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Вестник Челябинского государственного университета. 2015. № 3 (358). Математика. Механика. Информатика. Вып. 17. С. 26-40.
УДК 515.163 ББК В151.5
QUANTUM INVARIANTS OF 3-MANIFOLDS ARISING FROM NON-SEMISIMPLE CATEGORIES*
M. De Renzi
This survey article covers some of the results contained in the papers by Costantino, Geer and Patureau and by Blanchet, Costantino, Geer and Patureau. In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, Nr and N0, using non-semisi-mple categories of representations of a quantum version of sl2 at a 2r-th root of unity with r > 2. The secondary invariants N0 conjecturally extend the original Reshetikhin-Turaev quantum sl2 invariants. The authors also provide a machinery to produce invariants out of more general ribbon categories which can lack the semisimplicity condition. In the second paper a renormalized version of Nr for r ф 0 (mod 4) is extended to a TQFT, and connections with classical invariants such as the Alexander polynomial and the Reidemeister torsion are found. In particular, it is shown that the use of richer categories pays off, as these non-semisimple invariants are strictly finer than the original semisimple ones: indeed they can be used to recover the classification of lens spaces, which Reshetikhin-Turaev invariants could not always distinguish.
Keywords: q-binomial formula,dilogarithm identity.
1. Modular categories
A (strict) ribbon category C is a (strict) monoidal category equipped with a braiding c, a twist 9 and a compatible duality (*, b, d). We will tacitly assume that all the ribbon categories we consider are strict. The category RibC of ribbon graphs over C is the ribbon category whose objects are finite sequences (ys4),...,(V,sk) where Vi e Ob(C) and s{ = ±1 and whose morphisms are isotopy classes of C-colored ribbon graphs which are compatible with sources and targets.
Theorem 1. If C is a ribbon category then there exists a unique (strict) monoidal functor F: RibC ^ C such that (see Fig. 1):
1) F(y, +1) = y and F(y,-1) = V*;
2) F(XyW) = cyW, F(qy) = &y, F(1[-1] y) = by and F(^y) = dy;
3) F(r ) = f.
The functor F is the Reshetikhin-Turaev functor associated with C.
Wi
Wn
f
V!
ГХу \J~v
Fig. 1. Elementary C-colored ribbon graphs
Vm
Remark 1. Every i. e. Turaev functor F yields an invariant of framed oriented links colored with objects of C.
The author aknowledge support from Fondation Sciences Mathématiques de Paris.
A ribbon Ab-category is a ribbon category C whose sets of morphisms admit abelian group structures which make the composition and the tensor product of morphisms into Z-bilinear maps. Then K: = EndC(ll) becomes a commutative ring called the ground ring of C and all sets of morphisms are naturally endowed with K-module structures (the scalar multiplication being given by tensor products with elements of K on the left).
An object V e Ob(C) is simple if End(V) ~ K.
Remark 2. We will always suppose that all the ribbon Ab-categories we consider have a field K for ground ring.
A semisimple category is a ribbon Ab-category C together with a distinguished set of simple objects r(C) := {Vi}ieI such that:
1) there exists 0 e I such that V0 = l;
2) there exists an involution i ^ i* of I such that V* ~ V¡;
3) for all V e Ob(C) there exist i^...,in e I and maps aj : V ^ V, p■ : V ^ V{, such that idV = In ayPj (we say that the set r(C) dominates C);
4) for any distinct i, j e I we have HomC(Vi, Vj) = 0.
In a semisimple category we have the following results.
Lemma 1. For all V, W e Ob(C) :
1) HomC(V, W) is a finite-dimensional K-vector space;
2) HomC(Vi, V) = 0 for all but a finite number of i e I;
3) HomC(V, W) ~ 0 .eIHomC(V, v.) ®K HomC(V;,W) where the inverse isomorphism is given by f ® g ^ g ° f on direct summands;
4) the K-bilinear pairing HomC(V, W) ®K HomC(W, V) ^ K given by f ® g ^ trC(g o f) is non-degenerate.
Corollary 1. The quantum dimension of simple objects is non-zero.
Remark 3. Let (f)j,...,(f)n be a basis for the finite dimensional K-vector space HomC(V,V) and let (gi(gi)n denote the corresponding basis of HomC(Vi,V) defined by (f )h o (gi )k = Sh • idV,, which exists thanks to (iv) of the previous Proposition. Then we can write
idv = II a< )h -[(gi )k o (f )h ].
ieI h,k=1
But now we have bkh - idy. = (f )h o (g.)k = (f )h o idy o (g.)k = (X.)l - idy,. Therefore
idv = II(gi)j o (f;)y. (1)
ieI j=1
Equation (1) is called the fusion formula.
A premodular category is a semisimple category (C,r(C)) such that r(C) is finite. A modular category is a premodular category (C, r(C) = V}ieI) such that the matrix S = (F(Stj))ijeI with Sj given by Fig. 2 is invertible.
Sij
Fig. 2. Positive Hopf link colored with Vi and V■
2. The Reshetikhin—Turaev invariants
The construction of Reshetikhin and Turaev associates with every premodular category C an invariant tc of 3-manifolds (which will always be assumed to be closed and oriented) provided C satisfies some non-degeneracy condition. Let us outline the general procedure in this context: let C be a premodular category, let F: RibC ^ C be the associated i. e. Turaev functor and let Q be the associated Kirby color
Q := £ dimc(W) -W.
Wer(C)
It is known that every 3-manifold M3 can be obtained by surgery along some framed link L inside S3 (we write S3(L) for the result of this operation) and that two framed links yield the same 3-manifold if and only if they can be related by a finite sequence of Kirby moves. Therefore, in order to find an invariant of 3-manifolds, we can look for an invariant of framed links which remains unchanged under Kirby moves. For example let L c S3 be a framed link giving a surgery presentation for M3 and let L(Q) denote the C-colored ribbon graph obtained by assigning to each component an arbitrary orientation and the Kirby color Q. Then by evaluating F on L(Q) we get a number in K and, thanks to the closure (up to isomorphism) of r(C) under duality, we can prove that F(L(Q)) is actually independent of the chosen orientation for L. Therefore we have a number F(L(Q)) e K which depends only on the link L giving a surgery presentation for M. Let us see its behaviour under Kirby moves.
Proposition 1. [Slide]. Let (C,r(C)) be a pre-modular category and let T be a C-colored ribbon graph. If T is a C-colored ribbon graph obtained from T by performing a slide of an arc e c T over a circle component K c T colored by Q, then F(T) = F(T).
This result crucially relies on the semisimplicity of C, which enables us to establish the fusion formula 1, and on the finiteness of r(C), which enables us to define Kirby colors.
Now let us turn our attention towards blow-ups and blow-downs. Let A± denote the image under F of a ±1-framed unknot colored by Q. If L' c S3 is a link obtained from L c S3 by a ±1-framed blow-up then F(L'(Q)) = A± - F(L(Q)). At the same time we have that the positive and negative signatures of the linking matrices of L' and L satisfy a±(L') = a±(L) +1 and (L') = (L). Therefore we are tempted to consider the ratio
F(L(Q))
A°+(L) . A(L) ,
which is invariant under all Kirby moves. In order to be able to do so we must require from the premodular category C the following
Condition 1. A+ - A- * 0.
Therefore, let C be a premodular category satisfying Condition 1 and let (M, T) be a pair consisting of a 3-manifold M and a closed C-colored ribbon graph T c M. If L c S3 is any framed link yielding a surgery presentation for M and rT is a C-colored ribbon graph in S3 \ L representing T then the Reshetikhin-Turaev invariant associated with C is
F(L(Q) ufT )
Tc(M,T): =
■ a (L) .a (L)
A + - A -
Remark 4. The actual Reshetikhin-Turaev invariant is given by the renormalization d-b1(m)-1tc(M) where b1(M) is the first Betti number of M and D is an element of K satisfying D2 = F(u(Q)) with u(Q) the Q-colored 0-framed unknot. Note that such a D may not exists and we may have to manually adjoin it (compare with [3]).
Remark 5. The non-degeneracy condition is automatically satisfied by any modular category.
The most famous example of this construction, which yields the original invariants defined by Reshetikhin and Turaev, is obtained by considering a representation category of a quantum
version of sl2 at a root of unity. Let us recall the construction: fix an integer r > 2, set q := e%l/r and consider the quantum group Uq(sl2) generated (as a unital C-algebra) by E, F, K, K- with relations
KK-1 = K-{k = 1, KEK -1 = q2 E, KFK{ = q~2 F,
[E, F] = K - K. , Er = Fr =0
q - q
and with comultiplication, counit and antipode given by
A(E) = E ® K + 1 ® E, s(E) = 0, 5(E) = -EK A(F) = F ® 1 + K~l ® F, s(F) = 0, S(F) = -KF, A(K±4) = K± 1 ® K±4, s(K±4) = 1, S(K±4) = K*. A representation of Uq(sl2) is a weight representation, or a weight Uq(st2)-module, if it splits as a direct sum of eigenspaces for the action of K. The Hopf algebra structure on Uq(sl2) endows the category Uq(st2)-mod of finite-dimensional weight representations of Uq(sl2) with a natural monoidal structure and a compatible duality. Now let Uq (sl2) denote the quantum group obtained from Uq(sl2) by adding the relation K2r = 1. This condition forces all weights (eigenvalues for the action of K) to be integer powers of q for all representations of Uq(sl2). Therefore we can consider the operator qH®H/2 defined on V ® W for all weight Uq(5t2)-mod-ules V and W by the following rule:
qH®H/2(v ® w) = qmn/2v ® w
mnn ,
if Kv = qmv and Kw = qnw, where qmn/2 stands for eSet {m} := qm - q-m for all m e Z and define
[n]:={n},[n]!:=[n][n -1]-[1]. {1}
for all n e N. Consider the operator R defined on V ® W for all weight Uq(5t2)-modules V and W as
r-1 qn(n-1)/2
qH®h/2 y q-{1}nEn ® Fn.
n=0
[n]!
2
Finally consider the operator q~H /2 determined on each weight Uq(5t2)-module V by the rule q~H /2(v) = q~n /2v if Kv = qnv, define the operator u as
2 r-1 q3 n(n-1)/2
q-H /2 y q-{-1}n FnK-nEn
n=o [n]!
and set v := Kr-1u . Then the category Uq(5t2)-mod of finite dimensional weight representations of Uq(sl2) can be made into a ribbon Ab-category by considering the compatible braidings and twists given by
cV W = t o R : V ® W ^ W ® V, SV = v^1 : V ^ V, where t is the K-linear map switching the two factors of the tensor product. Moreover Uq(5t2)-mod is quasi-dominated by a finite number of simple modules, and thus it can be made into a modular category by quotienting negligible morphisms. The invariant we obtain is denoted Tr.
3. Relative G-premodular categories
To motivate the construction of non-semisimple invariants, let us consider the following different quantization of st2: let [/H(si2) denote the quantum group obtained by adding to [/q(si2) the additional generator H satisfying the following relations:
HK = KH, [H, E] = 2E, [H, F] = -2F,
A(H) = H ® 1 + 1 ® H, s(H) = 0, 5(H) = -H.
Remark 6. The new generator H should be thought of as a logarithm of K and, even though we will not require the relation to hold true at the quantum group level, we will restrict ourselves to representations where it is satisfied.
The category UH(sl2) -mod of finite dimensional weight representations of UH(sl2) where K acts like the operator qH can be made into a ribbon Ab-category by means of the same R-matrix and ribbon element used for Uq (st2)-mod.
Remark 7. The introduction of H is necessary in order to make sense of the formulas defining the operators R and u because the absence of the relation K2r = 1 makes room for weights which are not integer powers of q. The operator qH<SH/2 is then given by the rule
qH®H/2(v ® w) = qW2v ® w if Hv = Xv and Hw = ^w , where qa stands for exlll/r for all a e C . The definition of q~H /2 is analogous.
What is different in U^(st2)-mod is that simple objects are not in a finite number: indeed for any a e (C\Z) u r. Z the r-dimensional module Va generated by the highest weight vector va satisfying Ev0a = 0 and Hv[a = (a + r - 1)V is simple and projective, and is called a typical module (see [4] for details). If we could put this richer category into the Reshetikhin-Turaev machinery we would perhaps find more refined 3-manifold invariants. It is indeed the case, but we need to face (among other things) the following obstructions:
1) every typical module has zero quantum dimension;
2) we cannot quotient negligible morphisms as this would kill all typical modules, and thus we are forced to work with a non-semisimple category;
3) typical modules are pairwise non-isomorphic, and therefore we have to deal with infinitely many isomorphism classes of simple objects.
Let us see how we can work around these obstacles. The idea is to generalize the Reshetikhin-Turaev construction to more general ribbon Ab-categories which have the previous set of obstructions.
Facing obstruction (i): Modified quantum dimension
To begin with let us take care of the vanishing quantum dimension problem. The strategy will be to use categories C which admit a modified dimension which does not vanish. In order to do so we need an ambidextrous pair (A,d) , that is the given of a set of simple objects A c Ob(C) and a map d : A ^ K* with the following property: if T is an A-graph, i. e. a closed C-colored ribbon graph admitting at least one color in A, if e c T is an arc colored by V e A and if Te denotes the element of EndRlb ((V, +)) obtained by cutting open T at e, then F'(T) := d(V) •<Te} is independent of the chosen A-colored arc e (here <Te} denotes the unique element of K such that F(Te) = (Te} • idV ).
Definition 1. A ribbon Ab-category C admitting an ambidextrous pair (A, d) is said to have modified dimension d. F' is the modified A-graph invariant associated with (A, d).
Example 1. In the category UH(s[2)-mod considered before we have indeed an ambidextrous pair. It is obtained by taking A to be the set of typical modules and by defining
d(V ) = My-1!! qj - q j = (-1)r-1 - sin(an / r)
^ ^ ' XX „o.+ r-j -a-r +j ^ '
j=1 qa+r-j - + sin(an)
Facing obstruction (ii): G-grading relative to X
Moving on to the subject of semisimplicity, we will ask our categories to have a distinguished family of semisimple full subcategories nicely arranged, i. e. indicized by an abelian
group G in such a way that the tensor product respects the group operation. The aim of course is to work as much as possible in the semisimple part of the category and to leave aside the non-semisimple part.
Definition 2. Let C be a ribbon Ab-category. A full subcategory C' of C is said to be semisimple inside C if it is dominated by a set r(C') of simple objects of C' such that for any distinct V, W e r(C') we have HomC(V, W) = 0 .
Remark 8. We do not ask of r(C') to contain ¥ nor to be closed under duality up to isomorphism. In particular it may very well happen that the quantum dimension of a simple object of C' is zero.
Definition 3. We will say that a subset X of an abelian group G is small if G cannot be covered by any finite union of translated copies of X, i. e. if there exists no choice of g{,...,gk e G such that G c uk=1(gi + X). Let G be an abelian group and let X c G be a small subset. A family of full subcategories {C }geG of a ribbon Ab-category category C gives a G-grading relative to X for C if:
1) Cg is semisimple inside C for all g e G \ X;
2) V e Ob(Cg), V ' e Ob(Cg,) ^ V<g> V' e Ob(Cg+g,);
3) V e Ob(Cg) ^ V* e Ob(C_g);
4) V e Ob(Cg), V' e Ob(Cg), g * g' ^ HomC(V, V') = 0.
The elements of g which are not contained in X are called generic and a subcategory Cg in-dicized by a generic g is called a generic subcategory. A category C with a G-grading relative to X will be called a G-graded category for the sake of brevity.
Example 2. In the category U^(si2) -mod considered before we have a relative G-grading too. Indeed we can take G = C / 2Z, X = Z/ 2Z and set Ca equal to the full subcategory of modules whose weights are all congruent to a modulo 2. Then every Ca with a not integer is semisimple inside Uq (st2)-mod, being dominated by the typical modules it contains.
Facing obstruction (iii): Periodicity group
Finally, for the finiteness issue, we will proceed as follows: for a G-graded category C we will ask the sets of isomorphism classes of simple objects of all generic subcategories to be finitely partitioned in a way we can control.
Definition 4. A set C c Ob(C) of objects of a ribbon Ab-category is a commutative family if the braiding and the twist are trivial on C, i. e. if we have cWV ° cVW = idV<W and &V = idV for all V, W e C .
Definition 5. Let Z be an abelian group and C be a ribbon Ab-category. A realization of Z in C is a commutative family {s^ }teZ satisfying
s0 = =, S < ss = S+s, dimC(st) = 1 Vt, s e Z.
Any free realization of Z gives isomorphisms between the K-vector spaces HomC(V, W) and HomC(V <st ,W ) for all choices of V, W e Ob(C) and t e Z. Indeed the inverse of the map f ^ f < id t is simply given by g ^ g < id , . Therefore if V is simple then V <st is
s s
simple too for all t e Z. Thus any realization of Z induces an action of Z on (isomorphism classes of) objects of C given by the tensor product on the right with st. Such a realization is free if this action is free.
Definition 6. An abelian group Z is the periodicity group of the G-graded category C if there exists a free realization of Z in C0 whose action on r(C ) has a finite number of orbits for all g e G \ X.
In this case there exists some finite set of representatives of Z-orbits 0(Cg) c r(C ) for all generic g such that each simple module in r(C ) is isomorphic to some tensor product W < st for W e 0(Cg) and t e Z.
Example 3. Once again the category UH(st2)-mod considered before gives us an instance of this structure. Namely the periodicity group is Z = Z and its free realization in C^ is given by a 1-dimensional module s* for every t e Z which is spanned by the non-zero vector V such that Evt = Fvt =0 and Hvt = 2rtvt.
The categories which will allow us to extend the Reshetikhin-Turaev construction to the non-semisimple case admit all of the structures we just introduced.
Definition 7. A relative G-premodular category is (C,G 3 X,(A, d),Z) where C is a G-grad-ed category with modified dimension d and periodicity group Z satisfying the following compatibility conditions:
1) A 3r(Cg ) for all g e G\X;
2) c t = y(g,t) • c~l for all V e Ob(Cg), t e Z and for some Z-bilinear pairing y : G x Z ^ K" (see Fig. 3).
Example 4. UH(sl2) -mod is a relative C / Z-premodular category as it can be shown that a skein relation like the one required in condition 2 of the previous definition holds.
= Vto.i)
VeOb(%) e* V sf
Fig. 3. Skein-type relation for G and Z (the = sign stands for equality under F)
4. Construction of non-semisimple invariants
We are ready to sketch a construction analogous to the one of Reshetikhin and Turaev which associates with each relative G-premodular category C an invariant of 3-manifolds provided C satisfies some non-degeneracy conditions. The idea will be to use the modified invariant F' as a basis for this construction exactly as the functor F was used as a basis for the standard case. Remember however that in order to compute F' on a C-colored ribbon graph T we will need to make sure that T is actually an A-graph.
Let us fix a relative G-premodular category C. The first thing we did in the construction of Reshetikhin-Turaev invariants was to color a framed link giving a surgery presentation for a 3-man-ifold M with the Kirby color Q associated with some premodular category. Now, in C we do not have the concept of a Kirby color, but we can define an infinite family of modified Kirby colors.
Indeed if g e G is generic then the formal sum
Qg := £ d(W) •W
WeO(Cg )
is a modified Kirby color of degree g.
Remark 9. It can be easily proved using the properties of the periodicity group Z that the modified dimension d factorizes through a map defined on Z-orbits on all generic subcatego-ries, i.e. we have d(W ®el) = d(W) for all W e O(Cg) and all t e Z . In particular the coefficients in the formal sum Qg are independent of the choice of the representatives of Z-orbits in r(C ) . Of course W and W are not isomorphic if t ^ 0 but we will see that under certain circumstances this choice will not affect the value of F'.
Since we defined an infinite family of modified Kirby colors it is not clear which one should be used to color the components of a surgery link L for M3. The right choice is to let the coloring be determined by a cohomology class © e Hi(M \ T; G) ~ HomZ(H1(M \ T, G) which is compatible with the C-coloring which is already present on T.
Definition 8. Let I be a C-colored ribbon graph inside M and a be an element of Hi(M \ T; G). For every arc e c T let ^e denote the homology class of a positive meridian around e. The triple (M, T, ®) is compatible if the color of e is an object of C<a|^ >.
We will now look for an invariant of compatible triples (M, T, ®), where two triples (Mi, Ti, ai) for i = 1,2 are considered to be equivalent if there exists an orientation preserving diffeomorphism f : M1 ^ M2 such that f (7]) = T2 as C-colored ribbon graphs and f *(ra2) = a4.
Remark 10. We will have to be more careful and to keep track of the (isotopy class of the) diffeomorphism induced by each Kirby move.
The idea is to color each component L{ of a surgery link L with a modified Kirby color whose degree is determined by the evaluation (a, ^>, where ^ denotes the homology class corresponding to a positive meridian of L. Thus, since modified Kirby colors are defined only for generic degrees, not all surgery presentations can be used to define the new invariant.
Definition 9. A compatible triple (M, T, ®) admits a computable surgery presentation L = L u ... u Lm c S3 if one of the following holds:
1) L and (a,^> is generic for all i = 1,.,m;
2) L = 0 and T is an A-graph.
If L is a link yielding a computable surgery presentation for a compatible triple (M, T, ®) and we denote by L(®) the C-colored link obtained by coloring each component Li of L with Q<m^>, then F' can be evaluated on L(a) urT, where rT represents T inside S3 \ L.
Remark 11. It can be shown that a sufficient condition for the existence of a computable surgery presentation for a compatible triple (M, T, ®) is that the image of ® is not entirely contained in the critical set X (when we regard ® as a map from H1(M \ T) to G).
Let us see what happens when we perform Kirby moves.
Remark 12. In order to be able to evaluate F' we can only consider Kirby moves between computable surgery presentations of (M,T, a).
The slide of an arc e c L u rT over a component L{ of L corresponds to a change of basis in Hi(M\ T) which amounts to susbstituting ^ with ^ ± (depending on orientations). This operation preserves F' (L(ra) u rT).
Proposition 2. [Slide]. Let T be an A-graph,let e c T be an arc colored by V e Ob(Cg), let K c T be a knot component colored by Qk for some generic h e G and suppose that g + h is generic too. If T' is an A-graph obtained from T by sliding e along K and switching the color of K to Qg+h (like in Fig. 4) then F' (T' ) = F' (T).
To prove this proposition we need to establish a fusion formula (which can be done in the semisimple part of C exactly as before) and to use the skein-type relation in the definition of C in order to handle closed components colored with el for t e Z. The color of K changes because V ® W is an object of Cg+h for all W e O(Ch).
For what concerns blow-ups and blow-downs, we cannot compute F directly on a detached ±1-framed unknot as such a component should be colored with the modified Kirby color of degree 0 and it may very well happen that 0 e X (which is the case in our previous example).
Proposition 3. [Blow-up and blow-down]. Let T+ be the C-colored ribbon graph given by Fig. 5 (a) with g generic in G. Then A+ := (T+> does not depend on the generic g nor on the object U e Ob(C ). The same holds for the analogous graph T (obtained by turning each overcrossing of T+ into an undercrossing) and for A_ := (T_>.
V e Ob(Vg)
\
I
K
g+h
Fig. 4. Subtraction
U 6 Ob(SfB) f^J We A
Remark 13. The operation of blowing up a pos-V e A itive meridian of an arc in a ribbon graph T can replace the operation of blowing up an isolated unknotted componend provided T is non-empty. This is always the case for computable surgery presentations since there is always at least one arc colored in A. Thus what we need in order to be able to define (a) (b) the invariant is once again to ask the condition
Fig. 5. Blow-up of +1-framed meridian (a) A+ • A- * 0 . However, this time we need also an-and H-stabilization (b) other non-degeneracy condition which allows us
to perform an operation called H-stabilization which is needed in the proof of the invariance of our construction. Namely, let H(V,W) denote the C-colored ribbon graph given by Fig. 5 (b) for V,W e A. Then:
Condition 2.
1. A+ • A- * 0 .
2. <H(V, W)> * 0 for all V,W e A.
Now we can state our result.
Theorem 2. Let C be a relative G-premodular category satisfying the non-degeneracy Condition 2. Let L be a framed link giving a computable surgery presentation for a compatible triple (M, T, ®) and let rT be a C-colored ribbon graph inside S3 \ L representing T. Then
xw^t >> F'(L(®) utt )
NC(M, T, m):= (L) .„(/)
A++ • A--
is a well-defined invariant of (M, T, ®).
Remark 14. When C = UH(st,)-mod with q = em/r we write Nr instead of N H .
q 2 I r uHL (s[2)-mod
The subtlety in the proof of this result is the following: if we have two different computable surgery presentations L and L it may happen that the sequence of Kirby moves connecting them passes through some non-computable presentation. What we have to prove is that, up to passing to a different sequence of Kirby moves, we can make sure to get a computable presentation at each intermediate step.
This turns out to be true, but we have to allow an operation, called H-stabilization, which modifies the triple (M, T, ®) and which is defined as follows: let e c T be an arc colored by W e A, let a be a positive 0-framed meridian of e disjoint from T and colored by V e r(C ) for some generic g and let D2 c S3 be a disc intersecting e once and satisfying 3D2 = a . Now let TH denote the A-graph T u a and let raH be the cohomology class coinciding with ® on homology classes contained in M \ (T u D2) and satisfying <®H, ^a> = g where is the homology class of a positive meridian of a. Then the compatible triple (M, TH, ®H) is said to be obtained by H-stabilization of degree g from (M, T, ©), and a is called the stabilizing meridian. Now (M, TH, ®H) is not equivalent to (M,T,®) but we have the equality Nc (M, Th , ®h ) = <H(V, W)> • NC(M, T, ®) .
Returning to the proof of the Theorem, we split the argument into three steps: we begin by first proving the result in the case that T itself is an A-graph, that the initial and final surgery presentations are the same and that the sequence of Kirby moves involves only isotopies of rT inside S3(L), i. e. slides of arcs of rT over components of L (we call this sequence of moves an isotopy inside S3(L)). This case can be easily treated by performing a single H-stabilization on (M, T, ®) whose degree is sufficiently generic. Indeed if we slide a stabilizing meridian on some component Lj of the computable link L which is colored by Qh we change the color of
Lj, provided the degree g of the H-stabilization satisfies g + h. e G \ X. This would impose a condition on the choice of the degree, but there surely exists a g e G which satisfies it because X is small. More generally, if C c G denotes the finite set of (degrees of) colors appearing on L during the sequence of slides, we can choose the degree g of the H-stabilization in such a way that (g + C) o X = 0 . Thus, we can begin by sliding the stabilizing meridian a over all components of L, then we can follow the original sequence of Kirby moves and finally we can slide back a to its original position. What we will get is an equality of the form
F'(L(rn) urr) -<H(V,W)) = F'(L(rn) uVT) -<H(V,W))
(L) (L) ,
for some V, W e A, which proves the first step.
ro ro v v Tk rk The second step consists in proving the Theorem when T
L 'U 1 t —^ rs1 ''' —A rsk L ^^ 1 T
is an A-graph. If Fig. 6 is our sequence of Kirby moves, we perform an H-stabilization for each Kirby move sh which L u r L u r7- makes some non-generic color appear. If sh is a non-admis-
Fig' 6 sible slide over some component Lf1 we precede it by a
slide of the corresponding stabilizing meridian ah over L^-1. If sh is a non-admissible blow-up around some arc we perform it on the corresponding stabilizing meridian ah instead and then we slide the arc over the newly created component. All degrees can be chosen so to adjust all colors, and the use of different stabilizations ensures the independence of the conditions. The tricky point is that a move st which in the original sequence was a blow-down of a ±1-framed meridian of some arc or link component may now have become the blow-down of a component which is also linked to some of the stabilizing meridians we added. In this case, though, we can slide all these stabilizing meridians off, and this operation is an isotopy inside S 3(L'_1). Remark that the configuration we get at this point is not necessarily admissible, but problems can arise only for blow-downs of meridians of arcs in T^1. Thus in this case we can perform a new H-stabilization, slide the arc off and slide the new stabilizing meridian over. This operation is yet another isotopy inside S 3(L") which yields a computable presentation. Therefore, thanks to the first step, the invariant does not change. In the end we get the original final presentation plus some stabilizing meridian linked to the rest of the graph. All these meridians can be slid back to their initial positions, and once again this operation is an isotopy inside S 3(L' ).
The third step is the general case: now what we have to do is to blow-up two meridians of a component of L in such a way that its framing does not change. Then we can consider these new curves as part of T, falling back into the previous case, we can prove that we can undo our initial operation and that the result is not affected by our changes.
Extension to all compatible triples
There exist of course compatible triples which do not admit computable presentations. In order to include also this case in the construction we can build a second invariant NC which is defined for all compatible triples.
Remark 15. In categories where the quantum dimension of the objects of A is always zero (such as the categories in our example) this second invariant will vanish on all triples which admit computable presentations. Therefore in this case one should continue to use NC to get topological informations.
For the definition of N0C we will need the concept of connected sum of compatible triples. Let (M1, T1, ®1) and (M2, T2, ®2) be two compatible triples, let M3 = M1#M2 be the connected sum along balls Bi inside M{ \ Ti for i = 1,2 and set T3 = T1 U T2. Then we have the chain of isomorphisms
H.(M3 \ T3) ~ H.(M. \ (B. uT)) © H.(M, \ (B2 u T2)) ^ H.(M. \ T) © H.(M, \ T2)
where the first one is induced by a Mayer-Vietoris sequence and the second one comes from excision. These maps induce an isomorphism
H4(M3 \ T3; G) ~ H1(M1 \ T1; G) © H\M2 \ T2; G). Finally let ®3 be the unique element of Hi(M3 \ T3; G) which restricts to rai on Hi(Mi \ T; G) for i = 1,2 via the previous isomorphism. The connected sum of (M1, T1, ®1) and (M2, T2, ®2) is defined as (M3, T3, ®3) . Now if the compatible triple (M, T, «) does not admit any computable presentation consider the triple (S3, uV, ®V) where uV is a 0-framed unknot in S3 colored by V e A and ®V is the unique cohomology class in Hl(S3 \ uV; G) which makes the previous triple into a compatible one. Then we can define N£(M, T, ®) to be
Nc((M, T, a)#(S3, Uv , «v )) d(V) .
Remark 16.
1. Just like before, when C = UH(sQ-mod with q = em/r we write N0 instead of N0 H .
q 2 ^ r UH (sl2)-mod
As claimed earlier, N0 vanishes on computable presentations because in this category F' vanishes on split A-graphs, i. e. if T and T' are completely disjoint A-graphs then we have F(T U T') = F(T)F(T') = 0 .
2. As it was mentioned in the abstract, N 0 coincides with Tr in a lot of cases, though in general their equality remains conjectural.
5. Case r = 2: Alexander polynomial, Reidemeister torsion and lens spaces
For the special case r = 2 we have that q = i and the modified A-graph invariant F' associated with the category Uf (sl2) -mod can be related to the multivariable Alexander polynomial. This fact, which was first observed by Murakami in [5], is exposed in detail in Viro's paper [6]. He defines an Alexander invariant A2 for oriented trivalent graphs equipped with the following additional structure:
1) a half-integer framing (half-twists are allowed too);
2) a coloring with typical U"(sl2) -modules satisfying a condition like the ones shown in Fig. 7 around each vertex;
3) a cyclic ordering of the (germs of the) edges around each vertex.
a + p + y = ±1
a + p - y = ±1 a - p - y = ±1
Fig. 7. Admissible colorings
a + p + y = ±1
Viro's construction uses a functor which is similar to the Reshetikhin-Turaev one, though the source category is not the category of colored ribbon graphs. It is indeed a category G2 whose objects are the objects of Rib^ ( } d which feature only typical colors and whose
morphisms are (isotopy classes of) a non-closed version of the graphs mentioned above. In particular all vertices are either 3-valent (internal vertices) or 1-valent (boundary vertices). If such a graph r is closed, i.e. if it does not contain boundary vertices, and if its framing yields an orientable surface, then we can associate with it an A-graph Tr defined as follows:
consider the ordered basis
a+1
—-_Fva
0 ' 1 r 4 1 0
[a +1]
of the typical module Va and let {^j, } be its dual basis in Va* . Then we have an isomorphism wa : Va ^ V-a given by vj ^ i K 2 '9-J for j = 0,1. Moreover, every time a, p, y e C\ (2Z + 1) satisfy a + p + y = ±1, we can consider the morphism Wa,p,Y : C ^ Va ® Vp ® VY
mapping 1 to I
2( y -h)=a+p+Y+1
Clebsch-Gordan quantum coefficients (compare with [7]) and are defined as
jV ® V ® vl where the coefficients CJjft are derived from the
ß(k-1)-a( j+1)+2(k+h- j-1)+ j2-k2
r 1 -y 1 -1 r 1 -Y 1
1 -y-h a + ß - y + 2
t+s=h
(2t-h)(2-Y-h) 2
a + ß + y + 1 2
j -1
a - j + t +1 a - j +1
ß- k + s + 1" ß-k +1
Then we can construct Tr by replacing each edge of r which is not a connected component as shown in Fig. 8 (a) and each trivalent vertex of r as shown in Fig. 8 (b).
Ï-
V V -w
(a)
w
V
V
Fig. 8. C-colored ribbon graph Tr obtained from trivalent graph r
Proposition 4. F'(Tr ) = (-2i)1-v/2 A2(f) where obtained from r by inverting the orientation on each edge.
This result is obtained by checking that the two expressions coincide for a set of elementary graphs (the trivial one, the ©-graph and the tetrahedron graph) and by checking that they both satisfy the same set of relations which reduce the computation for an arbitrary graph to elementary ones (see [1] and [2] for details).
If L = L1 U ... U Lm is an oriented colored framed link whose j-th component Lj is colored with the typical module Va then Viro shows that
A2(L) = V, (i
.1+0
I jh
i1+am )ijh=1
2 ek(Lj ,Lh )
where VL is the Alexander-Conway function of L. Therefore, if the framing is integral, Proposition 4 immediately gives
F'(L) = (-2i)Vr (i
1—a
',..., i
')i
I
j',h=1
0 iah-1
jh- ?MLj Lh)
Now since CT is semisimple inside UH(st2)-mod we can take the critical set X cC/2Z to be just {0}. Therefore, thanks to Remark 11, every triple of the form (M, 0, ro) with ro ^ 0 is compatible and admits a computable presentation. In particular some computation (compare with [2]) yields
I m \
N2(M, 0, ra) = 2 • 4m-c+ (L)-c-(L) I- (L)-c+(L)-m-1 • n .a. 1 .-„■ VL(ia1,..„ %m ) • i
m / n \
Zaj(ah+2)
-^r- tk(Li L)
j ,h=1
v j=1 i j - i j y
where L = L1 U ... U Lm is a surgery presentation for M and a j := (ra, ) . Thus N2 recovers the Alexander - Conway function, which is known to be related to the Reidemeister torsion. Moreover N2 yields a canonical normalization of the Reidemeister torsion which fixes the scalar indeterminacy. Indeed recall that the refined abelian Reidemeister torsion of M defined by Turaev (see [8] for example) is determined by the choice of a homomorphism 9 : H1(M) ^ C*, of a homology orientation raM for M and of a Spinc-structure c e Spinc(M) (or equivalently of an Euler structure on M). We write t9 (M, raM , c) or, if M is oriented and we pick the canonical homology orientation associated with the orientation of M, simply t9 (M, c) . Now, if (M, 0, ra) is a compatible triple as above, we can use the non-zero cohomology class ra to define the homomorphism 9ra : H1(M) ^ C* given by h ^ ei<ra,h) = i2<ra h).
Theorem 3. Let M be a closed oriented 3-manifold endowed with a non-trivial cohomology class ra e H4(M;C/21). Then for any complex spin structure c e Spinc(M) we have
,61(M)+4VM c(ra)+1
T9ra (M, c) = i 2 ^ m) eN2(M, 0, ra),
where yMc: H4(M;C/2Z) ^ C/Z is the homomorphism obtained by first extending De-loup - Massuyeau's quadratic linking function yMc: H2(M;Q/Z) ^ Q/Z (compare with [9],Definition 2.2) to a homomorphism 9Mc: H2(M;C/Z) ^ C/l,and then by considering
the composition 9CMc D where D: H2(M;C/2Z) ^ H4(M;C/21) is Poincare duality, 1
— : H1(M; C / 22) ^ H1(M; C /Z) is induced by the "division by 2" isomorphism between C /2Z
and C/Z and c is the image of c under the standard involution of Spinc(M)
This is proven by using the surgery formula for the Reidemeister torsion (see [8], section VIII.2, equation (2.b)): if L is a computable surgery presentation for (M,0,ra) then
.aj (kj-1)
t9(M,c) = (-1 )2m-c+(L)eft ,a. .-„. •vl(ia1,.,ia),
j=1 i j - i j
where a j := (ra, ) and k1,., km are the charges of the Spinc-structure c (see [8], section VII.2.2 for a definition).
We conclude with a Proposition which gives the value of N2 for lens spaces and can be used to follow the path of the classical proof of their classification.
Proposition 5. Let p > q >0 be two coprime integers,let L(p, q) be a lens space and consider a non zero cohomology class ra e Hi(L(p, q);C / 22). Then
(-1)k(ra) gink(ra)2p/q
N2(L(p, q), ra) =
for some k(ra) e Z\ pZ.
Corollary 2. N2 classifies lens spaces.
„. . nk(ra)q . nk(ra)
2i sin-- sin-
p p
References
1. Costantino F., Geer N., Patureau-Mirand B. Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories. J. of Topology, 2014, vol. 7, is. 4, pp. 1005-1053.
2. Blanchet C., Costantino F., Geer N., Patureau-Mirand B. Non-semisimple TQFTs,Reide-meister torsion and Kashaev's invariants. Available at: http://arxiv.org/abs/1404.7289arX-iv:1404.7289.
3. Turaev V. Quantum Invariants of Knots and 3-Manifolds. Berlin, Walter de Gruyter Publ., 2010. 592 p.
4. Costantino F., Geer N., Patureau-Mirand B. Some remarks on the unrolled quantum group of sl(2). Available at: http://arxiv.org/abs/1406.0410arXiv:1406.0410.
5. Murakami J. The multi-variable Alexander polynomial and a one-parameter family of representations of Uq(sl(2,C)) at q2 = -1. Quantum Groups. Proceedings of Workshops held in the Euler International Mathematical Institute, Leningrad, Fall, 1990. Berlin, Springer Publ., 1992, pp. 350-353.
6. Viro O. Quantum relatives of the Alexander polynomial. St. Petersburg Math. J., 2007, vol. 18, no. 3, pp. 391-457.
7. Costantino F., Murakami J. On SL(2, C) quantum 6/-symbols and its relation to thehyper-bolic volume. Quantum Topology, 2013, vol. 4, is. 3, pp. 303-351.
8. Turaev V. Torsions of 3-dimensional manifolds. Berlin, Springer Publ., 2002. 196 p.
9. Deloup F., Massuyeau G. Quadratic functions and complex spin structures on three-manifolds. Topology, 2005, vol. 44, no. 3, pp. 509-555.
10. Geer N., Patureau-Mirand B., Turaev V. Modified Quantum dimensions and re-normalized link invariants. Compositio Mathematica, 2009, vol. 145, is. 1, pp. 196-212.
11. Ohtsuki T. Quantum Invariants,a Study of Knots,3-Manifolds and Their Sets. Singapore, World Scientific, 2002. 508 p.
About the author
Marco De Renzi, Paris Diderot University, Université Sorbonne Paris Cité, Institute of Mathematics of Jussieu (UMR 7586), French National Centre for Scientific Research, Pierre and Marie Curie University, Paris, France. derenzi@mail.dm.unipi.it.
Bulletin of Chelyabinsk State University. 2015. № 3 (358). Mathematics. Mechanics. Informatics. Issue 17. P. 26-40.
КВАНТОВЫЕ ИНВАРИАНТЫ ТРЕХМЕРНЫХ МНОГООБРАЗИЙ, ВОЗНИКАЮЩИЕ ИЗ НЕПОЛУПРОСТЫХ КАТЕГОРИЙ
М. Де Рензи
Эта обзорная статья охватывает некоторые из результатов, содержащихся в работах Костанти-но, Гир, Патуреау и Бланше. В первой работе авторы строят два семейства инвариантов типа Ре-шетихина — Тураева для трехмерных многообразий, Ц и Ц0, используя для этого неполупростые категории представлений квантовой версии з(2 в множество корней из единицы степени 2,, г > 2. Второе семейство инвариантов N0 предположительно обобщает оригинальные квантовые з(2 инварианты Решетихина — Тураева. Авторы также развивают технику для построения инвариантов, возникающих из более общих ленточных категорий, которые могут и не обладать свойством
полупростоты. Во второй работе перенормированная версия инварианта Nr при r Ф 0 (mod 4) продолжается до TQFT, а также устанавливаются связи с классическими инвариантами, такими как полином Александера и кручение Рейдемейстера. В частности показано, что использование более богатых категорий имеет смысл, так как эти неполупростые инварианты более информативны, чем оригинальные полупростые инварианты: в самом деле, они могут быть использованы для классификации линзовых пространств, в то время как инварианты Решетихина — Тураев не всегда их различают.
Ключевые слова: q-биномиальная формула,тождество дилогарифма.
Список литературы
1. Costantino, F. Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories / F. Costantino, N. Geer, B. Patureau-Mirand // J. of Topology. — 2014.
- Vol. 7, issue 4. — P. 1005-1053.
2. Non-semisimple TQFTs, Reidemeister torsion and Kashaev's invariants [Электронный ресурс] / C. Blanchet, F. Costantino, N. Geer, B. Patureau-Mirand. — URL: http://arxiv.org/ abs/1404.7289arXiv:1404.7289.
3. Turaev, V. Quantum Invariants of Knots and 3-Manifolds / V. Turaev. — Berlin : Walter de Gruyter, 2010. — 592 p.
4. Costantino, F. Some remarks on the unrolled quantum group of sl(2) [Электронный ресурс] / F. Costantino, N. Geer, B. Patureau-Mirand. — URL: http://arxiv.org/abs/1406.0410arX-iv:1406.0410.
5. Murakami, J. The multi-variable Alexander polynomial and a one-parameter family of representations of Uq (sl(2, C)) at q2 = -1 / J. Murakami // Quantum Groups. Proceedings of Workshops held in the Euler International Mathematical Institute, Leningrad, Fall, 1990. -Berlin : Springer, 1992. — P. 350-353.
6. Viro, O. Quantum relatives of the Alexander polynomial / O. Viro // St. Petersburg Math. J. — 2007. Vol. 18, № 3. — P. 391-457.
7. Costantino, F. On SL(2, C) quantum 6/-symbols and its relation to thehyperbolic volume / F. Costantino, J. Murakami // Quantum Topology. — 2013. — Vol. 4, issue 3. — P. 303-351.
8. Turaev, V. Torsions of 3-dimensional manifolds / V. Turaev. Berlin : Springer, 2002. — 196 p.
9. Deloup, F. Quadratic functions and complex spin structures on three-manifolds / F. De-loup, G. Massuyeau // Topology. — 2005. — Vol. 44, № 3. — P. 509-555.
10. Geer, N. Modified Quantum dimensions and re-normalized link invariants / N. Geer, B. Patureau-Mirand, V. Turaev // Compositio Mathematica. — 2009. — Vol. 145, issue 1. -P. 196-212.
11. Ohtsuki, T. Quantum Invariants, a Study of Knots, 3-Manifolds and Their Sets / T. Oht-suki. — Singapore : World Scientific, 2002. — 508 p.
Сведения об авторе
Де Рензи Марко, Университет Париж VII им. Дени Дидро, Университет Сорбонна города Париж, Математический институт им. Жюсье (иМ^ 7586), Национальный центр научных исследований Франции, Университет Пьера и Марии Кюри - Париж 6, Париж, Франция. derenzi@mail.dm.unipi.it.
