Appendix — Cochain Calculus Survival Kit

Cochain Calculus Survival Kit

The reference for the lattice toolkit introduced in Semester I Week 2 and used in every problem set thereafter. Kept operational: finite linear algebra on a lattice, no point-set topology. Every sign here is the one fixed in conventions (cells and ∂ in §1, δ, Δ and ⋆ in §2, the cup product in §8).

Read this once alongside Week 2, then keep it open as a lookup sheet.

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1. Cells, chains, cochains

A hypercubic lattice $\Lambda$ is a cell complex: 0-cells (sites), 1-cells (links), 2-cells (plaquettes), …, $p$-cells. Fix an orientation on every cell (links along $+\hat{\mu }$; a $p$-cell oriented by the ordered tuple of its directions ${\hat{\mu }}_1<\cdots <{\hat{\mu }}_p$).

• $p$-chain $c\in C_p$: a formal $\mathbb{Z}$-combination of oriented $p$-cells. “Where an extended object lives.”
• $p$-cochain $f\in C^p(\Lambda ,G)$: a $G$-valued function on oriented $p$-cells ($G=\mathbb{Z},\mathbb{R},U(1),{\mathbb{Z}}_N$). “A $p$-form field.”
• Pairing $⟨f,c⟩={\sum }_i{n}_{i}f({\sigma }_i)$ for $c={\sum }_i{n}_i{\sigma }_i$. This is ”${\int }_{c}f$.”

Fields live on cochains: a compact scalar is a $U(1)$ 0-cochain, a gauge field a $U(1)$ (or $\mathbb{R}$) 1-cochain, a ${\mathbb{Z}}_N$ 2-form background a ${\mathbb{Z}}_N$ 2-cochain.

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2. Boundary $\partial$ and coboundary $d$

• Boundary $\partial :C_p\to C_{p-1}$: the oriented boundary of a cell, signs from orientation. ${\partial }^2=0$.
• Coboundary $d:C^p\to C^{p+1}$ defined by the discrete Stokes theorem, taken as a definition: $⟨df,c⟩=⟨f,\partial c⟩.$ Then $d^2=0$ follows from ${\partial }^2=0$.

Concretely: $(df)$ on a $(p+1)$-cell = signed sum of $f$ over its boundary $p$-cells.

• 0-cochain (site field $\varphi$): $(d\varphi {)}_{\ell }={\varphi }_y-{\varphi }_x$ on link $\ell =(x\to y)$. (Lattice gradient.)
• 1-cochain (link field $a$): $(da{)}_p={\sum }_{\ell \in \partial p}\pm a_{\ell }$ around plaquette $p$. (Lattice curl / field strength.)
• 2-cochain: $(db)$ = signed sum over the faces of a cube. (Lattice "$dB$".)

Codifferential $\delta :C^p\to C^{p-1}$ is the adjoint of $d$ under the ${\ell }^2$ inner product $⟨f,g⟩={\sum }_{\mathrm{cells}}fg$ (not under the chain–cochain pairing of §1), with no further sign; on a 1-cochain $(\delta m{)}_x={\sum }_{\mu }[m_{\mu }(x-\hat{\mu })-m_{\mu }(x)]$, inflow minus outflow. With the Hodge map of §4 it reads $\delta =(-1{)}^{d(p+1)+1}\star d\star$. The lattice Laplacian is $\Delta =\delta d+d\delta \ge 0$; on 0-cochains $(\Delta f{)}_x={\sum }_{\mu }(2f_x-f_{x+\hat{\mu }}-f_{x-\hat{\mu }})$, minus the second difference.

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3. Homology and cohomology

• Closed / exact. $f$ is closed if $df=0$; exact if $f=dg$. Since $d^2=0$, exact $\Rightarrow$ closed.
• Cohomology $H^p(\Lambda ,G)=\ker d/\mathrm{im}d$ = closed mod exact. Measures “closed cochains that are not the coboundary of anything” = global (topological) field configurations.
• Homology $H_p(\Lambda ,G)=\ker \partial /\mathrm{im}\partial$ = cycles mod boundaries.

On the $d$-torus $T^d$ (the case the course needs): $H^p(T^d,G)\cong G^{\left(\genfrac{}{}{0ex}{}{d}{p}\right)},b_p=\left(\genfrac{}{}{0ex}{}{d}{p}\right)(\text{Betti numbers}).$

	$H^0$	$H^1$	$H^2$	$H^3$
$T^2$	$G$	$G^2$	$G$	—
$T^3$	$G$	$G^3$	$G^3$	$G$

With $G={\mathbb{Z}}_N$: replace $G$ by ${\mathbb{Z}}_N$ everywhere. $H^1(T^d,{\mathbb{Z}}_N)={\mathbb{Z}}_N^d$ is the counting behind ground-state degeneracy (Sem II Wk 2).

Generators. $H^1(T^d,\mathbb{Z})$ is generated by the cochains “counting crossings of the $\hat{\mu }$-cycle”; $H_1(T^d,\mathbb{Z})$ by the $d$ non-contractible loops. The pairing $H^1\times H_1\to \mathbb{Z}$ is the evaluation (winding) pairing, $⟨f,\gamma ⟩={\sum }_{\ell \in \gamma }f(\ell )$; linking is a different pairing, between cycles of complementary dimension, and appears in §4.

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4. The dual lattice and Poincaré duality

The dual lattice ${\Lambda }^{\ast }$ places a site at the center of each $d$-cell of $\Lambda$, and in general: $p\text{-cell of }\Lambda ⟷(d-p)\text{-cell of }{\Lambda }^{\ast }.$ So sites ↔ $d$-cells, links ↔ $(d-1)$-cells, etc. The dual of the cell $(x;S)$ sits at $x+\frac{1}{2}{\hat{e}}_S-\frac{1}{2}{\hat{e}}_{S^c}$ and is spanned by the complementary directions $S^c$; the Hodge map transports values with the sign $ϵ(S,S^c)$ of the shuffle, $(\star f)((x;S{)}^{\ast })=ϵ(S,S^c)f(x;S)$, so that $\star \star =(-1{)}^{p(d-p)}$ and $\delta =(-1{)}^{d(p+1)+1}\star d\star$ hold exactly on the lattice (conventions §2).

Poincaré duality $H^p(T^d,G)\cong H_{d-p}(T^d,G)$: a closed $p$-cochain on $\Lambda$ is the same data as a $(d-p)$-cycle on ${\Lambda }^{\ast }$. This is why a monopole ($m=dn$, a 3-cochain in 4d) is a worldline (1-cycle) on the dual lattice, and a vortex in 2d is a point.

Intersection pairing. On $T^d$, $H_p\times H_{d-p}\to \mathbb{Z}$ (or ${\mathbb{Z}}_N$) by counting intersections. On $T^2$: $H_1\times H_1\to {\mathbb{Z}}_N$, the two cycles intersect once — this single “1” is the $\omega =e^{2\pi i/N}$ in the clock–shift algebra.

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5. Poisson resummation — the duality engine

Every duality in the course is this identity, applied to an integer cochain: $\boxed{\sum _{n\in \mathbb{Z}}f(n)=\sum _{w\in \mathbb{Z}}{\int }_{-\infty }^{\infty }dxf(x)e^{2\pi iwx}}$

Periodic-Gaussian form (the one used constantly): ${\sum }_{n\in \mathbb{Z}}{e}^{-\frac{\beta }{2}(\varphi -2\pi n{)}^2}=\frac{1}{\sqrt{2\pi \beta }}{\sum }_{w\in \mathbb{Z}}{e}^{-\frac{1}{2\beta }{w}^2+iw\varphi }.$ Left: a periodic scalar with stiffness $\beta$. Right: a sum over integer “momenta”/charges $w$ with dual stiffness $1/\beta$. The coupling inverts, $\beta \leftrightarrow 1/\beta$ — the duality.

Recipe (works for XY Wk 3, compact QED Wks 9/11, modified Villain Wk 12):

1. Write the theory in Villain form with an integer cochain $n$.
2. Poisson-resum $n\to$ dual integer cochain $w$ on ${\Lambda }^{\ast }$.
3. The original field now appears linearly; integrating it over one period (it is compact) imposes an integer constraint ($\delta w=0$, current conservation, with a Kronecker delta).
4. Solve the constraint exactly: $w$ is the Hodge transport of $d$ of an integer dual field, plus winding sectors on the torus. No defects are visible yet, since $n$ has been summed away.
5. Poisson-resum the integer dual field a second time: it becomes a real dual field plus a new integer cochain, and these integers are the defects (vortices, monopoles), coupled linearly to the dual field (Week 3 §4).

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6. Cup product (needed Sem II Weeks 6–7)

The lattice analogue of the wedge product, $\cup :C^p\times C^q\to C^{p+q}$ (the cubical product of conventions §8; Chen–Tata order the two factors the other way): $(\alpha \cup \beta )(x;S)={\sum }_{A\subset S,|A|=p}ϵ(A,S\setminus A)\alpha (x;A)\beta (x+{\hat{e}}_A;S\setminus A),$ a sum over shuffles. For two 1-cochains on a plaquette it has two terms, $(\alpha \cup \beta )(P_{\mu \nu }(x))={\alpha }_{\mu }(x){\beta }_{\nu }(x+\hat{\mu })-{\alpha }_{\nu }(x){\beta }_{\mu }(x+\hat{\nu })$; with the first term alone Leibniz fails. Properties:

• Leibniz: $d(x\cup y)=dx\cup y+(-1{)}^{p}x\cup dy$.
• Graded commutativity up to exact terms, for cocycles: if $dx=dy=0$, then $x\cup y-(-1{)}^{pq}y\cup x$ is exact, $\pm d(x{\cup }_{1}y)$ with the higher cup product ${\cup }_1$. For general cochains the formula acquires the extra terms $dx{\cup }_{1}y$ and $x{\cup }_{1}dy$ (Chen–Tata eq. (29)). The non-commutativity carries physics (the anomaly), and we keep track of it.
• Pontryagin square $\mathcal{P}(B)$ for $B\in C^2(\cdot ,{\mathbb{Z}}_N)$: a refinement of $B\cup B$, valued in ${\mathbb{Z}}_{2N}$ ($N$ even) / ${\mathbb{Z}}_N$ ($N$ odd); it is the lattice object in the θ = π anomaly (Sem II Wk 6).

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7. The Villain dictionary (quick lookup)

Object	Cochain	Field strength / defect
Compact scalar $\theta$	$U(1)$ 0-cochain	vortex $v=dn$, a 2-cochain (points in 2d)
$U(1)$ gauge field $a$	$\mathbb{R}$ 1-cochain + integer $n\in C^2(\mathbb{Z})$	monopole $m=dn$, a 3-cochain (points in 3d, loops in 4d)
${\mathbb{Z}}_N$ $p$-form gauge field	${\mathbb{Z}}_N$ $p$-cochain	flatness $da=0$ mod $N$
${\mathbb{Z}}_N$ background	${\mathbb{Z}}_N$ $(p+1)$-cochain $B$	probes a $p$-form symmetry

• Villain scalar action: $S=\frac{\beta }{2}{\sum }_{\ell }(d\theta -2\pi n{)}^2$, $n\in C^1(\Lambda ,\mathbb{Z})$.
• Villain gauge action: $S=\frac{\beta }{2}{\sum }_p(da-2\pi n{)}^2$, $n\in C^2(\Lambda ,\mathbb{Z})$.
• Modified Villain: promote $n$ to a dynamical $\mathbb{Z}$ gauge field and impose $dn=0$ with a dual-lattice Lagrange multiplier — removes defects, makes symmetries and dualities exact (Sem II Wk 12).

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8. Worked micro-examples (do these by hand once)

1. $H^p(T^2,\mathbb{Z})$ from one cell each. Minimal complex: 1 site, 2 links ($x,y$), 1 plaquette. $\partial (\text{plaq})=0$ (opposite edges cancel on the torus) ⇒ $H_2=\mathbb{Z}$; $\ker {\partial }_1={\mathbb{Z}}^2$ with no boundaries ⇒ $H_1={\mathbb{Z}}^2$. On cochains the same cancellation reads $(da)(\text{plaq})=a_x+a_y-a_x-a_y=0$, so $d=0$ in every degree and $H^0=\mathbb{Z}$, $H^1={\mathbb{Z}}^2$, $H^2=\mathbb{Z}$. Betti $(1,2,1)$. ✓
2. Gradient is closed. $(d\varphi {)}_{\ell }={\varphi }_y-{\varphi }_x$; check $d(d\varphi )=0$ around any plaquette (telescoping sum). ✓
3. Poisson the Gaussian. Verify ${\sum }_n{e}^{-\frac{\beta }{2}(\varphi -2\pi n{)}^2}=(2\pi \beta {)}^{-1/2}{\sum }_w{e}^{-w^2/2\beta +iw\varphi }$ by completing the square inside the integral. ✓
4. Torus intersection = 1. The two 1-cycles of $T^2$ cross once; this is the seed of $ZX=\omega XZ$. ✓

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9. Common pitfalls

• Orientation signs. Fix them once (per conventions) and never re-derive mid-calculation; most “duality is off by a sign” bugs are orientation bookkeeping.
• Direct vs dual placement. A field and its dual live on different lattices; a monopole is a dual-lattice object. Draw both lattices.
• Villain vs cosine. Villain is exact for duality; the cosine is not. Do not mix the two inside one derivation.
• Compact measures. Integrate a compact variable over one period. Integrating it over $\mathbb{R}$ while also summing its integer branch label overcounts by the branch redundancy and gives a divergent measure.
• Cup non-commutativity. Do not “simplify” $x\cup y$ to $\pm y\cup x$ — the difference is the anomaly.

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This kit is the mathematical backbone of the course. Every problem set that says “dualize,” “count the ground states,” or “turn on a background” is an application of §§2–7. Corrected on 2026-09-27 (codifferential and Hodge-map signs, the duality recipe, the explicit cup product and its graded commutativity), after the review in reviews/2026-09-27-comprehensive-review.md.