The Chromatic Brauer Category and Its Linear Representations

The Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl’s framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl’s concrete positive TFT based on fold maps into the plane.


Introduction
The Brauer algebras D m have first appeared in work of Brauer [6] on representation theory of the orthogonal group O(n). In view of Schur-Weyl duality, they replace the role played by the group algebras of symmetric groups in representation theory of the general linear group. Generators of D m are the diagrams consisting of 2m vertices and m edges, where the vertices are arranged in two parallel rows of m vertices, and each vertex lies in the boundary of exactly one edge. Given a commutative ground ring k with unit, D m is the k(x)-algebra freely generated as k(x)-module by those diagrams. Multiplication is induced by concatenation of diagrams, where each arising free loop component gives rise to an additional multiplication with the indeterminate x. A signed variant of Brauer algebras has been studied in [22]. Brauer algebras play an important role in knot theory, where, for instance, Birman-Murakami-Wenzl algebras [5,21,26], which are the quantized version of Brauer algebras, have been used to construct generalizations of the Jones polynomial.
We are concerned with a natural (horizontal) categorification Br of Brauer's algebras that has been used by Banagl [3,4] in search of new topological invariants in the context of his framework of positive topological field theory (TFT). A similar category has been considered independently by Lehrer-Zhang [18] in a modern categorical approach to the invariant theory of the orthogonal and symplectic groups. Roughly speaking, morphisms in the so-called Brauer category Br are represented by 1-dimensional unoriented tangles in a high-dimensional Euclidean space. In particular, generators and relations for the strict monoidal category Br have been listed in [4] (compare also [18]) by adapting the methods that are used by Turaev [25] for deriving a presentation for the category of tangle diagrams.
Let us discuss the main ideas behind Banagl's notion of positive TFT, and the role of the Brauer category Br and its representation theory in this context. By definition, the axioms for positive TFT [3] differ from Atiyah's original axioms for TFT [1] in that they are formulated over semirings instead of rings. Recall that semirings are not required to have additive inverse ("negative") elements. In computer science, semirings and related structures have been studied by Eilenberg [7] in the context of automata theory and formal languages. The essential advantage of positive TFTs over usual TFTs is that so-called Eilenberg completeness of certain semirings can be used to give a rigorous construction of positive TFTs of arbitrary dimension. This construction is implemented by Banagl in a process he calls quantization that requires so-called fields and an action functional as input (see Sect. 5.1). Inspiration comes from theoretical quantum physics, where the state sum is expressed by fields and an action functional via the Feynman path integral.
In [4], Banagl applies his framework of quantization to produce in arbitrary dimension an explicit positive TFT for smooth manifolds. The construction uses singularity theory of so-called fold maps, and the resulting state sum invariants can distinguish exotic smooth spheres from the standard sphere. Now, in this concrete setting, the role of fields is played by certain fold maps into the plane, and the action functional assigns to such fields morphisms in the Brauer category Br by extracting the 1-dimensional patterns that arise from the singular locus of fold maps. However, as pointed out in Section 8 of [3], it is desirable to compose such a category-valued action functional with a symmetric strict monoidal functor from Br to the category Vect of real vector spaces and linear maps. Note that one requires the category Vect to be equipped with a symmetric strict monoidal structure, which is provided by using the Schauenburg tensor product [24]. In this way, the Brauer category serves only as an intermediary structure, and the state sum of the resulting positive TFT will become accessible through linear algebra. Of course, the loss of information should be kept at a minimum during this linearization process, which is motivation for studying faithfulness of such linear representations Br → Vect. This knowledge is required when it comes to the explicit computation of state sum invariants (compare Section 6.3 and Section 10.5 in [27] as well as Remark 9.5 in [28]).
In this paper we determine not only the faithful representations of Br, but also those of the chromatic Brauer category cBr which will be introduced in Sect. 3.2 as an enrichment of Br in which morphisms are component-wise labeled ("colored") by elements of a countable index set. Hence, in contrast to the Brauer category, isomorphic objects of the chromatic Brauer category need not be equal. Our reason for considering cBr is that it can be used to construct a refinement of Banagl's positive TFT based on fold maps in the following way (see Sect. 5). In analogy with the index of non-degenerate critical points in Morse theory, one can associate a (reduced) index to the singularities of a fold map. Those fold indices are intrinsically defined, locally constant along the singular set, and carry topological information about the source manifold. For fold maps from n-dimensional source manifolds into the plane, the set of possible fold indices is {0, . . . , (n − 1)/2 }. We will modify Banagl's original construction by defining a cBr-valued action functional which additionally remembers indices of fold lines as labels from the set N = {0, 1, 2, . . .} of natural numbers.
Concerning linear representations of Br, Banagl has shown in Proposition 2.22 of [4] that there exist linear representations that are faithful on loops. This suffices for his purpose to show that state sum invariants of the positive TFT are able to detect exotic smooth structures on spheres. As a much more general result, we have the following Theorem 1.1 [20,27]  What is more, we will show Theorem 1.2 below. Since the Brauer category is naturally (monochromatically) embedded in the chromatic Brauer category, Theorem 1.1 is implied by Theorem 1.2. (To conclude this, one has to use that any linear representation of Br can be extended to one of cBr by means of our structure results Theorem 3.5 and the corresponding result for Br.) In preparation of the statement of our result on linear representations of cBr, note that the objects of cBr that are mapped to the object [1] of Br under the forgetful functor cBr → Br are parametrized by the labels k ∈ N, say ( [1], k).
In particular, faithful linear representations of cBr exist because one can take d to be the sequence of prime numbers, and then apply Theorem 3.5 to construct a strict monoidal functor Y : cBr → Vect which realizes d k for k ∈ N as the dimension of the real vector space Y ( [1], k). We note, however, that the condition that implication (1.1) holds for all (l k ) k∈N ∈ ∞ k=0 Z is not equivalent to saying that every finite subset of the sequence d 0 , d 1 , . . . is relatively prime (e.g., take d 0 , d 1 , . . . to be the double 6, 10, 14, . . . of the sequence of odd prime numbers).
The paper is structured as follows. In Sect. 2 we recall fundamental facts about monoidal categories in general, and the Schauenburg tensor product in particular. The chromatic Brauer category is introduced in Sect. 3, where its linear representations are classified by Theorem 3.5. The proof of our main result Theorem 1.2 will be given in Sect. 4. Finally, in Sect. 5 we discuss our application to Banagl's positive TFT based on fold maps.

Notation
Throughout the paper, the natural numbers will be meant to be the set N = {0, 1, 2, . . . } (including zero).

Preliminaries on Strict Monoidal Categories
In this section, we introduce the definitions and notational conventions on strict monoidal categories (Sect. 2.1) and the Schauenburg tensor product (Sect. 2.2) that will be used throughout the paper.

Strict Monoidal Categories
We refer to Kassel [12] for the basic definitions that are recalled in this section.
A monoidal category (C, ⊗, I , α, λ, ρ) is a category C equipped with a bifunctor ⊗: C × C → C, an object I ∈ Ob(C), called unit with respect to the tensor product ⊗, and three isomorphisms which are functorial in X , Y , Z ∈ Ob(C). Furthermore, α, λ and ρ satisfy the coherence conditions given by the pentagon axiom and triangle axiom. These are Here, α is called associativity constraint, and λ and ρ are called left and right unit constraints, respectively. A monoidal category C is called strict if the associativity and unit constraints α, λ, ρ are given by identity morphisms of the category. If (C, ⊗, I , α, λ, ρ) is a (strict) monoidal category, then we will call the data (⊗, I , α, λ, ρ) a (strict) monoidal structure on C. In most of the paper, we work with strict monoidal categories. In that case, we will omit the associativity and unit constraints in the notation of a monoidal category, i.e. we will write (C, ⊗, I ) instead of (C, ⊗, I , α, λ, ρ).
Let C and D be monoidal categories. A monoidal functor is a functor F : C → D which respects the monoidal structure. To be more precise, it is a functor and an isomorphism ξ 0 : I D → F(I C ) such that the diagrams commute for all X , Y , Z ∈ Ob(C). The monoidal functor (F, ξ, ξ 0 ) is called strict if the isomorphisms ξ 0 and ξ are identity morphisms of D.
Let (C, ⊗, I , α, λ, ρ) be a monoidal category. A symmetric braiding b on C is a monoidal for any pair (X , Y ) of objects of the category C, satisfying the hexagon axiom, the unity coherence, and the inverse law which are given by the commutativity of the diagrams respectively, for all X , Y , Z ∈ Ob(C). A monoidal category C together with a symmetric braiding b is called symmetric monoidal category.

The Schauenburg Tensor Product
The usual tensor product ⊗ in the category Vect of real vector spaces and linear maps determines a monoidal structure on Vect (with unit object I = R) which is certainly not strict. When studying linear representations of strict monoidal categories, it is appropriate to endow Vect with a strict monoidal structure. As described in Theorem XI.5.3 in [12, p. 291], there is a general method for turning any given monoidal category C into a monoidally equivalent strict monoidal category C str . However, this procedure changes the category even on the object level, which is probably not convenient for studying properties of linear representations. Banagl employs an explicit strict monoidal structure on Vect, namely the Schauenburg tensor product introduced in [24]. The resulting strict monoidal category (Vect, , I ) is monoidally equivalent to the usual monoidal structure on Vect obtained by ⊗. The advantage of Schauenburg's construction is that the monoidal equivalence (F, ξ, ξ 0 ) can be chosen in such a way that F is the identity on Vect, and ξ 0 is the identity on I (see Theorem 4.3 in [24]). Via the natural isomorphism ξ : There is a standard symmetric braiding β V ,W : V ⊗ W → W ⊗ V on Vect for all V , W ∈ Ob(Vect) (with respect to the standard tensor product ⊗).
V ,W we obtain a symmetric braiding with respect to . All in all, the following proposition holds. For the rest of this paper, we will use the Schauenburg tensor product on Vect; thus, we will from now on write ⊗ instead of .

The Chromatic Brauer Category and Its Linear Representations
In this section, we first provide some background on compact closed categories (see Sect. 3.1), and then introduce the chromatic Brauer category as the certain quotient of a free strict compact closed category (see Sect. 3.2).

Compact Closed Categories
Let (C, ⊗, I , α, λ, ρ, b) be a symmetric monoidal category. An object X ∈ Ob(C) is called dualizable if there exists a triple (X , i X , e X ) consisting of an object X ∈ Ob(C), called a dual of X , and morphisms i X : I → X ⊗ X and e X : X ⊗ X → I , called unit and counit respectively, such that the "triangular equations" hold. A compact closed category is a symmetric monoidal category in which every object is dualizable. We use the notation (C, ⊗, I , α, λ, ρ, b, () , i, e) to include as data the assignment () : X → X , as well as the families i and e of unit morphisms and counit morphisms, respectively. As explained in Section 6 of [15], the axioms of a compact closed category determine three canonical isomorphism u X ,Y : (X ⊗ Y ) → Y ⊗ X , v : I → I , and w X : X → X for all objects X , Y in C which are uniquely determined by certain commutative diagrams. Following Section 9 of [15], We call a compact closed category monoidally strict if the underlying monoidal category is strict. Furthermore, we call a compact closed category strict if it is monoidally strict, and, in addition, the families of isomorphisms u, v, w described above are all identity morphisms.
In [15], Kelly and Laplaza give an explicit description of the free compact closed category FA on a given category A, thus solving the coherence problem for this structure. In Section 9 of [15], they discuss some related structures including the free monoidally strict compact closed category F A, as well as the free strict compact closed category F A on A. The latter category is relevant for the present paper, and has the following description in terms of generators and relations.
Theorem 3.1 [15] For any category A, the free strict compact closed category F A has as objects the tensor products X 1 ⊗ · · · ⊗ X n , n ≥ 0, where each X i is A or A for some object A of A. Moreover, F A is a strict compact closed category which is generated (as strict monoidal category) by the families of morphisms of its symmetric braiding b, its unit i, its counit e, and the morphisms of A (considered as morphisms in F A), and by the following relations: (R3) Triangular equations for i and e For all objects X in F A,  (6.2) in [15]), (6.4) in [15]).

The Chromatic Brauer Category
The Brauer category Br is considered in Section 10 of [3] and Section 2.3 of [4], where Banagl defines it as a natural (horizontal) categorification of Brauer algebras. Roughly speaking, isomorphism classes of finite sets serve as objects, and morphisms are isotopy classes of unoriented tangles in Euclidean 4-space. We shall next introduce the chromatic Brauer category cBr as a certain enrichment of the Brauer category. Namely, we equip the components of objects and morphisms with colorings using a countable number of colors (see Remark 3.3). However, we shal first give a more sophisticated definition of cBr in terms of the free strict compact closed category assigned to the discrete category N with Ob(N) = {(k); k ∈ N} a countable set indexed by the natural numbers.

Definition 3.2 (chromatic Brauer category)
The category cBr is the quotient of the free strict compact closed category F N (see Theorem 3.1) by the relation (k) = (k) for all objects (k) ∈ Ob(N). In other words, the objects of the strict compact closed category cBr are of the form and cBr is generated (as strict monoidal category) by the families of morphisms of its symmetric braiding b, its unit i, and its counit e, and by the following relations: (C3) Triangular equations for i and e For all objects X in cBr, (C4) Strictness For all objects X , Y in cBr, For an object in Ob(cBr) of the form (3.1), we introduce the alternative notation In later sections, we will mainly use the notation ([m], c) for objects in cBr because of its consistency with the notation [m] used for objects in the Brauer category Br in [3,4].
The elementary morphisms b (k),(l) , i (k) , and e (k) , k, l ∈ N, play an important role in the study of cBr, as the following remarks show.

Remark 3.3 (visualizing morphisms of cBr)
Similar to the discussion of morphisms of the Brauer category considered in [3,4], morphisms of cBr can be thought of as being ambient isotopy classes of 1-dimensional unoriented tangles in a high dimensional Euclidean space with the novelty that each component is now labeled by an object of N. By projecting the tangles into a plane, we can visualize morphisms of cBr by deformation classes of diagrams like in Fig. 1, where differently structured lines correspond to independent labels. We can use these diagrams to encode a decomposition of a morphism of cBr into elementary morphisms as follows. Horizontal lines in such a diagram represent identity morphisms, crossings represent elementary braidings b (k),(l) , and left and right half circles represent elementary units i (k) and counits e (k) , respectively. Moreover, composition of morphisms in cBr corresponds to horizontal composition of diagrams, and tensor product of morphisms in cBr is defined by vertical "stacking" of diagrams.

Remark 3.4 (generators and relations of cBr) It follows from the relations
• i X (C4) (and the analogous relation for the counit e) that we can take the elementary morphisms b (k),(l) , i (k) , and e (k) for k, l ∈ N as generators of cBr. Then, it can be shown that the following relations are sufficient to generate cBr as strict monoidal category (also compare with Theorem 2.6 in [18]): (A2)Sliding = .

Linear Representations of the Chromatic Brauer Category
For the purpose of constructing symmetric strict monoidal functors Y : cBr → Vect, where Vect is equipped with the Schauenburg tensor product (see Sect. 2.2), we use the notion of a duality structure on a finite dimensional real vector space V . Namely, a duality structure on V is a pair (i, e) whose components are a symmetric copairing i : R → V ⊗ V and a symmetric pairing e : V ⊗ V → R, also called unit and counit, respectively, satisfying the zig-zag equation In other words, V is dualizable with dual V , and is hence self-dual. Let d be the dimension of V , and let {v 1 , . . . , v d } be a basis of V . Then, the set of all duality structures on V is in 1-1 correspondence to the set of symmetric and invertible (d ×d)-matrices Sym(d, R)∩GL(d, R). Indeed, let e jk = e(v j ⊗v k ), and set X = Mat(e) = (e jk ) d j,k=1 . Then, X is symmetric due to the symmetry of e, and-by the zig-zag equation-X is invertible with inverse be symmetric and invertible. Then, the matrices vec(X −1 ) and vec(X ) T define a symmetric copairing and pairing such that the zig-zag equation is satisfied, where (−) T denotes the transposition of a matrix, and vec(−) denotes the vectorization of a matrix formed by stacking the columns of the matrix into a single column vector.
For a linear representation Y : cBr → Vect of cBr, the pair (Y (i (k) ), Y (e (k) )) forms a duality structure on Conversely, the following result shows that we may construct symmetric strict monoidal functors Y : cBr → Vect by choosing duality structures. Theorem 3.5 (Linear representations of cBr) Let V k be a finite dimensional real vector space and let the pair (i (k) , e (k) ) be a duality structure on V k for all k ∈ N. Then there exists a unique symmetric strict monoidal functor Y : (cBr, ⊗, Proof On object level, the values of Y are uniquely and unambiguously determined by the requirement that Y ((k)) = V k for all k ∈ N because the objects of cBr are just the finite tensor products of objects of the form (k), and Y has to respect the monoidal structure. (In particular, we have Y (I ) = R.) By Remark 3.4, cBr is generated by the elementary morphisms e (k) i (k) and b (k),(l) . Thus, the values of Y on morphisms are uniquely determined by the requirements Y (i (k) ) = i (k) and Y (e (k) ) = e (k) for all k ∈ N, and by the requirement Y (b (k),(l) ) = b V k ,V l , k, l ∈ N, of being a symmetric monoidal functor. To show that Y is unambiguously defined on morphisms, we have to verify that the relations (C1) to (C4) of Definition 3.2 are valid in Vect after applying Y to every elementary morphism and every identity morphism in these relations. In view of Remark 3.4, we can instead verify the relations (A1) to (A5) as follows. The zig-zag relation (A1) is satisfied by the definition of the duality structure. To verify relation (A2), we fix a bases {v d l } of the finite dimensional vector spaces V k and V l , respectively. Then, the desired relation follows by noting that, for η, ν ∈ {1, . . . , ). Relation (A3) is satisfied because i is a symmetric copairing. Moreover, the transposition b clearly satisfies relation (A4). Finally, b is a well-known solution of the Yang-Baxter equation so that relation (A5) is satisfied as well.

Proof of Theorem 1.2
The proof of Theorem 1.2 presented here is based on the proof of Theorem 1.1 given in Chapter 5 of [20]. It is divided into two parts as follows. The first part culminates in Corollary 4.4, which states that the given symmetric strict monoidal functor Y : cBr → Vect is faithful if and only if it is faithful on loops (a notion that will be defined below). As it turns out, Corollary 4.4 is a consequence of Proposition 4.2, which takes place solely in the chromatic Brauer category. Secondly, we prove Theorem 4.5 which classifies all symmetric strict monoidal functors Y which are faithful on loops.
For k ∈ N, we define the k-loop λ (k) = e (k) • i (k) , which is an endomorphism of the identity object I ∈ Ob(cBr). We call a morphism ϕ : Then, we obtain the following normal form for morphisms in cBr.
(In particular, note that the maps c 0 : [m] → N and c 0 : [m ] → N are monotone, and satisfy (4.1) Using the triangular equations (C3) one can equivalently define ϕ op by Intuitively, ϕ op can be obtained from ϕ by reflecting a diagram as considered in Remark 3.3 along a vertical axis. Note, that for isomorphisms α : Let us compute the expression ϕ op • ϕ seperately by using the normal form of ϕ from Lemma 4.1, Now, for the isomorphism α we have the relations valid in cBr given by (Indeed, if we interpret α as a bijection on [m] and forgetting about c, we can write α as a product of adjacent transpositions α = T 1 • · · · • T N . Then T i can be shifted along e ([m], ) (resp. i ([m], ) ) from T i ⊗ 1 to 1 ⊗ T i , but in the reverse order. Note that during this procedure, the coloring is changing after each step, after the last shift of T N it has become c 0 .) This leads to such that C and and C close up to S 1 . In other words, if P ∈ C and P ∈ C, then P ϕ = P ψ .
Let us now consider the given symmetric strict monoidal functor Y : cBr → Vect. Recall from the discussion in Sect. 3.3 that Y (( [1], k)) = V k is for all k ∈ N a finite dimensional vector space, whose dimension will be denoted by d k . By assumption, d k > 0 for all k.

Lemma 4.3 For all k
Proof By Proposition 2.9 in [4], the "trace formula" e • i = d k holds for any duality structure (i, e) on V k . In particular, this applies to the duality structure (Y (i (k) ), Y (e (k) )), and we have Y (λ (k) ) = Y (e (k) ) • Y (i (k) ).
The symmetric strict monoidal functor Y : cBr → Vect is called faithful on loops if for any two morphisms ϕ, ψ : that there are a sequence (l k ) k∈N ∈ ∞ k=0 N and loop-free morphisms ϕ 0 and ψ 0 such that ϕ = k λ ⊗l k (k) ⊗ ϕ 0 and ψ = k λ ⊗l k (k) ⊗ ψ 0 . As an immediate consequence of Proposition 4.2 we obtain the following corollary.

Corollary 4.4 Y : cBr → Vect is faithful on loops if and only if Y is faithful.
. Now, under the assumption that Y is faithful on loops, Proposition 4.2 implies that ϕ 0 = ψ 0 . Hence, Y is also faithful on loop-free morphisms. Now let ϕ, ψ : be morphisms (possibly containing loops) such that Y (ϕ) = Y (ψ), and rewrite them as Hence, we obtain i.e. ϕ 0 = ψ 0 and therefore ϕ = ψ.
If Y is faithful on loops, it is clear that the dimension d k of V k needs to satisfy for all (l k ) k∈N ∈ ∞ k=0 Z the implication (1.1), namely Recall that the trace is invariant under cyclic permutation, and the trace of the tensor product of two matrices is the product of their traces. We will also use the identity Thus, we obtain Note two things: The number Tr(Y (ϕ  1) and Y being a strict monoidal functor. Consequently, This completes the proof of Theorem 1.2.

Positive TFTs, Fold Maps, and Exotic Kervaire Spheres
Banagl [3,4] has employed the Brauer category Br as well as singularity theory of fold maps to construct a high-dimensional positive TFT which is defined on smooth cobordisms. As an application, he has shown that the state sum of his theory can distinguish exotic smooth structures on spheres from the standard smooth structure. Banagl's construction is sketched in Section 10 of [3] as an explicit example of the general framework of positive TFTs, and has been implemented in full detail in [4]. In the present section, we construct a refinement of Banagl's theory in which we replace the Brauer category Br by its chromatic enrichment cBr. Our Theorem 5.8 will illustrate the power of our state sum invariant by showing that the associated aggregate invariant can detect exotic Kervaire spheres in infinitely many dimensions. On the other hand, we point out that the gluing axiom of positive TFTs ensures that our state sum invariant is computable by chopping cobordisms into pieces and computing their state sum invariants.
The present section is structured as follows. In Sect. 5.1, we outline the features of Banagl's general framework of positive TFTs, and explain the abstract process of quantization within the framework. Next, Sect. 5.2 provides the concrete definitions of fold fields and the cBrvalued action functional. In particular, we point out the changes that arise from using the chromatic Brauer category instead of Br. Then, quantization of our data is discussed in Sect. 5.3, where we indicate carefully the necessary modifications in the algebraic process of so-called profinite idempotent completion. Finally, in Sect. 5.4, we define the aggregate invariant of homotopy spheres, and sketch our application to exotic Kervaire spheres.

General Framework
In [3], Banagl presents a new approach to the construction of certain TFTs in arbitrary dimension. The basic idea is to modify Atiyah's original axioms [1] by formulating them over semirings instead of rings. Compared to a ring, a semiring is not required to have additive inverses, i.e. "negative" elements. As a result, Banagl coins the notion of positive TFTs. He shows that any system of so-called fields and action functionals gives rise to a positive TFT by means of a process he calls quantization in analogy with theoretical physics. In order to avoid set theoretic difficulties that may arise in the definition of the Feynman path integral, Banagl employs the concept of complete semirings due to Eilenberg [7]. The reason is that a complete semiring has a summation law that allows to sum families of elements indexed by arbitrary index sets. Positive TFTs can motivate the construction of new invariants for smooth manifolds. For example, Banagl defines the aggregate invariant of homotopy spheres (see Section 10 in [4]).
In the following, we outline Banagl's construction of a n-dimensional positive TFT from given systems of fields and action functionals via the process of quantization (see Sections 4 to 6 in [3]).
By definition, a system F of fields assigns to every closed (n −1)-manifold M and to every n-cobordism W sets of fields F (M) and F (W ), respectively, such that certain properties hold. Fields on a cobordism can be restricted to subcobordisms and to codimension 1 submanifolds. Moreover, fields behave in a desirable way with respect to the action of homeomorphisms and disjoint union. Last but not least, fields are requested to glue under the gluing of cobordisms. Next, we consider a system T of action functionals (or action exponentials) on fields F with values in a fixed strict monoidal category C. In Sect. 5.2, we will specifically take C = cBr. The notion of action functional is inspired by the exponential of the action that appears in the integrand of the Feynman path integral, and satisfies the following axioms. To every n-cobordism W one associates a map T W : F (W ) → Mor(C) in such a way that disjoint union of cobordisms is reflected by tensor product of morphisms in C, and gluing of cobordisms is reflected by composition of morphisms in C. More precisely, one requires that for fields f on the gluing W = U ∪ N V along N of cobordisms U from M to N and V from N to P. Furthermore, the action functional is invariant under the action of homeomorphisms. Eventually, let us describe the process of quantization. For this purpose, we fix a system F of fields, a C-valued system T of action functionals on the fields, and a complete semiring S. Following Section 4 in [3], one first constructs a complete additive monoid Q = Q S (C) from the semiring S and the strict monoidal category C. The elements of Q are just maps Mor(C) → S. Then, one exploits the completeness of S to define two different multiplications on Q. As a result, one obtains a pair (Q c , Q m ) of generally non-commutative complete semirings. Multiplication in Q c is based on the composition of morphisms in C, whereas multiplication in Q m exploits the monoidal structure of C. Next, as explained in Section 6 of [3], one assigns to every n-cobordism W the composition T W : F (W ) → Q of T W : F (W ) → Mor(C) with the map Mor(C) → Q that assigns to every morphism γ in C its characteristic function χ γ . Finally, the state sum Z W : F (∂ W ) → Q is defined on a boundary condition f ∈ F (∂ W ) as where the sum ranges over all fields F on W that extend f , i.e., F| ∂ W = f . We note that Z W is well-defined due to the completeness of Q. In analogy with the quantum Hilbert state from physics, the state module Z (M) of a closed n-manifold M consists of all maps ("states") F (M) → Q that satisfy a certain constraint equation. It can be shown that Z W satisfies the constraint equation and is thus an element of the state module Z (∂ W ). Furthermore, the state modules and state sums thus defined can be shown to satisfy Banagl's axioms of a positive TFT, including the essential gluing axiom. For a topologically meaningful choice of fields and action functionals the state sum Z W provides an invariant of n-cobordisms W that is interesting for further investigation.

Fold Fields and cBr-Valued Actions
Fix an integer n ≥ 2. In this section, we specify the fields and actions that will determine our modification of the n-dimensional positive TFT constructed in [4]. All manifolds considered (with or without boundary) will be smooth, that is, differentiable of class C ∞ .

Cobordisms
We recall from Section 7.1 of [4] the terminology concerning manifolds and cobordisms.
From now on, we use the notation M, N , P etc. for closed (n − 1)-dimensional manifolds. Fix an integer D ≥ 2n + 1. We will always assume that any M is smoothly embedded in

System of Fold Fields
Our theory will use exactly the same definition of fold fields on n-cobordisms that is used by Banagl in the original construction. Thus, in this section we will outline the content of Section 7.2 of [4]. We also use the same sets of fields on closed (n − 1)-manifolds although their definition relies on our modified action functional (see the end of Sect. 5.2.3).
The construction of fold fields on an n-dimensional cobordism W is based on the notion of fold maps from W into the plane R 2 ∼ = C. By definition, a fold map of an n-manifold X without boundary into the plane is a smooth map F : X → R 2 such that for every point x ∈ X there exist coordinate charts centered at x and F(x) in which F takes one of the following two normal forms: Let S(F) denote the set of fold points of a fold map F : X → R 2 . It can be shown that S(F) ⊂ X is a smoothly embedded 1-dimensional submanifold that is closed as a subset, and that F restricts to an immersion S(F) → R 2 . In analogy with the Morse index of nondegenerate critical points, there is the following notion of an (absolute) index for fold points, which is a well-known application of the concept of intrinsic derivative (see Section VI.3 in [9, p. 149ff]).

Proposition 5.2
To any fold map F : X → R 2 one can associate a well-defined locally constant map where i ∈ {0, . . . , n − 1} is the number of minus signs that appear in the local normal form of fold points (5.1).
Let W be an n-dimensional cobordism from M to N in the sense of Definition 5.1. A smooth map F : W → R 2 is called fold map if F has for some ε > 0 an extension to a fold map Given a fold map F : W → R 2 , the intersection S( F) ∩ W does not depend on the choice of the fold map extension F, and will in the following be denoted by S(F).

3). Condition
Let F (W ) denote the set of all fold fields on W . If W = ∅, then one puts F (W ) = { * } (set with a single element). Fields on closed (n − 1)-dimensional manifolds will be introduced at the end of the following Sect. 5.2.3, which completes the definition of the system F of fields.

System of cBr-Valued Action Functionals
Banagl exploits singularity theory of fold maps into the plane to construct a system S of Br-valued action functionals (see Section 7.3 in [4]). Namely, for every n-cobordism W he constructs a function S : F (W ) → Mor(Br) assigning to every fold field on W a morphism in Br that encodes the combinatorial information of the 1-dimensional singular set of the fold map. In the present section, we modify the original construction by replacing the Brauer category Br with its chromatic enrichment cBr. The idea is to capture not only the singular patterns provided by the singular sets of fold fields, but also to remember the indices of fold lines by using labels from the set N. Hence, we construct a system S of cBr-valued action functionals which is a lift of S under the forgetful map Mor(cBr) → Mor(Br).
Let W be an n-cobordism from M to N . We construct the function S : F (W ) → Mor(cBr) as follows. If W is empty, then we set S( * ) = id ([0],c ∅ ) . Next suppose that W is non-empty and entirely contained in a set of the form  (F(k)). (Note that the tensor product is actually finite because W is compact.) This completes our construction of a system S of cBr-valued action functionals which lifts S under the forgetful map Mor(cBr) → Mor(Br). Note that Lemma 7.12 in [4] remains valid when replacing S with S in the formulation. That is, given a fold field F on W and some t ∈ (0, 1) such that t ∈ (F(k)) ∩ GenIm(F(k)) for all k ∈ {0, 1, . . . }, F restricts to fold fields Finally, fields on a closed (n − 1)-manifold M are defined to be certain fold fields on the cylinder [0, 1] × M ⊂ [0, 1] × R D , i.e., the trivial cobordism from M to M. Namely, when M is non-empty, we put  a single element). Hence, the set of fields on closed (n − 1)-manifold remains unchanged when replacing S with S. In particular, Lemma 7.13 and Lemma 7.14 (additivity axiom) in [4] remain valid when replacing S with our modified cBr-valued action functional S in the formulation.

Linearization
It can be advantageous to linearize a given category-valued system of action functionals because linear categories are easier to handle as pointed out in Section 8 of [3]. During the process of quantization in Sect. 5.3 below, we will employ a linearization T of our system S of cBr-valued action functionals from Sect. 5.2.3. Such a linearization T is a system of Vect-valued action functionals that is defined by means of a fixed linear representation Y : cBr → Vect by assigning to every n-cobordism W the composition Since the linear representation Y : cBr → Vect can be chosen to be faithful due to Theorem 1.2, there is no loss of informational content when working with the corresponding linearization of S. As a result, our theory yields a refinement of Banagl's aggregate invariant of homotopy spheres which will be studied in Sect. 5.4.

Quantization
The purpose of the present section is to apply the process of quantization over the Boolean semiring S = B (see Example 5.6 below) to the system F of fold fields of Sect. 5.2.2 and our system T of Vect-valued action functionals on F (see Sect. 5.2.4) along Banagl's general framework of positive TFTs outlined in Sect. 5.1. Hence, we obtain a refinement of Banagl's high-dimensional positive TFT defined on smooth cobordisms. In Sect. 5.4, we will study the associated aggregate invariant of homotopy spheres.
First of all, Sect. 5.3.1 below provides the necessary algebraic background on semirings. In Sect. 5.3.2, we will modify the algebraic process of profinite idempotent completion (see Section 6 in [4]) to represent loops of different colors in cBr by a countable family of loop parameters. Finally, we proceed to define our positive TFT Z in Sect. 5.3.3, where we will specify the state modules Z (M) of closed (n − 1)-manifolds M, and define the state sums of n-cobordisms W as certain elements Z W ∈ Z (∂ W ).

Semirings and Semimodules
We collect here a number of concepts from the theory of semirings and semimodules that are relevant for the process of quantization. For a detailed background, we refer to [7,8,11,16,23]. Our summary is based on the presentations in Section 2 of [3] and Section 4 of [4].
Recall that a (commutative) monoid is a triple M = (M,  * , e), where M is a set equipped with a (commutative) associative binary operation * and a two-sided identity element e ∈ M, that is, e * m = m * e = m for all m ∈ M. A semiring is a tuple S = (S, +, ·, 0, 1), where S is a set together with two binary operations + and · and two elements 0, 1 ∈ S such that (S, +, 0) is a commutative monoid, (S, ·, 1) is a monoid, the multiplication · distributes over the addition from either side, and 0 is absorbing, i.e. 0 · s = 0 = s · 0 for every s ∈ S. The semiring S is called commutative if the monoid (S, ·, 1) is commutative. A morphism of semirings sends 0 to 0, 1 to 1 and respects addition and multiplication. Fix a semiring S. Next, we discuss the important notion of Eilenberg-completeness [7, p. 125] for semirings and semimodules (see also [11,16]). A complete monoid is a commutative monoid (M, +, 0) together with an assignment , called a summation law, which assigns to every family (m i ) i∈I , indexed by an arbitrary set I , an element i∈I m i of M (called the sum of the m i ), such that and for every partition I = j∈J I j , we have A complete semiring is a semiring S for which (S, +, 0, ) is a complete monoid, and infinite distributivity holds, that is, i∈I ss i = s i∈I s i , i∈I s i s = i∈I s i s.
A semimodule M over a commutative semiring S is called complete if its underlying additive monoid is equipped with a summation law that makes it complete as a commutative monoid, and infinite distributivity i∈I sm i = s i∈I m i holds for every s ∈ S and every family (m i ) i∈I in M. If ϕ : S → T is a morphism of semirings and T is complete as a semiring, then T can be easily seen to be complete as an S-semimodule.
We will also need the following notion of continuity for idempotent complete semirings (cf. [8,11,16,23]). Here, we only state the definition, and refer to the summary preceding It is useful to note that the product i∈I M i of a family {M i } i∈I of continuous idempotent complete monoids is a continuous idempotent complete monoid.
Example 5. 6 The minimal example of a semiring that is not a ring is given by the Boolean semiring B. This is the set B = {0, 1} equipped with addition defined by 1 + 1 = 1 and multiplication given by 0 · 0 = 0 (where 0 and 1 serve as identity elements for addition and multiplication, respectively). Distributivity holds, but in B there exists no additive inverse for 1. We leave it to the reader to check that the commutative semiring B is idempotent, complete, and continuous.

Profinite Idempotent Completion
As mentioned before, our input data for the process of quantization are the Boolean semiring S = B (see Example 5.6), the system F of fold fields (see Sect. 5.2.2), and our system T of Vect-valued action functionals on F (see Sect. 5.2.4). Recall from Sect. 5.1 that the first step of quantization consists in the construction of a pair (Q c , Q m ) of generally noncommutative complete semirings. In principle, we could follow the general construction provided in Section 4 of [3]. However, we would like to take care of the additional structure given by the natural action of the polynomial semiring of colored loops on the morphism sets of the chromatic Brauer category. As a consequence, our state sum invariant will take values in power series in several variables with matrix coefficients. Banagl has implemented this algebraic process of profinite idempotent completion in Section 6 of [4] for the natural action of the polynomial semiring of loops on the morphism sets of the Brauer category Br. Since we work instead with the chromatic Brauer category cBr, we will in the following discuss the modifications of Banagl's construction in detail. Provided that the underlying functor U is chosen to be faithful on loops, the additive monoid (Q, +, 0) can be promoted to idempotent complete semirings Q c (the composition semiring, see Proposition 6.12 in [4]) and Q m (the monoidal semiring, see Proposition 6.14 in [4]). It can be shown that the semirings Q c and Q m are both continuous (see Proposition 6.15 in [4]). Continuity is exploited in [4] to check several axioms of positive TFTs, namely the behavior of state sums under disjoint union (Proposition 7.22), and the gluing axiom (Theorem 7.26).
We proceed to explain how the process of profinite idempotent completion applies to our setting. When using the chromatic Brauer category instead of Br, we need to replace the semimodule B  [4]). Set N = ∞ i=0 N, which is a commutative monoid with respect to component wise addition and identity element 0 = (0, 0, . . . ). In general, a (formal) power series in a countable number of indeterminates q = q 0 , q 1 , . . . and with coefficients in the Boolean semiring B is a function a : N → B, written as a formal sum ν∈N a(ν)q ν , where q ν denotes the (finite) product ∞ s=0 q ν s s . The element a(ν) is referred to as the coefficient of q ν . Let B[[q]] be the set of all power series over B having a countable number of indeterminates q = q 0 , q 1 , . . . . We write 0 for the power series a with a(ν) = 0 for all ν, and 1 for the power series a with a(0) = 1 and a(ν) = 0 for all ν = 0. Define an addition on power series by a + b = c, where c(ν) = a(ν) + b(ν) for all ν. Define a multiplication on power series by the Cauchy product, that is, Then, (B[[q]], +, ·, 0, 1) is a commutative idempotent semiring, the semiring of power series over B in a countable number of indeterminates. In a similar way, using finite sums rather than power series, one defines the polynomial semiring N[t] in a countable number of indeterminates t = τ 0 , τ 1  Returning to the category cBr, we observe that the k-loops λ (k) , k ∈ N, induce an action of the (multiplicatively written) commutative monoid N = {t ν ; ν ∈ N} on the morphism sets  (Y (([m], c)), Y (([m ], c ) inherits an action of the monoid N via t ν f = Using the algebraic tensor product of semimodules over the commutative semiring N[t] (see [13,14], and compare Section 4 in

Being the product of copies of B[[q]], each Q(H ([m],c),([m ]
,c ) ) is a continuous idempotent complete monoid. We conclude that the additive monoid (Q, +, 0) is continuous, idempotent and complete as well. Hence, (Q, +, 0) can be advanced to continuous idempotent complete semirings Q c and Q m in analogy with the construction in Section 6 in [4].

State Modules and State Sums
Let Q denote the profinite idempotent completion of the set Y (Mor(cBr)) associated to a fixed faithful linear representation Y : cBr → Vect as constructed in Sect. 5 is analogous to the discussion in Section 7.4 of [4]. Here, ⊗ denotes the complete tensor product of complete idempotent continuous semimodules (see Section 5 in [4]) rather than the algebraic tensor product ⊗ of function semimodules discussed in [2]. Let W n be a cobordism from M to N in the sense of Definition 5.1. The state sum will be defined as an element Z W ∈ Z (M) ⊗Z (N ). Fix a cylinder scale ε W for W . Given a boundary where the equivalence relation ≈ for fold fields on closed (n − 1)-manifolds X is defined as follows. Two smooth maps f : Definition 7.18 in [4]). On ( f M , f N ) the state sum Z W is then defined as In close analogy with the further steps in [4], one can prove that our assignment Z is in fact a positive topological field theory. Namely, following Section 7.6 in [4], one checks the correct behavior of our state sum under disjoint union. Moreover, following Section 7.7 in [4], one proves the essential gluing formula Z W = Z W , Z W (see Theorem 7.26 in [4]), where W is the result of gluing a cobordism W from M to N with a cobordism W from N to P along N . Note that the preparatory results Proposition 7.23, Lemma 7.24, and Proposition 7.25 in [4] need only be modified by replacing the Br-valued action functional S with the cBrvalued action functional S in the formulation. Diffeomorphism invariance ϕ * (Z W ) = Z W (see Theorem 9.16 in [4]) of our state sum under diffeomorphisms ϕ : ∂ W → ∂ W that can be extended to so-called time consistent diffeomorphisms ϕ : W → W can be shown along the lines of Section 9 in [4]. In particular, Lemma 9.12 and Lemma 9.14 in [4] remain valid when replacing S with S in the formulation. The map ϕ * : Z (∂ W ) → Z (∂ W ) can then be defined on a function z : F (∂ W ) → Q and a field g ∈ F (∂ W ) by ϕ * (z)(g) = z(g • (id [0,1] ×ϕ)) ∈ Q.

The Aggregate Invariant and Exotic Kervaire Spheres
Positive TFTs have been created by Banagl [3] with the intention to provide new topological invariants for high-dimensional manifolds. In this section, we explain how our positive TFT Z constructed in the previous section can be used to assign to every homotopy sphere M its aggregate invariant A(M), an element of the complete semiring Q from Sect. 5.3.2. The construction of A is analogous to that of the aggregate invariant A studied Section 10 in [4]. While the invariant A is known to distinguish exotic spheres from the standard sphere (see Corollary 10.4 in [4]), we will indicate briefly that the invariant A can distinguish exotic Kervaire spheres from other exotic spheres in infinitely many dimensions.
Fix a closed (n − 1)-manifold M which is homeomorphic (but not necessarily diffeomorphic) to the sphere S n−1 . Without loss of generality, we assume in the following that In conclusion, we outline an application to Kervaire spheres, which are a concrete family of homotopy spheres that can be obtained from a plumbing construction as follows (see [17, p. 162]). The unique Kervaire sphere n−1 K of dimension n − 1 = 4r + 1 can be defined as the boundary of the parallelizable (4r + 2)-manifold given by plumbing together two copies of the tangent disc bundle of S 2r +1 . According to the classification theorem of homotopy spheres (see Theorem 6.1 in [19, pp. 123f]), as well as recent work of Hill-Hopkins-Ravenel [10] on the Kervaire invariant one problem, it is known that n−1 K is an exotic sphere, i.e., homeomorphic but not diffeomorphic to S n−1 , except when n − 1 ∈ {5, 13, 29, 61, 125}.
Note that, according to Remark 6.3 in [29], there are infinitely many dimensions of the form n − 1 ≡ 13 (mod 16) in which there exist exotic spheres that are not diffeomorphic to the Kervaire sphere n−1 K . The following result shows that our aggregate invariant A can distinguish exotic Kervaire spheres from other exotic spheres in infinitely many dimensions. We give a sketch of the proof by referring to the results of [27]. A detailed proof is beyond the scope of this paper, and will appear elsewhere.  ) ). We choose ν such that ν j = 1 for j = n/2 and ν j = 0 for j = n/2. Then, it follows from Corollary 10.1.4 and Theorem 3.4.9 in [27] that the coefficient of q ν in ζ( n−1 ) is 1 whenever n−1 is diffeomorphic to n−1 K . Conversely, if n−1 is not diffeomorphic to n−1 K , then Corollary 10.1.4 in [27] implies that the coefficient of q ν in ζ( n−1 ) is 0.