Anomaly cancellation and modularity, II: The E 8 E 8 case

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1 SCIENCE CHINA Mathematis. ARTICLES. June 07 Vol. 60 No. 6: doi: 0.007/s Anomaly anellation and modularity, II: The E 8 E 8 ase In memory of Professor LU QiKeng HAN Fei, LIU KeFeng,3 & ZHANG WeiPing 4, Department of Mathematis, National University of Singapore, Singapore 9076, Singapore; Department of Mathematis, Capital Normal University, Beijing 00048, China; 3 Department of Mathematis, University of California at Los Angeles, Los Angeles, CA 90095, USA; 4 Chern Institute of Mathematis & LPMC, Nankai University, Tianjin 30007, China mathanf@nus.edu.sg, liu@math.ula.edu, weiping@nankai.edu.n Reeived November 7, 06; aepted February 0, 07; published online Marh 0, 07 Abstrat In this paper we show that both of the Green-Shwarz anomaly fatorization formula for the gauge group E 8 E 8 and the Hořava-Witten anomaly fatorization formula for the gauge group E 8 an be derived through modular forms of weight 4. This answers a question of Shwarz. We also establish generalizations of these fatorization formulas and obtain a new Hořava-Witten type fatorization formula. Keywords MSC00 E 8 bundle, anomaly anellation, Eisenstein series, modular form 9K56, 57R0, 53Z05, Z05 Citation: Han F, Liu K F, Zhang W P. Anomaly anellation and modularity, II: The E 8 E 8 ase. Si China Math, 07, 60: , doi: 0.007/s Introdution In [8, 9, 6], it has been shown that both of the Alvarez-Gaumé-Witten miraulous anomaly anellation formula [] and the Green-Shwarz anomaly fatorization formula [7] for the gauge group SO3 an be derived and extended through a pair of modularly related modular forms, whih are over the modular subgroups Γ 0 and Γ 0, respetively. In answering a question of Shwarz [9], we deal with the remaining ase of gauge group E 8 E 8 in this artile. Let Z X B be a fiber bundle with fiber Z being 0-dimensional. Let T Z be the vertial tangent bundle equipped with a metri g T Z and an assoiated Levi-Civita onnetion T Z see [3, Proposition 0.]. Let R T Z T Z be the urvature of T Z, whih we also for simpliity denote by R. Let T C Z be the omplexifiation of T Z with the indued Hermitian onnetion TCZ. Let P, ϑ and P, ϑ be two prinipal E 8 bundles with onnetions over X. Let ρ be the adjoint representation of E 8. Let W i P i ρ C 48, i, be the assoiated vetor bundles, whih are of rank 48. We equip both W and W with Hermitian metris and Hermitian onnetions, respetively. Let F i denote the urvature of the bundle W i. Let Tr denote the trae in the adjoint representation. The elementary fats about E 8 tells us that TrF n+ i 0, TrFi 4 00 TrF i and TrFi TrF i 3 see []. It is easy to see that W i π TrF i. Simply denote TrF n + TrF n by TrF n. Corresponding author Siene China Press and Springer-Verlag Berlin Heidelberg 07 math.sihina.om link.springer.om

2 986 Han F et al. Si China Math June 07 Vol. 60 No. 6 The Green-Shwarz anomaly formula [7] asserts that the following fatorization for the forms holds : I ÂT ZhW + W + ÂT ZhT CZ ÂT Z} 64π trr trr trr 3 + TrF 6 + TrF trr 30 TrF 4π 6π trr trr 58 TrF 4 trr 960 TrF 5 6 TrF trr TrF 5 6 trr trr. p T Z + 30 W + W I 8.. In [, ], Hořava and Witten observed, on the other hand, that the following anomaly fatorization formula holds for eah i, : Î i ÂT ZhW i + ÂT } ZhT CZ ÂT Z 64π trr trr trr 3 + TrFi 6 + TrFi 8 trr trr 54 TrF 4i trr 440 4π 4 trr 5 TrF i p T Z + 30 W i Îi 8 where Îi 8 an be written expliitly as and therefore Î i 8 6π 4 4 I Î + Î Îi 8,. 4 trr 30 TrF i 8 trr4 + 3 trr, p T Z + 30 W Î 8 + p T Z + 30 W Î 8. The purpose of this artile is to show that the above anomaly fatorization formulas an also be derived naturally from modularity as in the orthogonal group ase dealt with in [9]. This provides a positive answer to a question of Shwarz mentioned at the beginning of the artile. To be more preise, we onstrut in Setion a modular form QP i, P j, τ of weight 4 over SL, Z, for any i, j, }, suh that when i, j, the modularity of QP, P, τ gives the Green-Shwarz fatorization formula., while when i j, the modularity of QP i, P i, τ gives the Hořava-Witten fatorization formula.. Atually what we onstrut is a more general modular form QP i, P j, ξ, τ, whih involves a omplex line bundle or equivalently a rank two real oriented bundle and we are able to obtain generalizations of the Green-Shwarz formula and the Hořava-Witten formula by using the assoiated modularity. Our onstrution of the modular form QP i, P j, ξ, τ involves the basi representation of the affine Ka-Moody algebra of E 8. Inspired by our modular method of deriving the Green-Shwarz and Hořava-Witten fatorization formulas, we also onstrut a modular form RP i, ξ, τ of weight 0 over SL, Z, the modularity of whih will give us a new fatorization formula of Hořava-Witten type. See Theorem.4 for details. It would be interesting to ompare.8 and.9 with the Hořava-Witten fatorization. or.6. Atually, another interesting question of Shwarz is to onstrut quantum field theories assoiated to the generalized anomaly fatorization formulas in this paper and [9]. In the rest of this setion, we present our generalized Green-Shwarz and Hořava-Witten formula, as well as the new formulas of Hořava-Witten type obtained from RP i, ξ, τ. They are proved in Setion In what follows, we write harateristi forms without speifying the onnetions when there is no onfusion see [0].

3 Han F et al. Si China Math June 07 Vol. 60 No by using modularity after briefly reviewing some knowledge of the affine Ka-Moody algebra of E 8 in Setion. Let ξ be a rank two real oriented Eulidean vetor bundle over X arrying a Eulidean onnetion ξ. Let eξ, ξ be the Euler form anonially assoiated to ξ see [0, Subsetion 3.4]. Let T C Z be the omplexifiation of T Z and ξ C the omplexifiation of ξ. For a omplex vetor bundle E, denote Ẽ : E C rke. Theorem.. For i, j, the following identities hold: ÂT Ze hwi + W j + ÂT Ze htc Z ÂT Ze + ÂT Ze h ξc + 3 ξ C ξ C } p T Z W i + W j e 4 p T Z W i + W j p T Z W i + W j ÂT Ze ha + e 4 pt Z Wi+Wj ÂT Ze } 8,.3 where A W i + W j + T C Z + ξ C + 3 ξ C ξ C. Putting i j, one has for eah i, ÂT Ze hwi + ÂT Ze htc Z ÂT Ze + p T Z W i e 4 pt Z 3 + } ÂT Ze h ξc + 3 ξ C ξ C 5 Wi p T Z W i ÂT Ze hbi + e 4 pt Z Wi ÂT Ze } 8,.4 where B i W i + T C Z + ξ C + 3 ξ C ξ C. If ξ is trivial, we obtain the Green-Shwarz formula. for E 8 E 8 and the Hořava-Witten formula. for E 8 in the following orollary. Corollary.. One has ÂT ZhW + W + ÂT ZhT CZ p T Z + 30 W + W ÂT Z} e 4 pt Z+ 30 W+W ÂT ZhC p T Z + 30 W + W + e 4 p T Z+ 30 W + W ÂT Z} 8,.5 where C W + W + T C Z. In addition, for eah i, ÂT ZhW i + ÂT ZhT CZ ÂT Z } p T Z + 30 W i e 4 p T Z+ 5 W i p T Z + 5 ÂT ZhD i W i + e 4 p T Z+ 5 W i ÂT Z} 8,.6 where D i W i + T C Z. Remark.3. It an be heked by diret omputations that the seond fators in the right-hand sides of.5 and.6 are equal to I 8 and Îi 8, respetively. We now state a new fatorization formula, whih is of the Hořava-Witten type.

4 988 Han F et al. Si China Math June 07 Vol. 60 No. 6 Theorem.4. For eah i, the following identity holds: ÂT Ze hwi + ÂT Ze htc Z + 46ÂT Ze + ÂT Ze h ξc + 3 ξ C ξ C } p T Z W i e 4 p T Z W i p T Z ÂT Ze hei W i + e 4 pt Z Wi ÂT Ze } 8,.7 where E i W i + T C Z ξ C + 3 ξ C ξ C. If ξ is trivial, we have where F i W i + T C Z ÂT ZhW i + ÂT ZhT CZ + 46ÂT Z} p T Z + 30 W i e 4 p T Z+ 30 W i p T Z + 30 ÂT ZhF i W i + e 4 p T Z+ 30 W i ÂT Z} 8,.8 Remark.5. We an express.8 by diret omputations as follows: 64π trr trr trr 3 + TrFi 6 + TrFi 8 trr trr 54 TrF 4i trr 4π trr 30 TrF i 6π TrF i + 3 trr TrFi 5 6 trr trr p T Z + 30 W i Ĵ 8. i.9 Remark.6. As in [9], one may ask whether there is a physis model orresponding to.8 and.9. We point out that the appearane of the fator trr 30 TrF i in the Green-Shwarz anomaly anellation formulas being linked to the modularity of ertain q-series after being multiplied by e 4 E τ 8π trr 30 F was already realized by physiists see [5, 7], of whih we do not laim priority. However, no onrete and preise formulas were given in those work suh that later when needed one still derives the Green- Shwarz type anomaly anellation formulas by using the traditional method without applying modularity see [,, 8]. Atually, the motivation of our work as well as [9] is simply to derive expliitly all the above mentioned formulas in a unified framework of applying modularity. Moreover, the formulas derived in this unified framework make the Green-Shwarz formulas more transparent by showing expliitly what are the fators in the anomaly fatorization. For example, the seond fator in the right-hand side of.5 expliitly gives I 8 in.. This essentially omes from the appliation of the basi representation of affine E 8, whih was not expliitly used in [5, 7]. This unified framework an also inspire many new formulas. For example,.8 and.9 give a new Hořava-Witten type formula with one E 8 bundle involved;.3 and.7 give formulas with a omplex line bundle being involved, whih should deal with anomalies oming from nontriviality of determinant line bundles of twisted spin Dira operators. We hope these new formulas ould be physially meaningful. The basi representation of affine E 8 In this setion, we briefly review the basi representation theory for the affine E 8 by following [3] see also [4].

5 Han F et al. Si China Math June 07 Vol. 60 No Let g be the Lie algebra of E 8. Let, be the Killing form on g. Let g be the affine Lie algebra orresponding to g defined by g C[t, t ] g C, with braket dp t [P t x + λ, Qt y + µ] P tqt [x, y] + x, y Res t0 Qt. dt Let ĝ be the affine Ka-Moody algebra obtained from g by adding a derivation t d dt, whih operates on C[t, t ] g in an obvious way and sends to 0. The basi representation V Λ 0 is the ĝ-module defined by the property that there is a nonzero vetor v 0 highest weight vetor in V Λ 0 suh that v 0 v 0, C[t] Ct d dt v 0 0. Setting V k : v V Λ 0 t d dt kv} gives a Z +-gradation by finite spaes. Sine [g, d] 0, eah V k is a representation of g. Moreover, V is the adjoint representation of E 8. Let q e π τ. Fix a basis for the Cartan subalgebra and let z i } 8 i be the orresponding oordinates. The harater of the basi representation is given by hz, z,..., z 8, τ : hv k z, z,..., z 8 q k φτ 8 Θ g z, z,..., z 8, τ, k0 where φτ n qn so that ητ q /4 φτ is the Dedekind η funtion; Θ g z, z,..., z 8, τ is the theta funtion defined on the root lattie Q by Θ g z, z,..., z 8, τ γ Q q γ / e π γ z. It is proved in [6] see [0] that there is a basis for the E 8 root lattie suh that Θ g z,..., z 8, τ θz l, τ + θ z l, τ + where θ and θ i i,, 3 are the Jaobi theta funtions see [4, 8]. θ z l, τ + θ 3 z l, τ,. 3 Derivation of Green-Shwarz and Horava-Witten type anomaly fatorizations via modularity In this setion, we derive the Green-Shwarz and Hořava-Witten type fatorization formulas in Theorems. and.4 via modularity. For the prinipal E 8 bundles P i, i,, onsider the assoiated bundles V i P i ρk V k q k KX[[q]]. Sine ρ is the adjoint representation of E 8, we have W i P i ρ V. Following [5], set m k0 ΘT C Z, ξ C : S q m T C Z Λ q n ξ C KX[[q]], n Λ q u / ξ C u where ξ C is the omplexifiation of ξ. Clearly, ΘT C Z, ξ C admits a formal Fourier expansion in q as Λ q v / ξ C ΘT C Z, ξ C C + B q + B q +, 3. v

6 990 Han F et al. Si China Math June 07 Vol. 60 No. 6 where the B j s are elements in the semi-group formally generated by omplex vetor bundles over X. Moreover, they arry anonially indued onnetions denoted by B j. Let Θ be the indued onnetion with q-oeffiients on Θ. For i, j, set QP i, P j, ξ, τ : e 4 E τp T Z W i + W j ÂT Z Theorem 3.. osh hθt C Z, ξ C φτ 6 hv i hv j }. 3. QP i, P j, ξ, τ is a modular form of weight 4 over SL, Z. Proof. By the knowledge reviewed in Setion, we see that there are formal two forms yl i, l 8, i, suh that φτ 8 hv i θy i l, τ + θ yl, i τ + θ yl, i τ + θ 3 yl, i τ. 3.3 Sine θz, τ is an odd funtion about z, one an see that up to degree, the term 8 θyi l, τ an be dropped and we have φτ 8 hv i θ y i l, τ + θ yl, i τ + θ 3 yl, i τ. 3.4 Sine θ z, τ, θ z, τ and θ 3 z, τ are all even funtions about z, the right-hand side of the above equality only ontains even powers of y i j s. Therefore, hw i only onsists of forms of degrees divisible by 4 this is atually a basi fat about E 8. So On the other hand, θ y l, i τ + hv i + hw i q W i + q θ yl, i τ + From , we have θ 3 yl, i τ yl i + q + Oq. 3.6 y i l 30 W i. 3.7 Note that this is also a basi fat about representations of E 8, although it ould be dedued in this interesting way by playing with modular forms. Let ±π x k } k 5 be the formal Chern roots for T Z C, T Z C. Let eξ, ξ π u be the Euler form anonially assoiated to ξ. One has QP i, P j, ξ, τ e 4 EτpT Z Wi+Wj ÂT Z osh 5 e 4 E τp T Z W i + W j k } hθt C Z, ξ C φτ 6 hv i hv j x k θ 0, τ θx k, τ θ u, τ θ 0, τ θ u, τ θ 3 u, τ θ 0, τ θ 3 0, τ θ y i 4 l, τ + θ yl, i τ + θ 3 yl, i τ } θ y j l, τ + θ y j l, τ + θ 3 y j l, τ. 3.8

7 Han F et al. Si China Math June 07 Vol. 60 No Then we an perform the transformation formulas for the theta funtions and E τ see [4,8] to show that QP i, P j, ξ, τ is a modular form of weight 4 over SL, Z. Proof of Theorem.. Expanding the q-series, we have hθt C Z, ξ C φτ 6 hv i hv j e 4 EτpT Z Wi+Wj ÂT Z osh e 4 p T Z W i + W j e 4 p T Z W i + W j p T Z W i + W j q + Oq ÂT Z osh hc + B q + Oq 6q + Oq + hw i q + Oq + hw j q + Oq e 4 p T Z W i + W j ÂT Z osh + q e 4 p T Z W i + W j ÂT Z osh e 4 p T Z W i + W j + Oq. hb 6 + W i + W j ÂT Z osh p T Z W i + W j It is well known that modular forms over SL, Z an be expressed as polynomials of the Eisenstein series E 4 τ and E 6 τ, where 3.9 E 4 τ + 40q + 60q + 670q 3 +, 3.0 E 6 τ 504q 663q 976q Their weights are 4 and 6, respetively. Sine the weight of the modular form QP i, P j, ξ, τ is 4, it must be a multiple of So from 3.9 and 3., we have e 4 pt Z Wi+Wj ÂT Z osh E 4 τ E 6 τ 4q } hb 6 + W i + W j e 4 p T Z W i + W j p T Z W i + W j 4 e 4 p T Z W i + W j ÂT Z osh Therefore, ÂT Z osh } hw i + W j + B + 8 p T Z W i + W j e 4 p T Z W i + W j p T Z W i + W j + e 4 p T Z W i + W j ÂT Z osh ÂT Z osh } }. 3.3 Z osh hw i + W j + B + 8 ÂT }

8 99 Han F et al. Si China Math June 07 Vol. 60 No. 6 So To find B, we have ΘT C Z, ξ C S q m T C Z m n Λ q n ξ C u + T C Z 0q + Oq + ξ C q + Oq Λ q u / ξ C ξ C q / ξ C q + Oq 3/ + ξ C q / ξ C q + Oq 3/ v Λ q v / ξ C + T C Z 0 + ξ C + 3 ξ C ξ C q + Oq. 3.5 Plugging B into 3.4, we have ÂT Z osh B T C Z 0 + ξ C + 3 ξ C ξ C. hw i + W j + T C Z + ξ C + 3 ξ C ξ C p T Z W i + W j e 4 p T Z W i + W j p T Z W i + W j osh Z ÂT hw i + W j + T C Z + ξ C + 3 ξ C ξ C + e 4 pt Z Wi+Wj ÂT Z osh } } Sine hw i and hw j only ontribute degree 4l forms, we an replae osh by e. Then in.6, putting i, j gives.4 and putting i j gives.5. To prove Theorem.4, for eah i, set RP i, ξ, τ : Theorem 3.. Proof. e 4 EτpT Z Wi ÂT Z osh hθt C Z, ξ C φτ 8 hv i }. 3.7 RP i, ξ, τ is a modular form of weight 0 over SL, Z. This an be similarly proved to Theorem 3. by seeing that RP i, ξ, τ e 4 EτpT Z Wi ÂT Z osh e 4 E τp T Z W i θ y i l, τ + 5 θ yl, i τ + x l θ 0, τ θx l, τ and then applying the transformation laws of theta funtions. Proof of Theorem.4. e 4 EτpT Z Wi ÂT Z osh } hθt C Z, ξ C φτ 8 hv i θ u, τ θ 0, τ θ u, τ θ 3 u, τ θ 0, τ θ 3 0, τ θ 3 yl, τ} i, 3.8 Similar to that in the proof of Theorem., expanding the q-series, we have hθt C Z, ξ C φτ 8 hv i p T Z W i q + Oq e 4 p T Z W i e 4 p T Z W i

9 Han F et al. Si China Math June 07 Vol. 60 No ÂT Z osh hc + B q + Oq 8q + Oq + hw i q + Oq e 4 p T Z W i ÂT Z osh + q e 4 pt Z Wi ÂT Z osh hb 8 + W i e 4 p T Z W i + Oq. p T Z W i ÂT Z osh 3.9 However, modular form of weight 0 must be a multiple of E 4 τe 6 τ 64q +, so we have e 4 pt Z Wi ÂT Z osh Therefore, ÂT Z osh } hb 8 + W i e 4 p T Z W i p T Z W i 64 e 4 p T Z W i ÂT Z osh } hw i + B + 56 p T Z W i osh e 4 p T Z 3 + ÂT Z osh 30 W i p T Z ÂT Z W i hw i + B e 4 pt Z Wi ÂT Z osh Plugging in B, we have ÂT Z osh hw i + T C Z ξ C + 3 ξ C ξ C p T Z W i e 4 p T Z W i p T Z ÂT Z osh W i + e 4 pt Z Wi ÂT Z osh } } }. 3.0 } hw i + T C Z ξ C + 3 ξ C ξ C } Sine hw i only ontribute degree 4l forms, we an replae osh by e, and 3. gives.7. 4 Disussion Combining with the results in this paper and our previous work in [9], it is interesting to see that the fundamental anomaly anellation formulas in various string theories an be unified in the framework of modular forms and modular transformations. This phenomena has its roots in the hidden symmetry of the muh larger onfiguration spae of string theory, namely, loop spae, double loop spae, path spae, et. We expet that many other anomaly anellation formulas in string theory and M-theory an also be derived from the modular form method. On the other hand, the modular form method an help one detet new anellation formulas, for example, in [9] one finds similar anellation formulas for general gauge group SON, not restrited in

10 994 Han F et al. Si China Math June 07 Vol. 60 No. 6 SO3; and in this paper, for a single E 8 bundle, we find.9, whih is different from the Hořava- Witten s formula.. Moreover, we find formulas with a omplex line bundle involved, whih give anellations of anomalies oming from families of twisted spin Dira operators instead of spin Dira operators. We hope these new formulas an find appliations in physis. Aknowledgements This work was supported by a start-up grant from National University of Singapore Grant No. R , National Siene Foundation of USA Grant No. DMS-506 and National Natural Siene Foundation of China Grant No. 09. The authors are indebted to J. H. Shwarz for asking them the question onsidered in this paper. The authors also thank Siye Wu for helpful and inspiring ommuniations. Referenes Alvarez-Gaumé L, Witten E. Gravitational anomalies. Nulear Phys B, 983, 34: Avramis S D. Anomaly-free supergravities in six dimensions. ArXiv:hep-th/0633, Berline N, Getzler E, Vergne M. Heat Kernels and Dira Operators. Berlin-Tokyo: Springer, 00 4 Chandrasekharan K. Ellipti Funtions. Berlin-Heidelberg: Springer-Verlag, Chen Q, Han F, Zhang W. Generalized Witten genus and vanishing theorems. J Differential Geom, 0, 88: 39 6 Gannon T, Lam C S. Latties and θ-funtion identities, II: Theta series. J Math Phys, 99, 33: Green M B, Shwarz J H. Anomaly anellations in supersymmetri d 0 gauge theory and superstring theory. Phys Lett B, 984, 49: 7 8 Han F, Liu K, Zhang W. Modular forms and generalized anomaly anellation formulas. J Geom Phys, 0, 6: Han F, Liu K, Zhang W. Anomaly anellation and modularity. In Memory of Gu Chaohao. Frontiers in Differential Geometry, Partial Differential Equations and Mathematial Physis. Singapore: World Sientifi, 04, Harris C. The index bundle for a family of Dira-Ramond operators. PhD Thesis. Miami: University of Miami, 0 Hořava P, Witten E. Heteroti and Type I string dynamis from eleven dimensions. Nulear Phys B, 996, 460: Hořava P, Witten E. Eleven dimensional supergravity on a manifold with boundary. Nulear Phys B, 996, 475: Ka V G. An eluidation of Infinite-dimensional algebras, Dedekind s η-funtion, lassial Möbius funtion and the very strange formula : E 8 and the ube root of the modular funtion j. Adv Math, 980, 35: Ka V G. Infinite-Dimensional Lie Algebras, 3rd ed. Cambridge: Cambridge University Press, Lerhe W, Shellekens A N, Warner N P. Latties and strings. Phys Rep, 989, 77: 40 6 Liu K. Modular invariane and harateristi numbers. Comm Math Phys, 995, 74: Shellekens A N, Warner N P. Anomalies, haraters and strings. Nulear Phys B, 987, 87: Shwarz J H, Witten E. Anomaly analysis and brane-antibrane system. J High Energy Phys, 00, 3, Art. No. 03. ISSN Shwarz J H. Private ommuniations, 0 0 Zhang W. Letures on Chern-Weil Theory and Witten Deformations. Singapore: World Sientifi, 00