arXiv subjects

Mohamed M. Anber

Publications and source records attributed to Mohamed M. Anber.

At least 19 recordsLinked to original sources

arXiv2026

Anomalies on ALE spaces and phases of gauge theory

We show that certain 't Hooft anomalies not detected by standard closed four-dimensional probes can become visible when a quantum field theory is placed on asymptotically locally Euclidean (ALE) spaces. As a concrete example, we use the Eguchi-Hanson (EH) space, whose defining features are its nontrivial second cohomology generated by the self-intersecting two-sphere and its asymptotic boundary $\mathbb{RP}^3$, which carries torsion and thus furnishes additional cohomological data absent on conventional backgrounds. For a theory with symmetry $G_1\times G_2$, we turn on background flux for $G_1$ and probe potential anomalies by performing a global $G_2$ transformation; the resulting anomaly is captured by a five-dimensional mapping torus. The anomaly receives contributions from the four-dimensional characteristic classes on EH space as well as from the $η$-invariant associated with the $\mathbb{RP}^3$ boundary. The anomaly detected in this way imposes additional constraints on asymptotically free gauge theories with fixed trivial asymptotic boundary conditions. In particular, infrared composite spectra that match anomalies on standard closed manifolds may nevertheless fail to reproduce the EH anomaly, and thus cannot by themselves furnish a complete symmetry-preserving infrared realization.

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arXiv2026

$S$-duality, boundary states, and higher-form symmetries on ALE spaces

We study Abelian $S$-duality of Maxwell theory on $A$-type asymptotically locally Euclidean (ALE) spaces. Unlike on closed four-manifolds, the Maxwell path integral on an ALE space is not naturally a scalar partition function. Rather, it decomposes into theta-function blocks labeled by flat $U(1)$ holonomy sectors on the asymptotic lens-space boundary. We interpret these blocks as components of the Hilbert-space boundary state prepared by the ALE path integral. With this interpretation, the apparent failure of ordinary modularity is replaced by vector-valued modular covariance under the action of the modular group. We test this picture explicitly for Eguchi-Hanson space by gluing it to its orientation reversal. The resulting closed four-manifold is diffeomorphic to $S^2\times S^2$, and the natural pairing of the two ALE boundary states reproduces the standard Maxwell partition function on $S^2\times S^2$. We then refine the construction by turning on electric and magnetic $1$-form symmetry backgrounds. In their presence, the ALE theta blocks are not ordinary functions, but sections of a line bundle over the Cartan torus associated with the $A_{N-1}$ root lattice, reflecting the mixed electric-magnetic $1$-form anomaly. We also discuss gauging discrete $\mathbb Z_k$ subgroups of the $1$-form symmetries and show that the vector-valued boundary-state structure remains the natural covariant framework after gauging. In this sense, ALE spaces behave as chiral building blocks for four-dimensional Maxwell theory: individual ALE blocks carry sector-resolved boundary data, while gluing pairs these sectors to produce an ordinary closed-manifold partition function, much like the pairing of left- and right-moving conformal blocks in two-dimensional CFT.

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arXiv2026

Gauging the Standard Model 1-form symmetry via gravitational instantons

We investigate the fate of the Standard Model (SM) $\mathbb Z_6^{(1)}$ electric $1$-form global symmetry in the background of gravitational instantons, focusing on Eguchi-Hanson (EH) geometries. We show that EH instantons support quantized $\mathbb Z_6^{(1)}$ fluxes localized on their $S^2$ bolt, inducing fractional topological charge without backreacting on the geometry. The requirement that quark and lepton wavefunctions be globally well-defined under parallel transport imposes boundary conditions, removing ill-defined fermion zero modes; the surviving spectrum is confirmed by an explicit solution of the Dirac equation and by the Atiyah-Patodi-Singer index theorem. The Euclidean path integral in the EH background can be interpreted as a transition amplitude from an entangled state between two identical halves of space to the vacuum. Summing over all $\mathbb{Z}_6^{(1)}$ flux sectors in the path integral gauges the SM $1$-form symmetry; thus, it cannot persist as an exact global symmetry in the semiclassical limit of gravity. We further show that these fluxes induce baryon- and lepton-number violating processes, which are exponentially suppressed due to the smallness of the hypercharge coupling constant.

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arXiv2026

On the moduli space of multi-fractional instantons on the twisted $\mathbb T^4$

The moduli space of self-dual $SU(N)$ Yang-Mills instantons on $\mathbb T^4$ of topological charge $Q = r/N$, $1 \leq r \leq N-1$, is of current interest, yet is not fully understood. In this paper, starting from 't Hooft's constant field strength ($F$) instantons, the only known exact solutions on $\mathbb T^4$, we explore the moduli space via analytical and lattice tools. These solutions are characterized by two positive integers $k, \ell$, $k+\ell=N$, and are self-dual for $\mathbb T^4$ sides $L_μ$ tuned to $k L_1 L_2 = r \ell L_3 L_4$. For gcd$(k,r) = r$, we show, analytically and numerically (for $N = 3$) that the constant-$F$ solutions are the only self-dual solutions on the tuned $\mathbb T^4$, with $4r$ holonomy moduli. In contrast, when gcd$(k,r) \ne r$, we argue that the self-dual constant-$F$ solutions acquire, in addition to the $4\text{gcd}(k,r)$ holonomies, $4r - 4\text{gcd}(k,r)$ extra moduli, whose turning on makes the field strength nonabelian and non-constant. Thus, for gcd($k,r) \ne r$, 't Hooft's constant-$F$ solutions are a measure-zero subset of the moduli space on the tuned $\mathbb T^4$, a fact explaining a puzzle encountered in arXiv:2307.04795. We also show that, for $r = k = 2$, $N = 3$, the agreement between the approximate analytic solutions on the slightly detuned $\mathbb T^4$ and the $Q=2/3$ self-dual configurations obtained by minimizing the lattice action is remarkable.

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arXiv2025

Mass-deformed Super Yang-Mills theory on $\mathbb T^4$: sum over twisted sectors, $\mathbfθ$-angle, and CP violation

We study $SU(N)$ super Yang-Mills theory with a small gaugino mass $m$ and vacuum angle $θ$ on the four-torus $\mathbb{T}^4$ with 't Hooft twisted boundary conditions. Introducing a detuning parameter $Δ$, which measures the deviation from an exactly self-dual $\mathbb{T}^4$, and working in the limits $mLN \ll ΛLN \ll 1$ and $ \frac{(N-1) m^2 L^2}{4 π} \ll Δ\ll 1$, where $L$ is the torus size and $Λ$ the strong-coupling scale, we compute the scalar and pseudo-scalar condensates to leading order in $m^2L^2/Δ$. The twists generate fractional-charge instantons, and we show that summing over all such contributions is crucial for reproducing the correct physical observables in the decompactified strong-coupling regime. From a Hamiltonian perspective, the sum over twisted sectors, already at small torus size, projects in the $m=0$ limit onto a definite superselection sector of the $\mathbb{R}^4$ theory. In the massless limit, we recover the exact value of the gaugino condensate $|\langle λλ\rangle| = 16π^2 Λ^3$, and demonstrate how a spurious $U(1)$ symmetry eliminates all $CP$-violating effects. Our results are directly testable in lattice simulations, and our method extends naturally to non-supersymmetric gauge theories.

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arXiv2025

Higher-order gaugino condensates on a twisted $\mathbb T^4$: In the beginning was semi-classics

We compute the gaugino condensates, $\left\langle \prod_{i=1}^k \text{tr}(λλ)(x_i) \right\rangle $ for $1$ $\leq$ $k$ $\le$ $N-1$, in $SU(N)$ super Yang-Mills theory on a small four-dimensional torus $\mathbb{T}^4$, subject to 't Hooft twisted boundary conditions. Two recent advances are crucial to performing the calculations and interpreting the result: the understanding of generalized anomalies involving $1$-form center symmetry and the construction of multi-fractional instantons on the twisted $\mathbb T^4$. These self-dual classical configurations have topological charge $k/N$ and can be described as a sum over $k$ closely packed lumps in an instanton liquid. Using the path integral formalism, we perform the condensate calculations in the semi-classical limit and find, assuming gcd$(k,N)=1$, $\left\langle \prod_{i=1}^k \text{tr}(λλ)(x_i) \right\rangle = {\bf n}^{-1} \; N^2\left(16π^2 Λ^3\right)^k$, where $Λ$ is the strong-coupling scale and ${\bf n}$ is a normalization constant. We determine the normalization constant, using path integral, as ${\bf n} = N^2$, which is $N$ times larger than the normalization used in our earlier publication arXiv:2210.13568. This finding resolves the extra-factor-of-$N$ discrepancy encountered there, aligning our results with those obtained through direct supersymmetric methods on $\mathbb R^4$. The normalization constant ${\bf n}$ can be understood within the Euclidean path-integral framework as the Witten index $I_W$. From the Hamiltonian approach, it is well-established that $I_W = N$. While the value ${\bf n} = N^2$ correctly reproduces the condensate result, this discrepancy between the Hamiltonian and path-integral formulations calls for reconciliation. We attempt to provide a potential solution we outline in our discussion.

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arXiv2024

The Nahm transform of multi-fractional instantons

We embed the multi-fractional instantons of $SU(N)$ gauge theories on $\mathbb T^4$ with 't Hooft twisted boundary conditions into $U(N)$ bundles and use the Nahm transform to study the corresponding configurations on the dual $\widehat{\mathbb T}^4$. We first show that $SU(N)$ fractional instantons of topological charge $Q={r \over N}$, $r \in \{1, 2,...,N-1\}$, are mapped to fractional instantons of $SU(\widehat N)$ of charge $\widehat Q = {r \over \widehat N}$, where $\widehat N = N q_1 q_3 - r q_3 + q_1$ and $q_{1,3}$ are integer-quantized $U(1)$ fluxes. We then explicitly construct the Nahm transform of constant field strength fractional instantons of $SU(N)$ and find the $SU(\widehat N)$ configurations they map to. Both the $\mathbb T^4$ instantons and their $\widehat {\mathbb T}^4$ images are self-dual for appropriately tuned torus periods. The Nahm duality can be extended to tori with detuned periods, with detuning parameter $Δ$, mapping solutions with $Δ>0$ on $\mathbb T^4$ to ones with $\widehatΔ<0$ on $\widehat{\mathbb T}^4$. We also recall that fractional instantons appear in string theory precisely via the $U(N)$ embedding, suggesting that studying the end point of tachyon condensation for $Δ\ne 0$ is needed -- and is perhaps feasible in a small-$Δ$ expansion, as in field theory studies -- in order to understand the appearance and role of fractional instantons in $D$-brane constructions.

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arXiv2024

Global aspects of $3$-form gauge theory: implications for axion-Yang-Mills systems

We investigate the proposition that axion-Yang-Mills systems are characterized by a $3$-form gauge theory in the deep infrared regime. This hypothesis is rigorously examined by initially developing a systematic framework for analyzing $3$-form gauge theory coupled to an axion, specifically focusing on its global properties. The theory consists of a BF term deformed by marginal and irrelevant operators and describes a network of vacua separated by domain walls converging at the junction of an axion string. It encompasses $0$- and $3$-form spontaneously broken global symmetries. Utilizing this framework, in conjunction with effective field theory techniques and 't Hooft anomaly-matching conditions, we argue that the $3$-form gauge theory faithfully captures the infrared physics of the axion-Yang-Mills system. The ultraviolet theory is an $SU(N)$ Yang-Mills theory endowed with a massless Dirac fermion coupled to a complex scalar and is characterized by chiral and genuine $\mathbb{Z}_m^{(1)}$ $1$-form center symmetries, with a mixed anomaly between them. It features two scales: the vev of the complex scalar, $v$, and the strong-coupling scale, $Λ$, with $Λ\ll v$. Below $v$, the fermion decouples and a $U(1)^{(2)}$ $2$-form winding symmetry emerge, while the $1$-form symmetry is enhanced to $\mathbb Z_N^{(1)}$. As we flow below $Λ$, matching the mixed anomaly necessitates introducing a dynamical $3$-form gauge field of $U(1)^{(2)}$, which appears as the incarnation of a long-range tail of the color field. The infrared theory possesses spontaneously broken chiral and emergent $3$-form global symmetries. It passes several checks, among which: it displays the expected restructuring in the hadronic sector upon transition between the vacua, and it is consistent under the gauging of the genuine $\mathbb Z_m^{(1)}\subset \mathbb Z_N^{(1)}$ symmetry.

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arXiv2024

2-index chiral gauge theories

We undertake a systematic study of the $4$-dimensional $SU(N)$ $2$-index chiral gauge theories and investigate their faithful global symmetries and dynamics. These are a finite set of theories with fermions in the $2$-index symmetric and anti-symmetric representations, with no fundamentals, and they do not admit a large-$N$ limit. We employ a combination of perturbative and nonperturbative methods, enabling us to constrain their infrared (IR) phases. Specifically, we leverage the 't Hooft anomalies associated with continuous and discrete groups to eliminate a few scenarios. In some cases, the anomalies rule out the possibility of fermion composites. In other cases, the interplay between the continuous and discrete anomalies leads to multiple higher-order condensates, which inevitably form to match the anomalies. Further, we pinpoint the most probable symmetry-breaking patterns by searching for condensates that match the full set of anomalies resulting in the smallest number of IR degrees of freedom. Higher-loop $β$-function analysis suggests that a few theories may flow to a conformal fixed point.

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arXiv2023

Remarks on QCD$_4$ with fundamental and adjoint matter

We study 4-dimensional SU(N) gauge theory with one adjoint Weyl fermion and fundamental matter - either bosonic or fermionic. Symmetries, their 't Hooft anomalies, and the Vafa-Witten-Weingarten theorems strongly constrain the possible bulk phases. The first part of the paper is dedicated to a single fundamental fermion or boson. As long as the adjoint Weyl fermion is massless, this theory always possesses a $\mathbb{Z}_{2N}^χ$ chiral symmetry, which breaks spontaneously, supporting $N$ vacua and domain walls between them for a generic mass of the matter fields. We argue, however, that the domain walls generically undergo a phase transition, and we establish the corresponding 3d gauge theories on the walls. The rest of the paper is dedicated to studying the multi-flavor fundamental matter. Here, the phases crucially depend on the ratio of the number of colors and the number of fundamental flavors. We also discuss the limiting scenarios of heavy adjoint and fundamentals, which align neatly with our current understanding of QCD and $\mathcal{N}=1$ super Yang-Mills theory.

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arXiv2023

Noninvertible symmetries and anomalies from gauging $1$-form electric centers

We devise a general method for obtaining $0$-form noninvertible discrete chiral symmetries in $4$-dimensional $SU(N)/\mathbb Z_p$ and $SU(N)\times U(1)/\mathbb Z_p$ gauge theories with matter in arbitrary representations, where $\mathbb Z_p$ is a subgroup of the electric $1$-form center symmetry. Our approach involves placing the theory on a three-torus and utilizing the Hamiltonian formalism to construct noninvertible operators by introducing twists compatible with the gauging of $\mathbb Z_p$. These theories exhibit electric $1$-form and magnetic $1$-form global symmetries, and their generators play a crucial role in constructing the corresponding Hilbert space. The noninvertible operators are demonstrated to project onto specific Hilbert space sectors characterized by particular magnetic fluxes. Furthermore, when subjected to twists by the electric $1$-form global symmetry, these surviving sectors reveal an anomaly between the noninvertible and the $1$-form symmetries. We argue that an anomaly implies that certain sectors, characterized by the eigenvalues of the electric symmetry generators, exhibit multi-fold degeneracies. When we couple these theories to axions, infrared axionic noninvertible operators inherit the ultraviolet structure of the theory, including the projective nature of the operators and their anomalies. We discuss various examples of vector and chiral gauge theories that showcase the versatility of our approach.

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arXiv2023

Multi-fractional instantons in $SU(N)$ Yang-Mills theory on the twisted $\mathbb T^4$

We construct analytical self-dual Yang-Mills fractional instanton solutions on a four-torus $\mathbb{T}^4$ with 't Hooft twisted boundary conditions. These instantons possess topological charge $Q=\frac{r}{N}$, where $1\leq r< N$. To implement the twist, we employ $SU(N)$ transition functions that satisfy periodicity conditions up to center elements and are embedded into $SU(k)\times SU(\ell)\times U(1)\subset SU(N)$, where $\ell+k=N$. The self-duality requirement imposes a condition, $k L_1L_2=r\ell L_3L_4$, on the lengths of the periods of $\mathbb{T}^4$ and yields solutions with abelian field strengths. However, by introducing a detuning parameter $Δ\equiv (r\ell L_3L_4-k L_1 L_2)/\sqrt{L_1 L_2L_3L_4}$, we generate self-dual nonabelian solutions on a general $\mathbb{T}^4$ as an expansion in powers of $Δ$. We explore the moduli spaces associated with these solutions and find that they exhibit intricate structures. Solutions with topological charges greater than $\frac{1}{N}$ and $k\neq r $ possess non-compact moduli spaces, along which the $O(Δ)$ gauge-invariant densities exhibit runaway behavior. On the other hand, solutions with $Q=\frac{r}{N}$ and $k=r$ have compact moduli spaces, whose coordinates correspond to the allowed holonomies in the $SU(r)$ color space. These solutions can be represented as a sum over $r$ lumps centered around the $r$ distinct holonomies, thus resembling a liquid of instantons. In addition, we show that each lump supports $2$ adjoint fermion zero modes.

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arXiv2023

Breaking Free with AI: The Deconfinement Transition

Employing supervised machine learning techniques, we investigate the deconfinement phase transition within $4$-dimensional $SU(2)$ Yang-Mills (YM) theory, compactified on a small circle and endowed with center-stabilizing potential. This exploration encompasses scenarios both without and with matter in either the fundamental or adjoint representations. Central to our study is a profound duality relationship, intricately mapping the YM theory onto an XY-spin model with $\mathbb Z_p$-preserving perturbations. The parameter $p$ embodies the essence of the matter representation, with values of $p=1$ and $p=4$ for fundamental and adjoint representations, respectively, while $p=2$ corresponds to pure YM theory. The logistic regression method struggles to produce satisfactory results, particularly in predicting the transition temperature. Contrarily, convolutional neural networks (CNNs) exhibit remarkable prowess, effectively foreseeing critical temperatures in cases where $p=2$ and $p=4$. Furthermore, by harnessing CNNs, we compute critical exponents at the transition, aligning favorably with computations grounded in conventional order parameters. Taking our investigation a step further, we use CNNs to lend meaning to phases within YM theory with fundamental matter. Notably, this theory lacks conventional order parameters. Interestingly, CNNs manage to predict a transition temperature in this context. However, the fragility of this prediction under variations in the boundaries of the training window undermines its utility as a robust order parameter. This outcome underscores the constraints inherent in employing supervised machine learning techniques as innovative substitutes for traditional order parameters.

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arXiv2023

Noninvertible anomalies in $SU(N)\times U(1)$ gauge theories

We study $4$-dimensional $SU(N)\times U(1)$ gauge theories with a single massless Dirac fermion in the $2$-index symmetric/antisymmetric representations and show that they are endowed with a noninvertible $0$-form $\widetilde {\mathbb Z}_{2(N\pm 2)}^χ$ chiral symmetry along with a $1$-form $\mathbb Z_N^{(1)}$ center symmetry. By using the Hamiltonian formalism and putting the theory on a spatial three-torus $\mathbb T^3$, we construct the non-unitary gauge invariant operator corresponding to $\widetilde {\mathbb Z}_{2(N\pm 2)}^χ$ and find that it acts nontrivially in sectors of the Hilbert space characterized by selected magnetic fluxes. When we subject $\mathbb T^3$ to $\mathbb Z_N^{(1)}$ twists, for $N$ even, in selected magnetic flux sectors, the algebra of $\widetilde {\mathbb Z}_{2(N\pm 2)}^χ$ and $\mathbb Z_N^{(1)}$ fails to commute by a $\mathbb Z_2$ phase. We interpret this noncommutativity as a mixed anomaly between the noninvertible and the $1$-form symmetries. The anomaly implies that all states in the torus Hilbert space with the selected magnetic fluxes exhibit a two-fold degeneracy for arbitrary $\mathbb T^3$ size. The degenerate states are labeled by discrete electric fluxes and are characterized by nonzero expectation values of condensates. In an Appendix, we also discuss how to construct the corresponding noninvertible defect via the ``half-space gauging'' of a discrete one-form magnetic symmetry.

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arXiv2023

The gaugino condensate from asymmetric four-torus with twists

We calculate the gaugino condensate in $SU(2)$ super Yang-Mills theory on an asymmetric four-torus $\mathbb T^4$ with 't Hooft's twisted boundary conditions. The $\mathbb T^4$ asymmetry is controlled by a dimensionless detuning parameter $Δ$, proportional to $L_3 L_4 - L_1 L_2$, with $L_i$ denoting the $\mathbb T^4$ periods. We perform our calculations via a path integral on a $\mathbb T^4$. Its size is taken much smaller than the inverse strong scale $Λ$ and the theory is well inside the semi-classical weak-coupling regime. The instanton background, constructed for $Δ\ll 1$ in arXiv:hep-th/0007113, has fractional topological charge $Q=\frac{1}{2}$ and supports two gaugino zero modes, yielding a non-vanishing bilinear condensate, which we find to be $Δ$-independent. Further, the theory has a mixed discrete chiral/$1$-form center anomaly leading to double degeneracy of the energy eigenstates on any size torus with 't Hooft twists. In particular, there are two vacua, $|0\rangle$ and $|1\rangle$, that are exchanged under chiral transformation. Using this information, the $Δ$-independence of the condensate, and assuming further that the semi-classical theory is continuously connected to the strongly-coupled large-$\mathbb T^4$ regime, we determine the numerical coefficient of the gaugino condensate: $\langle 0| \mbox{tr}λλ|0\rangle=|\langle 1| \mbox{tr}λλ|1\rangle|=32π^2 Λ^3$, a result equal to twice the known $\mathbb R^4$ value. We discuss possible loopholes in the continuity approach that may lead to this discrepancy.

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arXiv2022

New Anomalies, TQFTs, and Confinement in Bosonic Chiral Gauge Theories

We study a class of 4-dimensional $SU(N)$ chiral gauge theories with fermions in the 2-index symmetric and antisymmetric representations and classify their infrared phases. The choice $N=4\mathbb{Z}$ corresponds to gauging the fermion number and makes the theory purely bosonic. We examine the most general background fields of the centers of the gauge, non-abelian flavor, and $U(1)$-axial groups that can be consistently activated, thereby determine the faithful global continuous and discrete symmetries of the theory. This allows us to identify new mixed 0-form/1-form `t Hooft anomalies on both spin and nonspin manifolds. If the theory confines, the absence of composite fermions implies that continuous symmetries must be broken down to anomaly-free subgroups. Anomalies associated with discrete symmetries can be saturated either by breaking the symmetry or by a symmetry-preserving topological quantum field theory (TQFT). The latter, however, is obstructed on spin manifold. The interplay between these features greatly restricts the possible infrared physics. We present two examples that demonstrate our approach. We argue that if the theory confines, the zoo of anomalies and TQFT obstruction greatly restrict the viable infrared condensates. We also discuss the possibility that some theories flow to a conformal fixed point.

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arXiv2021

Nonperturbative effects in the Standard Model with gauged 1-form symmetry

We study the Standard Model with gauged $\mathbb Z_{N=2,3,6}^{(1)}$ subgroups of its $\mathbb Z_6^{(1)}$ 1-form global symmetry, making the gauge group $SU(3) \times SU(2)\times U(1) \over \mathbb Z_N$. We show that, on a finite $\mathbb T^3$, there are self-dual instantons of fractional topological charge. They mediate baryon- and lepton-number violating processes. We compare their amplitudes to the ones due to the usual BPST-instantons. We find that the small hypercharge coupling suppresses the fractional-instanton contribution, unless the torus size is taken sub-Planckian, or extra matter is added above the weak scale. We also discuss these results in light of the cosmological bounds on the torus size.

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arXiv2021

Condensates and anomaly cascade in vector-like theories

We study the bilinear and higher-order fermion condensates in $4$-dimensional $SU(N)$ gauge theories with a single Dirac fermion in a general representation. Augmented with a mixed anomaly between the $0$-form discrete chiral, $1$-form center, and $0$-form baryon number symmetries (BC anomaly), we sort out theories that admit higher-order condensates and vanishing fermion bilinears. Then, the BC anomaly is utilized to prove, in the absence of a topological quantum field theory, that nonvanishing fermion bilinears are inevitable in infrared-gapped theories with $2$-index (anti)symmetric fermions. We also contrast the BC anomaly with the $0$-form anomalies and show that it is the former anomaly that determines the infrared physics; we argue that the BC anomaly lurks deep to the infrared while the $0$-form anomalies are just variations of local terms. We provide evidence of this assertion by studying the BC anomaly in vector-like theories compactified on a small spacial circle. These theories are weakly-coupled, under analytical control, and they admit a dual description in terms of abelian photons that determine the deep infrared dynamics. We show that the dual photons talk directly to the $1$-form center symmetry in order to match the BC anomaly, while the $0$-form anomalies are variations of local terms and are matched by fiat. Finally, we study the fate of the BC anomaly in the compactified theories when they are held at a finite temperature. The effective field theory that describes the low-energy physics is $2$-dimensional. We show that the BC anomaly cascades from $4$ to $2$ dimensions.

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