# From M-theory higher curvature terms to α' corrections in F-theory

\entrymodifiers=+!!¡0pt,¿

MPP-2013-299

IPhT-t13/266

From M-theory higher curvature terms

[.2cm] to corrections in F-theory

Thomas W. Grimm, Jan Keitel, Raffaele Savelli,
and Matthias Weissenbacher ^{1}

Max Planck Institute for Physics,

Föhringer Ring 6, 80805 Munich, Germany

Institut de Physique Théorique, CEA Saclay,

Orme de Merisiers, F-91191 Gif-surYvette, France

ABSTRACT

We perform a Kaluza-Klein reduction of eleven-dimensional supergravity on a Calabi-Yau fourfold including terms quartic and cubic in the Riemann curvature and determine the induced corrections to the three-dimensional effective action. We focus on the effective Einstein-Hilbert term and the kinetic terms for vectors. Dualizing the vectors into scalars, we derive the resulting Kähler potential and complex coordinates. The classical expressions for the Kähler coordinates are non-trivially modified, while the functional form of the Kähler potential is shown to be uncorrected. For elliptically fibered Calabi-Yau fourfolds the corrections can be uplifted to a four-dimensional F-theory compactification. We argue that also the four-dimensional Kähler coordinates receive non-trivial corrections. We find a simple expression for the induced corrections for different Abelian and non-Abelian seven-brane configurations by scanning over many Calabi-Yau fourfolds with resolved singularities. The interpretation of this expression leads us to conjecture that the higher-curvature corrections correspond to corrections that arise from open strings at the self-intersection of seven-branes.

###### Contents

## 1 Introduction

Compactifications of string theory to four-dimensional (4d) minimally supersymmetric theories are of particular phenomenological interest. The leading effective actions are often derived by dimensionally reducing the ten-dimensional supergravity actions with localized brane sources. Imprints of string theory arise from corrections that are at higher order in , which corresponds to the square of the string length. In 4d compactifications with minimal supersymmetry such corrections are in general difficult to compute. Nevertheless, they are crucial in determining the couplings and vacua of the effective theory and addressing the problem of moduli stabilization. A phenomenologically promising scenario for which the effective action has been studied intensively are Type IIB string compactifications with space-time filling seven-branes hosting non-Abelian gauge groups [1, 2, 3]. F-theory provides a formulation of such Type IIB string backgrounds at varying string coupling [4]. It captures string coupling dependent corrections in the geometry of an elliptically fibered higher-dimensional manifold. F-theory compactified on an elliptically fibered Calabi-Yau fourfold yields a 4d effective theory with supersymmetry. In this work we study certain corrections to the classical F-theory effective action determined in [5].

In order to study the general effective actions arising in F-theory compactifications one has to take a detour via M-theory [4, 2, 5]. While there is no fundamental twelve-dimensional low-energy effective action of F-theory, M-theory can be accessed through its long wave-length limit provided by eleven-dimensional (11d) supergravity. M-theory on a Calabi-Yau fourfold yields a three-dimensional (3d) effective theory with supersymmetry [6, 7, 8]. This theory can be lifted to four dimensions if the fourfold is elliptically fibered. Starting with the two-derivative 11d supergravity action, one derives the classical 4d F-theory effective action using this duality.

The aim of this work is to determine corrections to the classical 4d F-theory effective action using known higher curvature corrections to the 11d supergravity action. Indeed, following the M-theory to F-theory duality, one finds that terms that are of higher order in , the fundamental length scale of M-theory, can map to corrections in F-theory. One is thus able to derive corrections to the internal volume appearing in the 4d, Kähler potential of F-theory [9]. More precisely, one includes the eight derivative terms quartic in the Riemann tensor in a classical Kaluza-Klein reduction on a Calabi-Yau fourfold. The 11d -terms were determined and investigated in [10, 11, 12, 13, 14, 15] and were already argued to induce a correction to the 3d Einstein-Hilbert term on a Calabi-Yau fourfold in [7, 8]. It is important to stress that while determining the 3d Einstein-Hilbert action allows to infer corrections to the Kähler potential as argued in [9], the derivation of the Kähler coordinates requires a more extensive reduction.

As we show in this work, the Kähler coordinates can be determined by dimensionally reducing the recently found higher-derivative corrections quadratic in the M-theory four-form field strength and cubic in the Riemann tensor [16]. In the 3d, effective action these terms yield a modification of the kinetic terms of the vector fields that readily translates to a correction to the 3d Kähler coordinates. Both the corrections to the Kähler potential and the Kähler coordinates depend on the third Chern class of the internal manifold. Remarkably, we find that the functional dependence of the Kähler potential on the modified Kähler coordinates is not modified in comparison to the classical result. In particular, the Kähler potential still satisfies a strict no-scale condition as is already the case for the classical reduction without higher curvature terms. Let us stress, however, that in [17] it was found that a general M-theory reduction on a Calabi-Yau fourfold also includes a warp factor and we will neglect warping effects in this work.

Having derived the 3d, Kähler potential and Kähler coordinates, we proceed by discussing the F-theory limit to four space-time dimensions. In order to do that, one has to restrict to an elliptically fibered Calabi-Yau fourfold and separate the volume of the elliptic fiber. This volume modulus maps to the radius of a circle used in reducing a 4d, theory to three dimensions. Identifying the correct scaling limit, one finds that also the 4d Kähler coordinates and Kähler potential admit corrections that are now -dependent. As in three dimensions, however, the functional dependence of the 4d Kähler potential on the corrected coordinates is identical to the one found for the classical reduction. This implies the standard 4d no-scale condition.

It is an interesting question to interpret the corrections to the Kähler coordinates and Kähler potential in Type IIB string theory. In order to approach this, we argue for a simple formula that allows to express the third Chern class corrections in terms of seven-brane locations in the base of the elliptic fibration. While we do not have a general derivation of this formula, we are able to successfully test its validity for numerous seven-brane configurations with Abelian and non-Abelian gauge groups. In order to give an open string interpretation we then take the Type IIB weak string coupling limit [18, 19]. We argue that the identified F-theory corrections depend crucially on the topological properties of the self-intersection curve of the involved Abelian and non-Abelian D7-branes. A simple counting of powers of the string coupling suggests that the correction to the Kähler coordinates, identified as gauge coupling functions of D7-branes, arises at string one-loop level. Different corrections to F-theory effective actions and their weak coupling interpretations have been found in [20, 21].

The paper is organized as follows. In section 2 we perform a dimensional reduction of the recently found higher curvature terms [16] to determine the kinetic terms of the vectors in the 3d, effective action. This result allows us to derive the Kähler coordinates for the Kähler potential found in [9] and comment on the no-scale structure of the effective theory. The F-theory limit to four dimensions is carried out in section 3 for elliptically fibered Calabi-Yau fourfolds. Implementing the limit, we then derive the -corrected 4d, Kähler potential and Kähler coordinates. Finally, in section 4, we argue for a simple universal formula that allows to evaluate the corrections in F-theory using the seven-brane data. In the weak string coupling limit we find that the corrections seem to arise from open strings localized at the self-intersections of D7-branes. We test these statements for various Abelian and non-Abelian seven-brane configurations. In appendix A we summarize our conventions and give various useful identifies. A simple analytic computation of the third Chern class for setups is presented in appendix B.

## 2 Higher-derivative corrections in M-theory on Calabi-Yau fourfolds

In this section we derive the three-dimensional effective action of eleven-dimensional supergravity including a known set of eight-derivative corrections. More precisely, we dimensionally reduce higher curvature terms with four Riemann tensors found in [10, 11, 12, 13, 14, 15] and terms quadratic in the M-theory field strength and cubic in the Riemann tensors introduced in [16]. In subsection 2.1 we collect the relevant terms of the 11d supergravity action and recall the general form of a 3d, supergravity theory. Both are connected by a dimensional reduction that we carry out in subsection 2.2. Finally, in subsection 2.3 we determine the 3d, coordinates and the Kähler potential. We also comment on the no-scale properties of the resulting theory.

### 2.1 11d higher-curvature corrections and 3d supergravity

In order to set the stage for performing the dimensional reduction, let us first collect the relevant terms of the 11d supergravity theory. In the following we will focus only on the purely bosonic parts of the various supergravity theories. The two-derivative action of 11d supergravity [22] together with the relevant eight-derivative terms found in [10, 11, 12, 13, 14, 15, 16] reads

(2.1) |

where we have defined^{2}

(2.2) | |||||

(2.3) | |||||

(2.4) |

The constant is given by

(2.5) |

Since the explicit form of the higher-derivative corrections is rather lengthy, we summarize them in detail in appendix A.3. In particular, is defined in (A.17), in (A.3), in (A.19), in (A.3), and in (A.21).

In order to derive the 3d effective action, the terms summarized in (2.1) have to be reduced on a background of the form , where is the non-compact macroscopic space-time and is the internal compact space. Supersymmetric solutions including background fluxes for and certain higher-derivative corrections have been found in [17]. In general, these solutions include a warp factor multiplying the metric of that depends on the internal coordinates.

For a supersymmetric background, the resulting theory
admits four supercharges and can hence be matched with the
canonical form of the 3d, action. In general,
this action propagates a number of complex scalars in
chiral multiplets coupled to non-dynamical vectors.
In the following, we will only consider the ungauged case and can hence
start with a 3d theory with only gravity and chiral multiplets.^{3}

(2.6) |

Supersymmetry ensures that the metric is actually encoded in a real Kähler potential as . Even in the absence of gaugings, a scalar potential can arise from a holomorphic superpotential and takes the form

(2.7) |

where is the inverse of and is the Kähler covariant derivative.

In order to match the action (2.6) with the dimensional reduction of M-theory, it turns out to be useful to dualize some of the scalar multiplets into 3d vector multiplets. Therefore, we decompose and split the index as . If the real scalars have shift symmetries, it is possible to dualize them to vectors . The real parts of are redefined to real scalars that naturally combine with the vectors into the bosonic components of vector multiplets. The dual 3d, action reads

The new couplings can now be derived from a real function known as the kinetic potential according to

(2.9) |

The Kähler potential and kinetic potential as well as the fields and are related by a Legendre transform. Explicitly, the relations are given by

(2.10) |

In reverse, one finds that

(2.11) |

In the following we aim to read off the Kähler potential and metric from the dimensional reduction of the 11d action (2.1).

Neglecting higher-derivative terms, the Kähler potential arising from a reduction on a Calabi-Yau fourfold was derived in [6, 7]. For the Kähler structure moduli it was found to be

(2.12) |

where is the classical volume of , and is the Kähler form on . Note that the quantity in the logarithm, i.e. the volume , appears in front of the 3d Einstein-Hilbert term after dimensional reduction. In order to move to the standard Einstein frame, it has to be removed by a Weyl rescaling of the metric . In fact, due to the Weyl rescaling also the scalar potential is rescaled and by comparison with the factor in (2.7) one can heuristically infer (2.12).

Including the higher-derivative terms present in given by (2.2), one expects a correction to the classical Kähler potential (2.12). Neglecting warping, the precise form of the correction to was derived in [9]. Indeed, the reduction of gives the 3d Einstein-Hilbert term

(2.13) |

with the quantum corrected volume

(2.14) |

Applying the same strategy as above, one can then infer the corrected Kähler potential to be

(2.15) |

Here we have used the conventions^{4}

(2.16) |

It is important to emphasize that this derivation does not suffice to fix the 3d Kähler coordinates . This can be achieved by reading off the metric in front of the dynamical terms of the vectors in (2.1). More precisely, we perform the reduction of given in (2.3) on a Calabi-Yau fourfold , once again neglecting warping. The kinetic terms of the vectors arise as a subset of the terms induced by reduction of and take the form

(2.17) |

This chooses the frame where the vectors are dynamical and one can compare them to the canonical form of the action (2.1). To do this, one first has to Weyl rescale the action to get rid of the quantum volume in front of the Einstein-Hilbert term (2.13). In the process, one introduces a power of in front of the kinetic term of the vectors and one finds

(2.18) |

After comparing to (2.1) and using (2.16), one infers that . In order to find a consistent reduction, has to be compatible with as given in (2.15) and (2.14). This fixes the 3d Kähler coordinates as we discuss in more detail in subsection 2.3.

### 2.2 Dimensional reduction of higher-curvature terms

In this subsection we present the reduction of (2.3) on a Calabi-Yau fourfold to three dimensions with focus on the kinetic terms of the vectors. The variations of the Calabi-Yau metric split into Kähler structure and complex structure deformations. For simplicity we will consider geometries with in the following. Furthermore, we will not consider the complex structure deformations in the remainder of this work. In fact, one can check that the corrections analyzed in the following are indeed independent of the complex structure.

The Kähler structure deformations parametrize the variations of the Kähler form by expanding

(2.19) |

where is a basis of harmonic -forms on , and correspond to real scalar fields in the 3d effective theory. Let us define the intersection numbers

(2.20) |

which allow us to abbreviate

(2.21) |

These quantities can be expressed as integrals including powers of using (2.19). Furthermore, we define the topological quantities and their -contraction as

(2.22) |

where is the third Chern class of the tangent bundle of . Note that contains six internal derivatives.

In our reduction ansatz, the M-theory three-form is expanded into the harmonic -forms introduced in (2.19) with vector fields as coefficients. Hence, the field strength of takes the form

(2.23) |

where the are the field strengths of the 3d vector fields. Here we also introduced explicit real coordinates on and complex coordinates on .

Using (2.23), one performs the dimensional reduction of the classical part of (2.3), see [6, 7], and finds

(2.24) |

To rewrite expressions in terms of the quantities introduced in (2.21) and (2.22), one makes use of identities valid for the Hodge star evaluated on certain internal harmonic forms. The most important identities of this form are

(2.25) |

We will further discuss these equations in appendix A.5 and derive additional relations that straightforwardly follow from (2.25). These identities will be repeatedly used in the following. For example, applying the first equation in (2.25) one finds

(2.26) |

Let us now perform the dimensional reduction of the higher derivative
corrections in (2.3) by applying the same logic as for the classical
part discussed above.
This requires us to use (2.23), (2.25) and related identities summarized in appendix A.5.
We begin by discussing the reduction of and proceed with the reduction of .
We consider only terms that have two external derivatives and depend on the gauge fields .
Hence, is of the form (2.23) and has two external and two internal indices. All other remaining summed indices
are purely internal. The reduction of then yields^{5}

(2.27) |

Here, the symbols schematically represent all appearing permutations of internal indices dictated by the index structure of the tensor. Each of the terms in (2.27) is of the general form

(2.28) |

None of the terms in (2.27) arise from top forms containing the third Chern class , which can be seen by analyzing their index structure.

Similarly, one reduces and finds the following terms contributing to the kinetic terms of the vectors

(2.29) |

The term in the reductions of and cancels and only eight terms originating from the reduction of remain. They are of general type (2.28) and their explicit form is given in appendix A.3 in (A.24). The various index summations in (A.24) can be recast in terms of the following linear combination of top forms on the internal space, each containing the third Chern class and two -forms :

One uses the identities (2.25) and (A.27) - (A.31) to express the result in terms of the basic building blocks (2.21) and (2.22). Applying the identities (A.27) and (A.29), one finds

(2.31) |

In the next step, we relate this result to the canonical form of the 3d, action (2.1) as already outlined in subsection 2.1. Taking into account the contribution arising from the reduction of the classical kinetic term (2.24) and performing the Weyl rescaling with the quantum corrected volume (2.14), one can read off the couplings that arise from the reduction. We find an overall factor of for the contributions from (2.2). This is the same factor that appeared in the corrected volume given in (2.14). Due to the Weyl rescaling, the volume correction also contributes to in linear order in . Note that we will neglect quadratic corrections in to the Kähler metric in all of our computations. These corrections would contain six Riemann tensors of the internal space and would thus have twelve derivatives. Performing all outlined steps, we finally arrive at the result

(2.32) |

with the classical coupling function

(2.33) |

This concludes the dimensional reduction of the action given in (2.3). In the next step, we will use this result to infer the 3d, Kähler coordinates. Let us stress that in order to derive the fully reduced action one would also have to consider the kinetic terms of the by dimensional reduction of given in (2.2). However, as we will see next, the result (2.32) together with 3d, supersymmetry suffices to fix the Kähler coordinates.

### 2.3 Determining the 3d, coordinates and Kähler potential

As already noted above, the reduction of (2.2) performed in [9] to find the Kähler potential (2.15) does not suffice to fix the Kähler coordinates in the 3d, action (2.6). The Kähler coordinates can however be determined by using the relation of the Kähler potential given in (2.15) with the couplings found in (2.32). As a first step, one computes the general form of arising from a Kähler potential by Legendre transform. If the Kähler metric separates w.r.t. the coordinates , that is all mixed derivatives of vanish, one can compute using the identity

(2.34) |

In our reduction with , the separation into indeed takes place. Hence, one can compare the expression (2.34) to in order to read off .

The classical Kähler coordinates, which correspond to six-cycle volumes of the Calabi-Yau fourfold , are given by

(2.35) |

Performing the Legendre transform and using (2.34), one finds that the classical Kähler coordinates (2.35) together with the Kähler potential (2.15) do not suffice to arrive at the metric given in (2.32). Indeed, it is necessary to correct the Kähler coordinates as

(2.36) |

to achieve the match . This non-trivial field redefinition might also be interpreted as a quantum correction to the six-cycle volumes. We stress that the last term in (2.36) is constant, since are topological quantities, and cannot be inferred by using (2.34). In fact, this term could be removed by a trivial holomorphic Kähler transformation. The reason for including this shift will be explained below.

Having determined both the Kähler potential in (2.15) and the Kähler coordinates in (2.36), one can now show that a 3d no-scale condition holds. More precisely, one derives that

(2.37) |

This implies that the term in the scalar potential (2.7) will cancel precisely if is independent of .

The coordinates are the propagating complex scalars in the 3d, action (2.6). If one changes to different propagating degrees of freedom by dualizing and performing the Legendre transform for as described in subsection 2.1, one arrives at propagating real scalars in the dual version of the 3d action (2.1). It is convenient to perform all computations in this frame, since the Kähler potential , the Kähler form , and the geometric quantities (2.21) and (2.22) depend explicitly on the fields . These are real scalars in the 3d action and correspond to 2-cycle volumes of the internal space. By definition of the Legendre transform one has the relation

(2.38) |

To evaluate (2.38) one first needs to compute the partial derivative of the Kähler potential and the Kähler coordinates in (2.36) w.r.t. to the fields . Then one inverts the matrix . We neglect corrections that have more than six derivatives, which means that they are at least quadratic in . This implies that we assume the quantum corrections proportional to to be small compared to the classical contribution. Hence, we can expand the inverse matrix by using the formula for . Using (2.36) and applying the above steps one arrives at

(2.39) |

Furthermore, one can compute

(2.40) |

which is valid up to linear order in . The dual kinetic potential then takes the form

(2.41) |

Note that it is straightforward to evaluate the coordinates given in (2.36) as a function of given in (2.39) as

(2.42) |

Let us close this section with some further remarks. First of all, note that by using the field redefinition (2.39) one finds the same functional dependence of w.r.t. as in the classical reduction without higher curvature terms. This is equally true when evaluating the Kähler potential given in (2.15) as a function of the corrected given in (2.36). Clearly, this implies the no-scale condition (2.37) to linear order in the correction . Secondly, note that the redefinition of in (2.38) does not change if one varies the coefficient of the last term in given in (2.36). The convenient choice made in (2.36) implies that (2.40) and (2.41) do not have irrelevant linear terms of the form .

## 3 F-theory limit and the 4d effective action

In this section we examine the 4d effective theory obtained by taking the F-theory limit of the 3d results found in section 2. As in [9], we use the duality between M-theory and F-theory to lift the -corrections to -corrections of the 4d effective action arising from F-theory compactified on . In subsection 3.1 we formulate the F-theory limit in terms of the corrected Kähler coordinates and discuss the resulting 4d Kähler potential. Next, in subsection 3.2 we derive the quantum corrected expressions for the volume of the internal space and for the 4d Kähler coordinates in terms of two-cycle volumes. Analogously to the 3d case, the considered 4d effective couplings turn out to be identical to the classical ones when expressed in terms of the modified Kähler coordinates. We comment on the consequences of this observation.

### 3.1 F-theory limit and the effective 4d, effective action

To begin with, we require that admits an elliptic fibration over a three-dimensional Kähler base . We allow to accommodate both non-Abelian and gauge groups. A detailed discussion of its geometry will be given in section 4. The structure of the elliptic fibration allows us to split the divisors and Poincaré-dual two-forms , into three types: , , and . The two-form corresponds to the holomorphic zero-section, the two-forms to divisors obtained as elliptic fibrations over divisors of the base with , and the two-forms correspond to both the extra sections, i.e. Abelian factors, and the blow-up divisors, i.e. factors in the Cartan subalgebra of the non-Abelian gauge group. We can thus expand the Kähler form of the Calabi-Yau fourfold as

(3.1) |

where represents the volume of the elliptic fiber. Accordingly, one can also split the and introduced in (2.39) and (2.36) such that

(3.2) |

The field will play a special role in the uplift from three to four dimensions. In fact, one finds that is given by , where is the radius of the circle compactifying the 4d theory to three dimensions.

In the F-theory limit one sends , which translates to sending . Such an operation decompactifies the fourth dimension by sending the radius of the 4d/3d circle in string units to infinity: . Henceforth, all volumes of the base will be expressed in units of . In all 3d effective quantities one has to retain the leading order terms in such a limit. Therefore we introduce a small parameter and express the scaling of the dimensionless fields by writing . As explained in [5, 25], one shows that all scale to zero in the limit of vanishing , whereas . One then infers the scaling behavior of the classical and quantum volume of to be . In the following we use the letter to denote quantities of the base that are finite in the limit .

When compactifying a general 4d, supergravity theory on a circle, one can match the original 4d Kähler potential and gauge coupling functions with the 3d Kähler potential or kinetic potential . Since we have found that the dependence of and on the modified coordinates and is the same as in the classical case, we can perform the limit by simply following [5]. Firstly, we recall that the fields remain complex scalars in four dimensions, while the should be dualized already in three dimensions into vector multiplets with and and then uplifted to four dimensions. In fact, are parts of the 4d metric, while form the Cartan gauge vectors of the 4d gauge group. In this mixed frame one finds a kinetic potential , which can be computed for example by Legendre dualization of starting from (2.41). This kinetic potential has to be matched with the one arising in a dimensional reduction from four to three dimensions, which has the form

(3.3) |

where the are the Wilson line scalars from 4d Cartan vectors on a circle,
and is the holomorphic 4d gauge coupling function.
As a next step, one can implement the F-theory limit by
identifying the 3d fields with appropriate 4d fields.
In addition to and identifying the , we also set ^{6}

(3.4) |

which are the only and that are finite and non-zero in the limit . This is the same limit as taken in [5], but with the modified coordinates and .

It is now straightforward to determine , since in the modified coordinates this is just the classical analysis. First of all, one has to evaluate the intersection numbers for an elliptic fibration. One finds the always non-vanishing coupling , where we have introduced the base intersection numbers

(3.5) |

Second of all, one can split the kinetic potential (2.41) and coordinates (2.42) for an elliptic fibration. The terms of leading order in are given by

(3.6) | |||||

(3.7) |

where we have replaced the with by means of (3.4). Performing the Legendre transform in order to express everything in terms of and comparing the result with (3.3) setting one finds

(3.8) |

where one has to solve for and insert the result into . Analogously to the 3d case, one can compute

(3.9) |

In this case we also choose the constant shift in (3.14) in order to avoid irrelevant linear terms of the form in the kinetic potential.

The result (3.8) agrees with the classical result and hence, as in three dimensions, the functional dependence of on is not modified by the corrections. In particular one can trivially check that the no-scale condition

(3.10) |

is satisfied by this Kähler potential and Kähler coordinates. It should be stressed that the modifications arise when expressing and in terms of the finite two-cycle volumes as we discuss in detail in subsection 3.2.

Before closing this subsection we note that the gauge coupling function of the 4d gauge group can equally be determined by comparing (3.3) with the M-theory result (2.41). Clearly, one also just finds the classical result when working in the coordinates . More precisely, if the seven-brane supporting the gauge theory wraps the divisor dual to in , the gauge coupling is proportional to . As we will see in the next subsection, also this result differs from the classical expression when written in terms the two-cycle volumes of .

### 3.2 Volume dependence of the 4d, coordinates and Kähler potential

In this subsection we express the 4d, coordinates and Kähler potential given in (3.8) in terms of finite two-cycle volumes in the base . In these coordinates the corrections will reappear and we can comment on their structure.

To begin with, we introduce some additional notation. The base Kähler form is denoted by . The classical volume of the base and the volume dependent matrix are defined as

(3.11) |

where are the triple intersection numbers of defined in (3.5). All corrections to the 4d theory will be expressed in terms of the fundamental quantity

(3.12) |

Since the are inherited from the base there always exists a curve such that the middle equality in (3.12) is satisfied. An explicit expression for is derived in section 4 starting from for numerous singular configurations with extra sections. Let us note that we have defined in order to more easily distinguish and .

We now can relate the two-cycle volumes of to the two-cycle volumes of . Since both and scale with as discussed above, one is led to set

(3.13) |

This is the classical relation between the different two-cycle volumes.^{7}

(3.14) | |||||

(3.15) |

The only non-trivial step in this computation is to relate the inverse to the inverse of given in (3.11). We will discuss this in more detail momentarily. Before doing so, let us introduce the quantum base volume by setting

(3.16) |

This equation can be viewed as an extension of the relation between the classical volumes of and to a quantum shifted and . Inserting the identification (3.13) one finds