Contraction Algebras and their Properties Michael Wemyss www.maths.gla.ac.uk/ ∼ mwemyss 0 / 9
The Geometric Setup Consider C , a single contractible curve in a smooth CY 3-fold X . In cartoons, this means X or ➀ ➁ 1 / 9
The Geometric Setup Consider C , a single contractible curve in a smooth CY 3-fold X . In cartoons, this means X or ➀ ➁ The basic idea of this talk: associate an algebra A con C in X 1 / 9
How to do this? There are four ways of constructing this algebra. 1. Deformation Theory Probing how the curve deforms is one way to obtain good information about its behaviour. 2 / 9
How to do this? There are four ways of constructing this algebra. 1. Deformation Theory Probing how the curve deforms is one way to obtain good information about its behaviour. 2 / 9
How to do this? There are four ways of constructing this algebra. 1. Deformation Theory Probing how the curve deforms is one way to obtain good information about its behaviour. 2 / 9
How to do this? There are four ways of constructing this algebra. 1. Deformation Theory Probing how the curve deforms is one way to obtain good information about its behaviour. 2 / 9
How to do this? There are four ways of constructing this algebra. 1. Deformation Theory Probing how the curve deforms is one way to obtain good information about its behaviour. Noncommutative Deformation Theory (Laudal, Segal, ELO): there is a functor, giving rise to a noncommutative algebra... 2 / 9
...which is very difficult to control. 3 / 9
...which is very difficult to control. Via various isomorphisms (Donovan–W), it is possible to view A con in the following, explicit, form. 4. Superpotential Algebras There exists an f ∈ C � x , y � such that C � x , y � A con ∼ = ( δ x f , δ y f ) = J f where δ x is the formal derivative with respect to x etc. 3 / 9
...which is very difficult to control. Via various isomorphisms (Donovan–W), it is possible to view A con in the following, explicit, form. 4. Superpotential Algebras There exists an f ∈ C � x , y � such that C � x , y � A con ∼ = ( δ x f , δ y f ) = J f where δ x is the formal derivative with respect to x etc. Calibration: if f = x 4 + xyy + yxy + yyx , then δ x f = x 3 + y 2 and δ y f = xy + yx . 3 / 9
The Contraction Theorem Recall our setup: or ➀ ➁ 4 / 9
The Contraction Theorem Recall our setup: or ➀ ➁ Theorem (Donovan–W) 1. Situation ➀ ⇐ ⇒ A con is finite dimensional. 2. A con controls the symmetries, in both situations. 4 / 9
The Two Main Conjectures Rest of talk: situation ➀ (i.e. flopping contractions). The Classification Problem (Donovan–W) Let X → Spec R and Y → Spec S be two 3-fold flops, with associated contraction algebras A con and B con . Then ⇒ A con ∼ X ∼ Y ⇐ = B con . 5 / 9
The Two Main Conjectures Rest of talk: situation ➀ (i.e. flopping contractions). The Classification Problem (Donovan–W) Let X → Spec R and Y → Spec S be two 3-fold flops, with associated contraction algebras A con and B con . Then ⇒ A con ∼ X ∼ Y ⇐ = B con . The Realisation Problem (Brown–W) Every finite dimensional superpotential algebra J f = C � � x , y � � ( δ x f , δ y f ) can be constructed as the contraction algebra of some 3-fold flop. 5 / 9
Strange Behaviour 1 First, consider the following six algebras: C � x , y , z � . C � x , y , z � C � x , y , z � C , , , x + y + z = 0 x + y + z = 0 x + y + z = 0 x 2 = 0 x 2 = 0 x 2 = 0 y 2 = 0 y 3 = 0 y 3 = 0 z 2 = 0 z 3 = 0 z 4 = 0 C � x , y , z � C � x , y , z � , . x + y + z 2 = 0 x + y + z = 0 x 2 = 0 messy y 3 = 0 z 5 = 0 These have dimensions 1, 4, 12, 24, 40 and 60 respectively. 6 / 9
Now, consider the centre of J f ∼ = A con , with basis { 1 = c 1 , c 2 , . . . , c n } . Consider a generic central element s = � i λ i c i , which means that ( λ i ) belongs to a Zariski open subset of A n . 7 / 9
Now, consider the centre of J f ∼ = A con , with basis { 1 = c 1 , c 2 , . . . , c n } . Consider a generic central element s = � i λ i c i , which means that ( λ i ) belongs to a Zariski open subset of A n . Theorem (Donovan–W) A con / ( s ) is isomorphic to one of the six algebras on the last slide. Label the cases ℓ = 1 , . . . , 6 (where ℓ = 1 corresponds to the algebra of dimension one, and ℓ = 6 the algebra of dimension 60). 7 / 9
Strange Behaviour 2 Theorem (Hua–Toda) There is an equality ℓ � n i · i 2 , dim C A con = dim C A ab + con � �� � i =2 n 1 where ℓ is determined by the last slide, such that all n i � = 0. The n i are called the Gopakumar–Vafa (GV) invariants. The GV invariants are a property of the isomorphism class of A con , but it is still not known how to extract them intrinsically. 8 / 9
Upshot Given f ∈ C � � x , y � � with dim C J f < ∞ , the conjectures (and numerical evidence!) predict the following algebraic statements: ◮ A generic central cut J f / ( s ) is one of six algebras, so there is an ADE-type classification of such J f . ◮ The dimension of J f is a sum of squares, + n 2 · 2 2 + . . . + n ℓ · ℓ 2 dim C J f = dim C J ab f with all n i � = 0. ◮ J f is a symmetric algebra (Hom C ( J f , C ) ∼ = J f as bimodules). 9 / 9
Upshot Given f ∈ C � � x , y � � with dim C J f < ∞ , the conjectures (and numerical evidence!) predict the following algebraic statements: ◮ A generic central cut J f / ( s ) is one of six algebras, so there is an ADE-type classification of such J f . ◮ The dimension of J f is a sum of squares, + n 2 · 2 2 + . . . + n ℓ · ℓ 2 dim C J f = dim C J ab f with all n i � = 0. ◮ J f is a symmetric algebra (Hom C ( J f , C ) ∼ = J f as bimodules). ◮ Furthermore, 3-fold flops are classified by certain elements in the free algebra in two variables. 9 / 9
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