4D Theory Research Articles

Compactification of 6D minimal SCFTs on Riemann surfaces

We study compactifications on Riemann surfaces with punctures of N=(1,0) 6d SCFTs with a one dimensional tensor branch and no continuous global symmetries. The effective description of such models on the tensor branch is in terms of pure gauge theories with decoupled tensor. For generic Riemann surfaces, the resulting theories in four dimensions are expected to have N=1 supersymmetry. We compute the anomalies expected from the resulting 4d theories by integrating the anomaly polynomial of the 6d theory on the Riemann surface. For the cases with 6d gauge models with gauge groups SU(3) and SO(8) we further propose a field theory construction for the resulting 4d theories. For the 6d SU(3) theory, we argue that the theories in four dimensions are quivers with SU(3) gauge nodes and free chiral fields. The theories one obtains from the 6d SO(8) gauge theory are quivers with SU(4) gauge groups and chiral fields with R charge a half. In the last case the theories constructed for general Riemann surfaces involve gauging of symmetries appearing at strong coupling. The conformal manifolds of the models are constructed from gauge couplings and baryonic superpotentials. We support our conjectures by matching the dimensions of the conformal manifolds with complex structure moduli of the Riemann surfaces, matching anomalies between six and four dimensions, and checking the dualities related to different pair of pants decompositions of the surfaces. As a simple application of the results we conjecture that SU(3) gauge theory with nine flavors in four dimensions has a duality group acting on the seven dimensional conformal manifold which is the mapping class group of sphere with ten marked points.

Quiver subtractions

We study the vacuum structure of gauge theories with eight supercharges. It has been recently discovered that in the Higgs branch of 5d and 6d SQCD theories with eight supercharges, the new massless states, arising when the gauge coupling is taken to infinity, can be described in terms of Coulomb branches of 3d mathcal{N}=4 quiver gauge theories. The description of this new phenomenon draws from the ideas discovered in the analysis of nilpotent orbits as Higgs and Coulomb branches of 3d theories and promotes the Higgs mechanism known as the Kraft-Procesi transition to the status of a new operation between quivers. This is the quiver subtraction. This paper establishes this operation formally and examines some immediate consequences. One is the extension of the physical realization of Kraft-Procesi transitions from the classical to the exceptional Lie algebras. Another result is the extension from special nilpotent orbits to non-special ones. One further consequence is the analysis of the effect in 5d mathcal{N}=1 SQCD of integrating out a massive quark while the gauge coupling remains infinite. In general, the subtraction of quivers sheds light on the different types of singularities within the Coulomb branch and the structure of the massless states that arise at those singular points; including the nature of the new Higgs branches that open up. This allows for a systematic analysis of mixed branches of 3d mathcal{N}=4 quivers that do not necessarily have a simple embedding in string theory. The subtraction of two quivers is an extremely simple resource for the theoretical physicist interested in the vacuum structure of gauge theories, and yet its power is so remarkable that is bound to play a crucial role in the coming discoveries of new and exciting physics in 5 and 6 dimensions.

Vibration of Toroidal Shells with Hollow Circular Cross-Sections

This paper presents a three-dimensional (3D) analysis for the natural frequencies of completely free, toroidal shells of revolution with hollow circular cross-sections by the Ritz method. The displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in [Formula: see text], and of the ordinary algebraic simple polynomials in the [Formula: see text] and [Formula: see text] directions. The potential (strain) and kinetic energies of the torus are formulated, and the upper bound values of the frequencies are obtained by a minimization procedure. As the degree of the polynomials increases, the frequencies computed converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the torus. Comparisons are made between the frequencies from the present 3D method, a 3D finite element method, experimental methods, and thin and thick ring theories.

Three-dimensional hydroelasticity theory of ships in waves and ocean-acoustic environment with applications

The three-dimensional (3D) hydroelasticity theory of floating structures established in 1984 embodies the predictions of seakeeping behaviors, structural loads, dynamic strength, fatigue, and vibration of a surface ship or an underwater vehicle travelling in the sea in a unified generalized fluid-structure interaction theory. After more than thirty years of development, this theory has been further extended to two main branches: the 3D hydroelasticity theories for examining the wave excited dynamic responses of ships, and the 3D hydroelasticity theories for investigating the machinery or propeller excited vibrations and underwater acoustic radiations of a ship. The latter is also called the 3D sono-elasticity theory of ships. Drawn by the momentum gained from the comprehensive demand of marine applications, China Ship Scientific Research Center (CSSRC) has been devoted to the systematic innovative research in creating the linear and non-linear 3D hydroelasticity theories of ships, the 3D hydroelasticity theory of floating structures deployed in complicated ocean geographical environment near islands and reefs, the 3D sono-elasticity theory of ships in ocean acoustic environment etc. together with developing the numerical methods and corresponding software. These have been extensively employed to solve the key technical problems, such as motion and structural response predictions, behavior and safety assessment, design check and optimization of surface ships, underwater vehicles and other marine structures, This paper is not to present the state of the art of the worldwide development of hydroelasticity theories, but to concisely described the concept and framework of the 3D hydroelasticity and sono-elasticity theories of ships which are mainly developed by the research group of CSSRC. Typical application examples of the hydroelastic analyses of a 500000 DWT ultra large ore carrier and an 8-module very large floating structure, as well as the sono-elastic analysis of a simplified submarine model are also briefly illustrated. Finally, simply mentioned are the important directions of future research in the field of hydroelasticity and sono-elasticity of ships.

Punctures and dynamical systems

With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of N = 1 supersymmetric punctures for class S_Gamma theories, namely the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes probing a C^2 / Z_k singularity, the punctures are governed by a dynamical system in which evolution in time corresponds to motion to a neighboring node in an affine A-type quiver. Classification of punctures reduces to determining consistent initial conditions which produce periodic orbits. The study of this system is particularly tractable in the case of a single M5-brane. Even in this "simple" case, the solutions exhibit a remarkable level of complexity: Only specific rational values for the initial momenta lead to periodic orbits, and small perturbations in these values lead to vastly different late time behavior. Another difference from half BPS punctures of class S theories includes the appearance of a continuous complex "zero mode" modulus in some puncture solutions. The construction of punctures with higher order poles involves a related set of recursion relations. The resulting structures also generalize to systems with multiple M5-branes as well as probes of D- and E-type orbifold singularities.

Exotic massive 3D gravity

The linearized equations of “New Massive Gravity” propagate a parity doublet of massive spin-2 modes in 3D Minkowski spacetime, but a different non-linear extension is made possible by “third-way” consistency. There is a “Chern-Simons-like” action, as for other 3D massive gravity models, but the new theory is “exotic”: its action is parity odd. This “Exotic Massive Gravity” is the next-to-simplest case in an infinite sequence of third-way consistent 3D gravity theories, the simplest being the “Minimal Massive Gravity” alternative to “Topologically Massive Gravity”.

Universal features of BPS strings in six-dimensional SCFTs

In theories with extended supersymmetry the protected observables of UV superconformal fixed points are found in a number of contexts to be encoded in the BPS solitons along an IR Coulomb-like phase. For six-dimensional SCFTs such a role is played by the BPS strings on the tensorial Coulomb branch. In this paper we develop a uniform description of the worldsheet theories of a BPS string for rank-one 6d SCFTs. These strings are the basic constituents of the BPS string spectrum of arbitrary rank six-dimensional models, which they generate by forming bound states. Motivated by geometric engineering in F-theory, we describe the worldsheet theories of the BPS strings in terms of topologically twisted 4d mathcal{N}=2 theories in the presence of 1/2-BPS 2d (0, 4) defects. As the superconformal point of a 6d theory with gauge group G is approached, the resulting worldsheet theory flows to an mathcal{N}=left(0, 4right) NLSM with target the moduli space of one G instanton, together with a nontrivial left moving bundle characterized by the matter content of the six-dimensional model. We compute the anomaly polynomial and central charges of the NLSM, and argue that the 6d flavor symmetry F is realized as a current algebra on the string, whose level we compute. We find evidence that for generic theories the G dependence is captured at the level of the elliptic genus by characters of an affine Kac-Moody algebra at negative level, which we interpret as a subsector of the chiral algebra of the BPS string worldsheet theory. We also find evidence for a spectral flow relating the R-R and NS-R elliptic genera. These properties of the string CFTs lead to constraints on their spectra, which in combination with modularity allow us to determine the elliptic genera of a vast number of string CFTs, leading also to novel results for 6d and 5d instanton partition functions.

Anomalies of duality groups and extended conformal manifolds

A self-duality group $\cal G$ in quantum field theory can have anomalies. In that case, the space of ordinary coupling constants $\cal M$ can be extended to include the space $\cal F$ of coefficients of counterterms in background fields. The extended space $\cal N$ forms a bundle over $\cal M$ with fiber $\cal F$, and the topology of the bundle is determined by the anomaly. For example, the ${\cal G}=SL(2,\mathbb{Z})$ duality of the 4d Maxwell theory has an anomaly, and the space ${\cal F}=S^1$ for the gravitational theta-angle is nontrivially fibered over ${\cal M}=\mathbb{H}/SL(2,\mathbb{Z})$. We will explain a simple method to determine the anomaly when the 4d theory is obtained by compactifying a 6d theory on a Riemann surface in terms of the anomaly polynomial of the parent 6d theory. Our observations resolve an apparent contradiction associated with the global structure of the Kahler potential on the space of exactly marginal couplings of supersymmetric theories.

An mathcal{N}=1 3d-3d correspondence

M5-branes on an associative three-cycle M3 in a G2-holonomy manifold give rise to a 3d mathcal{N}=1 supersymmetric gauge theory, {T}_{mathcal{N}=1}left[{M}_3right] . We propose an mathcal{N}=1 3d-3d correspondence, based on two observables of these theories: the Witten index and the S3-partition function. The Witten index of a 3d mathcal{N}=1 theory {T}_{mathcal{N}=1}left[{M}_3right] is shown to be computed in terms of the partition function of a topological field theory, a super-BF-model coupled to a spinorial hypermultiplet (BFH), on M3. The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on M3. Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d (2, 0) theory. We also consider a correspondence for the S3-partition function of the {T}_{mathcal{N}=1}left[{M}_3right] theories, by determining the dimensional reduction of the M5-brane theory on S3. The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on M3, whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic G2-manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the S3-partition function of {T}_{mathcal{N}=1}left[{M}_3right] is given by the Witten-Reshetikhin-Turaev invariant of M3.

Mass and angular momentum of black holes in 3D gravity theories with first order formalism

We apply the Wald formalism to obtain masses and angular momenta of black holes in three dimensional gravity theories using the first order formalism. Wald formalism suggests that the entropy of a black hole can be defined by an integration of a conserved charge on the bifurcation horizon, and mass and angular momentum of a black hole as an integration of some charge variation form at spatial infinity. The action of three dimensional gravity theories can be represented by a form including some auxiliary fields. As well-known examples we have calculated masses and angular momenta of some black holes in topologically massive gravity and new massive gravity theories using the first order formalism. We have also calculated mass and angular momentum of BTZ black hole and new type black hole in minimal massive gravity theory with the action represented by the first order formalism. We have also calculated the entropy and central charges of new type black hole. According to AdS / CFT correspondence we suggest that the left and right moving temperatures should be equal to the Hawking temperature in the case of new type black hole in minimal massive gravity.

Localization for Chern–Simons on circle bundles via loop groups

We consider Chern–Simons theory on a 3-manifold M that is the total space of a circle bundle over a 2d base Σ . We show that this theory is equivalent to a new 2d TQFT on the base, which we call Caloron BF theory, that can be obtained by an appropriate type of push-forward. This is a gauge theory on a bundle with structure group given by the full affine level k central extension of the loop group L G . The space of fields of this 2d theory is naturally symplectic, and this provides a new formulation of a result of Beasley–Witten about the equivariant localization of the Chern–Simons path integral. The main tool that we employ is the Caloron correspondence, originally due to Murray–Garland, that relates the space of gauge fields on M with a certain enlarged space of connections on an equivariant version of the loop space of the G -bundle. We show that the symplectic structure that Beasley–Witten found is related to a looped version of the Atiyah–Bott construction in 2-dimensional Yang–Mills theory. We also show that Wilson loops that wrap a single circle fiber are also described very naturally in this framework.

RG flows with supersymmetry enhancement and geometric engineering

In this paper we study a class of mathcal{N}=2 SCFTs with ADE global symmetry defined via Type IIB compactification on a class of hypersurfaces in ℂ3 × ℂ*. These can also be constructed by compactifying the 6d (2,0) theory of type ADE on a sphere with an irregular and a full punctures. When we couple to the ADE moment map a chiral multiplet in the adjoint representation and turn on a (principal) nilpotent vev for it, all the theories in this family display enhancement of supersymmetry in the infrared. We observe that all known examples of theories which flow, upon the same type of deformation, to strongly coupled mathcal{N}=2 theories fit naturally in our framework, thus providing a new perspective on this topic. We propose an infrared equivalence between this RG flow and a manifestly mathcal{N}=2 preserving one and, as a byproduct, we extract a precise prescription to relate the SW curves describing the UV and IR fixed points for all theories with A or D global symmetry. We also find, for a certain subclass, a simple relation between UV and IR theories at the level of chiral algebras.

Nilpotent symmetries of a 4[formula omitted] model of Hodge theory: Augmented (anti-)chiral superfield formalism

We derive the continuous nilpotent symmetries of the four (3+1)-dimensional (4D) model of the Hodge theory (i.e. 4D Abelian 2-form gauge theory) by exploiting the beauty and strength of the symmetry invariant restrictions on the (anti-)chiral superfields. The above off-shell nilpotent symmetries are the Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-)co-BRST transformations which turn up beautifully due to the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral superfields that are defined on the (4, 1)-dimensional (anti-)chiral super-submanifolds of the general (4, 2)-dimensional supermanifold on which our ordinary 4D theory is generalized. The latter supermanifold is characterized by the superspace coordinates ZM=(xμ,θ,θ̄) where xμ(μ=0,1,2,3) are the bosonic coordinates and a pair of Grassmannian variables θ and θ̄ are fermionic in nature as they obey the standard relationships: θ2=θ̄2=0,θθ̄+θ̄θ=0). The derivation of the proper (anti-)co-BRST symmetries and proof of the absolute anticommutativity property of the conserved (anti-)BRST and (anti-) co-BRST charges are novel results of our present investigation (where only the (anti-)chiral superfields and their super-expansions have been taken into account).

Tinkertoys for the E7 theory

We classify the class S theories of type E7. These are four-dimensional mathcal{N}=2 superconformal field theories arising from the compactification of the E7 (2, 0) theory on a punctured Riemann surface, C. The classification is given by listing all 3-punctured spheres (“fixtures”), and connecting cylinders, which can arise in a pants-decomposition of C. We find exactly 11,000 fixtures with three regular punctures, and an additional 48 with one “irregular puncture” (in the sense used in our previous works). To organize this large number of theories, we have created a web application at https://golem.ph.utexas.edu/class-S/E7/. Among these theories, we find 10 new ones with a simple exceptional global symmetry group, as well as a new rank-2 SCFT and several new rank-3 SCFTs. As an application, we study the strong-coupling limit of the E7 gauge theory with 3 hypermultiplets in the 56. Using our results, we also verify recent conjectures that the T2 compactification of certain 6d (1, 0) theories can alternatively be realized in class S as fixtures in the E7 or E8 theories.

A Tower Weak Gravity Conjecture from Infrared Consistency

AbstractWe analyze infrared consistency conditions of 3D and 4D effective field theories with massive scalars or fermions charged under multiple U(1) gauge fields. At low energies, one can integrate out the massive particles and thus obtain a one‐loop effective action for the gauge fields. In the regime where charge‐independent contributions to higher‐derivative terms in the action are sufficiently small, it is then possible to derive constraints on the charge‐to‐mass ratios of the massive particles from requiring that photons propagate causally and have an analytic S‐matrix. We thus find that the theories need to contain bifundamentals and satisfy a version of the weak gravity conjecture known as the convex‐hull condition. Demanding self‐consistency of the constraints under Kaluza‐Klein compactification, we furthermore show that, for scalars, they imply a stronger version of the weak gravity conjecture in which the charge‐to‐mass ratios of an infinite tower of particles are bounded from below. We find that the tower must again include bifundamentals but does not necessarily have to occupy a charge (sub‐)lattice.

Teichm\xfcller TQFT vs. Chern-Simons theory

Teichmüller TQFT is a unitary 3d topological theory whose Hilbert spaces are spanned by Liouville conformal blocks. It is related but not identical to PSL(2, ℝ) Chern-Simons theory. To physicists, it is known in particular in the context of 3d-3d correspondence and also in the holographic description of Virasoro conformal blocks. We propose that this theory can be defined by an analytically-continued Chern-Simons path-integral with an unusual integration cycle. On hyperbolic three-manifolds, this cycle is singled out by the requirement of invertible vielbein. Mathematically, our proposal translates a known conjecture by Andersen and Kashaev into a conjecture about the Kapustin-Witten equations. We further explain that Teichmüller TQFT is dual to complex SL(2, ℂ) Chern-Simons theory at integer level k = 1, clarifying some puzzles previously encountered in the 3d-3d correspondence literature. We also present a new simple derivation of complex Chern-Simons theories from the 6d (2,0) theory on a lens space with a transversely-holomorphic foliation.

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.

Topological strings on singular elliptic Calabi-Yau 3-folds and minimal 6d SCFTs

We apply the modular approach to computing the topological string partition function on non-compact elliptically fibered Calabi-Yau 3-folds with higher Kodaira singularities in the fiber. The approach consists in making an ansatz for the partition function at given base degree, exact in all fiber classes to arbitrary order and to all genus, in terms of a rational function of weak Jacobi forms. Our results yield, at given base degree, the elliptic genus of the corresponding non-critical 6d string, and thus the associated BPS invariants of the 6d theory. The required elliptic indices are determined from the chiral anomaly 4-form of the 2d worldsheet theories, or the 8-form of the corresponding 6d theories, and completely fix the holomorphic anomaly equation constraining the partition function. We introduce subrings of the known rings of Weyl invariant Jacobi forms which are adapted to the additional symmetries of the partition function, making its computation feasible to low base wrapping number. In contradistinction to the case of simpler singularities, generic vanishing conditions on BPS numbers are no longer sufficient to fix the modular ansatz at arbitrary base wrapping degree. We show that to low degree, imposing exact vanishing conditions does suffice, and conjecture this to be the case generally.

Argyres-Douglas matter and S-duality. Part II

We study S-duality of Argyres-Douglas theories obtained by compactification of 6d (2,0) theories of ADE type on a sphere with irregular punctures. The weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points, among which three punctured sphere represents isolated superconformal theories. We also discuss twisted irregular punctures and their S-duality.

Chiral limit of mathcal{N} = 4 SYM and ABJM and integrable Feynman graphs

We consider a special double scaling limit, recently introduced by two of the authors, combining weak coupling and large imaginary twist, for the γ-twisted mathcal{N} = 4 SYM theory. We also establish the analogous limit for ABJM theory. The resulting non-gauge chiral 4D and 3D theories of interacting scalars and fermions are integrable in the planar limit. In spite of the breakdown of conformality by double-trace interactions, most of the correlators for local operators of these theories are conformal, with non-trivial anomalous dimensions defined by specific, integrable Feynman diagrams. We discuss the details of this diagrammatics. We construct the doubly-scaled asymptotic Bethe ansatz (ABA) equations for multi-magnon states in these theories. Each entry of the mixing matrix of local conformal operators in the simplest of these theories — the bi-scalar model in 4D and tri-scalar model in 3D — is given by a single Feynman diagram at any given loop order. The related diagrams are in principle computable, up to a few scheme dependent constants, by integrability methods (quantum spectral curve or ABA). These constants should be fixed from direct computations of a few simplest graphs. This integrability-based method is advocated to be able to provide information about some high loop order graphs which are hardly computable by other known methods. We exemplify our approach with specific five-loop graphs.