88 research outputs found
Tropical Theta Functions and Log Calabi-Yau Surfaces
We generalize the standard combinatorial techniques of toric geometry to the
study of log Calabi-Yau surfaces. The character and cocharacter lattices are
replaced by certain integral linear manifolds described by Gross, Hacking, and
Keel, and monomials on toric varieties are replaced with the canonical theta
functions which GHK defined using ideas from mirror symmetry. We describe the
tropicalizations of theta functions and use them to generalize the dual pairing
between the character and cocharacter lattices. We use this to describe
generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and
finite Fourier series expansions. We hope that these techniques will generalize
to higher-rank cluster varieties.Comment: 40 pages, 2 figures. The final publication is available at Springer
via http://dx.doi.org/10.1007/s00029-015-0221-y, Selecta Math. (2016
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Tropical theta functions and log Calabi-Yau surfaces
textWe describe combinatorial techniques for studying log Calabi-Yau surfaces. These can be viewed as generalizing the techniques for studying toric varieties in terms of their character and cocharacter lattices. These lattices are replaced by certain integral linear manifolds described in [GHK11], and monomials on toric varieties are replaced with the canonical theta functions defined in [GHK11] using ideas from mirror symmetry. We classify deformation classes of log Calabi-Yau surfaces in terms of the geometry of these integral linear manifolds. We then describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher rank cluster varieties.Mathematic
Bracelets bases are theta bases
The skein algebra of a marked surface, possibly with punctures, admits the
basis of (tagged) bracelet elements constructed by Fock-Goncharov and
Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta
basis of Gross-Hacking-Keel-Kontsevich, quantized by Davison-Mandel. We show
that these two bases coincide (with a caveat for notched arcs in once-punctured
tori). In unpunctured cases, one may consider the quantum skein algebra. We
show that the quantized bases also coincide. Even for cases with punctures, we
define quantum bracelets for the cluster algebras with coefficients, and we
prove that these are again theta functions. On the corresponding cluster
Poisson varieties (parameterizing framed -local systems), we prove in
general that the canonical coordinates of Fock-Goncharov, quantized by
Bonahon-Wong and Allegretti-Kim, coincide with the associated (quantum) theta
functions. Long-standing conjectures on strong positivity and atomicity follow
as corollaries. Of potentially independent interest, we examine the behavior of
cluster scattering diagrams under folding.Comment: 131 pages; v2: minor corrections in Appendix A; v3: corrected issues
in S3.3, proved the local digon relation, and added a new characterization of
the tagged arc skein algebra in S9.2.
Strong positivity for quantum theta bases of quantum cluster algebras
We construct "quantum theta bases," extending the set of quantum cluster
monomials, for various versions of skew-symmetric quantum cluster algebras.
These bases consist precisely of the indecomposable universally positive
elements of the algebras they generate, and the structure constants for their
multiplication are Laurent polynomials in the quantum parameter with
non-negative integer coefficients, proving the quantum strong cluster
positivity conjecture for these algebras. The classical limits recover the
theta bases considered by Gross-Hacking-Keel-Kontsevich. Our approach combines
the scattering diagram techniques used in loc. cit. with the Donaldson-Thomas
theory of quivers.Comment: 84 pages; comments welcome
Theta bases are atomic
Fock and Goncharov conjectured that the indecomposable universally positive
(i.e., atomic) elements of a cluster algebra should form a basis for the
algebra. This was shown to be false by Lee-Li-Zelevinsky. However, we find that
the theta bases of Gross-Hacking-Keel-Kontsevich do satisfy this conjecture for
a slightly modified definition of universal positivity in which one replaces
the positive atlas consisting of the clusters by an enlargement we call the
scattering atlas. In particular, this uniquely characterizes the theta
functions.Comment: Published versio
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