Wilson Surfaces and Higher Dimensional Knot Invariants

Cattaneo, Alberto S.; Rossi, Carlo A.

doi:10.1007/s00220-005-1339-0

Cited by 44 publications

(71 citation statements)

References 16 publications

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“…5 /. We will see here that (5) is equal to 1 2 times the second invariant in Cattaneo-Rossi [7], which, in this degree, was first obtained by Bott [4]:…”

Section: Example 23 Fromsupporting

confidence: 63%

“…In Cattaneo-Rossi [7] and Rossi [15], the invariance of the higher degree z k is claimed and in some part idea of the proof is given, while explicit proofs are given in the other parts even for higher degrees. The explicit descriptions of z k for higher degrees are not given there, while the complete definitions and proofs are given for degrees up to 3.…”

Section: Lemma 411mentioning

confidence: 99%

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Configuration space integral for longn–knots and the Alexander polynomial

Watanabe

2007

Algebr. Geom. Topol.

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There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n-knots and characterize the Bott-Cattaneo-Rossi invariant as a finite type invariant of long ribbon n-knots introduced in [HKS]. As a consequence, we obtain a non-trivial description of the Bott-Cattaneo-Rossi invariant in terms of the Alexander polynomial.The results for higher codimension knots are also given. In those cases similar differential forms to define Bott-Cattaneo-Rossi invariant yields infinitely many cohomology classes of Emb(R n , R m ) if m, n ≥ 3 odd and m > n + 2. We observe that half of these classes are non-trivial, along a line similar to Cattaneo-CottaRamusino-Longoni [CCL].

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“…5 /. We will see here that (5) is equal to 1 2 times the second invariant in Cattaneo-Rossi [7], which, in this degree, was first obtained by Bott [4]:…”

Section: Example 23 Fromsupporting

confidence: 63%

Section: Lemma 411mentioning

confidence: 99%

Configuration space integral for longn–knots and the Alexander polynomial

Watanabe

2007

Algebr. Geom. Topol.

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“…There are many subtleties in defining surface operators 30 which we do not discuss here. That aside, we expect that for such a 'line+point' defect to exist in the bulk, the following analogue of the delta function in Eq.…”

Section: The Bf + Bb Description Of Mtc'smentioning

confidence: 99%

Three-dimensional topological lattice models with surface anyons

Keyserlingk¹,

Burnell²,

Simon³

2013

Phys. Rev. B

126

186

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We study a class of three dimensional exactly solvable models of topological matter first put forward by Wang [arXiv:1104.2632v2]. While these are not models of interacting fermions, they may well capture the topological behavior of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a ν = 1/2 bosonic fractional quantum Hall effect in its topological limit, and these considerations help motivate an effective field theoretic description for the lattice models as variants of bF theories. Our special example of the 3D semion model captures much of the behavior of more general 'confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk and we discuss how more general models may have some confined and some deconfined excitations. Having seen that there exist lattice models whose surfaces have the same topological order as a bosonic fractional quantum Hall effect on a confining bulk, we construct a lattice model whose surface has similar topological order to a fermionic quantum hall effect. We find that in these models a fermion is always deconfined in the three dimensional bulk.

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“…Even though not justified in terms of the BV formalism, this choice of propagator was done before in [2] for Chern-Simons theory out of purely topological reasons, and later extended to BF theories in [7]. A propagator with these properties also appears in [13] for the Poisson sigma model on the interior of a polygon.…”

Section: Bv Formalism and Zero Modesmentioning

confidence: 99%

Effective Batalin–Vilkovisky Theories, Equivariant Configuration Spaces and Cyclic Chains

Cattaneo

Felder

2010

Progress in Mathematics

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Summary. Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L∞-quasi-isomorphic to its cohomology. The construction of the L∞-map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of R d with standard volume form, we obtain an L∞-morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin-Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.

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Wilson Surfaces and Higher Dimensional Knot Invariants

Cited by 44 publications

References 16 publications

Configuration space integral for longn–knots and the Alexander polynomial

Configuration space integral for longn–knots and the Alexander polynomial

Three-dimensional topological lattice models with surface anyons

Effective Batalin–Vilkovisky Theories, Equivariant Configuration Spaces and Cyclic Chains

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