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On Arithmetic Dijkgraaf-Witten Theory On arithmetic Dijkgraaf-Witten theory Hikaru HIRANO, Junhyeong KIM and Masanori MORISHITA Dedicated to the memory of Professor Toshie Takata Abstract: We present basic constructions and properties in arithmetic Chern-Simons theory with finite gauge group along the line of topological quantum field theory. For a finite set S of finite primes of a number field k, we construct arithmetic ana- logues of the Chern-Simons 1-cocycle, the prequantization bundle for a surface and the Chern-Simons functional for a 3-manifold. We then construct arithmetic ana- logues for k and S of the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten partition function in (2+1)-dimensional Chern-Simons TQFT. We show some basic and functorial properties of those arithmetic analogues. Finally we show decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions. Contents. Introduction 1. Preliminaries on torsors and group cochains 2. Classical theory 2.1. Arithmetic prequantization bundles and arithmetic Chern-Simons 1-cocycles 2.2. Arithmetic Chern-Simons functionals arXiv:2106.02308v2 [math.NT] 7 Jun 2021 3. Quantum theory 3.1. Arithmetic quantum spaces 3.2. Arithmetic Dijkgraaf-Witten partition functions 4. Some basic and functorial properties 4.1 Change of the 3-cocycle c 4.2. Change of number fields 2010 Mathematics Subject Classification: 11R, 57M, 81T. Key words: arithmetic Chern-Simons theory, arithmetic topology, Dijkgraaf-Witten theory, topological quantum field theory 1 4.3. The case that S is empty 4.4. Disjoint union of finite set of primes and reversing the orientation of ∂VS 5. Decomposition and gluing formulas 5.1. Arithmetic Chern-Simons functionals and arithmetic Dijkgraaf-Witten partition functions for VS 5.2. Decomposition formulas for arithmetic Chern-Simons invariants and gluing formulas for arithmetic Dijkgraaf-Witten invariants References Introduction In [Ki] Minhyong Kim initiated to study arithmetic Chern-Simons theory for number rings, which is based on the ideas of Dijkgraaf-Witten theory for 3-manifolds ([DW]) and the analogies between 3-manifolds and number rings, knots and primes in arithmetic topology ([Mo2]). We note that Dijkgraaf- Witten theory may be seen as a 3-dimensional Chern-Simons gauge theory with finite gauge group (cf. [FQ], [G], [Wa], [Y] etc). Among other things, Kim constructed an arithmetic analog of the Chern-Simons functional, which is defined on a space of Galois representations over a totally imaginary number field. In the subsequent paper [CKKPY] Kim and his collaborators showed a decomposition formula for arithmetic Chern-Simons invariants and applied it to concrete computations for some examples. Later, Kim’s construction was extended over arbitrary number field which may have real primes ([H], [LP]). Computations of arithmetic Chern-Simons invariants have also been carried out for some examples, by employing number-theoretic considerations in [AC], [BCGKPT], [CKKPPY], [H] and [LP]. In [CKKPPY], the arithmetic Chern- Simons correlation functions for finite cyclic gauge groups were computed in terms of arithmetic linking numbers. It should be noted that Kim also consid- ered arithmetic Chern-Simons functionals for the case where the gauge groups are p-adic Lie groups ([Ki; 3]). By arithmetic Dijkgraaf-Witten theory in the title, we mean arithmetic Chern-Simons theory with finite gauge group in the sense of Kim. The purpose of this paper is to add some basic constructions and prop- erties to Kim’s theory and lay a foundation for arithmetic Dijkgraaf-Witten theory along the line of topological quantum field theory, TQFT for short, in the sense of Atiyah ([A1]). TQFT is a framework to produce topological 2 invariants for manifolds. For example, the Jones polynomials of knots can be obtained in the context of (2+1)-dimensional Chern-Simons TQFT with com- pact connected gauge group (cf. [A2], [Ko], [Wi]). For the TQFT structure of Dijkgraaf-Witten theory, we consult [DW], [FQ], [G], [Wa], [Y]. In this paper, following Gomi’s treatment [G] and Kim’s original ideas [Ki], we construct an arithmetic analogue of Dijkgraaf-Witten TQFT in a certain special situation, namely, we construct arithmetic analogues, for a finite set S of finite primes of a number field k, of the prequantization bundles, the Chern-Simons 1-cocycle, the Chern-Simons functional, the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten partition function. Arithmetic Dijkgraaf- Witten invariants are new arithmetic invariants for a number field, which may be seen as variants of (non-abelian) Gaussian sums. We fix a finite group G and a 3-cocycle c Z3(G, R/Z). For an ori- ∈ ented compact manifold X with a fixed triangulation, let X be the space of gauge fields associated to G and let be the gauge groupF Map(X,G) GX acting on X . Note that X and X are finite sets and that the quotient space F:= / is identifiedF withG Hom(π (X),G)/G if X is connected, MX FX GX 1 where Hom(π1(X),G)/G is the quotient of the set of homomorphisms from the fundamental group π1(X) of X to G by the conjugate action of G. As for the classical theory in the sense of physics, we construct, using the 3-cocycle c, the following correspondences (0.1) oriented closed surface Σ 1-cocycle λ Z1( , Map( , R/Z)), Σ ∈ GΣ FΣ oriented compact 3-manifold M 0-chain CS C0( , Map( , R/Z)), M ∈ GM FM which satisfy (0.2) dCSM = res∗λ∂M , where res : M (resp. M ) ∂M (resp. ∂M ) is the restriction map and d : C0( , Map(F , R/GZ) →C F1( , Map( G , R/Z)) is the coboundary ho- GM FM → GM FM momorphism of group cochains. The key ingredient to construct λΣ and CSM i i d is the transgression homomorphism C (G, R/Z) C − ( , Map( , R/Z)) → GX FX with d = dim X and, in fact, λΣ and CSM are given by the images of c for i = 3, X = Σ and M, respectively ([G]). Then we can construct a -equivariant GΣ principal R/Z-bundle Σ and the associated complex line bundle LΣ over Σ, using λ , and hence theL complex line bundle L over . In fact, isF the Σ Σ MX LΣ product bundle Σ R/Z on which Σ acts by (ρΣ, m).g =(ρΣ.g, m+λΣ(g, ρΣ)) for ρ , m F R×/Z and g G. We call λ the Chern-Simons 1-cocycle. Σ ∈ FΣ ∈ ∈ GΣ Σ The line bundle LΣ (or LΣ) is called the prequantization complex line bundle 3 for a surface Σ. The 0-chain CSM is called the Chern-Simons functional for a 3-manifold M. We see that CSM is a M -equivariant section of res∗ Σ over . G L FM As for the quantum theory, the formalism of (2 + 1)-dimensional TQFT is given by the following correspondences (functor from the cobordism category of surfaces to the category of complex vector spaces) oriented closed surface Σ quantum Hilbert space , (0.3) Σ oriented 3-manifold M partition function Z H , M ∈ H∂M which satisfy several axioms (cf. [A1]). Here we notice the following two axioms: (0.4) functoriality: An orientation preserving homeomorphism f : Σ ≈ Σ′ → induces an isomorphism ∼ ′ of Hilbert quantum spaces. Moreover, HΣ → HΣ if f extends to an orientation preserving homeomorphism M ≈ M ′, with → ∂M = Σ, ∂M ′ = Σ′, then ZM is sent to ZM ′ under the induced isomorphism ∼ ′ . H∂M → H∂M (0.5) multiplicativity and involutority: For disjoint surfaces Σ1, Σ2 and the surface Σ∗ = Σ with the opposite orientation, we require Σ1 Σ2 = Σ1 Σ2 , Σ∗ =( Σ)∗, H ⊔ H ⊗ H H H where ( )∗ is the dual space of . Moreover, if ∂M = Σ Σ , ∂M = HΣ HΣ 1 1 ⊔ 2 2 Σ∗ Σ and M is the 3-manifold obtained by gluing M and M along Σ , 2 ⊔ 3 1 2 2 then we require <ZM1 ,ZM2 > = ZM , ∗ where < , >: Σ1 Σ2 Σ Σ3 Σ1 Σ3 is the natural gluing pairing · · H ⊔ × H 2⊔ → H ⊔ of quantum Hilbert spaces. This multiplicative property is indicative of the “quantum” feature of the theory (cf. [A1]). The construction of the Hilbert space is phrased as the geometric quan- HΣ tization. We note that Σ is known to be isomorphic to the space of conformal blocks for the surface ΣH when the gauge group is a compact connected group (cf. [Ko]). Elements of are called (non-abelian) theta functions (cf. [BL]). HΣ For Dijkgraaf-Witten theory, Σ is constructed, in an analogous manner, as the space of -equivariant sectionsH of the prequantization line bundle L over GΣ Σ Σ, in other words, the space of sections of LΣ over Σ: (0F .6) M 2π√ 1λΣ(g)(ϑ) = ϑ : C ϑ(̺ .g)= e − ϑ(̺ ) g , ̺ HΣ { FΣ → | Σ Σ ∀ ∈ GΣ Σ ∈FΣ} = Γ( , L ). MΣ Σ 4 In quantum field theories, partition functions are given as path integrals. In Dijkgraaf-Witten theory, the Dijkgraaf-Witten partition function ZM ∂M is defined by the following finite sum fixing the boundary condition: ∈ H 1 2π√ 1CSM (̺) (0.7) Z (̺ )= e − (̺ ). M ∂M #G ∂M ∈F∂M ̺ M ∈F res(̺X)=̺∂M The value Z (̺ ) is called the Dijkgraaf-Witten invariant of ̺ . M ∂M ∂M ∈ F∂M We note that when [c] is trivial and S is empty, then Σ = and the Dijkgraaf-Witten invariant Z ( ), denoted by Z(M), coincidesF with{∗} the (av- M ∗ eraged) number of homomorphism from π1(M) to G: #Hom(π (M),G) (0.8) Z(M)= 1 , #G which is the classical invariant for the connected 3-manifold M.
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