(<ref>) we see that the associated Bockstein homomorphism $\beta: H^n(G, U(1)_\rho) \rightarrow H^{n+1}(G, \z_\rho)$ is an isomorphism. For most discussions in this paper, especially when $G$ is a finite group (and $n>0$), this Bockstein homormorphism is indeed an isomorphism, and only in the example in Appendix <ref> it is not, on which we will comment explicitly.

The second short exact sequence that is important to us is

1 r     r×2      rmod 2     ℤ_2  r     1

For $x\in H^n(G, \z_2)$, the Bockstein homomorphism $\beta_2$ is sometimes written as
β_2(x) = 1/2dx.
When $H^n(G,\z_\rho) = (\z_2)^k$ with some non-negative integer $k$, $\tilde i$ maps $H^n(G, \z_\rho)$ to 0 in $H^n(G, \z_\rho)$. Therefore, from Eq. (<ref>), we see that $\tilde p$ is injective while $\beta_2$ is surjective.

We can also consider the natural map from Eq. (<ref>) to Eq. (<ref>), which is inclusion for every factor as follows,

1 r     r×2π       R  rmod 2π      U(1)  r     1

1 r     r×2 u≅    rmod 2 u×π    ℤ_2  r ui     1

where $i$ is again the inclusion of $\z_2=\{\pm 1\}$ into $U(1)$. As a result, we have a map of long exact sequences,

…r      H^n(G, ℤ_2) dĩ rβ_2     H^n+1(G, ℤ_ρ) [d, "≅"]r     H^n+1(G, _ρ) [d] [r]     …

…r      H^n(G, U(1)_ρ)   rβ       H^n+1(G, ℤ_ρ)r     H^n+1(G, R_ρ) [r]     …
Here we distinguish the first Bockstein homomorphism by denoting it by $\beta_2$, and $\tilde i$ denotes the map induced by $i:\mb{Z}_2\rightarrow U(1)$ specifically. Hence, we have $\beta_2 = \beta\circ \tilde i$. When $H^n(G,\z_\rho) = (\z_2)^k$, since $\beta$ is an isomorphism while $\beta_2$ is surjective , $\tilde i$ is surjective as well. It suggests that in this case every element $\Omega\in H^n(G, U(1)_\rho)$ can be written as $\tilde i(L)$ or $e^{i \pi L}$ for some $L\in H^n(G, \z_2)$. In fact, every element $\Omega\in H^n(G, U(1)_\rho)$ whose inverse is itself can be written as $e^{i \pi L}$ for some $L\in H^n(G, \z_2)$. We use this fact throughout the paper.

§.§ Cup product and $\z_2$ cohomology ring

In this sub-appendix, we review cup product and $\z_2$ cohomology ring in group cohomology that we will use <cit.>. We will specialize to the case where the module is $\mb{Z}_2 = \{0, 1\}$ and the group action $\rho$ is trivial. The special feature of $\mb{Z}_2$, countrary to e.g. $U(1)$, is the fact that $\mb{Z}_2$ is a ring. Note that here addition in $\mb{Z}_2$ is regarded as the group multiplication used in Eq. (<ref>), and we will use $+$ to denote this addition in this sub-appendix. There is another ring multiplication that will be important later, which should be distinguished with the group multiplication used in Appendix <ref>.

The cross product is defined as the following operation on group cohomology,

×:   H^m(G, Z_2) ⊗H^n(H, Z_2) →H^m+n(G×H, Z_2),
such that for $x\in H^m(G, \mb{Z}_2)$ and $y\in H^n(H, \mb{Z}_2)$, after choosing cochain representatives $\tilde x$ and $\tilde y$, we have the cochain representative of $x\times y$ as follows,
x×y((g_1, h_1),…, (g_m+n, h_m+n) )≡x̃(g_1,…, g_m)·ỹ(h_m+1,…,h_m+n),
where $g_i\in G, h_i\in H, i=1,\dots,m+n$.

The cup product is defined as the following operation on group cohomology,

H^m(G, _2) ⊗H^n(G, _2)  r×     H^m+n(G×G, _2)  rΔ^*     H^m+n(G, _2),

where $\Delta: G\rightarrow G\times G$ is the diagonal embedding $g\rightarrow (g, g)$. We can also define it at the cochain level, for $x\in H^m(G, \mb{Z}_2)$ and $y\in H^n(G, \mb{Z}_2)$, after choosing cochain representatives $\tilde x$ and $\tilde y$, we have the cochain representative of $x\cup y$ as follows,
x∪y(g_1, …, g_m+n)
≡x̃(g_1,…, g_m)·ỹ(g_m+1,…,g_m+n).
We can prove that cup product is commutative, $x\cup y = y\cup x$.

The cup product $\cup$ gives a multiplication on the direct sum of cohomology groups
H^*(G, ℤ_2)=⊕_k∈NH^k(G, ℤ_2),
Together with the fact that $1\cup x = x$ where $x$ is any element in $H^*(G, \z_2)$ and $1$ here denotes the nontrivial element in $H^0(G, \mathbb{Z}_2) = \z_2$, the cup product $\cup$ turns $H^*(G, \mb{Z}_2)$ into a ring that is naturally $\mb{N}$ graded and commutative. We call this ring the $\mb{Z}_2$ cohomology ring of $G$.

Moreover, $H^*(G, \z_2)$ is also a $\z_2$ algebra, and therefore can be presented by generators and relations, all elements in $H^n(G, \mathbb{Z}_2)$ for any $n>0$ are either generators or can be expressed as sum of (cup) products of generators, and generators satisfy some relations which dictate that certain sums of (cup) products actually yield a trivial cohomology element. We will call a generator in $H^n(G, \mb{Z}_2)$ a degree $n$ generator. Hence, the $\z_2$ cohomology ring of $G$, $H^*(G, \z_2)$, can be written as follows,
H^*(G, Z_2)=Z_2[A_∙, ⋯, B_∙, ⋯]/relations
with $A_\bullet$($B_\bullet$) generators in degree 1(2) belonging to $H^1(G, \mb{Z}_2)$($H^2(G, \mb{Z}_2)$), and $\bullet$ the name of the generator. Together with potential higher order generators, $C_\bullet$ in degree 3, they are supposed to form a complete list of generators of the entire cohomology ring.

For example, the $\mb{Z}_2$ cohomology ring of the group $\mb{Z}_2$ is
where $A_c$ is the nontrivial element in $H^1(\z_2, \z_2)$ and can be thought of as nothing but the gauge field of $C_2$ rotation when pulled back to the spacetime manifold. In other words, for the $\z_2$ cohomology ring of $\z_2$, there is a single generator $A_c$ in degree 1 and no relation. Accordingly, we can see that $H^n(G, \z_2) = \z_2$ for $n\in \mb{N}$, with the nontrivial element given by $A_c^n\equiv A_c\cup A_c\cup\cdots\cup A_c$, the cup product of $n$ $A_c$'s.

As another example, the $\mb{Z}_2$ cohomology ring of $\mb{Z}_4$ is
ℤ_2[A_c, B_c^2]/(A_c^2=0),
where here $A_c$ is the nontrivial element in $H^1(\z_4, \z_2)$ and can be thought of as (the $\z_2$ reduction of) the gauge field of $C_4$ rotation when pulled back to the spacetime manifold, while $B_{c^2}$ is the nontrivial element in $H^2(\z_4, \z_2)$, which corresponds to the fractionalization pattern of the $\z_4$ symmetry on an $SO(3)$ monopole, with $C_4^4 = C_2^2 = -1$. That is to say, for the $\z_2$ cohomology ring of $\z_4$, there are two generators at degree 1 and 2 respectively, with the square of degree 1 generator $A_{c}$ equal to 0. Then we see that $H^n(\z_4, \z_2) = \z_2$ for $n\in \mb{N}$ as well, and the nontrivial element is given by $B_{c^2}^k$ when $n=2k$ and $A_c B_{c^2}^k$ when $n=2k+1$ ($k\in \mb{N}$). Note that for both $G=\z_2$ and $G=\z_4$, $H^n(G, \mb{Z}_2)=\mb{Z}_2$ for any $n\in \mb{N}$, but the $\z_2$ cohomology rings give more information that differentiates the two groups.

For any two groups $G_1$ and $G_2$, we have $H^*(G_1\times G_2, \z_2) = H^*(G_1, \z_2) \otimes H^*(G_2, \z_2)$. Moreover, if $G$ can be written as $G_1 \rtimes G_2$, where $G_1$ is a normal subgroup of $G$ and $G_2$ acts on $G_1$ by conjugation, the calculation of the $\z_2$ cohomology ring of $G$ can be achieved with the help of
Lyndon–Hochschild–Serre spectral sequence <cit.> that also connects $H^*(G, \z_2)$ with $H^*(G_1, \z_2)$ and $H^*(G_2, \z_2)$ which we possibly already know. The general strategy for calculating the $\z_2$ cohomology ring of wallpaper groups $G$ is as follows:

* Identify all generators and elements in the $\z_2$ cohomology ring of $G$ through Lyndon–Hochschild–Serre spectral sequence.

* If there is no relation, given generators $A_1, A_2, B_1, C_1\dots$, all elements of the form $A_1^mA_2^nB_1^pC_1^q\dots,m,n,p,q\dots\in \mathbb{N}$ will appear explicitly as different elements in the $\z_2$ cohomology ring. Therefore, when some $A_1^2$ is missing, we should identify some relation that relates $A_1^2$ to elements that appear explicitly, which can be achieved through pulling back to (enough) subgroups of $G$.

To illustrate the strategy, in the following we calculate the $\z_2$ cohomology ring of three space groups in one or two spatial dimensions, including the generators and relations.

* $p1$: $\z_2[x]/(x^2=0)$.

Consider the line group $p1$, generated by a single translation $T$. The cohomology of $p1$ is $H^1(p1, \z_2)\cong \z_2$ while $H^n(p1, \z_2)\cong 0,~~n>1$. Denote the nontrivial element in $H^1(p1, \z_2)$ as $x$, which corresponds to (the $\z_2$ reduction of) the gauge field of translation, the $\z_2$ cohomology ring of $p1$ is given by $\z_2[x]/(x^2=0)$.

* $p1m$: $\z_2[x, m]/(x^2=xm)$.

Consider the line group $p1m$, generated by translation $T$ and mirror symmetry $M$ with relation $MTM = T^{-1}$. The cohomology of $p1m$ is $H^n(p1m, \z_2)\cong \z_2^2,~~n\geqslant 1$. Since $p1m\cong \z\rtimes\z_2$, with the help of the corresponding Serre spectral sequence, we know that $H^n(p1m, \z_2),~n\geqslant 1$ is spanned by 2 elements, $m^n$ and $m^{n-1} x$. where $m,x\in H^1(p1m, \z_2)$ are two generators that correspond to the gauge field of mirror symmetry and (the $\z_2$ reduction of) the gauge field of translation, respectively.

The next thing to do is to identify $x^2$, which does not explicitly appear as elements of the $\z_2$ cohomology ring. Write $x^2$ as $a_1xm+a_2m^2,a_{1,2}\in \{0,1\}$. By restricting to $\z_2$ subgroup generated by $M$, whose $\z_2$ cohomology ring can be denoted by $\z_2[m']$, we see that $x$ becomes 0 while $m$ becomes $m'$, and thus $a_2=0$. By restricting to the $\z_2$ subgroup generated by $TM$, whose $\z_2$ cohomology ring can be denoted by $\z_2[m'']$, we see that both $x$ and $m$ become $m''$, and thus $a_1=1$. Therefore, we have $x^2=xm$.

Therefore, the $\z_2$ cohomology ring of $p1m$ is $\z_2[x, m]/(x^2=xm)$.

* $cm$: $\z_2[A_{x+y}, A_m, B_{xy}]/(A_{x+y}A_m=0, A_{x+y}^2=0, B_{xy} A_{x+y} = 0, B_{xy}^2 = 0)$.

Consider 2$d$ wallpaper group $cm$, generated by two translation symmetries $T_1,T_2$ as well as mirror symmetry $M$ that interchanges the two translations, $M T_1 M=T_2$ and $M T_2 M=T_1$. The cohomology of $cm$ is $H^n(cm, \z_2)\cong (\z_2)^2,~~n\geqslant 1$. Since $cm\cong (\z\times\z)\rtimes \z_2$, with the help of the corresponding Serre spectral sequence, we know that $H^1(cm, \z_2)$ is spanned by $A_{x+y}$ and $A_m$, while $H^n(cm, \z_2), n\geqslant 2$ is spanned by $B_{xy}A_m^{n-2}$ and $A_m^n$. Here $A_m, A_{x+y}\in H^1(cm, \z_2)$ correspond to the gauge field of mirror symmetry and (the $\z_2$ reduction of) the sum of gauge fields of $T_1$ and $T_2$, respectively. Note that since $T_1$ and $T_2$ map to each other under conjugation by $M$, the gauge field of the two translations $x$ and $y$ individually is not invariant under conjugation by $M$, yet their sum that we denote by $A_{x+y}$ is invariant under conjugation by $M$, which is a necessary condition for it to be a cohomology element, as required by Eq. (<ref>). To conform to the notation, we also denote the gauge field of mirror symmetry by $A_m$ when considering wallpaper groups. Moreover, there is an extra degree-2 generator $B_{xy}$, an element belonging to $H^2(cm, \z_2)$ that cannot be written as sum of cup product of elements in $H^1(cm, \z_2)$. The name $xy$ comes from the fact that its restriction to subgroup $p1$ generated by $T_1,T_2$ is $A_xA_y$ (see Appendix <ref>).

To identify the relations, we note that there are now 4 missing elements: $A_{x+y}A_m, A_{x+y}^2, B_{xy}A_{x+y}, B_{xy}^2$. By restricting to the subgroup $p1$ generated by $T_1,T_2$ as well as the subgroup $\z_2$ generated by $M$, we see that $A_{x+y}A_m = A_{x+y}^2 = 0$. By restricting to the subgroup $pm$ generated by $T_1 T_2^{-1},T_1 T_2,M$, we see that $B_{xy}A_{x+y}=0$ as well as $B_{xy}^2=0$. Note that the pullback of $A_{x+y}$, $A_m$ and $B_{xy}$ to the subgroup $pm$ is 0, $A_m$ and $A_y A_m$, respectively.

Therefore, the $\z_2$ cohomology ring of $cm$ is

$$\z_2[A_{x+y}, A_m, B_{xy}]/\left(A_{x+y}A_m=0, A_{x+y}^2=0, B_{xy} A_{x+y} = 0, B_{xy}^2 = 0\right)$$

§.§ $\mc{SQ}^1$

In this sub-appendix, we define a new map we call $\mathcal{SQ}^1$, reminiscent of $Sq^1$ in regular Steenrod algebra, as follows

𝒮𝒬^1:H^n(G, _2)  rĩ     H^n(G, U(1)_ρ)  rβ     H^n+1(G, _ρ)  rp̃     H^n+1(G, _2),
𝒮𝒬^1 ≡p̃ ∘β∘ĩ,
where $\tilde i$ and $\tilde p$ are the map of group cohomology induced by the homomorphism of modules $i: \mb{Z}_2\rightarrow U(1)$ and $p: \mb{Z}\rightarrow \mb{Z}_2$, and $\beta$ is the Bockstein homomorphism associated with the short exact sequence $1\rightarrow\mb{Z}\rightarrow\mb{R}\rightarrow U(1)\rightarrow 1$. Note that $\beta\circ \tilde i$ is the Bockstein homomorphism $\beta_2$ associated with the short exact sequence $1\rightarrow\mb{Z}\rightarrow\mb{Z}\rightarrow \mb{Z}_2\rightarrow 1$, and therefore when the action $\rho$ is trivial, $\mc{SQ}^1$ is exactly $Sq^1$ in regular Steenrod algebra.

Moreover, $\mc{SQ}^1$ is related to $Sq^1$ via the following simple fact

For $x\in H^n(G, \mathbb{Z}_2)$, we have
SQ^1(x) = SQ^1(1)∪x + Sq^1(x).

According to Eq. (<ref>), choosing a cochain $\tilde x\in C^n(G, \z)$ such that the $\z_2$ reduction of $\tilde x$ is $x$, we have

SQ^1(x)    = 1/2
((-1)^ρ̃(g_1)x̃(g_2,…,g_n+1) +∑_j=1^n (-1)^j x̃(g_1,…,g_jg_j+1,…,g_n+1) +(-1)^n+1 x̃(g_1,…,g_n))

((-1)^ρ̃(g_1)-1)x̃(g_2,…,g_n+1) + Sq^1(x)

=SQ^1(1)∪x + Sq^1(x)2.

For example, for $\mb{Z}_2^T$ with nontrivial action on $U(1)$ or $\mb{Z}$, we have,

SQ^1(t^2n+1) = 0,      SQ^1(t^2n) = t^2n+1,
where $t\in H^1(\z_2^T, \z_2)$ is the generator of the $\z_2$ cohomology ring of $\z_2^T$. We see that in the presence of nontrivial $\rho$, the operation $\mc{SQ}^1$ is not distributive with respect to the cup product. Note that $\mc{SQ}^1(1)$ is nonzero and equals $t$, which when pulled back to the spacetime manifold $\mc{M}$ equals $w_1$ as well, the first Stiefel-Whitney class of $\mc{M}$. In contrast, for $\z_2$ with trivial action on $U(1)$ or $\z$, we have

SQ^1(A_c^2n) = 0,      SQ^1(A_c^2n+1) = A_c^2n+2,
where $A_c\in H^1(\z_2, \z_2)$ is the generator of the $\z_2$ cohomology ring of $\z_2$ as well.

As another example, consider $O(5)$ with $\tilde \rho: O(5)\rightarrow \z_2$ the determinant, an $O(5)$ element complex conjugates an $U(1)$ element or multiplies a $\z$ element by $-1$ if and only if the determinant of the $O(5)$ element is $-1$. From Lemma <ref> we immediately have,

SQ^1(w_4^O(5)) = w_5^O(5),
as suggested by the calculation in the context of DQCP in Refs. <cit.>.

Moreover, even if $\mc{SQ}^1$ is not distriutive with respect to the cup product, from Lemma <ref> $\mc{SQ}^1$ is still distributive with respect to the cross product involving two different groups, we have

For $x\in H^m(G, \mathbb{Z}_2)$ and $y\in H^n(H, \mathbb{Z}_2)$, we have $x\times y\in H^{m+n}(G\times H, \mathbb{Z}_2)$ and
𝒮𝒬^1(x×y) = 𝒮𝒬^1(x)×y + x ×𝒮𝒬^1(y).

This lemma is also important when calculating $\mc{SQ}^1$ because it decomposes the calculation into different pieces corresponding to different groups. For example, with the help of Eqs. (<ref>) and (<ref>), the lemma tells us how to calculate $\mc{SQ}^1$ for the group $(\z_2)^k$ with every $\z_2$ piece acting trivially or nontrivially on $U(1)$ or $\z$.

Now consider $O(5)$ with $\tilde \rho: O(5)\rightarrow \z_2$ associated to the action on $U(1)$ or $\z$ coefficient, which complex conjugates the $U(1)$ element or multiplies a $-1$ to the $\z$ element if and only if the determinant of the $O(5)$ element is $-1$. Motivated by the example of DQCP, we wish to prove that

SQ^1(w_4^O(5)) = w_5^O(5),
as suggested by Refs. <cit.>. Consider the diagonal $(\z_2)^5$ subgroup
ϕ: (_2)^5⟶O(5),
with every $\z_2$ piece acting nontrivially on $U(1)$ or $\z$. Since the pullback $\phi^*$ is injective, to prove Eq. (<ref>) we just need to prove that the two sides are equal under the pullback $\phi^*$. From the Whitney product formula, we have

ϕ^*(w_4^O(5))    = t_1 t_2 t_3 t_4 + t_1 t_2 t_3 t_5 + t_1 t_2 t_4 t_5 + t_1 t_3 t_4 t_5 + t_2 t_3 t_4 t_5,

ϕ^*(w_5^O(5))    = t_1 t_2 t_3 t_4 t_5,

where $t_i\in H^1(\z_2^{(i)}, \z_2), i=1,\dots,5$ are the generators of the $\z_2$ cohomology ring of $(\z_2)^5$. According to the lemma, we have
SQ^1(t_i1t_i2t_i3t_i4) = SQ^1(t_i1×t_i2×t_i3×t_i4×1_i5) = t_1 t_2 t_3 t_4 t_5,
and as a result the left hand side under the pullback equals $5t_1 t_2 t_3 t_4 t_5 = t_1 t_2 t_3 t_4 t_5$, which equals the right hand side. Therefore we establish Eq. (<ref>). The calculation of $\mc{SQ}^1$ acting on any Stiefel-Whitney class can be achieved likewise.

Finally, from the fact that
A_LH≅ker[ĩ: H^2(G_s, ℤ_2)→H^2(G_s, U(1)_ρ)],
as argued in Section <ref>, for LSM anomaly written as $\exp\left(i \pi \lambda \eta\right)$ where $\lambda\in H^2(G_s, \mb{Z}_2)$ and $\eta \in H^2(G_{int}, \mb{Z}_2)$, we have
SQ^1(λ) = 0.
This can also be mathematically checked by considering different representations $\rho: G_s\rightarrow O(n)$. For example, consider $G_s = p4m$. Then $A_{\rm LH}$ is spanned by $\lambda_1 = B_{xy} + A_{x+y}(A_{x+y} + A_m) + B_{c^2}$, $\lambda_2 = B_{xy}$, $\lambda_3 = A_{x+y}(A_{x+y} + A_m)$ in $H^2(p4m, \z_2)$ (see Appendix <ref>), corresponding to LSM anomaly associated to DOF at the site $a$, plaquette center $b$, and bond center $c$ as in Fig. <ref>, repectively. Consider the following three representations of $p4m$. The first one is

-1     0     0

0     -1    0

0     0     1
-1     0     0

0     1    0

0     0     -1

1     0     0

0     0    1

0     -1     0

1     0     0

0     1     0

0     0     -1

The pullback of $w_2^{O(3)}$ equals $B_{xy} + B_{c^2}$ while the pullback of $w_3^{O(3)}$ is zero (see sub-Section <ref> and especially Eq. (<ref>)). From  $\mathcal{SQ}^1\left(w_2^{O(3)}\right) = w_3^{O(3)}$, we establish that
𝒮𝒬^1(B_xy+ B_c^2) = 0.
The second representation is

1    0

0     1
1    0

0     1

0    1

-1     0

1     0

0     -1

The pullback of $w_2^{O(2)}$ equals $B_{c^2}$, and from $\mathcal{SQ}^1\left(w_2^{O(2)}\right) = 0$, we establish that
𝒮𝒬^1(B_c^2) = 0
The third representation is

-1    0

0     -1
-1    0

0     -1

1    0

0     1

1     0

0     -1

and we have
𝒮𝒬^1(A_x+y(A_x+y + A_m)) = 0.
Since $B_{xy}+B_{c^2}, B_{c^2}, A_{x+y}(A_{x+y} + A_m)$ span $A_{\rm LH}$ as well, indeed we mathematically show that $\mc{SQ}^1(\lambda) = 0$ for $\lambda\in A_{\rm LH}$ in $p4m$. Then according to Lemma <ref> we also have
SQ^1(λη) = SQ^1(λ) ×η+ λ×SQ^1(η) = λ×SQ^1(η),
This equation will be very useful in the analysis of anomaly-matching. Note that this equation holds for LSM constraints on lattices with any wallpaper group.

§ TOPOLOGICAL PARTITION FUNCTION CORRESPONDING TO LSM

In this appendix, we provide a more rigorous argument that the cocycle corresponding to the topological partition function (TPF) of the $(3+1)$-d $G_s\times G_{int}$ SPT, whose boundary has some LSM constraint, can indeed be written in the form of Eq. (<ref>). [Since the $(3+1)$-d $G_s\times G_{int}$ SPT is captured by an element in $H^4(G_s\times G_{int}, U(1)_\rho)$, one may attempt to show the validity of Eq. (<ref>) by combining the Kunneth decomposition $H^4(G_s\times G_{int}, U(1)_\rho)\cong \oplus_{i=0}^4 H^i(G_s, H^{4-i}(G_{int}, U(1)_\rho))$ and the fact that the relevant $(1+1)$-d $G_{int}$ SPT is captured by $H^2(G_{int}, U(1)_\rho)$, which suggests that in the Kunneth decomposition only the term $H^2(G_s, H^2(G_{int}, U(1)_\rho))$ is relevant to the LSM constraints. Although intuitively appealing, this argument is flawed, because there is generically no unambiguous way to determine whether an element in $H^4(G_s\times G_{int}, U(1)_\rho)$ is in $H^2(G_s, H^2(G_{int}, U(1)_\rho))$. Our argument below does not suffer from this ambiguity. Furthermore, even if we know that the relevant cocycle is in $H^2(G_s, H^2(G_{int}, U(1)_\rho))$, it requires an explanation why its representative cochain can necessarily be written as Eq. (<ref>).]

To start, first recall that the lattice homotopy picture indicates that all LSM constraints for a given wallpaper group $G_s$ are classified by a group $A_{\rm LH}=\mb{Z}_2^k$ with some integer $k$. This means that the sought-for cocycle in $H^4(G_s\times G_{int}, U(1)_\rho)$ can be written as
Ω(g_1, g_2, g_3, g_4)=e^iπκ(g_1, g_2, g_3, g_4)
with $\kappa$ taking values in $\{0,1\}$. This allows us to view $\kappa(g_1, g_2, g_3, g_4)$ as a representative cochain in $H^4(G_s\times G_{int}, \mb{Z}_2)$, where the multiplication between two elements is implemented by the mod 2 addition of their corresponding representative cochains. Since $H^4(G_s\times G_{int}, \mb{Z}_2)\simeq\oplus_{i=0}^4H^i(G_s, \mb{Z}_2)\otimes H^{4-i}(G_{int}, \mb{Z}_2)$, we can always write $\Omega$ as

Ω(g_1, g_2, g_3, g_4)=∏_i=0^4 e^iπλ_i(l_1, ⋯, l_i)η_4-i(a_i+1, ⋯, a_4)
where each $g_i\in G_s\times G_{int}$ is again written as $g_i=l_i\otimes a_i$, with $l_i\in G_s$ and $a_i\in G_{int}$. Both $\lambda_i$ and $\eta_{4-i}$ take values in $\{0, 1\}$, and they can be viewed as representative cochains in $H^i(G_s, \mb{Z}_2)$ and $H^{4-i}(G_{int}, \mb{Z}_2)$, respectively. Furthermore, we can view $e^{i\pi \eta_{4-i}(a_{i+1}, \cdots, a_4)}$ as a representative cochain in $H^{4-i}(G_{int}, U(1)_\rho)$, which can be physically interpreted as a $G_{int}$ SPT living in $3-i$ spatial dimensions.

Previous studies of $G_s\times G_{int}$ SPTs indicate that all these SPTs have a real-space construction, in which various lower dimensional SPTs (or invertible states) are decorated into various submanifolds of the entire crystal <cit.>. Indeed, the SPT relevant to LSM constraints can be constructed by putting copies of $(1+1)$-d $G_{int}$ SPTs at various IWP of the wallpaper group $G_s$. Combining these two observations together, we conclude that in Eq. (<ref>) only the factor with $i=2$ can possibly be related to LSM constraints, because only that factor can possibly be related to putting $(1+1)$-d $G_{int}$ SPTs at various positions, while other factors involve SPTs living in the wrong dimension (the $i=1$ term means that some $(2+1)$-d $G_{int}$ SPT is decorated into the system in some way). Moreover, for a given PR type of the system, $e^{i\pi\eta_2(a_3, a_4)}$ should be the cocycle corresponding to the $(1+1)$-d $G_{int}$ SPT whose boundary hosts this particular PR.

Therefore, the cocycle related to LSM constraints can always be written in a form given by Eq. (<ref>), and $\lambda(l_1, l_2)$, which is written as $\lambda_2(l_1, l_2)$ in Eq. (<ref>), can be viewed as a representative cochain in $H^2(G_s, \mb{Z}_2)$. Furthermore, according to the lattice homotopy picture, $\lambda$ or $\lambda_2$ should just encode the information of which IWP host $(1+1)$-d $G_{int}$ SPTs, so it should be completely determined by $G_s$ and the lattice homotopy class corresponding to each LSM constraint, and be the same for all $G_{int}$ and all PR types of the system.

We remark that the above argument does not show that all cocycles in the form of Eq. (<ref>) must be related to LSM constraints. In fact, in Sec. <ref> we have found that some of them are not. Those SPTs can be constructed by inserting a $(2+1)$-d $Z_2\times G_{int}$ SPT on the mirror plane, such that the $Z_2$ domain wall is decorated with a $(1+1)$-d $G_{int}$ SPT. See Appendix <ref> for more detail.

We also remark that although we have assumed that the projective representations of $G_{int}$ are $\z_2^k$-classified with $k$ some integer in the above argument, we expect that the topological partition functions corresponding to LSM constraints can always be written in a form similar to Eq. (<ref>), for any $G_{int}$. Specially, if a PR type of $G_{int}$ has order $n$, then the LSM-related cocycle takes the form
Ω(g_1, g_2, g_3, g_4)=e^i2π/nλ(l_1, l_2)η(a_3, a_4)
where $\lambda$ and $\eta$ take integral values, and $e^{\frac{2\pi i}{n}\eta(a_3, a_4)}$ is the cocycle corresponding to the relevant $(1+1)$-d $G_{int}$ SPT. Moreover, this statement, including its special form Eq. (<ref>), has been derived in the special cases where $G_s$ contains only translation or only point group, using equivariant homology <cit.>, and we expect that the method in Ref. <cit.> can be generalized to an arbitrary lattice symmetry group $G_s$. A systematic proof of this statement is beyond the scope of this paper and we leave it for future work.

§ FRACTIONALIZATION PATTERN INVOLVING BOTH TRANSLATION AND GLIDE SYMMETRIES

Among all 17 wallpaper groups, there is only one group, $pg$, in which the fractionalization pattern has to be specified in a way that necessarily invokes the glide symmetry. In this appendix, we present its corresponding physical picture.

The group $pg$ is generated by $T_1$ and $G$, a translation and a glide reflection. The translation vector of $T_1$ is flipped under $G$, and $G^2$ is another translation along a direction perpendicular to the translation vector of $T_1$. These generators satisfy $G^{-1}T_1GT_1=1$.

Acting $G^{-1}T_1GT_1$ on an $SO(3)$ monopole. The first $T_1$ action is marked in blue, the following $G$ action is marked in red, the next $T_1$ action is marked in green, and the last $G^{-1}$ action is marked in purple. The dashed line is the reflection axis of $G$. This figure shows that the operation $G^{-1}T_1GT_1$ moves an $SO((3)$ monopole along a trajectory that encloses a fundamental domain.

Consider the case where $G_{int}=SO(3)$. Just as in the main text, we gauge the $SO(3)$ symmetry and examine the fractionalization pattern of $pg$ on the $SO(3)$ monopole by applying the operation $G^{-1}T_1GT_1$ to an $SO(3)$ monopole, which moves the monopole around the fundamental domain (see Fig. <ref>). If the fundamental domain contains an odd (even) number of Haldane chains, this process results in a $-1$ (1) phase factor, which is a signature of nontrivial (trivial) symmetry fractionalization pattern of the $pg$ symmetry carried by the $SO(3)$ monopole. Because $H^2(pg, \mb{Z}_2)=\mb{Z}_2$, there is only one nontrivial symmetry fractionalization pattern. A topological invariant detecting the nontrivial element in $H^2(pg, \mb{Z}_2)$ is given in Eq. (<ref>), so this must be the topological invariant that diagnoses the fractionalization pattern of the $pg$ symmetry on an $SO(3)$ monopole.

§ NON-LSM FRACTIONALIZATION PATTERNS

In this appendix, we discuss in more detail the $M\times SO(3)$ SPT corresponding to $\alpha_{\rm non-LSM}=\frac{\omega(M, M)}{\omega(1, 1)}=-1$, which has TPF $\exp(i\pi\int (w_1^{TM})^2 w_2^{SO(3)})$. In particular, we will show that this SPT can be constructed by inserting into the mirror plane of $M$ a $(2+1)$-d $\z_2\times SO(3)$ SPT, whose $\z_2$ domain walls are decorated with Haldane chains. Moreover, we will show that for any $G_{int}$ with $\mb{Z}_2^k$-classified PR, the $(3+1)$-d $M\times G_{int}$ SPTs with $\alpha_{\rm non-LSM}=-1$ can always be constructed by inserting into the mirror plane of $M$ a $(2+1)$-d $\z_2\times G_{int}$ SPT, whose $\z_2$ domain walls are decorated with the relevant nontrivial $(1+1)$-d $G_{int}$ SPT.

Focusing on the case with $G_{int}=SO(3)$, let us enumerate all $(3+1)$-d $M\times SO(3)$ SPTs. According to the crystalline equivalence principle <cit.>, the classification of these SPTs is the same as the classification of $(3+1)$-d $\z_2^T\times SO(3)$ SPTs, where $\z_2^T$ is a time reversal symmetry. It is known that the latter are classified by $\mb{Z}_2^4$ (see Appendix F of Ref. <cit.> for the descriptions of the physical properties of these SPTs). So $(3+1)$-d $M\times SO(3)$ SPTs are $\mb{Z}_2^4$-classified. According to Ref. <cit.>, these SPTs can all be constructed by putting on the mirror plane of $M$ some $(2+1)$-d invertible states that have at most a $\z_2\times SO(3)$ symmetry (note that this $\z_2$ symmetry does not reverse the spacetime orientation). The $(2+1)$-d $\z_2\times SO(3)$ SPTs are classified by $H^3(\z_2\times SO(3), U(1))=\mb{Z}_2\times\mb{Z}\times\mb{Z}_2$, where the Kunneth formula is used in calculating this classification. It is easy to read off the physical meaning of the root states of these $(2+1)$-d SPTs: one $\mb{Z}_2$ factor represents SPTs protected purely by the $Z_2$ symmetry, the $\mb{Z}$ factor represents spin quantum Hall states <cit.>, which are SPTs protected purely by the $SO(3)$ symmetry and has a TPF given by $SO(3)$ Chern-Simons theories, and the other $\mb{Z}_2$ factor must represent an SPT protected by both $Z_2$ and $SO(3)$. The decorated-domain-wall method <cit.> allows us to construct the SPT by decorating the $\z_2$ domain walls with a Haldane chain.

All the $(2+1)$-d $\z_2\times SO(3)$ SPTs can be inserted into the mirror plane of $M$ to construct a $(3+1)$-d $M\times SO(3)$ SPT, and one can also insert an $E_8$ state to the mirror plane. In total, these give the $\mb{Z}_2^4$ classification of $(3+1)$-d $M\times SO(3)$ SPTs. Note inserting a $\z_2$ SPT or an $E_8$ state to the mirror plane results in a $(3+1)$-d SPT protected by $M$ only, so these states will not have a TPF $\exp(i\pi\int (w_1^{TM})^2 w_2^{SO(3)})$, which shows that this SPT requires both $M$ and $SO(3)$ for protection. Now we are only left with the cases where the bosonic spin quantum Hall state and/or the state constructed from decorated domain wall is inserted into the mirror plane. To understand the physical properties of these states, we can refer to the corresponding $(3+1)$-d $\z_2^T\times SO(3)$ SPTs. If a spin quantum Hall state is decorated into the time reversal domain wall, the resulting state will have fermionic $SO(3)$ monopoles. Using the correspondence between $\z_2^T\times SO(3)$ SPTs and $M\times SO(3)$ SPTs, this indicates that if the spin quantum Hall state is inserted into the mirror plane, the $SO(3)$ monopole will also be fermionic. However, the TPF $\exp(i\pi\int (w_1^{TM})^2 w_2^{SO(3)})$ means that the $SO(3)$ monopole is a boson (but carries nontrivial fractionalization pattern of the $M$ symmetry). This means that the $(3+1)$-d SPT of interest must be obtained from inserting to the mirror plane the $(2+1)$-d SPT constructed from decorated domain wall.

In fact, one can explicitly demonstrate that a $(3+1)$-d $M\times SO(3)$ SPT constructed in this way indeed has an $SO(3)$ monopole carring the nontrivial fractionalization pattern of the $M$ symmetry. To this end, it suffices to show a simpler version of this statement: suppose we break the $SO(3)$ symmetry in this SPT to $U(1)$, the $U(1)$ monopole in the resulting state will carry the nontrivial fractionalization pattern of $M$. This statement can be explicitly shown using the method in Ref. <cit.> (see Appendix B therein). This also means that upon this symmetry breaking, the resulting $M\times U(1)$ symmetric state is a nontrivial SPT. According to the general discussion in Sec. <ref>, this implies that $\alpha_{\rm non-LSM}$ is unrelated to LSM constraints of interest.

The above discussion concerns about the case where $G_{int}=SO(3)$. Now we argue that for any $G_{int}$ with $\z_2^k$-classified PR, $\alpha_{\rm non-LSM}$ can be triggered in a $(3+1)$-d $M\times G_{int}$ SPT constructed in a way similar to the above, and all we need to modify is to replace the Haldane chain decorated into the $\z_2$ domain wall by a $(1+1)$-d $G_{int}$ SPT. To this end, it suffices to show that the TPF of this $(3+1)$-d $M\times G_{int}$ SPT is $e^{i\pi\int (w_1^{TM})^2\eta}$, where $\eta\in H^2(G_{int}, \z_2)$ and $e^{i\pi\int \eta}$ is the TPF of the $(1+1)$-d $G_{int}$ SPT. This can be shown by noting i) this SPT is its own inverse, and ii) this construction works for all such $G_{int}$. Then an argument very similar to that in Appendix <ref> suggests that the TPF of this SPT can indeed be written as $e^{i\pi\int (w_1^{TM})^2\eta}$.

§ GROUP COHOMOLOGY AND $\Z_2$ COHOMOLOGY RING OF WALLPAPER GROUPS

In this appendix, we list the $\z_2$ cohomology rings of all 17 wallpaper groups. The calculation is done with the help of spectral sequence. See Appendix <ref> for some brief mathematical introduction of the relevant concepts, and Refs. <cit.> for more details.
It turns out that for all wallpaper groups $G_s$ except $p4g$, the cohomology ring can be written as
H^*(G_s, Z_2)=Z_2[A_∙, ⋯, B_∙, ⋯]/relations
with $A_\bullet$ and $B_\bullet$ the generators belonging to $H^1(G_s, \mb{Z}_2)$ and $H^2(G_s, \mb{Z}_2)$, respectively. Subscripts “$\bullet$" are the names of generators which differ for different $G_s$, and their meanings will often be clear in the context. As a result, all elements of the cohomology ring $H^*(G_s, \mb{Z}_2)$ can be written as cup product of the generators $A_\bullet$ and $B_\bullet$, but there are some relations that dictate that certain sums of cup products actually yield a trivial cohomology element. We will present all elements of $H^1(G_s, \mb{Z}_2)$ and $H^2(G_s, \mb{Z}_2)$ together with their representative cochains, as well as the complete set of the relations. This encodes the full information of the cohomology ring $H^*(G_s, \mb{Z}_2)$. The situation for $p4g$ is similar, but we need an extra degree-3 generator $C\in H^3(p4g, \z_2)$, which, together with the generators in $H^1(p4g, \z_2)$ and $H^2(p4g, \z_2)$, forms a complete set of generators of $H^*(p4g, \z_2)$.

For later usage, we define a set of functions that take integers as their arguments:

1,     x is odd

0,     x is even

[x]_a={y=x (mod a)|0⩽y<a},

1,     x=b (mod a)

0,     otherwise

* Wallpaper group 1: $p1$

This group is generated by $T_1$ and $T_2$, two independent translations which are commutative,
T_1 T_2 = T_2 T_1.
An arbitrary group element in $p1$ can be written as $g=T_1^x T_2^y$, with $x, y\in\mb{Z}$. For $g_1=T_1^{x_1} T_2^{y_1}$ and $g_2=T_1^{x_2} T_2^{y_2}$, the group multiplication rule gives
g_1 g_2=T_1^x_1+x_2T_2^y_1+y_2.

The $\z_2$ cohomology ring of $p1$ is
_2[A_x, A_y]/(A_x^2=0, A_y^2=0).
Here $H^1(p1, \z_2) = \z_2^2$, with generators $\xi_1=A_x,\quad\xi_2=A_y$, which have representative cochains,
ξ_1(g) = x,    ξ_2(g)=y.
$H^2(p1, \z_2) = \z_2$, with generators $\lambda_1=A_x A_y$, which have representative cochains,
λ_1(g_1, g_2)=y_1x_2
Indeed $\lambda_1$ generates the LSM constraint.

* Wallpaper group 2: $p2$

This group is generated by $T_1$, $T_2$ and $C_2$, two independent translations and a $C_2$ rotational symmetry, with the following relations among generators
C_2^2=1,      C_2 T_1 C_2 = T_1^-1,      C_2 T_2 C_2 = T_2^-1,      T_1 T_2 = T_2 T_1.
An arbitrary group element in $p2$ can be written as $g=T_1^x T_2^y C_2^c$, with $x, y\in\mb{Z}$ and $c\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_2^{c_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_2^{c_2}$, the group multiplication rule gives

The $\z_2$ cohomology ring of $p2$ is
_2[A_x, A_y, A_c]/(A_x^2=A_x A_c, A_y^2=A_y A_c)
Here $H^1(p2, \z_2) = \z_2^3$, with generators $\xi_1=A_x,\quad\xi_2=A_y,\quad\xi_3=A_c$, which have representative cochains,
ξ_1(g) = x,    ξ_2(g)=y,    ξ_3(g)=c.
$H^2(p2, \z_2) = \z_2^4$, with generators $\lambda_1 = (A_x+A_c)(A_y+A_c), \quad\lambda_2 = A_x(A_y+A_c), \quad\lambda_3 = (A_x+A_c)A_y, \quad\lambda_4 = A_x A_y$, which have representative cochains,

λ_1(g_1, g_2)=y_1 x_2 + c_1(x_2+y_2+c_2)

λ_2(g_1, g_2)=(y_1+c_1)x_2

λ_3(g_1, g_2)=y_1 x_2+c_1 y_2

λ_4(g_1, g_2)=y_1x_2

The generators are chosen so that they have a 1-1 correspondence with topological invariants presented in Appendix <ref>. There we will also see that all of them, $\lambda_1,\lambda_2,\lambda_3,\lambda_4$, generate LSM constraints.

* Wallpaper group 3: $pm$

This group is generated by $T_1$, $T_2$ and $M$, where $T_1$ and $T_2$ are translations with perpendicular translation vectors, and $M$ is a mirror symmetry such that
M^2=1,      MT_1M=T_1^-1,      MT_2M=T_2,      T_1 T_2 = T_2 T_1.
An arbitrary element in $pm$ can be written as $g=T_1^x T_2^y M^m$, with $x, y\in\mb{Z}$ and $m\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} M^{m_2}$, the group multiplication rule gives

The $\z_2$ cohomology ring of $pm$ is
_2[A_x, A_y, A_m]/(A_x^2=A_x A_m,  A_y^2=0)
Here $H^1(pm, \z_2) = \z_2^3$, with generators $\xi_1=A_x,\quad\xi_2=A_y,\quad\xi_3=A_m$, which have representative cochains,
ξ_1(g) = x,    ξ_2(g)=y,    ξ_3(g)=m.
$H^2(pm, \z_2) = \z_2^4$, with generators $\lambda_1 = (A_x+A_m)A_y, \quad\lambda_2 = A_x A_y, \quad\lambda_3 = (A_x+A_m)A_m, \quad\lambda_4 = A_x A_m$, which have representative cochains,

λ_1(g_1, g_2)=y_1 x_2+m_1y_2

λ_2(g_1, g_2)=y_1 x_2

λ_3(g_1, g_2)=m_1(x_2+m_2)

λ_4(g_1, g_2)=m_1x_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2$ generate LSM constraints, while $\lambda_3,\lambda_4$ correspond to non-LSM fractionalization patterns.

* Wallpaper group 4: $pg$

This group is generated by $T_1$ and $G$, where $T_1$ is a translation and $G$ is a glide reflection, such that
Note that $G^2$ is a translation along the direction perpendicular to the translation vector of $T_1$. An arbitrary element in $pg$ can be written as $g=T_1^xG^{s}$, with $x, s\in\mb{Z}$. For $g_1=T_1^{x_1}G^{s_1}$ and $g_2=T_1^{x_2}G^{s_2}$, the group multiplication rule gives

The $\z_2$ cohomology ring of $pg$ is
_2[A_x, A_s]/(A_x^2=A_x A_s,  A_s^2=0)
Here $H^1(pg, \z_2) = \z_2^2$, with generators $\xi_1=A_x,\quad\xi_2=A_s$, which have representative cochains,
ξ_1(g) = x,    ξ_2(g)=s.
$H^2(pg, \z_2) = \z_2$, with generators $\lambda_1 = A_xA_s$, which have representative cochains,
λ_1(g_1, g_2)=s_1x_2
In Appendix <ref>, we will see that $\lambda_1$ generates LSM constraints.

* Wallpaper group 5: $cm$

This group is generated by $T_1$, $T_2$ and $M$, two independent translations and a mirror symmetry whose mirror axis bisects the translation vectors of $T_1$ and $T_2$. They satisfy
M^2=1,      MT_1M=T_2,      MT_2M=T_1,      T_1 T_2 = T_2 T_1.
An arbitrary element of $cm$ can be written as $g=T_1^x T_2^y M^m$, with $x, y\in\mb{Z}$ and $m\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} M^{m_2}$, the group multiplication rule gives

The $\z_2$ cohomology ring of $cm$ is
_2[A_x+y, A_m, B_xy]/(A_x+y^2=0,  A_x+y A_m = 0,   B_xy A_x+y=0,  B_xy^2=0)
Here $H^1(cm, \z_2)=\z_2^2$, with two generators $\xi_1=A_{x+y},\quad\xi_2=A_m$, which have representative cochains
ξ_1(g)=x+y,     ξ_2(g)=m.
$H^2(cm, \z_2)=\z_2^2$, with generators $\lambda_1=B_{xy}, \quad\lambda_2=A_m^2$, which have representative cochains

λ_1(g_1, g_2)   =P_c(m_1) y_1 x_2+P(m_1) y_2(x_2+y_1)

λ_2(g_1, g_2)   =m_1m_2

In Appendix <ref>, we will see that $\lambda_1$ generates LSM constraints, while $\lambda_2$ corresponds to non-LSM fractionalization patterns.

* Wallpaper group 6: $pmm$

This group is generated by $T_1$, $T_2$, $C_2$ and $M$, two translations with perpendicular translation vectors, a $C_2$ rotation and a mirror symmetry such that

M^2=1,      M C_2 M = C_2,      M T_1 M = T_1^-1,      M T_2 M = T_2

C_2^2=1,      C_2 T_1 C_2=T_1^-1,      C_2 T_2 C_2=T_2^-1,      T_1T_2=T_2 T_1.

Note that $C_2 M$ is another mirror symmetry that flips the translation vector of $T_2$. An arbitrary element in $pmm$ can be written as $g=T_1^x T_2^y C_2^{c} M^{m}$, with $x, y\in\mb{Z}$ and $c, m\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_2^{c_2} M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_2^{c_2} M^{m_2}$, the group multiplication rule gives

The $\z_2$ cohomology ring of $pmm$ is
_2[A_x, A_y, A_c, A_m]/(A_x^2=A_x (A_m + A_c),  A_y^2=A_y A_c)
Here $H^1(pmm, \z_2) = \z_2^4$, with generators $\xi_1=A_x,\quad\xi_2=A_y,\quad\xi_3=A_c,\quad\xi_4=A_m$, which have representative cochains,
ξ_1(g) = x,    ξ_2(g)=y,    ξ_3(g)=c,    ξ_4(g)=m.
$H^2(pmm, \z_2) = \z_2^8$, with generators    $\lambda_1 = (A_x+A_c+A_m)(A_y+A_c),\quad\lambda_2 = A_x A_y,\quad\lambda_3 = A_x(A_y+A_c),\quad\lambda_4 = (A_x+A_c+A_m)A_y ,\quad\lambda_5 = (A_x+A_c+A_m)A_m,\quad\lambda_6=(A_y+A_c)A_m,\quad\lambda_7 = A_x A_m,\quad\lambda_8 = A_y A_m$, which have representative cochains,

λ_1(g_1, g_2)=(y_1+c_1)x_2 + (c_1+m_1)(y_2+c_2)

λ_2(g_1, g_2)=y_1 x_2

λ_3(g_1, g_2)=(y_1+c_1)x_2

λ_4(g_1, g_2)=y_1 x_2 + (c_1 + m_1)y_2

λ_5(g_1, g_2)=m_1(x_2+c_2+m_2)

λ_6(g_1, g_2)=m_1(y_2 + c_2)

λ_7(g_1, g_2)=m_1x_2

λ_8(g_1, g_2)=m_1y_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2,\lambda_3,\lambda_4$ generate LSM constraints, while $\lambda_5,\lambda_6,\lambda_7,\lambda_8$ correspond to non-LSM fractionalization patterns.

* Wallpaper group 7: $pmg$

This group is generated by $T_1$, $T_2$, $C_2$ and $M$, two translations with perpendicular translation vectors, a 2-fold rotation and a mirror symmetry with mirror axis parallel to the translation vector of $T_2$, and displaced from the $C_2$ rotation center by a quarter of the unit translation vector of $T_1$. They satisfy

M^2=1,      MC_2M=T_1 C_2,      MT_1M=T_1^-1,      MT_2M=T_2,

C_2^2=1,      C_2 T_1 C_2 = T_1^-1,      C_2 T_2 C_2 = T_2^-1,      T_1 T_2 = T_2 T_1.

An arbitrary element in $pmg$ can be written as $g=T_1^x T_2^y C_2^c M^m$, with $x, y\in\mb{Z}$ and $c, m\in\{0, 1\}$.
For $g_1=T_1^{x_1} T_2^{y_1} C_2^{c_1} M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_2^{c_2} M^{m_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Q(c_1+m_1)x_2+ Q(c_1)c_2m_1 T_2^y_1+Q(c_1)y_2 C_2^P(c_1+c_2) M^P(m_1+m_2).

The $\z_2$ cohomology ring of $pmg$ is
_2[A_y, A_c, A_m]/(A_y^2=A_c A_y,  A_c A_m=0)
Here $H^1(pmg, \z_2) = \z_2^3$, with generators $\xi_1=A_y,\quad\xi_2=A_c,\quad\xi_3=A_m$, which have representative cochains,
ξ_1(g) = y,    ξ_2(g)=c,    ξ_3(g)=m.
$H^2(pmg, \z_2) = \z_2^4$, with generators $\lambda_1 = A_c(A_y+A_c), \quad\lambda_2 = A_c A_y, \quad\lambda_3 = A_yA_m, \quad\lambda_4 = A_m^2$, which have representative cochains,

λ_1(g_1, g_2)=c_1(y_2+c_2)

λ_2(g_1, g_2)=c_1y_2

λ_3(g_1, g_2)=m_1y_2

λ_4(g_1, g_2)=m_1m_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2$ generate LSM constraints, while $\lambda_3,\lambda_4$ correspond to non-LSM fractionalization patterns.

* Wallpaper group 8: $pgg$

This group is generated by $T_1$, $T_2$, $C_2$ and $G_1$, two translations with perpendicular translation vectors, a $C_2$ rotation, and a glide reflection whose reflection axis is parallel to the translation vector of $T_2$, and displaced from the $C_2$ center by a quarter of the unit translation vector of $T_1$. They satisfy
C_2^2=1,      G_1 C_2 G_1^-1 = T_1 T_2 C_2,      C_2 T_1 C_2 = T_1^-1,      G_1 T_1 G_1^-1 = T_1^-1,      G_1^2 = T_2.
An arbitrary element in $pgg$ can be written as $g=T_1^x T_2^y C_2^c G_1^{s}$, with $x, y\in\mb{Z}$ and $c, s\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_2^{c_1} G_1^{s_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_2^{c_2} G_1^{s_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Q(c_1+s_1)x_2+Q(c_1)c_2s_1 T_2^y_1+Q(c_1)y_2+Q(c_1)c_2s_1+Q(c_1+c_2)s_1s_2C_2^P(c_1+c_2)G_1^P(s_1+s_2).

The $\z_2$ cohomology ring of $pgg$ is
_2[A_c, A_s, B_c(x+y)]/(A_s^2=0, A_s A_c=0, A_s B_c(x+y)=0, B_c(x+y)^2 = A_c^2 B_c(x+y))
Here $H^1(pgg, \z_2) = \z_2^2$, with generators $\xi_1=A_c,\quad\xi_2=A_s$, which have representative cochains,
ξ_1(g) = c,    ξ_2(g)=s.
$H^2(pgg, \z_2) = \z_2^2$, with generators $\lambda_1=B_{c(x+y)}+A_c^2, \quad \omega_2 = B_{c(x+y)}$, which have representative cochains,

λ_1(g_1, g_2)=c_1(x_2+y_2+c_2)+(c_1+c_2)s_1 s_2+s_1x_2

λ_2(g_1, g_2)=c_1(x_2+y_2)+(c_1+c_2)s_1 s_2+s_1x_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2$ generate LSM constraints.

* Wallpaper group 9: $cmm$

This group is generated by $T_1$, $T_2$, $C_2$ and $M$, two translations with translation vectors not perpendicular to each other, a $C_2$ rotation, and a mirror symmetry whose mirror axis bisects the translation vectors of $T_1$ and $T_2$. They satisfy

M^2=1,      M C_2 M = C_2,      M T_1 M=T_2,      M T_2 M=T_1,

C_2^2=1,      C_2 T_1 C_2 = T_1^-1,      C_2 T_2 C_2 = T_2^-1,      T_1 T_2 = T_2 T_1.

Note that $C_2M$ is another mirror symmetry whose mirror axis bisects the translation vectors of $T_1$ and $T_2^{-1}$. An arbitrary element in $cmm$ can be written as $g=T_1^x T_2^y C_2^{c} M^{m}$, with $x, y\in\mb{Z}$ and $c, m\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_2^{c_{1}} M^{m_{1}}$ and $g_2=T_1^{x_2} T_2^{y_2} C_2^{c_{2}} M^{m_{2}}$, the group multplication rule gives
g_1g_2=T_1^x_1+Q(c_1)X T_2^y_1+Q(c_1)Y C_2^P(c_1+c_2) M^P(m_1+m_2),
where $X$ and $Y$ are defined as

The $\z_2$ cohomology ring of $cmm$ is
_2[A_x+y, A_c, A_m, B_xy]/(A_x+y^2=A_c A_x+y,  A_x+y A_m = 0,   B_xy A_x+y=0,  B_xy^2=(A_c^2+A_c A_m)B_xy)
Here $H^1(cmm, \z_2) = \z_2^3$, with generators $\xi_1=A_{x+y},\quad\xi_2=A_c,\quad\xi_3=A_m$, which have representative cochains,
ξ_1(g) = x+y,    ξ_2(g)=c,    ξ_3(g)=m.
$H^2(cmm, \z_2) = \z_2^5$, with generators $\lambda_1 = B_{xy} + A_c A_{x+y} + A_c^2 + A_m A_c, \quad \lambda_2 = B_{xy}, \quad \lambda_3 = A_c A_{x+y}, \quad \lambda_4 = (A_c + A_m) A_m, \quad \lambda_5 = A_c A_m$, which have representative cochains,

λ_1(g_1, g_2)=P_c(m_1)y_1x_2+P(m_1)y_2(x_2+y_1) + c_1(x_2+y_2+c_2+m_2)

λ_2(g_1, g_2)=P_c(m_1)y_1x_2+P(m_1)y_2(x_2+y_1)

λ_3(g_1, g_2)=c_1(x_2+y_2)

λ_4(g_1, g_2)=m_1(c_2+m_2)

λ_5(g_1, g_2)=m_1c_2

In Appendix <ref>, we will that $\lambda_1,\lambda_2,\lambda_3$ generate LSM constraints, while $\lambda_4,\lambda_5$ correspond to non-LSM fractionalization pattern.

* Wallpaper group 10: $p4$

This group is generated by $T_1$, $T_2$ and $C_4$, two translations with perpendicular translation vectors that have equal length, and a 4-fold rotational symmetry, such that C_4^4=1,      C_4T_1C_4^-1=T_2,      C_4T_2C_4^-1=T_1^-1,      T_1 T_2 = T_2 T_1.
An arbitrary element in $p4$ can be written as $g=T_1^x T_2^y C_4^c$, with $x, y\in\mb{Z}$ and $c\in\{0, 1, 2, 3\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_4^{c_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_4^{c_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(x_2, y_2, c_1) T_2^y_1+Δy(x_2, y_2, c_1)C_4^[c_1+c_2]_4

Δx(x, y, c)=

x,     c=0

-y,     c=1

-x,     c=2

y,     c=3

Δy(x, y, c)=

y,     c=0

x,     c=1

-y,     c=2

-x,     c=3

The $\z_2$ cohomology ring of $p4$ is

ℤ_2[A_c, A_x+y, B_c^2, B_xy]/(   A_c^2=0,  A_cA_x+y = 0,

B_xyA_x+y = B_xyA_c,  B_c^2 A_x+y = A_x+y^3 + B_xyA_x+y,  B_xy^2 = B_c^2B_xy)

Here $H^1(p4, \z_2) = \z_2^2$, with generators $\xi_1=A_{x+y},\quad\xi_2=A_c$, which have representative cochains,
ξ_1(g) = x+y,    ξ_2(g)=c.
$H^2(p4, \z_2) = \z_2^3$, with generators $\lambda_1 = B_{xy} + A_{x+y}^2 + B_{c^2}, \quad \lambda_2 = B_{xy}, \quad \lambda_3 = A_{x+y}^2$, which have representative cochains,

λ_1(g_1, g_2)=λ_2(g_1, g_2) + λ_3(g_1, g_2) +[c_1]_4+[c_2]_4-[c_1+c_2]_4/4

λ_2(g_1, g_2)=P_c(c_1)y_1x_2+P(c_1)y_2(x_2+y_1)

λ_3(g_1, g_2)=P_41(c_1) x_2 + P_42(c_1)(x_2+y_2) + P_43(c_1)y_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2,\lambda_3$ generate LSM constraints.

* Wallpaper group 11: $p4m$

This group is generated by $T_1$, $T_2$, $C_4$ and $M$, where the first three generators have the same properties as those in $p4$, and the last generator $M$ is a mirror symmetry that flips the translation vector of $T_1$, such that

M^2=1,      MC_4 M =C_4^-1,      MT_1M=T_1^-1,      MT_2M=T_2,

C_4^4=1,      C_4T_1C_4^-1=T_2,      C_4T_2C_4^-1=T_1^-1,      T_1 T_2 = T_2 T_1.

An arbitrary element in $p4m$ can be written as $g=T_1^xT_2^yC_4^cM^m$, with $x, y\in\mb{Z}$, $c\in\{0, 1, 2, 3\}$ and $m\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_4^{c_1}M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_4^{c_2}M^{m_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(Q(m_1)x_2, y_2, c_1) T_2^y_1+Δy(Q(m_1)x_2, y_2, c_1)C_4^[c_1+Q(m_1)c_2]_4M^P(m_1+m_2)
where $\Delta x(x, y, c)$ and $\Delta y(x, y, c)$ are defined in Eq. (<ref>).

The $\z_2$ cohomology ring of $p4m$ is

ℤ_2[A_c, A_x+y, A_m, B_c^2, B_xy]/(   A_c(A_c+A_m) = 0,  A_cA_x+y = 0,

B_xyA_x+y = B_xy(A_c + A_m),  B_c^2 A_x+y = A_x+y^3 + A_m A_x+y^2 + B_xyA_x+y ,

B_xy^2 = B_c^2B_xy)

Here $H^1(p4m, \z_2) = \z_2^3$, with generators $\xi_1=A_{x+y},\quad\xi_2=A_c,\quad\xi_3=A_m$, which have representative cochains,
ξ_1(g) = x+y,    ξ_2(g)=c,    ξ_3(g)=m.
$H^2(p4m, \z_2) = \z_2^6$, with generators $\lambda_1 = B_{xy} + A_{x+y}(A_{x+y} + A_m) + B_{c^2}, \quad \lambda_2 = B_{xy}, \quad \lambda_3 = A_{x+y}(A_{x+y} + A_m),\quad\lambda_4 = A_m(A_m+A_{x+y} + A_c), \quad \lambda_5 = A_m A_{x+y}, \quad \lambda_6 = A_m A_c$, which have representative cochains,

λ_1(g_1, g_2)=λ_2(g_1, g_2) + λ_3(g_1, g_2) + [c_1]_4+Q(m_1)[c_2]_4-[c_1+Q(m_1)c_2]_4/4

λ_2(g_1, g_2)=P_c(c_1)y_1x_2+P(c_1)y_2(x_2+y_1)

λ_3(g_1, g_2)=P_41(c_1) x_2 + P_42(c_1)(x_2+y_2) + P_43(c_1)y_2+m_1 y_2

λ_4(g_1, g_2)=m_1(x_2+ y_2 +c_2 + m_2)

λ_5(g_1, g_2)=m_1 (x_2+y_2)

λ_6(g_1, g_2)=m_1 c_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2,\lambda_3$ generate LSM constraints, while $\lambda_4,\lambda_5,\lambda_6$ correspond to non-LSM fractionalization patterns.

* Wallpaper group 12: $p4g$

This group is generated by $T_1$, $T_2$, $C_4$ and $G$. The first three generators have the same properties as those in $p4$, and the last generator $G$ is a glide reflection whose reflection axis passes through the rotation center of $C_4$ and bisects the translation vectors of $T_1$ and $T_2$, such that
C_4^4=1,      C_4T_1C_4^-1=T_2,      C_4T_2C_4^-1=T_1^-1,      T_1 T_2 = T_2 T_1,

G^2=T_1T_2,      GT_1G^-1=T_2,       GT_2G^-1=T_1,      GC_4G^-1=T_2C_4^-1.
Note that there is also a mirror symmetry $M=T_1^{-1}G$. An arbitrary element in $p4g$ can be written as $g=T_1^x T_2^y C_4^c G^{s}$, with $x, y\in\mb{Z}$, $c\in\{0, 1, 2, 3\}$ and $s\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_4^{c_1} G^{s_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_4^{c_2} G^{s_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(X, Y, c_1) T_2^y_1+Δy(X, Y, c_1) C_4^[c_1+Q(s_1)c_2]_4 G^P(s_1+s_2)
where $\Delta x(x, y, c)$ and $\Delta y(x, y, c)$ are defined in Eq. (<ref>), and

X=P_c(s_1)x_2+P(s_1)y_2+(P_42(c_2)+P_43(c_2))s_1+Δx(s_1 s_2, s_1 s_2, [Q(s_1)c_2]_4)

Y=P_c(s_1)y_2+P(s_1)x_2+(P_41(c_2)+P_42(c_2))s_1+Δy(s_1 s_2, s_1 s_2, [Q(s_1)c_2]_4)

The $\z_2$ cohomology ring of $p4g$ is

ℤ_2[A_c, A_s, B_c^2, B_c(x+y), C_c^2(x+y)]/(   A_c^2 = 0,  A_cA_s = 0, A_c B_c(x+y)=0, A_s B_c(x+y)=A_s B_c^2,

B_c(x+y)^2 = B_c^2 B_c(x+y),  A_c C_c^2(x+y) = 0,

B_c(x+y) C_c^2(x+y)  = B_c^2 C_c^2(x+y),

C_c^2(x+y)^2 = B_c(x+y)^3 + A_s B_c^2 C_c^2(x+y) )

Here $H^1(p4g, \z_2) = \z_2^2$, with generators $\xi_1=A_c,\quad\xi_2=A_s$, which have representative cochains,
ξ_1(g) = c,    ξ_2(g)=s.
$H^2(p4g, \z_2) = \z_2^3$, with generators $\lambda_1 = B_{c(x+y)} + B_{c^2}, \quad \lambda_2 = B_{c(x+y)}, \quad \lambda_3 = A_s^2$, which have representative cochains,

λ_1(g_1, g_2)=λ_2(g_1, g_2) + [c_1]_4+Q(s_1)[c_2]_4-[c_1+Q(s_1)c_2]_4/4

λ_2(g_1, g_2)= P_40(c_1)s_1(P_43(c_2)+(c_1-c_2)s_2)+P_41(c_1)[P_c(s_1)y_2+s_1(x_2+1-P_40(c_2)+(c_1-c_2)s_2)]

λ_3(g_1, g_2)=s_1s_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2$ generate LSM constraints, while $\lambda_3$ corresponds to non-LSM fractionalization patterns.

Pay attention that there is a degree-3 generator $C_{c^2(x+y)}\in H^3(p4g, \z_2)$. We do not have the explicit form of its representative cochain, but it can be determined by its pullback to the subgroup $p4$ generated by $T_1$, $T_2$, $C_4$, which is $A_{x+y}^3$, as well as its pullback to the subgroup $cmm$ generated by $T_1$, $T_2$, $T_2 C_2$, $T_1^{-1}G$, which is $A_{x+y}^3+A_c^3 + B_{xy} A_m + A_c^2A_m$.

* Wallpaper group 13: $p3$

This group is generated by $T_1$, $T_2$ and $C_3$, two translations with translation vectors that have the same length and an angle of $2\pi/3$, and a 3-fold rotational symmetry, such that
C_3^3=1,      C_3T_1C_3^-1=T_2,      C_3T_2C_3^-1=T_1^-1T_2^-1,      T_1 T_2 = T_2 T_1.
An arbitrary element in $p3$ can be written as $g=T_1^x T_2^y C_3^c$, with $x, y\in\mb{Z}$ and $c\in\{0, 1, 2\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_3^{c_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_3^{c_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(x_2, y_2, c_1) T_2^y_1+Δy(x_2, y_2, c_1) C_3^[c_1+c_2]_3

Δx(x, y, c)=

x,     c=0

-y,     c=1

-x+y,     c=2

Δy(x, y, c)=

y,     c=0

x-y,     c=1

-x,     c=2

The $\z_2$ cohomology ring of $p3$ is
Here $H^1(p3, \z_2) = 0$, while $H^2(p3, \z_2) = \z_2$, with generator $\lambda_1 = B_{xy}$, which have representative cochain,

λ_1(g_1, g_2)=   P_30(c_1)y_1x_2+P_31(c_1)(y_2(y_2-1)/2+y_2(x_2+y_1))

+P_32(c_1)(x_2(x_2-1)/2+ y_1x_2 + y_2(x_2+y_1))

In Appendix <ref>, we will see that $\lambda_1$ generates LSM constraints.

* Wallpaper group 14: $p3m1$

This group is generated by $T_1$, $T_2$, $C_3$ and $M$, where the first three generators have the same properties as those in $p3$, and the last one is a mirror symmetry whose mirror axis passes through the $C_3$ center, and is perpendicular to the angle that bisects the two translation vectors of $T_1$ and $T_2$, such that

M^2=1,      MC_3M = C_3^-1,      MT_1M=T_2^-1,      MT_2M=T_1^-1,

C_3^3=1,      C_3T_1C_3^-1=T_2,      C_3T_2C_3^-1=T_1^-1T_2^-1,      T_1 T_2 = T_2 T_1.

An arbitrary element in $p3m1$ can be written as $g=T_1^x T_2^y C_3^c M^m$, with $x, y\in\mb{Z}$, $c\in\{0, 1, 2\}$ and $M\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_3^{c_1} M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_3^{c_2} M^{m_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(X, Y, c_1) T_2^y_1+Δy(X, Y, c_1)C_3^[c_1+Q(m_1)c_2]_3M^P(m_1+m_2)
where $\Delta x(x, y, c)$ and $\Delta y(x, y, c)$ are defined in Eq. (<ref>), and

The $\z_2$ cohomology ring of $p3m1$ is
_2[A_m, B_xy]/(B_xy^2=0)
Here $H^1(p3m1, \z_2) = 0$, with generator $\xi_1=A_m$, which have representative cochain,
ξ_1(g) = m.
$H^2(p3m1, \z_2) = \z_2^2$, with generators $\lambda_1 = B_{xy},\quad\lambda_2=A_m^2$, which have representative cochains,

λ_1(g_1, g_2)=

λ_2(g_1, g_2)=m_1m_2

In Appendix <ref>, we will see that $\lambda_1$ generates LSM constraints, while $\lambda_2$ corresponds to non-LSM fractionalization pattern.

* Wallpaper group 15: $p31m$

This group is generated by $T_1$, $T_2$, $C_3$ and $M$, where the first three generators have the same properties as those in $p3$ and $p3m1$, and the last one is a mirror symmetry whose mirror axis passes through the $C_3$ center and bisects the translation vectors of $T_1$ and $T_2$, such that

M^2=1,      MC_3M = C_3^-1,      MT_1M=T_2,      MT_2M=T_1,

C_3^3=1,      C_3T_1C_3^-1=T_2,      C_3T_2C_3^-1=T_1^-1T_2^-1,      T_1 T_2 = T_2 T_1.

An arbitrary element in $p31m$ can be written as $g=T_1^x T_2^y C_3^c M^m$, with $x, y\in\mb{Z}$, $c\in\{0, 1, 2\}$ and $M\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_3^{c_1} M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_3^{c_2} M^{m_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(X, Y, c_1) T_2^y_1+Δy(X, Y, c_1)C_3^[c_1+Q(m_1)c_2]_3M^P(m_1+m_2)
where $\Delta x(x, y, c)$ and $\Delta y(x, y, c)$ are defined in Eq. (<ref>), and

The $\z_2$ cohomology ring of $p31m$ is
_2[A_m, B_xy]/(B_xy^2=0)
Here $H^1(p31m, \z_2) = 0$, with generator $\xi_1=A_m$, which have representative cochain,
ξ_1(g) = m.
$H^2(p31m, \z_2) = \z_2^2$, with generator $\lambda_1 = B_{xy},\quad\lambda_2=A_m^2$, which have representative cochains,

λ_1(g_1, g_2)=P_30(c_1)[P_c(m_1)y_1x_2+m_1y_2(x_2+y_1)]

λ_2(g_1, g_2)=m_1m_2

In Appendix <ref>, we will see that $\lambda_1$ generates LSM constraints while $\lambda_2$ corresponds to non-LSM fractionalization pattern.

* Wallpaper group 16: $p6$

This group is generated by $T_1$, $T_2$ and $C_6$, two translations with translation vectors that have the same length and an angle of $2\pi/3$, and a 6-fold rotational symmetry, such that
C_6^6=1,      C_6T_1C_6^-1=T_1T_2,      C_6T_2C_6^-1=T_1^-1,      T_1 T_2 = T_2 T_1.
An arbitrary element in $p6$ can be written as $g=T_1^x T_2^y C_6^c$, with $x, y\in\mb{Z}$ and $c\in\{0, 1, 2, 3, 4, 5\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_6^{c_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_6^{c_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(x_2, y_2, c_1)T_2^y_1+Δy(x_2, y_2, c_1)C_6^[c_1+c_2]_6

Δx(x, y, c)=

x,     c=0

x-y,     c=1

-y,     c=2

-x,     c=3

-x+y,     c=4

y,     c=5

Δy(x, y, c)=

y,     c=0

x,     c=1

x-y,     c=2

-y,     c=3

-x,     c=4

-x+y,     c=5

The $\z_2$ cohomology ring of $p6$ is
_2[A_c, B_xy]/(B_xy^2=A_c^2 B_xy)
Here $H^1(p6, \z_2) = \z_2$, with generator $\xi_1=A_c$, which have representative cochain,
ξ_1(g) = c.
$H^2(p6, \z_2) = \z_2^2$, with generators $\lambda_1 = B_{xy} + A_{c}^2, \quad \lambda_2 = B_{xy}$, which have representative cochains,

λ_1(g_1, g_2)=λ_2(g_1, g_2) + +[c_1]_6+[c_2]_6-[c_1+c_2]_6/6

λ_2(g_1, g_2)=    P_60(c_1)y_1x_2+P_61(c_1)(x_2(x_2-1)/2+y_1x_2+y_2(x_2+y_1))

In Appendix <ref>, we will see that $\lambda_1,\lambda_2$ generate LSM constraints.

* Wallpaper group 17: $p6m$

This group is generated by $T_1$, $T_2$, $C_6$ and $M$, where the first three generators have the same properties as those in $p6$, and the last one is a mirror symmetry whose mirror axis passes through the $C_6$ center and bisects $T_1$ and $T_2$, such that

M^2=1,      MC_6M = C_6^-1,      MT_1M=T_2,      MT_2M=T_1,

C_6^6=1,      C_6T_1C_6^-1=T_1T_2,      C_6T_2C_6^-1=T_1^-1,      T_1 T_2 = T_2 T_1.

An arbitrary element in $p6m$ can be written as $g=T_1^x T_2^y C_6^c M^m$, with $x, y\in\mb{Z}$, $c\in\{0, 1, 2, 3, 4, 5\}$ and $m\in\{0, 1\}$. For $g_1=T_1^{x_1} T_2^{y_1} C_6^{c_1} M^{m_1}$ and $g_2=T_1^{x_2} T_2^{y_2} C_6^{c_2} M^{m_2}$, the group multiplication rule gives
g_1g_2=T_1^x_1+Δx(X, Y, c_1) T_2^y_1+Δy(X, Y, c_1) C_6^[c_1+Q(m_1)c_2]_6M^P(m_1+m_2)
where $\Delta x(x, y, c)$ and $\Delta y(x, y, c)$ are defined in Eq. (<ref>), and $X$ and $Y$ are defined in Eq. (<ref>).

The $\z_2$ cohomology ring of $p6m$ is
_2[A_c, A_m, B_xy]/(B_xy^2=(A_c^2+A_c A_m) B_xy)
Here $H^1(p6m, \z_2) = \z_2^2$, with generator $\xi_1=A_c,\quad\xi_2=A_m$, which have representative cochains,
ξ_1(g) = c,    ξ_2(g)=m.
$H^2(p6m, \z_2) = \z_2^4$, with generators $\lambda_1 = B_{xy} + A_{c}^2 +A_c A_m, \quad \lambda_2 = B_{xy}, \quad\lambda_3 = (A_c + A_m) A_m, \quad\lambda_4 = A_c A_m$, which have representative cochains,

λ_1(g_1, g_2)=λ_2(g_1, g_2) + [c_1]_6+Q(m_1)[c_2]_6-[c_1+Q(m_1)c_2]_6/6

λ_2(g_1, g_2) = P_60(c_1)[P_c(m_1)y_1x_2+m_1y_2(x_2+y_1)]

λ_3(g_1, g_2)=m_1(c_2+m_2)

λ_4(g_1, g_2)=m_1 c_2

In Appendix <ref>, we will see that $\lambda_1,\lambda_2$ generate LSM constraints, while $\lambda_3,\lambda_4$ correspond to non-LSM fractionalization patterns.

§ TOPOLOGICAL INVARIANTS FOR ALL LSM CONSTRAINTS

In this appendix, for each of the 17 wallpaper groups, we present the topological invariants for all LSM constraints and topological invariants for all non-LSM fractionalization patterns. These topological invariants can all be written down by simply inspecting the IWP and/or the mirror axes of the relevant wallpaper groups, and they correspond to various $(3+1)$-d $G_s\times G_{int}$ SPTs that can be constructed in a manner described in Sec. <ref>. This physics-based reasoning implies that the topological invariants we present here are complete and independent. In Appendix <ref>, we also provide explicit expressions of the representative cochains that correspond to each of the topological invariants, which show mathematically that the topological invariants here are indeed complete and independent.

* Wallpaper group 1: $p1$

All points in space correspond to the same IWP for $p1$. The fractionalization patterns of $p1$ are classified by $H^2(p1, \mb{Z}_2)=\mb{Z}_2$. There is only one nontrivial fractionalization pattern $T_1 T_2 = -T_2 T_1$, detected by the topological invariant
α_1[ω]=ω(T_1, T_2)/ω(T_2, T_1).
This fractionalization pattern is related to the first of the 3 basic no-go theorems in Sec. <ref>, so it corresponds to an LSM constraint, and the classification of LSM constraints is $\mb{Z}_2$. It is straightforward to check that $\alpha_1[(-1)^{\lambda_1}] = -1$, where $\lambda_1$ is defined in Appendix <ref>.

* Wallpaper group 2: $p2$

There are 4 different IWP for $p2$, which are rotation centers for $C_2$, $T_1C_2$, $T_2C_2$ and $T_1T_2C_2$, respectively. The fractionalization patterns of $p2$ are classified by $H^2(p2, \mb{Z}_2)=\mb{Z}_2^4$. All fractionalization patterns are generated by 4 root patterns, detected by the topological invariants,

α_1[ω]=ω(C_2, C_2)/ω(1, 1)

α_2[ω]=ω(T_1C_2, T_1C_2)/ω(1, 1)

α_3[ω]=ω(T_2C_2, T_2C_2)/ω(1, 1)

α_4[ω]=ω(T_1T_2C_2, T_1T_2C_2)/ω(1, 1)

corresponding to $C_2^2=-1$, $(T_1 C_2)^2=-1$, $(T_2 C_2)^2=-1$ and $(T_1 T_2 C_2)^2=-1$ respectively. All these topological invariants are related to the third of the 3 basic no-go theorems in Sec. <ref>, so they all correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^4$. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,4$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 3: $pm$

There are 2 different IWP for $pm$, which are the mirror axes for $M$ and $T_1M$, respectively. The fractionalization patterns of $pm$ are classified by $H^2(pm, \mb{Z}_2)=\mb{Z}_2^4$. All fractionalization patterns are generated by 4 root patterns, detected by the topological invariants

α_1[ω]=ω(T_2, M)/ω(M, T_2)

α_2[ω]=ω(T_2, T_1M)/ω(T_1M, T_2)

α_3[ω]=ω(M, M)/ω(1, 1)

α_4[ω]=ω(T_1M, T_1M)/ω(1, 1)

The first two topological invariants are related to the second of the 3 basic no-go theorems, so they correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^2$. The last two are non-LSM fractionalization patterns. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,4$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 4: $pg$

All points in space belong to one IWP of $pg$. The fractionalization patterns of $pg$ are classified by $H^2(pg, \mb{Z}_2)=\mb{Z}_2$. There is only one nontrivial fractionalization pattern, detected by the topological invariant

α_1[ω]=ω(T_1G^-1, T_1G)ω(T_1, G)/ω(G^-1, G)ω(T_1, G^-1)
corresponding to the symmetry fractionalization pattern $G^{-1} T_1 G T_1= -1$. This topological invariant is related to the first of the 3 basic no-go theorems, so it corresponds to an LSM constraint, and the classification of LSM constraints is $\mb{Z}_2$. It is straightforward to check that $\alpha_1[(-1)^{\lambda_1}] = -1$, where $\lambda_1$ is defined in Appendix <ref>.

* Wallpaper group 5: $cm$

The group $cm$ has one IWP, which includes points along the mirror axis of $M$. The fractionalization patterns of $cm$ are classified by $H^2(cm, \mb{Z}_2)=\mb{Z}_2^2$. All fractionalization patterns are generated by 2 root patterns, detected by the topological invariants

α_1[ω]=ω(T_1T_2, M)/ω(M, T_1T_2)

α_2[ω]=ω(M, M)/ω(1, 1)

The first topological invariant is related to the second of the 3 basic no-go theorems, so it corresponds to an LSM constraint, and the classification of LSM constraints is $\mb{Z}_2$. The second one is a non-LSM fractionalization pattern. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,2$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 6: $pmm$

There are 4 different IWP for $pmm$, which are the intersecting point of $M$ and $C_2M$, the intersecting point of $T_1M$ and $C_2M$, the intersecting point of $M$ and $T_2C_2M$, and the intersecting point of $T_1M$ and $T_2C_2M$. Note that these 4 IWP can also be respectively viewed as the rotation centers for the following four $C_2$ rotations: $C_2$, $T_1C_2$, $T_2C_2$ and $T_1T_2C_2$. The fractionalization patterns of $pmm$ are classified by $H^2(pmm, \mb{Z}_2)=\mb{Z}_2^8$. All fractionalization patterns are generated by 8 root patterns, detected by the topological invariants

α_1[ω]=ω(C_2, C_2)/ω(1, 1)

α_2[ω]=ω(T_1T_2C_2, T_1T_2C_2)/ω(1, 1)

α_3[ω]=ω(T_1C_2, T_1C_2)/ω(1, 1)

α_4[ω]=ω(T_2C_2, T_2C_2)/ω(1, 1)

α_5[ω]=ω(M, M)/ω(1, 1)

α_6[ω]=ω(C_2M, C_2M)/ω(1, 1)

α_7[ω]=ω(T_1M, T_1M)/ω(1, 1)

α_8[ω]=ω(T_2C_2M, T_2C_2M)/ω(1, 1)

The first four topological invariants are related to the third of the 3 basic no-go theorems, so they correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^4$. The last four are non-LSM fractionalization patterns. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,8$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 7: $pmg$

There are 3 different IWP for $pmg$. The first includes the rotation centers of $C_2$ and $T_1C_2$, the second includes the rotation centers of $T_2C_2$ and $T_1T_2C_2$, and the third includes the mirror axes of $M$ and $T_1M$. The fractionalization patterns of $pmg$ are classified by $H^2(pmg, \mb{Z}_2)=\mb{Z}_2^4$. All fractionalization patterns are generated by 4 root patterns, detected by the topological invariants

α_1[ω]=ω(C_2, C_2)/ω(1, 1)

α_2[ω]=ω(T_1T_2C_2, T_1T_2C_2)/ω(1, 1)

α_3[ω]=ω(T_2, M)/ω(M, T_2)

α_4[ω]=ω(M, M)/ω(1, 1)

The first two topological invariants are related to the third of the 3 basic no-go theorems, and the third is related to the second of the 3 basic no-go theorems, so they correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^3$. The fourth topological invariant is a non-LSM fractionalization pattern. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,4$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 8: $pgg$

There are 2 different IWP for $pgg$. The first includes the rotation centers of $C_2$ and $T_1T_2C_2$, and the second includes the rotation centers of $T_1C_2$ and $T_2C_2$. The fractionalization patterns of $pgg$ are classified by $H^2(pgg, \mb{Z}_2)=\mb{Z}_2^2$. All fractionalization patterns are generated by 2 root patterns, detected by the topological invariants

α_1[ω]=ω(C_2, C_2)/ω(1, 1)

α_2[ω]=ω(T_1C_2, T_1C_2)/ω(1, 1)

Both topological invariants are related to the third of the 3 basic no-go theorems, so they both correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^2$. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,2$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 9: $cmm$

There are 3 different IWP for $cmm$. The first is the 2-fold rotation center of $C_2$, the second is the 2-fold rotation center of $T_1T_2C_2$, and the third inlcudes the 2-fold rotation centers of $T_1C_2$ and $T_2C_2$.

The fractionalization patterns of $cmm$ are classified by $H^2(cmm, \mb{Z}_2)=\mb{Z}_2^5$. All fractionalization patterns are generated by 5 root patterns, detected by the topological invariants

α_1[ω]=ω(C_2, C_2)/ω(1, 1)

α_2[ω]=ω(T_1T_2C_2, T_1T_2C_2)/ω(1, 1)

α_3[ω]=ω(T_1C_2, T_1C_2)/ω(1, 1)

α_4[ω]=ω(M, M)/ω(1, 1)

α_5[ω]=ω(C_2M, C_2M)/ω(1, 1)

The first three topological invariants are related to the third of the 3 basic no-go theorems, so they correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^3$. The last two are non-LSM fractionalization patterns. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,5$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 10: $p4$

There are 3 different IWP for $p4$. The first is the 2-fold rotation center of $C_4^2$, the second is the 2-fold rotation center of $T_1T_2C_4^2$, and the third includes the 2-fold rotation centers of $T_1C_4^2$ and $T_2C_4^2$. Note that the first two IWP are also 4-fold rotation centers. The fractionalization patterns of $p4$ are classfied by $H^2(p4, \mb{Z}_2)=\mb{Z}_2^3$. All fractionalization patterns are generated by 3 root patterns, detected by the topological invariants

α_1[ω]=ω(C_4^2, C_4^2)/ω(1, 1)

α_2[ω]=ω(T_1T_2C_4^2, T_1T_2C_4^2)/ω(1, 1)

α_3[ω]=ω(T_1C_4^2, T_1C_4^2)/ω(1, 1)

All these topological invariants are related to the third of the 3 basic no-go theorems, so they all correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^3$. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,3$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 11: $p4m$

There are 3 different IWP for $p4m$, just like $p4$. The first is the 2-fold rotation center of $C_4^2$, the second is the 2-fold rotation center of $T_1T_2C_4^2$, and the third includes the 2-fold rotation centers for $T_1C_4^2$ and $T_2C_4^2$. All these IWP are also on some mirror axes, and the first two are also 4-fold rotation centers. The fractionalization patterns of $p4m$ are classified by $H^2(p4m, \mb{Z}_2)=\mb{Z}_2^6$. All fractionalization patterns are genereated by 6 root patterns, detected by the topological invariants

α_1[ω]=ω(C_4^2, C_4^2)/ω(1, 1)

α_2[ω]=ω(T_1T_2C_4^2, T_1T_2C_4^2)/ω(1, 1)

α_3[ω]=ω(T_1C_4^2, T_1C_4^2)/ω(1, 1)

α_4[ω]=ω(M, M)/ω(1, 1)

α_5[ω]=ω(T_1M, T_1M)/ω(1, 1)

α_6[ω]=ω(C_4M, C_4M)/ω(1, 1)

The first three topological invariants are related to the third of the 3 basic no-go theorems, so they correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^3$. The last three are non-LSM constraints. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,6$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 12: $p4g$

There are 2 different IWP for $p4g$. The first includes the 2-fold rotation centers of $C_4^2$ and $T_1T_2C_4^2$, and the second includes the 2-fold rotation centers of $T_1C_4^2$ and $T_2C_4^2$. Note that the first IWP are also 4-fold rotation centers, and they do not lie on any mirror axis. The second IWP lies on some mirror axes. The fractionalization patterns of $p4g$ are classified by $H^2(p4g, \mb{Z}_2)=\mb{Z}_2^3$. All fractionalization patterns are generated by 3 root patterns, detected by the topological invariants

α_1[ω]=ω(C_4^2, C_4^2)/ω(1, 1)

α_2[ω]=ω(T_1C_4^2, T_1C_4^2)/ω(1, 1)

α_3[ω]=ω(T_1^-1G, T_1^-1G)/ω(1, 1)

The first two topological invariants are related to the third of the 3 basic no-go theorems, so they correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^2$. The last one is a non-LSM fractionalization pattern. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,3$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 13: $p3$

There are 3 IWP for $p3$, and they are all 3-fold rotation centers. The fractionalization patterns of $p3$ are classified by $H^2(p3, \mb{Z}_2)=\mb{Z}_2$. All fractionalization patterns are generated by a root pattern, detected by the topological invariant
α[ω]=ω(T_1, T_2)/ω(T_2, T_1)
This topological invariant is related to the first of 3 basic no-go theorems, so it corresponds to an LSM constraint, and the classification of LSM constraints is $\mb{Z}_2$. It is straightforward to check that $\alpha_1[(-1)^{\lambda_1}] = -1$ for $i,j=1,\dots,8$, where $\lambda_1$ is defined in Appendix <ref>.

* Wallpaper group 14: $p3m1$

There are 3 different IWP for $p3m1$, and they are all 3-fold rotation centers, just as in $p3$, but they also lie on the mirror axes. The fractionalization patterns of $p3m1$ are classified by $H^2(p3m1, \mb{Z}_2)=\mb{Z}_2^2$. All fractionalization patterns are generated by 2 root patterns, detected by the topological invariants

α_1[ω]=ω(T_1, T_2)/ω(T_2, T_1)

α_2[ω]=ω(M, M)/ω(1, 1)

The first topological invariant is related to the first of the 3 basic no-go theorems, so it corresponds to an LSM constraint, and the classification of LSM constraints is $\mb{Z}_2$. The second one is a non-LSM fractionalization pattern. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,2$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 15: $p31m$

There are 3 different IWP for $p31m$, and they are all 3-fold rotation centers, just as in $p3$, but only one of them also lies on the mirror axes. The fractionalization patterns of $p31m$ are classified by $H^2(p31m, \mb{Z}_2)=\mb{Z}_2^2$. All fractionalization patterns are generated by 2 root patterns, detected by the topological invariants

α_1[ω]=ω(T_1T_2, M)/ω(M, T_1T_2)

α_2[ω]=ω(M, M)/ω( 1, 1)

The first topological invariant is related to the second of the 3 basic no-go theorems, so it corresponds to an LSM constraint, and the classification of LSM constraints is $\mb{Z}_2$. The second is a non-LSM fractionalization pattern. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,2$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 16: $p6$

There are 3 different IWP for $p6$, and they are centers of 6-fold, 3-fold and 2-fold rotations, respectively. The fractionalization patterns of $p6$ are classified by $H^2(p6, \mb{Z}_2)=\mb{Z}_2^2$. All fractionalization patterns are generated by 2 root patterns, detected by the topological invariants

α_1[ω]=ω(C_6^3, C_6^3)/ω(1, 1)

α_2[ω]=ω(T_1C_6^3, T_1C_6^3)/ω(1, 1)

Both topological invariants are related to the third of the 3 basic no-go theorems, so they both correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^2$. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,2$, where $\lambda_i$ is defined in Appendix <ref>.

* Wallpaper group 17: $p6m$

There are 3 different IWP for $p6m$. Just as $p6$, they are 6-fold, 3-fold and 2-fold rotation centers, respectively. Here all IWP also lie on some mirror axes. The fractionalization patterns of $p6m$ are classified by $H^2(p6m, \mb{Z}_2)=\mb{Z}_2^4$. All fractionalization patterns are generated by 4 root patterns, detected by topological invariants

α_1[ω]=ω(C_6^3, C_6^3)/ω(1, 1)

α_2[ω]=ω(T_1C_6^3, T_1C_6^3)/ω(1, 1)

α_3[ω]=ω(M ,M)/ω(1, 1)

α_4[ω]=ω(C_6^3M, C_6^3M)/ω(1, 1)

The first two topological invariants are related to the third of the 3 basic no-go theorems, and they correspond to LSM constraints, and the classification of LSM constraints is $\mb{Z}_2^2$. The last two are non-LSM fractionalization patterns. It is straightforward to check that $\alpha_i[(-1)^{\lambda_j}] = (-1)^{\delta_{ij}}$ for $i,j=1,\dots,4$, where $\lambda_i$ is defined in Appendix <ref>.

§ TOPOLOGICAL CHARACTERIZATION OF LSM CONSTRAINTS IN $(1+1)$-D

In this appendix, we present the derivation of the topological characterization of the LSM constraints for $(1+1)$-d $G_s\times G_{int}$ symmetric spin systems, where the results are already given in Sec. <ref>.

First, we note that an argument similar to the one in Appendix <ref> for the $(2+1)$-d case shows that in this case the relevant cocycle can be written as
Ω(g_1, g_2, g_3)=e^iπλ(l_1)η(a_2, a_3)
where $g_i\in G_s\times G_{int}$ is written as $g_i=l_i\otimes a_i$, with $l_i\in G_s$ and $a_i\in G_{int}$. The cocycle for the nontrivial $(1+1)$-d $G_{int}$ SPT is precisely $e^{i\pi \eta(a_1, a_2)}$, and $\lambda$ can be viewed as a cocycle in $H^1(G_s, \mb{Z}_2)$. Furthermore, $\lambda$ is determined completely by $G_s$ and the lattice homotopy class, and it is the same for all $G_{int}$ with $\mb{Z}_2^k$-classified PR and for all PR type of the system.

When $G_s=p1$, the line group that only contains translation generated by $T$, the lattice homotopy picture implies that the LSM constraints in this case are classified by $\mb{Z}_2$, and the only nontrivial LSM constraint corresponds to the case where the total PR inside each translation unit cell is nontrivial. On the other hand, $H^1(p1, \mb{Z}_2)=\mb{Z}_2$, so there is also only one nontrivial cocycle. Writing an elment in $p1$ as $T^x$ with $x\in\z$, $\lambda(T^x)=[x]_2$ is a representative cochain of the nontrivial element in $H^1(p1, \z_2)$. So we can identify the cocycle corresponding to the nontrivial LSM constraint as

Ω(g_1, g_2, g_3)=e^iπx_1η(a_2, a_3)

When $G_s=p1m$, the line group that contains a translation generated by $T$ and a mirror symmetry generated by $M$, with commutation relation $MTM = T^{-1}$, there are two IWP in each translation unit cell, which are the mirror centers of $M$ and $TM$, respectively. The lattice homotopy picture implies that the LSM constraints in this case are classified by $\mb{Z}_2^2$, and the two root LSM constraints can be taken to correspond to the cases where the total PR at one of the two IWP is nontrivial. On the other hand, $H^1(p1m, \mb{Z}_2)=\mb{Z}_2^2$, so all nontrivial cocycles in $H^1(p1m, \mb{Z}_2)$ must correspond to some nontrivial LSM constraint. These cocycles can be generated by two roots represented by $\lambda_1(T^xM^m)=x+m$ and $\lambda_2(T^xM^m)=x$, with $x\in\mb{Z}$ and $m\in\{0, 1\}$. So the cocycles corresponding to the LSM constraints can also be generated by

Ω_1(g_1, g_2, g_3)=e^iπ(x_1+m_1)η(a_2, a_3)

Ω_2(g_1, g_2, g_3)=e^iπx_1η(a_2, a_3)

where $g_i\in G_s\times G_{int}$ is written as $g_i=T^{x_i}M^{m_i}\otimes a_i$, with $a_i\in G_{int}$.

Now our task is just to identify $\Omega_1$ and $\Omega_2$ with the distributions of DOF that trigger the LSM constraint. To this end, first note that if $M$ is broken while $T$ is preserved, both $\Omega_1$ and $\Omega_2$ reduces to Eq. (<ref>), which implies that both of them correspond to a distribution of DOF with a net nontrivial PR inside each translation unit cell. So one of them must correspond to the case where the mirror center of $M$ hosts a nontrivial PR, while the other corresponds to the case where the mirror center of $TM$ hosts a nontrivial PR. Suppose the mirror center of $TM$ hosts a nontrivial PR, then after breaking the translation symmetry while keeping $M$ unbroken, the system should have no LSM constraint. Only $\Omega_2$ satisfies this condition, so this distribution of DOF is identified with $\Omega_2$, and $\Omega_1$ corresponds to the case where the mirror center of $M$ hosts a nontrivial PR.

§ MORE DETAILS OF THE STIEFEL LIQUIDS

In this appendix, we discuss more details of Stiefel liquids, some of which do not appear in Ref. <cit.>.

First we present some suggestive argument, but not rigorous proof, supporting that SL$^{(N>6)}$ are non-Lagrangian, they cannot be described by any weakly-coupled renormalizable Lagrangian in the UV. The key observation is that it appears unlikely for such Lagrangians to realize the $SO(N)$, $SO(N-4)$ and reflection symmetries of SL$^{(N)}$. To see it, let us start with even $N$. Usually in such a Lagrangian, symmetries like $SO(N)$ and $SO(N-4)$ are flavor symmetries, and there is a reflection symmetry that commutes with flavor symmetries. However, due to the locking between spacetime orientation reversals and improper rotations of $O(N)$ and $O(N-4)$, SL$^{(N)}$ has no such a reflection symmetry. This suggests that $SO(N)$ and $SO(N-4)$ cannot be simultaneously flavor symmetries. In the special case of $N=6$, which does have a renormalizable Lagrangian description, indeed only $SO(6)$ but not $SO(2)$ can be identified as a flavor symmetry. In this example, the $SO(2)$ is realized as the flux conservation symmetry in the gauge theoretic formulation. For $N>6$, there is no known generalization of the flux conservation symmetry that can give rise to symmetries like $SO(N-4)$. This indicates SL$^{(N>6)}$ with an even $N$ may be non-Lagrangian. Due to the cascade structure of SLs <cit.>, it also suggests all SL$^{(N>6)}$ are non-Lagrangian.

We emphasize that the above is just a suggestive argument, but not a rigorous proof. There can be ways to get around the above obstruction, by, implementing some symmetries via dualities, considering Lagrangians in very complicated forms, showing that Lagrangians with smaller symmetries can have emergent symmetries of the SLs, etc. After finding a Lagrangian that can realize the symmetries of a SL, one still needs to make sure that its anomaly and low-energy dynamics match with the SL, which appears also challenging. If all these nontrivial challenges can be overcome and a renormalizable Lagrangian can be found to describe the SL at the end, we believe this process can generate new insights and teach us some valuable general lessons of quantum field theories.

Next we discuss the anomalies of SLs, which should be captured by $\Omega_{\rm IR}$, an element in $H^4(G_{\rm IR}, U(1)_\rho)$, where $G_{\rm IR}=(O(N)^T\times O(N-4)^T)/\z_2$. Consider the projection: $p_{\rm SL}: \tilde G_{\rm IR}\equiv O(N)^T\times O(N-4)^T\rightarrow G_{\rm IR}$, which induces a pullback $p^*_{\rm SL}: H^4(G_{\rm IR}, U(1)_\rho)\rightarrow H^4(\tilde G_{\rm IR}, U(1)_\rho)$. The pullback of $\Omega_{\rm IR}$, $\tilde \Omega_{\rm IR}\equiv p^*_{\rm SL}\Omega_{\rm IR}$, is given by Eq. (<ref>), in a form $\tilde \Omega_{\rm IR}=e^{i\pi \tilde L_{\rm IR}}$, with $\tilde L_{\rm IR}\in H^4(\tilde G_{\rm IR}, \z_2)$ <cit.>. For even $N$, the structure of $\Omega_{\rm IR}$ is still not completely understood, but it is known that $\tilde \Omega_{\rm IR}$ misses some important information. In particular, the form of $\tilde \Omega_{\rm IR}$ suggests that two copies of SL$^{(N)}$ would be anomaly-free. However, only four copies of SL$^{(N)}$ is anomaly-free, while two copies is still anomalous <cit.>.

For odd $N$, because $O(N)^T=SO(N)\times\z_2^T$, $H^4(G_{\rm IR}, U(1)_\rho)$ has the structure of $\z_2^k$ with some $k\in\mb{N}$, and there exists $L_{\rm IR}\in H^4(G_{\rm IR}, \z_2)$ such that $\Omega_{\rm IR}=e^{i\pi L_{\rm IR}}$. Now notice that the pullback from $H^4(G_{\rm IR}, \z_2)$ to $H^4(\tilde G_{\rm IR}, \z_2)$ induced by $p_{\rm SL}$ is injective, and hence we can uniquely identify $L_{\rm IR}$ from $\tilde L_{\rm IR}$. The result is

L_IR=w_4^SO(N)+w_4^SO(N-4)+ (w_2^SO(N) + w_2^SO(N-4))w_2^SO(N-4)+

w_1^2w_2^SO(N),     N=1 ( mod 8)

w_1^2w_2^SO(N-4),     N=3 ( mod 8)

w_1^2(w_2^SO(N)+w_1^2),     N=5 ( mod 8)

w_1^2(w_2^SO(N-4)+w_1^2),     N=7 ( mod 8)

where $w_i^{SO(N)}$ and $w_i^{SO(N-4)}$ are the $i$-th Stiefel-Whitney class of the $SO(N)$ and $SO(N-4)$ gauge bundles. Considering enlarging $SO(N)$ and $SO(N-4)$ to $O(N)$ and $O(N-4)$, $w_1$ is sum of the first Stiefel-Whitney classes of the $O(N)$ and $O(N-4)$ gauge bundles. Due to the locking between spacetime orientation reversals and improper rotations of $O(N)$ and $O(N-4)$, $w_1$ can also be viewed as the first Stiefel-Whitney class of the tangent bundle of the spacetime manifold.

Finally, we discuss the effects of relevant operators on the DQCP (SL$^{(5)}$), DSL (SL$^{(6)}$) and SL$^{(7)}$. Because the low-energy dynamics of these states are not fully settled down, this discussion is also conjectural, and it is important to study these issues in a more rigorous manner in the future. However, given our understanding of these states, we believe the expectations below are reasonable.

For all SLs, the $(V_L, V_R)$ operator (or the $SO(5)$ vector for DQCP) should change the emergent order of the state. Due to the cascade structure among SLs <cit.>, it is natural that this operator will just drive SL$^{(N)}$ to SL$^{(N-1)}$ (for DQCP, it simply gaps out the state). The time-reversal breaking operator that is a flavor singlet is likely to drive the state into a semion topological order, and this expectation is supported by the gauge-theoretic formulations of DQCP and DSL, as well as the fact that the semion topological order can match the anomaly of SL$^{(N)}$ if time reversal is broken (for all $N\geqslant 5$) <cit.>. The $(A_L, A_R)$ operator (for all $N\geqslant 6$) is expected to convert SL$^{(N)}$ into certain spontaneous-symmetry-breaking state, as supported from the gauge-theoretic formulation of DSL <cit.>. For DQCP, the traceless symmetric rank-2 tensor of $SO(5)$ drives the state into a spontaneous-symmetry-breaking state, and this operator is the tuning operator of the Neel-VBS transition in the standard realization of DQCP <cit.>. All these operators change the emergent order of the states.

The remaining relevant operators to be discussed are the conserved current operators, whose effects on various states are complicated. It turns out that some of them can change the emergent order of the states, while others only shift the “zero momenta".

The simplest way to discuss it is perhaps to start from DSL, which has a relatively simple gauge-theoretic formulation in terms of $N_f=4$ QED$_3$:
L=∑_i=1^4ψ̅_i iD_aψ_i-1/4e^2f_μνf^μν
There are conserved currents due to the $SO(6)$ and $SO(2)$ symmetries, where a natural basis of the $SO(6)$ currents is $\bar\psi\gamma_\mu T^{su(4)}\psi$, with $\gamma_\mu$ the Dirac matrices and $T^{(su(4))}$ the generators of $su(4)$ in its fundamental representation, and the $SO(2)$ currents are $\epsilon_{\mu\nu\lambda}\partial^\nu a^\lambda/(2\pi)$, with $a$ the emergent $U(1)$ gauge field. If the time component of the $SO(6)$ currents is added as a pertubation to the DSL, the Dirac fermions will be doped and acquire a finite Fermi surface, so the emergent order of the state changes. If the time component of the $SO(2)$ currents is added, the Dirac fermions will experience magnetic fields, Landau levels will form, and the emergent order of the state also changes. Below we discuss the effect of the spatial components of the currents.

For the $SO(6)$ spatial currents, depending on the choice of $T^{(su(4))}$ and $\gamma_\mu$, various effects can be triggered. For example, the current $\bar\psi\gamma_x\sigma_{30}\psi$ merely shifts the positions of the Dirac cones in the momentum space in a flavor-dependent way, which does not really change the emergent order of DSL (here $\sigma_{ij}\equiv\sigma_i\otimes\sigma_j$, where $\sigma_{i=0,1,2,3}$ are the identity and standard Pauli matrices). The same is true for $\bar\psi(\gamma_x\sigma_{30}+\gamma_y\sigma_{03})\psi$. However, as another example, $\bar\psi(\gamma_x\sigma_{23}+\gamma_y\sigma_{33})\psi$ actually converts the 4 Dirac cones into 2 pairs of quadratic band touching (and another 2 pairs of gapped bands), which does change the emergent order of the state. By examining the effect of different spatial currents, one can see more complicated patterns. Although a systematic description of the effects of these spatial currents is lacking, it can be analyzed in a case-by-case manner. These spatial currents can all be converted into the language of SL$^{(6)}$, in terms of the $6\times 2$ matrix $n$. For example, using Appendix E of Ref. <cit.>, we see that $\bar\psi\gamma_x\sigma_{30}\psi\sim n_{3i}\partial_xn_{4i}$, $\bar\psi(\gamma_x\sigma_{30}+\gamma_y\sigma_{03})\psi\sim n_{3i}\partial_xn_{4i}+n_{1i}\partial_yn_{2i}$, and $\bar\psi(\gamma_x\sigma_{23}+\gamma_y\sigma_{33})\psi\sim n_{4i}\partial_xn_{6i}+n_{5i}\partial_yn_{6i}$.

Next we turn to the $SO(2)$ spatial current, which in the language of SL$^{(6)}$ is $n_{i1}\partial_{x,y}n_{i2}$, and in the gauge theory is the electric field of the emergent $U(1)$ gauge field. It is not obvious what this perturbation does to the DSL. However, we argue that its effect is also to shift the zero momenta. To see it, we consider $N_f=2$ QED$_3$, with Lagrangian
L=∑_i=1^2ψ̅_i iD_aψ_i-1/4e^2f_μνf^μν
This theory is argued to describe the easy-plane DQCP, which has an emergent $O(4)$ unitary symmetry (not to be confused with the DQCP we have been discussing, which has an emergent $SO(5)$ unitary symmetry) <cit.>. In this theory, $\bar\psi\gamma_{x,y}\sigma_3\psi$ clearly only shifts the zero momenta without changing the emergent order of the state. On the other hand, the improper $\z_2$ rotation of the $O(4)$ symmetry maps this operator into the electric fields of the emergent $U(1)$ gauge fields <cit.>, which means that the electric fields also play the role of shifting the zero momenta without changing the emergent order. So we propose that in DSL (SL$^{(6)}$), the $SO(2)$ spatial currents also only shift the zero momenta, but maintain the emergent order.

Now we turn to DQCP (SL$^{(5)}$), which has a couple of gauge-theoretic formulations <cit.>. From any of these formulations, one can see that the time component of the $SO(5)$
currents changes the emergent order of the state. The formulation that has a manifest $O(5)^T$ symmetry is an $SU(2)$ gauge theory with 2 flavors of Dirac fermions, where the $SO(5)$ symmetry is the flavor symmetry of these Dirac fermions. Under similar consideration of the $SO(6)$ spatial currents in DSL, we see that the effects of the $SO(5)$ spatial currents in DQCP are also complicated and need to be analysed in a case-by-case manner: some of them changes the emergent order of the states, while others only shift the zero momenta without changing the emergent order.

We remark that the effects of the spatial currents actually impose very strong constraints on the possible results of our anomaly-based framework of emergibility. Within this framework, it is easy to see that all realizations of DQCP and DSL on $p6m\times O(3)^T$ and $p4m\times O(3)^T$ symmetric lattice spin systems must have all entries of $n$ locating at some high-symmetry momenta in the Brillouin zone, because all possile symmetry embedding patterns satisfy this condition. This means that in all realizations, it is impossible to have a spatial current operator that is allowed by the microscopic symmetries and can shift the zero momenta. As we have explicitly checked, this is indeed true for all realizations obtained in our anomaly-based framework, which can be viewed as a highly nontrivial sanity check of this framework – It nicely corroborates the validity of the hypothesis of emergibility, the proposal that DSL can indeed be described by SL$^{(6)}$, and the dynamics of DSL.

Finally, we turn to SL$^{(7)}$, whose low-energy dynamics is poorly understood so far. It is still likely that the time component of the $SO(7)$ and $SO(3)$ currents will change the emergent order. For the spatial currents, we propose the following rule. Writing an $SO(7)$ spatial current operator as a sum of terms of the form $~n_{i_1j}\partial_{x,y}n_{i_2j}$, then we consider the effect of the same operator in DSL (it turns out that all such operators allowed by our microscopic symmetries only involve at most 4 rows of $n$, so its corresponding operator in DSL can always be found). If this operator changes the emergent order of DSL, then it also changes the emergent order of SL$^{(7)}$, and if it only shifts the zero momenta of DSL, it also only shifts the zero momenta of SL$^{(7)}$. For the $SO(3)$ spatial currents, it can be expanded as a sum as $\sim a_1n_{i1}\partial_xn_{i2}+a_2n_{i1}\partial_xn_{i3}+a_3n_{i2}\partial_xn_{i3}+b_1n_{i1}\partial_yn_{i2}+b_2n_{i1}\partial_yn_{i3}+b_3n_{i2}\partial_yn_{i3}$. We propose to first convert it into an $SO(7)$ spatial current $\sim a_1n_{1i}\partial_xn_{2i}+a_2n_{1i}\partial_xn_{3i}+a_3n_{2i}\partial_xn_{3i}+b_1n_{1i}\partial_yn_{2i}+b_2n_{1i}\partial_yn_{3i}+b_3n_{2i}\partial_yn_{3i}$. If this $SO(7)$ spatial current changes the emergent order (only shifts zero momenta) using the the above criterion, then the original $SO(3)$ spatial current also changes the emergent order (only shifts zero momenta).

The above proposal is of course conjectural, and more rigorous work is needed to fully settle it down. However, this proposal is supported by our results of anomaly-matching. We have checked all realizations of SL$^{(7)}$ obtained from the anomaly-based framework of emergibility, and found that the current operators that can shift zero momenta (according to the above proposal) are allowed by microscopic symmetries in a realization if and only if this realization belongs to a family where the momenta of some entries of $n$ can change continuously.

§ MORE EXAMPLES OF THE CALCULATION OF PULLBACK

In this appendix, we give three more examples of the analysis of anomaly matching, for $SU(2)_1$, DSL and SL$^{(7)}$. In Appendix <ref>, we also provide relevant formula for the calculation of pullback involving 5-dimensional representation of $SO(3)$.

§.§ $SU(2)_1$ and emergent anomaly

First let us consider a representative $(1+1)$-d quantum critical state, the $(1+1)$-d  $SU(2)_1$ conformal field theory, which describes the spin-1/2 antiferromagnetic Heisenberg chain at low energies <cit.>. The IR symmetry of the theory is $\frac{SU(2)\times SU(2)}{\z_2}\rtimes \z_2^T\cong O(4)$.

Ref. <cit.> works out the anomaly term of $SU(2)_1$ after gauging the $SO(4)$ part of $G_{\rm IR} = O(4)$, which corresponds to the interger Euler class of $SO(4)$, $e \in H^4(SO(4), \z)$. The bulk topological partition function capturing this anomaly is the Chern-Simons theory at level $(+1,-1)$ for the two $su(2)$ factors of $so(4)\cong su(2)\times su(2)$, which can be written in terms of two $su(2)$ gauge fields $A^{(1)},A^{(2)}$ as follows

It is straightforward to inspect that after gauging the $\z_2^T$ part of $G_{\rm IR}$, the anomaly term should correspond to the twisted Euler class of $O(4)$, and we denote it by $\tilde e\in H^4(O(4), \z_\rho)$. Note that this anomaly does not correspond to any element in $H^3(G_{\rm IR}, U(1)_\rho)$, the group cohomology (not the Borel cohomology in Ref. <cit.>) of $G_{\rm IR}$ acting nontrivially on the $U(1)$ coeffficient – this is the only example in this paper where the Bockstein homomorphism in Eq. (<ref>) is not an isomorphism. Hence we need some special care to write down the TPF of the bulk SPT theory.
[In this footnote we briefly review how to write down the TPF worked out in Ref. <cit.>. Suppose a (2+1)-d IR theory has gauge symmmetry $G$ and is defined on the manifold $\mc{M}_3$, which serves as the base space of some principal bundle of $G$. Given an element $\omega \in  H^4(G, \mathbb{Z})$, it is possible to define a 3d topological gauge theory of $G$ as follows

S=1/n(∫_B_4Ω-⟨γ^*ω, [B_4]⟩)1,
where $\Omega$ is the de Rham representative of the image of $\omega$ in $H^4(BG, \mathbb{R})$, $[\mc{B}_4]\in H_4(\mc{B}_4, \mathbb{Z})$ is the fundamental class of the manifold $\mc{B}_4$ that bounds $n$ copies of the manifold $\mc{M}_3$ with some extension of the principle bundle of $G$, and $\gamma$ is the classifying map of the extension. When $\omega$ is a torsion element, $\Omega=0$, and we retrieve the more familiar form of TPF
S = ⟨γ^*(β^-1(ω)), [M_3]⟩,
where $\beta$ is the Bockstein homomorphism associated to the short exact sequence $1\rightarrow\mathbb{Z}\rightarrow\mathbb{R}\rightarrow U(1)\rightarrow 1$. In particular, when $G=SO(4)$ and $\omega$ corresponds to the Euler class $e$, $\Omega$ can be explicitly written as follows,
In the presence of anti-unitary symmetries, the manifold $\mc{M}_3$ is assumed to be non-orientable. Then we have to choose $\mc{B}_4$ to be non-orientable as well, and demand $[\mc{B}_4]\in H^4(\mc{B}_4, \z_w)$ to be the fundamental class of the non-orientable manifold $\mc{B}_4$ twisted by the orientation character $w$ <cit.>.]

Consider the following homomorphism $\varphi$ from $G_{\text{UV}}=p1m\times O(3)$ to $G_{\rm IR}=O(4)$,

The LSM anomaly of a 1$D$ chain has been worked out in Appendix <ref>,
Ω_UV ≡exp(iπL_UV)=exp(iπ(x+m)w_2^O(3)^T).
We aim to prove that under the homomorphism $\varphi$, the pullback of the IR theory is the UV theory. Specifically, we need to prove that
[There are two terms in Eq. (<ref>). The first term will become 0 when  pulled back to $G_{\rm UV}$, which can be explicitly checked by considering the diagonal embedding of the Lie-algebra of $so(3)\cong su(2)$ into $so(4)\cong su(2)\times su(2)$. Then we just need to consider the pullback of the second term.]
β(Ω_UV) = φ^*(ẽ),
where $\beta$ is the Bockstein homomorphism associated to the short exact sequence $1\rightarrow\mathbb{Z}\rightarrow\mathbb{R}\rightarrow U(1)\rightarrow 1$.

From the commutativity of the square in the diagram below

H^4(G_IR, Z_ρ) dφ^*  rp̃     H^4(G_IR, ℤ_2) dφ^*

H^3(G_UV, ℤ_2)  rĩ      H^3(G_UV, U(1)_ρ)  rβ     H^4(G_UV, _ρ)  rp̃     H^4(G_UV, ℤ_2)

we just need to prove that
SQ^1(L_UV) = φ^*(p̃(ẽ)).
In particular, on the left hand side we have
SQ^1(L_UV) = (x+m)w_3^O(3)^T
according to Appendix <ref>, where $w_3^{O(3)^T} = w_3^{SO(3)} + t w_2^{SO(3)} + t^3$ and $t\in H^1(\mb{Z}_2^T, \mb{Z}_2)$ corresponds to the gauge field of time-reversal symmetry when pulled back to the spacetime manifold $\mc{M}_3$. On the right hand side we have $\tilde p(\tilde e) = w_4^{O(4)}$, and
φ^*(w_4^O(4)) = (x+m)(w_3^SO(3) + (t+x)w_2^SO(3) + (t+x)^3 )
Finally, using the cohomology relation $x^2=xm$, we see that both sides are equal to each other. Hence we establish that the pullback of the anomaly of IR CFT $SU(2)_1$ under the homomorphism $\varphi$ as in Eq. (<ref>) is the LSM anomaly of (1+1)-d spin chain.

Below we discuss the phenomenon of emergent anomalies. Following Ref. <cit.>, by imposing an extra constraint $T^2=1$, we can factorize $\varphi$ acting on $p1m$ into two pieces, a projection $p$ on $\z_2\times \z_2$ generated by $\tilde T$ or $M$, where $\tilde T$ acts trivially on $U(1)$ or $\z$ while $M$ acts nontrivially on $U(1)$ or $\z$, composed with an embedding $\tilde \varphi$ of the $\z_2\times \z_2$ into $O(4)$.

φ= φ̃∘p:      p1m = ⋊_2 [r, "p"]     _2×_2 [r, "φ̃"]     O(4)

With slight abuse of notation, we denote the gauge field of $\tilde T$ as $x$ as well. Then we have
φ̃^*(w_4^O(4)) = (x+m)(w_3^SO(3) + (t+x)w_2^SO(3) + (t+x)^3 ) =SQ^1((x+m)w_2^O(3)^T + (x+m)x^2)
in $H^4(\z_2\times \z_2\times O(3)^T, \z_2)$. According to the terminology in Ref. <cit.>, the first term $(x+m)w_2^{O(3)^T}$ as in Eq. (<ref>) is the intrinsic anomaly, while the second term $(x+m)x^2$ is identified as the emergent anomaly. The emergent anomaly should be absent when pulled back to $p1m$, which is guaranteed by the relation $(x+m)x=0$ present in $p1m$. As a sanity check, in the absence of mirror symmetry, in the line group $p1$, the intrinsic anomaly becomes $x w_2^{O(3)^T}$ and the emergent anomaly becomes $x^3$, consistent with the example in Ref. <cit.>.

We envision that similar emergent anomaly will be present in IR theories emerging from a 2d lattice system with wallpaper group $G_s$, because a lot of cohomology relations of $G_s$ will be absent when projected to a finite group by imposing $T_1^n=T_2^n=1$ for some integer $n$. More precisely, write $G_s = (\z\times \z)\rtimes O_s$, if we can find an integer $n$ such that $\varphi: G_s\rightarrow G_{\rm IR}$ factorizes as the composition of projection and another embedding

φ= φ̃∘p:      G_s = (×)⋊O_s [r, "p"]     G̃_s ≡(_n ×_n) ⋊O_s [r, "φ̃"]     G_IR,

then $\tilde \varphi^*(\Omega_{\rm IR}) \in H^4(\tilde G_s \times G_{int}, U(1)_\rho)$ will generically not be in the form of $\exp(i\pi \lambda\eta)$ with $\lambda\in H^2(\tilde G_s, \z_2)$ and $\eta\in H^2(G_{int}, \z_2)$, but contains a nonzero piece that nevertheless vanishes when pulled back to $G_s$, using certain cohomology relations of $G_s$ that is not present in $\tilde G_s$. Specifically, when $n=2$, of the 3 important relations displayed in Appendix <ref>, the first two relations, $x^2=0$ in $p1$ and $x^2=xm$ in $p1m$, will be absent when projected to $\tilde G_s$, while the third relation, $A_{x+y} A_m = 0$ in $cm$, will still be present when projected to $\tilde G_s$.

For example, when the IR effective theory is the DQCP emergent from a square lattice spin-1/2 system with wallpaper group $p4m$, we can choose $n=2$ and $\tilde G_s = (\z_2\times \z_2)\rtimes D_4$. The $\z_2$ cohomology ring of $\tilde G_s$ is

_2[A_x+y, A_m, A_c, B_xy, B_c^2,    B_c(x+y)]/(A_x+y A_c = 0,  (A_m+A_c) A_c=0, B_c(x+y) A_c = 0,

B_c(x+y)(B_c(x+y) + A_x+y(A_m + A_c)) = (A_m^2 + A_c^2) B_xy + A_x+y^2 B_c^2),

with the pullback of $B_{c(x+y)}$ equal to $A_{x+y}(A_{x+y} + A_m)$ in $H^*(p4m, \z_2)$, and the pullback of $A_{x+y}, A_m, A_c, B_{xy}, B_{c^2}$ their namesake. Then from the fact that the IR anomaly of DQCP corresponds to $w_5^{O(5)}\in H^5(O(5), \z_2)$, we have

φ̃^*(w_5^O(5))    = (B_xy + B_c(x+y) + B_c^2)(w_3^SO(3) + (t+A_x+y)w_2^SO(3) + (t+A_x+y)^3 )

= SQ^1((B_xy + B_c(x+y) + B_c^2)w_2^O(3)^T + (B_xy + B_c(x+y) + B_c^2) A_x+y^2).

The first term $\left(B_{xy} + B_{c(x+y)} + B_{c^2}\right)w_2^{O(3)^T}$ is again the intrinsic anomaly, while the second term $\left(B_{xy} + B_{c(x+y)} + B_{c^2}\right) A_{x+y}^2$ is the emergent anomaly that vanishes when pulled back to $G_s=p4m$. This is a slight generalization of the result in Ref. <cit.> to the whole group $p4m$.

§.§ DSL

Next consider DSL <cit.>, whose IR symmetry $G_\text{IR}$ is $\frac{O(6)\times O(2)}{\z_2}$, where an improper rotation of either $O(6)$ or $O(2)$ complex conjugates the $U(1)$ coefficient of $H^4(G_{\rm IR}, U(1)_\rho)$. The precise form of the anomaly term for $G_\text{IR}$ is unknown, yet it is possible to write down its pullback to $O(6)\times O(2)$
under the projection $p: O(6)\times O(2)\rightarrow \frac{O(6)\times O(2)}{\mathbb{Z}_2}$ <cit.>

Ω̃_IR    ≡exp(iπL̃_IR)

= exp[i π( w_4^O(6) + w_2^O(6)(w_2^O(2) + (w_1^O(2))^2) + ((w_2^O(2))^2 + w_2^O(2)(w_1^O(2))^2 + (w_1^O(2))^4))]

where $\tilde L_\text{IR}\in H^4(O(6)\times O(2),\mb{Z}_2)$. On a triangular lattice, we consider the following example embedding $\varphi$ of $G_{\rm UV} = p6m \times O(3)^T$ into $G_{\rm IR}$,

-1/2     -√(3)/2

√(3)/2      -1/2

-1/2     -√(3)/2

√(3)/2    -1/2

Note that $\varphi$ factorizes into an embedding $\tilde \varphi$ into $O(6)\times O(2)$ composed with the projection $p$, $\varphi = p\circ \tilde \varphi$. In fact, for $G_{\rm UV} = G_s\times O(3)^T$ with any $G_s$, if $\varphi$ satisfies the condition that some but not all entries of $n$ are left invariant under $SO(3)$, then $\varphi$ can always factorize into $p\circ \tilde \varphi$, where $\tilde \varphi$ is a homomorphism from $G_{\rm UV}$ to $O(6)\times O(2)$. Therefore, we can think of the IR symmetry as $O(6)\times O(2)$ for simplicity in the calculation of pullback. Moreover, we can always choose $\tilde \varphi$ such that $G_s$ acts as identity and $\z_2^T$ acts as minus identity in the block where $SO(3)$ acts.

The LSM anomaly of a triangular lattice spin-1/2 system has been obtained in Appendix <ref>, and we repeat it here
Ω_UV ≡exp(iπL_UV)= exp(i π(B_xy + A_c(A_c + A_m))w_2^O(3)^T),
where $L_\text{UV}\in H^4(G_\text{UV}, \mb{Z}_2)$. We wish to prove that $\Omega_\text{UV}=\varphi^*\Omega_\text{IR}$, which amounts to proving $\Omega_\text{UV}=\tilde \varphi^*\tilde \Omega_\text{IR}$. Again, from the commuting diagram Eq. (<ref>) (with $G_{\rm IR}$ changed to $O(6)\times O(2)$ and $\varphi$ changed to $\tilde\varphi$), we just need to prove that

SQ^1(L_UV) = φ̃^*(SQ^1(L̃_IR)).

According to Lemma <ref>, we have

𝒮𝒬^1(L̃_IR) =     w_5^O(6) + w_4^O(6)w_1^O(2) + w_3^O(6)(w_2^O(2) + (w_1^O(2))^2) + w_2^O(6)(w_1^O(2))^3

+ w_1^O(6)((w_2^O(2))^2 + w_2^O(2)(w_1^O(2))^2 + (w_1^O(2))^4) + ((w_2^O(2))^2 w_1^O(2) + (w_1^O(2))^5).

On the other hand,

SQ^1(L_UV) = ((B_xy + A_c(A_c + A_m))w_3^O(3)^T,
where $w_3^{O(3)^T} = w_3^{SO(3)} + t w_2^{SO(3)} + t^3$ and $t\in H^1(\mb{Z}_2^T, \mb{Z}_2)$ corresponds to the gauge field of time-reversal symmetry when pulled back to the spacetime manifold $\mc{M}_4$.

What remains is the calculation of the pullback $\tilde \varphi^*\left(\mc{SQ}^1(\tilde L_\text{IR})\right)$, which is a straightforward application of the Whitney product formula. In particular, considering the $O(2)$ block, the pullback gives

φ̃^*(w_1^O(2))    = A_c

φ̃^*(w_2^O(2))    = 0

On the other hand, $O(6)$ factorizes into two $3\times 3$ blocks, and for the lower $3\times 3$ block we have

φ̃^*(w_1^O(3))    = A_c + A_m

φ̃^*(w_2^O(3))    = B_xy + A_c^2

φ̃^*(w_3^O(3))    = A_cB_xy + A_c^2(A_c + A_m)

Assembling the Stiefel-Whitney class of the lower $O(3)$ and upper $O(3)^T$ into the Stiefel-Whitney class of $O(6)$, we have

φ̃^*(w_5^O(6))    = w_3^O(3)^T(B_xy + A_c^2) + w_2^O(3)^T(A_cB_xy + A_c^2(A_c + A_m))

φ̃^*(w_4^O(6))    = w_3^O(3)^T(A_c + A_m) + w_2^O(3)^T(B_xy + A_c^2) + t(A_cB_xy + A_c^2(A_c + A_m))

φ̃^*(w_3^O(6))    = w_3^O(3)^T + w_2^O(3)^T(A_c+A_m) + t(B_xy + A_c^2) + (A_cB_xy + A_c^2(A_c + A_m))

φ̃^*(w_2^O(6))    = w_2^O(3)^T + t(A_c+A_m) + (A_cB_xy + A_c^2(A_c + A_m))

φ̃^*(w_1^O(6))    = t+(A_c+A_m)

Combining Eqs. (<ref>), (<ref>), (<ref>) and (<ref>), indeed we get Eq. (<ref>). Hence we establish that $\Omega_\text{UV}=\varphi^*\Omega_\text{IR}$.

§.§ SL$^{(7)}$

The next examples we want to consider are two realizations of $N=7$ Stiefel liquid, i.e. SL$^{(7)}$, proposed in Ref. <cit.> (see Sec. VII D therein). The IR symmetry $G_{\rm IR}$ of the theory is $\frac{O(7)\times O(3)}{\z_2}$, and the precise form of the anomaly is given in Eq. (<ref>) for $N=7$. However, following the example in Appendix <ref>, for the sake of the analysis of anomaly-matching, we can again think of the IR symmetry as $O(7)\times O(3)$ and consider the pullback of the anomaly under the projection $p: O(7)\times O(3)\rightarrow\frac{O(7)\times O(3)}{\z_2}$,

Ω̃_IR    ≡exp(iπL̃_IR)

= exp(i π( w_4^O(7) + w_2^O(7)(w_2^O(3) + (w_1^O(3))^2) + ((w_2^O(3))^2 + w_2^O(3)(w_1^O(3))^2 + (w_1^O(3))^4)))

where $\tilde L_\text{IR}\in H^4(O(7)\times O(3),\mb{Z}_2)$. We will omit the tilde symbol in the following calculation.

On a triangular lattice, we consider the following embedding $\varphi$ of $G_{\rm UV} = p6m \times O(3)^T$ into $O(7)\times O(3)$,

-1/2     √(3)/2

-√(3)/2     -1/2

-1/2     √(3)/2

-√(3)/2     -1/2

-1/2     √(3)/2

-√(3)/2     -1/2

-1/2     √(3)/2

-√(3)/2     -1/2

Again, the LSM anomaly of a triangular lattice spin-1/2 system is
Ω_UV ≡exp(iπL_UV)= exp(i π(B_xy + A_c(A_c + A_m))w_2^O(3)^T),
where $L_\text{UV}\in H^4(G_\text{UV}, \mb{Z}_2)$. We wish to prove that $\Omega_\text{UV}=\varphi^*\Omega_\text{IR}$. From the commuting diagram Eq. (<ref>), we just need to prove that
SQ^1(L_UV) = φ^*(SQ^1(L_IR)).

According to Lemma <ref>, we have

𝒮𝒬^1(L_IR) =     w_5^O(7) + w_4^O(7)w_1^O(3) + w_3^O(7)(w_2^O(3) + (w_1^O(3))^2) + w_2^O(7)(w_3^O(3) + (w_1^O(3))^3)

+ w_1^O(7)((w_2^O(3))^2 + w_2^O(3)(w_1^O(3))^2 + (w_1^O(3))^4) + (w_3^O(3)(w_1^O(3))^2 + (w_2^O(3))^2 w_1^O(3) + (w_1^O(3))^5).

SQ^1(L_UV) = ((B_xy + A_c(A_c + A_m))w_3^O(3)^T,
where $w_3^{O(3)^T} = w_3^{SO(3)} + t w_2^{SO(3)} + t^3$ and $t\in H^1(\mb{Z}_2^T, \mb{Z}_2)$ corresponds to the gauge field of time-reversal symmetry when pulled back to the spacetime manifold $\mc{M}_4$.

What remains is the calculation of the pullback $\varphi^*\left(\mc{SQ}^1(L_\text{IR})\right)$, which is a straightforward application of the Whitney product formula. In particular, $O(7)$ factorizes into one $3\times 3$ block and two $2\times 2$ block, and for the $O(3)$ part and the $O(7)$ part seperately, the pullback gives

φ^*(w_1^O(3))   = A_m,

φ^*(w_2^O(3))   = B_xy,

φ^*(w_3^O(3))   =0,

φ^*(w_1^O(7))    = t,

φ^*(w_2^O(7))    = w_2^SO(3)+t^2+A_c^2+A_m^2+A_m A_c,

φ^*(w_3^O(7))   = (w_3^SO(3)+t w_2^SO(3) + t^3)+t(A_c^2+A_m^2+A_m A_c)+A_c A_m(A_c+A_m),

φ^*(w_4^O(7))   = (w_2^SO(3)+t^2)(A_c^2+A_m^2+A_m A_c)+tA_c A_m(A_c+A_m),

φ^*(w_5^O(7))   = (w_3^SO(3)+t w_2^SO(3) + t^3)(A_c^2+A_m^2+A_m A_c)+(w_2^SO(3)+t^2)A_c A_m(A_c+A_m),

Substituting them back into Eq. $\eqref{eq: SQL SL}$, and using the cohomology relation $B_{xy}^2 = B_{c^2} B_{xy}$, indeed we get Eq. (<ref>) as promised. Hence we establish that $\Omega_\text{UV}=\varphi^*\Omega_\text{IR}$.

On a Kagome lattice spin-1/2 system, we consider the following embedding $\varphi$ of $G_{\rm UV} = p6m \times O(3)^T$ into $O(7)\times O(3)$,

The LSM anomaly of a Kagome lattice spin-1/2 system is
Ω_UV ≡exp(iπL_UV)= exp(i πB_xyw_2^O(3)^T).
Again, we wish to prove that $\Omega_{\rm UV} = \varphi^* \Omega_{\rm IR}$ by proving $\mc{SQ}^1(L_{\rm UV}) = \varphi^*\left(\mc{SQ}^1(L_{\rm IR})\right)$. $\mc{SQ}^1(L_{\rm IR})$ is given in Eq. (<ref>), while for $\mc{SQ}^1(L_{\rm UV})$ we have

SQ^1(L_UV) = B_xy w_3^O(3)^T.
It is now straightforward to calculate the pullback of various Stiefel-Whitney classes in Eq. (<ref>),

φ^*(w_1^O(3))    = A_m+A_c

φ^*(w_2^O(3))    = B_xy+A_c^2

φ^*(w_3^O(3))    = A_c^3+A_c^2A_m + A_cB_xy

φ^*(w_1^O(7))    = t + A_c,

φ^*(w_2^O(7))    =w_2^SO(3)+t^2 + tA_c + A_c^2 +  A_cA_m + A_m^2,

φ^*(w_3^O(7))   = (w_3^SO(3)+t w_2^SO(3) + t^3) + (w_2^SO(3)+t^2)A_c + t(A_c^2 +  A_cA_m + A_m^2) + A_c^3,

φ^*(w_4^O(7))   = (w_3^SO(3)+ tw_2^SO(3) + t^3)A_c + (w_2^SO(3)+t^2)(A_c^2 +  A_cA_m + A_m^2) + t A_c^3 + A_c^2 A_m(A_c+A_m),

φ^*(w_5^O(7))   = (w_3^SO(3)+t w_2^SO(3) + t ^3)(A_c^2 +  A_cA_m + A_m^2) + (w_2^SO(3)+t^2)A_c^3 + t A_c^2 A_m(A_c+A_m).

Substituting them into $\eqref{eq: SQL SL}$ and using the cohomology relation $B_{xy}^2 = B_{c^2} B_{xy}$, indeed we get Eq. (<ref>), and thus establish that $\mc{SQ}^1(L_{\rm UV})=\varphi^*\left(\mc{SQ}^1(L_{\rm IR})\right)$.

§.§ Five dimensional representation of $SO(3)$

In all previous examples presented in this appendix, the $SO(3)$ spin rotation symmetry is embedded into the IR symmetry $G_{\rm IR}$ as a 3 dimensional representation. It is natural to consider embedding involving other representations of $SO(3)$, whose physical relevance is illustrated in Section <ref>. In this sub-appendix, we present formula relevant to mapping $SO(3)$ into $G_{\rm IR}$ as a 5 dimensional representation of $SO(3)$.

First consider the 5 dimensional representation $\varphi_5: SO(3)\rightarrow O(5)$ of $SO(3)$ alone, which can be thought of as a symmetric traceless tensor $V_5$, whose 5 basis are

where $n_{1,2,3}$ form an $SO(3)$ vector. Consider the $\z_2^2$ subgroup of $SO(3)$, generated by $\pi$-rotations around the $x$- and $y$-axes, respectively. Using the above 5 basis, these two $\pi$-rotations are mapped into ${\rm diag}(-1, 1, -1, 1, 1)$ and ${\rm diag}(-1, -1, 1, 1, 1)$, respectively, from which (or from the splitting principle <cit.>) we see that
φ_5^*(w_2^O(5)) = w_2^SO(3),   φ_5^*(w_3^O(5)) = w_3^SO(3),   φ_5^*(w_1^O(5)) = 0, φ_5^*(w_4^O(5)) = 0, φ_5^*(w_5^O(5)) = 0

Now go back to $G_{\rm UV} = SO(3)\times\tilde G$ and consider a 5 dimensional representation $\varphi_5: G_{\rm UV}\rightarrow O(5)$ that can be written as $V_5\otimes V_1$, where $V_5$ denotes the 5 dimensional representation of $SO(3)$ while $V_1$ denotes a 1 dimensional real representation of $\tilde G$ correponding to $x\in H^1(\tilde G, \z_2)$. Again from inspecting the action of the diagonal $\z_2^2$ subgroup, we have

φ_5^*(w_1^O(5)) = x,

φ_5^*(w_2^O(5)) = w_2^SO(3),

φ_5^*(w_3^O(5)) = w_3^SO(3) + x w_2^SO(3),

φ_5^*(w_4^O(5)) = x^2 w_2^SO(3) + x^4,

φ_5^*(w_5^O(5)) = x^2 w_3^SO(3) + x^3 w_2^SO(3) + x^5.

§ STRATEGY OF EXHAUSTIVE SEARCH OF SEP AND RESULTS

In this appendix, we briefly review our strategy of the exhaustive search of SEP. We also illustrate how to check all the SEPs from the csv data files we provide in the Data_and_Codes folder and the mathematica file embedding.m, which transforms the data in csv files into matrices representing generators $C_6/C_4$, $M$, $T_1$, $T_2$ and $\mc{T}$. Some interesting realizations have been shown in Sections <ref> and <ref>.

In order to enumerate all SEPs that match LSM constraints with IR anomaly, we just need to enumerate all embeddings from $G_{\rm UV}$ to $G_{\rm IR}$ and, following Section <ref> and Appendix <ref>, calculate the pullback $\varphi^*(\Omega_{\rm IR})$ to see if it is identical to $\Omega_{\rm UV}$ corresponding to a particular LSM constraint. Motivated by quantum magnetism, we assume that the IR theory will emerge as a consequence of the competition between a magnetic state and a non-magnetic state. Therefore, we only consider embeddings such that, in terms of the $N\times (N-4)$ matrix $n$ for SL$^{(N)}$, some but not all entries of $n$ transform under the $SO(3)$ symmetry.

For DQCP, since the IR symmetry is $O(5)$, all embeddings are just composed of representations of $G_{\rm UV}$. For DSL and SL$^{(7)}$, even though the IR symmetry is $\frac{O(6)\times O(2)}{\z_2}$ and $\frac{O(7)\times O(3)}{\z_2}$, respectively, because of the constraints on the embeddings, it suffices to only consider embeddings into $\frac{O(6)\times O(2)}{\z_2}$ and $\frac{O(7)\times O(3)}{\z_2}$ which can be respectively lifted to an embedding into $O(6)\times O(2)$ or $O(7)\times O(3)$, as discussed below Eq. (<ref>). Therefore, all embeddings we consider are just composed of real representations of $G_{\rm UV}$. In other words, our task to specify an embedding becomes finding appropriate irreducible representations of $G_{\rm UV}$, and fill them into the $O(N)$ and $O(N-4)$ matrices in a block diagonal form.

Hence let us make a detour and discuss representations of $G_{\rm UV} = G_s\times SO(3)\times \z_2^T$. For any two groups $G_{1,2}$, an irreducible representation $V$ of $G_1\times G_2$ is $V_1\otimes V_2$, where $V_{1,2}$ is an irreducible representation of $G_{1,2}$, respectively. So any irreducible representation $V$ of $G_{\rm UV}$ takes the form of $V=V^{2n+1}_{SO(3)}\otimes V_s\otimes V_T$, where $V^{2n+1}_{SO(3)}$ is a $(2n+1)$-dimensional irreducible representation of $SO(3)$ with $n\in \mb{N}$, $V_s$ is an irreducible representation of $G_s$, and $V_T=\pm 1$ is an irreducible representation of $\z_2^T$. The complete list of irreducible representations $V_s$ of $G_s$ can be found using the method of induced representations <cit.>, and we provide complete lists of irreducible representations of $p4m$ and $p6m$ in the Mathematica file Representation.nb.

To figure out which representations of $G_{\rm UV}$ are relevant to our discussions, it is useful to analyze in which blocks the $SO(3)$ can act nontrivially, with the assumption that some but not all entries of $n$ transform under the $SO(3)$ symmetry. For DQCP, $SO(3)$ must act nontrivially in a 3-d block, while the rest 2-d block should be a reducible or irreducible representation of $G_s\times \z_2^T$. That is, the relevant representation $V$ of $G_{\rm UV}$ schematically takes the form

where $V^1_s$ and $V^1_T$ are 1-d representations of $G_s$ and $\z_2^T$ respectively, and $V^2_{G_s\times\z_2^T}$ is a 2-d (reducible or irreducible) representation of $G_s\times\z_2^T$.

For DSL, the block involving nontrivial $SO(3)$ actions can be 3-d or 5-d, and it has to lie in $O(6)$. For SL$^{(7)}$, the block involving $SO(3)$ should embed into $O(7)$ and can be 3-d, 5-d or 6-d. The 6-d representation takes the form of $V_{SO(3)}\otimes V^2_{G_s\times\z_2^T}$, where $V_{SO(3)}$ is the 3-d representation of $SO(3)$, and $V^2_{G_s\times\z_2^T}$ involves either two 1-d representations of $G_s\times \z_2^T$ or one irreducible 2-d representation of $G_s\times \z_2^T$. However, it turns out that it is impossible to match the anomaly with any LSM constraint in the presence of some 6-d block involving $SO(3)$. Therefore, for DSL and SL$^{(7)}$, we consider two cases, either $SO(3)$ embeds as a 3-d representation, corresponding to deconfined quantum criticle points or quantum critical spin liquids in Section <ref>, or as a 5-d representation, corresponding to quantum critical spin-quadrupolar liquids in Section <ref>.

For DSL and SL$^{(7)}$, we still have freedom to choose the lifting to $O(6)\times O(2)$ or $O(7)\times O(3)$, and different embeddings into $O(6)\times O(2)$ or $O(7)\times O(3)$ may correspond to the same embedding into $\frac{O(6)\times O(2)}{\z_2}$ or $\frac{O(7)\times O(3)}{\z_2}$. For embeddings involving 3-d representation of $SO(3)$, we choose such that only $\mc{T}$ acts in the $3\times 3$ block as $-I_3$, while $G_s$ acts trivially in that block. That is, for DSL and SL$^{(7)}$, the relevant representations $V$ of $G_{\rm UV}$ schematically take the form  as follows

) ×(V^1_SO(3)⊗V^2_G_s×_2^T)^2×2

) ×(V^1_SO(3)⊗V^3_G_s×_2^T)^3×3
where $1_s$ is the 1-d trivial representation of $G_s$, and $-1_T$ is the 1-d non-trivial representation of $\z_2^T$. For embeddings involving 5-d representation of $SO(3)$, we choose such that both $\z_2^T$ and $G_s$ act trivially in the $5\times 5$ block. That is, the relevant representations $V$ of $G_{\rm UV}$ schematically take the form

) ×(V^1_SO(3)⊗V^2_G_s×_2^T)^2×2

) ×(V^1_SO(3)⊗V^3_G_s×_2^T)^3×3
where $1_s$ and $1_T$ are 1-d trivial representations of $G_s$ and $\z_2^T$, respectively.

Having identified all possible embeddings, it is a striaghtforward exercise to calculate the pullback in each case following Section <ref> and Appendix <ref>. We use Mathematica to automate the computation and store results in csv files in the ancillary folder <cit.>. For example, data.csv contains data for matching LSM constraints with IR anomaly of SL$^{(N=5,6,7)}$ when $SO(3)$ embeds into $O(6)$ as a 3-d representaion, while dataSL5Rep.csv contains data for matching LSM constraints of $p4m$ with IR anomaly of SL$^{(7)}$ when $SO(3)$ embeds into $O(7)$ as a 5-d representaion. Moreover, for both $p4m$ and $p6m$, there is a single embedding involving 5-d representation of $SO(3)$ that can match IR anomaly of DSL, shown in  Eq. (<ref>), which actually matches IR anomaly with zero LSM constraint.

To read the embeddings, to read the explicit image of the generators $C_4/C_6$, $M$, $T_1$, $T_2$ and $\mc{T}$ in $G_{\rm IR}$, we provide a wrapper file Embedding.m. When $SO(3)$ embeds into $G_{\rm IR}$ as a 3-d representation, it provides two functions

$$p4mPrintEmbedding[n\_Integer, lsm\_Integer, p\_Integer]$$

$$p6mPrintEmbedding[n\_Integer, lsm\_Integer, p\_Integer]$$

The arguments are $n=5,6,7$ corresponding to DQCP, DSL and SL$^{(7)}$ respectively, $lsm=1,\dots,8$ in $p4m$ or $lsm=1,\dots,4$ corresponding to a particular LSM constraint with the order shown in Table <ref>, and $p$ corresponding to a position in the array for a particular embedding/realization.
When $SO(3)$ embeds into $G_{\rm IR}$ as a 5-d representation and the IR theory is SL$^{(7)}$, it also provides two functions

$$p4m5dPrintEmbedding[lsm\_Integer, p\_Integer]$$

$$p6m5dPrintEmbedding[lsm\_Integer, p\_Integer]$$

with similar arguments and output. Note that in this scenario for DQCP there is no realization, and for DSL there is a single realization in $p4m$ or $p6m$ shown in Eq. (<ref>). For $p4m$ and SL$^{(7)}$, it also provides a function

$$IncommensuratePrintEmbedding[lsm\_Integer, p\_Integer]$$

to check whether some embedding corresponds to an incommensurate order, and if it does, output the corresponding incommensurate embedding. An illustration of how to use these functions is provided in ReadMe.nb.

§ STABLE REALIZATIONS ON VARIOUS LATTICE SPIN SYSTEMS

In this appendix, we list all stable realizations of DQCP, DSL and SL$^{(7)}$ on triangular, kagome, and square lattice half-integer spin systems, as well as those on $p6m$-anomaly-free systems (including honeycomb lattice half-integer spin systems and all integer-spin systems with $p6m$ lattice symmetry) and $p4m$-anomaly-free systems (including all integer-spin systems with $p4m$ lattice symmetry). For square lattice, we only list the realizations in lattice homotopy class with PR at the type-a IWP, from which the ones with PR at the type-b IWP can be obtained by redefining the $C_4$ center. As in the main text, here a stable DQCP means a realization with only a single relevant perturbation allowed by microscopic symmetries, so that it can be realized as a generic (pseudo-)critical point. A stable DSL means a realization with no relevant perturbation allowed by microscopic symmetries, so that it can be realized as a stable phase. A stable SL$^{(7)}$ means a realization with either no relevant perturbation allowed by microscopic symmetries, or a single symmetry-allowed relevant perturbation that does not change the emergent order but only shifts the “zero momenta", so that this realization can still be viewed as a stable phase. All stable realizations of these states, including those on lattice systems discussed here and also those on other lattice systems, are explicitly documented in ReadMe.nb.

§.§ Stable realizations of DQCP

On all these systems, there is a single new stable realization of DQCP, given by Eq. (<ref>), adjacent to ferromagnetic order on triangular lattice, kagome lattice integer spin systems or honeycomb lattice half-integer/integer spin systems. There is a known stable realization of DQCP on the square lattice half-integer spin system, given by Eq. (<ref>), adjacent to anti-ferromagnetic (Neel) order. There is another known stable realization of DQCP on $p6m$-anomaly-free system <cit.>, adjacent to anti-ferromagnetic order on honeycomb lattice half-integer/integer spin systems, given by

-1/2     √(3)/2

-√(3)/2     -1/2

C_6: n→(

O(3)^T: n→(

These three are all stable realizations of DQCP.

§.§ Stable realizations of DSL

On $p6m$-anomaly-free systems, there is a single stable realization of DSL where the most relevant spin fluctuations carry spin-1, given by Eq. (<ref>). On both $p6m$-anomaly-free systems and $p4m$-anomaly-free systems, there is also a single stable realization of DSL where the most relevant spinful fluctuations carry spin-2, given by Eq. (<ref>). Below we discuss the other systems.

§.§.§ Triangular lattice half-integer spin systems

On triangular lattice half-integer spin systems, there are 3 stable realizations of DSL. One of them is known <cit.>, given by Eq. (<ref>), adjacent to $120^\circ$ order. The other two have identical actions of $T_{1, 2}$, $C_6$ and $O(3)^T$:

T_1: n→(

T_2: n→(

C_6: n→(

O(3)^T: n→(

The action of the mirror symmetry $M$ in these two realizations are respectively

M: n→(

M: n→(

§.§.§ Kagome lattice half-integer spin systems

On kagome lattice half-integer spin systems, there are 3 stable realizations of DSL. One of them is known <cit.>, adjacent to $q=0$ order, given by

T_1: n→(

T_2: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

M: n→(

O(3)^T: n→(

The other two have the same actions of $T_{1, 2}$, $C_6$ and $O(3)^T$:

C_6: n→(

T_1: n→(

T_2: n→(

O(3)^T: n→(

And the action of $M$ in the two realizations are respectively
M: n→(

M: n→(

§.§.§ Square lattice half-integer spin systems

On square lattice half-integer spin systems, there are 3 stable realizations of DSLs. One of them is given by Eq. (<ref>). The other two have the same actions of $T_{1,2}$, $C_4$ and $O(3)^T$:

T_1: n→(

T_2: n→(

C_4: n→(

O(3)^T: n→(

The action of $M$ on these two realizations are respectively
M: n→(

M: n→(

§.§ Stable realizations of SL$^{(7)}$

Below we list the stable realizations of SL$^{(7)}$ on various systems.

§.§.§ $p6m$-anomaly-free systems

On $p6m$-anomaly-free systems, there are two stable realizations of SL$^{(7)}$, both of which have the most relevant spinful fluctuations carrying spin-2. The symmetry actions of one of them is given by Eq. (<ref>). The other one has symmetry actions:

SO(3): n→(

T: n→(

T_1,2: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

C_6: n→(

M: n→(

§.§.§ Triangular lattice half-integer spin systems

On triangular lattice half-integer spin systems, there are 8 stable realizations of SL$^{(7)}$. The first has appeared in Ref. <cit.>, given by Eq. (<ref>).

The second has symmetry actions:

T_1: n→n

T_2: n→n

C_6: n→(

M: n→(

O(3)^T: n→(

The third has symmetry actions:

T_1: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

T_2: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

C_6: n→(

M: n→(

O(3)^T: n→(

The fourth has symmetry actions:

T_1: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

T_2: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

C_6: n→(

M: n→(

O(3)^T: n→(

The first to the fourth realization are all adjacent to tetrahedral order.

The fifth has symmetry actions:

T_1: n→n

T_2: n→n

C_6: n→(

M: n→(

O(3)^T: n→(

The sixth has symmetry actions:

T_1: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

T_2: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

C_6: n→(

M: n→(

O(3)^T: n→(

The seventh has symmetry actions:

T_1: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

T_2: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

C_6: n→(

M: n→(

O(3)^T: n→(

The eighth has symmetry actions:

T_1: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

T_2: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

C_6: n→(

M: n→(

O(3)^T: n→(

§.§.§ Kagome lattice half-integer spin systems

On kagome lattice half-integer spin systems, there are 9 stable realizations of SL$^{(7)}$. The first has appeared in Ref. <cit.> and is given by Eq. (<ref>). The second is given by Eq. (<ref>). Both realizations are adjacent to cuboc1 order.

The third has symmetry actions

T_1: n→n

T_2: n→n

C_6: n→(

M: n→(

O(3)^T: n→(

The fourth has symmetry actions

T_1: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

T_2: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

C_6: n→(

M: n→(

O(3)^T: n→(

The fifth has symmetry actions

T_1: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

T_2: n→(

-1/2     √(3)/2

-√(3)/2     -1/2

C_6: n→(

1/2     √(3)/2

-√(3)/2     1/2

M: n→(

O(3)^T: n→(

The third to the fifth realization are all adjacent to cuboc2 order.

The sixth has symmetry actions:

T_1: n→(

T_2: n→(

C_6: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

M: n→(

The sixth realization is adjacent to $q=0$ umbrella order.

The seventh has symmetry actions:

T_1: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

T_2: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

C_6: n→(

M: n→(

O(3)^T: n→(

The seventh realization is adjacent to $q=\sqrt{3}\times \sqrt{3}$ umbrella order.

The eighth has symmetry actions:

T_1: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

T_2: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

C_6: n→(

M: n→(

The ninth has symmetry actions:

T_1: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

T_2: n→(

-1/2     -√(3)/2

√(3)/2     -1/2

C_6: n→(

M: n→(

§.§.§ $p4m$-anomaly-free systems

On $p4m$-anomaly-free systems, there are 4 stable realizations of SL$^{(7)}$, all of which has the most relevant spinful fluctuations carrying spin-2.

The first has symmetry actions:

SO(3): n→(

T: n→(

T_1, 2: n→(

C_4: n→(

M: n→(

The second has symmetry actions:

SO(3): n→(

T: n→(

T_1, 2: n→(

C_4: n→n

M: n→(

The third has symmetry actions:

T: n→(

T_1,2: n→(

C_4: n→(

M: n→(

The fourth has symmetry actions:

T: n→(

T_1,2: n→(

C_4: n→(

M: n→(

§.§.§ Square lattice half-integer spin systems

On square lattice half-integer spin systems, there are 2 stable realizations of SL$^{(7)}$ where the most relevant spinful fluctuations have spin-1 and all $n$ modes are at high-symmetry momenta in the Brillouin zone. There are also three realizations where some $n$ modes can have continuously changing momenta, among which two of them have only a single symmetric relevant perturbation that shifts the momenta of the $n$ modes, as long as these momenta are not tuned to high-symmetry point. This is the case for the third family of realizations for most non-high-symmetry momenta, except at two special momentum points (see below). Furthermore, there is also one stable realization where the most relevant spinful fluctuations have spin-2 and all $n$ modes are at high-symmetry momenta.

We start with the 2 realizations with the most relevant spinful fluctuations carrying spin-1 and all $n$ modes locating at high-symmetry momenta. The first has symmetry actions:

T_1: n→(

T_2: n→(

C_4: n→(

M: n→(

The second has symmetry actions:

T_1: n→(

T_2: n→(

C_4: n→(

M: n→(

Next, we turn to the three realizations with some $n$ modes at continuously changing momenta. The first has symmetry actions given by Eq. (<ref>), adjacent to tetrahedral umbrella order, and the second has symmetry actions

C_4: n→(

M: n→(

T_1: n→(

cosk     sink

-sink     cosk

T_2: n→(

cosk     -sink

sink     cosk

O(3)^T: n→(

where $k\in(-\pi, \pi)$ is a generic momentum. In both of these two realizations, the only relevant perturbation that is allowed by microscopic symmetries is the one that shifts the momenta of the $n$ modes.

The third has symmetry actions:

C_4: n→(

M: n→(

T_1: n→(

cosk     sink

-sink     cosk

cosk     sink

-sink     cosk

T_2: n→(

cosk     sink

-sink     cosk

cosk     -sink

sink     cosk

O(3)^T: n→(

where $k\in(-\pi, \pi)$ is a generic momentum. For this family of realizations, as long as $k\neq\pm\pi/2$, the only symmetric relevant perturbation is the one that shifts the momenta of the $n$ modes. When $k=\pm\pi/2$, besides this symmetric relevant perturbation, there is an additional one that can change the emergent order of SL$^{(7)}$ and make it unstable.

Finally, there is one realization where all $n$ modes are at high-symmetry momentum, and the most spinful fluctuations have spin-2. It has symmetry actions:

SO(3): n→(