You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We compute some arithmetic path integrals for $BF$-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to ${{\mathbb{G}}_{m}}$ and abelian varieties.'
address:
- 'M.C: Department of Mathematics, Hebrew University'
- 'M.K: Mathematical Institute, University of Oxford, and the Korea Institute for Advanced Study'
author:
- Magnus Carlson
- Minhyong Kim
title: 'A note on abelian arithmetic BF-theory'
---

Towards arithmetic BF theory
============================

$BF$-theory is a rare example of a topological field theory that can be defined in any dimension [@Cattaneo]. Let $M$ be an oriented $n$-manifold, $G$ a compact Lie group, and $P$ a principal $G$-bundle on $M$. Assume given also a finite-dimensional representation $\rho$ of the Lie algebra ${{\mathfrak{g}}}$ of $G$. The $BF$ functional depends on two fields, a connection $A$ on $P$, and an $(n-2)$-form $B\in \Om^{n-2}(M,adP)$ with values in the adjoint bundle of $P$. Then $$BF(B,A):=\int_MTr[\rho(B)\wedge \rho(F_A)],$$ where $F_A$ is the curvature form of $A$. (There seem to be some other possibilities for the invariant function on ${{\mathfrak{g}}}$ that goes into the integral.) A number of properties of $BF$-theories make them easier to deal with than Chern-Simons theories. Primary among them for our purposes is that the two variables allow us to extend the definition to non-orientable manifolds, simply by letting the field $B$ vary over $\Om^{n-2}(M,adP\otimes \omega_M)$, where $\omega_M$ is the orientation bundle of $M$.

In some earlier papers, a preliminary attempt to define and compute arithmetic analogues of Chern-Simons functions was made [@KimLinking; @KimCSI; @KimCSII; @AhlquistCarlson; @Bleher]. Also, moduli spaces of ‘arithmetic gauge fields’ have been applied to Diophantine geometry [@Balakrishnan; @KimGauge]. One of the obstructions to developing a full-fledged arithmetic topological field theory based on Chern-Simons theory is that natural arithmetic dualities involve a sheaf $\mu_n$ or $\hat{{{\mathbb{Z}}}}(1)$, which are not trivialisable in general. That is, arithmetic schemes are not orientable. There are various ways to circumvent this problem, all of which introduce some difficulties for working out interesting examples. The purpose of this paper is to suggest, by way of two brief computations, that BF-theory provides a simpler way to link number theory to topological field theory.

Let $X={{\operatorname{Spec}}}({{\cmcal{O}}}_F)$, where $F$ is a totally imaginary number field. Then

$$\sum_{(a,b)\in H^1(X, {{\mathbb{Z}}}/n)\times H^1(X, \mu_{n})} \exp(2\pi i BF(a,b))=|n\operatorname{Cl}_F[n^2]|   \cdot | \UX/(\UX)^{n}| \cdot |\operatorname{Cl}_F/n|.$$

Thus, for $n$ large enough, the ‘finite path integral’ will capture exactly $$| \UX/(\UX)^{n}| \cdot |\operatorname{Cl}_F|,$$a quantity of the form (regulator $\times$ class number). The precise notation and definitions that go into this proposition as well as the next one will be explained in subsequent sections.

Now let $A$ and $B$ be dual abelian varieties over $F$ with semi-stable reduction at all places. Let $n$ be an integer coprime to all the Tamagawa numbers of $A$ and $B$ as well as to the places of bad reduction for them. Assume that the Tate-Shafarevich groups $\Sha(A)$ and $\Sha(B)$ are finite and $\Sha(B)[n]=\Sha(B)[n^2]$. Then

$$\sum_{(a,b)\in H^1(X,{{\cmcal{A}}}[n])\times H^1(X,{{\cmcal{B}}}[n])} \exp(2\pi i BF(a,b))=|A(F)/n| \cdot |B(F)/n| \cdot |\Sha(A)[n]|$$

Here, ${{\cmcal{A}}}$ and ${{\cmcal{B}}}$ are the Neron models of $A$ and $B$ respectively. Note that the expression is symmetric in $A$ and $B$ because $|\Sha(A)[n]|=|\Sha(B)[n]|$ via the Cassel-Tate pairing.

We view these formulae as some (weak) evidence for the suggestion made in [@KimCSI] that an arithmetic topological functional on moduli spaces of Galois representations will have something to do with $L$-functions. In any case, it is rather striking to find expressions for the orders of class groups and Tate-Shafarevich groups as exponential sums, which perhaps have not appeared heretofore in the literature. Equally notable is that a path integral for the BF functional for three manifolds leads to the Alexander polynomial of a knot in the physics literature [@Cattaneo], which is well-known to be an analogue of the $p$-adic $L$-function [@Morishita].

In forthcoming work, we will develop $BF$-theory for arithmetic schemes with boundary, that is ${{\operatorname{Spec}}}({{\cmcal{O}}}_F[1/S])$ for a finite set $S$ of places, as well as a $p$-adic theory. Also interesting would be to develop arithmetic $BF$-theory for arithmetic schemes in higher dimension by way of the duality theory of [@Geisser]. However, in the present announcement, the primary goal is to illustrate with a minimum of clutter the relationship between a path integral in the sense of physicists and important arithmetic invariants.

Some finite path integrals for $\mathbb{G}_m$: Proof of Proposition 1.1.
========================================================================

As before, let $X={{\operatorname{Spec}}}({{\cmcal{O}}}_F)$ for $F$ a totally imaginary number field and let $\mu_{n}$ and ${{\mathbb{Z}}}/n$ be the usual finite flat group schemes viewed as sheaves in the flat topology. We have the Bockstein map $$\d: H^1(X, \mu_{n})\rTo H^2(X,\mu_{n})$$ coming from the exact sequence $$1\rTo\mu_{n}\rTo \mu_{n^2}\rTo \mu_{n}\rTo 1$$ and the invariant isomorphism [@MazurNotes] $$\int :H^3(X, \mu_{n})\rTo \nZ.$$ Define the BF-functional on $${{\cmcal{M}}}:=H^1(X, {{\mathbb{Z}}}/n)\times H^1(X, \mu_{n})$$ by $$BF(a,b)=\int(a \cup \d b).$$ (The class $\d b$ is the analogue of the curvature $F$.)

One can also define a BF-functional on ${{\cmcal{M}}}$ as $BF'(a,b) = \int(\d a \cup b),$ where $\d:H^1(X,{{\mathbb Z}}/n) \rightarrow H^2(X,{{\mathbb Z}}/n^2)$ is the Bockstein map coming from the exact sequence $$1 \rTo {{\mathbb Z}}/n \rTo {{\mathbb Z}}/n^2 \rTo {{\mathbb Z}}/n \rTo 1.$$ However, $BF' = BF,$ since we have an equality $\d a \cup b = a \cup \d b$ by Lemma 2.1 and the proof of Lemma 2.2 in [@KimLinking].

We will now calculate the path integral $$\sum_{(a,b)\in {{\cmcal{M}}}} \exp(2\pi i BF(a,b)).$$ First, let us calculate the groups $H^i(X,\mu_n).$ We define $\operatorname{Div}F$ to be the group of fractional ideals of $F$ and for $x \in F^*,$ we let $\operatorname{div}(x)$ be the associated principal ideal. We claim that $$H^i(X,\mu_n) = \begin{cases} \mu_n(F) & \text{for } i=0  \\ Z_1/B_1 & \text{for } i = 1 \\ \operatorname{Cl}_F/n & \text{for } i=2 \\ {{\mathbb Z}}/n & \text{for } i=3 \\ 0 & \text{for } i >3 \end{cases}$$ where $$Z_1 = \{ (x,I) \in F^* \oplus \operatorname{Div}F | \ nI = -\operatorname{div}(x)\}$$ and $B_1 = \{(x^n,-\operatorname{div}(x))\in F^* \oplus \operatorname{Div}F | \ x \in F^*\}.$ To see this, note that $\mu_n,$ is quasi-isomorphic to the complex $$\mathcal{C}_\bullet = (\mathbb{G}_{m,X} \xrightarrow{\cdot n} \mathbb{G}_{m,X})$$ with non-zero terms in degree $0$ and $1.$ Since $\mathbb{G}_{m,X}$ is representable by a smooth group scheme, flat cohomology groups with coefficients in $\mathbb{G}_{m,X}$ coincide with the corresponding étale cohomology groups [@MilneEtale III, Theorem 3.9]. Thus, to compute flat cohomology with coefficients in $\mu_n,$ it suffices to compute the hypercohomology of $\mathcal{C}_\bullet,$ viewed as a complex of sheaves on the étale site. To do this, we take the well-known resolution $j_* \mathbb{G}_{m,F} \xrightarrow{\operatorname{div}} \operatorname{Div}X$ of $\mathbb{G}_{m,X}.$ Here, $$j:{{\operatorname{Spec}}}F \rightarrow X$$ is the inclusion of the generic point and $$\operatorname{Div}X = \underset{x \in X^0}{\oplus} {{\mathbb Z}}_{/x},$$ where ${{\mathbb Z}}_{/x}$ is the skyscraper sheaf at the closed point $x.$ Resolving the complex $\mathcal{C}_\bullet$ levelwise with this resolution and taking the total complex of the resulting double complex, we get the complex $$j_* \mathbb{G}_{m,F} \xrightarrow{(\operatorname{div},n^{-1})} \operatorname{Div}X \oplus j_* \mathbb{G}_{m,F} \xrightarrow{n+\operatorname{div}} \operatorname{Div}X.$$ Now, as in Lemma 4.2 and Corollary 4.3 of [@CarlsonSchlank] one uses the local-to-global spectral sequence to conclude that the complex above computes the flat cohomology of $\mu_n$ for $i= 0,1,2;$ to conclude the calculation for $i \geq 3$ one applies flat duality as in III, Corollary 3.2 of [@MilneADT]. We now proceed to analyze the Bockstein map $$\d: H^1(X,\mu_n) \rTo H^2(X,\mu_n).$$ Using the complex given earlier, the Bockstein was computed in Lemma 4.1 of [@AhlquistCarlson], and it is the composite of two maps: the first is the surjective map which takes $(x,I) \in H^1(X,\mu_n) \cong Z_1/B_1$ to $I \in \operatorname{Cl}_F[n]$ and the second is the reduction map $\operatorname{Cl}_F[n] \rightarrow \operatorname{Cl}_F/n.$ By noting that the kernel of the first map is $\UX/(\UX)^n$ and that the kernel of the second map is $n \operatorname{Cl}_F[n^2],$ we see that our sum becomes $$|\UX/(\UX)^n| \cdot |n \operatorname{Cl}_F[n^2]| \sum_{(a,b) \in H^1(X,{{\mathbb Z}}/n{{\mathbb Z}}) \times \operatorname{Cl}_F[n]/n\operatorname{Cl}_F[n^2]} \exp(2 \pi i a \cup \bar{b})$$ where $\bar{b} \in \operatorname{Cl}_F/n$ is the reduction of $b.$ But for $b$ non-trivial, it is clear that this sum is zero, giving us $$\sum_{(a,b)\in {{\cmcal{M}}}} \exp(2\pi i BF(a,b))=|n\operatorname{Cl}_F[n^2]|   \cdot | \UX/(\UX)^{n}| \cdot |\operatorname{Cl}_F/n|.$$ In particular, if $\operatorname{Cl}_F[n] = \operatorname{Cl}_F,$ we see that the sum evaluates to $|\UX/(\UX)^{n}| \cdot |\operatorname{Cl}_F|.$ If one calculates the path integral $\sum_{(a,b) \in {{\cmcal{M}}}} \exp(2\pi i BF'(a,b))$ where $BF'$ is as in the above remark, one sees, since $BF'$ coincides with $BF,$ that $\sum_{(a,b) \in {{\cmcal{M}}}} \exp(2 \pi i BF(a,b))$ equals $|\operatorname{Cl}_F[n]| \cdot |\UX/(\UX)^n|$ times the number of isomorphism classes of unramified ${{\mathbb Z}}/n$-étale algebras which can be embedded into an ${{\mathbb Z}}/n^2$-étale algebra.

Some finite path integrals for abelian varieties: Proof of Proposition 1.2
==========================================================================

Let ${{\cmcal{A}}}$ be the Néron model of an abelian variety $A$ over $F$ and let ${{\cmcal{B}}}$ be the Néron model of the dual $B$. Assume that both ${{\cmcal{A}}}$ and ${{\cmcal{B}}}$ have semi-stable reduction and let $n$ be a sufficiently large positive integer. More precisely, suppose both $A$ and $B$ have good reduction at all places dividing $n$. Further assume that $n$ is coprime to $|\Phi_A |\cdot |\Phi_B|$, where $\Phi_A$ and $\Phi_B$ are the groups of connected components of ${{\cmcal{A}}}$ and ${{\cmcal{B}}}$. According to [@Ces Theorem 1.1(a)(ii)], we have isomorphisms $$H^1(X, {{\cmcal{A}}}[n])\cong Sel(F,A[n]),$$ $$H^1(X, {{\cmcal{B}}}[n])\cong Sel(F,B[n])$$ where the left hand side is flat cohomology (in the fppf site). We claim that our assumption that ${{\cmcal{B}}}$ has semi-stable reduction, together with our assumptions on $n$, implies that multiplication by $n$ on ${{\cmcal{B}}}$ is an epimorphism in the category of fppf sheaves. To see this, note that by [@Ces Lemma B.4.], ${{\cmcal{B}}}^0 \xrightarrow{\cdot n} {{\cmcal{B}}}^0$ is faithfully flat, thus an epimorphism of fppf sheaves. Further, we have a commutative diagram $$\begin{tikzcd} 
0 \arrow[r] & {{\cmcal{B}}}^0 \arrow[r] \arrow[d,"n"] & {{\cmcal{B}}}\arrow[r] \arrow[d,"n"] & \Phi_B \arrow[r] \arrow[d,"n"] & 0 \\
0 \arrow[r] & {{\cmcal{B}}}^0 \arrow[r] & {{\cmcal{B}}}\arrow[r] & \Phi_B \arrow[r] & 0  
\end{tikzcd}$$ and since $n$ is prime to $|\Phi_B|,$ multiplication by $n$ on $\Phi_B$ is actually an isomorphism. Thus, by the snake lemma, multiplication by $n$ on ${{\cmcal{B}}}$ is an epimorphism. By [@MilneADT Corollary 3.4] there is a perfect pairing $$\cup : H^1(X,{{\cmcal{A}}}[n]) \times H^2(X,{{\cmcal{B}}}[n]) \rightarrow H^3(X, \mathbb{G}_m) \cong \mathbb{Q}/\mathbb{Z}.$$ Using the exact sequence $$0\rTo {{\cmcal{B}}}[n]\rTo {{\cmcal{B}}}[n^2]\rTo^{n} {{\cmcal{B}}}[n]\rTo 0,$$ we get the Bockstein map $$\d: H^1(X,{{\cmcal{B}}}[n])\rTo H^2(X,{{\cmcal{B}}}[n]).$$ We now define the BF-functional $$BF: H^1(X,{{\cmcal{A}}}[n])\times H^1(X,{{\cmcal{B}}}[n])\rTo H^3(X,\mathbb{G}_m)[n] \cong H^3(X,\mu_{n}) \cong \dfrac{1}{n}{{\mathbb Z}}/{{\mathbb Z}}$$ $$BF(a,b) = \int (a\cup \d b),$$ where $\int$ is the isomorphism $H^3(X,\mu_{n}) \cong \dfrac{1}{n}{{\mathbb Z}}/{{\mathbb Z}},$ as in the previous section. Our goal is now to calculate the path integral $$\sum_{(a,b)\in H^1(X,{{\cmcal{A}}}[n])\times H^1(X,{{\cmcal{B}}}[n])} \exp(2\pi i BF(a,b)).$$

To this end, note that, by [@Ces Proposition 4.5 (c)-(d)] and our assumptions on $n,$ we have $$H^1(X,{{\cmcal{A}}})[n]\cong \Sha(A)[n], \ \ H^1(X,{{\cmcal{B}}})[n]\cong \Sha(B)[n].$$ Assuming the finiteness of the Tate-Shafarevich group, we can take $n$ large enough so that $$H^1(X,{{\cmcal{B}}})[n^2]=H^1(X,{{\cmcal{B}}})[n].$$

We now identify the kernel of $\d.$ By using the diagram $${\boldsymbol}0&\rTo & {{\cmcal{B}}}/n^2{{\cmcal{B}}}&\rTo& H^1(X,{{\cmcal{B}}}[n^2])&\rTo &H^1(X,{{\cmcal{B}}})[n^2]&\rTo 0\\
 && \dTo& & \dTo^{n}& &\dTo^{n}& \\
 0&\rTo & {{\cmcal{B}}}/n{{\cmcal{B}}}&\rTo& H^1(X,{{\cmcal{B}}}[n])&\rTo &H^1(X,{{\cmcal{B}}})[n]&\rTo  0\ed$$ and the fact that the rightmost map is zero, we see that $Ker(\d)=Im({{\cmcal{B}}}/n{{\cmcal{B}}}).$ Thus, we get an induced injection $$\bar{\d}:H^1(X,{{\cmcal{B}}})[n]\rInto H^2(X,{{\cmcal{B}}}[n]).$$ By an argument identical to the case of $\mu_{n}$ above, since $$|H^1(X,{{\cmcal{A}}}[n])| =  |A(F)/n| \cdot |\Sha(A)[n]|,$$ we get $$\sum_{(a,b)\in H^1(X,{{\cmcal{A}}}[n])\times H^1(X,{{\cmcal{B}}}[n])} \exp(2\pi i BF(a,b))=|B(F)/n| \sum_{(a,c)\in H^1(X,{{\cmcal{A}}}[n])\times Im(\d)}\exp(2\pi i a\cup c)$$

$$=|B(F)/n| \sum_{(a,c)\in H^1(X,{{\cmcal{A}}}[n])\times {\operatorname{Im}}(\d)}\exp(2\pi i a\cup c)$$

$$=|A(F)/n| \cdot |B(F)/n| \cdot |\Sha(A)[n]|.$$

Acknowledgements {#acknowledgements .unnumbered}
================

M.K. is grateful to Kevin Costello and Edward Witten for urging him to look at BF-theory, and to Nima Arkani-Hamed for an invitation to speak at the IAS high-energy theory seminar in the course of which the initial ideas for an arithmetic BF-theory came to mind. He is also grateful to Philip Candelas, Xenia de la Ossa, Tudor Dimofte, Rajesh Gopakumar, Sergei Gukov, Jeff Harvey, Theo Johnson-Freyd, Albrecht Klemm, Si Li, Tony Pantev, Ingmar Saberi, Johannes Walcher, Katrin Wendland, and Philsang Yoo for numerous illuminating conversations on quantum field theory. He was supported in part by the EPSRC programme grant EP/M024830/1, ‘Symmetries and Correspondences’.

M.C. would like to thank the Wallenberg foundation for their support. He is also grateful to Merton College for a visiting research scholarship in April of 2019, thanks to which it was possible to begin the research presented in this paper.

[10]{}

esearch Notices, Published electronically, November, 2017 Chung, H., Kim, D., Kim, M., Park, J., Yoo, H., Arithmetic Chern-Simons Theory II. (with ) To be published in p-adic Hodge Theory, Simons Symposia, Springer-Verlag.