A Definitions and properties of modular forms and weak Jacobi forms

USTC-ICTS-16-22

LPTENS 17/01


Refined BPS invariants of 6d SCFTs

from anomalies and modularity


Jie Gu, Min-xin Huang, Amir-Kian Kashani-Poor, Albrecht Klemm


Laboratoire de Physique Théorique de l’École Normale Supérieure, CNRS, PSL Research University, Sorbonne Universités, UPMC, 75005 Paris, France Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China Bethe Center for Theoretical Physics (BCTP), Physikalisches Institut, Universität Bonn, 53115 Bonn, Germany


F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds engineer six dimensional superconformal field theories and their mass deformations. The partition function of the refined topological string on these geometries captures the particle BPS spectrum of this class of theories compactified on a circle. Organizing in terms of contributions at base degree of the elliptic fibration, we find that these, up to a multiplier system, are meromorphic Jacobi forms of weight zero with modular parameter the Kähler class of the elliptic fiber and elliptic parameters the couplings and mass parameters. The indices with regard to the multiple elliptic parameters are fixed by the refined holomorphic anomaly equations, which we show to be completely determined from knowledge of the chiral anomaly of the corresponding SCFT. We express as a quotient of weak Jacobi forms, with a universal denominator inspired by its pole structure as suggested by the form of in terms of 5d BPS numbers. The numerator is determined by modularity up to a finite number of coefficients, which we prove to be fixed uniquely by imposing vanishing conditions on 5d BPS numbers as boundary conditions. We demonstrate the feasibility of our approach with many examples, in particular solving the E-string and M-string theories including mass deformations, as well as theories constructed as chains of these. We make contact with previous work by showing that spurious singularities are cancelled when the partition function is written in the form advocated here. Finally, we use the BPS invariants of the E-string thus obtained to test a generalization of the Göttsche-Nakajima-Yoshioka -theoretic blowup equation, as inspired by the Grassi-Hatsuda-Mariño conjecture, to generic local Calabi-Yau threefolds.

1

Contents:

1 Introduction

How well and how generally we can compute the topological string partition function serves as a benchmark for how well we understand topological string theory. The computational tools available depend sensitively on the class of geometries on which we consider the theory. The most computable class of geometries to date are local toric Calabi-Yau 3-folds. The key methods that exist for the computation of on these geometries rely on localization [1, 2], large N-dualities involving matrix models [3] or 3d Chern-Simons theories giving rise to the topological vertex [4, 5], as well as the modular approach [6, 7, 8] based on the holomorphic anomaly equations [9, 7]. In this paper, following [10], we will use modular methods in conjunction with vanishing conditions on 5d BPS invariants to compute the refined topological string partition function on a class of non-toric geometries, consisting of elliptically fibered local Calabi-Yau manifolds .

The are the multiplicities of 5d BPS states that arise upon M-theory compactification on the Calabi-Yau manifold . Such BPS states were first considered in [11, 12] and play a decisive role in our analysis. They fall into spin representations of the 5d little group and are labelled by classes determining the mass of the corresponding BPS particles. The determine the refined partition function , which depends on Kähler parameters of the geometry as fugacities for the classes , and the parameters which serve as fugacities for the spins and .

On general Calabi-Yau manifolds, a deformation invariant BPS index is obtained only upon summing over the right spin quantum number. This corresponds to setting to zero and leads to the conventional topological string partition function in which the string coupling constant — a genus counting parameter — is identified as  [11, 12]. A geometrical model for computing these 5d BPS invariants was proposed in [13]. If the Calabi-Yau manifold admits a isometry, as will be the case with the geometries that we consider in this paper, the integers are individually invariant [2, 14]. On local Calabi-Yau manifolds that engineer gauge theories in 4 and 5 dimensions, can be identified [15, 16, 5] with the Nekrasov partition function [17], with playing the role of the equivariant parameters introduced by Nekrasov in his localization calculation.

The refined plays a central role in the AGT correspondence [18] and its generalizations, via which the refined partition function is related to correlators in Liouville and Toda conformal field theories [19, 20, 21, 22]. In the limit, refined relates to integrable models underlying the space of vacua of supersymmetric gauge theories [23] . It also features prominently in a recent proposal [24] for a non-perturbative completion of the refined topological string on toric geometries which relies crucially on refinement for the so-called pole cancellation mechanism [25, 26].

In addition to the coupling constants and , refined depends on Kähler parameters  of the underlying geometry. The modular methods referred to in the opening paragraph yield as an asymptotic series in and with coefficients that are exact in , whereas topological vertex computations yield as an expansion in with coefficients that are exact in and [4, 5]. For the latter, on appropriate geometries, the sum over some [27] but not all of the Kähler parameters can be performed. The methods [10, 28] we will advance in this paper for elliptically fibered manifolds rely on modular considerations and the holomorphic anomaly equation in the form [29], and yield again as an expansion in Kähler parameters assigned to the base of the elliptic fibration, but now with coefficients that enjoy modular properties making the invariance of under fiber monodromies manifest. These coefficients are meromorphic Jacobi forms of vanishing weight and of index determined by the holomorphic anomaly equations, with modular parameter the Kähler parameter of the fiber, and elliptic parameters built from the parameters and additional Kähler parameters identified with masses in the six dimensional setting that we will discuss presently. Based on the pole structure of these forms, as implied by the form of in terms of 5d BPS states [11, 12, 30], we can argue that they must be quotients of weak Jacobi forms, with a universal denominator. Determining the numerator at each base degree then becomes a finite dimensional problem, which can be solved by imposing boundary conditions in the form of vanishing conditions on the refined BPS invariants . These methods can also be applied to compact geometries [10], in which case the vanishing conditions are however not sufficient to solve the theory, as the index of the denominator grows too rapidly with the base degree [28].

The geometries on which we will consider yield via F-theory compactification an intriguing class of chiral 6d supersymmetric field theories. These theories are exotic in that they generally do not admit a Lagrangian description, exhibit strings in their spectrum (which become tensionless in the IR), and defy the expectation based on power counting that they should be trivial in the infrared. Indeed, we will focus on geometries that lead to 6d theories with a superconformal fixed point in the infrared. The topological string on these geometries captures the Kaluza-Klein modes of the tensionless string upon circle compactification. As initiated in [31], studying these modes can yield insight into the nature of such strings. The elliptic genus of their 2d chiral worldsheet theory is closely related to , and provides a physical explanation for the transformation properties as Jacobi forms, based on the chiral anomaly of the theory.

A classification of geometries leading to 6d super conformal field theories upon F-theory compactification, conjectured to be complete, is presented in [32, 33], based on earlier work in [34]. They consist of elliptic fibrations over non-compact complex surfaces which contain tree-like configurations of intersecting ’s. In this work, we will focus on geometries leading to 6d theories without enhanced gauge symmetry; this requires the self-intersection number of these curves to be or . In upcoming work [35], we will take up the more general case.

The cases in which all curves in have self-intersection number are covered by the resolution of the quotient , with a discrete subgroup of . As in this case has trivial canonical class, the elliptic fibration yielding the Calabi-Yau 3-fold is trivial, and the corresponding 6d theories have supersymmetry. The series in this class yields superconformal theories which describe a stack of M5 branes. The case, corresponding to the geometry , is called the M-string [36].

The generic case requires a non-trivial elliptic fibration to yield a Calabi-Yau 3-fold over , and leads to a 6d theory with supersymmetry. The simplest example of this class is called the E-string [37, 38, 31], and consists of the elliptic fibration over the non-compact base surface . This geometry can also be constructed as the total space of the canonical bundle of the compact elliptic surface K3, a nine point blow-up of . According to the classification results in [32, 33], an ADE chain of curves with a single curve at one end also engineers a 6d theory without gauge symmetry with a superconformal fixed point.

Our methods yield closed results at a given base degree from which all refined BPS invariants can easily be extracted. We list some of these invariants for ease of reference for mathematicians approaching them by other means. We use this data to refine the vanishing conditions on these invariants which follow from application of the adjunction formula. For the E-string, this data also allows us to provide some circumstantial evidence that the E-string BPS spectrum is computable via the quantization of an appropriate underlying quantum curve: We show that a suitably generalized consistency condition [39] between two perspectives [24, 39] on the quantization of the mirror curve in the case of toric Calabi-Yau manifolds is satisfied by these invariants. For a special class of toric Calabi-Yau manifolds, the consistency condition is shown in [40] to be the Nekrasov-Shatashvili limit of the Göttsche-Nakajima-Yoshioka -theoretic blowup equation [41, 42, 43] for the gauge theoretic Nekrasov partition functions. The natural generalization of the blowup equation that we propose, and that is satisfied by the E-string, can be applied to any local Calabi-Yau geometry on which the refined topological string can be formulated.

This paper is organized as follows. In section 2, we outline the general strategy to obtain the refined topological string partition on elliptically fibered Calabi-Yau manifolds and introduce the coefficients of the expansion of in base Kähler parameters. We then review integer BPS invariants and their vanishing from a geometric point of view in section 3.1 and present the geometries we will discuss in this paper, centered around the E- and the M-string, in section 3.2. We discuss the geometric meaning of the mass parameters in section 3.3. In section 4.1, we show that Jacobi forms satisfy a differential equation, which we identify with a generalization of the holomorphic anomaly equations in wave function form to the refined case in subsection 4.2. The differential equation encodes the indices of the various elliptic parameters on which depends. These indices can be derived from the anomaly polynomial of the 2d chiral theory living on the worldsheet of the non-critical strings of the 6d theories, as we demonstrate in section 4.3. In section 4.4, we make a universal ansatz for the denominator of based on the pole structure of refined as implied by the form of in terms of 5d BPS numbers. Together with the information regarding the indices, this determines up to a finite number of coefficients. In section 5.1, we determine under what circumstances vanishing conditions on BPS invariants suffice to determine these coefficients. We conclude that for all geometries relevant to this paper, they suffice2. Our approach thus leads to a complete solution for the class of theories we are considering. We present concrete results for the M-string in section 5.2, for the E-string in section 5.3, and for the E-M string chain in section 5.4. Further results and examples for refined BPS invariants are relegated to appendix C. In section 6, we extract BPS numbers from our closed expressions for and discuss their structure and their vanishing. In section 7, we relate our results for the E- and M-string to the ones obtained by [44] using a domain wall argument in terms of ratios of theta function. Based on theta functions identities, proven in appendix B, we show that additional poles exhibited by these latter expressions are spurious. Finally, in section 8, we propose a generalization of the blowup equation and find that it is satisfied for the E-string. Some of the data needed for this check is presented in appendix D. Some background on the ring of modular and generalized Jacobi forms is provided in appendix A. As we summarize in the conclusions in section 9, our method has a wide range of applicability. We are confident that it can be adapted to encompass all geometries engineering 6d superconformal field theories [45, 35].

2 General strategy for the solution of the topological string partition function on elliptically fibered Calabi-Yau manifolds

We will compute the refined topological string partition function on elliptic Calabi-Yau spaces with at least one zero section recursively as an expansion in the base classes . The fundamental objects in this study are the expansion coefficients in these classes, defined via

(2.1)

We distinguish between three classes of Kähler parameters: those associated to the base, denoted , the exponentials of which provide our expansion parameters,3 the Kähler parameter of the elliptic fiber, with , which will play the role of modular parameter, and the remaining Kähler parameters associated to mass deformations of the theory (for geometries containing curves of self-intersection number less than -2, there will also be Kähler parameters associated to the resolution of singular elliptic fibers, see [45, 35]).

For the unrefined case , an ansatz for has been presented in [10] based on general arguments that is valid both for compact and non-compact elliptically fibered Calabi-Yau spaces. In this paper, we seek to generalize this ansatz to the refined case. The ingredients are the following:

  • Elliptically fibered Calabi-Yau manifolds exhibit special auto-equivalences of the derived category of coherent sheafs which can be identified with the and the transformation of the modular group . Via the Fourier-Mukai transform, they induce an action on the -theory charges . Via mirror symmetry, this action can be identified with the monodromy action of the symplectic group on the middle cohomology of the mirror manifold to . This action can also be computed directly on . One obtains

    (2.2)

    The projective nature of the first two transformations arises as the Kähler parameters are ratios of periods. The more non-trivial fact is the transformation of up to exponentially suppressed contributions. In particular gives rise to a multiplier system . It is characterized by , see [28] for a derivation.

  • The transformation properties of under this action have been determined in [29].

  • Demanding that (before taking the holomorphic limit!) be invariant under the monodromy transformations (2.2) requires also transforming as

    (2.3)

    Combining this with results on the refined holomorphic anomaly equations allows us to extend this transformation property to the refined case, yielding

    (2.4)
  • The invariance of under the joint transformations (2.2) and (2.4) implies that in the holomorphic limit, transform as Jacobi forms with the multiplier system , modular parameter and elliptic parameters .

  • The pole structure of as suggested by the form of the free energy expressed in terms of 5d BPS numbers (as reviewed in the next section, see formula (3.11)) implies that is a ratio of weak Jacobi forms, and allows us to make a universal ansatz for the denominator, see (4.31).

  • The indices related to the elliptic parameters can be calculated in the unrefined case from the topological terms that appear in the holomorphic anomaly equations of [29] as explained in [10, 28]. A derivation of the indices in the refined case should be possible based on the refined holomorphic anomaly equations. We provide such a derivation, modulo several constants which we fix by studying examples. We demonstrate that our final result also follows from anomaly considerations, following [45].4

  • The ring of weak Jacobi forms with given even weight and integer index in a single elliptic parameter is generated by the forms and , see appendix A. Lacking an analogous structure theorem for weak Jacobi forms exhibiting multiple elliptic parameters, we conjecture that the numerator of is an element of the ring generated by , where , together with an appropriate generating set of Weyl invariant Jacobi forms: these exhibit the mass parameters as tuples of elliptic parameters, and are invariant under the action of the Weyl group of the flavor symmetry group on these tuples. This fixes the numerator up to a finite number of coefficients.

  • Vanishing conditions on 5d BPS invariants suffice to uniquely fix the numerator, and therewith . We thus obtain explicit expressions for which, aside from passing the stringent test of integrality for all 5d BPS invariants encompassed, match all results available in the literature computed by other means.

3 Geometry

We begin this section by recalling some aspects of the geometric invariants associated to the topological string and its refinement. We then introduce the elliptically fibered geometries that will be the subject of this paper.

3.1 Integer BPS numbers and their vanishing

The free energy of the topological string on a Calabi-Yau manifold was initially defined from the world-sheet point of view as a sum over connected word-sheet instanton contributions, via the expansion

(3.1)

The rational invariants are called Gromov-Witten invariants. Mathematically, they can be defined as follows: Let be the moduli stack of holomorphic embedding maps such that the image lies in the curve class . Then is the degree of the virtual fundamental class. As the virtual dimension is zero on Calabi-Yau 3-folds, the determination of reduces to a point counting problem with rational weights, generically with a non-zero answer.

An important insight into the structure of the topological string free energy was provided by string-string duality. Comparing the expansion (3.1) with a BPS saturated heterotic one loop computation, Gopakumar and Vafa [11, 12] conjectured that the expansion takes the general form

(3.2)

The coefficients of this expansion are integers. They can be identified with the counting parameters in the BPS index

(3.3)

defined via a trace over the Hilbert space of BPS states in the compactification of M-theory on the Calabi-Yau manifold .5 The BPS states can be organized into representations of the Poincaré group; their mass is proportional to their charge , and each component of can be decomposed into irreducible representations of the little group ,

(3.4)

Generically, this decomposition is not invariant upon motion in moduli space. The invariant index (3.3) is obtained upon summing over the spin quantum numbers of with alternating signs. Using ideas of [12], this index was geometrically interpreted in [13] based on the Lefschetz decomposition of the moduli space of a - brane system. was realized in [13] as the Jacobian fibration over the geometric deformation space of a family of curves of maximal genus in the class wrapped by the brane. The brane number counts the degeneration of the Jacobian and is hence related to the geometric genus of the curve.

An important ingredient in our determination of the topological string partition function will be the vanishing of the invariants at fixed class for sufficiently large . For geometries which are the total space of the canonical bundle of a compact surface , as is the case for the E-string and the M-string geometry which we will review below, this vanishing can be argued for very simply: the only curves that contribute to the partition function lie inside the compact surface. Given a smooth representative of a class , the adjunction formula permits us to express the canonical class of in terms of its normal bundle in the surface, and the canonical class of the surface . Via the relation between the degree and the arithmetic genus of the curve, , this yields

(3.5)

providing an upper bound

(3.6)

on the geometric genus of any representative of the class . Note that a smooth representative of a class does not necessarily exist. More generally, we have to resort to Castelnuovo theory, which studies the maximal arithmetic genus of curves of given degree in projective space, see e.g. [46]. Either way, at fixed , there exists a bound beyond which vanishes, simply because the moduli space is empty. This is in contrast to the Gromov-Witten invariants , which due to multi-covering contributions do not vanish for fixed even at arbitrarily large genus. At large degree, the bound scales as . We will use the term Castelnuovo bound generically to refer to a bound beyond which BPS invariants vanish.

Mathematically, the BPS data of the - brane system is best described by stable pair or PT invariants [47]. If the curve is smooth at the bound and the obstructions vanish, the dimension of the deformation space follows from the Riemann-Roch theorem

(3.7)

and the evaluate to

(3.8)

The invariants at generically evaluate to much larger numbers due to the increasing degeneration of the Jacobian.

When the Calabi-Yau manifold admits a isometry, one can also give a geometrical interpretation to states with definite + Lefschetz number and the BPS numbers introduced in (3.4) which keep track of the degeneracy of states with definite and the spin. The corresponding index is

(3.9)

We have here assigned to every irreducible representation labeled by a Laurent polynomial,

(3.10)

where the summation index is increased in increments of starting at . These BPS numbers are computed by the refined topological string, whose free energy takes the form [30]

(3.11)

We have introduced the variables

(3.12)

with

(3.13)

For geometries that geometrically engineer gauge theories in 5 dimensions, (3.11) coincides with the -theoretic instanton partition function of Nekrasov [17], and map to equivariant parameters in the localization computation performed in the gauge theoretic setting.

To facilitate the transition between the refined and the unrefined free energy, it is also convenient to introduce the variables

(3.14)

Comparing (3.3) and (3.9) allows us to relate the BPS number to the invariants . In terms of the tensor representations , where the bracket indicates the irreducible representation of spin , we obtain

(3.15)

This relation implies that if at fixed , the maximal genus for which is , the maximum left spin for which the number is

(3.16)

To describe a generic bound on the right spin, we will need to review some aspects of PT theory of stable pairs [47, 48] and its relation to KKV theory [13] and refined KKV theory as outlined in [2]. As above, we will assume that there is a smooth curve in the class . Following the notation in [2], we denote by the moduli space of stable pairs , where is a free sheaf of pure dimension 1 generated by the section outside points, with and holomorphic Euler characteristic . The PT invariant is defined as the degree of the virtual fundamental class of . These invariants were related to in [48]. Following [2], we will review a generic model for based on which the decomposition of the Hilbert space of 5d BPS states can be computed. This will permit us to arrive at a bound for the maximal right spin of non-vanishing BPS numbers . Based on experience, this also serves as a bound below .

The basic identity relating the geometric Lefschetz decomposition to the decomposition of is motivated by the KVV approach [13, 2]. It is given, at sufficiently small Euler characteristic as we explain presently, by

(3.17)

here is the relative Hilbert scheme parameterizing curves of class with distinguished points. In the generic cases we will consider, such Hilbert schemes will be projective spaces, and the corresponding cohomology carries the standard Lefschetz decomposition. is an lowering operator, i.e. it acts as

(3.18)

with the understanding that for . is the map from representations to the diagonal representation. Corrections to (3.17) arise from reducible curves. At a given class , the minimal Euler characteristic of a reducible curve with components of class will be . At given , the minimal Euler characteristic at which corrections due to reducible curves can arise is thus given by

(3.19)

Below this bound, (3.17) holds unmodified.

We will argue for a bound on the number at given for geometries which are the total space of the canonical bundle over a Fano surface . In this case, for small bounded by a constraint linear in [48], the moduli space of stable pairs on coincides with that of , . The latter in turn is isomorphic to the appropriate relative Hilbert scheme of curves of class with distinguished points (cf. proposition B.8 of [48]),

(3.20)

For a curve of class , a model for this Hilbert scheme is given by over .

Returning to (3.17), we see that the contributions to highest left spin can be read off at . We obtain

(3.21)

i.e.

(3.22)

We reiterate that we have derived as the right spin at highest left spin, but expect it to present more generally a bound on right spin at given . This generic bound is quadratic in , just as the bound on . It is implicit in the formalism of [2], but not spelled out explicitly. Instead, more complicated calculations are performed there for , showing an asymptotic pattern of the refined BPS numbers that we also observe in our examples. The generic bound is satisfied for the large asymptotics in all known geometric examples, in particular for the -string for which is only semi-Fano.

Let us finally comment on geometries bases on chains of and curves. We can consider each such curve as the base of an elliptic surface. For curve classes lying in a single such surface, the bounds (3.16) and (3.22) should apply unaltered. More general classes will require a more detailed analysis. In these cases, the leading contributions must come from reducible curves which have been excluded by the condition (3.19), so the precise bound will have to include these. We still expect it to remain quadratic in the curve class. This expectation is confirmed by our explicit computations, see section 6.

3.2 Geometries underlying the E- and M-string and generalizations

The geometries we will consider in this paper consist of elliptically fibered Calabi-Yau manifolds over non-compact bases , such that

  • all compact curves in are contractible, and

  • the elliptic fiber does not degenerate over all of for any .

The first condition is required for the 6d theories engineered by compactifying F-theory on to have a superconformal fixed point [32, 33], and is a prerequisite, as we explain below, for the boundary conditions supplied by vanishing conditions on GV invariants to completely fix our ansatz. The second condition implies that the 6d theories do not exhibit gauge symmetry. We will treat the case with gauge symmetry in a forthcoming publication [35].

The simplest examples that we consider contain a single compact curve in the base . These geometries can be obtained via decompactification limits of elliptic fibrations over Hirzebruch surfaces. Recall that a Hirzebruch surface is a fibration over a base . Two irreducible effective divisors and can be chosen to span . corresponds to a fiber of , hence , and is a section of the fibration, thus , with . The canonical class of in terms of these two divisors is given by .

Calabi-Yau elliptic fibrations over the Hirzebruch surfaces , , can be constructed as hypersurfaces in an appropriate ambient toric variety. The constructions considered in [49] yield geometries with , corresponding to the lifts of and , and the elliptic fiber . Taking the defining equation of the hypersurface to be in Weierstrass form,

(3.23)

we must choose and as sections of particular line bundles over the base surface ,

(3.24)

to ensure that the total space of this fibration is a Calabi-Yau manifold. Here, is the canonical class of . The elliptic fibration is singular along the discriminant locus of the fibration, defined as the zero locus of the discriminant

(3.25)

The type of singularity along each component of the discriminant locus depends on the vanishing order of together with that of and on this component, and has been classified by Kodaira.

The minimal vanishing order of a section of the line bundle along a curve with negative self-intersection number can be computed by decomposing , with the smallest positive integer such that . Equivalently, we can consider a decomposition , where now, and we impose . We have , and by minimality of , ; hence the minimal vanishing order is given by the smallest integer larger or equal to . Either way, based on the canonical class of the Hirzebruch surface cited above, we can argue that for , sections of or generically do not vanish identically over . Thus, will generically vanish only at the intersection of with , hence at points. As and will generically not vanish at these points, the elliptic fibration above (recall that is a section of ) will generically exhibit isolated singularities, hence correspond to a surface, surface, and the trivial product respectively.

We decouple gravity by decompactifying these geometries along the direction of the fiber of the Hirzebruch surface. The fibration structure of these geometries is summarized in diagram (3.26).

(3.26)

We can engineer more general 6d superconformal theories by replacing the base of the non-compact base of the elliptic fibration by a string of intersecting ’s. Setting , with chosen among the discrete subgroups of , yields theories with supersymmetry, as the elliptic fibration here is trivial. The string of compact curves in and their intersections is encoded in the Dynkin diagram corresponding to the subgroups ; recall that these enjoy an ADE classification. In particular, all of these curves have the topology of and have normal bundle . We refer to them as curves. The massless M-string is the simplest member of this class of geometries, with .

The papers [32, 33] discuss which F-theory compactifications on elliptic fibrations over non-compact bases can lead to 6d SCFTs with generically only (1,0) supersymmetry. The only additional SCFTs without gauge symmetry contained in this classification arise upon including a single curve at the end of a string of curves; in the case , this corresponds to an E-M string chain bound state.

The topological data that we will need to determine the topological string partition function on these geometries consists of the intersection numbers of these curves, as well as their intersection product with the canonical line bundle of . The latter is once again simply determined by the adjunction formula (3.5), this time applied to . As all compact curves in have genus 0, we obtain

(3.27)

3.3 Turning on mass parameters

Above, we have identified the elliptic fibration over the total space of the bundle as local . Let us briefly recall the geometry of the surface. It can be obtained as the blow up of at nine points. One can construct a series of del Pezzos surfaces by blowing up generic points in . The canonical class is given by

(3.28)

Here, is the hyperplane class of and , , denote the exceptional curves resulting from the blowups. The intersection numbers between these divisors are

(3.29)

Let and

(3.30)

The intersection product induces an inner product on this lattice. One can readily show that can be equipped with a root (or co-root) basis whose negative Cartan matrix is for . The simple roots are always of the form or , where are taken to have distinct values.

For example, in the case , we obtain the root lattice of , as depicted in figure 1. We set , , and . The labels correspond to the upper indices in the figure (the lower indices are the Dynkin labels). The full set of positive roots is then all , of which there are 28, all (56), all (28) and (8), yielding .

Figure 1: The root lattice of affine .

It follows from the intersection numbers that . The surface is obtained by blowing up the unique base point of the elliptic pencil that models . It is elliptically fibered over this exceptional divisor . As , is no longer positive. Indeed, it can be identified with the elliptic fiber of this surface. Setting , the divisors can be mapped to simple roots of the affine Lie algebra, with constituting the affine root. The additional intersection numbers are , , and .

The explicit geometric interpretation of the mass parameter in the case of the M-string, and more generally in the case of elliptic fibrations over resolved ADE singularities, is not equally transparent to us. Following [50], one could attempt to interpret the mass parameter as deforming the geometry away from the trivial product by fibering the non-compact surface over . We leave the exploration of this possibility (and in particular the analysis of the and case, the mass deformation of which is excluded in [51] based on physical considerations) to future work. In section 6, we will discuss a related proposal for the mass deformation.

4 Holomorphic anomaly and Jacobi forms

The holomorphic anomaly equations for elliptic fibrations can be written in a form which implies that is a Jacobi form of weight 0 and index a fixed function of . For the relevant properties of Jacobi forms, see appendix A. In the following, we will use the notation

(4.1)

with , for the two generators of the ring of holomorphic weak Jacobi forms interchangeably. When only one argument is given, the elliptic parameter is meant.

4.1 A differential equation for Jacobi forms

A weak Jacobi form of weight has a Taylor expansion

(4.2)

with the being quasi-modular forms of weight (in fact, one can be more precise about the form of these coefficients, expressing them as functions of modular forms, their derivatives, and the weight and index of [52]). Considering (4.2) term by term, we deduce that the phase factor arising upon modular transformation of is due to the occurrence of factors of in the coefficients . Since transforms under modular transformations as

(4.3)

the exponential

(4.4)

generates the inverse of the phase factor occurring in the transformation of the Jacobi form. The product thus transforms without phase factor. Arguing again termwise, we conclude that the dependence in the Taylor coefficients is cancelled by this prefactor. Hence a Jacobi form of index satisfies the equation

(4.5)

Conversely, a power series in with quasi-modular coefficients that satisfies the equation

(4.6)

transforms under modular transformations as a Jacobi form of index . If it is also periodic under , its transformation behavior under as a Jacobi form follows by combining periodicity and modular transformations. In the following, we will use elliptic parameters with period ; they are related to by . For these,

(4.7)

4.2 The index from the holomorphic anomaly equations

For the geometries described in the previous section, the holomorphic anomaly equations have been motivated in the form [10]

(4.8)

Here, the modular parameter of is the Kähler parameter of the elliptic fiber, and , where denote compact divisors in the non-compact base surface of the elliptic fibration, whose canonical divisor is denoted as . Comparing to equation (4.7), we conclude that the are Jacobi forms with modular parameter , elliptic parameter the string coupling , and of index

(4.9)

This expression for the index can easily be evaluated using the intersection matrix of the and the adjunction formula in the form (3.27). For the E- and the M-string, with , we obtain

(4.10)

Already in [53], it was argued that the masses of the E-string should enter the partition function as the elliptic parameters of Jacobi forms, with index proportional to the number of strings. The same holds true for the M-string. This gives rise to the equation