602 research outputs found
A unified interpretation of several combinatorial dualities
AbstractSeveral combinatorial structures exhibit a duality relation that yields interesting theorems, and, sometimes, useful explanations or interpretations of results that do not concern duality explicitly. We present a common characterization of the duality relations associated with matroids, clutters (Sperner families), oriented matroids, and weakly oriented matroids. The same conditions characterize the orthogonality relation on certain families of vector spaces. This leads to a notion of abstract duality
Multiplicative duality, q-triplet and (mu,nu,q)-relation derived from the one-to-one correspondence between the (mu,nu)-multinomial coefficient and Tsallis entropy Sq
We derive the multiplicative duality "q1/q" and other typical mathematical
structures as the special cases of the (mu,nu,q)-relation behind Tsallis
statistics by means of the (mu,nu)-multinomial coefficient. Recently the
additive duality "q2-q" in Tsallis statistics is derived in the form of the
one-to-one correspondence between the q-multinomial coefficient and Tsallis
entropy. A slight generalization of this correspondence for the multiplicative
duality requires the (mu,nu)-multinomial coefficient as a generalization of the
q-multinomial coefficient. This combinatorial formalism provides us with the
one-to-one correspondence between the (mu,nu)-multinomial coefficient and
Tsallis entropy Sq, which determines a concrete relation among three parameters
mu, nu and q, i.e., nu(1-mu)+1=q which is called "(mu,nu,q)-relation" in this
paper. As special cases of the (mu,nu,q)-relation, the additive duality and the
multiplicative duality are recovered when nu=1 and nu=q, respectively. As other
special cases, when nu=2-q, a set of three parameters (mu,nu,q) is identified
with the q-triplet (q_{sen},q_{rel},q_{stat}) recently conjectured by Tsallis.
Moreover, when nu=1/q, the relation 1/(1-q_{sen})=1/alpha_{min}-1/alpha_{max}
in the multifractal singularity spectrum f(alpha) is recovered by means of the
(mu,nu,q)-relation.Comment: 20 page
Higher Cluster Categories and QFT Dualities
We present a unified mathematical framework that elegantly describes
minimally SUSY gauge theories in even dimension, ranging from to , and
their dualities. This approach combines recent developments on graded quiver
with potentials, higher Ginzburg algebras and higher cluster categories (also
known as -cluster categories). Quiver mutations studied in the context of
mathematics precisely correspond to the order dualities of the gauge
theories. Our work suggests that these equivalences of quiver gauge theories
sit inside an infinite family of such generalized dualities, whose physical
interpretation is yet to be understood.Comment: 61 pages, 30 figure
Quantum Gravity
General lectures on quantum gravity.Comment: Lectures given at Karpacz. 40 pages, submitted to Lecture Notes in
Physics. Bigger figure
Embedding Fractional Quantum Hall Solitons in M-theory Compactifications
We engineer U(1)^n Chern-Simons type theories describing fractional quantum
Hall solitons (QHS) in 1+2 dimensions from M-theory compactified on eight
dimensional hyper-K\"{a}hler manifolds as target space of N=4 sigma model.
Based on M-theory/Type IIA duality, the systems can be modeled by considering
D6-branes wrapping intersecting Hirzebruch surfaces F_0's arranged as ADE
Dynkin Diagrams and interacting with higher dimensional R-R gauge fields. In
the case of finite Dynkin quivers, we recover well known values of the filling
factor observed experimentally including Laughlin, Haldane and Jain series.Comment: Latex, 14 pages. Modified version, to appear in IJGMM
BPS states in the Omega-background and torus knots
We clarify some issues concerning the central charges saturated by the
extended objects in the SUSY gauge theory in the
-background. The configuration involving the monopole localized at the
domain wall is considered in some details. At the rational ratio
the trajectory of the monopole
provides the torus knot in the squashed three-sphere. Using the
relation between the integrable systems of Calogero type at the rational
couplings and the torus knots we interpret this configuration in terms of the
auxiliary quiver theory or theory with nontrivial boundary
conditions. This realization can be considered as the AGT-like representation
of the torus knot invariants.Comment: 39 pages, 6 figure
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