101 research outputs found
Kontsevich integral for knots and Vassiliev invariants
We review quantum field theory approach to the knot theory. Using holomorphic
gauge we obtain the Kontsevich integral. It is explained how to calculate
Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial
way which can be programmed on a computer. We discuss experimental results and
temporal gauge considerations which lead to representation of Vassiliev
invariants in terms of arrow diagrams. Explicit examples and computational
results are presented.Comment: 25 pages, 17 figure
Generalised Ordinary vs Fully Simple Duality for <i>n</i>-Point Functions and a Proof of the Borot–Garcia-Failde Conjecture
We study a duality for the n-point functions in VEV formalism that we call the ordinary vs fully simple duality. It provides an ultimate generalisation and a proper context for the duality between maps and fully simple maps observed by Borot and Garcia-Failde. Our approach allows to transfer the algebraicity properties between the systems of n-point functions related by this duality, and gives direct tools for the analysis of singularities. As an application, we give a proof of a recent conjecture of Borot and Garcia-Failde on topological recursion for fully simple maps..</p
Principal Roles, Work Demands, and Supports Needed to Implement New Teacher Evaluation
Policy makers at the federal level have embraced an educator effectiveness agenda, which in turn has driven many states across the country to rapidly develop and implement new and more complex teacher evaluation systems. It is increasingly clear that the success of these nascent teacher evaluation systems partly depends on the will, skill, and capacity of school principals, individuals who have historically been tasked with evaluating teachers. School principals have traditionally had, and will in most cases continue to have, primary responsibility for evaluating the 3.7 million public school teachers nationwide. While teacher evaluation innovations present several opportunities for improving instructional supervision and teacher quality, they also involve several challenges, especially on the part of principals. Time demands and cognitive challenges will be inevitable as principals learn about and implement new teacher evaluation systems. Simultaneously, other educational changes going to scale, including Common Core State Standards with aligned assessments and state school accountability systems, will compete for the attention of school leaders and teachers. Negotiating these changes to maximize the positive potential of evaluation reforms requires a commitment by states and districts to resources for training and support as well as policy coherence
Hypercommutative operad as a homotopy quotient of BV
We give an explicit formula for a quasi-isomorphism between the operads
Hycomm (the homology of the moduli space of stable genus 0 curves) and
BV/ (the homotopy quotient of Batalin-Vilkovisky operad by the
BV-operator). In other words we derive an equivalence of Hycomm-algebras and
BV-algebras enhanced with a homotopy that trivializes the BV-operator.
These formulas are given in terms of the Givental graphs, and are proved in
two different ways. One proof uses the Givental group action, and the other
proof goes through a chain of explicit formulas on resolutions of Hycomm and
BV. The second approach gives, in particular, a homological explanation of the
Givental group action on Hycomm-algebras.Comment: minor corrections added, to appear in Comm.Math.Phy
Classical theta constants vs. lattice theta series, and super string partition functions
Recently, various possible expressions for the vacuum-to-vacuum superstring
amplitudes has been proposed at genus . To compare the different
proposals, here we will present a careful analysis of the comparison between
the two main technical tools adopted to realize the proposals: the classical
theta constants and the lattice theta series. We compute the relevant Fourier
coefficients in order to relate the two spaces. We will prove the equivalence
up to genus 4. In genus five we will show that the solutions are equivalent
modulo the Schottky form and coincide if we impose the vanishing of the
cosmological constant.Comment: 21 page
Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian
A long-standing question in string theory is to find the explicit expression
of the bosonic measure, a crucial issue also in determining the superstring
measure. Such a measure was known up to genus three. Belavin and Knizhnik
conjectured an expression for genus four which has been proved in the framework
of the recently introduced vector-valued Teichmueller modular forms. It turns
out that for g>3 the bosonic measure is expressed in terms of such forms. In
particular, the genus four Belavin-Knizhnik "wonderful formula" has a
remarkable extension to arbitrary genus whose structure is deeply related to
the characterization of the Jacobian locus. Furthermore, it turns out that the
bosonic string measure has an elegant geometrical interpretation as generating
the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to
identify forms on the Siegel upper half-space that, if certain conditions
related to the characterization of the Jacobian are satisfied, express the
bosonic measure as a multiresidue in the Siegel upper half-space. We also
suggest that it may exist a super analog on the super Siegel half-space.Comment: 15 pages. Typos corrected, refs. and comments adde
Superpolynomials for toric knots from evolution induced by cut-and-join operators
The colored HOMFLY polynomials, which describe Wilson loop averages in
Chern-Simons theory, possess an especially simple representation for torus
knots, which begins from quantum R-matrix and ends up with a trivially-looking
split W representation familiar from character calculus applications to matrix
models and Hurwitz theory. Substitution of MacDonald polynomials for characters
in these formulas provides a very simple description of "superpolynomials",
much simpler than the recently studied alternative which deforms relation to
the WZNW theory and explicitly involves the Littlewood-Richardson coefficients.
A lot of explicit expressions are presented for different representations
(Young diagrams), many of them new. In particular, we provide the
superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k. The procedure is not
restricted to the fundamental (all antisymmetric) representations and the torus
knots, still in these cases some subtleties persist.Comment: 23 pages + Tables (51 pages
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