Intersection Spaces, Perverse Sheaves and Type IIB String Theory

LAURENTIU MAXIM University of Wisconsin-Madison (joint with Markus Banagl and Nero Budur)

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory singular space = n-dim. complex projective variety manifold = n-dim. complex projective manifold coeﬃcients: Q Manifolds have a nice hidden symmetry: Poincar´eDuality.

In particular: X manifold =⇒ bk (X ) = b2n−k (X ). Singular spaces do not posses such symmetry.

Motivation: Poincar´eDuality

Convention:

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory manifold = n-dim. complex projective manifold coeﬃcients: Q Manifolds have a nice hidden symmetry: Poincar´eDuality.

In particular: X manifold =⇒ bk (X ) = b2n−k (X ). Singular spaces do not posses such symmetry.

Motivation: Poincar´eDuality

Convention: singular space = n-dim. complex projective variety

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory coeﬃcients: Q Manifolds have a nice hidden symmetry: Poincar´eDuality.

In particular: X manifold =⇒ bk (X ) = b2n−k (X ). Singular spaces do not posses such symmetry.

Motivation: Poincar´eDuality

Convention: singular space = n-dim. complex projective variety manifold = n-dim. complex projective manifold

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Manifolds have a nice hidden symmetry: Poincar´eDuality.

In particular: X manifold =⇒ bk (X ) = b2n−k (X ). Singular spaces do not posses such symmetry.

Motivation: Poincar´eDuality

Convention: singular space = n-dim. complex projective variety manifold = n-dim. complex projective manifold coeﬃcients: Q

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory In particular: X manifold =⇒ bk (X ) = b2n−k (X ). Singular spaces do not posses such symmetry.

Motivation: Poincar´eDuality

Convention: singular space = n-dim. complex projective variety manifold = n-dim. complex projective manifold coeﬃcients: Q Manifolds have a nice hidden symmetry: Poincar´eDuality.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Singular spaces do not posses such symmetry.

Motivation: Poincar´eDuality

Convention: singular space = n-dim. complex projective variety manifold = n-dim. complex projective manifold coeﬃcients: Q Manifolds have a nice hidden symmetry: Poincar´eDuality.

In particular: X manifold =⇒ bk (X ) = b2n−k (X ).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Motivation: Poincar´eDuality

Convention: singular space = n-dim. complex projective variety manifold = n-dim. complex projective manifold coeﬃcients: Q Manifolds have a nice hidden symmetry: Poincar´eDuality.

In particular: X manifold =⇒ bk (X ) = b2n−k (X ). Singular spaces do not posses such symmetry.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory these are homology groups of a complex of allowed chains, deﬁned by imposing restrictions on how chains meet the singularities.

Intersection homology

much of the manifold theory (e.g., Poincar´eDuality, K¨ahler package) is recovered in the singular context if one uses instead the intersection (co)homology groupsIH ∗(X ) of Goresky-MacPherson.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Intersection homology

much of the manifold theory (e.g., PoincaréDuality, Kähler package) is recovered in the singular context if one uses instead the intersection (co)homology groupsIH ∗(X ) of Goresky-MacPherson. these are homology groups of a complex of allowed chains, defined by imposing restrictions on how chains meet the singularities.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory ∼ if Xe is a small resolution: IH∗(X ) = H∗(Xe).

IH∗(X ) carries the K¨ahlerpackage: Hodge structure, weak and hard Lefschetz.

IH∗ is a topological invariant, but is not functorial, and has no ring structure in general. IH∗(X ) is realized by a perverse, self-dual, constructible sheaf complex ICX , i.e.

i ∼ i IH (X ) = H (X , ICX [−n]).

IH∗ (like H∗) is unstable under smoothings of singularities.

Properties of intersection homology

if Xe is a resolution of singularities of X , then:

IH∗(X ) ⊂ H∗(Xe)

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory IH∗(X ) carries the K¨ahlerpackage: Hodge structure, weak and hard Lefschetz.

IH∗ is a topological invariant, but is not functorial, and has no ring structure in general. IH∗(X ) is realized by a perverse, self-dual, constructible sheaf complex ICX , i.e.

i ∼ i IH (X ) = H (X , ICX [−n]).

IH∗ (like H∗) is unstable under smoothings of singularities.

Properties of intersection homology

if Xe is a resolution of singularities of X , then:

IH∗(X ) ⊂ H∗(Xe) ∼ if Xe is a small resolution: IH∗(X ) = H∗(Xe).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory IH∗ is a topological invariant, but is not functorial, and has no ring structure in general. IH∗(X ) is realized by a perverse, self-dual, constructible sheaf complex ICX , i.e.

i ∼ i IH (X ) = H (X , ICX [−n]).

IH∗ (like H∗) is unstable under smoothings of singularities.

Properties of intersection homology

if Xe is a resolution of singularities of X , then:

IH∗(X ) ⊂ H∗(Xe) ∼ if Xe is a small resolution: IH∗(X ) = H∗(Xe).

IH∗(X ) carries the K¨ahlerpackage: Hodge structure, weak and hard Lefschetz.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory IH∗(X ) is realized by a perverse, self-dual, constructible sheaf complex ICX , i.e.

i ∼ i IH (X ) = H (X , ICX [−n]).

IH∗ (like H∗) is unstable under smoothings of singularities.

Properties of intersection homology

if Xe is a resolution of singularities of X , then:

IH∗(X ) ⊂ H∗(Xe) ∼ if Xe is a small resolution: IH∗(X ) = H∗(Xe).

IH∗(X ) carries the K¨ahlerpackage: Hodge structure, weak and hard Lefschetz.

IH∗ is a topological invariant, but is not functorial, and has no ring structure in general.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory IH∗ (like H∗) is unstable under smoothings of singularities.

Properties of intersection homology

if Xe is a resolution of singularities of X , then:

IH∗(X ) ⊂ H∗(Xe) ∼ if Xe is a small resolution: IH∗(X ) = H∗(Xe).

IH∗(X ) carries the K¨ahlerpackage: Hodge structure, weak and hard Lefschetz.

IH∗ is a topological invariant, but is not functorial, and has no ring structure in general. IH∗(X ) is realized by a perverse, self-dual, constructible sheaf complex ICX , i.e.

i ∼ i IH (X ) = H (X , ICX [−n]).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Properties of intersection homology

if Xe is a resolution of singularities of X , then:

IH∗(X ) ⊂ H∗(Xe) ∼ if Xe is a small resolution: IH∗(X ) = H∗(Xe).

IH∗(X ) carries the K¨ahlerpackage: Hodge structure, weak and hard Lefschetz.

IH∗ is a topological invariant, but is not functorial, and has no ring structure in general. IH∗(X ) is realized by a perverse, self-dual, constructible sheaf complex ICX , i.e.

i ∼ i IH (X ) = H (X , ICX [−n]).

IH∗ (like H∗) is unstable under smoothings of singularities.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory 2 For s 6= 0, Xs is an elliptic curve (i.e., a 2-torus T ). X = X0 has a nodal singularity (X is a pinched torus).

H1(X ) = Q, IH1(X ) = 0, but H1(Xs ) = IH1(Xs ) = Q ⊕ Q so neither H∗(−) nor IH∗(−) is invariant under the smoothing X ; Xs .

Example 2 2 Let Xs := {y = x(x − 1)(x − s)} ⊂ P , with |s| < 1.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory X = X0 has a nodal singularity (X is a pinched torus).

H1(X ) = Q, IH1(X ) = 0, but H1(Xs ) = IH1(Xs ) = Q ⊕ Q so neither H∗(−) nor IH∗(−) is invariant under the smoothing X ; Xs .

Example 2 2 Let Xs := {y = x(x − 1)(x − s)} ⊂ P , with |s| < 1. 2 For s 6= 0, Xs is an elliptic curve (i.e., a 2-torus T ).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory H1(X ) = Q, IH1(X ) = 0, but H1(Xs ) = IH1(Xs ) = Q ⊕ Q so neither H∗(−) nor IH∗(−) is invariant under the smoothing X ; Xs .

Example 2 2 Let Xs := {y = x(x − 1)(x − s)} ⊂ P , with |s| < 1. 2 For s 6= 0, Xs is an elliptic curve (i.e., a 2-torus T ). X = X0 has a nodal singularity (X is a pinched torus).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory but H1(Xs ) = IH1(Xs ) = Q ⊕ Q so neither H∗(−) nor IH∗(−) is invariant under the smoothing X ; Xs .

Example 2 2 Let Xs := {y = x(x − 1)(x − s)} ⊂ P , with |s| < 1. 2 For s 6= 0, Xs is an elliptic curve (i.e., a 2-torus T ). X = X0 has a nodal singularity (X is a pinched torus).

H1(X ) = Q, IH1(X ) = 0,

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory so neither H∗(−) nor IH∗(−) is invariant under the smoothing X ; Xs .

Example 2 2 Let Xs := {y = x(x − 1)(x − s)} ⊂ P , with |s| < 1. 2 For s 6= 0, Xs is an elliptic curve (i.e., a 2-torus T ). X = X0 has a nodal singularity (X is a pinched torus).

H1(X ) = Q, IH1(X ) = 0, but H1(Xs ) = IH1(Xs ) = Q ⊕ Q

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Example 2 2 Let Xs := {y = x(x − 1)(x − s)} ⊂ P , with |s| < 1. 2 For s 6= 0, Xs is an elliptic curve (i.e., a 2-torus T ). X = X0 has a nodal singularity (X is a pinched torus).

H1(X ) = Q, IH1(X ) = 0, but H1(Xs ) = IH1(Xs ) = Q ⊕ Q so neither H∗(−) nor IH∗(−) is invariant under the smoothing X ; Xs .

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Poincar´eDuality holds; it has a ring structure; it is more stable under deformations; it carries a K¨ahlerpackage; it is realized by a self-dual perverse sheaf?

Questions: Is there a (co)homology theory for singular spaces, so that:

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory it has a ring structure; it is more stable under deformations; it carries a K¨ahlerpackage; it is realized by a self-dual perverse sheaf?

Questions: Is there a (co)homology theory for singular spaces, so that: Poincar´eDuality holds;

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory it is more stable under deformations; it carries a K¨ahlerpackage; it is realized by a self-dual perverse sheaf?

Questions: Is there a (co)homology theory for singular spaces, so that: Poincar´eDuality holds; it has a ring structure;

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory it carries a K¨ahlerpackage; it is realized by a self-dual perverse sheaf?

Questions: Is there a (co)homology theory for singular spaces, so that: Poincar´eDuality holds; it has a ring structure; it is more stable under deformations;

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory it is realized by a self-dual perverse sheaf?

Questions: Is there a (co)homology theory for singular spaces, so that: Poincar´eDuality holds; it has a ring structure; it is more stable under deformations; it carries a K¨ahlerpackage;

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Questions: Is there a (co)homology theory for singular spaces, so that: Poincar´eDuality holds; it has a ring structure; it is more stable under deformations; it carries a K¨ahlerpackage; it is realized by a self-dual perverse sheaf?

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Theorem (Banagl): He∗(IX ) satisfies PoincaréDuality. Note: The intersection space cohomology H∗(IX ) has an internal ring structure defined by the usual cup product in cohomology, so H∗(IX ) 6= IH∗(X ) !!!

Intersection Spaces

Banagl (2010) uses homotopy theory to associate to a (certain) singular space X a cellular complex IX , the intersection space of X .

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Note: The intersection space cohomology H∗(IX ) has an internal ring structure deﬁned by the usual cup product in cohomology, so H∗(IX ) 6= IH∗(X ) !!!

Intersection Spaces

Banagl (2010) uses homotopy theory to associate to a (certain) singular space X a cellular complex IX , the intersection space of X .

Theorem (Banagl): He∗(IX ) satisﬁes Poincar´eDuality.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Intersection Spaces

Banagl (2010) uses homotopy theory to associate to a (certain) singular space X a cellular complex IX , the intersection space of X .

Theorem (Banagl): He∗(IX ) satisfies PoincaréDuality. Note: The intersection space cohomology H∗(IX ) has an internal ring structure defined by the usual cup product in cohomology, so H∗(IX ) 6= IH∗(X ) !!!

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory f∗ : Hr (K

Let K be a simply-connected CW complex, and ﬁx n ∈ Z≥0. Moore approximation of K is a CW complex K

Moore approximations exist (P. Hilton), and can often be constructed in the non-simply-connected case, e.g., if ∂n = 0, (n−1) choose K

Construction of Intersection Spaces

IX is deﬁned via spatial homology truncation, i.e., by replacing links of singularities of X by their Moore approximations.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory f∗ : Hr (K

Moore approximation of K is a CW complex K

Moore approximations exist (P. Hilton), and can often be constructed in the non-simply-connected case, e.g., if ∂n = 0, (n−1) choose K

Construction of Intersection Spaces

IX is deﬁned via spatial homology truncation, i.e., by replacing links of singularities of X by their Moore approximations.

Let K be a simply-connected CW complex, and ﬁx n ∈ Z≥0.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory f∗ : Hr (K

Construction of Intersection Spaces

IX is deﬁned via spatial homology truncation, i.e., by replacing links of singularities of X by their Moore approximations.

Let K be a simply-connected CW complex, and ﬁx n ∈ Z≥0. Moore approximation of K is a CW complex K

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Hr (K

Construction of Intersection Spaces

IX is deﬁned via spatial homology truncation, i.e., by replacing links of singularities of X by their Moore approximations.

Let K be a simply-connected CW complex, and ﬁx n ∈ Z≥0. Moore approximation of K is a CW complex K

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Moore approximations exist (P. Hilton), and can often be constructed in the non-simply-connected case, e.g., if ∂n = 0, (n−1) choose K

Construction of Intersection Spaces

IX is deﬁned via spatial homology truncation, i.e., by replacing links of singularities of X by their Moore approximations.

Let K be a simply-connected CW complex, and ﬁx n ∈ Z≥0. Moore approximation of K is a CW complex K

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Construction of Intersection Spaces

IX is deﬁned via spatial homology truncation, i.e., by replacing links of singularities of X by their Moore approximations.

Let K be a simply-connected CW complex, and ﬁx n ∈ Z≥0. Moore approximation of K is a CW complex K

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Assume X has only one isolated singularity x, with link K. ◦ Let M := X − c (K). Then ∂M = K, and X = M ∪K c(K). f Let g : K

H∗(IX ; Q) is well-deﬁned, i.e., independent of all choices.

Construction of Intersection Spaces, cont’d

IX is deﬁned by replacing links of singularities of X by their Moore approximations.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory ◦ Let M := X − c (K). Then ∂M = K, and X = M ∪K c(K). f Let g : K

H∗(IX ; Q) is well-deﬁned, i.e., independent of all choices.

Construction of Intersection Spaces, cont’d

IX is deﬁned by replacing links of singularities of X by their Moore approximations. Assume X has only one isolated singularity x, with link K.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory f Let g : K

H∗(IX ; Q) is well-deﬁned, i.e., independent of all choices.

Construction of Intersection Spaces, cont’d

IX is deﬁned by replacing links of singularities of X by their Moore approximations. Assume X has only one isolated singularity x, with link K. ◦ Let M := X − c (K). Then ∂M = K, and X = M ∪K c(K).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Set IX := cone(g) := M ∪g cone(K

H∗(IX ; Q) is well-deﬁned, i.e., independent of all choices.

Construction of Intersection Spaces, cont’d

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory If X has several isolated singularities, IX is deﬁned by performing this procedure on each of the links, simultaneously.

H∗(IX ; Q) is well-deﬁned, i.e., independent of all choices.

Construction of Intersection Spaces, cont’d

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory H∗(IX ; Q) is well-deﬁned, i.e., independent of all choices.

Construction of Intersection Spaces, cont’d

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Construction of Intersection Spaces, cont’d

H∗(IX ; Q) is well-deﬁned, i.e., independent of all choices.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory The intersection space IX of X is

H1(IX ) = Q ⊕ Q = H1(Xs ).

Example In the case of the nodal curve X , with n = 1, the link of the 1 1 singular point is two circles K = S t S , so K<1 = • t •.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory H1(IX ) = Q ⊕ Q = H1(Xs ).

Example In the case of the nodal curve X , with n = 1, the link of the 1 1 singular point is two circles K = S t S , so K<1 = • t •. The intersection space IX of X is

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Example In the case of the nodal curve X , with n = 1, the link of the 1 1 singular point is two circles K = S t S , so K<1 = • t •. The intersection space IX of X is

H1(IX ) = Q ⊕ Q = H1(Xs ).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory But which Calabi-Yau 3-fold should model the string? Conjecture [Green-H¨ubsch] Topologically distinct Calabi-Yau’s are connected to each other by conifold transitions, which induce a phase transition between the corresponding string models.

Conifold transitions in (10d) String Theory

In (10d) string theory, the 6 (real) dimensions needed for a string to vibrate form aCY space.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Conjecture [Green-H¨ubsch] Topologically distinct Calabi-Yau’s are connected to each other by conifold transitions, which induce a phase transition between the corresponding string models.

Conifold transitions in (10d) String Theory

In (10d) string theory, the 6 (real) dimensions needed for a string to vibrate form aCY space. But which Calabi-Yau 3-fold should model the string?

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Conifold transitions in (10d) String Theory

In (10d) string theory, the 6 (real) dimensions needed for a string to vibrate form aCY space. But which Calabi-Yau 3-fold should model the string? Conjecture [Green-H¨ubsch] Topologically distinct Calabi-Yau’s are connected to each other by conifold transitions, which induce a phase transition between the corresponding string models.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory S is obtained from Xs by deforming the complex structure. π is a small resolution. The deformation collapses embedded 3-spheres (vanishing cycles) to isolated o.d.p.’s. π resolves the singularities of S by replacing each of them 1 with a CP .

Conifold transition: s→0 π Xs ; S ←− Y , where:

Xs and Y are topologically distinct smooth Calabi-Yau 3-folds.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory π is a small resolution. The deformation collapses embedded 3-spheres (vanishing cycles) to isolated o.d.p.’s. π resolves the singularities of S by replacing each of them 1 with a CP .

Conifold transition: s→0 π Xs ; S ←− Y , where:

Xs and Y are topologically distinct smooth Calabi-Yau 3-folds.

S is obtained from Xs by deforming the complex structure.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory The deformation collapses embedded 3-spheres (vanishing cycles) to isolated o.d.p.’s. π resolves the singularities of S by replacing each of them 1 with a CP .

Conifold transition: s→0 π Xs ; S ←− Y , where:

Xs and Y are topologically distinct smooth Calabi-Yau 3-folds.

S is obtained from Xs by deforming the complex structure. π is a small resolution.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory π resolves the singularities of S by replacing each of them 1 with a CP .

Conifold transition: s→0 π Xs ; S ←− Y , where:

Xs and Y are topologically distinct smooth Calabi-Yau 3-folds.

S is obtained from Xs by deforming the complex structure. π is a small resolution. The deformation collapses embedded 3-spheres (vanishing cycles) to isolated o.d.p.’s.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Conifold transition: s→0 π Xs ; S ←− Y , where:

Xs and Y are topologically distinct smooth Calabi-Yau 3-folds.

S is obtained from Xs by deforming the complex structure. π is a small resolution. The deformation collapses embedded 3-spheres (vanishing cycles) to isolated o.d.p.’s. π resolves the singularities of S by replacing each of them 1 with a CP .

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory The intersection homology of the conifold accounts for all of these so it is the physically correct homology theory for type IIA string theory.

In type IIB string theory there are charged 3-branes wrapped around the vanishing cycles, and which become massless as these vanishing cycles are collapsed by the deformation of complex structure. Neither H∗(S) nor IH∗(S) account for these massless 3-branes, but H∗(IS) yields the correct count. So the homology of intersection spaces is the physically correct homology theory in the IIB string theory.

In type IIA string theory, there are charged 2-branes that wrap 1 around the CP 2-cycles, and which become massless when these 2-cycles are collapsed to points by the resolution map π.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory In type IIB string theory there are charged 3-branes wrapped around the vanishing cycles, and which become massless as these vanishing cycles are collapsed by the deformation of complex structure. Neither H∗(S) nor IH∗(S) account for these massless 3-branes, but H∗(IS) yields the correct count. So the homology of intersection spaces is the physically correct homology theory in the IIB string theory.

In type IIA string theory, there are charged 2-branes that wrap 1 around the CP 2-cycles, and which become massless when these 2-cycles are collapsed to points by the resolution map π. The intersection homology of the conifold accounts for all of these so it is the physically correct homology theory for type IIA string theory.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Neither H∗(S) nor IH∗(S) account for these massless 3-branes, but H∗(IS) yields the correct count. So the homology of intersection spaces is the physically correct homology theory in the IIB string theory.

In type IIB string theory there are charged 3-branes wrapped around the vanishing cycles, and which become massless as these vanishing cycles are collapsed by the deformation of complex structure.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory So the homology of intersection spaces is the physically correct homology theory in the IIB string theory.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory In type IIA string theory, there are charged 2-branes that wrap 1 around the CP 2-cycles, and which become massless when these 2-cycles are collapsed to points by the resolution map π. The intersection homology of the conifold accounts for all of these so it is the physically correct homology theory for type IIA string theory.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory If X and X ◦ are smooth, their Betti numbers are related by precise ◦ algebraic identities, e.g., b3(X ) = b2(X ) + b4(X ) + 2, etc. Conjecture: [Morrison] The mirror of a conifold transition is again a conifold transition, but performed in the reverse order (mirror symmetry should exchange resolutions and deformations). So, if S and S◦ are mirrored conifolds (in mirrored conifold transitions), the intersection space homology of one space and the intersection homology of the mirror space form a mirror-pair, i.e.,

◦ b3(IS ) = Ib2(S) + Ib4(S) + 2,

etc., where Ibi is the i-th intersection homology Betti number.

Relation to mirror symmetry

Given a Calabi-Yau 3-fold X , the mirror map associates to it another Calabi-Yau 3-fold X ◦ so that type IIB string theory on 4 4 ◦ R × X corresponds to type IIA string theory on R × X .

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Conjecture: [Morrison] The mirror of a conifold transition is again a conifold transition, but performed in the reverse order (mirror symmetry should exchange resolutions and deformations). So, if S and S◦ are mirrored conifolds (in mirrored conifold transitions), the intersection space homology of one space and the intersection homology of the mirror space form a mirror-pair, i.e.,

◦ b3(IS ) = Ib2(S) + Ib4(S) + 2,

etc., where Ibi is the i-th intersection homology Betti number.

Relation to mirror symmetry

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory So, if S and S◦ are mirrored conifolds (in mirrored conifold transitions), the intersection space homology of one space and the intersection homology of the mirror space form a mirror-pair, i.e.,

◦ b3(IS ) = Ib2(S) + Ib4(S) + 2,

etc., where Ibi is the i-th intersection homology Betti number.

Relation to mirror symmetry

Given a Calabi-Yau 3-fold X , the mirror map associates to it another Calabi-Yau 3-fold X ◦ so that type IIB string theory on 4 4 ◦ R × X corresponds to type IIA string theory on R × X . If X and X ◦ are smooth, their Betti numbers are related by precise ◦ algebraic identities, e.g., b3(X ) = b2(X ) + b4(X ) + 2, etc. Conjecture: [Morrison] The mirror of a conifold transition is again a conifold transition, but performed in the reverse order (mirror symmetry should exchange resolutions and deformations).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Relation to mirror symmetry

◦ b3(IS ) = Ib2(S) + Ib4(S) + 2,

etc., where Ibi is the i-th intersection homology Betti number.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Upshot: The above mirror symmetry considerations suggest that one could compute H∗(IX ) for a variety X in terms of the topology of a smoothing family Xs , by “mirroring” known results relating the intersection homology groups IH∗(X ) of X to the topology of a resolution of singularities Xe.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory n n Let Kx , Fx and Tx : H (Fx ) → H (Fx ) be the link, Milnor ﬁber and local monodromy operator of (X , x). h.e. F ' W Sn, µ = Milnor number x µx x (=number of vanishing cycles at x).

Hypersurface singularities

Let f be a homogeneous polynomial s.t.

n+1 X = {f (x) = 0} ⊂ CP

has only one isolated singularity x.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory h.e. F ' W Sn, µ = Milnor number x µx x (=number of vanishing cycles at x).

Hypersurface singularities

Let f be a homogeneous polynomial s.t.

n+1 X = {f (x) = 0} ⊂ CP

has only one isolated singularity x. n n Let Kx , Fx and Tx : H (Fx ) → H (Fx ) be the link, Milnor ﬁber and local monodromy operator of (X , x).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Hypersurface singularities

Let f be a homogeneous polynomial s.t.

n+1 X = {f (x) = 0} ⊂ CP

has only one isolated singularity x. n n Let Kx , Fx and Tx : H (Fx ) → H (Fx ) be the link, Milnor ﬁber and local monodromy operator of (X , x). h.e. F ' W Sn, µ = Milnor number x µx x (=number of vanishing cycles at x).

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Corollary n ∼ n H (IX ) = H (Xs ) ⇐⇒ Tx is trivial.

Remark Recall that IH∗(−) is invariant under small resolutions. We regard the local trivial monodromy condition as “mirroring” that of the existence of small resolutions.

Theorem A (Banagl-M.) n+1 Let X ⊂ CP be a hypersurface with only one isolated singular point x. Let Xs be a nearby smoothing of X . Then:

 i dim H (Xs ) if i 6= n, 2n i  i dim H (IX ) = dim H (Xs ) − rank(Tx − 1) if i = n  0 if i = 2n.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Remark Recall that IH∗(−) is invariant under small resolutions. We regard the local trivial monodromy condition as “mirroring” that of the existence of small resolutions.

Theorem A (Banagl-M.) n+1 Let X ⊂ CP be a hypersurface with only one isolated singular point x. Let Xs be a nearby smoothing of X . Then:

 i dim H (Xs ) if i 6= n, 2n i  i dim H (IX ) = dim H (Xs ) − rank(Tx − 1) if i = n  0 if i = 2n.

Corollary n ∼ n H (IX ) = H (Xs ) ⇐⇒ Tx is trivial.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Theorem A (Banagl-M.) n+1 Let X ⊂ CP be a hypersurface with only one isolated singular point x. Let Xs be a nearby smoothing of X . Then:

 i dim H (Xs ) if i 6= n, 2n i  i dim H (IX ) = dim H (Xs ) − rank(Tx − 1) if i = n  0 if i = 2n.

Corollary n ∼ n H (IX ) = H (Xs ) ⇐⇒ Tx is trivial.

Remark Recall that IH∗(−) is invariant under small resolutions. We regard the local trivial monodromy condition as “mirroring” that of the existence of small resolutions.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory i (b) H (X ; ISX ) carries a natural mixed Hodge structure, ∀i. (c) If the local monodromy Tx at x is semi-simple in the eigenvalue 1, the intersection-space complex ISX is self-dual. (d) If the local monodromy Tx at x is semi-simple in the ∗ eigenvalue 1, and the global monodromy T acting on H (Xs ) is ∗ semi-simple in the eigenvalue 1, then H (X ; ISX ) carry pure Hodge structures satisfying the Hard Lefschetz theorem.

Intersection space complex: Existence and Properties

Theorem B (Banagl-Budur-M., 2012)

(a) There exists a perverse sheaf complex ISX on X so that there are (abstract) isomorphisms ( Hi (IX ) if i 6= 2n i (X ; IS [−n]) ' H X 2n H (Xs ) = Q if i = 2n.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory (c) If the local monodromy Tx at x is semi-simple in the eigenvalue 1, the intersection-space complex ISX is self-dual. (d) If the local monodromy Tx at x is semi-simple in the ∗ eigenvalue 1, and the global monodromy T acting on H (Xs ) is ∗ semi-simple in the eigenvalue 1, then H (X ; ISX ) carry pure Hodge structures satisfying the Hard Lefschetz theorem.

Intersection space complex: Existence and Properties

Theorem B (Banagl-Budur-M., 2012)

(a) There exists a perverse sheaf complex ISX on X so that there are (abstract) isomorphisms ( Hi (IX ) if i 6= 2n i (X ; IS [−n]) ' H X 2n H (Xs ) = Q if i = 2n.

i (b) H (X ; ISX ) carries a natural mixed Hodge structure, ∀i.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory (d) If the local monodromy Tx at x is semi-simple in the ∗ eigenvalue 1, and the global monodromy T acting on H (Xs ) is ∗ semi-simple in the eigenvalue 1, then H (X ; ISX ) carry pure Hodge structures satisfying the Hard Lefschetz theorem.

Intersection space complex: Existence and Properties

Theorem B (Banagl-Budur-M., 2012)

(a) There exists a perverse sheaf complex ISX on X so that there are (abstract) isomorphisms ( Hi (IX ) if i 6= 2n i (X ; IS [−n]) ' H X 2n H (Xs ) = Q if i = 2n.

i (b) H (X ; ISX ) carries a natural mixed Hodge structure, ∀i. (c) If the local monodromy Tx at x is semi-simple in the eigenvalue 1, the intersection-space complex ISX is self-dual.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Intersection space complex: Existence and Properties

Theorem B (Banagl-Budur-M., 2012)

(a) There exists a perverse sheaf complex ISX on X so that there are (abstract) isomorphisms ( Hi (IX ) if i 6= 2n i (X ; IS [−n]) ' H X 2n H (Xs ) = Q if i = 2n.

i (b) H (X ; ISX ) carries a natural mixed Hodge structure, ∀i. (c) If the local monodromy Tx at x is semi-simple in the eigenvalue 1, the intersection-space complex ISX is self-dual. (d) If the local monodromy Tx at x is semi-simple in the ∗ eigenvalue 1, and the global monodromy T acting on H (Xs ) is ∗ semi-simple in the eigenvalue 1, then H (X ; ISX ) carry pure Hodge structures satisfying the Hard Lefschetz theorem.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Assume n > 2, so π1(Kx ) = 0. Let π : Xe → S ⊂ C be a family of hypersurfaces with −1 −1 X = π (0), s.t. Xe is smooth and Xs := π (s) for s 6= 0 is a n+1 smooth hypersurface in CP . b b Let ψπ, ϕπ : Dc (Xe) → Dc (X ) be the nearby and vanishing cycle functors of π, with monodromy T and resp. Te. Hi (X ) =∼ i (X ; ψ ), with compatible monodromies. s H πQXe If ix : {x} ,→ X is the inclusion, then:

Hi (F ) =∼ Hi (i ∗ψ ) , Hi (F ) = Hi (i ∗ϕ ) x x πQXe e x x πQXe with compatible monodromies. Supp(ϕ ) = Sing(X ) = {x}. πQXe

Nearby and vanishing cycles.

n+1 Let X ⊂ CP be a hypersurface with Sing(X ) = {x}.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Let π : Xe → S ⊂ C be a family of hypersurfaces with −1 −1 X = π (0), s.t. Xe is smooth and Xs := π (s) for s 6= 0 is a n+1 smooth hypersurface in CP . b b Let ψπ, ϕπ : Dc (Xe) → Dc (X ) be the nearby and vanishing cycle functors of π, with monodromy T and resp. Te. Hi (X ) =∼ i (X ; ψ ), with compatible monodromies. s H πQXe If ix : {x} ,→ X is the inclusion, then:

Hi (F ) =∼ Hi (i ∗ψ ) , Hi (F ) = Hi (i ∗ϕ ) x x πQXe e x x πQXe with compatible monodromies. Supp(ϕ ) = Sing(X ) = {x}. πQXe

Nearby and vanishing cycles.

n+1 Let X ⊂ CP be a hypersurface with Sing(X ) = {x}. Assume n > 2, so π1(Kx ) = 0.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory b b Let ψπ, ϕπ : Dc (Xe) → Dc (X ) be the nearby and vanishing cycle functors of π, with monodromy T and resp. Te. Hi (X ) =∼ i (X ; ψ ), with compatible monodromies. s H πQXe If ix : {x} ,→ X is the inclusion, then:

Hi (F ) =∼ Hi (i ∗ψ ) , Hi (F ) = Hi (i ∗ϕ ) x x πQXe e x x πQXe with compatible monodromies. Supp(ϕ ) = Sing(X ) = {x}. πQXe

Nearby and vanishing cycles.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Hi (X ) =∼ i (X ; ψ ), with compatible monodromies. s H πQXe If ix : {x} ,→ X is the inclusion, then:

Hi (F ) =∼ Hi (i ∗ψ ) , Hi (F ) = Hi (i ∗ϕ ) x x πQXe e x x πQXe with compatible monodromies. Supp(ϕ ) = Sing(X ) = {x}. πQXe

Nearby and vanishing cycles.

n+1 Let X ⊂ CP be a hypersurface with Sing(X ) = {x}. Assume n > 2, so π1(Kx ) = 0. Let π : Xe → S ⊂ C be a family of hypersurfaces with −1 −1 X = π (0), s.t. Xe is smooth and Xs := π (s) for s 6= 0 is a n+1 smooth hypersurface in CP . b b Let ψπ, ϕπ : Dc (Xe) → Dc (X ) be the nearby and vanishing cycle functors of π, with monodromy T and resp. Te.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory If ix : {x} ,→ X is the inclusion, then:

Hi (F ) =∼ Hi (i ∗ψ ) , Hi (F ) = Hi (i ∗ϕ ) x x πQXe e x x πQXe with compatible monodromies. Supp(ϕ ) = Sing(X ) = {x}. πQXe

Nearby and vanishing cycles.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Supp(ϕ ) = Sing(X ) = {x}. πQXe

Nearby and vanishing cycles.

Hi (F ) =∼ Hi (i ∗ψ ) , Hi (F ) = Hi (i ∗ϕ ) x x πQXe e x x πQXe with compatible monodromies.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Nearby and vanishing cycles.

Hi (F ) =∼ Hi (i ∗ψ ) , Hi (F ) = Hi (i ∗ϕ ) x x πQXe e x x πQXe with compatible monodromies. Supp(ϕ ) = Sing(X ) = {x}. πQXe

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory There are canonical morphisms:

can : ψπ → ϕπ and var : ϕπ → ψπ

s.t. can ◦ var = Te − 1, var ◦ can = T − 1.

There are decompositions:

ψπ = ψπ,1 ⊕ ψπ,6=1 and ϕπ = ϕπ,1 ⊕ ϕπ,6=1

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory There are decompositions:

ψπ = ψπ,1 ⊕ ψπ,6=1 and ϕπ = ϕπ,1 ⊕ ϕπ,6=1

There are canonical morphisms:

can : ψπ → ϕπ and var : ϕπ → ψπ s.t. can ◦ var = Te − 1, var ◦ can = T − 1.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Then C ∈ Perv(X ), Supp(C) = {x}, and ( i 0 , if i 6= 0, H (X ; C) = Image(Tx − 1) , if i = 0. Let ι := var ◦ ι : C −→ ψ [n]. ϕ πQXe Deﬁne: IS := Coker ι : C −→ ψ [n] ∈ Perv(X ) X πQXe ISX underlies a mixed Hodge module. If Tx is semi-simple in the eigenvalue 1, then: IS =∼ ψ [n]. X π,1QXe

Intersection space complex: Construction

Let ιϕ C := Image(T − 1) ,→ ϕ [n] ∈ Perv(X ) e πQXe

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Let ι := var ◦ ι : C −→ ψ [n]. ϕ πQXe Deﬁne: IS := Coker ι : C −→ ψ [n] ∈ Perv(X ) X πQXe ISX underlies a mixed Hodge module. If Tx is semi-simple in the eigenvalue 1, then: IS =∼ ψ [n]. X π,1QXe

Intersection space complex: Construction

Let ιϕ C := Image(T − 1) ,→ ϕ [n] ∈ Perv(X ) e πQXe Then C ∈ Perv(X ), Supp(C) = {x}, and ( i 0 , if i 6= 0, H (X ; C) = Image(Tx − 1) , if i = 0.

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Deﬁne: IS := Coker ι : C −→ ψ [n] ∈ Perv(X ) X πQXe ISX underlies a mixed Hodge module. If Tx is semi-simple in the eigenvalue 1, then: IS =∼ ψ [n]. X π,1QXe

Intersection space complex: Construction

Let ιϕ C := Image(T − 1) ,→ ϕ [n] ∈ Perv(X ) e πQXe Then C ∈ Perv(X ), Supp(C) = {x}, and ( i 0 , if i 6= 0, H (X ; C) = Image(Tx − 1) , if i = 0. Let ι := var ◦ ι : C −→ ψ [n]. ϕ πQXe

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory ISX underlies a mixed Hodge module. If Tx is semi-simple in the eigenvalue 1, then: IS =∼ ψ [n]. X π,1QXe

Intersection space complex: Construction

Let ιϕ C := Image(T − 1) ,→ ϕ [n] ∈ Perv(X ) e πQXe Then C ∈ Perv(X ), Supp(C) = {x}, and ( i 0 , if i 6= 0, H (X ; C) = Image(Tx − 1) , if i = 0. Let ι := var ◦ ι : C −→ ψ [n]. ϕ πQXe Deﬁne: IS := Coker ι : C −→ ψ [n] ∈ Perv(X ) X πQXe

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory If Tx is semi-simple in the eigenvalue 1, then: IS =∼ ψ [n]. X π,1QXe

Intersection space complex: Construction

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory Intersection space complex: Construction

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory THANK YOU !!!

LAURENTIU MAXIM Intersection Spaces, Perverse Sheaves, String Theory