Tropical Mirror Symmetry, Scattering Diagrams, and the Gross–Siebert Program¶
Job: develop the scattering diagrams / consistency (tropical walls / slabs), the broken lines / theta functions / GHK mirror, the log Calabi–Yau tropical mirror (the Gross–Siebert theorem), the intrinsic / family mirror and the cluster / higher-Teichmüller connection, and the tropical / log GW as the mirror map — with the \(\mathbb{P}^2\) scattering / Hori–Vafa mirror via the Gross–Siebert theta functions as the worked example.
Core¶
Scattering diagrams (tropical)¶
The scattering diagram \(\mathfrak{D}\) over an integral affine manifold \(B\) (a collection of walls (codimension-1 polyhedral cells) with attached functions \(f_\mathfrak{w}\in\mathbb{C}[x_1^{\pm},\dots,x_n^{\pm}]\) / the log-monodromy). The consistency (the path-ordered product around the singularities is trivial, the wall-crossing is consistent). The tropical / the polyhedral picture: the walls are the tropical curves / the slabs, the functions the disk / the instanton counts.
| Scattering diagram | Description |
|---|---|
| \(\mathfrak{D}\) over \(B\) | walls (codim-1 polyhedral cells) with functions \(f_\mathfrak{w}\in\mathbb{C}[x_1^{\pm},\dots,x_n^{\pm}]\) / the log-monodromy |
| Consistency | the path-ordered product around the singularities is trivial (the wall-crossing consistent) |
| Tropical / polyhedral | the walls are the tropical curves / the slabs, the functions the disk / the instanton counts |
| Gross–Siebert scattering | the scattering diagrams from the tropical / the log geometry, the consistent scattering / the cluster structure |
The broken lines and the theta functions¶
| Broken lines / theta functions | Description |
|---|---|
| Broken line | the piecewise-linear path in \(B\) bending at the walls (the monomial attached to the wall, the instanton-corrected straight line) |
| Theta function \(\vartheta_p\) | the canonical basis of the mirror: the sum over the broken lines ending at \(p\) (the Gross–Hacking–Keel / the theta function as the canonical / the positive basis) |
| GHK mirror | the Gross–Hacking–Keel construction of the mirror via the theta functions / the broken lines (the canonical basis of the cluster / the scattering) |
| Cluster / canonical basis | the theta functions form the cluster algebra of the mirror (the canonical basis, the higher Teichmüller / Fock–Goncharov connection) |
The log Calabi–Yau and the tropical mirror¶
A log CY variety \((X,D)\) (a pair with \(D\) an anticanonical divisor, the log-structure from \(D\)) is the algebro-geometric avatar of the tropical / SYZ picture.
| Log CY / tropical mirror | Description |
|---|---|
| Log CY \((X,D)\) | \(D\) anticanonical, the log-structure (the algebro-geometric avatar of the tropical / SYZ) |
| Tropical mirror of \((X,D)\) | the mirror reconstructed from the tropical / the log data (the scattering over the dual intersection complex \(B\), the theta functions, the GHK) |
| Gross–Siebert mirror theorem | the mirror family from the log GW invariants of \((X,D)\) (the mirror map / the Yukawa coupling the log-period computation) |
| Connection to SYZ & Gross–Siebert | the mirror symmetry / SYZ & Gross–Siebert (the tropical / the log skeleton as the SYZ base) |
The intrinsic mirror and the family / the relative¶
| Intrinsic / family mirror | Description |
|---|---|
| Intrinsic mirror (Gross–Siebert) | the mirror constructed intrinsically from the log / the tropical data (without the SYZ transcendental fibration, the algebro-geometric / the tropical mirror) |
| Family mirror | the mirror as a family over the scattering base (the relative / the family HMS) |
| Canonical basis / cluster | the mirror has a canonical basis of the theta functions (the cluster algebra, the higher Teichmüller / Fock–Goncharov) |
The tropical mirror and the enumerative invariants¶
| Tropical / log GW as the mirror map | Description |
|---|---|
| Tropical / log GW | the log GW invariants of \((X,D)\) give the mirror map (the period / the Yukawa) |
| Tropical / log B-model | the tropical side as the B-model (the periods / the integrals over the tropical / the scattering, the tropical mirror / the B-model) |
| Tropical / log descendants | the Gross–Siebert / the log descendant, the tropical / the gravitational descendants (the mirror / the B-model descendant) |
See it¶
\(\mathbb{P}^2\) scattering / the Hori–Vafa mirror via the Gross–Siebert theta functions. For the log CY \((\mathbb{P}^2,D)\) with \(D\) the toric boundary (three lines, the triangle), the dual intersection complex \(B\) is the triangle (the fan of \(\mathbb{P}^2\)). The scattering diagram over \(B\): the walls are the legs from the vertices (the rays of the fan), the functions \(f_\mathfrak{w}=(1+z)^{n_w}\) (the disk counts \(n_w\) the log GW invariants — the three coordinate-line disks, \(n_w=1\) for the three toric boundary disks of \(\mathbb{P}^2\)). The consistent scattering reconstructs the mirror as the total space of the superpotential $$ W=x+y+\frac{1}{xy} $$ over \((\mathbb{C}^*)^2\) (the Hori–Vafa mirror of \(\mathbb{P}^2\), recalled from mirror-symmetry/SYZ). The theta functions: the sum over the broken lines (the broken lines bending at the scattering walls, the monomials \((1+z)^{n_w}\)) give the canonical basis of the coordinate ring of the mirror (the \(\vartheta_p\) the theta functions, the GHK). The key point: the \(\mathbb{P}^2\) scattering / the Gross–Siebert theta functions reconstruct the Hori–Vafa mirror intrinsically from the tropical / log data — the tropical mirror symmetry as the intrinsic / the algebro-geometric construction, the scattering / the theta functions as the operational tropical mirror.
Check¶
Show that the \(\mathbb{P}^2\) scattering with the wall functions \(f_\mathfrak{w}=(1+z)\) reconstructs the Hori–Vafa \(W=x+y+1/(xy)\), and identify the consistency.
The log CY \((\mathbb{P}^2,D)\), the dual intersection complex \(B\) the triangle (the fan of \(\mathbb{P}^2\), the 3 rays). The scattering diagram \(\mathfrak{D}\) over \(B\): the walls are the legs from the vertices (the 3 rays of the fan), the wall functions \(f_\mathfrak{w}=(1+z)^{n_w}\) with \(n_w=1\) (the three toric boundary disks, the log GW invariants of \((\mathbb{P}^2,D)\), each disk class meeting one boundary component). The consistency: the path-ordered product around the vertices is trivial — the three walls with the functions \((1+z)\), the path-ordered product \(f_{\mathfrak{w}_1}\cdot f_{\mathfrak{w}_2}\cdot f_{\mathfrak{w}_3}=(1+z)^3\) must be consistent with the monodromy around the vertex, the consistency the non-trivial constraint (the three walls with \((1+z)\) are consistent because the relation \(\beta_1+\beta_2+\beta_3=0\) / the toric relation). The Hori–Vafa: the superpotential \(W=\sum n_i z^{\beta_i}=z^{\beta_1}+z^{\beta_2}+z^{\beta_3}=x+y+1/(xy)\) (the three monomials the three toric disk classes, the relation \(\beta_1+\beta_2+\beta_3=0\) giving \(z^{\beta_3}=1/(xy)\)). So the scattering / the consistency reconstructs the Hori–Vafa \(W=x+y+1/(xy)\) from the tropical / log data.
Explain the broken lines / the theta functions, and identify the canonical basis / the cluster structure of the mirror.
The broken line \(\gamma\) in \(B\) (the piecewise-linear path, bending at the walls, the monomial attached to each wall crossing — the instanton-corrected straight line, the tropical / the scattering-corrected geodesic). The theta function \(\vartheta_p\) at \(p\in B\): the sum over all broken lines ending at \(p\), \(\vartheta_p=\sum_{\gamma:\gamma(\infty)=p}\mathrm{mon}(\gamma)\) (the monomials / the wall functions, the sum over the broken lines). The GHK: the theta functions \(\vartheta_p\) form a canonical basis of the coordinate ring of the mirror (the canonical / the positive basis, the cluster algebra structure — the \(\vartheta_p\) the cluster variables, the scattering the cluster transformations). The cluster: the theta functions generate the cluster algebra of the mirror (the higher Teichmüller / the Fock–Goncharov cluster — the same cluster structure on the moduli, the scattering / the theta functions as the cluster). The canonical basis / the cluster: the theta functions are the canonical basis (the Lusztig / the canonical / the positive basis), the scattering the cluster transformations — the tropical mirror has the canonical / the cluster structure from the scattering.
Pitfalls¶
- The scattering diagrams require the consistency (not automatic — the consistent scattering is the non-trivial constraint, the path-ordered product trivial around the singularities). Not every collection of walls is consistent.
- The GHK mirror / the theta functions are for the log CY (the general / the non-log-CY mirror is not the GHK — the log CY is the input, the GHK the construction for the log CY).
- The Gross–Siebert mirror theorem is the log GW (the log / the tropical, not the classical GW — the mirror map / the Yukawa from the log-period, not the classical period).
- The intrinsic mirror (the Gross–Siebert) is the algebro-geometric / the tropical (not the transcendental SYZ — the two are related but distinct: the intrinsic is the log / the tropical, the SYZ is the transcendental / the special Lagrangian).
- The canonical basis / the cluster (the theta functions) is for the log CY mirror, not the general mirror — the cluster structure is the special / the log-CY feature.
- The tropical / log B-model (the periods / the integrals over the tropical / the scattering) is the tropical side as the B-model, the relation to the classical B-model (the periods of the mirror CY) via the degeneration / the tropical limit, not the direct equality.
Refs¶
- Gross–Siebert, "Mirror symmetry via logarithmic degeneration data I–IV" (J. Diff. Geom. 2007–2011); "From real geometry to complex geometry" (Annals 2011)
- Gross–Hacking–Keel, "Mirror symmetry for log Calabi–Yau surfaces I" (Publ. Math. IHÉS 2015); "Theta functions and the canonical basis"
- Kontsevich–Soibelman, "Affine structures and non-archimedean analytic spaces" (2006); the scattering / the wall-crossing
- Fock–Goncharov, "Moduli spaces of local systems and higher Teichmüller theory" (Publ. Math. IHÉS 2006); the cluster / the theta functions connection
- Hori–Vafa, "Mirror symmetry" (hep-th/0002222); the \(\mathbb{P}^2\) / the Hori–Vafa superpotential
Next¶
TG index · Foundations · Berkovich & Skeletons · Matroids & Linear Spaces · Moduli & Enumerative · Mirror symmetry / SYZ · Higher Teichmüller