Deformations of special geometry: in search of the topological string

Abstract

Abstract

The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus g [less than or equal to] 3 contributions are constructed explicitly for a general class of effective actions associated with a special-Kähler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identification of a topological string free energy from an effective action is primarily based on conceptual arguments and does not involve any of its more specific properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that non-holomorphic deformations of special geometry can be dealt with consistently.

Published for SISSA by Springer Received: July 17, 2014 Accepted: August 24, 2014 Published: September 18, 2014

G.L. Cardoso,a B. de Witb,c and S. Mahapatrad

aCenter for Mathematical Analysis, Geometry and Dynamical Systems,

Department of Mathematics, Instituto Superior Tcnico, Universidade de Lisboa,

Av. Rovisco Pais, 1049-001 Lisboa, Portugal

bNikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands

cInstitute for Theoretical Physics, Utrecht University,

Leuvenlaan 4, 3584 CE Utrecht, The Netherlands

dPhysics Department, Utkal University,

Bhubaneswar 751 004, IndiaE-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: The topological string captures certain superstring amplitudes which are also encoded in the underlying string e ective action. However, unlike the topological string free energy, the e ective action that comprises higher-order derivative couplings is not dened in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is dened in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the e ective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus g 3 contributions are constructed explicitly for a general class of

e ective actions associated with a special-Kahler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identication of a topological string free energy from an e ective action is primarily based on conceptual arguments and does not involve any of its more specic properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that nonholomorphic deformations of special geometry can be dealt with consistently.

Keywords: Supersymmetry and Duality, Extended Supersymmetry, Topological Strings, Supersymmetric E ective Theories

ArXiv ePrint: 1406.5478

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP09(2014)096

Web End =10.1007/JHEP09(2014)096

Deformations of special geometry: in search of the topological string

JHEP09(2014)096

Contents

1 Introduction 1

2 Real and deformed special geometry 8

3 The generic structure of the Hesse potential 12

4 Non-holomorphic deformations and the anomaly equation 20

5 Evaluating the third-order contributions 27

6 Summary and conclusions 31

A Non-holomorphic deformation of special geometry 33A.1 Theorem 33A.2 Proof 34A.3 Corollary 35

B The symplectic functions H(a)i for a 2 and some other functions that

do not initially appear in H 36 C Transformation rules of !I and !IJ to order 2 38

D Topological free energies for genus g 3 that satisfy the holomorphic

anomaly equation 39

E An application: the FHSV model 41

1 Introduction

As is well known, Lagrangians for N = 2 supersymmetric vector multiplets are encoded in a holomorphic function F (X) whose arguments correspond to the complex scalar elds XI of the vector multiplets. These Lagrangians often play a role as Wilsonian e ective eld theories that describe the physics below a certain mass scale. Homogeneity of the holomorphic function is required whenever the vector multiplets are coupled to supergravity [1]. The physical vector multiplet scalars are then projectively dened in terms of these variables as a result of the local scale and U(1) invariance of the description used in [1]. It is possible that the function depends, in addition, on one or more holomorphic elds, possibly associated with some other chiral multiplets. An example of this is the so-called Weyl multiplet that describes the pure supergravity degrees of freedom [2]. When the function F depends on the Weyl multiplet, then it will also encode a class of higher-derivative

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couplings.1 In case these higher-order derivative couplings are absent, we will denote the function by F (0), which is always holomorphic and homogeneous and encodes an action that is at most quadratic in space-time derivatives. This action will henceforth be referred to as the classical action, and its associated non-linear sigma model parametrizes a special-Kahler space.

The abelian vector elds in these actions are subject to electric/magnetic duality under which the electric eld strengths and their duals transform under symplectic rotations. It is then possible to convert to a di erent duality frame, by regarding half of the rotated eld strengths as the new electric eld strengths and the remaining ones as their duals. The latter are then derivable from a new action. To ensure that the characterization of the new action in terms of a holomorphic function remains preserved, the scalars of the vector multiplets are transformed correspondingly. This amounts to rotating the complex elds XI and the holomorphic derivatives of the underlying function F (X) by the same symplectic rotation as the eld strengths and their dual partners [1, 5]. For reasons that will be described shortly, we shall refer to the array (XI, FJ) as the period vector, where

FJ(X) = @JF (X). The indices I, J label the vector multiplets, and cover the range I, J = 0, 1, . . . , n, so that the period vector has (complex) dimension 2(n + 1). Electric/magnetic duality thus constitutes a group of equivalence transformations that relate two di erent Lagrangians (based on two di erent functions) giving rise to an equivalent set of equations of motion and Bianchi identities. A subgroup of these equivalence transformations may constitute an invariance group, meaning that the Lagrangian and its underlying function F (X) remain unchanged. We stress that the latter two quantities do not transform as a function under these equivalence transformations.

As it turns out one encounters a similar situation when studying Calabi-Yau threefolds. The moduli space of these three-folds is a local product of two submanifolds, describing the metric deformations of the complex structure and of the Kahler class, respectively. The complex structure moduli determine the shape, and the Kahler moduli the size of the Calabi-Yau three-folds. Usually, when referring to the Calabi-Yau moduli space, one refers to either one of these two submanifolds. As it turns out, the corresponding metric deformations are related to the odd and even harmonic forms, respectively, and the number of moduli is thus determined by the topology of the Calabi-Yau three-folds (see e.g. [6]). The latter is specied by the Hodge numbers hp,p which specify the number of independent (p, p) harmonic forms. The odd harmonic forms consist of a (3, 0) holomorphic form , and h2,1 (2, 1)-forms, as well as their conjugate (0, 3)- and (1, 2)-forms. Under an innitesimal change of the complex structure the (3, 0)-form changes into and the (2, 1)-forms, which leads to its periods, in the following way,

XI =

IAI , F (0)I = [contintegraldisplay]BI . (1.1)

Here AI and BI are an integral basis of homology 3-cycles (in a symplectic basis) dual to the three-forms. The index I takes the values I = 0, 1, . . . , h2,1, corresponding to the (3, 0)-

1These actions are all based on a chiral superspace density, but other N = 2 supersymmetric higher-derivative couplings are known to exist (see e.g. [3, 4]). The latter will not be considered in this paper.

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and the (2, 1)-forms and their conjugates. The XI (or alternatively the F (0)I) projectively parametrize the complex-structure deformations, so that the complex dimension of the corresponding moduli space equals h2,1.2

For the periods one can also show (at least in a suitable homology basis) that there exists a holomorphic, homogeneous, function F (0)(X), such that F (0)I = @IF (0)(X).3 The duality transformations on the periods simply arise from symplectic redenitions of the homology basis of the 3-cycles. It is worth pointing out that in this case one is dealing with discrete symplectic transformations, while in the supergravity case the transformations are continuous (unless one has to account for an integral lattice of electric and magnetic charges). This particular geometry with its associated period vectors and symplectic transformations is known as special geometry [7] (for a review, see [8]).

The Calabi-Yau moduli space and the supergravity action describing a Calabi-Yau string compactication are related, because the target space metric associated with the non-linear sigma model contained in the corresponding Wilsonian e ective action of vector multiplets coupled to supergravity, must be equal to the metric of the Calabi-Yau moduli space [9]. This target space is a so-called special-Kahler space, whose Kahler potential is proportional to [1],

K(t, t) / log [bracketleftBigg]

i XI F(0)I

XI F (0)I

[parenrightbig]

[bracketrightBigg]

JHEP09(2014)096

, (1.2)

and F (0)(X) is the holomorphic function that determines the supergravity action quadratic in space-time derivatives. Because F (0)(X) is homogeneous of second degree, this Kahler potential depends only on the special coordinates ti = Xi/X0 and their complex conjugates, where i = 1, . . . , n, so that we are dealing with a special-Kahler space of complex dimension n. In view of the homogeneity, the symplectic rotations acting on the period vector (XI, F (0)I) induce corresponding (non-linear) transformations on the special coordinates ti. Up to a Kahler transformations, the Kahler potential transforms as a function under duality.

Yet another quantity that reects the geometrical features of the Calabi-Yau moduli space is the topological string. Perturbative string theory is dened in terms of maps from Riemann surfaces g to a target space. When the worldsheet theory has (2, 2) supersymmetry and the target space is Ricci at, one may construct a topological version of perturbative string theory by a procedure called twisting [10]. The resulting theory is a cohomological theory, and correlation functions of observables are independent of the word-sheet metric on g. When the target-space is a Calabi-Yau three-fold, the twisted theory is called topological string theory [11]. There exist two versions of topological string theory, called the A- and the B-models. In the A-model, the correlation functions depend only

2For completeness we mention that there exists an analogous construction for the Kahler moduli. While the complex structure moduli are associated with the odd cohomology class, H(3,0) H(2,1) H(1,2) H(0,3), the Kahler moduli are associated with the even class, H(0,0) H(1,1) H(2,2) H(3,3). The corresponding 2 + 2 h1,1 coordinates projectively describe the h1,1 complex Kahler moduli [6].

3The functions F (0) encoding the moduli space geometry of Calabi-Yau three-folds correspond to a restricted class. This paper pertains to functions belonging to a more general class.

3

|X0|2

on the Kahler moduli, while in the B-model they only depend on the complex structure moduli.

The topological string is dened in terms of topological free energies F (g), which are computed from suitable correlators on orientable Riemann surfaces g of genus g. These free energies may be formally combined into one single object F (X, ), the free energy of topological string theory, which has the asymptotic expansion,

F (X, ) =

with playing the role of a formal (complex) expansion parameter. This expression, being a perturbative series in , is expected to receive non-perturbative corrections in [12 14]. Note that we have (tentatively) included F (0)(X) in (1.3), which is the function that

encodes the Calabi-Yau moduli space metric. The functions F (g)(X) are homogeneous

functions of the XI of degree 2(g 1), so that the X0-dependence can be scaled out

and subsequently be absorbed into the expansion parameter . In this way F (t, [prime]) =

(X0)2 F (X, ) with [prime] = /(X0)2, and F (g)(t) = (X0)2g2 F (g)(X). This suggests a role of [prime] as a loop-counting parameter with ( [prime])1 F (0)(t) equal to the classical free energy.

It may seem tempting to identify the expansion (1.3) in this way with the similar expansion of the e ective action in terms of W 2, the square of the lowest component of the Weyl multiplet, thus generating higher-derivative couplings in the action. However, this interpretation is inconsistent with the behaviour one expects for the free energy of the topological string. The reason is that the genus-g free energies should be consistent with dualities induced by symplectic rotations of the periods (XI, F (0)I) of the underlying

Calabi-Yau moduli space. In particular, the F (g)(X) (with g > 0) should transform as functions under these dualities, and the F (g)(t) as sections. Note that this does not apply to F (0), which does not transform as a function under electric/magnetic duality. At this point one concludes that it was premature to include F (0) into the free energy of the topological string, as the genus-g contributions with g > 0 behave as functions under duality, while F (0) does not.

On the other hand, the Wilsonian action encoded by the similar expansion,

F (X, W 2) =

is subject to di erent duality transformations, namely those induced by rotations of the full period vector (XI, FI) rather than of the classical period vector (XI, F (0)I). Consequently the corresponding coordinates XI will transform di erently under duality, so that one must conclude that the XI appearing in the topological string free energy and the XI appearing in the Wilsonian action cannot be identical variables. Hence the coe cient functions F (g) for the Wilsonian action appearing in (1.4) that multiply even powers of the Weyl multiplet are not transforming as functions under duality, unlike those of the topological string. This aspect is most striking when considering duality symmetries such as S- and T-duality. Under these dualities the functions F (g) of the topological string are invariant

4

1

Xg=0 g1 F (g)(X) , (1.3)

JHEP09(2014)096

1

Xg=0(W 2)g F (g)(X) , (1.4)

(possibly up to a scale factor), whereas the analogous coe cient functions of the Wilsonian action (1.4) transform non-linearly and are not invariant. Hence, in spite of the similarity of the expansions, there is no ground for assuming that the coe cient functions F (g) of the

topological string will coincide with the corresponding coe cient functions appearing in the expression (1.4) that encodes the e ective action. This observation was already made in e.g. [15, 16], where its consequences were investigated for dualities that dene symmetries of the model. Nevertheless, we should stress that there must exist a relation between the e ective action and the topological string in view of the fact that the topological string does capture certain contributions to string amplitudes, which must in turn be reected in the e ective action [11, 17].

We thus conclude that one seems to be dealing with two di erent series expansions of the form (1.3) and (1.4), one pertaining to the topological string free energy and another one to the e ective Wilsonian action with a class of higher-derivative coupings. In spite of their qualitatively di erent behaviour with respect to duality they should somehow describe the same physics. To make matters more subtle, it is known that both the topological string and the e ective supergravity action are subject to non-holomorphic modications. Hence it is reasonable to expect that these modications are therefore related as well. However, so far non-holomorphic deformations have not been incorporated in the standard treatment of special geometry. The non-holomorphic modication in the e ective action is due to the integration over massless modes [18], whereas those in the topological string free energy originate from the pinching of cycles of the Riemann surfaces [19]. The need for non-holomorphic corrections can often be deduced from the lack of invariance under integer-valued duality symmetries, which requires modular functions that are not fully holomorphic. This was also observed when calculating the entropy for BPS black holes with S-duality invariance [20].

In this paper we will systematically study the connection between the e ective action and the topological string.4 Here we should stress that we are just referring to functions that can potentially dene the topological string free energy in relation to an underlying e ective action. Whether these functions will actually have a topological string realization is a priori not known. But the connection that is proposed in this paper seems to be universal so that it will apply also to those cases where a topological string realization does exist. We will start from the holomorphic function that encodes the Wilsonian action, and construct another quantity that transforms in the same way under duality as the topological string free energy. Here we are inspired by previous work on BPS black holes [15, 16, 22], where the so-called Hesse potential emerged as the relevant quantity, dened in the context of real special geometry [2325]. The Hesse potential transforms as a (real) function under duality. It is related to the function F (X) that encodes the e ective action via a Legendre transform and it is expressed in terms of duality covariant variables, so that its behaviour under duality is comparable to what one observes for the topological string. In hindsight it is not so di cult to understand this relation by reecting on the more familiar case of four-dimensional abelian gauge elds, where the Lagrangian is a function of the abelian

4A preliminary account of our results was published in the proceedings of the Frascati School 2011 on Black Objects in Supergravity [21].

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eld strengths F and possibly other elds (that we assume to be electrically neutral). The expressions for the dual eld strengths, which are related to the derivative of the full Lagrangian with respect to the original eld strength, do depend on the specic interaction terms contained in the Lagrangian. Therefore the electric/magnetic duality transformation rules for the original eld strengths will depend on the details of the underlying Lagrangian. On the other hand the Hamilonian depends on di erent quantities, namely the spatial part of the gauge potential A and the electric displacement eld D, where the latter follows from taking the derivative of the Lagrangian with respect to E. The precise denition of D will thus implicitly depend on the details of the Lagrangian. Under electric/magnetic duality B / r A and D transform as a dual pair and the Hamiltonian is a function of

these duality covariant variables: they transform into each other under symplectic rotations in a way that is independent of the details of the Hamiltonian. In fact, for a theory without higher derivatives, A and D are the canonical variables.

Electric/magnetic duality transformations thus act as canonical transformations and the Hamiltonian will usually decompose into a number of di erent functions that transform consistently under them. When the canonical transformations constitute an invariance of the system then these functions will be invariant. As it turns out, the Hesse potential of real special geometry is the direct analogue of the Hamiltonian. Rather than depending on the elds XI, it depends on canonical variables I and ~I. As we shall see, these can again be combined into complex variables in a way that involves the classical period vector associated with the function F (0)(X). The duality covariant variables (I, ~J) transform under the same duality transformations as the classical period vector.

The Hesse potential is related to the function F (X, W 2) via a Legendre transform, and thus contains the same information as the e ective action. In principle, other relevant quantities that are related to the underlying Calabi-Yau moduli space, such as the topological string free energy, can be characterized by functions of (I, ~J). Precisely as the Hamiltonian discussed above, the Hesse potential decomposes into di erent functions that all transform consistently under duality. The central conjecture of this paper is that the topological string should coincide with (part of) the Hesse potential, as this is the only way to explain why it can reproduce (part of) the e ective action. To identify this particular function we will rst consider what happens when the e ective action is purely Wilsonian. As it turns out there is just one function belonging to the Hesse potential that is almost harmonic, where the meaning and implication of the term almost harmonic will be explained in due course. The Hesse potential is nevertheless harmonic in terms of the (holomorphic) function that encodes the Wilsonian action. Subsequently it is demonstrated that, upon relaxing the harmonicity constraint on the function that encodes the e ective action, the resulting almost harmonic contribution to the Hesse potential satises the same holomorphic anomaly equation that is known from the topological string.

The paper is organized as follows. In section 2 we characterize possible non-holomorphic deformations of special geometry in the context of the e ective action based on a theorem that is presented in appendix A. Subsequently we introduce the formulation of real special geometry in terms of the Hesse potential, which transforms as a function under symplectic rotations of its real variables, and derive a number of results that are important

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for what follows in subsequent sections. Section 3 is devoted to evaluating the Hesse potential in terms of complex duality covariant variables by carrying out the Legendre transform by iteration to fourth order, which is su cient to appreciate its general structure. At this stage the non-holomorphic contributions can be understood in the context of a diagrammatic representation of the Hesse potential as a sum over connected tree graphs. The Hesse potential decomposes into an innite number of terms, which arrange themselves into an innite set of functions, all transforming consistently under electric/magnetic duality. Some of the expressions for these terms are collected in appendix B up to the corresponding order in the iteration. When the corresponding e ective action is characterized in terms of a holomorphic function, precisely one of the functions contributing to the Hesse potential becomes almost harmonic in the moduli. This function is thus the only possible candidate for a topological string free energy, and in section 4 we demonstrate that it indeed satises a holomorphic anomaly equation which partially coincides with the holomorphic anomaly equation known for the topological string [11].

Subsequently we relax the harmonicity restriction on the e ective action by allowing a specic non-holomorphic term that transforms as a symplectic function up to a term that is harmonic. Introducing such a term induces quite a large variety of additional contributions to the Hesse potential that leave its characteristic properties intact, but the candidate function for the topological string free energy now satises the full holomorphic anomaly equation. Hence this function has now all the prerequisites for representing the generating function of the genus-g free energies of the topological string and we explicitly demonstrate this up to g 3. The calculation for g = 3 is rather involved and it is described

in section 5. We should stress here that the logic of our calculations is rather di erent from the one that is often followed for the topological string, where the non-holomorphic corrections are found by integrating the anomaly equation [11, 26, 27], with the holomorphic contributions playing the role of generalized integration constants. In this paper we construct the Hesse potential starting from holomorphic functions, which, in order to ensure that they transform consistently under duality transformations, will necessarily contain non-holomorphic contributions. These non-holomorphic contributions then turn out to satisfy the holomorphic anomaly equation. In this way it is obvious that the holomorphic anomaly arises due to an incompatibility between duality covariance and holomorphicity. On the other hand we demonstrate that the function that encodes the e ective action, which is holomorphic in the Wilsonian limit, must also contain corresponding non-holomorphic corrections of a specic form, which we evaluate order-by-order by iteration. All these results are established in the context of a generic special-Kahler space, but we do not wish to imply that in all these cases an actual topological string realization will exist.

A summary and a discussion of the results is presented in section 6. Here we also present a comparison of our present results with previous work [15, 16] on the FHSV model [28]. Furthermore we briey discuss some of the consequences of the results of this paper for BPS black hole entropy, especially in connection with its conjectured relation to the topological string [29].

There are ve appendices. The rst appendix A establishes the consistency of special geometry under non-holomorphic deformations. The second appendix B lists a number of

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symplectic functions that emerge when evaluating the Hesse potential by iteration. Appendix C lists some intermediate results that are relevant for the third-order calculation described in section 5. The explicit expressions for the twisted string free energies F (g)

of genus g 3 are presented in appendix D based on the construction presented in this

paper. Finally in appendix E we give further details about the comparison of the present results to earlier results obtained for the FHSV model.

2 Real and deformed special geometry

In the previous section we introduced holomorphic functions that encode either the Wilsonian action or the topological string free energy, as well as a real function known as the Hesse potential. While the rst two are initially holomorphic, they eventually acquire nonholomorphic terms caused by the underlying physics. In the Hesse potential there seems no immediate obstacle to include such modications as it is initially dened in terms of real variables. Let us now reiterate some of the distinctive features of these three structures and clarify the relevant issues.

The topological string free energy is a function of the Calabi-Yau moduli which are subject to dualities related to the homology group of the underlying holomorphic three-form. As explained in section 1, these moduli are associated with a holomorphic function F (0)(X), which also encodes a corresponding vector multiplet Lagrangian with at most two space-time derivatives coupled to supergravity. The dualities of this Lagrangian are generated by certain electric/magnetic dualities and they are related to the (discrete) homology group associated with the Calabi-Yau periods. When deforming the supergravity, for instance by introducing couplings to the square of the Weyl multiplet as specied in (1.4), the duality transformations of the moduli XI will change their form, whereas the variables XI in the topological string will still be associated with F (0)(X). Therefore, as explained in the previous section, the supergravity denition and the topological string denition of the variables XI will no longer be the same, and correspondingly the genus-g free energies cannot be identical to the higher-derivative supergravity couplings.

The topological string free energy contains non-holomorphic corrections related to the pinchings of cycles in the underlying Riemann surfaces. These corrections should presumably be related to the non-holomorphic contributions to the function F (X, W 2) which are induced by the integration over massless modes, in view of the fact that the two quantities are known to describe the same (on-shell) string amplitudes [11, 17]. Irrespective of this relationship the situation regarding the non-holomorphic corrections to the function F (X, W 2) is subtle. Integrating out the massless modes leads to interactions that are non-local in generic space-times and it is not known what the precise dictionary is between non-holomorphic terms in the function F and the non-local terms in the Lagrangian. In the supergravity context, non-holomorphic corrections are most likely related to chiral anomalies associated with the U(1) local symmetry that is an essential part of the superconformal multiplet calculus. These anomalies are cancelled by the non-holomorphic terms that emerge in the e ective action. Such a phenomenon has been claried in [30] for

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a number of situations. Another relevant observation is that non-holomorphic corrections are often required in order to have an exact duality invariance.

The variables XI and W 2 are only projectively dened, so that physically relevant results should not depend on uniform rescalings by a complex number. Hence we can replace the XI by uniformly rescaled variables Y I that di er by a uniform multiplicative complex factor or eld according to a prescription that may depend on the application that is being considered. Likewise one must also rescale the expression for W 2 by the square of the same factor as for the XI. The resulting expression is usually denoted by . However, in what follows we will regard as one of the generalized coupling constants that may play a role. As it turns out it is not necessary to refer explicitly to such coupling constants, so that we will suppress them henceforth.

The information encoded in F (X, W 2) can also be encoded in the context of real special geometry where the relevant quantity is the Hesse potential. As was already argued in the introductory section the Hesse potential represents the Hamiltonian form of the Wilsonian action and depends on real duality-covariant variables denoted by I and ~I. They can be dened by

I = Y I + Y, ~I = FI + FI . (2.1)

Note that the replacement of the original variables XI by Y I is now relevant, as it would not make sense to consider linear combinations of the original variables XI and their complex conjugates in view of the fact that they are projectively dened.5 As it turns out, nonholomorphic corrections can be encoded in a real function (Y, Y ), which is incorporated into the function F in the following way [32],

F (Y, Y ) = F (0)(Y ) + 2i (Y, Y ) , (2.2)

where F (0)(Y ) is holomorphic and homogeneous of second degree. Note that the decomposition (2.2) is subject to the equivalence transformation,

F (0)(Y ) ! F (0)(Y ) + g(Y ) , (Y,

Y ) ! (Y,

Y ) Im g(Y ) , (2.3) which amounts to a shift of F (Y, Y ) by an anti-holomorphic function: F (Y, Y ) ! F (Y,

Y )+

g( Y ). This change does not a ect the period vector (Y I, FI), which only involves holomorphic derivatives, which is the underlying reason for this equivalence. When the function is harmonic, i.e., when it can be written as the sum of a holomorphic and an anti-holomorphic function, then one may simply absorb the holomorphic part into the rst term according to (2.3). We usually refer to F (0)(Y ) as the classical contribution, because

it refers to the part of the Lagrangian that is quadratic in space-time derivatives. In that case only the function will depend on possible deformation parameters such as and

and it may contain harmonic and non-harmonic contributions. The ansatz (2.2) may seem somewhat ad hoc, but in fact it can be derived in a much more general context as

5The same strategy was followed previously in the study of BPS black holes (see, e.g. [31]). The same comment applies to the holomorphic derivatives FI. Note that FI equals the derivative of F with respect to Y I. At this point we refrain from distinguishing holomorphic and anti-holomorphic derivatives, @/@Y I and @/@ Y I, by the use of di erent types of indices.

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proven in the theorem presented in appendix A, which makes use of the analogue of the Hesse potential. The rst indication for these results came from the study of BPS black hole entropy [15, 16, 22, 32, 33].

The new variables (2.1) have the virtue of transforming linearly under duality by real symplectic rotations. At this point it is convenient to dene a quantity H of I and ~I,

which contains the same information as the F (Y, Y ) but transforms as a function under the duality transformations. This quantity is the Hesse potential. It is a generalization of the Hesse potential that was dened in the context of real special geometry [2325] and follows from the Legendre transform of 4(Im F (0) + ) with respect to the imaginary part of Y I,

H(, ~) = 4

Im F (0)(Y ) + (Y, Y )

[bracketrightbig]+ i~I (Y I Y I) . (2.4)

Its generic variation satises

H = i(FI

FI) I + i(Y I

Y I) ~I , (2.5)

where FI refers to the holomorphic derivative of (2.2), which conrms that H is indeed a

function of the duality-covariant variables (I, ~I). The theorem of appendix A demonstrates that many of the special geometry properties remain valid under non-holomorphic deformations. This result had already been indicated by earlier work on this subject in [15].

The classical function F (0) is assumed to be holomorphic and homogeneous of second degree in Y I. In the remainder of the section we summarize some results for such a function with respect to its behaviour under electric/magnetic duality that are needed in the next section. The electric/magnetic dualities are dened by Sp(2n+2, R) rotations of the period vector (Y I, FI), dened in the usual way,

Y I !

I = UIJ Y J + ZIJFJ ,

FI !

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~FI = VIJFJ + WIJY J , (2.6)

where U, V , Z and W are the (n + 1) (n + 1) real submatrices that constitute an element

of Sp(2n + 2, R). Applying these transformations to the case where F = F (0), so that we are dealing with a homogeneous and holomorphic function, it follows that ~F0)I can be

expressed as the holomorphic derivative of a new holomorphic function, ~F(0)( ), with the

latter equal to

~F(0)( ) = F (0)(Y )

1

2Y IF (0)I(Y ) +

1

2(UTW )IJ Y IY J

+ 12(UTV + W TZ)IJ Y I F (0)J(Y ) +

1

2(ZTV )IJ F (0)I(Y ) F (0)J(Y ) , (2.7)

which, in general, is di cult to solve explicitly. Note that, when the function is not homogeneous, there are integration constants corresponding to either a constant or terms proportional to the I. In the presence of non-holomorphic terms the proof of existence of a new function is much more complicated, but the arguments, presented in a more generic context in appendix A, indicate that this is indeed the case, although no explicit expression has been given in the general case in analogy to (2.7).

10

Finally we present the transformation rules of the rst multiple derivatives of the function F (0) under the dualities (2.6),

~F(0)IJ = (VILF (0)LK + WIK) [S10]KJ , ~F(0)IJK = [S10]LI [S10]MJ [S10]NK F (0)LMN , ~F(0)IJKL = [S10](MI [S10]NJ [S10]P K [S10]Q)L

F (0)MNPQ 3 F (0)MN Z0 F (0)PQ

[bracketrightbig], ~F(0)IJKLM = [S10](NI [S10]P J [S10]QK [S10]RL [S10]S)M

F (0)NPQRS 10 F (0)NPQ Z0 F (0)RS + 15 F (0)NP Z0 F (0)Q Z0 F (0)RS

[bracketrightbig]

~F(0)IJKLMN = [S10](P I [S10]QJ [S10]RK [S10]SL [S10]T M [S10]U)N

hF (0)PQRST U 15 F (0)PQRS Z0 F (0)TU

10 F (0)PQR Z0 F (0)STU

+ 60 F (0)PQR Z0 F (0)S Z0 F (0)TU

+ 45 F (0)PQ Z0 F (0)RS Z0 F (0)TU

90 F (0)PQ Z0 F (0)R Z0 F (0)S Z0 F (0)TU

15 F (0)XY Z

Z0X F (0)PQ

[bracketrightbig] [bracketleftbig]Z0Y F (0)RS

[bracketrightbig] [bracketleftbig]Z0Z F (0)TU

[bracketrightbig][bracketrightBig], (2.8)

which can be obtained by repeated di erentiation of the basic equation (2.6) or (2.7). Note that for clarity we have occasionally replaced indices by bullets in cases where the index contractions are unambiguous. In the above formulae the bullets are indices that are simply contracted with the nearest neighbour bullet as a string. Furthermore we have made use of the following denitions,

S0I J (Y ) = @

I@Y J = UIJ + ZIKF (0)KJ(Y ) ,

ZIJ0(Y ) = [S10]IK ZKJ . (2.9)

Because the matrices U and Z are submatrices of a (2n+2)-dimensional symplectic matrix, it follows that ZIJ0 is symmetric in (I, J). It also follows that

[S10]IJ = Z0IK F (0)KL [S10]LJ ,

Z0IJ = Z0IK F (0)KL Z0LJ . (2.10)

Dening

N(0)IJ(Y, Y ) = 2 Im [F (0)IJ(Y )], (2.11)

we derive the following expression for the behaviour of its inverse N(0)IJ under duality

transformations,

(0)IJ = S0IK

S0JL N(0)KL . (2.12)

11

JHEP09(2014)096

,

Using the identity [S10]IK [

written as,

N(0)KL + i ZKL0[bracketrightbig]. (2.13)

These identities will be relevant later on. Incidentally, from the results presented above one can straightforwardly construct tensors that transform covariantly under the symplectic transformations, such as

CIJKL = F (0)IJKL + 3i N(0)MN F (0)M(IJ F (0)KL)N . (2.14)

However, these tensors are not purely holomorphic in view of the appearance of the matrix N(0)IJ. We will encounter such almost holomorphic covariant functions throughout this paper.

Note that in the next section we will introduce di erent complex variables denoted by

YI. Since we will treat as a perturbation of the classical function F (0), the full func

tion F will no longer play a role in the various formulae. Therefore, in due course, we will simply suppress all sub- and superscripts 0 referring to the lowest-order quantities F (0), S0IJ , ZIJ0, and N(0)IJ. Note also that we are only distinguishing holomorphic and anti-holomorphic derivatives, I, J, . . . and, J, . . . with a bar when we are dealing with a real quantity. For instance, S0JK is anti-holomorphic, so we will just keep generic indices

J, K (rather than J, K), while I and need a holomorphic or anti-holomorphic index to distinguish between the derivative with respect to Y I and Y I. The reason for this convention is that we will often not have the situation where holomorphic and anti-holomorphic indices are contracted consistently.

3 The generic structure of the Hesse potential

In this section we study the generic structure of the Hesse potential, which requires to carry out a Legendre transform. In practice this can only be done by iteration. The results will then take the form of an innite power series in terms of and its derivatives. We will explicitly evaluate the rst terms in this expansion up to order 5, which we expect to su ce for uncovering the general structure of the full expression. The homogeneous and holomorphic function F (0)(Y ) will not be subject to further restrictions and will thus encode a generic special-Kahler space, while will be an arbitrary function of Y and Y . The actual calculations are rather laborious although in principle straightforward; we have relegated some relevant material to several appendices. Similar calculations have been performed for more specic cases some time ago [15, 16]. For instance, the rst two terms to quadratic order in ,

have been determined for the FHSV model [28]. As it turned out some of these terms were consistent with known results obtained from the topological string by integration of the holomorphic anomaly equation [27], but the Hesse potential contained additional terms at this order that were separately S- and T-duality invariant but did not have an interpretation in the topological string context. In this paper we will investigate the generic structure of the Hesse potential and clarify these partial results. We will return to a discussion of the results for the FHSV model in section 6.

12

N(0)KL iZKL0[bracketrightbig]= S0IK S0JL

S0]KJ = IJ iZIK0 N(0)KJ, it follows that (2.12) can also be

(0)IJ = S0IK S0JL

JHEP09(2014)096

To carry out the Legendre transform by iteration we rst choose convenient variables. Originally the Hesse potential was dened in terms of the variables (I, ~I) of real special

geometry, whose denition involves the full e ective action. It is, however, more convenient to convert them again to complex variables, subsequently denoted by YI, which coincide

precisely with the elds Y I that one would obtain from (I, ~I) upon using just the lowest-order holomorphic function F (0). The identication proceeds as follows [16],

2 Re Y I = I = 2 Re YI ,2 Re FI(Y, Y ) = ~I = 2 Re F (0)I(Y) . (3.1)

Since the relation between the variables YI and the real variables (I, ~I) involves only

F (0), their duality transformations will be directly related. Consequently we will refer to the variables YI as duality covariant variables. Under duality the variables YI transform

according to,

I = UIJ YJ + ZIJ F (0)J(Y) = SIJ (Y) YJ , (3.2) where we used the homogeneity of F (0)(Y) as well as the denition of SIJ given in (2.9),

except that the expression is now written in terms of the new variables YI. Furthermore

we have dropped the subscript by the replacement S0 ! S, as we had already indicated at

the end of section 2.At the classical level, where = 0, we obviously have YI = Y I, but in higher orders

the relation between these moduli is complicated and will involve . Let us therefore write

YI = Y I + Y I, where Y I is purely imaginary, and F = F (0) + 2i , so that we can

express (3.1) in terms of F (0), , YI and Y I. Because the equations will no longer involve F , we will henceforth drop the index 0 on F (0), as mentioned earlier. Consequently all the derivatives of F will be holomorphic. The equations (3.1) can then be written as,

FI(Y Y ) +

FI( Y + Y ) FI(Y)

FI( Y)

= 2i I(Y Y,Y + Y ) (Y Y,Y + Y )

[bracketrightbig], (3.3)

where we made use of (2.2). Upon Taylor expanding, this equation will lead to an innite power series in Y I, which we can solve by iteration. Retaining only the term of rst order in Y I shows that it is proportional to rst derivatives of . Proceeding to higher orders will then lead to an expression for Y I involving increasing powers of and F and their derivatives taken at Y I = YI. Up to fourth order in this iteration gives the following

expression for Y I,

Y I 2 ( I )

2i(F +

F)IJK( J

J)( K

K) 8 Re( IJ I J) ( J

+ 43i

+ 8i

JHEP09(2014)096

J)

(F F)IJKL + 3i(F + F)IJM(F + F)MKL [bracketrightbig]

( J

J)( K

K)( L

L)

h2 (F + F)IJKRe( KL K L) + Re( IK I K)(F +F)KJL [bracketrightBig]

( J

J)( L

13

L)

+ 32 Re( IJ I J) Re( JK J

K) ( K K) + 8i Im( IJK 2 IJ K + I JK)( J

J) ( K

K) + O( 4) . (3.4)

Here indices have been raised by making use of N(0)IJ, which was already dened in (2.11), where, for consistency, we will henceforth change notation and refer to N(0)IJ by NIJ. Here

we stress once more that all the derivatives of F and are taken at Y I = YI and

Y I = YI.

Using the same notation we obtain the following expression for the Hesse potential (2.4),

H(Y,

Y) = i [bracketleftbig]

YIFI(Y) YI

FI( Y)

[bracketrightbig]

+ 4 (Y,

Y)

I(Y, Y ) (Y, Y )

[parenrightbig][bracketrightbig]. (3.5)

Here we made use of the homogeneity of F (Y ) and of (3.3). Again this result must be Taylor expanded upon writing Y I = YI Y I and

Y I = YI + Y I. The last two lines of (3.5) then lead to a power series in Y , starting at second order in the Y ,

H(Y,

Y) i[

YIFI(Y) YI

FI( Y)] + 4 (Y,

Y)

NIJ Y I Y J

2

3i(F +

JHEP09(2014)096

i

YI

FI(Y ) FI(Y)

[parenrightbig]+ Y IFI(Y ) h.c.

+ 4

(Y, Y ) (Y,Y) + Y I

F)IJK Y I Y J Y K

4 Re( IJ I

J) Y I Y J + 14i(F

F)IJKL Y I Y J Y K Y L

+ 83i Im( IJK 3 IJ

K) Y I Y J Y K + . (3.6)

Inserting the result of the iteration (3.4) into the expression above leads to the following expression for the Hesse potential, up to terms of order 5,

H(Y,

Y) i[

YIFI(Y) YI

FI( Y)] + 4 (Y,

Y)

4

NIJzI zJ + 83i(F +

F)IJK NIL NJM NKNzL zM zN

43i[(F

F)IJKL + 3i(F + F)IJR NRS(F + F)SKL]

NIM NJN NKP NLQzM zN zP zQ

32

3 i Im( IJK 3 IJ

K) NIL NJM NKN zL zM zN + O( 5) , (3.7)

where zI = I

, and where NIJ is the inverse of the real, symmetric matrix NIJ,

dened by

J) . (3.8)

Upon expanding NIJ we straightforwardly determine the contributions to the Hesse po-

14

NIJ = NIJ + 4 Re( IJ I

tential up to fth order in ,

H = H| =0 + 4 4 NIJ( I J +

J) + 8 NIJ I J

+ 16 Re( IJ I

J)NIKNJL K L + K L 2 K

L

[parenrightbig]

16

3 (F +

F)IJKNILNJMNKN Im( L M N 3 L M

N )

64NIP Re PQ P

Q

[parenrightbig]

NQRRe ( RK R

K) NKJ ( I J + J 2 I

J)

+ 64(F + F)IJKNILNJMNKP Re PQ P

Q

[parenrightbig]

NQN Im( L M N 3 L M

N )

83i[(F

F)IJKL + 3i(F + F)R(IJNRS(F + F)KL)S]NIMNJNNKP NLQ

Re M N P Q 4 M N P

Q + 3 M N P Q

[parenrightbig]

JHEP09(2014)096

+ 64

3 Im( IJK 3 IJ

K)NILNJMNKN Im( L M N 3 L M

N ) + O( 5) .

(3.9)

We stress once more that all the quantities in (3.9) are taken at Y I = YI.

It is clear that the Hesse potential takes a complicated form, but we note two systematic features. First of all it turns out that the expression (3.9) can be understood diagrammatically. To appreciate this, let us return to the denition (2.4) of the Hesse potential and rewrite it in a di erent form,

H(, ~) = 4

Im F (Y ) + (Y, Y )

[bracketrightbig]+ i~I (Y I Y I)

= 2i F (Y Y ) + 2iF( Y + Y ) + 4 (Y Y,Y + Y )

+ i~I YI

2i ~I Y I , (3.10)

where the purely imaginary quantities Y I were introduced in the beginning of section 3 when dening the iteration procedure. We remind the reader that I = 2 Re Y I = 2 Re YI

and ~I = 2 Re FI(Y). Substituting these results we derive

H(Y,

Y) = H(Y,

Y)

[vextendsingle][vextendsingle]

=0 + NIJ(Y,

Y) Y I Y J

+ 2i

YI

[parenrightbig]

()n+1 FI1[notdef][notdef][notdef]In(Y) +FI1[notdef][notdef][notdef]In( Y)

[bracketrightbig] Y1 Yn

+ 4 (Y Y,

Y + Y ) . (3.11)

It thus follows that H(Y,

1

Xn=31 n!

Y) H(Y,

Y)| =0 4 (Y,

Y) can be written as a series expan

sion in positive powers of Y I. Integrating the exponential of this expression over the (purely imaginary) uctuations Y I, the result can be expressed as an innite sum over Feynman diagrams in the standard way with propagators given by NIJ and vertices by the derivatives of F and F as well as of the holomorphic and anti-holomorphic derivatives of . Because the Legendre transformation performed above should correspond to the tree diagrams, it follows that H(Y,

Y)H(Y,

Y)| =0 4 (Y,

Y) will comprise all the connected

tree diagrams. In this way one can account for all the terms in (3.9), including their combinatorial factors. Note that the diagrams are not 1PI: by removing the propagator NIJ

15

each term in the expression will factorize into two terms! Here we should mention that diagrammatic techniques have repeatedly played a role in the analysis of the holomorphic anomaly equation and of the topological string (see e.g. [11, 26, 27]).

Another feature is based on the fact that (3.9) transforms as a function under duality transformations of the elds YI, and so does the rst term H| =0. This observation enables

one to determine how will transform under dualities. Obviously the transformation behaviour of must be non-trivial in view of the non-linear dependence on of the Hesse potential. The evaluation of this transformation proceeds again by iteration.

To demonstrate the procedure, let us review the rst few steps. In lowest order, must transform as a function, which implies that

~

(,

~

Y) = (Y,

JHEP09(2014)096

Y) + O( 2) ,

~

I(,

Y) = [S1]JI (Y) J(Y,

Y) + O( 2) , (3.12) where the matrix S was already dened previously (note that we have suppressed the

subscript 0). Applying this result to the rst few terms of (3.9) and making use of the fact that H H| =0 transforms as a function, one deduces the next-order result,

~ IJ(~

I ~

J + ~

~

J) + 2IJ ~

I ~

J + O(

~

~

3)

= NIJ( I J +

J) + 2 NIJ I J + O( 3) , (3.13) where on the left-hand side the functions depend on the transformed eldsI, while on

the right-hand side they depend on the original elds YI.

Using the exact relations (2.9)(2.13), suppressing also the subscript 0 in the symmetric matrix ZIJ0, one discovers that the rst equation in (3.12) receives the following

correction in second order in ,

~

(,

~

Y) = i ZIJ I J

+ O( 3) , (3.14) which in turn gives rise to the following result for derivatives of ,

~

I(,

~

ZIJ

J

[parenrightbig]

Y) = [S1]JI [bracketleftBig] J + iFJKL ZKM M ZLN N 2i JKZKL L + 2i JK ZKL L

[bracketrightBig]

+ O( 3) ,

~

IJ(,

Y) = [S1]KI [S1]LJ[bracketleftBig] KL FKLM ZMN N[bracketrightBig]+ O( 2) ,

~

~

I J(,

~

Y) = [S1]KI [

S1]L

J K L + O( 2) . (3.15)

Here and henceforth we make frequent use of (2.10).

The iteration can be continued by including the terms of order 3, making use of (3.15) for derivatives of , to obtain the expression for ~

up to terms of order 4. In the next iterative step one then derives the e ect of a duality transformation on up to terms of order 5. Before presenting this result, we wish to observe that terms transforming as a proper function under duality, will not contribute to this result. This is precisely what happens to the term proportional to NIJ I J that appears in (3.9), which transforms as

16

a function under symplectic transformations in this order of the iteration. Consequently this term does not contribute to (3.14). As it turns out an innite set of contributions to the Hesse potential will be generated that transform separately as functions under duality. By separating those from (3.9), we do not change the transformation behaviour of but we can extract certain functions from the Hesse potential in order to simplify its structure. However, these functions must be constructed also by iteration, order by order in .

We have evaluated this decomposition in terms of separate functions in detail, which leads to

H = H(0) + H(1) + H(2) + H(3)1 + H(3)2 + h.c. [parenrightbig]

+ H(3)3 + H(4)1 + H(4)2 + H(4)3 + H(4)4 + H(4)5 + H(4)6 + H(4)7 + H(4)8 + H(4)9 + h.c.

[parenrightbig]

. . . , (3.16)

where the H(a)i are certain expressions to be dened below, whose leading term is of order

a. For higher values of a it turns out that there exists more than one function with the same value of a, and those will be labeled by i = 1, 2, . . .. Of all the combinations H(a)i appearing in (3.16), H(1) is the only one that contains , while all the other combinations

contain derivatives of . Obviously, H(0) equals,

H(0) = i[

YIFI(Y) YI

FI( Y)] , (3.17)

whereas H(1) at this level of iteration is given by,

H(1) = 4 4 NIJ( I J +

J)

JHEP09(2014)096

+ 16 Re

h IJ(N )I(N )J

[bracketrightbig]

+ 16 I J (N )I(N

)J

3 Im

16 FIJK(N )I(N )J(N )K[bracketrightbig]

4 3i

[bracketleftBig][parenleftBigg][parenleftBigg]F

IJKL + 3iFR(IJNRSFKL)S

[parenrightbig]

(N )I(N )J(N )K(N )L h.c.[bracketrightBig]

16 IJK(N )I(N )J(N )K + h.c. [bracketrightbig]

16

3 IJ K(N )I(N )J(N )K + h.c. [bracketrightbig]

16i

hFIJKNKP PQ(N )I(N )J(N )Q h.c.[bracketrightBig] 16

h(N )P PQ NQR RK (N )K + h.c.

[bracketrightBig]

16

h(N )P PQ NQR R K (N )K

+ NIP P Q NQR RK (N )K + R K (N

)K

[parenrightbig]

I + h.c.

[bracketrightBig]

hFIJKNKP P Q(N )I(N )J(N )Q h.c.[bracketrightBig]+ O( 5) . (3.18)

Here we have used the notation (N )I = NIJ J, (N

)I = NIJ J. Index symmetrizations, such as in FR(IJNRSFKL)S, are always of strength one. For instance, in this example, where there are three independent combinations, one includes a factor 1/3. The expressions for the higher-order functions H(a)i with a = 2, 3, 4 are given in appendix B. These

17

16i

expressions have been obtained by requiring that they constitute functions under symplectic transformations, order by order in . There exist other expressions that do transform as functions under duality in this approximation, but which do not appear in H. We

have included two examples of such functions in appendix B, one of which will be relevant later on.

Because H(1) transforms as a function under symplectic transformations, we can deduce

the transformation behavior of up to order 5 by generalizing (3.12) to higher orders. In this way one derives,

~

(,

~

Y) = i ZIJ I J

JHEP09(2014)096

ZIJ

J

[parenrightbig]

+ 23 FIJK ZIL L ZJM M ZKN N + h.c. [parenrightbig]

2 IJ ZIK KZJL L + h.c. [parenrightbig]

+ 4 I J ZIK K

ZJL

L

+

[bracketleftbigg]

1 3iFIJKL(Z )I(Z )J(Z )K(Z )L

+ 43i IJK(Z )I(Z )J(Z )K+ i FIJR ZRS FSKL (Z )I(Z )J(Z )K(Z )L

4i IJ

K (Z )I (Z )J(

Z

) K

4i FIJKZKP PQ (Z )I(Z )J (Z )Q + 4i FIJKZKP P

Q(Z )I(Z )J(

Z

) Q

+ 4i (Z )P PQ ZQR [parenleftBig]

RK (Z )K 2 R

K ( Z

) K

[parenrightBig]

4i (Z )P P

Q Z Q R

RK (Z )K + h.c.[bracketrightbigg]

+ O( 5) , (3.19)

where the functions on the right-hand side depend on the elds YI and

YI. We used

the obvious notation Z

I = ZIJ J andZ I = ZJ J. It is remarkable that the matrix NIJ no longer appears in this relation. This is due to a subtle interplay of the various contributions to this result, which involves the ones coming from (3.15). On closer inspection, the result (3.19) turns out to be identical (modulo an overall factor 4) to

H(1) given in (3.18) upon making the replacement NIJ ! iZIJ and/or NIJ ! i

ZJ,

where the precise form depends on the type of index contractions (i.e holomorphic or anti-holomorphic) to NIJ. The fact that none of the other functions H(a) contribute is perhaps

not surprising because those are manifestly symplectic functions where no Z dependent

variations are generated, whereas in H(1) such terms are generated and have to be absorbed

into the transformation rule of . Clearly this is an intriguing result for which we have not found a general proof within our approach, although we are aware of the fact that similar properties have been encountered in [26].

With the above result (3.19) one can continue with the iterations by extending (3.15) to the next order, nding

~

I(,

~

Y) = [S1]JI [bracketleftBig] J + iFJKL (Z )K (Z )L 2i JK(Z )K + 2i JK( Z

) K

18

+ 23FJKLP (Z )K(Z )L(Z )P + 2 FKLP (Z )KJ (Z )L(Z )P + 4 FJKL(Z )K(Z )LP (Z )P 4 FJKL(Z )K(Z )L

P( Z

) P

2 FJKLZLP FPQS(Z )K(Z )Q(Z )S + 2

FK L P( Z

) KJ ( Z

)L( Z

) P

2 JKL(Z )K(Z )L 4 KL(Z )KJ (Z )L 2 J

K L( Z

) K( Z

)L

4

K L( Z

) KJ ( Z

)L + 4 JK L(Z )K(

Z

)L + 4 K L(Z )KJ (

Z

)L

+ 4 K L(Z )K(

Z

)LJ

[bracketrightBig]

+ O( 4) ,

IJ = [S1]KI [S1]LJ[bracketleftBig]

KL FKLM ZMN N + iFKLMN(Z )M(Z )N

+ 2i FKMN(Z )ML(Z )N 2i FKMNZMP FPQL(Z )Q(Z )N 2i KLP (Z )P 2i KP (Z )P L + 2i KP ZPQFQLS(Z )S + 2i KL P( Z

~

JHEP09(2014)096

) P + 2i K P( Z

) PL

[bracketrightBig]

+ O( 3) ,

~

I J = [S1]KI [

S1]L

J

h K L + 2 iFKMN(Z )ML(Z )N 2iFL P N ( Z ) NK( Z ) P

2i KM

L(Z )M 2i KM(Z )M

L + 2i K L M ( Z

) M + 2i K M ( Z

) M L

[bracketrightBig]

+ O( 3) , (3.20) where (Z )ML = ZMN NL, (

Z

P, etc. These

results will then contribute to the determination of the next order contribution to (3.19). The fact that transforms non-linearly under dualities, while we are at the same time considering an expansion in terms of and its derivatives, suggests to introduce a formal expansion parameter and expand =

) PL = Z P N

NL, (Z )L

P = ZLK K

P1n=1 n1 (n), so that we are obtaining relations between products of di erent coe cient functions (n) order-in-order in . At this stage there is no direct need for this, but we will follow this strategy in the next section where matters become somewhat more involved.

Rather than proceeding with this iteration procedure, we will simply assume that all characteristic features noted in the results above, will continue to hold in higher orders as well. An obvious conclusion is then that the quantity does not transform as a function under symplectic transformations in view of the result (3.19). This is in agreement with our earlier claims, for instance in [15, 16]. On the other hand we expect that must belong to a restricted class and, in particular, it should have a well-dened harmonic limit that will dene the Wilsonian action. To understand how this may come about, let us rst start from an that is harmonic in the variables YI and their complex conjugates

YI. It is

then reasonable to expect that also ~

will be harmonic in the new variables, so that both

and ~

can be written as a sum of a holomorphic and an anti-holomorphic function in their respective variables. Indeed this property is conrmed by (3.19), or alternatively by the last equation in (3.20). Hence we conclude that the condition

I J = 0 , (3.21)

is preserved under symplectic transformations. In the next section we rst discuss the consequences of this harmonicity constraint, before considering modications thereof.

19

4 Non-holomorphic deformations and the anomaly equation

At the end of the previous section we mentioned the possibility that is an harmonic function of the variables YI and

YI, a property that is consistent with respect to symplectic

transformations. However, the function of interest is H(1)(Y,

Y), which is not harmonic

in this case but which can still be decomposed in a way that is rather similar to the decomposition of a harmonic function. Namely one can write,

(Y,

Y) = !(Y) +

!( Y) ,

H(1)(Y,

Y) = h(Y,

Y) +

h( Y, Y) , (4.1)

where the function h(Y,

h(Y,

3 !IJK(N!)I(N!)J(N!)K 16i FIJKNKP !PQ(N!)I(N!)J(N!)Q

16 (N!)P !PQ NQR!RK (N!)K + O(!5) . (4.2)

Because both H(1) and

~

, given in (3.18) and (3.19), are now harmonic in the !, it follows that ! must transform under symplectic transformations in direct correspondence with (3.19),

~

!() = ! i ZIJ!I !J +

2

3FIJK (Z!)I (Z!)J (Z!)K

2 !IJ (Z!)I (Z!)J

1 3iFIJKL(Z!)I(Z!)J(Z!)K(Z!)L

+ 43i !IJK(Z!)I(Z!)J(Z!)K

+ i FIJK ZKL FLMN (Z!)I(Z!)J(Z!)M(Z!)N

4i FIJKZKP !PQ (Z!)I(Z!)J (Z!)Q+ 4i(Z!)I!IJZJK!KL(Z!)L + O(!5) , (4.3)

so that ! transforms holomorphically. Obviously h(Y,

Y) must be a symplectic function.

However, its dependence on Y resides exclusively in the complex matrix NIJ. Therefore

we rst study the dependence of h(Y,

Y) on NIJ, and derive the following equation

@h@NIJ =

14@Ih @Jh , (4.4)

which we have veried up to terms of order !5. This result can easily be understood on the basis of the diagrammatic interpretation that we have presented in the previous section. From it one straightforwardly determines the non-holomorphic derivative of h(Y,

Y),

@h(Y,

Y) =

1 4i

FIKL NKMNLN @Mh(Y,

20

Y) depends holomorphically on ! and equals,

JHEP09(2014)096

Y) = 4 ! 4 NIJ !I!J

+ 8 !IJ(N!)I(N!)J + 83i FIJK(N!)I(N!)J(N!)K

43i FIJKL + 3iFR(IJNRSFKL)S

16

[parenrightbig]

(N!)I(N!)J(N!)K(N!)L

Y) @Nh(Y,

Y) . (4.5)

The equation (4.5) partially coincides with what is known as the holomorphic anomaly equation for the topological string and represents the terms that are induced by the pinching of a cycle of the underlying Riemann surface resulting in two disconnected Riemann surfaces [11]. This restricted anomaly equation is also obtained when considering the N = 2 chiral superspace action for abelian vector multiplets in the presence of a chiral background eld [34]. When expanding this action in terms of the background, the holomorphic expansion coe cient functions do not transform as functions under duality. This can be resolved by covariantizing the Taylor expansion with a suitable connection, but in that case the expansion coe cient functions are no longer holomorphic. As it turns out these modied coe cient functions then satisfy the same anomaly equation (4.5). Note that this equation is integrable, so that no additional constraints are implied. Clearly the holomorphic anomaly is due to the fact that there is a conict between symplectic covariance and holomorphicity.

The above result would justify the identication of H(1) with the topological string,

except that the holomorphic anomaly equation is still incomplete. This implies that we have to somehow relax the assumption that is harmonic in Y, while still expressing it

in terms of a holomorphic function !(Y) in such a way that H(1) will remain harmonic (possibly up to a separate non-harmonic function, as we shall see in section 5) in terms of !. However, modifying the ansatz (4.1) must be consistent with duality, in the sense that the modication will hold for the whole class of functions that are related by duality. Our task is therefore to demonstrate that the present framework can be extended so as to induce the remaining term in the anomaly equation that is related to pinchings of cycles of the Riemann surface that reduce the genus by one unit.

As it turns out, a consistent extension can be constructed by introducing non-harmonic terms whose variation under symplectic transformations is still harmonic. Such a modication does preserve the present framework in a way that is consistent with duality. For instance we could choose the following ansatz for ,

(Y,

Y) = !(Y) +

!( Y) + ln det[NIJ] + (Y,

Y) , (4.6) where and are arbitrary real parameters and a non-holomorphic function of Y and

Y. Note that we assume that the two deformations do not depend on the holomorphic

function ! or its complex conjugate, because we insist on harmonicity with respect to !.

The deformation proportional to is the easiest to deal with, so let us consider this one rst. Since is a given function one cannot simply substitute the ansatz (4.6) with = 0 into the expression for the function H(1), because this would require to change

non-trivially in order to satisfy (3.19). Hence we must introduce additional terms into to ensure that the modied will still transform according to (3.19) without modifying the holomorphicity of ! and leaving as a function. As its turns out, this can be done in the next order by including the following additional terms,

(Y,

Y) = ! +

! + (Y,

Y) + NIJ

2 @I! @J + 2 @I @J + h.c.

[bracketrightbig]. (4.7)

So far we have been performing an interation in powers of . It is formally consistent to treat and ! as being of the same order, but then we must assume that ! can obtain

21

JHEP09(2014)096

terms of even higher order in in view of the non-linear transformation rules, Hence we assume ! can be expanded in terms of the parameter ,

!(Y) ! !(Y, ) =

1

Xn=1!n(Y) n1 , (4.8)

although we will keep this expansion implicit in what follows. It is now straightforward to see that ! transforms as follows under symplectic transformations

~

! = ! i ZIJ!I !J + O(!3) , (4.9)

which agrees with (4.3). Subsequently we consider the function H(1) in this order of iteration,

H(1) = h +

h + 4 + O(!3) , (4.10) where h coincides with (4.2) in this order of iteration. Consequently the addition of such a deformation leaves the holomorphic anomaly equation (4.5) unaltered.

In view of this result we continue with the rst modication in (4.6) proportional to . In principle the analysis proceeds in a similar way as in the previous case, but here one has to also investigate the consistency in rst order. However, it is easy to see how consistency can be achieved because we have

ln det[IJ] = ln det[NIJ] ln det S[bracketrightbig] ln det[bracketleftbig]S[bracketrightbig], (4.11)

where S was dened in (2.9). Because S is holomorphic, the e ect of the non-harmonic

modication ln det[NIJ] under duality can simply be absorbed by assigning the following transformation to !,

~

!() = !(Y) + ln det S[bracketrightbig], (4.12)

up to terms of higher order in (or !) and . Hence in lowest order (Y,

Y) transforms

as a function, so that our previous analysis remains una ected as ~

= + O( 2). This

is conrmed by the following. First of all, derivatives of the holomorphic function !(Y)

remain holomorphic but they do acquire extra terms in their transformation rules, as is

shown in

~

!I = [S1]JI !J + FJKLZKL [parenrightbig]

+ O( 2) ,

JHEP09(2014)096

!IJ = [S1]KI [S1]LJ [bracketleftBig]

!KL FKLMZMN(!N + FNPQ ZPQ)

+ FKLMN FKMP FLNQ ZPQ

~

ZMN[bracketrightBig]+ O( 2) . (4.13)

Furthermore the rst few derivatives of are now equal to

I = !I i FIJK NJK + O( 2) , IJ = !IJ + FIKL FJMN NKMNLN i FIJKL NKL + O( 2) , I J = FIKL

FJMN NKMNLN + O( 2) , (4.14)

22

which, in leading order, transform consistently under duality transformations (i.e. according to (3.20)). To show this one makes use of (4.13) and (2.8). Hence we conclude that to lowest order in , the non-holomorphic deformation (4.6) is consistent. Observe that the last equation (4.14) constitutes a deviation from the harmonicity condition (3.21).

In due course we will also need the result for the transformation of the third- and fourth-derivatives of !, implied by (4.12),

~

!IJK = [S1]M(I[S1]NJ [S1]PK) !P + FPQR ZQR [parenrightbig]

FMNP 3 FMN ZFP Z ! + FQR ZQR + FMNPQR 3 FMNQ Z FPR + 3 FMN Z FQ Z FPR ZQR

+ 2 FMQ Z FN Z FPRZQR

!IJKL = [S1]M(I[S1]NJ [S1]PK[S1]QL) !PQ + FPQUV ZUV[parenrightbig]

4 FMNP 3 FMN ZFP [parenrightbig]

Z !Q + FQUV ZUV

Z FPQ

FMNPQ 4FMNP ZFQ 6FMN ZFPQ

12 FMN Z FPZ FQ Z ! + FUV ZUV

FPQ

FMNPQUV ZUV 4FMNPUV ZUFQZV 3FMNXU FPQY V ZXY ZUV + 4

3 FMNZFPZFQZ

[parenrightbig]+ FMNPZX FXZFQZ [parenrightbig][bracketrightbig]

2 FMNZX

6 FM Z FN Z FP Z FQ Z

12 FMN Z FP ZX FX ZFQ Z

12 FMN ZX FX ZFP ZFQ Z

3 FMN ZX (FXZFY Z) ZY FPQ + O( 2) . (4.15) To continue this scheme to higher orders in is not an easy task. So far we have been working order-by-order in powers of , but now we are dealing also with additional terms that are proportional to the parameter . Within the iterative procedure that we have been following it is consistent to formally treat ! and as being of the same order as . Counting in this way shows that the corrections in (4.15) are of rst order in . Because the equations that we are dealing with are non-linear it is therefore imperative that the ! itself can in principle contain contributions of arbitrary order in , a possibility that we have already been alluding to below equation (3.20). Therefore we will in addition assume that ! can be expanded in terms of ,

!(Y) ! !(Y, ) =

1

[bracketrightbig]

+ O( 2) ,

Z

! + FUV ZUV[parenrightbig]

JHEP09(2014)096

~

+ 3 FMN Z ! + FUV ZUV

3 ZR FMN

+

[parenrightbig][parenleftBigg][parenleftBigg]ZS

[parenrightbig]

FRST ZT ! + FUV ZUV

+ 6

FXPZFQZ

[parenrightbig]+ FMNZX

FXZFPQZ [parenrightbig][bracketrightbig]

[parenrightbig]

[parenrightbig]

[parenrightbig]

Xn=1!n(Y) n1 , (4.16)

although we will keep this expansion implicit in what follows. Assuming that ! incorporates the higher-order terms in we can proceed to higher orders by iteration to obtain the

23

extension of the original almost-harmonic ansatz (4.6), possibly up to terms that separately constitute proper functions under symplectic transformations.

We thus continue to derive the terms of order 2, which can be found from (3.19),

~

= i ZIJ I J

ZIJ

!I !J 2i !I FJKLNKL 2 FIKL FJMN NKL NMN[bracketrightbig]+ h.c.+ O( 3) , (4.17)

where we note that the right-hand side is of order 2, as both and ! are counted as being of the same order as . Our task is now to include further modications into , just as we did earlier in (4.6), so that the change of under a symplectic transformation becomes consistent with the right-hand side of (4.17), up to a term that is harmonic, which can then be absorbed into the variation of !. The problem here is, however, that the expression (4.17) is linear in ZIJ but involves also terms that are linear or quadratic in

NIJ. It is clear that one cannot construct a suitable addition to depending exclusively on !I, NIJ and FIJK. There exists, however, an alternative, namely to include higher derivatives of !. According to (4.13) the second derivative !IJ leads to similar variations, suggesting another possible modication. Indeed, it follows that NIJ!IJ, FIJKLNIJNKL

and FIJK FLMN NILNJMNKN (and complex conjugates where appropriate) are the terms that one may add to so that only a holomorphic variation will remain in (4.17). And indeed, one can verify by explicit calculation that should be written as (up to a symplectic function of Y and

Y),

= iZIJ

Y) = ln N

+ h! + 2 NIJ!IJ 2 [bracketleftbigg]iFIJKL 23FIKMFJLNNMN[bracketrightbigg]

NIJNKL + h.c. [bracketrightBig]

+ O( 3) , (4.18) where here and henceforth we use the denition N det[NIJ]. Hence the holomorphic

function ! is now accompanied by a variety of specic non-holomorphic modications which will contribute to the e ective action. Indeed, with this result for (Y,

Y), ex

plicit evaluation shows that (4.17) is satised up to order 3 provided that ! transforms (holomorphically) according to

~

! = ! + ln det

S[bracketrightbig]+ iZIJ

FIJKL 23FIKMFJLNZMN[bracketrightbigg] ZIJZKL+ O( 3) . (4.19)

To see this one makes use of the transformations of multiple derivatives of the holomorphic function F (Y), listed in (2.8), the transformation of NIJ as given in (2.13), and the

transformation rule for !IJ specied in the second equation of (4.13). Incidentally, the result (4.18) is in line with the ansatz (4.16) according to which the function ! is expanded in powers of . Furthermore it turns out that the result (4.18) takes the form of a sum over connected 1PI diagrams, unlike the corresponding result for H(1).

24

J

[parenrightbig]

+ O( 3)

JHEP09(2014)096

(Y,

2 !IJ

!I + FIKLZKL[parenrightbig]!J + FJMNZMN [parenrightbig][bracketrightbig]

+ i 2

Let us now return to the almost harmonic function H(1)(Y,

Y; N), which now decom-

poses according to

H(1)(Y,

Y; N) = 4 ln N + h(Y,

Y) +

h( Y, Y) , (4.20) which turns out to be a harmonic function of !. Indeed, making use of (4.18) one nds that h(Y,

Y) now takes the form, h(Y,

Y) = 4 ! 4 !I i FIKLNKL [parenrightbig]

NIJ !J i FJMNNMN

+ 8 !IJNIJ 4 2 [bracketleftbigg]

[parenrightbig]

iFIJKL

2

3FIKMFJLNNMN[bracketrightbigg]

NIJNKL + O( 3) . (4.21)

We stress that, by construction, H(1) remains a symplectic function in the presence of the

term 4 ln N in (4.20). Furthermore h will separately transform as a function beyond linear order in and derivatives of h + 4 ln N will still transform as proper tensors. This must be the case because the transformations of h beyond the linear order depend only on S

through the tensor ZIJ, and likewise

h depends only on S through the tensor

ZIJ. As

we shall see, the non-holomorphic derivative of h will transform as a vector as it is not of rst order in . We also observe that the transformation of ! as specied in (4.19) follows precisely from the expression for 14h(Y,

Y) upon replacing NIJ by iZIJ, with the exception

of the term ln det

S[bracketrightbig]that is related to the explicitly non-harmonic term in (4.18). This is in line with the phenomenon noted below equation (3.19).

Now we return to the holomorphic anomaly equation. Following the discussion at the beginning of this section we rst determine,

@h @NIJ =

14@I h + 4 ln N

JHEP09(2014)096

[parenrightbig]

@J h + 4 ln N

[parenrightbig]

+ O( 3) . (4.22)

Here we have introduced a covariant derivative which ensures covariance under the symplectic transformations. On a holomorphic vector, VI, this covariant derivative takes the form,

DIVJ = @IVJ IJK VK , (4.23) where IJK is Christo el connection associated with the Kahler metric gI J = @I@ JK(Y,

Y)

= NIJ, with K the Kahler potential6

K(Y,

Y) = i

[bracketleftbig]

+ 2 DI@J h + 4 ln N

[parenrightbig]

. (4.24)

Observe that, for a Kahler space the non-vanishing connection components are IJK and

its complex conjugate J

K. The non-vanishing (up to complex conjugation) connection and curvature components are then equal to

IJ K = gK L@IgJ L = iFIJL NLK , RJK

L = @ JKL = NLM

YFI(Y) YI

F( Y)

[bracketrightbig]

F M N NNP FPJK . (4.25)

6Note that there is no uniformity in the literature regarding the overall sign of K. See, e.g. [15].

25

We remind the reader that combinations of higher derivatives of the holomorphic function F (Y) that involve also the matrix NIJ can transform covariantly under symplectic

transformations, as was pointed out at the end of section 2 (see e.g. (2.14)).

We should also stress that the above discussion pertains to the underlying Kahler geometry. Obviously the special di eomorphism related to the symplectic transformations form a subgroup of the group of holomorphic di eomorphisms. This is conrmed by evaluating the transformation of the connection under a symplectic transformation, using the results presented in section 2,

IJ K ! [S1]LI [S1]MJ LMNSKN @LSKM[bracketrightbig]. (4.26)

After these comments and clarications we return to equation (4.22). Noting the lack of holomorphicity in h resides in NIJ, we can now determine the anti-holomorphic derivative of h,

@h = i FIKLNKMNLN

+ O( 3) , (4.27)

where the right-hand side contains no terms linear in . However, one could consider the mixed derivative of h + 4 ln N, which does contain terms linear in given by

@@J h + 4 ln N

[parenrightbig]

FILN . (4.28)

The expression on the right-hand side is precisely equal to 4 RJ , where RJ = RKJ K

equals the Ricci tensor of the special Kahler manifold whose value follows from the second equation in (4.25). The equations (4.27) and (4.28) are the familiar holomorphic anomaly equations of the topological string.

In this section we introduced a deformation of proportional to the parameter which induced further corrections to of higher orders in . This deformation was not itself a proper function, but its variation under a symplectic transformation was harmonic. Obviously this deformation was not unique because the e ect of the symplectic transformation would remain the same upon adding a proper symplectic function to the deformation. Therefore we also considered adding a separate non-harmonic function to (cf. (4.6)). We concluded that this modication must lead to new terms in , as shown in (4.7), but they contribute only to the Hesse potential by an additive contribution of the original non-harmonic function. However, when dening H(1) we agreed that such additions should

be included as a separate symplectic functions in the expansion of the Hesse potential in terms of independent functions. Hence this additive term will not a ect H(1). This aspect is essential for deriving the holomorphic anomaly equation. However, when the non-harmonic function is of rst order in the deformation parameter, it will not contribute to the holomorphic anomaly equation, because it does not generate higher-order terms under the iteration, while (4.27) only receives contributions beyond the rst order. The rst-order contributions are instead governed by the separate equation (4.28).

26

JHEP09(2014)096

14@M h + 4 ln N

[parenrightbig]

[parenrightbig]

@N h + 4 ln N

2 DM@N h + 4 ln N

[parenrightbig][bracketrightbigg]

= 4 NKLNMNFJKM

We will not analyze this issue in further detail here, but there is one type of deformation of the anomaly equation that is worth recalling. Suppose that on the right-hand side of (4.27) we change h + 4 ln N by adding a function of the Kahler potential dened in (4.24) (which equals the Hesse potential with = 0). This Kahler potential satises the following identities,

@IK = NIJ YJ , @I@

JK = NIJ , DI@JK = 0 , (4.29)

which all have a geometrical meaning. Adding a function of K to h+4 ln N will introduce terms proportional to NMP YP or NNP

YP on the right-hand side of (4.27) that cancel when

contracted with the overall factor FIKLNKMNLN.

Finally, as is shown in (4.19), the transformation rule of ! has now acquired new terms of order 2. The derivatives of ! will thus receive corresponding contributions. In particular (4.13) and (4.15) will change. For the calculations in section 5 it is relevant to present the full expressions for the variations of !I and !IJ up to order 3. In view of their length we have listed these equations in appendix C.

5 Evaluating the third-order contributions

There are good reasons for evaluating also the contributions of order 3. One of them is that and H(1) are no longer obviously partially harmonic in higher orders. Another one

is that the contributions of third order have never been fully worked out explicitly for the topological string.

Let us again start with (3.19), but now approximated to terms of order 3,

~

(,

~

Y) = i ZIJ I J

JHEP09(2014)096

ZIJ

J

[parenrightbig]

+ 23 FIJK ZIL L ZJM M ZKN N + h.c. [parenrightbig]

2 IJ ZIK KZJL L + h.c. [parenrightbig]

+ 4 I J ZIK K

L + O( 4) , (5.1)

which must hold irrespective of the precise form of . To evaluate the right-hand side we must rst determine I to order 2, which follows from (4.18),

I = !I i FIJK NJK+ 2

!IJK NJK + i !JK +

!JK NJLFILMNMK[bracketrightbig]

2 [bracketleftbigg]

iFIJKLM

ZJL

43FIJLNNNP FKMP [bracketrightbigg]

NJKNLM

FJKLM

NJKNLNNMP FINP + 2i 2 FJKLFMNP + FJKL FMNP

NJMNKNNLQNPRFIQR + O( 3) = !I i FIJK NJK + 2 !IJK NJK

+ 2i FIJK NJMNKN

!MN + FMPR FNQSNPQNRS i FMNPQNPQ

[bracketrightbig]

2 [bracketleftbigg]

iFIJKLM

+ 2 2 FJKLM

43FIJLNNNP FKMP [bracketrightbigg]

NJKNLM

27

!MN + FMPR FNQSNPQNRS + i FMNPQNPQ +

O( 3) . (5.2)

We now observe that the above expression is no longer almost holomorphic and thus deviates from the results obtained before. The troublesome terms are contained in the last line of (5.2), which turns out to be equal to

2i FIJK NJMNKN M N + O( 3) , (5.3)

where we made use of the second equation in (4.14). It will be convenient to keep writing these non-holomorphic contributions in terms of non-holomorphic derivatives of . The crucial point to note is, however, that we have extracted an explicit factor of , whereas so far appeared only implicitly in . As we will demonstrate shortly, one consequence of our analysis is that the Hesse potential will involve additional symplectic functions, but now multiplied by explicit powers of .

Substituting the above result (5.2) and the last two equations of (4.14) into (5.1), we obtain,

~

(,

~

Y) (Y,

+ 2i FIJK NJMNKN

Y)

JHEP09(2014)096

=

i !I i FI N

ZIJ

!J i FJ N [parenrightbig]

+ 23FIJK ZIL !L i FL N

[parenrightbig]

ZJM !M i FM N

[parenrightbig]

ZKN !N i FN N

[parenrightbig]

4i !I i FI N

ZIJ

ZIJ [bracketleftbigg]iFJKLMN 43FJKMP NPQFLNQ[bracketrightbigg]NKLNMN

!J N + i!K NFJNK

[bracketrightbig]

+ 2i 2 !I i FI N

4i 2 !I i FI N

ZIJFJPQNMP NNQNKL FKLMN + 4 2 !I i FI N

ZIJFJRSNMRNQSNKNNLP FKLMFNPQ

2 !I i FI N

ZIJ !JK + FJMNFKPQNMP NNQ i FJKN [parenrightbig]

ZKL !L i FL N

[parenrightbig]

+ h.c.

I ZIJ FJKL NKMNLN M N + h.c. [bracerightbig]

4 I ZIJ FJKL NKMNLN

FPMN ZPQ

Q + O( 4) . (5.4)

The last two lines are not almost harmonic. The rst of these two lines arises as a result of the non-holomorphic terms noted in (5.3), and the last line originates from the manifestly non-harmonic term present at the end of the expression (5.1) (which has been included above upon replacing I J by the corresponding expression given in (4.14)).

It is now straightforward to verify with the help of (3.20) that these two lines are precisely generated upon assuming that will contain a term 4 IJ NIKNJL

K L at

+ 4

this order of iteration. This is quite a non-trivial result, because we are not just rewriting the expression (5.1) that was originally expressed in terms of and its derivatives, into a

28

similar expression! Rather, as already mentioned, we have now extracted an explicit power of , whereas so far the parameter only appeared implicitly in . This signals a new pattern that will become more manifest shortly.

Using the previous results of the transformation rules of the function ! and its derivatives, exhibited in (4.15) and (C.1), as well as the transformation rules (2.8) for multiple derivatives of the holomorphic function F , one can, after a fair amount of non-trivial manipulations, determine the expression for (Y,

Y), up to a symplectic function of Y and

Y,

(Y,

Y) = ln N

+ ! + 2 NIJ!IJ 2 NIJNKL !IK !JL !IJKL

[bracketrightbig]

+ 83i 2 FIJKNILNJMNKN !LMN

2 [bracketleftbigg]

2

3FIKMFJLNNMN[bracketrightbigg]

NIJNKL

+ 4i 2 NPQ FPQIJ + iFPIM NMNFQJN

NIKNJL !KL

+ 2 3

43 3FIJKFLPQFRST FUV W NILNJRNKUNPSNQV NTW + h.c.

4 IJNIKNJL

K L + O( 4) . (5.5) It is clear that the terms that are independent of the holomorphic function !(Y) are

becoming more and more numerous in higher orders. Note that the above result is almost harmonic, with the exception of the last term. Furthermore the almost harmonic terms take again the form of a sum over 1PI diagrams.

In the limit ! 0 the expression (5.5) for reduces to the original harmonic expression

that we started from initially in (4.1). With the exception of the last term in (5.5), which will recombine with other terms in due course, the almost harmonic terms have to be included into the expression for the function that encodes the e ective action. Hence they imply that the original non-harmonic modication ln N in (4.6) is incomplete and must be modied order-by-order by additional non-harmonic term. These terms will thus contribute to the e ective action, where they are expected to encode non-local interactions associated with the massless modes.

We remind the reader that is not a symplectic function and the next step is to determine the symplectic function H(1) in third order of , which follows upon substitution of

29

JHEP09(2014)096

iFIJKL

2

3 3NIJ

iFIJKLMNNKLNMN 4 FIJKLM NKNNLP NMQ FNPO

[bracketrightbig]

NIJFIJKLNKP NLQFPQRS NRS

+ 13FIJKL NIMNJNNKP NLQ FMNPQ[bracketrightbigg] + 4i 3FIJKL

NIJNKMNLNFMPQNPRNQSFRSN+ NIMNJNNKP NLQFMNRNRSFPQS

[bracketrightbig]

2 3FIJKFLPQFRST FUV W NILNJP NKRNQUNSV NTW

the above result for into (3.18). Let us rst concentrate on the terms that are not almost harmonic. They originate from three di erent sources. First there is the last term in (5.5) (which appears with an additional factor 4 in H(1)), then there are explicit non-harmonic terms in the expression (3.18) for H(1), and nally there are the non-holomorphic contri

butions in (5.2) that were summarized in (5.3), which induce corresponding modications in H(1). These three contibutions are

16 IJNIKNJL

K L

FPQKNKL L

16 i INIJFJKL NKMNLN

M N + h.c. (5.6)

and they combine into

16 IJ + iFIJMNMP P

16 INIJFJMNNMP NNQ

K L iFKLNNNQ Q[parenrightbig], (5.7)

which equals precisely 16 times the non-harmonic symplectic function G1(Y,

Y) that

has been listed in appendix B, up to terms of order 3. The function H(1) thus acquires

the form

H(1)(Y,

Y; N) = 4 ln N + h(Y,

Y) +

h( Y, Y) 16 G1(Y,

Y) + O( 4) , (5.8)

where

h(Y,

Y) = 4 ! 4 !I i FIKLNKL

JHEP09(2014)096

NIK NJL

NIJ !J i FJMNNMN

+ 8 !IJNIJ 8 NIJNKL !IK !JL !IJKL

[bracketrightbig]

+ 32

3 i 2 FIJKNILNJMNKN !LMN

+ 16i 2 N FIJ + iFIMNMNFJN

NIKNJL !KL

[parenrightbig]

16 !IN NIJ !J i FJN + 8 !I i FIN

NIJ !JK NKL

!L i FLN [parenrightbig]

+ 83i FIJKNILNJMNKN !L i FLN [parenrightbig]

!M i FMN[parenrightbig][parenleftBigg]!

N

i FNN

16i !I i FIN

NIJFJKL NKMNLN !MN + 8i 2 !I i FIN

[parenrightbig]

NIJ

FJKL +43iFJKM NMNFLN[parenrightbigg]NKLN

8i !I i FIN

NIJ

FJK + iFJPNPQFKQ

N

NKL !L i FLN

[parenrightbig]

16 2 !I i FIN

NIJFJKL NKMNLN

FMN + iFMPNPQFNQ

N

[parenrightbig]

4 2 [bracketleftbigg]

iFIJKL

2

3FIKMFJLNNMN[bracketrightbigg]

NIJNKL

30

8 3 3NIJ

iFIJKLMNNKLNMN 4 FIJKLM NKNNLP NMQ FNPQ

[bracketrightbig]

+ 8 3

NIJFIJKLNKP NLQFPQRS NRS

+ 13FIJKL NIMNJNNKP NLQ FMNPQ[bracketrightbigg] + 16i 3FIJKL

NIJNKMNLNFMPQNPRNQSFRSN+ NIMNJNNKP NLQFMNRNRSFPQS

[bracketrightbig]

8 3FIJKFLPQFRST FUV W NILNJP NKRNQUNSV NTW

16

3 3FIJKFLPQFRST FUV W NILNJRNKUNPSNQV NTW

+ O( 4) . (5.9)

As before the !(Y) will transform holomorphically such that its explicit transformation

rule follows from (5.9) upon making the substitution NIJ ! iZIJ. We have veried by

explicit calculation that this is indeed the case, which provides an explicit check on the calculation.

At this point one can again determine the holomorphic anomaly equation following the same steps as before. As it turns out the result coincides with (4.27), but now valid up to order 4. This can be seen as an indication that the holomorphic anomaly equation will not acquire further corrections in higher orders.

6 Summary and conclusions

Based on the observation that the duality transformations act di erently on the function that encodes the e ective action than on the topological free energy, we have proposed a conceptual framework based on the Hesse potential of real special geometry to understand the relation between the two. Subsequently we have studied the Hesse potential by iteration for a generic e ective action, rst starting from a Wilsonian e ective action and subsequently by considering the e ect of non-harmonic deformations. The Hesse potential decomposes into an innite series of symplectic functions and we established that the topological string free energy could reside in precisely one of them. This function is then subject to the holomorphic anomaly equation, irrespective of its dynamical content.

The results of an explicit iteration of the genus g 3 topological string free energy

fully conrms the correctness of the proposal. We should again stress that we concentrate on the generic features of this relationship, rather than on specic models. The relations that we nd are thus universal, but it is not assumed that the resulting expression for the topological string free energy will have an actual realization as a topological string model. This is the reason that we do not make contact with specic aspects of the topological string, such as the wave function approach and the issue of background dependence [35].

One implication of our result is that we are also able to relate the non-holomorphic terms associated to the e ective action to the ones that appear in the topological string free energy. This is perhaps not so surprising in view of the fact that there is a qualitative

31

JHEP09(2014)096

relation between the pinching of a cycle that decreases the genus of the Riemann surface in the topological string and the integration over massless modes in the e ective action! But it is important to realize that, while our construction demonstrates how to construct the topological string free energy from a given e ective action, the inverse is clearly not possible because the e ective action is equivalent to the full Hesse potential, while the topological string free energy constitutes only part of the Hesse potential.

At several occasions we already mentioned that the results of this paper are consistent with our previous work [15, 16], where we analyzed the same issues by using a variety of di erent strategies. It is therefore of interest to compare the present results with the results of the past. To highlight some interesting issues we therefore reconsider the earlier results on the FHSV model, which were based on imposing the exact S- and T-dualities of this model on the e ective action. There we used a slightly di erent perturbative procedure and we worked in a parametrization based on special coordinates. Subsequently we determined the Hesse potential by iteration, in a way that is similar to what was done in the present paper. We then discovered that the Hesse potential did indeed contain terms that cannot belong to the topological string free energy at genus-2, because they do not depend (anti-)holomorphically on the topological string coupling. As we now know, those are the contributions that do not belong to the function H(1), but at that stage such a systematic classication was not available. Nevertheless, the terms that did depend (anti-)holomorphically on the topological string coupling were consistent with the results obtained from the FHSV topological string [27], except that the proportionality constant remained ambiguous in view of the fact that the corresponding expression was duality invariant, so that its leading contribution could be changed by a corresponding change in the e ective action where we had only imposed the requirement of invariance. Interestingly, the present approach which emphasizes covariance rather than invariance, claries this result. To appreciate all this we have summarized some of details of the derivation of the genus-2 FHSV topological string free energy in appendix E.

Finally we wish to return to the issue of BPS black hole entropy in supersymmetric theories with eight supercharges, which formed a major motivation for the present work. In [31, 33] a general formula for BPS black hole entropy was given based on Walds denition of black hole entropy [36], which was covariant under dualities and incorporated the higher-derivative corrections to the Weyl multiplet that we already referred to in section 1. (Incidentally, there is now increasing evidence that other higher-derivative couplings will not contribute to BPS black hole entropy by virtue of certain non-renormalization theorems [3, 4].) The formula of [33] was reinterpreted in [29] in terms of a mixed partition function which was subsequently related to the topological string. However, this relationship depended crucially on the assumption that the topological free energy and the function that encodes the supergravity action are directly related, or perhaps even identical! As we have been trying to emphasize in this paper, the topological string does capture certain string amplitudes that should also follow from the e ective action. But this does not imply that the topological string and the action are given by the same function.

We should perhaps add here that it is possible to present the supergravity input in the form of the Hesse potential (analogous to converting a Lagrangian into a Hamiltonian

32

JHEP09(2014)096

description), for which one can dene a modied black hole partition function associated with the canonical ensemble [22]. This would o er an e ective way to make contact with the topological string, were it not for the fact that the black hole solutions from which one starts in supergravity are, by denition, solutions of the full e ective action. Therefore they should involve the full Hesse potential, which, as we have shown in this paper, consists of an innite series of symplectic functions of which just one will correspond to the topological string free energy. Finally we note that the work of this paper pertains specically to theories with eight supercharges, while a substantial part of the literature on BPS black holes is based on theories with sixteen supercharges (although often treated in a reduction to eight supercharges). An extension of the work of this paper for theories to sixteen supersymmetries should therefore be of interest.

Acknowledgments

We acknowledge helpful discussions with Murad Alim, Michele Ciraci, Edi Gava, Thomas Grimm, Babak Haghighat, Albrecht Klemm, Thomas Mohaupt, Kumar Narain, Hirosi Ooguri,lvaro Osorio, Ashoke Sen, Samson Shatashvili, Marcel Vonk, Edward Witten and Maxim Zabzine. The work of G.L.C. is partially funded by Fundao para a Cincia e a Tecnologia (FCT/Portugal) through project PEst-OE/EEI/LA0009/2013 and through the grants PTDC/MAT/119689/2010 and EXCL/MAT-GEO/0222/2012. The work of B.d.W. is supported by the ERC Advanced Grant no. 246974, Supersymmetry: a window to non-perturbative physics. This work is also supported by the COST action MP1210 The String Theory Universe.

We thank our respective institutes for hospitality during the course of this work. S.M. thanks the Alexander von Humboldt Stiftung for a reinvitation grant which enabled her to visit the Max Planck Institut, Mnchen, and Dieter Lst for o ering hospitality. We also thank the Max Planck Institut fr Gravitationsphysik (Albert-Einstein-Institute) for hospitality extended to us during the completion of this work.

A Non-holomorphic deformation of special geometry

In this appendix we prove the following theorem.

A.1 Theorem

Given a Lagrangian L(,

responding Hamiltonian H(, ) =

i) and a complex function F (x, x), such that,

2 Re xi = i ,

2 Re Fi(x, x) = i , where Fi = @F (x, x)

@xi . (A.1) The function F (x, x) is dened up to an anti-holomorphic function and can be decomposed into a holomorphic and a purely imaginary non-harmonic function,

F (x, x) = F (0)(x) + 2i (x, x) . (A.2)

33

JHEP09(2014)096

) depending on n coordinates i and n velocities

i i L(,

i, with cor-

), there exists a description in terms of

complex coordinates xi = 12(i + i

The equivalence transformations take the form,

F (0) ! F (0) + g(x) , ! Im g(x) , (A.3)

which results in F (x, x) ! F (x, x) + g(x).

The Lagrangian can then be expressed in terms of F and ,

L = 4[Im F ] , (A.4)

so that the Hamiltonian takes the form

H = 4 Im F

[bracketrightbig]+ 2 i Im xi . (A.5)

This expression is identical to the expression for the Hesse potential given in (2.4), up to an overall minus sign. Alternatively the Hamiltonian can be written as

H = i(xi

F xFi) 4 Im [bracketleftbigg]

F (0)

1

2xi F (0)i[bracketrightbigg]

2(2 xi i x ) , (A.6)

where Fi = @F/@xi, F = @ F/@x, and similarly for the functions F (0) and . When the function F (0)(x) is homogeneous of second degree, the second term will vanish. The third term is a measure of the deviation from homogeneity of . This decomposition is known from the entropy function for BPS black holes [22].

Furthermore, the 2n-vector (xi, Fi) turns out to dene a complexication of the phase-space coordinates (i, i) that transforms precisely as (i, i) under canonical (symplectic) reparametrizations,

xi

Fi(x, x)

JHEP09(2014)096

[parenrightBigg] ![parenleftBigg][parenleftBigg]

~xi ~Fi(~x, ~x)

[parenrightBigg]

Uij Zij

= Wij Vij

[parenrightBigg][parenleftBigg][parenleftBigg]

xj

Fj(x, x)

[parenrightBigg]

, (A.7)

where the real matrix is an element of Sp(2n, R). Observe that for the real part of the vector (xi, Fi), the above transformation is the standard canonical transformation on coordinates and momenta. The equation (A.7) is integrable so that the symplectic transformation leads to new functions ~F(0) and ~

.

A.2 Proof

The proof of this theorem proceeds as follows. First note the following complex vectors,

xi = 12 [parenleftbigg]

i + i@H @i

[parenrightbigg]

, yi = 12 [parenleftbigg]

i i

@H

@i

[parenrightbigg]

, (A.8)

constructed out of two canonical pairs, one comprising the variables i, i and the other one the derivatives of the Hamiltonian, which transform in the same way under canonical transformations (here we use that the Hamiltonian transforms as a function under canonical transformations).

34

In view of the inverse Legendre relation,

i = @H/@i, the complex xi in (A.8) coincide

with the xi dened previously. Furthermore, when writing the Lagrangian as a function of the xi and x, it follows that

@L(x, x)

@xi = 2iyi . (A.9)

Here we used that the Legendre transformation leading to the Hamiltonian yields @L/@i =

@H/@i (where on the right-hand side i is kept constant and on the left-hand side

i is

kept constant). Observe that we did not make use of the equations of motion.

Subsequently we write L as the sum of a harmonic and a non-harmonic function,

L = 2i

F (0)(x) F(0)(x)

[bracketrightbig]+ 4 (x, x) = 4[Im F ] , (A.10)

so that (A.9) reads

yi = @

@xi

~

xi,i + i) = H(xi + xi, yi + yi). The dual quantities (~xi,i) and the new Hamiltonian will satisfy the same relation as the original quantities.

The new Lagrangian ~L which follows from an inverse Legendre transformation of the new

Hamiltonian, will depend on ~xi and ~xi. Applying the same steps as before we then nd the new function ~F(~x, ~x).

There is one subtlety here and that is that the decomposition of the function F into F (0) and is ambiguous. The ambiguity is resolved by noting that the symplectic transformation (A.7) can also be applied to the the vector (xi, Fi(0)(x)). In that case the new function F (0) can be determined separately, as the holomorphic case is known to be integrable, and it is given in (2.7), up to a constant and terms linear in ~xi. The latter terms can be determined explicitly, for instance, by using that F (0) 12xiFi(0) transforms as a function under duality. Having determined the functions ~F(0) and ~F, the non-harmonic

function ~

follows. This completes the proof of the theorem.

A.3 Corollary

Let us derive the well-known result (see, for instance, [37]) that the rst-order derivative of the Lagrangian with respect to some parameter (such as a coupling constant) transforms as a function under symplectic transformations (A.7). We denote this parameter by g and note that @gH(, ; g) transforms as a function under canonical transformations (which do

not depend on g, but they act on g-dependent quantities as shown in (A.7)) for any value

35

F (0)(x) + 2i (x, x)

[bracketrightbig]. (A.11)

Thus yi = @iF (x, x) with F (x, x) = F (0)(x)+2i (x, x), up to an arbitrary anti-holomorphic function, and Re yi = i. The Hamiltonian then follows from (A.5), which leads to the expression (A.6). Hence we now have shown that (xi, Fi) equals the vector (xi, yi) which transforms under canonical transformations according to (A.7).

What remains to be proven is that the result of the transformation (A.7) is integrable. The vector (xi, yi) transforms according to (A.7) into a vector (~xi,i) while the

Hamiltonian, which depends on (xi + xi, yi + yi), transforms as a function under canonical transformations, so that(~xi +

JHEP09(2014)096

of g. Subsequently, take the derivative of the Hamiltonian with respect to g keeping i and i xed. Consequently one derives

@H(, ; g)

@g = i

@L(,

; g) @

i

[parenrightBigg]

@

i @g

@L(,

; g)@g =

@L(,

; g)@g , (A.12)

which proves the assertion.

B The symplectic functions H(a)i for a 2 and some other functions

that do not initially appear in HHere we collect the explicit results for various functions H(a)i that appear in (3.16). These

functions have been determined by iteration in orders of and its derivatives. We present the terms of the iterative expansion up to O( 5). At the end of this appendix we will be

presenting two more functions, G1 and G2, that did not initially show up in the iterative

procedure for the Hesse potential carried out in this paper. These will only be given up to order 3. Note that here and elsewhere we only use indices, J, . . . when they are necessary. For instance, we will write FIJK and FIJK because F is holomorphic and F is anti-holomorphic, so that there is no need for using holomorphic or anti-holomorphic indices, whereas for the derivatives of the real quantity we write I and to distinguish holomorphic and anti-holomorphic derivatives. The reason is that NIJ has no unique assignment of (anti)holomorphic indices, so that there will never be a consistent pattern of contractions based on holomorphic and anti-holomorphic indices. Denoting (N )I

NIJ J and (N

)I NIJ

J, we have obtained the following expressions,

H(2) = 8 NIJ I

JHEP09(2014)096

J 16

IJ(N )I(N )J + I J(N )I(N )J + h.c. [bracketrightbig]

FIJK(N )I(N )J(N )K h.c. [bracketrightbig]

+ 16

3 i[bracketleftbig][parenleftBigg][parenleftBigg]F

8i

IJKL + 3iFIJMNMNFNKL

[parenrightbig]

(N )I(N )J(N )K(N

)L h.c.

[bracketrightbig]

+ 16

IJK (N )I(N )J(N )K + h.c.

+ 16

[bracketleftbig][parenleftBigg][parenleftBigg]

IJ K + iFIJMNMN N K

[parenrightbig][parenleftBigg][parenleftBigg](N

)I(N )J(N )K + 2(N )I(N

)J(N

)K

[parenrightbig] [bracketrightbig]

+ 16

[bracketleftbig][parenleftBigg][parenleftBigg]

JK i

FIJMNMN NK

[parenrightbig][parenleftBigg][parenleftBigg](N

[parenrightbig] [bracketrightbig]

)I(N

)J(N

)K + 2(N

)I(N )J(N )K

+ 32

h IK NKL LJ (N )I(N )J + h.c. [bracketrightBig]

+ 32 IK NKL L J (N )I(N

)J

+ 16i

hFIJK NKL LM (N )I(N )J(N

)M + 2(N )M(N )I(N

)J

[parenrightbig]

h.c.

[bracketrightBig]

+ 16i

hFIJK NKL L M (N )I(N )J(N )M h.c.[bracketrightBig]

+ 8 (N )I (N )J FIJMNMN FNKL(N

)K(N

)L

+ 32

h(N )I IJ NJK K L(N )L + h.c.

[bracketrightBig]

+ 32

h(N )I IJ NJK K L(N )L + h.c. [bracketrightBig]

+ 32

h(N )I IJ NJK KL(N )L + h.c.

[bracketrightBig]

36

+ 16i

h(N )I(N )JFIJKNKL LM (N )M h.c.[bracketrightBig]

+ 32

h(N )I I J NJK KL (N )L + h.c. [bracketrightBig]

+ 32

h(N )I I J NJK K L (N )L[bracketrightBig], (B.1)

H(3)1 =

8 3iFIJK(N

)I(N

)J(N

)K

+ 8i FIJK(N

)I(N

)J NKL

2 L M (N )M + 2 LM (N )M iFLMN(N )M(N )N[bracketrightbig], (B.2)

H(3)2 = 8 IJ + iFIJK(N )K [parenrightbig]

(N

)I(N

)J

JHEP09(2014)096

83i FIJKL + 3iFM(IJNMNFKL)N

[parenrightbig]

h3(N )I(N )J(N )K(N )L 2(N )I(N )J(N )K(N )L [bracketrightBig]

16

3 IJK 3 (N

[parenrightbig]

)I(N

)J(N )K (N

)I(N

)J(N

)K

16 IJ

K(N

)I(N

)J(N

)K

16i FIJKNKL LM

[bracketleftbig]

(N

)I(N

)J(N

)M + (N

)I(N

)J(N )M + 2(N

)I(N )J(N

)M

[bracketrightbig]

16 (N

)I IJ NJK KL (N

)L

32 (N )I IJ + iFIJK(N )K

[parenrightbig]

NJL L M i

FLMN(N

)N

[parenrightbig]

(N )M

+ 16i(N )I(N )JFIJKNKL

L M iFLMN(N ) N[parenrightBig] (N )M

16 (N )I

J NJK K M (N )M

32 (N

[parenrightbig]

)I IJ + iFIJK(N )K

NJL LM (N )M

16i (N

)I (N

)JFIJKNKL L M (N

)M , (B.3)

H(3)3 = 16 I

J(N

)I(N )J

+ 16

h2(N )I(N )J IKNKL L J + I JM (N )M

[parenrightbig]

+ (N

)I I JNJK iFKLM(N )L(N )M + 2 KL(N )L + 2 K L(N

)L

[parenrightbig]

+ 2i(N

)I(N )JFIJKNKL L M (N )M + h.c.

[bracketrightBig]

, (B.4)

H(4)1 = 32 (N

[parenrightbig]

)I IJ + iFIJK(N )K

NJL L M i

FL M N (N

)N

[parenrightbig]

(N )M , (B.5)

H(4)2 = 32 (N )I

J NJK KL (N

)L (B.6)

H(4)3 = 8 FIJMNMN

FNKL (N

)I(N

)J(N )K(N )L , (B.7)

H(4)4 =

43i FIJKL + 3i FMIJNMNFKLN

[parenrightbig]

(N

)I(N

)J(N

)K(N

)L , (B.8)

H(4)5 = 16i FIJKNKQ

QL(N

)L (N

)I(N

)J , (B.9)

H(4)6 = 16i FIJKNKL

L M i

FLMN(N

)N

[parenrightbig]

(N

)I(N

)J(N )M , (B.10)

H(4)7 = 16 IJ

K + iFIJL NLM M K

[parenrightbig]

(N

)I(N

)J(N )K , (B.11)

37

H(4)8 = 32 (N

)I IJ + iFIJK(N )K

[parenrightbig]

NJL LM (N

)M , (B.12)

H(4)9 = 16i (N

)M . (B.13)

As indicated above there are also other functions that do not initially appear in H.

We give two examples below up to terms of order 3.

G1 = IJ + iFIJMNMP P

NIK NJL

)I(N

)JFIJKNKL LM (N

K L iFKLNNNQ Q[parenrightbig], (B.14)

G2 = I JNIL NJK K L . (B.15)

Note that the functions G1,2 take the form of 1PI connected diagrams, whereas the functions H(0)i do not.

C Transformation rules of !I and !IJ to order 2

In this appendix we list the transformation rules of some of the derivatives of the function !. For the rst four multiple derivatives those were already given in (4.13) and (4.15) to order . However, the transformation rule of ! itself is known to order 2 (cf. (4.19)) so that also the derivatives can be determined in that order. In section 5 we in fact need the transformation rules for !I and !IJ to order 3. In view of their length we display these transformations in this appendix. The results read as follows,

~

!I = [S1]JI [bracketleftBig]

JHEP09(2014)096

!J + FJKLZKL + 2i !JKL ZKL

2i !J + FJKLZKL [parenrightbig]

Z ! + FMNZMN

+ i ! + FKLZKL

Z FJ Z

! + FMNZMN[parenrightbig]

! + FMNZMN [parenrightbig][bracketrightbig]

2i 2ZK FJ ZL FKLMN FKM Z FLN [parenrightbig]

ZMN

2i ZK FJ ZL

!KL FKL Z

+ i 2

FJKLMN 43FJKM Z FLN[bracketrightbigg] ZKL ZMN[bracketrightBig]+ O( 3) ,

~

!IJ = [S1]K(I [S1]LJ)

h!KL FKLMZMN(!N + FNPQ ZPQ)

+ FKLMN FKMP FLNQ ZPQ ZMN

+ 2i

[bracketleftbig][parenleftBigg]!

KLP Q

FKL Z !PQ

ZPQ 2 (!K ZFLZ) [bracketrightbig]

2i !K + FKMNZMN

Z

!L + FLPQ ZPQ

+ 2i FKLZ ! + FMNZMN

Z ! + FPQ ZPQ

2i !KL + FKLMNZMN

[parenrightbig]

Z ! + FPQZPQ

+ 4i !K + FKMNZMN

Z FL Z ! + FPQ ZPQ

+ i ! + FMN ZMN

Z

FKL 2 FKZFL

[bracketrightbig] Z

iFRST (ZR FKL

[parenrightbig] [bracketleftbig]ZS[parenleftBigg]!

+ FMN ZMN

[parenrightbig][bracketrightbig] [bracketleftbig]ZT

! + FPQ ZPQ [parenrightbig]

! + FPQ ZPQ

[parenrightbig][bracketrightbig]

!L + FLMN ZMN[parenrightbig]

38

+ 4i FXZFKZ

ZX

+ 4i FXKZFLZ

ZX

! + FMN ZMN[parenrightbig]

4i FXZFKZ

ZX FL Z

! + FMN ZMN[parenrightbig]

2i (FKLRS FKLZFRS 2FKRZFLS) ZRT ZSU ! + FXY ZXY[parenrightbig][bracketrightbig]

+ i 2

FKLMNZ 4

FKMNZFLZ

[parenrightbig][bracketrightbig] ZMN

+ 2i 2 ZMFKZN

[parenrightbig] [bracketleftbig]F

MNP Q

FMPZFNQ

FLZQ

ZXY FY ZFLZ

+ 8i 2 FKMNP FQRS ZMQ ZNR ZPFLZS

2i 2 ZMFKLZN 2ZMFKZFLZN [parenrightbig]

FMNPQ FMPZFNQ ZPQ

43i 2 FKLMNP FQRS + FKMNP FLQRS

[parenrightbig]

[bracketrightbig][parenleftBigg]ZP

[parenrightbig]

2i 2 ZFKZFX

JHEP09(2014)096

ZMQ ZNR ZPS

+ 2i 2 FKLZX ZM FX ZN FMNPQ FMP Z FNQ

[parenrightbig]

ZPQ

i 2 FKLZ [bracketleftbigg]

[parenrightbig]

43FMP Z FNQ[bracketrightbigg] ZMN ZPQ

2i 2 Z FK Z FM Z FL Z FN ZMN[bracketrightBig]+ O( 3) . (C.1)

Again we have sometimes represented indices by bullets whenever they are contracted in an ope or closed stringlike fashion and there is no ambiguity.

D Topological free energies for genus g 3 that satisfy the holomorphic

anomaly equation

In this appendix we list the topological free energies F (g)(Y,

Y) that follow from expanding H(1), given in (4.20), order-by-order in . To order 3 we obtain

H(1)(Y,

Y; N) = 4[bracketleftBig]

F (1)(Y,

FMNPQ

Y) + F (2)(Y,

Y) + F (3)(Y,

Y) + h.c.

[parenrightbig][bracketrightBig]

16 G1(Y,

Y) + O( 4) , (D.1) where the function G1 is given in (B.14),. The symplectic functions F (g)(Y,

Y) that appear

at order g are given by (for g = 1, 2, 3)

F (1)(Y,

Y) = !(1) +

!(1) + ln det NIJ ,

F (2)(Y,

Y) = !(2) NIJ !(1)I i FIKLNKL

[parenrightbig][parenleftBigg]

!(1)J i FJPQNPQ

+ 2 NIJ!(1)IJ 2[bracketleftbigg]

[parenrightbig]

iNIJNKLFIJKL

2

3NIJFIKLNKP NLQFJPQ[bracketrightbigg]

,

F (3)(Y,

Y) = !(3) 2 NIJ !(2)I !(1)J + 2 !(1)IJ NIK !(1)K NJL !(1)L

+ 2

3i FIJKNIP !(1)P NJQ !(1)Q NKL!(1)L

+

2i NIJ!(2)IFJKLNKL 4 NIJ!(1)IKLNKL!(1)J

4 iNIJNKLFILMNMN!(1)KN !(1)J 2i FIJKLNKLNIP !(1)PNJQ!(1)Q

39

+ 2 FIKP NKLFLQJNPQNIR!(1)RNJS!(1)S

4i !(1)IJNIK!(1)KNJLFLPQNPQ+ 2 FIJKNIP !(1)PNJQ!(1)QNKRFRST NST

+ 2 NIJ !(2)IJ 2 NIJNKL !(1)IK !(1)JL [bracketrightbig]

2i NIJFIKLMNNKLNMN !(1)J 4NIJNKLFILMNMNFKNPQNPQ !(1)J

83NIJFIMNP NMKNNLNPQFKLQ !(1)J

4i NIJNKLFILMNMNFKRT NRP NTQFNPQ !(1)J+ 4i NIJ!(1)IMNNMNFJKLNKL 4NIJNMNFINP NPQ!(1)MQFJKLNKL

2 !(1)IJNIKFKPQNPQNJLFLRSNRS

4 FIJMNNMNNIK!(1)KNJLFLRSNRS

4i FIMNNMP FJPQNQNNIK!(1)KNJLFLRSNRS

2 i FIJKNIP !(1)PNJQFQST NST NKRFRUV NUV

+ 2NIJNKL !(1)IJKL +

83i FIJKNILNJP NKQ !(1)LPQ

+ 4i FIJRS NIJNRKNSL !(1)KL 4 FIPQFJRSNIKNJLNPRNQS !(1)KL[bracketrightbigg]

2 NIJFIMNPQNMNNPQFJKLNKL

+ 4i NIJNPMFIMNNNQFPQRSNRSFJKLNKL

+ 83i NIJFIMNP NMKNNLNPQFKLQFJRSNRS

4 NIJNPMFIMNNNQFPT U NTRNUSFQRSFJKLNKL + 2i FIJMNNMNNIKFKPQNPQNJLFLRSNRS

2 FIMNNMT FJTU NUNNIKFKPQNPQNJLFLRSNRS

2

3 FIJKNIP FPMN NMNNJQFQST NST NKRFRUV NUV

83NIJ FIJKLP NKRNLSNPT FRST

+ 2 NIJ FIJKL NKP NLQ FPQRS NRS

+ 23FIKPS NIJNKLNPQNSR FJLQR

+ 4i NIJFIJKL NKP NLQ FPST NSUNTV FQUV

+ 4i FIJKL NIP NJQNKRNLSFPQU NUV FV RS

2 FIJKFLPQFRST FUV W NILNJP NKRNQUNSV NTW

2

3i FIJKLPQ NIJNKLNPQ +

43FIJKFLPQFRST FUV W NILNJRNKUNPSNQV NTW [bracketrightbigg]

40

+ 2

JHEP09(2014)096

+ 3

. (D.2)

P1n=1 !(n)(Y), where we count !(n)(Y) as being of order n, following (4.16). The non-holomorphicity of F (g)(Y,Y) is entirely contained in

the quantities NIJ. Observe that F (1) is real, while the higher F (g) (g 2) are not.

The expressions for F (g) given above were obtained by explicit construction and they satisfy the holomorphic anomaly equations (4.27) of perturbative topological string theory (g 2),

@F (g) = i FIJK NJMNKN"2 DM@NF (g1) + g1

Xr=1@MF (r) @NF (gr)

Here we expanded !(Y) as !(Y) =

[bracketrightBigg]

, (D.3)

where DM denotes the covariant derivative introduced in (4.23). The expression for F (2) has been obtained before by other methods [11, 26, 27] based on a direct integration of (D.3). Partial results for F (3) have been given in [11].

E An application: the FHSV model

In this appendix we illustrate our results in the context of the FHSV model [28] and compare them to earlier results obtained in [16] by means of a related but slightly di erent approach. Here we restrict ourselves to second order. In the type-II description, the FHSV model corresponds to the compactication on the Enriques Calabi-Yau three-fold, which is described as an orbifold (T2 K3)/Z2, where Z2 is a freely acting involution.

The massless sector of the four-dimensional theory comprises 11 vector supermultiplets, 12 hypermultiplets and the N = 2 graviton supermultiplet. The classical moduli space of the vector multiplet sector equals the special-Kahler space,

Mvector = SL(2)

SO(2)

O(10, 2)

O(10) O(2)

JHEP09(2014)096

, (E.1)

which is encoded in the classical holomorphic function

F (0)(Y) = Y1 Ya

abYb

Y0

, (E.2)

where a, b = 2, . . . , 11, and the symmetric matrix ab is an SO(9, 1) invariant metric of indenite signature. The two factors of the special-Kahler space are associated with T2/Z2 and the K3 ber, and special coordinates for these two spaces are denoted by S = iY1/Y0 and T a = iYa/Y0. This leads to the following expression for N det[2 Im[F (0)IJ]] and H| =0,

N = c (S + S)10 (T + T)2

2 , H| =0 = (S +S)(T + T)2 |Y0|2 , (E.3)

where c is an irrelevant constant. Here we use the notation that T 2 T a abT b and likewise for |T |2. Observe that N is not covariant under symplectic reparametrizations, while H| =0

is covariant. Note that in the present approach we are making use of a specifc parametrization dened by (E.2). Therefore the covariance under symplectic reparametrizations is not always clear, and instead we may have to rely on the S- and T-duality invariances that we will discuss below. This was also the strategy used in [15, 16].

41

Y) into powers of as ! = !(1) + !(2) +

O( 3) and = (1) + (2) + O( 3), respectively. Following the discussion in [16], we start

with the expression for (1), known from threshold corrections and from the topological string side [38, 39]. In the conventions of [15], it is given by

(1)(Y,

Y) =

1 4

Subsequently, we expand both !(Y) and (Y,

12 ln[ 24(2S) (T )] +12 ln[ 24(2S) ( T)]

+ 2 ln[(S + S)3(T + T)2]

[bracketrightbigg]

. (E.4)

It is invariant under S-duality transformations belonging to the (2) subgroup of SL(2; Z), and also invariant under the T-duality group O(10, 2; Z), since (T ) is a holomorphic automorphic form of weight 4 [40], transforming under the T-duality transformation T a !

T a [T 2]1 as

(T ) ! [T 2]4 (T ) . (E.5)

We can now recast (E.4) in the form of (4.6),

(1)(Y,

Y) = !(Y) +

!( Y)

18 ln N + (Y,

Y) , (E.6)

so that = 1/(8) and = 1, with

!(1)(Y) =

32 ln 2(2S)

18 ln (T ) +

14 ln Y0 ,

JHEP09(2014)096

(Y,

Y) =

14 ln

(S + S)(T + T)2|Y0|2[bracketrightbig], (E.7)

where we note that (Y,

Y) transforms as a function because it is equal to the logarithm

of H| =0, the classical part of the Hesse potential.

Next, we insert these expressions into (2). Here we recall that in the presence of a function (Y,

Y), the expression for (2) is not simply obtained by the second line of (4.18),

but to this we also have to add -dependent terms, as shown in (4.7). Thus, we have

2)(Y,

Y) = [bracketleftBig]

!(2) + 2 NIJ!(1)IJ 2[bracketleftbigg]

iFIJKL

2

3FIKMFJLNNMN[bracketrightbigg]

NIJNKL + h.c.

[bracketrightBig]

!(1) + ln N[parenrightBig] J + I J + h.c.

[bracketrightBig]. (E.8)

Then, direct evaluation of this results in

(2) =

!(2) + 1(Y0)2[bracketleftbigg]1642 G2(2S)@ ln (T )@T a@ ln (T )@Ta 1322 G2(2S)@2 ln (T ) @T a@Ta[bracketrightbigg]

+ 1

(Y0)2

42(2S, 2 S) @ (1)@T a@ (1)@Ta +13222(2S, 2 S)

+ NIJ

h2 @I

@2 ln (T )@T a@Ta +@ ln (T )@T a@ ln (T ) @Ta[parenrightbigg][bracketrightbigg]

1 G2(2S) @ (1)@T a@ (1)@Ta +14@ ln (T )@Ta@ (1)@T a@ (1)@S[bracketrightbigg]+ h.c.

, (E.9)

where

G2(2S) = 12@S ln 2(2S) , 2(2S, 2 S) = G2(2S) + 1

2(S + S) . (E.10)

42

The rst line of (E.9) contains purely holomorphic terms, while the second line contains terms that are invariant under S- and T-duality. The last line contains the terms that are neither holomorphic nor invariant under S- and T-dualities. They were already obtained in [15] by requiring invariance of the model under S- and T-duality, and thus were determined up to invariant terms. Here, the duality invariant terms are unambiguously determined and given by the second line of (E.9), as we just established. The reason is that the scheme presented in this paper ensures the validity of the holomorphic anomaly equation. This implies that invariant terms cannot be arbitrarily included, as we discussed in section 4. Earlier results obtained in [15, 16] are fully consistent with the ones given above.

Next, we compute the symplectic function F (2), which is constructed from (2) as follows. Recalling (E.8), we write (2) as (2) = +

. Then, from (D.2) we infer the relation F (2) = NIJ (1)I (1)J, where however (and di erently from (D.2)) (1) now

also contains (Y,

Y), cf. (4.6). As we have observed in the text below (4.10), the terms

depending on cancel in the higher order result for H(1), and therefore F (2) will not

depend on . We obtain

F (2) = !(2) + 1 (Y0)2

[bracketleftbigg]

1642 G2(2S)

@ ln (T ) @T a

@Ta

1322 G2(2S)

@2 ln (T ) @T a@Ta

[bracketrightbigg]

JHEP09(2014)096

@ ln (T )

+ 1

322 (Y0)2

2(2S, 2 S) @2 ln (T )

@T a@Ta +

@ ln (T ) @T a

@ ln (T ) @Ta

[parenrightBigg]

(E.11)

2(2S, 2 S) @ log

(T ) [(T + T)2]4 @T a

@ log

(T ) [(T + T)2]4 @T a .

The rst line contains purely holomorphic terms, while the second and third lines are given in terms of non-holomorphic combinations that are S- and T-duality invariant. The holomorphic contributions in the rst line should, however, be invariant as well. We can verify this by making use of the transformation rule (4.19) for !(2). We have checked that

the rst line is indeed invariant under S-duality, and we expect the same for T-duality. At this stage we are not able to give an explicit representation of !(2) as a function of Y0, S

and T a that generates the desired transformations. The expression for F (2) given above is in agreement with the nding of [27].

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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