Hecke-Langlands Duality and Witten's Gravitational Moonshine

We show that there is a dual description of conformal blocks of $d=2$ rational CFT in terms of Hecke eigenfields and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related $d=2$ spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten's idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of $c=24$ Monster vertex algebra of Frenkel-Lepowsky-Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (cf. references).

1 Introduction. Number-theoretical dualities, S-dualities and gravitational cosmology In this paper we discuss an amazing interplay between number-theoretical dualites and black holes in Anti-de Sitter spaces. It passes through modular invariance of conformal field theories and AdS/CFT correspondences.
Recall that electric E and magnetic field B in the vacuum (in regions without charges and currents) are related by Maxwell equations, invariant by transformations: where c is the speed of light. This is the basic form of the electromagnetic duality. We can already interpret it as the modular S-invariance corresponding to the matrix S = 0 −1 1/c 2 0 , exchanging fields and reversing coupling constants. It extends to S-dualities in CFT and string theories.
In recent years [GW07,KW01], it was realized that the Langlands correspondence is the number-theoretical counterpart of the electromagnetic duality in gauge field theories. Representations of the absolute Galois group correspond to modular objects on the moduli space of curves. Moreover, these modular objects are eigensheaves of Hecke operators with meaningful eigenvalues. In particular, the coefficients of the L-function L(E, s) = of weight 2. This is a consequence of the famous modularity theorem related to the Fermat's Last Theorem.
In this paper we define vertex operator algebras for all Galois number fields K/Q, equipped with integral trace forms on their rings of integers OK (cf. sect. 5). Thus, the modular S-invariance of lattice CFTs is closely related to the arithmetic Langlands program.
In addition, Witten [Wit07] has associated, via AdS/CFT correspondence, the pure d = 3 gravity to the Monster Moonshine Module V ♮ . It transports the Hecke-Langlands duality for the extremal c = 24 CFT to modular dualities for BTZ black holes on AdS3 (cf. sect. 11).

Virasoro algebra and rational CFT
For general mathematical definitions, related to vertex algebras and conformal field theories (CFT), see [FBZ01,Kac98].
Recall that a vertex algebra V = ⊕Vn is conformal of central charge c ∈ C if it contains a conformal vector ω ∈ V2 such that Fourier coefficients Ln = L V n of the corresponding vertex operator satisfy the defining relations of the Virasoro algebra: In addition, L V −1 = T : V → V should be a translation operator of degree 1, L V 0 |V n = n · Id the grading operator and L2ω = c 2 |0 . The Virasoro algebra is the central extension of C((t))∂t with topological basis given by Ln = −t n+1 ∂t, n ∈ Z, and c ∈ C.
A conformal vertex algebra V is called rational if it is completely reducible with finitely many inequivalent simple modules having finitedimensional graded components.

Lattice vertex superalgebras
Let Λ be a lattice in R n equipped with a non-degenerate Z-valued symmetric bilinear form ·, · . There is a well-known quantization of the space of maps from the circle S 1 to the torus R n /Λ called lattice vertex superalgebra.
For any α ∈ L, mutually local fields Γα(z) = Y (1 ⊗ e α ) are of the form they generate a vertex operator algebra with the space of state VL and the vacuum vector |0 = 1 ⊗ 1 [ Kac98,prop. 5.4].
Putting cα s ⊗ e β = c α,β s ⊗ e β for s ∈ Sym h <0 and β ∈ Λ, we get the following equations for numbers c α,β ∈ C: It defines a cohomology class in H 2 (Λ, C × ), asserting the existence of a unique (up to isomorphism) vertex superalgebra structure on Moreover, it is a rational conformal superalgebra with central charge c = rank(Λ).
Let Λ be an integral lattice of rank n. We define the theta series of Λ as The lattice theta-series satisfies the functional equation where Λ * is the dual lattice and disc(Λ) = #(Λ * /Λ) is the discriminant of Λ.
Consider θΛ(τ ) = ΘΛ(e 2πiτ ) as a function on the upper half-plane H+. If Λ = Λ * is self-dual (integral and unimodular) then we get functional equations: Recall that the full modular group PSL2(Z) is generated by matrices S = 0 −1 1 0 and T = 1 1 0 1 . Then, for any γ ∈ Γ+ = S, T 2 , we have So, θΛ(τ ) becomes a modular form of weight n/2 for Γ+ with multiplier system ε c,d ∈ C * satisfying ε 8 satisfies functional equations Thus, η(τ ) is a modular form of weight 1/2 and level 1 with a multiplier system of order 24. The general case of lattices with arbitrary discriminants, related to modular forms for congruence subgroups, needs additional investigations. However, when Λ is an even lattice, the modularity follows from Zhu's results on characters of rational vertex algebras [Zhu96] (see sect. 6).

Vertex algebras for number fields
be a Galois number field over Q of degree n = r1 + 2r2 where r1 is the number of real embeddings and 2r2 the number of complex embeddings of K in C. In this case of Galois extension over Q, K is either totally real (ξ ∈ R) with r2 = 0 or totally imaginary (ξ ∈ C\R) with r1 = 0.
The important thing here is to use the nondegenerate symmetric bilinear form tr : K × K → Q, x, y → tr K/Q (xy), (5.1) called the trace form. When K is totally real, it is positive-definite. In the totally imaginary case, it is negative-definite and we should take rather the anti-trace form x, y → −tr K/Q (xy). Let OK be the ring of integers in K/Q corresponding to a lattice Λ, constructed from real and complex embeddings of K (see [Koc92, ch. 1, § 1]). Recall that OK is the unique maximal order in K. In our numbertheoretical setting, there is a finite number of orders preserving Λ, equal to the class number h(K) of K. The class number measures the deviation of OK from being a principal ideal domain. Our bilinear form ·, · restricts to an integral form on OK . Now, choosing a basis {ω 1 , . . . , ω n } of OK, we can extend ·, · to a positivedefinite integral form on Λ.
So, we can construct a vertex superalgebra VK , associated to any Galois number field K/Q. It opens the whole area of investigations in order to express the class field theory, aritmetical reciprocity laws etc. in terms of modularity theorems for lattice vertex superalgebras. As we have seen in section 3, the vertex superalgebra structure on VΛ is determined by a cocycle c α,β ∈ H 2 (Λ, C × ). Kac  Let C2(Vc) be the subspace generated by elements of the form A (−2) ·B for all A, B ∈ Vc. Then Vc is said to satisfy Zhu's finiteness condition if dim Vc/C2(Vc) < ∞ and any vector can be written as Ln 1 · · · Ln k A, ni < 0, where LnA = 0 for all n > 0.  ((a1, z1), . . . , (an, zn), τ ) (6.2) as meromorphic continuations (q = e 2πiτ ) of limits  , z1), . . . , (an, zn), τ ) (6.5) where Sα(i, j) are constants depending only on α, i, j. In particular, if Vc has a unique simple module M , and a is a highest weight vector of weight w for the Virasoro algebra, then SM (a, τ ) is a modular form of weight w with a certain multiplier system.
Here we see the appearance of the modular S-matrix S(α, i, j) for any α ∈ SL2(Z). The functional equation for correlation functions corresponds to α = 0 −1 1 0 .

Minimal models and S-matrix of the critical Ising model
It is known that Virc is reducible as the module over the Virasoro algebra if and only if c = c(p, r) = 1 − 6(p − r) 2 /pr, p, r > 1, (p, r) = 1 (7.1) In this case, the irreducible quotient L c(p,r) of Vir c(p,r) is a rational vertex algebra, called minimal model of Belavin-Polyakov-Zamolodchikov [BPZ84,dFMS97,Wan93].
The characters are given by the following formulas (q = e 2πiτ ): Here we used 3 remarquable theta functions (where q = e πiτ ): The functional equation for characters: is expressed in terms of the modular S-matrix: corresponding to α = 0 −1 1 0 in the Zhu's theorem.
Finally, notice that the partition function is modular invariant.

Hecke eigenforms and Wiles modularity theorem
The mth Hecke operator Tm acts on lattice functions by taking sums over all sublattices of index m. If f (z) = a n q n a weakly holomorphic (with possible poles at cusps) modular form of weight k then When f (τ ) is normalized, we have λm = a m .
Consider the field K f = Q[λ m , m 1], generated by Hecke eigenvalues, that will be called Hecke eigenfield, associated to f . It is an algebraic extension of Q.
Proposition 9.1 Let N = 1. The Hecke eigenfield K f is totally real or CM-field (when a 1 ∈ C\R).
Proof. It follows from the fact that Hecke operators are self-adjoint with respect to the Petersson inner product.
Shimura has attached to f (τ ) of weight 2 an abelian variety Sh f of dimension [K f : Q]. In the Wiles case of elliptic curve E/Q, K f = Q and Sh f = E (up to an isogeny). Hecke eigenvalues correspond to the monodromy of the Galois action on f (τ ) (as a horizontal section of an appropriate line bundle with a flat connection) around cusps. This is one of basic keypoints of the Beilinson-Drinfeld geometric Langlands duality.
The following Maeda's conjecture [RS17, sect. 3.6.1] is important in order to to study the Langlands duality in terms of Hecke eigenfields.
where a−m, 0 m k, come from the formula In this way we obtain and so on. This construction is consistent with the Bekenstein-Hawking entropy for black holes. However, it is not yet clear to what extremal CFT correspond partitions functions Z k (q) for k > 1.
It looks relatively simple at the level of partition functions but this is misleading. Zhu [Zhu96] shows that V ♮ can be decomposed as the direct sum of tensor products of highest weight modules of 48 Virasoro algebras with central charge c = 1/2. Actually, it also proves the rationality of M.

Perspectives
We have indicated just the beginning of the story. First of all, numbertheoretical questions can be treated as a particular case of the study of lattice vertex superalgebras for number fields. It demonstrates an amazing unification of the arithmetic with lattice CFT theories.
On the one hand, arithmetic generalizations would include Drinfeld associators, motivic Galois groups and Grothendieck-Teichmüller groupoids. In positive characteristic, an upcoming article [Pot01] will treat vertex talgebras, generalizing Drinfeld modules and Anderson's t-motives.
On the other hand, physical generalisations would include the S-duality between electric and magnetic branes of the unifying M -theory. As Witten have already noticed, Hecke operators are related to 't Hooft operators and Wilson loops.
The holographic preimage of the Monstrous Moonshine, giving a spacetime with BTZ black holes, is inspiring and requires additional investiations.