Hessian geometry and the holomorphic anomaly

Abstract

Abstract

We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kähler geometry which we characterise. The holomorphic anomaly equation arises in this framework from the integrability condition for the existence of a Hesse potential.

Published for SISSA by Springer

Received: December 20, 2015 Accepted: February 12, 2016 Published: February 24, 2016

JHEP02(2016)161

Hessian geometry and the holomorphic anomaly

G.L. Cardosoa and T. Mohauptb

aCenter for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior T ecnico, Universidade de Lisboa,Av. Rovisco Pais, 1049-001 Lisboa, Portugal

bDepartment of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, U.K.

E-mail: mailto:[email protected]

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

Web End [email protected]

Abstract: We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kahler geometry which we characterise. The holomorphic anomaly equation arises in this framework from the integrability condition for the existence of a Hesse potential.

Keywords: Di erential and Algebraic Geometry, Supergravity Models, Topological Strings

ArXiv ePrint: 1511.06658

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP02(2016)161

Web End =10.1007/JHEP02(2016)161

Contents

1 Introduction 1

2 Review of special Kahler geometry 52.1 A ne special Kahler manifolds 52.2 Conical a ne special Kahler manifolds 7

3 The holomorphic deformation 83.1 Deformation of the immersion 83.2 Real coordinates and the Hesse potential 103.3 Deformed special Kahler geometry 123.4 Stringy complex coordinates 143.5 The symplectic covariant derivative 163.6 Anomaly equation from the Hessian structure 19

4 The non-holomorphic deformation 214.1 Non-holomorphic deformation of the prepotential 214.2 Real coordinates and the Hesse potential 234.3 The symplectic covariant derivative 254.4 The holomorphic anomaly equation 264.5 From Hessian structure to the full anomaly equation 28

5 Concluding remarks 29

A Connections on vector bundles 31

B Special coordinates 32

C Symplectic transformations and functions 33

1 Introduction

Quantum gravity is expected to manifest itself in an e ective eld theory framework through higher derivative terms. Supersymmetry provides some control over such terms, and in a theory of N = 2 vector multiplets coupled to supergravity a certain class of higher derivative terms can be described by generalizing the prepotential which encodes all the couplings at the two-derivative level. While this function remains holomorphic within a Wilsonian framework, the inclusion of threshold corrections due to massless particles is known to induce non-holomorphic corrections to the couplings. In fact these corrections

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JHEP02(2016)161

are required for consistency with electric-magnetic duality and are essential for incorporating higher derivative corrections to black hole entropy. While supergravity provides a powerful tool to organise an e ective action for quantum gravity, the actual computation of couplings requires a speci c theory. String theory is the natural candidate, and in particular the higher derivative corrections to N = 2 vector multiplets are captured by the topological string. However, the relation between supergravity and the topological string is subtle, and non-holomorphic corrections are incorporated di erently in the respective formalisms. In this paper we develop a new geometrical description of the higher derivative corrections on the supergravity side, by showing that they can be understood in terms of an extended scalar manifold which carries a deformed version of special geometry. We also derive various exact relations between the variables used in supergravity and in the topological string. The most interesting result we obtain is that the holomorphic anomaly equation which controls the non-holomorphic corrections in both the supergravity and topological string formalism can be derived from the integrability condition for the existence of a Hesse potential on the extended scalar manifold.

Let us next introduce our topic in more technical terms. In four dimensions, the complex scalar elds residing in N = 2 vector multiplets parametrize a scalar manifold which is the target space of the non-linear sigma-model that enters in the Wilsonian Lagrangian describing the couplings of N = 2 vector multiplets at the two-derivative level. The scalar manifold is an a ne special Kahler manifold in global supersymmetry, and a projective special Kahler manifold in local supersymmetry; both types of target space geometry are referred to as special geometry [1{7]. Special geometry, when formulated in terms of complex variables Y I, is encoded in a holomorphic function F (0)(Y ), called the prepotential. When formulated in terms of special real coordinates, it is the Hesse potential that plays a central role. For a ne special Kahler manifolds, the Hesse potential is related to the prepotential by a Legendre transform [8].

When coupling the N = 2 vector multiplets to the square of the Weyl multiplet, the resulting Wilsonian Lagrangian, which now contains higher derivative terms proportional to the square of the Weyl tensor, in encoded in a generalized prepotential F (Y; ), where denotes a complex scalar eld residing in the lowest component of the square of the Weyl multiplet. The complex scalar elds (Y I; ) will be called supergravity variables in the following. The prepotential F (0)(Y ) is obtained from F (Y; ) by setting = 0. Electric-magnetic duality, a central feature of N = 2 systems based on vector multiplets, then acts by symplectic transformations of the vector (Y I; FI), where FI = @F=@Y I. While F (Y; )

itself does not transform as a function under symplectic transformations, F = @F=@ does [9]. The associated Hesse potential, obtained by a Legendre transform of Im F (Y; ), is also a symplectic function.

Away from the Wilsonian limit, the coupling functions encoded in F receive nonholomorphic corrections, in general. In supergravity models arising from string theory, these modi ed coupling functions can be derived in the context of topological string theory [10, 11]. The precise relation between these two computations is subtle, however [12]. The coupling functions computed in topological string theory depend on stringy variables (YI; ) that do not coincide with the supergravity variables (Y I; ) (unless = 0). The

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precise relation between these two sets of coordinates was discussed in [12] and was used to express the supergravity Hesse potential (which is a symplectic function) in terms of stringy variables. The Hesse potential is not any longer obtained from the holomorphic generalized prepotential F (Y; ) that characterizes the Wilsonian Lagrangian. Instead, it is computed from a deformed version of F that is not any longer holomorphic. It was then laboriously shown, by means of power law expansions, that the Hesse potential contains a unique subsector H(1) that comprises coupling functions F (n)(Y;

Y) that, for n 2, satisfy

Xr=1@JF (r)@KF (nr) 2 DJ@KF (n1)!: (1.1)

Here, denotes the deformation parameter that characterizes the deviation from the Wilsonian limit. The superscript (0) in

F (0)JKI indicates that this quantity has been formed by taking derivatives of F (0)(Y ), and that indices have been raised using the inverse NIJ(0) corresponding metric N(0)IJ = 2ImF (0)IJ. If no such superscript is present, as in

F JKI, then

it is understood that we take derivatives of the generalized prepotential F (Y; ), and that indices are lowered and raised using NIJ = 2ImFIJ. This convention is applied throughout the paper.

While in topological string theory = 12 [10], one may ask a more general question,

namely whether irrespective of the value of , the holomorphic anomaly equation (1.1) can be understood in terms of Hessian structures and Hessian geometry. This is indeed the case, as we will show in this paper. Namely, the anomaly equation (1.1) may be viewed as the integrability condition for the existence of a Hesse potential in supergravity. This is simplest to establish in the case when = 0, as we will explain next (when = 0, the coupling functions are encoded in F (Y; ) on the supergravity side, and hence still holomorphic in supergravity variables).

Any a ne special Kahler manifold M can be realised as an immersion into a complex symplectic vector space V [7], as we will review in section 2.1. When passing from the prepotential F (0) to the generalized prepotential F (Y; ), this construction gets extended, giving rise to a holomorphic family of immersions and deformed a ne special Kahler manifolds, which combine into a complex manifold ^

M = M [notdef]

C. By pulling back the standard

M becomes equipped with a Kahler metric g and a at torsion free connection r which we use to de ne special real coordinates. Taking the Leg

endre transform of the generalized prepotential F (Y; ) as a Hesse potential, we can then de ne a Hessian metric gH. When analysing the integrability condition for the existence of a Hesse potential, namely that rgH must be a completely symmetric rank three tensor

in complex coordinates, one infers the anomaly equation (1.1) with = 0, as we will show in subsection 3.6. This anomaly equation can also be viewed as a consequence of a tension between preserving holomorphicity and symplectic covariance, as follows. We introduce stringy variables YI, and we derive various properties of the di erence Y I = YI Y I.

We then express the symplectic function F (Y; ) in terms of stringy variables. By taking multiple derivatives @ [notdef]Y of F we obtain symplectic functions @n F [notdef]Y that we express

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the holomorphic anomaly equation of topological string theory,

@F (n)

@

YI

= i

F (0)JKI

n1

JHEP02(2016)161

Hermitian form of V , the space ^

back in terms of supergravity variables. Then, setting = 0, we obtain the set of symplectic functions F (n) = 2i Dn1 F =n![notdef] =0 introduced in [9] that, for n 2, satisfy the

anomaly equation (1.1) with = 0. The symplectic covariant derivative D introduced in [9] is related to @ Y I=@ , and thus it is simply a consequence of the passage from stringy to supergravity variables. The non-holomorphicity induced by the coordinate transformation re ects the tension between holomorphicity and symplecticity, and is thus a universal feature of the deformation induced by the passage from the prepotential F (0)(Y ) to the

generalized prepotential F (Y; ).The aforementioned Hessian structure condition (namely that rgH is totally symmet

ric) gives a master equation for F ,

@D F =

F JKIF JF K ;

which, upon applying Dn to it and setting = 0, yields, by induction, the anomaly equation (1.1) with = 0 for the functions F (n) de ned above. This master equation for

F is on par with the one derived for the topological free energy Ftop,

Ftop(Y;

where Q is an expansion parameter related to the topological string coupling, which satis es

@Ftop = i

F (0)JKI@JFtop @KFtop :

Next, let us discuss the case [negationslash]= 0. When turning on , F ceases to be holomorphic.

Thus, starting from a non-holomorphic generalized prepotential F as in [12], we investigate the consequences for the master equation for F that result from the Hessian structure condition. The equation we obtain is quite complicated. To compare it with (1.1), we specialise to a particular deformation, proportional to NIJ(0), where NIJ(0) is the inverse of

N(0)IJ = i(F (0)IJ

Xn=1

QnF (n)(Y;

Y) ;

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Y; Q) =

1

F (0)IJ). Working at lowest order, we show that when setting = 0, the master equation for F equals the anomaly equation (1.1) with n = 2. The anomaly equation for the higher F (n) can, in principle, be obtained from this master equation by acting with multiple covariant derivatives D on it. Here, D denotes the symplectic covariant derivative introduced in [13], which is based on a non-holomorphic generalized prepotential F . We note that while the speci c -deformation we picked is tied to the topological string, the framework presented in this paper is quite general and can be applied to other deformed systems, such as those discussed in [13].

The paper is organised as follows. In section 2 we review the extrinsic construction of special Kahler manifolds through immersion into a model vector space. In section 3 we deform this construction by passing from the prepotential F (0) to the holomorphic

generalized prepotential F (Y; ). We introduce the Hessian structure based on special real coordinates, and use the latter to introduce stringy variables (YI; ), as in [12]. We relate

the di erence @ Y I=@ to the symplectic covariant derivative of [9], which we subsequently

use to derive a master equation for F . Next, we use the Hessian structure to derive a

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di erent equation for F , which we then relate to the holomorphic anomaly equation (1.1) with = 0. In section 4 we redo the analysis, but now based on a non-holomorphic generalized prepotential F . In the concluding section we compare the approach of [12] for obtaining the holomorphic anomaly equation with the approach taken here. In the appendices we have collected some standard material to facilitate the reading of the paper.

2 Review of special Kahler geometry

2.1 A ne special Kahler manifolds

We start by reviewing the intrinsic de nition of (a ne) special Kahler geometry given in [6]: a Kahler manifold (M; g; !) with complex structure J is a ne special Kahler if there exists a at, torsion-free, symplectic connection r such thatdrJ = 0 : (2.1)

We will refer to r as the special connection. Our convention for the relation between

metric g, Kahler form ! and complex structure J is

!([notdef] ; [notdef]) = g([notdef] ; J[notdef]) ;

or, in local coordinates

!ac = gabJbc :

The de nition of the exterior covariant derivative dr is reviewed in appendix A. As shown

in appendix B, in r-a ne coordinates qa the condition (2.1) becomes@[aJbc] = 0 ; (2.2)

while the coe cients !ab are constant. This in turn implies that

@agbc = @bgac ;

which by applying the Poincar e lemma twice shows that the Kahler metric is Hessian,

gab = @2a;bH ;

where the real function H is called a Hesse potential. The coordinate-free version of this local de nition of a Hessian metric is as follows: given a Riemannian metric g and a at, torsion-free connection r, the pair (g; r) is called a Hessian structure, and g is called a

Hessian metric, if the rank-3 tensor rg is totally symmetric. It is easy to see that, given

a Kahler manifold with a at, torsion-free, symplectic connection r, the condition (2.1) is

equivalent to the requirement that the metric g is Hessian (that is rg is totally symmetric).

On an a ne special Kahler manifold one can choose the r-a ne coordinates (qa) = (xI; yI) to be Darboux coordinates, i.e. such that the Kahler form takes the standard form

! = 2dxI ^ dyI = abdqa ^ dqb ; ( ab) =

1

1 0 !

JHEP02(2016)161

0

:

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The coordinates qa are called special real coordinates, and are unique up to a ne transformations with symplectic linear part.1

As shown in [6] the above de nition is equivalent to the well known alternative de -nition in terms of special holomorphic coordinates Y I and of a holomorphic prepotential F (Y I). We will now review the holomorphic formulation of special Kahler geometry in the context of the universal extrinsic construction of [7], which allows to realize any a ne special Kahler manifold, at least locally. For simply connected a ne special Kahler manifold this construction in fact works globally. The universal construction realises special Kahler manifolds M as immersions into the standard complex symplectic vector space V = T

where dim V = 2 dim M = 4n. We now review some details, which we are going to use for our later generalised construction.

Let (Y I; WI) be complex Darboux coordinates on V = T

= dY I ^ dWI

is the standard complex symplectic form on V , and

V = i ([notdef]; [notdef]) = i

dY I dW I dWI dY I = gV + i!V (2.3)

is the associated Hermitian form, gV is a at (inde nite) Kahler metric, and !V the corresponding Kahler form.

Next, let

: M ! Vbe a non-degenerate, holomorphic, Lagrangian immersion of a complex manifold M of (real)

dimension 2n into V . We can assume, without loss of generality, that the image (M) is realized as a graph, that is the immersion has been chosen such that, when identifying M locally with its image, we can take Y I as coordinates on the locally embedded M, so that in terms of coordinates (Y I; WI) the immersion takes the form

: M ! V ; (Y I) [mapsto]!(Y I; FI(Y )) :

This situation is generic, and can always be achieved, at least locally, by a symplectic transformation. Since the immersion is Lagrangian, we have = 0, which is readily seen to be the integrability condition for the existence of a holomorphic function F such that FI = @F=@Y I. In the non-generic situation where (M) is not realized as a graph, the immersion is still well de ned and can be described using a complex symplectic vector (Y I(Z); WI(Z)), where Z = (ZI) are holomorphic coordinates on M. However, the components Y I cannot be used as coordinates on M, and the components WI fail to satisfy the integrability condition for the existence of a prepotential. This is well known in the literature as a symplectic vector (or holomorphic section) without prepotential [14]. We will assume in the following that we are in a generic symplectic frame where a prepotential exists.

1We remark that the special coordinates (qa) di er from standard Darboux coordinates by a conventional normalization factor, see appendix B for details.

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Cn,

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[similarequal]

Cn

C2n. Then

Since the immersion is non-degenerate, the pull back M = V of the Hermitian

form V to M is a non-degenerate Hermitian form, which by decomposition into real and imaginary part de nes a non-degenerate metric and two-form:

M = gM + i!M :

The explicit form of M is

M = NIJdY I d

Y J ;

where

JHEP02(2016)161

NIJ = 2ImFIJ = i(FIJ

FIJ) = @2

@Y I@

Y J [i(Y K

FK FK

Y K)] :

From this expression for NIJ it is manifest that the metric gM is Kahler, with Kahler potential

K = i(Y I

FI FI

Y I) ; (2.4)

and a ne special Kahler with prepotential F .

Special holomorphic and special real coordinates are related as follows. Given special holomorphic coordinates Y I on M, the corresponding special real coordinate are given by the real part of the complex symplectic vector (Y I; FI):

Y I = xI + iuI(x; y) ;

FI = yI + ivI(x; y) :

Moreover, the holomorphic prepotential and the Hesse potential are related by a Legendre transform [8]:

H(x; y) = 2ImF (x + iu(x; y)) 2yIuI(x; y) :

We remark that special real coordinates are well de ned, at least locally, in any symplectic frame (including those without a prepotential) as a consequence of the non-degeneracy of the symplectic form. For simply connected special Kahler manifolds they are even globally de ned functions, since the immersion is global, though not necessary global coordinates, since the immersion need not be an embedding.

2.2 Conical a ne special Kahler manifolds

While a ne special Kahler manifolds are the scalar manifolds of generic rigid N = 2 vector multiplets, conical a ne special Kahler manifolds are the scalar manifolds of rigid super-conformal vector multiplets. These are in turn the starting point for the construction of the coupling of vector multiplets to Poincar e supergravity, which proceeds as follows:2 (i) start with a theory of n + 1 superconformal vector multiplets, (ii) gauge the superconformal algebra; this introduces various connections which reside in the Weyl multiplet, (iii) partially gauge x the superconformal transformations to obain a theory of n vector multiplets coupled to Poincar e supergravity. In this construction the projective special Kahler manifold

M of the supergravity theory arises as a Kahler quotient of a conical a ne special

2This is reviewed in [15, 16].

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Kahler manifold. Since we will not use this construction, we refer the interested reader to the literature.

The additional condition implied by superconformal symmetry is, in terms of special holomorphic coordinates, that the prepotential is homogeneous of degree two under complex scale transformations,

F ( XI) = 2F (XI) ; 2

This is equivalent to the statement that the Hesse potential is homogeneous of degree two under real scale transformations of the special real coordinates, and invariant under the U(1) part of

C . The condition can also be formulated in a coordinate-free way [7]: a conical a ne special Kahler manifold3 is a special Kahler manifold equipped with a homothetic Killing vector eld satisfying

r = D = Id ;

where r is the special connection, D the Levi-Civita connection, and Id the identity en

domorphism on T M. One can then show that this implies the existence of an in nitesimal holomorphic homothetic

C action on M, which is generated by and J. To obtain a projective special Kahler manifold by a Kahler quotient, one needs to assume that this group action is free and proper.

3 The holomorphic deformation

3.1 Deformation of the immersion

One possible deformation of the vector multiplet action is to give it an explicit dependence on a background chiral multiplet [9], see [15] for a review. By identifying this chiral multiplet with the Weyl multiplet W 2, one can describe a particular class of higher derivative terms. Compatibility with superconformal symmetry determines the scaling behaviour of the chiral multiplet, while insisting on a local supersymmetric action implies that the dependence is holomorphic, that is the standard F-term vector multiplet action is deformed by allowing the prepotential, as a function on superspace, to depend explicitly on the chiral multiplet. After integration over superspace, the action is a local functional of the elds, which contains additional terms involving holomorphic derivatives of the prepotential with respect to the background. When identifying the chiral multiplet with the Weyl multiplet W 2, one nds that the auxiliary elds cannot any longer be eliminated in closed form, but only iteratively, thus generating an expansion in derivatives. Such an action is naturally interpreted as a Wilsonian e ective action.

In the following we will investigate how the introduction of a background eld can be interpreted as a deformation of special geometry. Since we focus on the scalar geometry, the background chiral eld enters through its lowest component, a complex scalar denoted

3Apart from conical the term conic is also use in the literature.

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C :

JHEP02(2016)161

C ;

. The generalized prepotential F (Y; ) is holomorphic in Y I and , and (graded) homogeneous of degree two, that is

F ( Y; w ) = 2F (Y; ) ; 2

C

where w is the weight of under scale transformations. If is the lowest component of the Weyl multiplet W 2, then w = 2. Our geometric model for the deformation parameterized by is a map

: ^

M := M [notdef]

! V ; (Y I; ) [mapsto]!(Y I; FI(Y; )) ; (3.1) which can be interpreted as a holomorphic family of immersions : M ! V ; (Y I) [mapsto]!

(YI; FI(Y; )), that de ne a family of a ne special Kahler structures on M. While is a scalar under symplectic transformations, it enters into the transformation of the complex symplectic vector (Y I; FI(Y; )), and other objects, through the generalized prepotential.

Our set-up is consistent with [9], in particular we can draw on the various formulae for symplectic transformations derived there.

We de ne a metric and a two-form on ^

M = M [notdef]

C by pulling back the canonical

hermitian form V :

= V = g + i! = NIJdY I d

Y J + i

FI dY I d

iFI d d

Y I ;

JHEP02(2016)161

FIJ). We assume that is non-degenerate, which certainly is true for su ciently small .4 In the following, holomorphic coordinates on ^

M are denoted

(vA) = (Y I; ). Using the conventions

da db = 12 (da db + db da) ; da ^ db = da db db da ;

we obtain the metric

g = gABdvAd

vB = NIJdY Id

where NIJ = i(FIJ

Y J + i

FI dY Id

iFJ d d

Y J ;

which is a Kahler metric gAB = @2A; BK with Kahler potential

K = i

Y IFI(Y; )

Y ;

)Y I

; (3.2)

and

! =

i2NIJdY I ^ d

Y J + 12

FI dY I ^ d

1

2FI d ^ d

Y I

is the associated Kahler form. The Kahler metric gAB has occured in the deformed sigma model [17], which provides a eld theoretic realization of our set-up.

4In applications will not necessarily be small, but it is reasonable to expect that is non-degenerate, at least generically.

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3.2 Real coordinates and the Hesse potential

Following [18], we now de ne special real coordinates and a Hesse potential in presence of the deformation. Special real coordinates are de ned by

Y I = xI + iuI(x; y; ;

) ; FI = yI + ivI(x; y; ;

) ;

and the (generalized) Hesse potential is related to the (generalized) prepotential by a Legendre transform:

H(x; y; ;

) = i(F

F ) 2yIuI(x; y; ;

) ;

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where F = F (Y (x; u(x; y; ;

)); ).

We are interested in the coordinate transformation between special complex and special real coordinates5

(x; u; ;

) [mapsto]!(x; y(x; u; ;

); ;

)

and its inverse

(x; y; ;

) [mapsto]!(x; u(x; y; ;

); ;

) :

When rewriting derivatives between the coordinate systems, one needs to carefully use the chain rule: when di erentiating a function f = f(x; y(x; u; ;

); ;

) the following

formulae are useful

@f @xI

u

= @f

@xI

y

+ @f

@yK

x

@yK

@xI ;

1 0 0 0

@u @x

@f @uI

= @f

@yK

@yK

@uI ;

@f @

x;u

x = @f

@

x;y

x + @f

@yK

x

@yK

@ :

The Jacobians for the coordinate transformations take the form

D(x; u; ;

)

D(x; y; ;

) =

0

B

B

B

B

@

1 0 0 0

@y @x

1 0 0 0 0

1

1 0 0 0 0

1

y

@u

@y

x

x;y

@u @

x;y

1

C

C

C

C

A

@u @

0 0

0

B

B

B

B

@

and

D(x; y; ;

)

D(x; u; ;

) =

u

x

1

C

C

C

C

A

@y @u

@y @

x;u

@y @

x;u

:

0 0

5We nd it convenient to work with and

when using special real coordinates instead of decomposing them into their real and imaginary parts.

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1 0 0 0

1

2 R 12N

1

2 FI

By the chain rule it is straightforward to evaluate

D(x; y; ;

)

D(x; u; ;

) =

1 0 0 0 0

1

1 0 0 0 N1R 2N1 N1FI N1

FI

0

B

B

B

@

1

2

FI

;

0 0

1

C

C

C

A

where 2FIJ = RIJ + iNIJ. This matrix can easily be inverted, with the result:

D(x; u; ;

)

D(x; y; ;

) =

0 0

1 0 0 0 0

1

0

B

B

B

@

1

C

C

C

A

JHEP02(2016)161

:

In order to transform the Kahler metric to special real coordinates, the following relations are useful:

@H@xI = 2vI ;

@H@yI = 2uI :

Moreover, using the chain rule one computes:

@vI @xJ

y

= 12 N + RN1R

IJ ;

@vI @yJ

=

@uJ

@xI

x

IJ ;

y

= 2 N1

= 12RIJ :

Using the notation (qa) = (xI; yI), the Kahler metric g expressed in special real variables takes the form

g = @2H@qa@qb dqadqb +

@2H

@qa@ dqad +

@vI @uJ

x

@2H

@qa@

dqad

;

where

@2H

@qa@qb

= N + RN1R 2RN1

2N1R 4N1

!

;

and

FIMNMNFN ; @2H

@xI@

= 2FIMNMN

@2H

@xI@ = 2

FN ;

@2H

@yI@

= 2NIJ

@2H

@yI@ = 2NIJFJ ;

FJ :

In the undeformed case the Kahler metric is simultaneously Hessian. To see whether this is still the case, we rst note that ^

M can be equipped with an a ne structure and thus a Hessian metric gH with Hesse potential H can be de ned. This requires the existence of a at, torsion-free connection. For xed we know that the special connection r is

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C by imposing

rdxI = 0 ; rdyI = 0 ; rd = 0 ; rd

such a connection, with a ne coordinates xI; yI. We can extend r to a at, torsion-free

connection on ^

M = M [notdef]

= 0 :

If xI; yI are not global coordinates on M, we use that M can be covered by special real coordinate systems, which are related by a ne transformations with symplectic linear part. Since for xed [negationslash]= 0 the map still induces an a ne special Kahler structure, special

real coordinate systems extend to ^

M and provide it with the a ne structure required to de ne a at torsion-free connection.

Upon computing the components of the Hessian metric gH explicitly, we realize that is not equal to the Kahler metric g. The di erence between the two metrics is

gH g = @2H|x;y =

@2H

@ @ d d + 2

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@2H

@ @

d d

+ @2H@

@

d

;

where

@2H

@ @

= NIJFI

FJ ; @2H

@ @ = iF + NIJFI FJ ;

@2H

@

@

= i

F + NIJ

FI

FJ :

We remark that these metric coe cients are symplectic functions, see [9], which is necessary in order that gH g is a well de ned tensor eld (which we know to be the case, because

gH and g are both metric tensors). We further remark that

2H = K 2i F + 2i

F

di ers from the Kahler potential (3.2) by a Kahler transformation. Therefore 2H, taken as a Kahler potential, de nes the same Kahler metric g = gK as K. However, when taking K as a Hesse potential one does not get the Hessian metric gH. While a Kahler potential is unique up to Kahler transformations, a Hesse potential is unique up to a ne transformations. Moreover, since our Hessian metric has de nite scaling properties, we can impose that the Hesse potential is homogeneous of degree two, which is automatic in the way we have de ned it as the Legendre transform of the generalized prepotential. If homogeneity is imposed on top of using special real coordinates, then the Hesse potential is unique up to symplectic transformations. We remark that the Hesse potential is the sum of two symplectic functions. Di erent linear combinations of these two functions de ne di erent metrics. By inspection one nds that de ning the (generalized) Hesse potential as the Legendre transform of the generalized prepotential leads to a particularly simple form of the coe cients @2H|x;y. We will see later how the Hessian metric gH encodes the

holomorphic anomaly equation.

3.3 Deformed special Kahler geometry

We are now in position to demonstrate that ^

M carries itself a deformed version of a ne special Kahler geometry. We have already seen that g is a Kahler metric with Kahler form

{ 12 {

!. To compare this with the two-form 2dxI ^ dyI, which is the Kahler form on M, we

compute

2dxI ^ dyI =

i2NIJdY I ^ d

Y J

1

2FI d ^ d

Y I + 12

FI dY I ^ d

+12FI dY I ^ d +

1

2

FI d

Y I ^ d

; (3.3)

and therefore the Kahler form can be written as

! = 2dxI ^ dyI

1

2FI dY I ^ d

1

2

FI d

Y I ^ d

:

JHEP02(2016)161

This shows in particular that 2dxI ^ dyI, when considered as a form on

^

M, is not of type

(1; 1) (since ! is, and both di er by pure forms). Using the rewriting

FI dY I ^ d = dF ^ d = d( dF ) ;

we nd

! = 2dxI ^ dyI +

1

2d( dF ) +

1

2d(

F ) : (3.4)

Thus the di erence between the Kahler forms ! of ^

M and 2dxI ^ dyI of M is exact, so

that both forms are homologous. The deformation involves the function F = @ F , which

plays a central role in describing the deformation and should be viewed as the supergravity counterpart of the topological free energy Ftop. First, note that while the generalized prepotential F , and its higher derivatives @n F with n > 1, are not symplectic functions,

F is a symplectic function [9]. Moreover, it is independent of the undeformed (two-derivative) prepotential F (0)(Y ) = F (Y; = 0), but contains all the information about the

deformation. We remark that while within the present construction F is holomorphic, this condition will be relaxed later.

Next we compute

r! =

12d(FI ) (dY I ^ d ) + c:c: (3.5) which shows that ! is not parallel, and the connection r is not a symplectic connection on

^

M. This shows that while ( ^

M; g; !; r) is Kahler, it is not special Kahler. The deformation

is controlled by an exact form, which is determined by the symplectic function F .

The fourth condition on a special connection is that the complex structure is covariantly closed. To compute the exterior covariant derivative of the complex structure J, we note that the vector elds @xI ; @yI ; @ ; @

de ne a r-parallel frame which is dual to the r-

parallel co-frame dxI; dyI; d ; d

. Using this one veri es that

12@@xI +12FIJ@@yJ = 12dFIJ @ @yJ :

Using that drJ = dJaea Ja ^ drea where ea is any basis of sections of T

^

M, so that

r

@@Y I = r

drea = rea, we nd

drJ =

idY I ^12dFIJ + c:c: @ @yJ :

{ 13 {

Note the rewriting

dY I ^ dFIJ = dY I ^ FIJ d = d(FIJdY I) = d(FI d ) ; where we used symmetry of FIJ and the chain rule. Therefore

drJ = (id(FI d ) + c:c:)

@@yI = iFIJ dY J ^ d + c:c:

C; J; g) is a Kahler manifold with Kahler form !, equipped with a at, torsion-free connection, such that r! and drJ are given by (3.5) and (3.6). We will

call such manifolds deformed a ne special Kahler manifolds. Since our de nition involves the map , this is not an intrinsic de nition, but the name for a speci c construction.

For completeness we remark that the pullback of the complex symplectic form of V is non-vanishing:6

= FI dY I ^ d = d( dF ) :As we by now expect, the right hand side is exact and controlled by F .

3.4 Stringy complex coordinates

The framework introduced so far is based on a generalized holomorphic prepotential F (Y; ), a complex symplectic vector (Y I; FI(Y; )) and a map : ^

M ! V which in

troduces a Kahler metric g on ^

M = M [notdef]

@@yI ; (3.6)

which is non-vanishing. As a consistency check, observe that it is manifest that d2r = 0, which must be true because r is at. Since the complex structure J of

^

M is not covariantly closed, the fourth condition required on the connection r in order to de ne a special Kahler

manifold is also violated. Again the deformation involves an exact form constructed out of the function F .

In summary, ( ^

M = M[notdef]

C, which deviates from being special Kahler if F [negationslash]= 0. Although is a symplectic scalar, symplectic transformations of data derived

from F or the symplectic vector (Y I; FI) depend on . If we expand F in a power series

F (Y; ) =

1

JHEP02(2016)161

Xg=0F (g)(Y ) g ;

then the functions F (g)(Y ) are holomorphic and homogeneous of degree 2 2g, but they

are not symplectic functions, and transform in a complicated way under symplectic transformations.

When the background is identi ed with the Weyl multiplet W 2, our formalism describes an Wilsonian e ective action for vector multiplets which includes a certain class of higher derivative terms. The same class of terms can be described using the topological string, but the formalism used in this context is di erent. There is no generalized prepotential, but instead one works with an undeformed complex symplectic vector (YI; F (0)I(Y)). The information which is encoded in the symplectic function F in the supergravity formalism

6It is of course clear already for dimensional reasons that ^

M cannot be a (locally immersed) Lagrangian

submanifold of V .

{ 14 {

is then di erently encoded in a hierarchy of genus-g topological free energies F (g)(Y;

Y)

which individually are symplectic functions, at the expense of not being holomorphic. The deviation from holomorphicity is controlled by the holomorphic anomaly equation. Elaborating on [12], we will now show that the relation between the two frameworks can be understood as a coordinate transformation. This will proceed in two steps. First we will show that when starting from the holomorphically deformed special geometry introduced so far, one obtains a hierarchy of free energies, where F (1) is holomorphic, while the F (g) with g > 1 are non-holomorphic and satisfy a version of the holomorphic anomaly equation where the two-derivative term is absent. This is not quite the situation for the topological string, where already F (1) is non-holomorpic and the anomaly equation requires an additional two-derivative term. In the next section we will generalize our deformed special geometry by making it explicitly non-holomorphic, and then show that by a coordinate transformation we obtain the full anomaly equation.

The relation between the supergravity coordinates Y I and the stringy coordinates YI

is de ned by imposing that the corresponding special real coordinates agree [12]:

2xI

2yI

!

= Y I +

JHEP02(2016)161

Y I FI(Y; ) +

FI(

Y ;

)

!

= YI +

YI

!

:

F (0)I(Y) +

F (0)I(

Y)

This implicitly de nes a non-holomorphic coordinate transformation between complex coordinates on ^

M,

(Y I; ) [mapsto]!(YI; ) ; (3.7) which we parametrize as [12]

YI = Y I + Y I(Y;

Y ; ;

) :

Note that by construction YI = Y I for = 0. In particular, the YI still are holomorphic

coordinates on M. If [negationslash]= 0 the coordinate transformation can be constructed itera

tively [12]. Since this gets complicated very soon, we will focus on statements that can be made without expansion or iteration.

To this end, let us consider the two-form 2dxI ^ dyI given in (3.3). Using (3.7), we

express this two-form in the new complex variables (YI;

YI), obtaining

2dxI ^ dyI =

i 2NIJ

@ Y J@YKdYK ^ dYI @ Y J @

YK

d

YK ^ d

YI

+ IK JL @ Y I@YK JL + IK @ Y J @

YL

dYK ^ d YL

+2@ Y I

@ dxJ ^ d + 2

@ Y I @

dxJ ^ d

+FI dxI ^ d +

FI dxI ^ d

; (3.8)

where in the last two lines we combined various terms into terms containing dxI. We now

convert all di erentials appearing in (3.8) to the real at frame (dxI; dyI; d ; d

) using

dYI = dxI + iNIK(0)R(0)KJ dxJ 2iNIJ(0) dyJ ;

{ 15 {

where here and in the following we use a notation where the subscript or superscript (0) indicates that a quantity has been calculated using the undeformed prepotential F (0)(Y) =

F (Y; = 0). Then, by comparing the di erentials on both sides of the resulting expression, we obtain the relations

@ Y J@ = iNJKFK ; (3.9)

NIJ =

IK @ Y I@YK+ @ Y I @

YK

NKJ(0) ;

N[IK @ Y K @YJ]

= 0 ;

where in the last equation the square bracket denotes antisymmetrization of the uncontracted indices.

3.5 The symplectic covariant derivative

The advantage of the stringy coordinates is that the variable does not enter into symplectic transformations. Thus given any symplectic function G(Y; ) (not necessarily holomor

phic), then the symplectic transformation behaviour is not modi ed when taking partial derivatives with respect to . In particular, if G(Y; ) is a symplectic function, then so

is @G=@ . In contrast, when using the supergravity variables Y I, then partial derivatives with respect to change the symplectic transformation behaviour. For example, while the derivative F of the generalized prepotential is a symplectic function, its derivatives like

F are not [9]. Using (3.9) there is a systematic way to compensate for this behaviour. Suppose G(Y; ) is a symplectic function, given in supergravity variables, which for the time being we assume to be holomorphic.7 Expressing G in stringy variables, we obtain G = G(Y (Y; ); ).8 When regarding G as a function of Y and , the partial derivative

with respect to is again a symplectic function. Now apply the chain rule:

@G @

Y

= @G

@

JHEP02(2016)161

Y

+ @G

@Y I

@Y I

@

and use (3.9)

@Y I

@ Y I

@ = +iNIJFJ :

Therefore, if G(Y; ) is a symplectic function, then

D G := @G

@

Y

@ =

+ iNIJFJ @G

@Y I

is also symplectic. The expression

D = @

@

Y

+ iNIJFJ @

@Y I ; (3.10)

7This restriction will be lifted later.

8Note that, though we do not indicate this by notation, Y (Y, ) is not holomorphic.

{ 16 {

which we have derived from the coordinate transformation between supergravity and stringy variables, is the symplectic covariant derivative which was introduced in [9] based on studying the symplectic transformation behaviour of derivatives of symplectic functions. We remark that while G was assumed holomorphic, D G is not holomorphic, due to the presence of the inverse metric NIJ. By taking higher covariant derivatives Dn G, one can create a whole tower of symplectic functions. We remark that when the initial function G is non-holomorphic, the covariant derivative needs to be modi ed, as will be discussed later.

The main application of this result is to show how one can obtain, starting from F (Y; ) a hierarchy of functions F (n)(Y;

Y) which can be interpreted as topological free energies, because they satisfy the holomorphic anomaly equation. While this is result known from [9] we brie y explain how this works and how the hierarchy of equations for the functions F (n)(Y;

Y) can be consolidated into a master anomaly equation.

First, following [9] we de ne9 a hierarchy of symplectic functions through covariant derivatives of the holomorphic symplectic function F (Y; ):

(n)(Y;

Y ; ;

) = 1

n!Dn1 F ;

for n = 1; 2; : : :, and (0) = 0. Then

(1) = F

(2) = 12D F ;

etc. (1) is the only holomorphic function in this hierarchy. One computes

@ (2)

@

Y I =

i 2

JHEP02(2016)161

@NJK

@

Y I FJ FK =

1

2

F JKI@J (1)@K (1) ;

where

F JKI =

FIPQNPJNQK. From this starting point it is straightforward to obtain the holomorphic anomaly equation

@ (n)

@

Y I =

1

2

F JKI

n1

Xr=1@J (r)@K (nr) ; n 2 (3.11)

by complete induction. Next we de ne

F (n)(Y;

Y) = (n)(Y;

Y ; = = 0) (3.12)

where we used that Y I = YI for = 0. Explicit expressions for F (1); F (2) and F (3) are

given in (C.3) (the normalization used there di ers by a factor 2i).

Setting = 0, one obtains a holomorphic anomaly equation

@F (n)

@

YI

= 12

Xr=1@JF (r)@KF (nr) ; n 2 (3.13)

9Note that here we use a di erent normalisation from the one used in section 1.

F JKI(0)

n1

{ 17 {

for the functions F (n)(Y;

Y). This is not the full anomaly equation for the genus n topological free energies of the topological string. The reason is that, so far, F and hence (1) and

F (1) are holomorphic, while for the topological string they are not. This will be addressed in the next step where we extend our formalism to the case of a non-holomorphic F . For

terminological convenience we will refer to the functions F (n) as genus n topological free energies, or free energies for short.10

The hierarachy of equations (3.13) can be re-organised into a master anomaly equation for the topological free energy

Ftop(Y;

Y; Q) =

1

Xn=1QnF (n)(Y;

Y) ;

where the expansion parameter Q is, for the topological string, related to the topological string coupling. Taking into account that F (1) is (so far) holomorphic, it is straightforward to verify that (3.13) follows from

@Ftop

@

YI

= 12

F JKI(0)@JFtop@KFtop (3.14)

by expansion in Q.

Since F is the natural master function in the supergravity formalism, one would like to have a master anomaly equation for it. This is not straightforward, since the Taylor coe cients of F (Y; ) with respect to are not symplectic functions. We proceed by expressing F in stringy variables and introducing a shift in :

F (Y;

Y; ; ; Q) := F (Y;

JHEP02(2016)161

Y; + Q; ) :

Then we make a Taylor expansion with respect to the uctuation Q. As indicated by notation, we need to treat and

as independent variables, and the shifted F is not any more holomorphic in supergravity variables. Also note that when using supergravity variables the dependence on Q is not any more of the form + Q.

Now we express the expansion coe cients in supergravity variables:

F (Y;

Y; ; ; Q) =

1

Xn=11 n!Qn@n F (Y;

Y; ; ) =

1

Xn=11 n!QnDn F (Y; )

=

1

Xn=1Qnn! (n + 1)! (n)(Y;

Y ; ; )

=

1

Xn=1(n + 1)Qn (n)(Y;

Y ; ; ) :

Next we integrate with respect to Q:

G(Y;

Y ; ;

; Q) =

1

Xn=0Qn+1 (n+1)(Y;

Y ; ; ) ;

10We remark that our formalism is independent of an explicit realization by a concrete topological string model, and in this sense independent of the topological string. Our formalism is a general framework, for which the topological string is one (important) application.

{ 18 {

and by setting = 0 =

we obtain the topological free energy Ftop.

G(Y;

Y ; =

= 0; Q) = G(Y;

Y; = = 0; Q) =

1

Xn=0Qn+1F (n+1)(Y;

Y)

= Ftop(Y;

Y; Q) :

The function G satis es the master anomaly equation

@ @

Y I G =

1

2

F JKI@JG@KG ;

which for =

= 0 becomes the master anomaly equation (3.14) for the topological free energy.

We note that the relation between the topological free energies F (g) and the function F is complicated. This is of course to be expected from [12]. The reason is that in a

Taylor expansion of F (Y; ) the coe cients are not symplectic functions. The topological free energies can be regarded as coe cients in a symplectically covariant Taylor expansion, which in practice we cannot manage in closed form but only by evaluating derivatives at = 0. We also remark that the use of two complementary set of complex coordinates, re ects that there is a tension between holomorphicity and symplecticity. In the supergravity variables we have manifest holomorphicity, but only the full symplectic vector (Y I; FI)

and the full function F are symplectically covariant. If one wants to organise data in a hierarchy of symplectic function, holomorphicity is violated, albeit in a systematic way controlled by the anomaly equation. One can then either work with the covariant derivatives (n+1) [similarequal] Dn F or use the stringy variables and work with the free energies F (n).

Above we obtained a master equation for F from the master anomaly equation for Ftop. The result is not quite satisfactory, as we need the background shift Q as a device. But in the next section we will see that a master equation for F can be obtained directly within the supergravity formalism.

3.6 Anomaly equation from the Hessian structure

We will now show that the holomorphic anomaly equation arises as an integrability condition for the existence of a Hesse potential on ^

M. The metric gH being Hessian means that S = rgH is a completely symmetric rank three tensor. In r-a ne coordinates

Qa = (xI; yI; ;

JHEP02(2016)161

) the components of the tensor S are simply the third partial derivatives of the Hesse potential, or equivalently the rst partial derivatives of the metric, and therefore proportional to the Christo el symbols of the rst kind (which for a Hessian metric are completely symmetric with respect to r-a ne coordinates):

Sabc = @3abcH = @agHbc :

One particular relation is

SxI = S xI ;

or

@xI gH = @ gHxI ; (3.15)

{ 19 {

where

gH = iD F

and

gHxI = 2

FIJNJKFK :

We now evaluate equation (3.15) in supergravity coordinates (Y I;

Y I; ;

), using the

corresponding Jacobian to obtain

SxI = @gH

@xI

y

= @gH @xI

@gH

@uK ; where

@ @xI

u

+ @uK

@xI

JHEP02(2016)161

u

= @

@Y I +

@ @

Y I

x;y

and

x;u

S xI =

@gHxI @

= @gHxI @

+ @uK

@

@gHxI

@uK :

We nd for the various terms,

@gH @xI

u

= i

@ @

Y I D F i

KLFK FL ;

@gH @uK

@@Y K

@ @

Y K

F

+ iNKLFK FL

x

=

= FK FKPQFP FQ 2NPQFKP FQ

FKPQFP FQ ;

@gHxI @

x;u

= iFI + F IJFJ + FIJ +

FIJ

iF

JK FK + F J

;

@gHxI @uK

x

= FIK + iFIKLFL + i FIL +

FIL

iF

KLP FP + F KL

i

FIKLFL FIL +

FKLP FP ; (3.16)

where indices are raised using NIJ. Then, the Hessian condition (3.15) results in

@ @

Y I D F =

FIJKNJP NKQFP FQ : (3.17)

In the holomorphic case at hand (@F = 0), this equation can be regarded as a master anomaly equation in supergravity variables. First note that for = 0 (3.17) reduces to the anomaly equation for F (2). The anomaly equations for F (n) with n > 2 are obtained by covariant di erentiation of (3.17). Here one uses that holomorphicity of the generalized prepotential implies

@F = 0 ; D

FIJK = 0

and one also uses the identity [13]

[D ; NIJ@J] = 0 :

{ 20 {

For example, to derive the anomaly equation for (3) and, hence, for F (3) we need to evaluate

@D2 F = D @D F + i(@NJK)FJ @KD F

= 3

F JKI@JF @KD F :

Using that Dn1 F = n! (n) this becomes

@ (3) =

F JKI@J (1)@K (2) = 1

2

F JKI

2

Xr=1 @J (r)@K (3r) ;

which for = 0 is the anomaly equation for F (3). Proceeding by induction one obtains the full hierarchy (3.11) of anomaly equations.

One may ask whether other components of S will give rise to additional non-trivial di erential equations. To investigate this, we now consider the component SxI

= @xI gH ,

which is constructed out of the metric component gH = NIJFI

FJ . Evaluating the

xI = @

gHxI in supergravity variables we nd that it is identically satis ed. Thus, the only non-trivial di erential equation resulting from gH and gH is encoded in the relation SxI = S xI .

4 The non-holomorphic deformation

So far we have assumed that F (Y; ) and, hence, F are holomorphic in the supergravity variables, which implies that F (1)(Y) is also holomorphic, while F (g)(Y;

JHEP02(2016)161

relation SxI

= S

Y) with g > 1

satisfy the anomaly equation (3.13). For the topological string the situation is more complicated since already F (1)(Y;

Y) is non-holomorphic. We therefore now generalize our framework and induce geometric data on ^

M using a non-holomorphic map : ^

M ! V ,

which then corresponds to a non-holomorphic generalized prepotential F = F (Y;

Y ; ;

).

This explicit non-holomorphicity will in turn modify the anomaly equation (3.13) satis ed by the topological free energies F (g)(Y;

Y). The precise form of the modi cation depends on the details of the non-holomorphic deformation. We will rst keep the discussion general, and later show that when chosing a particular non-holomorphic deformation we obtain the correct full anomaly equation (at least to leading order in a formal expansion we will explain later). As discussed in [13] there are other types of non-holomorphic deformations that are, for example, relevant for non-linear deformations of electrodynamics. Any such deformation could be analyzed in the framework of our formalism.

Since F and F are no longer holomorphic, they will have non-vanishing derivatives with respect to

Y I and

. In the following we will use a notation involving unbarred indices I; J; : : : and barred indices

I;

J; : : :.

4.1 Non-holomorphic deformation of the prepotential

We generalize the map (3.1) to

: ^

M = M [notdef]

C

! V ; (Y I; ) [mapsto]!(Y I; FI(Y;

Y ; ;

)) ; (4.1)

{ 21 {

where FI = @F=@Y I, can be obtained from a generalized prepotential F . We assume that F has the form [18]

F (Y;

Y ; ;

) = F (0)(Y ) + 2i (Y;

Y ; ;

) ; (4.2)

where F (0) is the undeformed prepotential, and where is real-valued.11 The holomorphic deformation is recovered when is harmonic. This makes use of the observation that the complex symplectic vector (Y I; FI) does not uniquely determine the prepotential F [12].

If we make a transformation

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

(Y;

Y ; ;

) ! (Y;

Y ; ;

JHEP02(2016)161

)

1

2i(g(Y; )

g(

Y ;

)) ;

where g(Y; ) is holomorphic, then F changes by an antiholomorphic function, F ! F +

g,

and the symplectic vector (Y I; FI) and the map are invariant. If is harmonic,

(Y;

Y ; ;

) = f(Y; ) +

f(

Y ;

) ;

we can make a transformation with g = 2if and obtain

F ! F (0)(Y ) + 2if(Y; ) =: F (Y; ) ;

which is a holomorphically deformed prepotential, as considered in the previous section. If, however, is not harmonic, then we have a genuine generalization which requires us to consider non-holomorphic generalized prepotentials. For the case of the topological string, it is convenient to split the non-holomorphic generalized prepotential as in (4.2) into the undeformed prepotential F (0) and a real-valued non-harmonic function which encodes all higher derivative e ects, holomorphic as well as non-holomorphic.

We proceed by analysing the geometry induced by pulling back the standard hermitian form V of V given by (2.3) to ^

M using (4.1):

= i(F (0)IJ

F (0)IJ)dY I d

Y J + 2( IJ + J)dY I d

Y J + 2 J dY I dY J

+2 I Jd

Y I d

Y J + 2

dY I d

+ 2 I d d

Y I + 2 dY I d

+2 I

d

d

Y I :

By decomposing = g + i!, we obtain the following metric on ^

M:

g = i(F (0)IJ

F (0)IJ)dY Id

Y J + 2( IJ + J)dY Id

Y J + 2 J dY IdY J + 2 I Jd

Y Id

Y J

+2

dY Id

+ 2 I d d

Y I + 2 dY Id + 2 I

d

d

Y I :

From this expression it is manifest that g is not Hermitian, and hence not Kahler with respect to the natural complex structure J. The non-Hermiticity is encoded in the mixed

11This function is not to be confused with the complex symplectic form on the vector space V introduced in subsection 2.1.

{ 22 {

derivatives I J, which makes it manifest that it is related to the non-harmonicity of . This metric occurs in the sigma model discussed in [17].

The imaginary part of de nes a two-form on ^

M:

! = 1

2i(i(F (0)IJ

F (0)IJ))dY I ^ d

Y J i(

J + IJ)dY I ^ d

Y J

i J dY I ^ dY J + i IJd

Y I ^ d

Y J i

dY I ^ d

i I d ^ d

Y I

i dY I ^ d + i I d

Y I ^ d

: (4.3)

This two-form is no longer of type (1; 1) with respect to the standard complex structure, which is consistent with the non-Hermiticity of g. However, ! is still closed

d! = 0 ;

so that ( ^

M; !) is at least a symplectic manifold.

Comparing ! to dxI ^ dyI, we nd:2dxI ^ dyI = ! + 2i I

JdY I ^ d

Y J + i I dY I ^ d i

JHEP02(2016)161

d

Y I ^ d

+i I

dY I ^ d

i

d

Y I ^ d : (4.4)

As a consistency check, we verify that 2dxI ^ dyI ! is closed:2dxI ^ dyI = !

1

2d( dF )

1

2d(

dF

) @@F ;

where @ = dY I @Y

I + d @ . Note that the di erence between the symplectic form

2dxI ^ dyI of M and the symplectic form ! of

^

M is still exact. Compared to (3.4) we have an additional term which measures the non-holomorphicity of the generalized prepotential.

4.2 Real coordinates and the Hesse potential

To convert from complex supergravity variables we need to generalize our previous calculation of the Jacobian and its inverse:

D(x; y; ;

)

D(x; u; ;

) =

1 0 0 0

1

2 R+ 12N

1

2 (FI +

F ) 12 (

F

+ FI

)

1 0 0 0 N1R+ 2N1 N1(FI +

F ) N1(

0 0

1 0 0 0 0

1

0

B

B

B

@

1

C

C

C

A

and, by a straigtforward matrix inversion

D(x; u; ;

)

D(x; y; ;

) =

0 0

1 0 0 0 0

1

0

B

B

B

@

1

C

C

C

A

+ FI

)

:

This reduces to the previous result when switching o the non-holomorphic deformation. When restricting to the left upper block, the result agrees with [17]. We have used the following de nitions [17]:

NIJ = NIJ [notdef] 2ImFI

J = i(FIJ

F J [notdef] FI

J

FJ )

{ 23 {

and

RIJ = RIJ [notdef] 2ReFI

J = FIJ +

F J [notdef] FI

J [notdef]

FJ :

Note that NT = N, while RT = R .

As already observed in [18], in the presence of explicit non-holomorphic deformations the Hesse potential is not to be de ned as the Legendre transform of 2ImF but rather as the Legendre of

L = 2ImF 2 = 2ImF (0) + 2 : (4.5)

As explained in [12, 13], the function L can be interpreted as a Lagrange function, and the Hesse potential as the corresponding Hamilton function.12 Thus the Hesse potential associated to a non-holomorphically deformed prepotential is

H(x; y; ;

) = i(F

JHEP02(2016)161

F ) 2 2uIyI :

By taking derivatives with respect to the real coordinates (QA) = (qa; ;

), where (qa) =

(xI; yI) we obtain the components of a Hessian metric:

@H @qa@qb =

N+ + RN1R+ 2RN1

2N1R+ 4N1 !

;

@2H

@xI@ = i(FI

F ) + RIKNKJ(FJ +

F J ) ;

@2H

@yI@ = 2NIK(FK +

F K ) ;

together with their complex conjugates and

@2H

@ @

= iF

+ NIJ(

)(F J F

J) = iD F

;

@2H

@

@

= iD F ;

@2H

@ @ = iD F ;

where

D = @ + iNIJ(F J F

J)

@@Y I

@ @

Y I

(4.6)

is the symplectic covariant derivative introduced in [13]. This covariant derivative is a generalization of (3.10) which generates a hierarchy of symplectic functions starting from a non-holomorphic symplectic function. For holomorphic symplectic functions it reduces to (3.10). We will show below that D can be derived by transforming the partial derivative @ [notdef]Y from stringy variables to supergravity variables.

As before, the Hessian metric gH di ers from the metric g induced by pulling back gV using by di erentials involving derivatives of H with respect to ;

:

gH = g + @2H

x;y

where

@2H

x;y =

@2H

@ @ d d + 2

@2H

@ @

d d

+ @2H

@

@

d

:

12In [12, 13] the Hesse potential is normalized di erently by a factor 2 compared to [18] and the present paper.

{ 24 {

4.3 The symplectic covariant derivative

As before we can use (4.3) and (4.4) to obtain exact information about the coordinate transformation Y I = YI Y I between supergravity variables and stringy variables. By

proceeding as in subsection 3.4, namely converting the expression (4.4) from supergravity variables to stringy variables, and then converting the result to real variables, we nd the following consistency condition:

@ Y I@ = i

^

NIK(FK +

F K ) ; (4.7)

where ^

NIJ is the inverse of the matrix

^

NIJ = NIJ + iFJ i

JHEP02(2016)161

F JI = i(FIJ

F J F

J +

FI J)

= i(FIJ

F J 2i J

2i

JI )

de ned in [13]. Note that ^

NIJ = NIJ, which was de ned before. The for

mula (4.7) can be used to derive a modi ed symplectically covariant derivative, which allows to generate new symplectic functions given a non-holomorphic symplectic function G(Y;

Y ; ;

). Indeed, if such a function is given we can express it in stringy variables, G = G(Y (Y;

Y; ; );

Y (Y;

Y; ; ); ; ), and we know that @G=@ [notdef]Y is a symplectic

function. Expressing this function in supergravity variables we obtain

@ @

Y

G = @

@

Y I )G =: D G ;

where D denotes the symplectically covariant derivative introduced in (4.6), as can be readily veri ed by using F = 2i , where is real valued, which implies

F = F

.

@ Y I

@ (@Y

G

I

@

Y

This derivative operator was already found in [13] based on studying the symplectic transformation of derivatives of a non-holomorphic generalized prepotential. We have now derived this covariant derivative from a coordinate transformation.

The symplectically covariant derivative D can be applied to any non-holomorphic symplectic function. Thus, we can now construct a hierarchy of symplectic functions starting from a non-holomorphic F :

(n+1) = 1

(n + 1)!Dn F (Y;

Y ; ;

) : (4.8)

As before, we de ne topological free energies by

F (n)(Y;

Y) := (n)(Y;

Y ; ; )

=0

; n 2 :

These functions will satisfy a holomorphic anomaly equation, whose precise form depends on the details of the non-holomorphic deformation.

{ 25 {

4.4 The holomorphic anomaly equation

We would like to show that for a suitable choice of a non-holomorphic deformation we obtain the holomorphic anomaly equation of the topological string. As anticipated from [12] this is laborious to do explicitly, since an explicit non-holomorphic deformation leads to a proliferation of non-holomorphic terms. As in [12] we will resort to a (formal) series expansion in parameters which control the non-holomorphicity, and rely on results about the symplectic transformation behaviour of various quantites.

The non-holomorphic dependence of the topological free energies F (g) is entirely encoded in NIJ(0) (which, we recall, is the inverse of N(0)IJ = i(F (0)IJ

f(n)(Y ) + 2 NIJ(0) f(n1)IJ : (4.11)

To rst order in and in NIJ(0), the function F = 2i , given by

F (Y;

Y ; ) = 2i

= 2i

F (0)IJ)). The higher

F (g) (with g 2) are polynomials of degree 3g 3 in NIJ(0), while F (1) depends on the

logarithm of det NIJ(0). In the following, we will focus on the polynomial dependence on N(0)IJ of the higher F (g), keeping F (1) holomorphic for the time being. We thus consider the deformation

(Y;

Y ; ;

) = f(Y; ) + 2 NIJ(0) fIJ(Y; ) + c:c :; (4.9)

where the departure from harmonicity is encoded in NIJ(0). We will work to rst order in the deformation parameter and in NIJ(0) to avoid a proliferation of new terms compared to the holomorphic case. Note that in applications and NIJ(0) are not necessarily small, so that the expansion is formal. The above de nes a toy model that, as we will see, reproduces the holomorphic anomaly equation for the topological free energies F (g) for g 2, to leading

order in and in NIJ(0). Since F (1) is still holomorphic, this toy model does not fully capture the topological string. We will address this issue at the end of this section.

Expanding

f(Y; ) =

we obtain from (4.9)

= f(1) +

1

JHEP02(2016)161

1

1

Xn=1 n f(n)(Y ) ) f =

Xn=1n n1f(n)(Y ) ; (4.10)

Xn=2 n n1

hf + 2 NIJ(0) fIJ(Y; ) + 2 NIJ(0) f IJ(Y; )

i

(4.12)

hf(1)(Y ) + 2

f(2)(Y ) + 2 NIJ(0) f(1)IJ(Y )

f(3)(Y ) + 2 NIJ(0) f(2)IJ(Y ) + [notdef] [notdef] [notdef]i;

transforms as a function under symplectic transformations provided we modify the transformation behaviour of f(Y; ) to (note that we are using supergravity coordinates Y I),

f(Y; ) ! f(Y; ) + 2i ZIJ0 fIJ(Y; ) ; (4.13)

{ 26 {

+3 2

to rst order in and in ZIJ0. Here, ZIJ0 denotes the transformation matrix given in (C.2).

Note that NIJ(0) fIJ transforms as follows [12] under symplectic transformations, to rst order in and in NIJ(0) (or ZIJ0),

NIJ(0) fIJ !

NIJ(0) iZIJ0

fIJ F (0)IJLZLP0 fP

= NIJ(0) iZIJ0 fIJ + O(N1Z0; Z20) : (4.14)

Using (4.13) and (4.14), it follows that (4.12) is a symplectic function at this order.We now observe that

F / @

NJK(0) (4.15)

which is of higher order in N1(0), and hence will be dropped. Thus

Dn F [notdef] =0 = Dn(0) F =0

+ O((N1(0))2) ;

JHEP02(2016)161

where by

D(0) = @

@ + iNIJFI

@ @Y J

we denote the symplectically covariant derivative (3.10). Thus, while when starting with a non-holomorphic function (3.10) must normally be replaced by (4.6), we neglect the additional terms in the present context because they will necessarily bring in higher powers of N1(0).

Hence, by working to order and neglecting terms of order (N1(0))2, we obtain

Dn F [notdef] =0 = Dn(0) (2if ) =0

+ 4i NIJ(0)@n

1

Xm=1(m + 1) mf(m)IJ

! =0

= Dn(0) (2if )

=0

+ 4i (n + 1)! NIJ(0) f(n)IJ : (4.16)

Now we consider the hierarchy (4.8),

F (n+1) = Dn F (n + 1)!

=0

(n + 1)! Dn(0) (2if ) =0

+ 2 NIJ(0) (2if(n)IJ)

=0

r + O( 2; (N1(0))2) :

For = 0 only the rst term F (n+1)holo = [(n + 1)!]1Dn(0) (2if )[notdef] =0 is present, which satis es the anomaly equation (3.13). Including the deformation term of order we get (note that Y I = YI when = 0)

@ @

YK

F (n+1) = @

@

= 1

YK

F (n+1)holo 2i

F (0)IJK (2if(n)IJ)

=0

= 12

F (0)IJK

n

Xr=1@IF (r)holo@JF (n+1r)holo 4i F (n)(holo)IJ !:

{ 27 {

Rede ning F (n) ! 2iF (n) (with n 1), we obtain@ @

YK

F (n+1) = iF (0)IJK

n

Xr=1@IF (r)@JF (n+1r) 2 DI@JF (n)!; (4.17)

up to terms of higher order in N1(0). Here we used that the Levi-Civita connection DI and the non-holomorphic deformation terms of the F (n) involve at least one further power of

N1(0), which can be dropped at the order we are working at. Note that (4.17) is the full holomorphic anomaly equation for the higher F (n) in big moduli space [12]. The standard normalization of the anomaly equation is obtained by setting 2 = 1.

Let us return to the transformation law (4.13). Inserting the expansion (4.10) into it, we see that f(1) remains invariant under symplectic transformations. This is not what happens in topological string theory, where f(1) transforms into f(1) ! f(1) + ln det S0,

with 2 R. This transformation behavior is, in turn, compensated for by the presence of

an additional term ln det N(0)IJ, which ensures that the topological free energy F (1), given by

F (1)(Y;

Y) = 2i

JHEP02(2016)161

f(1)(Y) + f(1)(

Y) + ln det N(0)IJ

;

is invariant under symplectic transformations. If we now insist that F (1) and are related

by F (1) = 2i [notdef] =0, then this is only possible if we take to be real, in which case

given in (4.9) gets modi ed to

(Y;

Y ; ) = f(Y; ) + 2 NIJ(0) fIJ(Y; ) + c:c: + ln det N(0)IJ ; (4.18)

to rst order in . Thus, while the deformation did not enforce any restriction on , the presence of the deformation does.

So far, we restricted ourselves to working at rst order in , and N1(0). At higher order, the analysis in [12] shows that and get locked onto the same value = . This is a consequence of the requirement that should transform consistently under symplectic transformations. In this way we recover the non-holomorphic deformation relevant for the topological string.

4.5 From Hessian structure to the full anomaly equation

We now show how to recover the holomorphic anomaly equation (4.17) from the underlying Hessian structure. We proceed as in subsection 3.6, and consider the totally symmetric rank three tensor S = rgH, where gH denotes the Hessian metric computed in subsection 4.2.

As before, we consider the components

SxI = S xI :

Since S is a tensor, we can evaluate this relation in other coordinate systems, in particular in complex supergravity coordinates. As before, using

gH = iD F ;

SxI = @xI [notdef]y gH = @x

I

|u gH +

@uK

@xI

@gH

@uK ; where @x

I

|u =

@@Y I +

@ @

Y I ;

x;y

= @gHxI

@

x;u

S xI =

@gHxI @

+ @uK

@xI

@gHxI

@uK ;

{ 28 {

we obtain, after some rearrangements, an expression for the antiholomorphic derivative

@ @

Y I gH =

@gHxI @

u

+ @uK

@

@gHxI

@uK

@gH

@Y I

@uK

@xI

@gH

@uK :

After a lengthy but straightforward calculation similar to the one in subsection 3.6, we nd

i @

@

Y I D F =

@gH

@

Y I (4.19) = i

F + 2(FS +

FS )NSL

FL

FL

JHEP02(2016)161

i(FS +

FS )NSL(FQ +

F Q )NQP

FPL

FP L +

FP L 2

:

In the holomorphic case, this reduces to (3.17). In the non-holomorphic case based on (4.9) it can be readily veri ed that when setting = 0, one obtains the holomorphic anomaly equation (4.17) for F (2) to leading order in and N1(0). Namely, using (4.9) and setting = 0, (4.19) reduces to

@ @

Y I D F

=0

= 2 @

@

YI

F (2) =

F + FS NSLFQ NQP FP L

=0

:

Using

F [notdef] =0 = 2i

[notdef] =0 = 8

F (0)KLIf(1)KL = 4i

F (0)KLIF (1)KL, and rede ning

F (n) ! 2iF (n), we obtain (4.17) for F (2).

We note that, in principle, one may now proceed to derive the holomorphic anomaly equation for the higher F (n) (with n 3) by applying covariant derivatives D to (4.19),

and subsequently setting = 0, as in subsection 3.6.

We nish by verifying that the component SxI

= @xI gH , which is constructed out

of the metric component gH = iD F

, does not give rise to an additional non-trivial di erential equation. Evaluating the relation SxI

= S

xI = @

gHxI in supergravity variables we nd that it is identically satis ed. Thus, the only non-trivial di erential equation resulting from gH and gH is encoded in the relation SxI = S xI .

5 Concluding remarks

Let us conclude with a comparison of the approach taken in [12] and the one taken here for obtaining the holomorphic anomaly equation. Both are based on the Hesse potential, which is obtained by a Legendre transform, see (4.5). In the approach of [12] one works directly with the Hesse potential, while here we work with the associated Hessian structure (gH; r) on the extended scalar manifold

^

C. The Hesse potential, which is akin to a Hamiltonian [12], is a symplectic function of the special real coordinates. It can also be expressed either in terms of supergravity variables (Y I; ) (and their complex conjugate) or in terms of stringy (or covariant) variables (YI; ) (and their complex conjugate), see (3.7).

The approach in [12] consisted in rst expressing the Hesse potential in terms of covariant variables by means of a power series expansion in Y I, and then expressing Y I in terms of a power series in derivatives of , which we recall was introduced in (4.2). In this way it was shown in [12] that the Hesse potential, when expressed in terms of covariant variables,

{ 29 {

M = M [notdef]

equals an in nite sum of symplectic functions, which were denoted by H(a)i. The label a

indicates that the leading term is of order a. For higher values of a there are several functions (labelled by i = 1; 2; : : : ) with the same value of a. This decomposition is unique. In this decomposition there is only one function, namely H(1), whose leading term is

itself, while the leading term of all the other H(a)i (with a 2) involves derivatives of .

In [12] it was shown that H(1) comprises a subsector of the full Hesse potential that encodes

the holomorphic anomaly equation. This was achieved by using the explicit expression for

H(1), which consists of an in nite sum that starts with 4 , and that involves terms of higher

and higher powers of derivatives of . By using the fact that depends on , this in nite sum was in turn rewritten as a series expansion in , with coe cient functions that are again symplectic functions. When is taken to be harmonic, these symplectic functions, denoted by F (n) (the rst three of which we display in (C.3)), satisfy the holomorphic anomaly equation (1.1) with = 0. Subsequently, by deforming by -dependent terms, as in (4.18), it was shown [12] that the resulting functions F (n) satisfy the full holomorphic anomaly equation (1.1).

Thus, summarizing, it was shown in [12] that the full Hesse potential, when expressed in terms of covariant variables, contains a subsector H(1) that, in turn, contains an in nite

set of functions F (n) that satisfy the full holomorphic anomaly equation (1.1). The other sectors, described by the other functions H(a)i, are constructed out of derivatives of , and

thus contain derived information. They are nevertheless important, since they are needed to build up the full Hesse potential.

In the approach taken in this paper, we instead work with the Hessian metric gH associated with the full Hesse potential. We work in supergravity variables, and we focus on particular components of gH, namely on gH and gH . These two components (both of which are given in terms of the symplectic covariant derivative introduced in [9, 13]) encode di erent information. For instance, when evaluated at = 0, gH gives the symplectic function F (2), while gH gives the symplectic function H(2) evaluated at = 0. This is

reminiscent of the decomposition of the full Hesse potential into symplectic functions H(a)i discussed above. We then consider the totally symmetric rank three tensor S = rgH, and we rst focus on its component SxI = @xI gH . Evaluating the relation SxI =

S xI in supergravity variables and subsequently setting = 0 we obtain the holomorphic anomaly equation for F (2). One may then ask whether other components of S will lead to additional non-trivial di erential equations. To address this, we consider the component SxI

= @xI gH . Evaluating the relation SxI

= S

JHEP02(2016)161

xI = @

gHxI in supergravity variables we nd that it is identically satis ed. Thus, we conclude that the only non-trivial di erential equation resulting from gH and gH is encoded in the relation SxI = S xI .

Acknowledgments

We would like to thank Vicente Cort es and Bernard de Wit for valuable discussions. The work of G.L.C. was partially supported by FCT/Portugal through UID/MAT/04459/2013 and through EXCL/MAT-GEO/0222/2012. The work of T.M. was, in part, supported by STFC and by a BCC visiting fellowship associated to the excellency grant EXCL/MAT-

{ 30 {

GEO/0222/2012. This work was also supported by the COST action MP1210 \The String Theory Universe".

T.M. would like to thank CAMGSD (Department of Mathematics, IST) for the BCC fellowship which enabled him to visit IST, and for the kind hospitality. G.L.C. would like to thank the Max-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institute) for kind hospitality during the completion of this work.

A Connections on vector bundles

We review some standard facts about connections in vector bundles. Let E ! M be a

vector bundle over a manifold M. Then a connection or covariant derivative r on E is

a map

r : X(M) [notdef] (E) ! (E) ; (X; s) [mapsto]! rXswhich is a linear derivation (satis es the product rule) with respect to sections s 2 (E)

of E, and which is C1(M)-linear with respect to vector elds in X 2 X(M).

For vector bundles of the form p(M; E) = pT M E, that is for bundles of p-forms

with values in a vector bundle E, the covariant exterior derivative

dr : p(M; E) ! p+1(M; E)is uniquely determined by its action on sections of E. For a given basis [notdef]sa[notdef] of sections

one sets

drsa = rsa = !basb ;where !ba is the connection one-form of r. The derivative of a general section s = fasa 2

0(M; E), where fa 2 C1(M) is thendrs = dfa sa + fa!basb :

The extension to forms of degree p > 0 is completely determined by linearity and the product rule

where 2 p(M). The exterior covariant derivative dr of ! 2 p(M; E) can be expressed

in terms of the covariant derivative r by

(dr!)(X0; : : : ; Xp) =

p

The curvature of the connection r is given by Rr(s) = dr(drs). If the connection r

is at, then d2r = 0, so that dr de nes an exact sequence. In this case a version of the

Poincar e lemma exists.

{ 31 {

JHEP02(2016)161

dr( s) = d s + (1)deg ^ drs ;

Xl=0(1)lrXl(!(: : :^

Xl : : :))

+

Xi<j(1)i+j!([Xi; Xj]; : : :^

Xi : : : ^

Xj : : :) ;

where X0; : : : Xp are vector elds, and where ^

X indicates that the corresponding vector

eld is omitted.

B Special coordinates

One of the de ning conditions of a ne special Kahler geometry is drJ = 0, where J is the

complex structure. We can apply the above results since J is a section of End(E) [similarequal] E E, where E = T M. In the following we derive the local form (2.2) of (2.1), and also explain how the existence of special coordinates can be derived.

Let A be a section of End(T M). Choosing dual bases [notdef]ea[notdef]; [notdef]ea[notdef] of sections, and

regarding A as a T M-valued one-form, we have

A = Aaea = Aabeb ea 2 1(M; T M)

and

drA = dAa ea Aa ^ !

ba eb 2 2(M; T M) :

If the connection r is at, we can choose sections ea such that the connection one-form

vanishes, ! b

a = 0 and in such a frame drA = 0 reduces to dAa = 0. Thus the one-forms

Aa are locally exact, Aa = da.Since E = T M, the torsion of the connection r is de ned by

T r(X; Y ) = rXY rY X [X; Y ] :

One can show thatT r(X; Y ) = drId(X; Y ) ;

where

Id = ea ea = baea eb

is the identity endomorpism of T M. If r is both at and torsion free, then drId = 0 implies

that in a frame where the connection vanishes, the one-forms Ia = ea are locally exact ea = dta. This de nes a set of r-a ne coordinates. In such coordinates the condition (2.1)

becomes

drJ = 0 ) dJa = 0 ) @[bJbc] = 0 :If the manifold M is in addition equipped with a non-degenerate, closed two-form ! 2 2(M), d! = 0, then a connection r is called symplectic if the symplectic form ! is

parallel:

r! = 0 , rX! = X(!ab)ea ^ eb + !ab(rXea) ^ eb + !abea ^ rXeb = 0 ;

for all vector elds X. If the connection r is in addition at, we can choose sections

ea such that rXea = 0. With respect to such a basis the coe cients of ! are constant,

X(!ab) = 0. If the connection r is in addition torsion-free, the co-frame ea comes from an

a ne coordinate system ta, and

! = 12!abdta ^ dtb

{ 32 {

JHEP02(2016)161

where !ab is a constant, antisymmetric, non-degenerate matrix. Using the linear part of the a ne transformation that we can apply to ta, the matrix !ab can be brought to the standard form

(!ab)Standard = 0

This form is still invariant under a ne transformations where the linear part is symplectic. The associated coordinates are called Darboux coordinates. Thus we have seen that for a at, torsion-free, symplectic connection the r-a ne coordinates can be chosen to be

Darboux coordinates. In the context of a ne special Kahler geometry such coordinates are called special real coordinates.

We remark that in the main part of this paper we use special real coordinates (qa) =

(xI; yI) where the Kahler form takes the form

! = 2dxI ^ dyI = abdqa ^ dqb :Note that the components of ! with respect to the coordinates (qa) are !ab = 2 ab = 2!Standardab. In other words the special real coordinates di er from standard Darboux coordinates by a conventional factor p2.

C Symplectic transformations and functions

Symplectic transformations acts as follows on a symplectic vector (Y I; FI) (I = 0; : : : n),

Y I ! UIJ Y J + ZIJFJ ; FI ! VIJFJ + WIJY J ;

where U; V; Z; W are the (n + 1) [notdef] (n + 1) real submatrices that give rise to an element of

Sp(2n + 2; R). When F equals the prepotential F (0), then NIJ(0) transforms as follows under symplectic transformations [9],

NIJ(0) !

S0I K S0JLNKL(0) = S0IK S0JL

NKL(0) iZKL0 ; (C.1)

S0I K = UIK + ZIJF (0)JK ;

ZIJ0 = [S10]IKZKJ : (C.2) Consider a generalized prepotential F (Y;

Y ; ;

) = F (0)(Y ) + 2i (Y;

Y ; ;

), where

is taken to be harmonic, (Y;

Y ; ;

) = f(Y; ) +

f(

Y ;

). Expanding f(Y; ) as in (4.10) and inserting this into (3.12), yields explicit expressions for the symplectic functions F (n). The rst three read as follows,

F (1) = 2i f(1) ; (C.3)

F (2) = 2i

f(2) NIJ(0) f(1)If(1)J ;

F (3) = 2i

f(3) 2NIJ(0) f(2)If(1)J + 2f(1)INIJ(0)f(1)JKNKL(0)f(1)L

+2i

3 F (0)IJKNIP(0) NJQ(0)NKR(0)f(1)Pf(1)Qf(1)R

{ 33 {

1 0 !

1

:

JHEP02(2016)161

where

;

in accordance with [12]. These expressions get modi ed when is not any longer harmonic. The resulting expressions for the topological string can be found in appendix D of [12].

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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