Noncommutative Geometry of Instanton Moduli Spaces

Noncommutative Geometry of Instanton Moduli Spaces 

Simon Brain 

(with Giovanni Landi and Walter D. van Suijlekom) 

IMPAN, 10th January 2011 

S. Brain (UL) NCG of Instanton Moduli Spaces IMPAN, 10th January 2011 1 / 28

Motivation 

What is the difference between a classical four-manifold and a 

noncommutative four-manifold? 


Motivation 



To find out, we’d like to ‘probe’ their geometric structures somehow. 

Donaldson did this for a classical four-manifold M by looking at the moduli 

space of self-dual gauge fields (instantons) on M. Can we do the same for a 

noncommutative manifold? 


Motivation 



To find out, we’d like to ‘probe’ their geometric structures somehow. 

Donaldson did this for a classical four-manifold M by looking at the moduli 

space of self-dual gauge fields (instantons) on M. Can we do the same for a 

noncommutative manifold? 

In this talk, we ask: 

◮ how do we construct instantons on a noncommutative manifold? 

◮ can we use them to ‘detect’ something about the noncommutative differential 

structure? 


Instantons in Classical Geometry 

Let (M, g) be a compact Riemannian four-manifold. 

The Hodge operator ∗ : Ω 2 (M) → Ω 2 (M) obeys ∗ 2 = id and there is an 

eigenspace decomposition 

Ω 2 (M) = Ω 2 +(M) ⊕ Ω 2 −(M) 

into self-dual and anti-self-dual two-forms. 


Instantons in Classical Geometry 

Let (M, g) be a compact Riemannian four-manifold. 

The Hodge operator ∗ : Ω 2 (M) → Ω 2 (M) obeys ∗ 2 = id and there is an 

eigenspace decomposition 

Ω 2 (M) = Ω 2 +(M) ⊕ Ω 2 −(M) 

into self-dual and anti-self-dual two-forms. 

Now fix a smooth (S)U(2) vector bundle E over M. 

We say that a connection ∇ : Γ(E) → Ω 1 (E) is an instanton if its curvature 

F = ∇ 2 is a self-dual two-form, i.e. it obeys ∗F = F . 

NB: On M = S 4 , U(2) and SU(2) instantons are ”equivalent” objects. 


The Moduli Space of Instantons 

The gauge group of E is the group 

G ⊂ Γ(End(E)) 

of (S)U(2) endomorphisms of E which cover the identity on M. 

The gauge group G acts on the set C of connections on E by 

(U, ∇) ↦→ U∇U ∗ , U ∈ G, ∇ ∈ C. 


The Moduli Space of Instantons 

The gauge group of E is the group 

G ⊂ Γ(End(E)) 

of (S)U(2) endomorphisms of E which cover the identity on M. 

The gauge group G acts on the set C of connections on E by 

(U, ∇) ↦→ U∇U ∗ , U ∈ G, ∇ ∈ C. 

The set C/G of equivalence classes has the structure of an 

(infinite-dimensional) Banach manifold. 

If non-empty, the submanifold 

M := {[∇] ∈ C/G | ∇ is an instanton} 

is finite-dimensional (AHS). This is the moduli space of instantons on E 


Instantons on the Euclidean Four-Sphere 

For simplicity, we focus on studying instantons on the Euclidean four-sphere S 4 . 

Note that (S)U(2) bundles over S 4 (with ch1(E) = 0) are indexed by their 

‘topological charge’ 

k = ch2(E) ∈ H 4 (S 4 , Z). 

Let us write Mk for the moduli space of instantons on an (S)U(2) vector 

bundle E over S 4 with fixed topological charge k ∈ Z. 


Instantons on the Euclidean Four-Sphere 

For simplicity, we focus on studying instantons on the Euclidean four-sphere S 4 . 

Note that (S)U(2) bundles over S 4 (with ch1(E) = 0) are indexed by their 

‘topological charge’ 

k = ch2(E) ∈ H 4 (S 4 , Z). 

Let us write Mk for the moduli space of instantons on an (S)U(2) vector 

bundle E over S 4 with fixed topological charge k ∈ Z. 

We’re interested in moduli spaces of instantons on noncommutative 

four-spheres.... 

....so let’s begin by studying the construction of instantons on classical S 4 

from the point of view of noncommutative geometry (i.e. in terms of 

function algebras, projective modules,...). 


The SU(2) Hopf Fibration 

Define A(C 4 ) := A[zi, z ∗ j 

Write 

| i, j = 1, . . . , 4], then take unit sphere 

A(S 7 ) := A[zi, z ∗ j | 

Ψ = 

z1 z2 z3 z4 

−z ∗ 2 z ∗ 1 −z ∗ 4 z ∗ 3 

µ z∗ µzµ = 1]. 

tr 

; 

then Ψ ∗ Ψ = 2. Define a right action of w ∈ SU(2) by Ψ ↦→ Ψw. 


The SU(2) Hopf Fibration 

Define A(C 4 ) := A[zi, z ∗ j 

Write 

| i, j = 1, . . . , 4], then take unit sphere 

A(S 7 ) := A[zi, z ∗ j | 

Ψ = 

z1 z2 z3 z4 

−z ∗ 2 z ∗ 1 −z ∗ 4 z ∗ 3 

µ z∗ µzµ = 1]. 

tr 

; 

then Ψ ∗ Ψ = 2. Define a right action of w ∈ SU(2) by Ψ ↦→ Ψw. 

The invariant subalgebra is A(S 4 ) := A[α, α ∗ , β, β ∗ , x | αα ∗ + ββ ∗ + x 2 = 1] 

generated by the entries of the projection 

ΨΨ ∗ = 1 

⎛ 

1 + x 

⎜ 

2 ⎝ 

0 α −β∗ 0 1 + x β α∗ α∗ β∗ 1 − x 

⎞ 

⎟ 

0 ⎠ 

−β α 0 1 − x 

Then A(S 4 ) ↩→ A(S 7 ) is a principal SU(2)-bundle. 


Quaternionic Structure 

Note that the A(C 4 )-valued matrix 

Ψ = 

tr z1 z2 z3 z4 

−z ∗ 2 z ∗ 1 −z ∗ 4 z ∗ 3 

is made from a pair of quaternion-valued functions. 




Ψ = 


−z ∗ 2 z ∗ 1 −z ∗ 4 z ∗ 3 


This quaternion structure is encoded on A(C4 ) by the ∗-algebra map 

J : A(C 4 ) → A(C 4 ), J z1 z2 z3 

 

∗ 

z4 = −z2 z∗ 1 −z∗ 4 z∗ 

3 . 

This gives an identification of C 4 with H 2 . 




Ψ = 


−z ∗ 2 z ∗ 1 −z ∗ 4 z ∗ 3 


This quaternion structure is encoded on A(C4 ) by the ∗-algebra map 

J : A(C 4 ) → A(C 4 ), J z1 z2 z3 

 

∗ 

z4 = −z2 z∗ 1 −z∗ 4 z∗ 

3 . 

This gives an identification of C 4 with H 2 . 

The J-invariant subalgebra of A(S 7 ) is precisely A(S 4 ), which is a 

coordinate-algebraic interpretation of the statement that S 4 HP 1 . 


Construction of Instantons 

Instantons on S4 are constructed from monads on C4 , i.e. homomorphisms of free 

right A(C4 )-modules 

H ⊗ A(C 4 ) σz 

−→ K ⊗ A(C 4 ) 

for complex vector spaces H, K of dimensions k, 2k + 2, such that: 





H ⊗ A(C 4 ) σz 

−→ K ⊗ A(C 4 ) 


σz is linear in the generators z1, . . . , z4 of A(C 4 ); 





H ⊗ A(C 4 ) σz 

−→ K ⊗ A(C 4 ) 



σz is injective and σ ∗ J(z) 

is surjective; 





H ⊗ A(C 4 ) σz 

−→ K ⊗ A(C 4 ) 



σz is injective and σ∗ J(z) 

the composition σ∗ J(z) σz = 0. 






H ⊗ A(C 4 ) σz 

−→ K ⊗ A(C 4 ) 






Given such a monad, set V := 

σz σJ(z) ∈ Mat2k+2,2k(A(C4 )) and then 

P := 2k+2 − V (V ∗ V ) −1 V ∗ ∈ Mat2k+2(A(S 4 )). 

This P is clearly a projection, P 2 = P ∗ = P . 





H ⊗ A(C 4 ) σz 

−→ K ⊗ A(C 4 ) 






Given such a monad, set V := 

σz σJ(z) ∈ Mat2k+2,2k(A(C4 )) and then 

P := 2k+2 − V (V ∗ V ) −1 V ∗ ∈ Mat2k+2(A(S 4 )). 

This P is clearly a projection, P 2 = P ∗ = P . 

Moreover, E := P A(S 4 ) 2k+2 is a rank two vector bundle over S 4 and 

∇ := P ◦ d is an instanton with topological charge k. 


Gauge Freedom 

We say that a pair of monads are equivalent if there is a commuting diagram 

H ⊗ A(C4 ) 

⏐ 

U⊗id 

H ⊗ A(C 4 ) 

σz 

−−−−→ K ⊗ A(C4 ) 

⏐ 

W ⊗id 

˜σz 

−−−−→ K ⊗ A(C 4 ) 

for invertible linear maps U : H → H and W : K → K. The effect on the resulting 

projection is to map P ↦→ W P W ∗ . 


Gauge Freedom 

We say that a pair of monads are equivalent if there is a commuting diagram 

H ⊗ A(C4 ) 

⏐ 

U⊗id 

H ⊗ A(C 4 ) 

σz 

−−−−→ K ⊗ A(C4 ) 

⏐ 

W ⊗id 

˜σz 

−−−−→ K ⊗ A(C 4 ) 

for invertible linear maps U : H → H and W : K → K. The effect on the resulting 

projection is to map P ↦→ W P W ∗ . 

Theorem (ADHM) There is a bijective correspondence 

Mk ∼ = {Monads with index k}/ ∼ . 

So these equivalences of monads account for all of the gauge freedom in the 

construction of instantons. 


The Space of Monads 

Note that, since σz : H ⊗ A(C 4 ) → K ⊗ A(C 4 ) is linear in the generators 

z1, . . . , z4, it can be written 

σz = 

4 

j=1 

M j 

ab ⊗ zj, σ ∗ z = 

4 

j=1 

for constant matrices M j ∈ Mat2k+2,k(C), j = 1, . . . , 4. 

M j ∗ ∗ 

ab ⊗ zj , 





σz = 

4 

j=1 

M j 

ab ⊗ zj, σ ∗ z = 

4 

j=1 


M j ∗ ∗ 

ab ⊗ zj , 

As we allow σz to vary, we can think of the matrix elements M j j 

ab , M 

coordinate functions on the space of all monads with index k. 

ab ∗ as 





σz = 

4 

j=1 

M j 

ab ⊗ zj, σ ∗ z = 

4 

j=1 


M j ∗ ∗ 

ab ⊗ zj , 


, M 


The monad condition σ ∗ J(z) σz = 0 translates into the relations 

 

for all j, l = 1, . . . , 4 and c, d = 1, . . . , k, where 

b 

ab 

ab ∗ as 

 

N j 

abM l bd + N l abM j 

 

bd = 0, (1) 

σ ∗ J(z) = N j ⊗ zj = M j ∗ ⊗ J(zj) ∗ . 





σz = 

4 

j=1 

M j 

ab ⊗ zj, σ ∗ z = 

4 

j=1 


M j ∗ ∗ 

ab ⊗ zj , 


, M 


The monad condition σ ∗ J(z) σz = 0 translates into the relations 

 

for all j, l = 1, . . . , 4 and c, d = 1, . . . , k, where 

b 

ab 

ab ∗ as 

 

N j 

abM l bd + N l abM j 

 

bd = 0, (1) 

σ ∗ J(z) = N j ⊗ zj = M j ∗ ⊗ J(zj) ∗ . 

We write A(Mk) for the algebra generated by the functions M j j 

ab , M 

modulo the relations (1). 

S. Brain (UL) NCG of Instanton Moduli Spaces IMPAN, 10th January 2011 10 / 28 

ab ∗ ,

Functorial Cocycle Deformation 

Now we pass to noncommutative geometry. We view the deformation as a functor 

LF : H C → HF C 

from a certain category H C (wherein lives our classical geometry) to a new 

category HF C, both to be determined. 







Let H be a commutative unital Hopf algebra. Let F : H ⊗ H → C be a 

convolution-invertible unital two-cocycle on H. 









Then there is a new Hopf algebra HF which is the same as H as a coalgebra, 

but has the new product 

h •F g := F (h (1), g (1))h (2)g (2)F −1 (h (3), g (3)). 









Then there is a new Hopf algebra HF which is the same as H as a coalgebra, 

but has the new product 

h •F g := F (h (1), g (1))h (2)g (2)F −1 (h (3), g (3)). 

In fact, since H = HF as a coalgebra, H-comodules are exactly the same 

thing as HF -comodules. 

So there is an invertible functor LF : H C → HF C which simultaneously 

converts all H-covariant constructions into HF -covariant ones (it’s just the 

identity functor). 


The Deformation Functor 

More interesting is the fact that the trivial braiding on H C, 

Ψ : A ⊗ B → B ⊗ A, Ψ(a ⊗ b) = b ⊗ a, 

is twisted into a new braiding on HF C, 

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

If A is a left H-comodule algebra, 

∆L : A → H ⊗ A, ∆(a) = a (−1) ⊗ a (0) , 

then the algebra AF := LF (A) with product a ·F b := F (a (−1) , b (−1) )a (0) b (0) 

is a left HF -comodule algebra. 


The Deformation Functor 

More interesting is the fact that the trivial braiding on H C, 

Ψ : A ⊗ B → B ⊗ A, Ψ(a ⊗ b) = b ⊗ a, 

is twisted into a new braiding on HF C, 

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

If A is a left H-comodule algebra, 

∆L : A → H ⊗ A, ∆(a) = a (−1) ⊗ a (0) , 

then the algebra AF := LF (A) with product a ·F b := F (a (−1) , b (−1) )a (0) b (0) 

is a left HF -comodule algebra. 

Theorem (Majid-Oeckl) The functor LF : H C → HF C is an isomorphism of 

braided monoidal categories. 


Example: Twisting by a Torus Action 

Take H = A(T 2 ), generated by t1, t2 with tjt ∗ j = t∗ j tj = 1, 

∆(tj) = tj ⊗ tj, S(tj) = t ∗ j , ɛ(tj) = 1. 




Define a cocycle by 


F (ti, ti) = 1, F (t1, t2) := exp( 1 

2 iπθ), 

extended as a Hopf bicharacter. Then H = HF as a Hopf algebra, but the 

category of H-comodules is twisted. 




Define a cocycle by 


F (ti, ti) = 1, F (t1, t2) := exp( 1 

2 iπθ), 

extended as a Hopf bicharacter. Then H = HF as a Hopf algebra, but the 

category of H-comodules is twisted. 

Example: The coproduct ∆ : H → H ⊗ H makes H into an H-comodule algebra 

in the category H C. The comodule-twisted torus has algebra relations 

t1 ·F t2 = F (t1, t2)t1t2 = F 2 (t1, t2)t2 ·F t1 = µt2 ·F t1, µ = exp(iπθ) 

i.e. we get the noncommutative torus A(T 2 θ ) as an algebra in HF C. 


Example: The Connes-Landi Sphere 

Similarly, we have a coaction 

A(C 4 ) → H ⊗ A(C 4 ), zj ↦→ τj ⊗ zj, (τj) := (t1, t ∗ 1, t2, t ∗ 2). 

This gives twisted algebras A(C 4 θ ), A(S7 θ ), generated by zj, z ∗ l 

zjzl = ηljzlzj, 

zjz ∗ l = ηjlz ∗ l zj, 

⎛ 

1 

⎜ 

where (ηjl) = ⎜1 

⎝µ 

1 

1 

¯µ 

¯µ 

µ 

1 

⎞ 

µ 

¯µ ⎟ 

1⎠ 

, 

subject to 

µ := exp (iπθ). 

¯µ µ 1 1 


Example: The Connes-Landi Sphere 

Similarly, we have a coaction 

A(C 4 ) → H ⊗ A(C 4 ), zj ↦→ τj ⊗ zj, (τj) := (t1, t ∗ 1, t2, t ∗ 2). 

This gives twisted algebras A(C 4 θ ), A(S7 θ ), generated by zj, z ∗ l 

zjzl = ηljzlzj, 

zjz ∗ l = ηjlz ∗ l zj, 

⎛ 

1 

⎜ 

where (ηjl) = ⎜1 

⎝µ 

1 

1 

¯µ 

¯µ 

µ 

1 

⎞ 

µ 

¯µ ⎟ 

1⎠ 

, 

subject to 

µ := exp (iπθ). 

¯µ µ 1 1 

The SU(2)-invariant subalgebra A(S4 θ ) generated by entries of the projection 

ΨΨ ∗ = 1 

2 

⎛ 

1 + x 0 α −¯µ β 

⎜ 

⎝ 

∗ 

⎞ 

⎟ 

⎠ . 

0 1 + x β µ α ∗ 

α ∗ β ∗ 1 − x 0 

−µ β ¯µ α 0 1 − x 

The inclusion A(S4 θ ) ↩→ A(S7 θ ) is a noncommutative principal SU(2)-bundle. 


Differential and Hodge Structure on S 4 θ 

The coaction A(S 4 ) → H ⊗ A(S 4 ) is by isometries, i.e. it is an intertwiner 

for the exterior derivative d : A(S 4 ) → Ω 1 (S 4 ). 

This means that the differential calculus Ω • (S 4 ) is also an object in H C. So 

we can deform it to get a differential calculus Ω • (S 4 θ ) on S4 θ . 







Similarly, the Hodge operator ∗ : Ω 2 (S 4 ) → Ω 2 (S 4 ) is H-equivariant and so 

it is a morphism in H C. Similarly for the map J : A(C 4 ) → A(C 4 ). Their 

images under LF are a Hodge operator and a quaternion structure 

∗θ : Ω 2 (S 4 θ ) → Ω 2 (S 4 θ ), J : A(C 4 θ) → A(C 4 θ). 











This is all we need to discuss instantons on S 4 θ . 











This is all we need to discuss instantons on S 4 θ . 

Remark: In fact this works for the action of any locally compact Abelian group - 

so all of the following works in particular for the Moyal plane R4 as well. 

Can we use this to find a construction of instantons on S 4 θ ? 


A Noncommutative Family of Monads over C 4 θ 

The map 

σz : H ⊗ A(C 4 ) → A(Mk) ⊗ K ⊗ A(C 4 ), σz = 

is a morphism in H M provided A(Mk) carries the H-coaction 

A(Mk) → H ⊗ A(Mk), M j 

ab ↦→ τ ∗ j ⊗ M j 

ab . 

j 

j 

Mab ⊗ zj 



The map 



A(Mk) → H ⊗ A(Mk), M j 

ab ↦→ τ ∗ j ⊗ M j 

ab . 

j 

j 

Mab ⊗ zj 

Under the deformation functor, we get a new algebra A(Mk,θ) generated by 

j 

, M 

the matrix elements M j 

ab 

ab ∗ , but now subject to the twisted relations 

M j 

ab M l cd = ηljM l cdM j 

ab . 



The map 



A(Mk) → H ⊗ A(Mk), M j 

ab ↦→ τ ∗ j ⊗ M j 

ab . 

j 

j 

Mab ⊗ zj 

Under the deformation functor, we get a new algebra A(Mk,θ) generated by 

j 

, M 

the matrix elements M j 

ab 

ab ∗ , but now subject to the twisted relations 

M j 

ab M l cd = ηljM l cdM j 

ab . 

We interpret the underlying ‘space’ Mk,θ as parameterising a 

noncommutative family of monads over C4 θ with index k. 


The Noncommutative ADHM Construction 

Although the space Mk,θ has fewer classical points than Mk, we can nevertheless 

work with the whole noncommutative family of monads at once. 

The algebra-valued matrix σz = 

j M j ⊗ zj is now thought of as a map 

σz : H ⊗ A(C 4 θ) → A(Mk,θ) ⊗ K ⊗ A(C 4 θ). 








For each character ɛ : A(Mk,θ) → C (i.e. for each classical point of Mk,θ) 

there is a corresponding monad 

(ɛ ⊗ id) ◦ σz : H ⊗ A(C 4 θ) → K ⊗ A(C 4 θ). 








For each character ɛ : A(Mk,θ) → C (i.e. for each classical point of Mk,θ) 

there is a corresponding monad 

(ɛ ⊗ id) ◦ σz : H ⊗ A(C 4 θ) → K ⊗ A(C 4 θ). 

The ADHM construction goes through just as before. This time take 

 

∈ Mat2k+2,2k(A(Mk,θ) ⊗ A(C 4 θ)). 

V := σz σ ∗ z 

Then set P := 2k+2 − V (V ∗ V ) −1 V ∗ ∈ Mat2k+2(A(Mk,θ) ⊗ A(S 4 θ )). 


A Noncommutative Family of Instantons 

From this family of projections P parameterised by the space Mk,θ we obtain a 

noncommutative family of instantons. 

Theorem (SB-GL) (Generalises k = 1 case of Landi-Pagani-Reina-van Suijlekom) 

The finitely generated projective right A(Mk,θ) ⊗ A(S 4 θ )-module 

E := P A(Mk,θ) ⊗ A(S 4 θ ) 2k+2 

is a noncommutative family of rank two vector bundles over S 4 θ , 

parameterised by the noncommutative space Mk,θ. 


A Noncommutative Family of Instantons 

From this family of projections P parameterised by the space Mk,θ we obtain a 

noncommutative family of instantons. 

Theorem (SB-GL) (Generalises k = 1 case of Landi-Pagani-Reina-van Suijlekom) 

The finitely generated projective right A(Mk,θ) ⊗ A(S 4 θ )-module 

E := P A(Mk,θ) ⊗ A(S 4 θ ) 2k+2 

is a noncommutative family of rank two vector bundles over S 4 θ , 

parameterised by the noncommutative space Mk,θ. 

The operator ∇ := P ◦ (id ⊗ d) is a noncommutative family of instantons 

with topological charge k, parameterised by the noncommutative space Mk,θ. 

The latter statement means that the curvature F = ∇ 2 of the family obeys 

(id ⊗ ∗θ)F = F, 

where ∗θ : Ω 2 (S 4 θ ) → Ω2 (S 4 θ ) is the Hodge operator on S4 θ . 


Gauge Freedom in the Noncommutative Case 

Even though the map 

σz : H ⊗ A(C 4 θ) → A(Mk,θ) ⊗ K ⊗ A(C 4 θ) 

is a noncommutative family of module homomorphisms, we still have the 

freedom to change bases in the vector spaces H and K. This is the same 

gauge freedom that we had in the classical case. 








However, we now have more gauge freedom. The deformation functor 

canonically equips A(Mk,θ) with a Z 2 -action defined by 

(m1, m2) ⊲ a = F −2 (a (−1) , t m1 

1 tm2 

2 ) a(0) , (m1, m2) ∈ Z 2 , a ∈ A(Mk,θ). 

This action becomes trivial in the classical limit. 

This action is by gauge transformations (i.e. it generates projections which 

are unitarily equivalent to P ). 








However, we now have more gauge freedom. The deformation functor 

canonically equips A(Mk,θ) with a Z 2 -action defined by 

(m1, m2) ⊲ a = F −2 (a (−1) , t m1 

1 tm2 

2 ) a(0) , (m1, m2) ∈ Z 2 , a ∈ A(Mk,θ). 

This action becomes trivial in the classical limit. 

This action is by gauge transformations (i.e. it generates projections which 

are unitarily equivalent to P ). 

Thus the correct (?) parameter space for the construction is the quotient of 

the space Mk,θ by Z 2 , described by the crossed product A(Mk,θ)>⊳ Z 2 . 


Bosonisation 

There is another way to arrive at this crossed product algebra. 

The construction of instantons and, in particular, the parameter algebra 

A(Mk,θ) live in the braided tensor category HF C. 

To get back to the category of vector spaces, we apply Majid’s ‘bosonisation’ 

construction, which just means taking the smash product with HF = A(T 2 ), 

yielding the algebra A(Mk,θ)>⊳ A(T 2 ). 

Now applying the Fourier transform on T 2 gives an isomorphism 

since Z 2 is the Pontryagin dual of T 2 . 

A(Mk,θ)>⊳A(T 2 ) → A(Mk,θ)>⊳ Z 2 , 


Understanding the Noncommutative Parameter Spaces 

How can we make sense of these noncommutative parameter spaces? 




Theorem (SB-GL-WvS) 

The algebra A(Mk,θ)>⊳ Z 2 has a commutative subalgebra, denoted A(Mk). 

There is a canonical action of Z 2 on A(Mk) by gauge transformations 

There is an isomorphism 

A(Mk,θ)>⊳ Z 2 A(Mk)>⊳ ′ Z 2 . 




Theorem (SB-GL-WvS) 

The algebra A(Mk,θ)>⊳ Z 2 has a commutative subalgebra, denoted A(Mk). 

There is a canonical action of Z 2 on A(Mk) by gauge transformations 

There is an isomorphism 

A(Mk,θ)>⊳ Z 2 A(Mk)>⊳ ′ Z 2 . 

The algebra A(Mk) is the (commutative) coordinate algebra of a space Mk of 

bona fide monads on C 4 θ , i.e. homomorphisms of free A(C4 θ )-modules 

H ⊗ A(C 4 θ) σz 

−→ K ⊗ A(C 4 θ) 

for complex vector spaces H, K of dimensions k, 2k + 2. 


The Noncommutative ADHM Equations 

Now that we have discovered a classical space of instantons, we can describe it 

more explicity. 

Theorem (SB-WvS) The space Mk/ ∼ of equivalence classes of monads on C 4 θ is 

given by the set of matrices 

satisfying the equations 

B1, B2 ∈ Matk(C), I ∈ Mat2,k(C), J ∈ Matk,2(C) 

¯µB1B2 − µB2B1 + IJ = 0, 

[B1, B ∗ 1] + [B2, B ∗ 2] + II ∗ − J ∗ J = 0 

(where µ = exp(iπθ)), modulo the action of g ∈ U(k) given by Bj ↦→ gBjg −1 , 

I ↦→ gI, J ↦→ Jg −1 . 

The classical limit recovers the ordinary ADHM equations on classical S 4 . 


The Moyal-Groenewold Plane R 4 

If we work in a local patch R4 of S4 and twist by the group of translation 

symmetries, we get the Moyal plane A(R4 ). The twisting Hopf algebra is 

H = A(R4 ). 

The deformation is again functorial, so we get a noncommutative family of 

monads A(Mk,) on the deformed space C 4 . 

This time there is a canonical action of R 4 by gauge A(Mk,) by gauge 

transformations, so we obtain the cross product A(Mk,)>⊳ R 4 . 

We can also get this by bosonisation and Fourier transform, 

A(Mk,)>⊳A(R 4 ) → A(Mk,)>⊳ R 4 . 

Again we recover a classical space of monads; the resulting ADHM equations are 

those of Nekrasov-Schwarz: 

[B1, B2] + IJ = 0, 

[B1, B ∗ 1] + [B2, B ∗ 2] + II ∗ − J ∗ J = −i 


A Geometric Interpretation 

In both cases, we’re twisting the geometry by the action of a locally compact 

Abelian group Γ. Then we find isomorphic crossed product spaces 

A(Mk,Γ)>⊳ Γ ∼ = A(Mk)>⊳ Γ. 





A(Mk,Γ)>⊳ Γ ∼ = A(Mk)>⊳ Γ. 

Morally, we could view this as a pair of quotient maps 

Mk,Γ 

Mk 

 

 

Mk,Γ/ Γ = Mk/ Γ 


.




A(Mk,Γ)>⊳ Γ ∼ = A(Mk)>⊳ Γ. 

Morally, we could view this as a pair of quotient maps 

Mk,Γ 

Mk 

 

 

Mk,Γ/ Γ = Mk/ Γ 

So the moduli problem has a ”stacky” behaviour - both Mk,Γ and Mk are valid 

parameter spaces of charge k instantons, but one is quantum and the other is 

classical. 


.

U(2) or SU(2) Instantons? 

On classical S 4 , the ADHM construction produces either U(2) instantons or 

SU(2) instantons - they are equivalent. 

This means ∇ = d + ω, where ω ∈ Ω 1 (P ) ⊗ g for either g = u(2) or su(2). 






For U(2) instantons, the gauge group is just 

U(E) := 

 

U ∈ End A(S 4 θ )(E) | U ∗ U = UU ∗ = idE 

It turns out that the moduli space is computable in this case. 

Theorem (SB-GL-WvS) The moduli space of U(2) instantons with charge k on 

the quantum sphere S 4 θ is a smooth manifold of dimension 8k − 3. 


 

.





For U(2) instantons, the gauge group is just 

U(E) := 

 

U ∈ End A(S 4 θ )(E) | U ∗ U = UU ∗ = idE 

It turns out that the moduli space is computable in this case. 

Theorem (SB-GL-WvS) The moduli space of U(2) instantons with charge k on 

the quantum sphere S 4 θ is a smooth manifold of dimension 8k − 3. 

What happens when you try to look for SU(2) instantons? Recall that, in the 

classical case, it doesn’t matter. 


 

.

The Trouble with the Trace 

If you want connections with ω ∈ Ω 1 (P ) ⊗ su(2), things get tricky. What is 

the gauge group here? 

We could try to define it as 

SU(E) = U ∈ U(E) | UωU ∗ + U(dU ∗ ) ∈ Ω 1 (P ) ⊗ su(2) for all ω . 







But there aren’t so many of these (only the torus-invariant part of the 

classical SU(2) gauge group survives). 

To get the full ‘set’ of SU(2) gauge symmetries, one needs a quantum group 

- this partly explains the origin of the noncommutative gauge parameters. 







But there aren’t so many of these (only the torus-invariant part of the 

classical SU(2) gauge group survives). 

To get the full ‘set’ of SU(2) gauge symmetries, one needs a quantum group 

- this partly explains the origin of the noncommutative gauge parameters. 

In fact the new ‘quantum’ gauge symmetries given by the Pontryagin dual Z 2 

seem to interfere with the U(1) part of the gauge group - further evidence 

that you have to build them in to the ADHM construction. 


Summary 

In obtaining an ADHM construction of instantons on S4 θ , we were naturally 

led to noncommutative parameter spaces. 

Nevertheless, by incorporating the ‘quantum’ gauge symmetries into the 

construction, we were able to recover a classical space of parameters. 

The quotient of this parameter space by the ‘classical’ gauge symmetries 

gives the ’coarse’ moduli space. 

But it’s not the full story: we should not ignore the ‘quantum’ gauge 

symmetries. Dividing by these as well leads to the noncommutative quotient 

space A(Mk)>⊳ Z 2 . 

There seems to be a big difference between the moduli problem on classical 

spaces and on noncommutative spaces...but we’re getting closer to 

understanding it! 


Relevant Papers 

S.Brain, G.Landi: Families of Monads and Instantons from a 

Noncommutative ADHM Construction. Clay Math. Proc 11, 55-84 (2010). 

arXiv:math.qa/0901.0772 

S. Brain, G. Landi: Moduli Spaces of Noncommutative Instantons: Gauging 

Away Noncommutative Parameters. Quart. J. Math., to appear 


S. Brain, W.D. van Suijlekom: The ADHM Construction of Instantons on 

Noncommutative Spaces. 

arXiv:math.ph/1008.4517 

G. Landi, C. Pagani, C. Reina, W.D. van Suijlekom: Noncommutative 

Families of Instantons. IMRN 12 (2008), Art. ID rnn038 


S. Brain, G.Landi, W.D. van Suijlekom: Moduli Spaces of Instantons on Toric 

Noncommutative Manifolds. In preparation.

Noncommutative Geometry of Instanton Moduli Spaces

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