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Analogies between the Langlands correspondence and topological quantum field theory. (English) Zbl 0858.11062

Gindikin, Simon (ed.) et al., Functional analysis on the eve of the 21st century. Volume I. In honor of the eightieth birthday of I. M. Gelfand. Proceedings of the conference, held at Rutgers University, New Brunswick, NJ, October 24-27, 1993. Boston, MA: Birkhäuser. Prog. Math. 131, 119-151 (1995).
The starting point of the paper consists of two generalizations of traditional class field theory. On the one hand one has the generalization considered by Parshin, Kato, Bloch and Saito in terms of higher Milnor \(K_n\)-groups of suitably defined rings of adèles, and on the other hand, there is the Langlands program concerned with higher dimensional representations of Galois groups in terms of representations of groups of adelic matrices. To come to a unified generalization one is led to some kind of higher non-abelian cohomology. Recently, motivated by mathematical physics, topics on non-abelian cohomology and 2-categories (or higher \(n\)-categories) have been put forward, e.g. in extended topological quantum field theories.
Let \(F\) be a finite extension of \(\mathbb{Q}_p\) and let \({\mathcal M} = {\mathcal M}_F(F)\) be the category of motives over \(F\) with complex multiplication by \(F\). For \({\mathcal M}\) one defines a Langlands correspondence for \(F\) as an assignment of a complex vector space \({\mathcal L}(V)\) to each \(V\in {\mathcal M}\), satisfying a list of appropriate conditions. One also defines a Langlands correspondence for motives over a finite field \(\mathbb{F}_q\), but in this case one assigns a numerical datum \(L(V)\) to such a motive \(V\). To fix ideas, let \(X=\text{Spec} ({\mathcal O}_F)\), where \(F\) is a number field, be a one-dimensional scheme, and let \(V\) be a motive over \(F\). For almost all primes \({\mathfrak p} \in {\mathcal O}_F\), \(V\) has good reduction and restricts to a motive \(V |_{\mathfrak p}\) over a finite field \(\mathbb{F}_{\mathfrak p}\). Then to the whole motive \(V\) the Langlands correspondence associates a vector space datum \({\mathcal L} (V)\), and to the restriction \(V |_{\mathfrak p}\) the Langlands correspondence associates a numerical datum, the Euler factor \(L_{\mathfrak p} (V,s)\). These constructions bear a striking resemblance to topological quantum field theory (TQFT) as advocated by M. Atiyah. The resemblance goes even further. The use of Shimura varieties as moduli spaces for motives (of a particular kind) and taking the spaces of ‘multiplicities’ can be compared to the calculation of the Feynman action integral in TQFT. Also, inverse Hecke theory, i.e. the reconstruction of an automorphic representation from its \(L\)-function, looks very much like the reconstruction of the TQFT out of its Green functions.
To go further and find a way to compare some ‘Langlands correspondence’ with some extended (or higher order) TQFT, i.e. taking into account manifolds of arbitrary dimension and associating to them objects of higher \(n\)-categories, the notions of charade and Waldhausen space are introduced. For an exact functor \(f:{\mathcal A}\to{\mathcal B}\) between abelian categories \({\mathcal A}\) and \({\mathcal B}\) (with a suitable notion of admissibility for short exact sequences in \({\mathcal A})\), a \(k\)-linear charade \((k\) a field) over \(f\) associates a \(k\)-vector space to any object of \({\mathcal A}\), again satisfying a list of nice properties. For \(k=\mathbb{C}\) (or \(\overline \mathbb{Q}_\ell)\) and \(f:{\mathcal M} \to \text{Vect}_F\) the realization functor, one regains the Langlands correspondence mentioned above. One can give several examples of known charades (Steinberg modules, determinantal vector spaces, operads). Also, to \(f:{\mathcal A} \to {\mathcal B}\) as before, one may associate a simplicial category \(S_\bullet (f)\) leading to a CW-complex, the Waldhausen space \(S(f)\). On such spaces one can construct combinatorial cellular stacks. One has a bijection between \(k\)-linear charades over \(f\) and cellular stacks over \(S(f)\) that associate to the unique point the category \(\text{Vect}_k\), and to 1-cells functors \(\text{Vect}_k \to\text{Vect}_k\) of the form \(V\mapsto X \otimes V\) (i.e. functors respecting the structure of \((\text{Vect}_k, \oplus, \otimes)\)-module category on \(\text{Vect}_k)\). Thus, the Langlands correspondence gives rise to a certain stack on the Waldhausen space associated to the category of motives with respect to the realization functor (and the class of admissible exact sequences).
Parshin introduced the notion of \(n\)-dimensional local field for any \(n\geq 0\). In particular, a zero-dimensional local field is just a finite field and a one-dimensional local field is just an ordinary local field. Thus one covers the cases discussed so far. An \(n\)-dimensional local field is defined as a complete discrete valued field such that its residue field is an \((n-1)\)-dimensional field. Thus, \(n\)-dimensional local fields arise as completions of function fields of rational functions on schemes of absolute dimension \(n\). The notion of higher dimensional local field may be applied to higher dimensional Langlands correspondences. Here dimension 2 is considered in some detail, thus leading to 2-categories (the most important being \(2\text{-Vect}_k\), the 2-category of 2-vector spaces) and combinatorial 2-stacks on Waldhausen spaces. In particular, to a motive over a 2-dimensional scheme there is associated a 2-vector space (with group action), …, etc. The 2-dimensional Langlands correspondence contains (conjecturally) a wealth of data and extends Parshin’s and Kato’s class field theory for two-dimensional local fields. For a group \(G\) one defines a 2-representation as an action \(\rho\) of \(G\) on a category which is a 2-vector space such that the \(\rho(g)\), \(g \in G\), are Vect-module functors. One obtains a relationship, going back to A. Grothendieck and J. Giraud, between suitable equivalence classes of ‘one-dimensional’ 2-representations of \(G\) and non-abelian \(H^2(G,k^*)\).
For the entire collection see [Zbl 0830.00037].

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
11S31 Class field theory; \(p\)-adic formal groups
19D45 Higher symbols, Milnor \(K\)-theory
14F45 Topological properties in algebraic geometry
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