A TQFT associated to the LMO invariant of three-dimensional manifolds
Dorin Cheptea, Thang T Q Le
备注：
28 pages, 13 postscript figures. Version 2 (Section 1 has been considerably shorten, and section 3 has been slightly shorten, since they will constitute a separate paper. Section 4, which contained only announce of results, has been suprimated; it will appear in detail elsewhere. Consequently some statements have been re-numbered. No mathematical changes have been made.)
We construct a Topological Quantum Field Theory (in the sense of Atiyah) associated to the universal finite-type invariant of 3-dimensional manifolds, as a functor from the category of 3-dimensional manifolds with parametrized boundary, satisfying some additional conditions, to an algebraic-combinatorial category. It is built together with its truncations with respect to a natural grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The TQFT(s) induce(s) a (series of) representation(s) of a subgroup ${\cal L}_g$ of the Mapping Class Group that contains the Torelli group. The N=1 truncation produces a TQFT for the Casson-Walker-Lescop invariant.
Let Z^{LMO} be the 3-manifold invariant of [LMO]. It is shown that Z^{LMO}(M)=1, if the first Betti number of M, b_{1}(M), is greater than 3. If b_{1}(M)=3, then Z^{LMO}(M) is completely determined by the cohomology ring of M. A relation of Z^{LMO} with the Rozansky-Witten invariants Z_{X}^{RW}[M] is established at a physical level of rigour. We show that Z_{X}^{RW}[M] satisfies appropriate connected sum properties suggesting that the generalized Casson invariant ought to be computable from the LMO invariant.
Let C be a spherical fusion category. We prove that the Turaev-Viro-Barrett-Westbury state sum invariant of 3-manifolds derived from C is equal to the Reshetikhin-Turaev surgery invariant of 3-manifolds derived from Z(C), where Z(C) is the Drinfeld-Joyal-Street center of C.
In this article we prove that any unitary, axiomatic topological quantum field theory in four-dimensions can not detect changes in the smooth structure of M, a simply connected, closed (compact without boundary), oriented smooth manifold. However, as Donaldson-Witten theory (a topological quantum field theory but not an axiomatic one) is able to detect changes in the smooth structure of such an M, this seemingly leads to a contradiction. This seeming contradiction is resolved by introducing a new set of axioms for a "differential quantum field theory", which in truth only slightly modify the n…
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum $sl(2)$ were obtained by the last three authors in arXiv:1202.3553 . They are invariants of $3$-manifolds together with a cohomology class which can be interpreted as a line bundle with flat connection. In arXiv:1404.7289 we have extended those invariants to graded TQFTs on suitable cobordism categories. Here we give an overview of constructions and results, and describe the TQFT vector spaces. Then we provide a new, algebraic, approach to the computation of these vector spaces.
A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a p_1-structure, or an extended manifold structure. We perform the universal construction of Blanchet, Habegger, Masbaum and Vogel on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to t…
I give a formula for computing the number of regular $Γ$-coverings of closed orientable Seifert 3-manifolds, for a given finite group $Γ$. The number is computed using a 3d TQFT with finite gauge group, through a cut-and-glue process.
The object of this paper is to define a subcategory of the category of 3-cobordisms to which invariants of rational homology 3-spheres should generalize. We specify the notion of Topological Quantum Field Theory (in the sense of Atiyah) to this case, and prove two interesting properties that these TQFTs always have. In the case of the LMO invariant these properties amount to saying that the TQFT is anomaly-free.
We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra ${\cal N}=\Z/2\ltimes \ext *\R^2$ on the other side. We find…
In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category $C$, to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center $Z(C)$. In the present paper we give a partial proof of this statement.
We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',Γ') into a larger sutured manifold (M,Γ), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties.
We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to the p-modular representation theory of the symmetric groups S_n. We construct resolutions of simple p-modular TQFT's and S_n-modules over Z/pZ using powers of Lefschetz operators. These yield expansions of the irreducible p-modular S_n-characters in terms of the ordianry ones as …
Adapting a construction of D Salamon involving the U(1) vortex equations, we explore the properties of a Floer theory for 3-manifolds that fiber over S^1 which exhibits several parallels with monopole Floer homology, and in all likelihood coincides with it. The theory fits into a restricted analogue of a TQFT in which the cobordisms are required to be equipped with Lefschetz fibrations, and has connections to the dynamics of surface symplectomorphisms.
We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple quantum invariants of closed 3-manifolds decorated with ribbon graphs and cohomology classes. When we consider the zero cohomology class, these quantum invariants are shown to coincide with the renormalized Hennings invariants coming from the corresponding small quantum groups.
In this paper, we establish the general theory of (2+1)-dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev-Viro-Ocneanu unitary TQFT's obtained from subfactors satisfy the axioms of TQFT's with Verlinde bases. Hence, in a Turaev-Viro-Ocneanu TQFT, we have a Dehn surgery formula for 3-manifolds. It turns out that this Dehn surgery formula is nothing but the formula of the Reshetikhin-Turaev invariant constructed from a tube system,…
Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed intersections of such Lagrangians, we supply the analytic details of Donaldson's TQFT construction of the Seiberg-Witten invariants of a closed 3-manifold.
In this paper, we begin constructing a new finite-dimensional topological quantum field theory (TQFT) for three-manifolds, based on group PSL(2,C) and its action on a complex variable by fractional-linear transformations, by providing its key ingredient -- a new type of chain complexes. As these complexes happen to be acyclic often enough, we make use of their torsion to construct different versions of manifold invariants. In particular, we show how to construct a large set of invariants for a manifold with boundary, analogous to the set of invariants based on Euclidean geometric values and us…
We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev theory.
Gluing two manifolds M_1 and M_2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x=Sum_i(a_i M_i) yields a sesquilinear pairing p=<,> with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing p onto a finite dimensional quotient pairing q with values in C which in physically motivated cases is positive definite. To see if such a "unitary" TQFT can potentially detect any nontrivial x, we ask if <x,x> is non-zero whenever x is non-zero. If this is the case, we call the pa…
We discuss ways that the ring of coefficients for a TQFT can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers :Z [zeta_{2p}], if p \equiv -1 mod{4}, and Z [zeta_{4p}], if p \equiv 1 \pmod{4}, where zeta_k is a primitive kth root of unity. We study the quantum invariants of prime power order simple cyclic covers of 3-manifolds. We define new invariants arisi…
We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in $SL(2,\Z)$ and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and lastly by Ng and Schauenburg for the Drinfeld doubles of spherical fusi…
In this article, we discuss a (2+1)-dimensional topological quantum field theory, for short TQFT, with a Verlinde basis. As a conclusion of this general theory, we have a Dehn surgery formula. We show that Turaev-Viro-Ocneanu TQFT has a Verlinde basis. Several applications of this theorem are exposed. Based on Izumi's data of subfactors, we list several computations of Turaev-Viro-Ocneanu invariants for some 3-manifolds.
We define a "sutured topological quantum field theory", motivated by the study of sutured Floer homology of product 3-manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach to make computations of contact elements in sutured Floer homology over $\Z$ of sutured manifolds $(D^2 \times S^1, F \times S^1)$ where $F$ is finite. This generalises previous results of the author over $\Z_2$ coefficients. Our approach elaborates upon the quantum field theoreti…
We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic $q$. Both types of invariants are labeled by extra data which plays an important role in the proposed relation. Bridging the two sides - which until recently were developed independently, using very different methods - opens many new avenues. In one direction, it allows to study (and perhaps even to formulate) $q$-series invariants labeled by spin$^c$ structures in terms of non-semisimple invariants. In the opposite direction, it offers new insight…
Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quotient of P. Morita defined a homomorphism from the k-th term of the Johnson filtration to the third homology group of the k-th nilpotent quotient of P. In this paper, we replace groups by their Malcev Lie algebras and we study the "infinitesimal" version of the k-th Morita homomorp…
Let $Σ$ be a compact connected oriented surface with one boundary component and let $\mathcal{M}$ denote the mapping class group of $Σ$. By considering the action of $\mathcal{M}$ on the fundamental group of $Σ$ it is possible to define different filtrations of $\mathcal{M}$ together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of $\mathcal{M}$ introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by $Σ$. We shall call it the "alternative Johnson filtration", and the corresponding h…
We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of…
We give a presentation of the $n$-dimensional oriented cobordism category $\text{Cob}_n$ with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor $F$ from the category of smooth oriented manifolds and diffeomorphisms to an arbitrary category $C$, and morphisms induced by surgeries along framed spheres, we obtain a necessary and sufficient set of relations these have to satisfy to extend to a functor from $\text{Cob}_n$ to $C$. If $C$ is symmetric and monoidal, then we also characterize when the extension is a T…
We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 to an invariant that `counts' gradient flow lines--including closed orbits--of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the Seiberg-Witten invariants of 3-manifolds by making use of a `topological quantum field theory,' which makes the calculation completely explicit. We also realize a version of the Seiberg-Witten invariant of X as the intersection number of a pair of totally real submanifolds of a pr…
In this paper, we shall be concerned with a relation between TQFTs and cut and paste invariants introduced by Karras, Kreck, Neumann and Ossa. Cut and paste invariants, or SK invariants, are functions on the set of smooth manifolds that are invariant under the cutting and pasting operation. Central to the work in this paper are also SKK invariants, whose values on cut and paste equivalent manifolds differ by an error term depending only on the glueing diffeomorphism. Here we investigate a surprisingly natural group homomorphism between the group of invertible TQFTs and the group of SKK invaria…
The free energy of a closed 3-manifold is a 2-parameter formal power series which encodes the perturbative Chern-Simons invariant (also known as the LMO invariant) of a closed 3-manifold with gauge group U(N) for arbitrary $N$. We prove that the free energy of an arbitrary closed 3-manifold is uniformly Gevrey-1. As a corollary, it follows that the genus $g$ part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of Bender-Gao-Richmond on the asymptotics o…
We work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Furthermore, we construct refined Turaev-Viro and Reshetikhin-Turaev TQFT's and study the relationship between them.
A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of…
We define a universal state sum construction which specializes to most previously known state sums (Turaev-Viro, Dijkgraaf-Witten, Crane-Yetter, Douglas-Reutter, Witten-Reshetikhin-Turaev surgery formula, Brown-Arf). The input data for the state sum is an n-category satisfying various conditions, including finiteness, semisimplicity and n-pivotality. From this n-category one constructs an n+1-dimensional TQFT, and applying the TQFT gluing rules to a handle decomposition of an n+1-manifold produces the state sum.
We show that, associated with any complex root of unity $ω$, there exists a particularly simple 4d-TQFT model $M_ω$ defined on the cobordism category of Delta complexes. For an oriented closed 4-manifold $X$ of Euler characteristic $χ(X)$, it is conjectured that the quantity $N^{3χ(X)/2}M_ω(X)$, where $N$ is the order of $ω$, takes only finitely many values as a function of $ω$. In particular, it is equal to 1 for $S^4$, $\left(3+(-1)^{N}\right)/2$ for $S^2\times S^2$, and $N^{-1/2}\sum_{k=1}^Nω^{k^2}$ for $\mathbb{C} P^2$.
We discuss topological quantum field theories that compute topological invariants which depend on additional structures (or decorations) on three-manifolds. The $q$-series invariant $\hat{Z}(q)$ proposed by Gukov, Pei, Putrov and Vafa is an example of such an invariant. We describe how to obtain these decorated invariants by cutting and gluing, and make a proposal for Hilbert spaces that are assigned to two-dimensional surfaces in the $\hat{Z}$-TQFT.
Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its sign…
Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witt…
Given a TQFT in dimension d+1, and an infinite cyclic covering of a closed (d+1)-dimensional manifold M, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams' work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of M has a circle factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a…
