Higher-form symmetries in 5d

We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and ’t Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as G2-holonomy manifolds, which give rise to 4d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 theories.


Introduction
Higher-form symmetries [1]  In this paper we will initiate the study of higher-form symmetries in 5d supersymmetric gauge theories. Five-dimensional gauge theories are not renormalizable, however much evidence has been accumulated that points towards the existence of UV fixed points, which are interacting super-conformal field theories (SCFTs) that flow after perturbation to the gauge theory. Recently much progress has been made in the classification of 5d and 6d SCFTs . The case for the existence of such strongly-coupled fixed points relies on geometric constructions in M-theory and F-theory on non-compact Calabi-Yau threefolds, which model the entire parameter space of the gauge theory, including the strongly coupled limit. For 5d N = 1 gauge theories, the Calabi-Yau geometries are resolutions of so-called canonical singularities, which characterize the SCFT [2,3]. The extended Coulomb branch, which incorporates both the vacuum expectation values of the scalars in the vector multiplet as well as masses of hypermultiplet matter fields, is geometrically realized in terms of the relative Kähler cone of the resolved Calabi-Yau geometry [4]. The compact 4-cycles, i.e., divisors, characterize the Cartan subalgebra of the gauge group, where their volume sets 1/g 2 Y M . This geometric approach has advantages as it manifestly encodes properties of the UV-fixed point, such as the enhanced flavor symmetry [3,[5][6][7][8], which in the geometry is encoded in the intersections of compact and non-compact divisors in the Calabi-Yau geometry. Furthermore, this approach does not rely on the existence of a weakly-coupled gauge theory description, and is applicable in cases where the UV fixed point does not have a Lagrangian description, such as in the case of the rank one P 2 theory [2,3].
The main observation of this paper is that 5d gauge theories can have discrete higherform symmetries, which we first study in the gauge theory description and provide criteria for their existence. 1 Specifically, we will focus on the case of 1-form symmetries, which in the case of gauge theories can be understood as electric or center symmetries arising from the action of the center of the gauge group. In general, these theories may also have 2-form magnetic symmetries, which arise when the gauge group is not simply connected. However, we may always pass from 1-form to 2-form symmetries and back by gauging these higherform symmetries, and so in this paper we will focus for concreteness on theories with 1-form symmetries only. In the gauge theory language, this means we typically take the gauge group to be simply connected. The field theoretic approach is complemented with an analysis in M-theory, where we find a geometric characterization of the 1-form symmetry in terms of the intersection theory on the non-compact Calabi-Yau threefold. This is shown to agree with the 1-form symmetry computed from the gauge theory description, whenever such a formulation exists, but is applicable more generally, e.g., to the rank one P 2 theory, which we show to have a Z 3 1-form symmetry.
Although the computation of the 1-form symmetry relies on the resolved geometry, we anticipate that the 1-form symmetry will be present also at the UV fixed point, where the gauge theory description breaks down, but the geometric description still applies. This is also supported by the fact that the 1-form symmetry is invariant under so-called 'UV dualities', i.e., two IR descriptions that share the same UV SCFT, have the same 1-form symmetry. Such dualities can be inferred geometrically by determining distinct rulings of compact surfaces in the Calabi-Yau geometry, and we show that permuting these leaves invariant the intersection data that determines the 1-form symmetry. The 1-form symmetry is also invariant under changes in the Coulomb branch, which do not alter the weakly-coupled gauge theory description, which in turn correspond in the geometry to flop transitions between the compact 1 1-form symmetries in 5d and 6d SCFTs were discussed in [9] (and upcoming work [10]) using representation theory of the superconformal algebra. It was shown that there cannot be any continuous 1-form symmetries, but those works do not exclude discrete higher-form symmetries in 5d and 6d, which will be the focus of our attention.
surfaces. In the case of toric Calabi-Yau threefolds this is particularly manifest, as the 1-form symmetry depends on the structure of the non-compact divisors (the external vertices in the toric diagram), and not on the specific triangulation of the toric diagram.
In the M-theory realization, the line operators that are charged under the 1-form symmetry are realized by M2-branes wrapping non-compact curves, which are infinite mass objects whose world-lines are line operators in the theory. However, the charges of such operators can be screened by dynamical particles, which correspond to M2 branes wrapping compact cycles . This perspective implies that the 1-form symmetry can be characterized in terms of the cohomology of the non-compact Calabi-Yau threefold M 6 relative to its boundary, H 2 (M 6 , ∂M 6 ). Specifically, we find the 1-form symmetry group, Γ, is given schematically by 2 Γ = H 2 (M 6 , ∂M 6 ; Z)/H 2 (M 6 ; Z) . (1.1) The magnetic dual 't Hooft operators are similarly understood in terms of M5-branes wrapped on non-compact divisors, which are labeled by the Poincaré-Lefschetz-dual homology groups.
We can also understand this formula from the viewpoint of asymptotic fluxes of the M-theory three-form, C. The quotient group in (1.1) can be straightforwardly computed from the intersection matrix of compact divisors and compact curves in M 6 , which we use to determine it in many examples. This point of view also allows generalization to other M-theory compactifications such as higher dimensional Calabi-Yau manifolds and G 2 and Spin(7) holonomy spaces.
All 5d SCFTs that have thus far been constructed have an origin in a 6d SCFT, by circlecompactification and mass deformations (or equivalently, holonomies in the flavor symmetry).
In 6d theories, one can have non-trivial 2-form symmetries, giving rise to the "defect group" of the theory, which is determined by the base of the elliptic fibration in F-theory models [11,12].
There are also discrete 1-form symmetries in 6d and together with the 2-form symmetry they contribute to the 1-form symmetry in the dimensional reduction to 5d. We provide evidence for this by determining the higher-form symmetries in 6d for the building blocks, namely the non-Higgsable clusters (NHCs) and conformal matter theories, and find agreement with the 1-form symmetry in 5d computed by intersection theory in the resolved Calabi-Yau threefold.
We begin the paper with a brief review of higher-form symmetries in section 2, where we discuss also the 5d gauge theory description of 1-form symmetries. In section 3 we find a characterization of the 1-form symmetry in terms of the geometry of the M-theory realization, 2 We refer to section 2.2 for the precise formula, which involves the long exact sequence in relative homology. In addition, here and in most of this paper we assume that M6 does not have torsion in its homology groups, which is the case in most of the examples considered in physics. It would be interesting to extend these arguments to the torsion case. and find an explicit formula in terms of the intersection theory on the Calabi-Yau threefold.
In section 4 we apply this formalism to a large class of 5d theories, starting with the N = 2 theories, and pure Super-Yang Mills. We furthermore give a prescription for how to determine the 1-form symmetry for any collection of glued compact surfaces which are (blowups of) Hirzebruch surfaces. This allows us to check numerous UV dualities in section 4.5, which are shown to be consistent with the 1-form symmetry. Another large class of 5d SCFTs are obtained from toric Calabi-Yau threefolds, which we discuss in section 5. To provide a more systematic framework that applies to all 5d SCFTs, it is key to determine the relation between the higher-form symmetry in 6d and in 5d, which we provide in section 6. The higherform symmetry for NHCs and conformal matter theories are computed and shown to be in agreement with the dimensional reductions to 5d. In section 7 we consider generalizations to other M-theory compactifications. We also discuss the relation of the 1-form symmetry to the fluxes of the three-form in M-theory, and the associated surface operators. We close in section 8 with conclusions and an outlook to future directions. The appendices contain computational details.
While we were completing this paper we were informed of related work to appear in [13] after one of us presented our results in [14].

Higher-Form Symmetries in Gauge Theories
In this section we briefly review higher-form symmetries in general, and then focus on the case of most interest in this paper, which are 5d N = 1 gauge theories.

Generalities about Higher-Form Symmetries
Higher-form symmetries [1] can be defined in terms of a set of topological surface operators, which obey various axioms. We refer to [1] for details, and only sketch some general features, mostly to fix notations.
A q-form symmetry with group Γ implies the existence of a set of (d − q − 1)-dimensional topological surface operators labeled by an element of Γ, which are called the "charge operators." These fuse according to the group law of Γ. They may have non-trivial correlation functions with operators of dimension q with which they are linked in spacetime, and these operators may be decomposed into representations of Γ. The case of q = 0 corresponds to ordinary symmetries, where the charge operators measure the charge of all local operators contained in the surface. For q > 0, Γ must be abelian. If the group Γ has a continuous part then there is a higher-form conserved current, but if Γ is purely discrete, there is not. We will mostly focus on the discrete case.
Gauge theories provide natural examples of theories with higher-form symmetries. Let G be the gauge group of the theory, which we may assume has some charged matter content transforming in a (generally reducible) representation, R. Let Z G be the center of G, and let Γ e be the subgroup of Z G , which acts trivially on R. If this is non-trivial, then we say the theory has an "electric" 1-form symmetry with group Γ e . Similarly, if G is non-simply connected, with π 1 (G) ∼ = Γ m , then the theory has an "magnetic" (d − 3)-form symmetry with group Γ m . The charged operators of the electric symmetry are Wilson loops, which are always 1-dimensional, and those of the magnetic symmetry are 't Hooft operators, which have Given a discrete q-form symmetry with group Γ, we may "gauge" this symmetry, 3 which produces a new theory with a "dual" (d − q − 2)-form symmetry, Γ, where Γ ≡ Hom(Γ, U (1)) is the Pontryagin dual group. Gauging the dual group returns us to the original theory. More generally, we may gauge a subgroup H of Γ, which produces a theory with q-form symmetry Γ/H and (d − q − 2)-form symmetry H. For example, given a gauge theory with simply connected group G and a 1-form electric symmetry Γ e , gauging this symmetry gives rise to the theory with gauge group G/Γ e , which now has a Γ m = π 1 (G/Γ e ) ∼ = Γ e (d − 3)-form symmetry.
If we deform the action of a QFT by a relevant local operator, this will break ordinary (0-form) symmetries the operator is charged under. However, there may also be accidental symmetries in the IR. In the case of higher-form symmetries (q > 0), the former effect does not happen, since local operators are not charged under such a symmetry, and so we typically expect that the higher-form symmetry group can only increase under RG flow. For example, integrating out massive charged matter in a gauge theory can increase the electric 1-form symmetry, as now there can be a larger subgroup of the center acting trivially on the remaining matter. We will see this monotonicity property is satisfied in examples below.

5d N = 1 Gauge Theories
Let us now specialize to the case of gauge theories with d = 5, and in particular to theories with N = 1 supersymmetry, which will be the main focus of this paper. First, consider a Yang-Mills theory with a simple gauge group, G, which we take to be simply connected. 4 Let Z G be the center of the group. If there are no further matter representations or interaction 3 Here we assume the symmetry is not anomalous. We will discuss anomalies shortly. 4 As discussed above, we always have the option to take the gauge group to be simply connected, which may lead to 1-form electric symmetries. Gauging some or all of these symmetries gives rise to the possible non-simply connected versions of the gauge group, and these will have 2-form magnetic symmetries. We may characterize this in general as follows. We start with the center of the gauge group, Z G , which is isomorphic to the quotient of the coweight lattice by the coroot lattice of the Lie algebra The center is furthermore isomorphic to the automorphism group of the extended Dynkin diagram associated to G [15]. Each irreducible representation R of G determines a character, The set of all such characters generate a subgroup, H, of Z G , the group of characters of Z G , i.e., if the matter representation decomposes into irreducible representations Then the 1-form symmetry, Γ, preserved by the matter is the annihilator of this subgroup, i.e., For the simple gauge groups we list the centers in table 1 and the transformations of some of the representations; see, e.g., Theorem 23.12 in [16] and [17]. For E n the representations refer to F = (1, 0, · · · , 0), C = (0, · · · , 0, 1, 0), S = (0, · · · , 0, 1), where the roots are labeled α 1 · · · , α n−1 and α n attaches to α n−3 . For E 6 this is F = 27, S = 78 and C = 27, whereas for E 7 , F = 56 and S = 912 and C = 133.
This applies also to semi-simple gauge groups. For example, consider a quiver with bifundamental matter connecting the two gauge groups. Without this matter, the 1form symmetry would be the center, Z N 1 × Z N 2 . The matter is charged under the center as is the greatest common divisor, and the 1-form symmetry is given by the annihilator, Z d , generated by

Chern-Simons Terms and Theta Angles
The above prescription determines the 1-form symmetry for all gauge groups besides SU (N ), for N ≥ 3, and Sp(N ). However, in these cases, there are additional interaction terms we may include that can affect the 1-form symmetry, namely, the Chern-Simons and theta angle terms, respectively. These also interact with the matter content as, e.g., integrating out charged matter can shift the effective CS term. Thus we must take care with how we specify these terms in the presence of charged matter.
First consider a pure SU (N ) gauge theory, for which we include a level k Chern-Simons term. Then we claim the 1-form symmetry is broken from Z N to Z gcd(N,k) . This follows from the fact that an instanton particle in the presence of such a CS term gains an effective charge of k under the Z N center of the group, and so breaks the 1-form symmetry as would a charged matter field. Similarly, we claim for Sp(N ), a non-trivial theta angle (i.e., θ = π) breaks the 1-form symmetry from Z 2 to the trivial group. 6 Next let us include matter. First, consider an SU (N ) theory with fundamental matter.
We may integrate out the matter with a large positive or negative mass, and the theory we obtain after doing so will have some CS levels differing by 1, say, k and k + 1, respectively. Normally this is described by saying the level of the theory with the fundamental matter is k + 1 2 , and integrating it out leads to a shift by ± 1 2 . However, the actual CS term appearing in the action must be integer quantized, and the effective shift by 1 2 from the matter is an effect of the fermion determinant, which is not really a half-integer CS term. This is discussed in [18] (see also [19] for an analogous discussion in the 3d case).
For the purposes of determining the 1-form symmetry, it is always the "bare," integerquantized CS term appearing in the action that is relevant. To make this manifest, we will take 5 Here we denote characters of ZN by the elements of U (1) ∼ = R/Z to which they map the generator of ZN . 6 Another way to see that the symmetry is broken is to note that if we place the SU (N )/Z d theory on a spacetime M4 × S 1 , the presence of fractional instantons means the CS term is not invariant under large transformations along the S 1 direction. A similar argument holds for the theta angle in Sp(N ) theories.
the "U (1) −1/2 " convention of [18], which means that, e.g., for SU (N ) k with a fundamental hypermultiplet, integrating out the CS term with a large positive (respectively, negative) mass leads to the pure SU (N ) k (respectively, SU (N ) k−1 ) theory. More generally, integrating out a representation R with positive mass does not shift the CS level, while doing it with negative mass may shift the CS level by an integer, −A R , given by the cubic Casimir of the representation. For example, for matter in the mth antisymmetric power of the fundamental representation, R = Λ m , we have [20], A useful way to characterize the effect of the CS term is to assign it a character in Z G , as we did for the matter above. Namely, to each SU (N ) factor with N > 3, and with a CS term k, we assign the character χ k = k N . Then the 1-form symmetry is the annihilator of the subgroup of Z G generated by the matter characters and all the CS characters, However, our choice of U (1) −1/2 quantization is an arbitrary choice, and we must check that the 1-form symmetry is independent of this choice. In effect, this means we must show 1-form symmetry is unchanged under shifts of the form where R is any representation present in the theory. For the corresponding characters to generate the same group, H, it suffices to show that Namely, if this is true, then and so the 1-form symmetry is the same for both conventions. For example, for the mth antisymmetric representation, Λ m of SU (N ), which has χ R = m N , it suffices to show that: (2.10) We have checked this for many values of N and m and this appears to hold, as well as the analogous statement for more general representations. It would be interesting to prove this in full generality.
Let us consider again the quiver: where we now make explicit the CS terms. Above we saw this has 1-form symmetry Z d , where d = (N 1 , N 2 ). But had we taken the opposite convention for the CS levels, we would instead find: Now the CS levels break the 1-form symmetry in each node to Z d before accounting for the matter, and then the matter breaks this to the diagonal subgroup. Thus we indeed obtain the same answer using either convention.
Similar considerations apply for the theta angle in Sp(N ) theories. Here we take the convention that integrating out fundamental matter with a positive mass does not affect the theta angle, while doing it with a negative mass shifts it by π.
Finally, let us comment on anomalies. We claim that the above 1-form symmetries are non-anomalous, and so may be freely gauged to pass between the electric 1-form symmetries and magnetic 2-form symmetries, or equivalently, between simply and non-simply connected versions of the gauge group. Namely, a q-form symmetry has an anomaly when the charge operators are themselves charged under the symmetry. For this to be possible, they must have the same dimension as the charged operators, i.e., q = d − q − 1. For example, this can happen for 1-form symmetries in 3d, and the anomaly is typically contributed by 3d Chern-Simons terms. In contrast, 1-form symmetries cannot have such an anomaly in 5d, and we claim that 5d Chern-Simons terms do not just contribute an anomaly, but actually break the 1-form symmetry. We can understand this difference from the fact that, in 5d, there are dynamical particles (the instanton particles) that gain a charge in the presence of the 5d Chern-Simons terms, and so break the symmetry, while an analogous process does not occur in 3d. We note, however, that there could, in principle, be an anomaly for 2-form symmetries in 5d. However, all 2-form symmetries we encounter in this paper arise by gauging a non-anomalous 1-form symmetry and so are not anomalous.
In the next section, we will determine the 1-form symmetry of 5d N = 1 theories engineered from CY3 compactifications in terms of the topology of the Calabi-Yau manifold. In many cases, these theories have effective descriptions as 5d gauge theories, in some cases having more than one such description. We will check below that the above field-theoretic prescription and the geometric prescription for the 1-form symmetry agree.

Higher-Form Symmetries in 5d M-theory Compactifications
In this section we derive the higher-form symmetry of QFTs that arise from compactification of M-theory on non-compact manifolds. In this section we will focus on the case of 5d N = 1 theories arising by compactification on a Calabi-Yau threefolds. However, many of the arguments we will use are more general, and we comment on extensions to other compactifications in section 7 below.

Canonical Calabi-Yau Threefold Singularities
To characterize a 5d N = 1 gauge theory, and its UV SCFT fixed points, we consider Mtheory on a canonical Calabi-Yau threefold singularity [3,4]. A crepant resolution, i.e., one which keeps the canonical class trivial, introduces a collection of complex surfaces S i , i = 1, · · · , r, dual to (1, 1)-forms. These are identified in M-theory by expanding the three- Along special loci in the Coulomb branch, the theories can have non-abelian gauge theory descriptions. In the geometry this corresponds to the existence of rulings of the surfaces S i .
The surfaces can be collapsed to curves of singularities along the sections of the rulings. The gauge couplings are set by vol(S i ) = 1/g 2 i so that collapsing these curves further to zero volume results in the UV fixed point. Much recent progress has been made in studying 5d gauge theories and SCFTs using this geometric approach [5-8, 18, 20-32].
There are various approaches to studying the geometry of the canonical Calabi-Yau threefold singularity and its resolutions. As we will see shortly, what is important in determining the higher-form symmetry is the intersection theory of compact and non-compact cycles in the resolved geometry. The simplest class of geometries are toric Calabi-Yau threefolds, for which the intersection matrix can be computed simply from the toric data. This is the topic of section 5.
A complementary approach starts with collections of compact, intersecting surfaces and characterizes when these yield 5d gauge theories and SCFTs [23,[25][26][27][28][29]. This last approach is quite useful in computing the 1-form symmetry, although it does not encode manifestly the flavor symmetry, and we discuss it in section 4.2.
A top down approach, which is applicable to all 5d theories that descend from 6d by S 1reduction and mass deformation, is to study the resolutions of singular elliptic Calabi-Yau threefolds that underlie the F-theory realization of the 6d theory. Resolving these geometries results in non-flat fibrations (i.e. fibrations where the dimension of the fiber can jump due to the presence of complex surface components) [5-8, 24, 31]. In this case the intersections can be computed from the explicit resolution divisors and curves. This approach is particularly useful in studying the (zero-form) flavor symmetry of the SCFTs, using structures called the combined fiber diagrams (CFDs). We will apply this approach in various instances to cross-check against the toric and surface approaches.

1-form Symmetry from Effective Coulomb Branch Description
In order to derive the 1-form symmetry in these theories, we will first consider each theory in a particularly simple phase, corresponding to an effective abelian gauge theory description on the Coulomb branch. Namely, consider the compactification of M-theory on a smooth, noncompact (respectively, compact) Calabi-Yau 3-fold M 6 , giving rise to an effective 5-dimensional QFT (respectively, quantum gravity theory) at low energies. Let us denote by S (5) the spacetime of this effective theory, so that the total space is M 6 × S (5) . Then we find the following field content on S (5) [4]: .., r, where r = rank(H 2 cpt (M 6 ; R)), run over a basis of the 2-cocycles in de Rham cohomology with compact support. Then we may take the following ansatz for the C 3 field, where A i is an ordinary (1-form) gauge field on S (5) . This leads to a U (1) r gauge group in the effective theory, with Lie algebra H 2 cpt (M 6 ; R).
• These gauge fields are subject to large gauge transformations, and the coweight lattice can be identified with: 7 We note for later convenience that Poincaré-Lefschetz duality allows us to also write so that the generators of the gauge group can also be identified with (compact) 4-cycles, i.e., divisors of the Calabi-Yau threefold.
• For each compact 2-cycle, C ∈ H 2 (M 6 ; Z), we may wrap an M2 brane on C, leading to a particle in the effective QFT. The charge of this particle under the gauge group generator corresponding to a divisor, S, is given by where the RHS is the intersection number in M 6 . The mass of the particle is determined by the symplectic volume of C.
An additional ingredient in the effective description are Chern-Simons interactions, which are controlled by triple intersection numbers for the compact divisors. We will discuss these in more detail below. Now let us derive the 1-form symmetry of this effective theory. We will consider two distinct, but related, arguments. First, consider the line operators in the theory. Recall that wrapping M2 branes on compact 2-cycles gives dynamical, finite mass particles. On the other hand, wrapping them on non-compact cycles gives infinite mass probe particles, whose world lines define line operators in the theory. These correspond to elements in the relative homology group (we will henceforth take the coefficient group to be Z unless otherwise specified), where we have introduced a shorthand notation for (co)homology relative to the boundary.
These line operators can be screened by dynamical particles, and so the group labeling the distinct classes of line operators modulo screening is where we quotient by the image of H 2 (M 6 ) under the map f 2 in the long exact sequence associated to the pair (M 6 , ∂M 6 ), (3.7) As Γ classifies the line operators of the theory, it is natural to conjecture that Γ is the 1-form symmetry group. By exactness of the sequence (3.7), we can also identify That it, it is given by the 1-cycles in the boundary, ∂M 6 , which become trivial when included into the bulk.
For another derivation, recall that a pure U (1) r gauge theory has a U (1) r 1-form symmetry, but this can be broken to a subgroup by charged matter. In general, the 1-form symmetry preserved in the presence of matter is the subgroup of the center, Z G , of the gauge group, which acts trivially on the matter. 8 Suppose the matter weights generate a lattice, Ω * , in t * , and let Ω be the dual lattice inside t. Then Ω contains the co-weight lattice Λ cw , and the subgroup of the center acting trivially on the matter is given by 9) or, by duality, This latter relation can also be understood as the statement that the absence of matter fields of a given weight implies there are Wilson lines that are not screened by the matter, so whose 1-form charge is non-trivial, as above.
It remains to identify the lattice Ω * of matter weights from the topology of M 6 . From The matter fields come from M2 branes wrapping cycles in H 2 (M 6 ), and, by Poincaré-Lefschetz duality, this group can also be identified with H 4 cpt (M 6 ). Then, from (3.4), the weights in Λ w which are realized by matter are those maps which arise by taking the intersection numbers with 2-cycles. One can show these lie in the image of the mapf 4 fitting into the long exact Thus we have and so But the sequence (3.12) is precisely what is obtained by applying Poincaré-Lefschetz duality to the sequence (3.7), and the quotient in (3.14) is isomorphic to the quotient in (3.6), giving another derivation of this formula.
The above derivation only strictly applies for a smooth Calabi-Yau manifolds, while the cases of most interest often have some singularities. For example, we may associate to a singular manifold, X, a CFT, while its regularizations,X, may correspond to effective descriptions on its (extended) Coulomb branch. We conjecture that the 1-form symmetry associated to a theory, T X , engineered by a singular manifold, X, is the same as that for any point on the extended Coulomb branch, i.e., the group Γ determined by (3.6) is the same for any resolution, 10 X, of X. In subsequent sections we will present several pieces of evidence for this statement.
For example, in section 4.3 we will argue the 1-form symmetry is invariant under flop transitions. In the case of a singular geometry with a non-abelian gauge theory description, we can also compare to the field theory result of section 2.2, and this is the subject of section 4. In addition, in section 7 we will consider an alternative derivation of the 1-form symmetry that depends on the behavior at the asymptotic boundary of the non-compact Calabi-Yau, and so is less sensitive to the de-singularization in the interior.

Explicit Formula from Intersections
To give a more explicit formula for the 1-form symmetry, let us continue to assume, for simplicity, that M 6 has no torsion in any of its integer homology groups. Then we may write where b i is the ith Betti number. For compact M 6 , we would have b i = b 6−i , but more generally, we have from Poincaré-Lefschetz duality, Now the intersection pairing between H i and H 6−i can be represented by Then the map f i in the exact sequence (3.7) above is explicitly Thus we have Here we recall b 4 = r is the rank of the effective gauge group. Moreover, we have that b 2 = r+f with f ≥ 0 the rank of the flavor group. 11 Thus M 4 is an r × (r + f ) matrix.
To compute the quotient, it is convenient to write M 4 in Smith normal form, as where S and T are invertible r × r and (r + f ) × (r + f ) integer matrices, respectively, and D is diagonal, where α i ∈ Z ≥0 and α i divides α i+1 . Then we find, Incidentally, we note the transpose of this matrix, M 2 = M T 4 , appears in the quotient, Namely, by a similar argument, this group is given by Then using the same Smith normal form decomposition above, we see this group is given by In section 7 we will understand this identification from the point of view of the asymptotic fluxes of the C 3 field.

Compact M 6 and the Swampland
Let us consider this formula when M 6 is compact. Then ∂M 6 is empty, so f i is an isomorphism, and so the RHS of (3.6) is trivial, meaning there is no 1-form symmetry. Equivalently, by such states would all be of finite mass, and break the 1-form symmetry. Both statements 11 As discussed in, e.g., [18], we may identify f = rk(H4(M6))−rk(H2(M6)) with the rank the flavor symmetry.
are consistent with the fact that we expect, given the compactness of M 6 , that this theory describes a quantum gravity theory, and then the expectation from the weak gravity conjecture (see, e.g., [33,34] 12 ) is that such a theory (i) has particles with all possible gauge charges, and (ii) has no global symmetry, including higher-form symmetry. On the other hand, if M 6 is non-compact, we may have a non-trivial 1-form symmetry. This is consistent with the fact that for non-compact M 6 , the theory is expected to be a QFT, and so may have non-trivial global symmetries.
It is tempting to conjecture a converse to the above statement, namely, that any QFT not coupled to gravity has a global symmetry. In particular, we observe that any theory with no 0-form global symmetry has a non-trivial 1-form symmetry. In rank 1 this the P 2 Seiberg theory, which has Z 3 1-form symmetry; in rank 2, there are two theories with no 0-form global symmetry, which are also non-Lagrangian (with geometries F 3 ∪ P 2 and F 6 2 ∪ P 2 [23], or models 67 and 68 in [6]), but have 1-form symmetries Z 5 and Z 4 as can be computed from the geometry. Finally, in section 5, we will see there are theories that are descendants of T N , which have G F = 1 but also non-trivial 1-form symmetry. It would be interesting to explore this observation further.

5d Gauge Theories and SCFTs from Calabi-Yau Threefolds
We saw in section 3 that the higher-form symmetry of an M-theory compactification is determined by the intersection pairing of the compactification manifold. In this section, we will consider some examples of Calabi-Yau geometries engineering 5d N = 1 gauge theories, and compute the 1-form symmetry by studying their topology. We can then compare this result to the 1-form symmetry of the corresponding field theories, as described in section 2.2, which provides a strong consistency check of our results.

5d N = 2 Theories
Let us start by considering M-theory on a Calabi-Yau threefold of the form where Γ ADE is a finite subgroup of SU (2), with ADE classification, and T 2 is given a complex structure. This engineers the 5d N = 2 SYM theory for the corresponding ADE Lie algebra.
The higher-form symmetry of this theory depends on the choice of global form of the gauge group: for the simply connected choice there is a 1-form symmetry given by the center of the group, which is the abelianization, Ab[Γ ADE ]. This theory is equivalent to the 6d N = (2, 0) theory compactified on a circle, where the radius of the circle is inversely proportional to the volume of T 2 . We will keep the T 2 at finite volume, so that we are considering a 5d rather than a 6d theory.
Let us verify this is the 1-form symmetry predicted by (3.6). Let us consider the A 1 case for simplicity, the others are analogous. We first resolve the singularity, which amounts to going onto the Coulomb branch of the gauge theory, and consider Then we find where the former is generated by the P 1 and T 2 cycles, and the latter by the non-compact fiber, F , of the line bundle. Now, two copies of the fiber are homologous to the base, P 1 , and so the map f 2 in (3.7) sends T 2 to zero and P 1 to twice the generator of H ∂ 2 (X). Thus we find which is the expected result.
More generally, this can also be seen from the identification of Γ with the subgroup of H 1 (∂X, Z) projecting to zero in bulk. In the present case, the boundary is The 1-cycles on T 2 are non-trivial in the bulk, while the torsion cycles are trivial, and so we find which is the expected result for this ADE gauge group in 5d.
In this case, we also compute with the former generated by T 2 × P 1 , and the latter by T 2 × F and S 2 × F , which is the base of this (trivial) elliptic fibration. The map between them sends T 2 × P 1 to twice T 2 × F , and so the quotient in (3.25) is Here the Z factor, which comes from the base, B = S 2 × F , of the fibration, implies the flavor rank is f = 1, corresponding to the U (1) instanton symmetry, which corresponds in the 6d uplift to the U (1) KK symmetry.

Intersecting Surfaces
A useful method for describing the local geometry of a non-compact Calabi-Yau manifold is to specify a set of intersecting complex surfaces inside the Calabi-Yau, and study their local neighborhood. In more detail, one considers a set of blown up Hirzebruch surfaces, F b n , of degree n and with b blow-ups, intersecting along curves, and consistency conditions can be derived for these to embed into a Calabi-Yau threefold. In particular the interest here is in collections of surfaces that can be collapsed to zero volume, thus describing a 5d SCFT.
First we recall some basic properties of the blown up Hirzebruch surface 13 F b n . This has curves generated by the base, e, fiber, f , and curves x i from blown-up points, i = 1, ..., b, which satisfy The canonical divisor can be written as Given a collection of such surfaces, S i , we specify which curves they intersect along by saying S i is glued to S j by identifying the curve C ij in S i and C ji in S j . Consistency dictates that the genus is and the Calabi-Yau condition implies that the degrees of the normal bundles have to satisfy where C 2 ij is the self-intersection of the curve inside S i . In many cases we can read off an effective 5d gauge theory description from the collection of intersecting surfaces. We first recall that the generators of the gauge group, G, correspond to the compact divisors, S i , i = 1, ..., r G . The fibers, f i give rise to the W-bosons when we wrap M2 branes on them, and we may define: which is (minus) the Cartan matrix of the gauge group [4,37]. Note that to compute the intersection of curves with surfaces, we use: (4.13) 13 If b = 0, we suppress the b in the notation.
Example SU (3) 0 . To take a simple example of a geometry, consider the following diagram F 1 e e F 1 , (4.14) where the notation means that we glue the two surfaces along the base curves, e, in the respective surfaces. Then the matrix N ij above is computed to be the Cartan matrix of SU (3), and in fact this geometry generates the SU (3) 0 pure N = 1 gauge theory.
Let us consider the 1-form symmetry of this theory using the formulas of section 3. A basis of the curves is given by where we recall e 1 = e 2 due to the gluing. A basis of the compact divisors are given by the two Hirzebruch surfaces. Then we compute the matrix M 4 in (3.17) as One finds the Smith normal form of this matrix to be which, comparing to (3.22) and (3.25), implies the 1-form symmetry is Z 3 , and the flavor rank is 1, which is as expected.
General Formula for the 1-Form Symmetry. More generally, we proceed as follows.
Our goal is to compute the quotient where X is the Calabi-Yau containing the intersecting surfaces. We recall, by Poincaré-Lefschetz duality To compute this group, we note that H 2 (X) is generated by the curves {e a , f a , x i,a }, where a indexes the blown-up Hirzebruch surfaces, S a = F ba na . There are a total of such generators. However, these are subject to the relations implied by the gluing curves, Suppose there are N g such relations, then we may organize them into an N g × N c matrix, G. Then we have: (4.21) In both cases these are isomorphic to Z Nc−Ng , but the bases above will be important in the computation.
The second line in (4.21) represents the space of all homomorphisms from H 2 (X) into the integers. The divisors provide a special class of such homomorphisms via taking the intersection pairing. If there are N d divisors, this gives rise to another matrix, Ω of size where S a runs over the N d (blown up) Hirzebruch surfaces, and C α runs over the N c generators, e a , f a , x i,a , and the intersection numbers can be computed in the above basis using (4.13) along with (4.8) and (4.9). Then it can be shown using the gluing consistency conditions above that the image of this matrix lies inside ker G, and we have, We may equivalently summarize this by saying we have a complex, i.e., G · Ω = 0, and Γ ⊕ Z f is given by the homology of this complex at the central term.
One can compute this homology purely in terms of the "elementary divisors" of Ω -i.e., the diagonal entries, ω j , in its Smith normal form -and the ranks of G and Ω, as In appendix A we argue that the 1-form symmetry determined by the formula above agrees precisely with that derived in section 2.2 field theoretically. Let us briefly sketch the argument here; we refer to appendix A.2 for more details. Consider the flop of a rational curve C a , with C 2 a | Sa = −1. Furthermore let C a · S b = 1 and for all other surfaces, C a · S c = 0. This corresponds to a row in the matrix Ω

Invariance under Flops
Note that, for another divisor, S a , which contains the curve C a , there is a corresponding row with a curve C a , which is identified with C a , which is identical to the row for C a . In the Smith normal form of Ω, this just adds a row of zeros, and does not affect the group in (4.25).
After the flop, the row for C a is removed, and a new row for C b appears with all the other rows unchanged. The divisors,Ŝ a/b are the corresponding blown-up/down versions of S a/b . We see that the only effect is to change the sign of one row, which clearly does not affect the Smith normal form of the matrix, and so preserves the quotient group in (4.25).
Although the flops do not change the 1-form symmetry, if in addition to flopping the curve, we also take its area to infinity (which in the field theory corresponds to decoupling the associated hypermultiplet), then the 1-form symmetry can change. For more general flops, we claim the effect is still to flip the sign of a row in this matrix, and so this will still preserve the group.
From the toric perspective, which we will discuss in section 5, the invariance under internal flops is straightforward, as the 1-form symmetry only depends on the external vertices, and does not depend on the internal vertices and the triangulation.
There are several other operations in the geometry which do not change the theory, which are described in appendix A.2.

Pure Super-Yang Mills
Pure 5d N = 1 SYM for SU (N ) k , for 15 N > 2, can be constructed from a collection of N − 1 surfaces. There are various description: toric geometry, intersecting surfaces, and non-flat 15 For SU (2) we should use the description in terms of Sp(1) below.
resolutions of elliptic Calabi-Yau threefolds, as well as from the homology of the five-manifold that is the boundary of the non-compact Calabi-Yau.
All these have in common that there is an intersection of N − 1 compact surfaces S i , which have a ruling along which they can be collapsed to curves of an A N −1 singularities.
The prepotential can be computed from the triple intersection numbers in the geometry, in particular the Chern-Simons level is determined by intersecting the compact surfaces The surfaces S i (blown up) Hirzebruch surfaces and are glued to realize SU (N ) k as follows We can apply the rules for computing the intersection matrix M 4 in the last section. The details of this analysis are shown in appendix A and we find that the 1-form symmetry is and θ = (N + 1)π mod 2π The 1-form symmetry in these cases is These surface configurations arise from a resolution of a canonical singularity in certain Calabi-Yau threefolds, specifically, a class of geometries that are non-flat resolutions of the elliptically fibered threefolds associated to the (D k , D k ) conformal matter theory [5]. The 5d IR description is an SU (N ) 0 + (2N + 2)F . After decoupling the flavors entirely, which in the geometry corresponds to flopping rational curves out of the compact surfaces, this realizes The surface geometry is (equivalent to) the one in (4.29). Note also that this class of theories have a UV dual description in terms of Sp(N − 1) + (2N + 1)F , and after decoupling fundamental hypermultiplets, realize also the theories Sp(N − 1) θ .
The toric description of SU (N ) k was discussed in [18]. We will discuss general toric Calabi-Yau threefolds in detail in section 5, but let us briefly sketch the argument for this special case. Consider the toric diagram with external verticesv i = (w i , 1), where To ensure convexity we require that 0 < a + b < 2N . The CS level is identified with As we will show in section 5 the 1-form symmetry in the toric cases is determined simply by taking the matrix and computing its Smith normal form. One again finds the 1-form symmetry is Z gcd(N,k) .
The last point of view is to consider the five-manifold that bounds the Calabi-Yau threefold.
As discussed in (3.8) Γ = H 1 (∂M ). For SU (N ) k the boundary 5-manifold is the Sasaki-Einstein space Y N,k , which are circle fibrations over S 2 × S 2 with c 1 given by N and k, respectively. The first homology can be computed, e.g., as explained in [46], to be

Dualities
Distinct 5d gauge theories can share the same UV completions in 5d or 6d. Such theories are at times referred to as "UV duals". A classic example is the duality between SQCD theories with SU and Sp gauge groups with fundamental matter. For example for rank 2 there is the duality In the geometric realization these can be seen by constructing the associated rulings from a given resolved Calabi-Yau geometry. These UV dual gauge theories share parts of their extended Coulomb branch, and we therefore expect them to have the same 1-form symmetries.
E.g., the duality above will have trivial 1-form symmetry, from both gauge theory descriptions, due to the fundamental matter. At rank 2 there are no UV dualities that relate two theories with non-trivial one-form symmetry (for a complete list with all dual descriptions see [6,23,47]). However, a large class of dualities of 5d gauge theories were conjectured in [20]. The consistency with respect to the 1-form symmetry provides a non-trivial test for these dualities.
Here we summarize those that have a non-trivial 1-form symmetry and show how it is realized in terms of the geometry and the dual gauge theory descriptions (equation labels on the lefthand side are those in [20]). The 1-form symmetries are determined from the gauge theory rules in section 2.2 and, complementarily, from the surface intersections, as we have verified in many cases by explicit computation.
Duality (1.62) The theories on both sides of this duality have a Z 2 1-form symmetry. Gauging this symmetry implies a new duality: This duality has a Z 2 1-form symmetry.

Duality (3.74) The geometry of surfaces is
which results in the gauge theory duality For m odd, both sides have 1-form symmetry Γ = Z 2 .
Duality (3.80) With the CS level 16 where A m,n is the anomaly coefficient from integrating out Λ m for SU (n) (see [48]) Then there is a duality (4.47) For m, p both odd this has a Γ = Z 2 1-form symmetry. The surface configuration for example This duality has a Z 2 1-form symmetry for m odd, which can be confirmed from the geometry (4.50) Duality (3.258) This duality has a Z 3 1-form symmetry on both sides, as S is uncharged under the center.

Duality (3.285)
This duality has a Z 2 1-form symmetry on both sides, as C is uncharged under the center.
Duality (3.296) This duality has a Z 2 1-form symmetry on both sides.

Duality (3.300)
This duality has a Z 2 1-form symmetry on both sides.
There are in addition numerous dualities with Spin(N ) and vector matter, which exhibit non-trivial one-form symmetries.

1-Form Symmetry from Toric Data
An interesting class of examples of Calabi-Yau singularities comes from toric geometry (see [18,31] for a recent discussion in the context of 5d SCFTs). These can be constructed from toric diagrams, which are sets of points, v i ∈ Z 3 , i = 1, ..., n, whose hull forms a convex polytope. The Calabi-Yau condition forces them to lie in a plane, which we may take to be at v 3 = 1, and so write v i = (w i , 1), where w i ∈ Z 2 . In general there may be internal points, which correspond to compact divisors, and external points, which correspond to non-compact divisors. We label these respectively aŝ where n = n I + n E . The compact divisors associated to the internal points give rise to the gauge group generators, and the non-compact divisors associated to the external points give rise to flavor symmetry generators (after quotienting by some relations). Explicitly, we find: where r and f are the ranks of the gauge and flavor groups. Edges correspond to cycles, and taking a maximal triangulation gives rise to a complete resolution of the singularity, with internal edges giving the resulting exceptional curves.
The cohomology groups can be straightforwardly computed from this data. For example, the group H ∂X 4 (X) is generated by all compact and non-compact divisors. If we define the n × 3 matrix where To compute the (compact) homology group, H 4 (X), we note that this is generated by the internal points, of the form  which generate a Z n I subgroup of Z n . The divisors which are equivalent to these are ones related by an element in im(A). Let us denote this group by Z n I + im(A). Then, where in the second equality we used the second isomorphism theorem, and in the third we observe that Z n I ∩ im(A) is trivial, since A E has trivial kernel. 17 From (3.25), the 1-form symmetry group and flavor rank are given by where f 4 is defined in (3.7). In the present case, H 4 (X) is a subgroup of H ∂ 4 (X), and this map is simply the inclusion map. Then we have, using the third isomorphism theorem H ∂ 4 (X)/f 4 (H 4 (X)) ∼ = (Z n /im(A))/ ((Z n I + im(A))/im(A)) ∼ = Z n /(Z n I ⊕ im(A)) . (5.9) Note that both Z n ∼ = Z n I × Z n E and Z n I + im(A) ∼ = Z n I × im(A E ) are product groups with a common factor, and so their quotient is simply given by In particular, note that this is independent of the internal points, A I .
We may compute this quotient by writing A E in Smith normal form as where D is diagonal, explicitly, Figure 1: Toric diagrams for SU (2) 0 , SU (2) π and P 2 theory (which is obtained from SU (2) π by an external flop). The internal vertex (blue) indicates the Cartan of the rank one gauge group, the external vertices correspond to non-compact divisors. Whenever there is a ruling, i.e. a partial singularization to an A 1 singularity, this is indicated by a green line.
with n E − 3 = f zero rows (and α 1 |α 2 |α 3 ). Then, from (5.8), we find the 1-form symmetry group is and we see that f is the rank of the flavor symmetry, as expected.
Note that this prescription is manifestly independent of the resolution,X (which would correspond to a choice of triangulation of the toric diagram). One way to see this is to observe that, for any choice of triangulation, we have: The different choices of triangulation simply identify different curves with different generators of ker(A T ). As long as the external vertices remain the same, the choice of triangulation of the convex region they enclose will not change the quotient, and so neither the 1-form symmetry.

Examples
The first class of examples we consider are the pure SYM theories, which were discussed in This theory has no flavor symmetry, but a non-trivial 1-form symmetry.
As another, more complicated example, consider a toric diagram with external points at shown in figure 2. This engineers an SU (N ) N 2 +k 1 × SU (N ) N 2 +k 2 linear quiver theory. The 1-form symmetry of this theory can be computed as follows: before considering the matter and CS terms, it is Z N × Z N from the two SU (N ) factors. The bifundamental matter then breaks this to the diagonal Z N subgroup, and the two CS terms finally break it to: 18 Γ = Z gcd(N,k 1 ,k 2 ) . (5.17) This is in agreement with the 1-form symmetry computed from the toric diagram. The quivers above appear to be the most general that are (i) strictly convex, (ii) give rise to an "allowed vertical reduction," in the notation of [18], leading to a standard gauge theory description, and (iii) have non-trivial 1-form symmetry.
Finally in the case N = 2, recall that there is not a proper CS term, but only a discrete theta angle. Roughly speaking, we may identify the theta angle with θ = kπ (mod π). As discussed in section 2.2, the Z 2 1-form symmetry of an SU (2) theory is broken for θ = π, so (5.17) will continue to hold in this case. For example, for the above quiver in the case N = 2, k 1 = k 2 = 1, which is referred to as the "beetle quiver," [18], which is also a descendant of the (D 5 , D 5 ) conformal matter theory [6,23], we find θ = π for the two gauge groups, which breaks the putative 1-form symmetry. This is consistent with the fact that this theory has a dual description as an SU (3) theory with N f = 2, which has no 1-form symmetry.

T N and Descendants
Another class of toric Calabi-Yau threefolds corresponding to 5d gauge theories with interacting UV fixed points are the theories T N . They are constructed as M-theory on the singularity They have a description in terms of toric geometry and (dual) brane-webs [49]. More recently, their descendants, that are obtained by decoupling hypermultiplets, were studied, and their flavor symmetries and Higgs branches were determined [31]. In particular it was shown that there is a large class of non-Lagrangian descendants, which turn out to have non-trivial 1-form symmetries (as well as 0-form symmetries).

(5.18)
Applying the toric formalism to this, it is easy to check using (5.13) that there is no 1-form symmetry for any of the T N theories. A field theory way to see this is to recall that there is a linear quiver description of T N as where we see the 1-form symmetry is completely broken by the fundamental and bifundamental matter.
However, once flavors are decoupled the theories can have higher-form symmetries. For instance, one of the descendants of T 3 is the theory with geometry P 2 , which has Z 3 oneform symmetry. Similarly for T N the endpoints of the decoupling tree (where we successively   Figure 4: The non-Lagrangian theories determined in [31] obtained from T N after decoupling all hypermultiplets, as well as their flavor symmetries G F . decouple hypermultiplets) has a non-trivial one-form symmetry: these theories are denoted by B N (as in "bottom of T N ") [31], and like the rank one case, do not admit an IR description in terms of a non-abelian gauge theory. 19 Since these models are toric, we can apply the general formalism above. These models have external vertices, which are precisely the vertices of the CFD of these theories, given by where each is a curve of self-intersection N − 2 [31]. The CFD is shown in figure 3. The Smith normal form results in the above 1-form symmetry Γ: These are generically not the only theories that are descendants of T N and have a nontrivial 1-form symmetry. For rank 1 we know already the theory SU (2) 0 which has Γ = Z 2 .
For T 4 there are three more descendants with non-trivial 1-form symmetry. The matrix A E is obtained from the CFD, simply by recalling which coordinates the vertex has in the initial toric diagram figure 3.
Here we have only shown one toric triangulation as well as the CFD, which are the curves that are the intersections of compact with non-compact divisors. The descendants are obtained by toric flops, or more efficiently, CFD-transitions. The models with non-trivial 1-form symmetry are shown in tables 2 and 3. The general case is shown in figure 4. Computing the 1-form symmetry, we find for these non-Lagrangian models (N > 2)

(5.22)
It would indeed be interesting to find alternative methods to confirm these higher-form symmetries.  Table 2: Descendants of the T 4 theory, which have non-trivial 1-form symmetry. First we give the toric vectors for the vertices of the curves in the toric diagram and its boundary, which is the CFD, that is shown in the second column. The flavor symmetry is indicated by G F and Γ is the 1-form symmetry.  Table 3: Descendants of the T 6 theory, which have non-trivial 1-form symmetry. First we give the CFD, flavor symmetry is indicated by G F and Γ is the 1-form symmetry.

Higher-Form Symmetries from 6d to 5d
In this section we investigate the relation between the higher-form symmetries of 5d theories and their 6d parent theories. All known 5d SCFTs are obtained by circle-reduction from 6d with additional mass deformation (and possibly the action of automorphisms). To study this question, we will use F-theory realizations of 6d theories in terms of elliptically fibered Calabi-Yau threefolds, and use the F-theory/M-theory duality to realize their 5d compactifications as M-theory on the same elliptic Calabi-Yau. We saw a simple example of this in section 4.1.
One interesting aspect of this correspondence is that the 6d theories themselves may have higher-form symmetries, which we can relate to those of their 5d reductions. In particular, many 6d theories are not genuine QFTs, but instead only defined as so-called "relative" 6d theories [51,52]. Instead of a partition function, they have a partition vector, analogous to the set of conformal blocks of a 2d CFT. This data can be characterized by a finite abelian group assigned to each 6d theory, called the "defect group" [11], which can be thought of as a "selfdual" 2-form symmetry [12,53]. We briefly review this structure below. Upon dimensional reduction to 5d, this gives rise to a 1-form symmetry. 20 In addition, the 6d theory may itself have a 1-form symmetry, in which case this can also contribute to the 5d 1-form symmetry.
Schematically, we have Below we will demonstrate that the 1-form symmetry on the LHS can be read off from the topology of the elliptic Calabi-Yau of the 6d F-theory model according to the discussion of section 3.
The circle-reduction of theories with non-trivial defect group, which are so-called non-very Higgsable theories in 6d, gives rise to 5d theories with non-trivial 1-form symmetry. In general a 6d theory will be a combination of minimal conformal matter theories (which typically have matter content which breaks the 1-form symmetry in 6d), and a collection of curves with possibly tuned gauged groups. These can give rise to 1-form symmetries in 6d, if the matter is invariant under the action of the center of the total gauge group. We will encounter examples of this type.

Defect Group in 6d
As was first understood for the 6d N = (2, 0) theories [51,52,54], and later for more general 6d N = (1, 0) theories [11], many 6d theories are not genuine QFTs, but have some subtle properties which make them more analogous to chiral CFTs in 2d. For example, rather than having a unique partition function on a closed spacetime manifold, M 6 , they have a "partition vector," similar to a set of conformal blocks. The elements of the partition vector are labeled by a (non-canonically) chosen Lagrangian subgroup of H 3 (M 6 , C), where the coefficient group, C, is a finite abelian group known as the "defect group." This is a similar structure to a 2-form symmetry with group Γ, whose observables would be labeled by the entire group H 3 (M 6 , Γ).
For this reason we may refer to this structure as a "self-dual" 2-form symmetry [12,53]. Upon dimensional reduction to 5d, this descends to either a 1-form or 2-form symmetry, depending on the choice of Lagrangian subgroup.
Six-dimensional N = (1, 0) theories have a realization in terms of F-theory on a singular elliptic Calabi-Yau threefold, where the base is The defect group C of the associated theory was determined in [11] as the abelianization of The defect groups in 6d were classified and have the following types Here A refers to a linear quiver of curves A : n 1 · · · n m (6.6) in the tensor branch of the geometry, where the curves have self-intersections −n j . The defect group is Z p where p/q is the continued fraction of n j .
The D-type quivers are curves in the base arranged in shape of a D-type Dynkin diagram with negative self-intersection numbers D : 2 2 n n 1 · · · , n m . Finally, E type corresponds to −2 curves arranged in the Dynkin diagram of the associated Lie algebra. For the 6d (2, 0) theories of ADE type, the defect group is C = Z(g ADE ), i.e., the center of the associated ADE algebras. If the defect group in 6d is non-trivial, the theories are called non-very Higgsable, and we will see that the 5d theory inherits the defect group as part of its 1-form symmetry.

NHCs
The non-Higgsable clusters (NHCs) [55] are essential building blocks for 6d SCFTs and examples of non-very Higgsable theories. The single node NHCs have a tensor branch description as g n , (6.9) where the base is a single −n self-intersection curve and g is the non-Higgsable gauge algebra The base is of A-type and contributes a defect group C = Z n . In addition, there is a 1-form symmetry in 6d, given by the center, Z G , of the simply connected gauge group with algebra g (except for n = 7, where we instead have trivial 1-form symmetry due to the matter).
From (6.1), we then expect that 1-form symmetry of the 5d theory is given by for the NHC (6.9). To reproduce this using the 5d description, note that in the dimensional reduction, it is observed in [8,22] that -after decoupling of a single compact surface -there is a pure G gauge theory without matter, except for the −7 case which is an E 7 with 1 2 56. For the SCFT sector, we then expect that the center of the NHC group leads to a Z G 1-form symmetry in 5d.
However, we also expect a Z n 1-form symmetry, which we claim comes from the decoupled sector, as follows. First consider starting with the pure N = 1 G gauge theory in 5d, which has a U (1) inst symmetry. We may then gauge this symmetry, which amounts to including a new U (1) gauge multiplet with a topological coupling involving the instanton density of the G gauge field. Specifically, we suppose that we gauge this symmetry with charge n, i.e., we include n times the minimal topological coupling. Then we claim this theory provides a 5d description of the NHC associated to a G gauge group on a −n curve. Namely, as we describe below, the geometry of the NHC is given by a configuration of intersecting (blown up) Hirzebruch surfaces associated to the affine Dynkin diagram of the associated group, as shown in table 4. Correspondingly, we may define the prepotential of the theory on the extended Coulomb branch in terms of parameters, ϕ i , i = 0, ..., r G associated to the Dynkin nodes.
Then one checks that if we identify ϕ 0 with the Coulomb parameter of the U (1) gauge field and ϕ i , i = 1, ..., r G with those of the G gauge field in the theory above, and make the redefinitions where d i are the Dynkin indices, then the prepotential of this gauge theory precisely agrees with that computed from the geometry of the NHC. We may then take a limit where the gauge coupling of the U (1) goes to infinity, which decouples the gauge field, and leaves the pure G gauge theory, as above. The upshot of this interpretation is that, in addition to the Z G 1-form symmetry from the G gauge field, the U (1) gauge field also contributes a 1-form symmetry.
Due to the coupling to the instanton symmetry with charge n, the U (1) 1-form symmetry of the U (1) gauge field is broken to a Z n subgroup, which is precisely the contribution to the 5d 1-form symmetry from the defect group in 6d.
We can now compare this to the result from studying the NHC geometries and applying the results of section 3. For even n, the surfaces, F i , are glued together in the form of an affine g Dynkin diagram, as shown in table 4. The Hirzebruch surfaces intersect as 14) The intersection matrix of compact surfaces F n i = S i and compact curves gives in Smith normal form the 1-form symmetries of the theory. This is summarized in table 4 and agrees with the expression (6.12).
An alternative approach, which applies in the cases of even n, is to note that the total space of the singular elliptically fibered Calabi-Yau, can be presented as a quotient [56] Here Z n acts as: (z 1 , z 2 , w) ∼ (ω z 1 , ω z 2 , ω −2 w) , (6.16) where ω = e 2πi/n , z i are the coordinates of C 2 , and w is the coordinate on T 2 . Then we argue in appendix B.1 that studying the homology of the boundary of this quotient space also gives the expected result for the 1-form symmetry.
There are also NHCs with reducible gauge group on a collection of curves in the base: The last line is the 1-form symmetry that the gauge theory with matter preserves. The 1-form symmetry of the theory in 5d is then The order of the defect group is computed as ord(C) = p, with C = Z p determined by The geometry of the NHCs with multiple curves is discussed in appendix B.2, where we confirm the above 1-form symmetry groups, by computing the resolution of the Tate model of the elliptic fibration and computing the intersection matrix between compact divisors and compact curves.
Whenever we tune the gauge group on a single curve of self-intersection −n to a larger group than the one forced in the NHC, 6d anomaly cancellation requires introducing matter, which breaks the 1-form symmetry that the gauge algebra would contribute (see, e.g., table 3 in [57]). This matter can break the 1-form symmetry that arises from the center of the gauge group. The only case where the matter is compatible with some remnant 1-form symmetry is Field theoretically this would result in a theory with where the Z 2 is the subgroup of the center of so(N ) that acts trivially on the vector matter.
This result can be checked from the geometry by noting that the surfaces are [26] F 2 (6.22) where the boxed entry 2r − 8 means the surfaces are glued with this multiplicity along curves.
Application of the intersections of the (blown up) Hirzebruch surfaces confirms the 1-form symmetry (6.21). For all other higher self-intersection number curves n > 2 with gauge groups g ⊃ g NHC the 1-form symmetry is Γ = Z n .
An alternative check is to compute the resolution of the elliptic fibration with an I * p singularity tuned above (−n), similar to the analysis in appendix B.2. The resolution for both codimension one and two (in the base) singularities was obtained in [58]. The starting point is the Tate form [59,60] and with the varnishing orders ord U =0 (b i ) = (1, 1, k, 2k+1) for I * s 2k−3 corresponding to SO(4k+ 2) (a similar analysis for the other groups). U = 0 here corresponds to the NHC curve with self-intersection number −4. Resolving the singularity results in the Cartan divisors D α i , labeled by the roots of SO(4k + 2), which intersect in the Cartan matrix, and correspond to the compact surfaces S i . The curves are obtained as complete intersections from the ambient space, and are given by the rulings of the Cartan divisors, as well as additional curves above the codimension two loci, where the vector matter is located. Intersecting the curves and compact surfaces results in the matrix M 4 , which confirms the above 1-form symmetry.

Non-Minimal Conformal Matter
A simple class of models are the conformal matter theories in 6d. The minimal conformal matter theories of type (G 1 , G 2 ) have a smooth base B 2 , and so a trivial defect group, but non-trivial flavor symmetry G 1 × G 2 . The 6d 2-form symmetry and 1-form symmetries are trivial and upon reducing to 5d the theories indeed do not have any 1-form symmetry. Only after decoupling hypermultiplets does a 1-form symmetry emerge -e.g. starting with the (D N , D N ) minimal conformal matter theories and decoupling, results in pure SYM theories Non-minimal conformal matter theories are non-very Higgsable, i.e. have a singular base.
The compactification of a NHV theory on S 1 results in a 5d SCFT coupled to a 5d N = 1 vector multiplet, where the type of the vector multiplet is determined from the tensor branch geometry of the 6d theory [61]. For (G, G) non-minimal conformal matter, the flavor symmetry is G × G and a tensor branch geometry is given by . (6.24) There are N − 1 (−2) curves in the base, which is a base of generalized type A. Dimensional reduction on S 1 results in a 5d SCFT, coupled to an SU (N ) vector multiplet [8,61]. The SCFT part (and its IR description as a quiver) does not carry a higher-form symmetry, as the theory has hypermultiplets in the fundamental. The IR descriptions are quivers in terms of the affine Dynkin diagrams of G with SU (N d i ) nodes, where d i are the Dynkin labels of G, with N fundamentals attached at at least one SU (N ) node (see, e.g., table 3 of [8]): 25) where G N corresponds to the quiver that is the Dynkin diagram of G with each node of type SU (N d i ). The vector multiplet corresponds to a pure G SYM theory, which has Z G 1-form symmetry, which correctly reproduces the contribution of the 6d defect group to the 5d 1-form symmetry, as in (6.1).

General M-theory Compactifications
In We expect that, for smooth compactification manifolds, the low energy description is given by an abelian gauge theory, with U (1) gauge fields associated to the 2nd cohomology generators, and with matter content given by wrapping M2 branes on 2-cycles. Since these were the only ingredients we used to derive the 1-form symmetry in the Calabi-Yau threefold case, we may make essentially the same argument for more general compactification manifolds.
After appropriately modifying the degrees of the various homology groups in the arguments leading to (3.6) and (3.14), we find To give further evidence for this formula, we now turn to an alternative argument, which considers the fluxes of the M-theory C-field at the asymptotic boundary of spacetime. A similar approach was used in [12] to understand 2 form symmetries in F-theory compactifications, and we comment on the relation to their results.

Asymptotic Fluxes in M-theory Compactifications
An alternative perspective on the higher-form symmetry in M-theory compactifications of general dimensions can be obtained by studying electric and magnetic fluxes for the 3-form C-field of M-theory. In particular, the choice of higher-form symmetry can be traced to a noncanonical choice of boundary condition for torsion fluxes of this field. A similar perspective was taken in [12] in the case of type IIB string theory. 21 When placing M-theory on a non-compact 11d spacetime, X, we must also specify the asymptotic behavior of the C-field. Specifically, there are various superselection sectors of the theory which can be labeled by the asymptotic "electric" and "magnetic" fluxes of the C-field, which, to a first approximation, take values in H 7 (∂X) and H 4 (∂X), respectively. However, there are several caveats to this statement.
First, we expect that the correct mathematical formulation of the C-field fluxes would not be in terms of ordinary cohomology classes. In type II string theory, it is known that the proper formulation of RR fluxes is in terms of the K-theory, rather than cohomology, of spacetime [63,64]. For the purposes of classifying the boundary fluxes, however, it was argued in [12] that in many examples the K-theoretic analysis agrees with that of ordinary cohomology, and so it is sufficient to consider the latter. Since we do not know of the proper mathematical formulation of the C-field in M-theory, we will instead consider the fluxes as valued in ordinary cohomology classes, and we leave it as an open question to find the appropriate modifications to the discussion below using a more precise formulation.
Second, recall from the exact sequence in (3.12) there is a map, Only elements in the image of this map can be created by excitations in the bulk, so the only possible fluxes at the boundary lie in the subgroup (in the notation of (3.12)), and similarly for H 7 (X).
Finally, as argued in [65], in the case where ∂X has torsion in its 4th cohomology group (and so, by Poincaré duality, also in its 7th cohomology group), it is not possible to simultaneously specify the torsion part of both H 4 (∂X) and H 7 (∂X). A similar issue arose in [12] in type IIB string theory, and these issues are related via the circle compactification considered in section 6.
Let us now specialize to a spacetime of the form As mentioned above, we may take the following ansatz for the C-field, where ω a runs over a basis of the 2nd cohomology of M d , but now we do not demand that ω a have compact support. Suppose we take A a to lie in a bundle labeled by c a ∈ H 2 (S D ). Then the C-field has a flux given by This then projects to a class in H 4 (∂X), as in (7.2), defining the asymptotic magnetic flux of the C field. Note this indeed lives in the subgroup (7.3). In fact, since S D is compact,ĝ 4 only acts on the component from M d , and we may usê where we have used the Poincaré-Lefschetz dual of (3.25). Then we may label the fluxes by On the other hand, the second factor in (7.8) corresponds to cocycles, ω a , which have non-compact support, but such that some power of them have (some gauge-equivalent representative with) compact support. For example, suppose we take a U (1) gauge field, A a , associated to such a cocycle, ω a and suppose that nω a is gauge equivalent to a compact cocycle. If S D ∼ = R D , then A a is topologically trivial, and so we may write which is equivalent to a gauge field with compact support, and so does not influence the asymptotic behavior of the C field. On the other hand, if S D is topologically non-trivial, then there may be an obstruction to defining 1 n A a as living in a well-defined gauge bundle. This obstruction lives in (7.10) More generally, we see that the obstruction to redefining a gauge field as a compact gauge field lives in the group 11) which is precisely the first factor in (7.8). In other words, a non-zero choice in this group instructs us not to integrate over ordinary U (1) r gauge fields on S D , but rather to those with a given obstruction in (7.11). This is precisely the prescription for turning on a background 2-form gauge field coupled the electric 1-form symmetry group Γ of the gauge theory. Thus we learn that the factor H 2 (S D , Γ) in (7.8) labels the 1-form symmetry backgrounds.
Finally, we observe that we have the option to gauge this 1-form symmetry, as discussed where T H i denotes the torsion subgroup, we may identify Then we claim that, after gauging, the backgrounds are labeled by fluxes in

Surface Operators
The above argument shows that the labels of an observable implied by the properties of the asymptotic flux agree with those we expect from the higher-form symmetries derived in the previous subsection. However, a more direct way to see the higher-form symmetries is to construct the topological surface operators measuring the charge under these symmetries. Let us briefly describe this construction.
In M-theory, associated to the 3-form gauge field, C, we may define the following "electric" and "magnetic" charge operators, where Ω 4 and Ω 7 are 4-and 7-cycles in spacetime, and: Since there are both dynamical electrically and magnetically charged objects of unit charge (M2 branes and M5 branes, respectively), α and β above must be integers for these to be welldefined. Then for compact cycles, these operators are actually trivial. However, the situation is more subtle for non-compact cycles, which we take to mean relative cycles in H ∂ i (X). For concreteness, let us consider a spacetime of the form R D × M d , where M d is the compactification manifold, which we take to be non-compact, and d + D = 11. We consider wrapping an M2 brane and a U E operator on curves, To see this, first note that the M2 brane creates a unit G 7 flux, so if we integrate this flux over a cycle which is linking the M2 brane with linking number , we find the flux is equal to . Normally this linking number is an integer, and then, since α is an integer in (7.15), the expectation value will be trivial. However, in the present case, the linking number can be computed as where the linking number on the RHS is computed in R D , and the intersection number in M d . The former is an integer, but the intersection number of two non-compact cycles is more subtle to define. We claim that a natural definition assigns it not an integer, but a rational number (see, e.g., appendix B of [18]). Specifically, the fractional part of this rational number can be computed using the maps g i in (3.7) as where is where we defined a map, µ i , As an example, let us consider compactifying M-theory on a 7-dimensional G 2 manifold, leading to an effective 4d theory with N = 1 supersymmetry.

Pure SYM and Mass Gap
Starting with the Bryant-Salamon metric [66] on the local G 2 manifold R 4 ×S 3 , by quotienting with a discrete group of ADE type, we are left with a G 2 holonomy singular space, X 7 , whose M-theory compactification gives rise to the 4d N = 1 SYM theories with gauge group G = ADE [67,68]. This theory has only adjoint matter, and so preserves a 1-form symmetry Ab[Γ ADE ], equal to the center of the corresponding simply-connected ADE Lie group. This geometry characterizes the UV of the gauge theory, and is given by a K3-fibration over S 3 and is simply-connected. We can compute the 1-form symmetry from the geometry using (7.1) and (3.8), and find where we have used that X 7 is simply-connected. As discussed in [67] this model is related by a geometric transition, a G 2 -flop, to a theory on the space X 7 which is an R 4 bundle over S 3 /Γ ADE . X 7 and X 7 are connected by varying the complexified coupling, τ . The theory exhibits confinement, and the 1-form symmetry is unbroken in that phase. The confining strings were described in [69] as M2 branes wrapping 1-cycles in the geometry. These are precisely charged under the Γ given in (7.24). The mass gap can be understood from considering the metric fluctuations of the G 2 -manifold X 7 [70], which are not normalizable and thus the spectrum has a mass gap.
Another simple class of examples of manifolds with (at most) G 2 holonomy are simply products of a Calabi-Yau threefold and S 1 . These lead to 4d theories with N = 2 supersymmetry. For simply connected Calabi-Yau manifolds, the formula (7.1) gives the same 1-form symmetry as the 5d theory, which is as expected as we expect the 5d 1-form symmetry to reduce to a 1-form symmetry in 4d. Unlike for N = 1 compactifications, in this case there is a notion of calibrated 2-cycles on which we may wrap M5 branes, and so correspondingly a notion of supersymmetric loop operators, which can be charged under the 1-form symmetry.

Models with Chiral Matter
Extending this to theories with matter, recall that chiral matter in 4d is created by conical singularities of codimension 7 in the G 2 -holonomy manifold [71,72]. The local geometry that we wish to model is ALE-fibration over M 3 , which degenerates at points. To realize fundamental matter for SU (N ), the geometry corresponds to an unfolding of an SU (N + 1) singularity using the hyper-Kähler quotient construction. The geometry constructed in [71] has M 3 = R 3 , and shows that chiral matter in the N + of U (N ) = SU (N )×U (1) Z N is realized by the cone over a weighted projective space It follows that Γ = H 1 (∂X 7 , Z) = H 1 (CP 1,1,N,N, ) = 0 (e.g. [73]). Similarly, the unfolding of is captured by the cone over CP p,p,q,q , which also has Γ = 0 -consistently with the field theory expectation of bi-fundamental matter for SU (n) groups.
Generalizing to Spin(2n) gauge theories, the field theory expectation is that there is a Z 2 1-form symmetry preserved for vector matter. In [72] the corresponding hyper-Kähler quotient was discussed and shown to be a Z 2 -quotient of the one for A n xy = z 2n , (x, y, z) → (y, x, −z) . This theory preserves a Z 2 1-form symmetry, which is confirmed from the geometry, as the Z 2 quotient the space has now non-trivial first homology and Γ = Z 2 . Similar considerations can be studied using the Higgs bundle approach to G 2 -holonomy manifolds [74,75], to generalize this to theories with more general matter content.

Summary and Outlook
In this paper we studied 1-form symmetries in 5d gauge theories and identified the geometric counterpart in their M-theory realization. Whenever possible we cross-checked the 1-form symmetry in all available descriptions: gauge theory, M-theory geometry, and directly in terms of the geometry of surfaces that realize the gauge theory in M-theory -and find agreement.
1-form symmetries have been observed to increase (and never decrease) under RG-flows. In particular starting with a 6d model, the circle compactification to 5d (and mass deformation) yields a theory with 1-form symmetry at least as large as that in 6d, but generically enhanced by the 2-form symmetry in 6d, which is the defect group in 6d. Note that not all dimensional reductions correspond to 5d SCFTs, e.g., when the theory has a non-trivial defect group in 6d (and is a so-called non-very Higgsable theory in 6d), the theory in 5d is an SCFT coupled to a gauge sector [8,61]. The 1-form symmetry of the combined theory agrees with the one in 6d. The 5d SCFT can thus in general have a smaller 1-form symmetry than the 6d theory.
There are various extensions which might be made of this work. The most interesting direction is to construct observables, which are sensitive to the 1-form symmetry -such as was discussed in 3d in [53] in the context of the Witten index. One class of such observables would be supersymmetric partition functions on spaces with non-trivial H 2 (S (5) ); e.g., Sasaki-Einstein manifolds (see [76] for a review), Σ g 1 × Σ g 2 × S 1 [77] or S 3 b × Σ g [78], should be of interest.
It would also be interesting to define the 1-form symmetry more intrinsically and develop a method to determine how it changes along mass deformations/RG-flows. For the 0-form symmetries, i.e., flavor symmetries, this is achieved by the combined fiber diagrams (CFDs) [5][6][7][8]31]  For some theories we do not have a gauge theoretic description, such as the toric B N theories and the P 2 theory at rank 1, which do not have any weakly coupled gauge theory description, although on the Coulomb branch we have some support for the 1-form symmetries in terms of that of the abelian gauge theory. It would be interesting to test these by further compactification to lower dimensions, e.g., 4d or 3d, where the theories may have a Lagrangian description and a field theory analysis can confirm the 1-form symmetry.
Finally, we have only briefly discussed the generalizations, and a more in depth analysis for higher dimensional Calabi-Yau manifolds, as well as G 2 or Spin(7) manifolds, would be of interest. Furthermore we assumed that our Calabi-Yau threefolds have no torsion. It would be interesting to extend the analysis by relaxing this condition.
Another generalization is to consider non-supersymmetric theories, such as those proposed in [79], where the rank 1 SU (2) 0 theory is mass deformed into a non-supersymmetric theory, which then has a conjectural UV completion in a 5d non-supersymmetric CFT. The mass deformations do not affect the theta angle, which remains 0, and thus we expect the 1-form symmetry to be present also in the non-supersymmetric case. This would be interesting to extend and consider in other instances, e.g., for SU (N ) kN pure SYM theories in 5d, which for certain k (e.g., SU (3) 3 ) also exhibit a global SU (2) F 0-form symmetry.
3. This places us in the phase where the matter has been "minimally integrated in," i.e., the first phase after the one where the mass of the hypermultiplet can be taken to −∞.
Other phases can be obtained by flopping curves, as described below.
Let us now argue that the 1-form symmetry from the geometry engineering a gauge theory agrees with that read off from the gauge theory description, as described in section 2.2 above.
First we must check that the pure gauge theories have the appropriate 1-form symmetry. For example, consider the geometry engineering SU (N ) k for N ≥ 3: where h ≡ e + nf in F n . In general, the curves e a and f a , as a = 1, , ..., r ranges over the divisors, generate the curves in the geometry, but, for simple G, the gluing curves allow us to eliminate all but one of the e a , say 22 e 1 . Dropping these redundant rows, the matrix Ω in this case can be written as: . . .
where C ab is the Cartan matrix, and we defined γ a = e 1 · S a . For the example of SU (N ) k above, this is given by: Now the Cartan matrix, C, of SU (N ) can be diagonalized as: where, . . .
(A.6) 22 For semi-simple G, there will be an equivalence class of e curves for each simple factor.
Recall Ω has the form, where A is an × (N − 1) matrix, in this case with = 1. Then if we act on the left with S 0 0 1 and on the right with T , we obtain Since all but the last diagonal entry of D is 1, we may act again on the left to eliminate all but the last column of AT to get (writing D explicitly), where α is a length column vector with components which may be considered mod N , since we can shift by a multiple of N by adding the last row of the diagonalized Cartan matrix. In the present case, A 1,b = γ b , and this gives: Finally, using additional row operations on the last column, we obtain: which gives, using (4.25), as expected. A similar analysis holds for other pure gauge theories, showing that their 1-form symmetry is equal to their center. 23 23 In the case of Sp(N ), the 1-form symmetry is equal to the Z2 center for θ = 0, and is trivial for θ = π.
Next we add matter. We expect the matter will break the 1-form symmetry of the pure gauge theory to the subgroup of the center acting trivially on the matter. Continuing with the SU (N ) k example for concreteness, suppose we add matter in a representation with Dynkin labels δ a , a = 1, ..., N − 1, where we take δ a ∈ {0, 1} as mentioned above. This is included by blowing up a curve in the corresponding divisors, and adds a row to the matrix Ω as: Ω = · · · S a · · · . . . . . .
x · · · −δ a · · · . (A.14) This is again a matrix of the form (A.7), and so reasoning as above, we find the Smith normal form can be written as: where α is now a 2-component column vector, with first entry −k, as above, and second entry: This is precisely the Z N center charge of the representation with these Dynkin labels, and we have correspondingly denoted this by q δ . Including N f total representations, we find that: and applying further row operations, we finally obtain: so that, from (4.25), which is the expected result. One may repeat this argument for other gauge groups.
Finally, we should check the 1-form symmetry is the same in other phases of the gauge theory, which are related by flops, and we turn to this next.

A.2 Invariance under Surface Equivalences and Internal Flops
Next we check that the 1-form symmetry is invariant under various equivalences in the geometry.
The first is (local) S-duality, which exchanges the e and f curves in a divisor F b 0 . In other words, if one of the divisors in our geometry is of this form, we exchange e and f in any gluing curves involving this divisor. Recall the intersection matrix for curves (e, f, x i ) is Finally, we consider the effect of a (−1, −1) flop. Let us first describe how a flop acts in a collection of intersecting surfaces. We will flop a curve, C, which may lie in multiple divisors.
We first partition the set of divisors, S, into three groups, S = A ∪ B ∪ C, as follows: • A -those divisors containing the curve, C, to be flopped. We assume that (after possibly applying the isomorphisms above) the curve to be flopped in each divisor is one of the blown up curves, x i . These are then all identified with each other by gluing curves. Note then that the intersection of each of these divisors with the curve is −1. 24 In other words, the rows of the matrix corresponding to the e and f curves inside this divisor will be swapped, which will not affect the Smith normal form. 25 We write this in block form, with the first block acting on the curves in divisor which we apply the isomorphism to, and the second block acting on all the others • B -Those divisors, which intersect the curve. We assume 26 that they all have intersection number 1. This implies that C ab = · · · − x, where a ∈ A, b ∈ B, and x is the curve in a to be flopped.
• C -Those divisors, which do not intersect the curve.
Then the effect of the flop is essentially to exchange the role of the divisors in sets A and B. Namely, we blow down the curve sitting inside the divisors in A, and then blow up a curve in each divisor in B, and these blow ups are all identified. The divisors in A then meet this common curve with intersection number 1. The divisors in C are unchanged. At the level of the gluing curves, we have, schematically: 27 Note that flopping this new curve brings us back to the original geometry. Note also that if B is empty, the curve C is flopped out of the collection of surfaces, but after the flop it still meets the curves in A with intersection number 1. This takes us outside the formalism of [20], as the geometry now involves compact curves not contained in any of the Hirzebruch surfaces. We may now take a limit where the curve becomes decompactified, and then lose this compact curve from the geometry, taking us to a new geometry which again lies within the formalism of [20].
In the case described above, where B is empty, the curve flops out of the collection of surfaces, but we still retain a row as in (4.27), although this curve does not sit inside any surface. If we now take a decompactification limit of this curve, we simply lose this row from Ω. Note that the flop itself does not change the 1-form symmetry, but taking this decompactification limit can do so.

B NHC details B.1 Quotient space construction
As mentioned in the main text, we may construct the NHC geometries for even n as a quotient [56], where Z n acts as (z 1 , z 2 , w) ∼ (ω z 1 , ω z 2 , ω −2 w) , (B.2) where ω = e 2πi/n , z i are the coordinates of C 2 , and w ∼ w + 1 ∼ w + τ is the coordinate on T 2 . Here we should choose τ to be compatible with the action, eg, τ = i for n = 8, and τ = e 2πi/3 for n = 6, 12. The boundary of this space is then a quotient: which is a T 2 fibration over the lens space, L(n, 1).
To determine the 1-form symmetry, we must compute the homology of the boundary. We will focus on computing H 1 (∂X), which is the abelianization of π 1 (∂X). To compute this, we note that the universal cover of ∂X is S 3 ×R 2 . To obtain ∂X, we first quotient by Z 2 to obtain S 3 × T 2 , and then quotient by Z n as above. Equivalently, we may quotient the universal cover where (Z 2 ) Zn is the quotient of Z 2 by the subgroup generated by elements of the form The 1-form symmetry is then given by the subgroup of this homology group which is trivial when included into the bulk manifold, X. For n > 2, the bulk manifolds are simply connected, and so this is the entire group. For n = 2, the total space is C 2 /Z 2 × T 2 , and so the Z 2 factors above are still non-trivial in the bulk, and only the Z 2 factor remains. This is indeed the expected result for the 1-form symmetry of these theories.

B.2 Multi-Curve NHCs
In this appendix we compute the geometry of surfaces and curves for the multi-curve NHCs in (6.17) and determine from the geometry the 1-form symmetry.
The geometry of the NHC g 2 ⊕ su(2) on (−3)(−2) curves is obtained by resolving the Tate model (a discussion of the Weierstrass model has appeared in [80]) Here above U = 0 corresponds to the curve with self-intersection −3, upon which sits the g 2 with a Kodaira fiber I * ns 0 , and V = 0 is the −2-curve, with the su(2). The base has four coordinates U, V as well as the non-compact divisors W, Z, which are linearly arranged as The Cartan divisors for the affine g 2 are (U, u 3 , u 2 ) and for the affine su(2) (V, v 1 ) . These are the compact divisors. The curves are complete intersections in the ambient space and are given by the following pair-wise intersections 14) The matrix of intersections between compact divisors and curves is then