Factorization structures via the non-commutative Hilbert scheme of - PowerPoint PPT Presentation

Factorization structures via the non-commutative Hilbert scheme of points in C 3 Emily Cliff University of Illinois at Urbana–Champaign 17 March, 2018

Section 1 The question

Let X be a smooth complex surface (e.g. C 2 ).

Let X be a smooth complex surface (e.g. C 2 ). The Hilbert scheme of n points of X parametrizes 0-dimensional subschemes of X of length n .

Let X be a smooth complex surface (e.g. C 2 ). The Hilbert scheme of n points of X parametrizes 0-dimensional subschemes of X of length n . n ≥ 0 Hilb n Write Hilb X = ⨆︁ X and H = H * (Hilb X ) = ⨁︂ H * (Hilb n X ) . n ≥ 0 .

It follows from the work of many people in geometry and in algebra that

It follows from the work of many people in geometry and in algebra that 1 H is an irreducible representation of the Heisenberg Lie algebra h X . [Nakajima, Grojnowski]

It follows from the work of many people in geometry and in algebra that 1 H is an irreducible representation of the Heisenberg Lie algebra h X . [Nakajima, Grojnowski] 2 H is isomorphic to the Heisenberg vertex algebra. [Frenkel–Lepowski–Meurmann]

It follows from the work of many people in geometry and in algebra that 1 H is an irreducible representation of the Heisenberg Lie algebra h X . [Nakajima, Grojnowski] 2 H is isomorphic to the Heisenberg vertex algebra. [Frenkel–Lepowski–Meurmann] 3 On any smooth curve C , there is associated to Hilb X the Heisenberg chiral algebra. [Huang–Lepowski, Frenkel–Ben-Zvi]

It follows from the work of many people in geometry and in algebra that 1 H is an irreducible representation of the Heisenberg Lie algebra h X . [Nakajima, Grojnowski] 2 H is isomorphic to the Heisenberg vertex algebra. [Frenkel–Lepowski–Meurmann] 3 On any smooth curve C , there is associated to Hilb X the Heisenberg chiral algebra. [Huang–Lepowski, Frenkel–Ben-Zvi] 4 On any smooth curve C , there is a Heisenberg factorization algebra ℋ C . [Beilinson–Drinfeld, Francis–Gaitsgory]

Open problem: Given a smooth curve C and a smooth surface X , find a way to construct the factorization algebra ℋ C directly from the geometry of X and C and the Hilbert scheme, without passing through all of the formal algebra.

Open problem: Given a smooth curve C and a smooth surface X , find a way to construct the factorization algebra ℋ C directly from the geometry of X and C and the Hilbert scheme, without passing through all of the formal algebra. Strategy:

Open problem: Given a smooth curve C and a smooth surface X , find a way to construct the factorization algebra ℋ C directly from the geometry of X and C and the Hilbert scheme, without passing through all of the formal algebra. Strategy: 1 Construct a factorization space over C whose fibres are built from copies of the Hilbert scheme.

Open problem: Given a smooth curve C and a smooth surface X , find a way to construct the factorization algebra ℋ C directly from the geometry of X and C and the Hilbert scheme, without passing through all of the formal algebra. Strategy: 1 Construct a factorization space over C whose fibres are built from copies of the Hilbert scheme. 2 Linearize (e.g. taking by cohomology along the fibres) to obtain a factorization algebra with fibres copies of H .

Section 2 The physics

The AGT correspondence 4d TFT 2d CFT

The AGT correspondence 4d TFT 2d CFT In math:

The AGT correspondence 4d TFT 2d CFT Moduli space of In math: G -instantons on X

The AGT correspondence 4d TFT 2d CFT Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L

The AGT correspondence 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L

The AGT correspondence 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L G = U (1):

The AGT correspondence 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L Hilb X G = U (1):

The AGT correspondence 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L Hilb X Heisenberg G = U (1): vertex algebra

The AGT correspondence Why? 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L Hilb X Heisenberg G = U (1): vertex algebra

The AGT correspondence (2,0)-6d field theory Why? 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L Hilb X Heisenberg G = U (1): vertex algebra

The AGT correspondence (2,0)-6d field theory Why? “Theory X ” 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L Hilb X Heisenberg G = U (1): vertex algebra

The AGT correspondence (2,0)-6d field theory Why? “Theory X ” X × C 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L Hilb X Heisenberg G = U (1): vertex algebra

The AGT correspondence (2,0)-6d field theory Why? “Theory X ” X × C Dimensional reduction 4d TFT 2d CFT X C Moduli space of Vertex algebra: In math: G -instantons on X 𝒳 -algebra for g L Hilb X Heisenberg G = U (1): vertex algebra

New strategy:

New strategy: 1 Build a factorization space over X × C .

New strategy: 1 Build a factorization space over X × C . 2 Use dimensional reduction to get a space over C .

New strategy: 1 Build a factorization space over X × C . 2 Use dimensional reduction to get a space over C . 3 Linearize.

Section 3 The math

Factorization spaces Let Z be a separated scheme.

Factorization spaces Let Z be a separated scheme. The Ran space of Z parametrizes non-empty finite subsets S ⊂ Z .

Factorization spaces Let Z be a separated scheme. The Ran space of Z parametrizes non-empty finite subsets S ⊂ Z . Definition A factorization space over Z is a space living over the Ran space, 𝒵 → Ran Z ,

Factorization spaces Let Z be a separated scheme. The Ran space of Z parametrizes non-empty finite subsets S ⊂ Z . Definition A factorization space over Z is a space living over the Ran space, 𝒵 → Ran Z , whose fibres 𝒵 S are equipped with compatible factorization isomorphisms :

Factorization spaces Let Z be a separated scheme. The Ran space of Z parametrizes non-empty finite subsets S ⊂ Z . Definition A factorization space over Z is a space living over the Ran space, 𝒵 → Ran Z , whose fibres 𝒵 S are equipped with compatible factorization isomorphisms : ∙ Given some points { S i } n i =1 ⊂ Ran Z

Factorization spaces Let Z be a separated scheme. The Ran space of Z parametrizes non-empty finite subsets S ⊂ Z . Definition A factorization space over Z is a space living over the Ran space, 𝒵 → Ran Z , whose fibres 𝒵 S are equipped with compatible factorization isomorphisms : ∙ Given some points { S i } n i =1 ⊂ Ran Z such that, as subsets of Z, the S i are pairwise disjoint,

Factorization spaces Let Z be a separated scheme. The Ran space of Z parametrizes non-empty finite subsets S ⊂ Z . Definition A factorization space over Z is a space living over the Ran space, 𝒵 → Ran Z , whose fibres 𝒵 S are equipped with compatible factorization isomorphisms : ∙ Given some points { S i } n i =1 ⊂ Ran Z such that, as subsets of Z, the S i are pairwise disjoint,we have n ∏︂ 𝒵 S i − ∼ → 𝒵 ⊔ S i . F { S i } : i =1

The Hilbert scheme factorization space

The Hilbert scheme factorization space In this case, we have Z = C , a smooth complex curve.

The Hilbert scheme factorization space In this case, we have Z = C , a smooth complex curve. We define a space ℋ ilb X × C , whose fibre over S = { c 1 , . . . , c n } ∈ Ran C is given by

The Hilbert scheme factorization space In this case, we have Z = C , a smooth complex curve. We define a space ℋ ilb X × C , whose fibre over S = { c 1 , . . . , c n } ∈ Ran C is given by ⃒ n ⃒ ⨆︂ ℋ ilb X × C , S = { ξ ∈ Hilb X × C ⃒ Supp ξ ⊂ ( X × { c i } ) } ⃒ ⃒ i =1 n ∼ ∏︂ ℋ ilb X × C , { c i } . = i =1

The Hilbert scheme factorization space In this case, we have Z = C , a smooth complex curve. We define a space ℋ ilb X × C , whose fibre over S = { c 1 , . . . , c n } ∈ Ran C is given by ⃒ n ⃒ ⨆︂ ℋ ilb X × C , S = { ξ ∈ Hilb X × C ⃒ Supp ξ ⊂ ( X × { c i } ) } ⃒ ⃒ i =1 n ∼ ∏︂ ℋ ilb X × C , { c i } . = i =1

The Hilbert scheme factorization space C

The Hilbert scheme factorization space C S

The Hilbert scheme factorization space C S

The Hilbert scheme factorization space C S

The Hilbert scheme factorization space C S

The Hilbert scheme factorization space C S

The Hilbert scheme factorization space C X × S S

The Hilbert scheme factorization space C X × S S ξ

The Hilbert scheme as a critical locus

The Hilbert scheme as a critical locus e.g. when X = C 2 , C = C 3 , we can write Hilb n X × C as a critical locus inside the non-commutative Hilbert scheme as follows: