ADHM construction

3.2 Instanton counting

3.2.1 ADHM construction

Let E be a rank N complex vector bundle onR⁴ with a connection A and a framing at infinity. The framing is an isomorphism of the fiber at infinity with C^N. The ADHM construction studies the moduli spaceM_k of connections Aon the bundleEthat satisfy the self-dual instanton equation F^`pAq “ 0, up to gauge transformations that are trivial at infinity. It turns out that this moduli space can be realized as a hyperk¨ahler quotient of linear data.

UpNqgauge group

Figure 3.1: Quiver representation of theUpNqADHM quiver. The vector spacesVandW arek and N-dimensional, respectively, with a natural action of the dual group Upkq and the framing groupUpNq. The mapsB₁,B₂,I andJare linear.

For the gauge groupG“UpNqthe linear data consists of four linear maps

pB₁,B₂,I,Jq PX“HompV,Vq ‘HompV,Vq ‘HompW,Vq ‘HompV,Wq, (3.11) whereVandW are two complex vector spaces of dimensionkand N, respectively. This linear data is summarized in an ADHM quiver diagram in Figure 3.1. The vector spaceW is isomorphic to the fiber ofE(which in our case is of rankN). It is best thought of as the fiber at infinity, as there is a natural action of the framing groupUpNqon it, which physi- cally can be thought of as the large gauge transformations at infinity. The tensor product of the vector spaceVwith the half canonical bundleK^1{2_C2 onC²–R⁴, on the other hand, can be identified with the space of normalizable solutions to the Dirac equation in the back- ground of the instanton gauge fieldA. Since the instanton numberk “1{8π²ş

F_A^F_Ais given by the second Chern class ofE, it follows from index theorems that this space has dimensionk, as we advertised above. In particular it carries in a natural way the action of the dual groupUpkq. More algebraically, the vector spaceVitself is isomorphic to the

cohomology groupH¹pEq.

The framing group UpNq and the dual group Upkq thus act naturally on the linear ADHM data. Setting the three real moment maps

µ_R “ rB₁,B^:₁s ` rB<sub>2</sub>,B<sub>2</sub><sup>:</sup>s `I I^:´J^:J (3.12)

µ_C “ rB₁,B₂s `I J, (3.13)

to zero gives the so-called ADHM equations. The ADHM construction identifies the in- stanton moduli space M^UpNq_k with the hyperk¨ahler quotient of the solutions X to those equations by the dual group,

M^Upkq_k “X{{Upkq ”µ^´1p0q{Upkq. (3.14) From a physical perspective the ADHM construction can be most natural understood using D-branes. We can engineer the moduli space ofkinstantons in the four-dimensional UpNqtheory by putting k Dpp´4q-branes on top of N Dp-branes. The Dpp´4q-branes appear as zero-dimensional instantons on the transverse four-dimensional manifold. The mapspB₁,B₂,I,Jqcan be understood as the zero-modes of Dpp´4q´Dpp´4q), Dp´Dpp´ 4qand Dpp´4q´Dp open strings, respectively, and the ADHM equations are the D-term conditions. The ADHM quotient can thus be identified with the moduli space of the Higgs branch of theUpkqgauge theory on the Dpp´4q-branes.

Since the above quotient is highly singular due to small instantons, we change it by giving non-zero value to the Fayet-Illiopolous termζ. This is equivalent to turning on NS 2-form field on the Dp-branes, and the resulting desingularized quotient can be interpreted as a moduli space of non-commutative instantons [72].

To perform our computations another, equivalent way of representing the ADHM con- struction will be useful. Let us introduce the spinor bundlesS^˘ of positive and negative chirality onR⁴, and for brevity denote the half canonical bundle by L “ K^1{2_C2. Consider

the sequence

VbL^´1 Ñ^σ

VbS^´

‘ W

Ñ^τ VbL, (3.15)

whereS^´andLare the fibers of the bundles S^´ andL, respectively. Although these can all be trivialized, they are non-trivial equivariantly. We thus need to keep track of them for later. The mappingsσandτare defined by

σ“

¨

˚

˚

˚

˝

z₁´B₁ z₂´B₂

J

˛

‹

‹

‹

‚

, τ“

´

´z₂`B₂, z₁´B₁, I

¯

, (3.16)

wherepz₁,z₂qare coordinates onC². From the ADHM equations it follows thatτ˝σ“0, so that the sequence (3.15) is a chain complex. Sinceσ is injective andτsurjective, it is a so-called monad.

Notice that the vector spaceVbL^´1at the first position of the sequence (3.15) fixes the vector spaces at the remainder of the sequence. The fieldsB₁andB₂are coordinates onC² and thus mapVbL^´1 Ñ VbS^´. The fieldsI and Jare the two scalar components of an N “2 hypermultiplet, that properly speaking transform as sections of the line bundleL.

To recover the vector bundle E, we vary the cohomology space pKer øq{pIm œq over C², which gives indeed a vector bundle whose fiber at infinity is equal toW. One can also show that the curvature of this bundle is self-dual and that it has instanton numberk. Even better, every solution of the self-dual instanton equations can be found in this way.

We are now ready to construct the main tool in our computation. This is theuniversal bundleE over the instanton moduli spaceM^UpNq_k ˆR⁴. The universal bundle is obtained by varying the ADHM-parameters of the maps in the complex (3.15). It has the property that

E_{A,z “}E_z, (3.17)

i.e., its fiber over an element A P M^UpNq_k is the total space of the bundle Ewith connec-

tionA. Remember that the bundleEhas fiberW at infinity inR⁴and that the vector space Vof solutions to the Dirac equations is related to its first cohomology H¹pEq. The vector spacesVandWcan be extended to bundlesVandWover the instanton moduli space. We can then easily compute the Chern character of the universal bundleE from its defining complex (3.15) as

ChpE_{q “Chp}W_{q `Chp}V_q^´_ChpS^´_{q ´Chp}L_{q ´Chp}L^´1_q^¯_{. (3.18)} SO{Spgauge groups

The construction forSOpNqandSppNqgauge groups is very similar. We defineSppNqto be the special unitary transformations on C^2N that preserve its symplectic structure Φs, andSOpNqthe special unitary transformations onC^N that preserve its real structureΦr.

Figure 3.2: Quiver representation of theSppNqADHM quiver. The vector spacesVandW arekand 2N-dimensional, respectively. V has a real structure Φrand a natural action of the dual groupSOpkq, whereasWhas a symplectic structureΦsand a natural action of the framing groupSppNq. The mapsB₁,B₂andJare linear.

ForSppNqthe linear data that is needed to define the ADHM complex consists of pB₁,B₂,Jq PY“HompV,Vq ‘HompV,Vq ‘HompV,Wq, (3.19) whereV andW are a complex kand 2N-dimensional vector space, resp., together with a real structureΦr onVand a symplectic structureΦsonW. This is illustrated as a quiver diagram in Figure 3.2. The dual group is given byOpkq, so that the moduli space ofSppNq instantons is given by

M^SppNq_k “ tpB₁,B₂,Jq |ΦrB₁,ΦrB₂PS²V^˚, ΦrrB₁,B₂s ´J^˚ΦsJ“0u{Opkq. (3.20) ForSOpNqwe just need to replaceVandWby a complex 2kandN-dimensional vector space, resp., as well as change the role of symplectic structure and the real structure. This

Figure 3.3: Quiver representation of theSOpNq ADHM quiver. The vector spacesV and W are 2kandN-dimensional, respectively. Vhas a symplectic structure Φsand a natural action of the dual groupSppkq, whereasW has a real structureΦrand a natural action of the framing groupSOpNq. The mapsB₁,B₂andJare linear.

is illustrated as a quiver diagram in Figure 3.3. The dual group is given bySppkq, so that the moduli space ofSOpNqinstantons is given by

M^SOpNq_k “ tpB₁,B2,Jq |ΦsB₁,ΦsBP ^²V^˚,ΦsrB₁,B2s ´J^˚ΦrJ “0u{Sppkq. (3.21) A subtle issue for the above moduli spaces is that there is no appropriate Gieseker desingularization which resolves the singularity due to the zero-sized instantons (as in the case ofUpNq). One way to understand this is by considering the string theory em- bedding. The above ADHM constructions can be obtained by considering Dp-Dpp´4q system and also adding anO^˘p plane on the top of the Dpbranes. In the case ofUpNq, the non-commutativity parameter that we introduce is coming from the NS 2-form field on theDp-brane. But, the orientifold makes it impossible to turn on the background NS 2-form field. So we cannot resolve the singularity in the same way. An alternative way to resolve the singularity was studied by [73], but a physical understanding of this proce- dure is still lacking. Nevertheless, we will see that the equivariant volume of the moduli space can be obtained without explicitly resolving the singularity [53]. This formula is ver- ified mathematically using Kirwan’s formula of the equivariant volume of the symplectic quotient [69].

We can then again represent anySppNqinstanton solutionEas the cohomology bundle of the sequence

VbL^´1ÝÑ^σ

VbS^´

‘ W

ÝÑ^σ˚β˚ V^˚bL, (3.22)

where the mappingsσandβare defined by

σ “

¨

˚

˚

˚

˝

z₁´B₁ z₂´B₂

J

˛

‹

‹

‹

‚

, β“

¨

˚

˚

˚

˝

0 Φr 0

´Φr 0 0

0 0 Φs

˛

‹

‹

‹

‚

. (3.23)

The ADHM equations ensure thatσ^˚β^˚σ “0, and the sequence (3.22) is another monad.

Analogously to the UpNq example, when varying the ADHM-parameters in the com- plex (3.22) we find the universal bundle E_SppNq. V is the k-dimensional solution space of theSppNqDirac operator onC², which carries a real structure, andW is the fiber ofEat infinity, and hence carries a symplectic structure.

Similarly, anySOpNqinstanton solutionEcan be represented as the cohomology bun- dle of the complex (3.22) as well, once we exchangeΦrwithΦsin the definition of the map β. The resulting complex is a short exact sequence, since according to the ADHM equa- tionsσ^˚β^˚σ “0. By varying the ADHM-parameters we find theSOpNquniversal bundle E_SOpNq. Note thatV is the 2k-dimensional solution space of theSOpNqDirac operator on C², which carries a symplectic structure, whereasWis the fiber ofEat infinity, and hence carries a real structure.