474 research outputs found
Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem
The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction
Problem over a fixed template is solvable in polynomial time if the algebra of
polymorphisms associated to the template lies in a Taylor variety, and is
NP-complete otherwise. This paper provides two new characterizations of
finitely generated Taylor varieties. The first characterization is using
absorbing subalgebras and the second one cyclic terms. These new conditions
allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the
authors) and the characterization of locally finite Taylor varieties using weak
near-unanimity terms (proved by McKenzie and Mar\'oti) in an elementary and
self-contained way
Generic Expression Hardness Results for Primitive Positive Formula Comparison
We study the expression complexity of two basic problems involving the
comparison of primitive positive formulas: equivalence and containment. In
particular, we study the complexity of these problems relative to finite
relational structures. We present two generic hardness results for the studied
problems, and discuss evidence that they are optimal and yield, for each of the
problems, a complexity trichotomy
On stratified Mukai flops
We construct a resolution of stratified Mukai flops of type A, D, E_{6, I} by
successively blowing up smooth subvarieties. In the case of E_{6, I}, we
construct a natural functor which induces an isomorphism between the Chow
groups.Comment: use diagrams.st
Quantum geometry of moduli spaces of local systems and representation theory
Let G be a split semi-simple adjoint group, and S an oriented surface with
punctures and special boundary points. We introduce a moduli space P(G,S)
parametrizing G-local system on S with some boundary data, and prove that it
carries a cluster Poisson structure, equivariant under the action of the
cluster modular group M(G,S), containing the mapping class group of S, the
group of outer automorphisms of G, and the product of Weyl / braid groups over
punctures / boundary components. We prove that the dual moduli space A(G,S)
carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S))
is a cluster ensemble. These results generalize the works of V. Fock & the
first author, and of I. Le.
We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or
|h|=1. First, we define a *-algebra structure on the Langlands modular double
A(h; X) of the algebra of functions on X. We construct a principal series of
representations of the *-algebra A(h; X), equivariant under a unitary
projective representation of the cluster modular group M(X). This extends works
of V. Fock and the first author when h>0.
Combining this, we get a M(G,S)-equivariant quantization of the moduli space
P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series
representations. We construct realizations of the principal series
*-representations. In particular, when S is punctured disc with two special
points, we get a principal series *-representations of the Langlands modular
double of the quantum group Uq(g).
We conjecture that there is a nondegenerate pairing between the local system
of coinvariants of oscillatory representations of the W-algebra and the one
provided by the projective representation of the mapping class group of S.Comment: 199 pages. Minor correction
Cohomological Hall algebras, vertex algebras and instantons
We define an action of the (double of) Cohomological Hall algebra of
Kontsevich and Soibelman on the cohomology of the moduli space of spiked
instantons of Nekrasov. We identify this action with the one of the affine
Yangian of . Based on that we derive the vertex algebra at
the corner of Gaiotto and Rapcak. We conjecture
that our approach works for a big class of Calabi-Yau categories, including
those associated with toric Calabi-Yau -folds.Comment: 72 pages, 4 figures, v2: Added some clarifications and updated
references 73 pages, 4 figures, v3: Corrected typos and clarified some minor
point
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