aa r X i v : . [ h e p - t h ] S e p Prepared for submission to JHEP
Not Doomed to Fail
Anne Taormina a and Katrin Wendland b a Centre for Particle Theory, Department of Mathematical Sciences, University of DurhamLower Mountjoy, Stockton Road, Durham DH1 3LE, U.K. b Mathematics Institute, University of FreiburgErnst-Zermelo-Straße 1, D-79104 Freiburg, Germany
E-mail: [email protected] , [email protected] Abstract:
In their recent manuscript “An Uplifting Discussion of T-Duality”, 1707.08888,J. Harvey and G. Moore have reevaluated a mod two condition appearing in asymmetricorbifold constructions as an obstruction to the description of certain symmetries of toroidalconformal field theories by means of automorphisms of the underlying charge lattice. Therelevant “doomed to fail” condition determines whether or not such a lattice automorphism g may lift to a symmetry in the corresponding toroidal conformal field theory withoutintroducing extra phases. If doomed to fail, then in some cases, the lift of g must havedouble the order of g . It is an interesting question, whether or not “geometric” symmetriesare affected by these findings. In the present note, we answer this question in the negative,by means of elementary linear algebra: “geometric” symmetries of toroidal conformal fieldtheories are not doomed to fail. Consequently, and in particular, the symmetry groupsinvolved in symmetry surfing the moduli space of K3 theories do not differ from their lifts. Keywords: discrete symmetries, conformal field theory, sigma models
ArXiv ePrint: DCPT-17/27 ntroduction
Symmetries are a driving force in many areas of mathematics and theoretical physics.They are, in particular, central in the investigation of the recent phenomenon of MathieuMoonshine and its relations to K3 theories [1–11]. The Torelli theorems for complex tori[12] and K3 surfaces [13–17] allow a description of the symmetries of these complex surfacesin terms of automorphisms of the respective lattices of integral cohomology. Inspired bythese theorems, the use of lattice automorphisms has been extended to the investigationand classification of symmetries of superconformal field theories whose targets are complexsurfaces [18, 19].At the basis of this approach lies, roughly speaking, the description of the modulispaces of K3 theories [20–23], on the one hand, and of toroidal conformal field theories[24, 25], on the other, as Grassmannians that are modelled on the even or odd part of thetotal cohomology of the complex two-dimensional target manifold. This may be viewedas an extension of the description of the moduli spaces of hyperk¨ahler structures for therespective complex surfaces in terms of Grassmannians, which are modelled on the secondcohomology. It is thus natural to expect the properties of automorphisms of the underlyinglattices of integral cohomology to allow for a generalization to the conformal field theorysetting. Though well rooted in mathematics, the very description of general symmetriesboth of toroidal and of K3 theories in terms of such lattice automorphisms is not immediate.The more important are the recent results [26] by J. Harvey and G. Moore, which show thatthe symmetries of toroidal conformal field theories cannot, in general, be fully describedin terms of their action on the underlying charge lattice. Applied to toroidal N = (4 , c = c = 6, where the charge lattice maybe directly related to the lattice of odd integral cohomology of the target [23, 27], thisraises some interesting questions about the traditional discussions of symmetries for thesemodels.More precisely, this discussion concerns symmetries of a toroidal conformal field theorywhich induce an action on the underlying charge lattice fixing the respective parameterpoint in the Grassmannian description of the moduli space. In [26], it is shown that viceversa, for certain toroidal conformal field theories, there exist automorphisms of the chargelattice which preserve the parameter point in the moduli space, but which can only lift tosymmetries of at least double the order for the corresponding conformal field theory. Inother words, the charge lattice of such a theory does not fully capture the symmetry group ofthe conformal field theory. There are also cases where a lift of the same order exists, whichhowever acts non-trivially on some winding-momentum fields associated to invariant chargevectors . Both phenomena could potentially cast some doubt on traditional descriptionsof symmetries in conformal field theory. Whether or not either of these phenomena occursfor a given lattice automorphism is encoded by equation (2.17) of the manuscript [26],simply dubbed the “doomed to fail” condition. This condition had previously been statedwith different interpretation by K.S. Narain, M.H. Sarmadi and C. Vafa [28, 29], and in aslightly different context by J. Lepowsky [30, 31]. Hai Siong Tan used a similar approach We thank G. Moore for emphasizing this point to us. – 2 –n [32] to determine consistent asymmetric orbifold group actions. The condition can betraced back to the properties of involutions on the charge lattice which fix the parameterpoint: whenever there exists a charge vector which has an odd scalar product with its imageunder such an involution, the lift is “doomed to fail”, i.e. either only a lift of order fourexists, or there are invariant charge vectors whose associated momentum-winding fields aremultiplied by ( − Z -orbifolding,any toroidal N = (4 ,
4) superconformal field theory at central charges c = c = 6 givesrise to a K3 theory. The Z -orbifolding procedure thereby induces a map between therespective moduli spaces which was determined in [23]. This map is described in termsof the underlying lattices of integral cohomology, and it requires a transition between theodd and the even integral cohomology of complex two-tori by means of triality [23]. Thesymmetries of the underlying toroidal conformal field theories thus induce symmetries ofthe resulting K3 theories. A failure of a lattice automorphism to fully capture a symmetryof a toroidal theory may descend to the corresponding K3 theory.We are not able to predict the full scope of consequences for Mathieu Moonshine thatmay follow from [26]. However, in this note we come back to a remark made by J. Harveyand G. Moore in the first version of their recent paper, addressing symmetry surfing .The latter is a technique that we have proposed first in [34] and that allows to combine geometric symmetry groups from distinct points in the moduli space of K3 theories [6].In [9], we have shown that the maximal subgroup Z : A of M is the group which, bymeans of symmetry surfing, combines all geometric symmetries induced by the symmetriesof complex tori in the corresponding Z -orbifold conformal field theories. By constructingthe leading order massive representation which contributes to Mathieu Moonshine, forthis maximal subgroup, our work [8] provides the first piece of evidence that the relevantrepresentations of Mathieu Moonshine might arise intrinsically from conformal field theory.Our arguments have, in the meantime, been vastly generalized by M. Gaberdiel, Ch. Kellerand H. Paul [35], yielding additional evidence in favour of the proposal of symmetry surfing.Whether or not the symmetries that are relevant for the works [6, 8, 9, 35] are doomedto fail is an important question for our programme. In this note we show that thanks to thevery elegant “doomed to fail” condition, this question may be answered in the negative bymeans of elementary linear algebra: symmetry surfing is not doomed to fail by the uplifting Note that at this point, one needs to work on a 2 : 1 cover of the moduli space of toroidal superconformalfield theories in order to keep the target space orientation, as is required for the resulting K3 theories [23,(1.17)]. – 3 –roperties of lattice automorphisms. It is important to appreciate that this statement isclosely tied to the fact that symmetry surfing only proposes to combine geometric symme-tries of K3 theories. In fact, we show more generally that geometric symmetries of toroidalconformal field theories are never doomed to fail. The very definition of geometric symme-tries needs to be treated with great care. It refers to symplectic automorphisms of finiteorder that also leave invariant the B-field, viewed as a real-valued two-form. In particular,this notion excludes the identification of a B-field with its shifts by integral cohomologyclasses, thus excluding some symmetries that are certainly in the realm of geometry . Theadjective geometric thus solely means that such symmetries leave invariant a geometricinterpretation. In the context of symmetry surfing, our notion of geometric symmetries isinseparably connected with the idea that there is a space of states that generically exist inall K3 theories, which bears all the structure that is relevant to Mathieu Moonshine. Asalready predicted in [9, 36], the cohomology of the chiral de Rham complex of [37–42] isexpected to model this “space of generic states”. In identifying the cohomology of the chi-ral de Rham complex with the large volume limit of a topological half-twist of K3 theories[43], we need to require compatibility of the symmetries in question with a large volumelimit. We do so by requiring that they leave invariant a geometric interpretation of themodel. Recently, the idea of explaining Mathieu Moonshine by means of such a space ofgeneric states has been further substantiated by the observation that the cohomology ofthe chiral de Rham complex for K3 surfaces indeed decomposes into irreducible represen-tations of the “small” N = 4 superconformal algebra at central charge c = 6 with propermultiplicity spaces of every massive representation, i.e. without the occurrence of virtualrepresentations [10, 11].We emphasize that symmetry surfing has not been proved to explain Mathieu Moon-shine, so far, and that this proposal may still ultimately fail, as is extensively discussed in[11, § geometric symmetries . We describe the properties ofthe symmetries that enter the symmetry surfing proposal of [6, 9, 34] and that are thusrelevant for [8, 35]. In particular, the properties of the underlying symmetries of toroidalconformal field theories and their induced actions on the respective charge lattices arediscussed. Section 2 is devoted to the proof of our claim that geometric symmetries oftoroidal conformal field theories, in particular those that enter symmetry surfing, are notdoomed to fail. We remark that an alternative, just as immediate proof follows from thediscussion around equation (4.54) of [32]. A lower-dimensional toroidal example is a reflection in a simple root in the su (3)-point for two freebosons, discussed in [26, § – 4 – Geometric symmetries of K3 theories
The Mathieu Moonshine phenomenon, discovered by T. Eguchi, H. Ooguri and Y. Tachikawa[1], predicts the existence of a Z × Z graded representation of the “small” N = 4 super-conformal algebra of [44] at central charge c = 6, whose diagonally Z -graded characteryields the complex elliptic genus of a K3 surface, and which simultaneously furnishes arepresentation of the Mathieu group M . This should yield the corresponding twistedtwining genera with their strongly restrictive modular properties. The existence of such an M -module was proved by T. Gannon [7]. However, in addition one expects a compatiblestructure of a super vertex operator algebra on this M -module. The latter has not beenconstructed, so far. Neither has a satisfactory explanation been found for the existence ofan M -module with all this structure.Our quest for an explanation of these phenomena has led us to propose that the M -module in question should arise as a subspace of the spaces of states of K3 theories whichis common to all such theories. The representations constructed in [8, 35] arise preciselyas such from the spaces of states of Z -orbifold conformal field theories on K3. We suggestthat the action of M might then be explained by means of certain symmetry groups ofK3 theories on this space of generic states, combined from distinct points of the modulispace. The latter is our proposal of symmetry surfing .As explained in [9, 11, 36], we expect such a space of generic states to arise as thelarge volume limit of a topological half-twist of the space of states of our K3 theories(denoted X in [9, § geometric in the sense thatthey are induced by geometric symmetries of the underlying K3 surface, independently ofthe volume. More precisely, we require these symmetries to fix a geometric interpretationof our theories according to [22].To characterize such symmetries more specifically, let us recall the description of themoduli space of K3 theories, following [22, 23]. Indeed, a K3 theory may be specified bydata that determine a hyperk¨ahler structure on a K3 surface X , its volume V ∈ R , V > B -field B ∈ H ( X, R ). Here, the cohomology H ∗ ( X, R ) of K3 is equipped withthe scalar product h· , ·i of signature (4 ,
20) that is induced by the intersection form. Ahyperk¨ahler structure on X may then be uniquely specified by an oriented, positive definitethree-dimensional subspace Σ ⊂ H ( X, R ). Denoting by υ ∈ H ( X, Z ), υ ∈ H ( X, Z ) agenerating pair of vectors for the hyperbolic lattice H ( X, Z ) ⊕ H ( X, Z ) with h υ , υ i =1, the K3 theory in question is uniquely specified by the positive definite oriented four-dimensional subspace of H ∗ ( X, R ) which is generated by n σ − h B, σ i υ (cid:12)(cid:12)(cid:12) σ ∈ Σ o ∪ n υ + B + (cid:16) V − h B,B i (cid:17) υ o . As is explained, for example, in [18], the symmetries in question in particular induce latticeautomorphisms of H ∗ ( X, Z ) which leave the above four-dimensional subspace of H ∗ ( X, R )– 5 –nvariant, point-wise. To be compatible with a large volume limit, this property must holdindependently of the value of V . It follows that the vector υ must be invariant under ourlattice automorphisms. Since our large volume limit is not only independent of the valueof V but solely depends on a choice of complex structure, in all our works we have beeneven more restrictive on the symmetries that enter symmetry surfing. To call a symmetry geometric , we require it to fix the geometric interpretation, i.e. we require that the inducedlattice automorphism fixes both υ and υ . Thus B must also be fixed, see [9, footnotes 18,19], [8, § § R d / Λ, d = 4, with B-field e B . Here, e B is given by a real, skew-symmetric d × d matrix, Λ ⊂ R d is a lattice of rank d and by Λ ∗ ⊂ R d we denote its dual after identification of R d with ( R d ) ∗ by means of theEuclidean metric · , that is, Λ ∗ = n µ ∈ R d | µ · λ ∈ Z ∀ λ ∈ Λ o . The corresponding charge lattice then isΓ(Λ , e B ) = n √ ( µ − e Bλ + λ ; µ − e Bλ − λ ) | ( µ, λ ) ∈ Λ ∗ ⊕ Λ o ⊂ R d,d , (1.1)where we use the standard conventions as for example in [23, (1.11)], [33, (A.3)-(A.5)], [26,(3.3)], and R d,d = R d ⊕ R d is equipped with the scalar product •∀ ( p l ; p r ) , ( p ′ l ; p ′ r ) ∈ Γ(Λ , e B ) : ( p l ; p r ) • ( p ′ l ; p ′ r ) = p l · p ′ l − p r · p ′ r of signature ( d, d ). The geometric symmetries of the toroidal superconformal field theorythat might potentially be affected by the findings of [26] are thus given by linear maps g ∈ O ( d ) with g Λ = Λ and g e B = e Bg . The induced action of g on the charge lattice is ∀ p = ( p l ; p r ) ∈ Γ(Λ , e B ) : g ( p ) := ( gp l ; gp r ) . (1.2)For concrete examples relevant to symmetry surfing, the reader is referred to [9, § Consider a toroidal conformal field theory with charge lattice Γ ⊂ R d,d , and a latticeautomorphism γ of Γ which fixes the parameter point of the theory, i.e. which acts on thecharge lattice by means of ∀ p = ( p l ; p r ) ∈ Γ : γ ( p ) = ( g l p l ; g r p r )with g l , g r ∈ O ( d ). Assume that γ has order ℓ . Then, reevaluating obstructions previouslydiscussed in different interpretations or contexts [28–32], J. Harvey and G. Moore find– 6 –he following obstruction for γ to lift to an automorphism of order ℓ of the correspondingtoroidal conformal field theory that leaves invariant winding-momentum fields associatedto γ -invariant charge vectors [26, (2.17)]: such a lift is doomed to fail if ℓ is even and ∃ p ∈ Γ : p • γ ℓ/ ( p ) / ∈ Z . (2.1)To show that the geometric symmetries of toroidal conformal field theories are not doomedto fail, we may thus assume without loss of generality that ℓ = 2. As explained in Section1 above, by (1.2) we furthermore assume that g l = g r = g ∈ O ( d ), Γ = Γ(Λ , e B ) as in (1.1)with e B T = − e B , and that g Λ = Λ, g e B = e Bg . We thus have g = g − = g T , and we find( g e B ) T = − e Bg = − g e B. In particular, we have ∀ λ, λ ′ ∈ R d : λ · ( g e B ) λ ′ + λ ′ · ( g e B ) λ = 0 . (2.2)For charge vectors p = √ ( µ − e Bλ + λ, µ − e Bλ − λ ) , p ′ = √ ( µ ′ − e Bλ ′ + λ ′ , µ ′ − e Bλ ′ − λ ′ ) (2.3)with arbitrary λ, λ ′ ∈ Λ and µ, µ ′ ∈ Λ ∗ we thus have p ′ • γ ( p ) = ( µ ′ − e Bλ ′ ) · gλ + λ ′ · g ( µ − e Bλ ) g = g T = µ ′ · gλ + µ · gλ ′ − λ · ( g e B ) λ ′ − λ ′ · ( g e B ) λ ( . ) = µ ′ · gλ + µ · gλ ′ . In particular, ∀ p = √ ( µ − e Bλ + λ, µ − e Bλ − λ ) ∈ Γ(Λ , e B ) : p • γ ( p ) = 2 µ · gλ ∈ Z since by assumption, gλ ∈ Λ and µ ∈ Λ ∗ . In other words, the “doomed to fail” condition(2.1) does not hold. In particular, at d = 4 we learn that the symmetries that have beenrelevant for symmetry surfing, so far, are not doomed to fail.We remark that the “doomed to fail condition” of [26] is solely testing the cyclicsubgroups of a given symmetry group. However, as noted at the end of Section 1, symmetry-surfing does involve non-cyclic, in fact even non-abelian symmetry groups. Actually, allexamples of symmetries that are doomed to fail and that are discussed in [26] arise in non-abelian global symmetry groups of special conformal field theories with enhanced symmetry.This raises the question of whether every geometric symmetry group G has a lift to asymmetry group of the respective conformal field theory which is isomorphic to G . Thatthis is indeed the case follows from the existence of an invariant 2-cocycle ε : Γ × Γ −→ {± } – 7 –n the charge lattice Γ which governs the operator product expansions between vertexoperators (see, for example, [45–48] for the classical results). Indeed, with notations as in(2.3), one may use the cocycle ∀ p, p ′ ∈ Γ : ε (cid:0) p, p ′ (cid:1) := ( − µ · λ ′ . For a geometric symmetry γ as above, g ∈ O ( d ) thus implies ∀ p, p ′ ∈ Γ : ε (cid:0) γ ( p ) , γ ( p ′ ) (cid:1) = ε (cid:0) p, p ′ (cid:1) . In terms of [26, Appendix A], this means that a lift G −→ b G , g T g , of our symmetrygroup G exists which obeys T g ◦ T g = T g g for all g , g ∈ G (see [26, (A.5)–(A.11)]),thus yielding G ∼ = b G . Acknowledgements
We thank J. Harvey and G. Moore for communicating a preliminary version of theirwork [26] to us very early on, and for very useful discussions. This gave us the chanceto set the record straight quickly, regarding symmetry surfing not being doomed to fail.We also thank an anonymous referee for carefully reading the manuscript and for raising anumber of questions whose answers will certainly help the readers to put our work in theright context.
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