Denis Erkal, David Kutasov
We construct the a-function of [1] for general F-term perturbations of a 3+1 dimensional N=1 SCFT. We use this construction to argue that the central charge a always decreases along the corresponding RG flows, and discuss some other applications.
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A generalization of a-maximization is proposed that maximizes a subject to inequalities rather than equalities. The implication of this conjecture is that in the absence of emergent symmetries, there is a maximum R-charge for fields appearing in the path integral. This maximum R-charge leads to a novel way of detecting emergent Abelian symmetries and non-Abelian gauge symmetries.
The global counterpart of infinitesimal symmetries of noncommutative space-time is discussed.
We report preliminary results on the recently proposed F-maximization principle in 3D SCFTs. We compute numerically in the large-N limit the free energy on the three-sphere of an N=2 Chern-Simons-Matter theory with a single adjoint chiral superfield which is known to exhibit a pattern of accidental symmetries associated to chiral superfields that hit the unitarity bound and become free. We observe that the F-maximization principle produces a U(1) R-symmetry consistent with previously obtained bounds but inconsistent with a postulated Seiberg-like duality. Potential modifications of the princip…
We investigate the relation between gauge theories and brane configurations described by brane tilings. We identify U(1)_B (baryonic), U(1)_M (mesonic), and U(1)_R global symmetries in gauge theories with gauge symmetries in the brane configurations. We also show that U(1)_MU(1)_B^2 and U(1)_RU(1)_B^2 't Hooft anomalies are reproduced as gauge transformations of the classical brane action.
It is shown that non-trivial topological sectors can prevent the quantum mechanical implementation of the symmetries of the classical field equations of sigma models with torsion. The associated anomaly is computed, and it is shown that it depends on the homotopy class of the topological sector of the theory and the group action on the sigma model manifold that generates the symmetries of the classical field equations.
S-dualities in scale invariant N=2 supersymmetric field theories are derived by embedding those theories in asymptotically free N=2 theories with higher rank gauge groups. S-duality transformations on the coupling of the scale invariant theory follow from global symmetries acting on the Coulomb branch of the higher rank theory. Since these global symmetries are exact in the asymptotically free theory, this shows that S-duality is an exact equivalence of N=2 theories and not just a property of their supersymmetric states.
We prove that N=2 theories that arise by taking n free hypermultiplets and gauging a subgroup of Sp(n), the non-R global symmetry of the free theory, have a remaining global symmetry which is a direct sum of unitary, symplectic, and special orthogonal factors. This implies that theories that have SU(N) but not U(N) global symmetries, such as Gaiotto's T_N theories, are not likely to arise as IR fixed points of RG flows from weakly coupled N=2 gauge theories.
We revisit general arguments for the absence of exact continuous global symmetries in string compactifications and extend them to D-brane models. We elucidate the various ways approximate continuous global symmetries arise in the 4-dimensional effective action. In addition to two familiar methods - axionic Peccei-Quinn symmetries and remnant global abelian symmetries from Green-Schwarz gauge symmetry breaking - we identify new ways to generate approximate continuous global symmetries. Two methods stand out, both of which occur for local brane constructions within the LARGE volume scenario of m…
In this note we revisit Hitchin's prescription \cite{Hitchin} of the Fisher metric as a natural measure on the moduli space of instantons that encodes the space-time symmetries of a classical field theory. Motivated by the idea of the moduli space of supersymmetric instantons as an emergent space in the sense of the gauge/gravity duality, we extend the prescription to encode also global symmetries of the underlying theory. We exemplify our construction with the instanton solution of the $\mathbb{CP}^{N}$ sigma model on $\mathbb{R}^{2}$.
We analyze the global symmetries of ${\cal N}=4$ supersymmetric mechanics involving $4n$-dimensional Quaternion-Kähler (QK) $1D$ sigma models on projective spaces $\mathbb{H}{\rm H}^n$ and $\mathbb{H}{\rm P}^n$ as the bosonic core. All Noether charges associated with global worldline symmetries are shown to vanish as a result of equations of motion, which implies that we deal with a severely constrained hamiltonian system. The complete hamiltonian analysis of the bosonic sector is performed.
This is a general introduction to duality in field theories. The existence and breaking of global symmetries is used as a guideline to systematically prove duality between different field theories. Systems discussed include abelian and non-abelian T-duality in string theory, abelian and nonabelian bosonization, and duality for massless and massive antisymmetric tensor field theories in arbitrary number of dimensions. Open questions regarding these techniques are also discussed. (Lectures given at 33rd Karpacz Winter School `Duality: Strings and Fields' .)
We study four dimensional N=1 gauge theories that arise on the worldvolume of D3-branes probing complex cones over del Pezzo surfaces. Global symmetries of the gauge theories are made explicit by using a correspondence between bifundamental fields in the quivers and divisors in the underlying geometry. These global symmetries are hidden, being unbroken when all inverse gauge couplings of the quiver theory vanish. In the broken phase, for finite gauge couplings, only the Cartan subalgebra is manifest as a global symmetry. Superpotentials for these models are constructed using global symmetry in…
We present a method to solve the master equation for the Wilsonian action in the antifield formalism. This is based on a representation theory for cutoff dependent global symmetries along the Wilsonian renormalization group (RG) flow. For the chiral symmetry, the master equation for the free theory yields a continuum version of the Ginsparg-Wilson relation. We construct chiral invariant operators describing fermionic self-interactions. The use of canonically transformed variables is shown to simplify the underlying algebraic structure of the symmetry. We also give another non-trivial example, …
A relation between dimensional reduction and space-time symmetry gauging is outlined.
In this paper, two things are done. (i) First, it is shown that any global symmetry of a gauge-invariant theory can be extended to the ghosts and the antifields so as to leave invariant the solution of the master-equation (before gauge fixing). (ii) Second, it is proved that the incorporation of the rigid symmetries to the solution of the master-equation through the introduction of a constant ghost for each global symmetry can be obstructed already at the classical level whenever the theory possesses higher order conservation laws. Explicit examples are given.
We revisit universal features of duality in linear and nonlinear relativistic scalar and Abelian 1-form theories with single or multiple fields, which exhibit ordinary or generalized global symmetries. We show that such global symmetries can be interpreted as generalized Killing isometries on a suitable, possibly graded, target space of fields or its jet space when the theory contains higher derivatives. This is realized via a generalized sigma model perspective motivated from the fact that higher spin particles can be Nambu-Goldstone bosons of spontaneously broken generalized global symmetrie…
Consequences of Symmetry Fractionalization without 1-Form Global Symmetries
本文研究无1-form对称性理论中0-form对称性在线算符上的分数化，通过圆盘算符刻画射影作用并导出RG不变约束、选择定则与有限体积真空简并。
We study the holographic duals of four-dimensional field theories with 1-form global symmetries, both discrete and continuous. Such higher-form global symmetries are associated with antisymmetric tensor gauge fields in the bulk. Various different realizations are possible: we demonstrate that a Maxwell action for the bulk antisymmetric gauge field results in a non-conformal field theory with a marginally running double-trace coupling. We explore its hydrodynamic behavior at finite temperature and make contact with recent symmetry-based formulations of magnetohydrodynamics. We also argue that d…
Comments on Global Symmetries and Anomalies of $5d$ SCFTs
本研究探讨五维超对称共形场论的整体对称性，揭示相关形变下红外规范理论的1-形式与瞬子对称性的混合't Hooft反常对味对称群及RG流的约束，辅以三维案例验证，并分析共形不变性与高阶整体对称性的相容性。
We consider the general gauge theory with a closed irreducible gauge algebra possessing the non-anomalous global (super)symmetry in the case when the gauge fixing procedure violates the global invariance of classical action. The theory is quantized in the framework of BRST-BV approach in the form of functional integral over all fields of the configuration space. It is shown that the global symmetry transformations are deformed in the process of quantization and the full quantum action is invariant under such deformed global transformations in the configuration space. The deformed global transf…
In these notes, we emphasize the rôle of spontaneous broken global discrete symmetries acting on the moduli space of $N=2$ susy Yang-Mills theories and show how they can be used, together with the BPS condition, as a spectrum generating symmetry. In particular, in the strong-coupling region, all BPS states come in multiplets of this broken symmetry. This played a key rôle in the determination of the strong-coupling spectra.
Recently, the superconformal index for a class of 5-dimensional superconformal quantum field theories with gauge group SU(2) and $N_f$ flavors was computed in Kim, Kim, and Lee (2012) to a few low orders in the chemical potential and enhancement of the global symmetry to $E_{N_f+1}$ was confirmed to this low order. In this note I provide an argument that the information contained in these few low order terms in the expansion of the index is sufficient to prove the enhancement of global symmetries in the theories. So, in particular the expression for the indices must display the enhancement to …
We examine six-dimensional quantum field theories through the lens of higher-form global symmetries. Every Yang-Mills gauge theory in six dimensions, with field strength $f^{(2)}$, naturally gives rise to a continuous 1-form global symmetry associated with the 2-form instanton current $J^{(2)} \sim * \text{Tr} \left( f^{(2)} \wedge f^{(2)}\right)$. We show that suitable mixed anomalies involving the gauge field $f^{(2)}$ and ordinary 0-form global symmetries, such as flavor or Poincaré symmetries, lead to continuous 2-group global symmetries, which allow two flavor currents or two stress tenso…
Non-Invertible Symmetries from Discrete Gauging and Completeness of the Spectrum
本文研究非连通规范群下的不可逆对称性，提出弦论嵌入并发现扭转涡旋破缺全局对称性，支持量子引力无全局对称性的猜想。
It is outlined how deformations of field theoretical rigid symmetries can be constructed and classified by cohomological means in the extended antifield formalism. Special attention is devoted to deformations referring only to a subset of the rigid symmetries of a given model and leading to a nontrivial extension of the graded Lie algebra associated with that subset. The method is illustrated for a D=4, N=2 supersymmetric model where the central extension of the supersymmetry algebra emerges via a deformation. Deformations of gauge fixed actions with a BRST symmetry are discussed too and illus…
Fermionic Higher-form Symmetries
本文提出费米子高阶形式对称性这一新型全局对称性，构造了多个相关场论实例，并探讨了其规范、破缺及与沼泽地猜想的关联。
2-group symmetries arise when 1-form symmetries and 0-form symmetries of a theory mix with each other under group multiplication. We discover the existence of 2-group symmetries in 5d N=1 abelian gauge theories arising on the (non-extended) Coulomb branch of 5d superconformal field theories (SCFTs), leading us to argue that the UV 5d SCFT itself admits a 2-group symmetry. Furthermore, our analysis determines the global forms of the 0-form flavor symmetry groups of 5d SCFTs, irrespective of whether or not the 5d SCFT admits a 1-form symmetry. As a concrete application of our method, we analyze …
It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled `generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as special cases of more general 2-groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one-form and higher-form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topolog…
We study global symmetry groups of six-dimensional superconformal field theories (SCFTs). In the Coulomb branch we use field theoretical arguments to predict an upper bound for the global symmetry of the SCFT. We then analyze global symmetry groups of F-theory constructions of SCFTs with a one-dimensional Coulomb branch. While in the vast majority of cases, all of the global symmetries allowed by our Coulomb branch analysis can be realized in F-theory, in a handful of cases we find that F-theory models fail to realize the full symmetry of the theory on the Coulomb branch. In one particularly m…
On Global Symmetries and Fayet-Iliopoulos Terms
本文重新审视四维 N=1 超引力中的 Fayet-Iliopoulos 项，发现存在显式破坏整体对称性但保持规范不变性的超对称项，并通过实例和一般构造处方加以说明。
In earlier work, we (KI and BW) gave a two line "almost proof" (for supersymmetric RG flows) of the weakest form of the conjectured 4d a-theorem, that a_{IR}<a_{UV}, using our result that the exact superconformal R-symmetry of 4d SCFTs maximizes a=3Tr R^3-Tr R. The proof was incomplete because of two identified loopholes: theories with accidental symmetries, and the fact that it's only a local maximum of \it{a}. Here we discuss and extend a proposal of Kutasov (which helps close the latter loophole) in which a-maximization is generalized away from the endpoints of the RG flow, with Lagrange mu…
In arXiv:0811.1909 Spiridonov and Vartanov, using the superconformal index technique, found that 4-dimensional N=1 SQCD theory with SU(2) gauge group and four flavors has 72 dual representations. Recently in arXiv:1209.1404 the authors showed that these dual theories, when coupled to 5d hypermultiplets with specific boundary conditions have an extended E_7 global symmetry. In this work we find that for a reduced theory with 3 flavors the explicit SU(6) global symmetry is enhanced to an E_6 symmetry in the presence of 5d hypermultiplets. We also show connections between indices of different the…
We consider toy models of holography arising from 3d Chern-Simons theory. In this context a duality to an ensemble average over 2d CFTs has been recently proposed. We put forward an alternative approach in which, rather than summing over bulk geometries, one gauges a one-form global symmetry of the bulk theory. This accomplishes two tasks: it ensures that the bulk theory has no global symmetries, as expected for a theory of quantum gravity, and it makes the partition function on spacetimes with boundaries coincide with that of a modular-invariant 2d CFT on the boundary. In particular, on wormh…
We analyze in detail the global symmetries of various (2+1)d quantum field theories and couple them to classical background gauge fields. A proper identification of the global symmetries allows us to consider all non-trivial bundles of those background fields, thus finding more subtle observables. The global symmetries exhibit interesting 't Hooft anomalies. These allow us to constrain the IR behavior of the theories and provide powerful constraints on conjectured dualities.
We study the realisation of global symmetries in a polynomial formulation of the non-linear sigma-model. We show that there are global symmetries whose corresponding Noether currents are the topological currents in the usual formulation. The usual Noether currents associated with the internal symmetry group are reproduced, but part of them become non-local in terms of the dynamical variables of the polynomial formulation.
Subregion duality, wedge classification and no global symmetries in AdS/CFT
本文辨析AdS/CFT中背景楔形与四类算符重构楔形，解决既有悖论，并据此强化几何态无全局对称性的证明。
Symmetries, anomalies, and dualities of two-dimensional Non-Linear Sigma Models
本文研究了二维非线性σ模型的对称性结构，探讨了Wess-Zumino项对连续和离散对称性的影响，并分析了非可逆对称性与自对偶性之间的关系。
Symmetry invariant local interaction of a many body system leads to global constraints. We obtain explicit forms of the global macroscopic condition assuring that at the microscopic level the evolution respects the overall symmetry.
We discuss our attempts to generalize the known examples of dualities in N=1 supersymmetric gauge theories to exceptional gauge groups. We derive some dual pairs from known examples connected to exceptional groups and find an interesting phenomenon: sometimes the full global symmetry is ``hidden'' on the magnetic side. It is not realized as a symmetry on the fundamental fields in the Lagrangian. Rather, it emerges as a symmetry of the quantum theory. We then focus on an approach based on self-dual models. We construct duals for some very special matter content of $E_ 7$, $E_6$ and $F_4$. Again…
We present all NSR superstring and super-D-string actions invariant under a set of prescribed gauge transformations, and characterize completely their global symmetries. In particular we obtain locally supersymmetric Born-Infeld actions on general backgrounds in a formulation with extra target space dimensions. The nontrivial global symmetries of the superstring actions correspond to isometries of the background, whereas super-D-string actions can have additional symmetries acting nontrivially also on the coordinates of the extra dimensions.
Continuous Generalized Symmetries in Three Dimensions
提出一类三维量子场论，其普通与高形式对称性混合形成连续 2-group，并通过规范操作揭示 't Hooft 反常及无穷多不可逆对称性，最后探讨了其全息实现。
Generalized Symmetries, Gravity, and the Swampland
该研究在M理论的7维、5维紧致卡拉比-丘真空背景下，探究广义整体对称性的命运，揭示引力背景的额外自由度破缺局域高阶对称性的机制，导出规范相关对称性的粘合条件及优选规范群结构，并得到对应F理论模型的约束。
Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection …
Global Symmetries, Code Ensembles, and Sums Over Geometries
本文证明三维阿贝尔TQFT中可逆对称性的规范产生加性码，并将码CFT与全息对偶联系起来，进而提出一般三维TQFT中最大规范系综的边界CFT具有引力全息描述。
We present a general prescription for determining the global U(1) symmetries of six-dimensional superconformal field theories (6D SCFTs). We use the quiver-like gauge theory description of the tensor branch to identify candidate U(1) symmetries which can act on generalized matter. The condition that these candidate U(1)'s are free of Adler-Bell-Jackiw (ABJ) anomalies provides bottom-up constraints for U(1)'s. This agrees with the answer obtained from symmetry breaking patterns induced by Higgs branch flows. We provide numerous examples illustrating the details of this proposal. In the F-theory…
A $q$-form global symmetry is a global symmetry for which the charged operators are of space-time dimension $q$; e.g. Wilson lines, surface defects, etc., and the charged excitations have $q$ spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries ($q$=0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either complete…
In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related 't Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we prov…
We draw attention to a class of generalized global symmetries, which we call "Chern-Weil global symmetries," that arise ubiquitously in gauge theories. The Noether currents of these Chern-Weil global symmetries are given by wedge products of gauge field strengths, such as $F_2 \wedge H_3$ and $\text{tr}(F_2^2)$, and their conservation follows from Bianchi identities. As a result, they are not easy to break. However, it is widely believed that exact global symmetries are not allowed in a consistent theory of quantum gravity. As a result, any Chern-Weil global symmetry in a low-energy effective …
Discrete gauge symmetries in global intersecting D-brane models constrain the exact form of the perturbative as well as non-perturbative superpotential. We derive the complete set of conditions on the existence of discrete Zn gauge symmetries on toroidal orbifolds, T6/Z(N) and T6/Z(2)xZ(2M}, with fractional or rigid D6-branes on tilted tori, for which global models of particle physics are known. Several examples of global left-right symmetric and Pati-Salam models are presented. Some discrete `stringy' Zn symmetries are trivial from the field theory point of view, while others have not been id…
Global symmetries: locality, unitarity, and regularity
研究量子场论中对称性局部性与幺正性之间的张力，提出通过可观测量度非局部对称操作的性质，并展示其与对称性融合代数的数据关联。
SYK Model with global symmetries in the double scaling limit
本文研究具有全局对称性的SYK模型在双重标度极限下的行为，发展弦图技术计算哈密顿量矩和两点函数，并给出经典群对称性下的显式结果与固定荷配分函数。
Introduction to Generalized Global Symmetries in QFT and Particle Physics
本文是一份面向高能现象学家的广义（范畴）对称性入门讲义，避免范畴论语言，重点介绍离散对称性及其规范。
The question whether global symmetries can be realized in quantum-gravity-matter-systems has far-reaching phenomenological consequences. Here, we collect evidence that within an asymptotically safe context, discrete global symmetries of the form $\mathbb{Z}_n$, $n>4$, cannot be realized in a near-perturbative regime. In contrast, an effective-field-theory approach to quantum gravity might feature such symmetries, providing a mechanism to generate mass hierarchies in the infrared without the need for additional fine-tuning.
We introduce an effective action for non-dissipative magnetohydrodynamics. A crucial guiding principle is the generalized global symmetry of electrodynamics, which naturally leads to introducing a "dual photon" as the degree of freedom responsible for the electromagnetic component of the fluid. The formalism includes additional degrees of freedom and symmetries which characterize the hydrodynamic regime. By suitably enhancing one of the symmetries, the theory becomes force-free electrodynamics. The symmetries furthermore allow to systematize local and non-local conserved helicities. We also di…
We discuss O'Raifeartaigh models with general R-charge assignments, introduced by Shih to break R-symmetry spontaneously. We argue that most of these models have runaway directions related to the R-symmetry. In addition, we study the simplest model with a U(N) global symmetry and show that in a range of parameters R-symmetry is spontaneously broken in a metastable vacuum.
Can global internal and spacetime symmetries be connected without supersymmetry? To answer this question, we investigate Minkowski spacetimes with d space-like extra dimensions and point out under which general conditions external symmetries induce internal symmetries in the effective 4-dimensional theories. We further discuss in this context how internal degrees of freedom and spacetime symmetries can mix without supersymmetry in agreement with the Coleman-Mandula theorem. We present some specific examples which rely on a direct product structure of spacetime such that orthogonal extra dimens…
Wet Hair: Global Symmetries in Entanglement Islands
本文通过纠缠岛屿机制，探讨了量子引力中非存在全局对称性与信息丢失的关联，提出黑洞毛发在浴中可检测，并解决一个谜题。
The existence of abelian gauge symmetries in four-dimensional F-theory compactifications depends on the global geometry of the internal Calabi-Yau fourfold and has important phenomenological consequences. We study conceptual and phenomenological aspects of such U(1) symmetries along the Coulomb and the Higgs branch. As one application we examine abelian gauge factors arising after a certain global restriction of the Tate model that goes beyond a local spectral cover analysis. In SU(5) GUT models this mechanism enforces a global U(1)_X symmetry that prevents dimension-4 proton decay and allows …
We study field theories with global dipole symmetries and gauge dipole symmetries. The famous Lifshitz theory is an example of a theory with a global dipole symmetry. We study in detail its 1+1d version with a compact field. When this global symmetry is promoted to a $U(1)$ dipole gauge symmetry, the corresponding gauge field is a tensor gauge field. This theory is known to lead to fractons. In order to resolve various subtleties in the precise meaning of these global or gauge symmetries, we place these 1+1d theories on a lattice and then take the continuum limit. Interestingly, the continuum …
We study three-dimensional gauge theories based on orthogonal groups. Depending on the global form of the group these theories admit discrete $θ$-parameters, which control the weights in the sum over topologically distinct gauge bundles. We derive level-rank duality for these topological field theories. Our results may also be viewed as level-rank duality for $SO(N)_{K}$ Chern-Simons theory in the presence of background fields for discrete global symmetries. In particular, we include the required counterterms and analysis of the anomalies. We couple our theories to charged matter and determine…
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