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

◮ Ik(X, β) = Hilbert scheme of curves Y ⊂ X with no component of

codim 1, k = χ(OY ), β = [Y ] ∈ H2(X); parametrizes rank 1 torsion free sheaves T with det T trivial

◮ Donaldson–Thomas partition function:

ZDT(X) =

• β∈H2(X)

Qβ DTβ(X; q) , DTβ(X; q) =

• k∈Z

qk

• [Ik(X,β)]vir 1

◮ DT0(X; q) = M(q)χ(X) = ∞

• n=1

1

• 1 − qnn χ(X)

M(q) enumerates plane partitions π (3D Young diagrams)

Topological vertex formalism

◮ Trivalent planar toric graph Γ with:

(1) 3D Young diagram πv at each vertex v (2) 2D Young diagram λe at each edge e (asymptotics of πv)

◮ “Topological string” partition function (Aganagic et al. ’05; Okounkov, Reshetikhin & Vafa ’06; Maulik et al. ’06):

ZDT(X) =

• Young diagrams

λe

• edges e

Q|λe|

e

• vertices

v=(e1,e2,e3)

Mλe1,λe2,λe3(q)

◮ Mλ,µ,ν(q) =

• π : ∂π=(λ,µ,ν)

q|π| 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

A

= 0 = F 0,2

A

, F 1,1

A

∧ ω ∧ ω = 0

◮ Donaldson–Uhlenbeck–Yau equations:

BPS D6–D2–D0 states ≡ (generalized) instantons

◮ Localization of path integral onto instanton moduli space M

computes “ ZX =

• 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 C3 θ have

“ADHM form”

• Z i , Z j

= 0,

• Z i , Z †

i

• = 3 on Fock module

H = C

• ¯

z1, ¯ z2, ¯ z3 |0

◮ Solutions parametrized by monomial ideals I ⊂ C[z1, z2, z3],

HI = I(¯ z1, ¯ z2, ¯ z3)|0; correspond to plane partitions π with k := ch3(E) = |π|

◮ In “Coulomb branch” U(1)r noncommutative instantons correspond

to coloured partitions π = (π1, . . . , πr); after toric localization: Z r

gauge

• C3

=

• π

(−1)(r+1) |

π| q| π| = M

• (−1)r+1 q

r Degenerate central charge limit of higher-rank invariants

(Stoppa ’09);

not dual to topological string theory

Stacky gauge theories

◮ G-equivariant instantons on C3 for finite G ⊂ (C×)3 ⊂ SL(3, C)

with weights ρi, natural rep Q = C3; count G-equivariant closed subschemes of C3 (substacks of

• C3/G
• )

◮ Instanton equations Z (ρ+ρj) i

Z (ρ)

j

= Z (ρ+ρi)

j

Z (ρ)

i

• n

H =

ρ∈b G Hρ, Zi = ρ∈b G Z (ρ) i

, Z (ρ)

i

∈ HomC(Hρ, Hρ+ρi ); solutions parametrized by G-coloured plane partitions π = (πρ)ρ∈b

G ◮ Framed moduli space of torsion free sheaves E on P3/G,

ch0(E) = r, ch3(E) = k ≡ reps (V = Ck, W = Cr; B, I), B ∈ HomG(V , Q ⊗ V ), I ∈ HomG(W , V ) of framed McKay quiver

◮ McKay correspondence: ch(E) determined by exceptional curves on

crepant resolution X = HilbG(C3) via Beilinson’s theorem

Instanton quantum mechanics

◮ Topological matrix model with stability condition and

“orbifold ADHM equations” B(ρ+ρj )

i

B(ρ)

j

= B(ρ+ρi )

j

B(ρ)

i ◮ In “Coulomb branch” BRST fixed points correspond to

coloured plane partitions π = (π1, . . . , πr) with | π| = k and πl = (πl,ρ)ρ∈b

G, l |πl,ρ| = dimC(Vρ) ◮ Local model for instanton moduli space near fixed point of

• T = (C×)3 × (C×)r:

HomG(V

π, V π)

• HomG (V

π, V π ⊗ Q)

⊕ HomG(W

π, V π)

⊕ HomG(V

π, V π ⊗ V3 Q)

• HomG(V

π, V π ⊗ V2 Q)

⊕ HomG (V

π, W π ⊗ V3 Q)

G-equivariant version of instanton deformation complex

Orbifold invariants

◮ Partition function:

Z r

gauge

• [C3/G]
• =
• π

(−1)K(

π;r) qch3(E

π) Qch2(E π)

r = (dimC(W1), . . . , dimC(Wr)) Expressed in terms of intersection theory on X = HilbG(C3)

◮ Simple change of variables (q, Q) −

→ (pρ)ρ∈b

G with

• ρ∈b

G

pρ = q: Z r

gauge

• [C3/G]
• =
• π

(−1)K(

π;r) ρ∈b G

p

PN

l=1 |πl,ρ|

ρ

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 ] ∈ H2(X), k = χ(OY ):

Ik(X, β) ∼ = Ikβ(X, β) × X [k−kβ] kβ = − 1

2 β · (β + KX) (divisorial part)

dimC

• X [m]

= 2m (punctual part)

◮ Partition function:

Zcurve(X) =

• k∈Z
• β∈H2(X)

qk Qβ

• Ik(X,β)

e

• TIk(X, β)
• ◮ G¨
• ttsche’s formula:
• n≥0

qn χ

• X [n]

= ˆ η(q)−χ(X) =

∞

• n=1

1

• 1 − qnχ(X)

ˆ η(q)−1 enumerates Young diagrams λ = (λ1, λ2, . . . )

Curve counting — Torus fixed points

◮ Localization theorem in equivariant Chow theory (Edidin & Graham ’98): ◮

{∞ Young diagrams} ∼ = Z2

≥0 × {finite Young diagrams} ◮ For compact toric invariant divisor D = i λi Di, λi ∈ Z≥0

with ai = −D2

i :

χ(OD) = − 1

2 D · (D + KX) =

• i
• ai

λi (λi − 1) 2 + λi − λi λi+1

Vertex formalism for toric surfaces

◮ Partition function on bivalent planar toric graph Γ:

Zcurve(X) =

• λe∈Z≥0
• edges e

Gλe(q, Qe)

• vertices

v=(e1,e2)

Vλe1 ,λe2(q) Vλe1 ,λe2(q) = ˆ η(q)−1 q−λe1 λe2 , Gλe(q, Qe) = qae

λe (λe −1) 2

+λe Qλe 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 = ch2(E) ∈ H4(X, Z) , u = c1(E) ∈ H2(X, Z)

◮ Path integral computes Euler character of moduli space of

U(1) instantons on X (anti-self-dual connections ⋆FA = −FA)

◮ Conjectural exact expression on Hirzebruch–Jung spaces (Fucito, Morales & Poghossian ’06; Griguolo et al. ’07) ◮ Conjectured factorization for rank r > 1:

Z r

gauge(X) =

• Zgauge(X)

r

Instanton moduli spaces MX(β, n)

◮ Moduli space of rank 1 torsion free sheaves T (“noncommutative

instantons”), k = ch2(T ), β = ch1(T ) ∈ H2(X): MX(β, k) ∼ = Picβ(X) × X [k−kβ] ch2

• OX(D) ⊗ IZ
• =

1 2 D · D − χ(OZ) ◮ Partition function:

Zgauge(X) =

• k∈Q
• β∈H2(X)

q−k Qβ

• MX (β,k)

e

• TMX(β, k)
• ◮ Using linear equivalence, complete set of non-compact torically

invariant divisors to integral generating set for Picard group

(Kronheimer & Nakajima ’90):

ei =

• j
• C −1ij Dj ,

Cij = Di · Dj

Example — ALE spaces

◮ Resolution of An singularity C2/Zn+1: ◮ Curve counting: Zcurve(A1) =

1 ˆ η(q)2

∞

• λ=0

qλ2 Qλ

◮ Gauge theory: Zgauge(A1) =

1 ˆ η(q)2

∞

• u=−∞

q− 1

4 u2 Qu

◮ 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

◮ HF

– new Hopf algebra, H = HF 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: QF : HM − →

HF 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 HM:

Ψ : A ⊗ B − → B ⊗ A , Ψ(a ⊗ b) = b ⊗ a

◮ Twist into new braiding on HF M:

ΨF : AF⊗BF − → BF ⊗AF , ΨF(a⊗b) = F −2 b(−1), a(−1) b(0)⊗a(0)

◮ A — H-comodule algebra =

⇒ AF = QF(A) — HF-comodule algebra with new product: a · b := F

• a(−1) , b(−1)

a(0) b(0)

Noncommutative algebraic torus Tθ = (C×

θ )n ◮ H := C(t1, . . . , tn) = A(T) generated by tp := tp1 1 · · · tpn n ,

p ∈ Zn with: ∆(tp) = tp ⊗ tp , ǫ(tp) = 1 , S(tp) = t−p

◮ Cocycle: F(ti, tj) = exp

i

2 θij

• =: qij, θij = −θji ∈ C

H = HF as Hopf algebras, but category of H-comodules twisted

◮ ∆ : H −

→ H ⊗ H makes H into comodule algebra in HM, so cotwisted torus has: ti · tj = F(ti, tj) ti tj = F 2(ti, tj) tj · ti = q2

ij tj · ti

Noncommutative torus A(Tθ) as object of HF M

Quantization of toric varieties X − → Xθ

(Ingalls) ◮ Noncommutative affine toric varieties σ −

→ A

• Uθ[σ]
• finitely-generated HF-comodule subalgebras of A(Tθ)

◮ Example: A(Cn θ) = Cθ[z1, . . . , zn], zi zj = q2 ij zj zi

“Algebraic Moyal plane”; Generally modulo relations

◮ Gluing rules =

⇒ algebra automorphisms in category HF M

◮ Uses same fan, deforms coordinate algebra of each cone σ

Noncommutative projective plane P2

θ ◮ Maximal cones: Uθ[σi] ∼

= C2

θ, i = 1, 2, 3 ◮ Edges: Uθ[σi ∩ σi+1] ∼

= noncommutative projective line P1

θ:

w1 w2 = q2 w2 w1 , w1 w −1

2

= q−2 w −1

2

w1 , q := q12

◮ Homogeneous coordinate algebra: A = Cθ[w1, w2, w3]

graded algebra object in HF M

(Auroux, Katzarkov & Orlov ’08):

w1 w2 = q2 w2 w1 , w1 w3 = w3 w1 , w2 w3 = w3 w2

◮ Degree 0 left Ore localization A

• w −1

i

• 0 ∼

= A

• Uθ[σi]

Instanton moduli spaces Mθ(r, k)

◮ Mθ(r, k) = isomorphism classes of framed torsion-free A-modules

M with fixed trivialization MP1

θ := M/M · w3 ∼

= W ⊗ AP1

θ,

W = Cr, AP1

θ := A/A · w3, and dimC Ext1(A, M(−1)) = k

◮ Invariants: rank(M) = r,

χ(M) =

• p≥0

(−1)p dimC Extp(A, M) = r − k

◮ Noncommutative ADHM construction: V = Ck, W = Cr

Mθ(r, k) =

• B1 , B2 , I , J
• where B1, B2 ∈ EndC(V ),

I ∈ HomC(W , V ), J ∈ HomC(V , W ) satisfy: [B1, B2]θ + I J = 0 [B1, B2]θ := B1 B2 − q−2 B2 B1 braided commutator modulo stability and free proper action of GL(k, C)

Instanton moduli spaces — Properties

◮ Mθ(r, k) = fine moduli space (Nevins & Stafford ’07) ◮ Smooth of dimension 2 r k, T[M]Mθ(r, k) = Ext1

M , M(−1)

• ◮ At [M] = [(B1, B2, I, J)], T[M]Mθ(r, k) = cohomology H1 of

instanton deformation complex: 0 − → EndC(V ) − → EndC(V )⊕2 ⊕ HomC(W , V ) ⊕ HomC(V , W ) − → EndC(V ) − → 0

Instanton moduli spaces — Torus fixed points

◮

T = (C×)2 × (C×)r, coaction of HF = HF ⊗ C(ρ1, . . . , ρr): ∆L(B1, B2, I, J) = (t1 ⊗1⊗B1, t2 ⊗1⊗B2, t1 t2 ⊗ρ−1 ⊗I, 1⊗ρ⊗J) Makes V , W objects, (B1, B2, I, J) morphisms in e

HF M ◮ Coequivariant modules [M] ∈ Mθ(r, k)e T ∼

= finite set of length r sequences λ = (λ1, . . . , λr) of Young diagrams of size | λ| = k

◮ Restriction of instanton deformation complex to fixed point

λ complex in e

HF M with (Nakajima & Yoshioka ’05)

che

T(V λ) = r

• l=1
• p∈λl

ρl t1−p1

1

t1−p2

2

, che

T(W λ) = r

• l=1

ρl Hence equivariant instanton counting and (pure) gauge theory partition functions same as in classical case θ = 0