Generalized Charges, Part I: Invertible Symmetries and Higher Representations

$q$-charges describe the possible actions of a generalized symmetry on $q$-dimensional operators. In Part I of this series of papers, we describe $q$-charges for invertible symmetries; while the discussion of $q$-charges for non-invertible symmetries is the topic of Part II. We argue that $q$-charges of a standard global symmetry, also known as a 0-form symmetry, correspond to the so-called $(q+1)$-representations of the 0-form symmetry group, which are natural higher-categorical generalizations of the standard notion of representations of a group. This generalizes already our understanding of possible charges under a 0-form symmetry! Just like local operators form representations of the 0-form symmetry group, higher-dimensional extended operators form higher-representations. This statement has a straightforward generalization to other invertible symmetries: $q$-charges of higher-form and higher-group symmetries are $(q+1)$-representations of the corresponding higher-groups. There is a natural extension to higher-charges of non-genuine operators (i.e. operators that are attached to higher-dimensional operators), which will be shown to be intertwiners of higher-representations. This brings into play the higher-categorical structure of higher-representations. We also discuss higher-charges of twisted sector operators (i.e. operators that appear at the boundary of topological operators of one dimension higher), including operators that appear at the boundary of condensation defects.


Introduction
The recent developments on non-invertible symmetries hold promising potential to open an exciting chapter in the study of non-perturbative phenomena in quantum field theory (QFT).
To study systems with conventional group-like symmetries, representation theory is of course indispensable, as it describes the action of these symmetries on physical states and local operators. Likewise, we will argue that the key to unlocking the full utility of generalized, in particular non-invertible, symmetries is to understand their action on local and extended operators of various dimensions. Said differently, the key is to determine the generalized charges carried by operators in a QFT with generalized global symmetries. This will be laid out in this series of papers, where the present one is the first, with subsequent followups in Parts II [1] and III [2]. The role that representation theory plays for groups, is replaced here by higher-representations, which intimately tie into the categorical nature of the symmetries.
Although at this point in time firmly established as a central tool in theoretical physics, historically, group theory and representation theory of groups has faced an upwards battle. Eugene Wigner, who was one of the first to use group theory in the description of quantum mechanics [3], recalls that the advent of group theory in quantum mechanics was referred to by some as the "Gruppenpest" (German for "group plague"), a term allegedly coined by Schrödinger [4]. This sentiment was born out of the conviction that formal mathematicsin this case group theory -had no place in physics. Clearly history has proven Wigner and friends right, with group theory now a firmly established part of theoretical physics. Category theory has faced a similar battle in the past, justified or not. The case we would like to make here is that much like group theory is indispensable in describing physics, so is higher-category theory. In short we will make the case that what was group representation theory for physics in the 1920s, is higher-categories and the higher charges (as will be defined in this series of papers) for generalized symmetries 100 years later. extended operators. Indeed, this case will be familiar, in that 0-form symmetries can act on extended operators by permuting them, as in the following example.

Example 1.1: Higher-charges of charge conjugation 0-form symmetry
Consider 4d pure Maxwell theory, which has a charge conjugation 0-form symmetry The theory also has a G (1) = U (1) (1) e × U (1) (1) m 1-form symmetry, with U (1) e being the electric 1-form symmetry whose 1-charges are furnished by Wilson lines and U (1) m being the magnetic 1-form symmetry whose 1-charges are furnished by 't Hooft lines.
This theory has no local operators transforming non-trivially under charge conjugation 0-form symmetry. Thus, the theory does not furnish any 0-charges for this 0-form symmetry. However, the theory furnishes 1-charges and 2-charges of the 0-form symmetry G (0) = Z In the above example, the only non-trivial structure about the higher-charges is encoded in the Z 2 exchange action. However, for a general 0-form symmetry the structure of higher-charges is pretty rich, and will be elucidated in depth in this paper. For now, we note that the general structure of q-form charges of 0-form symmetries is encapsulated in the following statement.
For q = 0, we obtain the well-known statement that 0-charges of a 0-form symmetry group G (0) are representations (also referred to as 1-representations) of G (0) . However, for q > 0, this statement takes us into the subject of higher-representations, which are extremely natural higher-categorical generalizations of the usual (1-)representations. We will denote (q + 1)representations by We describe the mathematical definition of higher-representations in appendix B. In the main text, instead of employing a mathematical approach, we will take a physical approach exploring the possible ways a 0-form symmetry group G (0) can act on q-dimensional operators.
This naturally leads us to discover, that the physical concepts describing the action of G (0) on q-dimensional operators, correspond precisely to the mathematical structure of (q + 1)representations of G (0) . It is worthwhile emphasizing that this applies equally to finite but also continuous G (0) : the definition of higher-representations and statement 1.2 is equally applicable for finite and for continuous 0-form symmetries.
Naturally we should ask whether this extends to higher-form symmetries. Indeed, we find the following statement analogous to the statement 1.2:

Statement 1.3: Generalized Charges for Higher-Form Symmetries
q-charges of a G (p) p-form symmetry are (q + 1)-representations of the associated (p + 1)group G (p+1) In order to explain this statement, we need to recall that a (p + 1)-group G (p+1) is a mathematical structure describing r-form symmetry groups for 0 ≤ r ≤ p along with possible interactions between the different r-form symmetry groups. Now, a p-form symmetry group G (p) naturally forms a (p + 1)-group G (p+1) G (p) whose component r-form symmetry groups are all trivial except for r = p. We will discuss the above statement 1.3 at length for p = 1-form symmetries in this paper. For the moment, let us note that the statement 1.1 is obtained as the special case q = p of the above statement 1.3 because we have the identity (p + 1)-representations of the (p + 1)-group G (1.4) As a final generalization in this direction, while remaining in the realm of invertible symmetries, we have the following general statement whose special cases are the previous two statements 1.2 and 1.3.

Statement 1.4: Generalized Charges for Higher-Group Symmetries
q-charges of a G (p) p-group symmetry are (q + 1)-representations of the p-group G (p) .
For higher-groups (q + 1)-representations will be denoted by ρ (q+1) . (1.5) This covers all possible invertible generalized symmetries, taking into account interactions between different component r-form symmetry groups. We will discuss the above statement 1.4 at length for p = 2-group symmetries in this paper. As in the 0-form symmetry case, these statements apply equally to finite or to continuous higher-groups (and higher-form) symmetries.

Non-Genuine Generalized Charges
The considerations of the above subsection are valid only for q-charges furnished by genuine q-dimensional operators. These are q-dimensional operators that exist on their own and do not need to be attached to any higher-dimensional operators in order to be well-defined. In this subsection, we discuss the q-charges that can be furnished by non-genuine q-dimensional operators which need to be attached to higher-dimensional operators in order to be welldefined.

Example 1.2: Operators Carrying Gauge Charges are Non-Genuine
Examples of non-genuine operators are provided by operators which are not gauge invariant. Take the example of a U (1) gauge theory with a scalar field φ of gauge charge q. An insertion φ(x) of the corresponding local operator is not gauge invariant and hence not a well-defined genuine local operator. However, one can obtain a gauge-invariant configuration φ(x)exp iq can be displayed diagrammatically as Thus we have obtained a well-defined non-genuine local operator lying at the end of a line operator.
Most importantly, non-genuine operators form a layered structure: 1. We begin with genuine q-dimensional operators, which we denote by O with the superscript x distinguishing different such 1-morphisms. For a usual category the story ends here, but for higher-categories it continues further as follows. . We denote such 2-morphisms by M (a,b;A,B;x) 2 with the superscript x distinguishing different such 2-morphisms. 4. Continuing iteratively in the above fashion, given an ordered pair (M r−1 , M r−1 ) of two (r − 1)-morphisms in Hom(M r−2 , M r−2 ), a set Hom(M r−1 , M r−1 ) of rmorphisms from the (r − 1)-morphism M r−1 to the (r − 1)-morphism M r−1 . In this way, in a (q + 1)-category we have r-morphisms for 0 ≤ r ≤ q + 1.

Given an ordered pair (M
For q = 1, i.e. 2-categories, this layered structure can be depicted as in figure 2. Given that layering structure of genuine and non-genuine operators is so similar to the layering structure inherent in the mathematics of higher-categories, one might then wonder whether genuine and non-genuine higher-charges can be combined into the structure of a higher-category. This indeed turns out to be the case. In order to motivate what this highercategory should be, let us first note that (q + 1)-representations, which as discussed above describe genuine q-charges, form objects of a (q + 1)-category. We denote these (q + 1)-categories depending on the type of invertible symmetry as: (1.8) Indeed, for the simplest case p = q = 0, it is well known that representations of a group are objects of a category (also referred to as 1-category).
Thus, we are led to propose the following statement, whose various special sub-cases are justified in the bulk of this paper. The following statement is for a general p-group G (p) , but the reader can easily recover statements for a G (0) 0-form symmetry by simply substituting p = 1 and G (p=1) = G (0) , and for a G (p−1) (p − 1)-form symmetry by simply substituting to be the p-group associated to the (p − 1)-form group G (p−1) .

Statement 1.5: Non-Genuine Generalized Charges
(q − r)-charges of (q − r)-dimensional operators that can be embedded within genuine q-dimensional operators are r-morphisms in the (q + 1)-category (q + 1)-Rep (G (p) ). In more detail, we have the following correspondence: with q-charges associated to objects M of (q + 1)-Rep (G (p) ). The (q − 1)-charges of (q − 1)-dimensional operators changing O are elements of the set of 2-morphisms Hom(M (a,b;A) 1

Higher (q + 1)-Category
Higher-Charges Objects Genuine q-dimensional operators 1-Morphisms Non-genuine (q − 1)-dimensional operators r-Morphisms; r ≤ q Non-genuine (q − r)-dimensional operators Table 1: Correspondence between the layering structure of higher-categories and layering structure of non-genuine higher-charges. Note that the layer formed by (q + 1)-morphisms does not participate in this correspondence.

Twisted Generalized Charges
Finally, let us note that the statements of the previous subsection are valid only if the nongenuine operators furnishing higher-charges are not in twisted sectors for the symmetry, which are defined as follows:

Definition: Twisted Sectors
We say that an operator lies in a twisted sector of the symmetry G (p) if it lies at the end of one of the following types of topological operators related to the symmetry G (p) : 1. Symmetry Generators: These are topological operators generating the r-form symmetry groups inside the p-group G (p) for 0 ≤ r ≤ p − 1.

Condensation Defects:
These are topological operators obtained by gauging the above symmetry generators on positive codimensional sub-manifolds in spacetime [16,62].
The reason for the inclusion of condensation defects is because we want to discuss operators living at the ends of topological operators in the symmetry fusion (d − 1)-category that arises whenever we have G (p) symmetry in a d-dimensional QFT. See [17] for more details on higher-categories associated to symmetries, and [46] or the end of appendix B for more details on the higher-categories (1.9) associated to invertible symmetries. The elements of this (d − 1)-category include both symmetry generators and condensation defects.
In this paper, we will study in detail twisted q-charges for a G (0) 0-form symmetry lying in twisted sectors associated to symmetry generators of G (0) . Here we will encounter the following statement:

Statement 1.6: Twisted Generalized Charges for 0-Form Symmetries
Let D (g) d−1 be a codimension-1 topological operator generating g ∈ G (0) . Then the higher-charges of codimension-n operators, n ≥ 2, that can be placed at the boundary of D This category has is obtained from the information of the 't Hooft anomaly [ω] ∈ H d+1 (G (0) , C × ) for the G (0) 0-form symmetry and the symmetry element g ∈ G (0) by performing what is known as a slant product. See section 5.2 for more details.
Expanding out this statement in more detail, we have The above statement only incorporates the action of the centralizer H g . Consider a (d−r −2)dimensional g-twisted sector operator O d−r−2 carrying a (d − r − 2)-charge described by an We will also study twisted generalized charges in a couple of other contexts: • Twisted 1-charges in 4d when there is mixed 't Hooft between 1-form and 0-form symmetries. This reproduces a hallmark action of non-invertible 0-form symmetries on line operators in 4d theories.
• Generalized charges for operators lying in twisted sectors associated to condensation defects of non-anomalous higher-group symmetries. An interesting physical phenomena that arises in this context is the conversion of a non-genuine operator in a twisted sector associated to a condensation defect to a genuine operator and vice versa, which induces maps between twisted and untwisted generalized charges. In the language of [63], this can be phrased as the relationship between relative and absolute defects in an absolute theory. In Part II of this series of papers [1], we will upgrade this relationship to incorporate relative defects in relative theories.
Part II [1] will also extend the discussion to non-invertible, or more generally, categorical symmetries. The central tool to achieve this is the Drinfeld center of a given higher fusion category. We will formulate the current paper in the context of the Drinfeld center and see how this allows a generalization to non-invertible symmetries and their action on charged objects.

Organization of the Paper
This paper is organized as follows.
Section 2 discusses generalized charges for standard global symmetries, also known as 0form symmetries. The aim of this section is to justify the statement 1.2. After reviewing in section 2.1 why 0-charges are described by representations of the 0-form symmetry group, we encounter the first non-trivial statement of this paper in section 2.2, where it is argued that 1charges are described by 2-representations of the 0-form symmetry group. The arguments are further generalized in section 2.3 to show that q-charges are described by (q+1)-representations of the 0-form symmetry group. An important physical phenomenon exhibited by q-charges for q ≥ 2 is that of symmetry fractionalization, which we discuss in detail with various examples. Section 3 discusses generalized charges for 1-form symmetries. The aim of this section is to justify the statement 1.3, at least for p = 1. In section 3.1, we review why 1-charges are described by 2-representations of the 2-group associated to the 1-form symmetry group, which coincide with representations of the 1-form symmetry group. In section 3.2, we discuss 2-charges under 1-form symmetries. These exhibit many interesting physical phenomena involving localized symmetries (which are possibly non-invertible), induced 1-form symmetries, and interactions between localized and induced symmetries. Ultimately, we argue that all these physical phenomena are neatly encapsulated as information describing a 3-representation of the 2-group associated to the 1-form symmetry group. In section 3.3, we briefly discuss qcharges of 1-form symmetries for q ≥ 3, which in addition to the physical phenomena exhibited by 2-charges, also exhibit symmetry fractionalization. Section 4 discusses non-genuine generalized charges. The aim of this section is to justify the statement 1.5. In section 4.1, we study non-genuine 0-charges of 0-form symmetries and argue that they are described by intertwiners between 2-representations of the 0-form symmetry group. In section 4.2, we discuss non-genuine 0-charges under 1-form symmetries, which recovers the well-known statement that two line operators with different charges under a 1form symmetry cannot be related by screening. This is consistent with the corresponding mathematical statement that there are no intertwiners between two different irreducible 2representations of the 2-group associated to a 1-form symmetry group. Section 5 discusses twisted generalized charges. The aim of this section is to justify the statement 1.6 and it explores a few more situations. In section 5.1, we study generalized charges formed by operators living in twisted sectors correspond to symmetry generators of a non-anomalous 0-form symmetry. We argue that g-twisted generalized charges are described by higher-representations of the stabilizer H g of g. In section 5.2, we allow the 0-form symmetry to have anomaly. The g-twisted generalized charges are now described by twisted higher-representations of H g . In section 5. Lastly, we have a couple of appendices. We collect some important notation and terminology in appendix A. In appendix B, we discuss the mathematical definition of higherrepresentations of groups and higher-groups.
Before we begin the main text of the paper, let us make a technical disclaimer aimed mostly at experts.

Disclaimer: Restrictions on Dimensions of Operators and 't Hooft Anomalies
Throughout this paper, 1. We consider action of G (p) only on operators of co-dimension at least 2.
2. We allow G (p) to have a 't Hooft anomaly associated to an element of H d+1 (BG (p) , C × ), where BG (p) denotes the classifying space of G (p) .
Both these assumptions go hand in hand. The above type of 't Hooft anomaly is localized only on points, and so does not affect the action of G (p) on untwisted sector operators of co-dimension at least 2. Thus, while discussing the action of G (p) on operators of these co-dimensions, we can effectively forget about the anomaly. It should be noted though that the anomaly does affect the action of G (p) on twisted sector operators of co-dimension 2. We discuss in detail the modification caused by an anomaly for a 0-form symmetry.

Generalized Charges for 0-Form Symmetries
In this section we study physically the action of a 0-form symmetry group G (0) on operators of various dimensions, and argue that q-charges of these operators are (q + 1)-representations of G (0) , justifying the statement 1.2.

0-Charges
Here we reproduce the well-known argument for the following piece of statement 1.2. operator O as (2.1) One can now move two topological operators across sequentially, or first fuse them and then move them across, leading to the consistency condition Moreover, the topological operator corresponding to 1 ∈ G (0) is the identity operator, which clearly has the action 1 : Consequently, 0-charges furnished by local operators are representations of G (0) .

1-Charges
In this subsection we study 1-charges of 0-form symmetries, i.e. the possible actions of 0-form symmetries on line operators. Similar analyses have appeared recently in [64,65].
Consider now a simple line operator 1 L, where simplicity of an operator is defined as follows.

Definition
A q-dimensional operator O q is called simple if the vector space formed by topological 1 In this paper, the term 'q-dimensional operator' almost always refers to a simple q-dimensional operator.
Consequently, we denote a line operator in M L obtained by acting g ∈ G (0) on L by L [g] , and the action of g is g :  as follows where σ is a representative of the class [σ], and so σ(h 1 , h 2 ) is a C × factor in the above equation. In more detail, there are three fundamental properties of a 1-charge associated to a 2representation (which we denote with an appropriate superscript)

Example 2.1: Simplest Example of Non-trivial 1-Charge
The simplest non-trivial irreducible 2-representation arises for which is To realize it physically, we need a multiplet of two simple line operators L and L , or in other words a non-simple line operator The action of (2.13) then exchanges L and L . In fact, Z 2 has only two irreducible 2-representations, with the other 2-representation being the trivial one which is physically realized on a single simple line operator L , (2.17) which is left invariant by the action of (2.13).
Both of these 2-representations arise in the following example Quantum Field Theory.
This theory has (2.13) 0-form symmetry arising from the outer-automorphism of the gauge algebra so(2N ). The non-trivial 2-representation is furnished by where W S is Wilson line in irreducible spinor representation S of so(2N ) and W C is Wilson line in irreducible cospinor representation C of so(2N ). Indeed, the outerautomorphism exchanges the spinor and cospinor representations, and hence the two Wilson lines are exchanged under the action of (2.13).
On the other hand, there are representations of so(2N ) left invariant by the outerautomorphism, e.g. the vector representation V . The corresponding Wilson lines are left invariant by (2.13), e.g. we have and hence W V transforms in the trivial 2-representation of (2.13).
In the above example, neither of the 1-charges involved a 't Hooft anomaly for the 0-form symmetry induced on the line operators furnishing the 1-charge. Below we discuss an example where there is a non-trivial 't Hooft anomaly.

Example 2.2: 1-Charges having Anomalous Induced Symmetry
The simplest group G (0) having non-trivial H 2 (G (0) , C × ) is Here we present an example of a QFT T which contains a line operator L on which all of the bulk (2.22) 0-form symmetry descends to an induced 0-form symmetry H L = Z 2 × Z 2 along with the 't Hooft anomaly [σ] for the induced 0-form symmetry being given by the non-trivial element of (2.23). In other words, L transforms in the 2-representation For this purpose, we take This theory has a 1-form symmetry which arises from the center of the gauge group SO(4N ). The theory also has a 0-form symmetry The Z m 2 0-form symmetry, also known as the magnetic symmetry, acts non-trivially on monopole operators inducing monopole configurations (on a small sphere S 2 around the operator) of SO(4N ) that cannot be lifted to monopole configurations of Spin(4N ). On the other hand, the Z o 2 0-form symmetry arises from the outer-automorphism of the so(4N ) gauge algebra.
The 1-charge associated to the 2-representation (2.24) is furnished by any solitonic line defect for the 1-form symmetry [67,68], i.e. any line defect which induces a non-trivial background for 1-form symmetry on a small disk intersecting the line at a point: This is a consequence of anomaly inflow: in the bulk we have the following mixed anomaly between G (0) and G (1) where B 2 is the background field for Z where W can be taken to be any Wilson line operator. A line D

Higher-Charges
We now consider the extension to higher-dimensional operators, i.e. q-charges for q ≥ 2, for 0form symmetries. There is a natural extension of the discussion in the last section on 1-charges, to higher-dimensions, which will be referred to as higher-charges of group cohomology type. However, we will see that higher-representation theory forces us to consider also a generalization thereof, namely non-group cohomology type, which in fact have a natural physical interpretation as symmetry fractionalization.

Higher-Charges: Group Cohomology Type
There is an extremely natural generalization of the actions of G (0) on line operators to actions of G (0) on higher-dimensional operators.
These give rise to a special type of q-charges that we refer to as of group cohomology type, which are described by special types of (q + 1)-representations of the form which is comprised of a subgroup and a cocycle The action of G (0) is captured in this data quite similar to as in previous subsection.
The multiplet is irreducible in the sense that we can obtain any q-dimensional operator Mathematically, the size n of M (O q ) is known as the dimension of the (q + 1)representation ρ (q+1) . As we discussed in the previous subsection, for q = 1, the group cohomology type 1-charges are the most general 1-charges. However, for q ≥ 2, the group cohomology type q-charges only form a small subset of all possible q-charges. We describe the most general 2-charges in the next subsection.

Example 2.3: Simplest Example of a non-trivial q-Charge
The simplest (q + 1)-representation of group cohomology type is q , generating Z 2 1-form symmetries that do not act on spinor and cospinor representations respectively.

Example 2.4: q-Charges with Anomalous Induced Symmetry: Anomaly Inflow
Examples of group cohomology type q-charges carrying non-trivial [σ] can be obtained via anomaly inflow from the bulk d-dimensional QFT, as in an example discussed in the previous subsection. For example, the q-charge can be furnished by a q-dimensional solitonic defect inducing a background (on a small (d − q)-dimensional disk intersection its locus) for a (d − q − 1)-form symmetry having a mixed 't Hooft anomaly with 0-form symmetry in the bulk roughly a of the form a For brevity, we are suppressing many details that need to be specified for the following expression for the anomaly to make sense.

Higher-Charges: (Non-Invertible) Symmetry Fractionalization Type
At first sight, one might think that group cohomology type q-charges provide all possible q-charges. There are at least two reasons for believing so: 1. First of all, the mathematical structure of group cohomology type q-charges is a nice, uniform generalization of the mathematical structure of general 1-charges.
2. Secondly, the mathematical data of group cohomology type q-charges described in the previous subsection seems to incorporate all of the relevant physical information associated to the action of G (0) 0-form symmetry on q-dimensional operators.
However, if one believes statement 1.2, then one should check whether (q + 1)-representations are all of group cohomology type. It turns out that this is not the case for q ≥ 2, for which generic q-charges are in fact not of this type. In this way, the mathematics of higherrepresentations forces us to seek new physical phenomena that only start becoming visible when considering the action of G (0) 0-form symmetry on a q ≥ 2-dimensional operator O q .
In turn, physically we will see that these non-group cohomology type higher representations have concrete realizations in terms of symmetry fractionalization. Perhaps the most intriguing implication is that invertible symmetries can fractionalize into non-invertible symmetries, as we will see in the example of a Z (0) 2 factionalizing into the Ising category. Additionally, we also have induced localized symmetries. These are generated by (q − 1)dimensional and lower-dimensional topological defects arising in the (q −1)-dimensional worldvolume of a junction between O q and a bulk codimension-1 topological defect D

Definition
We refer to localized symmetries induced by a bulk 0-form symmetry g ∈ G (0) as induced localized symmetries in the g-sector.
Then, induced localized symmetries in the identity sector are just the localized symmetries discussed in the previous paragraph.
Note that we can compose an induced localized symmetry in the g-sector with an induced localized symmetry in the g -sector to obtain an induced localized symmetry in the gg -sector.
Mathematical Structure. As discussed in the introduction, the mathematical structure to encapsulate defects of various dimensions layered and embedded in each other is that of higher-categories. Thus, we can describe the induced localized symmetries in the g-sector by a (non-fusion) (q − 1)-category In total, all induced localized symmetries are described by a (q − 1)-category The composition of induced localized symmetries lying in different sectors discussed in the previous paragraph becomes a fusion structure on the (q − 1)-category it into a fusion (q − 1)-category. Moreover, since the fusion respects the group multiplication of the underlying sectors This is because we have only been studying a special class of q-charges: a q-charge in this class describes a multiplet of size 1 of q-dimensional operators, i.e. there is only a single q-dimensional operator O q in the multiplet. Indeed, while discussing above the structure of induced localized symmetries, we assumed that all the elements g ∈ Allowing more q-dimensional operators to participate in the multiplet, we obtain q-charges Such a q-charge is specified by two pieces of data and is realized by a multiplet comprising of a single q-dimensional operator O q . The physical information of the q-charge is obtained from these two pieces of data as follows 1. Localized Symmetries: There is a 0-form symmetry localized on the world-volume of O q given by the kernel of π, ker(π) ⊆ G (0) .
2. Induced Localized Symmetries: Additionally, we have induced localized symmetries. In the g-sector, these are in one-to-one correspondence with the elements of the subset Mathematically, these q-charges correspond to 1-dimensional (q +1)-representations whose of G (0) -graded (q − 1)-vector spaces with a non-trivial coherence relation (also known as associator) described by the class [ω].

Symmetry Fractionalization
A simple example of such a q-charge is provided by where there is a unique possible surjective map π: it maps the two generators of Z 4 to the generator of Z 2 . A q-dimensional operator O q realizing this q-charge has a localized symmetry ker(π) = Z 2 .
q−1 living in the worldvolume of O q . Now let us look at induced localized symmetries lying in the non-trivial sector in G (0) = Z 2 . These are in one-to-one correspondence with the two generators of G (0) = Z 4 . That is, there are two (q − 1)-dimensional topological operators that can arise at the junction of O q with the bulk codimension-1 topological defect D d−1 that generates The statement of symmetry fractionalization is now as follows. We try to induce the In order to implement this symmetry on O q , we need to make a choice of a topological defect lying at the intersection of O q and the symmetry generator D d−1 . We can either choose this topological defect to be D q−1 without loss of generality. Now we check whether the symmetry is still Z 2 valued by performing the fusion of these topological defects. As we fuse D d−1 with itself, it becomes a trivial defect, which means the symmetry is Z 2 -valued in the bulk. However along the worldvolume of O q we have to fuse D (2.50) See figure 8. Thus we see explicitly that the bulk G (0) = Z 2 symmetry fractionalizes to

along with a mixed 't Hooft anomaly of the form
and no pure 't Hooft anomaly for Z . QFTs having such a symmetry and anomaly structure are ubiquitous: simply take a d-dimensional QFT T which has a non-anomalous Z 4 0-form symmetry, and gauge the Z 2 subgroup of this 0-form symmetry. The resulting QFT after gauging can be identified with T. The Z This was shown in section 2.5 of [47] for d = 3, but the same argument extends to general d.

Example: Non-Invertible Symmetry Fractionalization
Generalizing the above story, the physical structure of a general q-charge can be understood as the phenomenon of the bulk G (0) 0-form symmetry fractionalizing to a non-invertible induced symmetry on the world-volume of an irreducible multiplet of q-dimensional operators furnish- ing the q-charge. When the irreducible multiplet contains a single q-dimensional operator O q , the non-invertible induced symmetry on O q is described by the symmetry (q − 1) category [17] Below we provide a simple example exhibiting non-invertible symmetry fractionalization, where a Z (0) 2 0-form symmetry fractionalizes on a surface defect O 2 to a non-invertible induced symmetry described by the Ising fusion category.

Example 2.6: Symmetry Fractionalization to Ising Category
Let us conclude this section by providing an illustrative example of non-invertible symmetry fractionalization. This is in fact the simplest example of non-invertible symmetry fractionalization. It is furnished by a surface operator (2.53) is the identity line operator on O 2 . See figure 9. Since this is a non-invertible fusion rule, this means that the bulk G (0) = Z 2 0-form symmetry fractionalizes to a non-invertible symmetry on O 2 . In fact, the non-invertible symmetry can be recognized as the well-known Ising symmetry generated by the Ising fusion category, which is discussed in more detail below.
Mathematically, the 2-charge carried by O 2 is described by a 1d 3-representation corresponding to a Z 2 -graded fusion category whose underlying non-graded fusion category is the Ising fusion category. This fusion category has three simple objects along with fusion rules (2.56) It is converted into a Z 2 -graded fusion category by assigning {D to the non-trivial grade.

Generalized Charges for 1-Form Symmetries
In this section, we discuss generalized charges of a 1-form symmetry group G (1) . As for 0form symmetries the simplest instance is the case of 1-charges, upon which the symmetry acts simply as representations. However, we will see again that higher q-charges, i.e. q-dimensional operators upon which the 1-form symmetry acts, are associated to higher-representations. Let us emphasize that these are not higher-representations of the group G (1) , but rather higherrepresentations of the 2-group G G (1) associated to the 1-form group G (1) . We will denote the generators of the 1-form symmetry group by topological codimension-2 operators

1-Charges
The action of a 1-form symmetry on line operators is similar to the action of a 0-form symmetry on local operators [5]. We can move a codimension-2 topological operator labeled by g ∈ G (1) across a line operator L. In the process, we may generate a topological local operator D Since L is assumed to be simple, the operators D (L,g) 0 can be identified with elements of C × , and then the above condition means that L corresponds to an irreducible representation (or character) of the abelian group G (1) . This is simply the special case p = 1 of statement 1.1.
In fact, mathematically, representations of the 1-form symmetry group G (1) are the same as 2-representations of the 2-group G G (1) based on the 1-form group G (1) . What we mean by G G (1) is simply the 2-group which is comprised of a trivial 0-form symmetry and said 1-form symmetry G (1) .
Thus, we recover the p = q = 1 version of the statement 1.3:

2-Charges
In this subsection, we want to understand how a simple surface operator O 2 interacts with a G (1) 1-form symmetry. , X ∈ C . (3. 3) The invertible part of localized symmetries described by C will play a special role in the discussion that follows. This is described by a group H C which we refer to as the 0-form symmetry localized on O 2 . We label the corresponding topological line operators as D  The composition rule of 1-form symmetries needs to be obeyed by the sourced localized symmetries (3.5) As a consequence of this, only the invertible localized 0-form symmetries described by group H C can be sourced by the induced 1-form symmetries. We thus have a homomorphism describing the localized 0-form symmetry sourced by each induced 1-form symmetry element and we can write (3.7) This results in the following non-trivial constraint on the possible background fields where (3.9) But we have already established that the junction can only source D (τ (g)) 1 , which implies that the line (3.9) must equal D (τ (g)) 1 . Restricting to the invertible part D we learn that the image of the homomorphism (3.6) is contained in the center Z(H C ) of H C .
That is, induced 1-form symmetries can only source localized symmetries lying in Z(H C ).
The action of the H C localized 0-form symmetry is This means that the junction topological local operator D (g) 0 furnishes a 1-dimensional representation of H C . Thus we have an homomorphism where χ(H C ) is the character group, namely the group formed by 1-dimensional representa- The action of H C on induced 1-form symmetries can be viewed as a mixed 't Hooft anomaly between the localized 0-form and the induced 1-form symmetries of the form where the notation for background fields has been discussed above. More concretely, A 3 is a C × valued 3-cocycle whose explicit simplicial form is where v i are vertices in a simplicial decomposition, Note that, for consistency we must demand that which is a non-trivial condition to satisfy due to the non-closure condition (3.8).
Moreover, the associativity of (3.4) imposes the condition that α is a 2-cocycle. In fact, using the freedom to rescale topological local operators D (g) 0 , only the cohomology class Physically, the class [α] describes a pure 't Hooft anomaly for the G (1) induced symmetry taking the form where the Bockstein is taken with respect to the short exact sequence specified by the extension class [α].
Mathematical Structure. All of the above information describing a 2-charge of 1-form symmetry can be neatly encapsulated using category theory. First of all, as we have already been using in the above physical description, the localized symmetries are described by a fusion category C. The interactions of localized and induced symmetries, along with the pure 't Hooft anomaly of induced symmetries is mathematically encapsulated in the information of a braided monoidal functor where Vec(G (1) ) is the braided fusion category obtained by giving a trivial braiding to the fusion category formed by G (1) -graded vector spaces and Z(C) is the modular (in particular braided) fusion category formed by the Drinfeld center of C.
Let us expand on how this mathematical structure encodes all of the physical information discussed above. As we will argue in Part II [1], we have the following general statement.

Statement 3.2: 0-Charges of a Non-Invertible Categorical Symmetry
The 0-charges of a possibly non-invertible symmetry described by a fusion category C are objects of its Drinfeld center Z(C).
Then, the functor (3.20) assigns to every 1-form symmetry element g ∈ G (1) a 0-charge for the localized symmetry C on O 2 . More concretely, an object of Drinfeld center Z(C) can be expressed as (X, β) , (3.21) where X is an object in C and β is a collection of morphisms in C involving X. The functor (3.20) thus assigns to g ∈ G (1) a simple object (X g = τ (g), β g ) ∈ Z(C) where the simple object X g = τ (g) ∈ Z(H C ) describes the localized symmetry sourced by the corresponding induced 1-form symmetry and the morphisms β g encode the action of localized symmetries on induced symmetries. This encoding will be described in Part II [1]. Finally, the fact that the functor is monoidal encodes the condition (3.16) along with the characterization (3.17).
Such functors capture precisely 1-dimensional 3-representations of the 2-group G G (1) based on the 1-form group G (1) , and general 3-representations are direct sums of these 1-dimensional ones. Thus, we recover the p = 1, q = 2 piece of statement 1.3:  First of all, the induced 1-form symmetry cannot carry a pure 't Hooft anomaly because Thus, we have the following possible 2-charges: 1. There is no interaction between the localized and induced symmetries.
2. The generator of the induced Z 2 1-form symmetry is charged under the localized symmetry (3.24).

This corresponds to a 't Hooft anomaly
3. The generator of the induced Z 2 1-form symmetry is in the twisted sector for the generator of the localized symmetry (3.24). In other words, the induced symmetry sources the localized symmetry.
In terms of background fields, we have the relationship Note that the generator of the induced Z 2 1-form symmetry cannot be both charged and be in the twisted sector at the same time, because in such a situation the relationship (3.27) would force the mixed 't Hooft anomaly (3.26) to be non-closed which is a contradiction.
Categorical Formulation. We can also recover the above three possibilities using the more mathematical approach outlined above. Mathematically, we want to enumerate braided monoidal functors from the braided fusion category Vec G (1) =Z 2 (with trivial braiding) to the modular tensor category Z(C = Vec Z 2 ). The latter can be recognized as the category describing topological line defects of the 3d Z 2 Dijkgraaf-Witten gauge theory, or in other words the 2+1d toric code. In other words, we are enumerating different ways of choosing a non-anomalous Z 2 1-form symmetry of the above 3d TQFT. 3. Choose the "magnetic" line m as the generator of the Z 2 1-form symmetry. This corresponds to the 2-charge in which the induced symmetry sources the localized symmetry.
Note that we cannot choose the "dyonic"/"fermionic" line ψ as the generator of Z 2 1form symmetry, because the ψ line is a fermion (recall θ(ψ) = −1) and hence generates a Z 2 1-form symmetry with a non-trivial 't Hooft anomaly. This corresponds to the fact that one cannot have a 2-charge in which induced symmetry is both charged under the localized symmetry and also sources the localized symmetry. Below we describe a concrete field theory which realizes the above discussed 2-charges.

Example 3.2: 4d O(4N ) gauge theory
The two non-trivial 2-charges exhibiting properties (3.27) and (3.26) are realized in 4d pure O(4N ) gauge theory. This can be easily seen if we begin with the 4d pure Pin + (4N ) gauge theory, which as discussed in [26] has topological surface operators described by 2-representations of a split 2-group. We will only use two surface operators D being gauged. This means that we have a 't Hooft anomaly (3.26) and hence the proposed surface operator O 2 indeed furnishes the desired non-trivial 2-charge.
A surface operator O 2 furnishing the other non-trivial 2-charge is simply obtained from O 2 by gauging its Z 2 localized symmetry along its whole world-volume. As explained above, O 2 then exhibits (3.27).

Higher-Charges
Continuing in the above fashion, one may study q-charges for q ≥ 3. The interesting physical phenomenon that opens up here is the possibility of fractionalization of 1-form symmetry, i.e. the induced 1-form symmetry on a q-dimensional operator O q may be a larger group G (1) , or may be a larger higher-group, or the induced symmetry may actually be non-invertible.
Mathematically, such q-charges are expected to form (q + 1)-representations ρ (q+1) of the 2-group G G (1) associated to the 1-form symmetry group G (1) .  tries. This information about the charge is encoded mathematically in the braiding of an arbitrary object of the graded category B G (1) with an object of the trivially graded part B ⊆ B G (1) .

1-Form Symmetry Fractionalization in Special
A generic choice of B G (1) corresponds to a non-invertible fractionalization of G (1) 1-form symmetry, quite similar to the non-invertible fractionalization of 0-form symmetry discussed in section 2.3.2.

Example 3.3: Invertible and Non-Invertible 1-Form Symmetry Fractionalization
Let us provide examples of B G (1) corresponding to both invertible and non-invertible symmetry fractionalization for For invertible symmetry fractionalization, take with trivial braiding, and grading specified by surjective homomorphism 1-form localized symmetry, which is extended to a total of G (1) ind-loc = Z 4 1-form symmetry by 1-form symmetries induced on O 3 from the bulk G (1) = Z 2 1-form symmetry: In other words, the bulk G (1) = Z 2 1-form symmetry is fractionalized to G (1) ind-loc = Z 4 1-form symmetry on the worldvolume of O 3 .
For non-invertible symmetry fractionalization, take B G (1) to be Ising modular fusion category, and grading that assigns trivial grade to {D

Non-Genuine Generalized Charges
So far we considered only genuine q-charges. As we will discuss now, non-genuine charges arise naturally in field theories and require an extension, to include a higher-categorical structure.
The summary of this structure can be found in statement 1.5. In this section, we physically study and verify that the statement is correct for 0-charges of 0-form and 1-form symmetries.

Non-Genuine 0-Charges of 0-Form Symmetries
We have discussed above that genuine 0-charges for G (0) 0-form symmetry are representations of G (0) . Similarly, genuine 1-charges are 2-representations of G (0) . In this subsection, we discuss non-genuine 0-charges going from a genuine 1-charge corresponding to a 2-representation ρ (2) to another genuine 1-charge corresponding to a 2-representation ρ (2) . These non-genuine 0-charges are furnished by non-genuine local operators changing a line operator L having 1-charge ρ (2) to a line operator L having 1-charge ρ (2) : Let ρ (2) and ρ (2) be the following irreducible 2-representations as explained in figure 14.
Because of the factor σσ −1 (h 2 , h 1 ) ∈ C × , non-genuine 0-charges from genuine 1-charge ρ (2) to genuine 1-charge ρ (2) are not linear representations of H LL in general. Such non-genuine 0-charges are linear representations only if  In this situation, we say the non-genuine 0-charges are twisted representations of H LL lying in the class [σσ −1 ] ∈ H 2 (H LL , C × ).

Aside: Difference between Twisted and Projective Representations
Two solutions of (4.4) give rise to isomorphic twisted representations if they are related by a basis change on the space of local operators. Note that twisted representations with trivial twist are equivalent to linear representations of H LL . Also note that two nonisomorphic In fact, mathematically, all these twisted representations combine together to form a 1-
When ρ (2) and ρ (2) are both trivial 2-representations, then the intertwiners are the same as representations of G (0) . Since the identity line operator necessarily transforms in trivial 2-representation, we hence recover the statement 2.1 regarding genuine 0-charges.

Example 4.1: Fractional Monopole Operators
Consider the example 2.2 of 3d pure SO(4N ) gauge theory. As we discussed earlier, the topological line operator D We can see this explicitly for special examples of such non-genuine local operators known as fractional gauge monopole operators [67]. In our case, these are local operators that induce monopole configurations for P SO(4N ) = SO(4N )/Z 2 on a small sphere S 2 surrounding them that cannot be lifted to monopole configurations for SO(4N ). Such fractional monopole operators can be further divided into two types: 1. The associated monopole configuration for P SO (4N ) can be lifted to a monopole configuration for Ss(4N ) = Spin(4N )/Z S 2 but not to a monopole configuration for 2. The associated monopole configuration for P SO (4N ) can be lifted to a monopole configuration for Sc(4N ) but not to a monopole configuration for Ss (4N ) or SO(4N ).
On the other hand, the monopole operators associated to monopole configurations for P SO(4N ) that can be lifted to monopole configurations for SO(4N ) but not to monopole configurations for Ss(4N ) or Sc(4N ) are non-fractional monopole operators, which are genuine local operators charged under Z m 2 0-form symmetry. Here Z S 2 × Z C 2 is the center of Spin(4N ). The generator of Z S 2 leaves the spinor representation invariant, but acts non-trivially on the cospinor and vector representations. Similarly, the generator of Z C 2 leaves the cospinor representation invariant, but acts non-trivially on the spinor and vector representations. Finally, the diagonal Z 2 subgroup is denoted as Z V 2 whose generator leaves the vector representation invariant, but acts non-trivially on the cospinor and spinor representations. Now, these two types of fractional monopole operators are exchanged by the outerautomorphism 0-form symmetry Z o 2 . On the other hand, only one of the two types of operators are non-trivially charged under Z m 2 0-form symmetry. This is because the two types of fractional monopole operators are interchanged upon taking OPE with nonfractional monopole operators charged under Z m 2 . Thus, fractional monopole operators indeed furnish representations twisted by the non-trivial element of (2.23) because the actions of Z m 2 and Z o 2 anti-commute on these operators.

Non-Genuine 0-Charges of 1-Form Symmetries: Absence of Screening
In the previous subsection, we saw that there exist 0-charges between two different irreducible 1-charges for a 0-form symmetry. However, the same is not true for the 1-form symmetry.
There are no possible 0-charges between two different irreducible 1-charges. This means that there cannot exist non-genuine local operators between two line operators carrying two differ-  G (1) associated to a 1-form symmetry group G (1) . Physically, this is the statement of charge conservation for 1-form symmetry as explained in figure 15. This explains the p = q = r = 1 piece of statement 1.5.
This fact is usually presented by saying that L 1 cannot be screened to another line operator L 2 , if L 1 and L 2 have different charges under the 1-form symmetry. In particular, a line operator L carrying a non-trivial charge under 1-form symmetry cannot be completely screened, i.e. cannot be screened to the identity line operator.

Twisted Generalized Charges
In this section we study higher-charges formed by operators living in twisted sectors of invertible symmetries. These, as defined in section 1.4, arise at the end of symmetry generators or condensation defects. We will see that the structure of twisted charges is sensitive to the 't Hooft anomalies of the symmetry, even for operators of codimension-2 and higher, which is unlike the case of untwisted charges.

Non-Anomalous 0-Form Symmetries
In this subsection, we begin by studying twisted higher-charges that can arise at the ends of symmetry generators of a G (0) 0-form symmetry group. 0-form symmetries are generated by Figure 16: Action of h ∈ G (0) not in the stabilizer group of g, maps a g-twisted operator to an hgh −1 -twisted operator.
be the stabilizer subgroup of g. We can act on O by an element As explained in figure 16, this maps O to a local operator O living in twisted sector for where M g is the vector space formed by local operators participating in the multiplet M and lying in the g-twisted sector. Moreover, the stabilizer H g acts as linear maps from M g to itself, implying Similarly, M g for any g ∈ [g] forms an irreducible representation of the corresponding stabi- . This representation is obtained simply by transporting the representation of H g formed by M g using an isomorphism The line operators in M lying in g -twisted sector for g = g gg −1 form a 2-representation ρ (2) g of the isomorphic group Similarly, considering lower-dimensional operators, one recovers the full categorical statement made for [ω g ] = 0 in statement 1.6.

Anomalous 0-Form Symmetry
Let us now turn on a 't Hooft anomaly of the form for the bulk G (0) 0-form symmetry and revisit the analysis of the previous subsection.
Two Dimensions. Just as for the non-anomalous case, the twisted sector operators form multiplets parametrized by conjugacy classes [g] ∈ G (0) , and g-twisted sector operators in a [g]-multiplet are acted upon by the stabilizer H g . However, instead of forming linear representations of H g , the g-twisted operators now form [ω g ]-twisted representations of H g , where for all h 1 , h 2 ∈ H g . See figure 17 for explanation and section 4.1 for more details on twisted representations. The map induced by Figure 17: The above chain of equalities provides a formula for the twist ω g = ω(h −1 1 , gh 1 , h 2 )ω(g, h 1 , h 2 )ω(g −1 , h −1 2 , h −1 1 ) which the reader can verify matches the expression shown in (5.13). The various ω factors arise by performing associativity/F-moves on the topological line operators generating an anomalous 0-form symmetry in 2d.
is often referred to as slant product in the literature (see e.g. [69]). This justifies d = 2 piece of the statement 1.6.
Three Dimensions. Again, as in the non-anomalous case, the line operators in g-twisted sector form multiplets M . The stabilizer H g still sends g-twisted sector lines into each other.
The associativity of the action of H g is governed by The class [ω g ] is again referred to as the slant product of [ω] and g. To see these constraints, pick an arbitrary topological local operator D As we fuse these operators, we will in general generate factors The action of the induced H L 0-form symmetry must be associative, which means that the non-associativity factor arising from ω g must be cancelled by the σ factors as follows where h 1 , h 2 , h 3 ∈ H L . See figure 18. In particular H L must be such that There is additional information in the factors σ. Note that by redefining topological local operators D (h) 0 , we can redefine σ as for a C × valued 1-cochain α on H L . This means that two 1-charges differentiated only by having 2-cochains σ and σ such that  (d) [ωg] is specified by and is actually a 1-dimensional d-representation of group cohomology type. Here is obtained by performing a slant product of g with [ω] where h i ∈ H g , s(i) = 1 for even i and s(i) = −1 for odd i.
Thus twisted g-sector generalized charges are specified by the (d−1)-category of morphisms from ρ (d) [ωg] to identity d-representation in the d-category d-Rep(H g ) formed by d-representations of H g . We denoted this (d − 1)-category as Figure 19: The mixed 0-form/1-form symmetry anomaly (5.28) as seen from the topological defects. The junction D 0 between the two topological surface defects D 2 that generate the 1-form symmetry is charged under the 0-form symmetry generated by the codimension-1 topological defect D 3 .
in statement 1.6 and called its objects as '[ω g ]-twisted (d − 1)-representations of H g '. This is because for low d, this matches the more well-known notion of twisted representations and twisted 2-representations, which has been discussed in detail above.

Mixed 't Hooft Anomaly Between 1-Form and 0-Form Symmetries
We have seen above that in the presence of 't Hooft anomaly, the structure of twisted generalized charges is quite different from the structure of untwisted generalized charges. In this subsection, we will see another example of this phenomenon, while studying the structure of 1-charges in the presence of 1-form and 0-form symmetries with a mixed 't Hooft anomaly in 4d QFTs.
In particular, we consider in 4d, 0-form and 1-form symmetries with mixed 't Hooft anomaly Let D 2 and D 3 be the topological operators generating Z    Figure 22: D q+1 is a (q + 1)-dimensional condensation defect, which means that it admits at least one topological non-genuine q-dimensional defect living at its boundary. In the figure, D q is one such topological boundary of D q+1 . On the other hand O q is a possibly non-topological operator living on the boundary of D q+1 . In other words, O q is in the twisted sector for the condensation defect D q+1 . We can perform an interval compactification involving O q , D q+1 and D q as shown in the figure to obtain an untwisted sector possibly non-topological It is straightforward to generalize to general G (0) and G (1) , but the expression for the anomaly (5.28) takes a more complicated form involving a cohomological operation combining the cup product and Pontryagin square operations appearing in (5.28) into a single operation, which takes in A 1 and B 2 to output the anomaly A 5 directly.
We will see in Part II [1] that this fact leads to a well-known action cite of a non-invertible symmetry on line operators, permuting untwisted sector and twisted sector lines for a 1-form symmetry into each other.

Condensation Twisted Charges
In this subsection, we study generalized charges appearing in twisted sectors associated to condensation defects. As described in the definition in section 1.4, condensation defects are topological defects obtained by gauging invertible symmetry generating topological defects on submanifolds in spacetime.
Twisted to Untwisted. The first interesting physical observation here is that a q-dimensional operator in twisted sector for a (q + 1)-dimensional condensation defect can always be converted into a q-dimensional untwisted sector operator. This is because a condensation defect always admits a topological end, which allows us to perform the above transition as explained in figure 22. There might be multiple such topological ends and hence multiple ways of performing the above transition. However, one should note that there is a canonical topological end as well corresponding to Dirichlet boundary conditions for the gauge fields localized on the (q + 1)-dimensional locus occupied by the condensation defect. Below we assume that we have performed this transition using this canonical boundary condition. Oq symmetry living on the world-volume of O q . Since topological operators generating induced symmetries extend into the bulk, we need to also specify a background for the G (p) bulk p-form symmetry.
There is a canonical way of specifying such a background by restricting the bulk topological operators generating G (p) to only lie in a (q+1)-dimensional submanifold Σ q+1 of d-dimensional spacetime, whose boundary is the world-volume of O q . This gives rise to an r-form symmetry background B r+1 | Σ q+1 on Σ q+1 whose restriction to the world-volume of O q gives rise to the background B r+1 | Oq . Now, performing gauge transformations of the background B r+1 | Σ q+1 , we might find a 't Hooft anomaly for the induced G (r) Oq symmetry taking the form where Θ q+1 B r+1 | Oq is an R/Z-valued cochain on Σ q+1 , which is a function of the background field B r+1 | Σ q+1 . This can be canceled by adding along Σ q+1 a G

(r)
Oq protected SPT phase whose effective action is A q+1 .
Once the anomaly has been cancelled in this fashion, we can promote B r+1 | Σ q+1 to a dynamical gauge field that we denote as b r+1 | Σ q+1 , thus gauging the G    1-form symmetry of the Pin + (4N ) theory on a codimension-1 manifold in spacetime.

1-Charges for 2-Group Symmetries
In this section, we study possible 1-charges that can be furnished by line operators under an arbitrary 2-group symmetry. A 2-group symmetry combines 0-form and 1-form symmetries, encapsulating possible interactions between the two types of symmetries.
We will proceed by studying 2-groups of increasing complexity. Let us begin by addressing "trivial" 2-groups, in which there are no interactions between the 0-form and 1-form symmetries. Then a 1-charge of the 2-group is a tuple formed by an arbitrary 1-charge of the G (0) 0-form symmetry and an arbitrary 1-charge of the G (1) 1-form symmetry without any correlation between these two pieces of data.

Split 2-Group Symmetry
The simplest possible interaction between 0-form symmetry and 1-form symmetry arises when 0-form symmetry acts on 1-form symmetry generators by changing their type. See figure 23.
That is we have a collection of automorphisms of G (1) labeled by elements g ∈ G (0) such that If this is the only interaction between 0-form and 1-form symmetries, then such a 2-group is known as a split 2-group.
for all h ∈ H L and γ ∈ G (1) . See figure 24. Such characters form a subgroup G (1) . An equivalent mathematical characterization of G (1) H L is as follows. First, note that the action α of G (0) on G (1) induces a dual action α of G (0) on G (1) satisfying for all g ∈ G (0) , χ ∈ G (1) and γ ∈ G (1) . Then G (1) H L is the subgroup of G (1) formed by elements left invariant by α h for all h ∈ H L .
Using the dual action, it is straightforward to describe the character of G (1) carried by another line operator L [g] in the multiplet M L . If L carries character χ L ∈ G (1)   . (6.5) Thus the action of the split 2-group on a multiplet M L of line operators is described by which precisely specifies an irreducible 2-representation of the split 2-group, thus justifying a a part of the p = 2, q = 1 piece of statement 1.4.

2-Group Symmetry With Untwisted Postnikov Class
A different kind of 2-group symmetry arises when there is no action of 0-form symmetry on 1-form symmetry, but the associativity of 0-form symmetry is modified by 1-form symmetry.

This modification is captured by an element
[Θ] ∈ H 3 (G (0) , G (1) ) , (6.7) which is known as the Postnikov class associated to the 2-group symmetry. Since there is no action of G (0) on G (1) , the Postnikov class is an element of the untwisted cohomology group.
In understanding the action of such a 2-group on line operators, we follow the same proce-  [Θ] under discussion [26,66], justifying a part of the p = 2, q = 1 piece of statement 1.4.

General 2-Group Symmetry
A general 2-group G (2) has the following information where G (0) is a 0-form symmetry group, G (1) is a 1-form symmetry group, α is an action of G (0) on G (1) , and This is precisely the information describing irreducible 2-representations of the 2-group G (2) [ 26,66], thus fully justifying the p = 2, q = 1 piece of statement 1.4.

Conclusions and Outlook
In this paper we answered the question, what the structure of charges for invertible generalized global symmetries is. The main insight that we gained is that these higher charges, or qcharges, fall into higher-representations of the symmetries.
This applies to standard 0-form symmetries (continuous and finite), but also higher-form symmetries and more generally higher-group symmetries. Thus, even when restricting one's attention to invertible symmetries, a higher-categorical structure emerges naturally. We have argued for the central relevance of higher-representations from a physical perspective -thus making their natural occurrence (and inevitability) apparent. The standard paradigms of extended p-dimensional operators being charged under p-form symmetries G (p) , i.e. forming representations of these groups, are naturally obtained as specializations of the general structure presented here. The important insight is however, that this is by far only a small subset of generalized charges!
We discussed charged operators that are genuine and those that are non-genuine (e.g. operators appearing at the ends of higher-dimensional operators), including twisted sector operators. There is a natural higher-categorical structure that organizes these non-genuine charges.
We provided several examples in various spacetime dimensions (d = 2, 3, 4). However the full extent of higher-representations of invertible symmetries deserves continued in depth study.
For instance, our examples of higher-charges of higher-form/group symmetries were focused on finite symmetries, but as we pointed out, the results should equally apply to continuous symmetries.
In view of the existence of non-invertible symmetries in d ≥ 3, a natural question is to determine the higher-charges in such instances as well. This is the topic of Part II of this series [1]. Already here we can state the main tool to study these, which is the Symmetry TFT (SymTFT) [70][71][72][73][74] or more categorically, the Drinfeld center of the symmetry category.
• Twisted Sector Operator: A term used to refer to a non-genuine operator arising at the boundary of a topological operator of one higher dimension.
• Local Operator: An operator of dimension 0.
• Extended Operator: An operator of dimension bigger than 0.

B Higher-Representations
In this appendix, we introduce the mathematics of higher-representation theory for groups and higher-groups.

B.1 Representations of Groups
Let us begin with usual representations of a group G (0) . Recall that a representation ρ on a finite dimensional vector space V is a map where End(V ) is the set of endomorphisms of V , i.e. the set of linear maps from V to itself.
In order for it to be a representation, the map ρ needs to satisfy the following additional To see that this definition matches the usual one, note that the functor ρ maps the single object of C G (0) to an object V of Vec, which is the underlying vector space for the representation ρ. moreover, the functor ρ maps the endomorphisms of the single object of C G (0) to endomorphisms of V .
Note that we could just replace BG (0) by any topological space X construct in this way a (q + 1)-category associated to it. In fact, in discussing (q + 1)-representations of a p-group G (p) we will need the category C Since the essential information of the classifying space BG (0) is in its first homotopy group, the essential information of the (q + 1)-category C (q+1) G (0) is in its 1-morphisms.