arXiv:2103.13636v1 [math.AT] 25 Mar 2021 ti xlie o hsrsl a eitrrtda ihra higher as interpreted be can result this how explained is It etost eaeteter ftplgclmdlrforms modular topological of theory the relate to rections 94B05. itcgnso h itngnscnb eoee.Ti objec This in recovered. class be Euler can from genus the Witten complex, the Rham d or de genus of chiral liptic the sheaf as a ch known constructed of algebras, they terms operator precisely, in More genus Witten theories. the field a of [GMS00] interpretation Schechtman an and provided Malikov Gourbonov, theories. field hoyi tl isn (cf.[Seg88][ST04]). missing still is theory h ups oetbihagoercitrrtto fthe of interpretation geometric a establish to purpose the epnec ewe ersnain ftemnindlatt of mentioned coefficients the the of representations par w between the is respondence via It forms work. modular c this to modern related of are part more representations not the respect, the is to this which comparison In algebras in factorization advantage understood. well an have is genu algebras lattices Witten even the to and ciated theory group index string uses the only which to one candidates usual cocycle the or perspe the from of new connection tower Whitehead theory This Its arithmetic representation the the to algebras. algebras relates lattice which operator theory vertex coding of from sheaves of terms uigteps eae hr aebe eea tepsi v in attempts several been have there decades past the During 2020 Date hssre ril switnb leri oooit who topologists algebraic by written is article survey This h ae otisanwrsl hoe . hc ie n t one a gives which 5.2 Theorem result new a contains paper The operator vertex of theory representation the Fortunately, OE,VRE PRTR N TOPOLOGICAL AND OPERATORS VERTEX CODES, ac 6 2021. 26, March : ahmtc ujc Classification. Subject Mathematics ycranltie.Tecntuto smtvtdb h ar the theory. disclos by representation tower motivated The in is codes groups. of orthogonal construction the The of tower Whitehead lattices. algeb operator certain vertex by of representations and forms modular Abstract. edsrb e ikbtenteter ftopological of theory the between link new a describe We tmf tmf OAGNE N EDLAURES GERD AND GANTER NORA ( u ulgoercdsrpino h cohomology the of description geometric full a but p ,tplgclmdlrfrswt Γ( with forms modular topological ), OUA FORMS MODULAR 1. Introduction 1 rmr 53;Scnay5R5 17B69, 55R35, Secondary 55N34; Primary a obtained ras tmf hgnlgroups. thogonal ffrnilvertex ifferential c lersand algebras ice sterole the es tmf etxoperator vertex ih present might t iinfunction. tition fvr specific very of l nw that known ell ithmetic aou fthe of nalogue lersasso- algebras rlconformal iral r uddby guided are iesfrom differs s hc h el- the which oconformal to p ogothers mong tv comes ctive ccce in -cocycles )-structure. s. n cor- one o rosdi- arious netof oncept 2 NORAGANTERANDGERDLAURES result of Atiyah-Bott-Shapiro which relates representations of Clifford alge- bras to the coefficients of K-theory. It turns out that codes already play an important role in K-theory. Phenomena like periodicity and triality of the spin groups can be best understood using coding theory. The article starts with a short introduction to coding theory and θ-series. The details can be found in [Ebe13]. The section does not contain new results but it formulates the van der Geer-Hirzebruch result for non linear codes. This will be important when it comes to partition functions of vertex operator algebra representations. The next section deals with the arithmetic Whitehead tower. It is shown how codes can be used to construct the in- teresting representation of the Clifford groups. Then its string extensions are studied in the complex case and the role of lattices L is demonstrated. A model of the loop space extension of the torus BL is given in the fourth section in terms of vertex operator algebras and its representations are spec- ified. The main Theorem 5.2 gives a one-to-one correspondence of the rep- resentation ring with the ring of Hilbert modular forms. The relation of the theorem to topological modular forms with level structures is described in the last section. It is a survey of the ideas to describe the cocycles by sheaves of vertex algebra modules. Corollary 4 is the promised analogue of the Atiyah-Bott-Shapiro theorem. Acknowledgments. The first author was supported by the DFG Priority Programme 1786 as well as ARC Discovery Grant DP160104912 during this project. The second author likes to thank the Max Planck Institute in Bonn for its hospitality. He also benefited from a workshop at the Perimeter Institute. The authors also like to thank G. H¨ohn, M. Kreck, A. Libgober, A. Linshaw, L. Meier, V. Ozornova, N. Scheithauer, B. Schuster for helpful conversations.
2. Codes and Lattices Let F be a field. A code of length n is a subset of F n. Let C and C′ be codes over F with the same length n. A (strict) morphism from C to C′ is a monomial transformation g such that g(C) ⊂ C′. This means that g is a linear endomorphism of F n which in terms of the standard basis takes the form g : ei 7→ cieσ−1(i) × for some ci ∈ F and some σ ∈ Σn. In particular, a strict automorphism of C is an element in × n (F ) ⋊ Σn. In addition, there is a coordinate-wise action of the automorphism group Γ = Aut(F ) on F n. An extended morphism from C to C′ is a pair (g, ψ) where g is a monomial transformation and ψ ∈ Γ with ψ(g(C)) ⊂ C′. A code is called linear if it is a linear subspace. CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 3
2.1. The (8,4)-Hamming Code. The (8,4)-Hamming code is a doubly even self-dual simply error correcting binary code, and it is the smallest code with these properties. As we shall see in Section 4.2 below, these features are responsible for a particularly nice structure of the real Clifford algebra Cl8, which is at the heart of real Bott periodicity, the A-genus and the triality symmetries of the group Spin(8). To recall the construction of H8, we introduce the Fano Plane b 2 F = P (F2), viewed as a finite geometry, with lines given by the 2-dimensional subspaces 3 of F2. Its power set P(F) endowed with the symmetric difference operation is a seven dimensional F2-vector space. Inside it, we have the three dimensional linear subspace C of line complements. Let 1 = F ∈ P(F). For any vector S ∈ P(F), the set-theoretic complement is F \ S = 1 + S. The (7,4)-Hamming code is the subspace
H7 = C ⊕ h1i of P(F). More precisely, any choice of numbering of the elements of the 7 ∼ Fano plane defines an identification F2 = P(F) with different choices giving strictly isomorphic codes. We give two such numberings in Figure 1. The parity check isomorphism 7 =∼ 8 ev F2 −−→ (F2) v 7−→ (|v|, v) transports H7 into a doubly even code H8 of length eight. This is the (8,4)-Hamming Code. The Fano plane is the smallest interesting example of a finite projective geometry, and as such it is dually isomorphic to itself, meaning that there is a bijection between points and lines preserving the incidence relation. We can realize this duality in a manner that decomposes 7 ev (F2) as the sum of two (strictly isomorphic) copies of C. To do so, we number the points of F in two different ways, as depicted in Figure 1. The dual isomorphism is then obtained by numbering the lines on either side as in Figure 2. Writing ci for the complement of Line i in the right-hand picture and bi for the complement of Line i in the left-hand picture, we have
ci + bi = {i} + 1. As a result, we obtain the decompositions 7 ev F2 = B⊕C and 8 ev ∼ 7 F2 = F2 = B ⊕ h1i⊕C, 4 NORAGANTERANDGERDLAURES
7 7
1 2 1 2 5 4
3 4 6 6 3 5
Figure 1. Two numberings of the Fano plane
1234567
Line 1
Line 2
Line 3
Line 4
Line 5
Line 6
Line 7
Figure 2. Incidence table relating points and lines in Figure 1. The dual isomorphism is reflected in the diagonal symme- try. where B and C are the subspaces of line complements consisting of the bi (respectively ci) and the zero vector. A further feature of point-line duality is that lines on the right govern addition in B and vice versa: three distinct numbers i, j and k form a line in the right-hand picture, if and only if
bi + bj + bk = 0. CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 5
Similarly, ci + cj + ck = 0 in C if and only if i, j and k form a line on the left. This will become important in our explicit calculations of spinor representations.
2.2. The Standard Ternary Code. A code is called ternary if F = F3. A famous ternary code is
T = {(s, a, a + s, a + 2s)| a, s, ∈ F3}. 4 It is a linear code of dimension 2 in F3 which is self dual. Its automorphism group is GL2(F3). This group coincides with the automorphism group of the Lubin-Tate curve C : y2 + y = x3 over F4 = {0, 1, ω, ω}. This curve plays an important role in the construction of the spectrum tmf. It will be reconsidered in Section 6. 2.3. The ternary Golay Code. The ternary Golay code is a self dual 12 linear code C in F3 with 729 words. Its automorphism group is the Mathieu group 2.M12. This code seems to be related to the triality of spin groups as considered in 4.3 but the authors have not yet been able to make this connection precise.
3. Lattices over integers of number fields Let p be an odd prime. Let F be the number field Q(ζ) for some primitive pth root of unity. It is a field extension over Q of degree p − 1. For r = r 1,...,p − 1 there are embeddings σr of F in C given by σr(ζ) = ζ . The trace of an element α ∈ F is defined by p−1 Tr(α)= σr(α). r=1 X We write O for the ring of integers in F . It is a free abelian group of rank p − 1 generated by ζ0,...,ζp−2. Let P be the principal ideal of O generated by 1 − ζ. It is the kernel of the map ρ : O → Z/pZ given by p−2 (1) (a0 + a1ζ + · · · + ap−2ζ ) 7→ (a0 + a1 + · · · + ap−2). The pairing xy¯ hx,yi = Tr p defines a bilinear form on O with values in the rationals. It makes P into an even lattice, that is, hx,yi∈ Z and hx,xi∈ 2Z. It is isomorphic to the root lattice Ap−1. 6 NORAGANTERANDGERDLAURES
n n 3.1. Lattices from Codes. More generally, let ρ : O → Fp be the reduc- tion map modulo the principal ideal P in each coordinate. For a code C set −1 n ΓC = ρ (C) ⊂ O . Assume that C is a self-orthogonal linear code. Then the symmetric bilinear form n x y¯ (2) hx,yi = Tr i i . p Xi=1 n−2m turns ΓC to an even lattice of rank n(p − 1) with discriminant p . It is unimodular if C is self dual. Moreover, the dual lattice Γ∨ = Hom(Γ, Z)= {x ∈ F n| hx,yi∈ Z} satisfies ∨ ∼ (3) ΓC = ΓC⊥ (see f.i. [Ebe13, Proposition 5.2 and Lemme 5.5] for details).
Example 3.1. The standard ternary code gives the root lattice E8. The ternary Golay code produces an even modular lattice of rank 24 which con- tains 12A2. 3.2. Theta series. Lattices come with theta series, and these are key to elliptic genera. The usual theta series of the lattice ΓC is xx¯ xx¯ πizTr p 2πizTrk p ϑC(z)= e = e . xX∈ΓC xX∈ΓC where k = Q(ζ + ζ−1) is the real subfield of F . There is an refinement of ϑC which is a holomorphic function on r = (p − 1)/2-copies of the upper half-plane H: set r xx¯ 2πiTrk z xx¯ σl(xx¯) θ (z)= e p with Tr z = z . C k p l p xX∈ΓC Xl=1 Set θj = θP+j for j ∈ Fp. These functions are symmetric Hilbert modular forms of weight m = 1 with respect to the congruence group a b 1 0 Γ(p)= {A = ∈ Sl (O)| A = mod p} c d 2 0 1 where p = P ∩ k. In detail, this means they are holomorphic functions f which transform as r m f(σ1(A)z1,...,σr(A)zr)= f(z) (σl(c)zl + σl(d)) . Yl=1 The symmetry means that they are invariant under the action of the Galois group Gal(k, Q) which permutes the coordinates of Hr. It turns out that every function θC is a sum of monomials in the θjs and hence is a symmetric CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 7
Hilbert modular form. In order to formulate this result more precisely, recall that the weight enumerator of a code C is the polynomial
l0(w) l1(w) lr(w) WC (x0,x1,...xr)= x0 x1 · · · xr wX∈C where l0(w) is the number of zeroes in w and lj(w) is the number of ±j. The following result is the Theorem of Alpbach by van der Geer-Hirzebruch. It is formulated there for linear self orthogonal codes in [Ebe13][Theorem 5.3]. We repeat the proof below because the assumption is unnecessary. Theorem 3.2. Let C be arbitrary (not necessary linear). Then
θC = WC (θ0,θ1,...,θr) In particular, each code of length n gives a symmetric Γ(p)-Hilbert modular form of weight n. Proof. For c ∈ C calculate x x¯ x x¯ 2πiTr z i i 2πiTr z 1 1 2πiTr z xnx¯n e k p = e k p . . . e k p −1 −1 −1 x∈Xρ (c) x1∈Xρ (c1) xn∈Xρ (cn) l0(c) l1(c) lr(c) = θ0 θ1 · · · θr (z) since xx¯ xx¯ 2πiTrkz p 2πiTrkz p θj(z)= e = e . P P x∈X+j x∈X−j The claim follows after summing over all code words.
Corollary 1. The usual theta series ϑC of C is a modular form for Γ(p) of weight nr. Self dual codes give Sl2(Z)-invariant modular forms. Proof. The first statement follows from the theorem after taking the diagonal map from H to Hr. The second statement uses (3) and the well known fact that θ-series of self dual lattices are invariant under the full modular group. It can also be obtained from the theorem by analyzing the action of Sl2(Fp) (see [Ebe13, Section 5] for p = 3, 5). Remark 1. Doi-Naganuma have constructed a map from the group of mod- ular forms for the congruence group Γ0(p) to Hilbert modular forms for Γ(p) which preserves the weight and Hecke eigenforms. It would be interesting to analyze those symmetric Hilbert modular forms which are in the image of the DN lift in terms of code words.
4. Arithmetic Whitehead towers The organizing principle for the different cohomology theories considered in this paper comes from the Whitehead towers of the orthogonal and the unitary groups. These are the towers Z/2 PU(H) O(n) ←− SO(n) ←−−− Spin(n) ←−−−−− String(n) ←− . . . 8 NORAGANTERANDGERDLAURES and PU(H) U(n) ←− SU(n) ←−−−−− String(n) ←− . . . defined for large n. The step to the right is obtained by killing off the lowest homotopy group. For instance, SO(n) is a connected component of O(n) and Spin(n) is its simply connected cover, which turns out to be 2-connected. For n ≥ 3 the smallest non-trivial homotopy group of Spin(n) is Z =∼ hSpin(3)i for n 6= 4 π3(Spin(n)) =∼ (Z ⊕ Z for n = 4, [MT91, Chapter VI, Theorem 4.17]. The next step in the tower, String(n), is classified by a generator of the integral cohomology H4(BSpin(n)). To be specific, we recall the low degree cohomology groups: (i) Let n ≥ 3. Then Hk(BSpin(n)) vanishes for k ≤ 3 and is cyclic of infinite order for k = 4 except for n = 4 where it is two copies of the integers. (ii) For n ≥ 4, one generator is p1/2 where p1 is the first Pontryagin class of the canonical bundle associated to the adjoint representation. (iii) For n = 3, the Euler class of the spinor representation generates. It coincides with p1/4. An explicit model realizing String(n) as topological group can be found in [ST04, Section 5]. This method can be continued, but for the purposes of the paper at hand, we are only interested in the first steps of the tower. For n ≥ 4, String(n) fibers over Spin(n) with fibre PU(H) = K(Z, 2), the projective unitary group of an infinite dimensional separable Hilbert space. There is also a realization of Stringn(R) as compact 2-group with fibre BU(1), whose geometric realization gives the topological group, see for instance [SP11]. 4.1. The Clifford groups and codes. We revisit the (8, 4)-Hamming Code in the context of Whitehead tower. If we make the identifications 8 ∼ 8 ∼ F2 = O(1) = O(8) ∩ Diag then the Fano line complements can be realized as the the three-fold tensor products of the diagonal Pauli matrices 1 0 1 0 σ = and σ = . 0 0 1 3 0 −1 More precisely, in the second picture in Figure 1, the [a : b : c] ∈ F has number 4a + 2b + c. The resulting isomorphism ∼ 7 ∼ 8 ev P(F) = F2 = (F2) CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 9 identifies the linear map 3 C : F2 −→ P(F) p 7−→ F \ p⊥ with the map 3 8 F2 −→ O(1) a b c (a, b, c) 7−→ σ3 ⊗ σ3 ⊗ σ3. Here p⊥ denotes the orthogonal complement, so that the image of C is the subspace of line complements. The vector 1 corresponds to − id, so the the full Hamming code H8 is identified with the subgroup a b c H = ±σ3 ⊗ σ3 ⊗ σ3 | a, b, c ∈ {0, 1} .
The various properties makingn H8 suitable for coding theoryo are, to a topolo- gists, statements about the Whitehead tower. The pre-image of O(1)n inside P in(n) is the extraspecial 2-group Fn of Clifford words. Writing −1 ∈ Fn for the central element, this group consists of words in the symbols e0,...en−1 subject to the relations 2 ei = −1 and eiej = −ejei.
The code H8 is even, meaning H⊂ SO(8).
Moreover, H8 is self dual, so its spin-cover H⊂ Spin(8) is abelian. The fact that H8 is doubly even makes H elementary abelian, e H =∼ h±1i×H, e meaning that H lifts to a subgroup of Spin(8). Indeed, H is a maximal e ev abelian subgroup of the group F8 of even Clifford words. The fact that H8 is simply error correcting means that the Clifford symbols e0,...e7 form a ev system of coset representatives for F8 /H. The orthogonal decomposition ev F8 = B⊕H8 e ev discussed in Section 2.1 allows us to write F8 as semi-direct product of two elementary abelian groups, ev ∼ F8 = B ⋉ H. Here B acts on H by e b: h 7−→ (−1)hb,hi h, where h ∈ H is ae lift of h ∈H. e e e e 10 NORAGANTERANDGERDLAURES
4.2. Spinors. The twisted group algebra of Fn is the Clifford algebra Cln of Rn with its standard quadratic form
Cln = R[Fn]/(−1)R ∼−1. In particular, Clifford modules coincide with centre-faithful representations ρ of Fn. Because of the inclusion ev ∼ tw ev Spin(n) ⊂ Cln = R [Fn ] ev [ABS64], it follows that representations of Fn give rise to representations of the spin groups. Let now n = 8. The spinor representations ∆+ and − ∆ are the two irreducible modules over the Clifford algebra Cl8. Each has dimension 8. Following Pressley and Segal [PS86, Section 12.6?], the spinor representations are constructed by inducing up centre-faithful representa- tions of a maximal abelian subgroup. This it the point where it becomes essential that our maximal abelian subgroup is, in fact, elementary abelian, i.e., that H8 is a Type II code. Let + χ : h±1i×H8 −→ O(1) −1 7−→ −1 h 7−→ 1 and − χ : h±1i×H8 −→ O(1) −1 7−→ −1
h 7−→ h0, then the eight dimensional spinor representations ∆± are given as F ev ∆± = ind 8 χ±. H e ev Due to the semi-direct product decomposition F8 = B ⋉ H, elements of B act as permutation matrices of order two, which turn out to be three-fold tensor products of the 2 × 2 permutation matrices e 1 0 0 1 σ = and σ = . 0 0 1 1 1 0 CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 11
More precisely, B acts by addition on itself, which is governed by C and given by the doubly even permutations
b1 7−→ (01) (23) (45) (67) = σ0 ⊗ σ0 ⊗ σ1,
b2 7−→ (02) (13) (46) (57) = σ0 ⊗ σ1 ⊗ σ0,
b3 7−→ (03) (12) (47) (56) = σ0 ⊗ σ1 ⊗ σ1,
b4 7−→ (04) (15) (26) (37) = σ1 ⊗ σ0 ⊗ σ0,
b5 7−→ (05) (14) (27) (36) = σ1 ⊗ σ0 ⊗ σ1,
b6 7−→ (06) (17) (24) (35) = σ1 ⊗ σ1 ⊗ σ0,
b7 7−→ (07) (16) (25) (34) = σ1 ⊗ σ1 ⊗ σ1. The subgroup H is fixed by ∆+ and acted upon by χ− ⊗ id for ∆−. 4.3. Triality. Together with the fundamental representation π : Spin(8) −→ SO(8), the spinor representations ∆± form the eight dimensional irreducible repre- sentations of Spin(8). All three are double covers, their respective kernels generated by the three non-trivial centre elements: ker(∆+) = hωi, ker(∆−) = h−ωi, ker(π) = h−1i, where ω = e0 · · · e7 is the second generator of the centre of Spin(8). The spin covers of the spinor representations ∆± are the extraordinary outer automorphisms of Spin(8) depicted in Figure 3.
(ω, −ω) σ
Spin(8) Spin(8)
∆+ ∆− SO(8) (−1,ω) τ + τ − (−1, ω, −ω) π Spin(8)
Figure 3. Triality symmetry of the Dynkin diagram D4. The extraordinary automorphisms τ ± are the lifts of the spinor representations, while σ is conjugation with e0. The action of the symmetric group S3 on the centre is indicated in red. 12 NORAGANTERANDGERDLAURES
4.4. Bott periodicity. The same objects that define triality are also re- sponsible for real Bott periodicity. The Bott element is given by the full spinor representation F8 + + + ∆ = ind χ =∆ ⊕ e0∆ , H combining ∆+ and ∆− in the sensee that
ev ∼ + − resF8 ∆ = ∆ ⊕ ∆ . If we allow ourselves to work over the complex numbers, the doubly even condition can be dropped, and an even self-dual code will suffice, since unlike O(1), the group U(1) can accommodate elements of order 4. This is the reason why complex Bott periodicity already occurs at n = 2, where we have the first even self-dual code. The Pauli matrix point of view has the advantage that it gives an explicit description of the Bott element, which is more direct than the approach in the standard literature [ABS64] [LM89]. One checks directly that the matrices −1 −1 1 1 −1 1 1 −1 E1 = E2 = 1 −1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 E3 = E4 = −1 1 −1 1 1 −1 1 −1 −1 −1 1 −1 −1 −1 1 −1 E5 = E6 = −1 1 1 1 −1 1 1 1 CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 13
−1 1 1 −1 E7 = 1 −1 −1 1 satisfy the Clifford relations. Hence ± ∆ (e0ei)= ±Ei defines two modules over the even Clifford algebra, which are readily checked to have the desired effect on centers. Bott periodicity is the statement that the representation ∆ : Cl8 −→ Mat16×16 is an isomorphism. This is now an easy consequence of the fact that the real Pauli matrices σ0, σ1, σ1σ0, and σ3. form a basis of the 2 × 2-matrices, taking into account that ∆ sends the 256 positive Clifford words to the 256 possible 4-fold tensor products of the real Pauli matrices. Indeed, the 64 three-fold tensor products are the possible products of the matrices E1,...,E7 above, which are easily read off from the Pauli matrix realizations of H and B. Combined with ω 7→ σ3 ⊗ id ⊗ id ⊗ id and e1 7→ σ1 ⊗ id ⊗ id ⊗ id, these determine all of ∆. 4.5. The integral spin and string groups. There is a version of the Whitehead tower for finite groups, where the successive killing of lower de- gree group cohomology has the meaning of forming first the commutator subgroup then the universal central extension (Schur extension), then the categorical Schur extension. This process requires the commutator subgroup of the original group to be perfect, so we will limit ourselves to considering only such groups. For details we refer the reader to [EG17]. As before, a tower of topological groups can be obtained by taking classifying spaces of the fibers, so that the categorification interpretation is really a matter of taste. We saw above how codes fit into the Whitehead tower for O(n)= On(R). Since codes, however, are finite objects, a more natural home is the arith- metic Whitehead tower for On(Z), the group of orthogonal matrices with integral entries. These group are known as the hyper-octahedral groups. They are generated by permutation matrices and diagonal matrices with ±1 entries on the diagonal. It is not hard to verify the isomorphism ∼ n (4) On(Z) = Σn ⋉F2 . ± ev The spinor representations ∆ discussed above map the group F8 to SO8(Z), or rather its spin double cover, which we will denote Spin8(Z). 14 NORAGANTERANDGERDLAURES
In the stable range, the arithmetic Whitehead tower of the symmetric groups is governed by the homotopy groups of spheres [EG17], while that of GLn(Z) is governed by the K-theory of the integers, see Section 4.6 below. The inclusions Σn ⊂ On(Z) ⊂ GLn(Z). suggest to think of the Whitehead tower of the hyper-octahedral groups as interpolating between these two. Theorem 1. For large enough n, the arithmetic Whitehead tower of the hyper-octahedral groups starts with the terms 2 ev (Z/2) ev On(Z) ←− An ⋉ (Fn ) ←−−−−− An ⋉ Fn ←− . . . Proof. For n ≥ 2, the abelisation of O (Z) is a Klein 4-group. This is n e discussed, for instance, in [Ste92]. The index 2 subgroups of the hyper- octahedral group consist of (i) the group SO(Z) of elements with even determinant, n (ii) the group An ⋉F2 of signed even permutations and n (iii) the group Dn =Σn ⋉ (F2 )ev of evenly signed permutations. The latter is a Weyl group of type D. The intersection of these three groups is the group n ev H = An ⋉ (F2 ) of evenly signed even permutations. This is the commutator subgroup of On(Z). We shall see below that it is perfect and hence possesses a universal n ev central extension. Write E for the elementary 2-group (F2 ) . Recall that the universal coefficient theorem identifies the Pontryagin dual of the integral homology with cohomology with circle coefficients i Hi(G) =∼ H (G; U(1)), and we will switch between these two as convenient for calculating the Schur multiplier of H. Consider theb Lyndon-Hochschild-Serre spectral sequence for the U(1)-cohomology of the extension
0 → E −→ H → An −→ 0. We have 0,1 ∗ An E2 = (E ) = 0 for n ≥ 3, while 0,2 2 ∗ 2 ∗ An E2 ((Λ E)An ) = (Λ E ) has two elements. The non-trivial element is the bilinear form
β = xi ∧ xj, 1≤Xi
0 H2(H) HH 0
e 2 ev n ev 0 {±1} An ⋉ Fn An ⋉ (F2 ) 0
e The arrow between the central subgroups must be an isomorphism, since the bottom extension cannot be reduced to an extension by {±1}. It follows that the middle arrow is an isomorphism, as well. This completes the proof of the theorem. ev As the Schur extension of the perfect group An ⋉E, the group An ⋉Fn is super perfect and hence possesses a categorical Schur extension, giving the next step in the arithmetic Whitehead tower of the hyper-octahedrale groups (for n ≥ 8). In the stable range, we have maps 0 ev π3(S ) −→ An ⋉ Fn −→ π3(KZ). It is therefore natural to conjecture that the categorical Schur multiplier of ev e An ⋉ Fn also stabilizes for large enough n and that its order is a multiple of 24. e 4.6. Arithmetic Whitehead tower for GLn(Z). Mason and Stothers [MS74, Theorem 5.1] have shown that for n ≥ 3 the subgroup En(Z) of SLn(Z) generated by the elementary matrices is perfect and coincides with the commutator subgroup of GLn(Z). We have
SOn(Z) ⊂ En(Z). The elementary matrices satisfy the following relations
(5) eij(a)eij (b) = eij(a + b)
(6) eij(a)ekl(b) = ekl(b)eij (a), j 6= k and i 6= l −1 −1 (7) eij(a)ejk(b)eij (a) ejk(b) = eik(ab); i, j, k distinct −1 −1 (8) eij(a)eki(b)eij (a)eki(b) = ekl(−ba); i, j, k distinct
The universal central extension of En(Z) is known as Steinberg group Stn(Z). It is defined by symbols xij(a) with i 6= j, 1 ≤ i, j, ≤ n, a ∈ Z subject to the relations (5)(6)(7)(8). The kernel of the map
Stn(Z) −→ En(Z); xij(a) 7→ eij(a) is the algebraic K homology group K2(Z). In particular, the first two ho- mology groups of Stn(Z) vanish and the Schur multiplier is
H2(En(Z)) =∼ K2(Z) =∼ Z/2. CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 17
Using standard techniques from algebraic K-theory [Ros94, 5.2.7] it is not hard to see that H3(Stn(Z)) =∼ K3(Z) =∼ Z/48. So, we have the tower (Z/2)2 BZ/48 GLn(Z) ←− En(Z) ←−−−−− Stn(Z) ←−−−−−St(Z). ←− . . .
The group K3(Z) plays a key role in the computation of the fourth coho- mology of the Conway group 4 H (Co1; Z) =∼ Z/24Z by Johnson-Freyd and Treumann [JFT20]. 4.7. Integers in cyclotomic fields and string extensions. We have seen that good codes lead to interesting representations of the integral spin groups. We like to see similar phenomena for the integral string groups. For this purpose, we consider the complex Whitehead tower and restrict it to integers in cyclotomic fields. Let Un(C) be the group of unitary matrices, that is, complex isometries n of C . Since the fourth cohomology of BSUn(C) is generated by the second Chern class we obtain a PU(H)-extension SU n(C) in the same way as in the real case. Let O be the ring of integers in the number field K = Q(ζ) considered in Section 3. The integral unitaryg group
Un(O)= Un(C) ∩ Gln(O) coincides with the semi direct product of diagonal with permutation matrices i n Un(O)= {±ζ | i = 0,...,p − 1} ⋊ Σn.
A code of length n can be considered as subset of Un(O) by identifying Fp with the multiplicative group of roots ζi. Consider the pullback diagram
n (Fp)det=1 SU n(O) SU n(C)
PUf(H) g PU(H) g PU(H)
n (Fp)det=1 SUn(O) SUn(C). There are several ways to deal with representations of PU(H)-extensions of topological groups G. One way is to take based loops Ω and observe that ΩPU(H) =∼ S1. Hence, ΩG is equivalent to an extension of ΩG by a compact Lie group. Then one looks for representations which mimic the structure of the loop space of a representatione of G. This approach does not work in the arithmetic setting since the base spaces are discrete. In our case, however, we maye employ the short exact ρ sequence P → O → Fp of Section 3 to obtain the fibre sequence n n n Fp −→ BP −→ BO . 18 NORAGANTERANDGERDLAURES
Explicitly, the topological group BP is the (p − 1)-dimensional torus
TP = (O ⊗ R)/(P ⊗ Z) ∼ and Fp sits in TP as O ⊗ {1}/P ⊗ {1}. The inclusion of Fp in U1(C) = R/Z factorizes through TP by the map p−2 (9) (a0 + a1ζ + · · · + ap−2ζ ) 7→ (a0 + a1 + · · · + ap−2)/p. with ai ∈ R. When choosing the basis (10) 1 − ζ,ζ − ζ2,...,ζp−2 − ζp−1 of P then the induced map p−1 ρ (R/Z) =∼ R/Z ⊗ P = TP −→ R/Z is the projection to the last factor. n Lemma 2. The extension of (Fp)det=1 defined by the quadratic form of the n n lattice P coincides with (Fp)det=1 which was defined by the second Chern class. f Proof. The quadratic form of the lattice P with respect to the basis (10) has the form p−1 2 q(x)= xi + xixi+1. i=0 X 2 4 As a 4-dimensional cohomology class it restricts to a generator z in H BFp by what we said before. The computation 2 2 zi = σ1 − 2σ2 Xi in the elementary symmetric polynomials σi implies that the quadratic form n of orthogonal sum P corresponds to c2 = σ2 up to the unit -2 mod p once c1 vanishes. n In the pull back diagram above, one may hence replace the group Fp with n n the torus TP and look for loop representation of the extension ΩTP with the center S1 acting non trivially. This will be done in the next section in the n infinitesimal setting. The role of codes in Fp will then be disclosed.g 5. Lattice vertex operator algebras and their representations The mathematical notion of a vertex algebra goes back to the work of Borcherds in 1986 [Bor86] who axiomatized the relations amongst lattice vertex operators. The definition looks quite technical but it will become explicit when we look at the example ΩTL where L is a lattice.
g CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 19
5.1. Vertex operator algebras and modules. A vertex algebra consists of the following data:
(i) a Z+-graded vector space V (ii) a vacuum vector 1 ∈ V0, 1 6= 0 (iii) a linear map D : V → V of degree one, D1 = 0 −1 −n−1 (iv) a linear map Y (·, z) : V → End(V )[[z, z ]], Y (v, z)= vnz The endomorphisms v are called modes of v. These data should satisfy the n P following properties for all v, w ∈ V (see [Fre02][MT10] for details) :
(i) there is an integer N = N(w) with vn(w) = 0 for all n > N (ii) Y (v, z) ∈ V [[z, z−1]][z−1] and there is a k ≥ 0 with k (z1 − z2) [Y (v1, z1),Y (v2, z2)] = 0. This property is called locality and one writes Y (v, z) ∼ Y (w, z)
if a power of (z1 − z2) annihilates the bracket. (iii) Y (v, z)1 = v + O(z) (iv) [D,Y (v, z)] = ∂Y (u, z) where ∂ is the formal derivative with respect to z Note that the operator D is completely determined by the state-field corre- spondence Y : property (iv) implies Dv = v−21. A vertex operator algebra is a vertex algebra which is equipped with a distinguished state ω ∈ V . Its modes Ln = ωn+1 of ω should satisfy (i) Vn = {v ∈ V | L0v = nv} (ii) Y (L−1v, z)= ∂Y (v, z) m3−m (iii) [Lm,Ln] = (m − n)Lm+n + 12 δm,−nc idV for some number c. In addition, one requires dim Vn < ∞, Vn = 0 for n << 0. The state ω also carries the name conformal vector. (It is responsible for the action of the Virasoro algebra, that is, the reparametrization action of loops.) A module over a vertex operator algebra is a vector space M together with a linear map −1 M −n−1 YM : V → End(M)[[z, z ]]; v → YM (v, z)= vn z n X with the properties:
(i) Y (1, z)= idM (ii) YM (v, z) ∼ YM (w, z) (iii) for k >> 0 associativity holds, that is, k k (z1 + z2) YM (v, z1 + z2)YM (w, z2) = (z1 + z2) YM (YM (v, z1)w, z2).
Usually one requires another grading M = λ∈C Mλ such that dim Mλ < ∞, Mλ+n = 0 for n << 0 and L0m = λm for m ∈ Mλ. It turns out that irreducible modules over tame vertex operatorL algebras have this property. Here, a V -module is irreducible if it contains no proper, nonzero submodule M (invariant under all modes vn ). 20 NORAGANTERANDGERDLAURES
Note that if M is irreducible and if Mλ+n 6= 0 for some n ∈ Z then
M = Mλ+n Z Mn∈ because the right module is invariant under all modes. Set h = λ + n with the smallest n such that Mλ+n 6= 0. This number turns out to be rational and it is called the conformal weight of M. The partition function of a representation M is the function on the upper half plane given by the formula M L0 −c/24 h−c/24 n Z(z)= trM q = q dim(Mh+n)q nX≥0 where q = e2πiz. 5.2. Lattice algebras. We now specify to the vertex algebra associated to a d-dimensional lattice L with a non degenerate symmetric bilinear form q. 1 We will construct an infinitesimal model for ΩTL where TL = L ⊗ S . d Set h = C and think of h as the abelian complexified Lie algebra of TL. Define the Heisenberg algebra g ˆh = h ⊗ C[t,t−1] ⊕ CK with central element K and brackets m n [v ⊗ t , w ⊗ t ]= m hv, wi δm,−nK.
The first summand in ˆh corresponds to loops in TL and the second one gives 1 a central extension by S . This already is a Lie algebra model for ΩTL. However, the loop space of a group and its extension has more structure than just being a group. For example, one can reparameterize loops. Thisg action should be incorporated into the model. The Heisenberg algebra comes with a triangular decomposition given by ± n 0 ˆh = ⊕±n>0h ⊗ t ; ˆh = h ⊕ CK. + 0 Let M be a h-module X. Let l be a scalar. Extend the action of h to ˆh ⊕ˆh + by letting ˆh annihilate M and by letting K act by multiplication with the level l. Let Vh(l, M) be the induced ˆh-module. Explicitly, − − (11) V (l, M)= U(ˆh) ⊗ 0 + M =∼ U(ˆh ) ⊗ M =∼ S(ˆh ) ⊗ M h U(ˆh ⊕ˆh ) where U denotes the universal enveloping algebra and S is the symmetric algebra. Set −n−1 Y (v, z)= v(n)z n n X and v(n) is the action of v ⊗ t on Vh(l, M). This state-field correspondence Y makes d def M0 = Vh(1, C1) CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 21 into a vertex algebra. The conformal vector is d 1 ω = (v ) (vi) 1 2 i (−1) (−1) Xi=1 d i and central charge is d. Here, {vi} is a basis of C and {v } is the dual basis d with respect to Q. Moreover, Vh(1, M) is a module over M0 (see [MT10] for details). Now assume that L is an even lattice in h = Rd with respect to a positive definite real quadratic form qh and let h and q be the complexifications. The irreducible representations Mα of h are one-dimensional and indexed by elements of the dual space α ∈ h∗. Set − ∨ ∼ ˆ VL = Vh(1, Mα ) = S(h ) ⊗ C[L] αM∈L ∨ where α denotes the dual of α. The construction makes VL into a module d over the Heisenberg vertex algebra M0 . With some effort, the vector space VL can itself be given the structure of α a vertex operator algebra: for e ∈ Mα∨ set α(−n) α(−n) Y (eα, z) = exp( zn) exp( zn)eαzα. n n n>0 n<0 X X d α β q(α,β) β Here, αn are the modes of Y (α, z) in M0 , z : v ⊗ e 7→ z v ⊗ e for v ∈ hˆ− and eα : v ⊗ eβ 7→ ǫ(α, β)v ⊗ eα+β for a certain bilinear 2-cocycle ǫ : L ⊗ L → {±1}. This imposes the structure of a vertex algebra on VL. See the textbook [Kac98] or [FLM88] for details.
Proposition 1. (see [MT10, Equation (75)]) The partition function of VL is −d hα,αi/2 −d ZVL (z)= η(q) q = η(q) ϑL(q) αX∈L where η is the Dedekind η-function η(q)= q1/24 (1 − qn). nY≥1 The irreducible modules of VL were computed by Dong. They are indexed ∨ ∨ by elements of the set Cf = L /L where L denotes the dual lattice L∨ = {β ∈ Rd| Q(α, β) ∈ Z for all α ∈ L}.
For λ ∈ Cf the module takes the form ∼ ˆ− VL+λ = Vh(1, M(α+λ)∨ ) = S(h ) ⊗ C[L + λ] αM∈L and its partition function is −d hα+λ,α+λi/2 ZVL+λ (z)= η(q) q . αX∈L 22 NORAGANTERANDGERDLAURES
Theorem 5.1 ([Don93]). The VL-modules {VL+λ} for λ ∈ C are all distinct and they provide a complete list for the isomorphism classes of irreducible VL-modules. 5.3. The Eisenstein vertex algebra and its representations. Let O be the ring of integers of the cyclotomic field Q[ζ] as considered before. Let n ⊥ C be a linear code in Fp with C ⊂ C . In section 3 we defined the even n lattice ΓC ⊂ O with bilinear form n x y¯ hx,yi = Tr i i . p i=1 X n Its dual is the lattice ΓC⊥ . In particular, for L =Γ0 = P this means (Pn)∨ = On.
The associated vertex operator algebra Vn = VPn will be called Eisenstein vertex algebra in the sequel. Theorem 5.1 tells us that the isomorphism classes of irreducible representations of Vn are indexed by words w of the n n ∼ n full code Cf = O /P = Fp . Let Rep(Vn) be the free group generated by these isomorphism classes. Lattice algebras associated to positive-definite even lattices are rational, that is, any module is a direct sum of irreducible modules (see [DM04]). These sums are necessary finite because of the restrictions on the dimensions. We conclude that Rep(Vn) coincides with the Grothendiek group of the category of Vd-modules. Set Rep(V )= Rep(Vn) nX≥0 with Rep(V0)= Z. It has a ring structure since
′ ∼ ′ Rep(Vn+n ) = Rep(Vn) ⊗ Rep(Vn ). × n The automorphism group Aut(Cf ) = (Fp ) ⋊ Σn acts on Rep(Vn) in the obvious way. It contains the subgroup n AutR(Cf )= {±1} ⋊ Σn of real automorphisms. Before formulating the main result we need to introduce some notation. For a ring R let RepR(V ) be the product Rep(V ) ⊗ R. It is the free R- module generated by the irreducibles. Let mf(p) be the ring of modular 1 1/p forms for the congruence group Γ(p) with q-expansion at ∞ in O[ p ][[q ]]. Σ 1 Similarly, write hmf (p) for the O[ p ]-algebra of symmetric Hilbert modular forms for Γ(p) generated by θj for j = 0, . . . , , r. CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 23
Example 1. For p = 3, the ring mf(3)C is freely generated by θ0,θ1 (see f.i.[Ebe13, Theorem 5.4]). They have q-expansions at ∞ x2−xy+y2 2 4 7 θ0 = q = 1 + 6(q + q + q + 2q . . .) 2 (x,yX)∈Z 1 x2−xy+y2+x−y 1 2 4 θ1 = q 3 q = 3q 3 (1 + q + 2q + 2q + . . .). 2 (x,yX)∈Z The first two coefficients of θ0 and θ1 imply that mf(3) itself is freely gen- erated by θ0,θ1and hence Σ 1 (12) hmf (3)= mf(3) = O[ 3 ][θ0,θ1]. For p = 5, Hirzebruch has proved in [Hir77] that the ring of complex sym- metric Hilbert modular forms is a polynomial ring in θ0,θ1,θ2. This implies Σ ∼ 1 (13) hmf (5) = O[ 5 ][θ0,θ1,θ2]. Theorem 5.2. (i) The partition function induces a well defined ring map
Z : Rep 1 (V )/AutR(Cf ) −→ mf(p) O[ p ] n(p−1) [M] 7→ η ZM (ii) The map Z is the composite of a map of graded rings Σ Z˜ : Rep 1 (V )/AutR(Cf ) −→ hmf (p) O[ p ] with the map induced by the diagonal. (iii) The map Z˜ is an isomorphism for p = 3 and p = 5. n Proof. Irreducible representations of Vn are indexed by words w ∈ Fp and −n(p−1) have the partition function η ϑ{w} by Proposition 1. This expression coincides with the diagonal of
l0(c) l1(c) lr(c) θ{w} = θ0 θ1 · · · θr (z) by Theorem 3.2 and hence is invariant under the action of AutR(Cf ). Since Σ each θj lies in hmf (p) we obtain a well defined map Z˜ and the first two assertions are shown. For the last claim, observe that the target is a polynomial ring in the θjs by Equations (12)(13) and the map sends a representation indexed by symmetrized word to the corresponding monomial. n Corollary 3. For a code C ⊂ Fp define the Vn-module MC = w∈C VPn+w. Suppose C is self dual. P (i) The partition function Z of MC is a SL2(Z)-invariant modular form. (ii) The refined partition function Z˜ is a SL2(O)-invariant Hilbert mod- ular form for p = 5. 24 NORAGANTERANDGERDLAURES
Proof. The first statement follows from Theorem 5.2 and Corollary 1. The second statement for p = 5 follows from Theorem 5.2 and a result of Gleason- Pierce and Sloane (see [MMS72],[Ebe13, Corollary 5.4,5.5]). Remark 2. For p ≥ 5 the refined partition function Z˜ should have a more intrinsic description in terms of traces of Eisenstein vertex operators. Also, it would be worth knowing the representations which allow for the Doi- Naganuma lift (c. Remark 1).
Remark 3. There is an action of SL2(Z) on the ring of integral Γ(3)- modular forms: if θ is a modular form of weight n and level 3 then az + b θ (cz + d)−n cz + d is again a modular form of weight n and level 3. Since the subgroup Γ(3) acts trivially it is an action of
G1 = SL2(F3) =∼ SL2(Z)/Γ(3).
Explicitly, the generators S and T act on θ0 by −1 z(−1 − 2ζ) θ = (θ (z) + 2θ (z)) 0 z 3 0 1 θ0(z +1) = θ0(z) and on θ1 by −1 z(−1 − 2ζ) θ = (θ (z) − θ (z)) 1 z 3 0 1 θ1(z +1) = ζθ1(z). The fix point ring
G1 (14) mf(3) = mf 1 O[ 3 ] coincides with the ring of integral modular forms for the full group SL2(Z). 6. Topological modular forms with level structure Atiyah-Bott-Shapiro have discovered the role of Clifford algebra represen- tations in K-theory [ABS64]. We have seen in Section 4 that these corre- spond to representations of the finite group Fn in the arithmetic Whitehead tower and are best understood using coding theory. In Section 5 we in- vestigated the representations of the string extension of the corresponding group in the complex arithmetic Whitehead tower. Theorem 5.2 gave a correspondence of its representation ring with modular forms for the con- gruence subgroup Γ(N). In this section we like to interpret this result as a higher version of the Atihay-Bott-Shapiro construction. The higher ver- sion of K-theory will be T MF (N), the cohomology theory of topological modular forms with full level N structure. CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 25
6.1. The spectrum T MF (N). Elliptic cohomology has its origin in the study of elliptic genera. F. Hirzebruch [Hir88] generalized the earlier notion of S. Ochanine from level 2 to higher levels and constructed ring maps
χ∗ : MU∗ −→ mf(N)∗ where MU∗ is the bordism ring of stably almost complex manifolds. These elliptic genera provided rigidity results for the equivariant index of the Dirac operator on manifolds with circle actions. E. Witten related them to index theory on loop spaces of manifolds with string structures. The Witten genus is a ring map W : MString∗ −→ mf∗ where MString denotes the cobordism theory of manifolds with string struc- tures. In the late 80s Landweber-Ravenel-Stong [LRS95] introduced a cohomol- ogy theory, nowadays denoted by T MF1(2), for the congruence group Γ1(2). It was generalized to other levels by A. Baker [Bak94], J.L. Brylinski [Bry90] (with 2 inverted) and by the second author with the help of a result by J. Franke [Fra92]. The method given in [Lau99] shows that the Hirzbruch genera define periodic homology and cohomology theories given by −1 (15) T MF (N)∗(X) = MU∗(X) ⊗χ∗ mf(N)∗[∆ ]. The corresponding ring spectra come with ring maps χ : MU → T MF (N) but these complex orientations are not compatible amongst each other along the maps T MF (N) → T MF (NM). A family version of the Witten genus W : MString −→ tmf was achieved by M. Hopkins et al.[Hop02][DFHH14]. Its target is the spec- trum of topological modular forms tmf (or T MF for the periodic version). Its coefficient ring maps to the ring mf∗ of integral modular forms. This map is neither surjective nor injective but rationally it is an isomorphism. There are basically two different constructions of the spectrum tmf as a highly structured ring spectrum. One construction is based on its local- izations with respect to Morava K-theories (see [DFHH14, Chapter 11]). It uses the fact that its K(2)-localization is the Lubin-Tate theory E2 coming from deformations of the elliptic curve mentioned in Section 2.2. Its K(1)- localization is related to the theory of p-adic modular forms (see [Lau04]). The other construction uses derived algebraic geometry, that is, algebraic geometry with ordinary rings replaced by highly structured ring spectra. The reader is referenced to the survey article [Lur09]. For a construction of the connective theories tmf(N) along these lines see [HL16]. There is an equivalence of homotopy fixed point spectra T MF [1/p] → T MF (p)hGl2(Fp) 26 NORAGANTERANDGERDLAURES which should be regarded as a homotopy version of Equation (14). The spectral sequences s H (Gl2(Fp),π2tT MF (p)) =⇒ π2t−sT MF [1/p] where computed by Mahowald-Rezk in [MR09] for p = 3 and by Behrens- Omsky in [BO16] for p = 5. For p = 3, the proposed generalization of the Atiyah-Bott-Shapiro theo- rem is the following: Corollary 4. (i) The partition function Z gives an isomorphism =∼ 0 2n Rep 1 (V )n/AutR(Cf ) −→ tmf(3) (S ). O[ 3 ] (ii) If C is a self dual ternary code of length n then there is an element 0 2n 0 2n of tmf (S ) whose image in tmf(3) (S ) is Z(MC ). Proof. The first claim only is a reformulation of Theorem 5.2 for the case p = 3. The second claim follows from Equation (3) and [Hop02, Proposition 5.12]. H. Tamanoi argued in [Tam99] that the equivariant index of the formal Dirac operator on lopp spaces is a virtual module over some vertex operator algebra. Of course, this is a consequence of Corollary 4 for p = 3 since the index coincides with the elliptic genus and hence takes values in the ring of Γ(3)-modular forms. This suggests that there is a geometric construction of the cohomology theory tmf(3) along these lines. It turns out that bundles whose fibers are representations of the Eisen- stein vertex algebra lead to a cohomology theory KV by the usual construc- tion: one takes the Grothendieck group of isomorphism classes and inverts the representation which is attached to the discriminant ∆ to get a peri- odic theory. The corollary then says that KV has the same coefficients as T MF (3). However, using results from Dong-Liu-Ma-Zhou [DLMZ04] it is not hard to see that one only gets a form of K-theory −1 KV (X)= K(X) ⊗ mf(3)∗[∆ ] since the associated genus is equivalent to the Todd genus. Bundles of representations hence do not lead to the correct elliptic cocycles. On the other hand, Equation (15) implies that a cohomology theory with the same coefficients and the correct Euler classes has to coincide with T MF (3). 6.2. Sheaves of vertex operator algebras and Euler classes. Bundles correspond to quasi-coherent sheaves. Gorbounov-Malikov-Schechtman- Vaintrob [GMS00] constructed a more general sheaf MSV (X) of vertex operator algebras for each complex manifold X. It comes with a multipli- cation map O(X) ⊗ MSV (X) −→ MSV (X) CODES, VERTEX OPERATORS AND TOPOLOGICAL MODULAR FORMS 27 by functions on X which, however, is not associative. There is a filtration of MSV (X) compatible with the multiplication such that the associated graded is the coherent sheaf (cf.[BL00]) ∞ ∞ ∞ ∞ ∗ ∗ ∗ ellX = Λy−1 T X⊗ Λyqn T X⊗ Sqn T X⊗ Λy−1qn T X⊗ Sqn T X. n=1 n=1 n=1 n=1 O O O O Take y = −ζ and consider ellX as an element of 1/N KT ate(N)(X)= K(X)O[1/N]((q )).
Proposition 2. Up to a normalization factor, ellX coincides with the Euler class of T X in KT ate(N) once KT ate(N) is equipped with the orientation coming from the level N-elliptic character [Lau99] [Lau00]
λ : T MF (N) −→ KT ate(N). Proof. This can be easily verified with a Chern character computation (see [HBJ92, p.117f]). Proposition 2 suggests that MSV (X) should rather be considered as an 2n element of T MF (X), the elliptic Euler class, which maps under λ to ellX . Up to a normalization factor, the elliptic character hence is the map which takes the associated graded object. The normalization factor also played a role in Corollary 4 and the objects considered there have the same spirit. In fact, the construction of MSV (X) given in [LL07] resembles Equation (15) where the module M is replaced by a quasi-coherent sheaf. Lian-Linshaw even provided equivariant versions of MSV (X) in the same article for X which are equipped with an action of a compact Lie group. Needless to say, such geometric interpretations in terms vertex operator algebras would enhance the theory of topological modular forms and conformal field theories tremendously.
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School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria, 3010, Australia E-mail address: [email protected]
Fakultat¨ fur¨ Mathematik, Ruhr-Universitat¨ Bochum, NA1/66, D-44780 Bochum, Germany E-mail address: [email protected]