Fibrations of Elliptic Curves in F-Theory.

F-theory compactifications are by definition compactifications of the type IIB string with non-zero, and in general non-constant string coupling – they are thus intrinsically non-perturbative. F-theory may also seen as a construction to geometrize (and thereby making manifest) certain features pertaining to the S-duality of the type IIB string.

Let us first recapitulate the most important massless bosonic fields of the type IIB string. From the NS-NS sector, we have the graviton g_μν, the antisymmetric 2-form field B as well as the dilaton φ; the latter, when exponentiated, serves as the coupling constant of the theory. Moreover, from the R-R sector we have the p-form tensor fields C^(p) with p = 0,2,4. It is also convenient to include the magnetic duals of these fields, B⁽⁶⁾, C⁽⁶⁾ and C⁽⁸⁾ (C⁽⁴⁾ has self-dual field strength). It is useful to combine the dilaton with the axion into one complex field:

τ_IIB ≡ C⁽⁰⁾ + ie^-φ —– (1)

The S-duality then acts via projective SL(2, Z) transformations in the canonical manner:

τ_IIB → (aτ_IIB + b)/(cτ_IIB + d) with a, b, c, d ∈ Z and ad – bc = 1

Furthermore, it acts via simple matrix multiplication on the other fields if these are grouped into doublets (^B(2)_C⁽²⁾), (^B(6)_C⁽⁴⁾), while C⁽⁴⁾ stays invariant.

The simplest F-theory compactifications are the highest dimensional ones, and simplest of all is the compactification of the type IIB string on the 2-sphere, P¹. However, as the first Chern class does not vanish: C₁(P¹) = – 2, this by itself cannot be a good, supersymmetry preserving background. The remedy is to add extra 7-branes to the theory, which sit at arbitrary points z_i on the P¹, and otherwise fill the 7+1 non-compact space-time dimensions. If this is done in the right way, C₁(P¹) is cancelled, thereby providing a consistent background.

Encircling the location of a 7-brane in the z-plane leads to a jump of the perceived type IIB string coupling, τ_IIB →τ_IIB  +1.

To explain how this works, consider first a single D7-brane located at an arbitrary given point z₀ on the P¹. A D7-brane carries by definition one unit of D7-brane charge, since it is a unit source of C⁽⁸⁾. This means that is it magnetically charged with respect to the dual field C⁽⁰⁾, which enters in the complexified type IIB coupling in (1). As a consequence, encircling the plane location z₀ will induce a non-trivial monodromy, that is, a jump on the coupling. But this then implies that in the neighborhood of the D7-brane, we must have a non-constant string coupling of the form: τ_IIB(z) = 1/2πiIn[z – z₀]; we thus indeed have a truly non-perturbative situation.

In view of the SL(2, Z) action on the string coupling (1), it is natural to interpret it as a modular parameter of a two-torus, T², and this is what then gives a geometrical meaning to the S-duality group. This modular parameter τ_IIB = τ_IIB(Z) is not constant over the P¹ compactification manifold, the shape of the T² will accordingly vary along P¹. The relevant geometrical object will therefore not be the direct product manifold T² x P¹, but rather a fibration of T² over P¹

Fibration of an elliptic curve over P¹, which in total makes a K3 surface.

The logarithmic behavior of τ_IIB(z) in the vicinity of a 7-brane means that the T² fiber is singular at the brane location. It is known from mathematics that each of such singular fibers contributes 1/12 to the first Chern class. Therefore we need to put 24 of them in order to have a consistent type IIB background with C₁ = 0. The mathematical data: “T² fibered over P¹ with 24 singular fibers” is now exactly what characterizes the K3 surface; indeed it is the only complex two-dimensional manifold with vanishing first Chern class (apart from T⁴).

The K3 manifold that arises in this context is so far just a formal construct, introduced to encode of the behavior of the string coupling in the presence of 7-branes in an elegant and useful way. One may speculate about a possible more concrete physical significance, such as a compactification manifold of a yet unknown 12 dimensional “F-theory”. The existence of such a theory is still unclear, but all we need the K3 for is to use its intriguing geometric properties for computing physical quantities (the quartic gauge threshold couplings, ultimately).

In order to do explicit computations, we first of all need a concrete representation of the K3 surface. Since the families of K3’s in question are elliptically fibered, the natural starting point is the two-torus T². It can be represented in the well-known “Weierstraβ” form:

W_T² = y² + x³ + xf + g = 0 —– (2)

which in turn is invariantly characterized by the J-function:

J = 4(24f)³/(4f³ + 27g²) —– (3)

An elliptically fibered K3 surface can be made out of (2) by letting f → f₈(z) and g → g₁₂(z) become polynomials in the P¹ coordinate z, of the indicated orders. The locations z_i of the 7-branes, which correspond to the locations of the singular fibers where J(τ_IIB(z_i)) → ∞, are then precisely where the discriminant

∆(z) ≡ 4f₈³(z) + 27g₁₂²(z)

=: ∏_i=1²⁴(z –  z_i) vanishes.

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