Vector bundles on moduli of curvesfrom representations of vertex algebras

Angela Gibney

Madison Moduli Weekend

September 2020

This talk is about joint work with

Chiara Damiolini and Nicola Tarasca.

Today: vector bundles onMg,n

depend on:

a vertex operator algebra V , and on n V -modules W i .

V = V(V ;⊗iW i)

��Mg,n

V(C,p•)

��(C,p•)

Fibers are vector spaces of coinvariants

V(C,p•)∼= ⊗iW

i/L(C,p•)(V ) · ⊗iWi .

Today: vector bundles onMg,n

depend on:

a vertex operator algebra V , and on n V -modules W i .

V = V(V ;⊗iW i)

��Mg,n

V(C,p•)

��(C,p•)

Fibers are vector spaces of coinvariants

V(C,p•)∼= ⊗iW

i/L(C,p•)(V ) · ⊗iWi .

Today: vector bundles onMg,n

depend on:

a vertex operator algebra V , and on n V -modules W i .

V = V(V ;⊗iW i)

��Mg,n

V(C,p•)

��(C,p•)

Fibers are vector spaces of coinvariants

V(C,p•)∼= ⊗iW

i/L(C,p•)(V ) · ⊗iWi .

You may have questions:

1. What are these bundles and why are they interesting?

2. Why care about vector bundles anyway?

3. What are vertex operator algebras?

You may have questions:

1. What are these bundles and why are they interesting?

2. Why care about vector bundles anyway?

3. What are vertex operator algebras?

Interesting since: (1) there are many such bundles

Examples have been construct using:

1. VOAs from Lie algebras:

(a) Vg(`), giving Verlinde bundles [TUY], and(b) Virc, giving Virasoro bundles [BFM].

2. Even lattice VOAs;

3. Holomorphic VOAs like the moonshine module V \.

4. VOAS from tensor products, invariants, commutants.

See [DGT3] for more details about (2), (3), and (4).

Interesting since: (1) there are many such bundles

Examples have been construct using:

1. VOAs from Lie algebras:

(a) Vg(`), giving Verlinde bundles [TUY], and(b) Virc, giving Virasoro bundles [BFM].

2. Even lattice VOAs;

3. Holomorphic VOAs like the moonshine module V \.

4. VOAS from tensor products, invariants, commutants.

See [DGT3] for more details about (2), (3), and (4).

Interesting since: (2) generalize Verlinde bundles

[1987] Tsuchiya and Kanie: defined coinvariants forsmooth g = 0 with coordinates.

[1991] Tsuchiya, Ueno, and Yamada define V on cover ofMg,n, showing exists connection and factorization holds.

[1993] Tsuchimoto: the vector bundles and theconnection descend toMg,n.

[2010] Fakhruddin: glob. gen. and first Chern class g = 0.

[2015] Marian, Oprea, Pandharipande: first Chern class inthe tautological ring.

[2017] Marian, Oprea, Pandharipande, Pixton, Zvonkine:Chern characters give semi-simple CohFT, deriveexpression, all Chern classes tautological.

Why should one care about vector bundles?

Here are two reasons:

1. Sections of vector bundles are like (twisted) functions.

2. Globally generated bundles give morphisms.

For instance, onMg the Hodge bundle is globallygenerated and has been a source of a lot of information.

Why should one care about vector bundles?

Here are two reasons:

1. Sections of vector bundles are like (twisted) functions.

2. Globally generated bundles give morphisms.

For instance, onMg the Hodge bundle is globallygenerated and has been a source of a lot of information.

Why should one care about vector bundles?

Here are two reasons:

1. Sections of vector bundles are like (twisted) functions.

2. Globally generated bundles give morphisms.

For instance, onMg the Hodge bundle is globallygenerated and has been a source of a lot of information.

Why should one care about vector bundles?

Here are two reasons:

1. Sections of vector bundles are like (twisted) functions.

2. Globally generated bundles give morphisms.

For instance, onMg the Hodge bundle is globallygenerated and has been a source of a lot of information.

To say more about why potentially interesting:

I am going to state our main results

(and what we are trying to show).

DGT I, II, III (on the arXiv, and my website)

Theorem (Damiolini, G, Tarasca)

For V of CohFT-type, and W i simple V-modules,

V = V(V ;W •) is a vector bundle,

V satisfies the factorization property, and

Chern characters define a semi-simple CohFT,so Chern classes are tautological.

Factorization

Factorization enables one to transform fibers of thebundle to fibers defined on simpler curves. This leads torecursions, and allows one to make inductive arguments.This is the crucial ingredient allowing ranks of suchbundles, and their Chern characters to be given explicitly.

Factorization

Factorization enables one to transform fibers of thebundle to fibers defined on simpler curves. This leads torecursions, and allows one to make inductive arguments.This is the crucial ingredient allowing ranks of suchbundles, and their Chern characters to be given explicitly.

At curve with non-separating node:

Theorem (Factorization)For V a conformal vertex algebra of CohFT-type,

V (V ;W •)(C,P•)∼=⊕

W∈W

V(V ;W • ⊗W ⊗W ′)(

C̃,P•tQ•) ,

whereW is set of simple V-modules, and Q• = (Q+,Q−).

At a curve with a separating node:

V(V ;W •)(C,P•)∼=⊕

W∈W

V(V ;W •

+ ⊗W)

X+ ⊗ V(V ;W •

− ⊗W ′)X−

whereX± = (C±,P±• tQ±),

and W •± are the modules at the P±• on C±.

Chern characters form a CohFT [DGT3]

Instead of describing a semisimple cohomological fieldtheory, we mention some consequences of this resultextending [MOPPZ 2017] to new examples.

Consequence 1: information about ranks

The ranks of Vg(V ;W •), which are the degree zero Chernclasses of the bundles, form a topological quantum fieldtheory [TQFT], and can be computed recursively from 3point ranks [fusion rules].

Examples [DGT3]

Any bundle produced from a holomorphic vertex algebraof CohFT-type has rank one.

Let L be a positive definite even lattice such that

L′/L ∼= Z/mZ, m ≥ 2.

Then

rank Vg

(VL;W

⊗n00 ⊗ · · · ⊗W⊗nm−1

m−1

)= mg δ∑m−1

j=0 jnj ≡m 0.

for g = 0 these have rank one.

Let L be a positive definite even lattice such that

L′/L ∼= Z/mZ, m ≥ 2.

Then

rank Vg

(VL;W

⊗n00 ⊗ · · · ⊗W⊗nm−1

m−1

)= mg δ∑m−1

j=0 jnj ≡m 0.

for g = 0 these have rank one.

Consequence 2: information about Chern classes

Corollary (DGT3)

1. From the Chern character one can explicitly solve forthe Chern classes; and

2. since the CohFT is semi-simple, these Chern classes liein the tautological ring.

First Chern class

For V a VOA of CohFT-type with central charge c,n simple V -modules M i of conformal dimension ai ,c1 (Vg(V ;M•)) is

rkVg(V ;M•)

(c2λ+

n∑i=1

aiψi

)− birrδirr −

∑i,I

bi:Iδi:I ,

birr =∑

W∈W

aW · rkVg−1(V ;M• ⊗W ⊗W ′)

bi:I =∑

W∈W

aW · rkVi

(V ;M I ⊗W

)· rkVg−i

(V ;M Ic ⊗W ′

).

Simplest example:

V any holomorphic VOA of central charge c:

c1 (Vg(V ;V •)) =c2λ.

In particular, for the monster V \:

c1

(Vg(V \; (V \)•)

)= 12λ.

These are all trivial onM0,n.

Simplest example:

V any holomorphic VOA of central charge c:

c1 (Vg(V ;V •)) =c2λ.

In particular, for the monster V \:

c1

(Vg(V \; (V \)•)

)= 12λ.

These are all trivial onM0,n.

Simplest example:

V any holomorphic VOA of central charge c:

c1 (Vg(V ;V •)) =c2λ.

In particular, for the monster V \:

c1

(Vg(V \; (V \)•)

)= 12λ.

These are all trivial onM0,n.

Family of bundles from lattices, notation

From before let (L,q) be a positive definite even lattice ofrank d such that

L′/L ∼= Z/mZ, m ≥ 2.

The VOA VL has central charge d.

Simple modules W i = VL+mi, with mi ≤m− 1,

ci =q(mi)

2 ≥ 0, conformal weight;∑

1≤i≤n mi = km, k ≥ 1.

We next give c1(V0(VL;W •)).

Family of bundles from lattices, notation

From before let (L,q) be a positive definite even lattice ofrank d such that

L′/L ∼= Z/mZ, m ≥ 2.

The VOA VL has central charge d.

Simple modules W i = VL+mi, with mi ≤m− 1,

ci =q(mi)

2 ≥ 0, conformal weight;∑

1≤i≤n mi = km, k ≥ 1.

We next give c1(V0(VL;W •)).

Family of bundles from lattices, notation

From before let (L,q) be a positive definite even lattice ofrank d such that

L′/L ∼= Z/mZ, m ≥ 2.

The VOA VL has central charge d.

Simple modules W i = VL+mi, with mi ≤m− 1,

ci =q(mi)

2 ≥ 0, conformal weight;∑

1≤i≤n mi = km, k ≥ 1.

We next give c1(V0(VL;W •)).

Family from lattices, first Chern class onM0,n

With notation from prior slide, onM0,n one has

c1 (V0(VL;W•)) =

n∑i=1

ciψi −∑

I⊂[n],p1∈I

q(∑

i∈I mi

)2

δ0,I ,

where q(∑

i∈I mi

)= 0 if

∑i∈I mi ≡ 0 (mod m).

Note that if on g > 0 then the rank of the bundle and thelattice would make the class more complicated.

Family from lattices, first Chern class onM0,n

With notation from prior slide, onM0,n one has

c1 (V0(VL;W•)) =

n∑i=1

ciψi −∑

I⊂[n],p1∈I

q(∑

i∈I mi

)2

δ0,I ,

where q(∑

i∈I mi

)= 0 if

∑i∈I mi ≡ 0 (mod m).

Note that if on g > 0 then the rank of the bundle and thelattice would make the class more complicated.

Before stating what we hope to show, I want to mentionwork done by others for g ≤ 1 and for smooth curves.

Our approach was different than in prior research in that:(1) we made different finiteness assumptions, and(2) used a different sheaf of Lie algebras for coinvariants.

Surprisingly, as we checked, in case both coinvariants aredefined, they are isomorphic.

Zhu’s coinvariants(generalizing [TUY],different than ours)

[1994] Zhu defined coinvariants, conjectured factorizationfor V quasi primary generated (qpg).

[2005] Abe and Nagatomo: smooth genus g withcoordinates, coinvariants finite dimensional,V qpg +.

[2005] Nagatomo and Tsuchiya: vector bundles genuszero stable with coordinates, factorization, V qpg +.

[2005] Huang: genus g ∈ {0, 1} curves with coordinatessatisfy factorization (and more), if V satisfies hypotheses.

[2019] Codogni: factorization at curves with coordinates,for holomorphic VOAs

Virasoro & FBZ coinvariants

[1991] Beilinson, Feigin, and Mazur construct coinvariantsand prove factorization for coordinatized curves and theVirasoro VOA.

[2004] Frenkel and BenZvi define sheaves coinvariants forsmooth co-ordinatized pointed curves, generalizing[BFM]. Show support proj flat connection, mentionfactorization expected if V is rational.

NEXT: Damiolini and G: (checking details)

Conjecture: [D, G] For V of CohFT-type, W i simple, if

1. g = 0 and V is generated in degree 1; or

2. g ≥ 0 and V has rank 1;

then V = Vg(V ;W •) is globally generated.

Some (potential) globally generated examples:

1. Bundles defined by VOAs from Lie algebras. eg:

[Fakh] V(Vg(`),W •) onM0,n[DG] V(Vg1(`1)⊗ · · · ⊗ Vgk(`k),W •) onM0,n

[DG] V(Ve8(1),W•) onMg,n, g > 0.

[DG] rank one Vir bundles V(Virc,W •) onMg,n.

2. Bundles defined by even lattice VOAs: For instance, ifL′/L ∼= Z/m Z, choosing modules so bundles arenontrivial, we get bundles of rank 1 onM0,n by[DGT3], so are globally generated by [DG].

3. Holomorphic VOAs like V \ define bundles of rank oneonMg,n [DGT3], so are globally generated by [DG].

Examples of bundles not globally generated

The discrete series VOAs are of CohFT-type (see [DGT3]).

For V = V(V3,4, {W1,3,W41,2})

Rk(V) = 2;

D = c1(V) = ψ1 +18

∑i∈{2345}

ψi −∑

ij∈{2345}

δ0,{ij}−18

∑i∈{2345}

δ0,{1i};

One can check that V is not globally generated, since Dis not nef. The intersection of D with F-curves:

D1 · F1,i,j,{k,`} = 1; and D1 · Fi,j,k,{1,`} = −1.

One can generate other examples like this, of higher rankdiscrete series bundles with non-nef first Chern classes.

Examples of bundles not globally generated

The discrete series VOAs are of CohFT-type (see [DGT3]).

For V = V(V3,4, {W1,3,W41,2})

Rk(V) = 2;

D = c1(V) = ψ1 +18

∑i∈{2345}

ψi −∑

ij∈{2345}

δ0,{ij}−18

∑i∈{2345}

δ0,{1i};

One can check that V is not globally generated, since Dis not nef. The intersection of D with F-curves:

D1 · F1,i,j,{k,`} = 1; and D1 · Fi,j,k,{1,`} = −1.

One can generate other examples like this, of higher rankdiscrete series bundles with non-nef first Chern classes.

Surprise: V = V(V4,5,W 51,3)

One can use the fusion rules to check: Rk(V) = 3.And our formula to verify that:

D = c1(V) =95

5∑i=1

ψi −65

∑ij

δ0,{ij},

By intersecting with F-curves one can check that D is nef:

D · F1,2,3,{4,5} = 395+

65− 3

65=

65> 0.

THE UPSHOT: There is potentially more to this story.

Surprise: V = V(V4,5,W 51,3)

One can use the fusion rules to check: Rk(V) = 3.And our formula to verify that:

D = c1(V) =95

5∑i=1

ψi −65

∑ij

δ0,{ij},

By intersecting with F-curves one can check that D is nef:

D · F1,2,3,{4,5} = 395+

65− 3

65=

65> 0.

THE UPSHOT: There is potentially more to this story.

Now can answer: Why study these bundles?

In analogy with the Verlinde bundles:

1. Give elements in the tautological ring, so may beuseful in testing Pixton’s conjectures.

2. May help study birational geometry ofMg,n.

Now can answer: Why study these bundles?

In analogy with the Verlinde bundles:

1. Give elements in the tautological ring, so may beuseful in testing Pixton’s conjectures.

2. May help study birational geometry ofMg,n.

OK: What is a vertex operator algebra?

V = (V ,1, ω,Y (·, z))

C-vector space V = ⊕i∈NVi , with dimVi <∞;vacuum vector 1 ∈ V0;conformal vector ω ∈ V2;vertex operators Y (·, z) : V → End(V )[[ z, z−1]]

A 7→ Y (A, z) :=∑i∈Z

A(i)z−i−1.

satisfying a list of axioms.

OK: What is a vertex operator algebra?

V = (V ,1, ω,Y (·, z))

C-vector space V = ⊕i∈NVi , with dimVi <∞;vacuum vector 1 ∈ V0;conformal vector ω ∈ V2;vertex operators Y (·, z) : V → End(V )[[ z, z−1]]

A 7→ Y (A, z) :=∑i∈Z

A(i)z−i−1.

satisfying a list of axioms.

To give the flavor, I’ll state a few axioms. For a full list, seeany of the books of my colleagues Huang, Lepowsky andcoauthors, or our papers which also give somealgebro-geometric constructions.

The vacuum vector 1 acts like the identity:

Y (1, z) =∑j∈Z

1(j) z−j−1 = idV ,

which means in particular, that

1(−1) = idV and 1(j) = 0, for j 6= −1.

The vacuum vector 1 acts like the identity:

Y (1, z) =∑j∈Z

1(j) z−j−1 = idV ,

which means in particular, that

1(−1) = idV and 1(j) = 0, for j 6= −1.

The vacuum vector 1 acts like the identity:

Y (1, z) =∑j∈Z

1(j) z−j−1 = idV ,

which means in particular, that

1(−1) = idV and 1(j) = 0, for j 6= −1.

The conformal vector gives the action of a Virasoro Liealgebra on V :

Y (ω, z) =∑i∈Z

ω(i)z−i−1 ∈ End(V )[[ z, z−1]].

Endomorphisms Lp := ω(p+1), satisfy the Virasoro relations:

[Lp, Lq] = (p − q) Lp+q +c12

δp+q,0 (p3 − p) idV ,

c ∈ C is the central charge of V .

L0 acts like a grading operator: L0|Vd= d · idVd

, for all d

L−1 acts like a derivative: Y (L−1A, z) = ∂zY (A, z).

The conformal vector gives the action of a Virasoro Liealgebra on V :

Y (ω, z) =∑i∈Z

ω(i)z−i−1 ∈ End(V )[[ z, z−1]].

Endomorphisms Lp := ω(p+1), satisfy the Virasoro relations:

[Lp, Lq] = (p − q) Lp+q +c12

δp+q,0 (p3 − p) idV ,

c ∈ C is the central charge of V .

L0 acts like a grading operator: L0|Vd= d · idVd

, for all d

L−1 acts like a derivative: Y (L−1A, z) = ∂zY (A, z).

V-modules (also incomplete)

An (admissible) V -module is a pair (W ,Y W ( − , z)) withvector space W = ⊕d∈NWd , and vertex operators

Y W (·, z) : V → End(W )[[z, z−1]],

A 7→ Y W (A, z) :=∑i∈Z

AW(i)z−i−1,

satisfying a list of hypothesis.

Can think of the vertex operators as giving infinitely manyactions of V on W .

Notice: No assumption that Wd are finite dimensional!

V-modules (also incomplete)

An (admissible) V -module is a pair (W ,Y W ( − , z)) withvector space W = ⊕d∈NWd , and vertex operators

Y W (·, z) : V → End(W )[[z, z−1]],

A 7→ Y W (A, z) :=∑i∈Z

AW(i)z−i−1,

satisfying a list of hypothesis.

Can think of the vertex operators as giving infinitely manyactions of V on W .

Notice: No assumption that Wd are finite dimensional!

V-modules (also incomplete)

An (admissible) V -module is a pair (W ,Y W ( − , z)) withvector space W = ⊕d∈NWd , and vertex operators

Y W (·, z) : V → End(W )[[z, z−1]],

A 7→ Y W (A, z) :=∑i∈Z

AW(i)z−i−1,

satisfying a list of hypothesis.

Can think of the vertex operators as giving infinitely manyactions of V on W .

Notice: No assumption that Wd are finite dimensional!

V-modules (also incomplete)

An (admissible) V -module is a pair (W ,Y W ( − , z)) withvector space W = ⊕d∈NWd , and vertex operators

Y W (·, z) : V → End(W )[[z, z−1]],

A 7→ Y W (A, z) :=∑i∈Z

AW(i)z−i−1,

satisfying a list of hypothesis.

Can think of the vertex operators as giving infinitely manyactions of V on W .

Notice: No assumption that Wd are finite dimensional!

CohFT-type

A VOA of CohFT type if

(a) V is of CFT type

V0∼= C;

(b) V is rational

finitely generated admissible modulesare completely reducible;

(c) V is C2-cofinite

dim(V/C2(V )) <∞,C2(V ) = SpanC{A(−2)B : A,B ∈ V}.

Vertex algebras of CohFT-type have good properties:

I If V is rational (or V is C2-cofinite) there are just finitelymany simple modules.

I If V is rational and C2-cofinite, the simple admissiblemodules are the same as the simple ordinary modules[DLM, 1997 Remark 2.4].

I Ordinary modules satisfy additional finitenessconditions, including

1. graded pieces Wd are finite dimensional;2. for fixed d, one has Wd+` = 0 for ` >> 0.

Construction: Step 1

Given V -modules (W i ,Y W i( − , zi)), let zi be a formal

coordinate τi at pi ∈ C and consider the constant bundle

W• :=(⊗iW i

)⊗O4

Mg,n

��4Mg,n

W(C,p•,τ•) = ⊗iW i

��(C,p•, τ•)

4Mg,n moduli of stable n-pointed curves with coordinates.

Step 2: Sheaf of Lie algebras L acts onW•

L

��4Mg,n

L(C,p•,τ•) ∼= H0(C \ p•, (VC ⊗ ωC) /∇)

��(C,p•, τ•)

H0(C \ p•, (VC ⊗ ωC) /∇)→⊕

i

H0(Dxpi, (VC ⊗ ωC) /∇)

∼=⊕

i

V ⊗ C((τi))/∇. (1)

Step 3

Have the sheaf of coinvariants:

W•/L · W• on4Mg,n,

and the forgetful map:

π :4Mg,n →Mg,n, an Aut(O)n torsor.

And we set

V = V(V ;⊗iWi) = (π∗ (W•/L · W•))Aut(O)n

.

For global generation

For each i, have W i = ⊕d∈NW id . Consider the composition

W •0 ⊗O4

Mg,n

↪→W• := W • ⊗O4Mg,n

�W•/L · W•

which induces a map

W•0 :=

(π∗

(W •

0 ⊗O4Mg,n

))Aut(O)n

→ V = (π∗ (W•/L · W•))Aut(O)n.

W•0 is a constant sheaf. The idea is to show the map issurjective under the given hypotheses.

For global generation

For each i, have W i = ⊕d∈NW id . Consider the composition

W •0 ⊗O4

Mg,n

↪→W• := W • ⊗O4Mg,n

�W•/L · W•

which induces a map

W•0 :=

(π∗

(W •

0 ⊗O4Mg,n

))Aut(O)n

→ V = (π∗ (W•/L · W•))Aut(O)n.

W•0 is a constant sheaf. The idea is to show the map issurjective under the given hypotheses.

For global generation

For each i, have W i = ⊕d∈NW id . Consider the composition

W •0 ⊗O4

Mg,n

↪→W• := W • ⊗O4Mg,n

�W•/L · W•

which induces a map

W•0 :=

(π∗

(W •

0 ⊗O4Mg,n

))Aut(O)n

→ V = (π∗ (W•/L · W•))Aut(O)n.

W•0 is a constant sheaf. The idea is to show the map issurjective under the given hypotheses.

The End (thank you).