The four-dimensional (4D) Lorentz group $SL(2,\mathbb{C})$ acts as the two-dimensional (2D) global conformal on celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of flat space amplitudes and 2D correlators obscured by fact that former usually expressed in terms wavefunctions which transform simply under spacetime translations rather than $SL(2,\mathbb{C})$. In this paper we construct on-shell massive scalar Minkowski primaries....

We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in ${\mathbb{R}}^{1,d+1}$ that transform as $d$-dimensional conformal primaries under Lorentz group $SO(1,d+1)$. Such solutions, called primary wavefunctions, are labeled by a dimension $\mathrm{\ensuremath{\Delta}}$ point ${\mathbb{R}}^{d}$, rather than an on-shell ($d+2$)-dimensional momentum. show continuum scalar wavefunctions on principal continuous series...

For any quantum system invariant under gauging a higher-form global symmetry, we construct non-invertible topological defect by in only half of spacetime. This generalizes the Kramers-Wannier duality line 1+1 dimensions to higher spacetime dimensions. We focus on case one-form symmetry 3+1 dimensions, and determine fusion rule. From direct analysis protected phases, show that existence certain kinds defects is intrinsically incompatible with trivially gapped phase. give an explicit...

Recently, spin-one wave functions in four dimensions that are conformal primaries of the Lorentz group $SL(2,\mathbb{C})$ were constructed. We compute low-point, tree-level gluon scattering amplitudes space these primary functions. The answers have same covariance as correlators a $2d$ CFT. Britto--Cachazo--Feng--Witten (BCFW) recursion relation between three- and four-point is recast into this basis.

We discuss nonstandard continuum quantum field theories in 2+1 dimensions. They exhibit exotic global symmetries, a subtle spectrum of charged excitations, and dualities similar to systems 1+1 These models represent the low-energy limits certain known lattice systems. One key aspect these is important role played by discontinuous configurations. In two companion papers, we will present 3+1-dimensional versions particular, some fractons.

We extend our exploration of nonstandard continuum quantum field theories in 2+1 dimensions to 3+1 dimensions. These exhibit exotic global symmetries, a peculiar spectrum charged states, unusual gauge and surprising dualities. Many the systems we study have known lattice construction. In particular, one them is gapless fracton model. The novelty here their theory description. this paper, focus on models with $U(1)$ symmetry followup paper will $\mathbb{Z}_N$ symmetry.

We identify infinitely many noninvertible generalized global symmetries in QED and QCD for the real world massless limit. In QED, while there is no conserved Noether current U(1)A axial symmetry because of Adler-Bell-Jackiw anomaly, every rational angle 2πp/N, we construct a gauge-invariant topological operator. Intuitively, it composition rotation fractional quantum Hall state coupled to electromagnetic U(1) gauge field. These operators do not obey group multiplication law, but fusion...

Symmetry plays a central role in quantum field theory. Recent developments include symmetries that act on defects and other subsystems, are categorical rather than group-like. These generalized notions of symmetry allow for new kinds anomalies constrain dynamics. We review some transformative instances these novel aspects theory, give broad-brush overview recent applications.

We study field theories with global dipole symmetries and gauge symmetries. The famous Lifshitz theory is an example of a symmetry. in detail its $1+1\mathrm{D}$ version compact field. When this symmetry promoted to $U(1)$ symmetry, the corresponding tensor This known lead fractons. To resolve various subtleties precise meaning these or symmetries, we place on lattice then take continuum limit. Interestingly, limit not unique. Different limits different theories, whose operators, defects,...

In gauge theory, it is commonly stated that time-reversal symmetry only exists at $\theta=0$ or $\pi$ for a $2\pi$-periodic $\theta$-angle. this paper, we point out in both the free Maxwell theory and massive QED, there non-invertible every rational $\theta$-angle, i.e., $\theta= \pi p/N$. The implemented by conserved, anti-linear operator without an inverse. It composition of naive transformation fractional quantum Hall state. We also find similar symmetries non-Abelian theories, including...

A bstract In axion-Maxwell theory at the minimal axion-photon coupling, we find non-invertible 0- and 1-form global symmetries arising from naive shift center symmetries. Since Gauss law is anomalous, there no conserved, gauge-invariant, quantized electric charge. Rather, using half higher gauging, a associated with symmetry, which related to Page These act invertibly on axion field Wilson line, but non-invertibly monopoles strings, leading selection rules Witten effect. We also derive...

We study the symmetries of closed Majorana chains in 1+1d, including translation, fermion parity, spatial and time-reversal symmetries. The algebra symmetry operators is realized projectively on Hilbert space, signaling anomalies lattice, constraining long-distance behavior. In special case free Hamiltonian (and small deformations thereof), continuum limit 1+1d CFT. Its chiral parity (-1)^{F_L} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"...

Using the numerical modular bootstrap, we constrain space of $1+1\mathrm{d}$ conformal field theories (CFTs) with a finite noninvertible global symmetry described by fusion category $\mathcal{C}$. We derive universal and rigorous upper bounds on lightest $\mathcal{C}$-preserving scalar local operator for categories such as Ising Fibonacci categories. These possible robust gapless phases protected symmetry, which commonly arise from microscopic lattice models anyonic chains. also...

The authors provide a comprehensive study of the role boundary conditions in presence generalized non-invertible symmetries, beginning with question when are symmetric under those symmetries. They find different categories (which would be identical normal symmetries) and discuss their relation to 't Hooft anomalies, gauging, RG flows.

We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space qubits. This is associated with topological defect and conserved operator, latter can be presented as matrix operator. Importantly, unlike its continuum counterpart, algebra involves translations. Consequently, it not described by fusion category. In presence this defect, involving parity/time-reversal realized projectively, which reminiscent an anomaly. Different...

We derive model-independent quantization conditions on the axion couplings (sometimes known as anomaly coefficients) to standard model gauge group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mo stretchy="false">[</a:mo><a:mrow><a:mi>SU</a:mi></a:mrow><a:mo stretchy="false">(</a:mo><a:mn>3</a:mn><a:mo stretchy="false">)</a:mo><a:mo>×</a:mo><a:mrow><a:mi>SU</a:mi></a:mrow><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mo...

We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by fusion category Rep(D_{8}). Therefore, state is not only symmetry protected topological (SPT) phase, but also SPT phase. further find two new commuting Pauli Hamiltonians for other Rep(D_{8}) phases on tensor product Hilbert space of qubits, matching classification in field theory and mathematics. identify edge modes local projective algebras at interfaces between these phases. Finally,...

We explore exact generalized symmetries in the standard 2+1d lattice \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mstyle mathvariant="double-struck"><mml:mi>ℤ</mml:mi></mml:mstyle><mml:mn>2</mml:mn></mml:msub></mml:math> gauge theory coupled to Ising model, and compare them with their continuum field counterparts. One model has a (non-anomalous) non-invertible symmetry, we identify two distinct symmetry protected topological phases....

We conjecture a precise relationship between the Schur limit of superconformal index four-dimensional $$ \mathcal{N}=2 field theories, which counts local operators, and spectrum BPS particles on Coulomb branch. verify this for special case free QED, SU(2) gauge theory coupled to fundamental matter. Assuming validity our proposal, we compute all Argyres-Douglas theories. Our answers match expectations from connection operators with two-dimensional chiral algebras. Based results propose that...

A bstract We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries Verlinde lines, TDLs together with their attached operators provide models of fusion categories without braiding. study the crossing relations TDLs, discuss relation to ’t Hooft anomaly, use them constrain renormalization group flows either critical points or quantum theories (TQFTs). show that if certain non-invertible are preserved along a...

We explore the large spin spectrum in two-dimensional conformal field theories with a finite twist gap, using modular bootstrap lightcone limit. By recursively solving crossing equations associated to different $PSL(2,\mathbb{Z})$ elements, we identify universal contribution density of states from vacuum dual channel. Our result takes form sum over whose leading term generalizes usual Cardy formula wider regime. Rather curiously, becomes negative specific limit, which can be canceled by that...

Following our earlier analyses of nonstandard continuum quantum field theories, we study here gapped systems in 3+1 dimensions, which exhibit fractonic behavior. In particular, present three dual theory descriptions the low-energy physics X-cube model. A key aspect constructions is use discontinuous fields theory. Spacetime continuous, but are not.

We study the implications of 't Hooft anomaly (i.e. obstruction to gauging) on conformal field theory, focusing case when global symmetry is $\mathbb{Z_2}$. Using modular bootstrap, universal bounds (1+1)-dimensional bosonic theories with an internal $\mathbb{Z_2}$ are derived. The bootstrap depend dramatically anomaly. In particular, there a upper bound lightest odd operator if anomalous, but no non-anomalous. non-anomalous case, we find that state and defect ground cannot both be...