2,435 research outputs found
Matrix factorizations and link homology
For each positive integer n the HOMFLY polynomial of links specializes to a
one-variable polynomial that can be recovered from the representation theory of
quantum sl(n). For each such n we build a doubly-graded homology theory of
links with this polynomial as the Euler characteristic. The core of our
construction utilizes the theory of matrix factorizations, which provide a
linear algebra description of maximal Cohen-Macaulay modules on isolated
hypersurface singularities.Comment: 108 pages, 61 figures, latex, ep
Counting factorizations of Coxeter elements into products of reflections
In this paper, we count factorizations of Coxeter elements in well-generated
complex reflection groups into products of reflections. We obtain a simple
product formula for the exponential generating function of such factorizations,
which is expressed uniformly in terms of natural parameters of the group. In
the case of factorizations of minimal length, we recover a formula due to P.
Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex
case. For the symmetric group, our formula specializes to a formula of D. M.
Jackson.Comment: 38 pages, including 18 pages appendix. To appear in Journal of the
London Mathematical Society. v3: minor changes and corrected references; v2:
added extended discussion on the definition of Coxeter element
Hurwitz equivalence of braid monodromies and extremal elliptic surfaces
We discuss the equivalence between the categories of certain ribbon graphs
and subgroups of the modular group and use it to construct
exponentially large families of not Hurwitz equivalent simple braid monodromy
factorizations of the same element. As an application, we also obtain
exponentially large families of {\it topologically} distinct algebraic objects
such as extremal elliptic surfaces, real trigonal curves, and real elliptic
surfaces
Higher melonic theories
We classify a large set of melonic theories with arbitrary -fold
interactions, demonstrating that the interaction vertices exhibit a range of
symmetries, always of the form for some , which may be .
The number of different theories proliferates quickly as increases above
and is related to the problem of counting one-factorizations of complete
graphs. The symmetries of the interaction vertex lead to an effective
interaction strength that enters into the Schwinger-Dyson equation for the
two-point function as well as the kernel used for constructing higher-point
functions.Comment: 43 pages, 12 figure
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