4d quivers are also dual to each other. The 4d dualities are infrared dualities which can be checked matching global symmetry anomalies and in certain cases can be proven using basic Seiberg dualities sequentially. We also consider dualities obtained by Higgsing in two different ways the same 5d theory, in some simple examples.

We study the class of AdS3 × ℂℙ3 solutions to massive Type IIA supergravity with osp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{osp} $$\\end{document}(6|2) superconformal algebra recently constructed in [1]. These solutions are foliations over an interval preserving N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = (0, 6) supersymmetry in two dimensions, that in the massless limit can be mapped to the AdS4 × ℂℙ3 solution of ABJM/ABJ. We show that in the massive case extra NS5-D8 branes, that we interpret as 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}-BPS surface defect branes within the ABJ theory, backreact in the geometry and turn one of the 3d field theory directions onto an energy scale, generating a flow towards a 2d CFT. We construct explicit quiver field theories that we propose flow in the IR to the (0, 6) SCFTs dual to the solutions. Finally, we show that the AdS3 solutions realise geometrically, in terms of large gauge transformations, an extension to the massive case of Seiberg duality in ABJ theories proposed in the literature.

The classical theory of Fuchsian differential equations is largely equivalent to the theory of Seiberg dualities for quiver SUSY gauge theories. In particular: all known integral representations of solutions, and their connection formulae, are immediate consequences of (analytically continued) Seiberg duality in view of the dictionary between linear ODEs and gauge theories with 4 supersymmetries. The purpose of this divertissement is to explain "physically" this remarkable relation in the spirit of Physical Mathematics. The connection goes through a "mirror-theoretic" identification of irreducible logarithmic connections on P 1 with would-be BPS dyons of 4d N = 2 SU(2) SYM coupled to a certain Argyres-Douglas "matter". When the underlying bundle is trivial, i.e. the log-connection is a Fuchs system, the world-line theory of the dyon simplifies and the action of Seiberg duality on the Fuchsian ODEs becomes quite explicit. The duality action is best described in terms of Representation Theory of Kac-Moody Lie algebras (and their affinizations).

We introduce unsupervised machine learning techniques in order to identify toric phases of 4d N ¼ 1 supersymmetric gauge theories corresponding to the same toric Calabi-Yau 3-fold. These 4d N ¼ 1 supersymmetric gauge theories are world volume theories of a D3-brane probing a toric Calabi-Yau 3-fold and are realized in terms of a type IIB brane configuration known as a brane tiling. It corresponds to the skeleton graph of the coamoeba projection of the mirror curve associated to the toric Calabi-Yau 3-fold. When we vary the complex structure moduli of the mirror Calabi-Yau 3-fold, the coamoeba and the corresponding brane tilings change their shape, giving rise to different toric phases related by Seiberg duality. We illustrate that by employing techniques such as principal component analysis and t-distributed stochastic neighbor embedding, we can project the space of coamoeba labeled by complex structure moduli down to a lower-dimensional phase space with phase boundaries corresponding to Seiberg duality. In this work, we illustrate this technique by obtaining a 2-dimensional phase diagram for brane tilings corresponding to the cone over the zeroth Hirzebruch surface F 0 .

We consider the quiver Yangians associated to general affine Dynkin diagrams. Although the quivers are generically not toric, the algebras have some similar structures. The odd reflections of the affine Dynkin diagrams should correspond to Seiberg duality of the quivers, and we investigate the relations of the dual quiver Yangians. We also mention the construction of the twisted quiver Yangians. It is conjectured that the truncations of the (twisted) quiver Yangians can give rise to certain \U0001d4b2-algebras. Incidentally, we give the screening currents of the \U0001d4b2-algebras in terms of the free field realization in the case of generalized conifolds. Moreover, we discuss the toroidal and elliptic algebras for any general quivers.

We focus on quiver Yangians for most generalized conifolds. We construct a coproduct of the quiver Yangian following the similar approach by Guay–Nakajima–Wendlandt. We also prove that the quiver Yangians related by Seiberg duality are indeed isomorphic. Then we discuss their connections to -algebras analogous to the study by Ueda. In particular, the universal enveloping algebras of the -algebras are truncations of the quiver Yangians, and therefore they naturally have truncated crystals as their representations.

We apply the technique of sequential deconfinement to the four dimensional mathcal{N} = 1 U sp(2N) gauge theory with an antisymmetric field and 2F fundamentals. The fully deconfined frame is a length-N quiver. We use this deconfined frame to prove the known self-duality of U sp(2N) with an antisymmetric field and 8 fundamentals. Along the way we encounter a subtlety: in certain quivers with degenerate holomorphic operators, a naive application of Seiberg duality rules leads to an incorrect superpotential or chiral ring.We also consider the reduction to 3d mathcal{N} = 2 theories, recovering known fully deconfined duals of U sp(2N) and U(N) gauge theories, and obtaining new ones.

We demonstrate that interacting ultraviolet fixed points in four dimensions exist at strong coupling, and away from large-$N$ Veneziano limits. This is established exemplarily for semi-simple supersymmetric gauge theories with chiral matter and superpotential interactions by using the renormalisation group and exact methods from supersymmetry. We determine the entire superconformal window of ultraviolet fixed points as a function of field multiplicities. Results are in accord with the $a$-theorem, bounds on conformal charges, Seiberg duality, and unitary. We also find manifolds of Leigh-Strassler models exhibiting lines of infrared fixed points. At weak coupling, findings are confirmed using perturbation theory up to three loop. Benchmark models with low field multiplicities are provided including examples with Standard~Model-like gauge sectors. Implications for particle physics, model building, and conformal field theory are indicated.

Adopting the Mahler measure from number theory, we introduce it to toric quiver gauge theories, and study some of its salient features and physical implications. We propose that the Mahler measure is a universal measure for the quiver, encoding its dynamics with the monotonic behaviour along a so-called Mahler flow including two special points at isoradial and tropical limits. Along the flow, the amoeba, from tropical geometry, provides geometric interpretations for the dynamics of the quiver. In the isoradial limit, the maximization of Mahler measure is shown to be equivalent to a-maximization. The Mahler measure and its derivative are closely related to the master space, leading to the property that the specular duals have the same functions as coefficients in their expansions, hinting the emergence of a free theory in the tropical limit. Moreover, they indicate the existence of phase transition. We also find that the Mahler measure should be invariant under Seiberg duality.

We consider the matrix model of U(N) refined Chern–Simons theory on S^3 for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q-difference operator for a similar refinement of ABJ theory and show that the corresponding q-Virasoro constraints are equal to those of refined Chern–Simons for a gauge super-group U(N|M). Our equations and solutions are manifestly symmetric under Langlands duality qleftrightarrow t^{-1} which correctly reproduces 3d Seiberg duality when q is a specific root of unity.

We further explore a recent proposal that the vector mesons in QCD have a special role as Chern-Simons fields on various QCD objects such as domain walls and the one flavored baryons. We compute contributions to domain wall theories and to the baryon current coming from a generalized Wess-Zumino term including vector mesons. The conditions that lead to the expected Chern-Simons terms and the correct spectrum of baryons, coincide with the conditions for vector meson dominance. This observation provides a theoretical explanation to the phenomenological principle of vector dominance, as well as an experimental evidence for the identification of vector mesons as the Chern-Simons fields. By deriving the Chern-Simons theories directly from an action, we obtain new results about QCD domain walls. One conclusion is the existence of a first order phase transition between domain walls as a function of the quarks' masses. We also discuss applications of our results to Seiberg duality between gluons and vector mesons and provide new evidence supporting the duality.

We compute the partition functions of [Formula: see text] gauge theories on [Formula: see text] using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of [Formula: see text] and at the origin of [Formula: see text]. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the [Formula: see text] partition functions on the [Formula: see text]-background, and show that the Nekrasov partition functions can be recovered from these building blocks.

Considered is the direct $$\mathcal{N}=1$$ SQCD (i.e. supersymmetric QCD)-like $$\Phi$$ -theory with $$SU(N_{c})$$ colors and $$3N_{c}/2<N_{F}<2N_{c}$$ flavors of light quarks $${\overline{Q}}^{b}_{j}$$ , $$Q^{i}_{a}$$ , $$a,b=1...N_{c}$$ , $$i,j=1...N_{F}$$ , with small mass parameter $$0<m_{Q}\ll\Lambda_{Q}$$ in the superpotential. Besides, it includes $$N^{2}_{F}$$ additional colorless but flavored fields $$\Phi_{i}^{j}$$ , with the large mass parameter $$\mu_{\Phi}\gg\Lambda_{Q}$$ , interacting with quarks through the Yukawa coupling in the superpotential. In parallel, is considered its Seiberg’s dual variant, i.e. the $$d\Phi$$ -theory with $${\overline{N}}_{c}=(N_{F}-N_{c})$$ dual colors and $$3N_{c}/2<N_{F}<2N_{c}$$ flavors of dual quarks $${\overline{q}}_{d}^{j}$$ , $${q}_{i}^{c}$$ , $$c,d=1...{\overline{N}}_{c}$$ . The multiplicities of various vacua and values of the quark and gluino condensates in all vacua are found. It is shown that in considered vacua of both the direct and dual theories the quarks are in the conformal regimes at scales $$\mu<\Lambda_{Q}$$ . The dynamics of these regimes is sufficiently simple and well understood, so that no additional dynamical assumptions were needed to calculate the mass spectra in Sections 4 and 5. It is shown that mass spectra of the direct $$\Phi$$ and dual $$d\Phi$$ -theories are different, in disagreement with the Seiberg hypothesis about equivalence of two such theories. Besides it is shown in the direct $$\Phi$$ -theory that a qualitatively new phenomenon takes place: the seemingly heavy and dynamically irrelevant fields $$\Phi$$ ‘return back’ and there appear two additional generations of light $$\Phi$$ -particles with small masses $$\mu^{\textrm{pole}}_{2,3}(\Phi)\ll\Lambda_{Q}$$ .