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Register a new userTorsion functions on moduli spaces in view of the cluster algebra
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We introduce non-acyclic $PGL_n(mathbb{C})$-torsion of a 3-manifold with toroidal boundary as an extension of J. Portis $PGL_2(mathbb{C})$-torsion, and present an explicit formula of the $PGL_n(mathbb{C})$-torsion of a mapping torus for a surface with punctures, by using the higher Teichm{u}ler theory due to V. Fock and A. Goncharov. Our formula gives a concrete rational function which represents the torsion function and comes from a concrete cluster transformation associated with the mapping class.
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The Hecke algebra H_n contains well known idempotents E_{lambda} which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_{lambda} was given by Aiston and Morton. The closure of E_{lambda} becomes an element Q_{lambda} of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s_{lambda}. But previous proofs of this fact used results about quantum groups which were far beyond the scope of skein theory. Our elementary proof uses only skein theory and basic algebra.
We study two sorts of actions on the space of conjugacy classes of irreducible $SU_2$-representations of a knot group. One of them is an involution which comes from the algebraic structure of $SU_2$ and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization.
We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkaehler 2g-folds that are not birational, for every g >= 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivalent.
Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between the quantum Teichmuller space and the generalized Kashaev algebra.
We examine moduli spaces of locally homogeneous surfaces of Type~$mathcal{B}$ with torsion where the symmetric Ricci tensor is non-degenerate. We also determine the space of affine Killing vector fields in this context.
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