Categorical Tori

We give explicit and elementary constructions of the categorical extensions of a torus by the circle and discuss an application to loop group extensions. Examples include maximal tori of simple and simply connected compact Lie groups and the tori associated to the Leech and Niemeyer lattices. We obtain the extraspecial 2-groups as the isomorphism classes of categorical fixed points under an involution action.


Acknowledgments
It is a pleasure to thank David Roberts for very helpful conversations and correspondence. The idea for Construction 2.1 came from a conversation with him, and I understand that he will also write about it elsewhere. The idea to look for crossed module extensions of tori, picked up in Section 6.2, is also due to David. Many thanks go to Matthew Ando for very inspiring conversations, to Konrad Waldorf for patiently answering a long list of emails full of technical questions and to Arun Ram, who helped with some of the references.

Constructions of categorical tori
To build a categorical torus, we need a finite dimensional lattice 1 , which we denote Λ ∨ , and an integer-valued bilinear form J on Λ ∨ . Up to isomorphism, our constructions will depend only on the even symmetric bilinear form I(m, n) = J(m, n) + J(n, m).
We will also write J for the bilinear form J ⊗ R on t. The exponential map is and we make the identification Λ ∨ = ker(exp) ⊆ t.
We write t/ /Λ ∨ for the action groupoid of Λ ∨ on t. We have a canonical equivalence of Lie groupoids 2 where T is viewed as groupoid with only identity arrows. We will give three equivalent constructions of the categorical torus associated to (Λ ∨ , J). The first is as a strict monoidal Lie groupoid.
So, T has objects t and arrows t × Λ ∨ × U (1), the source of (x, m, z) is x, and the target of (x, m, z) is x + m. We equip T with the following multiplication x • y = x + y on objects (x, m, z) • (y, n, w) = (x + y, m + n, z · w · exp(−J(m, y))) on arrows.
The unit object is 0, and associativity and unit isomorphisms are identities. This makes (T, •) a strict monoidal Lie groupoid, i.e., a group object in the category of Lie-groupoids.
The second construction re-interprets the data of Construction 2.1 as a compact Lie 2-group in the sense of Schommer-Pries [SP11].
Construction 2.2. Let T be the Lie groupoid i.e., T has objects T and arrows T × U (1), source and target are projection onto the first factor and composition of arrows is multiplication in the second factor. Then we have an equivalence of Lie groupoids Note that p × id does not possess a continuous inverse equivalence. So, one interprets the data of Construction 2.1 as those of a 2-group object in a suitable localization of the bicategory of Lie-groupoids, namely the bicategory of bibundles Bibun. Different communities have different language for the 1-morphisms in Bibun. Depending on your taste, you may think of multiplication on T as the zig-zag, span, orbifold map, or anafunctor Our third construction is as a multiplicative bundle gerbe in the sense of [Bry08] and [CJM + 05].
Construction 2.3. Let I be the trivial bundle gerbe over T . Recall that a multiplication on I is a stable isomorphism µ : pr * 1 I ⊗ pr * 2 I −→ m * I of bundle gerbes over T ×T , where the pr i are the projections to the factors and m is multiplication in T . Such a stable isomorphism of trivial bundle gerbes is the same as a line bundle, and we take µ to be for (m, n) ∈ Λ ∨ × Λ ∨ . Finally, we need to specify the associativity isomorphism where pr ij is the projection onto factors i and j and m ij is multiplication of these two factors, e.g., m 12 = m × id. We take α to be the canonical isomorphism resulting from the fact that its source and target have identical multipliers Remark 2.4. Construction 2.3 takes the sum of two bilinear forms, J 1 + J 2 to the tensor product of multiplicative bundle gerbes.
By [Wal12,Thm.3.2.5], the data of a multiplicative bundle gerbe over T are equivalent to those of an extension of T by pt / /U (1). In the case of Construction 2.3, we may think of the objects of I as pairs (t, L) with t ∈ T and L a hermitian line, and of the arrows (t, L) → (t, L ′ ) as the unitary isomorphisms from L to L ′ . Then I is a monoidal groupoid with multiplication (s, L 1 ) • (t, L 2 ) = (s · t, L J s,t ⊗ L 1 ⊗ L 2 ). The associativity isomorphisms for • are encoded in α, α r,s,t : L J r·s,t ⊗ L J r,s ∼ = L J r,s·t ⊗ L J s,t . Proposition 2.5. The three constructions yield equivalent 2-group extension of T .
Proof. It is clear that Construction 2.1 and Construction 2.2 are equivalent. To see their equivalence with Construction 2.3, let F be the functor To make F a monoidal equivalence, we need a natural isomorphism This is a map of the form or, equivalently, a trivialization of L J over t × t. We have such a trivialization by construction of L J . One checks that (F, φ) is a monoidal equivalence from T to I.
Definition 2.6. We will write J t for the bilinear form J t (m, n) = J(n, m) on Λ ∨ . We say that J is symmetric if J = J t and that J is skew symmetric if J = −J t . A symmetric bilinear form is called even if Proposition 2.7. The bilinear forms J and J t yield equivalent 2-group extensions of T .
Corollary 2.8. (i) If I is an even symmetric bilinear form on Λ ∨ , then the multiplicative bundle gerbe associated to I possesses a square root. (ii) If B is a skew symmetric integral bilinear form on Λ ∨ , then B yields a trivial 2-group extension of T .
Proof. (i) follows from the fact that every even symmetric bilinear form I can be written in the form I = J + J t for an integer-valued bilinear form J. For instance, fix a basis (b 1 , . . . , b r ) of Λ ∨ and set else.
(ii) follows from the fact that, similarly, every skew symmetric bilinear form B can be written in the form B = J − J t for an integer-valued bilinear form J.
Corollary 2.9. Let J be an integer-valued bilinear form on Λ ∨ . Then, up to equivalence over T , the 2-group (T, • J ) only depends on the even bilinear form Proof. Let J 1 and J 2 be two integer-valued bilinear forms on Λ ∨ , and assume that Then J 1 − J 2 is skew symmetric. By Corollary 2.8, the multiplicative bundle gerbe obtained from J 1 − J 2 is trivial. Using Remark 2.4, we conclude that the multiplicative bundle gerbes obtained from J 1 and J 2 are isomorphic.

The example of the circle
Let t = R and Λ ∨ = Z. Any even symmetric bilinear form on Z is an integer multiple of I(m, n) = 2mn.
For this I, there is a unique choice of J, namely J(m, n) = mn.
The basic circle extension U (1) of the circle group U (1) consists of the following data: for (m, n) ∈ Z 2 , (iii) the canonical isomorphism For k ∈ Z, the kth circle extension U (1) k of U (1) is obtained by replacing the multipliers with exp(kmy).
Here are a few words of explanation about these choices: gerbes on U (1) are classified, up to stable isomorphism, by their Dixmier-Douady class in So, any bundle gerbe over U (1) is trivializable, and we might as well start with the trivial bundle gerbe I. Line bundles on U (1) × U (1) are classified, up to isomorphism, by their first Chern class in This is isomorphic to the group of skew symmetric bilinear forms on Z 2 , which is infinite cyclic, generated by the determinant. To construct a line bundle L with Chern class we use Chern-Weil theory: Figure 1. Chern-Weil theory: the multipliers can be read off from this cocycle in the truncatedČech-Deligne double complex for U (1) × U (1).
Note that this argument builds L as a line bundle with connection, given by the 1-form Similarly, the bundle L ⊗k with multipliers exp(kmy) has connection kxdy and Chern class k ·det.

The classification
Let T be a compact torus with Lie algebra t and coweight lattice Λ ∨ = ker(exp). Up to equivalence, the 2-group extensions of T by pt / /U (1) are classified by , where the left-hand side is Lie group cohomology as in [SP11] [WW13]. There are a number of different, but equivalent definitions of Lie group cohomology. We choose to work with thě Cech-simplicial double complexČ * (BT • ; U (1)) as in the classification of multiplicative bundle gerbes in [Bry08] and [CJM + 05, Prop 5.2]. The goal of this section is to analyze the degree four part of the composite of isomorphisms where S * (Λ) is the symmetric algebra of the weight lattice Λ = Hom(Λ ∨ , Z).
Weights are given degree 2, so that the degree four part is We may think of elements of S 2 Λ as homogeneous polynomials of degree 2 in the weights, and we have the symmetrization map identifying S 2 (Λ) with the group of even symmetric bilinear forms on Λ ∨ . So, (2) establishes a group homomorphism from the even symmetric bilinear forms on Λ ∨ to the isomorphism classes of multiplicative bundle gerbes over T .
Theorem 4.1. Let I be an even symmetric bilinear form on Λ ∨ , and let J be an integral bilinear form on Λ ∨ satisfying I = J + J t . If we apply Construction 2.3 to (Λ ∨ , J), then the resulting multiplicative bundle gerbe is classified by I.
Proof. Let λ ∈ Λ be a weight of T with character e λ , and write for the line bundle on BT classified by Be λ . The first isomorphism in (2) goes back to Borel, and is defined as where on the left-hand side, the weights have degree 2.
To define the second isomorphism in (2), let ET be a contractible free T -space, BT = ET /T , and and recall that ET × BT · · · × BT ET ∼ = ET × T × · · · × T.

So, the maps
form a hypercover of BT whoseČech double complex can be identified withČ * (BT • ; Z). Under this identification, the cup product becomes where r and s are theČech degrees of f and g, pr 1 is the projection onto the first deg simp (f ) factors and pr 2 is the projection onto the last deg simp (g) factors.
Note that these first two isomorphisms are actually isomorphisms of graded rings. To determine the image of λµ in H 4 (BT • ; Z), we determine the images of λ and µ and then take the cup product. The first Chern class of L λ inČech hypercohomology is given by the multipliers Hence, λ maps to the degree (1, 1) cocycle in theČech-simplicial double complex. Given two weights, λ and µ, their cup product is represented by the cocycle inČ 2 (BT 2 ; Z). The last isomorphism in (2) is the inverse of the connecting homomorphism for the short exact sequence of presheaves It is easy to read off that Construction 2.3 associates to J = µ ⊗ λ the multiplicative bundle gerbe corresponding to the U (1)-valuedČech-simplicial 3-cocycle (1, λ(m)µ(y), 1, 1).

The image of this cocycle under the connecting homomorphism is indeed the integral 4-cocycle
(1, 1, λ(k)µ(n), 1, 1), see the picture: This concludes the proof. in the Leray-Serre spectral sequence for the universal bundle EG → BG. For a compact, connected Lie group G, this map was calculated (in all degrees) by Chern and Simons. In the relevant degree, their result is summarized by the commuting diagram where ν is the Cartan 3-form 3 associated to I, ν(x, y, z) = I([x, y], z).
Since the Lie bracket on a torus is zero, it follows that the Dixmier-Douady class of the underlying bundle gerbe of any multiplicative bundle gerbe on T vanishes.

Connections
If J = λ ⊗ µ, then the Chern-Weil discussion analogous to that of Section 3 produces the connection ∇ on L J with connection 1-form ω t×t = λdµ and curvature 2-form For arbitrary J, we introduce the maps Then ∇ is defined by the 1-form The pair (L J , ∇) makes a multiplicative bundle gerbe with connection (in the sense of [Wal10]) out of the trivial bundle gerbe I on T (with the remaining data trivial).

Symmetries
In many interesting examples, we have a finite group Γ of linear isometries of (Λ ∨ , I). In this case, I may be interpreted as a Γ-invariant cohomology element, and the action of Γ on T preserves the multiplicative bundle gerbe classified by I up to isomorphism. As a consequence, we obtain a 2-group extension of Γ, namely the automorphism 2-group of the categorical torus. There are two variations worth exploring, depending on whether or not we require our symmetries to preserve the connection. We will come back to this topic at a different occasion.

Maximal tori
Let G be a compact connected Lie group with maximal torus T and Weyl group W . Then we have The formula [CS74, (3.10)] is often cited in its original form where ω is the right-invariant Maurer-Cartan form on G. A look at the definitions on page 50 of [CS74] identifies I(ω ∧ [ω, ω]) with the bi-invariant 3-form on G whose restriction to g = T1G equals 12 ν.
and we can choose I as a multiple of the Killing form. If G is simple and simply connected, then we have The positive definite generator I bas is called the basic bilinear form on Λ ∨ . The form kI bas ∈ H 4 (BG; Z) classifies the 2-group denoted String k (G) in [SP11]. The restriction of String k (G) to T is equivalent to the categorical torus constructed from (Λ ∨ , kI bas ). This property determines String k (G) uniquely. In other words, if we are given a 2-group extension G of G by pt / /U (1), and wonder which of the String k (G) it is equivalent to, it suffices to identify its restriction to T . This recognition principle promises to be useful for a comparison of the many equivalent constructions of the String 2-groups.

The Leech lattice
Another interesting example is given by the Leech lattice Λ ∨ = Λ Leech ⊆ R 24 , together with the symmetric bilinear form on R 24 making Λ ∨ an even unimodular lattice. Then T = R 24 /Λ Leech is the Leech torus, and the group of linear isometries of (Λ ∨ , I) is the Conway group Co 0 . So I can be interpreted as an element I ∈ H 4 (BT, Z) Co 0 .

Niemeyer lattices
Similarly, we can choose Λ ∨ as one of the Niemeyer lattices, e.g., A 24 1 or A 12 2 , and I as the symmetric bilinear form on R 24 making Λ ∨ an even unimodular lattice. Then T = R 24 /A 24 1 (respectively, T = R 24 /A 12 2 ). The group of linear isometries of (Λ ∨ , I) is the Mathieu group M 24 or M 12 , and we have I ∈ H 4 (BT, Z) M 24 respectively I ∈ H 4 (BT, Z) M 12 .

Conway, Mathieu and Weyl 2-groups
To return to our remarks in Section 4.2, the symmetries of categorical tori form 2-group extensions of Weyl and Mathieu groups as well as Conway's group Co 0 . The low dimensional (co)homology of the Mathieu groups was calculated in [DSE09], the generator of the cohomology group 6. Other things classified by (Λ ∨ , I) Besides our categorical tori, the degree 4 cohomology class classifies a number of important objects. In this section, we list a few of these and offer some comments on their relationship to T .

From categorical groups to loop groups
I classifies a central extension LT of the loop group of T by U (1). If T is a maximal torus of G and I is invariant under the Weyl group, as in Section 5.1, then I classifies a central extension LG of the loop group of G.
The relationship between loop groups and 2-groups is well studied, see for instance [BSCS07] and, more recently, [Wal13] and [Wal12].
We will work in the context of Waldorf's transgression-regression machine. The term 'transgression' in this context does not refer to the map τ mentioned above, but to a recipe for turning multiplicative bundle gerbes with connection into central extensions of loop groups. Applied to (I, (L J , ∇), α), transgression yields the central extension of LT with trivial underlying principal bundle and 2-cocycle where Hol stands for holonomy, and (f, g) is a lift of (ϕ, γ) to t × t. Proof. Pressley and Segal denote our I by −, − , their b can be taken to be our J, and they write Λ for the cocharacter latticeŤ = Hom(U (1), T ). We first note that c describes the correct extension of this lattice namely c Λ ∨ (m, n) = (−1) J(m,n) .

The commutators in LT are
(integration by parts). Over the identity component of LT , our extension is completely described by its Lie-algebra cocycle, (5) (6), it sends (g, r) ∈ Lt to Waldorf's regression machine reconstructs a Lie 2-group extension G of G by pt / /U (1) from a central extension LG equipped with an extra piece of data, called fusion product. Let P 1 G be the space of paths in based at 1 in G, and π the evaluation map π: P 1 G −→ G γ −→ γ(1).
Then the arrows in theČech groupoid P 1 G × G P 1 G P 1 G are identified with the space ΩG of based loops in G, and replacing ΩG with the central extension ΩG, one obtains the pt / /U (1) extension of G ΩG). G := (P 1 G The composition of arrows in ΩG is the fusion product mentioned above. Let us study this in detail for our torus. The construction of c in (4) as holonomy implies that c is a fusion map, i.e., for triples (ϕ 1 , ϕ 2 , ϕ 3 ) and (γ 1 , γ 2 , γ 3 ) with identical start and end points, ϕ 1 (0) = ϕ 2 (0) = ϕ 3 (0), and ϕ 1 (1) = ϕ 2 (1) = ϕ 3 (1), and likewise for the γ i . Therefore, we have a canonical fusion product on LT . To compare the regression of LT to Construction 2.1, we consider the commuting diagram Here exp [0,x]  To compare this expression with the factor turning up in Construction 2.1, we note that and δ simp exp • J 2 = 0. Let T be the groupoid of Construction 2.1, and let ⊙ be the monoidal structure on T obtained like •, but with the factor exp(−J(m, y)) replaced by c(γ (x,m) γ (y,n) ). Then the map makes the identity functor a monoidal equivalence from (T, •) to (T, ⊙).

Extraspecial 2-groups
Given an even symmetric bilinear form I on Λ ∨ , we have the integer-valued quadratic form on Λ ∨ . The form I can be recovered from φ by the identity i.e., I(m, n) = φ(m + n) − φ(m) − φ(n).
Let T [2] be the F 2 -vectorspace of points of order 2 in T . Then φ induces a quadratic form on which we will also denote φ: T [2] −→ F 2 .
Such a φ classifies an extraspecial 2-group 5 , i.e., a central extension An explicit 2-cocycle for this extension is given by J ⊗ F 2 , for any integer-valued J on Λ ∨ with I = J + J t . In the situation of Section 5.2, there is a prominent subgroup of the Monster, isomorphic to where Co 1 is the Conway group Co 1 = Co 0 /±1. This subgroup is typically the first step in the construction of the Monster, see for instance [Tit85]  Not unexpectedly, it turns out that the Looijenga line bundle also plays a key role in the character theory of representation of categorical tori. This leads to a theta function formalism for 2-characters, resembling that of loop group characters, but developed directly from the categorical picture, without any mention of loops. We will come back to this topic at a different occasion.