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arXiv:1412.5739v2 [hep-th] 28 Jan 2015 ocmatfiain nlwrsaetm iesosas dimensions space-time lower generalizes well. in duality compactifications F-theory–heterotic suitable to the over duality spaces, this base fibering adiabatically Upon 4]. fibered elliptically an K on F-theory dimensions space-time a enls xlrdo h ee fmdl pcsfor spaces the moduli to of it due level is [7], This the spectra interactions. on effective particle explored (quantum-exact) less of) been (indexes has for established de- well F-theory dual the that in — description [9] scription. states geometric NS5-branes a duality as the enjoy such of the — aspects side non-perturbative string certain heterotic heterotic on the light shed on under- has well Furthermore, rather is effec- spectrum energy stood. particle low theory the gauge and tive space compactification geo- theory to toolbox [7]. powerful theories a gauge cover supersymmetric to (minimal) spectral led engineer the metrically has via 6] string [2, heterotic groups construction the gauge to non-Abelian F-theory us compacti- of in given F-theory relationship has of The physics string fications. the heterotic into the insights to valuable duality the theory, ∗ 1 2 † yeIAhtrtcdaiyi i pc-iedmnin [5 dimensions space-time six relates in duality duality IIA–heterotic F-theory–heterotic type torus, two duali eight-dimensional additional the compactifying particular, In [email protected] [email protected] o eiwo hstpc e o ntnerf [8]. ref. instance for see topic, this on review a For ufc otehtrtcsrn ntetotrs[1– torus two the on string heterotic the to surface 3 -hoyhtrtcdaiyrmral eae neight in relates remarkably duality F-theory–heterotic hl -hoyhtrtcdaiydcinr sfairly is dictionary duality F-theory–heterotic While F- of description microscopic a have not do we As 1 2 sarsl h itoaybtenteF- the between dictionary the result a As fnngoerchtrtccmatfiain capture dime compactifications six heterotic for particular non-geometric sem In of the dimensions. to gen limit space-time (fibered) lower precise of and the terms establish in we models — of compactifications he class particular the a in whic for limit — side Malmendier–Morrison fiber and large Clingher–Doran by and proposal F-theory in limit degeneration tig hc erfrt stehtrtcSnlmt nti l Val this and In Pioline limit. Louis, Sen Alexandrov, effects. the heterotic by of the established regi structure as recently quantum fibrational two-staged to the s semi-classical refer in the geometric we deep a limit which with another string, combined analyze forms we modular dimensions Siegel two genus of eeoi hoy h eicasclpaeo the on phase semi-classical the theory, heterotic elliptically on compactifications supergravity heterotic nti okw td h ult ewe -hoyadtehet the and F-theory between duality the study we work this In .INTRODUCTION I. o-emti -hoyHtrtcDuality F-Theory–Heterotic Non-Geometric hsklshsIsiu,Uiesta Bonn Universität Institut, Physikalisches eh etrfrTertclPhysics Theoretical for Center Bethe i Gu Jie 31 on Germany Bonn, 53115 yo an on ty othe to ∗ ]. n asJockers Hans and K in,wihdsrbsteprilhgsn ftegauge the of higgsing partial the group dimen- describes space-time which eight in sions, pair simple spaces particular F-theory–heterotic moduli a Malmendier of dual the interactions) — effective identify the to Doran thus (and achieve and [22] Morrison Clingher and by pursued gram elliptically-fibered certain of K structure Shioda–Inose the esos—telmtt h eisal eeeainge- di- degeneration space-time semi-stable lower the in to and limit eight the — in mensions in both establish — explicitly picture We this [22]. F-theory– correspondence heterotic Clingher–Doran–Malmendier–Morrison the which varieties, varieties. Calabi–Yau projective not fibered are genus-two the- of description string geometric terms heterotic effective in an These pose 22]. nevertheless ories [16, F-theory of string geometric compactifications heterotic non-geometric simple the to such rise give that it scenarios argued dimensions, Upon further space-time [23]. lower is forms to action modular effective compatifying Siegel energy two further low genus the in and encoded curve is quantum-exact captured two the genus geometrically a that is by find description they heterotic duality effective the Result- from ing theory. supergravity eight-dimensional effective ecie h ult nya h onayo h moduli theories. the dual of which the boundary — the of 10] of at spaces only 3, limit duality [2, fiber the large space describes a compactification and heterotic geometry degeneration the stable F-theory a the in formu- of namely — often limit limit is certain a duality in F-theory–heterotic lated the that fact ufc srcvrdfo eakbelimit remarkable a from recovered is surface 3 3 e,frisac,rf.[,1–5,weepriua inter particular where limit. this 11–15], beyond [4, described refs. are instance, for See, ufcs[72]—bsdo h ahmtclpro- mathematical the on based — [17–21] surfaces 3 α uligo ale ok[1 2 6 n yexploiting by and 16] 12, [11, work earlier on Building h i fti ril st ute analyze further to is article this of aim The fibered ′ unu orcin otesemi-classical the to corrections quantum eeoi yemliltmdl pc — space moduli hypermultiplet heterotic E soa hois eageta hsclass this that argue we theories, nsional stocre snngoercheterotic non-geometric as curves two us -lsia eeoi tigi oheight both in string heterotic i-classical 8 eoi hoy uliguo recent a upon Building theory. terotic nr escretdb quantum by corrected gets — andro K × hae h ult nteheterotic the on duality the phrases h ufcs rmtenon-geometric the From surfaces. 3 † eo h o-emti heterotic non-geometric the of me mtw a xlctyagethat argue explicitly can we imit E reyoeain ial,i four in Finally, operation. urgery 8 rtcsrn eodtestable the beyond string erotic to E 7 × E 8 3 fteascae o energy low associated the of BONN–TH–2014–17 actions 2 ometry on the F-theory side and the large fiber limit on where f8(t) and f12(t) are polynomials of degrees 6 8 the heterotic side. Establishing this semi-classical limit and 6 12 respectively of the affine coordiante t on the to the F-theory–heterotic duality provides for a natural P1 base. These are 22 parameters. After modding out starting point to study heterotic quantum corrections to the SL(2, C) diffeomorphism on the base P1 as well as its semi-classical large fiber approximation, as described one overall rescaling, one finds 18 complex moduli in the by the non-geometric effective heterotic compactification F-theory as well. space. Beyond the simple match of dimensionalities, an exact Deep in the non-geometric heterotic regime, we dis- dictionary between the moduli spaces of the two theories cover in the moduli spaces of genus two fibered effective is only known in the large volume limit on the heterotic heterotic compactifications the loci of constant (but non- side, which corresponds to the stable degeneration limit trivial) genus two fibration. In analogy to the Sen limit in on the F-theory side [2, 3, 10]. In ref. [11] a dictionary was F-theory [24], we call such loci the heterotic Sen limit. In given when all the Wilson lines are turned off. Clingher– particular, for non-geometric heterotic compactifications Doran and Malmendier–Morrison recently extended this to six and four space-time dimensions, we connect the result to one non-trivial Wilson line in the heterotic the- heterotic Sen limit to the structure of the hypermultiplet ory [20–22]. Our first task is to analyze their proposal in moduli space, as recently discussed in the semi-classical further detail by exhibiting its relation to the duality in approximation in refs. [25, 26]. By comparing with the the large volume/stable degeneration limit. heterotic Sen limit, we give evidence that their proposed hierarchical fibrational structure in the hypermultiplet E × E moduli space is a property of the semi-classical approxi- A. Moduli space of the 8 7 heterotic string mation in the absence of quantum corrections. 2 The organization of this article is as follows. In Sec- The E8×E8 heterotic theory on T with one non-trivial tion II we discuss the F-theory–heterotic correspondence Wilson line has the unbroken gauge group E8 × E7. Its in eight space-time dimensions. In particular, we es- moduli space is parametrized by two complex moduli τ,ρ tablish the limit to the semi-classical approximation of and the complex Wilson line z, and it is given by [22, 27] the F-theory–heterotic duality in the F-theory stable de- 4 generation limit and the heterotic large volume limit in + 2,3 terms of the Siegel operator acting on genus two Siegel Mhet = D2,3/O (L ) . (2) modular forms. In Section III we consider the F-theory– with the Teichmüller space heterotic correspondence in the non-geometric phase adi- 1 abatically fibered over a P base. Again we establish the D2,3 = (O(2) × O(3))\O(2, 3) . (3) limit to the F-theory–heterotic duality in the stable de- The action of the U-duality group O+(L2,3) mixes all generation and the large fiber limit, where we recover 2,3 the semi-classical heterotic geometric compactification on moduli τ,ρ,z, where the U-duality lattice L of sig- the elliptically-fibered K3 surface. Deep in the heterotic nature (2, 3) is the orthogonal complement of the sublattice E8(−1) ⊕ E7(−1) in the unique even unimod- quantum regime, we define the heterotic Sen limit, which 2,18 allows us to compare with the semi-classical two-staged ular lattice Λ of signature (2, 18). The sublattice fibrational structure of the hypermultiplet moduli space E8(−1) ⊕ E7(−1) is associated to the remaining 17 Wil- of four-dimensional heterotic compactifications studied son lines to be set to zero. by Alexandrov, Louis, Pioline and Valandro. In Sec- The Teichmüller space D2,3 is isomorphic to the genus tion IV we present our conclusions. two Siegel upper half-space [28] τ z H = τ = Im(τ) pos. definite , (4) 2 z ρ

II. 8D N=2 F-THEORY–HETEROTIC DUALITY where τ,ρ,z ∈ C. The duality asserts that the Siegel H Z It was proposed in refs. [1–3] that F-theory compacti- threefold A2 = 2/Sp(4, ), which is a partial compact- fied on an elliptic K3 surface with a section is dual to the ification of the moduli space of genus two curves, is isomorphic to the moduli space Mhet. E8 × E8 heterotic string theory compactifed on a torus 2 The Siegel threefold A2 has a boundary H0 and two T . Since then much work has been devoted to under- Z stand this duality. The moduli space of the heterotic singular surfaces of 2-orbifold singularities known as the string has complex dimension 18, combining the com- Humbert surfaces H1 and H4. We refer to them as di- plex structure modulus τ and the complexified Kähler visors H0, H1 and H4, respectively, in both the com- modulus ρ = B + iJ of the torus as well as the 16 compactified moduli space A2 of genus two curves and the plex Wilson lines. In the F-theory one can write down the Weierstrass model for the elliptic K3 surface with a section 4 Here we follow ref. [22] and use the index 2 subgroup O+(Λ2,3) of the full duality group O(Λ2,3), as the former is the maximal 2 3 subgroup for which modular forms are holomorphic. y = x + f8(t)x + f12(t) (1) 3 compactified heterotic moduli space Mhet. The bound- weight by [31] ary H0 at ρ → i∞ is associated to a singular genus two 1 curve with a nodal point. At the Humbert surface H1 χ2 = χ 224 315 χ5 − 213 39 ψ3χ4 − 213 39 ψ2 χ4 at z = 0 the genus two curve degenerates into two genus 35 212 39 10 12 4 12 6 12 3 6 3 3 3 2 3 14 8 2 3 one components transversely intersecting in a single dou- +3 ψ4 χ12 − 2 · 3 ψ4 ψ6 χ12 − 2 3 ψ4 ψ6 χ10 χ12 ble point, whereas the Humbert surface H at τ = ρ 4 − 223 312 52 ψ χ2 χ3 +33 ψ4 χ3 +211 36 37 ψ4 χ2 χ2 describes a (generically) smooth genus two curve with a 4 10 12 6 12 4 10 12 11 6 2 2 2 23 9 3 3 2 non-generic Z2-automorphism (in addition to the hyper- +2 3 5 · 7 ψ4 ψ6 χ10 χ12 − 2 3 5 ψ6 χ10χ12 elliptic involution). In the heterotic theory, the divisor 2 7 2 2 4 2 2 − 3 ψ4 χ10 χ12 +2 · 3 ψ4 χ6 χ10 χ12 H0 maps to the large volume limit, at the surface H1 +211 35 5 · 19 ψ3 ψ χ3 χ +220 38 53 11 ψ2 χ4 χ the Wilson line is turned off, and at the surface H4 a 4 6 10 12 4 10 12 Z 2 4 2 11 5 2 3 3 6 3 2-quantum symmetry emerges. − 3 ψ4 ψ6 χ10 χ12 +2 3 5 ψ6 χ10 χ12 − 2 ψ4 ψ6 χ10 As the Siegel threefold A2 is parametrized by the ring − 212 34 ψ5 χ4 +22 ψ3 ψ3 χ3 +212 34 52 ψ2 ψ2 χ4 of genus two Siegel modular forms of even weight gener- 4 10 4 6 10 4 6 10 +221 37 54 ψ ψ χ5 − 2 ψ5χ3 +232 39 55 χ6 . ated by the modular forms ψ4, ψ6,χ10,χ12 with weights 4 6 10 6 10 10 4, 6, 10, 12, respectively [29], it is tempting to identify (8) the heterotic moduli space Mhet with the weighted To further study our moduli spaces, we now locate the 3 P3 three divisors H0,H1,H4 in the space P . The sur- projective space (2,3,5,6) of homogeneous coordinates (2,3,5,6) face H1 at z = 0 is given by χ10 = 0, while the surface ψ4, ψ6,χ10,χ12. We will illustrate momentarily that this 2 description is not quite complete. H at τ = ρ becomes χ35 = 0, which is a polynomial 4 χ10 Let us recall the definition of the modular forms ψ2k constraint in the (even) generators ψ4, ψ6,χ10,χ12. of the classical Siegel Eisenstein series of weight 2k The large volume divisor H0 is more complicated. The Siegel operator sends a Siegel cusp form to zero (c.f., eq. (7)) and maps a Siegel Eisenstein series to an elliptic ψ (τ)= det(Cτ + D)−2k, k ∈ Z . (5) 2k + Eisenstein series of the same weight (C,DX) τ 0 (Φψ2k)(τ) = lim ψ2k = E2k(τ) . (9) Here the sum is taken over all equivalence classes of pairs ρ→i∞ 0 ρ (C,D) of symmetric 2×2 integral matrices subject to the equivalence relation (C,D) ∼ (M · C,M · D) in terms of Furthermore, due to the periodicity of ρ, a Siegel modular Z any SL(2, )-matrix M. Furthermore, χ10,χ12 are two form φ of weight k enjoys the Jacobi–Fourier development Siegel cusp forms defined by [21, 30] ∞ 2πimρ 43867 φk = φk,m(τ,z)e . (10) χ10 = 12 5 2 (ψ4 · ψ6 − ψ10) , mX=0 2 3 5 7 · 53 (6) 131 · 593 2 2 3 3 2 φ (τ,z) is a Jacobiform of weight k and index m, which χ12 = (3 7 ψ4 +2 · 5 ψ6 −691ψ12) , k,m 213 37 53 72 337 satisfies the two modular transformation properties

2πimcz2 which — as Siegel cusp forms — are mapped to zero aτ+b z k cτ d φk,m( , ) = (cτ + d) e + φk,m(τ,z) , under the Siegel operator cτ+d cτ+d (11) −2πim(λ2τ+2λz) φk,m(τ,z + λτ + µ)= e φk,m(τ,z) , τ 0 (Φχk)(τ) = lim χk =0 . (7) ρ→i∞ 0 ρ a b with λ, µ ∈ Z and ∈ SL(2, Z). Since φk, (τ,z) c d 0 The ring of all genus two Siegel modular forms has one of index 0 is independent of z (it is an elliptic modular form of weight k), we arrive with eqs. (7) and (9) at more generator namely a cusp form χ35 of odd weight [31].5 It is related to the Siegel modular forms of even lim ψ2k(τ)= E2k(τ) , lim χk(τ)=0 . (12) ρ→i∞ ρ→i∞

The coefficients in the full Jacobi–Fourier development of the Siegel Einstein series are given in ref. [32] 5 The ring of all genus two Siegel modular forms describes a dou- A ble cover of the Siegel threefold 2 as the modular forms of odd 2πi Tr(Nτ) degree reverse their sign with respect to the modular transforma- E2k(τ)= a(N)e . (13) tion of the matrix Diag(+1, −1, +1, −1) ∈ Sp(4, Z). This implies XN that a given genus two curve uniquely specifies the value of all even modular forms, but determines the odd modular forms only The sum is taken over all half-integral symmetric matri- up to a sign. ces N with integral entries on the diagonal. Combined 4 with the definition (6), we find B. Clingher–Doran–Malmendier–Morrison construction 2 2 χ12 ζ + 10ζ +1 4(ζ − 1) q lim = − 2 − ρ→i∞ χ10 3(ζ − 1) ζ Using the Shioda–Inose structure [19–21] on K3 sur- (14) faces with Picard lattices of rank 17, in ref. [22] Mal- 4(ζ +2)(2ζ +1)(ζ −1)2q2 3 mendier and Morrison determined the dictionary be- − 2 + O(q ) , ζ tween the moduli space of F-theory compactifed on an elliptic K3 surface with II∗ and III∗ singular fibers where q = e2πiτ and ζ = e2πiz. This series agrees with and that of the E8 × E8 heterotic string compactified on ℘(z;τ) 2 the expansion of π2 in q and ζ, where ℘(z; τ) is the T with one non-trivial Wilson line. Starting from the χ12 ℘(z;τ) Weierstrass ℘-function. As lim and 2 have Weierstrass model for the elliptically fibered K3 surface ρ→i∞ χ10 π the same modular properties, they must be identical, i.e., y2 = x3 + (at4 + bt3)x + (ct7 + dt6 + et5) , (19)

χ12(τ) ℘(z; τ) with affine coordinate t on the base P1, the singular fibers lim = 2 . (15) ρ→i∞ χ10(τ) π III∗ and II∗ are located over t = 0 and t = ∞, respectively. Then the five coefficients can be identified with Therefore, in the large volume limit the homoge- the Siegel modular forms P3 neous coordinates (ψ4, ψ6,χ10,χ12) of (2,3,5,6) become 1 (E4, E6, 0, 0), which furnishes a subvaritiety h0 of codi- a = − ψ (τ) , b = −4χ (τ) , c =1 , 48 4 10 mension two. It is located within the2 non-transverse in- (20) χ35 tersection of the divisors χ10 and of the Humbert 1 χ10 d = − ψ6(τ) , e = χ12(τ) . surfaces H1 and H4. As h0 has not the correct dimen- 864 sionality to be identified with the boundary divisor H0, Together with a choice of symplectic basis, they uniquely we consider the blow-up of P3 along the subvariety (2,3,5,6) fix the periods τ of a genus two curve, which describe the h0 moduli (τ,ρ,z) of the discussed heterotic string theory. Here, τ and ρ are the complex structure and the volume P3 P3 2 π : Blh0 (2,3,5,6) → (2,3,5,6) , (16) modulus of T , whereas z is the Wilson line modulus. We want to trace this correspondence to the stable de- explicitly realized by the incidence correspondence generation limit of the K3 surface, so as to argue that the blow-up (16) describes the physical moduli space Mhet P3 of the dual heterotic string theory. In this limit the P1- Blh0 (2,3,5,6) base of the K3 surface splits into two P1s intersecting = {(I , I , ψ , ψ ,χ ,χ ) | I χ − I χ =0} . 0 2 4 6 10 12 2 10 0 12 transversely at one point. Correspondingly, the elliptic (17) K3 surfaces splits into two elliptic rational surfaces S1 Here (I0, I2, ψ4, ψ6,χ10,χ12) denote the homogeneous co- and S , which intersect at an elliptic curve E [3]. This ordiantes of the total space P(O ⊕ O(1)) → P3 , 2 (2,3,5,6) elliptic curve E becomes the two torus of the heterotic where the projective fiber coordinates (I0, I2) have weight theory, while S1 and S2 determine the bundle data over P3 2 (0, 1) with respect to the base space (2,3,5,6). T . To extract E and S1,S2, we perform the change of Taking now the limit ρ → i∞ in the blown-up mod- variables P3 uli space Blh0 (2,3,5,6), we approach the fiber of the 2 3 −1 t x t y exceptional divisor π (h0) with the coordinate ratio (x,y,t) 7→ , ,πt , (21) χ12 I2 ℘(z;τ) 4 16 lim = = 2 . We claim that the excep- ρ→i∞ χ10 I0 π −1 tional divisor π (h0) maps to the boundary divisor H0, and the Weierstrass model becomes [13] 3 and that the blow-up space Blh P describes the 0 (2,3,5,6) −1 compactified heterotic string moduli space, i.e., tp+1 + p0 + t p−1 =0 . (22)

P3 where Blh0 (2,3,5,6) ≃ Mhet . (18) 8 7 p+1 =2 π , While blowing-up the codimension two locus h0 is a 4 8 p = −y2 +4x3 − π4ψ (τ) x − π6ψ (τ), (23) natural operation to arrive at the moduli space Mhet, a 0 3 4 27 6 detailed matching of moduli spaces would require a com- 8 3 8 5 p− = −2 π χ (τ) x +2 π χ (τ) . parison to the compactification procedure of the Siegel 1 10 12 threefold A2 as developed in refs. [33, 34]. This is beyond Here, p0 is the elliptic curve E, whereas p±1 encode the the scope of this note. Instead, we will give a physics bundle data. In the discussed limit p0 simplifies to argument in the next subsection that justifies the identi- P3 2 3 fication of Blh0 (2,3,5,6) with the moduli space Mhet. y =4x − g2(τ)x − g3(τ) , (24) 5

4π4 8π6 ′ with g2(τ) = 3 E4(τ) and g3(τ) = 27 E6(τ), and de- α corrections to the hypermultiplet moduli space play scribes an elliptic curve with period τ. Furthermore, with an important role. The dual F-theory description arises eq. (15) the intersection E ∩{p−1 =0} yields two points from an elliptically-fibered and K3-fibered Calabi–Yau (x, y) = (℘(z; τ), ±℘′(z; τ)) on the elliptic curve E that threefold X. Malmendier and Morrison show that the add up to zero, which characterize the SU(2) bundle over threefold X is an elliptic fibration over the Hirzebruch E in the heterotic string [2], associated to the Wilson line surface F12 [22]. This F-theory compactification of X is modulus z. actually familiar, as it appears in the conventional formu- On the heterotic side, the stable degeneration limit lation of the F-theory–heterotic duality [35]. This comes corresponds to the large volume limit ρ → i∞ of the as a surprise, namely both phases of the heterotic string two torus. Hence, we can compare this limit to our pro- — the geometric semi-classical and the non-geometric posal (18) for the compactified heterotic moduli space. quantum regime — are dual to the same semi-classical P3 In the space Blh0 (2,3,5,6), this limit maps to the excep- geometric F-theory compactification, and there is no ap- −1 tional divisor H0 = π (h0), where the point in h0 yields pearance of a non-geometric F-theory phase. the complex structure of E, whereas — up to a trivial To resolve this mystery, we consider the corresponding I2 ℘(z;τ) type IIA–heterotic duality [5], i.e., type IIA on the same rescaling — the ratio of the coordinates = 2 de- I0 π termines via the two points (℘(z; τ), ±℘′(z; τ)) on E the Calabi–Yau threefold X is dual to the geometric heterotic string on K3 × T 2 or — as we claim — to the non- SU(2) bundle data of the heterotic string. 2 This verifies that the proposed compactification (18) geometric heterotic string on S × T . The discussed 6d hypermultiplets become 4d hypermultiplets. In type IIA of the moduli space Mhet agrees with the physical data in the large volume limit of the heterotic string. Fur- on the threefold X, the hypermultiplet sector receives thermore, it demonstrates explicitly that the Clingher– corrections in the string coupling gs [36]. However, as we Doran–Malmendier–Morrison construction (20) is consis- fiberwise apply the 8d type IIA–heterotic duality in the P1 tent with the F-theory–heterotic duality in the stable de- adiabatic limit, the volume of the heterotic is taken to generation/large volume limit. be large. On the type IIA side this amounts to working in the weak string coupling limit gs → 0 [5]. As a consequence, the tree level result of the type IIA compactifi- III. 6D N=1 F-THEORY-HETEROTIC DUALITY cation is a good approximation for the hypermultiplets. In particular, it is legitimate to approximate the hypermultiplet sector metric ds2,tree by the classical c-map, In eight dimensions the Clingher–Doran–Malmendier– IIA HM which (in type IIA) yields the tree level metric from the Morrison description encodes the complex structure mod- Weil–Petersson metric of the complex structure moduli ulus τ, the Kähler modulus ρ of the torus as well as the space of X [37, 38]. As the Weil–Petersson metric of the bundle modulus z of the 8d heterotic theory in the mod- complex structure moduli space does not receive any α′ uli space of a genus two curve. To arrive at a 6d heterotic corrections, we also do not expect any α′ effects in the string theory, we can now adiabatically fiber this 8d con- hypermultiplet sector — at least so long as g -corrections struction over a suitable base. Hence, either we fiber the s are suppressed. torus together with the bundle data over the base — to 2 obtain the heterotic string on an elliptically fibered sur- Therefore, in decompactifying the torus T , we expect face with a bundle — or we directly fiber the three mod- that the absence of quantum corrections carries over to uli (τ,ρ,a) over the same base to describe the heterotic the 6d hypermultiplets of the F-theory compactification. string in terms of a genus two fibered surface. In the lat- In this way two heterotic phases of rather distinct na- ter case the Sp(4, Z) transition functions of the fiberation ture enjoy a unifying semi-classical geometric F-theory would mix up the three moduli in going from one coor- description. Therefore, the discussed dual string theories dinate patch to another. Therefore, there is no global furnish a remarkable example, where a non-trivial string distinction anymore between the complex structure, the duality gives rise to a conventional geometric description Kähler and the bundle moduli. In other words, the com- for a complicated dual non-geometric quantum phase. plex structure moduli space of the genus two fibration combined with the Kähler moduli space of the base, con- stitutes the total moduli space of a non-geometric com- A. Genus two fibration pactification of the heterotic theory. Let us fiber both sides of the Clingher–Doran– To arrive at the non-geometric heterotic theory in six Malmendier–Morrison correspondence over a common P1 dimensions, we start with the Weierstrass model on the base. In the non-geometric description of the heterotic F-theory side [22, 31] string — given in terms of the genus two fibration S → P1 — the complex structure moduli of S together with the 1 volume of the base P1 are the heterotic moduli fields y2 = x3− ψ (s)t4 +4χ (s)t3 x 48 4 10 appearing in hypermultiplets. This non-geometric com- (25) pactification is deep in the quantum regime of the het- 1 + t7 − ψ (s)t6 + χ (s)t5 , erotic string on a K3 surface with finite volume, and 864 6 12 6 with the affine coordinate s on the P1 base. Instead, we proceed by examining the properties of the ψ4(s), ψ6(s),χ10(s), ψ12(s) are now sections of the line surface S in the limit that is dual to the stable degener- bundles O(8), O(12), O(20), O(24) over P1, respectively. ation limit in F-theory. For this approach it suffices to Note that the degrees of the line bundles are twice the study the structure of generic and non-generic genus two weights of the associated modular forms. fibers of S, as characterized by local models of possible The fibered modular forms ψ4(s), ψ6(s),χ10(s), ψ12(s) genus two fibrations classified by Namikawa and Ueno in determine a genus two fibration over the same P1 base by ref. [39]. assigning to the quadruple of modular forms the associated genus two curve. To arrive at an explicit description for this genus two fibered surface, we start with the hy- B. Stable degeneration limit perelliptic form of genus two curves

6 To go to the stable degeneration limit — which re- y2 = x6 + c (s)x5 + ... + c (s)= (x − ξ (s)) . (26) lates to the heterotic large volume limit ρ → i∞ — we 5 0 ℓ would naively consider the Jacobi–Fourier expansion of ℓY=1 the modular sections Here the coefficients cℓ are elementary symmetric poly- 2 nomials of the roots ξℓ, which are both fibered over the ψ2i = E2i(τ)+ ψ2i,1q2 + O(q2 ) , 1 P base. Then the roots ξℓ define the Igusa–Clebsch in- 2 χ2j = χ2j,1q2 + O(q2 ) , variants I2, I4, I6 and I10 of respective weights 2, 4, 6, and 10 [29, 31]6 2πiρ with q2 = e . Consistently, we can only send q2 to zero, if it is a global section. This, however, is not the 1 I = (12)(34)(56) , case, as globally there is a non-trivial mixing with the 2 48 σX∈S6 other fiber moduli τ and z. 1 We know from Section II A that the compactified mod- I = (12)(23)(31)(45)(56)(64) , 4 72 uli space of 8d heterotic theory is the blown-up weighted σX∈S6 P3 projective space Blh0 (2,3,5,6) parametrized by the ho- 1 mogeneous coordinates (I , I , ψ , ψ ,χ ,χ ). Further- I6 = (12)(23)(31)(45)(56)(64)(14)(25)(36) , 0 2 4 6 10 12 12 more, the stable degeneration locus is identified with the σX∈S6 exceptional divisor H ≡ π−1(h ) at χ = χ = 0. 2 0 0 10 12 I10 = (ξi − ξj ) . The non-geometric compactification of heterotic string Yihyperelliptic curve [40], we determine is beyond the scope of this work. the structure of genus two fibers in the limit σ → 0. A

7 6 In the original references the numerical factors are absent in the This parameter is called the smoothing parameter in ref. [25] 8 Formally, the homotopy H : P1 × [0, 1] → Bl P3 with deﬁnitions of I2,I4,I6, because there the sums are taken over h0 (2,3,5,6) 1 1 1 subsets of S6 with indices 48, 72, 12, respectively. H(P , 1) = ι(P ) and H(P , 0) ⊂ H0 realizes the limit. 7

in the two points p±, which add up to zero. The described surgery process is illustrated in FIG. 1. spectral cover Let us now turn to the non-generic fibers in the limit double point P1 topological σ → 0, which arise at the intersection of the image ι( ) surgery P3 in Blh0 (2,3,5,6) with (the proper transforms of) the divi- 10 sors H1 and H4. The former intersection points correspond to singular fiber of type I1-I0-1 of ref. [39], which is a degenerate reducible singular fiber of two elliptic curve components singular fiber I1−0−0 smooth elliptic fiber with self-intersection number −1. In addition, one elliptic curve component has developed a nodal point. As a FIG. 1. Depicted is the topological surgery operation per- formed in the limit σ → 0. It maps the singular genus two consequence after performing the described topological fibers of type I1−0−0 (left side) to smooth elliptic fibers with surgery on the entire surface the nodal component turns the spectral cover data of an SU(2) bundle (right side). into a rational curve of self-intersection −1. We blow- down these rational curves to arrive at a minimal surface. Since the two elliptic components of the reducible fibers generic genus two fiber degenerates in the limit σ → 0 to along H1 intersect in the zeros of the two elliptic curves, the equation the spectral cover points coincide after the blow-down with zero of the maintained elliptic component. 2 3 4 4 8 6 2 y = 4x − π ψ4(s) x − π ψ6(s) (x − γ(s)) . To examine the intersection points of H4 with the ex- 3 27 ceptional fiber H0, we determine the proper transform (31) of the divisor H4 for the blow-up (16). With the affine Such a singular fiber is of type I1−0−0 in the classifica- I2 P3 coordinate u = in the patch I0 6= 0 of Blh0 , , , tion of ref. [39], which is an elliptic fiber together with I0 (2 3 5 6) the defining equation for the divisor H4 becomes a double point. Using surgery techniques we remove the neighborhood of the singular point at x = γ(s) and glue 2 χ35 3 3 2 2 3 4 in two disjoint smooth patches so as to arrive at a smooth = χ10(ψ4 −ψ6 ) (27u −9uψ4 −2ψ6)+O(χ10) . (35) χ10 elliptic fiber. In this way, we have reduced the singular fiber of arithmetic genus two to a smooth elliptic curve of Therefore, on the exceptional divisor H0 — given by genus one. Furthermore, in recording the two pointsp ˜± χ10 = 0 — the proper transform of H4 restricts to on the elliptic fiber, where the patches have been glued in, 3 2 2 3 we can deduce the spectral cover for the heterotic SU(2) H4|H0 = (ψ4 − ψ6) (27u − 9uψ4 − 2ψ6) . (36) bundle. Given the Weierstrass function ℘( · ; τ) associated to the modulus τ of the elliptic curve in Weierstrass In the stable degeneration limit σ → 0 of the analyzed form, we find genus two fibered surface, with eqs. (12) and (15) these two components turn into z(s) γ(s)= ℘( 2 ; τ(s)) , (32) 4π4 8π6 D =4℘(s)3 − E (s)℘(s) − E (s) , with the periods τ(s) in the limit σ → 0. Thus, up to 1 3 4 27 6 (37) an overall factor of two the loci of the surgery encode D = (E (s)3 − E (s)2)2 . the Wilson line modulus z of the SU(2) bundle,9 and 2 4 6 the spectral cover is given by the two points p = 2˜p , ± ± The elliptic Eisenstein functions E4 and E6 are now holo- where p± arise from the zeros of the Weierstrass func- morphic sections of O(8) and O(12), respectively, while tion ℘(z; τ). Thus, altogether the topological surgery ℘ is a meromorphic section of O(4) over the base P1. amounts to replacing the degenerate genus two curve (31) At the points D1 = 0, we again obtain singular genus with the genus one Weierstrass equation two fibers of the type I1−0−0, which turn into smooth 4 8 elliptic fibers via the surgery. At such points the Wilson y2 = 4x3 − π4ψ (s) x − π6ψ (s) . (33) 11 3 4 27 6 line modulus z becomes a half elliptic period, , which shows the appearance of a non-generic Z2 fiber symmetry. Furthermore, with eq. (15) the spectral cover becomes At the points with D2 = 0 we find singular genus ′ 2 ′ two fibers of type I1−1−0 in the classification of ref. [39]. 0 = χ10(s) x − π χ12(s) , (34) which — as required for a spectral cover of an SU(2)- bundle — indeed intersects the constructed elliptic fiber 10 In Section II A, we have introduced the divisors H1 and H4 in the P3 context of the projective space (2,3,5,6), but we use in the follow- ing the same letters for their proper transforms in the blown-up 3 9 1 projective space Blh P . It is tempting to relate the factor of 2 in eq. (32) to the square of 0 (2,3,5,6) 11 2 the appearance of the spectral cover factor x − γ(s) in eq. (31). This is because D1 is identified with (d℘(s)/dz(s)) . 8

These are genus two fibers with two (Z2 symmetric) nodal Here the bundle moduli space MF (g) is fibered over the points, where the two nodal points in the genus two fibers moduli space Mg of the metric of the K3 surface due relate to the square in the component D2. The topologi- to the anti-self-dual field strength ⋆gF = −F given in cal surgery removes one nodal point, and we are left with terms of the metric-dependent Hodge star ⋆g of the K3 a singular elliptic fiber of Kodaira type I0. surface. Then the hypermultiplet moduli space MH is Altogether — after implementing the topological completed by the B-field moduli space MB(g, F ), which surgery and performing the blow-down to a minimal sur- in turn is fibered over the moduli space Mg,F . The B- face — we arrive at a smooth elliptically fibered surface field is governed by the supergravity equation of motion P1 3 2 1 over with the discriminant ∆ = E4(s) −E6(s) , which H = dB+ 4 (ωG−ωL) and d⋆H = 0 with the gravitational is a section of O(24). As a consequence the constructed and gauge Chern–Simons terms ωG,ωL, which depend on surface has 24 singular elliptic fibers of Kodaira type I0, the metric and the gauge connection. each of which descends from a degenerate genus two fiber By construction the two-staged fibration arises from P1 over the intersection of ι( ) and the divisor H4. How- the supergravity approximation of the heterotic Kalaza– ever, this is nothing else but an elliptically fibered K3 Klein reduction. As a consequence, deep inside the hy- surface of the Weierstrass form permultiplet moduli space — where in particular α′- 2 3 corrections are sizeable — we expect this structure to 0= −y +4x − g2(s)x − g3(s) , (38) break down. With the help of the non-geometric het- 4 6 erotic compactification, we present an argument here where g (s) = 4π E (s) and g (s) = 8π E (s) are now 2 3 4 3 27 6 that this fibrational structure indeed disappears in the sections of O(8) and O(12). Furthermore, it comes with interior of the heterotic hypermultiplet moduli space. the spectral cover (34) of a SU(2) bundle. Thus, the stable degeneration limit in F-theory indeed realizes the Let us suppose that in the Weierstrass model of the limit of the non-geometric heterotic compactification to F-theory compactification (25) is determined by the sec- the geometric semi-classical large volume heterotic com- tions pactification on an elliptic K3 surface together with an 2 3 SU(2) bundle. ψ4(s)= α4h(s) , ψ6(s)= α6h(s) , 5 6 (40) χ10(s)= α10h(s) , χ12(s)= α12h(s) ,

C. Heterotic Sen limit where h(s) is a section of the line bundle O(4) over the 1 base P while α4, α6, α10, α11 are constants. In the dual As the non-geometric F-theory–heterotic duality for non-geometric heterotic string, any generic point on the 4d and 6d theories gives us insight in the hypermulti- P1 base describes the same genus two curve in the (com- plet moduli space beyond the semi-classical supergravity pactified) moduli space A2 of genus two curves. As a Kaluza–Klein reduction of the heterotic string, we can consequence the genus two period τ is constant over the use this correspondence to study quantum effects in the entire base. Nevertheless, there are still degenerate genus hypermultiplet sector of the heterotic string. For con- two fibers, appearing at the zero loci of the section h(s). creteness we focus now on the 4d theories. We call this limit the heterotic Sen limit, as this picture In ref. [26] Alexandrov, Louis, Pioline and Valandro realizes the analog for genus two fibrations to the con- study the heterotic hypermultiplet sector in a similar ventional Sen limit for elliptic fibrations in the context context. They utilize the duality between the heterotic of F-theory [24]. string on K3 × T 2 and the type IIB string on the mirror Y of the dual elliptically fibered Calabi–Yau threefold For our purpose, in the heterotic Sen limit the three X. They work in a limit, where the volume of K3 is constant periods τ of the genus two fibers are regarded very large to suppress α′ corrections. Furthermore, they as three moduli fields in the heterotic theory. Subse- demand that the base P1 of the elliptically fibered K3 is quently, we identify the complex structure moduli field τ, much larger than the elliptic fiber. In the dual type IIB the complexified Kähler moduli field ρ = B + iJ, and the 4d gauge bundle moduli field z with a subset of two scalar theory this renders both the 4d string coupling gs and 10d degrees of freedom of three hypermultiplets [25]. In the the 10d string coupling gs small. As a consequence all types of string corrections become negligible. As a semi-classical limit these three fields are distributed over result, using the classical c-map metric of the hypermul- the entire two-staged fibrational structure (39). Never- tiplet sector in the type IIB theory [37, 38], they find that theless, in the heterotic Sen limit, this property clearly the metric of the heterotic hypermultiplet moduli space breaks down, because neither is the metric for z fibered exhibits a two-staged fibrational structure: over τ or J, nor is the metric of the B-field fibered over the remaining fields. Instead, there is a Z2-symmetry MB(g, F ) −→ MH exchanging τ and ρ = B + iJ. This can be seen for- ↓ maly, as the SO(2, 3)-isometric metric on the sub-moduli MF (g) −→Mg,F (39) space Msub parametrized by τ,ρ, and z is the same as ↓ the metric on the Kähler symmetric space D2,3, which is Mg explicitly spelt out in ref. [41]. For the pairing η of the 9 lattice L2,3, represented by the matrix the analysis of non-geometric heterotic compactifications in the quantum regime — in particular at special loci 0100 0 in the moduli space such as the heterotic Sen limit — 1000 0  may shed light on conceptual questions concerning the ηAB = 0001 0 , (41) quantum hypermultiplet moduli space.   0010 0  To make further progress in analyzing the discussed 0000 −2   class of non-geometric heterotic string compactifications in four and six dimenions, we need a better understand- we parametrize a null vector X in terms of the three ing of the relevant genus-two fibered surfaces. In this moduli τ,ρ,z as [21, 41] work, we mainly analyzed the local structure of such XA = 1,z2 − τρ,τ,ρ,z . (42) genus two fibered surfaces by studying the structure of their singular fibers, using the classification of Namikawa A B A B and Ueno [39]. It would be interesting to study the global Then one finds that ηABX X = 0 and ηABX X¯ > 0, features of the relevant genus-two fibered surfaces — for and the metric on D2,3 can be written as [41] instance by comparing to the general properties of genus 2 i ¯ i two fibered surfaces [43] — and to give more detailed dsM = Ki¯dt dt¯ , t = (τ,ρ,z) , (43) sub physical interpretations of the geometric structures of with Ki¯ = ∂i∂¯K and the Kähler potential such surfaces. While the studied F-theory–heterotic quantum duality 1 K = − log η XAX¯ B . (44) is based on the rather special class of F-theory–heterotic 4 AB models with a single Wilson line modulus, it provides for an explicit and detailed toy model for non-geometric The obtained Kähler metric (43) with SO(2, 3) isometry string compactifications in general. The technique — — representing a subspace of the quanternionic hyper- to combine the compactification space with the spectral Kähler metric of the hypermultiplet moduli space — cover data by surgery operations — may also open up a clearly does not fit in the two-staged fibration (39). new method to arrive at more general non-geometric heterotic string theories, which describe quantum corrected IV. CONCLUSIONS heterotic strings beyond a single non-trivial Wilson line modulus. Using the non-geometric duality correspondence pro- Although our discussion starts from the F-theory– posed by Clingher–Doran and Malmendier–Morrison [20– heterotic duality, the general philosophy to use local du- 22], we studied F-theory–heterotic duality beyond the ality transformations in order to arrive at non-trivial non- semi-stable degeneration limit and the semi-classical het- geometric global string compactifications [44], is applied erotic limit in eight and lower space-time dimensions. By here as well. The possibility to explicitly describe the geometric means, we analyzed in detail the limit from the transition to a conventional semi-classical heterotic com- non-geometric quantum phase to the semi-classical het- pactification in terms of topological surgery makes our erotic phase. This allowed us to argue that in six and non-geometric heterotic model appealing and calculable. four space-time dimensions, which arose from adiabati- It would be interesting to see if such non-geometric mod- cally fibering the eight dimensional duality over a com- els also relate to recent proposals on non-geometric string mon base, the non-geometric heterotic compactifications compactifications [45–48]. captured α′-corrections to the semi-classical large fiber compactification. Furthermore, we shed light on the puz- zling phenomenon discovered in ref. [22] that even though the heterotic string moduli space continuously interpo- lated between the non-geometric quantum phase and the ACKNOWLEDGEMENTS semi-classical large fiber phase that the dual F-theory description remained geometric over the entire dual moduli space. We would like to thank Per Berglund, Fabrizio Analyzing the heterotic theory in four dimensions in Catanese, Chuck Doran, Daniel Huybrechts, Albrecht the heterotic Sen limit, we observed that α′-corrections Klemm, Andreas Malmendier, Stephan Stieberger, and modified the semi-classical two-staged fibrational struc- Thomas Wotschke for interesting discussions and use- ture derived by Alexandrov, Louis, Pioline and Valandro ful correspondences. J.G. is supported by the BCGS [26]. While such alterations to the hypermultiplet sec- program, and H.J. is supported by the DFG grant KL tor are expected on general grounds [42], we believe that 2271/1-1. 10

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