C.3. THE GOPAKUMAR–VAFA CONJECTURE 107
or “conifold transition” in this case. Since the partition function for the open theory is known for all genera (from Chern–Simons theory), this can be used to postulate the full closed string partition function on the resolved conifold, which does have a K¨ahler parameter J and would in general be hard to compute to all orders.
The agreement of the partition function on both sides has been shown in [63] for arbitrary ’t Hooft couplingλ=N gs and to all orders in 1/N. In this sense, this duality is an example of a large N duality which has been suggested by ’t Hooft: for large N holes in the Riemann surface of Feynman diagrams are “filled in” or “condensed”, where one takes N → ∞ with gs = fixed.
The authors of [63] matched the free energy Fg at every genus g via the identification of the ’t Hooft coupling
iλ = N gs (open) ←→ iλ = t (closed), (C.31)
where t is the complexified K¨ahler parameter of the S2 in the resolved conifold and the iden-tification of the ’t Hooft coupling for open strings is dictated by the Chern–Simons theory on S3.
Beyond that, it was also shown that the coupling to gravity (to the metric)4 and Wilson loops take the same form for the open and closed theory. The two topological string theories described here correspond to the different limits λ→ 0 and λ→ ∞, but they are conjectured to describe the same string theory (with the same smallgs) only on different geometries.
Embedding in Superstrings and Superpotential
This scenario has an embedding in “physical” type IIA string theory. Starting with N D6 branes on the S3 of the deformed conifold we find a dual background with flux through the S2 of the resolved conifold. The Calabi–Yau breaks 3/4 of the supersymmetry (which leaves 8 supercharges), therefore the theory on the worldvolume of the branes has N = 1 (the branes break another half of the supersymmetry). There is a U(N) gauge theory on the branes (in the low energy limit of the string theory the U(1) factor decouples and we have effectively SU(N)).
As described in the last section, these wrapped branes create flux and therefore a superpotential.
This superpotential is computed from topological strings, but we need a gauge theory parameter in which it is expressed. The relevant superfield forN = 1 SU(N) isS, the chiral superfield with gaugino bilinear in its bottom component. We want to express the free energy Fg in terms of S. Since there will be contributions from worldsheet with boundaries, we can arrange this into a sum over holesh
Fg(S) =
∞
X
h=0
Fg,hSh. (C.32)
It turns out that the genus zero term computes the pure gauge theory, i.e. pure SYM, higher genera are related to gravitational corrections.
As discussed above, the open topological string theory is given by Chern Simons on T∗S3, which has no K¨ahler modulus. The superpotential created by the open topological amplitude of genus zero is given by [6]
λsWopen = Z
d2θ∂F0open(S)
∂S +α S+β (C.33)
4It might seem contradictory that there can be a coupling to the metric when we are speaking about topological models. The classical Chern–Simons action is indeed independent of the background, but at the quantum level such a coupling can arise. In the closed side there are possible IR divergences, anomalies for non–compact manifolds that depend on the boundary metric of these manifolds.
C.3. THE GOPAKUMAR–VAFA CONJECTURE 109 withα, β = constant,α S being the explicit annulus contribution (h= 2).
Although the topological model is not sensitive to any flux through a 4– or 6– cycle, in the superstring theory the corresponding RR formsF4andF6can be turned on. In the closed string side this corresponds to a superpotential
λsWclosed = Z
F2∧k∧k+i Z
F4∧k+ Z
F6. (C.34)
The topological string amplitude is not modified by these fluxes [6]. The genus zero topological string amplitudeF0 determines the size of the 4– and 6–cycle to be ∂F∂t0 and 2F0−t∂F∂t0, respec-tively, wheretis the usual complexified K¨ahler parameter (of the resolved conifold). If we have N, L, P units of 2–, 4– and 6–form flux, respectively, the superpotential yields after integration
λsWclosed = N ∂F0
∂t +itL+P . (C.35)
Note that requiring W = 0 and ∂tW = 0 fixes P and L in terms of N and t. N is of course fixed by the number of branes in the open string theory.
This looks very similar to the superpotential for the open theory (C.33). We have already discussed that the topological string amplitudes agree
Fopen = Fclosed (C.36)
if one identifies the relevant parameters as in (C.31). In this case we have to identify S witht and α, β with the flux quantum numbers iL, P. It is clear from the gauge theory side that α (orL) is related to a shift in the bare coupling of the gauge theory. In particular, to agree with the bare coupling to all orders we require iL=V /λs, whereV is the volume of the S3 that the branes are wrapped on. This gives an interesting relation between the size V of the blown–up S3 (open) and the sizetof the blown–up S2 (closed):
et−1N
= const·e−V /λs. (C.37)
This indicates that for small N (N gs/V 1) the D–brane wrapped on S3 description is good (since t → 0), whereas for large N (N gs 1) the blown–up S2 description is good (since V → −∞ does not make sense). It should be clear from our discussion that after the S3 has shrunk to zero size there cannot be any D6–branes in the background, but RR fluxes are turned on.
To summarize the superstring picture of the conifold transition: In type IIA we start with N D6–branes on the S3 of the deformed geometry and find as its dual N units if 2–form flux through the S2 of the resolved conifold. In the mirror type IIB, N D5 branes wrapping the S2 of the resolved conifold are dual (in the large N limit) to a background without D–branes but 3–form flux turned on. The geometry after transition is given by the deformed conifold with blown upS3. In both cases we have to identify the complex structure modulus of the deformed conifold with the K¨ahler modulus of the resolved conifold or, roughly speaking, the size of the S3 with the size of theS2.
Let us finish this section with the explicit derivation of the Veneziano–Yankielowicz super-potential in type IIA [6]. To lowest order the type IIA supersuper-potential is given by
W(S) = Z
d2θ 1
λsτ S+iN2αe−τ /N
(C.38)
whereτ is the chiral superfield withiC+V /λsin its bottom component,Cbeing the background value of the 3–form gauge field in IIA andV the size of theS3 as before. Ctakes the role of theθ angle and the gauge coupling is promoted to a superfield. The second term in (C.38) stems from instanton corrections. It might seem surprising thatV enters into this superpotential (although the A model is independent of the complex structure modulus), the reason lies within quantum corrections. But the linear term in S is the only coupling that V has to this theory.
We now use that τ is actually a dynamical superfield and can be integrated out from its equation of motion. Requiring ∂τW = 0 gives
S = iλsN αe−τ /N. (C.39)
Solving this forτ and plugging the result back into (C.38) gives an effective superpotential for S
Weff = −N λs
S log
S iN αλs
−S
. (C.40)
We do indeed recover the Veneziano–Yankielowicz superpotential [59]. The scale of the gauge theory Λ can be identified with (N αλs)1/3. The vacuum of the theory exhibits all the known phenomena of gaugino condensation, chiral symmetry breaking and domain walls. This is a remarkable result and the first example where string theory produces the correct superpotential of a gauge theory.