aa r X i v : . [ m a t h . QA ] S e p EVALUATING THIN FLAT SURFACES
MIKHAIL KHOVANOV, YOU QI, AND LEV ROZANSKY
Abstract.
We consider recognizable evaluations for a suitable category of oriented two-dimensional cobordisms with corners between finite unions of intervals. We call such cobor-disms thin flat surfaces. An evaluation is given by a power series in two variables. Rec-ognizable evaluations correspond to series that are ratios of a two-variable polynomial bythe product of two one-variable polynomials, one for each variable. They are also in a bi-jection with isomorphism classes of commutative Frobenius algebras on two generators witha nondegenerate trace fixed. The latter algebras of dimension n correspond to points onthe dual tautological bundle on the Hilbert scheme of n points on the affine plane, with acertain divisor removed from the bundle. A recognizable evaluation gives rise to a functorfrom the above cobordism category of thin flat surfaces to the category of finite-dimensionalvector spaces. These functors may be non-monoidal in interesting cases. To a recognizableevaluation we also assign an analogue of the Deligne category and of its quotient by the idealof negligible morphisms.
Contents
1. Introduction 12. The category of thin flat surfaces 42.1. Category TFS 42.2. Classification of thin flat surfaces 72.3. Endomorphisms of 1 and homs between 0 and 1 in TFS 113. Linearizations of the category TFS 133.1. Categories k TFS and VTFS α for recognizable α α . 173.3. Quotient by negligible morphisms and Karoubi envelopes. 213.4. Summary of the categories and functors 234. Hilbert scheme and recognizable series 245. Modifications 285.1. Adding closed surfaces 285.2. Coloring side boundaries of cobordisms 33References 371. Introduction
Universal constructions of topological theories [BHMV, Kh1, RW] that are not necessar-ily multiplicative [FKNSWW] are interesting even in dimension two [Kh2, KS], providing
Date : September 4, 2020. examples somewhat different from commutative Frobenius algebras for the invariants of two-dimensional cobordisms. In this note we consider the analogue of the latter construction fororiented two-dimensional cobordisms with corners. For simplicity we restrict to cobordismsbetween finite unions of intervals; boundary points of the intervals give rise to corners ofcobordisms. Furthermore, we require that each connected component of a cobordism hasnon-empty boundary, which is a natural condition when excluding cobordisms with cornersthat have circles as some boundary components.Cobordisms that we consider can be “thinned” to consist of ribbons glued to disks andcan be depicted in the plane as regular neighbourhoods of immersed graphs, see Figure 2.1.2below for an example. For this reason we refer to these cobordisms as thin flat surfaces or tf-surfaces throughout the paper. When viewed as a morphism in the appropriate categoryTFS of thin flat cobordisms, a particular immersion of the surface into the plane is inessential,and morphisms are equivalence classes of such cobordisms modulo diffeomorphisms that fixthe boundary.The category TFS admits an analogue of α -evaluations from [Kh2, KS, KKO]. This timeclosed connected morphisms S (connected endomorphisms of the unit object 0, the emptyunion of intervals) are parametrized by two non-negative integers ( ℓ, g ), where ℓ + 1 is thenumber of boundary components of S and g is the genus. Assigning an element α ℓ,g ofthe ground field k (or a ground commutative ring R ) to such a component and extendingmultiplicatively to disjoint unions gives an evaluation α on endomorphisms of the unit object0. Evaluation α can be conveniently encoded as power series(1) Z α ( T , T ) = X k,g ≥ α k,g T k T g , α = ( α k,g ) k,g ∈ Z + , α k,g ∈ k , where the degree of the first variable T counts “holes” in a cobordism (a disk has no holesand an annulus has one hole) and T keeps track of the genus.With α as above and n ≥
0, one can define a bilinear form on a k -vector space with a basisgiven by equivalence classes of thin flat surfaces with n boundary intervals. The quotientby the kernel of the bilinear form is a vector space A α ( n ). The collection of quotient spaces { A α ( n ) } n ≥ is what we refer to as the universal construction for the category TFS, given α .The spaces A α ( n ) rarely satisfy the Atiyah factorization axiom, that is, the relation A α ( m + n ) ∼ = A α ( m ) ⊗ A α ( n )does not hold. From the quantum field theory (QFT) perspective, this violation may happenif the 2-dimensional QFT is embedded as a 2-dimensional defect inside a higher-dimensionalQFT.It is straightforward to see that A α ( n ) is finite-dimensional for all n iff A α (1) is finite-dimensional iff the series (1) is recognizable or rational (terms from the control theory andthe theory of noncommutative power series). Recognizable power series in this case have theform(2) Z α ( T , T ) = P ( T , T ) Q ( T ) Q ( T ) , that is, Z α can be written as a ratio of a polynomial in T , T and two one-variable polynomials,see Proposition 3.1 and Fliess [F]. VALUATING THIN FLAT SURFACES 3
Constructions of [KS] go through for the category of thin flat surfaces and any recognizable α as above. We define the category STFS α (skein thin flat surfaces) where homs are finitelinear combinations of cobordisms, closed components evaluate to coefficients of α , and thereare skein relations given by adding holes and handles to a component of a cobordism andequating to zero linear combinations corresponding to elements of the kernel ideal I α ⊂ k [ T , T ] associated to α and also known as the syntactic ideal of rational series α . A twovariable polynomial z = z ( T , T ) is in I α iff α ( zf ) = 0 for any polynomial f ∈ k [ T , T ], with α ( T ℓ T g ) = α ℓ,g extended to a linear map k [ T , T ] α −→ k .For the rest of the paper we change our terminology and call connected components of a thinflat surface that have neither top nor bottom boundary intervals floating components insteadof closed components, since they otherwise have boundary, what we call side boundary, thatis present inside the cobordism but not at its top or bottom. This avoid possible confusionwith the usual notion of a closed surface. A non-empty thin flat surface is never closed in thelatter sense.The category STFS α has finite-dimensional hom spaces. Taking the additive Karoubienvelope of this category to form DTFS α := Kar(STFS ⊕ α )gives an idempotent-complete k -linear rigid symmetric monoidal category DTFS α which isthe analogue of the Deligne category [D, CO, EGNO] for TFS and recognizable series α intwo variables.Once we pass to k -linear combinations of cobordisms, and α is available to evaluate floatingcobordisms, there is a trace map on endomorphisms of any object n . It is given by closingeach term in the linear combination of tf-surfaces describing the endomorphism via n stripsinto a floating tf-surface and evaluating it via α . Consequently, one can form the ideal J α ofnegligible morphisms [D, CO, EGNO, KS] and quotient the category by that ideal.We call the quotient category gligible quotient to avoid the awkward-sounding word “non-negligible quotient” and mirroring the terminology from [KKO]. The gligible quotient TFS α of the skein category STFS α carries non-degenerate bilinear forms on its hom spaces andotherwise shares key properties of STFS α : objects are non-negative integers, category TFS α is rigid symmetric tensor, and the hom spaces are finite-dimensional over k .Likewise, the Deligne category DTFS α has the gligible quotient DTFS α by the ideal ofnegligible morphisms. The same category can be recovered as the additive Karoubi closure ofTFS α .Section 3.4 and diagram (12) contain a summary of these categories and key functorsrelating them.Similar to [EO, EGNO, KKO], it is natural to ask under what conditions will DTFS α besemisimple. Unlike [KKO, Kh2], we do not work out any specific examples of these categorieshere and leave that to an interested reader or another time.Our evaluation α is encoded by a power series Z α in two variables (1), and the recognizableseries Z α gives rise to a finite-codimension ideal I α in k [ T , T ], the largest ideal contained inthe hyperplane ker( α ). Such an ideal defines a point on the Hilbert scheme of the affine plane A . We discuss the relation to the Hilbert schemes in Section 4 and explain a bijection betweenrecognizable power series with the ideal I α of codimension k and points in the complement MIKHAIL KHOVANOV, YOU QI, AND LEV ROZANSKY T ∨ k \ D k of the dual tautological bundle T ∨ k on the Hilbert scheme and a suitable divisor D k on it.It is not clear whether the appearance of the Hilbert scheme of A is a bug or a feature. InSection 5 we explain two generalizations of our construction. One of them involves “coloring”side boundary components of a thin flat surface into r colors. For the resulting category,recognizable series depends on r + 1 parameters (generalizing from 2 parameters for r = 1),and one would get a generalization of our construction from the Hilbert scheme of A to thatof A r +1 , with the appropriate divisor removed from the dual tautological bundle in both cases.Of course, the Hilbert scheme has vastly different properties and uses in the case of algebraicsurfaces versus higher dimensional varieties.The other generalization considered in that section is given by extending the TFS (thin flatsurfaces) category by allowing closed components and circles as boundaries. This correspondsto the usual category of two-dimensional oriented cobordisms with boundary and cornersstudied in [MS, LP, C, SP] and other papers. Objects of that category are finite disjointunions of intervals and circles. We briefly touch on this generalization and explain encodingof recognizable series via certain rational power series in this case as well.Relations between Frobenius algebras, recognizable power series, codes and two-dimensionalTFTs are considered in Friedrich [Fr], which is quite close in spirit to this paper.A possible relation between moduli spaces of SU ( m ) instantons on R (the Hilbert schemeof C corresponds to U (1) case) and control theory is explored in [H, S] and the follow-uppapers. We do not know how to connect it to the constructions in the present paper. Acknowledgments.
M.K. was partially supported by the NSF grant DMS-1807425 whileworking on this paper. Y. Q. was partially supported by the NSF grant DMS-1947532. L.R.was partially supported by the NSF grant DMS-1760578.2.
The category of thin flat surfaces
Category
TFS . We introduce the category TFS of thin flat surfaces . Its objects are non-negative integers n ∈ Z + = { , , , . . . } . An object is represented by n intervals I , . . . , I n placed along the x -axis in the xy -plane. A morphism from n to m is a “thin” surface S immersed in R × [0 ,
1] connecting n intervals on the line R × { } with m intervals on the line R × { } . The immersion map S −→ R × [0 ,
1] is a local diffeomorphism, but the image of S may have overlaps, that can be thought of as virtual overlaps and ignored. The surface S inherits an orientation from its immersion into R × [0 . S , the immersion is open.Alternatively, the immersion can be perturbed to an embedding of S into R × [0 ,
1] byturning overlaps into over- and under-crossings of strips of a surface. This can be done justfor aesthetic purposes, and whether one chooses an over- or an under-crossing does not matterfor the morphism associated to the surface.The boundary of S consists of several circles (at least one circle unless S is the emptysurface) and decomposes into n + m disjoint intervals that constitute horizontal boundary ∂ h S and n + m intervals and some number of circles that constitute side , or vertical , or inner boundary ∂ v S of S : ∂S = ∂ h S ∪ ∂ v S. VALUATING THIN FLAT SURFACES 5
Horizontal intervals that constitute ∂ h S are the intersections of S with R × { , } ⊂ R × [0 , ∂ v S is the closure of the intersection of ∂S with R × (0 , ∂ h S ∩ ∂ v S consists of 2( n + m ) boundary points of the horizontal intervals. These are alsothe corners of the surface S .In the graphical depicitions of thin flat surfaces below, we will draw horizontal boundarysegments as red intervals, and vertical boundary components as green arcs for better visual-ization (Figure 2.1.1 and Figure 2.1.2 right), but the figures can also be viewed and carry fullinformation in greyscale. Starting from Figure 2.1.2, we depict tf-surfaces in light aquamarine. ...... PSfrag replacements 11 223 mn Figure 2.1.1.
A thin flat surface in R × [0 , S is to immerse a finite unoriented graph Γ, possibly with multipleedges and loops, into the strip R × [0 , : Γ −→ R × [0 , v of valency 1 such that ( v ) ∈ R × { , } . The remainingvertices are mapped inside the strip. The immersion is disjoint on vertices. Edges of Γ mayintersect in R × [0 , is shown in Figure 2.1.2 left.Taking a regular neighbourhood N (Γ , ) of Γ under , locally in Γ, results in a thin flatsurface N (Γ). Vice versa, any thin flat surface S can be deformed to the surface N (Γ) forsome Γ.Take a thin flat surface S and forget the embedding into R × [0 , S asa cobordism between ordered collections of oriented intervals (induced by the orientation of R , say from left to right). The cobordism S has corners (unless n = m = 0) and two types ofboundary, as discussed. By definition, two cobordisms S , S represent the same morphismif they are diffeomorphic rel horizontal boundary, that is, keeping all horizontal boundarypoints fixed.The category TFS is symmetric monoidal, and a possible set of generating morphismsis shown in Figure 2.1.5. We have included the identity morphism id into the Figure toemphasize that the identity morphism id n is represented by the surface which is the directproduct of the disjoint union of n intervals (representing object n ) and [0 , MIKHAIL KHOVANOV, YOU QI, AND LEV ROZANSKY ...... ......
A virtual crossing
PSfrag replacements 1111 2222 33 mm nn
Figure 2.1.2.
An immersed graph Γ in R × [0 ,
1] and associated thin flatsurface N (Γ , ).morphism P of 2 = 1 ⊗ = PSfrag replacements ι ǫ m ∆ id P Figure 2.1.3.
A set of generating morphisms. From left to right: ι, ǫ, m, ∆ aremorphisms from 0 to 1, from 1 to 0, from 2 to 1 and from 1 to 2, respectively.The rightmost morphism P is the permutation morphism on 1 ⊗ of object 1 is shown for completeness.The elements ι, ǫ, m, ∆ , P constitute a set of monoidal generators of TFS. Together withthe identity morphism id they can be used to build any morphism in TFS, via horizontaland vertical compositions. In particular, from these generators we can build the self-dualitymorphisms for the object 1, see Figure 2.1.4. == Figure 2.1.4.
Self-duality morphisms ǫ m : 1 ⊗ −→ ι : 0 −→ ⊗ S representing a morphism from n to m in TFS a thin flat cobordism from n to m . A thin flat cobordism S is a disjoint union of its connected components S , . . . , S k .Consider one such component S ′ . It necessarily has non-empty boundary, and we can assign VALUATING THIN FLAT SURFACES 7 = = == = = ==== Figure 2.1.5.
Some relations in TFS.to S ′ non-negative integers ℓ, g , where ℓ + 1 is the number of boundary components and g ≥ S ′ . The surface S ′ carries an orientation, inherited via an immersion from theorientation of the plane.We will also call a thin flat surface a tf-surface and, when viewed as a cobordism, a tf-cobordism .The morphisms ι : 0 −→ , ǫ : 1 −→ , m : 1 ⊗ −→ , ∆ : 1 −→ ⊗ Classification of thin flat surfaces.
By a closed or floating tf-surface S we mean onewithout horizontal boundary. A floating tf-surface necessarily has side boundary, unless it isthe empty surface. Diffeomorphism equivalence classes of floating tf-surfaces are in a bijectionwith endomorphisms of the object 0 of TFS. Such a surface is a disjoint union of its connectedcomponents, and a component is uniquely determined by its pair ( ℓ + 1 , g ), ℓ, g ∈ Z + , thenumber of boundary components and the genus, respectively. Any such pair is realized bysome surface, since pairs (0 + 1 , , ,
1) are realized by a disk, an annulus,a flat punctured torus, see Figure 2.2.1, and taking band-connected sum of surfaces with
MIKHAIL KHOVANOV, YOU QI, AND LEV ROZANSKY = Figure 2.1.6.
Left: object 1 is not commutative Frobenius. That the twodiagrams on the left are not diffeomorphic rel horizontal boundary can be seeneasily by examining the matchings on the six corner points in each diagramprovided by side boundaries. The two matchings of the six points are differ-ent, a sufficient condition for the two cobordisms not to be diffeomorphic relboundary. Right: a diagram that’s not a morphism in TFS.invariants ( ℓ + 1 , g ) , ( ℓ + 1 , g ) yields a surface with the invariant ( ℓ + ℓ + 1 , g + g ).Choose a closed connected tf-surface S ℓ +1 ,g , one for each value of these parameters. (0+1,0) (1+1,0) (0+1,1) (2+1,0) (1+1,1) Figure 2.2.1.
Examples of closed connected tf-surfaces S ℓ +1 ,g for small valuesof ℓ and g . We explicitly write ℓ + 1 to remember that a surface always haveat least one boundary component.Connected morphisms from 0 to 0 in TFS are in a bijection with S ℓ +1 ,g as above. Endo-morphisms of 0 in TFS is a free commutative monoid on generators S ℓ +1 ,g , over all ℓ, g ∈ Z + ,End TFS (0) ∼ = h S ℓ +1 ,g i ℓ,g ≥ . An element a ∈ End
TFS (0) has a unique presentation as a finite product of S ℓ +1 ,g ’s withpositive integer multiplicities, a = k Y i =1 S r i ℓ i +1 ,g i , r i ∈ { , , . . . } . Consider a tf-surface S describing a morphism from n to m in TFS. It may have some floating connected components, that is, those that are disjoint from the horizontal boundary of S .Each of these components is homeomorphic to S ℓ +1 ,g as above for a unique ℓ, g . Componentsof S that have non-empty horizontal boundary are called viewable or visible components.Any component of S is either floating or viewable . We call S viewable if it has no floatingcomponents. The empty cobordism is viewable.The commutative monoid End TFS (0) acts on the set Hom
TFS ( n, m ) by taking a cobordismto its disjoint union with a floating cobordism. Any morphism S ∈ Hom
TFS ( n, m ) has a unique VALUATING THIN FLAT SURFACES 9 presentation S = S · S where S ∈ End
TFS (0), S is a viewable cobordism in Hom TFS ( n, m )and dot · denotes the monoid action. In particular, Hom TFS ( n, m ) is a free End TFS (0)-setwith a “basis” of viewable cobordisms.Let us specialize to viewable cobordisms S . All connected components of S are viewableand determine a set-theoretic partition of n + m horizontal boundary intervals of S . Let uslabel these boundary intervals from left to right by 1 , , . . . , n for the bottom intervals and1 ′ , . . . , m ′ for the top intervals.Each viewable component contains a non-empty subset of this set of intervals and togetherviewable components give a decomposition λ of this set into disjoint sets. We denote by D mn the set of partitions of these n + m intervals, so that λ ∈ D mn . To further understandthe structure of morphisms, we restrict to the case of connected S , thus a surface with oneviewable connected component. All horizontal intervals are in S .The surface S and its horizontal boundary segments inherit orientation from R × [0 ,
1] andfrom induced orientations of the top and bottom boundary of R × [0 , + (cid:0) ...... PSfrag replacements 1 2 n ′ m ′ ∂ − ∂ R × [0 , S Figure 2.2.2.
Orientation convention for R × [0 , S and its horizontal and side boundary.We use the convention of reversing the orientation on the source (bottom) part of theboundary of a cobordism, see Figure 2.2.2. Consequently, bottom intervals I , . . . , I n in ∂S are oppositely oriented from the rest of the boundary, while top intervals I ′ , . . . , I m ′ areoriented compatibly with the side boundary orientations, inherited from that of S and in turninherited from the orientation of R × [0 , S into the “core” of S to make it easier to see compatible and reverse orientations of thehorizontal boundary segments of S .We can now classify isomorphism classes of connected tf-cobordisms S from n to m . Sucha cobordism has ℓ + 1 boundary circles and genus g . On ℓ + 1 boundary circles choose n + m non-overlapping intervals and label them 1 , . . . , n, ′ , . . . , m ′ . Choose an orientation of theinterval 1 ′ or, if m = 0, orientation of interval 1.The orientation of the interval 1 ′ induces an orientation of that boundary component of S and hence of S itself. One then gets induced orientations for all boundary components of S . Horizontal parts of ∂S for the intervals 2 ′ , . . . , m ′ are then oriented compatibly with the boundary, while those corresponding to the intervals 1 , . . . , n in the opposite way from thatfor the boundary.Horizontal intervals on the ℓ + 1 boundary components determine a partition of N mn := { , . . . , n, ′ , . . . , m ′ } into ℓ + 1 disjoint subsets, possibly with some subsets empty. Orientations of boundary com-ponents induce a cyclic order on elements of each subset, where one goes along a componentin the direction of its orientation and records horizontal intervals that one encounters. We callan instance of this data a locally cyclic partition of N mn together with a choice of genus g ≥ N mn by D mn,cyc and by D mn,cyc ( ℓ ) if the number ofcomponents is fixed to be ℓ + 1. This time, empty components are allowed. They correspondto components of ∂S disjoint from the boundary R × { , } of the strip. We have D mn,cyc = G ℓ ≥ d mn,cyc ( ℓ ) . For the example in Figure 2.2.2 we have n = 3, m = 2, the set of horizontal intervals is { , , , ′ , ′ } , there are two components ( ℓ = 1), and the cyclic orders are (1 ′ , , ′ ,
3) and(2).PSfrag replacements 1 12 23 34 42 ′ ′ ′ ′ ′ ′ Figure 2.2.3.
Left: converting the partition and genus data into a surfacewith boundary and labelled edges on the boundary. One boundary component(inner right) does not carry labelled edges, since the partition contains onecopy of the empty set. Genus two is indicated by schematically showing twohandles. Right: stretching out labelled edges into corresponding horizontalintervals to produce a morphism in TFS.Vice versa, suppose given ( ℓ, g ) as above, a locally cyclic partition λ ∈ D mn,cyc ( ℓ ) of N mn into ℓ + 1 subsets, possibly with some subsets empty, with a cyclic order on each subset. Tosuch data we can assign a connected thin flat surface S ( λ, g ) of genus g with the horizontalboundary these n + m intervals, ℓ + 1 boundary components, and horizontal intervals placedaccording to the cyclic order for the subset along each component. VALUATING THIN FLAT SURFACES 11
For another example, for n = 4, m = 3, the partition { (1 ′ , , ′ , , (3 ′ , , , () } , whichincludes one copy of the empty set, with cyclic orders as indicated and genus g = 2 theresulting tf-surface is shown in Figure 2.2.3 right.This bijection between connected morphisms from n to m and elements of the set D mn,cyc × Z + leads to a classification of morphisms in TFS. An arbitrary morphism S ∈ Hom
TFS ( n, m ) isthe union of the viewable subcobordism of S and the floating subcobordism. The latter areclassified by elements of Hom TFS (0 ,
0) and admit a very explicit description, via pairs ( ℓ, g ) ofthe number of circles minus one and the genus of each connected component. The viewablesubcobordism S ′ of S determines a partition of N mn by with the set of horizontal intervals foreach component of S ′ being a part of that partition. Each part of this partition is non-empty.Next, for each part of the partition, remove the connected components of S ′ for all otherparts, downsizing to just one component S ′′ . Relabel the horizontal intervals for S ′′ into1 , , . . . , n ′′ and 1 , , . . . , m ′′ . Then such components S ′′ are classified by data as above: alocally cyclic partition of N m ′′ n ′′ (possibly with empty subsets included) and a choice of genus g ≥ n to m in TFS.2.3. Endomorphisms of and homs between and in TFS . The category TFS is rigid symmetric monoidal, with the unit object 0 and the generatingself-dual object 1, with all objects being tensor powers of the generating object, n = 1 ⊗ n .In the rest of this section, since we only consider the category TFS, we may write Hom( n, m )instead of Hom TFS ( n, m ), End( n ) instead of End TFS ( n ), etc. Connected endomorphisms of : Endomorphisms End(1) = End
TFS (1) of the object 1 in thecategory TFS constitute a monoid. Consider the submonoid End c (1) of End(1) that consistsof connected endomorphisms of 1. Define endomorphisms b , b , b ∈ End c (1) via diagrams inFigure 2.3.1.PSfrag replacements b b b Figure 2.3.1.
Endomorphisms b , b , b .Note that b has equivalent presentations, as shown in Figure 2.3.2. The last diagram isnot a tf-surface, but describes a diffeomorphism class of one (rel boundary). The tf-cobordism b has genus one and one boundary component, with two horizontal segments labelled 1 and1 ′ on it, which uniquely determines it as an element of End(1).We refer to b as the “hole” cobordism, b as the “handle” cobordism, b as the “cross”cobordism. = = Figure 2.3.2.
Presentations of b . The diagram on the right is not a thin flatpresentation but shows a cobordism that can be deformed to a diagram inTFS . Proposition 2.1.
The endomorphisms b , b , b ∈ End
TFS (1) pairwise commute: b b = b b , b b = b b , b b = b b . Proof.
Note that the product with b just adds a hole with no horizontal segments on it to aconnected cobordism. Product with b adds a handle to a connected cobordism. (cid:3) Proposition 2.2. • End c (1) is a commutative monoid generated by commuting ele-ments b , b , b with an additional defining relation b = b b . • End c (1) consists of the following distinct elements: b n b m , b n b m b , n, m ≥ . Proof.
A cobordism S ∈ End c (1) is a connected surface with ℓ + 1 boundary circles, genus g ,and two horizontal intervals on it. If the intervals are on the same connected component ofthe boundary, S = b ℓ b g . If the intervals lie on distinct boundary components then ℓ ≥ S = b ℓ − b g b . (cid:3) Spaces Hom (0 , and Hom (1 , : An element y ∈ Hom(0 ,
1) is a tf-cobordism with onehorizontal interval, at the top. It is a product y y of one viewable component y ∈ Hom(0 , y ∈ Hom(0 , y is viewable, thus connected, since ithas a unique horizontal segment. Then y is determined by the number ℓ + 1 of its boundarycomponents and the genus g and can be written as y = b ℓ b g ι, where ι is the morphism 0 −→ b ι = b ι . Proposition 2.3.
A morphism y ∈ Hom
TFS (0 , has a unique presentation y = b ℓ b g ι · y ,where y ∈ End(0) is a floating cobordism.
Reflecting cobordisms about the horizontal line, we obtain a classification of elements inHom
TFS (1 , Proposition 2.4.
A morphism y ∈ Hom
T F S (1 , has a unique presentation y = y · ǫb ℓ b g ,where y ∈ End(0) is a floating cobordism.
VALUATING THIN FLAT SURFACES 13
Endomorphism monoid
End(1) . Recall that we continue with a minor abuse of notation,where we denote by 1 the generating object of TFS, also use it as the label for the bottomleft horizontal interval of a cobordism in Hom( n, m ), and use it convenionally as the label forthe first natural number.An element y of End TFS (1) may be one of the two types:1). Horizontal intervals 1 and 1 ′ belong to the same connected component of y .2). Intervals 1 and 1 ′ belong to different connected components of y .Denote by U i the set of elements of type i ∈ { , } , so that(3) End(1) = U ⊔ U . The set U is closed under left and right multiplication by elements of End(1), thus constitutesa 2-sided ideal in this monoid. The set U is a unital submonoid in End(1). These maps U −→ End(1) ←− U upgrade decomposition (3). The monoid U is commutative and naturally decomposes U ∼ = End c (1) × End(0)into the direct product, both terms of which we have already described. The direct productcorresponds to splitting an element of U into the viewable connected component and a floatingcobordism.Likewise, an element y of U splits into a floating cobordism y and a viewable one y . Aviewable element y of U consists of two connected components, one bounding horizontalinteval 1, the other bounding 1 ′ . Such an element can be written as y = b ℓ b g ι · ǫb ℓ b g , with a general y ∈ U given by y = b ℓ b g ι · y · ǫb ℓ b g . Multiplication of two viewable elements as above produces an additional connected compo-nent, see Figure 2.3.3, where by the ( ℓ, g ) coupon we denote the endomorphism b ℓ b g of 1. Remark:
Unlike the monoids End(0), End c (1), and their direct product U , monoid End(1)and its subsemigroup U are not commutative.3. Linearizations of the category
TFSIn this section we work over a field k , but the construction and some results may begeneralized to an arbitrary commutative ring R (or a commutative ring with additional con-ditions, such as being noetherian). A definitive starting reference for recognizable series withcoefficients in commutative rings is Hazewinkel [Ha]. = = PSfrag replacements l , g l , g l , g l , g l , g l , g l , g l , g l , g l , g l + l , g + g Figure 2.3.3.
Product of two viewable elements of U produces a floatingcomponent ǫ b ℓ + ℓ b g + g ι = S ℓ + ℓ +1 ,g + g , in addition to the componentsbounding intervals 1 and 1 ′ .3.1. Categories k
TFS and
VTFS α for recognizable α . Category k TFS . Starting with TFS we can pass to its preadditive closure k TFS. Objectsof k TFS are the same as those of TFS , that is, non-negative integers n ∈ Z + . A morphismin k TFS from n to m is a finite k -linear combination of morphisms from n to m in TFS. Inparticular, Hom k TFS ( n, m ) is a k -vector space with a basis Hom TFS ( n, m ). Composition ofmorphisms is defined in the obvious way.Category k TFS is a k -linear preadditive category. It is also a rigid symmetric monoidalcategory. Power series α . The ring Hom k TFS (0 ,
0) of endomorphisms of the unit object 0 of k TFS isnaturally isomorphic to the monoid algebra of Hom
TFS (0 , . The latter is a free commutativemonoid on generators S ℓ +1 ,g , over all ℓ, g ∈ Z + , so thatHom k TFS (0 , ∼ = k [ S ℓ +1 ,g ] ℓ,g ∈ Z + is the polynomial algebra on countably many generators, parametrized by pairs ( ℓ, g ) of non-negative integers. Homomorphisms of k -algebrasHom k TFS (0 , −→ k are in a bijection with doubly-infinite sequences α = ( α ℓ,g ) ℓ,g ∈ Z + , α ℓ,g ∈ k . The bijection associates to a sequence α the homomorphism, also denoted α ,Hom TFS (0 , ∼ = k [ S ℓ +1 ,g ] ℓ,g ∈ Z + α −→ k , α ( S ℓ +1 ,g ) = α ℓ,g . Sequences α are also in a bijection with multiplicative k -valued evaluations of floating cobor-disms in TFS. These evaluations are maps from the set of floating cobordisms (endomorphismsof object 0) in TFS to k that take disjoint union of cobordisms to the product of evaluations, α ( S ⊔ S ′ ) = α ( S ) · α ( S ′ ) . VALUATING THIN FLAT SURFACES 15
Thus, α is a map of sets α : Z + × Z + −→ k that we can think of a Z + × Z + -matrix with coefficients in k α = α , α , α , α , . . .α , α , α , α , . . .α , α , α , α , . . .α , α , α , α , . . . ... ... ... ... . . . We encode α into power series in two variables T , T :(4) Z α ( T , T ) = X k,g ≥ α k,g T k T g , α = ( α k,g ) k,g ∈ Z + , α k,g ∈ k . A doubly-infinite sequence α can also be thought of as a linear functional on the space ofpolynomials in two variables: α ∈ k [ T , T ] ∗ := Hom k ( k [ T , T ] , k ) . We assume that α is not identically zero (the theory is trivial otherwise). Then ker( α ) ⊂ k [ T , T ] is a codimension one subspace. Category
VTFS α . Given α , we can form the quotient VTFS α of category k TFS by addingthe relation that a floating surface S ℓ +1 ,g of genus g with ℓ + 1 boundary components evaluatesto α ℓ,g ∈ k . Objects of VTFS α are still non-negative integers n . Morphisms from n to m arefinite k -linear combinations of viewable cobordisms from n to m . Composition of cobordismsfrom n to m and from m to k is a cobordism from n to k which may have floating components.These components are removed simultaneously with multiplying the viewable cobordism thatremains by the product of α ℓ,g ’s, for every component S ℓ +1 ,g .The space of homs from n to m in this category has a basis of viewable cobordisms from n to m . Letter V in the notation VTFS α stands for viewable . Recognizable series.
Borrowing terminology from control theory [F, FM], we say that alinear functional or series α is recognizable if ker( α ) contains an ideal I ∈ k [ T , T ] of finitecodimension. Proposition 3.1. α is recognizable iff the power series Z α has the form (5) Z α ( T , T ) = P ( T , T ) Q ( T ) Q ( T ) , where Q ( T ) , Q ( T ) are one-variable polynomials and P ( T , T ) is a two-variable polynomial,all with coefficients in the field k . We assume that Q (0) = 0 , Q (0) = 0, otherwise at least one of these polynomials is notcoprime with P ( T , T ) and either T or T cancels out from the numerator and denominator.With the denominator not zero at T , T = 0 the power series expansion makes sense. Proof.
See [F] for a proof. This result is also mentioned in [FM, Remark 2]. To prove it,assume that α is recognizable. We start with the case when k is algebraically closed. A finite codimension ideal I ⊂ k [ T , T ] necessarily contains a sum,(6) I ⊗ k [ T ] + k [ T ] ⊗ I ⊂ I ⊂ k [ T , T ]for some finite codimension ideals I ⊂ k [ T ] and I ⊂ k [ T ]. To see this, note that the finiteaffine scheme Spec( k [ T , T ] /I ) is supported over finitely many points of the affine plane A . Projecting these points onto the coordinate lines and counting them with multiplicitiesproduces two one-variable polynomials U ( T ) , U ( T ) such that I contains the ideal ( U ( T )) +( U ( T )) of k [ T , T ]. We can now take principal ideals I i = ( U i ( T i )), i = 1 , k [ T ] / ( U ( T )) ⊗ k [ T ] / ( U ( T )) −→ k [ T , T ] /I lifting to the identity map on k [ T , T ]. Existence of such finite codimension ideals I , I overan arbitrary field k follows as well.Hence, recognizable series α has the property that α ( U ( T ) T k T m ) = 0 for any k, m ≥ U ( T ) is a polynomial of some degree r with the lowest degree term u s T s for s ≤ r and write U ( T ) = u r T r + u r − T r − + · · · + u s +1 T s +1 + u s T s , ≤ s ≤ r, u r , u s = 0 , u j ∈ k . Then, for any k, m ≥ u r α r + k,m + u r − α r − k,m + · · · + u s +1 α s + k +1 ,m + u s α s + k,m = 0 . We obtain a similar relation on the coefficients with U and T in place of U and T andvarying the second index. Let us write U ( T ) = v r ′ T r ′ + v r ′ − T r ′ − + · · · + v s ′ +1 T s ′ +1 + v s ′ T s ′ , ≤ s ′ ≤ r ′ , v r ′ , v s ′ = 0 , v j ∈ k . Then, for any k, m ≥ v r ′ α r ′ + k,m + v r ′ − α r ′ − k,m + · · · + v s ′ +1 α s ′ + k +1 ,m + v s α s ′ + k,m = 0 . Consequently, α is eventually recurrent in both T and T directions and its values aredetermined by α i,j with 0 ≤ i < r, ≤ j < r ′ .Consider polynomials b Q ( T ) = T r U ( T − ) = u s T r + u s +1 T r − + u s +2 T r − + · · · + u r T r − s , b Q ( T ) = T r ′ U ( T − ) = v s ′ T r ′ + v s ′ +1 T r ′ − + v s ′ +2 T r ′ − + · · · + v r ′ T r ′ − s ′ . Form the product b P ( T , T ) := Z α ( T , T ) b Q ( T ) b Q ( T ) = X i,j ≥ w i,j T i T j and examine coefficients of its power series expansion. Formulas (7), (8) show that w i,j = 0if i ≥ r of j ≥ r ′ . Therefore, b P ( T , T ) is a polynomial with T , T degrees bounded by r − r ′ −
1, respectively. We can then form the quotient b P ( T , T ) b Q ( T ) b Q ( T )The numerator and denominator may share common factors, including T r − s T r ′ − s ′ . Aftercanceling those out, we arrive at the presentation (5) for Z α ( T , T ). VALUATING THIN FLAT SURFACES 17
We leave the proof of the opposite implication of the proposition to the reader or referto [F].Note that the proof works for any finite number of variables T , . . . , T c , not only for two. (cid:3) The condition that α is recognizable can also be expressed via its Hankel matrix H α . Thelatter matrix has rows and columns enumerated by pairs ( m, k ) ∈ Z + × Z + , equivalently bythe monomial basis elements T m T k . The (( m , k ) , ( m , k ))-entry of H α is α m + m ,k + k .The following result is proved in [F]. Proposition 3.2.
The series α is recognizable iff the Hankel matrix H α has finite rank. Note that H α has finite rank iff there exists M such that any M × M minor of H α hasdeterminant zero. The rank is M − M − × ( M −
1) minor witha non-zero determinant.3.2.
Skein category
STFS α . Recognizable series and commutative Frobenius algebras.
Assume that α is recognizable.Among all finite-codimension ideals I ⊂ ker( α ) there is a unique largest ideal I α , given by thesum over all such I . Equivalently, it can be described as follows. There is a homomorphismof k [ T , T ]-modules(9) h : k [ T , T ] −→ k [ T , T ] ∗ given by sending 1 to α and z ∈ k [ T , T ] to zα ∈ k [ T , T ] ∗ with ( zα )( f ) = α ( zf ). The ideal I α is the kernel of h .Notice that α descends to a nondegenerate bilinear form on the quotient algebra(10) A α := k [ T , T ] /I α . In particular, A α is a commutative Frobenius algebra on two generators T , T with a nonde-generate trace form α .Vice versa, assume given a commutative Frobenius k -algebra B with the nondegeneratetrace form β : B −→ k and a pair of generators g , g . To such data we can associate asurjective homomorphism ψ : k [ T , T ] −→ B, ψ ( T i ) = g i , i = 1 , , the trace map α = β ◦ ψ on k [ T , T ] given by composing ψ with β , and recognizable series α β = X ℓ,g ≥ β ( g ℓ g g ) T ℓ T g . Thus, recognizable power series on k [ T , T ] are classified by isomorphism classes of data( B, g , g , β ): a commutative Frobenius algebra B generated by g , g ∈ B and a non-degenerate trace β . Category
STFS α . We can now define the category STFS α (where first S stands for “skein”)to be a quotient of VTFS α by the skein relations in the ideal I α . The category STFS α has thesame objects as all the other cobordism categories we’ve considered so far, that is, nonnegative integers n . Morphisms from n to m are k -linear combinations of viewable cobordisms modulothe relations in I α . Precisely, let(11) p ( T , T ) = X i,j p i,j T i T j ∈ I α be a polynomial in the ideal I α . Given a viewable cobordism x choose a component c of x anddenote by x c ( i, j ) the cobordism given by inserting i holes and adding j handles to x at thecomponent c . We now mod out the hom space Hom VTFS α ( n, m ), which is a k -vector spacewith a basis of all viewable cobordisms from n to m , by the relations X i,j p i,j x c ( i, j ) = 0 , one for each component c of x , over all viewable cobordisms x .It is easy to see that these “skein” relations are compatible with α -evaluation of floatingcobordisms. Namely, if instead of a viewable cobordism x we consider a floating cobordism y and choose a component c of y to add holes and handles, resulting in cobordisms y c ( i, j ),then X i,j p i,j α ( y c ( i, j )) = 0 . This compatibility condition, immediate from our definition of I α as the kernel of the modulemap (9), ensures non-triviality of this quotient and its compatibility with the composition ofmorphisms.Viewing VTFS α as a tensor category, it is enough to write down corresponding relations onhoms from 0 to 1 and then mod out by them in the tensor category (by gluing each term inthe resulting linear combination of products of holes and handles on a disk to any componentalong a segment on its side boundary). Choose a generating set v , . . . , v r of I α viewed as k [ T , T ]-module. Specializing to a single basis element v j , assume that it is given by thepolynomial p on the right hand side of (11). Form the element b ( v j ) := X i,j p i,j b i b j ι ∈ Hom(0 , . The skein category STFS α can be defined as the quotient of VTFS α by the tensor idealgenerated by elements b ( v ) , . . . , b ( v r ). Figure 3.2.1 shows an example of an element b ( v ). Figure 3.2.1. b ( v ), for v = 3 T T − T + T T ; handles are shown schematically. VALUATING THIN FLAT SURFACES 19
Remark:
For a recognizable series α there are unique minimal degree monic polynomials q α, , q α, , q α, ( x ) = x t + a t − x t − + · · · + a , q α ( x ) = x t ′ + a ′ t ′ − x t ′ − + · · · + a ′ , such that q α, ( T ) ∈ I α , q α, ( T ) ∈ I α . Among skein relations associated to elements of I α in STFS α there is a polynomial relationthat utilizes only adding holes to a component of the cobordism. This relation is given by thepolynomial q α, ( T ): b t + a t − b t − + · · · + a = 0 , describing an equality in the ring of endomorphisms of object 1 of STFS α , where b is the hole cobordism, see Figure 2.3.2. Equivalently, it can be rewritten as a relation in Hom(0 , b t + a t − b t − + · · · + a ) ι = 0 , Likewise, there is a skein relation on cobordisms that differ only by genus of a given com-ponent. The relation is given by the polynomial q α, ( T ): b t ′ + a ′ t ′ − b t ′ − + · · · + a ′ = 0 , where b is the handle morphism, see Figure 2.3.2. Minimal viewable cobordisms, B α -companions, and bases of hom spaces of STFS α . Consider a connected viewable cobordism x . We say that x is minimal if it has genus zeroand no holes , that is, each boundary component of x contains at least one horizontal segment.Equivalently x is minimal if it cannot be factored into x ′ b x ′′ or x ′ b x ′′ for some morphisms x ′ , x ′′ . Note that if such a factorization exists, then there exists one with x ′′ the identitycobordism and one with x ′ the identity cobordism. Any viewable connected cobordism x from n to m with m > b i b j ⊗ id m − ) y for some minimal y and, if n > y ( b i b j ⊗ id n − ) for the same y , see Figure 3.2.2. If one of n or m is zero, only one of thesetwo presentations exist. Equivalently, a connected viewable cobordism x is minimal if it is = = = PSfrag replacements x yyy b i b j b i b j b i b j Figure 3.2.2.
Factorization of a connected cobordism x into a coupon and aminimal cobordism is shown schematically. Since y is connected, b i b j couponcan be moved to any leg of y . handless and has no holes.A viewable cobordism y is called minimal if each connected component of y is minimal. Aviewable cobordism x factors into a product of a minimal cobordism and “coupons” carrying powers of b , b , one for each connected component of x . That is, for each connected compo-nent c of x count holes and handles on it and then remove them to get a minimal connectedcomponent c ′ . The original component can be recovered by inserting holes and handles backanywhere along c ′ . For instance, they may be inserted at one of its top or bottom legs bymultiplying c ′ by the corresponding powers of b and b there.To any viewable x we can associate its minimal counterpart y by removing holes and handlesfrom each connected component of x . Given y , we can recover x by multiplying by appropriatepowers of b and b at horizontal intervals for different components of y .Denote by M ( n, m ) the set of minimal viewable cobordisms from n to m . Proposition 3.3. M ( n, m ) is a finite set.Proof. From our classification of morphisms in TFS it is clear that minimal cobordisms from n to m are in a bijection with partitions λ of the set N mn of n + m horizontal intervals, togetherwith a choice of a partition µ i of each part λ i of λ and a cyclic order on each part of µ i . (cid:3) Recall finite codimension ideal I α (the syntactic ideal) associated with recognizable series α . Let d α = dim( k [ T , T ] /I α ) . Choose a set of pairs P α = { ( i t , j t ) } d α t =1 , i t , j t ∈ Z + such that monomials T i t T j t constitute a basis of the algebra k [ T , T ] /I α . Denote this basisby B α . It is well-known [MiS] that a basis can always be choosen so that the exponents ( i t , j t )of the monomials, when placed into corresponding points of the square lattice, constitute apartition of d α , but we do not need this result here.Choose a minimal cobordism y and assign an element v c ∈ B α to each connected component c of y . This assignment gives rise to a cobordism x obtained from y by inserting cobordisms b ( v c ) at all components c of y . For v c = T i T j we add i holes and j handles to the component c or, equivalently, multiply it at one of its horizontal boundary intervals by b i b j .In this way to y ∈ M ( n, m ) there are assigned d rα cobordisms x , where r is the number ofcomponents of y . These x are called B α -companions of y . Denote the set of such x by B α ( y ) . Proposition 3.4.
Elements of sets B α ( y ) , over all y ∈ M ( n, m ) , constitute a basis ofHom STFS α ( n, m ) . In other words, to get a basis of homs from n to m in the skein category STFS α we take allminimal cobordisms y from n to m and insert a basis element from B α into each componentof y . Proof.
The proposition follows immediately from our construction of STFS α . One needs tocheck consistency, that our rules do not force additional relations when composing cobordisms.This is straightforward. (cid:3) Corollary 3.5.
Hom spaces in the category
STFS α are finite dimensional.Remark: In a seeming discrepancy, object 1 of the category TFS is a symmetric Frobeniusobject but not a commutative Frobenius object, see Figure 2.1.6 left, since the multiplicationmap 1 ⊗ −→ ⊗
1. Yet, in thecategory STFS α the state space Hom(0 ,
1) of the interval is a commutative Frobenius algebra
VALUATING THIN FLAT SURFACES 21 A α , defined in (10), with the multiplication on Hom(0 ,
1) given by the thin flat pants cobordismin Figure 3.2.3 left. This is explained by the observation that the thin flat pants multiplicationis commutative in the categories we consider, including TFS and VTFS α and STFS α . Indeed,viewable morphisms from 0 to 1 in TFS have the form b n b m ι , and the product of two suchmorphisms does not depend on their order, see Figure 3.2.3 right. Adding floating components(or passing to linear combinations, or taking quotients) does not break commutativity.Later, in Section 5.1, in a similar situation we also denote b n b m ι by b n b m . = = PSfrag replacements b n b n b n b n b m b m b m b m b n + m b n + m Figure 3.2.3.
Left: thin flat pants cobordism from 1 ⊗ , Quotient by negligible morphisms and Karoubi envelopes.
Category
TFS α . Consider the ideal J α ⊂ STFS α of negligible morphisms, relative to thetrace form tr α associated with α , and form the quotient categoryTFS α := STFS α /J α . The trace form is given on a cobordism x from n to n by closing it via n annuli connecting n top with n bottom circles of the horizontal boundary of x into a floating cobordism b x andapplying α , tr α ( x ) := α ( b x ) . This operation is depicted in Figure 3.3.1.PSfrag replacements x x b x = α α ( b x ) Figure 3.3.1.
The trace map: closing endomorphism x of n into b x and ap-plying α .A morphism y ∈ Hom( n, m ) is called negligible if tr α ( zy ) = 0 for any morphism z ∈ Hom( m, n ). Negligible morphisms constitute a two-sided ideal in the pre-additive categorySTFS α . The quotient category TFS α has finite-dimensional hom spaces, as does STFS α (recall that α is recognizable). The trace is nondegenerate on TFS α and defines perfect bilinear pairingsHom( n, m ) ⊗ Hom( m, n ) −→ k on its hom spaces. We may call TFS α the gligible quotient of STFS α , having modded out bythe ideal of negligible morphisms.Let us go back to the category TFS and its linear version k TFS. Fix the number n ofintervals and consider the vector space V n with a basis of all viewable tf-surfaces with thatboundary, that is viewable cobordisms in TFS from 0 to n . Given α , define a bilinear formon V n via its values on pairs of basis vectors:( x, y ) = α ( yx ) ∈ k , where y is given by reflecting y about a horizontal line to get a cobordism from n to 0, and yx is a floating cobordism from 0 to 0 given by composing y and x . This bilinear form on V n is symmetric. Define A α ( n ) as the quotient of V n by the kernel of this bilinear form. Thenthere is a canonical isomorphism A α ( n ) ∼ = Hom TFS α (0 , n )as well as isomorphisms A α ( n + m ) ∼ = Hom TFS α (0 , n + m ) ∼ = Hom TFS α ( m, n )given by moving m invervals from bottom to top via the duality morphism.The symmetric group S n acts by permutation cobordisms on A α ( n ). Furthermore, at eachcircle there is an action of the endomorphism algebra End(1) = End TFS α (1). Consequently,the cross-product algebra k S n ⋉ End(1) ⊗ k acts on A α ( n ).Multiplication maps A α ( n ) ⊗ A α ( m ) −→ A α ( n + m )turn the direct sum A α := M n ≥ A α ( n )into a unital commutative associative graded algebra, with A α (0) ∼ = k . All of this data,including the power series P n ≥ dim A α ( n ) z n encoding dimensions of A α ( n ), are invariantsof recognizable series α .In the diagram of five categories and four functorsTFS −→ k TFS −→ VTFS α −→ STFS α −→ TFS α one can get from k TFS to TFS α in one step, bypassing VTFS α and STFS α , by taking theideal of negligible morphisms in k TFS (for essentially the same trace map, shown in Fig-ure 3.3.1) and modding out by it. It is convenient to introduce those intermediate categories,though. For instance, STFS α already has finite-dimensional hom spaces and allows to definethe analogue of the Deligne category in our case. The Deligne category
DTFS α and its gligible quotient DTFS α . The skein category STFS α is a rigid symmetric monoidal k -linear category with objects n ∈ Z + and finite-dimensional VALUATING THIN FLAT SURFACES 23 hom spaces. We form the additive Karoubi closureDTFS α := Kar(STFS ⊕ α )by allowing formal finite direct sums of objects in STFS, extending morphisms correspond-ingly, and then adding idempotents to get a Karoubi-closed category. Category DTFS α playsthe role of the Deligne category in our construction.In the Deligne category DTFS α endomorphisms of an object ( n, e ), where e is an idempotentendomorphism of n , inherit the trace map tr α into the ground field. Consequently, categoryDTFS α also has a two-sided ideal of negligible morphisms J D,α . Taking the quotient by thisideal DTFS α := DTFS α /J D,α gives us an idempotent-complete category with non-degenerate symmetric bilinear formson hom spaces Hom(0 , ( n, e )), where ( n, e ) is an object as above, and more generally non-degenerate bilinear pairings on hom spacesHom(( n, e ) , ( m, e ′ )) ⊗ Hom(( m, e ′ ) , ( n, e )) −→ k where e ′ is an idempotent endomorphism of object m . Due to the symmetry between homsgiven by the contravariant involution on all categories that we have considered so far (reflectionabout a horizontal line), the above bilinear pairings can be converted into non-degeneratesymmetric bilinear forms on Hom(( n, e ) , ( m, e ′ )) in DTFS α .Category DTFS α is also equivalent to the additive Karoubi closure of the category TFS α ,see the commutative square in (12).3.4. Summary of the categories and functors.
Below is a summary for each categorythat has been considered. • TFS: the category of thin flat surfaces (tf-surfaces). Its objects are non-negativeintegers and morphisms are thin flat surfaces. • k TFS: this category has the same objects as TFS; its morphisms are formal finite k -linear combinations of morphisms in TFS. • VTFS α : in this quotient category of k TFS we reduce morphisms to linear combina-tions of viewable cobordisms. Floating connected components are removed by evalu-ating them via α . • STFS α : to define this category, specialize to recognizable α and add skein relations bymodding out by elements of the ideal I α in k [ T , T ] along each connected componentof a surface ( T is a hole, T a handle). Hom spaces in this category are finite-dimensional. • TFS α : the quotient of STFS α by the ideal J α of negligible morphisms. This categoryis also equivalent (even isomorphic) to the quotients of k TFS and VTFS α by thecorresponding ideals of negligible morphisms in them. The trace pairing in TFS α between Hom( n, m ) and Hom( m, n ) is perfect. • DTFS α : it is the analogue of the Deligne category obtained from STFS α by allowingfinite direct sums of objects and then adding idempotents as objects to get a Karoubi-closed category. • DTFS α : the quotient of DTFS α by the two-sided ideal of negligible morphisms. Thiscategory is equivalent to the additive Karoubi closure of TFS α and sits in the bottomright corner of the commutative square below. We arrange these categories and functors, when α is recognizable, into the following dia-gram:(12) TFS −−−−→ k TFS −−−−→
VTFS α −−−−→ STFS α −−−−→ DTFS α y y TFS α −−−−→ DTFS α All seven categories are rigid symmetric monoidal. All but the leftmost category TFS are k -linear. Except for the two categories on the far right, the objects of each category are non-negative integers. The four categories on the right all have finite-dimensional hom spaces.The two categories on the far right are additive and Karoubi-closed. The four categories inthe middle of the diagram are pre-additive but not additive.The arrows show functors between these categories considered in the paper. The square iscommutative. An analogous diagram of functors and categories can be found in [KS] for thecategory of oriented 2D cobordisms in place of TFS.It is possible to go directly from k TFS to TFS α by modding out by the ideal of negligiblemorphisms in the former category. We found it convenient to get to this quotient in severalsteps, introducing categories VTFS α and STFS α along the way. Remark:
For possible future use, it may be convenient to relabel the categories above usingshorter strings. For instance, writing S (for “surfaces”) in place of TFS we can rename thecategories as follows:(13) S −−−−→ k S −−−−→ VS α −−−−→ SS α −−−−→ DS α y y S α −−−−→ DS α For convenience we wrote below short reminders of what these categories are:(14) cobordisms / / k -linear / / viewable / / skein (cid:15) (cid:15) / / Deligne (Karoubian) (cid:15) (cid:15) gligible / / gligible and KaroubianIf α is not recognizable, we can still define categories VTFS α , TFS α and DTFS α (in thestreamlined notation, categories VS α , S α and DS α ), but it is not clear whether these cate-gories may be interesting for some such α .4. Hilbert scheme and recognizable series
Recognizable series and points on the dual tautological bundle.
Recognizable series α givesrise to the ideal I α in k [ T , T ] of finite codimension k = d α and the quotient algebra A α bythis ideal, see formula (10) in Section 3.2. This algebra is commutative Frobenius and comeswith two generators T , T and a non-degenerate trace. The ideal I α describes a point in the VALUATING THIN FLAT SURFACES 25
Hilbert scheme of codimension k ideals in A = Spec k [ T , T ], where k = d α = dim A α . Let us specialize to the ground field k = C . Denote by Rec k the set of recognizable serieswith the syntactic ideal I α of codimension k and refer to α ∈ Rec k as a recognizable series ofcodimension k . Let also Rec := G k ≥ Rec k , Rec ≤ n := G k ≤ n Rec k . Consider the Hilbert scheme H k = Hilb k ( C ) of k points in C or, equivalently, the schemeof codimension k ideals in C [ T , T ], see [N].Denote by T k the tautological bundle over H k whose fiber over the point associated to anideal I of codimension k is the space A I = C [ T , T ] /I . Points of the dual bundle T ∨ k above apoint I ∈ H k describe elements of A ∗ I = Hom C ( A I , C ), that is, linear functionals on A I . Let π : T ∨ k −→ H k be the projection of the bundle onto its base. For a point p ∈ T ∨ k the element π ( p ) ∈ H k is the projection of p onto the base of the bundle, and we denote by I π ( p ) the correspondingcodimension k ideal of C [ T , T ].The point p also defines a linear functional α p on A π ( p ) := C [ T , T ] /I π ( p ) , α p : A π ( p ) −→ C , associated to p . This functional lifts to a functional on C [ T , T ], which is recognizable,contains I π ( p ) in its kernel, and has codimension at most k . The latter functional (equivalently,recognizable power series) is also denoted α p .This functional has the associated ideal I p = I α p ⊂ C [ T , T ] of finite codimension, thelargest ideal in the kernel of functional α p on C [ T , T ]. There is an inclusion of ideals I π ( p ) ⊂ I p . For a generic point p on T ∨ k this inclusion is the equality I π ( p ) = I p , but for some points p theinclusion is proper.Another way to describe the ideal I p is to consider the symmetric bilinear form ( , ) p on A π ( p ) given by ( x, y ) p := α p ( xy ) , x, y ∈ A π ( p ) . The kernel of the form ( , ) p is an ideal I ′ p in A π ( p ) that lifts to the above ideal I p in C [ T , T ],and there is an isomorphism I ′ p ∼ = I p /I π ( p ) . The inclusion I π ( p ) ⊂ I p is proper precisely when I ′ p is a nonzero ideal, that is, when the bilinear form ( , ) p is degenerate.These ideals are shown in the diagram below, where the two squares on the left are pull-backs. The bottom sequence is short exact, and the top row becomes exact upon replacing I π ( p ) by 0. I π ( p ) (cid:31) (cid:127) / / (cid:15) (cid:15) I p (cid:15) (cid:15) (cid:31) (cid:127) / / C [ T , T ] / / / / (cid:15) (cid:15) A p / / (cid:15) (cid:15) (cid:31) (cid:127) / / I ′ p / / A π ( p ) / / / / A p / / Denote by D k the subset of T ∨ k that consists of points p such that the inclusion I π ( p ) ⊂ I p is proper: D k := { p ∈ T ∨ k | I π ( p ) = I p } . Recognizable power series α p for p ∈ T ∨ k has codimension k (in our notations, α p ∈ Rec k )precisely when p ∈ T ∨ k \ D k .If p ∈ D k so that codim( I p ) = m < k = codim( I π ( p ) ) , then recognizable power series α p has codimension m strictly less than k and α p ∈ Rec m . Forexample, if p ∈ H k ⊂ T ∨ k is a point in the zero section of T ∨ k , so that the linear map α p isidentically zero, the ideal I p = C [ T , T ] has zero codimension and m = 0. A mild confusionexists in our notations in this case (and in this case only), for then p = π ( p ).Going the other way, to a recognizable series α with the associated ideal I α of codimension d α = k as above we associate a point p α of T ∨ k . It is the point in the fiber of T ∨ k over theideal I α which describes functional α on C [ T , T ] and the induced functional on the quotientalgebra A α = A I α .The above discussion implies the proposition below. Proposition 4.1.
Assigning p α to α ∈ Rec k and α p to p ∈ T ∨ k \ D k establishes a bijection Rec k ∼ = T ∨ k \ D k . In particular, p α p = p and α p α = α for p and α as in the proposition, so the two assignmentsare mutually-inverse bijections. (cid:3) Note that the two ideals coincide, I π ( p ) = I p , precisely when α p is a nondegenerate trace mapon A π ( p ) . In particular, in this case A π ( p ) is Frobenius. We obtain the following statement. Proposition 4.2.
Points p ∈ T ∨ k \ D k classify isomorphism classes of data ( A, ǫ, t , t ) : acommutative Frobenius algebra A over C of dimension k with a non-degenerate trace ǫ andgenerators t , t of A . Not every commutative Frobenius algebra can be generated by two elements, of course.Taking codimension m ≤ k of I p into account, one gets the following statement. Proposition 4.3.
Associating α p to p ∈ T ∨ k gives a surjective map T ∨ k −→ k G m =0 Rec m . Restricting this map to D k gives a surjective map D k −→ k − G m =0 Rec m , while on the complement to D k this map is the bijection in Proposition 4.1. Example:
The set Rec is a single point corresponding to the zero series α , α i,j = 0 , i, j ∈ Z + . The ideal for this point is the entire algebra C [ T , T ]. Points of Rec correspond tohyperplanes (codimension one subspaces) that are ideals J = ( T − λ , T − λ ) together with VALUATING THIN FLAT SURFACES 27 a nonzero functional on C ∼ = C [ T , T /J , determined by its value λ on 1. Consequently, wecan identify Rec ∼ = C × C × C × by sending a point in Rec to the triple ( λ , λ , λ ). Set-theoretic divisor D k . Quasi-projective variety H k admits an open cover by affine sets U λ , over all partitions λ of k, see Theorem 18.4 in [MiS, Section 18.4], for example. Placepartition λ in the corner of the first quadrant of the plane so that it covers nodes ( i, j ) of thesquare lattice with 0 ≤ i < λ j +1 . In particular, it covers λ nodes on the x -axis.Let T λ be the set of monomials T i T j with ( i, j ) ∈ λ (in particular, | T λ | = k ) and T ′ λ be theset of complementary monomials, for pairs ( i, j ) ∈ Z + × Z + \ λ . Open set U λ ⊂ H k consists ofideals I such that monomials in T λ descend to a basis of A I = C [ T , T ] /I , see [MiS, Section18.4] for details.The vector bundle T ∨ k −→ H k can be trivialized over U λ , being naturally isomorphic tothe trivial bundle of functions on the set T λ . A functional p on C [ T , T ] /I π ( p ) is determinedby its values on the basis elements t ∈ T λ of this quotient space.To describe the points p ∈ T ∨ k with π ( p ) ∈ U λ consider an arbitrary linear functional α ∈ ( C T λ ) ∗ , given by its values α ( T i T j ) ∈ C , for T i T j ∈ T λ , and an ideal I ∈ U λ . Such pair ( α, I ) trivializes a pair ( p, π ( p )) with π ( p ) ∈ U λ . For a pair u, v ∈ T λ take the product uv , view it as an element of A I = C [ T , T ] /I , and then write it asa linear combination of elements in T λ , allowing to apply α to it explicitly.Consider a matrix M α where rows and columns are labelled by elements of T λ and put α ( uv ) as the entry at the intersection of row u and column v . Proposition 4.4.
Point p with π ( p ) ∈ U λ is in the subset D k iff det ( M α ) = 0 .Proof. Matrix M α is the Gram or Hankel matrix of the bilinear form ( x, y ) = α ( xy ) on theassociative algebra A I in the basis T λ . A bilinear form on a finite-dimensional algebra B given by the composition of the multiplication with a fixed linear functional on B is non-degenerate exactly when its Hankel matrix with respect to some (equivalently, any) basis isnon-degenerate, that is, has a non-zero determinant. (cid:3) Condition that det ( M α ) = 0 is locally a codimension one condition (given by a singleequation), unless the determinant is identically zero on points ( p, π ( p )) with π ( p ) on someirreducible component of the open subset U λ of the Hilbert scheme. To see that the lattercase does not happen, observe that a “generic” point I on the Hilbert scheme H k correspondsto a semisimple quotient (no nilpotent elements in C [ T , T ] /I ), with the quotient algebraisomorphic to the product of k fields, C [ T , T ] /I ∼ = C × C × · · · × C . On this quotient an open subset of linear functionals are non-degenerate, with the associatedbilinear forms having trivial kernels. Indeed, a functional α on the algebra Q ki =1 C is non-degenerate iff each of its k coefficients is non-zero.These observations imply the following result. Proposition 4.5. D k is a set-theoretic divisor on the variety T ∨ k . It is straightforward to check that D k comes from an actual divisor on T ∨ k . For a finite-dimensional C -vector space V define a one-dimensional vector space det V := (Λ top V ) ∨ = Λ top ( V ∨ ) . The determinant det b α of a bilinear form b α : V ⊗ V → C is an element det b α ∈ ( det V ) ⊗ definedas the determinant of the matrix of b α . Namely, if e , . . . , e k is a basis of V and e , . . . , e k isthe dual basis in V ∨ , then e ∧ · · · ∧ e k is a basis in the one-dimensional space det V and det b α := det || b α ( e i , e j ) || ( e ∧ · · · ∧ e k ) ⊗ . A point p ∈ T ∨ k defines a symmetric bilinear form b α p ( x, y ) := α p ( xy ) on the fiber T π ( p ) = I π ( p ) of the tautological bundle. The determinant of this form is an element of ( det T π ( p ) ) ⊗ .Hence the pullback line bundle π ∗ (cid:0) ( det T ) ⊗ (cid:1) −→ T ∨ k over T ∨ k has a canonical section σ det given by σ det ( p ) := det b α p . The set D k is the divisor ofzeroes of this section. Corollary 4.6. D k is the divisor of zeros of the section σ det . Each point of T ∨ k \ D k gives rise to recognizable series α in two variables and to the corre-sponding rigid symmetric monoidal categories, as discussed in the Section 3 and summarizedin Section 3.4, including category TFS α , the Deligne category DTFS α and its gligible quo-tient DTFS α . It may be interesting to understand these categories for various α ’s, includingfinding the analogue of the classification result from [KKO] on when the category DTFS α issemisimple. 5. Modifications
Adding closed surfaces.
Category TFS can be enlarged to a category C with morphisms – oriented 2D cobordisms(surfaces) with corners between oriented 1D manifolds with corners. Extensions of 2D TQFTsto this category have been widely studied [MS, LP, C, SP]. An oriented 1D manifold withcorners is diffeomorphic to a disjoint union of finitely-many oriented intervals and circles. Weadopt a minimalist approach and choose one manifold for each such diffeomorphism class.Consequently, objects of C are pairs n = ( n , n ) of non-negative integers, and an object n is represented by a fixed disjoint union W ( n ) = W ( n , n ) of n intervals and n cir-cles. Morphisms from n to m = ( m , m ) are compact oriented 2D cobordisms M , possiblywith corners, with both horizontal and side boundary and corners where these two differentboundary types meet: ∂M = ∂ h M ∪ ∂ v M, ∂ h M = W ( m , m ) ⊔ ( − W ( n , n )) . Cobordisms that are diffeomorphic rel boundary define the same morphisms. Category C contains TFS as a subcategory. C is a rigid symmetric monoidal category, with self-dual objects. The unit object is theempty one-manifold W (0 , S ℓ,g with ℓ boundary components and of genus g , one for each pair ( ℓ, g ), ℓ, g ∈ Z + . The difference VALUATING THIN FLAT SURFACES 29 with endomorphisms of the unit object of TFS is that in C closed surfaces are allowed, whichcorresponds to ℓ = 0 and surfaces S ,g , over all g ∈ Z + .Multiplicative evaluations β of endomorphisms of the unit object are again encoded by apower series(15) e Z β ( T , T ) = X k,g ≥ β k,g T k T g , β = ( β k,g ) k,g ∈ Z + , α k,g ∈ k , with the first index shifted by 1 compared to evaluations for TFS. We changed the labelfrom α in evaluations in TFS to β in C to make it easier to compare evaluations in these twocategories. Now the coefficient β k,g = β ( S k,g )is the evaluation of connected genus g surface with k boundary components rather than with k + 1 components as in the TFS case, see earlier.To relate these two power series encodings, in formulas (1) and (4) versus (15), start with Z α ( T , T ) as in (4) and also form a one-variable power series Z γ ( T ) = X k ≥ γ k T k , γ k ∈ k . To the pair ( Z α , Z γ ) assign the series(16) e Z β ( T , T ) = T Z α ( T , T ) + Z γ ( T ) . Adding coefficients of Z γ to the data provided by Z α precisely means that we now includeevaluations of closed surfaces, via coefficients γ k (for a closed surface genus k ). The scalingfactor T in the formula is needed to match the discrepancy in the evaluation conventions inthe two categories TFS and C . Formula (16) gives a bijection between series encoded by β and those encoded by ( α, γ ). Starting from e Z β , one recovers Z α and Z γ as Z γ ( T ) = e Z β (0 , T ) Z α ( T , T ) = ( e Z β ( T , T ) − e Z β (0 , T )) /T . From C pass to its k -linearization k C by allowing finite k -linear combinations of morphismsin C . Given series β , we can define analogues of categories VTFS α and TFS α in (12). Denotethese new categories by VC β and C β : • In VC β one evaluates floating components to elements of k via β . A connected com-ponent is floating if it has no horizontal boundary. • To form category C β we quotient category VC β (alternatively, category k C ) by thetwo-sided ideal of negligible morphisms , defined in the same way as for TFS.We say that evaluation β (or series e Z β ) is recognizable if category C β have finite-dimensionalhom spaces. Proposition 5.1. β is recognizable iff the power series e Z β has the form (17) e Z β ( T , T ) = e P ( T , T ) e Q ( T ) e Q ( T ) , where e Q ( T ) , e Q ( T ) are one-variable polynomials and e P ( T , T ) is a two-variable polynomial,all with coefficients in the field k . It is easy to see that series β is recognizable iff hom spacesHom( , (1 , , (0 , C β are finite-dimensional. These are the hom spaces from the empty 1-manifold W (0 , ) to an interval W (1 ,
0) and a circle W (0 , (cid:3) Corollary 5.2.
Series β is recognizable iff the corresponding series α and γ are both recog-nizable. Note that, when α and γ are recognizable, their rational function presentation may havevery different denominators, Z α ( T , T ) = P ( T , T ) Q ( T ) Q ( T ) , Z γ ( T ) = P γ ( T ) Q γ ( T ) , so that e Z β ( T , T ) = T P ( T , T ) Q ( T ) Q ( T ) + P γ ( T ) Q γ ( T )= T P ( T , T ) Q γ ( T ) + Q ( T ) Q ( T ) P γ ( T ) Q ( T ) Q ( T ) Q γ ( T ) . For generic polynomials, there are no cancellations and e Q ( T ) = Q ( T ) , e Q ( T ) = Q ( T ) Q γ ( T )are the denominators in the minimal rational presentation (17) for e Z β .For recognizable β , the state spaces A β (1 ,
0) := Hom C β ( , (1 , , A β (0 ,
1) := Hom C β ( , (0 , , of homs from the unit object = (0 ,
0) to the interval and the circle objects, respectively,are both commutative Frobenius algebras. Annuli, viewed as morphisms between (1 ,
0) and(0 , δ : A β (1 , −→ A β (0 , , δ : A β (0 , −→ A β (1 , hole and handle endomorphisms b , b of the interval and c , c of the circle,respectively, in Figure 5.1.2 top.Multiplications in algebras A β (1 ,
0) and A β (0 ,
1) are given by pants and flat pants cobor-disms, see Figure 5.1.3, where the cobordisms for the unit and trace morphisms on A β (1 , A β (0 ,
1) are shown as well.Take endomorphisms b , b , c , c of the interval and circle and cap them off at the bot-tom with the unit morphisms ι and ι ′ for the interval and circle (see Figure 5.1.3) to getelements b = b ι, b = b ι in A β (1 ,
0) and elements c = c ι ′ , c = c ι ′ in A β (0 , VALUATING THIN FLAT SURFACES 31 (0,1)(1,0) (1,0)(0,1)
PSfrag replacements δ δ Figure 5.1.1.
Maps δ and δ .PSfrag replacements b b c c b b c c Figure 5.1.2.
Endomorphisms b , b of the interval, endomorphisms c , c ofthe circle and corrresponding elements b , b of A β (1 ,
0) = Hom( , (1 , c , c ∈ A β (0 ,
1) = Hom( , (0 , m ι ǫ m ′ ι ′ ǫ ′ Figure 5.1.3.
Flat pants and pants cobordisms, together with the other struc-ture maps ι, ǫ and ι ′ , ǫ ′ (units ι, ι ′ and counits ǫ, ǫ ′ ) of commutative Frobeniusalgebras A β (1 ,
0) and A β (0 , C , and the “interval” Frobenius algebra A β (1 ,
0) isgenerated by commuting elements b , b (the hole and handle elements). Likewise, the “circle”Frobenius algebra A β (0 ,
1) is generated by commuting hole and handle elements c and c . Endomorphisms b , b of the interval in the category C are different (endomorphism b isalso shown in Figure 2.3.2), but they induce the same map on A β (1 , x ∈ A β (1 ,
0) can be written as a linear combination of monomials b n b m , with b actingby b b n b m = b n +11 b m = b b n b m . = = PSfrag replacements xx b b Figure 5.1.4. b x = b x for any x ∈ A β (1 , b = b as End((1 , C (and in C β , in general).Trace maps(18) ǫ : A β (0 , −→ k , ǫ ′ : A β (1 , −→ k , given by capping off the interval with a disk, respectively the circle with a cap, turn thesetwo commutative algebras into Frobenius algebras (for recognizable β ).Compositions of δ and δ are endomorphisms of the interval and the circle in C (and in C β ) and satisfy δ δ = b , δ δ = c ,δ c = b δ , δ c = b δ ,δ b = c δ , δ b = c δ . In particular, maps δ , δ intertwine the hole endomorphisms b , c of the interval and thecircle. They also intertwine the handle endomorphisms b , c of the interval and the circle.Their two compositions produce the hole endomorphisms of the interval and the circle.The map δ is a surjective unital homomorphism of commutative algebras, while the map δ is an injective homomorphism of cocommutative coalgebras, with comultiplications given bythe dual of the multiplications on these Frobenius algebras. In particular, δ respects traces,in the sense that ǫ ′ δ = ǫ. A recognizable power series β is encoded by a commutative Frobenius algebra (the statespace of a circle) A β (0 ,
1) with generators c , c and non-degenerate trace map ǫ ′ such that(19) β ℓ,g = ǫ ′ ( c ℓ c g ) , ℓ, g ∈ Z + . Further unwrapping this data, to a recognizable power series β we can associate • Two commutative Frobenius algebras A (1 ,
0) = A β (1 ,
0) and A (0 ,
1) = A β (0 ,
1) withgenerators b , b and c , c , respectively (hole and handle elements). VALUATING THIN FLAT SURFACES 33 • Non-degenerate traces ǫ and ǫ ′ as in (18), subject to (19) and β ℓ +1 ,g = ǫ ( b ℓ b g ) , ℓ, g ∈ Z + . • Linear maps δ , δ : A β (1 , A β (0 , δ δ that intertwine the action of handle elements b and c . The hole elements are givenby b = δ δ (1) , c = δ δ (1) . • δ is a surjective unital homomorphism of commutative algebras.The reader may want to constrast the data coming from a recognizable series β as above,with both algebras A β (0 ,
1) and A β (1 ,
0) commutative Frobenius, with that given by a 2-dimensional TQFT with corners [MS, LP, C, SP] where the Frobenius algebra B associatedto the interval is not necessarily commutative and the algebra associated to the circle is relatedto the center of B .To a recognizable series β there is associated a finite codimension ideal I β ⊂ k [ T , T ]describing relations on the hole and handle endomorphisms along any component of a surface.Starting with the viewable category VC β , described earlier, where floating components areevaluated via β , we impose relations in I β on hole and handle endomorphisms along anycomponent. The resulting category, denoted SC β (the skein category) has finite-dimensionalhom spaces.From the skein category we can pass to the already defined gligible quotient C β by takingthe quotient of SC β by the ideal of negligible morphisms. This ideal comes from the trace on SC β or, equivalently, from the bilinear form given by pairing of cobordisms.Taking the additive Karoubi closure of SC β results in the Deligne category DC β .Taking the quotient of DC β by the ideal of negligible morphisms produces the category DC β . Alternatively, this category is equivalent to the additive Karoubi closure of C β , and thesquare below is commutative.The following diagram summarizes these categories and functors (compare with (12), (14),and [KS]).(20) C −−−−→ k C −−−−→ VC β −−−−→ SC β −−−−→ DC β y y C β −−−−→ DC β Each of the four categories in the vertices of the commutative square has finite-dimensionalhom spaces between its objects.5.2.
Coloring side boundaries of cobordisms.
Fix a natural number r ≥ ( r ) of the category TFS whereside boundaries of cobordisms are colored by numbers from 1 to r . Let N r = { , . . . , r } be theset of colors. A morphisms in TFS ( r ) is a tf-surface x , up to rel boundary diffeomorphisms,such that each side (or vertical) boundary component of x carries a label from N r . Coloring of x induces a coloring on the set of corners of x , that is, on endpoints of the one-manifold ∂ h x which is the horizontal boundary of x , see Figure 5.2.1. i j j i ij (cid:4) (cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10) (cid:11) Figure 5.2.1.
Left: A morphism in TFS ( r ) from the colored interval (3 ,
1) tothe union (3 , ⊔ (2 ,
1) of two colored intervals. Middle: the dual of object( i, j ) is the object ( j, i ). Right: a connected floating component of genus 1 andthe sequence (1 , , I of x , being oriented, gets an induced ordered se-quence ( r ( I ) , r ( I )) of two colors. We consider a skeletal version of TFS ( r ) , choosing onlyone object for each isomorphism class. An object a then is determined by the r × r matrix M = M ( a ) with the ( i, j )-entry the number of intervals in a colored ( i, j ).Thus, objects a are described by r × r matrices of non-negative integers counting numberof colored intervals in a . We can call these objects r -colored or r -labelled thin one-manifoldsor r -boundary colored thin one-manifolds. An object can also be described by a list of coloredintervals in it.This skeletal version is still rigid tensor, with the obvious tensor product. The unit object = ∅ corresponds to the matrix of size 0 × ∅ ) of endomorphisms of the empty one-manifold ∅ is a free abelian monoid generated by diffeomorphism classes of connected floating r -colored tf-surfaces. Such a surface S is classified by its genus g ≥ r non-negative integers n = ( n , . . . , n r ), where n i is the number of boundary components ofcolor i . Denote such component by S n ,g . Figure 5.2.1 right shows the component S (1 , , , .For each color i ≤ r there is an embedding of TFS into TFS ( r ) by coloring each sideboundary of morphisms in TFS by i . Each horizontal interval is then an ( i, i )-interval.For a morphisms between two objects in TFS ( r ) to exist, there must exist a suitable match-ing between the colorings of their endpoints. For instance, there are no morphisms from theempty object ∅ to ( i, j ) interval if i = j , since the i and j endpoints must belong to the sameside interval and have the same coloring. There are morphisms from ∅ to ( i, j ) ⊔ ( j, i ) but nomorphisms from ∅ to ( i, j ) ⊔ ( i, j ) for i = j , since matching the two i ’s via a side interval isnot possible with our orientation setup.As usual, denote by k TFS ( r ) the k -linear version of TFS ( r ) , with the same objects as TFS ( r ) and morphisms – kk -linear combinations of morphisms in TFS ( r ) . VALUATING THIN FLAT SURFACES 35
The construction of evaluation categories and recognizable (or rational) series can be ex-tended from TFS to TFS ( r ) in a direct way.An evaluation α is a multiplicative homomorphism from the monoid End( ∅ ) of floatingcolored tf-surfaces to a field k . Such an evaluation is determined by its values on connectedfloating surfaces S n ,g . Let(21) Z α ( T , . . . , T r ) = X n ,g α n ,g T g T n , α n ,g ∈ k be a formal power series in r + 1 variables, with T n := T n . . . T n r r , n = ( n , . . . , n r ) , n i ∈ Z + where T n is a monomial in T , . . . , T r . Thus, T is the genus variable and T , . . . , T r are color variables. Coefficient α n ,g at T g T n . . . T n r r encodes the evaluation of floating connectedsurface S n ,g .Since each component of a tf-surface has non-empty boundary, coefficients at T g , with n = = (0 , . . . ,
0) do not appear in this formal sum. We set them to zero and extend thesum to these indices by setting(22) α ,g = 0 , g ∈ Z + . Thus, our power series has the property that(23) Z α ( T , , . . . ,
0) = 0 . We can also view α as a linear map of vector spaces α : k [ T , . . . , T r ] −→ k subject to condition (22), that is, α ( T g ) = 0, g ≥ α we assign the category VTFS ( r ) α , the quotient of k TFS ( r ) by the relations that aconnected floating component diffeomorphic to S n ,g evaluates to α n ,g . This is the category of viewable r -colored tf-surfaces with the α -evaluation.Categor VTFS ( r ) α carries a natural trace form given on an endomorphism x of an object a by closing x into a floating surface b x and evaluating this surface via α , see Figure 3.3.1, wherenow side boundaries are r -colored. If x is not a single cobordism but a linear combination,we use linearity of the trace to define tr α ( x ) = α ( b x ).Denote by J α the two-sided ideal of negligible morphisms in VTFS ( r ) α for this trace map.Define the gligible cobordism category TFS ( r ) α as the quotient of VTFS ( r ) α by the ideal J α :TFS ( r ) α := VTFS ( r ) α /J α . We say that evaluation α is rational or recognizable if category TFS ( r ) α has finite-dimensionalhom spaces. Proposition 5.3.
The following properties are equivalent. (1) α is recognizable. (2) Hom spaces Hom ( ∅ , ( i, i )) from the empty one-manifold to the ( i, i ) -interval are finite-dimensional in TFS ( r ) α for all i = 1 , . . . , r . (3) Power series Z α has the form Z α ( T , . . . , T r ) = P ( T , . . . , T r ) Q ( T ) Q ( T ) . . . Q r ( T r ) , where P is a polynomial in r +1 variables and Q , . . . , Q r are one-variable polynomials,with Q i (0) = 0 , i = 0 , . . . , r . Polynomials Q i can be normalized so that Q i (0) = 1 for all i . Power series Z α also satisfiesequation (23). Proof.
The proof is essentially the same as in r = 1 case, when all side components carry thesame color and there is no need to mention colors. Proof of Proposition 3.1 carries directlyto the case of arbitrary r . (cid:3) Take any floating component S and a monomial T = T g T n . . . T n r r . Define S ( T ) as thesurface S with additional g handles and additional n i holes with boundary colored i , for i = 1 , . . . , r. Given a linear combination y = P µ i T i of monomials, define S ( y ) = P µ i S ( T i ) as thelinear combination of corresponding floating surfaces. Evaluation α ( S ( y )) is an element ofthe ground field k .Given α , we can then define the syntactic ideal I α ⊂ k [ T , . . . , T r ]. Namely, y ∈ I α if α ( S ( y )) = 0 for any floating S . Proposition 5.4. α is recognizable iff the ideal I α has finite codimension in k [ T , . . . , T r ] . Thus, for recognizable α , one can define the skein category STFS ( r ) α as the quotient ofVTFS ( r ) α by the relations that inserting any y ∈ I α into a cobordism is zero. CategorySTFS ( r ) α has finite-dimensional hom spaces. It also has the ideal of negligible morphisms, withthe quotient category isomorphic to TFS ( r ) α . One can then define the analogue of the Delignecategory for STFS ( r ) α by taking its additive Karoubi closure and define the glibigle quotientof the latter. The resulting diagram of categories and functors below mirrors diagrams (12),(20), and the corresponding diagram in [KS]. The square is commutative.(24) TFS ( r ) −−−−→ k TFS ( r ) −−−−→ VTFS ( r ) α −−−−→ STFS ( r ) α −−−−→ DTFS ( r ) α y y TFS ( r ) α −−−−→ DTFS ( r ) α Condition (23) on the power series Z α seems rather unnatural. It can be removed by passingto the larger category, as in Section 5.1, where now closed components are allowed. Objects ofthe new category that extends TFS ( r ) are disjoint unions of oriented intervals (with endpointscolored by elements of N r ) and circles. Morphisms are two-dimensional oriented cobordismsbetween these collections, with side boundary intervals and side circles colored by elementsof N r . In the definition of evaluation α we can now omit condition (22) or, equivalently,restriction (23) on the power series Z α .Definition and basic properties or recognizable series now work as in the TFS ( r ) case.In the analogue of Proposition 5.3 for this modification, property (2) is replaced by the VALUATING THIN FLAT SURFACES 37 condition that the state space of the circle is finite-dimensional (hom space Hom( ∅ , S ) isfinite-dimensional). This is due to the surjection from the state space of the circle to thatof the interval ( i, i ) induced by the map δ in Figure 5.1.2 with the side (vertical) intervalcolored i . It is straightforward to set up the analogue of the diagram (24) of categories andfunctors for this case as well, for recognizable α . References [BHMV] C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel,
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