Two-dimensional algebra in lattice gauge theory

# Two-dimensional algebra in lattice gauge theory

Arthur J. Parzygnat Mathematics Department, University of Connecticut Storrs, CT 06269, USA
###### Abstract

We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1- and 2-form gauge theory. In particular, we utilize string diagrams, tools similar to tensor networks, to compute the parallel transport along a surface using approximations on a lattice. Although this work is mainly intended as expository, we prove a convergence theorem for the surface transport in the continuum limit. Locality is used to define infinitesimal parallel transport and two-dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data. The correct surface ordering is dictated by two-dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice. The surface ordering is inherently complicated, but we prove a simplification theorem confirming earlier results of Schreiber and Waldorf. Assuming little background, we present a simple way to understand some abstract concepts of higher category theory. In doing so, we review all the necessary categorical concepts from the tensor network point of view as well as many aspects of higher gauge theory.

## 1 Introduction

We use string diagrams to express many concepts in gauge theory in the broader context of two-dimensional algebra. By two-dimensional algebra, we mean the manipulation of algebraic quantities along surfaces. Such manipulations are dictated by 2-category theory and we include a thorough and visual introduction to 2-categories based on string diagrams. Such string diagrams, including their close relatives known as tensor networks, have been found to provide exceptionally clear interpretations in areas such as open quantum systems [WBC15], foundations of quantum mechanics [AC04], entanglement entropy [Or14], and braiding statistics in topological condensed matter theory [Bo17] to name a few.

We postulate simple rules for associating algebraic data to surfaces with boundary and use the rules of two-dimensional algebra to derive non-abelian surface transport from infinitesimal pieces arising from a triangulation/cubulation of the surface. One of the novelties in this work is an analytic proof for the convergence of surface transport together with a more direct derivation of the iterated surface integral than what appears in [SW2] for instance. To be as self-contained as possible, we include discussions on gauge transformations, orientation data on surfaces, and a two-dimensional calculation of a Wilson cube deriving the curvature 3-form. We also review ordinary transport for particles to make the transition from one-dimensional algebra to two-dimensional algebra less mysterious.

Ordinary algebra, matrix multiplication, group theory, etc. are special cases of one-dimensional algebra in the sense that they can all be described by ordinary category theory. For example, a group is a type of category that consists of only a single object. Thanks to the advent of higher category theory, beginning with the work of Bénabou on 2-categories [Be], it has been possible to conceive of a general framework for manipulating algebraic quantities in higher dimensions. In particular, monoidal categories and the string diagrams associated with them [JSV] can be viewed as 2-categories with a single object. The special case of this where all algebraic quantities have inverses are known as 2-groups, with a simple review given in [BH] and a more thorough investigation in [BL]. We do not expect the reader is knowledgeable of these definitions and we only assume the reader knows about Lie groups (even a heuristic knowledge will suffice since our formulas will be expressed for matrix groups).

While there already exist several articles [BH], [Pf], [GP], [SW4], introducing the conceptual basic ideas of higher gauge theory and parallel transport for strings in terms of category theory and even a book by Schreiber describing the mathematical framework of higher-form gauge theories [Sc], there are few articles that provide explicit and computationally effective methods for calculating such parallel transport [Pa]. Although Girelli and Pfeiffer explain many ideas, most results useful for computations are infinitesimal and it is not clear how to build local quantities from the infinitesimal ones [GP]. Baez and Schreiber [BS] focus on similar aspects as we do in this article, but our presentation is significantly simplified since we assume certain results on path spaces without further discussion, such as relationships between differential forms on a manifold and smooth functions on its path space, and therefore do not deal with the delicate analytical issues on such path spaces. Our goal is to provide tools and visualizations to perform more intuitive calculations involving mainly calculus and matrix algebra.

### 1.1 Some background and history

In 1973, Kalb and Ramond first introduced the idea of coupling classical abelian gauge fields to strings in [KR]. Actions for interacting charged strings were written down together with equations of motions for both the fields and the strings themselves. Furthermore, a little bit of the quantization of the theory was discussed. The next big step took place in 1985 with the work of Teitelboim (aka Bunster) and Henneaux, who introduced higher form abelian gauge fields that could couple to higher-dimensional manifolds [Te], [HT]. In [Te], Teitelboim studied the generalization of parallel transport for higher dimensional surfaces and concluded that non-abelian -form gauge fields for cannot be coupled to -dimensional manifolds in order to construct parallel transport. The conclusion was that the only possibilities for string interactions involved abelian gauge fields. As a result, it seemed that only a few tried to get around this in the early 1980’s. For example, the non-abelian Stoke’s theorem came from analyzing these issues in the context of Yang-Mills theories and confinement [Ar] (see also for instance Section 5.3 of [Ma]). Although such calculations led people to believe defining non-abelian surface parallel transport is possible, the expressions were not invariant under reparametrizations and they did not seem well-controlled under gauge transformations. Without a different perspective, interest in it seemed to fade.

The crux of the argument of Teitelboim is related to the fact that higher homotopy groups are abelian. This is sometimes also known as the Eckmann-Hilton argument [BH]. However, J. H. C. Whitehead in 1949 realized that higher relative homotopy groups can be described by non-abelian groups [Wh]. In fact, it was Whitehead who introduced the concept of a crossed module to describe homotopy 2-types. This work was in the area of algebraic topology and the connection between crossed modules and higher groups were not made until much later. A review of this is given in [BH]. Eventually, non-abelian generalizations of parallel transport for surfaces were made using category theory and ideas from homotopy theory stressing that one should also associate differential form data to lower-dimensional submanifolds beginning with the work of Girelli and Pfeiffer [GP]. Before this, most of the work on non-abelian forms associated to higher-dimensional objects did not discuss parallel transport but developed the combinatorial and cocycle data [At],[Pf] building on the foundational work of Breen and Messing [BrMe]. This cocycle perspective eventually led to the field of non-abelian differential cohomology [Sc], [Wo11], [Wal1]. The idea of decorating lower-dimensional manifolds is consistent with the explicit locality exhibited in the extended functorial field theory approach to axiomatizing quantum field theories [Se88], [Ati88], [BD], [Lu]. Recently, in a series of four papers, Schreiber and Waldorf axiomatized parallel transport along curves and surfaces [SW1], [SW2], [SW3], [SW4], building on earlier work of Caetano and Picken [CP].

### 1.2 Motivation

We have already indicated one of the motivations of pursuing an understanding of parallel transport along surfaces, namely in the context of string theory. Strings can be charged under non-abelian groups and interact via non-abelian differential forms. Just as parallel transport can be used to described non-perturbative effects in ordinary gauge theories for particles, parallel transport along higher-dimensional surfaces might be used to describe non-perturbative effects in string theory and M-theory. Yet another use of parallel transport is in the context of lattice gauge theory where it is used to construct Actions whose continuum limit approaches Yang-Mills type Actions [Wi74].

Higher form symmetries have also been of recent interest in high energy physics and condensed matter in the exploration of surface operators and charges for higher-dimensional excitations [GKSW]. However, the forms in the latter are strictly abelian and the proper mathematical framework for describing them is provided by abelian gerbes (aka higher bundles) [MP02],[TWZ] and Deligne cohomology. Higher non-abelian forms appear in many other contexts in physics, such as in a stack of D-branes in string theory [My], in the ABJM model [PaSa], and in the quantum field theory on the M5-brane [FSS]. In fact, the authors of [PaSa] show how higher gauge theories provide a unified framework for describing certain M-brane models and how the 3-algebras of [BagLam] can be described in this framework. Further work, including an explicit Action for modeling M5-branes, was provided recently in [SS17].

Although a description of the non-abelian forms themselves is described by higher differential cohomology [Sc], parallel transport seems to require additional flatness conditions on these forms [BH], [BS], [GP], [Pf], [Sc], [Wal2]. For example, in the special case of surfaces, this condition is known as the vanishing of the fake curvature. Some argue that this condition should be dropped and the existence of parallel transport is not as important for such theories [Ch]. However, our perspective is to take this condition seriously and work out some of its consequences. Indeed, since higher-dimensional objects can be charged in many physical models besides just string theory, parallel transport might be used to study non-perturbative or effective aspects of these theories, an important tool to understand quantization (see the discussion at the end of [Sctalk]). Because it is not yet known how to avoid these flatness conditions, further investigation is necessary, with some recent progress by Waldorf [Wal1], [Wal2].

Therefore, because of the subject’s infancy, it is a good idea to devote some time to understanding how to calculate surface transport explicitly to better understand how branes of different dimensions can be charged under various gauge groups. Here, we focus on the case of two-dimensional surfaces such as strings, or D1-branes. However, we make no explicit reference to any known physical models. For these, we refer the reader to other works in the literature such as [SS17] and the references therein.

Higher category theory is notoriously, and inaccurately, thought to be too abstract of a theory to be useful for calculations or describing physical phenomenon. We hope to dispel this misconception in our work and show how it can be used to expand our perspectives on algebra, geometry, and analysis.

### 1.3 Outline

In Section 2, we describe how categorical ideas can be used to express a mix of algebraic and geometric concepts. Namely, in Section 2.1, we review in detail “string diagrams” for ordinary categories and how group theory arises as a special case of ordinary category theory. In Section 2.2, we define 2-categories and other relevant structures providing a two-dimensional visualization of the algebraic quantities in terms of string diagrams. In Section 2.3, we specialize to the case where the algebraic data are invertible. We restrict attention to strict 2-groups, which is sufficient for many interesting applications [GKSW], [GuKa13], [PaSa], [SS17], [Sh15].

In Section 3, we describe how gauge theory for 0-dimensional objects (particles) and 1-dimensional objects (strings) can be expressed conveniently in the language of two-dimensional algebra. In detail, in Section 3.1, we review how classical gauge theory for particles is described categorically. We include a review of the formula for parallel transport describing it in terms of one-dimensional algebra as an iterated integral obtained from a lattice discretization and a limiting procedure. In Section 3.2, we include several crucial calculations for gauge theory for 1-dimensional objects (strings) expressing everything in terms of two-dimensional algebra. In particular, we derive the local infinitesimal data of a higher gauge theory. To our knowledge, these ideas seem to have first been analyzed in [At], [GP], and [BS], though our inspiration for this viewpoint came from [CT]. Furthermore, we use the rules of two-dimensional algebra to derive an explicit formula for the discretized and continuous limit versions of the local parallel transport of non-abelian gauge fields along a surface. Although such a formula appears in the literature [BS], [SW2], we provide a more intuitive derivation as well as a useful expression for lattice computations. We provide a picture for the correct surface ordering needed to describe parallel transport along surfaces with non-abelian gauge fields in Proposition 3.57 and the discussion surrounding this new result. We then proceed to prove that the surface ordering can be dramatically simplified in Theorem 3.78. In Remark 3.87, we show our resulting formula agrees with the one given by Schreiber and Waldorf that was obtained through different means [SW2]. In Section 3.3, we study the gauge covariance of the earlier expressions and derive the infinitesimal counterparts in terms of differential forms. In Section 3.4, we discuss the subtle issue of orientations of surfaces and how our formalism incorporates them. In Section 3.5, we again use two-dimensional algebra to calculate a Wilson cube on a lattice and from it obtain the 3-form curvature. We then study how it changes under gauge transformations showing consistency with the results of Girelli and Pfeiffer [GP].

Finally, in Section 4 we discuss some indication as to how these ideas might be used in physical situations and indicate several open questions.

### 1.4 Acknowledgements

We express our sincere thanks to Urs Schreiber and Radboud University in Nijmegen, Holland, who hosted us for several productive days in the summer of 2012 during which a preliminary version of some ideas here were prepared and presented there. We also thank Urs for many helpful comments and suggestions. We would like to thank Stefan Andronache, Sebastian Franco, Cheyne Miller, V. P. Nair, Xing Su, Steven Vayl, Scott O. Wilson, and Zhibai Zhang, for discussions, ideas, interest, and insight. Most of this work was done when the author was at the CUNY Graduate Center under the NSF Graduate Research Fellowship Grant No. 40017-01-04 and during a Capelloni Dissertation Fellowship. The present work is an updated version of a part of the author’s Ph.D. thesis [Pa16].

## 2 Categorical algebra

### 2.1 Categories as one-dimensional algebra

We do not assume the reader is familiar with categories in this paper. We will present categories in terms of what are known as “string diagrams” since we find that they are simpler to manipulate and compute with when working with 2-categories. Therefore, we will define categories, functors, and natural transformations in terms of string diagrams. Afterwards, we will make a simplification and discuss special examples of categories known as groups.

###### Definition 2.1.

A category, denoted by consists of

1. a collection of 1-d domains (aka objects)

(labelled for now by some color),

2. between any two 1-d domains, a collection (which could be empty) of 0-d defects (aka morphisms)111Technically, 0-d defects have a direction/orientation. In this paper, the convention is that we read the expressions from right to left. Hence, is thought of as “beginning” at and “ending” at or transitioning from to In many cases, as in the theory of groups, we will always be able to go back by an inverse operation. However, in general, will merely be a transformation from to If at any point confusion may arise as to the direction, we will signify with an arrow close to the 0-d defect. See Remark 2.2 for further details.

(labelled by lower-case Roman letters),

3. an “in series” composition rule

whenever 1-d domains match,

4. and between every 1-d domain and itself, a specified 0-d defect

called the identity.

These data must satisfy the conditions that

1. the composition rule is associative and

2. the identity 0-d defect is a left and right identity for the composition rule.

###### Remark 2.2.

For the reader familiar with categories, we are defining them in terms of their Poincaré duals. The relationship can be visualized by the following diagram.

 \xy0;/r.25pc/:(17,0)∗+A="1";(−17,0)∗+R="2";\ar"1";"2"g;\endxy (2.3)

instead and denote the 1-d domains as “objects” and the 0-d defects as “morphisms.” The motivation for using the terminology of domains and defects comes from physics (see Remark 2.13 for more details).

###### Example 2.4.

Let be a group. From one can construct a category, denoted by consisting of only a single domain (say, red) and the collection of 0-d defects from that domain to itself consists of all the elements of The composition is group multiplication. The identity at the single domain is the identity of the group.

The previous example of a category is one in which all 0-d defects are invertible.

###### Definition 2.5.

Let and be two categories. A functor is an assignment sending 1-d domains in to 1-d domains in and 0-d defects in to 0-d defects in satisfying

1. the source-target matching condition

2. preservation of the identity

3. and preservation of the composition in series

This last condition can be expressed by saying that the following triangle of defects commutes

meaning that going left along the top two parts of the triangle and composing in series is the same as going left along the bottom.

There are several ways to think about what functors do. On the one hand, they can be viewed as a construction in the sense that one begins with data and from them constructs new data in a consistent way. Another perspective is that functors are invariants and give a way of associating information that only depends on the isomorphism class of 1-d defects. Another perspective that we will find useful in this article is to think of a functor as attaching algebraic data to geometric data. We will explore this last idea in Section 3.1 and generalize it in Section 3.2. Yet another perspective is to view categories more algebraically and think of a functor as a generalization of a group homomorphism since the third condition in Definition 2.5 resembles this concept. We will explore this last perspective in in the following example.

###### Example 2.6.

Let and be two groups and let and be their associated one-object categories as discussed in Example 2.4. Then functors are in one-to-one correspondence with group homomorphisms

###### Definition 2.7.

Let and be two categories and be two functors. A natural transformation is an assignment sending 1-d domains of to 0-d defects of in such a way so that

and to every 0-d defect

the condition

 =

must hold.

The last condition in the definition of a natural transformation can be thought of as saying both ways of composing in the following “square”

are equal (the arrows have been drawn to be clear about the order in which one should multiply), i.e. as an algebraic equation without pictures

 σ(R)F(g)=G(g)σ(A). (2.8)

Natural transformations can be composed though we will not need this now and will instead discuss this in greater generality for 2-categories later.

###### Example 2.9.

Let be a group and its associated category. Let be the category of vector spaces over a field Namely, the 1-d domains are vector spaces and the 0-d defects are -linear operators between vector spaces. Let us analyze what a functor is. To the single 1-d domain of assigns to it some vector space, To every group element i.e. to every 0-d defect of assigns an invertible operator This assignment satisfies and Thus, the functor encodes the data of a representation of Now, let and be two representations, where the vector space associated to is denoted by A natural transformation consists of a single linear operator satisfying the condition that

 σρ(g)=ρ′(g)σ (2.10)

for all In other words, a natural transformation encodes the data of a intertwiner of representations of 222For the physicist not familiar with the terminology “intertwiners,” these are used to relate two different representations. For instance, the Fourier transform is a unitary intertwiner between the position and momentum representations of the Heisenberg algebra in quantum mechanics. As another example, all tensor operators in quantum mechanics are intertwiners [Ha13].

### 2.2 2-categories as two-dimensional algebra

2-categories provide one realization of manipulating algebraic data in two dimensions.

###### Definition 2.11.

A 2-category, also denoted by , consists of

1. a collection of 2-d domains (aka objects)

(labelled for now by some color),

2. between any two 2-d domains, a collection (which could be empty) of 1-d defects (aka 1-morphisms or domain walls)

(labelled by lower-case Roman letters),

3. between any two 1-d defects that are themselves between the same two 2-d domains, a collection (which could be empty) of 0-d defects (aka 2-morphisms or excitations)333Technically, both 1-d defect and 0-d defects have direction as explained later in Remark 2.13. Our convention in this paper is that 1-d defects are read from right to left and 0-d defects are read from top to bottom on the page. Occasionally, it will be convenient to move diagrams around and draw them sideways or in other directions for visual purposes. In these cases, we will label the directionality when it might be unclear.

(labelled by lower case Greek letters),

4. an “in parallel” composition (aka horizontal composition) rule for 1-d defects

5. an “in series” composition (aka vertical composition) rule for 0-d defects

6. an “in parallel” composition (aka horizontal composition) rule for 0-d defects

7. Every 2-d domain has both an identity 1-d defect and an identity 0-d defect

respectively, and every 1-d defect has an identity 2-d defect

These data must satisfy the following conditions.

1. All composition rules are associative.444This will be implicit in drawing the diagrams as we have.

2. The identities obey rules exhibiting them as identities for the compositions.

3. The composition in series and in parallel must satisfy the “interchange law”

meaning that the final diagram is unambiguous, i.e.

 (μτ)(λσ)=(μλ)(τσ). (2.12)

These laws guarantee the well-definedness of concatenating defects in all allowed combinations.

###### Remark 2.13.

The above depiction of 2-categories is related to the usual presentation of 2-categories via

and are called “string diagrams.” We prefer the string diagram approach as opposed to the “globular” approach because they are used in more areas of physics such as in condensed matter [KiKo12] and open quantum systems [WBC15]. The terminology of domains, domain walls, defects, and excitations comes from physics [KiKo12].

Using this definition, we can make sense of combinations of defects such as

interpreting it as the composition in parallel of the top two 1-d defects along the common 2-d domain (drawn in green)

In fact, a 0-d defect can have any valence with respect to 1-d defects

but it is important to keep in mind which 1-d defects are incoming and outgoing. Our convention is that all incoming 1-d defects come from above the 0-d defect and all outgoing 1-d defects go towards the bottom of the page. Occasionally, we will go against this convention, and we will rely on the context to be clear, or to be cautious, we may even include arrows to indicate the direction. For example, this last 4-valence diagram might be drawn as

Furthermore, we can define composition in parallel between a 1-d defect and a 0-d defect as in

by viewing the 1-d defect with an identity 0-d defect and then use the already defined composition of 0-d defects in parallel

A similar idea can be used if the right side was just a 1-d defect. Using these rules, we can make sense of diagrams such as

by extending the left “dangling” 1-d defect to the bottom and the right “dangling” 1-d defect to the top as follows

Then we can compose in parallel to obtain

and finally compose in series

One must be cautious in such an expression. It does not make sense to compose with alone in series because is an outgoing 1-d defect from Therefore, the expression must be calculated by first composing in parallel and then one can compose the results in series as we have done. It may be less ambiguous to write this expression as More details can be found in Joyal and Street’s seminal paper on the invariance of string diagrams under continuous deformations [JoSt91] or in many introductory accounts of string diagrams in 2-categories. Examples of 2-categories related to groups will be given in Section 2.3.

###### Example 2.14.

Let Hilb be the category of Hilbert spaces, i.e. 1-d domains are Hilbert spaces and 0-d defects are bounded linear operators. Let be the subcategory whose 1-d domains are Hilbert spaces and whose 0-d defects are isometries. Finally, let be the 2-category whose 2-d domains are Hilbert spaces, 1-d defects are isometries, and 0-d defects are elements of More precisely, given two Hilbert spaces and and two isometries a 0-d defect from to is an element such that The composition in series is given by the product of elements in

and the in parallel composition is also defined by the product of elements in

The products and are given by the composition of linear operators. The reader should check that this is indeed a 2-category.

###### Example 2.15.

A common 2-category that appears in tensor networks in quantum information theory is [WBC15]. In this 2-category, there is only a single object (2-d domain). The 1-d defects are Hilbert spaces and 0-d defects are bounded linear transformations. The parallel composition of Hilbert spaces and bounded linear transformations is the tensor product. The series composition of linear transformations is the functional composition of these operators. It is a basic property of the tensor product and functional composition that if and are given, then

 (g⊗g′)∘(f⊗f′)=(g∘f)⊗(g′∘f′). (2.16)

This equality is precisely the interchange law for the compositions in 2-categories, but writing the composition in two dimensions, namely vertically and horizontally, makes it more clear that these expressions are equal. Note that the identity Hilbert space for the parallel composition, the tensor product, is the Hilbert space of complex numbers Technically, this is not an identity on the nose, nor is the tensor product strictly associative, but one can safely ignore this issue due to MacLane’s coherence theorem on monoidal categories [Ma63].

Kitaev and Kong provide more examples of 2-categories in their discussion of domains, defects, and excitations in the context of condensed matter [KiKo12]. In their language, we are viewing excitations as generalized defects.

###### Definition 2.17.

Let and be two 2-categories. A (normalized) weak functor is an assignment sending -dimensional domains/defects of to -dimensional domains/defects of together with an assignment that associates to every pair of parallel composable 1-d defects and in an invertible 0-d defect in interpolating from to as in

These assignments must satisfy the following conditions.

1. The assignment is such that all sources and targets are respected, i.e.

2. All identities are preserved (this is the “normalized” condition).

3. For any 1-d defect

the equalities

i.e.

 cFf,idV=idF(f)=cFidR,f (2.18)

must hold.

4. To every triple of parallel composable 1-d defects

the equality

i.e.

 [0.9]cFf,gidF(h)cFfg,h=[0.9]idF(f)cFg,hcFf,gh (2.19)

must hold.

If is the identity for all and in then is said to be a strict functor.

###### Remark 2.20.

For each pair of composable 1-d defects and the 0-d defect can be thought of as filling in the triangle from the comments after Definition 2.5 by enlarging the 1-d domains to 2-d domains and enlarging the 0-d defects to 1-d defects. Condition (d) resembles associativity. In fact, it is an example of a cocycle condition and will be discussed more in the following example (in particular, this definition allows one to define higher cocycles for non-abelian groups). Condition (d) can also be re-written as

which illustrates more of a connection to Pachner moves for triangulations of surfaces. However, this latter presentation requires arrows to keep track of incoming versus outgoing directions.

Examples of weak functors abound. For example, projective representations are described by weak functors that are not strict functors as will be explained in the following example. Weak functors can also be used to define the local cocycle data of higher bundles [Wo11]. Since we will be working locally for simplicity, we will make little use of weak functors, but have included their discussion here for completeness and so that the standard definitions of higher bundles may be less mysterious [Pa], [SW4], [Wo11]. Strict functors will be used as a means of defining parallel transport along surfaces in gauge theory in Section 3.2. Natural transformations will be used to define gauge transformations of such functors and their infinitesimal counterparts will be derived from these definitions.

###### Example 2.21.

Let be a group and its associated category (see Example 2.4). Every category, such as can be given the structure of a 2-category by adding only identity 0-d defects. Namely, there is only a single 2-d domain, the 1-d defects are elements of and the 0-d defects are all identities. This 2-category will also be denoted by Let be the 2-category introduced in Example 2.14. A weak normalized functor encodes the data of a Hilbert space a function and a function in such a way so that to every pair of elements

i.e.

 ρ(gh)=cρg,hρ(g)ρ(h) (2.22)

and also

 ρ(e)=idH. (2.23)

Furthermore, satisfies the condition that to every triple

 cρgh,kcρg,h=cρg,hkcρh,k. (2.24)

This provides the datum of a (normalized) projective unitary representation of on a Hilbert space (ignoring any continuity conditions).

###### Definition 2.25.

Let be two weak functors between two 2-categories. A natural transformation is an assignment sending -d domains/defects of to -d defects of for satisfying the following conditions.

1. The assignment is such that

and555The diagram on the right can be thought of as filling in the square from the comments after Definition 2.7 (rotate the square by counterclockwise to see this more clearly).

2. To every pair of parallel composable 1-d defects

the equality

i.e.

 [0.9]σ(f)idF(g)idG(f)σ(g)cGf,gσ(A)=[0.9]σ(R)cFf,gσ(fg) (2.26)

must hold.

3. To every identity 1-d defect the equality

 σ(idR)=idσ(R) (2.27)

must hold.

4. To every 0-d defect

the equality

i.e.

 [0.9]σ(f)G(λ)idσ(A)=[0.9]idσ(R)F(λ)σ(g) (2.28)

must hold.

Such string diagram pictures facilitate certain kinds of computations [PoSh] (for instance, compare the definition of natural transformation in Figure 10 of said paper). Natural transformations between functors can be thought of as symmetries. For example, just as natural transformations of functors between ordinary categories describe intertwiners for ordinary representations, natural transformations of functors between 2-categories describe intertwiners of projective representations.

###### Example 2.29.

Using the notation of Example 2.21, let be two projective unitary representations on and with cocycles and respectively. A natural transformation provides an isometry and a function whose value on is denoted by and fits into

which in particular says

 π(g)σHK=σgσHKρ(g), (2.30)

satisfying the condition

 σghcρg,h=cπg,hσgσh (2.31)

for all This provides the data of an intertwiner of projective unitary representations.

It will be important to compose natural transformations. This will correspond to iterating gauge transformations successively.

###### Definition 2.32.

Let be two weak functors between two 2-categories and let and be two natural transformations. The vertical composition of with written as (read from top to bottom)

 [0.5]λσ, (2.33)

is a natural transformation defined by the assignment

on 2-d domains and

on 1-d domains.

Technically, one should check this indeed defines a natural transformation. This is a good exercise in two-dimensional algebra. There are actually similar symmetries between natural transformations, called modifications, which we define for completeness.

###### Definition 2.34.

Let be two weak functors between two 2-categories and two natural transformations. A modification assigns to every 2-d domain of a 0-d defect in such that the following conditions hold.

1. The assignment is such that

2. To every 1-d defect

the equality

i.e.

 [0.9]m(R)idF(g)ρ(g)=[0.9]σ(g)idG(g)m(A) (2.35)

must hold.

### 2.3 Two-dimensional group theory

A convenient class of 2-categories are those for which there is only a single 2-d domain and all defects are invertible under all compositions. Such a 2-category is called a 2-group. 2-groups therefore only have labels on 1-d and 0-d defects. They can be described more concretely in terms of more familiar objects, namely ordinary groups.

###### Definition 2.36.

A crossed module is a quadruple of two groups, and group homomorphisms and satisfying the two conditions

 ατ(h)(h′)=hh′h−1 (2.37)

and

 τ(αg(h))=gτ(h)g−1 (2.38)

for all and Here is the automorphism group of i.e. invertible group homomorphisms from to itself. If the groups and are Lie groups and the maps and are smooth, then is called a Lie crossed module.

Examples of crossed modules abound.

###### Example 2.39.

Let be any group, and let be conjugation.

###### Example 2.40.

Let be any group, let be the automorphism defined by for all and set

###### Example 2.41.

Let be a normal subgroup of Set the inclusion, and conjugation.

###### Example 2.42.

Let be a Lie group, a covering space, and conjugation by a lift. For instance, and the quotient map give examples. Here is the set of special unitary matrices and is its center, i.e. elements of the form with

###### Example 2.43.

Let the trivial group, any abelian group, the trivial map, and the trivial map.

###### Remark 2.44.

It is not possible for to be a non-abelian group if is trivial. In fact, for an arbitrary crossed module is always a central subgroup of

We now use crossed modules to construct examples of 2-categories, specifically 2-groups.

###### Example 2.45.

Let be a crossed module. From one can construct a 2-category, denoted by consisting only of a single 2-d domain, the 1-d defects are labelled by elements of and the 0-d defects are labelled by elements of However, such labels must be of the form

Composition of 1-d defects in parallel is the group multiplication in just as in (see Example 2.4). Composition of 0-d defects in series is defined by

Composition of 0-d defects in parallel is defined by

Notice that the outgoing 1-d defect is consistent with our definitions because

 τ(h2αg2(h1))g2g1=τ(h2)g2τ(h1)g−12g2g1=τ(h2)g2τ(h1)g1 (2.46)

due to (2.38).

The identities are given as follows. The 1-d defect identity associated to the single 2-d domain is the 1-d defect labelled by the identity of The identity 0-d defect associated to a 1-d defect labelled by is labelled by slight abuse of notation the identity of It follows from these two definitions that the identity 0-d defect associated to the single 2-d domain is labelled by the identity on both the 1-d and 0-d defects. These three identities are depicted visually as

respectively.

The inverse of the 1-d defect labelled by for the parallel composition of 1-d defects is just the 1-d defect labelled by Inverses for 0-d defects are depicted for series composition by

and parallel composition by

and similarly on the left. Notice that 0-d defects have two inverses for the two compositions.

This last class of examples of 2-groups from crossed modules will be used throughout this paper. In fact, all 2-groups arise in this way.

###### Theorem 2.47.

For every 2-group, let be the set of 1-d defects and let be the set of 0-d defects of the form

(i.e. 0-d defects whose source 1-d defect is ). Define by from 0-d defects of the above form. Set to be the resulting 0-d defect obtained from the composition

The product in is obtained from the composition of 1-d defects in parallel and the product in is obtained from the composition of 0-d defects in series. With this structure, is a crossed module. Furthermore, this correspondence between crossed modules and 2-groups extends to an equivalence of 2-categories [BH].

We now provide some examples of 2-groups along with weak functors between them to illustrate their meaning.

###### Example 2.48.

Let be a group and a Hilbert space. Let denote the unitary operators of Let be the crossed module where the stand for the trivial map and trivial action, respectively. Let be the crossed module with and the trivial action. By definition, a weak functor consists of a function and a function of the form sending to

which in particular says

 cρg,hρ(g)ρ(h)=ρ(gh), (2.49)

satisfying

 cρg,e=1=cρe,g (2.50)

for all and

 cρgh,kcρg,h=cρg,hkcρh,k (2.51)

for all This is the definition of a (normalized) projective representation of on and is really a special case of Example 2.21, where the Hilbert space is fixed from the start. The crossed module introduced here is actually the automorphism crossed module (in analogy to the automorphism group) of the Hilbert space viewed as a 2-d domain in the 2-category

The following fact will be used in distinguishing two types of gauge transformations. It allows one to decompose an arbitrary gauge transformation into a composition of these two types.

###### Proposition 2.52.

Let be a category viewed as a 2-category so that its 1-d domains become 2-d domains, its 0-d defects become 1-d defects, and its 0-d defects are all identity 0-d defects. a crossed module with associated 2-group and two strict functors (so that and are identities). A natural transformation consists of a function from 2-d defects of to denoted by

and a function from 1-d defects of to denoted by

which says that

 τ(h(γ))g(z)F(γ)=F′(γ)g(y), (2.53)

satisfying the axioms in the definition of a natural transformation. Thus, can be written as the pair Furthermore, there exists a strict functor such that the natural transformation decomposes into a vertical composition (recall Definition 2.32) of the natural transformations and i.e.

 σ=[0.9](g,e)(e,h) (2.54)

namely, for any 1-d defect