Course Conventions

Conventions

Notation for the generalized-symmetries course. Read before Week 1; the cochain conventions below appear in every problem set. This file is the single source of every sign and normalization the notes use. No note fixes a convention locally: a note that needs something missing here adds it here, in the same commit (template §5). Where this file is silent on a matter no note uses yet, the default references are Gorantla–Lam–Seiberg–Shao (GLSS, arXiv:2103.01257) for Villain and modified-Villain constructions and Chen–Tata (arXiv:2106.05274) for higher cup products.

Course-local override. In this course, lowercase $d$ is the Euclidean spacetime dimension (the lattice-literature convention), so a $d$-dimensional theory has $D=d-1$ spatial directions in its Hamiltonian presentation. This differs from the owner’s global default ($d$ = spatial dimensions), and the difference is the probable origin of erratum E1 (a $(d-2)$ rule for spatial $d$ evaluated at spacetime $d$); every degree formula in this course is therefore written with $d$ = spacetime dimension.

Reserved symbols.

• $d$: the spacetime dimension, and the coboundary operator on cochains; context always separates the two. A greatest common divisor is written $\gcd (N,q)$, or $r=\gcd (N,q)$ when a short symbol is needed, never $d$ (the group’s manuscript, version of 2026-09-20, writes $g=\gcd (N,k)$ and uses $d$ for another greatest common divisor; quotations from it translate $g\to r$).
• $a$: the $U(1)$ lattice gauge field (a real 1-cochain) in the gauge weeks; in the 2d scalar weeks (Weeks 1–4) $a(x)=G(0)-G(x)$ is the subtracted Green function of §2, always written with its argument. The two never occur in one formula. In the RG of Week 4, $a$ without an argument is the running short-distance cutoff, rescaled as $a\to a{e}^{\ell }$. The lattice spacing is $a_{\mathrm{lat}}$ ($a_t$, $a_s$ on anisotropic lattices), set to 1 unless dimensional analysis is the point; in Weeks 1–4, where the letter $a$ is taken by the Green function, the same quantity is written $a_0$, as in $\ln (r/a_0)$.
• $\beta$: the inverse coupling of a lattice action; an inverse temperature only when a transfer matrix says so explicitly. ${\beta }_R$ is the renormalized (screened) stiffness of the BKT analysis.
• $G$: a lattice Green function. The spin–spin correlator is $C(x)=⟨e^{i({\theta }_x-{\theta }_0)}⟩$, always with its argument; in Week 4, $C$ without an argument is the invariant of the Kosterlitz flow.
• $e$: the $U(1)$ gauge coupling; $g$: the non-abelian and Hamiltonian coupling. For $U(1)$ the Kogut–Susskind $g$ equals $e$.
• $K$: context-dependent, and each note says which: the Ising coupling (Weeks 4–5), the plaquette coupling of the ${\mathbb{Z}}_N$ Hamiltonians (Weeks 7 and 13–15), and the dual-photon stiffness $K\equiv e^2/8{\pi }^2$ of compact QED₃ (Week 10; §5).
• κ: the 2d Green-function constant of §2 in Weeks 1–4; the matter hopping coupling in Weeks 13–14, and the matter stiffness of the Villain abelian-Higgs model of Sem II Week 3; ${\kappa }_a$, always with its label, the Frobenius–Schur sign of Sem II Week 11. In Sem II Week 14, following the group’s manuscript (§7), κ is the cost per unit length of a wall current, $\tilde{\kappa }=4{\pi }^2/\kappa N^{\prime 2}$ the dual clock stiffness and ${\beta }_V=1/\kappa$ its Villain form, and the matter stiffness is written ${\kappa }_m$. $N^{\prime }=N/r$ with $r=\gcd (N,k)$, and $\bar{\alpha }={\min }_{m\in \mathbb{Z}}|\alpha +N^{\prime }m|$ is the minimal lift of $\alpha \in {\mathbb{Z}}_{N^{\prime }}$. ${\tau }_v^{x,z}$: Pauli operators of ℤ₂ matter on sites (Week 13); τ alone is Euclidean time or the modular parameter of §10, as the context says.

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1. Lattice and cell complex

• Hypercubic lattice $\Lambda \subset {\mathbb{Z}}^d$. Euclidean signature throughout Semester I; Hamiltonian (continuous-time) presentations name their spatial lattice explicitly.
• A $p$-cell is $c=(x;S)$ with $x$ a site and $S=\{{\mu }_1<\cdots <{\mu }_p\}$ a set of directions: sites are 0-cells, links ${\ell }_{\mu }(x)=(x;\{\mu \})$ run from $x$ to $x+\hat{\mu }$, plaquettes $P_{\mu \nu }(x)$ ($\mu <\nu$) are oriented by $(\hat{\mu },\hat{\nu })$, and so on. We write ${\hat{e}}_S={\sum }_{\mu \in S}\hat{\mu }$.
• $p$-chains are formal integer combinations of $p$-cells, with boundary

$$\partial (x;S)=\sum _{i=1}^p(-1{)}^{i-1}[(x+{\hat{\mu }}_i;S\setminus {\mu }_i)-(x;S\setminus {\mu }_i)],{\partial }^2=0.$$

For a plaquette this is bottom + right − top − left: $\partial P_{\mu \nu }(x)={\ell }_{\mu }(x)+{\ell }_{\nu }(x+\hat{\mu })-{\ell }_{\mu }(x+\hat{\nu })-{\ell }_{\nu }(x)$.

• $p$-cochains with coefficients in an abelian group $G$: $C^p(\Lambda ,G)$, the $G$-valued functions on oriented $p$-cells. The coboundary is $(df)(c)=f(\partial c)$, so $d^2=0$; this is the discrete Stokes theorem, by definition. For a 0-cochain $(df)({\ell }_{\mu }(x))=f(x+\hat{\mu })-f(x)$; for a 1-cochain $(d\alpha )(P_{\mu \nu }(x))={\alpha }_{\mu }(x)+{\alpha }_{\nu }(x+\hat{\mu })-{\alpha }_{\mu }(x+\hat{\nu })-{\alpha }_{\nu }(x)$.
• Cohomology $H^p(\Lambda ,G)=\ker d/\mathrm{im}d$; on the torus $T^d$ computed by hand in Week 2. Coefficients used in the course: $\mathbb{Z}$, $\mathbb{R}$, $U(1)$, ${\mathbb{Z}}_N$.

2. Inner product, codifferential, Laplacian, dual lattice

• Inner product. $⟨f,g⟩={\sum }_{p\text{-cells}}f(c)g(c)$ on real cochains; $\Vert f{\Vert }^2=⟨f,f⟩$.
• Codifferential. $\delta :C^p\to C^{p-1}$ is the adjoint of $d$ under this inner product, with no further sign: $⟨\delta f,g⟩=⟨f,dg⟩$. On a 1-cochain,

$$(\delta m{)}_x=\sum _{\mu }[m_{\mu }(x-\hat{\mu })-m_{\mu }(x)](\text{inflow minus outflow at }x).$$

The physical divergence is therefore $(\mathrm{div}m{)}_x=-(\delta m{)}_x$.

• Laplacian. $\Delta =\delta d+d\delta \ge 0$. On 0-cochains $(\Delta f{)}_x={\sum }_{\mu }(2f_x-f_{x+\hat{\mu }}-f_{x-\hat{\mu }})$; the lattice second difference is $-\Delta$. The spin-wave action is $\frac{\beta }{2}\Vert d\theta {\Vert }^2=\frac{\beta }{2}⟨\theta ,\Delta \theta ⟩$.
• Green functions. $\Delta G={\delta }_{x,0}$ on the infinite lattice; on a finite torus $G^{\prime }$ denotes the inverse of $\Delta$ on the complement of the constants. In Fourier space

$$G(x)={\int }_{-\pi }^{\pi }\frac{d^{d}k}{(2\pi {)}^d}\frac{e^{ik\cdot x}}{{\hat{k}}^2},{\hat{k}}^2=\sum _{\mu }4{\sin }^2\frac{k_{\mu }}{2},$$

and the free scalar propagator is $⟨{\theta }_x{\theta }_y⟩=G(x-y)/\beta$.

• $d=2$: $G(0)$ diverges in the infrared; the subtracted function is finite,

$$a(x)=G(0)-G(x)=\frac{1}{2\pi }\ln |x|+\kappa +O(|x{|}^{-2}),\kappa =\frac{2{\gamma }_E+\ln 8}{4\pi }=0.257343\dots ,$$

with $a(1,0)=\frac{1}{4}$ and $a(1,1)=1/\pi$ exactly.

• $d=3$: $G(0)=0.252731\dots$, $G(x)\to 1/(4\pi |x|)$.
• $d=4$: $G(0)=0.154933\dots$, $G(x)\to 1/(4{\pi }^2|x{|}^2)$.
• Nearest-neighbour Green functions. $G_d(0)-G_d(\hat{e})=1/2d$ on the infinite hypercubic lattice, so $G_3(\hat{e})=G_3(0)-\frac{1}{6}$ and $G_4(\hat{e})=G_4(0)-\frac{1}{8}$.
• Sums over a sublattice (Sem II Week 3 §4). Summing $G_d$ over the translates of a point along $p$ of the $d$ lattice directions gives the Green function of the transverse displacement, ${\sum }_{k\in {\mathbb{Z}}^p}{G}_d(x_{\perp }+k)=G_{d-p}(x_{\perp })$, for $d-p\ge 3$; it follows from the Bessel representation of §5, since ${\sum }_k{I}_k(2t)=e^{2t}$.
• Dual lattice. ${\Lambda }^{\ast }$ has a site at the center of every $d$-cell of $\Lambda$. The dual of the $p$-cell $(x;S)$ is the $(d-p)$-cell of ${\Lambda }^{\ast }$ based at $x+\frac{1}{2}{\hat{e}}_S-\frac{1}{2}{\hat{e}}_{S^c}$ and spanned by the complementary directions $S^c$ in increasing order.
• Hodge map. $\star :C^p(\Lambda )\to C^{d-p}({\Lambda }^{\ast })$ transports values with the sign of the shuffle,

$$(\star f)((x;S{)}^{\ast })=ϵ(S,S^c)f((x;S)),$$

where $ϵ(S,S^c)$ is the sign of the permutation that sorts $(S,S^c)$. The same rule maps ${\Lambda }^{\ast }$ back to $\Lambda$. With it, exactly as in the continuum,

$$\star \star =(-1{)}^{p(d-p)},\delta =(-1{)}^{d(p+1)+1}\star d\star \text{on }{C}^p,$$

checked cell by cell on random cochains for all $p$ in $d=2,3,4$. In $d=2$ the rule rotates a link by $+90°$ (the dual of ${\ell }_1(x)$ runs from $x+(\frac{1}{2},-\frac{1}{2})$ to $x+(\frac{1}{2},\frac{1}{2})$, the dual of ${\ell }_2(x)$ from $x+(\frac{1}{2},\frac{1}{2})$ to $x+(-\frac{1}{2},\frac{1}{2})$), sends a site to the counter-clockwise dual plaquette around it, and gives $\delta =-\star d\star$ on 1-cochains, i.e. $(d\star m)(x^{\ast })=\text{out}-\text{in}=-(\delta m{)}_x$ (Week 2 §5.2).

• Poincaré duality $H^p(\Lambda )\cong H_{d-p}(\Lambda )$ is used freely after Week 2.

• Hodge map and intersection pairing (Sem II Week 1 §5). ${\star }^{-1}\tilde{\delta }=(-1{)}^{d+q+1}d{\star }^{-1}$ on $C^q({\Lambda }^{\ast })$, and for chains $X$ of Λ and $\tilde{Y}$ of ${\Lambda }^{\ast }$ of complementary dimensions, with $X$ a $(p+1)$-chain, $I(\partial X,\tilde{Y})=(-1{)}^{p+1}I(X,\partial \tilde{Y})$. Together they give $d{\lambda }_{\tilde{V}}={\star }^{-1}{J}_{\partial \tilde{V}}$ for the ${\lambda }_{\tilde{V}}$ of §6.

3. Poisson resummation and the Villain identities

The duality “move” of the course is always the same pair: (i) Poisson resummation of an integer cochain, (ii) solving the resulting constraint on the dual lattice. The identities, with the normalizations used everywhere:

$$\sum _{n\in \mathbb{Z}}f(n)=\sum _{w\in \mathbb{Z}}{\int }_{-\infty }^{\infty }dxf(x)e^{2\pi iwx},\sum _{n\in \mathbb{Z}}{e}^{-2\pi ibn}=\sum _{m\in \mathbb{Z}}\delta (b-m),$$ $$\sum _{n\in \mathbb{Z}}{e}^{-\frac{\beta }{2}(\varphi -2\pi n{)}^2}=\frac{1}{\sqrt{2\pi \beta }}\sum _{m\in \mathbb{Z}}{e}^{-\frac{m^2}{2\beta }+im\varphi },{\int }_{-\pi }^{\pi }\frac{d\theta }{2\pi }{e}^{ik\theta }={\delta }_{k,0}(k\in \mathbb{Z}).$$

The last identity is why the integration over a compact variable must run over one period: integrating an angle over $\mathbb{R}$ while also summing its integer branch label overcounts by the branch redundancy and gives a divergent measure.

4. Compact fields, lattice actions and Hamiltonians

• Compact scalar. ${\theta }_x\in (-\pi ,\pi ]$. Villain action $S=\frac{\beta }{2}\Vert d\theta -2\pi n{\Vert }^2$ with $n\in C^1(\Lambda ,\mathbb{Z})$, redundancy $\theta \to \theta +2\pi k$, $n\to n+dk$. Vorticity $v=dn\in C^2(\Lambda ,\mathbb{Z})$. Since $d(d\theta -2\pi n)=-2\pi v$, the branch-reduced angle winds by $-2\pi v_P$ around a plaquette $P$: a counter-clockwise $+2\pi$ vortex has $v_P=-1$.
• Compact gauge field. $a_{\ell }\in (-\pi ,\pi ]$, measure ${\prod }_{\ell }{\int }_{-\pi }^{\pi }d{a}_{\ell }/2\pi$. Villain action $S=\frac{\beta }{2}\Vert da-2\pi n{\Vert }^2$ with $n\in C^2(\Lambda ,\mathbb{Z})$, redundancies $a\to a+d\lambda$ and $a\to a+2\pi k$, $n\to n+dk$ ($k\in C^1(\Lambda ,\mathbb{Z})$). Monopole number $m=dn\in C^3(\Lambda ,\mathbb{Z})$; the flux of the Villain field strength out of a cube $c$ is ${\sum }_{\partial c}(da-2\pi n)=-2\pi m_c$.
• Couplings and continuum limits.
  • $U(1)$: Wilson action $S=\beta {\sum }_P(1-\cos (da{)}_P)$ or the Villain action above, with $\beta =1/(e^2{a}_{\mathrm{lat}}^{4-d})$; both reduce to $\frac{1}{2e^2}\int F\wedge \star F=\frac{1}{4e^2}\int F_{\mu \nu }{F}_{\mu \nu }$, and $e^2$ has mass dimension $4-d$. The $SU(N)$ formula below is never applied to $U(1)$.
  • $SU(N)$: $S=\frac{\beta }{N}{\sum }_P\mathrm{Re}\mathrm{tr}(1-U_P)$ with $\beta =2N/(g^2{a}_{\mathrm{lat}}^{4-d})$ and $\mathrm{tr}{T}^a{T}^b=\frac{1}{2}{\delta }^{ab}$; the continuum limit is $\frac{1}{2g^2}\int \mathrm{tr}{F}_{\mu \nu }{F}_{\mu \nu }=\frac{1}{4g^2}\int F_{\mu \nu }^a{F}_{\mu \nu }^a$. Links are $U_{\ell }=P\exp \left(i{\int }_{\ell }A\right)$ with earlier points to the left, so that $U_P=\exp (i{a}_{\mathrm{lat}}^2{F}_{\mu \nu }+O(a_{\mathrm{lat}}^3))$ with $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+i[A_{\mu },A_{\nu }]$ (coupling absorbed in $A$). The owner’s continuum papers write $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$, the same convention after $A\to -gA$.
  • ${\mathbb{Z}}_2$ (Wegner): $S=-\beta {\sum }_P{\sigma }_P$, ${\sigma }_P={\prod }_{\ell \in \partial P}{\sigma }_{\ell }$.
• Character expansion. $e^{-S_P(U)}=a_0(\beta )[1+{\sum }_{r\ne 0}{d}_r{\tilde{c}}_r(\beta ){\chi }_r(U)]$ with ${\tilde{c}}_r=a_r/(d_r{a}_0)$ and $a_r=\int dU{\chi }_r(U{)}^{\ast }{e}^{-S_P(U)}$. At strong coupling $⟨W_r(C)⟩={\tilde{c}}_r^{A(C)}(1+\dots )$ and ${\sigma }_r=-\ln {\tilde{c}}_r$ is the leading area coefficient; for large loops, representations of zero $N$-ality are screened (the adjoint tube, Week 6 §7.5). With the Wilson action ${\tilde{c}}_r=O({\beta }^{n_r})$, where $n_r$ is the minimal number of factors $F$ and $\bar{F}$ whose product contains $r$; $n_r$ equals the number of boxes of the Young diagram for $SU(2)$ and for representations first reached through $F^{\otimes k}$, and the adjoint of $SU(N)$ has $n=2$. Values: $U(1)$, ${\tilde{c}}_n=I_n(\beta )/I_0(\beta )$; $SU(2)$, ${\tilde{c}}_{1/2}=I_2(\beta )/I_1(\beta )=\beta /4+O({\beta }^3)$ (the fundamental is pseudo-real, so $\mathrm{tr}U$ and $\mathrm{tr}{U}^{†}$ both feed it); $SU(N\ge 3)$, ${\tilde{c}}_F=\beta /2N^2+O({\beta }^2)$.
• Kogut–Susskind Hamiltonians ($a_s=1$).
  • $U(1)$: $H=\frac{g^2}{2}{\sum }_{\ell }{E}_{\ell }^2-\frac{1}{g^2}{\sum }_P\cos {\theta }_P$, with $E_{\ell }\in \mathbb{Z}$, $[E_{\ell },U_{{\ell }^{\prime }}]={\delta }_{\ell {\ell }^{\prime }}{U}_{\ell }$, $U_{\ell }=e^{i{\theta }_{\ell }}$, and $g=e$. It is the $a_t\to 0$ limit of the anisotropic Wilson action with ${\beta }_t=1/(g^2{a}_t)$, ${\beta }_s=a_t/g^2$, and gives photons of unit speed. Gauss law: ${\sum }_{\mu }[E_{\mu }(x)-E_{\mu }(x-\hat{\mu })]=q_x$, i.e. $\delta E=-q$. Here $E_{\ell }$ is the momentum conjugate to the link angle; with the $a=-\phi dt+A\cdot dx$ of §10 it is minus the physical field, $E_{\ell }\leftrightarrow -E/e^2$, so the Kogut–Susskind charge $q_x$ is minus the physical charge of §10: the static Wilson line of a charge $+1$, running forward in time, carries $q_x=-1$ (Week 7 §4.2; Sem II Week 1 §5). Statements about the direction of electric flux in Weeks 7, 13 and 14 use the Kogut–Susskind sign. Strong-coupling string tension $\sigma =g^2/2$.
  • $SU(N)$: $H=\frac{g^2}{2}{\sum }_{\ell }{E}_{\ell }^a{E}_{\ell }^a-\frac{1}{g^2}{\sum }_P(\mathrm{tr}{U}_P+\mathrm{tr}{U}_P^{†})$, with $E^a{E}^a$ the quadratic Casimir of the link representation; strong-coupling tension $\sigma =\frac{g^2}{2}{C}_2(F)$, $C_2(F)=\frac{N^2-1}{2N}$. Gauge transformations act as $U_{\ell }\to {\Lambda }_x^{†}{U}_{\ell }{\Lambda }_{x+\hat{\mu }}$ for $\ell =(x\to x+\hat{\mu })$; the right and left generators obey $[E_R^a,U]=U{T}^a$ and $[E_L^a,U]=-T^{a}U$, and the Gauss generator is $G_x^a={\sum }_{\ell \mathrm{in}}{E}_R^a+{\sum }_{\ell \mathrm{out}}{E}_L^a$. These reduce to the $U(1)$ conventions above.
  • ${\mathbb{Z}}_2$ in $2+1$ dimensions: $H=-\Gamma {\sum }_{\ell }{\sigma }_{\ell }^x-K{\sum }_P{B}_P$, $B_P={\prod }_{\ell \in \partial P}{\sigma }_{\ell }^z$. Here ${\sigma }_{\ell }^z=U_{\ell }$ is the link variable, which flips the electric flux, and ${\sigma }_{\ell }^x=(-1{)}^{E_{\ell }}$ is the electric field; Gauss law ${\prod }_{\ell \ni v}{\sigma }_{\ell }^x=(-1{)}^{q_v}$.
• ℤ₂ gauge–Higgs model (Week 13). $S=-\beta {\sum }_P{\sigma }_P-\kappa {\sum }_{\ell ={\ell }_{\mu }(x)}{s}_x{\sigma }_{\ell }{s}_{x+\hat{\mu }}$ with $s\in C^0$, $\sigma \in C^1$; unitary gauge $Z=2^{N_s}{\sum }_{\sigma }\exp (\beta {\sum }_P{\sigma }_P+\kappa {\sum }_{\ell }{\sigma }_{\ell })$; abbreviations $t=\tanh \beta$, $u=\tanh \kappa$, $y=e^{-2\beta }$, $v=e^{-2\kappa }$. At strong coupling the string tension is $\sigma =\ln \coth \beta$ and the rest energy of a probe bound to a dynamical charge is $E_M=\ln \coth \kappa$, so the string breaks at $R_{\ast }=2E_M/\sigma$ (Week 13 §5). In $d=3$ the model is self-dual under $(\beta ,\kappa )\to ({\kappa }^{\ast },{\beta }^{\ast })$ with $x^{\ast }=-\frac{1}{2}\ln \tanh x$: $Z(\beta ,\kappa )=(\sinh 2\beta \sinh 2\kappa {)}^{3N_s/2}Z({\kappa }^{\ast },{\beta }^{\ast })$ on the $L^3$ torus, with self-dual line $\sinh 2\beta \sinh 2\kappa =1$. Hamiltonian form with matter: Gauss law ${\tau }_v^x{\prod }_{\ell \ni v}{\sigma }_{\ell }^x=1$; its unitary form is the toric code in fields, $-h\sum A_v-K\sum B_P-\Gamma \sum X-J\sum Z$, with the $e$–$m$ duality $(h,K,\Gamma ,J)\to (K,h,J,\Gamma )$.
• ${\mathbb{Z}}_N$ gauge theory with charge-$q$ matter (Week 14). $a\in C^1(\Lambda ,{\mathbb{Z}}_N)$, $\varphi \in C^0(\Lambda ,H)$ with $H=⟨q⟩=r{\mathbb{Z}}_N$, $r=\gcd (N,q)$, and

$$S[a,\varphi ]=-\beta \sum _P\cos \frac{2\pi (da{)}_P}{N}-\kappa \sum _{\ell }\cos \frac{2\pi (d\varphi -qa{)}_{\ell }}{N},$$

invariant under $a\to a+d\lambda$, $\varphi \to \varphi +q\lambda$; for $N=2$, $q=1$ it is Wegner’s action plus the hopping term $-\kappa \sum s_x{\sigma }_{\ell }{s}_y$ of Week 13. The ${\mathbb{Z}}_N$ character coefficients of the plaquette weight are ${\tilde{c}}_k=a_k/a_0$ with $a_k=\frac{1}{N}{\sum }_j{e}^{\beta \cos (2\pi j/N)}{\omega }^{-kj}$: ${\tilde{c}}_1=\tanh \beta$ for $N=2$; ${\tilde{c}}_1=\tanh (\beta /2)$ and ${\tilde{c}}_2={\tanh }^2(\beta /2)$ for $N=4$; ${\tilde{c}}_1={\tilde{c}}_2=(e^{3\beta /2}-1)/(e^{3\beta /2}+2)$ for $N=3$. For comparison, the Villain $U(1)$ weight has ${\tilde{c}}_q=e^{-q^2/2\beta }$. In the Hamiltonian limit of the ${\mathbb{Z}}_N$ transfer matrix, $e^{-{\beta }_t(1-\cos 2\pi /N)}=a_t\Gamma /2$ for $N\ge 3$ (Week 7 §7.1 gives the ${\mathbb{Z}}_2$ case). The matter weight expands as $e^{\kappa \cos (2\pi q{a}_{\ell }/N)}=b_0[1+{\sum }_{j=1}^{N^{\prime }-1}{h}_j{\omega }^{jq{a}_{\ell }}]$ with $N^{\prime }=N/r$ and $h_j={\tilde{c}}_j^{(N^{\prime })}(\kappa )$, the coefficients of the ${\mathbb{Z}}_{N^{\prime }}$ weight: $h_1=\tanh \kappa$ for $N^{\prime }=2$ and $h_1=\kappa /2+O({\kappa }^2)$ for $N^{\prime }\ge 3$ (Week 14 §4.3). Finite temperature: $N_{\tau }$ time slices, $T=1/N_{\tau }$, couplings ${\beta }_t$, ${\beta }_s$ on time-like and space-like plaquettes, and the Polyakov loop $L_e(\vec{x})={\omega }^{e{\sum }_{\tau }{a}_0(\vec{x},\tau )}$ ($P$ stays reserved for plaquettes). Hamiltonian form: $H=-\Gamma {\sum }_{\ell }\mathrm{Re}{X}_{\ell }-K{\sum }_P\mathrm{Re}{B}_P$ with clock and shift on each link, $Z_{\ell }|k⟩={\omega }^k|k⟩$ ($Z_{\ell }=U_{\ell }$, the link variable), $X_{\ell }|k⟩=|k+1⟩$ and $Z_{\ell }{X}_{\ell }=\omega X_{\ell }{Z}_{\ell }$, $G_x={\prod }_i{X}_{(x-\hat{i},i)}{X}_{(x,i)}^{†}$ and physical states $G_x=1$ (with matter, $G_x{\Pi }_x=1$); electric-flux operators $U_i(n)={\prod }_{x:x_i=n}{X}_{(x,i)}$. The charge-2 $U(1)$ model of Hansson–Oganesyan–Sondhi in the London limit is $S=-\beta {\sum }_P\cos (da{)}_P-\kappa {\sum }_{\ell }\cos (d\varphi -2a{)}_{\ell }$.

• Villain abelian-Higgs model of charge $q$ (Sem II Week 3 §5). $S=\frac{\beta }{2}\Vert da-2\pi n{\Vert }^2+\frac{\kappa }{2}\Vert d\varphi -qa-2\pi \ell {\Vert }^2$ with $a\in C^1$, $\varphi \in C^0$ over one period, $n\in C^2(\Lambda ,\mathbb{Z})$, $\ell \in C^1(\Lambda ,\mathbb{Z})$; invariant under $(a,n,\ell )\to (a+2\pi k,n+dk,\ell -qk)$, $(\varphi ,\ell )\to (\varphi +2\pi j,\ell +dj)$ and $(a,\varphi )\to (a+d\lambda ,\varphi +q\lambda )$. With $F=da-2\pi n$ and $D\varphi =d\varphi -qa-2\pi \ell$, $d(D\varphi )=-qF-2\pi v$ with the vortex number $v=qn+d\ell \in C^2(\Lambda ,\mathbb{Z})$; the photon mass is $m_A^2=q^2\kappa /\beta$, and $\kappa =\infty$ is the ${\mathbb{Z}}_q$ Villain theory of Sem II Week 2 (6.1).
• Spin chains (Sem II Week 5). $S^a={\sigma }^a/2$ and $n_j=S_j^z+\frac{1}{2}$ on sites $j=1,\dots ,L$; translation $T{S}_j^a{T}^{-1}=S_{j+1}^a$ and momentum $T|\psi ⟩=e^{iP}|\psi ⟩$. The flux-threaded XXZ ring is $H(\phi )={\sum }_j\left[\frac{J}{2}\right(e^{i\phi /L}{S}_{j+1}^{+}{S}_j^{-}+h.c.)+J\Delta S_j^z{S}_{j+1}^z]$, with the large gauge transformation $U=\exp \left(\frac{2\pi i}{L}{\sum }_{j}j{n}_j\right)$. The cluster chain is $H=-{\sum }_j{Z}_{j-1}{X}_j{Z}_{j+1}$, with the edge operators of Sem II Week 5 (5.5) and the twist rule $Z_{j+L}\equiv U{Z}_j{U}^{-1}$.
• Dualities of the ${\mathbb{Z}}_2$ models.
  • 2d Ising, Kramers–Wannier: $\sinh 2K\sinh 2K^{\ast }=1$, self-dual point $K_c=\frac{1}{2}\ln (1+\sqrt{2})=0.440687$.
  • 3d ${\mathbb{Z}}_2$ gauge theory ↔ 3d Ising on ${\Lambda }^{\ast }$: $\tanh \beta =e^{-2K^{\ast }}$. Strong coupling (small β, confined) maps to low temperature (large $K^{\ast }$, ordered Ising); weak coupling (deconfined) maps to the disordered Ising phase. ${\beta }_c=0.76141$ ↔ $K_c^{\ast }=0.221654$. The dual Ising spin is the endpoint of a string of flipped plaquettes (a vison), a local operator, and it condenses in the confined phase. Prefactors: on an open box with $+$ boundary spins, $Z_g(\beta )=2^{N_{\ell }}(2\sinh 2K^{\ast }{)}^{-N_P/2}{Z}_I^{+}(K^{\ast })$; on the 3-torus the eight boundary-condition sectors are summed, $Z_g(\beta )=2^{N_{\ell }-1}(2\sinh 2K^{\ast }{)}^{-N_P/2}{\sum }_{\epsilon \in {\mathbb{Z}}_2^3}{Z}_I^{(\epsilon )}(K^{\ast })$ (Week 5 §7.1 and F6). The value of $K_c^{\ast }$ is from Ferrenberg–Xu–Landau (2018).
  • In the Hamiltonian picture, ${\sigma }_{\ell }^x={\mu }_P^z{\mu }_{P^{\prime }}^z$ and $B_P={\mu }_P^x$ map the $2+1$d ${\mathbb{Z}}_2$ gauge theory to the transverse-field Ising model $-\Gamma \sum {\mu }^z{\mu }^z-K\sum {\mu }^x$ on dual sites; $\Gamma \gg K$ (confined) is the ferromagnet.
  • 4d ${\mathbb{Z}}_2$ gauge theory is self-dual, $\sinh 2\beta \sinh 2{\beta }^{\ast }=1$, with the self-dual point at $0.440687$.
  • Kramers–Wannier operator on the periodic chain of $N$ sites (Week 4 §4.4): $H(g)=-{\sum }_i({\sigma }_i^z{\sigma }_{i+1}^z+g{\sigma }_i^x)$ and $\eta ={\prod }_i{\sigma }_i^x$. The relations $D{\sigma }_i^x={\sigma }_i^z{\sigma }_{i+1}^{z}D$ and $D{\sigma }_i^z{\sigma }_{i+1}^z={\sigma }_{i+1}^{x}D$ fix $D$ up to a scale, chosen real; with the translation $T{\sigma }_i{T}^{-1}={\sigma }_{i+1}$ they give $D^{†}D=1+\eta$, $D^2=(1+\eta )T$ (with $T$, not $T^{-1}$), $D^{†}=T^{-1}D$ and $DH(g)=gH(1/g)D$. The closed form (3.12) of Sem II Week 4, $D=2^{-L/2}{\sum }_{s,s^{\prime }}(-1{)}^{{\sum }_i{s}_i(s_i^{\prime }+s_{i+1}^{\prime })}|s^{\prime }⟩⟨s|$ on a chain of $L$ sites, fixes the overall sign. Transfer matrix of the classical Ising model (Sem II Week 13 §2): $V_1(K_v)={\prod }_i(e^{K_v}+e^{-K_v}{\sigma }_i^x)=(2\sinh 2K_v{)}^{L/2}{e}^{K_v^{\ast }{\sum }_i{\sigma }_i^x},V_2(K_h)=e^{K_h{\sum }_i{\sigma }_i^z{\sigma }_{i+1}^z}$, with $\tanh K^{\ast }=e^{-2K}$ and the symmetric row transfer matrix $\mathbb{T}=V_2^{1/2}{V}_1{V}_2^{1/2}$, the only form with which the dressed duality line $V_2^{1/2}D{V}_2^{1/2}$ commutes ($Z=\mathrm{Tr}{\mathbb{T}}^M$ is unchanged; the letter $T$ stays the translation). A duality line on the ring is $H_D^{(d)}$ of Week 13 (3.6), moved by the two-site unitary $U_d=H\cdot CZ$; gauging on an interval uses Dirichlet ends, and the end qubits are written $\tilde{\sigma }$ (τ is reserved). For ${\mathbb{Z}}_N$ the duality operator carries the phase ${\omega }^{{\sum }_i{s}_i(s_i^{\prime }-s_{i+1}^{\prime })}$. Gauging the chain puts link qubits $X,Z$ at $i+\frac{1}{2}$ with Gauss law $G_i=X_{i-\frac{1}{2}}{\sigma }_i^x{X}_{i+\frac{1}{2}}$ ($G_i=X_{i-\frac{1}{2}}^{†}{X}_i{X}_{i+\frac{1}{2}}$ for ${\mathbb{Z}}_N$), and the dual chain sits at the shifted sites $i+\frac{1}{2}\mapsto i+1$.
• Modified Villain (Sem II Wk 12): promote $n$ to a $\mathbb{Z}$ gauge field with its own constraint, $dn=0$, enforced by a compact Lagrange multiplier on the dual lattice with the weight $e^{+i⟨\tilde{\theta },\star dn⟩}$ (XY) or $e^{+i⟨\tilde{\varphi },\star dn⟩}$ (gauge); this sign makes the multiplier the dual field of §5, so that defects would enter as $e^{-i⟨v,\tilde{\theta }⟩}$, and it is opposite to GLSS’s default. The defects are removed instead of summed, and the dualities become exact: $Z_p[\Lambda ;\beta ]=(2\pi \beta {)}^{-N_{p+1}/2}{Z}_{d-p-2}[{\Lambda }^{\ast };\tilde{\beta }]$ with $\tilde{\beta }=1/4{\pi }^2\beta$, where $N_{p+1}$ is the number of $(p+1)$-cells (Sem II Week 12 (2.7), (3.4)–(3.5)); the dual integer is $\tilde{n}=\star m$ in $d=2$ and $\tilde{n}=-\star m$, $a^{\vee }=-a$ in $d=4$. Summation by parts on the two lattices: $⟨\tilde{\varphi },\star d\omega ⟩=(-1{)}^{q+1}⟨\tilde{d}\tilde{\varphi },\star \omega ⟩$ (Week 12 (2.3)). The exact magnetic operator of monopole-free compact QED is $U_{\alpha }^{\mathrm{m}}(\Sigma )=e^{\frac{i\alpha }{2\pi }{\sum }_{\Sigma }F}=e^{-i\alpha {\sum }_{\Sigma }n}$, the 4d ‘t Hooft line is $T_m=e^{-im\sum \tilde{a}}$ (against $e^{+iq\sigma }$ for the 3d monopole), and a ${\mathbb{Z}}_N$ electric background enters as $da-2\pi n-\frac{2\pi }{N}B$ with the constraint $d(Nn+B)=0$.

5. Compact scalars and compact QED: normalizations of the dual descriptions

• XY and BKT (Weeks 1–4). Spin-wave exponent $\eta =1/(2\pi \beta )$ and amplitude $C(x)=e^{-\kappa /\beta }|x{|}^{-\eta }$ at the Gaussian level. Vortex pair energy $2\pi \beta \ln (r/a_{\mathrm{lat}})+2E_{\mathrm{core}}$ with $E_{\mathrm{core}}=2{\pi }^2\beta \kappa$; fugacity $y=e^{-2{\pi }^2\beta \kappa }$, a long-distance matching convention with vertex operators normal-ordered at the lattice scale, not an exact microscopic identity. Sine-Gordon dual: $S=\frac{1}{8{\pi }^2\beta }\int (\partial \chi {)}^2-2y\int \cos \chi$, with scaling dimensions $\pi \beta$ for $e^{i\chi }$ and $q^2/4\pi \beta$ for $e^{iq\theta }$. The universal statement is $\pi {\beta }_R=2$ at the transition for the renormalized stiffness; the bare critical couplings are model-dependent, ${\beta }_c\approx 1.12$ for the cosine model (Hasenbusch 2005) and ${\beta }_c\approx 0.75$ for the Villain model (Janke–Nather 1993), both above $2/\pi$ because screening lowers the stiffness.
• Continuum compact QED. $S=\frac{1}{2e^2}\int F\wedge \star F$; the unit electric charge defines the Wilson line $e^{i\oint a}$, and Dirac quantization reads ${\oint }_{{\Sigma }_2}F\in 2\pi \mathbb{Z}$.
• Dual photon in $d=3$ (erratum E2). $\sigma \sim \sigma +2\pi$, with

$$S[\sigma ]=\frac{e^2}{8{\pi }^2}\int d^{3}x(\partial \sigma {)}^2,B_{\mu }\equiv \frac{1}{2}{ϵ}_{\mu \nu \rho }{F}_{\nu \rho }=+\frac{i{e}^2}{2\pi }{\partial }_{\mu }\sigma (\text{Euclidean}),$$

derived by adding $+\frac{i}{2\pi }\int \sigma (\partial \cdot B-2\pi {\rho }_m)$ to $\frac{1}{2e^2}\int B^2$ and integrating out $B$; the $2\pi$-periodicity of σ is Dirac quantization. With this sign a monopole of magnetic charge $q$ ($\oint B=2\pi q$) is the insertion $e^{iq\sigma }$, and $⟨B_{\mu }(x)e^{iq\sigma (0)}⟩/⟨e^{iq\sigma (0)}⟩=q{x}_{\mu }/2|x{|}^3$ carries outward flux $2\pi q$. (The opposite pairing, $B=-\frac{i{e}^2}{2\pi }\partial \sigma$ with $e^{-iq\sigma }$, is the same convention after $\sigma \to -\sigma$; sources that use it are translated before a formula is imported.) Lattice monopoles, whose flux is $-2\pi m_c$ (§4), enter as $e^{-i{m}_c\sigma }$, which is immaterial once $\pm$ charges are summed. The magnetic current is $j_{\mathrm{mag}}^{\mu }=\frac{1}{4\pi }{ϵ}_{\mu \nu \rho }{F}_{\nu \rho }=B_{\mu }/2\pi$.

• Monopole gas in $d=3$. Pair interaction $\frac{4{\pi }^2}{e^2}{\sum }_{i<j}{q}_i{q}_j{G}_3(x_i-x_j)$, equivalently $\frac{2{\pi }^2}{e^2}{\sum }_{i\ne j}$. Villain monopole action $S_{\mathrm{mono}}=2{\pi }^2\beta G_3(0)=4.99\beta$ with $\beta =1/(e^2{a}_{\mathrm{lat}})$; fugacity $\zeta \propto e^{-S_{\mathrm{mono}}}$ per unit volume. Dilute-gas effective action $S=\int [\frac{e^2}{8{\pi }^2}(\partial \sigma {)}^2-2\zeta \cos \sigma ]$, photon mass $m_{\gamma }^2=8{\pi }^2\zeta /e^2$, equal to the Debye mass $m_D$ of the gas. The small parameter that controls the dilute-gas and semiclassical steps is $\zeta /e^6$. Vertex operators are normal-ordered at the lattice scale, $V_{\pm }=e^{S_{\mathrm{mono}}}{e}^{\pm i\sigma }$; in the monopole-free theory $|⟨e^{i\sigma }⟩|=e^{-S_{\mathrm{mono}}}$. Plasma scales (Week 9 §2.5): mean separation $\bar{r}=(2\zeta {)}^{-1/3}$, Bjerrum length ${\ell }_B=\pi /e^2$, Debye length ${\lambda }_D=1/m_D$, with ${\lambda }_D/\bar{r}=2^{5/6}e/(4\pi {\zeta }^{1/6})$ and ${\ell }_B/{\lambda }_D=2\sqrt{2}{\pi }^2(\zeta /e^6{)}^{1/2}$; on the Villain lattice the ordered regime holds for $\beta \gtrsim 4$–5.
• The Wilson loop in compact QED₃ (Week 10). $K\equiv e^2/8{\pi }^2$ is the dual-photon stiffness, $S=\int [K(\partial \sigma {)}^2-2\zeta \cos \sigma ]$, so $m_{\gamma }^2=\zeta /K$ (the letter $A$ stays reserved for the area of a loop). Through the moves of Week 8 a charge-$q$ Wilson loop on $C$ becomes $d\tilde{n}=-q\star J_C$: σ winds by $2\pi q$ around $C$ in the right-handed sense. Each monopole sees the loop through ${\eta }_C=2\pi {\omega }_{\Sigma }={\Omega }_C/2$, with the solid angle ${\Omega }_C$ positive on the side of the right-handed normal of the spanning surface, and in the plasma $\cos \sigma \to \cos (\sigma -q{\eta }_C)$. Pinned values: the kink ${\sigma }_{\mathrm{k}}=4\arctan e^{m_{\gamma }\xi }$; the string tension ${\sigma }_{\mathrm{str}}=16K{m}_{\gamma }=2e^2{m}_{\gamma }/{\pi }^2$; the number of monopoles per Debye volume $N_D=2\zeta /m_{\gamma }^3=2K/m_{\gamma }$, whose inverse sets the size of loop corrections; the deconfinement temperature $T_c=2e^2/\pi$, with ${\beta }_{XY}=T/e^2$ for the dual-photon XY model of the thermal theory.
• Duality in $d=4$. Villain $U(1)$ at coupling $\beta$ is dual to Villain $U(1)$ on ${\Lambda }^{\ast }$ at $\tilde{\beta }=1/(4{\pi }^2\beta )$, i.e. $\tilde{e}=2\pi /e$; the dual field has no dual monopoles and couples to the integer monopole current, which is conserved.
• Dual variables on the lattice (Week 8). In the gauge weeks the Hubbard–Stratonovich integer, the electric flux, is $b\in C^2(\Lambda ,\mathbb{Z})$ with $\delta b=0$; $m=dn$ is reserved for the monopole number (in the XY weeks the current integer is $m\in C^1(\Lambda ,\mathbb{Z})$ and the vorticity is $v=dn$). With $\tilde{\phi }$ and $\tilde{A}$ the Poisson conjugates of the integer dual fields in the form of §3, the compact dual fields are $\sigma =-2\pi \tilde{\phi }$ ($d=3$) and $\tilde{a}=-2\pi \tilde{A}$ ($d=4$); then $F\simeq \frac{i}{2\pi \beta }\star d\sigma$, i.e. $B=+\frac{i{e}^2}{2\pi }\partial \sigma$, and $F\simeq 2\pi i\tilde{\beta }\star d\tilde{a}$, and the monopoles couple as $e^{-i{m}_c\sigma }$ and $e^{-i⟨j,\tilde{a}⟩}$. The fugacity in the lattice normal-ordering scheme is $\zeta =e^{-S_{\mathrm{mono}}}/a_{\mathrm{lat}}^3$.
• Monopole current in $d=4$. $j=\star dn$ on dual links, $j_4=+m_{123}$, $j_3=-m_{124}$, $j_2=+m_{134}$, $j_1=-m_{234}$ (the shuffle signs of §2), and $\delta j=0$ follows from $\star \star =-1$ on $C^3$ in $d=4$.
• Four-dimensional compact QED (Week 11). Villain–Wilson matching by the first character, $e^{-1/2{\beta }_V}=I_1({\beta }_W)/I_0({\beta }_W)$ (good to a few per cent). The twisted-sheet operator $T_{\alpha }$, the plaquette weights shifted by α on the plaquettes dual to a closed dual surface, is the generator of the electric $U(1{)}^{(1)}$; in the Villain theory with dynamical monopoles the integer ‘t Hooft loop is a relabeling of the Villain integers and equals 1 identically, since dynamical monopoles screen it. Dual Ginzburg–Landau normalization: $\frac{1}{4{\tilde{e}}^2}{\tilde{F}}^2+|\tilde{D}\varphi {|}^2+\frac{\lambda }{4}(|\varphi {|}^2-v^2{)}^2$, with $m_V^2=2{\tilde{e}}^2{v}^2$ and $m_H^2=\lambda v^2$. Julia–Toulouse/Kalb–Ramond form of the monopole condensate: $\frac{1}{2e^2}(da-\Lambda {)}^2+\frac{1}{2g^2}(d\Lambda {)}^2$, with mass $m^2=g^2/e^2$, dual to the BF form $-\frac{i}{2\pi }\int \Lambda \wedge d\tilde{a}$ with the dual photon. Continuum inner product $⟨\alpha ,\beta ⟩=\int \alpha \wedge \star \beta$. Further lattice values: $G_4(\hat{e})=G_4(0)-\frac{1}{8}$, and the Bessel representation $G_d(x)={\int }_0^{\infty }dt{e}^{-2dt}{\prod }_{\mu }{I}_{x_{\mu }}(2t)$.
• Magnetic current in $d=4$ (Sem II Week 3 §3). $J_m^{\nu \mu }=\frac{1}{4\pi }{ϵ}_{\nu \mu \rho \sigma }{F}_{\rho \sigma }$ with ${ϵ}_{\tau 123}=+1$; a ‘t Hooft line of charge $m$ on $C$ is defined by ${\partial }_{\mu }{J}_m^{\nu \mu }=m{J}_C^{\nu }$, outward magnetic flux $2\pi m$.
• Transport of sources. For $\eta \in C^3(\Lambda )$, $\star (\delta \eta )=-d(\star \eta )$ in both $d=3$ and $d=4$, a case of $\delta =(-1{)}^{d(p+1)+1}\star d\star$.

6. Extended operators, symmetry operators and the form-degree master table

• Wilson operator of charge $q$ on a closed $p$-cycle $C$: $W_q(C)=\exp (iq{\sum }_{C}a)$, or ${\omega }^{q{\sum }_{C}a}$ in the ${\mathbb{Z}}_N$ case.
• ‘t Hooft and disorder operators live on cycles of ${\Lambda }^{\ast }$ and are defined by their modification of the path-integral sum (Kadanoff–Ceva style in Sem I Wk 4; the 4d ‘t Hooft loop, which flips the plaquettes dual to a dual surface bounded by $\tilde{C}$, in Wk 11).
• Symmetry and ‘t Hooft operators of ${\mathbb{Z}}_N$ lattice gauge theory. Replace $(da{)}_P$ by $(da{)}_P-1$ in the plaquette weights (reverse β for ${\mathbb{Z}}_2$) on the plaquettes dual to a $(d-2)$-chain $\tilde{c}$ of ${\Lambda }^{\ast }$. If $\tilde{c}$ is closed and bounds, $\tilde{c}=\partial \tilde{V}$, let ${\lambda }_{\tilde{V}}$ be the signed indicator of the links dual to the cells of $\tilde{V}$, oriented so that $d{\lambda }_{\tilde{V}}=+1$ on the plaquettes dual to $\tilde{c}$; the change of variables $a\to a+{\lambda }_{\tilde{V}}$ removes the insertion, and the result is the generator $U(\tilde{c})$ of the electric 1-form symmetry, with $⟨U⟩=1$ in every phase and $⟨W(C)U(\tilde{c})⟩={\omega }^{\mathrm{Link}(C,\tilde{c})}⟨W(C)⟩$. This defines the linking number of a loop of $\Lambda$ and a cycle of ${\Lambda }^{\ast }$: $\mathrm{Link}(C,\partial \tilde{V})\equiv ⟨J_C,{\lambda }_{\tilde{V}}⟩=\#(C\cap \tilde{V})$, the signed number of links of $C$ dual to cells of $\tilde{V}$ (Weeks 5, 11 and 14). For $U(1)$ the twisted sheet replaces $da-2\pi n$ by $da-2\pi n-\alpha d{\lambda }_{\tilde{V}}$ and gives $e^{iq\alpha \mathrm{Link}(C,\tilde{c})}$ on $W_q(C)$ (Week 11 §5.1). On a closed $\tilde{c}$ that does not bound, such as a noncontractible cycle of a torus, the same insertion is a twisted boundary condition, and $⟨U⟩=Z_{\mathrm{tw}}/Z$ need not equal 1 (Week 5 F6; Week 15 §3.4). If $\tilde{c}$ has a boundary, the result is a ‘t Hooft (disorder) operator of dimension $d-3$ supported on $\partial \tilde{c}$: the vison, a local operator, in $d=3$, and the ‘t Hooft loop $\tilde{W}$ in $d=4$ (Week 5 §8).
• A $q$-form symmetry acts on $q$-dimensional charged objects. Its symmetry operator is $U_{\alpha }(\Sigma )=\exp \left(i\alpha {\oint }_{\Sigma }\omega \right)$ on a closed $(d-q-1)$-dimensional $\Sigma$, with $\omega$ the closed form listed below; the charge is measured by linking, $U_{\alpha }(\Sigma )W(C)=e^{i\alpha \mathrm{Link}(\Sigma ,C)}W(C)U_{\alpha }(\Sigma )$ as a statement about correlators. On a time slice the operator statement is $U_{\alpha }{W}_q(\Gamma )U_{\alpha }^{-1}=e^{-iq\alpha I_s}{W}_q(\Gamma )$, with $I_s$ the intersection number of the closed surface and the line in the slice; the minus sign is the jump $-I_s$ of the spacetime linking number as the slice passes the line. In the continuum, Link $=+1$ for a worldline running in $+\tau$ and the outward-oriented sphere around it ($d=4$), and for two loops by the right-hand rule ($d=3$). The closed forms listed below are the real-time ones; in Euclidean signature they carry a factor of $i$, ${\omega }_E=i\beta \star d\theta$ for the shift symmetry of the compact scalar and $-\frac{i}{e^2}\star F$ for the electric symmetry of Maxwell theory (Sem II Week 1 §§3, 6).
• Ground-state degeneracy bookkeeping on $T^d$ and ${\Sigma }_g$ uses $H_1$ with the intersection pairing; clock–shift pairs $(Z,X)$ obey $ZX=\omega XZ$.

Form-degree master table (erratum E1). Every statement about a higher-form symmetry in the notes is checked against this table.

Theory ($d$ Euclidean dimensions)	Symmetry	Degree	Closed form $\omega$ (closed by)	Charged objects
compact scalar θ	shift (“electric”)	0	$\beta \star d\theta$ (equation of motion)	local $e^{iq\theta }$
compact scalar θ	winding (“magnetic”)	$d-2$	$d\theta /2\pi$ (identically; vortices break it explicitly)	vortex operators, dimension $d-2$
$U(1)$ gauge field $a$	electric	1	$\star F/e^2$ (equation of motion)	Wilson lines
$U(1)$ gauge field $a$	magnetic	$d-3$	$F/2\pi$ (Bianchi; dynamical monopoles break it explicitly)	‘t Hooft operators, dimension $d-3$
$U(1)$ $p$-form gauge field	electric / magnetic	$p$ / $d-p-2$	$\star F/e^2$ / $F/2\pi$	$p$- / $(d-p-2)$-dimensional operators
${\mathbb{Z}}_N$ $p$-form gauge theory (BF)	electric / magnetic	$p$ / $d-p-1$	topological	Wilson $p$-surfaces / $(d-p-1)$-surfaces

• Lattice versus TQFT. The ${\mathbb{Z}}_N$ BF row describes the deconfined topological limit. The Wegner lattice model has only the electric ${\mathbb{Z}}_2^{(1)}$ as an exact symmetry; its magnetic symmetry emerges in the deconfined phase, since visons are dynamical at finite β.
• Low dimensions. For Maxwell theory in $d=3$ the magnetic symmetry is a 0-form $U(1)$ whose charged operators are the local monopole operators $e^{i\sigma }$, and the dual photon is the Goldstone boson of its spontaneous breaking in the monopole-free theory. In $d=4$ both symmetries are 1-form and act on Wilson and ‘t Hooft lines.
• Projective representations and anomalies (Sem II Week 5). $U(g)U(h)=\omega (g,h)U(gh)$; a rephasing $U\to \beta U$ changes ω by the coboundary $\delta \beta (g,h)=\beta (g)\beta (h)/\beta (gh)$; for abelian $G$ the class is detected by $c(g,h)=\omega (g,h)/\omega (h,g)$. For ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ the Pauli representation $U(a^{g_1}{b}^{g_2})=X^{g_1}{Z}^{g_2}$ has $\omega (g,h)=(-1{)}^{g_2{h}_1}$; for ${\mathbb{Z}}_N\times {\mathbb{Z}}_N$ class $k$ is $U(a)=X$, $U(b)=Z^{-k}$, $c(a,b)=e^{2\pi ik/N}$ (with $ZX=\omega XZ$, $X{Z}^k={\omega }^{-k}{Z}^{k}X$). An ‘t Hooft anomaly is a phase $Z[A^{\lambda }]=e^{i\mathcal{A}(A,\lambda )}Z[A]$ under background gauge transformations that no local counterterm removes, $\mathcal{A}\to \mathcal{A}+S_{\mathrm{ct}}[A^{\lambda }]-S_{\mathrm{ct}}[A]$. The ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ SPT weight is $(-1{)}^{\sum A\cup B}$; its edge obeys $Z_{\mathrm{edge}}[A,B+d\beta ]=(-1{)}^{{\sum }_t{A}_t{\beta }_{t+1}}{Z}_{\mathrm{edge}}[A,B]$ with transfer matrices $M_t=e^{-ϵH}V(a{)}^{A_t}V(b{)}^{B_t}$, later times to the left.
• Group cohomology, MPS and anyon data (Sem II Week 7). Inhomogeneous $U(1)$-valued cochains of $G$ with coboundary

$$(\delta \omega )(g_1,\dots ,g_{n+1})=\omega (g_2,\dots ,g_{n+1})\prod _{i=1}^n\omega (g_1,\dots ,g_i{g}_{i+1},\dots ,g_{n+1}{)}^{(-1{)}^i}\omega (g_1,\dots ,g_n{)}^{(-1{)}^{n+1}},$$

normalized ($\omega =1$ whenever an argument is the identity); in degree 2 this is the coboundary of the projective-representation bullet above. Homogeneous form: $\nu (g_0,\dots ,g_d)=\omega (g_0^{-1}{g}_1,\dots ,g_{d-1}^{-1}{g}_d),\nu (g{g}_0,\dots ,g{g}_d)=\nu (g_0,\dots ,g_d)$, with simplex weights ${\nu }^{s_{\Delta }}$. Representatives: $(-1{)}^{xyz}$ for $H^3({\mathbb{Z}}_2,U(1))={\mathbb{Z}}_2$ and ${\omega }_k(x,y,z)=\exp (\frac{2\pi ik}{N}x\mathrm{carry}(y,z))$ for ${\mathbb{Z}}_N$. Matrix-product states: $E(X)={\sum }_s{A}^{s}X{A}^{s†},E^{\ast }(Y)={\sum }_s{A}^{s†}Y{A}^s,E_O(X)={\sum }_{s,s^{\prime }}{O}_{s{s}^{\prime }}{A}^{s^{\prime }}X{A}^{s†}$, normalized by $E(1)=1$ and $E^{\ast }(\rho )=\rho >0$, $\mathrm{tr}\rho =1$; a symmetry acts as $(u\cdot A{)}^s={\sum }_{s^{\prime }}{u}_{s{s}^{\prime }}{A}^{s^{\prime }}$ and on the virtual space as $u\cdot A=e^{i\theta }{V}^{†}AV$. Anyons: $F$ maps $(X\otimes Y)\otimes Z\to X\otimes (Y\otimes Z)$, $R$ maps $X\otimes Y\to Y\otimes X$, ${\theta }_X=R(X,X)$; for Dijkgraaf–Witten theories $F=\omega$ on fluxes and $R(X,Y)={\chi }_X(y)$. The lattice ${\mathbb{Z}}_2$ theory of class $k$ (the letter $k$ is also the Chern–Simons level of §10; context separates them) is

$$Z_k(M)=\frac{1}{2^{N_0}}\sum _{a\in Z^1(\Lambda ,{\mathbb{Z}}_2)}(-1{)}^{k{\sum }_{\mathrm{cubes}}a\cup a\cup a}=\frac{1}{|H^0|}\sum _{[a]\in H^1(M,{\mathbb{Z}}_2)}(-1{)}^{k⟨a^3,[M]⟩},$$

and the double semion has labels $\{1,b,s,\bar{s}\}$ with ${\theta }_s=+i$. Six-index $F$-symbols and the pentagon are Sem II Week 11 (2.1)–(2.2); for abelian data they reduce to $F(a,b,c)$, and $[F_a^{aaa}{]}_{00}={\kappa }_a/d_a$, so that a closed loop of label $a$ evaluates to ${\kappa }_a{d}_a$, which is not a quantum dimension when ${\kappa }_a=-1$; the label 0 is the null string and 1 the trivial anyon.

• Gauging. Gauging a ${\mathbb{Z}}_N^{(q)}$ symmetry in $d$ dimensions produces a dual ${\mathbb{Z}}_N^{(d-q-2)}$ symmetry (for $q=0$, $d=2$: Kramers–Wannier). The degree $d-p-1$ of the ${\mathbb{Z}}_N$ magnetic symmetry and the degree $d-q-2$ of the dual symmetry are different statements. With the normalization of Sem II Week 4 (2.4)–(2.5), $Z_{T/{\mathbb{Z}}_N^{(p)}}=\frac{|H^{p-1}||H^{p-3}|\cdots }{|H^p||H^{p-2}|\cdots }{\sum }_{[B]\in H^{p+1}}{Z}_T[B]$ (prefactor $1/|H^0|$ for $p=0$ and $|H^0|/|H^1|$ for $p=1$); the dual background $C\in C^{d-p-1}$ couples as ${\omega }^{\sum B\cup C}$, and gauging the dual symmetry with ${\omega }^{-\sum B^{\prime }\cup C}$ returns $T$ up to $N^{\pm \chi (X)}$, which is 1 on tori. Aharony–Seiberg–Tachikawa label lines by $(z_e,z_m)\in {\mathbb{Z}}_N\times {\mathbb{Z}}_N$ with mutual locality $z_e{z}_m^{\prime }-z_m{z}_e^{\prime }\equiv 0$ mod $N$, and θ → θ + 2π acts as $(z_e,z_m)\to (z_e+z_m,z_m)$, the direction of §10.
• Higher-form Coleman–Mermin–Wagner. A continuous $q$-form symmetry cannot be spontaneously broken if $q\ge d-2$ (GKSW §5). So a continuous 1-form symmetry can break only for $d\ge 4$; $d=3$ is the marginal case, where the charged loop has a logarithmic potential. Discrete symmetries break in lower dimensions: ${\mathbb{Z}}_2^{(0)}$ in $d=2$ (Ising), ${\mathbb{Z}}_2^{(1)}$ in $d=3$ (the deconfined ${\mathbb{Z}}_2$ gauge theory); a discrete $q$-form symmetry cannot break if $q\ge d-1$ (Sem II Week 3 §4).
• SSB criterion for 1-form symmetries. A perimeter law for the charged loop means the symmetry is spontaneously broken; an area law means it is unbroken. Precisely (Sem II Week 3 (2.1)): dilate a fixed shape $C$ by λ; the symmetry is unbroken if $|⟨W(C_{\lambda })⟩|e^{{\int }_{C_{\lambda }}\mu ds}\to 0$ for every local counterterm μ, and broken if it stays bounded away from zero for some μ. The product need not converge: corners add powers of λ that no local term removes. The criterion presupposes an exact symmetry: charged dynamical matter breaks the electric 1-form symmetry explicitly, and then Wilson loops obey a perimeter law in every phase (string breaking).
• Realizations in Semester I. (In the Villain lattice theory, where monopoles are dynamical, the integer ‘t Hooft loop is screened and equals 1 identically; statements about ‘t Hooft loops below refer to their perimeter behaviour or to the continuum limit.) 3d compact QED: electric $U(1{)}^{(1)}$ unbroken (area law at every coupling), magnetic $U(1{)}^{(0)}$ explicitly broken by monopoles, photon gapped. 4d compact QED, Coulomb phase: both loops obey perimeter laws; the electric 1-form symmetry and the emergent magnetic one are spontaneously broken. 4d confining phase: Wilson loops obey an area law and ‘t Hooft loops a perimeter law; the electric symmetry is unbroken and the magnetic one explicitly broken. Gauge–Higgs with fundamental matter: the electric symmetry is explicitly broken, so no 1-form symmetry distinguishes Higgs from confinement (Fradkin–Shenker). ${\mathbb{Z}}_N$ gauge theory with charge-$q$ matter keeps the residual ${\mathbb{Z}}_{\gcd (N,q)}^{(1)}$.

7. ${\mathbb{Z}}_N$ fields and BF presentation

• ${\mathbb{Z}}_N$ $p$-form field: $a\in C^p(\Lambda ,{\mathbb{Z}}_N)$, represented by integers mod $N$; $\omega =e^{2\pi i/N}$.
• ${\mathbb{Z}}_N$ BF weight in $d=4$ for a 1-form ${\mathbb{Z}}_N$ gauge field: $\exp (\frac{2\pi i}{N}\sum b\cup da)$ with $a\in C^1(\Lambda ,{\mathbb{Z}}_N)$ and $b\in C^2(\Lambda ,{\mathbb{Z}}_N)$ both on $\Lambda$ and the cup product of §8; in $d=3$ the same with $b\in C^1$. The GLSS presentation with $b$ on ${\Lambda }^{\ast }$ is equivalent; Sem II Week 2 §6 gives the dictionary. With ${\Lambda }^{\ast }=\Lambda +h$, $h=\frac{1}{2}(1,\dots ,1)$, and $\tilde{b}(y+h)$ the dual-lattice copy of $b$, $\sum b\cup da=⟨\tilde{b},\star da{⟩}_{{\Lambda }^{\ast }}=⟨a,\star d\tilde{b}{⟩}_{\Lambda }$: the cup product places $b$ on $\Lambda +h$ and $a\cup db$ on $\Lambda -h$, a lattice framing of the two kinds of line. GLSS’s integer is $\tilde{n}=-\tilde{b}$ up to their orientation convention (their weight is $e^{-S}$). Normalized, $Z=N^{-N_0-N_1}\sum {\omega }^{\sum b\cup da}=|H^1|/|H^0|$ in $d=3$. In the continuum, with orientation $dt\wedge dx\wedge dy$ and ${ϵ}^{xy}=+1$, the bracket is $[a_i(x),b_j(y)]=\frac{2\pi i}{N}{ϵ}_{ij}{\delta }^2(x-y)$; the generator of the electric ${\mathbb{Z}}_N^{(1)}$ is the $b$-line $U_s=V_{-s}=\exp (-is\oint b)$, i.e. ${\omega }_e=-\frac{N}{2\pi }b$, so that $⟨W_e{U}_s⟩={\omega }^{es\mathrm{Link}}$ with the Link of §6, and the $a$-line generates the magnetic one. In the Hamiltonian lattice form, $e^{i\int b}$ across a link is $X^{†}$ and $U$ is a product of $X$. In the continuum the weight is $\exp (\frac{iN}{2\pi }\int b\wedge da)$.
• When comparing with the group’s manuscript on higher-gauging walls, that manuscript’s normalizations take precedence; discrepancies get a margin note in the lecture notes rather than a silent fix.
• Higher-gauging walls (Sem II Week 14). In four-dimensional ${\mathbb{Z}}_N$ gauge theory, $H=⟨k⟩=r{\mathbb{Z}}_N\cong {\mathbb{Z}}_{N^{\prime }}$ is a subgroup of the electric ${\mathbb{Z}}_N^{(1)}$, a group of fluxes labelling the surfaces $U_h$, $h\in H$, and the wall on a closed 3-manifold $Y$ is $D_H(Y)={\mathcal{N}}_Y^{-1}{\sum }_{S\in Z_2(Y;H)}U[S]$, with the manuscript’s ${\mathcal{N}}_Y$ fixed by letting a transparent line cross with amplitude 1 (${\mathcal{N}}_Y=|Z_2(Y;H)|$ when every closed surface bounds). The slab normalization of Sem II Week 13 (4.2) gives $C_1=Z_H(Y)D_H$ with $Z_H(Y)=|H^1(Y;H)|/|H^0(Y;H)|$. A charge-$q$ Wilson line crosses intact iff $q\in \mathrm{Ann}(H)=N^{\prime }{\mathbb{Z}}_N$; otherwise it carries the endpoint charge $\alpha ={\mathrm{res}}_H(q)=q$ mod $N^{\prime }\in \hat{H}\cong {\mathbb{Z}}_{N^{\prime }}$. Two meanings of $H$: in Sem I Week 14 §3, Sem II Week 13 §5 and Sem II Week 14 §4.1, $H$ is a group of condensed charges, whose Wilson lines end on the sheet, and a flux $s$ passes iff $s\in \mathrm{Ann}(H)$; the pairing ${\omega }^{qh}$ exchanges the two readings, and each note says which it uses. Wall currents ${\ell }_e\in {\mathbb{Z}}_{N^{\prime }}$, ${\ell }_{-e}=-{\ell }_e$, follow the manuscript’s signs: the outgoing divergence $({\partial }_Y\ell {)}_v={\sum }_{e\text{ out of }v}{\ell }_e=-(\partial \ell {)}_v$, the spin difference $\Delta {\theta }_e={\theta }_v-{\theta }_{v^{\prime }}=-(d\theta {)}_e$ for $e:v\to v^{\prime }$, the source $J=\alpha ({\delta }_x-{\delta }_{x^{\prime }})$, and the junction $E_{\alpha }$ represented by the clock operator ${\omega }_{N^{\prime }}^{-\alpha \theta }$. The periodic-Gaussian weight is $W_{\kappa }(\ell )={\sum }_{m\in \mathbb{Z}}{e}^{-\frac{\kappa }{2}(\ell +N^{\prime }m{)}^2}$, $w_{\kappa }=W_{\kappa }/W_{\kappa }(0)$. Exchange charges of bulk monopoles entering the wall are written ν, with $r_{\mathrm{ex}}=\gcd (N^{\prime },\{\nu \})$ (the manuscript’s $r$ and $d$).

8. Cup products

• Definition (the cubical cup product). For $\alpha \in C^p$, $\beta \in C^q$ and a $(p+q)$-cell $(x;S)$,

$$(\alpha \cup \beta )(x;S)=\sum _{\begin{array}{c}A\subset S\\ |A|=p\end{array}}ϵ(A,S\setminus A)\alpha (x;A)\beta (x+{\hat{e}}_A;S\setminus A),$$

a sum over the shuffles of $S$ with their signs: α sits on the face at the base corner, β on the complementary face at the far corner. Chen–Tata (arXiv:2106.05274, eqs. (25)–(26)) order the factors the other way, so their product is $(-1{)}^{pq}$ times ours with the factors exchanged; their formulas, eq. (29) in particular, are translated before use. In low degrees: $(f\cup \beta )(\ell )=f(\text{tail})\beta (\ell )$, $(\beta \cup f)(\ell )=\beta (\ell )f(\text{head})$, $(f\cup \omega )(P(x))=f(x)\omega (P)$, $(\omega \cup f)(P_{\mu \nu }(x))=\omega (P)f(x+\hat{\mu }+\hat{\nu })$, and for two 1-cochains the rule has two terms,

$$(\alpha \cup \beta )(P_{\mu \nu }(x))={\alpha }_{\mu }(x){\beta }_{\nu }(x+\hat{\mu })-{\alpha }_{\nu }(x){\beta }_{\mu }(x+\hat{\nu })(\mu <\nu ).$$

• Leibniz. $d(\alpha \cup \beta )=d\alpha \cup \beta +(-1{)}^p\alpha \cup d\beta$, checked on random integer cochains for every degree pair up to $d=5$. With the single first term alone, Leibniz fails.
• Higher cup product ${\cup }_1$ (Sem II Week 6 §3). For $\alpha \in C^p$, $\beta \in C^q$ and a $(p+q-1)$-cell $(x;S)$,

$$(\alpha {\cup }_1\beta )(x;S)=\sum _{i\in S}\sum _{A\bigsqcup C=S\setminus \{i\}}(-1{)}^{|C_{>}|}ϵ(\{i\}\cup A,C\left)\alpha \right(x+{\hat{e}}_{C_{>}};\{i\}\cup A\left)\beta \right(x+{\hat{e}}_{A_{<}};\{i\}\cup C),$$

with the notation of the cup product above ($C_{>}$, $A_{<}$ the subsets of directions that follow or precede, ε the shuffle sign). It equals $(-1{)}^{pq}$ times the ${\cup }_1$ of Chen–Tata with the factors exchanged, and it obeys

$$d(\alpha {\cup }_1\beta )=d\alpha {\cup }_1\beta +(-1{)}^p\alpha {\cup }_{1}d\beta +(-1{)}^{p+q-1}\alpha \cup \beta +(-1{)}^{pq+p+q}\beta \cup \alpha ,$$

so that for cocycles $\alpha \cup \beta -(-1{)}^{pq}\beta \cup \alpha =(-1{)}^{p+q-1}d(\alpha {\cup }_1\beta )$ (checked in $d=3,4,5$ for every degree pair). The overall sign is fixed by Chen–Tata’s recursion, their (29).

• Pontryagin square. For a lift $\tilde{B}\in C^2(\Lambda ,\mathbb{Z})$ of $B\in Z^2(\Lambda ,{\mathbb{Z}}_N)$, $\mathcal{P}(\tilde{B})=\tilde{B}\cup \tilde{B}+\tilde{B}{\cup }_{1}d\tilde{B}\in C^4(\Lambda ,{\mathbb{Z}}_{2N})$ for $N$ even, and $\tilde{B}\cup \tilde{B}$ mod $N$ for $N$ odd; $\int \mathcal{P}(B)$ is independent of the lift and of gauge transformations. A twisted $SU(N)$ bundle on $T^4$ has fractional instanton number $Q\equiv \frac{N-1}{2N}\int \mathcal{P}(B)$ mod 1 ($\frac{1}{4}\int \mathcal{P}(B)$ for $SU(2)$).
• Graded commutativity. On the lattice $\alpha \cup \beta \ne (-1{)}^{pq}\beta \cup \alpha$ in general, and the non-commutativity is physical. For cocycles the difference is exact, $\alpha \cup \beta -(-1{)}^{pq}\beta \cup \alpha =(-1{)}^{p+q-1}d(\alpha {\cup }_1\beta )$, with the ${\cup }_1$ defined in the bullet above (Sem II Week 6 §3); for general cochains the formula acquires the extra terms $d\alpha {\cup }_1\beta$ and $\alpha {\cup }_{1}d\beta$ of the coboundary formula there. Check on $T^2$: for the harmonic 1-cochains of Week 2, ${\sum }_P{h}^x\cup h^y=L^2$ and ${\sum }_P{h}^y\cup h^x=-L^2$, the intersection form.

9. Hamiltonian conventions (Semester II Block 3)

• Qubits on links; Pauli operators $X,Z$; ${\mathbb{Z}}_N$ generalization by clock and shift.
• Toric code: $A_v={\prod }_{\ell \ni v}{X}_{\ell }$ (star), $B_p={\prod }_{\ell \in \partial p}{Z}_{\ell }$ (plaquette), $H=-{\sum }_v{A}_v-{\sum }_p{B}_p$. It is the $\Gamma \to 0$ point of the ${\mathbb{Z}}_2$ Kogut–Susskind Hamiltonian of §4 with the magnetic term rescaled and Gauss’s law imposed energetically.
• Anyons: $e$ (star violation), $m$ (plaquette violation), $\epsilon =e\times m$; braiding phase of $e$ around $m$: $-1$ (${\omega }^{q{q}^{\prime }}$ in ${\mathbb{Z}}_N$).
• Entanglement conventions: von Neumann entropy $S({\rho }_A)=-\mathrm{tr}{\rho }_A\ln {\rho }_A$ (natural log; the TEE of the toric code is $\gamma =\ln 2$).
• Toric code (Sem II Week 8). On the $L_1\times L_2$ square torus, $h(x,y)$ and $v(x,y)$ are the horizontal and vertical links at $(x,y)$ and $p(x,y)$ the plaquette with lower-left corner $(x,y)$; the logical operators are ${\bar{Z}}_1=W(C_x)$, ${\bar{X}}_1=U({\tilde{C}}_y)$, ${\bar{Z}}_2=W(C_y)$, ${\bar{X}}_2=U({\tilde{C}}_x)$. The fermion ε is $e\times m$ with the $m$ in the plaquette north-east of the $e$ site, and $W_{\epsilon }$ uses a push-off of the dual path. For ${\mathbb{Z}}_N$, $X_{\ell }={\omega }^{-E_{\ell }}$ and $G_x={\omega }^{q_x}$, and an open $Z$-string from $x$ to $y$ creates Kogut–Susskind charges $+1$ at $x$ and $-1$ at $y$ (physical charges $-1$ and $+1$, §4). Modular data: $S$ and $T$ are those of Kitaev (2006), eq. (223), with $T=\mathrm{diag}({\theta }_a)$ and no factor $e^{-2\pi ic/24}$, so that $(ST{)}^3=\Theta C$ with $\Theta =D^{-1}{\sum }_a{d}_a^2{\theta }_a$; the monodromy (full braiding) is written $M_{ab}$ in Sem II Weeks 7–8 and $B(a,b)$ in Week 10, the same object.
• String nets (Sem II Week 11). String basis $n_{\ell }=(1-X_{\ell })/2$, as in Levin–Wen §II; vertex and plaquette projectors $Q_v=(1+A_v)/2$ and $B_p=(1+d_1{B}_p^1)/2$, so that the toric-code string net is half of the toric-code Hamiltonian plus a constant; the double-semion plaquette operator is Sem II Week 11 (4.4), with the projector $P_p$ onto closed strings; the Wilson loop is $W=\prod Z$. A string tension in the wave function is written ζ, with fugacity $t=e^{-2\zeta }$ and the auxiliary Ising coupling $\tanh K=t$ (ζ is also the monopole fugacity of Semester I, and $K$ is context-dependent).
• Entanglement, circuits and codes (Sem II Week 9). For a bipartition of the links, $|\partial A|$ is the number of stars the cut separates; the toric-code disk entropy is $S=|\partial A|\ln 2-\ln 2$, and the topological entropies are $S_{\mathrm{topo}}=-\gamma$ in the Kitaev–Preskill combination and $-2\gamma$ in the Levin–Wen one, with $\gamma =\ln \mathcal{D}$, ${\mathcal{D}}^2={\sum }_a{d}_a^2$. A local-unitary circuit has depth $M$ and range $l$, with light-cone radius $R=Ml$ measured between link midpoints in the max norm. SRE means connected to a product state by a local-unitary evolution (quasi-local, with tails that decay faster than any power), which circuits of fixed depth approximate; exact preparation by such a circuit is the strict version, used in the model proof of Sem II Week 9 (2.5). Codes are written $[[n,k,d]]$ (double brackets, to avoid the wikilink syntax), with $d$ the minimal weight of a nontrivial logical operator; the planar code with its rough and smooth edges and its logical operators is Sem II Week 9 (5.6)–(5.8). Decoding: the reference chain carries $\eta =-1$ on its links, the random-bond Ising coupling is $e^{-2K}=p/(1-p)$ on the Nishimori line, and the seam is τ.
• Condensation sheets (Sem II Week 13 §5). For a subgroup $H$ of the 1-form symmetry of a 2+1d ${\mathbb{Z}}_N$ gauge theory, $S_H(\Sigma )=\frac{1}{|H^0(\Sigma ;H)|}{\sum }_{\gamma \in H_1(\Sigma ;H)}W(\gamma )$, with fusion $S_H\times {\bar{S}}_H=|H{|}^{2g-1}{S}_H$ on ${\Sigma }_g$ and lattice form $S_H=|H{|}^{-N_P}{\sum }_{c\in Z_1(\Lambda ;H)}W(c)$; Roumpedakis–Seifnashri–Shao normalize by $1/\sqrt{|H_1|}$, which agrees on the torus and differs elsewhere by an Euler counterterm (the group’s manuscript takes precedence where it fixes its own normalization, §7). On $T^2$ the anyon basis is $|e^a{m}^b⟩$.

10. θ-terms and the Witten effect

• $\theta \in [0,2\pi )$. Minkowski, with orientation $dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$: $S_{\theta }=\frac{\theta }{8{\pi }^2}\int F\wedge F=-\frac{\theta }{4{\pi }^2}\int d^{4}xE\cdot B$, independently of the metric signature. Euclidean, with $\tau =it$ and orientation $d\tau \wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$: the θ-term contributes the phase $e^{i\theta Q}$ to the weight, i.e. $S_E\supset -i\theta Q$, with $Q=\frac{1}{8{\pi }^2}\int F\wedge F$. For $U(1)$, $Q=\frac{1}{2}\int c_1\wedge c_1$ with $c_1=F/2\pi$, an integer on closed spin 4-manifolds (such as $T^4$) and in $\frac{1}{2}\mathbb{Z}$ in general. The electric potential enters as $a=-\phi dt+A\cdot dx$, so that $W_q=e^{iq\oint a}$ is the worldline of a positive charge $q$. On the 4d lattice, when a formula distinguishes Euclidean time, time is axis 1, so the orientation (1234) is $d\tau \wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ and $Q_{\mathrm{lat}}=\frac{1}{8{\pi }^2}\sum (F\cup F)(x;1234)$; a note that puts time on axis 4 (Week 8 §5.3 labels its current components that way) flips the sign of $Q$ and says so.
• Witten effect. With $\oint B=2\pi n_m$, Gauss’s law $\nabla \cdot (E/e^2-\theta B/4{\pi }^2)=n_e{\delta }^3(x)$ gives the electric charge

$$q_e\equiv \oint \frac{E}{e^2}\cdot dS=n_e+\frac{\theta }{2\pi }{n}_m,$$

so the spectrum at $\theta +2\pi$ coincides with the spectrum at θ: the dyon labelled $(n_e,n_m)$ at $\theta +2\pi$ carries the charge of $(n_e+n_m,n_m)$ at θ. With $\tau =\frac{\theta }{2\pi }+\frac{2\pi i}{e^2}$, the duality group acts by $T:\theta \to \theta +2\pi$ and $S:(n_e,n_m)\to (-n_m,n_e)$; the Dirac pairing is $⟨v,w⟩=n_e{n}_m^{\prime }-n_m{n}_e^{\prime }$; CP at $\theta =\pi$ acts as $(n_e,n_m)\to (-n_e-n_m,n_m)$. A θ-wall carries a Chern–Simons term of level $\Delta \theta /2\pi$ (orientation $dt\wedge dx\wedge dy$) and the Hall conductivity ${\sigma }_{xy}=\frac{\Delta \theta }{2\pi }\frac{e^2}{2\pi }$ (Week 12 §8). Dirac pairing on the lattice: integer linking of electric and magnetic worldobjects via the cochain pairing.

• Yang–Mills with a ${\mathbb{Z}}_N$ 2-form background (Sem II Week 6). $S_W[U,B]=\frac{\beta }{N}{\sum }_P\mathrm{Re}\mathrm{tr}(1-{\omega }^{-B_P}{U}_P)$ with $B\in Z^2(\Lambda ,{\mathbb{Z}}_N)$; the continuum statements read the background as $da-B$. With the counterterm $\frac{2\pi ip}{N}\int \frac{\mathcal{P}(B)}{2}$, time reversal at θ = π maps the counterterm label as $p\to -p-(N-1)$ and an infrared vacuum label as $p_{\mathrm{IR}}\to -p_{\mathrm{IR}}+N-1$.
• Chern–Simons term (Euclidean, $d=3$): the weight carries the phase $\exp (\frac{ik}{4\pi }\int a\wedge da)$, the same orientation rule as the θ-term above.
• K-matrix Chern–Simons theory (Sem II Week 10). Weight $\exp (\frac{i}{4\pi }{K}_{IJ}\int a^I\wedge d{a}^J)$ with the orientation of the Chern–Simons bullet above; the theory is bosonic iff $K$ has an even diagonal. Bracket $[a_i^I(\mathbf{x}),a_j^J(\mathbf{y})]=2\pi i(K^{-1}{)}^{IJ}{ϵ}_{ij}{\delta }^2(\mathbf{x}-\mathbf{y})$; a quasiparticle $l\in {\mathbb{Z}}^n$ carries flux $f=-2\pi K^{-1}l$ and charge $Q_l=-t^T{K}^{-1}l$, and ${\sigma }_{xy}=-t^T{K}^{-1}t{e}^2/2\pi$ (a response term $\frac{k}{4\pi }\int AdA$ means ${\sigma }_{xy}=k{e}^2/2\pi$, as for the θ-wall). Braiding and spin: $B(l,l^{\prime })=e^{-2\pi i{l}^T{K}^{-1}{l}^{\prime }}$ and, for a counterclockwise exchange (Lk $=+1$), ${\theta }_l=e^{-i\pi l^T{K}^{-1}l}$; positive $K$ has a clockwise edge, with $[{\varphi }^I(x),{\varphi }^J(y)]=i\pi (K^{-1}{)}^{IJ}\mathrm{sgn}(x-y)$. Much of the quantum-Hall literature writes $-\frac{K}{4\pi }$, the mirror image, in which ${\theta }_l=e^{+i\pi l^T{K}^{-1}l}$. Labels: the toric code is $K=\left(\begin{array}{cc}0& 2\\ 2& 0\end{array}\right)$ with $e=(1,0)$, $m=(0,-1)$, so that $e^a{m}^b$ has $\theta ={\omega }^{ab}$; the double semion is $K=\mathrm{diag}(2,-2)$ with $s=(0,1)$, ${\theta }_s=+i$, as in Sem II Week 7. For a quantum double $D(G)$ the anyons are pairs (conjugacy class, irreducible representation of a centralizer), with $\theta ={\chi }_{\alpha }(g)/\dim \alpha$.
• Lattice θ-term. It is built from the Villain field strength with cup products. It is $2\pi$-periodic only in the monopole-free (modified-Villain) theory, $dn=0$; with monopoles present the lattice $Q$ is in general not an integer (on an odd torus, $Q_{\mathrm{lat}}\in \mathbb{Z}$ for every $a$ if and only if $dn=0$). With $F=da-2\pi n$, $\frac{1}{8{\pi }^2}\sum F\cup F=\frac{1}{2}\sum n\cup n-\frac{1}{4\pi }\sum (a\cup dn-dn\cup a)$, and a static monopole contributes the lattice Witten phase $e^{i(\theta /2\pi )\oint a}$ (Week 12 §6). Sem II Week 12 §5 seals this: in the modified Villain theory the θ-term is an integer and θ exactly $2\pi$-periodic; with a ${\mathbb{Z}}_N$ background, $Z(\theta +2\pi N;B)=\exp \left(\frac{2\pi i}{N}\frac{\int \mathcal{P}(B)}{2}\right)Z(\theta ;B)$, and θ → θ + 2π alone is not a symmetry.

11. Units and miscellany

• $\hslash =c=1$; $k_B=1$.
• “SSB of a $p$-form symmetry” always means the statement about the realization of $U_{\alpha }(\Sigma )$ on the ground-state sector, made precise in Sem II Week 3.
• Numerical values quoted from simulations or rigorous bounds carry their reference in the note that quotes them and in bibliography-and-paper-map.
• Abbreviations: LGT (lattice gauge theory), BKT (Berezinskii–Kosterlitz–Thouless), KW (Kramers–Wannier), KS (Kogut–Susskind), FS (Fradkin–Shenker), GKSW, GKKS, GLSS, RSS (per the paper map), TEE (topological entanglement entropy), SPT, DW (Dijkgraaf–Witten), GSD (ground-state degeneracy), JTA (Julia–Toulouse approach).