There is a large body of results and conjectures on explicit semiorthogonal decompositions of (derived categories of) Fano manifolds. A unifying theme, motivated by homological mirror symmetry, is that there should exist 'canonical' decompositions related by mutations. The blocks of these decompositions are indexed by eigenvalues of the quantum multiplication by the first Chern class. Indeed, this statement holds true for the Fukaya-Seidel category of the mirror, essentially by definition. Canonical decompositions are expected to have strong properties such as compatibility with coarser 'standard' decompositions. As an application, one should be able to propagate canonical decompositions from one Fano manifold to another using two-ray games. Concretely, let X be a smooth Fano variety, and consider two extremal contractions, X ⇢ Y and X ⇢ Z. The expectation is that X admits two 'canonical' decompositions (related by mutations), one compatible with Y and one compatible with Z. The notion of compatibility depends on the type of extremal contraction. For example, if X → Y is a regular morphism and Y is smooth, then the canonical decomposition of X compatible with Y should contain a pullback of the canonical decomposition of Y. Conversely, knowing a canonical decomposition of X compatible with Z allows us to construct a canonical decomposition of Z by discarding unnecessary blocks.
If X → Y is a (twisted) projective bundle, then the canonical decompositions of X and Y should be related by Orlov's theorem on projective bundles. For example, the Beilinson exceptional collection for the projective space should be its canonical decomposition. Likewise, if X → Y is a blow-up with a smooth indecomposable center B (for example the canonical divisor of B is trivial or globally generated), then the canonical decomposition of X compatible with Y should be the one from the Orlov blow-up theorem. This generalizes naturally to the case where X → Y factors into a sequence of blow-ups with smooth indecomposable centers and standard flips of (twisted) projective bundles over smooth indecomposable bases. We call a contraction of this form a standard contraction.
The mutation between the decompositions compatible with Y and Z should correspond to the monodromy braid of eigenvalues of the quantum multiplication by the first Chern class of X, acting on the quantum cohomology of X. This monodromy arises as the base of the small quantum cohomology varies along a path in the ample cone of X, with a small purely imaginary perturbation added to avoid collisions of eigenvalues. When the path approaches the walls of the ample cone, the eigenvalues agglomerate into groups that depend on the structure of the birational contractions. If the contraction X → Y is standard, this provides a way to determine a canonical decomposition of Z.
We present several examples where this conjecture can be verified by comparing the monodromy braid with a mutation braid constructed algebraically. In most cases, we compute the eigenvalues of the quantum multiplication by the first Chern class indirectly, by computing the critical values of the LG superpotential (as a function of the parameters q1 and q2, which describe the variation of the base of the small quantum cohomology) using the Givental-Hori-Vafa mirror construction. It may differ from the actual quantum spectrum by a linear transformation.
\[ q_1 t^3 + t^4 - 27q_1^2 q_2 - 36q_1 q_2 t - 8q_2 t^2 + 16q_2^2 = 0 \]
Narrated movie of dancing eigenvalues.
\[ W = q_1 (1 + x_1 + x_2) + x_3 (1 + x_1 + x_2) + \frac{q_2 (1 + x_1 + x_2)}{x_1 x_2 x_3} \]
\[ q_1^4 t - 4q_1^3 t^2 + 6q_1^2 t^3 - 4q_1 t^4 + t^5 - 32q_1^2 q_2 t - 192q_1 q_2 t^2 - 32q_2 t^3 + 256q_2^2 t=0 \]
Narrated movie of dancing eigenvalues.\[ W=x_3 (1 + x_1)(1 + x_2) + \frac{q_2 (1 + x_1)(1 + x_2)}{x_1 x_2} + \frac{q_1 q_2 (1 + x_1)(1 + x_2)}{x_1 x_2 x_3} \]
\[ 64q_1^3 q_2 t - 48q_1^2 q_2^2 t + 12q_1 q_2^3 t - q_2^4 t + 136q_1^2 q_2 t^2 + 166q_1 q_2^2 t^2 + 4q_2^3 t^2 - q_1^2 t^3 + 80q_1 q_2 t^3 - 6q_2^2 t^3 - 2q_1 t^4 + 4q_2 t^4 - t^5=0 \]
Narrated movie of dancing eigenvalues.
\[ W=q_1(1 + x_1 + x_2) + x_3(1 + x_1 + x_2)^2 + \frac{q_2 q_1 (1 + x_1 + x_2)}{x_1 x_2 x_3} \]
\[ q_1^4 t + 4q_1^3 q_2 t - 4q_1^3 t^2 - 132q_1^2 q_2 t^2 - 432q_1 q_2^2 t^2 + 6q_1^2 t^3 - 132q_1 q_2 t^3 - 4q_1 t^4 + 4q_2 t^4 + t^5=0 \]
Narrated movie of dancing eigenvalues.
\[ W = q_1 (1 + x_1 + x_2) + x_3 (1 + x_1 + x_2) + x_4 (1 + x_1 + x_2) + \frac{q_2 q_1 (1 + x_1 + x_2)}{x_1 x_2 x_3 x_4} \]
\[ q_1^5 t - 5q_1^4 t^2 + 10q_1^3 t^3 - 10q_1^2 t^4 + 5q_1 t^5 - t^6 + 216q_1^3 q_2 t + 2052q_1^2 q_2 t^2 + 873q_1 q_2 t^3 - 16q_2 t^4 + 11664q_1 q_2^2 t=0 \]
Narrated movie of dancing eigenvalues.