822 research outputs found
Twisted Alexander polynomials of 2-bridge knots
We investigate the twisted Alexander polynomial of a 2-bridge knot associated
to a Fox coloring. For several families of 2-bridge knots, including but not
limited to, torus knots and genus-one knots, we derive formulae for these
twisted Alexander polynomials. We use these formulae to confirm a conjecture of
Hirasawa and Murasugi for these knots.Comment: 29 pages, 2 figure
Polynomial Invariants for Arbitrary Rank Weakly-Colored Stranded Graphs
Polynomials on stranded graphs are higher dimensional generalization of Tutte
and Bollob\'as-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we
deepen the analysis of the polynomial invariant defined on rank 3
weakly-colored stranded graphs introduced in arXiv:1301.1987. We successfully
find in dimension a modified Euler characteristic with
parameters. Using this modified invariant, we extend the rank 3 weakly-colored
graph polynomial, and its main properties, on rank 4 and then on arbitrary rank
weakly-colored stranded graphs.Comment: Basic definitions overlap with arXiv:1301.198
Majorana Fermion Quantum Mechanics for Higher Rank Tensors
We study quantum mechanical models in which the dynamical degrees of freedom
are real fermionic tensors of rank five and higher. They are the non-random
counterparts of the Sachdev-Ye-Kitaev (SYK) models where the Hamiltonian
couples six or more fermions. For the tensors of rank five, there is a unique
symmetric sixth-order Hamiltonian leading to a solvable large
limit dominated by the melonic diagrams. We solve for the complete energy
spectrum of this model when and deduce exact expressions for all the
eigenvalues. The subset of states which are gauge invariant exhibit
degeneracies related to the discrete symmetries of the gauged model. We also
study quantum chaos properties of the tensor model and compare them with those
of the SYK model. For there is a rapidly growing number of
invariant tensor interactions. We focus on those of them that are
maximally single-trace - their stranded diagrams stay connected when any set of
colors is erased. We present a general discussion of why the tensor
models with maximally single-trace interactions have large limits dominated
by the melonic diagrams. We solve the large Schwinger-Dyson equations for
the higher rank Majorana tensor models and show that they match those of the
corresponding SYK models exactly. We also study other gauge invariant operators
present in the tensor models.Comment: 36 pages, 19 figures, 2 tables, v3: some clarifications and
references adde
Parametric Representation of Rank d Tensorial Group Field Theory: Abelian Models with Kinetic Term
We consider the parametric representation of the amplitudes of Abelian models
in the so-called framework of rank Tensorial Group Field Theory. These
models are called Abelian because their fields live on . We concentrate
on the case when these models are endowed with particular kinetic terms
involving a linear power in momenta. New dimensional regularization and
renormalization schemes are introduced for particular models in this class: a
rank 3 tensor model, an infinite tower of matrix models over
, and a matrix model over . For all divergent amplitudes, we
identify a domain of meromorphicity in a strip determined by the real part of
the group dimension . From this point, the ordinary subtraction program is
applied and leads to convergent and analytic renormalized integrals.
Furthermore, we identify and study in depth the Symanzik polynomials provided
by the parametric amplitudes of generic rank Abelian models. We find that
these polynomials do not satisfy the ordinary Tutte's rules
(contraction/deletion). By scrutinizing the "face"-structure of these
polynomials, we find a generalized polynomial which turns out to be stable only
under contraction.Comment: 69 pages, 35 figure
A reverse Sidorenko inequality
Let be a graph allowing loops as well as vertex and edge weights. We
prove that, for every triangle-free graph without isolated vertices, the
weighted number of graph homomorphisms satisfies the inequality
where denotes the degree of vertex in . In particular, one has for every -regular
triangle-free . The triangle-free hypothesis on is best possible. More
generally, we prove a graphical Brascamp-Lieb type inequality, where every edge
of is assigned some two-variable function. These inequalities imply tight
upper bounds on the partition function of various statistical models such as
the Ising and Potts models, which includes independent sets and graph
colorings.
For graph colorings, corresponding to , we show that the
triangle-free hypothesis on may be dropped; this is also valid if some of
the vertices of are looped. A corollary is that among -regular graphs,
maximizes the quantity for every and ,
where counts proper -colorings of .
Finally, we show that if the edge-weight matrix of is positive
semidefinite, then This implies that among -regular graphs,
maximizes . For 2-spin Ising models, our results give a
complete characterization of extremal graphs: complete bipartite graphs
maximize the partition function of 2-spin antiferromagnetic models and cliques
maximize the partition function of ferromagnetic models.
These results settle a number of conjectures by Galvin-Tetali, Galvin, and
Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture
by Kahn.Comment: 30 page
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