Basics of F-theory Constructions

F-theory provides a geometrization ofN = 1 Type IIB backgrounds with backreacted seven- branes and a holomorphically varying axio-dilaton [72]

τ = C0+

i gs

= C0+ ie−φ (4.27)

Due to the genuine description of non-perturbative effects like (p, q)-strings and string junc- tions rich gauge dynamics can be obtained allowing even for exceptional groups Ek for

k > 6 [203, 204], that have been accessible before only in the heterotic theory.

Starting with the simplest setup of a backreacted D7-brane in flat space it is the basic question of F-theory to find a compact geometry, which is a solution to the equations of motion of Type IIB string theory, in particular the Einstein equations, and that extends the local solution of a single D7-brane. This compact solution is found in [205] in the context of stringy cosmic strings and applied to D7-branes in the seminal work [72]. The D7-brane is a magnetic source of the axio-dilaton τ through the R–R-form C0 and a source of gravity.

Consequently a solution to the Type IIB effective action is determined by a solution of τ and the metric g, which for symmetry reasons is of the form of a direct product R1,7× B1 where

the space B1 transverse to the D7-brane worldvolume is determined in the following. The

solution of τ is given in complex coordinates on B1 by

τ (z)∼ 1

2πilog(z) (4.28)

which is consistent with the monodromy C0 7→ C0 + 1 due to the C0-charge of the D7-

the supersymmetries are preserved [72]. The metric g is found to describe a conical space centered at z = 0 with a deficit angle of π/6 [72, 205]. It is crucial to note that the SL(2, Z)- symmetry of Type IIB acting on τ renders the energy of the solution (4.28) finite15. The SL(2, Z)-action in particular implies, that τ (z) is not a well-defined function on B1, but a

section of an SL(2, Z)-bundle over B1. More invariantly, τ can be described by the modular

parameter of an elliptic curve _{E, i.e. a two-torus, that is fibered holomorphically over B}1.

To obtain a space B1 of finite volume one considers a multicenter solution of 24 D7-branes

yielding a deficit angle of 4π, i.e. B1 curls itself up to form the compact space of S2 ∼= P1.

Physically this is consistent since the metric is well-defined around 1/z and the net D7-charge on the S2 is zero since a loop encircling all 24 D7-branes is contractible to a point yielding a trivial monodromy for C0. Furthermore, this already indicates that we are no more allowed to

think of 24 D7-branes since the total monodromy and charge are zero. The resolution of this paradox is again the SL(2, Z)-invariance of the solution that allows for more general seven- branes, denoted (p, q) 7-branes16_{, having a different monodromy and charge. In particular}

a seven-brane is only specified up to its conjugacy class under SL(2, Z). We note that τ (z) varies holomorphically over P1 where gs is not necessarily small and diverges, τ → i∞, at the

location of a seven-brane. This and the presence of non-perturbative seven-branes indicates that this eight-dimensional Type IIB vacuum is non-perturbative. It is denoted as an F-theory compactification to eight dimensions.

Geometrically the F-theory setup is described by a fibration of an elliptic curve over B1 = P1 where the generic elliptic fiber E degenerates at the 24 loci of the seven-branes.

However, the total space of the fibration remains smooth. A smooth complex surface which is an elliptic fibration over P1 is a two-dimensional Calabi-Yau manifold, which is K3. This can be seen as follows. First we note that every two-torus T2 _{is algebraic. Indeed by means of the}

Weierstrass p-function, every point u on the lattice quotient T2= C/L is mapped bijectively to the projective plane curve _{E = {y}2 = 4x3_{− g}2x− g3} via u 7→ (p(u), p′(u)) ≡ (x, y), due

to the algebraic differential equation obeyed by p [206]. In a fibration, the curve_Ez depends

on the point z in P1. Then the discriminant ∆ = g₂3− 27g2

3 of Ez has to vanish to first order

at 24 points, which fixes the degree of g2, g3 as polynomials in the local coordinate z on

P1 _{to be eight and twelve so that the total space of the fibration, denoted as X2, obeys the} Calabi-Yau condition. Thus, we see that an F-theory vacuum in eight dimensions is in one- to-one correspondence with an elliptic K3-surface. Analogously, lower-dimensional F-theory vacua are obtained using the adiabatic argument [207] by compactifying on n-dimensional elliptically fibered Calabi-Yau manifolds Xn, where the base B1 is replaced by a complex

n_{−1-dimensional Fano variety Bn−1}[72–74]. This explains the relevance of elliptically fibered Calabi-Yau manifolds as discussed next in section 4.2.2. In all these cases, the relation of the F-theory setup specified by the elliptic Calabi-Yau manifold Xn to the Type IIB physics is

made precise in [75, 76] by identifying the Type IIB manifold as the double cover of the base

15_{The domain of integration is reduced from the complex plane to F, the fundamental domain of the torus.} 16_{A (p, q)7-brane is an object on which a (p, q)-string can end, cf. sections 2.2.1 and 3.1.}

4.2. F-THEORY COMPACTIFICATIONS 79 B_n−1 branched over the divisor wrapped by the O7-plane.

We conclude this general discussion by noting the M-theory description of F-theory, see [29] for a detailed derivation. Using the adiabatic argument [207] for M-theory on an elliptically fibered Calabi-Yau n-fold Xn and for the application of fiberwise T-duality, F-theory is iden-

tified with M-theory on Xnin the limit vol(T2)→ 0, where T2 denotes the class of the generic

elliptic fiber. This M-theory description in particular yields an alternative and independent explanation why F-theory vacua have N = 1 spacetime supersymmetry.