Figure 1: Spectrum for $\mathrm{Gr}(3,8)$

Here's another update for Grassmannian.info: it now contains spectra (as in, the distribution of eigenvalues) of quantum multiplication with $\mathrm{c}_1(G/P)$, for $q=1$. This is interesting for various reasons, but before explaining why, let me tell you that the picture you're seeing now is giving the eigenvalues for $\mathrm{Gr}(3,8)$. There are 56 eigenvalues, because the (quantum) cohomology is 56-dimensional.

Spectral decomposition

The Fukaya category of a Fano symplectic manifold has a decomposition into orthogonal summands (I'm glancing over many details here). This is the spectral decomposition from the title, and you can read about it in many places. Let's rather focus on other aspects of the spectrum.

Conjecture $\mathcal{O}$

In Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures Galkin–Golyshev–Iritani introduce property $\mathcal{O}$, which relates the (real) eigenvalue of largest absolute value to the structure sheaf (explaining the name $\mathcal{O}$), and the other eigenvalues of the same absolute value to twists by an ample $\mathcal{O}(1)$ up to the index $m$ (so that $\omega_X^\vee\cong\mathcal{O}(m)$). The precise statement says that, denoting $T$ the spectral radius:

  • $T$ is an eigenvalue of multiplicity one;
  • all other eigenvalues with absolute value $T$ are of the form $\zeta T$, where $\zeta$ is an $m$th root of unity.

This property is conjectured to hold for the quantum cohomology of all Fano varieties, and is in fact checked for all $G/P$ in On the conjecture $\mathcal{O}$ of GGI for $G/P$. So the pictures don't provide any new information, only a pretty experimental check in low ranks.

Kuznetsov–Smirnov conjecture

Dubrovin's conjecture (the first part at least) states the equivalence of

  • the existence of a full exceptional collection in $\mathbf{D}^{\mathrm{b}}(G/P)$
  • the generic semisimplicity of (big) quantum cohomology $\mathrm{BQH}(G/P)$

But there are other flavours of quantum cohomology to consider. The small quantum cohomology (which is an algebra over $\mathbb{C}[q]$ in this context) and the quantum cohomology at $q=1$, which is truly a finite-dimensional algebra. Standard arguments tell us that semisimplicity at $q=1$ implies it generic semisimplicity for the small quantum cohomology, which in turn implies it for the big quantum cohomology. But examples show that the opposite does not hold.

So one could ask what non-semisimplicity of small quantum cohomology, when the big quantum cohomology is generically semisimple, tells us. Or phrased differently, when is the small quantum cohomology semisimple?

In Alexander Kuznetsov, Maxim Smirnov: On residual categories for Grassmannians the authors suggest that (at least for Fano varieties of Picard rank 1) if

  • the small quantum cohomology is generically semisimple

then

  • $\mathbf{D}^{\mathrm{b}}(X)$ has a Lefschetz exceptional collection (an exceptional collection with additional symmetries), whose residual category is generated by a completely orthogonal collection.

Whilst not explicitly phrased in op. cit. (but see Residual categories for (co)adjoint Grassmannians in classical types for more information), the number of objects in the residual category is given by the multiplicity of the eigenvalue 0.

Figure 2: Spectrum for $\mathrm{Q}^{10}$

This blogpost is already more than long enough, so let's just consider the example from Figure 2. A 10-dimensional quadric has 12 exceptional objects, and has index 10. One possible Lefschetz collection is of the form \begin{equation} \mathbf{D}^{\mathrm{b}}(\mathrm{Q}^{10})= \left\langle \mathcal{S}_+,\mathcal{S}_-,\mathcal{O}; \mathcal{O}(1); \ldots; \mathcal{O}(9) \right\rangle. \end{equation}

Here the residual category is the part which is not common to all twists of the Lefschetz block, which is the subcategory generated by $\mathcal{S}_+,\mathcal{S}_-$. These being the spinor bundles one can show that they are indeed completely orthogonal.

There exist refinements for the case where the small quantum cohomology is not generically semisimple, and the structure of the residual category is related to the structure of the subalgebra of quantum cohomology with eigenvalue 0, but that is for another time.

The eigenvalue computation is based on the data provided by Sergey Galkin, which goes up to rank 7 (with some missing cases for $\mathrm{E}_7$). At least in type $\mathrm{A}$ there are actually closed formulae, so maybe I can improve the computation at some point. And next week I hope to run some computations on a bigger computer than my laptop, to get more pictures. For now you can admire the beauty of the spectra of