Deformations and Supersymmetry Conditions

7.1. FIVE-BRANEN = 1 EFFECTIVE DYNAMICS 173 discarded sincee₃ is not well-defined at the positionr= 0 of the brane. We note further that e⁽⁰⁾₂ is not a global form since it is not gauge invariant under the SO(4) action on the normal bundle. Then, we obtain the global expressions for the field strength F₃ and H₃ as

F₃=hF₃i+dC₂−dρ∧e⁽⁰⁾₂ /2, H₃=hH₃i+dB₂−dρ∧e⁽⁰⁾₂ /2 +ω₃, (7.10) where ω₃ =ω₃^L−ω₃^G denotes the Chern-Simons form for trR²−₃₀¹TrF² and hF₃i, hH₃i are background fluxes inH³(Z₃,Z). With these formulas at hand we immediately check that the reasoning of (7.7) and the localization (7.8) to the boundary of the open manifold Z₃ −Σ applies globally. Furthermore, the expansion (7.10) formally unifies the superpotentials as we will discuss in detail below in section 7.1.3.

Finally, we note that (7.10) implies thatC₂ respectivelyB₂ have an anomalous transfor-mation under the SO(4) gauge transfortransfor-mations of N_Z3Σ. This is necessary to compensate the anomalous transformation δe⁽⁰⁾₂ so that F₃ respectively H₃ are gauge invariant. This anomalous transformation plays a crucial role for anomaly cancellation in the presence of five-branes [267].

heterotic string. Thus, this part of the vacuum energy cancels which is a necessary condition for supersymmetry. Indeed, this is easily seen, for example, in orientifold setups. As we have seen in chapter 2 the orientifold compactification preserve N = 1 supersymmetry in the effective theory if the geometric part of the orientifold projection is a holomorphic and isometric involutionσ acting on Z₃. Hence, theO5-planes, being the fix-point set of σ, wrap holomorphic curves insideZ₃ and are also calibrated with respect toJ. Thus, they contribute the same potential in the vacuum but with opposite sign as argued in section 3.2 [60].

Let us now consider a general fluctuation of the supersymmetric Σ ≡ Σ₀ to a nearby curve Σs. From the above one expects the generation of a positive potential when deforming Σ non-holomorphically. A deformation is described by a complex section s of the normal bundleN_Z3Σ≡N_Z^1,0

3Σ. The split of the complexified normal bundle has been performed in a background complex structure ofZ₃. Clearly, the space of such sections is infinite dimensional as is the space of all Σ_s. To make the identification between Σ_sandsmore explicit, one recalls that in a sufficiently small neighborhood of Σ₀ the exponential mapexp_sis a diffeomorphism of Σ₀ onto Σ_s. Roughly speaking, one has to consider geodesics through each pointp on Σ₀ with tangent s(p) and move this point along the geodesic for a distance of||s||to obtain the nearby curve Σsas depicted in figure 7.1. That the holomorphic curve Σ0is of minimal volume

Σ₀ Σ_s

exp(p)s

p s(p)

Figure 7.1: Deformations of Σ0along geodesics in Z3. The tangent vectors(p) along the geodesic passingpis a normal vec-tor of Σ0 atp.

can now be seen infinitesimally. For any normal deformation Σ_ǫs to Σ₀, i.e. a deformation with infinitesimal displacement ǫs, the volume increases quadratically [169, 170] as

d²

dǫ²Vol(Σ_ǫs)

ǫ=0

= 1 2

Z

Σk∂s¯ k² vol_Σ, (7.12) wherek∂s¯ k²denotes the contraction of indices, both on the curve as well as on its normal bun-dle N_Z3Σ via the metric. In the effective action, (7.12) is the leading F-term potential when deforming the D5-brane curve non-holomorphically as we derived explicitly in section 3.4. A quadratic term of the form (7.12) in the scalar potential implies that the four-dimensional fields corresponding to these non-holomorphic deformations s acquire masses given by the value of the integral (7.12)³. When integrating out the massive deformations with ¯∂s6= 0 the

3The first variation of vol(Σu) vanishes by the First Cousin Principle [170].

7.1. FIVE-BRANEN = 1 EFFECTIVE DYNAMICS 175 remaining sections are elements of

H⁰(Σ, N_Z3Σ)≡H_∂⁰_¯(Σ, N_Z^1,0

3Σ) ⊂ C^∞(N_Z3Σ) . (7.13) Reversely only holomorphic sections s ∈ H⁰(Σ, N_Z3Σ) deforming Σ into a nearby curve Σ_s can lead to massless or light fields in the effective theory. It is crucial to note that even for an s∈H⁰(Σ, N_Z3Σ) the deformation might be obstructed at higher order and hence not yield a massless deformation. The higher order mass terms for these deformations can be studied by computing the superpotential as we will discuss throughout the next sections.

Before delving into the discussion of these holomorphic deformations, let us conclude with a discussion of the effect of complex structure deformations on the F-term potential (7.12) and the hierarchy of masses of fields associated to brane deformations. Deformations generated by non-holomorphic vector fields sdo not obey the classical equations of motion⁴. The main complication is that there are infinitely many such off-shell deformations and it would be very hard to compute their full scalar potential. In contrast toC^∞(N_Z3Σ), the spaceH⁰(Σ, N_Z3Σ) is finite dimensional. However, there is a distinguished finite dimensional subset ofC^∞(N_Z3Σ) that should not be integrated out in the effective action. This is related to the fact that the dimension of H⁰(Σ, NZ3Σ) is not a topological quantity and will generically jump when varying the complex structure of Z₃. For example, this can lift some of the holomorphic deformationss∈H⁰(Σ, N_Z3Σ) since the notion of a holomorphic section is changed. Indeed, by deforming ¯∂ by A inH¹(Z3, T Z3) we obtain

d²

dǫ²Vol(Σǫs)

ǫ=0

= 1 2

Z

ΣkAsk²volΣ= 1 2|t|²

Z

ΣkA1sk²volΣ+O(t⁴), (7.14) where A₁ is the first order complex structure deformation of Z₃ as introduced above. Here we used that the complex structure on Σ is induced from Z₃ and s is in H⁰(N_Z3Σ) in the unperturbed complex structure onZ₃, ¯∂s= 0. This result is clear from the point of view of the new complex structure ¯∂^′ = ¯∂+A, since ¯∂^′s=As6= 0. Thussis a section in C^∞(NZ3Σ) in the new complex structure unless s is in the kernel of A. Similarly, the corresponding field acquires a mass given by the integral (7.14). However, the main difference to a generic massive mode inC^∞(NZ3Σ) with mass at the compactification scale, cf. eq. (7.12) and (3.88), is the proportionality to the square of the VEV oft. Consequently the mass of this field can be made parametrically small tuning the value oft. Thus, we can summarize our approach to identify the light fields as follows: (1) drop an infinite set of deformations s which are massive via (7.12) at each point in the complex structure moduli space, (2) include any brane deformation that has vanishing (7.12) at some point in the closed string moduli space.

These remaining deformations are not necessarily massless at higher orders in the complex structure deformations, or at higher ǫ order when expanding Vol(Σ_ǫs). This induces a five-brane superpotentialW which can be computed using the blow-up proposal as we will show for a number of examples in section 8.⁵

4This can be seen readily by varying (7.12) with respect tos.

5The critical locus ofW will either set the VEVtback to zero promotingsto a unobstructed deformation

Brane Deformations II: Analytic Families of Holomorphic Curves

Let us now present the standard account on deformations of holomorphic curves [259]. The basic question in this context is, as in the case of complex structure deformations, whether a given infinitesimal deformation can be integrated. Mathematically, finite deformations are described by the existence of an analytic family of compact submanifolds, in our context of curves. An analytic family of curves is a fiber bundle over a complex base or parameter manifoldM with fibers of holomorphic curves Σ_u inZ₃ over each pointu∈M.

Given a single curve Σ in Z₃ one can ask the reverse question, namely under which condi-tions does an analytic family of curves exist? The answer to this question was formulated by Kodaira [259]. In general an analytic family of holomorphic curves⁶ exists if the obstructions ψ, that are elements inH¹(Σ, N_Z3Σ), vanish at every orderm, which is of course trivially the case ifH¹(Σ, NZ3Σ) = 0. Thenuare coordinates of pointsuinM and a basis of holomorphic sections in H⁰(Σ_u, N_Z3Σ_u) is given by the tangent space of T M_u via the isomorphism⁷

ϕ^z_∗: ∂

∂u^a 7−→ ∂ϕⁱ(zⁱ;u)

∂u^a (7.15)

at every point u inM. Here, ϕⁱ,i= 1,2, are local normal coordinates to Σ_u, cf. eqn. (7.16).

In other words, in this case every deformation H⁰(Σ, NZ3Σ) corresponds to a finite direction u^a in the complex parameter manifoldM of the analytic family of curves.

This theorem can be understood locally [259] but is somewhat technical. Starting with the single holomorphic curve Σ we introduce patches U_i on Z₃ covering Σ with coordinates yⁱ₁,y₂ⁱ, zⁱ. Then Σ is described as yⁱ₁ =y₂ⁱ = 0 and zⁱ is tangential to Σ. A deformation Σ_u of Σ = Σ₀ is described by finding functions ϕⁱ_l(zⁱ;u), l= 1,2, with the boundary condition ϕⁱ_l(zⁱ; 0) = 0 such that Σ_u reads

y₁ⁱ =ϕⁱ₁(zⁱ;u), y₂ⁱ =ϕⁱ₂(zⁱ;u) (7.16) upon introducing parametersufor convenience chosen in polycylinders||u||< ǫ. Furthermore, the first derivatives _∂u^∂

aϕⁱ_k|u=0 should form a basis s^a of H⁰(Σ, N_Z3Σ). In addition, these functions have to obey specific consistency conditions, that we now discuss. As in the complex structure case, these functions are explicitly constructed as a power series

ϕⁱ(zⁱ;u) =ϕⁱ(0) +ϕⁱ₁(u) +ϕⁱ₂(u) +. . . , kuk< ǫ , (7.17) where we suppress the dependence onzⁱ and further denote a homogeneous polynomial in u of degreenby ϕⁱ_n(u). The first order deformation is defined as

ϕⁱ₁(u) =X

a

u^as⁽ⁱ⁾_a (zⁱ), (7.18)

or will leave a discrete set of holomorphic curves.

6Kodaira considered the general case of a compact complex submanifold in an arbitrary complex manifold.

7This map is called the infinitesimal displacement of Σualong _∂u^∂a [259].

7.1. FIVE-BRANEN = 1 EFFECTIVE DYNAMICS 177 where a = 1, . . . , h⁰(NΣ) in the basis s_a of H⁰(Σ, N_Z3Σ) so that (7.15) is obviously an isomorphism.

Then them^thobstructions ψ^ik(z^k;u) are homogeneous polynomials of orderm+ 1 taking values in ˇCech 1-cocycles on the intersection U_i∩U_j ∩U_k of the open covering of Σ with coefficients in N_Z3Σ. This means that the collection of local section ψ^ik(z^k;u) defines an element in the ˇCech-cohomology H¹(Σ, NZ3Σ). It expresses the possible mismatch in gluing together theϕⁱ(zⁱ;u) defined on open patches U_i∩Σ_u consistently to a global section on Σ_u at orderm+ 1 inu. In other words if the obstruction atm^th order is trivial and we consider (7.16) onUi and U_k,

U_i : yⁱ_l =ϕⁱ_l(zⁱ;u), U_k: y_l^k=ϕ^k_l(z^k;u), (l= 1,2), (7.19) then there exist functionsf^ik and g^ik withy_lⁱ=f_l^ik(y^k, z^k), zⁱ =g^ik(y^k, z^k) so that

yⁱ = ϕⁱ(zⁱ;u) =ϕⁱ(g^ik(y^k, z^k);u) =ϕⁱ(g^ik(ϕ^k(z^k;u), z^k);u) yⁱ = f^ik(y^k, z^k) =f^ik(ϕ^k(z^k;u), z^k)

⇒ ϕⁱ(g^ik(ϕ^k(z^k;u), z^k);u) =f^ik(ϕ^k(z^k;u), z^k) (7.20) holds at order m+ 1 in u. Here we suppressed the index l labeling the coordinates y₁ⁱ, y₂ⁱ. Thenψ^ik(z;u) is the homogeneous polynomial of degree m+ 1

ψ^ik(z^k;u) :=

ϕⁱ(g^ik(ϕ^k(z^k;u), z^k);u)−f^ik(ϕ^k(z^k;u), z^k)

m+1 (7.21)

where we expandϕⁱ,ϕ^k to order m inu. It can be shown to have the transformation ψ^ik(z^k;u) =ψ^ij(z^j;u) +F^ij(z^j)·ψ^jk(z^k;u), (7.22) where F^ij(z^j) is the complex 2×2 transition matrices on N_Z3Σ at a point z^j in Σ_u that acts on the two-component vector ψ^jk ≡ (ψ₁^jk, ψ₂^jk). This equation identifies the ψ^ik as elements inH¹(N_Z3Σ) which can be identified by the Dolbeault theorem with ¯∂-closed (0, 1)-forms taking values in N_Z3Σ, H¹(N_Z3Σ) = H^(0,1)_¯

∂ (N Z₃). Assuming that all ψ^ik are trivial in cohomology and further proving the convergence of the power series (7.17), the analytic family of holomorphic curves is constructed.

In principle one can calculate the obstructions ψ^ik according to this construction at any orderm. However, the obstructions are precisely encoded in the superpotential of the effective theory of a five-brane on Σ. This superpotential is in general a complicated function of both the brane and bulk deformations. Thus, determining the superpotential is equivalent to solving the deformation theory of a pair given by the curve Σ and the Calabi-Yau threefold Z3 containing it. It is this physical ansatz that we will take in the following.