5d Gauge Theories Research Articles

Basic curvature & the Atiyah cocycle in gauge theory

Abstract Motivated from target space covariant formulations of topological sigma models and from a graded-geometric approach to higher gauge theory, we study connections on Lie and Courant algebroids and on their description as differential graded (dg) manifolds. We revisit the notion of basic curvature for a connection on a Lie algebroid, which measures its compatibility with the Lie bracket and it appears in the BV-BRST differential of 2D gauge theories such as (twisted) Poisson and Dirac sigma models. We define a basic curvature tensor for connections on Courant algebroids and we show that in the description of a Courant algebroid as a QP2 manifold it appears naturally as part of the homological vector field together with the Gualtieri torsion of an induced generalised connection. Furthermore, we consider connections on dg manifolds and revisit the structure of gauge transformations in the graded-geometric approach to higher gauge theories from a manifestly covariant perspective. We argue that it is governed by the brackets of a Kapranov L ∞ [ 1 ] algebra, whose binary bracket is given by the Atiyah cocycle that measures the compatibility of the connection with the homological vector field. We also revisit some aspects of the derived bracket construction and show how to extend it to connections and tensors.

Large AdS6 black holes from CFT5

We study supersymmetric AdS6 black holes at large angular momenta, from the index of 5d SCFTs on S4 × ℝ in the large N and Cardy limit. Our examples are the strong coupling limits of 5d gauge theories on the D4-D8-O8 system. The large N free energy scales like N5/2, statistically accounting for the entropy of large black holes in AdS6. Instanton solitons play subtle roles to realize these deconfined degrees of freedom.

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 dualities from 5d dualities

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.

Exact lattice chiral symmetry in 2D gauge theory

We construct symmetry-preserving lattice regularizations of 2D QED with one and two flavors of Dirac fermions, as well as the “3450” chiral gauge theory, by leveraging bosonization and recently proposed modifications of Villain-type lattice actions. The internal global symmetries act just as locally on the lattice as they do in the continuum, the anomalies are reproduced at finite lattice spacing, and in each case we find a sign-problem-free dual formulation. Published by the American Physical Society 2024

More holographic M5 branes in AdS7 × S4

We study classical M5 brane solutions in the probe limit in the AdS7×S4 spacetime geometry with worldvolume 3-form flux. These solutions describe the holography of codimension-4 defects in the 6d boundary dual N=(0,2) supersymmetric gauge theories. Starting from some half-BPS solutions which follow stricter BPS conditions, we find a general 116-BPS solution. We show how some of the giant-like M5 brane solutions here may be related to the codimension-2 surface operators in the 4d gauge theory.

Anomalies of generalized symmetries from solitonic defects

We propose the general idea that ’t Hooft anomalies of generalized global symmetries can be understood in terms of the properties of solitonic defects, which generically are non-topological defects. The defining property of such defects is that they act as sources for background fields of generalized symmetries. ’t Hooft anomalies arise when solitonic defects are charged under these generalized symmetries. We illustrate this idea for several kinds of anomalies in various spacetime dimensions. A systematic exploration is performed in 3d for 0-form, 1-form, and 2-group symmetries, whose ’t Hooft anomalies are related to two special types of solitonic defects, namely vortex line defects and monopole operators. This analysis is supplemented with detailed computations of such anomalies in a large class of 3d gauge theories. Central to this computation is the determination of the gauge and 0-form charges of a variety of monopole operators: these involve standard gauge monopole operators, but also fractional gauge monopole operators, as well as monopole operators for 0-form symmetries. The charges of these monopole operators mainly receive contributions from Chern-Simons terms and fermions in the matter content. Along the way, we interpret the vanishing of the global gauge and ABJ anomalies, which are anomalies not captured by local anomaly polynomials, as the requirement that gauge monopole operators and mixed monopole operators for 0-form and gauge symmetries have non-fractional integer charges.

Anomaly enforced gaplessness and symmetry fractionalization for SpinG symmetries

Symmetries and their anomalies give strong constraints on renormalization group (RG) flows of quantum field theories. Recently, the identification of a theory’s global symmetries with its topological sector has provided additional constraints on RG flows to symmetry preserving gapped phases due to mathematical results in category and topological quantum field theory. In this paper, we derive constraints on RG flows from ℤ2-valued pure- and mixed-gravitational anomalies that can only be activated on non-spin manifolds. We show that such anomalies cannot be matched by a unitary, symmetry preserving gapped phase without symmetry fractionalization. In particular, we discuss examples that commonly arise in 4d gauge theories with fermions.

Saw varieties

We introduce a new class of quiver varieties, called saw (quiver) varieties. These can be seen as a generalization of the three existing theories: the quiver gauge theory of monopoles, Kac's representation theory of Dynkin types, and Donaldson's theory on the based maps of P1. E.g., in our terminology, the handsaw (quiver) varieties and chainsaw varieties are finite and affine A type saw varieties respectively with the maximal symmetry breaking. These varieties are the monopole spaces and the caloron spaces in 3d gauge theory, respectively. We expand the theory over to the DE type quiver gauge theory.The saw varieties are constructed via a Hamiltonian formalism, so that the moment maps defining these varieties naturally arise. The main assertion of the paper is that the moment map μ defining a saw variety is a flat morphism for arbitrary dimension vector, if and only if the underlying graph is of finite ADE types. Furthermore under the assumption of maximal symmetry breaking or all the higher frame dimension vectors, the affine ADE types also attain the flatness property of μ. We give an algebro-geometric description of the momentum-zero scheme μ−1(0) and the saw variety defined as the Hamiltonian reduction.Various partition functions given rise to by the above mentioned theories are the byproducts of the geometric structure of the corresponding saw varieties.

Remarks on QCD4 with fundamental and adjoint matter

We study 4-dimensional SU(N) gauge theory with one adjoint Weyl fermion and fundamental matter — either bosonic or fermionic. Symmetries, their ’t Hooft anomalies, and the Vafa-Witten-Weingarten theorems strongly constrain the possible bulk phases. The first part of the paper is dedicated to a single fundamental fermion or boson. As long as the adjoint Weyl fermion is massless, this theory always possesses a ℤ2Nχ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{Z}}_{2N}^{\\chi } $$\\end{document} chiral symmetry, which breaks spontaneously, supporting N vacua and domain walls between them for a generic mass of the matter fields. We argue, however, that the domain walls generically undergo a phase transition, and we establish the corresponding 3d gauge theories on the walls. The rest of the paper is dedicated to studying the multi-flavor fundamental matter. Here, the phases crucially depend on the ratio of the number of colors and the number of fundamental flavors. We also discuss the limiting scenarios of heavy adjoint and fundamentals, which align neatly with our current understanding of QCD and 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} = 1 super Yang-Mills theory.

The Schrödinger representation and 3D gauge theories

In this paper, we consider the Hamiltonian analysis of Yang–Mills theory and some variants of it in three space–time dimensions using the Schrödinger representation. This representation, although technically more involved than the usual covariant formulation, may be better suited for some nonperturbative issues. Specifically for the Yang–Mills theory, we explain how to set up the Hamiltonian formulation in terms of manifestly gauge-invariant variables and set up an expansion scheme for solving the Schrödinger equation. We review the calculation of the string tension, the Casimir energy and the propagator mass and compare with the results from lattice simulations. The computation of the first set of corrections to the string tension, string breaking effects, extensions to the Yang–Mills–Chern–Simons theory and to the supersymmetric cases are also discussed. We also comment on how entanglement for the vacuum state can be formulated in terms of the BFK gluing formula. This paper concludes with a discussion of the status and prospects of this approach.

(2,0) theory on S5 × S1 and quantum M2 branes

The superconformal index Z of the 6d (2,0) theory on S5×S1 (which is related to the localization partition function of 5d SYM on S5) should be captured at large N by the quantum M2 brane theory in the dual M-theory background. Generalizing the type IIA string theory limit of this relation discussed in arXiv:2111.15493 and arXiv:2304.12340, we consider semiclassically quantized M2 branes in a half-supersymmetric 11d background which is a twisted product of thermal AdS7 and S4. We show that the leading non-perturbative term at large N is reproduced precisely by the 1-loop partition function of an “instanton” M2 brane wrapped on S1×S2 with S2⊂S4. Similarly, the (2,0) theory analog of the BPS Wilson loop expectation value is reproduced by the partition function of a “defect” M2 brane wrapped on thermal AdS⊂3 AdS7. We comment on a curious analogy of these results with similar computations in arXiv:2303.15207 and arXiv:2307.14112 of the partition function of quantum M2 branes in AdS×4S7/Zk which reproduced the corresponding localization expressions in the ABJM 3d gauge theory.

Mass deformations of brane brick models

We investigate a class of mass deformations that connect pairs of 2d (0, 2) gauge theories associated to different toric Calabi-Yau 4-folds. These deformations are generalizations to 2d of the well-known Klebanov-Witten deformation relating the 4d gauge theories for the ℂ2/ℤ2 × ℂ orbifold and the conifold. We investigate various aspects of these deformations, including their connection to brane brick models and the relation between the change in the geometry and the pattern of symmetry breaking triggered by the deformation. We also explore how the volume of the Sasaki-Einstein 7-manifold at the base of the Calabi-Yau 4-fold varies under deformation, which leads us to conjecture that it quantifies the number of degrees of freedom of the gauge theory and its dependence on the RG scale.

Study of gapped phases of 4d gauge theories using temporal gauging of the ℤN 1-form symmetry

To study gapped phases of 4d gauge theories, we introduce the temporal gauging of ℤN 1-form symmetry in 4d quantum field theories (QFTs), thereby defining effective 3d QFTs with overset{sim }{mathbb{Z}} N × ℤN 1-form symmetry. In this way, spatial fundamental Wilson and ’t Hooft loops are simultaneously genuine line operators. Assuming a mass gap and Lorentz invariant vacuum of the 4d QFT, the overset{sim }{mathbb{Z}} N × ℤN symmetry must be spontaneously broken to an order-N subgroup H, and we can classify the 4d gapped phases by specifying H. This establishes the 1-to-1 correspondence between the two classification schemes for gapped phases of 4d gauge theories: one is the conventional Wilson-’t Hooft classification, and the other is the modern classification using the spontaneous breaking of 4d 1-form symmetry enriched with symmetry-protected topological states.

Bethe/Gauge correspondence for ABCDEFG-type 3d gauge theories

In this paper, we give a new effective superpotential that makes clear Bethe/Gauge correspondence between 2d (and 3d) SO/Sp gauge theories and open XXX (and XXZ) spin chains with diagonal boundary conditions, and also works in the case of 2d (and 3d) BCN-type gauge theories which is not previously discussed in the literature. Especially, for exceptional Lie algebras F4, G2, we give the effective superpotential and vacuum equations. For E6,7,8, we only give theirs effective superpotential for convenience.

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.

Non-invertible higher-categorical symmetries

We sketch a procedure to capture general non-invertible symmetries of a dd-dimensional quantum field theory in the data of a higher-category, which captures the local properties of topological defects associated to the symmetries. We also discuss fusions of topological defects, which involve condensations/gaugings of higher-categorical symmetries localized on the worldvolumes of topological defects. Recently some fusions of topological defects were discussed in the literature where the dimension of topological defects seems to jump under fusion. This is not possible in the standard description of higher-categories. We explain that the dimension-changing fusions are understood as higher-morphisms of the higher-category describing the symmetry. We also discuss how a 0-form sub-symmetry of a higher-categorical symmetry can be gauged and describe the higher-categorical symmetry of the theory obtained after gauging. This provides a procedure for constructing non-invertible higher-categorical symmetries starting from invertible higher-form or higher-group symmetries and gauging a 0-form symmetry. We illustrate this procedure by constructing non-invertible 2-categorical symmetries in 4d gauge theories and non-invertible 3-categorical symmetries in 5d and 6d theories. We check some of the results obtained using our approach against the results obtained using a recently proposed approach based on ’t Hooft anomalies.

Local observables in SUq(2) lattice gauge theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\mathrm{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities that live on the vertices of the lattice and are labeled by pairs of incident edges. Any function on the classical phase space, e.g., Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$ matrix, which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$ deformation of ${\mathfrak{so}}^{*}(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.

On continuous 2-category symmetries and Yang-Mills theory

We study a 4d gauge theory U(1)N−1 ⋊ SN obtained from a U(1)N−1 theory by gauging a 0-form symmetry SN. We show that this theory has a global continuous 2-category symmetry, whose structure is particularly rich for N > 2. This example allows us to draw a connection between the higher gauging procedure and the difference between local and global fusion, which turns out to be a key feature of higher categorical symmetries. By studying the spectrum of local and extended operators, we find a mapping with gauge invariant operators of 4d SU(N) Yang-Mills theory. The largest group-like subcategory of the non-invertible symmetries of our theory is a {mathbb{Z}}_N^{(1)} 1-form symmetry, acting on the Wilson lines in the same way as the center symmetry of Yang-Mills theory does. Supported by a path-integral argument, we propose that the U(1)N−1 ⋊ SN gauge theory has a relation with the ultraviolet limit of SU(N) Yang-Mills theory in which all Gukov-Witten operators become topological, and form a continuous non-invertible 2-category symmetry, broken down to the center symmetry by the RG flow.

0-form, 1-form, and 2-group symmetries via cutting and gluing of orbifolds

Orbifold singularities of M-theory constitute the building blocks of a broad class of supersymmetric quantum field theories (SQFTs). In this paper we show how the local data of these geometries determines global data on the resulting higher symmetries of these systems. In particular, via a process of cutting and gluing, we show how local orbifold singularities encode the 0-form, 1-form and 2-group symmetries of the resulting SQFTs. Geometrically, this is obtained from the possible singularities which extend to the boundary of the non-compact geometry. The resulting category of boundary conditions then captures these symmetries, and is equivalently specified by the orbifold homology of the boundary geometry. We illustrate these general points in the context of a number of examples, including 5D superconformal field theories engineered via orbifold singularities, 5D gauge theories engineered via singular elliptically fibered Calabi-Yau threefolds, as well as 4D SQCD-like theories engineered via M-theory on non-compact $G_2$ spaces.

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Important illustration to the principle “partition functions in string theory are τ-functions of integrable equations” is the fact that the (dual) partition functions of 4d mathcal{N} = 2 gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve q-difference equations of non-autonomous dynamics of the “cluster-algebraic”integrable systems.We explain in details the “solutions” side of the proposal. In the simplest non-trivial example we show how 3d box-counting of topological string partition function appears from the counting of dimers on bipartite graph with the discrete gauge field of “flux” q. This is a new form of topological string/spectral theory type correspondence, since the partition function of dimers can be computed as determinant of the linear q-difference Kasteleyn operator. Using WKB method in the “melting” q → 1 limit we get a closed integral formula for Seiberg-Witten prepotential of the corresponding 5d gauge theory. The “equations” side of the correspondence remains the intriguing topic for the further studies.