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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Topological symmetry in quantum field theory

We introduce a definition and framework for internal topological symmetries in quantum field theory, including “noninvertible symmetries” and “categorical symmetries”. We outline a calculus of topological defects which takes advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called “gauging” and “condensation defects”), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.

Mathematics↗
DOE OSTI2024

Characterization and classification of interacting ( 2 + 1 )-dimensional topological crystalline insulators with orientation-preserving wallpaper groups

While free fermion topological crystalline insulators have been largely classified, the analogous problem in the strongly interacting case has been only partially solved. In this work, we develop a characterization and classification of interacting, invertible fermionic topological phases in (2+1) dimensions with charge conservation, discrete magnetic translation and M-fold point group rotation symmetries, which form the group G f = U(1) f × Φ [Z 2 $\rtimes$Z M ] for M = 1,2,3,4, and 6. Φ is the magnetic flux per unit cell. We derive a topological response theory in terms of background crystalline gauge fields, which gives a complete classification of different phases and a physical characterization in terms of quantized response to symmetry defects. We then derive the same classification in terms of a set of real space invariants {$Θ^±_o$} that can be obtained from ground state expectation values of suitable partial rotation operators. We explicitly relate these real space invariants to the quantized coefficients in the topological response theory, and find the dependence of the invariants on the chiral central charge c – of the invertible phase. Finally, when Φ = 0 we derive an explicit map between the free and interacting classifications.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

The Branes Behind Generalized Symmetry Operators

The modern approach to m-form global symmetries in a d-dimensional quantum field theory (QFT) entails specifying dimension d–m–1 topological generalized symmetry operators which non-trivially link with m-dimensional defect operators. In QFTs engineered via string constructions on a non-compact geometry X, these defects descend from branes wrapped on non-compact cycles which extend from a localized source / singularity to the boundary ∂X. The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in ∂X. This provides a systematic way to read off various properties of such topological operators, including their worldvolume topological field theories, and the resulting fusion rules. We illustrate these general features in the context of 6D superconformal field theories, where we use the F-theory realization of these theories to read off the worldvolume theory on the generalized symmetry operators. Defects of dimension 3 which are charged under a suitable 3-form symmetry detect a non-invertible fusion rule for these operators. We also sketch how similar considerations hold for related systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

On the holographic dual of a symmetry operator at finite temperature

Topological symmetry operators of holographic large 𝑁 CFT 𝐷 ’s are dual to dynamical branes in the gravity dual AdS 𝐷+1 . We use this correspondence to establish a dictionary between thermal expectation values of symmetry operators in the Euclidean CFT 𝐷 and the evaluation of gravitational saddles in the presence of a dynamical brane. Expectation values of 0-form symmetry operators in the CFT 𝐷 are then related to branes wrapped on volume minimizing cycles in the bulk, i.e., the Euclidean continuation of a black hole horizon. We illustrate with some representative examples, including gravity in AdS 3 , duality/triality defects in four-dimensional 𝒩 = 4 super Yang-Mills theory, and the dual of R-symmetry operators probing five-dimensional Bogomol’nyi–Prasad–Sommerfield black holes.

Anomalies↗
DOE OSTI2023

Obstructions to gapped phases from noninvertible symmetries

Quantum systems in 3+1 dimensions that are invariant under gauging a one-form symmetry enjoy novel noninvertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain dynamics. We show that such noninvertible symmetries often forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, we prove that a self-dual theory with $\mathbb{Z}^{(1)}_N$ one-form symmetry is gapless or spontaneously breaks the self-duality symmetry unless N = k 2 ⁢ℓ where –1 is a quadratic residue modulo . We also extend these results to noninvertible symmetries arising from invariance under more general gauging operations including, e.g., triality symmetries. Along the way, we discover how duality defects in symmetry-protected topological phases have a hidden time-reversal symmetry that organizes their basic properties. These noninvertible symmetries are realized in lattice gauge theories, which serve to illustrate our results.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

On the holographic dual of a topological symmetry operator

We study the holographic dual of a topological symmetry operator in the context of the AdS/CFT correspondence. Symmetry operators arise from topological field theories localized on a subspace of the boundary conformal field theory spacetime. We use bottom up considerations to construct the topological sector associated with their bulk counterparts. In particular, by exploiting the structure of entanglement wedge reconstruction we argue that the bulk counterpart has a nontopological world volume action, i.e., it describes a dynamical object. As a consequence, we find that there are no global 𝑝-form symmetries for 𝑝 ≥ 0 in asymptotically anti–de Sitter spacetimes, which includes the case of noninvertible symmetries. Provided one has a suitable notion of subregion-subregion duality, our argument for the absence of bulk global symmetries applies to more general spacetimes. These considerations also motivate us to consider for general QFTs (holographic or not) the notion of lower-form symmetries, namely, (−𝑚)-form symmetries for 𝑚 ≥ 2.

quantum gravity↗
DOE OSTI2020

Berry phase in quantum field theory: Diabolical points and boundary phenomena

We study aspects of Berry phase in gapped many-body quantum systems by means of effective field theory. Once the parameters are promoted to spacetime-dependent background fields, such adiabatic phases are described by Wess-Zumino-Witten (WZW) and similar terms. In the presence of symmetries, there are also quantized invariants capturing generalized Thouless pumps. Consideration of these terms provides constraints on the phase diagram of many-body systems, implying the existence of gapless points in the phase diagram which are stable for topological reasons. We describe such diabolical points, realized by free fermions and gauge theories in various dimensions, which act as sources of "higher Berry curvature" and are protected by the quantization of the corresponding WZW terms or Thouless pump terms. Here, these are analogous to Weyl nodes in a semimetal band structure. We argue that in the presence of a boundary, there are boundary diabolical points---parameter values where the boundary gap closes---which occupy arcs ending at the bulk diabolical points. Consideration of the topological effective action for the parameters also provides some new checks on conjectured infrared dualities and deconfined quantum criticality in 2+1d.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Mott insulators with boundary zeros

The topological classification of electronic band structures is based on symmetry properties of Bloch eigenstates of single-particle Hamiltonians. In parallel, topological field theory has opened the doors to the formulation and characterization of non-trivial phases of matter driven by strong electron-electron interaction. Even though important examples of topological Mott insulators have been constructed, the relevance of the underlying non-interacting band topology to the physics of the Mott phase has remained unexplored. Here, we show that the momentum structure of the Green’s function zeros defining the “Luttinger surface" provides a topological characterization of the Mott phase related, in the simplest description, to the one of the single-particle electronic dispersion. Considerations on the zeros lead to the prediction of new phenomena: a topological Mott insulator with an inverted gap for the bulk zeros must possess gapless zeros at the boundary, which behave as a form of “topological antimatter” annihilating conventional edge states. Placing band and Mott topological insulators in contact produces distinctive observable signatures at the interface, revealing the otherwise spectroscopically elusive Green’s function zeros.

36 MATERIALS SCIENCE↗
NASA NTRS2007

Bifurcation and Hysteresis of the Magnetospheric Structure with a varying Southward IMF: Field Topology and Global Three-dimensional Full Particle Simulations

Using a three-dimensional full electromagnetic particle model (EMPM), we have performed global simulations of the interaction between the solar wind and the terrestrial magnetosphere, and have investigated its asymptotic stability. The distance between the dayside magnetopause subsolar point and the Earth center, R(sub mp) is measured, as the intensity of southward IMF |B(sub z)| is slowly varying. Based on the field topology theory, one analyzes the variation of R(sub mp) as a reference index of the dynamics of this interaction, when IMF |B(sub z)| successively increases and decreases to its original value. Two striking results are observed. First, as the IMF |B(sub z)| increases above a critical value, the variation of R(sub mp) suddenly changes (so called 'bifurcation' process in field topology). Above this critical value, the overall magnetic field topology changes drastically and is identified as being the signature of magnetic reconnection at the subsolar point on the magnetopause. Second, this subsolar point recovers its original location R(sub mp) by following different paths as the IMF |B(sub z)| value increases (from zero to a maximum fixed value) and decreases (from this maximum to zero) passing through some critical values. These different paths are the signature of 'hysteresis' effect, and are characteristic of the so-called 'subcritical-type' bifurcation. This hysteresis signature indicates that dissipation processes take place via an energy transfer from the solar wind to the magnetosphere by some irreversible way, which leads to a drastic change in the magnetospheric field topology. This hysteresis is interpreted herein as a consequence of the magnetic reconnection taking place at the dayside magnetopause. The field topology reveals to be a very powerful tool to analyze the signatures of three-dimensional magnetic reconnection without the obligation for determining the mechanisms responsible for, and the consequences of the reconnection on the overall magnetospheric dynamics.

Cai, DongSheng↗
DOE OSTI2025

Frozen generalized symmetries

M-theory frozen singularities are (locally) D - or E -type orbifold singularities with a background fractional C 3 -monodromy surrounding them. In this paper, we revisit such backgrounds and address several puzzling features of their physics. We first give a top-down derivation of how the D - or E -type 7D N = 1 gauge theory directly “freezes” to a lower-rank gauge theory due to the C 3 background. This relies on a Hanany-Witten effect of fractional M5 branes and the presence of a gauge anomaly of fractional D p probes in the circle reduction. Additionally, we compute defect groups and 8D symmetry topological field theories of the 7D frozen theories in several duality frames. We apply our results to understanding the evenness condition of strings ending on O 7 + planes, and calculating the global forms of supergravity gauge groups of M-theory compactified on T 4 / Γ with frozen singularities. We also revisit IIA A D E singularities with a C 1 -monodromy along a 1-cycle in the boundary lens space and show that this freezes the gauge degrees of freedom via confinement. Published by the American Physical Society 2025

Cvetič, Mirjam↗
DOE OSTI2025

Generalized symmetries of nonsupersymmetric orbifolds

We determine generalized symmetries for 4D theories engineered via type II strings on nonsupersymmetric orbifold backgrounds ℝ 3,1 × ℝ 6 /Γ. Probe branes detect generalized symmetries via the adjacency matrix for fermionic degrees of freedom in an associated quiver gauge theory. In situations where the tachyons are sequestered away from the boundary 𝑆 5 /Γ, this exactly matches the result extracted from singular homology. In situations with an unsequestered tachyon which stretches out to the boundary, the presence of tachyonic pulses partitions up the space into several distinct sectors, and the net contribution again matches with the answer expected via quiver methods. For IIA backgrounds, the presence of a localized closed string tachyon leads to transitions in the spectrum of states, generalized symmetries, higher-group symmetries, as well as the level matrix of the associated symmetry topological field theory (SymTFT). For IIB backgrounds with a stack of spacetime filling probe D3-branes, the onset of a radiatively generated potential leads to similar considerations involving scale dependent transitions in the symmetries of the theory, including structures such as duality defects/interfaces.

compactification↗
DOE OSTI2025

Multipartite entanglement structure of fibered link states

We study the patterns of multipartite entanglement in Chern-Simons theory with compact simple gauge group 𝐺 and level 𝑘 for states defined by the path integral on “link complements,” i.e., compact manifolds whose boundaries consist of 𝑛 topologically linked tori. We focus on link complements which can be described topologically as fibrations over a Seifert surface. We show that the entanglement structure of such fibered link complement states is controlled by a topological invariant, the monodromy of the fibration. Thus, the entanglement structure of a Chern-Simons link state is not simply a function of the link, but also of the background manifold in which the link is embedded. In particular, we show that any link possesses an embedding into some background that leads to Greenberger–Horne–Zeilinger state (GHZ)-like entanglement. Furthermore, we demonstrate that all fibered links with periodic monodromy have GHZ-like entanglement, i.e., a partial trace on any link component produces a separable state. These results generalize to any three dimensional topological field theory with a dual chiral rational conformal field theory.

conformal field theory↗
DOE OSTI2025

Phases of Theories with ℤ 𝑁 1-Form Symmetry, and the Roles of Center Vortices and Magnetic Monopoles

We analyze the phases of theories having a microscopic ℤ 𝑁 1-form symmetry, starting with a topological BF theory and deforming it so that only the microscopic symmetry is preserved. These theories have a well-defined notion of confinement, prototypical examples being pure SU⁡(𝑁) and ℤ 𝑁 gauge theories in the continuum and on the lattice. Our analysis shows that the generic phases are in 𝑑 = 2, only the confined phase; in 𝑑 = 3, both the confined phase and the topological BF phase; and in 𝑑 = 4, the confined phase, the topological BF phase, and a Coulomb phase. We construct a ℤ𝑁 lattice gauge theory with a deformation that, surprisingly, produces up to (𝑁−1) photons. We give an interpretation of these findings in terms of the behaviors of two competing drivers of confinement—magnetic monopoles and center vortices—and conclude that proliferation of center vortices is necessary but insufficient for confinement, while proliferation of magnetic monopoles is both necessary and sufficient.

Color confinement↗
DOE OSTI2026

Anyonic Membranes and Pontryagin Statistics

Anyons, unique to two spatial dimensions, underlie extraordinary phenomena such as the fractional quantum Hall effect, but their generalization to higher dimensions has remained elusive. The topology of Eilenberg-MacLane spaces constrains the loop statistics to be only bosonic or fermionic in any dimension. In this work, we introduce the novel anyonic statistics for membrane excitations in four dimensions. Analogous to the $\mathbb{Z}_N$-particle exhibiting $\mathbb{Z}_{N\times \gcd(2,N)}$ anyonic statistics in two dimensions, we show that the $\mathbb{Z}_N$-membrane possesses $\mathbb{Z}_{N\times \gcd(3,N)}$ anyonic statistics in four dimensions. Given unitary volume operators that create membrane excitations on the boundary, we propose an explicit 56-step unitary sequence that detects the membrane statistics. We further analyze the boundary theory of $(5{+}1)$D 1-form $\mathbb{Z}_N$ symmetry-protected topological phases and demonstrate that their domain walls realize all possible anyonic membrane statistics. We then show that the $\mathbb{Z}_3$ subgroup persists in all higher dimensions. In addition to the standard fermionic $\mathbb{Z}_2$ membrane statistics arising from Stiefel-Whitney classes, membranes also exhibit $\mathbb{Z}_3$ statistics associated with Pontryagin classes. We explicitly verify that the 56-step process detects the nontrivial $\mathbb{Z}_3$ statistics in 5, 6, and 7 spatial dimensions. Furthermore, in 7 and higher dimensions, the statistics of membrane excitations stabilize to $\mathbb{Z}_{2} \times \mathbb{Z}_{3}$, with the $\mathbb{Z}_3$ sector consistently captured by this process.

Abstract algebra↗
DOE OSTI2024

Creating kinks with quantum mediation

We consider the creation of kink-antikink pairs of a scalar field ϕ by the scattering of classical wave packets of a second scalar field ψ when there are no direct interactions between ϕ and ψ . The creation becomes possible only due to a quantum field that interacts with both ϕ and ψ . We scan parameter space and find it favorable for kink production when the initial wave packets have large total energy and wide spatial extent but scatter at low velocities. Published by the American Physical Society 2024

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Ordering the topological order in the fractional quantum Hall effect

Here, we discuss the possible topological order/topological quantum field theory of different quantum Hall systems. Given the value of the Hall conductivity, we constrain the global symmetry of the low-energy theory and its anomaly. Specifically, the one-form global symmetry and its anomaly are presented as the organizing principle of these systems. This information is powerful enough to lead to a unique minimal topological order (or a small number of minimal topological orders). Almost all of the known experimentally discovered topological orders are these minimal theories. Since this work is interdisciplinary, we made a special effort to relate to researchers with different backgrounds by providing translations between different perspectives.

Anyons↗
DOE OSTI2022

Topological Invariant of Manifolds in Slawianowski’s Field Theory [Slides]

Witten constructed topological invariants of manifolds by introducing actions depending only on smooth structures of the manifold without using a metric. He focused on an example of Chern-Simons’ theory on 3D manifolds. Slawianowski based his field theory on frame/coframe fields without a metric. A Witten-type topological invariant can be introduced in Slawianowski’s framework, using an integral over the reper fields. Such invariant behaves differently for group manifolds than for all others. It additionally distinguishes semi-simple Lie group manifolds from other group manifolds. Further study of this new invariant is needed, including possible generalizations to other field theories.

42 ENGINEERING↗
DOE OSTI2020

Symmetry-enriched quantum spin liquids in (3 + 1)d

We use the intrinsic one-form and two-form global symmetries of (3+1)d bosonic field theories to classify quantum phases enriched by ordinary (0-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the 0-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in (2 + 1)d. We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative ’t Hooft anomaly for different couplings. We discuss several examples including the gapless pure U(1) gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in (3 + 1)d for SU(2) gauge theory with two adjoint Weyl fermions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗