SLIDE 1
Contraction Algebras and their Properties
Michael Wemyss
www.maths.gla.ac.uk/∼mwemyss
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The Geometric Setup
Consider C, a single contractible curve in a smooth CY 3-fold X. In cartoons, this means X ➀
- r
➁
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Contraction Algebras and their Properties Michael Wemyss - - PowerPoint PPT Presentation
Contraction Algebras and their Properties Michael Wemyss - - PowerPoint PPT Presentation
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 1 / 9 The Geometric
SLIDE 2
SLIDE 3
The Geometric Setup
Consider C, a single contractible curve in a smooth CY 3-fold X. In cartoons, this means X ➀
- r
➁ The basic idea of this talk: C in X an algebra Acon
associate
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SLIDE 4
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.
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SLIDE 5
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.
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SLIDE 6
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.
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SLIDE 7
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.
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SLIDE 8
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...
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SLIDE 9
...which is very difficult to control.
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SLIDE 10
...which is very difficult to control. Via various isomorphisms (Donovan–W), it is possible to view Acon in the following, explicit, form.
- 4. Superpotential Algebras
There exists an f ∈ Cx, y such that Acon ∼ = Cx, y (δxf , δyf ) = Jf where δx is the formal derivative with respect to x etc.
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SLIDE 11
...which is very difficult to control. Via various isomorphisms (Donovan–W), it is possible to view Acon in the following, explicit, form.
- 4. Superpotential Algebras
There exists an f ∈ Cx, y such that Acon ∼ = Cx, y (δxf , δyf ) = Jf where δx is the formal derivative with respect to x etc. Calibration: if f = x4 + xyy + yxy + yyx, then δxf = x3 + y2 and δyf = xy + yx.
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SLIDE 12
The Contraction Theorem
Recall our setup: ➀
- r
➁
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SLIDE 13
The Contraction Theorem
Recall our setup: ➀
- r
➁
Theorem (Donovan–W)
- 1. Situation ➀ ⇐
⇒ Acon is finite dimensional.
- 2. Acon controls the symmetries, in both situations.
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SLIDE 14
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 Acon and Bcon. Then X ∼ Y ⇐ ⇒ Acon ∼ = Bcon.
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SLIDE 15
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 Acon and Bcon. Then X ∼ Y ⇐ ⇒ Acon ∼ = Bcon.
The Realisation Problem (Brown–W)
Every finite dimensional superpotential algebra Jf = C x, y
- (δxf , δyf )
can be constructed as the contraction algebra of some 3-fold flop.
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SLIDE 16
Strange Behaviour 1
First, consider the following six algebras: C,
Cx,y,z. x + y + z = 0 x2 = 0 y2 = 0 z2 = 0
,
Cx,y,z x + y + z = 0 x2 = 0 y3 = 0 z3 = 0
,
Cx,y,z x + y + z = 0 x2 = 0 y3 = 0 z4 = 0 Cx,y,z x + y + z2 = 0 messy
,
Cx,y,z x + y + z = 0 x2 = 0 y3 = 0 z5 = 0
. These have dimensions 1, 4, 12, 24, 40 and 60 respectively.
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SLIDE 17
Now, consider the centre of Jf ∼ = Acon, with basis {1 = c1, c2, . . . , cn}. Consider a generic central element s =
i λici, which means that
(λi) belongs to a Zariski open subset of An.
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SLIDE 18
Now, consider the centre of Jf ∼ = Acon, with basis {1 = c1, c2, . . . , cn}. Consider a generic central element s =
i λici, which means that
(λi) belongs to a Zariski open subset of An.
Theorem (Donovan–W)
Acon/(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).
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SLIDE 19
Strange Behaviour 2
Theorem (Hua–Toda)
There is an equality dimC Acon = dimC Aab
con
- n1
+
ℓ
- i=2
ni · i2, where ℓ is determined by the last slide, such that all ni = 0. The ni are called the Gopakumar–Vafa (GV) invariants. The GV invariants are a property of the isomorphism class of Acon, but it is still not known how to extract them intrinsically.
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SLIDE 20
Upshot
Given f ∈ C x, y with dimC Jf < ∞, the conjectures (and numerical evidence!) predict the following algebraic statements:
◮ A generic central cut Jf /(s) is one of six algebras, so there is
an ADE-type classification of such Jf .
◮ The dimension of Jf is a sum of squares,
dimC Jf = dimC Jab
f
+ n2 · 22 + . . . + nℓ · ℓ2 with all ni = 0.
◮ Jf is a symmetric algebra (HomC(Jf , C) ∼
= Jf as bimodules).
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SLIDE 21
Upshot
Given f ∈ C x, y with dimC Jf < ∞, the conjectures (and numerical evidence!) predict the following algebraic statements:
◮ A generic central cut Jf /(s) is one of six algebras, so there is
an ADE-type classification of such Jf .
◮ The dimension of Jf is a sum of squares,
dimC Jf = dimC Jab
f
+ n2 · 22 + . . . + nℓ · ℓ2 with all ni = 0.
◮ Jf is a symmetric algebra (HomC(Jf , C) ∼
= Jf as bimodules).
◮ Furthermore, 3-fold flops are classified by certain elements in
the free algebra in two variables.
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