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JHEP04(2014)069 Springer s a role for April 10, 2014 March 13, 2014 olds. We find : : February 20, 2014 : mmetries, Confor- = 1 N Published Accepted 10.1007/JHEP04(2014)069 Received s of representations of the doi: tic function can be expanded ersity, Published for SISSA by = 1 . These theories are obtained N . 3 . heterotic theory by compactifying on toroidal orbif 24 1402.2973 8 The Authors. E c Superstrings and Heterotic Strings, Discrete and Finite Sy

We show that the recently discovered Mathieu moonshine play × , 8 [email protected] E Stanford Institute for Theoretical ,Stanford, Stanford CA Univ 94305, U.S.A. E-mail: theories Keywords: Mathieu moonshine in four dimensional Open Access Article funded by SCOAP Abstract: certain four dimensional theories with Timm Wrase Mathieu group M that a universal contributionin to such the a holomorphic way gauge that kine the expansion coefficients are the ArXiv ePrint: mal Models in from the JHEP04(2014)069 ] 3 7 1 2 7 11 12 13 10 K 3 [ K ing theory tain BPS states this question was . Another connec- e positive sums of 2 fficients are exactly 24 ], where the authors T 7 × ne may ask whether this = 2 theories have gauge athieu moonshine. re Mathieu moonshine is 3 owards a deep connection elliptic genus in terms of irreducible representations K N II string theory compactified obtains from string compacti- are related to Gromov-Witten 24 ted to the elliptic genus of = 2 theories N = 2 spacetime theories. Furthermore, 3 and therefore to M ]. N K 6 3 manifold that yet needs to be understood. ] observed that the elliptic genus of the 1 K – 1 – . This observation was further checked and confirmed = 1 theories and the 24 N string theory compactified on = 4 , they find that the first few 24 representations [ 8 E N 24 × 8 E 3. One of these connections was discussed in [ ], where the authors studied compactifications of type II str K compactified on toroidal 8 ] that these of irreps of M 9 8 ]. The authors find that the four dimensional E 9 × in the presence of NS5-. There the authors find that cer 8 3 are related to the elliptic genus of 1 S E K 3 manifolds play a crucial role in string compactifications o ], before Gannon proved that all the expansion coefficients ar × × 5 K 3 2 – For the heterotic This Mathieu moonshine is very interesting since it points t 2 3.1 Threshold corrections3.2 in four-dimensional Mathieu moonshine in T K Since theories whose couplings receive corrections that are rela observation is also relevantfications for involving the spacetime theories one in [ are counted by a mock modular form that isaddressed closely in related [ to M it was shown in [ expansion coefficients are related(irreps) to of the the sum Mathieu of group dimensions M of manifold exhibits ‘MathieuVirasoro moonshine’: characters when of expanding the the tion was made in [ and can always bethe same expanded as in theclearly such ones important a appearing for in way certain Mathieu that moonshine. four the dimensional expansion Therefo coe on on between the M showed that certain BPS saturated 1-loop amplitudes in type dimensions of irreducible M 1 Introduction Recently Eguchi, Ooguri and Tachikawa [ A Conventions 3 1-loop threshold corrections and moonshine 4 Conclusion Contents 1 Introduction 2 The JHEP04(2014)069 ] is G 11 . α (2.1) (2.3) (2.2) A λ scuss + , α  , where A β ] is not only F ibit Mathieu /G 1 → 6 manifolds that ∧ manifolds. e functions: the α T α 3 . 3 let, a number of F A β ) CY D CY = 1 supersymmetry, αβ α f ]. This class of toroidal D N achikawa [ 9 αβ Im( 1 1 2 s that Mathieu moonshine − unction in toroidal rticular way that the expansion coef- ) articular d exhibit Mathieu moonshine + f β s include GUT-like models with . ound in [ ⋆F t preserve Re( we discuss some basic facts about and the holomorphic gauge-kinetic ∧ 1 2 W 2 α W α W δ + F i )  2 + | αβ f heterotic string theory on W and chiral multiplets with (complex) scalar K should not be confused with the D-terms | = 2 subsectors which give corrections to the I 8 3 representation. α φ Re( manifolds but also plays a role in four dimen- ∂ E K N A – 2 – = 2 supersymmetry. In this work we show that − 1 2 I I 3 φ φ × representations. We do that by recalling from [ N α W 1+ 8 CY J iδ 24 E φ W ∂ . Our conventions are summarized in appendix = 2 subsectors. There is a large class of toroidal orb- V ⋆ 4 = D + + N ¯ α J = 1 four dimensional theories whose 1-loop corrections to W that have ¯ φ I . The resulting low-energy effective theory is four dimen- D W φ N I M M ⋆d φ D Z Z ¯ ∂ ∧ . J I I × × W under the infinitesimal gauge transformations = K dφ N N ¯ I string theory compactified on toroidal orbifolds. Then we di J I Z φ Z K 8 / W e K heterotic string theory compactified on toroidal orbifolds 6 I E , the holomorphic superpotential or φ T = 8 1+ × K D N E V 8 Z . The two derivative action is completely determined by thre R ⋆ and E I 3 manifolds and certain × how the 1-loop corrections to the gauge kinetic function exh 1 2 φ . The two derivative action is 8 K N 3 − αβ = 1 . This theory consists of one gravity multip E Z  f / and similarly for 6 Z N I T − φ The outline of the paper is as follows: in section It is thus clear that the observations of Eguchi, Ooguri and T α gauge group and chiral matter in the 27 = δ 6 α S function ¨he potential K¨ahler components sional denotes either vector multiplets that include the vectors there is a large class of related sional spacetime theories that preserve where the scalar potential is given by Here the variation of 2 The the heterotic moonshine. We conclude in section orbifolds therefore leads to four dimensional theories tha that the moduli dependentcompactifications corrections arise only to from the gauge kinetic f relevant to We discuss compactifications of the the gauge kinetic function canficients likewise are be the expanded same in dimensions such a of M λ gauge kinetic function that are closely related to the ones f ifolds invariants in the dual type II string theory compactified on p which are are elliptic fibrationsactually over teaches a us something Hirzebruch about surface. the geometry This of these mean pa have a variety of different gaugein groups and the matter gauge content an kineticE coupling. In particular, these theorie The derivatives in section JHEP04(2014)069 by rus and , for 2 (2.4) j T αβ T f × 2 . T and ’s by rested reader 2 A × ˜ λ T W 2 , A ˜ γ T , which are taken N ] for many details K πi 2 2 12 = e 6 3 and as of → T hat will be important in 3 3 3 1 7 of [ ,U A f the three 3 2 2 3 omplex fermions which we ˜ which we take to be either λ ,U ,U ,U e gauge kinetic function to − − T T T 3 3 3 ,T : T T T 2 G heterotic string have long been = 2 moduli T he right-moving complex fermions to 8 N E = 1 supersymmetry we restrict to leads to a four dimensional theory × (as well as the unfixed moduli that ) , g 8 N 6 3 6 E A T T is then fixed by specifying the action of . , ϕ λ 2 A A Γ. Furthermore, in order to ensure that N ˆ ˜ γ / , ϕ N Z 6 πiγ 1 7) 6) 2 1) 2) 4) 3) 4) 5) 4) , , and the gauge bundle ϕ R , , , , , , , e ( 4 5 ’s denote the complex structure moduli of the 1 1 1 2 2 2 3 and 1 , , ]. j 1 , , , , , , , – 3 – N = 1 → A U (1 (1 (1 (1 (1 (1 (1 (1 (1 6 γ 1 3 1 4 1 6 1 6 1 7 1 8 1 8 1 1 A 12 12 , ˆ T λ j . Cyclic orbifold groups. ϕ : ) are constraint by the requirement that they preserve M Generator Table 1 , g N (and j we need to specify the action of a second generator which we Z I I I II z II II − N 7 3 4 j − − − − − 6 8 M Z Z Z 12 6 8 12 N Z Z Z πiϕ Z Z Z Z 2 . The orbifold action for . e Group j × , where the complex A z j ˜ = 4. In order to break the supersymmetry we orbifold the six to λ 2 gauge bundle by a discrete N → . We define three complex coordinates on the y Z 8 j j N M z E and on the spacetime coordinates Z iU = 2 subsectors and play a crucial role below). : g A × × + g λ that are contained in SU(3) but not in SU(2). We refer the inte 8 N 1 with corresponding angles ˆ N E ], we list the possible orbifold actions on − G g Z j 2 14 ] for a detailed discussion of toroidal orbifolds. In tables y The actual values of Compactifying the heterotic string theory on The derivation of the gauge group and matter content, as well or 13 The right-moving world sheet supersymmetry fixes the action on t = ’s. We will denote the moduli K¨ahler that control the sizes o 1 2 N j textbook material and we refer the reader to chapters 16 and 1 a generator that preserves denote ˆ combine the current algebradenote fermions by into two sets of eight c For the case of appear in from [ and a worked outthe example. next Here section we whenthe recall we moonshine a connect phenomena few the discovered relevant 1-loop in facts [ corrections t of th groups be the same as on the T the resulting four dimensional theory preserves only the case of toroidal orbifold compactifications of the Z the Γ which defines the torus to [ z and the JHEP04(2014)069 A γ 3 , ˜ dent (2.7) (2.6) (2.5) A . ,U γ I 3 3 3 − replaced 3 3 3 3 3 6 ,T ,T ,T while the 2 2 2 Z ,T ,T ,T ,T ,T N / 2 2 2 2 2 2 = 1 theory 6 ,T ,T ,U 1 1 T 2 . ,T ,T ,T ,T ,T , 1 1 1 1 1  N ,T ,U ,U = 2 moduli T T T T T U(1) } } 1 1 1 and 0 0 T T N , , 4 × ,U 0 0 1 Z 6 , , / T 0 0 6 E , , T root lattice and modulo , N, 0 0 N, ) 8 , , 3 0 0 ˆ E ϕ , , SO(16) symmetry that acts d embedding for which one , ) are trivially satisfied and . It is straight forward but 0 0 2 g × , , mod ˆ ϕ y consistent choices of mod 2 0 0 2.6 U(1) for , , , 1 modular invariance of the string 0 0 × broken to ϕ nt algebra fermions, so we refrain { { ( ˆ 1) 3) 5) 4) 2) 5) 3) 5) , , , , , , , , 2 1 M 1 1 1 1 1 1 1 1 = = ) or ( , , , , , , , , 2 2 ) A A (0 (0 (0 (1 (0 (0 (0 (0 ) ˜ ˆ ˜ 2.5 γ γ A 1 2 1 4 1 6 1 6 1 3 1 6 1 4 1 6 A γ = 0 mod 2 γ (˜ (˜ A 8 =1 ˜ γ and to SU(2) 8 =1 X A generator 3 X A 8 =1 Z – 4 – + nd / X A + , , ]. These conditions are 6 2 2 2 } } ) T = so the resulting four dimensional ) 0 0 15 A , , A ) γ A 6 3 0 0 γ ( . Product orbifold groups. γ ( , , E we have to impose the same conditions (with 0 0 , ϕ 8 =1 , since the manifest SO(16) . Furthermore they have to satisfy certain constraints in , , 2 8 8 =1 =1 X A 8 8 of 0 0 M X X A A , , , ϕ E E Z = 1 0 0 = = , , × ϕ 2 × Table 2 1) 2) 2) 3) 5) 1) 1) 1) 3 3 ( ) j 2 , , , , , , , , 8 N ˆ j ) ϕ 1 N 0 0 0 0 0 0 0 0 ϕ j Z , , ϕ E ϕ , , , , , , , , gauge groups gets generically 2 2 ( ϕ 3 (1 (1 (1 (1 (1 (1 (1 (1 ˆ =1 ( ϕ j 8 X 3 2 1 2 1 3 1 3 1 4 1 6 1 2 1 2 1 =1 , , ϕ j 3 X 1 1 =1 E remains unbroken. The chiral matter spectrum is model depen j X ˆ ϕ ϕ are only defined up to shifts by vectors in the 8 { { A E generator = = γ st A A 1 ˆ γ γ factor is enhanced to SU(3) for ). and ˜ and 2 A A 6 ′ 6 3 6 4 6 2 4 M . For the case of ˆ ˜ γ γ N Z Z Z Z Z Z Z Z ’s is enhanced to Z , N λ A × × × × × × × × × ) for the hatted angles corresponding to the action of ˆ γ 3 4 6 2 2 2 2 3 The For the standard embedding the conditions ( N Z Z Z Z Z Z Z Z The U(1) M Z 2 but contains matter in the 27 the action of the of second (hidden) However, we like to stress that our results hold for arbitrar path integral measure at 1-loop order [ order to ensure left-right level matching which guarantees the first of the two by on the (and ˆ pretty lengthy to classify allfrom possible doing actions so on the here. curre chooses One simple choice is the so called standar for odd for even JHEP04(2014)069 ′ . ]. ], 2 × at U /G 11 16 6 (2.9) (2.8) SU(2) T we see , where and ′ j SU(2) ⊂ 2 does not , we have ′ 2 G × g } T ) from [ 3 manifold. G . 2 I . In the case and they are and K φ 24 ′ j , g er ( 1 1 are and ˆ 1 whenever there G gauge group. orbifold models. { U 2 2 g T M T = 1-loop α Z = 2 subsector when- ′ f and focus on the simplest G × 1 -th N to be the empty group. he gauge couplings and j T N gauge group by including is again fixed. Z ns to {} subsectors’ of the orbifold acts on the three complex . it can be expanded in such etic function at tree-level is 6 6 are that we can write entations of M ) ectrum from certain toroidal } = E 2 3 ng to get an SU(3) is fixed under rt is the axion obtained by du- πi/ ′ j has an rk out the connection between πS e T 2 j , g G z − ections at 1-loop so that we have 2 However, to make the connection = so that it does not appear in the e 2 , ( 3 6 Z ) O U I , g, g whose real part is the (inverse) string πi/ φ 1 such that the compactification 7 and for all e ( { , ia which is generated by . Due to a renormalization theorem [ ) + α and 2 being an orbifold limit of a I G f = , we take 6 + = ′ φ j } Z 6= 3 ( 2 ). The φ = 2 subsectors which we label by 3 4 dB αβ ⊂ 2 3 ˆ 4 U g δ /G Z × z − ′ N ⋆ 4 N e 2 , g . Therefore they both appear in the last column – 5 – − G T of 6 1-loop 1 , α = Z ) = = 2 and the moduli of the second { f Z = 2 subsector. Looking at tables with coordinates I g φ S da g 2 × = + ( , i z N 2 T 2 1 S Z αβ , with Z ′ / f i z whenever ( 6 -th = ) = j /G T N for a particular orbifold fixes I ′ 3 4 → Z φ ′ j 6 T G ( ) G Z α 3 × f × 2 the generator , z 2 2 T ) is the key player in this paper. We show in the next section th 4 . For example for I Z , z is generated by Z ′ j = 1 / φ / ( 6 2 z G ′ 6 Z T . The corresponding moduli of the first T :( /G > list in the last column the unfixed moduli of the 6 = g 1-loop ′ α field in four dimensions T 2 ′ 2 f 2 = 2 supersymmetry in four dimensions. This is the case whenev G < g B and therefore leads to an . is the same for all gauge bosons that belong to the same simple N = and 2 3 α 6 1 z f Z subscript means that the In this paper we are mostly interested in the corrections to t For compactifications of the heterotic string, the gauge kin Before we do so we recall several facts about the contributio j = ′ 1 Tables that there is no non-trivial the exists a non-trivial last column. Lastly we have that such subsectors exist for To that end itcompactification: is we useful say that to the introduce orbifold the compactification concept of ‘different the gauge kinetic function receives only perturbative corr However, the full Actually most of the models have multiple The function in which case ever there exists a non-trivial subgroup closely resembles a GUT model. One can further break the its dependence on neutral scalar fields. To that end we recall preserves coupling that also sets the gaugealizing coupling. the The imaginary pa Wilson lines and it isorbifold possible compactifications to get of the the exact heteroticto chiral string Mathieu MSSM theory. sp moonshine transparent, we will refrain from tryi universally given by the axion- Mathieu moonshine and fully realistic models. U(1) gauge group withtoroidal orbifold the models. standard It model would be spectrum very and interesting to rather wo where G coordinates as act on both moduli of the full orbifold a way that the expansion coefficients are dimensions of repres whenever it has a non-trivial dependence on the moduli, then For example, for of table JHEP04(2014)069 ) ′ j 2 and /G ,U 4  2 3 factor (2.11) (2.10) ) . This . Fur- T T  j Z } 2 (i.e. for ( ) U / 4 {} ×  3 T 6 q = 2 com- ) 6 2 U ( , g 2 T q = T T 2 J U ( 1-loop 2 N q ′ 1 J = 4 subsector ( , g h − ) function and G 1 J ) q − 6 { . j πi N ( , 6 ) T e − J . Before we do so, 3 q  = . = ) ] have furthermore T ( πi/ 2  2 = 4 supersymmetry, 24 q 3  since e J  U T ( 11 )  ensional Z In particular (up to a q  J j  N ( were the action on the n. In our setup we give =  ) U ) to show how the gauge 3 = J  3 q ively) exact at tree-level, 2 log ky [ 2 ) ) implies that ( ctification on U ′ 3 1-loop 2 Z α 2 q U η log erve U / G ( f 2.10 1 π 2 q 4 log ( η 8 ( 2.10 1 π T 2 η 8 denotes the 1-loop correction to eterotic string, the only moduli are the moduli of the 1 π − and × 8 2 ) + log − } j 2 U ) 3  ]) − + log T 3 ) ,U 1-loop j ) j  , g + log 19 j 2 T h ) ) for one explicit case. Let us consider 1 ,U – T  q 3 and . = 2 subsectors. { 3 ( ) ( T action to be ,U 2 2 17 T A q η 2 ( T ( = T N , 2.10 T q II η ( 2 ( − , only applies to the dependence on untwisted moduli. 11 1-loop j η 6 Z . log 3 h and our conventions for the and 1-loop 3 Z j  24 – 6 – log = h U j 1-loop 2 A 3  log ′ 2 ∂ h 2 γ U j =2) πU 2  G T π ∂ 2 U 2 N 3 4 ∂ , =2) − ( α,j ∂ 3 T π 2 b 2 1 e 2 N =2) 4 ∂ ) arise from {} α, ( 2 T π and 3˜ I 1 2 b N 4 ∂ − − = α, ( = 2 subsectors. These are the only orbifolds with gauge = 2 theory obtained by a compactification on φ = 2 b 1  A ( j − − | ′ 1 γ | U ′ j  N N − − q G G 6 3 G |  , | j 1-loop is not fixed, it does not appear in 2 6 α + 3 , f πT 1 2 , 2 T = 2 sectors, all orbifold compactifications have an = 1 toroidal compactifications are related to M − X ) = =1 e j 3 N N = ,U modulus is fixed by the 3 j ) = ) function are given in appendix T 2 j q q ,T ( U 2 η ,U T j ( are determined by a compactification on is the corresponding beta function. Note that ( T ( for which we have being the trivial group). fermions is given by 3 =2) =2) } II 1-loop 2 8 orbifolds have gauge kinetic functions that are (perturbat α = 2 prepotential of the 1 N N − f = 4 subsectors do not contribute to the gauge kinetic functio α, ( α,j ( 1-loop E { 6 7 α b b Similarly to the In the next section, we work out the prepotential for four dim The gauge coupling is not renormalized in theories that pres f N Z Z Their argument, which we recall below in section × = N / / 3 ′ 8 6 6 and it might be illuminating to explicitly spell out ( the since these orbifolds havekinetic no functions that are not related to M leads to kinetic functions in T shown that, for toroidal orbifolddependent compactifications corrections of to the h constant) they are given by (cf. for example [ the Dedekind T a simple argument for this below. Dixon, Louis and Kaplunovs Although the modulus where we used thermore, the E pactifications, connect it to Mathieu moonshine and then use and G so which is just the untwisted sector that arises from the compa JHEP04(2014)069 3 2 τ T = 2 (3.3) (3.2) (3.1) (3.4) and N the new = 4 BPS 3 (cf. for A ′ et and . the 1-loop γ K # N /G 4 × =2) =2) 2 T N and 2˜ N ( α T ( α × b b A 2 γ . − T  , ! ¯ c ] 24   , ¯ 2 − n the threshold corrections 21 part of the 1-loop correc- 0 ¯ q  1 ¯ L n πτ theories ¯ c ¯ q 24 ions but only over the right q 8 , the 1-loop string threshold 0 ′ − J bsectors that preserve 0 ] that the new supersymmetric − y ¯ L = 2 us /G 2 α c ¯ = 2 vectormultiplet, which then q 10 24 4 c Q − 24 N X T ] 0 . N  − 3 L are determined by a compactification 0 × 20 ¯ c BPS hypers q 24 Z L 2 R / q − − 4 F 0 0 T =2) = 4 does not contribute since fermions is given by 2 ¯ ¯ L + n 3 = 2 theories. We have schematically T ¯ ¯ N q q L 8 iπJ ( α, gauge bundle. We use convention for which N F n e × c b N 24 E 9) internal CFT theory associated with the q 8 0 2 , 1) − J E × 0 arises from the two additional four dimensional – 7 – T − denotes the gauge charge and  L 8 ( 2 × q α ) R  E 0 ) and 8 q 3 Q X ] that for compactifications on ( Tr E iπJ RR 2 ) = (22 ,U e 10 /η ) ¯ BPS vectors c 3 0 q 1 T J  factor in ( c, ( i η

2 2 ) = Tr T R − = 1-loop 3 Tr = q, y h ( 2 new ) ). = 2 hypermultiplet and one i q Z -th gauge group is given by [ = 2 theories obtained from compactifications on new ( α η Z 3.3 N elliptic N − Z the complex structure modulus of the 1-loop string worldshe " τ τ were the action on the 2 ]). 2 τ for the . Similarly, 3 d 2 19 α Z Z , F / / with Z 4 4 10 T T = are the moduli of the πiτ × α 3 2 × 2 e U 2 Furthermore, it was shown in [ The integrand is essentially the new supersymmetric index [ ∆ T T = example [ which shows that a subsector which preserves As we have seentions in the to previous the section, gauge the kinetic moduli functions dependent arises entirely from su 3 1-loop threshold corrections and moonshine spacetime supersymmetry. Thus wein review four the dimensional general form of cancel each other in ( states split into one weighted by the gauge charge squared. It was shown in [ Here the trace ismoving taken Ramond over sector all left-moving of boundary the condit ( and index counts BPS states in four dimensional q 3.1 Threshold corrections in four-dimensional its imaginary part. The prefactor 1 on For compactifications of the heterotic string on in supersymmetric index is closely related to the elliptic gen toroidal orbifold and the left-moving spacetime bosons in lightcone gauge. correction ∆ beta function. JHEP04(2014)069 ′ . ) e ’s = 8 new /G L/R E g, h 4 N (3.7) (3.6) (3.5) Z F T , × | 2 2 , m T iquely given  . The answer + ′ ) 1 q o demand that , /G √  iUm 4 2 − q, T  1 ( 1) ) ent algebra fermions ndary conditions for 1) − iT n 3 cannot be explained by q, q, y q, + . There could be also moduli ( ( ( 2 K 4 elliptic 4 4 mation properties of T ary conditions, twisted by θ θ = 4 subsector gives a vanishing Z menta are such that ( new 6 3 T Un ndence can only arise in  requires us to embed a total  Z N is the only contribution that K elliptic ) |− ) f 2 + ) 0). q 2 q Z , , ( 2 U 5 ht-moving Ramond sectors, ( 2 6 4  Γ dB η  θ 2 ) ) ) ) Re( 1)  πτ q = q T ( 4 1 ( .Θ ) = (24 q, 2 q, y 3 q η 2 2 ( ( Re( θ 3 3 H a chemical potential for the U(1)-charge T − − ) for all compactifications on , n θ θ ) 1 ). The only dependence on the untwisted )  2 ) is an index an therefore does not change n  q n ) 3.5 πiz 2 q 3.1 2 + √ ( q, y m e 4 ( The transitions between models with different 2 − + – 8 – = 2 subsectors.  1 E = ) has ( ) n 6 q, ) 1 ( 3 y 1) N ¯ U 2.7 4 m K ] elliptic ( q, q, y ¯ ( ( T, 1) but as argued above this Z 3 10 , 2 2 12 = 24. πiτ K elliptic , θ ) θ 2 9 q 2 e Z ( ) appears in ( n Z 6 and we denote the number of in the two T, U,  η ∈  ; ) = (1 i 8 for states with non-trivial momenta and/or winding numbers + ) ¯ q q, y ) ,n q ( E X 0 q 1 i ( g, h q, ( ¯ n 3 L m ) = 8 ( η × θ 2 factor, i.e. for 3 and , 3 = ’s). These choices determine how we embed instantons into th 8 2  2 for our conventions. K q, y elliptic Γ K A E 1 4 2 R and ]). ( T γ p Z A Θ q 0 2 1 22 i 2 + ¯ L q 3 3 , even when going to the orbifold limit , 2 L SU(3) contains no non-trivial element that preserves a K p elliptic = K 18 . One finds that [ 1 2 ⊂ Z 0 , q ) together with its pole structure to argue that it has to be un J 2 G 9 , new Z 2 Γ Z ). For a supersymmetry preserving compactification we have t X ∈ 0 in addition to gauge group. The Bianchi identity for is also the same for all possible orbifold actions on the curr 2 p 8 ) (cf. [ = , n ≥ E 1 2 new , 2 ) along the two cycles of the string world-sheet. The only bou n The reason that 3.5 2 Z For example the standard embedding in ( Recall that Please see appendix × Γ 6 5 4 , n 8 1 Θ g, h both do not act on a n by ( which the trace receives contributions from windings and mo depends on the moduli of the toroidal orbifold and such a depe by ( 2 subsectors. This can be nicely seen from ( under SL(2; (i.e. for all different ( is the sum over windings and momenta on the For toroidal orbifolds we have to sum over all different bound measured by dependent contributions when ( are the fermion number operators and of 24 instantons into is the elliptic genus of numbers is non-perturbative, so the invariance o moduli arises from contribution. for E where the fact that it is an index. However, one can use the transfor The trace in the elliptic genus is taken over the left- and rig can be understood by the fact that when moving in JHEP04(2014)069 4) , = 4 were (3.8) (3.9) n the (3.10) (3.12) (3.11) (3.13) . N 24 = (0 3 24 . ) K elliptic N ]. However, ena relating Z ) 1 q q, y , ( ,... √ as its symmetry 2 1 articular, it was and ,  = − 24 n ,l n 24 q 4 1 q, y ( n 2277 + ,l · n A (or at least a subgroup 4 1 +1) y = n = n ( h =1 24 ∞ h q n = 2 X n 2 1 ch sums of irreps of M 4 tween M ch − q nd Tachikawa [ n mbeddings with 6 n 1 to M 2 + A  are positive sums of dimension . 1) ) ) − n q ) =1 q − ∞ ,A ( q ( X n (  A 3 ( ) η θ 6 q 24 ( ) E  ) + −∞ , 1 2 q ) 3 target space has M ∞ 4 1 X ( = ) q = q , 3 can be expanded in terms of n ( q η K ,l q, y 4 2 ) 4 1 ( √ K ) 1 2 E 3 = ) 2 q, h = , 1) + 3 q, y q ( 2 g ,l ) = 770 + 770 q, y ( ( ,l − Γ 1 4 ( q 1 η 3 1 4 1 – 9 – ( = Θ q, θ η = i ) + yθ h ( h 8 1 2 q ,l √ ( 4 1 − i − ch ). For the particular case of the standard embedding n 2 ch ,A = =0 6 − q h = ,l  − 1 4 ) ) ch 3.10 ) q = q 6 ( new h ) = ) = ( 4  g = 4 world-sheet supersymmetry and the relation between η Z q, y θ ) denote the two different 45 dimensional irreps of M ) 4) sigma model with ( q q, y q, y 24 , q  ( ) one finds ( ( N (  1 4 =0 2 1 2 η ) ,l θ q =0 q = 3.5 1 4 = 231 + 231 ], the elliptic genus of ] it was then shown that all the ,l ( ,l = (4  − 4 1 1 4 6 = 12 2 4 1 h ) = ) is such that the above coefficients are literally the same as i E + q N h 2 ( n , ) = 2 η = q ch q, y Γ 2 h ) into ( ]. In [ ( ( ,A Θ 5 ,l i ch , which provides very strong evidence for a moonshine phenom ]. 4 1 – 3.6 2 are again given by ( = 24 ) = 24 ch 25 = h ] that no n elliptic , g 3. This connection is however currently not understood. In p A Z q, y 23 24 K new ( Z = 45 + 45 and 3 1 and Plugging ( Several non-trivial checks that confirmed the connection be As was shown in [ K elliptic A Z 24 new thereof) [ M performed in [ of irreps of M where for example 45 and 45 In particular, one finds world sheet supersymmetry the above coefficients are related in such a way that first few expansion coefficients are positive there is strong evidence that even for arbitrary instanton e the left-moving sector has group. Z Virasoro characters we can write original Mathieu moonshine observation by Eguchi, Ooguri a shown in [ Defining where the JHEP04(2014)069 ′ . ) ). q 24 ( /G ,l 3.1 4 M 1 4 (3.17) (3.16) (3.14) (3.15) to the T or not × = h × g 24 . new 2 .  Z T gauge group )   SO(12)  and the Math- 8 lU n in ) of × + E q seems somewhat 8 , 2 m n new ( kT SO(12) symmetry E 1-loop ×  T ( 24 A c Z h π 7 m × 2  q =1 E ∞ − 8 ) U X n e ∂ E m ) ebra. This explains the  ) to M  ( 3  c U 2 + m uantity 1 mines the vector multiplet )] Li ( 3 has embedding) by the fact that edding, it is not so clear for ) − c − es there is clear evidence for U ously restored in ∞ q = K gauge group leads to a factor kl  X ( ( ) m + Re( η 8 c × q ) (  0 T Z E 2 T 1 2 2 ∂ , ∈ q T 2 ) ,l> ,l = ( Γ 0 ,l η T X 4 1 =0 Θ ], however the k k> i ] for details) = 2 h 4 log[ Re( 24 ) into representations of 10 ) g [ −  ) are exactly the same as the ones in the 1 π  m ) ( m − = 24 (2 c ) + ( U ) c q q − ( q – 10 – ) that connects the ( ( J 6 =0 24 1-loop (3) ,l ) − E )] + 4 Re h ζ 3.15 4 1 q ) ) )  ( U q is not manifest, we have (for the standard embedding) ) T = ( η (0) q h 7 = 2 spacetime theory. This then connects M 4 U = 2 prepotential is then determined by the following c ( g 3.13 4 E E J ) to the prepotential one has to perform the integral ( ) = 4 Virasoro character in the definition of the 2 ) Re( 24 N , N π  2 1 1 T ) Re( Γ N ) 3.15 12 log T 2(2 Θ ) q i  ( q ] 2 4 − ( moonshine. In particular the 3 η Re( 10 . The coefficients E − ) in ( Re 4 3 , n U 2 + n 24 x m 9 = log[2 Re( − ) π ( ). While the 1 c 1 π  12 = =1 new (2 ∞ n 2 − m 1-loop =2) are the and K¨ahler complex structure moduli of the 3.15 Z π q P + h N 8 ) , we are now going to connect the expansion coefficients α ( U 2 U in front of the b α ) = in ( 24 m ∂ π ( 6 ∆ 8 T 8 c + ) = ) ) ∂ 1 x q q and T,U ( ( = (  − ] for further results). ). It is thus natural to decompose the ( = 2 theory obtained by compactifications on 3 ∞ = X T /η /η 25 Re m Li ) 6 To connect the Having established the connection between the worldsheet q N ) q 3.12 ( q − 1-loop ( 4 h i and (see [ a connection to (atis generically least broken a and subgroup it of) is M unclear whether it is miracul While this conclusion seemsarbitrary inevitable instanton for embeddings. the standard For these emb more generic cas differential equation [ The 1-loop correction to the in ( prepotential since we have from ( ieu group M Here in addition to the M E which is solved by the 1-loop prepotential (see [ convoluted. This can bethe explained (at least for the standard from the left movingθ free fermions an affine SO(12) current alg sector of the four dimensional to a spacetime quantity, namely the prepotential that deter We like to mention that the equation ( JHEP04(2014)069 to 8 = 1  E (3.21) (3.22) (3.18) (3.20) (3.19) ) N j × U q 8 , ( , E sults above J    ! . −  kl ) j 24 ) l U j kl . ! ( q T c j ! q    k T ( )   =2) j j q ) J  l U j N ) lU ( α,j q − . b j lU j + 1 2 2 U k j T + q − ) that breaks n q log j ake it clear that the first (  . 2 kT r four dimensional ( 2 ) η − kT eterotic string theory that ) rection to the gauge kinetic j π log ( 1 π 1 , n 2 π U 8 1 πS 2 q − kl = 2 prepotential receives only 2 0 n ( ) e − 60 + 6 − − η e  e ) − kl N ) ) to find 3 + log  ( j ( k,l> j 3 T c  = O Li q ) 0 Q ,U ) ( Li j 3.17 j  ) + T η ) ] kl T q X 1) = 1. We can now rewrite the 1-loop =2) j k,l> ( kl ( ( 2  c U ( N 19 − η + ( H j j c q ( 1 T ( 0 U c Z −  1-loop q q 0 . The full ∈ J j j = h 1-loop j ,l> ,l log T ,l U 24 0 − − h X q X q + j ) k,l>  =0 1 j =1 – 11 – , b U k k> − k T  2 and ∂ 1 4 theories 2 q 1 j =2) ) n (

T π  j 1 π N J STU 4 ∂ ( α,j ) make the contributions to the 1-loop correction for U ≤ − b 2 1 (2 − = 1 ∂ . log    j m T − − = 2 prepotential and is thus related to M − 24 60 + 6 N 3.20 − ∂  ) = log

| 4 − |

N j ′ j ) | 2 U G 1 π | = G ) | G ∂ | ′ j | T,U 1 π j (2 2 ( 3 G T , | π h (2 2 =2) ∂ − , ) = 0 for 8 1 N X 3 H ( =1 m = = , b j ( 2 , ) = c j X =1 j ) = ,U j = 2 beta functions are given by [ j ) and thus still related to M T ) = ) which he repeat here ( ,U N j j = 2 supersymmetry in four dimension. We can now combine the re 3.15 T ,U ( 2.10 j the N 1-loop j T 2 h ( j H 1-loop U α ∂ f × j 1-loop α T 1 f ∂ correction as the sum over a logarithm where we used that however, the different terms in ( the gauge kinetic functionsterm somewhat is directly more related transparent to and the m Lastly we note that for a particular instanton embedding ( perturbative corrections at one loop and is given by H To get a more explicit expression we use the result ( expansion ( This concludes ourpreserve excursion into compactifications of the h with the previous section to connect the gauge coupling in ou 3.2 Mathieu moonshine in theories to the sporadic group M We now use the resultsfunction above in ( the equation for the 1-loop cor JHEP04(2014)069 . 24 tions s the man- mani- 3 4 CY CY and me physics with 3 = 1 theories, so the CY 3 manifolds with the -loop corrected gauge 3, N K arch Fellowship (Grant , J. Harvey, S. Kachru, coefficients, hints at the e often dual to the het- K . Using the fact that the have Gromov-Witten in- ld be interesting to study ] several new connections . derstanding such a larger 24 24 ar, F-theory compactifica- n ing the connection to M 24 26 F or decades, it is likely that as d S. Kachru and X. Dong for e M htly involved expansion of the otic string is connected to the ce t await us and that will connect actifications only receives moduli ety of interconnections that await ] that connects 1 = 1 supersymmetry whose 1-loop corrections . Furthermore, in [ N – 12 – 24 3 are also related to M ] that Mathieu moonshine implies that 9 K × 2 T manifolds lead to four dimensional = 2 subsectors, we have been able to present a large class of 4 N CY . One can show that this implies that string threshold correc ]. 24 25 . The starting point that allowed us to make this connection i 24 manifolds and sporadic groups are presented. Since ] are presumable also relevant for four dimensional spaceti 4 26 CY While our class of models contains GUT-like theories, it wou Using it was shown [ = 1 supersymmetry. Since these F-theory compactification ar presence of a largerstructure (sporadic) are taken group. in [ A few steps towards un results of [ interesting four dimensional theories with tions on elliptically fibered folds have been key componentsa in consequence string there compactifications are f aspacetime variety of physics further and discoveries the tha moonshine phenomena. In particul gauge kinetic function, that is needed in order to extract th It would also be interesting to investigate whether the slig whether it is possible to obtain the MSSM while still preserv to the gauge couplings are controlled by the Mathieu group M variants that are likewise connected to M discovery made by Eguchi, Ooguri, and Tachikawa [ I would like toS. thank Stieberger M. and Cheng, D. X.comments Whalen Dong, on for J. the illuminating Duncan, manuscript. discussions S.number This an Harrison WR work 166/1-1) was of supported the by German a Research Rese Foundation (DFG). Acknowledgments N erotic constructions discussed here, thereour should be discovery. a vari between in heterotic compactifications on largest Mathieu group M ifolds that are elliptically fibered over a Hirzebruch surfa In this paper wekinetic have function shown in that toroidalsporadic the compactifications group moduli of M dependence the of heter the 1 4 Conclusion gauge kinetic function in heteroticdependent toroidal corrections orbifold from comp JHEP04(2014)069 .   (A.7) (A.6) (A.5) (A.3) (A.2) (A.4) (A.1) 4 ) q ommons , ( 3 ) 3, θ 1 K − + n 4 q ) 1 , q ( − ) 2 1 y θ ]. − − ... n 8 , . q . ) function ) + 1 ) q ) q 24 )(1 ( 2 1 ( − 2 n n 4 η SPIRE ) that can be written in q y q − θ = 1, so that for example q n yq IN ( , − q 6 − y ][ ) − 1 redited. ]. 1 2 E  − (1 4 )(1 + y − )(1 ) surface and the Mathieu group n =1 n ∞ q n Y n − q 3 ( q yq 1 4 SPIRE ]. 1 K 24 θ − − ) and + 21493760 IN )(1 q y q − (1 2 1 ( q ][ 4 4 = ]. ) − )(1 + =1 ∞ n q E SPIRE Y Mathieu moonshine in the elliptic genus of Mathieu twining characters for n n ( 1) q 2 2 IN )(1 + ) are − yq / 2 θ 2 1 n 1 ][ − arXiv:1005.5415 SPIRE (3 − − [ y 8 q, y n n ) 8 IN ( (1 + 196884 / q q i Notes on the 1 – 13 – ( )(1 n ][ θ yq q 1 =1 3 ∞ n iq Y n θ 1) q 2 − dyons and the Mathieu group M − , / + − ( 1  =  )(1 + y -dependence, we have set 8 arXiv:1004.0956 4 [ n (1 (2010) 623 8 ) y 1 2 = 4 ) −∞ / q q ∞ q 1 4 − X ( = =1 + 984 = ), which permits any use, distribution and reproduction in ( ∞ q N n 4 Y n − n 4 2 θ y θ ) 1 arXiv:1008.3778 = 24 24 2 [ = (1 ) q ) + + q ( 2 1 2 1 q n 4 2 8 6 =1 ( ∞ ) − − y ) Y (2011) 91 n arXiv:1006.0221 η n q E n [ q 2 ) function ( 2 ( ) = ( n y q 3 q surfaces, = 3 q ( 2 20 q θ n ( ) θ 3 n n CC-BY 4.0 J 2 1 ) = η 2 8 y 1) q + K − ) 1) 2 ( (2010) 062 n 2 This article is distributed under the terms of the Creative C − q 8 n ( − J ( ( ) ) as follows ( q q 2 q q 10 θ ( ( −∞ (2010) 058 i 2 ∞  −∞ −∞ −∞ X θ = θ ∞ ∞ ∞ 1). We use the standard definition for the Dedekind 1 2 n X X X = = = 09 i n n n q, 2 1 − Exper. Math. ( JHEP i , θ ) = ) = ) = ) = 3, 24 ) = ) = M Commun. Num. Theor. Phys. JHEP K q q ( ( ) = q, y q, y q, y q, y 6 4 q ( ( ( ( ( 1 4 3 2 [1] T. Eguchi, H. Ooguri and Y. Tachikawa, [2] M.C.N. Cheng, [3] M.R. Gaberdiel, S. Hohenegger and R. Volpato, [4] M.R. Gaberdiel, S. Hohenegger and R. Volpato, E E i θ θ θ θ It is also convenient to use the Eisenstein series Whenever we do not specify the References any medium, provided the original author(s) and source are c Lastly we define the terms of the Our conventions for the Jacobi functions A Conventions θ Open Access. Attribution License ( JHEP04(2014)069 3, ] ]. and K ] = 2 SPIRE (1986) 592 N , IN ]. ][ ]. , , ifications on B 273 SPIRE (2014) 146 hep-th/9204030 [ IN elliptic genus and 01 ][ , , SPIRE ]. hep-th/9608034 3 [ -models IN supersymmetric heterotic string σ K (2009) 086011 2 ][ , Cambridge University ]. 4) 3 ]. T , ations in string JHEP SPIRE surface , K hep-th/9504006 (1992) 93 , (0 × Nucl. Phys. [ IN 3 , 3 D 79 ][ SPIRE K SPIRE ]. K ]. ]. ]. (1997) 141 IN IN [ [ Universality properties of B 383 A new supersymmetric index string compactifications arXiv:1108.0323 SPIRE [ (1995) 53 SPIRE SPIRE SPIRE IN Perturbative couplings of vector multiplets B 483 ]. IN IN IN hep-th/9608145 Searching for slow-roll moduli inflation in Phys. Rev. [ = 2 ][ Symmetries of ]. ][ ][ ][ , N (1991) 649 ]. B 451 Higher genus string corrections to gauge Moduli dependence of string loop corrections to SPIRE Twining genera of Nucl. Phys. ]. IN arXiv:1106.4315 SPIRE , [ (2012) 413 Moduli corrections to gauge and gravitational – 14 – IN ][ ]. [ SPIRE arXiv:1211.5531 Nucl. Phys. B 355 (1996) 187 , IN , SPIRE [ IN Threshold corrections in [ B 856 BPS-saturated string amplitudes: SPIRE Nucl. Phys. (2012) 1 hep-th/9807124 hep-th/9204102 hep-th/9510182 IN , Algebras, BPS states and strings [ [ [ arXiv:1008.4924 B 482 6 [ (1991) 37 ]. ]. Moonshine in fivebrane spacetimes Note on twisted elliptic genus of Nucl. Phys. , SPIRE SPIRE arXiv:1306.4981 B 267 Mathieu moonshine and [ Nucl. Phys. IN IN , heterotic gauge couplings and their M-theory origin (1999) 109 (1992) 405 (1996) 315 (2011) 446 ][ ][ Nucl. Phys. 10 arXiv:1309.0510 2) , χ hep-th/0609040 , Twining genera and moonshine of heterotic orbifold compact 3, , String theory. Vol. 2: and beyond K Much ado about Mathieu B 541 B 386 B 463 B 694 Toroidal orbifolds: resolutions, orientifolds and applic Phys. Lett. (2013) 030 heterotic string vacua , Modular invariance and discrete torsion on orbifolds ]. ]. ]. heterotic threshold corrections 09 = 2 N = 1 SPIRE SPIRE SPIRE -models on IN IN arXiv:0812.3886 IN arXiv:1307.7717 to appear. couplings Press, Cambridge U.K. (1998) [ Igusa cusp form Nucl. Phys. Commun. Num. Theor. Phys. Nucl. Phys. couplings in four-dimensional superstrings [ in [ massive type IIA supergravity with[ metric fluxes [ gauge coupling constants JHEP Nucl. Phys. Phys. Lett. [ compactifications phenomenology σ N [9] M.C.N. Cheng et al., [6] T. Gannon, [7] S. Hohenegger and S. Stieberger, [8] J.A. Harvey and S. Murthy, [5] T. Eguchi and K. Hikami, [26] M.C.N. Cheng et al., to appear. [21] S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, [19] S. Stieberger, (0 [16] I. Antoniadis, K.S. Narain and T.R. Taylor, [24] S. Harrison, S. Kachru and N.M. Paquette, [25] X. Dong et al., [22] M. Henningson and G.W. Moore, [23] M.R. Gaberdiel, S. Hohenegger and R. Volpato, [20] I. Antoniadis, E. Gava and K.S. Narain, [17] B. de Wit, V. Kaplunovsky, J. Louis and[18] D. E. L¨ust, Kiritsis, C. Kounnas, P.M. Petropoulos and J. Rizos, [15] C. Vafa, [12] J. Polchinski, [13] S. Reffert, [14] R. Flauger, S. Paban, D. Robbins and T. Wrase, [10] J.A. Harvey and G.W. Moore, [11] L.J. Dixon, V.S. Kaplunovsky and J. Louis,