Invariants Of Three-manifolds Research Articles

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

$${\hat{Z}}$$-Invariant for SO(3) and OSp(1|2) Groups

Three-manifold invariants $${\hat{Z}}$$ (“Z-hat”), also known as homological blocks, are q-series with integer coefficients. Explicit q-series form for $${\hat{Z}}$$ is known for SU(2) group, supergroup SU(2|1) and orthosymplectic supergroup OSp(2|2). We focus on $${\hat{Z}}$$ for SO(3) group and orthosymplectic supergroup OSp(1|2) in this paper. Particularly, the change of variable relating SU(2) link invariants to the SO(3) and OSp(1|2) link invariants plays a crucial role in explicitly writing the q-series.

Relationship of the Hennings and Witten–Reshetikhan–Turaev invariants for higher rank quantum groups

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Witten–Reshetikhin–Turaev (WRT) three-manifold invariant arising from Chern–Simons field theory for the same Lie algebra and the same root of unity on all integer homology three-spheres, at roots of unity where both are defined. This partially generalizes the work of Chen et al. [On the relation between the WRT invariant and the Hennings invariant, Math. Proc. Cambridge Philos. Soc. 146(1) (2009) 151–163; Three-manifold invariants associated with restricted quantum groups, Math. Z. 272(3–4) (2012) 987–999] which relates the Hennings and WRT invariants for [Formula: see text] and [Formula: see text] for arbitrary rational homology three-spheres.

Quantum invariants of three-manifolds obtained by surgeries along torus knots

We study the asymptotic behavior of the Witten–Reshetikhin–Turaev invariant associated with the square of the $n$-th root of unity with odd $n$ for a Seifert fibered space obtained by an integral Dehn surgery along a torus knot. We show that it can be described as a sum of the Chern–Simons invariants and the twisted Reidemeister torsions both associated with representations of the fundamental group to the two-dimensional complex special linear group.

Higher depth false modular forms

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra [Formula: see text], [Formula: see text], and from [Formula: see text]-invariants of three-manifolds associated with gauge group SU(3).

Three-manifold quantum invariants and mock theta functions.

Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding, we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere Σ(2, 3, 7). This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'.

$\mathrm{S}$ duality and framed BPS states via BPS graphs

We study a realization of S dualities of four-dimensional $\mathcal{N}=2$ class $\mathcal{S}$ theories based on BPS graphs. S duality transformations of the UV curve are explicitly expressed as a sequence of topological transitions of the graph, and translated into cluster transformations of the algebra associated to the dual BPS quiver. Our construction applies to generic class $\mathcal{S}$ theories, including those with non-maximal flavor symmetry, generalizing previous results based on higher triangulations. We study the the action of S duality on UV line operators, and show that it matches precisely with the mapping class group, by a careful analysis of framed wall-crossing. We comment on the implications of our results for the computation of three-manifold invariants via cluster partition functions.

A TQFT of Turaev–Viro Type on Shaped Triangulations

A shaped triangulation is a finite triangulation of an oriented pseudo-three-manifold where each tetrahedron carries dihedral angles of an ideal hyperbolic tetrahedron. To each shaped triangulation, we associate a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3–2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges through the Neumann–Zagier Poisson bracket. Similarly to Turaev–Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions in a way that they enjoy a manifest tetrahedral symmetry. We conjecture that for shaped triangulations of closed three-manifolds, our partition function is twice the absolute value squared of the partition function of Techmuller TQFT defined by Andersen and Kashaev. This is similar to the known relationship between the Turaev–Viro and the Witten–Reshetikhin–Turaev invariants of three-manifolds. We also discuss interpretations of our construction in terms of three-dimensional supersymmetric field theories related to triangulated three-dimensional manifolds.

Rozansky–Witten-Type Invariants from Symplectic Lie Pairs

We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams.

Knot Homology and Refined Chern–Simons Index

We formulate a refinement of SU(N) Chern–Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern–Simons theory is defined on any M with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern–Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern–Simons theory are similar in many ways; for example, the Verlinde formula holds in both. Refined Chern–Simons theory gives rise to new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the invariants are certain indices on knot homology groups. For torus knots in S3 colored by fundamental representation, the index equals the Poincare polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S3 has a large-N dual which is the refined topological string on \({X=\mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow {\rm I\!P}^1}\) ; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to SLN knot homology. We also provide a matrix model description of some amplitudes of the refined Chern–Simons theory on S3.

A topological Chern-Simons sigma model and new invariants of three-manifolds

We construct a topological Chern-Simons sigma model on a Riemannian three-manifold M with gauge group G whose hyperkahler target space X is equipped with a G-action. Via a perturbative computation of its partition function, we obtain new topological invariants of M that define new weight systems which are characterized by both Lie algebra structure and hyperkahler geometry. In canonically quantizing the sigma model, we find that the partition function on certain M can be expressed in terms of Chern-Simons knot invariants of M and the intersection number of certain G-equivariant cycles in the moduli space of G-covariant maps from M to X. We also construct supersymmetric Wilson loop operators, and via a perturbative computation of their expectation value, we obtain new knot invariants of M that define new knot weight systems which are also characterized by both Lie algebra structure and hyperkahler geometry.

Three-manifold invariants associated with restricted quantum groups

We show a simple relation between Witten–Reshetikhin–Turaev SU(2) invariant and the Hennings invariant associated with the restricted quantum \({{\mathfrak{sl}_{2}}}\) . These invariants are defined in very different methods: the former uses the representation theory of quantum \({{\mathfrak{sl}_{2}}}\) while the latter uses the integral of the Hopf algebra. But they turn out to be the same for most rational homology 3-spheres up to a sign. This relation can be used to prove the integrality of the former invariant.

Quasi-topological gauged sigma models, the geometric Langlands program, and knots

We construct and study a closed, two-dimensional, quasi-topological (0, 2) gauged sigma model with target space a smooth G-manifold, where G is any compact and connected Lie group. When the target space is a flag manifold of simple G, and the gauge group is a Cartan subgroup thereof, the perturbative model describes, purely physically, the recently formulated mathematical theory of “Twisted Chiral Differential Operators”. This paves the way, via a generalized T -duality, for a natural physical interpretation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups. In particular, the Hecke eigensheaves and Hecke operators can be described in terms of the correlation functions of certain operators that underlie the infinite-dimensional chiral algebra of the flag manifold model. Nevertheless, nonperturbative worldsheet twisted-instantons can, in some situations, trivialize the chiral algebra completely. This leads to a spontaneous breaking of supersymmetry whilst implying certain delicate conditions for the existence of BeilinsonDrinfeld D-modules. Via supersymmetric gauged quantum mechanics on loop space, these conditions can be understood to be intimately related to a conjecture by Hohn-Stolz [1] regarding the vanishing of the Witten genus on string manifolds with positive Ricci curvature. An interesting connection to Chern-Simons theory is also uncovered, whence we would be able to (i) relate general knot invariants of three-manifolds and Khovanov homology to “quantum” ramified D-modules and Lagrangian intersection Floer homology; (ii) furnish physical proofs of mathematical conjectures by Seidel-Smith [2] and Gaitsgory [3, 4] which relate knots to symplectic geometry and Langlands duality, respectively. ∗E-mail: tan@ias.edu ar X iv :1 11 1. 06 91 v1 [ he pth ] 2 N ov 2 01 1

Estimating Turaev-Viro three-manifold invariants is universal for quantum computation

The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds. We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting. The algorithm is motivated by the relationship between topological quantum computers and (2+1)-D topological quantum field theories. Its accuracy is shown to be nontrivial, as the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer. Thus approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation. This establishes a novel relation between the task of distinguishing non-homeomorphic 3-manifolds and the power of a general quantum computer.

Invariants in Low-Dimensional Topology

Invariants in Low-Dimensional Topology

ON THE TOPOLOGICAL ORIGIN OF ENTANGLEMENT IN ISING SPIN GLASSES

The origin of entanglement in a class of three-dimensional spin models, at low momenta, is traced to topological reasons. The establishment of the result is facilitated by the gauge principle which, in conjunction with the duality mapping of the spin models, enables us to recast them as lattice Chern–Simons theories. The entanglement measures are expressed in terms of the correlators of Wilson lines, loops, and their generalisations. For continuous spins, these yield the invariants of knots and links. For Ising-like models, they can be expressed in terms of three-manifold invariants obtained from finite group cohomology — the so-called Dijkgraaf–Witten invariants.

Quantum superalgebras at roots of unity and topological invariants of three-manifolds

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra U_q(osp(1|2n)) over C is considered with q a primitive N^th root of unity for all integers N >= 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U_q^(N)(osp(1|2n)) = U_q(osp(1|2n)) / I is a Z_2-graded ribbon Hopf algebra. For all n and all N >= 3, a finite collection of finite dimensional representations of U_q^(N)(osp(1|2n)) is defined. Each such representation of U_q^(N)(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U_q^(N)(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N >= 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.

U( N) framed links, three-manifold invariants, and topological strings

Three-manifolds can be obtained through surgery of framed links in S 3. We study the meaning of surgery procedures in the context of topological strings. We obtain U( N) three-manifold invariants from U( N) framed link invariants in Chern–Simons theory on S 3. These three-manifold invariants are proportional to the Chern–Simons partition function on the respective three-manifolds. Using the topological string duality conjecture, we show that the large N expansion of U( N) Chern–Simons free energies on three-manifolds, obtained from some class of framed links, have a closed string expansion. These expansions resemble the closed string A-model partition functions on Calabi–Yau manifolds with one Kahler parameter. We also determine Gopakumar–Vafa integer coefficients and Gromov–Witten rational coefficients corresponding to Chern–Simons free energies on some three-manifolds.

BRANCHED CYCLIC COVERS AND FINITE TYPE INVARIANTS

This work identifies a class of moves on knots which translate to $m$-equivalences of the associated $p$-fold branched cyclic covers, for a fixed $m$ and any $p$ (with respect to the Goussarov-Habiro filtration.) These moves are applied to give a flexible (if specialised) construction of knots for which the Casson-Walker-Lescop invariant (for example) of their $p$-fold branched cyclic covers may be readily calculated, for any choice of $p$. In the second part of this paper, these operations are illustrated by some theorems concerning the relationship of knot invariants obtained from finite type three-manifold invariants, via the branched cyclic covering construction, with the finite type theory of knots.

Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary

In Ozsváth and Szabó (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in Ozsváth and Szabó (Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math. (2001), to appear). Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the “complexity bounds” derived in Ozsváth and Szabó (Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math. (2001), to appear), restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF + for a variety of three-manifolds. Moreover, we show how the structure of HF + constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given three-manifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for CP 2 .