Posets and fractional Calabi-Yau categories¶

Frédéric Chapoton, C.N.R.S. et Université de Strasbourg

à Paris le 19 juin 2023

🇺🇦¶

Rough sketch¶

My aim is to explain some ideas relating

combinatorics and partial orders
isolated singularities of polynomial functions
derived categories

⚠️ Some pieces are missing or outside my reach.

Derived categories serve as a meeting point between combinatorics and singularity theory.

On the combinatorial side¶

In the field of enumerative combinatorics, one aims at counting things.

Sometimes, one gets closed formulas, of various sorts.

Maybe the nicest sort is a product formula.

For example, all lattice paths from $(0,0)$ to $(m,n)$ using only steps $(1,0)$ and $(0,1)$ .

These are counted by the binomial coefficient $\binom{m+n}{m}$ .

This formula can be written as follows

$\binom{m+n}{m} = \prod_{i=1}^{m} \frac{m+n+1-i}{i},$

which has the general shape

$\prod_{e \in E} \frac{D - e}{e},$

for some multi-set $E$ and integer $D$ (here $E = \{1,2,\ldots,m\}$ and $D = m+n+1$ ).

Fact: Many famous enumeration results in combinatorics involve formulas of this precise shape !

Dyck paths, Dyck paths of slope $m$ (Catalan and Fuss-Catalan numbers) A108
Plane partitions (MacMahon), symmetric plane partitions A6366
Alternating sign matrices (Mills-Robbins-Rumsey, Zeilberger, Kuperberg) A5130
Various kinds of planar maps (Tutte)
2-stack sortable permutations (West, Zeilberger) A139
intervals in the Tamari lattices A260
clusters and tilting modules for Dynkin quivers (Fomin and Zelevinsky) A1700

und so weiter...

No obvious general reason is known for these formulas to exist. Some partial reasons explain some cases.

For example, the number of clusters in a cluster algebra is given by

$\prod_{i=1}^{m} \frac{h+d_i}{d_i} ,$

where $h$ and the $(d_1,\dots,d_m)$ are the Coxeter number and the degrees of the associated Weyl group.

Using the symmetry of the degrees $d_i \leftrightarrow h+2 - d_{\varphi(i)}$ for some bijection $\varphi$ , this is the same as

$\prod_{i=1}^{m} \frac{2 h + 2 - d_i}{d_i} ,$

which has exactly the expected shape with $D = 2h+2$ .

On the singularity side¶

We will look at isolated quasi-homogeneous singularities of functions $f : \mathbb{C}^m \to \mathbb{C}$ .

These were studied by Milnor, in famous and classical works.

Recall that $z = (z_1,\ldots,z_m) \in \mathbb{C}^m$ is a singular point of $f$ if all partial derivatives of $f$ vanish at $z$ .

The word "isolated" means that $f$ has an isolated singular point, that we assume to be $0 \in \mathbb{C}^m$ .

We will also assume that $f(0) = 0$ .

The word "quasi-homogeneous" means that there exists integers $d_1,\ldots,d_m$ and $D$ such that:

the total degree of $f$ , when giving weight $d_i$ to the variable $z_i$ , is given by $D$ .

Example: $f = x^3 + y^4$ has total degree $12$ if $x$ has weight $4$ and $y$ has weight $3$ .

So every quasi-homogeneous isolated singularity comes with the data of $(d_1,\ldots,d_m),D$ .

Conversely, fix $(d_1,\ldots,d_m),D$ and pick $f$ as a generic quasi-homogeneous polynomial w.r.t. this data.

Fact: There exists necessary and sufficient conditions on $(d_1,\ldots,d_m),D$ in order to ensure that $f$ has an isolated singularity.

I will not need these rather technical conditions. We always assume that they hold.

Fact (A): The product $\prod_{i=1}^{m} \frac{D-d_i}{d_i}$ is an integer.

Fact (B): The product $\prod_{i=1}^{m} \frac{[D-d_i]_q}{[d_i]_q}$ is a polynomial in $\mathbb{N}[q]$ .

Here $[d]_q$ denotes the $q$ -integer $1+q+q^2+\cdots+q^{d-1}$ .

Both facts are not clear a priori. They only belong clearly to $\mathbb{Q}$ and $\mathbb{Q}(q)$ .

Milnor fiber and Milnor number¶

A quick little piece of geometry now, from Milnor

Choose $f$ with quasi-homogeneous isolated singularity. Let $B$ be a small-enough ball around $0 \in \mathbb{C}^m$ .

the fibers $f^{-1}(z) \cap B$ for $z$ in a small circle around $0 \in \mathbb{C}$ are all diffeomorphic
they form a locally-trivial fibration over the small circle
they have the homotopy type of a bouquet of $\mu$ spheres of dimension $m-1$

Sketch picture on the board if there is a board.

Fact (A2): the integer $\mu = \prod_{i=1}^{m} \frac{D-d_i}{d_i}$ (Milnor number of the singularity)

Fact (B2): the polynomial $\prod_{i=1}^{m} \frac{[D-d_i]_q}{[d_i]_q}$ describes a filtration on the homology of the fibers.

Homology and monodromy¶

Because fibers are bouquets 💐 of spheres 🌎 of the same dimension, only one interesting homology group $H^{m-1}$ isomorphic to $\mathbb{Z}^{\mu}$ .

Turning once around the small circle and following cycles by local triviality, one obtains a linear map

$H^{m-1} \to H^{m-1}$ which is called the monodromy map of the singularity.

You can think of the monodromy as a $\mu \times \mu$ matrix of integers.

Derived categories as a mediating object¶

Main idea : relate product formulas in combinatorics to Milnor formula for Milnor number $\mu$ of singularities

HOW ? Using derived categories

on the combinatorial side, enrich combinatorial objets with partial orders

and consider modules over their incidence algebras over a field

on the singularity side, consider a categorification $\mathscr{D}_{\operatorname{Mil}}$ of the Milnor fiber homology and monodromy

This $\mathscr{D}_{\operatorname{Mil}}$ should be a triangulated category, recovering the geometric data when passing to $K_0$ .

⚠️ WARNING: Apparently, this kind of category has not yet been constructed in any kind of generality. I am not sure.

This $\mathscr{D}_{\operatorname{Mil}}$ should be Something Like a Directed Fukaya Category 🦄, whatever it is

Here some hand-waving about A-Model, B-model, mirror symmetry ?

Motto¶

Suppose that you have a familly of combinatorial objects $(P_n)_n$ counted for each index $n$ by a combinatorial formula of the shape

$\prod_{e \in E} \frac{D - e}{e}$

for some multi-sets $E$ and integers $D$ depending on the index $n$ in some regular way.

(Example : Dyck paths and Catalan numbers)

THEN 🎁

There should exist partial orders $\leq$ on the combinatorial objects such that the derived category $\mathscr{D}_P$ of modules over the incidence algebra of $(P_n, \leq)$ is triangle-equivalent to the derived category $\mathscr{D}_{\operatorname{Mil}}$ attached to the quasi-homogeneous singularity of a generic quasi-homogeneous polynomial with weights $E$ and total weight $D$ .

(Example: the natural partial order on Dyck paths by inclusion)

Claim: all the derived categories involved should be fractional Calabi-Yau.

Calabi-Yau (CY) and fractional Calabi-Yau (fCY) categories¶

Let $T$ be triangulated category with finite-dimensional $\operatorname{Hom}$ spaces over a field

A Serre functor on $T$ is an auto-equivalence of $T$ such that

$\operatorname{Hom}(X,Y)^{*} \simeq \operatorname{Hom}(Y,S X)$

(functorially in both arguments)

This name comes from the Serre duality functor on coherent sheaves in algebraic geometry. Unique up to isomorphism.

The category $T$ is Calabi-Yau if $S$ is isomorphic to a shift functor $[D]$ .

This names comes from the properties of coherent sheaves on Calabi-Yau manifolds.

The category $T$ is fractional Calabi-Yau if a power of $S$ is isomorphic to a shift functor. (Kontsevich, around 2000)

meaning that $S^q \simeq [p]$ for some integers $p$ and $q$ . Abusingly, $p/q$ is called the Calabi-Yau dimension.

Examples of fCY categories¶

some examples come from algebraic geometry : pieces in semi-orthogonal decompositions of derived categories coming from Fano manifolds
derived categories of representations of Dynkin quivers (types $\mathbb{ADE}$ )
some examples from singularity theory using categories of matrix factorisations

Combinatorics of posets-with-product-formula as a new source of examples !

Monodromy on both sides¶

On the geometry side, the expected categorification $\mathscr{D}_{\operatorname{Mil}}$ of Milnor's fibration should be such that

$K_0(\mathscr{D}_{\operatorname{Mil}})$ is a free abelian group of rank $\mu$ ,
on which the Serre functor $S$ induces the monodromy matrix.
$\mathscr{D}_{\operatorname{Mil}}$ depends only on the data $(d_1,\ldots,d_m),D$

On the combinatorial side, the derived category $\mathscr{D}_P$ of modules over an incidence algebra of a partial order $P$ has an Auslander-Reiten functor $\tau$ (equivalent to having a Serre functor $S$ )

$K_0(\mathscr{D}_P)$ is a free abelian group of rank the cardinality of $P$ ,
$\tau$ induces a linear map on $K_0(T)$ with an easy matrix, computed directly from the partial order $\leq_P$

The Coxeter criterion¶

Suppose you have combinatorial objects counted by a product formula.

Suppose moreover that you have found partial orders on these objects.

How to convince yourself that they are "good" in the sense of our motto 🎁 ?

💡 IDEA : compare the characteristic polynomials of monodromy !

Product formula $\implies$ weights $E$ and degree $D$ $\implies$ formula for characteristic polynomial (Milnor-Orlik)

So one has a guess for the characteristic polynomial, to compare with the one from the partial order (called the Coxeter polynomial of $P$ )

If they match, one can hope to be on a good track !

Concrete example (almost for babies 👶🏼👶🏾👶🏿)¶

Consider the partial orders on Dyck paths by inclusion (being always below)

In size $3$ , there are $5$ Dyck paths.

The general formula is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$ .

For $n=3$ , this gives the product formula

$5 = \frac{6}{2}\frac{5}{3}$

so that $E = \{2,3\}$ and $D = 8$ .

From this, one finds that the monodromy of a generic singularity has char. polynomial $t^5 - t^4 + t - 1$ .

same concrete example, combinatorial side¶

On the other hand, one computes the matrix of the Auslander-Reiten translation $\tau$ and finds

$\left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 1 & 0 & 0 & -1 \\ -1 & 1 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -1 \end{array}\right)$

which has the same char. polynomial (up to technical details about shifts and signs).

Bonus track : factorising derived categories¶

Thom-Sebastiani sum of singularities : $f$ and $g$ with disjoint variables, consider $f+g$

in terms of weight data $(d_1,\ldots,d_m);D$ and $(e_1,\ldots,e_n);E$ , this means:

scale both data so that $E = D$ , then take the disjoint union

For example $(1);3$ and $(1);4$ together give $(4,3);12$

The the monodromy of Milnor fibers is the tensor product of monodromies

Then the associated Fukaya-style categories shoud be tensor product of the smaller Fukaya-style categories.

On the combinatorial side, for the cartesian product of posets, one also get that the derived category is the tensor product of categories for factor posets.

$\implies$ if you can factorise the weight data, you should be able to factorise the poset (up to derived equivalence)

☕¶

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