432,304 research outputs found
A note on the algebraic engineering of 4D super Yang-Mills theories
Some BPS quantities of 5D quiver gauge theories, like
instanton partition functions or qq-characters, can be constructed as algebraic
objects of the Ding-Iohara-Miki (DIM) algebra. This construction is applied
here to super Yang-Mills theories in four dimensions using a
degenerate version of the DIM algebra. We build up the equivalent of horizontal
and vertical representations, the first one being defined using vertex
operators acting on a free boson's Fock space, while the second one is
essentially equivalent to the action of Vasserot-Shiffmann's Spherical Hecke
central algebra. Using intertwiners, the algebraic equivalent of the
topological vertex, we construct a set of -operators acting on the
tensor product of horizontal modules, and the vacuum expectation values of
which reproduce the instanton partition functions of linear quivers. Analysing
the action of the degenerate DIM algebra on the -operator in the
case of a pure gauge theory, we further identify the degenerate version
of Kimura-Pestun's quiver W-algebra as a certain limit of q-Virasoro algebra.
Remarkably, as previously noticed by Lukyanov, this particular limit reproduces
the Zamolodchikov-Faddeev algebra of the sine-Gordon model.Comment: 13 pages (v3: references added
On the computability of some positive-depth supercuspidal characters near the identity
This paper is concerned with the values of Harish-Chandra characters of a
class of positive-depth, toral, very supercuspidal representations of -adic
symplectic and special orthogonal groups, near the identity element. We declare
two representations equivalent if their characters coincide on a specific
neighbourhood of the identity (which is larger than the neighbourhood on which
Harish-Chandra local character expansion holds). We construct a parameter space
(that depends on the group and a real number ) for the set of
equivalence classes of the representations of minimal depth satisfying some
additional assumptions. This parameter space is essentially a geometric object
defined over \Q. Given a non-Archimedean local field \K with sufficiently
large residual characteristic, the part of the character table near the
identity element for G(\K) that comes from our class of representations is
parameterized by the residue-field points of . The character values
themselves can be recovered by specialization from a constructible motivic
exponential function. The values of such functions are algorithmically
computable. It is in this sense that we show that a large part of the character
table of the group G(\K) is computable
Trading inference effort versus size in CNF Knowledge Compilation
Knowledge Compilation (KC) studies compilation of boolean functions f into
some formalism F, which allows to answer all queries of a certain kind in
polynomial time. Due to its relevance for SAT solving, we concentrate on the
query type "clausal entailment" (CE), i.e., whether a clause C follows from f
or not, and we consider subclasses of CNF, i.e., clause-sets F with special
properties. In this report we do not allow auxiliary variables (except of the
Outlook), and thus F needs to be equivalent to f.
We consider the hierarchies UC_k <= WC_k, which were introduced by the
authors in 2012. Each level allows CE queries. The first two levels are
well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in
KC, that is, f is represented by the set of all prime implicates, while UC_1 =
WC_1 is the same as UC, the class of unit-refutation complete clause-sets
introduced by del Val 1994. We show that for each k there are (sequences of)
boolean functions with polysize representations in UC_{k+1}, but with an
exponential lower bound on representations in WC_k. Such a separation was
previously only know for k=0. We also consider PC < UC, the class of
propagation-complete clause-sets. We show that there are (sequences of) boolean
functions with polysize representations in UC, while there is an exponential
lower bound for representations in PC. These separations are steps towards a
general conjecture determining the representation power of the hierarchies PC_k
< UC_k <= WC_k. The strong form of this conjecture also allows auxiliary
variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the
separation results from the discontinued arXiv:1302.442
Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations
Functional relation for commuting quantum transfer matrices of quantum
integrable models is identified with classical Hirota's bilinear difference
equation. This equation is equivalent to the completely discretized classical
2D Toda lattice with open boundaries. The standard objects of quantum
integrable models are identified with elements of classical nonlinear
integrable difference equation. In particular, elliptic solutions of Hirota's
equation give complete set of eigenvalues of the quantum transfer matrices.
Eigenvalues of Baxter's -operator are solutions to the auxiliary linear
problems for classical Hirota's equation. The elliptic solutions relevant to
Bethe ansatz are studied. The nested Bethe ansatz equations for -type
models appear as discrete time equations of motions for zeros of classical
-functions and Baker-Akhiezer functions. Determinant representations of
the general solution to bilinear discrete Hirota's equation and a new
determinant formula for eigenvalues of the quantum transfer matrices are
obtained.Comment: 32 pages, LaTeX file, no figure
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