Multiplicity-free Quantum 6 j-Symbols for
Nawata, Satoshi; Pichai, Ramadevi; Zodinmawia
2013-12-01
We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6 j-symbols for . The expression is a natural generalization of the quantum 6 j-symbols for obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.
Multiplicity-free Quantum 6j-Symbols for
Nawata, Satoshi; Ramadevi, P.; Zodinmawia
2013-01-01
We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for U_q(sl_N). The expression is a natural generalization of the quantum 6j-symbols for U_q(sl_2) obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.
Dilogarithme Quantique et 6j-Symboles Cycliques
Baseilhac, Stephane
2002-02-01
Let {W}_N be a quantized Borel subalgebra of U_q(sl(2,mc)), specialized at a primitive root of unity omega = exp(2iπ/N) of odd order N >1. One shows that the 6j-symbols of cyclic representations of {W}_N are representations of the canonical element of a certain extension of the Heisenberg double of {W}_N. This canonical element is a twisted q-dilogarithm. In particular, one gives explicit formulas for these 6j-symbols, and one constructs partial symmetrizations of them, the c-6j-symboles. The latters are at the basis of the construction of the quantum hyperbolic invariants of 3-manifolds.
Exact computation of the 3-j and 6-j symbols
A simple FORTRAN program for the exaxt computation of 3-j and 6-j symbols has been written for the VAX with VMS version v5.1 in our university's computing center. It goes beyond and contains all of the 3-j and 6-j symbols evaluated in the book by M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr. The 3-j symbols up to (30/m1 30/m2 30/m3) and 6-j symbols up to {20/20 20/20 20/20} can be computed exactly by this program. Approximate values for larger j's up to (200/m1 200/m2 200/m3) and {200/200 200/200 200/220} can also be computed by this program. (orig.)
The 6j-symbol: recursion, correlations and asymptotics
We study the asymptotic expansion of the {6j}-symbol using the Schulten-Gordon recursion relations. We focus on the particular case of the isosceles tetrahedron and we provide explicit formulas for up to the third-order corrections beyond the leading order. Moreover, in the framework of spinfoam models for 3D quantum gravity, we show how these recursion relations can be used to derive Ward-Takahashi-like identities between the expectation values of graviton-like spinfoam correlations.
6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories
Teschner, J.; Vartanov, G.S.
2012-02-15
We revisit the definition of the 6j-symbols from the modular double of U{sub q}(sl(2,R)), referred to as b-6j symbols. Our new results are (i) the identification of particularly natural normalization conditions, and (ii) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. We furthermore observe a close relation with the problem to quantize natural Darboux coordinates for moduli spaces of flat connections on Riemann surfaces related to the Fenchel-Nielsen coordinates. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of three-dimensional N=2 supersymmetric gauge theories. (orig.)
Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions
We present the first complete derivation of the well-known asymptotic expansion of the SU(2) 6j symbol using a coherent state approach, in particular we succeed in computing the determinant of the Hessian matrix. To do so, we smear the coherent states and perform a partial stationary point analysis with respect to the smearing parameters. This allows us to transform the variables from group elements to dihedral angles of a tetrahedron resulting in an effective action, which coincides with the action of first order Regge calculus associated to a tetrahedron. To perform the remaining stationary point analysis, we compute its Hessian matrix and obtain the correct measure factor. Furthermore, we expand the discussion of the asymptotic formula to next to leading order terms, prove some of their properties and derive a recursion relation for the full 6j symbol
On the fusion in SL(2)-WZNW models and 6j symbols of Uqsl(2) x Uq'osp(1 vertical stroke 2)
We introduce a novel method to determine 6j-symbols of quantum groups. This method is inspired by the methods used in the determination of fusing matrices of WZNW models. With this method we determine the 6j-symbols of the quantum group Uqsl(2) and the super quantum group Uqosp(1 vertical stroke 2). We present the 6j-symbols as a recurrence relation and its initial values. The 6j-symbols transform between the s-channel and the u-channel decomposition of the invariants of the four-fold tensor product of modules of a quantum group. These invariants fulfil certain difference equations. We set one of the representations in the invariant to the fundamental representation, and deduce a system of linear equations for the initial values of the recurrence relation determining the 6j-symbols. (orig.)
Koesling, Jens
2010-06-15
We introduce a novel method to determine 6j-symbols of quantum groups. This method is inspired by the methods used in the determination of fusing matrices of WZNW models. With this method we determine the 6j-symbols of the quantum group U{sub q}sl(2) and the super quantum group U{sub q}osp(1 vertical stroke 2). We present the 6j-symbols as a recurrence relation and its initial values. The 6j-symbols transform between the s-channel and the u-channel decomposition of the invariants of the four-fold tensor product of modules of a quantum group. These invariants fulfil certain difference equations. We set one of the representations in the invariant to the fundamental representation, and deduce a system of linear equations for the initial values of the recurrence relation determining the 6j-symbols. (orig.)
New algebraic tables of SU(2) quantities
Formulas for Clebsch-Gordan Coefficients, 6-j symbols and 9-j symbols of SU(2) are presented in a ready-to-program way for obtaining algebraic tables. An excerpt of the complete tables are also presented. (Author)
Uq(sl(2)) invariant operators and minimal theories fusion matrices
The existence of Uq(sl(2)) invariant operators for qp=1 leads to relations for the quantum Clebsch-Gordan kernels and for the quantum 6j-symbols (= fusion matrices). These relations effectively reduce some equalities, inherited from the generic q case, and imply, in particular, that the polynomial identities for the quantum 6j-symbols are consistent with the minimal theories chiral fusion rules. (author). 26 refs
Weak quasitriangular Quasi-Hopf algebra structure of minimal models
Teschner, J. A.
1995-01-01
The chiral vertex operators for the minimal models are constructed and used to define a fusion product of representations. The existence of commutativity and associativity operations is proved. The matrix elements of the associativity operations are shown to be given in terms of the 6-j symbols of the weak quasitriangular quasi-Hopf algebra obtained by truncating $\\usl$ at roots of unity.
Traces on ideals in pivotal categories
Geer, Nathan; Virelizier, Alexis
2011-01-01
We extend the notion of an ambidextrous trace on an ideal (developed by the first two authors) to the setting of a pivotal category. We show that under some conditions, these traces lead to invariants of colored spherical graphs (and so to modified 6j-symbols).
Marinelli, Dimitri; Aquilanti, Vincenzo; Anderson, Roger W; Bitencourt, Ana Carla P; Ragni, Mirco
2014-01-01
A unified vision of the symmetric coupling of angular momenta and of the quantum mechanical volume operator is illustrated. The focus is on the quantum mechanical angular momentum theory of Wigner's 6j symbols and on the volume operator of the symmetric coupling in spin network approaches: here, crucial to our presentation are an appreciation of the role of the Racah sum rule and the simplification arising from the use of Regge symmetry. The projective geometry approach permits the introduction of a symmetric representation of a network of seven spins or angular momenta. Results of extensive computational investigations are summarized, presented and briefly discussed.
Modular data for the extended Haagerup subfactor
Gannon, Terry
2016-01-01
We compute the modular data (that is, the $S$ and $T$ matrices) for the centre of the extended Haagerup subfactor. The full structure (i.e. the associativity data, also known as 6-$j$ symbols or $F$ matrices) still appears to be inaccessible. Nevertheless, starting with just the number of simple objects and their dimensions (obtained by a combinatorial argument in arXiv:1404.3955) we find that it is surprisingly easy to leverage knowledge of the representation theory of $SL (2, \\mathbb Z)$ into a complete description of the modular data. We also investigate the possible character vectors associated with this modular data.
Maslov indices, Poisson brackets, and singular differential forms
Esterlis, Ilya; Hedeman, Austin; Littlejohn, Robert G
2014-01-01
Maslov indices are integers that appear in semiclassical wave functions and quantization conditions. They are often notoriously difficult to compute. We present methods of computing the Maslov index that rely only on typically elementary Poisson brackets and simple linear algebra. We also present a singular differential form, whose integral along a curve gives the Maslov index of that curve. The form is closed but not exact, and transforms by an exact differential under canonical transformations. We illustrate the method with the $6j$-symbol, which is important in angular momentum theory and in quantum gravity.
Maslov indices, Poisson brackets, and singular differential forms
Esterlis, I.; Haggard, H. M.; Hedeman, A.; Littlejohn, R. G.
2014-06-01
Maslov indices are integers that appear in semiclassical wave functions and quantization conditions. They are often notoriously difficult to compute. We present methods of computing the Maslov index that rely only on typically elementary Poisson brackets and simple linear algebra. We also present a singular differential form, whose integral along a curve gives the Maslov index of that curve. The form is closed but not exact, and transforms by an exact differential under canonical transformations. We illustrate the method with the 6j-symbol, which is important in angular-momentum theory and in quantum gravity.
Simplification of the spectral analysis of the volume operator in loop quantum gravity
The volume operator plays a crucial role in the definition of the quantum dynamics of loop quantum gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulae for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which in turn are given in terms of Racah's formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far not even numerically the spectrum could be accessed. In this paper, we demonstrate that by means of the Elliot-Biedenharn identity one can get rid of all the 6j symbols for any valence of the gauge-invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge-invariant 4-vertex. The techniques derived in this paper could also be of use for the analysis of spin-spin interaction Hamiltonians of many-particle problems in atomic and nuclear physics
Conformal bootstrap, universality and gravitational scattering
Steven Jackson
2015-12-01
Full Text Available We use the conformal bootstrap equations to study the non-perturbative gravitational scattering between infalling and outgoing particles in the vicinity of a black hole horizon in AdS. We focus on irrational 2D CFTs with large c and only Virasoro symmetry. The scattering process is described by the matrix element of two light operators (particles between two heavy states (BTZ black holes. We find that the operator algebra in this regime is (i universal and identical to that of Liouville CFT, and (ii takes the form of an exchange algebra, specified by an R-matrix that exactly matches the scattering amplitude of 2+1 gravity. The R-matrix is given by a quantum 6j-symbol and the scattering phase by the volume of a hyperbolic tetrahedron. We comment on the relevance of our results to scrambling and the holographic reconstruction of the bulk physics near black hole horizons.
The Barrett-Crane model: measure factor
Kaminski, Wojciech
2013-01-01
The original spin foam model construction for 4D gravity by Barrett and Crane suffers from several fatal issues. In the simple examples of the vertex amplitude they can be summarized as the existence of contributions to the asymptotics from non geometric configurations. Even restricted to geometric contributions the amplitude is not completely worked out. While the phase is known to be the Regge action, the so called measure factor has remained mysterious for a decade. In the toy model case of the 6j symbol this measure factor has a nice geometric interpretation of V^{-1/2} leading to speculations that a similar interpretation should be possible also in the 4D case. In this paper we provide the first geometric interpretation of the geometric part of the asymptotic for the spin foam consisting of two glued 4-simplices (decomposition of the 4-sphere) in the Barrett-Crane model in the large internal spin regime.
The Barrett-Crane model: asymptotic measure factor
Kamiński, Wojciech; Steinhaus, Sebastian
2014-04-01
The original spin foam model construction for 4D gravity by Barrett and Crane suffers from a few troubling issues. In the simple examples of the vertex amplitude they can be summarized as the existence of contributions to the asymptotics from non-geometric configurations. Even restricted to geometric contributions the amplitude is not completely worked out. While the phase is known to be the Regge action, the so-called measure factor has remained mysterious for a decade. In the toy model case of the 6j symbol this measure factor has a nice geometric interpretation of V-1/2 leading to speculations that a similar interpretation should be possible also in the 4D case. In this paper we provide the first geometric interpretation of the geometric part of the asymptotic for the spin foam consisting of two glued 4-simplices (decomposition of the 4-sphere) in the Barrett-Crane model in the large internal spin regime.
The Barrett–Crane model: asymptotic measure factor
The original spin foam model construction for 4D gravity by Barrett and Crane suffers from a few troubling issues. In the simple examples of the vertex amplitude they can be summarized as the existence of contributions to the asymptotics from non-geometric configurations. Even restricted to geometric contributions the amplitude is not completely worked out. While the phase is known to be the Regge action, the so-called measure factor has remained mysterious for a decade. In the toy model case of the 6j symbol this measure factor has a nice geometric interpretation of V−1/2 leading to speculations that a similar interpretation should be possible also in the 4D case. In this paper we provide the first geometric interpretation of the geometric part of the asymptotic for the spin foam consisting of two glued 4-simplices (decomposition of the 4-sphere) in the Barrett–Crane model in the large internal spin regime. (paper)
Bitencourt, A C P; Ragni, M; Anderson, R W; Aquilanti, V
2012-01-01
Increasing interest is being dedicated in the last few years to the issues of exact computations and asymptotics of spin networks. The large-entries regimes (semiclassical limits) occur in many areas of physics and chemistry, and in particular in discretization algorithms of applied quantum mechanics. Here we extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient, enlightening the insight gained by exploiting its self-dual properties and studying it as a function of two (discrete) variables. This arises from its original definition as an (orthogonal) angular momentum recoupling matrix. Progress also derives from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational advances -based on traditional and new recurrence relations- which allow an interpretation of the...
The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second order difference equation which, by a complex phase change, we turn into a discrete Schroedinger-like equation. The introduction of discrete potential-like functions reveals the surprising crucial role here of the Regge symmetries, first discovered for the quantum mechanical 6j symbols; insight is provided into the underlying geometric features. The spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary ''quantum of space'', and a transparent asymptotic picture emerges of the semiclassical and classical regimes. The definition of coordinates adapted to Regge symmetry allows the construction of a novel set of discrete orthogonal polynomials.
Generating function method and its applications to Quantum, Nuclear and the Classical Groups
Hage-Hassan, Mehdi
2012-01-01
The generating function method that we had developing has various applications in physics and not only interress undergraduate students but also physicists. We solve simply difficult problems or unsolved commonly used in quantum, nuclear and group theory textbooks. We find simply: the generating function of the harmonic oscillator, the Feynman propagators of the oscillator and the oscillator in uniform magnetic field. We derive the invariants of SU(2) and the expressions of 3-j,6-j symbols. We find also the octonions or Hurwitz quadratic transformations. We show that the cross-product exist only in E3 and E7. We determine the {p} representation of hydrogen atom in three and n-dimensions. We generalize the Cramer's rule for the calculation of the rotational spectrum of the nucleus. We find the expression of the Hamiltonian in terms of quasi-bosons for study the collective vibration. We determine the basis and the expressions of 3-j symbols of SU (3) and SU(n).We find the Schr\\"odinger equation from Hamilton-Ja...
Recurrence relations for spin foam vertices
We study recurrence relations for various Wigner 3nj-symbols and the non-topological 10j-symbol. For the 6j- and the 15j-symbols which correspond to basic amplitudes of 3d and 4d topological spin foam models, recurrence relations are obtained from the invariance under Pachner moves and can be interpreted as quantizations of the constraints of the underlying classical field theories. We also derive recurrences from the action of holonomy operators on spin network functionals, making a more precise link between the topological Pachner moves and the classical constraints. Interestingly, our recurrence relations apply to any SU(2) invariant symbol, depending on the cycles of the corresponding spin network graph. Another method is used for non-topological objects such as the 10j-symbol and pseudo-isosceles 6j-symbols. The recurrence relations are also interpreted in terms of elementary geometric properties. Finally, we discuss the extension of the recurrences to take into account boundary states which leads to equations similar to Ward identities for correlation functions in the Barrett-Crane model.
Braiding knots with topological strings
Gu, Jie
2015-08-15
For an arbitrary knot in a three-sphere, the Ooguri-Vafa conjecture associates to it a unique stack of branes in type A topological string on the resolved conifold, and relates the colored HOMFLY invariants of the knot to the free energies on the branes. For torus knots, we use a modified version of the topological recursion developed by Eynard and Orantin to compute the free energies on the branes from the Aganagic-Vafa spectral curves of the branes, and find they are consistent with the known colored HOMFLY knot invariants a la the Ooguri-Vafa conjecture. In addition our modified topological recursion can reproduce the correct closed string free energies, which encode the information of the background geometry. We conjecture the modified topological recursion is applicable for branes associated to hyperbolic knots as well, encouraged by the observation that the modified topological recursion yields the correct planar closed string free energy from the Aganagic-Vafa spectral curves of hyperbolic knots. This has implications for the knot theory concerning distinguishing mutant knots with colored HOMFLY invariants. Furthermore, for hyperbolic knots, we present methods to compute colored HOMFLY invariants in nonsymmetric representations of U(N). The key step in this computation is computing quantum 6j-symbols in the quantum group U{sub q}(sl{sub N}).
Simple-current symmetries, rank-level duality, and linear skein relations for Chern-Simons graphs
A previously proposed two-step algorithm for calculating the expectation values of arbitrary Chern-Simons graphs fails to determine certain crucial signs. The step which involves calculating tetrahedra by solving certain non-linear equations is repaired by introducing additional linear equations. The step which involves reducing arbitrary graphs to sums of products of tetrahedra remains seriously disabled, apart from a few exceptional cases. As a first step towards a new algorithm for general graphs we find useful linear equations for those special graphs which support knots and links. Using the improved set of equations for tetrahedra we examine the symmetries between tetrahedra generated by arbitrary simple currents. Along the way we describe the simple, classical origin of simple-current charges. The improved skein relations also lead to exact identities between planar tetrahedra in level K G(N) and level N G(K) Chern-Simons theories, where G(N) denotes a classical group. These results are recast as WZW braid-matrix identities and as identities between quantum 6j-symbols at appropriate roots of unity. We also obtain the transformation properties of arbitary graphs, knots, and links under simple-current symmetries and rank-level duality. For links with knotted components this requires precise control of the braid eigenvalue permutation signs, which we obtain from plethysm and an explicit expression for the (multiplicity-free) signs, valid for all compact gauge groups and all fusion products. (orig.)
Holographic hierarchy in the Gaussian matrix model via the fuzzy sphere
The Gaussian Hermitian matrix model was recently proposed to have a dual string description with worldsheets mapping to a sphere target space. The correlators were written as sums over holomorphic (Belyi) maps from worldsheets to the two-dimensional sphere, branched over three points. We express the matrix model correlators by using the fuzzy sphere construction of matrix algebras, which can be interpreted as a string field theory description of the Belyi strings. This gives the correlators in terms of trivalent ribbon graphs that represent the couplings of irreducible representations of su(2), which can be evaluated in terms of 3j and 6j symbols. The Gaussian model perturbed by a cubic potential is then recognised as a generating function for Ponzano–Regge partition functions for 3-manifolds having the worldsheet as boundary, and equipped with boundary data determined by the ribbon graphs. This can be viewed as a holographic extension of the Belyi string worldsheets to membrane worldvolumes, forming part of a holographic hierarchy linking, via the large N expansion, the zero-dimensional QFT of the Matrix model to 2D strings and 3D membranes
Holographic hierarchy in the Gaussian matrix model via the fuzzy sphere
Garner, David; Ramgoolam, Sanjaye
2013-10-01
The Gaussian Hermitian matrix model was recently proposed to have a dual string description with worldsheets mapping to a sphere target space. The correlators were written as sums over holomorphic (Belyi) maps from worldsheets to the two-dimensional sphere, branched over three points. We express the matrix model correlators by using the fuzzy sphere construction of matrix algebras, which can be interpreted as a string field theory description of the Belyi strings. This gives the correlators in terms of trivalent ribbon graphs that represent the couplings of irreducible representations of su(2), which can be evaluated in terms of 3j and 6j symbols. The Gaussian model perturbed by a cubic potential is then recognised as a generating function for Ponzano-Regge partition functions for 3-manifolds having the worldsheet as boundary, and equipped with boundary data determined by the ribbon graphs. This can be viewed as a holographic extension of the Belyi string worldsheets to membrane worldvolumes, forming part of a holographic hierarchy linking, via the large N expansion, the zero-dimensional QFT of the Matrix model to 2D strings and 3D membranes. Note that if, after removing the white vertices, the graph contains a blue edge connecting to the same black vertex at both ends, then the triangulation generated from the black edges will contain faces that resemble cut discs. These faces are triangles with two of the edges identified.
Holographic hierarchy in the Gaussian matrix model via the fuzzy sphere
Garner, David, E-mail: d.p.r.garner@qmul.ac.uk; Ramgoolam, Sanjaye, E-mail: s.ramgoolam@qmul.ac.uk
2013-10-01
The Gaussian Hermitian matrix model was recently proposed to have a dual string description with worldsheets mapping to a sphere target space. The correlators were written as sums over holomorphic (Belyi) maps from worldsheets to the two-dimensional sphere, branched over three points. We express the matrix model correlators by using the fuzzy sphere construction of matrix algebras, which can be interpreted as a string field theory description of the Belyi strings. This gives the correlators in terms of trivalent ribbon graphs that represent the couplings of irreducible representations of su(2), which can be evaluated in terms of 3j and 6j symbols. The Gaussian model perturbed by a cubic potential is then recognised as a generating function for Ponzano–Regge partition functions for 3-manifolds having the worldsheet as boundary, and equipped with boundary data determined by the ribbon graphs. This can be viewed as a holographic extension of the Belyi string worldsheets to membrane worldvolumes, forming part of a holographic hierarchy linking, via the large N expansion, the zero-dimensional QFT of the Matrix model to 2D strings and 3D membranes.
Holographic Hierarchy in the Gaussian Matrix Model via the Fuzzy Sphere
Garner, David
2013-01-01
The Gaussian Hermitian matrix model was recently proposed to have a dual string description with worldsheets mapping to a sphere target space. The correlators were written as sums over holomorphic (Belyi) maps from worldsheets to the two-dimensional sphere, branched over three points. We express the matrix model correlators by using the fuzzy sphere construction of matrix algebras, which can be interpreted as a string field theory description of the Belyi strings. This gives the correlators in terms of trivalent ribbon graphs that represent the couplings of irreducible representations of su(2), which can be evaluated in terms of 3j and 6j symbols. The Gaussian model perturbed by a cubic potential is then recognised as a generating function for Ponzano-Regge partition functions for 3-manifolds having the worldsheet as boundary, and equipped with boundary data determined by the ribbon graphs. This can be viewed as a holographic extension of the Belyi string worldsheets to membrane worldvolumes, forming part of ...
A 2-categorical state sum model
Baratin, Aristide, E-mail: abaratin@uwaterloo.ca [Department of Applied Mathematics, University of Waterloo, 200 University Ave W, Waterloo, Ontario N2L 3G1 (Canada); Freidel, Laurent, E-mail: lfreidel@perimeterinstitute.ca [Perimeter Institute for Theoretical Physics, 31 Caroline Str. N, Waterloo, Ontario N2L 2Y5 (Canada)
2015-01-15
It has long been argued that higher categories provide the proper algebraic structure underlying state sum invariants of 4-manifolds. This idea has been refined recently, by proposing to use 2-groups and their representations as specific examples of 2-categories. The challenge has been to make these proposals fully explicit. Here, we give a concrete realization of this program. Building upon our earlier work with Baez and Wise on the representation theory of 2-groups, we construct a four-dimensional state sum model based on a categorified version of the Euclidean group. We define and explicitly compute the simplex weights, which may be viewed a categorified analogue of Racah-Wigner 6j-symbols. These weights solve a hexagon equation that encodes the formal invariance of the state sum under the Pachner moves of the triangulation. This result unravels the combinatorial formulation of the Feynman amplitudes of quantum field theory on flat spacetime proposed in A. Baratin and L. Freidel [Classical Quantum Gravity 24, 2027–2060 (2007)] which was shown to lead after gauge-fixing to Korepanov’s invariant of 4-manifolds.
Braiding knots with topological strings
For an arbitrary knot in a three-sphere, the Ooguri-Vafa conjecture associates to it a unique stack of branes in type A topological string on the resolved conifold, and relates the colored HOMFLY invariants of the knot to the free energies on the branes. For torus knots, we use a modified version of the topological recursion developed by Eynard and Orantin to compute the free energies on the branes from the Aganagic-Vafa spectral curves of the branes, and find they are consistent with the known colored HOMFLY knot invariants a la the Ooguri-Vafa conjecture. In addition our modified topological recursion can reproduce the correct closed string free energies, which encode the information of the background geometry. We conjecture the modified topological recursion is applicable for branes associated to hyperbolic knots as well, encouraged by the observation that the modified topological recursion yields the correct planar closed string free energy from the Aganagic-Vafa spectral curves of hyperbolic knots. This has implications for the knot theory concerning distinguishing mutant knots with colored HOMFLY invariants. Furthermore, for hyperbolic knots, we present methods to compute colored HOMFLY invariants in nonsymmetric representations of U(N). The key step in this computation is computing quantum 6j-symbols in the quantum group Uq(slN).
Refined Chern-Simons Theory in Genus Two
Arthamonov, Semeon
2015-01-01
Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117, opening a question if it extends to any genus. Our construction is built upon a q,t-deformation of the square of q-6j symbol of U_q(sl_2), which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply ...