Arithmetic Chern–Simons Theory II

Abstract

In this paper, we apply ideas of Dijkgraaf and Witten [6, 32] on 3 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern–Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications.

with “Appendix 2: Conjugation Action on Group Cochains: Categorical Approach” by Behrang Noohi.

Appendices

7 Appendix 1: Conjugation on Group Cochains

We compute cohomology of a topological group G with coefficients in a topological abelian group M with continuous G-action using the complex whose component of degree i is $C^i(G, M)$, the continuous maps from $G^i$ to M. The differential

$$d: C^i(G,M)\rightarrow C^{i+1}(G,M)$$

is given by

$$df(g_1, g_2, \ldots , g_{i+1})=g_1f(g_2, \ldots , g_{i+1})$$

$$ +\sum _{k=1}^i f(g_1, \ldots , g_{k-1}, g_k g_{k+1}, g_{k+2}, \ldots , g_{i+1})+ (-1)^{i+1} f(g_1, g_2, \ldots , g_i).$$

We denote by

$$B^i(G,M)\subset Z^i(G,M)\subset C^i(G,M)$$

the images and the kernels of the differentials, the coboundaries and the cocycles, respectively. The cohomology is then defined as

$$H^i(G,M):=Z^i(G,M)/B^i(G,M).$$

There is a natural right action of G on the cochains given by

$$a: c\mapsto c^a:=a^{-1}c\circ \mathrm {Ad}_a,$$

where $\mathrm {Ad}_a$ refers to the conjugation action of a on $G^i$.

Lemma 7.1

The G action on cochains commutes with d:

$$d(c^a)=(dc^a)$$

for all $a\in G$.

Proof

If $c\in C^i(G,M)$, then

$$d(c^a)(g_1, g_2, \ldots , g_{i+1})=g_1a^{-1}c(\mathrm {Ad}_a(g_2), \ldots , \mathrm {Ad}_a(g_{i+1}))$$

$$+ \sum _{k=1}^i a^{-1}c(\mathrm {Ad}_a(g_1), \ldots , \mathrm {Ad}_a( g_{k-1}), \mathrm {Ad}_a( g_k) \mathrm {Ad}_a( g_{k+1}), \mathrm {Ad}_a(g_{k+2}), \ldots , \mathrm {Ad}_a(g_{i+1}))$$

$$+ (-1)^{i+1} a^{-1}c(\mathrm {Ad}_a(g_1), \mathrm {Ad}_a(g_2), \ldots , \mathrm {Ad}_a(g_i))$$

$$= a^{-1}\mathrm {Ad}_a(g_1)c(\mathrm {Ad}_a(g_2), \ldots , \mathrm {Ad}_a(g_{i+1}))$$

$$+ \sum _{k=1}^i a^{-1}c(\mathrm {Ad}_a(g_1), \ldots , \mathrm {Ad}_a( g_{k-1}), \mathrm {Ad}_a( g_k) \mathrm {Ad}_a( g_{k+1}), \mathrm {Ad}_a(g_{k+2}), \ldots , \mathrm {Ad}_a(g_{i+1}))$$

$$+ (-1)^{i+1} a^{-1}c(\mathrm {Ad}_a(g_1), \mathrm {Ad}_a(g_2), \ldots , \mathrm {Ad}_a(g_i))$$

$$=a^{-1}(dc)(\mathrm {Ad}_a(g_1), \mathrm {Ad}_a(g_2), \ldots , \mathrm {Ad}_a(g_{i+1}))$$

$$=(dc)^a(g_1, g_2, \ldots , g_{i+1}).$$

We also use the notation $(g_1, g_2, \ldots , g_i)^a:=\mathrm {Ad}_a(g_1, g_2, \ldots , g_i)$. It is well-known that this action is trivial on cohomology. We wish to show the construction of explicit $h_a$ with the property that

$$c^a=c+dh_a$$

for cocycles of degree 1, 2, and 3. The first two are relatively straightforward, but degree 3 is somewhat delicate. In degree 1, first note that $c(e)=c(ee)=c(e)+ec(e)=c(e)+c(e)$, so that $c(e)=0$. Next, $0=c(e)=c(gg^{-1})=c(g)+gc(g^{-1})$, and hence, $c(g^{-1})=-g^{-1}c(g).$ Therefore,

$$c(aga^{-1})=c(a)+ac(ga^{-1})=c(a)+ac(g)+agc(a^{-1})=c(a)+ac(g)-aga^{-1}c(a).$$

From this, we get

$$c^a(g)=c(g)+a^{-1}c(a)-ga^{-1}c(a).$$

That is,

$$c^a=c+dh_a$$

for the zero cochain $h_a(g)=a^{-1}c(a).$

Lemma 7.2

For each $c\in Z^i(G,M)$ and $a\in G$, we can associate an

$$h^{i-1}_a[c]\in C^{i-1}(G,M)/B^{i-1}(G,M)$$

in such a way that

$$\begin{aligned} \begin{aligned} (1) \quad&c^a-c=dh^{i-1}_a[c]; \qquad \qquad \\ (2) \quad&h_{ab}^{i-1}[c]=(h^{i-1}_a[c])^b+h^{i-1}_b[c]. \end{aligned} \end{aligned}$$

Proof

This is clear for $i=0$ and we have shown above the construction of $h^0_a[c]$ for $c\in Z^1(G,M)$ satisfying (1). Let us check the condition (2):

$$h^0_{ab}[c](g)=(ab)^{-1}c(ab)$$

$$=b^{-1}a^{-1}(c(a)+ac(b))=b^{-1}h^0_a[c](\mathrm {Ad}_b(g))+h^0_b[c](g)=(h^0_a[c])^b(g)+h^0_b[c](g).$$

We prove the statement using induction on i, which we now assume to be $\ge 2$. For a module M, we have the exact sequence

$$0\rightarrow M\rightarrow C^1(G,M)\rightarrow N\rightarrow 0,$$

where $C^1(G,M)$ has the right regular action of G and $N=C^1(G,M)/M$. Here, we give $C^1(G,M)$ the topology of pointwise convergence. There is a canonical linear splitting $s: N\rightarrow C^1(G,M)$ with image the group of functions f such that $f(e)=0$, using which we topologise N. According to [24, Proof of 2.5], the G-module $C^1(G,M)$ is acyclic,¹³ that is,

$$H^i(G, C^1(G,M))=0$$

for $i>0$. Therefore, given a cocycle $c\in Z^i(G, M)$, there is an

$$F\in C^{i-1}(G, C^1(G,M))$$

such that its image $f \in C^{i-1}(G,N)$ is a cocycle and $dF=c$. Hence, $d(F^a-F)=c^a-c$. Also, by induction, there is a $k_a\in C^{i-2}(G,N)$ such that $f^a-f=dk_a$ and $k_{ab}=(k_a)^b+k_b+dl$ for some $l\in C^{i-3}(G,N)$ (zero if $i=2$). Let $K_a=s\circ k_a$ and put

$$h_a=F^a-F-dK_a.$$

Then the image of $h_a$ in N is zero, so $h_a$ takes values in M, and $dh_a=c^a-c$. Now we check property (2). Note that

$$K_{ab}=s\circ k_{ab} =s\circ (k_a)^b+s\circ k_b+s\circ dl.$$

But $s\circ (k_a)^b-(s\circ k_a)^b$ and $s\circ dl-d(s\circ l)$ both have image in M. Hence, $K_{ab}=K_a^b+K_b+d(s\circ l)+m$ for some cochain $m\in C^{i-2}(G,M)$. From this, we deduce

$$dK_{ab}=(dK_a)^b+dK_b+dm,$$

from which we get

$$h_{ab}=F^{ab}-F-dK_{ab}=(F^a)^b-F^b+F^b-F-(dK_a)^b-dK_b-dm=(h_a)^b+h_b+dm.$$

8 Appendix 2: Conjugation Action on Group Cochains: Categorical Approach

In this section, an alternative and conceptual proof of Lemma 7.2 is outlined. Although not strictly necessary for the purposes of this paper, we believe that a functorial theory of secondary classes in group cohomology will be important in future developments. This point has also been emphasised to M.K. by Lawrence Breen. More details and elaborations will follow in a forthcoming publication by B.N.

1.1 8.1 Notation

In what follows G is a group and M is a left G-module. The action is denoted by ${}^{a}\!{m}$. The left conjugation action of $a\in G$ on G is denoted ${\text {Ad}}_a(x)=axa^{-1}$. We have an induced right action on n-cochains $f \, G^{n} \rightarrow M$ given by

$$\begin{aligned} f^a(\mathbf {g}):={}^{a^{-1}}\!{(f({\text {Ad}}_{a}\mathbf {g}))}. \end{aligned}$$

Here, $\mathbf {g} \in G^n$ is an n-chain, and ${\text {Ad}}_a\mathbf {g}$ is defined componentwise.

In what follows, [n] stands for the ordered set $\{0,1,\ldots ,n\}$, viewed as a category.

1.2 8.2 Idea

The above action on cochains respects the differential, hence passes to cohomology. It is well known that the induced action on cohomology is trivial. That is, given an n-cocycle f and any element $a\in G$, the difference $f^a - f$ is a coboundary. In this appendix we explain how to construct an $(n-1)$-cochain $h_{a,f}$ such that $d(h_{a,f})=f^a- f$. The construction, presumably well known, uses standard ideas from simplicial homotopy theory [26, Sect. 1]. The general case of this construction, as well as the missing proofs of some of the statements in this appendix will appear in a separate article.

Let $\mathcal {G}$ denote the one-object category (in fact, groupoid) with morphisms G. For an element $a \in G$, we have an action of a on $\mathcal {G}$ which, by abuse of notation, we will denote again by ${\text {Ad}}_{a}: \mathcal {G}\rightarrow \mathcal {G}$; it fixes the unique object and acts on morphisms by conjugation by a.

The main point in the construction of the cochain $h_{a,f}$ is that there is a “homotopy” (more precisely, a natural transformation) $H_a$ from the identity functor id: $\mathcal {G}\rightarrow \mathcal {G}$ to ${\text {Ad}}_a: \mathcal {G}\rightarrow \mathcal {G}$. The homotopy between ${\text {id}}$ and ${\text {Ad}}_a$ is given by the functor $H_a: \mathcal {G}\times [1] \rightarrow \mathcal {G}$ defined by

$$\begin{aligned} H_a|_0= {\text {id}}, \ \ H_a|_1={\text {Ad}}_a, \text { and } H_a(\iota ) = a^{-1}. \end{aligned}$$

It is useful to visualise the category $\mathcal {G}\times [1]$ as

1.3 8.3 Cohomology of Categories

We will use multiplicative notation for morphisms in a category, namely, the composition of g: $x \rightarrow y$ with h: $y \rightarrow z$ is denoted gh: $x \rightarrow z$.

Let $\mathcal {C}$ be a small category and M a left $\mathcal {C}$-module, that is, a functor M : $\mathcal {C}^{\text {op}} \rightarrow \mathbf {Ab}$, $x \mapsto M_x$, to the category of abelian groups (or your favorite linear category). Note that when $\mathcal {G}$ is as above, this is nothing but a left G-module in the usual sense. For an arrow g: $x \rightarrow y$ in $\mathcal {C}$, we denote the induced map $M_y \rightarrow M_x$ by $m \mapsto {}^{g}\!{m}$.

Let $\mathcal {C}^{[n]}$ denote the set of all n-tuples $\mathbf {g}$ of composable arrows in $\mathcal {C}$,

$$\mathbf {g}\ = \ \bullet \xrightarrow {g_1} \bullet \xrightarrow {g_2} \cdots \xrightarrow {g_n}\bullet .$$

We refer to such a $\mathbf {g}$ as an n -cell in $\mathcal {C}$; this is the same thing as a functor $[n] \rightarrow \mathcal {C}$, which we will denote, by abuse of notation, again by $\mathbf {g}$.

An n-chain in $\mathcal {C}$ is an element in the free abelian group ${\text {C}}_n(\mathcal {C},\mathbb {Z})$ generated by the set $\mathcal {C}^{[n]}$ of n-cells. For an n-cell $\mathbf {g}$ as above, we let $s\mathbf {g}\in {\text {Ob}}\mathcal {C}$ denote the source of $g_1$.

By an n-cochain on $\mathcal {C}$ with values in M we mean a map f that assigns to any n-cell $\mathbf {g}\in \mathcal {C}^{[n]}$ an element in $M_{s\mathbf {g}}$. Note that, by linear extension, we can evaluate f on any n-chain in which all n-cells share a common source point.

The n-cochains form an abelian group ${\text {C}}^n(\mathcal {C},M)$. The cohomology groups ${\text {H}}^n(\mathcal {C},M)$, $n\ge 0$, are defined using the cohomology complex ${\text {C}}^{\bullet }(\mathcal {C},M)$:

$$ 0 \xrightarrow {}{\text {C}}^0(\mathcal {C},M) \xrightarrow {d} {\text {C}}^1(\mathcal {C},M) \xrightarrow {d} \cdots \xrightarrow {d} {\text {C}}^n(\mathcal {C},M) \xrightarrow {d} {\text {C}}^{n+1}(\mathcal {C},M) \xrightarrow {d} \cdots $$

where the differential

$$\begin{aligned} d\!:\,{\text {C}}^n(\mathcal {C},M) \rightarrow {\text {C}}^{n+1}(\mathcal {C},M) \end{aligned}$$

is defined by

$$\begin{aligned} df(g_1,g_2,\ldots ,g_{n+1}) = {}^{g_1}\!{(}f(g_2,\ldots ,g_{n+1}))&+ \underset{1\le i \le n}{\sum }(-1)^if(g_1,\ldots ,g_ig_{i+1},\ldots ,g_{n+1}) \\&\qquad + (-1)^{n+1}f(g_1,g_2,\ldots ,g_{n}). \end{aligned}$$

A left G-module M in the usual sense gives rise to a left module on $\mathcal {G}$, which we denote again by M. We sometimes denote ${\text {C}}^{\bullet }(\mathcal {G},M)$ by ${\text {C}}^{\bullet }(G,M)$. Note that the corresponding cohomology groups coincide with the group cohomology ${\text {H}}^n(G,M)$.

The cohomology complex ${\text {C}}^{\bullet }(\mathcal {C},M)$ and the cohomology groups ${\text {H}}^n(\mathcal {C},M)$ are functorial in M. They are also functorial in $\mathcal {C}$ in the following sense. A functor $\varphi$ : $\mathcal {D}\rightarrow \mathcal {C}$ gives rise to a $\mathcal {D}$-module $\varphi ^*M:=M\circ \varphi \, \mathcal {D}^{op} \rightarrow \mathbf {Ab}$. We have a map of complexes

$$\begin{aligned} \varphi ^*\!:\, {\text {C}}^{\bullet }(\mathcal {C},M) \rightarrow {\text {C}}^{\bullet }(\mathcal {D},\varphi ^*M), \end{aligned}$$

(8.1)

which gives rise to the maps

$$\begin{aligned} \varphi ^*\!:\,{\text {H}}^{n}(\mathcal {C},M) \rightarrow {\text {H}}^{n}(\mathcal {D},\varphi ^*M) \end{aligned}$$

on cohomology, for all $n\ge 0$.

1.4 8.4 Definition of the Cochains $h_{a,f}$

The flexibility we gain by working with chains on general categories allows us to import standard ideas from topology to this setting. The following definition of the cochains $h_{a,f}$ is an imitation of a well known construction in topology.

Let $f \in {\text {C}}^{n+1}(G,M)$ be an $(n+1)$-cochain, and $a \in G$ an element. Let $H_a$ :$\, \mathcal {G}\times [1] \rightarrow \mathcal {G}$ be the corresponding natural transformation. We define $h_{a,f} \in {\text {C}}^{n}(G,M)$ by

$$\begin{aligned} h_{a,f}(\mathbf {g})=f(H_a(\mathbf {g}\times [1])). \end{aligned}$$

Here, $\mathbf {g}\in \mathcal {C}^{[n]}$ is an n-cell in $\mathcal {G}$, so $\mathbf {g}\times [1]$ is an $(n+1)$-chain in $\mathcal {G}\times [1]$, namely, the cylinder over $\mathbf {g}$.

To be more precise, we are using the notation $\mathbf {g}\times [1]$ for the image of the fundamental class of $[n]\times [1]$ in $\mathcal {G}\times [1]$ under the functor $\mathbf {g}\times [1] \, [n]\times [1] \rightarrow \mathcal {G}\times [1]$. We visualize $[n]\times [1]$ as

Its fundamental class is the alternating sum of the $(n+1)$-cells

in $[n]\times [1]$, for $0\le r \le n$. Therefore,

$$\begin{aligned} h_{a,f}(\mathbf {g}) = \underset{0\le r \le n}{\sum } (-1)^r f(g_1,\ldots ,g_r,a^{-1},{\text {Ad}}_a{g_{r+1}},\ldots ,{\text {Ad}}_a{g_{n}}). \end{aligned}$$

(8.2)

The following proposition can be proved using a variant of Stokes’ formula for cochains.

Proposition 8.1

The graded map  $h_{-,a}\!:\, {\text {C}}^{\bullet +1}(G, M) \rightarrow {\text {C}}^{\bullet }(G,M)$ is a chain homotopy between the chain maps

$$\begin{aligned} {\text {id}},(-)^a\!: {\text {C}}^{\bullet }(G,M) \rightarrow {\text {C}}^{\bullet }(G,M). \end{aligned}$$

That is,

$$\begin{aligned} h_{a,df} + d(h_{a,f}) = f^a - f \end{aligned}$$

for every $(n+1)$-cochain f. In particular, if f is an $(n+1)$-cocycle, then $d(h_{a,f}) = f^a - f$.

1.5 8.5 Composing Natural Transformations

Given an $(n+1)$-cochain f, and elements $a,b \in G$, we can construct three n-cochains: $h_{a,f}$, $h_{b,f}$ and $h_{ab,f}$. A natural question to ask is whether these three cochains satisfy a cocycle condition. It turns out that the answer is yes, but only up to a coboundary $dh_{a,b,f}$. Below we explain how $h_{a,b,f}$ is constructed. In fact, we construct cochains $h_{a_1,\ldots ,a_k,f}$, for any k elements $a_i \in G$, $1\le i \le k$, and study their relationship.

Let $f \in {\text {C}}^{n+k}(G,M)$ be an $(n+k)$-cochain. Let $\mathbf {a}=(a_1,\ldots ,a_k)\in G^{\times k}$. Consider the category $\mathcal {G}\times [k]$,

Let $H_{\mathbf {a}}: \, \mathcal {G}\times [k] \rightarrow \mathcal {G}$ be the functor such that $\iota _i \mapsto a_{k-i}^{-1}$ and $H_{\mathbf {a}}|_{\{0\}}={\text {id}}_G$. (So, $H_{\mathbf {a}}|_{\{k-i\}}={\text {Ad}}_{a_{i+1}\cdots a_k}$.) Define $h_{\mathbf {a},f} \in {\text {C}}^{n}(G,M)$ by

$$\begin{aligned} {} h_{\mathbf {a},f}(\mathbf {g})=f(H_{\mathbf {a}}(\mathbf {g}\times [k])). \end{aligned}$$

(8.3)

Here, $\mathbf {g}\in \mathcal {C}^{[n]}$ is an n-cell in $\mathcal {G}$, so $\mathbf {g}\times [k]$ is an $(n+k)$-chain in $\mathcal {G}\times [k]$.

To be more precise, we are using the notation $\mathbf {g}\times [k]$ for the image of the fundamental class of $[n]\times [k]$ in $\mathcal {G}\times [k]$ under the functor $\mathbf {g}\times [k] \, [n]\times [k] \rightarrow \mathcal {G}\times [k]$. We visualize $[n]\times [k]$ as

Its fundamental class is the $(n+k)$-chain

$$\begin{aligned} \underset{P}{\sum }(-1)^{|P|}P, \end{aligned}$$

where P runs over (length $n+k$) paths starting from (0, 0) and ending in (n, k). Note that such paths correspond to (k, n) shuffles; |P| stands for the parity of the shuffle (which is the same as the number of squares above the path in the $n\times k$ grid).

The most economical way to describe the relations between various $h_{\mathbf {a},f}$ is in terms of the cohomology complex of the right module

$$\begin{aligned} \mathbb {M}^{\bullet }:=\underline{{{\text {Hom}}}}\left( {\text {C}}^{\bullet }(G,M),{\text {C}}^{\bullet }(G,M)\right) . \end{aligned}$$

Here, $\underline{{{\text {Hom}}}}$ stands for the enriched hom in the category of chain complexes, and the right action of G on $\mathbb {M}^{\bullet }$ is induced from the right action $f \mapsto f^a$ of G on the ${\text {C}}^{\bullet }(G,M)$ sitting on the right. The differential on $\mathbb {M}^{\bullet }$ is defined by

$$d_{\mathbb {M}^{\bullet }}(u)=(-1)^{|u|} u\circ d_{{\text {C}}^{\bullet }(G,M)}- d_{{\text {C}}^{\bullet }(G,M)}\circ u,$$

where |u| is the degree of the homogeneous $u \in {\text {C}}^{\bullet }(G,M)$.

Note that, for every $\mathbf {a}\in G^{\times k}$, we have $h_{\mathbf {a},f} \in \mathbb {M}^{-k}$. This defines a k-cochain on G of degree $-k$ with values in $\mathbb {M}^{\bullet }$,

$$\begin{aligned} h^{(k)}\!:\, \mathbf {a}\ \mapsto h_{\mathbf {a},-}, \ \mathbf {a}\in G^{\times k}. \end{aligned}$$

We set $h^{(-1)}:=0$. Note that $h^{(0)}$ is the element in $\mathbb {M}^{0}$ corresponding to the identity map id: ${\text {C}}^{\bullet }(G,M) \rightarrow {\text {C}}^{\bullet }(G,M)$.

The relations between various $h_{\mathbf {a},f}$ can be packaged in a simple differential relation. As in the case $k=0$ discussed in Proposition 8.1, this proposition can be proved using a variant of Stokes’ formula for cochains.

Proposition 8.2

For every $k \ge -1$, we have $d_{\mathbb {M}^{\bullet }}(h^{(k+1)})=d(h^{(k)})$.

In the above formula, the term $d_{\mathbb {M}^{\bullet }}(h^{(k+1)})$ means that we apply $d_{\mathbb {M}^{\bullet }}$ to the values (in $\mathbb {M}^{\bullet }$) of the cochain $h^{(k+1)}$. The differential on the right hand side of the formula is the differential of the cohomology complex ${\text {C}}^{\bullet }(G,\mathbb {M}^{\bullet })$ of the (graded) right G-module $\mathbb {M}^{\bullet }$.

More explicitly, let $f\in {\text {C}}^{n+k}(G,M)$ be an $(n+k)$-cochain. Then, Proposition 8.2 states that, for every $\mathbf {a}\in G^{\times (k+1)}$, we have the following equality of n-cochains:

$$\begin{aligned} (-1)^{(k+1)} h_{a_1,\ldots ,a_{k+1},df}-dh_{a_1,\ldots ,a_{k+1},f} =&\qquad \qquad h_{a_2,\ldots ,a_{k+1},f} + \\&\underset{1\le i \le k}{\sum }(-1)^ih_{a_1,\ldots ,a_ia_{i+1},\ldots ,a_{k+1},f} + \\&\qquad (-1)^{k+1}h_{a_1,\ldots ,a_{k},f}^{a_{k+1}}. \end{aligned}$$

Corollary 8.3

Let $f\in {\text {C}}^{n+k}(G,M)$ be an $(n+k)$-cocycle. Then, for every $\mathbf {a}\in G^{\times (k+1)}$, the n-cochain

$$h_{a_2,\ldots ,a_{k+1},f} + \underset{1\le i \le k}{\sum }(-1)^ih_{a_1,\ldots ,a_ia_{i+1},\ldots ,a_{k+1},f} + (-1)^{k+1}h_{a_1,\ldots ,a_{k},f}^{a_{k+1}}$$

is a coboundary. In fact, it is the coboundary of $-h_{a_1,\ldots ,a_{k+1},f}$.

Example 8.4

Let us examine Corollary 8.3 for small values of k.

• (i) For $k=0$, the statement is that, for every cocycle f, $f-f^a$ is a coboundary. In fact, it is the coboundary of $-h_{f,a}$. We have already seen this in Proposition 8.1.

• (ii) For $k=1$, the statement is that, for every cocycle f, the cochain

  $$\begin{aligned} h_{b,f}-h_{ab,f}+h_{a,f}^b \end{aligned}$$

  is a coboundary. In fact, it is the coboundary of $-h_{a,b,f}$.

1.6 8.6 Explicit Formula for $h_{a_1,\ldots ,a_k,f}$

Let f:$\, G^{\times (n+k)} \rightarrow M$ be an $(n+k)$-cochain, and $\mathbf {a}:=(a_1,a_2,\ldots ,a_{k}) \in G^{\times k}$. Then, by Eq. (8.3), the effect of the n-cochain $h_{a_1,\ldots ,a_k,f}$ on an n-tuple $\mathbf {x}:=(x_0,x_1,\ldots ,x_{n-1}) \in G^{\times n}$ is given by:

$$h_{a_1,\ldots ,a_k,f}(x_0,x_1,\ldots ,x_{n-1}) =\underset{P}{\sum }(-1)^{|P|}f(\mathbf {x}^{P}),$$

where $\mathbf {x}^{P}$ is the $(n+k)$-tuple obtained by the following procedure.

Recall that P is a path from (0, 0) to (n, k) in the n by k grid. The $l^{\text {th}}$ component $\mathbf {x}^{P}_l$ of $\mathbf {x}^{P}$ is determined by the $l^{\text {th}}$ segment on the path P. Namely, suppose that the coordinates of the starting point of this segment are (s, t). Then,

$$\begin{aligned} \mathbf {x}^{P}_l=a_{k-t}^{-1} \end{aligned}$$

if the segment is vertical, and

$$\begin{aligned} \mathbf {x}^{P}_l=(a_{k-t+1}\cdots a_k)x_s(a_{k-t+1}\cdots a_k)^{-1}, \end{aligned}$$

if the segment is horizontal. Here, we use the convention that $a_0=1$.

The following example helps visualize $\mathbf {x}^{P}$:

The corresponding term is

$$-f(x_0,x_1,a_4^{-1},a_4x_2a_4^{-1}, a_3^{-1},(a_3a_4)x_3(a_3a_4)^{-1},(a_3a_4)x_4(a_3a_4)^{-1}, a_2^{-1},a_1^{-1}). $$

The sign of the path is determined by the parity of the number of squares in the n by k grid that sit above the path P (in this case 15).