5d Gauge Theories Research Articles

A 3d gauge theory/quantum K-theory correspondence

The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kahler manifold X, which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the N=2 supersymmetric 3d gauge theory and the IR limit is given by Givental's permutation equivariant quantum K-theory on X. This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.

Antiparticles as Particles: QED_{2+1} and Composite Fermions at ν=1/2.

We present a continuum operator map that inverts the role of particles and antiparticles in three-dimensional QED. This is accomplished by the attachment of specific holomorphic Wilson lines to Dirac fermions. We show that this nonlocal map provides a continuum realization of Son's composite fermions at ν=1/2. The inversion of Landau levels and the Dirac-cone-composite-fermion duality is explicitly demonstrated for the case of slowly varying magnetic fields. The role of Maxwell terms as well as the connection of this construction to a gauge-invariant formulation of 3D gauge theories is also elaborated upon.

An exact quantization of Jackiw-Teitelboim gravity

We propose an exact quantization of two-dimensional Jackiw-Teitelboim (JT) gravity by formulating the JT gravity theory as a 2D gauge theory placed in the presence of a loop defect. The gauge group is a certain central extension of PSL(2, ℝ) by ℝ. We find that the exact partition function of our theory when placed on a Euclidean disk matches precisely the finite temperature partition function of the Schwarzian theory. We show that observables on both sides are also precisely matched: correlation functions of boundary- anchored Wilson lines in the bulk are given by those of bi-local operators in the Schwarzian theory. In the gravitational context, the Wilson lines are shown to be equivalent to the world-lines of massive particles in the metric formulation of JT gravity.

Dual infrared limits of 6d [formula omitted] theory

Compactifying type AN−1 6d N=(2,0) supersymmetric CFT on a product manifold M4×Σ2=M3×S˜1×S1×I either over S1 or over S˜1 leads to maximally supersymmetric 5d gauge theories on M4×I or on M3×Σ2, respectively. Choosing the radii of S1 and S˜1 inversely proportional to each other, these 5d gauge theories are dual to one another since their coupling constants e2 and e˜2 are proportional to those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU(N) Yang–Mills theories on M4×I and on M3×Σ2, where M4⊃M3=Rt×Tp2 with time t and a punctured 2-torus, and I⊂Σ2 is an interval. In the first case, shrinking I to a point reduces to Yang–Mills theory or to the Skyrme model on M4, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on M3 and employing the adiabatic method, we derive in the infrared limit a non-linear SU(N) sigma model with a baby-Skyrme-type term on Σ2, which can be reduced further to AN−1 Toda theory.

6d SCFTs, 5d dualities and Tao web diagrams

We propose 5d descriptions of 6d mathcal{N}=left(1, 0right) superconformal field theories arising from Type IIA brane configurations with an O8−-plane. We T-dualize the brane diagram along a compactification circle and obtain a 5-brane web diagram with two O7−- planes. The gauge theory description of the resulting 5d theory for a given 6d superconformal field theory is not unique, and we argue that the non-uniqueness leads to various dual 5d gauge theories. There are three sources which lead to the 5d dualities. One type comes from either resolving both or one of the two O7−-planes. The two situations give us two different ways to read off a 5d gauge theory from essentially the same web diagram. The second type originates from different distributions of D5 or D7-branes, shifting the gauge group ranks of the 5d quiver theory. The last one comes from the 90 or 45 degree rotations of the 5-brane web diagram, which is a part of the SL(2, ℤ) duality of Type IIB string theory, leading to completely different group structure. These lead to a very rich class of dualities between 5d gauge theories whose UV completion is the same 6d superconformal field theory. We also explore Higgsing of the 6d theories and their 5d counterparts. Decoupling the same flavors from the dual 5d theories gives rise to another dual 5d theories whose UV completion is the same 5d superconformal field theory. Finally we propose the 6d description of 5d theories which is obtained by a generalization of 5d TN theories with additional flavors, which turns out not to be in the class of Type IIA brane construction generically.

Quantum mirror curve of periodic chain geometry

The mirror curves enable us to study B-model topological strings on noncompact toric Calabi-Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with a single brane. In this paper, we discuss two types of geometries: one is the chain of N ℙ1’s which we call “N-chain geometry,” the other is the chain of N ℙ1’s with a compactification which we call “periodic N-chain geometry.” We calculate the partition functions of the open topological strings on these geometries, and obtain the mirror curves and their quantization, which is characterized by (elliptic) hypergeometric difference operator. We also find a relation between the periodic chain and ∞-chain geometries, which implies a possible connection between 5d and 6d gauge theories in the larte N limit.

Comments on one-form global symmetries and their gauging in 3d and 4d

We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on \mathbb{Z}_NℤN one-form symmetries. A 3d topological quantum field theory (TQFT) \mathcal{T}𝒯 with such a symmetry has NN special lines that generate it. The braiding of these lines and their spins are characterized by a single integer pp modulo 2N2N. Surprisingly, if \gcd(N,p)=1gcd(N,p)=1 the TQFT factorizes \mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}𝒯=𝒯′⊗𝒜N,p. Here \mathcal{T}'𝒯′ is a decoupled TQFT, whose lines are neutral under the global symmetry and \mathcal{A}^{N,p}𝒜N,p is a minimal TQFT with the \mathbb{Z}_NℤN one-form symmetry of label pp. The parameter pp labels the obstruction to gauging the \mathbb{Z}_NℤN one-form symmetry; i.e. it characterizes the ’t Hooft anomaly of the global symmetry. When p=0p=0 mod 2N2N, the symmetry can be gauged. Otherwise, it cannot be gauged unless we couple the system to a 4d bulk with gauge fields extended to the bulk. This understanding allows us to consider SU(N)SU(N) and PSU(N)PSU(N) 4d gauge theories. Their dynamics is gapped and it is associated with confinement and oblique confinement – probe quarks are confined. In the PSU(N)PSU(N) theory the low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a space-dependent \thetaθ-parameter, which leads to interfaces. Typically, the theory on the interface is not confining. Furthermore, the liberated probe quarks are anyons on the interface. The PSU(N)PSU(N) theory is obtained by gauging the \mathbb{Z}_NℤN one-form symmetry of the SU(N)SU(N) theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the PSU(N)PSU(N) theory.

IR duality in 2D N=(0,2) gauge theory with noncompact dynamics

Searching for the simplest non-abelian 2d gauge theory with $\mathcal{N}=(0,2)$ supersymmetry and non-trivial IR physics, we propose a new duality for $SU(2)$ SQCD with $N_f = 4$ chiral flavors. The chiral algebra of this theory is found to be $\mathfrak{so}(8)_{-2}$, the same as in 4d $\mathcal{N}=2$ $SU(2)$ gauge theory with four hypermultiplets.

Fiber-base duality from the algebraic perspective

Quiver 5D mathcal{N}=1 gauge theories describe the low-energy dynamics on webs of (p, q)-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping (p, q)-branes to branes of charge (−q, p), and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the Ding-Iohara-Miki (DIM) algebra. Specifically, the instanton partition function is obtained as the vacuum expectation value of an operator mathcal{T} constructed by gluing the algebra’s intertwiners (the equivalent of topological vertices) following the rules of the toric diagram/brane web. Intertwiners and mathcal{T} -operators are deeply connected to the co-algebraic structure of the DIM algebra. We show here that S-duality can be realized as the twist of this structure by Miki’s automorphism.

A tale of exceptional 3d dualities

We consider interesting Seiberg dualities for Usp gauge theories with an antisymmetric, 8 fundamentals and no superpotential. We reduce to three dimensions and systematically analyze deformations triggered by real and complex masses, reaching a plethora of mathcal{N} = 2 dualities for U(N) and Usp(2N) gauge theories, possibly with monopole superpotentials and Chern-Simons interactions. Special cases of these “exceptional dualities” are: supersymmetry enhancement dualities, “duality appetizers” and many known dualities relating rank-1 gauge groups. The 4d mathcal{N} = 1 Usp dualities provide a unified perspective on many curious phenomena of 3d and 4d gauge theories with four supercharges.Finally, we propose a free mirror for A2N Argyres-Douglas, with its related adjoint-Usp(2N) duality, and we construct a mirror for adjoint-U(N), with an arbitrary number flavors and zero superpotential.

Walking, weak first-order transitions, and complex CFTs

We discuss walking behavior in gauge theories and weak first-order phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling. We discuss what distinguishes a real theory from a complex theory and call these fixed points complex CFTs. By using conformal perturbation theory we show how observables of the walking theory are computable by perturbing the complex CFTs. This paper discusses the general mechanism while a companion paper [1] will treat a specific and computable example: the two-dimensional Q-state Potts model with Q > 4. Concerning walking in 4d gauge theories, we also comment on the (un)likelihood of the light pseudo-dilaton, and on non-minimal scenarios of the conformal window termination.

4D gauge theories with conformal matter

One of the hallmarks of 6D superconformal field theories (SCFTs) is that on a partial tensor branch, all known theories resemble quiver gauge theories with links comprised of 6D conformal matter, a generalization of weakly coupled hypermultiplets. In this paper we construct 4D quiverlike gauge theories in which the links are obtained from compactifications of 6D conformal matter on Riemann surfaces with flavor symmetry fluxes. This includes generalizations of super QCD with exceptional gauge groups and quarks replaced by 4D conformal matter. Just as in super QCD, we find evidence for a conformal window as well as confining gauge group factors depending on the total amount of matter. We also present F-theory realizations of these field theories via elliptically fibered Calabi-Yau fourfolds. Gauge groups (and flavor symmetries) come from 7-branes wrapped on surfaces, conformal matter localizes at the intersection of pairs of 7-branes, and Yukawas between 4D conformal matter localize at points coming from triple intersections of 7-branes. Quantum corrections can also modify the classical moduli space of the F-theory model, matching expectations from effective field theory.

Symmetry enhancement and closing of knots in 3d/3d correspondence

We revisit Dimofte-Gaiotto-Gukov’s construction of 3d gauge theories associated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d mathcal{N}=left(2,0right) theory of A1-type on a 3-manifold, we propose a topological criterion for SU(2)/SO(3) flavor symmetry enhancement for the u(1) symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the construction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising SU(3) symmetry enhancement which is unexpected from the 6d viewpoint.

Medium generated gap in gravity and a 3D gauge theory

It is well known that a physical medium that sets a Lorentz frame generates a Lorentz-breaking gap for a graviton. We examine such generated "mass" terms in the presence of a fluid medium whose ground state spontaneously breaks spatial translation invariance in $d = D+1$ spacetime dimensions, and for a solid in $D = 2$ spatial dimensions. By requiring energy positivity and subluminal propagation, certain constraints are placed on the equation of state of the medium. In the case of $D = 2$ spatial dimensions, classical gravity can be recast as a Chern-Simons gauge theory and motivated by this we recast the massive theory of gravity in AdS$_3$ as a massive Chern-Simons gauge theory with an unusual mass term. We find that in the flat space limit the Chern-Simons theory has a novel gauge invariance that mixes the kinetic and mass terms, and enables the massive theory with a non-compact internal group to be free of ghosts and tachyons.

Surface operators in 5d gauge theories and duality relations

We study half-BPS surface operators in 5d mathcal{N} = 1 gauge theories compactified on a circle. Using localization methods and the twisted chiral ring relations of coupled 3d/5d quiver gauge theories, we calculate the twisted chiral superpotential that governs the infrared properties of these surface operators. We make a detailed analysis of the localization integrand, and by comparing with the results from the twisted chiral ring equations, we obtain constraints on the 3d and 5d Chern-Simons levels so that the instanton partition function does not depend on the choice of integration contour. For these values of the Chern-Simons couplings, we comment on how the distinct quiver theories that realize the same surface operator are related to each other by Aharony-Seiberg dualities.

ADHM and the 4d quantum Hall effect

Yang-Mills instantons are solitonic particles in d = 4 + 1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions.It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function.

Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

We study Abelian braiding statistics of loop excitations in three-dimensional (3D) gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic symmetry-protected topological (FSPT) phases with unitary symmetries. It is known that the two problems are related by turning FSPT phases into gauge theories through gauging the global symmetry of the former. We show that there exist certain types of Abelian loop braiding statistics that are allowed only in the the presence of fermionic particles, which correspond to 3D "intrinsic" FSPT phases, i.e., those that do not stem from bosonic SPT phases. While such intrinsic FSPT phases are ubiquitous in 2D systems and in 3D systems with anti-unitary symmetries, their existence in 3D systems with unitary symmetries was not confirmed previously due to the fact that strong interaction is necessary to realize them. We show that the simplest unitary symmetry to support 3D intrinsic FSPT phases is $\mathbb{Z}_2\times\mathbb{Z}_4$. To establish the results, we first derive a complete set of physical constraints on Abelian loop braiding statistics. Solving the constraints, we obtain all possible Abelian loop braiding statistics in 3D gauge theories, including those that correspond to intrinsic FSPT phases. Then, we construct exactly soluble state-sum models to realize the loop braiding statistics. These state-sum models generalize the well-known Crane-Yetter and Dijkgraaf-Witten models.

Supersymmetric interactions of a six-dimensional self-dual tensor and fixed-shape second quantized strings

"Curvepole (2,0)-theory" is a deformation of the (2,0)-theory with nonlocal interactions. A "curvepole" is defined as a two-dimensional generalization of a dipole. It is an object of fixed two-dimensional shape whose boundary is a charged curve that interacts with a two-form gauge field. Curvepole theory was previously only defined indirectly via M-theory. Here we propose a supersymmetric Lagrangian, constructed explicitly up to quartic terms, for an "abelian" curvepole theory, which is an interacting deformation of the free (2,0) tensor mutliplet. This theory contains fields whose quanta are curvepoles (i.e., fixed-shape strings). Supersymmetry is preserved (at least up to quartic terms) if the shape of the curvepoles is (2d) planar. This nonlocal 6d QFT may also serve as a UV completion for certain (local) 5d gauge theories.

Higher AGT correspondences, $\mathcal{W}$-algebras, and higher quantum geometric Langlands duality from M-theory

We further explore the implications of our framework in [arXiv:1301.1977, arXiv:1309.4775], and physically derive, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent, (i) a 5d AGT correspondence for any compact Lie group, (ii) a 5d and 6d AGT correspondence on ALE space of type ADE, and (iii) identities between the ordinary, q-deformed and elliptic affine W-algebras associated with the 4d, 5d and 6d AGT correspondence, respectively, which also define a quantum geometric Langlands duality and its higher analogs formulated by Feigin-Frenkel-Reshetikhin in [3,4]. As an offshoot, we are led to the sought-after connection between the gauge-theoretic realization of the geometric Langlands correspondence by Kapustin-Witten [5,6] and its algebraic CFT formulation by Beilinson-Drinfeld [7], where one can also understand Wilson and 't Hooft-Hecke line operators in 4d gauge theory as monodromy loop operators in 2d CFT, for example. In turn, this will allow us to argue that the higher 5d/6d analog of the geometric Langlands correspondence for simply-laced Lie (Kac-Moody) groups G (G_k), ought to relate the quantization of circle (elliptic)-valued G Hitchin systems to circle/elliptic-valued ^LG (^LG_k)-bundles over a complex curve on one hand, and the transfer matrices of a G (G_k)-type XXZ/XYZ spin chain on the other, where ^LG is the Langlands dual of G. Incidentally, the latter relation also serves as an M-theoretic realization of Nekrasov-Pestun-Shatashvili's recent result in [8], which relates the moduli space of 5d/6d supersymmetric G (G_k)-quiver SU(K_i) gauge theories to the representation theory of quantum/elliptic affine (toroidal) G-algebras.

Asymptotic symmetries, holography and topological hair

Asymptotic symmetries of AdS4 quantum gravity and gauge theory are derived by coupling the holographically dual CFT3 to Chern-Simons gauge theory and 3D gravity in a “probe” (large-level) limit. Despite the fact that the three-dimensional AdS4 boundary as a whole is consistent with only finite-dimensional asymptotic symmetries, given by AdS isometries, infinite-dimensional symmetries are shown to arise in circumstances where one is restricted to boundary subspaces with effectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS4 quantum gravity. An AdS4 analog of Minkowski “super-rotation” asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT2 structure (in a large central charge limit), via AdS3 foliation of AdS4 and the AdS3/CFT2 correspondence. The maximal asymptotic symmetry is however probed by 3D conformal gravity. Both 3D gravities have Chern-Simons formulation, manifesting their topological character. Chern-Simons structure is also shown to be emergent in the Poincare patch of AdS4, as soft/boundary limits of 4D gauge theory, rather than “put in by hand” as an external probe. This results in a finite effective Chern-Simons level. Several of the considerations of asymptotic symmetry structure are found to be simpler for AdS4 than for Mink4, such as non-zero 4D particle masses, 4D non-perturbative “hard” effects, and consistency with unitarity. The last of these in particular is greatly simplified because in some set-ups the time dimension is explicitly shared by each level of description: Lorentzian AdS4, CFT3 and CFT2. Relatedly, the CFT2 structure clarifies the sense in which the infinite asymptotic charges constitute a useful form of “hair” for black holes and other complex 4D states. An AdS4 analog of Minkowski “memory” effects is derived, but with late-time memory of earlier events being replaced by (holographic) “shadow” effects. Lessons from AdS4 provide hints for better understanding Minkowski asymptotic symmetries, the 3D structure of its soft limits, and Minkowski holography.