INSTANTONS AND CURVE COUNTING Richard Szabo HeriotWatt University, - PowerPoint PPT Presentation

INSTANTONS AND CURVE COUNTING Richard Szabo Heriot–Watt University, Edinburgh Maxwell Institute for Mathematical Sciences Noncommutative Algebraic Geometry and D-Branes Simons Center for Geometry and Physics Stonybrook 2011

Outline I. Generalized instantons and curve counting on toric Calabi–Yau 3-folds II. Instantons and curve counting on toric surfaces III. Instanton counting on noncommutative toric varieties with Michele Cirafici, Lucio Cirio, Amir Kashani-Poor, Giovanni Landi & Annamaria Sinkovics

Part I Generalized instantons and curve counting on toric Calabi–Yau 3-folds

Curve counting on toric Calabi–Yau 3-folds X ◮ I k ( X , β ) = Hilbert scheme of curves Y ⊂ X with no component of codim 1, k = χ ( O Y ), β = [ Y ] ∈ H 2 ( X ); parametrizes rank 1 torsion free sheaves T with det T trivial ◮ Donaldson–Thomas partition function: � � � Q β DT β ( X ; q ) , q k Z DT ( X ) = DT β ( X ; q ) = [ I k ( X ,β )] vir 1 k ∈ Z β ∈ H 2 ( X ) ∞ � 1 ◮ DT 0 ( X ; q ) = M ( q ) χ ( X ) = � 1 − q n � n χ ( X ) n =1 M ( q ) enumerates plane partitions π (3D Young diagrams)

Topological vertex formalism ◮ Trivalent planar toric graph Γ with: (1) 3 D Young diagram π v at each vertex v (2) 2 D Young diagram λ e at each edge e (asymptotics of π v ) ◮ “Topological string” partition function (Aganagic et al. ’05; Okounkov, Reshetikhin & Vafa ’06; Maulik et al. ’06) : � � � Q | λ e | Z DT ( X ) = M λ e 1 ,λ e 2 ,λ e 3 ( q ) e edges e vertices Young diagrams λ e v =( e 1 , e 2 , e 3 ) � q | π | ◮ M λ,µ,ν ( q ) = π : ∂π =( λ,µ,ν ) Generating function for plane partitions π with boundaries λ, µ, ν ◮ GW/DT correspondence ≡ gauge/string theory duality

6D cohomological gauge theory (Iqbal et al. ’06) ◮ N = 2 topologically twisted U (1) Yang–Mills on K¨ ahler 3-fold ( X , ω ) localizes at BRST fixed points: F 2 , 0 = 0 = F 0 , 2 F 1 , 1 , ∧ ω ∧ ω = 0 A A A ◮ Donaldson–Uhlenbeck–Yau equations: BPS D6–D2–D0 states ≡ (generalized) instantons ◮ Localization of path integral onto instanton moduli space M � computes “ Z X = M e ( N ) ” e ( N ) = Euler characteristic class of obstruction bundle N ◮ Stability in D ( X )? B -field/noncommutative deformation, non-linear/higher-derivative corrections, worldsheet instantons, . . .

Singular instanton solutions ◮ Instanton equations on noncommutative deformation C 3 θ have � Z i , Z j � � � Z i , Z † “ADHM form” = 0, = 3 on Fock module i � z 3 � z 1 , ¯ z 2 , ¯ H = C ¯ | 0 � ◮ Solutions parametrized by monomial ideals I ⊂ C [ z 1 , z 2 , z 3 ], z 1 , ¯ z 2 , ¯ z 3 ) | 0 � ; correspond to plane partitions π with H I = I (¯ k := ch 3 ( E ) = | π | ◮ In “Coulomb branch” U (1) r noncommutative instantons correspond to coloured partitions � π = ( π 1 , . . . , π r ); after toric localization: � � C 3 � � � r π | = M π | q | � ( − 1) r +1 q Z r ( − 1) ( r +1) | � = gauge � π Degenerate central charge limit of higher-rank invariants (Stoppa ’09) ; not dual to topological string theory

Stacky gauge theories ◮ G -equivariant instantons on C 3 for finite G ⊂ ( C × ) 3 ⊂ SL (3 , C ) with weights ρ i , natural rep Q = C 3 ; count G -equivariant closed � � subschemes of C 3 (substacks of C 3 / G ) ◮ Instanton equations Z ( ρ + ρ j ) Z ( ρ ) = Z ( ρ + ρ i ) Z ( ρ ) on i j j i H = � G H ρ , Z i = � G Z ( ρ ) , Z ( ρ ) ∈ Hom C ( H ρ , H ρ + ρ i ); ρ ∈ b ρ ∈ b i i solutions parametrized by � G -coloured plane partitions π = ( π ρ ) ρ ∈ b G ◮ Framed moduli space of torsion free sheaves E on P 3 / G , ch 0 ( E ) = r , ch 3 ( E ) = k ≡ reps ( V = C k , W = C r ; B , I ), B ∈ Hom G ( V , Q ⊗ V ), I ∈ Hom G ( W , V ) of framed McKay quiver ◮ McKay correspondence: ch ( E ) determined by exceptional curves on crepant resolution X = Hilb G ( C 3 ) via Beilinson’s theorem

� � Instanton quantum mechanics ◮ Topological matrix model with stability condition and “orbifold ADHM equations” B ( ρ + ρ j ) B ( ρ ) = B ( ρ + ρ i ) B ( ρ ) i j j i ◮ In “Coulomb branch” BRST fixed points correspond to π = ( π 1 , . . . , π r ) with | � π | = k and coloured plane partitions � G , � l | π l ,ρ | = dim C ( V ρ ) π l = ( π l ,ρ ) ρ ∈ b ◮ Local model for instanton moduli space near fixed point of T = ( C × ) 3 × ( C × ) r : � Hom G ( V � π ⊗ Q ) π , V � π ⊗ V 2 Q ) ⊕ Hom G ( V � π , V � Hom G ( W � π ) Hom G ( V � π , V � π ) π , V � ⊕ π ⊗ V 3 Q ) ⊕ Hom G ( V � π , W � π ⊗ V 3 Q ) Hom G ( V � π , V � G -equivariant version of instanton deformation complex

Orbifold invariants ◮ Partition function: � � � π ; r ) q ch 3 ( E � π ) Q ch 2 ( E � Z r [ C 3 / G ] ( − 1) K ( � π ) = gauge � π r = (dim C ( W 1 ) , . . . , dim C ( W r )) Expressed in terms of intersection theory on X = Hilb G ( C 3 ) � ◮ Simple change of variables ( q , Q ) �− → ( p ρ ) ρ ∈ b G with p ρ = q : ρ ∈ b G � π ; r ) � P N � � l =1 | π l ,ρ | Z r [ C 3 / G ] ( − 1) K ( � = p ρ gauge � π ρ ∈ b G G -equivariant instanton charges are relevant variables in noncommutative crepant resolution chamber (Bryan & Young ’10; Joyce & Song ’11)

Part II Instantons and curve counting on toric surfaces

Curve counting on toric surfaces X ◮ Hilbert scheme of curves Y ⊂ X , β = [ Y ] ∈ H 2 ( X ), k = χ ( O Y ): I k ( X , β ) ∼ = I k β ( X , β ) × X [ k − k β ] k β = − 1 2 β · ( β + K X ) (divisorial part) � X [ m ] � dim C = 2 m (punctual part) ◮ Partition function: � � � � � q k Q β Z curve ( X ) = e TI k ( X , β ) I k ( X ,β ) k ∈ Z β ∈ H 2 ( X ) ◮ G¨ ottsche’s formula: ∞ � � � X [ n ] � 1 q n χ η ( q ) − χ ( X ) = = ˆ � 1 − q n � χ ( X ) n =1 n ≥ 0 η ( q ) − 1 enumerates Young diagrams λ = ( λ 1 , λ 2 , . . . ) ˆ

Curve counting — Torus fixed points ◮ Localization theorem in equivariant Chow theory (Edidin & Graham ’98) : {∞ Young diagrams } ∼ = Z 2 ≥ 0 × { finite Young diagrams } ◮ ◮ For compact toric invariant divisor D = � i λ i D i , λ i ∈ Z ≥ 0 with a i = − D 2 i : � � � λ i ( λ i − 1) χ ( O D ) = − 1 2 D · ( D + K X ) = + λ i − λ i λ i +1 a i 2 i

Vertex formalism for toric surfaces ◮ Partition function on bivalent planar toric graph Γ: � � � Z curve ( X ) = G λ e ( q , Q e ) V λ e 1 ,λ e 2 ( q ) λ e ∈ Z ≥ 0 edges e vertices v =( e 1 , e 2 ) η ( q ) − 1 q − λ e 1 λ e 2 λ e ( λ e − 1) + λ e Q λ e G λ e ( q , Q e ) = q a e V λ e 1 ,λ e 2 ( q ) = ˆ , 2 e ◮ Question: Is there a 4D “topological string theory” that reproduces this counting?

Vafa–Witten theory (Vafa & Witten ’94) ◮ N = 4 topologically twisted U (1) Yang–Mills on K¨ ahler surface X , with instanton and monopole charges k = ch 2 ( E ) ∈ H 4 ( X , Z ) , u = c 1 ( E ) ∈ H 2 ( X , Z ) ◮ Path integral computes Euler character of moduli space of U (1) instantons on X (anti-self-dual connections ⋆ F A = − F A ) ◮ Conjectural exact expression on Hirzebruch–Jung spaces (Fucito, Morales & Poghossian ’06; Griguolo et al. ’07) ◮ Conjectured factorization for rank r > 1: � � r Z r gauge ( X ) = Z gauge ( X )

Instanton moduli spaces M X ( β, n ) ◮ Moduli space of rank 1 torsion free sheaves T (“noncommutative instantons”), k = ch 2 ( T ), β = ch 1 ( T ) ∈ H 2 ( X ): M X ( β, k ) ∼ = Pic β ( X ) × X [ k − k β ] � � 1 O X ( D ) ⊗ I Z = 2 D · D − χ ( O Z ) ch 2 ◮ Partition function: � � � � � q − k Q β Z gauge ( X ) = T M X ( β, k ) e M X ( β, k ) k ∈ Q β ∈ H 2 ( X ) ◮ Using linear equivalence, complete set of non-compact torically invariant divisors to integral generating set for Picard group (Kronheimer & Nakajima ’90) : � � C − 1 � ij D j , e i = C ij = D i · D j j

Example — ALE spaces ◮ Resolution of A n singularity C 2 / Z n +1 : ∞ � 1 q λ 2 Q λ ◮ Curve counting: Z curve ( A 1 ) = η ( q ) 2 ˆ λ =0 ∞ � 1 4 u 2 Q u q − 1 ◮ Gauge theory: Z gauge ( A 1 ) = η ( q ) 2 ˆ u = −∞ ◮ Problems related but not identical in 4D!

Part III Instanton counting on noncommutative toric varieties

Cocycle twist quantization (Majid ’95) ◮ H commutative Hopf algebra F : H ⊗ H − → C convolution-invertible unital two-cocycle on H ◮ H F – new Hopf algebra, H = H F as coalgebra, but with: h × F g := F ( h (1) , g (1) ) ( h (2) g (2) ) F − 1 ( h (3) , g (3) ) ◮ Simultaneously deforms all H -covariant constructions as functorial isomorphism of categories of left comodules: H F M Q F : H M − → Notation: ∆ L : A − → H ⊗ A left coaction of H on A , ∆ L ( a ) := a ( − 1) ⊗ a (0)

Comodule twisting of algebras ◮ Trivial “flip” braiding on monoidal category H M : Ψ : A ⊗ B − → B ⊗ A , Ψ( a ⊗ b ) = b ⊗ a ◮ Twist into new braiding on H F M : Ψ F ( a ⊗ b ) = F − 2 � b ( − 1) , a ( − 1) � � b (0) ⊗ a (0) � Ψ F : A F ⊗ B F − → B F ⊗ A F , ◮ A — H -comodule algebra = ⇒ A F = Q F ( A ) — H F -comodule algebra with new product: � a ( − 1) , b ( − 1) � � a (0) b (0) � a · b := F