Unification diagram. Assume that the group G is simply laced. In a series of works [GMN1]-[GMN5], Gaiotto, Moore and Neitzke studied 4d N “ 2 SUYM theories of class S related to a Riemann surface with punctures Σ. The S alludes to “six dimensional”: the theories are
“defined” as compactifications of the hypothetical p2,0q theories XG, related to ADE Dynkin diagrams, on the Riemann surface Σ, with defects at the punctures.
The origins of the 4d theory can be perceived as follows. Let ΓG be a finite subgroup of SUp2q corresponding to G by the McKay correspondence. The theory XG itself is “defined” as a compactification of the ten dimensional type IIB superstring theory on the Klein singularity C2{ΓG. One should have a “commutative diagram”:
10d type IIB superstring theory
The vertical arrow is the compactification of the 10d type IIB superstring theory on the six-dimensional Riemannian manifold given by the complex open CY threefoldYG,Σ,t. The pointt belongs to the Hitchin base. The latter is the Coulomb branch of the space of vacua of the 4d theory TG,Σ.
Gaiotto-Moore-Neitzke count of BPS states. The Hilbert space H of the 4d N “ 2 SUYM class S theory is a huge representation of the N “ 2 Poincare super Lie algebra P “ P0‘P1. Its even partP0is the Poincare Lie algebra of the flat Minkowski spaceR3,1 plus a one dimensional center with the generator Z. The odd part is 8-dimensional. As a representation of the Lorenz group, it is a sum of two copies of the spinor representation S`‘S´.
The Hilbert space H is the symmetric algebra of a 1-particle Hilbert space H1. The n-th symmetric power of H1 is the “n-particle part”. The H1 should have a discrete spectrum, i.e.
be a sum rather then integral of unitary representations of the Poincare super Lie algebra P.
The Hilbert spaceH has the following structures, inherited on the subspaceH1:
1. It depends on a pointt of the Hitchin base. The Hitchin base is the Coulomb branch of the moduli space of vacua in the theory.
2. It is graded by acharge lattice Γ. In particular, there is a decomposition H1 “ ‘γPΓH1,γ.
The lattice Γ is equipped with an integral valued skew symmetric bilinear form x˚,˚y.
Irreducible unitary representations of the super Lie algebra P are parametrized by three parameters: the mass M P r0,8q, the spinjP t0,12,1,32, ...u, and the central charge Z PC.
The pairs pM, jq parametrize “positive” unitary representations of the Lorenz group.
The crucial fact is the inequality M ě |Z|. We are interested in the BPS part HBPS1 of the space, defined by the M “ |Z| condition. Let nj be the multiplicity of the irreducible representations of spin j in HBPS1 . The integers ΩGMNt pγq, “counting the BPS states” of the central charge γ, are not the integersnj but rather the “ second helicity supertrace”:
ΩGMNt pγq:“ÿ
j
p´1q2jp2j`1qnj.
Let us now discuss the Gaiotto-Moore-Neitzke approach to calculate these numbers. A point t of the Hitchin base determines a spectral curve Σt ĂT˚Σ, and a spectral cover πt : ΣtÑΣ.
It determines a lattice Γt. When Σ is compact it is given by Γt“Ker´
H1pΣt,ZqÝÑπt H1pΣ,Zq¯
. (37)
Then γ PΓt. Integrating the canonical 1-form α on T˚Σ over the homology classes from (37) we get a linear map, called the central charge map:
Zt: ΓtÝÑC, γ ÞÝÑ ż
γ
α.
Gaiotto, Moore and Neitzke introduced a spectral network related to a generict. They use it to develop an algorithm to calculate the numbers Ωtpγq. The algorithm has some mathematical issues for higher rank groups.4 Let us assume that they are resolved.
4One of them is a possibility of having an infinite number of “two side roads” in a spectral network, making the algorithm problematic.
The Gaiotto-Moore-Neitzkespectral generatoris a transformtion of the Hitchin moduli space.
It tells the cumulative result of the wall crossings which one encounters rotating a Higgs field Φ projecting to a point t by eiθΦ, with 0 ďθď 8. It turned out5 that our map CPGLm,S acting on the moduli space XPGLm,S coincides with the result of calculation of the spectral generator.
So formula (24) implies that
The DT-transformation DTm,S = The Gaiotto-Moore-Neitzke spectral generator. (38) Let us assume that a pointt of the universal Hitchin base determines a quiver q, and that the lattice Λqof this quiver is identified with the lattice Γt. This is known for G“SL2 [GMN2].
Examples were worked out in [GMN5] for G“SLm, mď9. They produce quivers of the type discussed above. Then the central charge map Zt translates into a central charge map
Zq: ΛqÝÑC.
Letteiube the basis of Λ provided by the quiverq. Assuming thatZqpeiq PH, we arrive at a stability condition s determined by t.
So if all mentioned above assumptions were satisfied, the formula (38) would imply that KS numerical DT-invariants ΩKSs pγq = GMN invariants ΩGMNt pγq. (39) Let us stress that the origins and definitions of the two sides of (39) are entirely different.
The numbers ΩKSs pγq came from 3d CY categories related to quiversq.
The numbers ΩGMNt pγqcame from a quantum field theory, and calculated using the geometry of a Riemann surface Σ.
Gaiotto-Moore-Neitzke algorithm for counting the numbers Ωtpγq can be interpreted as a count of certain type of branes in 10d type IIB superstring theory on
YtˆR3,1. (40)
These are theD3-branes supported onLˆl where LĂYt is a special Lagrangian sphere, and l Ă R3,1 is the world line of a BPS particle. The mass M of the particle is the Riemannian volume of L. Its central chargeZ is the integral ZpLq “ş
LΩ over L of the holomorphic 3-form Ω on Yt. So the BPS conditionM “Z just means that we count specialLagrangian spheres.
Conjecture 1.19 connects the two approaches. Namely, among the “combinatorial” 3d CY categories assigned to a quiver q there is a distinguished one, Cm,S, provided by the canonical potentials on ideal bipartite graphs on S. Conjecture 1.19 predicts that the category Cm,S has a geometric realisation as a Fukaya category of an open CY threefold Yt. And this is threefold needed to get the 4dN “2 SUYM theory from the IIB superstring theory in (40).
So now the combinatorial 3d CY categories and the 4d theories are linked directly to each other. The common structure visible in both is the cluster structure.
It is interesting to note that formula (24) involves both the 3d CY category used to define the DT-transformation and the moduli spaceXG,Son which the element CG,S acts naturally.
5We thank Davide Gaiotto who pointed this to us at the 6d conference at Banff, and to Andy Neitzke for providing some details.
Notice that although the moduli spaceXG,Sis closely related to the moduli space of G-local systems on S, the element CG,S does not act on the latter.
In general there is no canonical moduli space linked to the DT-transformation. We have only a cluster variety: it describes a lot of features of the space, yet it is different.