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

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

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

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 $PGL_2$-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.

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