separately. But on $C_{6}$-symmetric lattices such as triangular and Kagome,
one typically only need to specify one inversion center and the others will be
determined by symmetries. For example, consider triangular lattice. There are
four inversion centers per unit cell: the $C_{6}$ center (lattice site) and
three $C_{2}$ centers (bond center) that are related to each other through
$C_{6}$ rotations. The parity of $2S$ ($S$ being the spin moment) of the
entire unit cell is given by the sum of the parity on each inversion center.
This means that if we have anomaly $aw_{2}^{SO(3)}xy$ and
$bw_{2}^{SO(3)}c^{2}$, where $a,b\in\\{0,1\\}$ and $c$ probes the site-
centered inversion, then the anomaly associated with the bond-centered
inversion (probed by $c^{\prime}$) will be given by
$(a+b)w_{2}^{SO(3)}(c^{\prime})^{2}$. For this reason we will only focus on
one inversion center in these lattices.

The above situations (summarized in Eq. (57)) happen for a variety of $2d$
lattice system, including square, triangular and Kagome lattices 141414In
Appendix H, an alternative expression for the LSM anomaly on a square lattice
is given. We have checked that all the following statements about states on a
square lattice hold if either the expression here or the alternative one there
is used.. These are common playgrounds for studying frustrated quantum
magnetism, and we will see some examples next.

###  VII.3 Warm-up: anomaly-matching for DQCP

Given the variety of anomaly terms presented above, it is rather nontrivial
for a theory to exactly match all the anomalies. Below we show how this works
for the DQCP on square lattice. In appendix I, we show this for the slightly
more complicated case of $U(1)$ DSL on triangular lattice. Since these states
admit explicit parton mean-field constructions on the lattices, we expect them
to be emergible and the anomaly-matching should go through. So these exercises
serve as a benchmark for the anomaly-matching approach.

Figure 4: Square lattice and the relevant symmetries. Each filled red circle
represents an odd number of spin-1/2’s. The $C_{2}$ rotation is around a site
that hosts the spins, and the dashed line is the reflection axis of $R_{y}$.

As we reviewed in Sec. III.1, the DQCP corresponds to $S^{(5)}$, with IR
anomaly

$i\pi\int_{X_{4}}w_{4}^{O(5)},$ (58)

together with the restriction $w_{1}^{O(5)}=w_{1}^{TM}$.

We represent the microscopic symmetry implementations by their actions on the
$n$ field, which for DQCP is simply a $5$-component vector. The $SO(3)$ spin
rotation is implemented as

$n\to\left(\begin{array}[]{cc}SO^{s}(3)&0\\\ 0&I_{2}\end{array}\right)n.$ (59)

Time-reversal symmetry acts as

$n\to\left(\begin{array}[]{cc}-I_{3}&0\\\ 0&I_{2}\end{array}\right)n,\hskip
10.0pti\to-i.$ (60)

On square lattice the translation symmetries $T_{x},T_{y}$ are implemented as
(see Fig. 4)

$\displaystyle T_{x}:$ $\displaystyle
n\to\left(\begin{array}[]{ccccc}-1&0&0&0&0\\\ 0&-1&0&0&0\\\ 0&0&-1&0&0\\\
0&0&0&-1&0\\\ 0&0&0&0&1\end{array}\right)n,$ (66) $\displaystyle T_{y}:$
$\displaystyle n\to\left(\begin{array}[]{ccccc}-1&0&0&0&0\\\ 0&-1&0&0&0\\\
0&0&-1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&-1\end{array}\right)n.$ (72)

As for the lattice rotation, since only the site-centered inversion ($C_{2}$)
participates in the anomaly, we should just focus on it:

$C_{2}:n\to\left(\begin{array}[]{cc}I_{3}&0\\\ 0&-I_{2}\end{array}\right)n.$
(73)

Finally, for reflection $\mathcal{R}_{y}$:

$n\to\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&-1\end{array}\right)n.$ (74)

We can now pull back the $w_{4}^{O(5)}$ anomaly to the physical symmetries.
The calculation proceeds as follows. First, since none of the microscopic
symmetries mixes $n_{1,2,3}$ with $n_{4,5}$, we can decompose the $O(5)$
bundle into $O(3)\times O(2)$, where

$\displaystyle w_{1}^{O(3)}$ $\displaystyle=$ $\displaystyle t+x+y,$
$\displaystyle w_{1}^{O(2)}$ $\displaystyle=$ $\displaystyle r+x+y,$
$\displaystyle w_{2}^{O(3)}$ $\displaystyle=$ $\displaystyle
w_{2}^{SO(3)}+t^{2},$ $\displaystyle w_{2}^{O(2)}$ $\displaystyle=$
$\displaystyle xy+c^{2}+xr+cr,$ $\displaystyle w_{3}^{O(3)}$ $\displaystyle=$
$\displaystyle{\rm Sq}^{1}(w_{2}^{O(3)})+w_{1}^{O(3)}w_{2}^{O(3)}$ (75)
$\displaystyle=$ $\displaystyle{\rm
Sq}^{1}(w_{2}^{SO(3)})+(t+x+y)(w_{2}^{SO(3)}+t^{2}),$

and all higher SW classes vanish. Notice that we have used the fact that $x,y$
live in $H^{1}(X_{4},\mathbb{Z})$ so $x^{2}=y^{2}=0$. We also set
$cx=cy=ry=0$, based on the physical understanding that non-commuting
crystalline symmetries do not simultaneous contribute to the same anomaly term
– this can be seen, for example, from the dimension-reduction approach Song
_et al._ (2017). We can then write the $w_{4}^{O(5)}$ using Whitney product
formula:

$\displaystyle\begin{split}w_{4}^{O(5)}=&\sum_{i}w_{i}^{O(3)}w_{4-i}^{O(2)}\\\
=&(w_{2}^{SO(3)}+t^{2})(xy+c^{2}+xr+cr)\\\ &+r({\rm
Sq}^{1}(w_{2}^{SO(3)})+tw_{2}^{SO(3)}+t^{3})\\\ &+(x+y)[{\rm
Sq}^{1}(w_{2}^{SO(3)})+w_{1}^{TM}(w_{2}^{SO(3)}+t^{2})]\end{split}$ (76)

where we have used the constraint $r+t=w_{1}^{TM}$ to obtain the last line.
The first line above is exactly what we expect from LSM constraints from Eqs.
(52)-(55), so our task now is to show that the last two lines vanish. This
follows from the following relations:

$\displaystyle r{\rm Sq}^{1}(w_{2}^{SO(3)})$ $\displaystyle=$ $\displaystyle
w_{1}^{TM}rw_{2}^{SO(3)}+r^{2}w_{2}^{SO(3)}$ $\displaystyle=$ $\displaystyle
trw_{2}^{SO(3)},$ $\displaystyle rt^{3}$ $\displaystyle=$ $\displaystyle
w_{1}^{TM}t^{3}+t^{4}=0,$ $\displaystyle w_{1}^{TM}(x+y)w_{2}^{SO(3)}$
$\displaystyle=$ $\displaystyle{\rm Sq}^{1}[(x+y)w_{2}^{SO(3)}]$
$\displaystyle=$ $\displaystyle(x+y){\rm Sq}^{1}(w_{2}^{SO(3)}),$
$\displaystyle w_{1}^{TM}(x+y)t^{2}$ $\displaystyle=$ $\displaystyle{\rm
Sq}^{1}[(x+y)t^{2}]=0.$ (77)

###  VII.4 $N=7$: intertwining non-coplanar magnets with valence-bond solids

In spin systems, as we reviewed in Sec. III, the DQCP (SL(N=5)) naturally
describes the competition (or intertwining) between collinear magnetic and
valence-bond solid (VBS) orders, while the $U(1)$ Dirac spin liquid (SL(N=6))
naturally describes the intertwining between coplanar magnetic and VBS orders.
The natural extension to the intertwining between non-coplanar magnetic and
VBS orders is the $N=7$ Stiefel liquid state. Below we discuss two lattice
realizations of the SL(7) theory, one on triangular lattice and one on Kagome
lattice.

####  VII.4.1 Triangular lattice

We consider a triangular lattice with an odd number of half-integer spins per
site, so the LSM anomalies are given by Eqs. (52), (53), (54) and (55). These
conditions impose strong constraints on the allowed lattice realizations of
the $N=7$ SL theory. We now describe an embedding of the microscopic
symmetries to the $N=7$ SL theory that matches the anomaly. We specify the
embedding by the symmetry actions on the $SO(7)/SO(4)$ field $n_{ji}$, where
the $SO(7)$ symmetry acts on the left and the $SO(3)$ symmetry acts on the
right.

Figure 5: Triangular lattice and the relevant symmetries. Each filled red
circle represents an odd number of spin-1/2’s. The $C_{6}$ rotation is around
a site that host the spins, and the dashed line is the reflection axis of
$R_{y}$.

First, the on-site spin rotation $SO^{s}(3)$ symmetry acts as an $SO^{s}(3)$
subgroup of $SO(7)$:

$n\to\left(\begin{array}[]{cc}SO^{s}(3)&0\\\ 0&I_{4}\end{array}\right)n.$ (78)

Next we specify time-reversal symmetry as

$n\to\left(\begin{array}[]{cc}-I_{3}&0\\\ 0&I_{4}\end{array}\right)n,\hskip
10.0pti\to-i.$ (79)

For translations along the three unit vectors $T_{\bm{a}_{1}}$,
$T_{\bm{a}_{2}}$ and $T_{-\bm{a}_{1}-\bm{a}_{2}}$ (see Fig. 5), we have (we
shall use $\sigma_{\mu}^{i,j}$ to denote the $\mu$-th Pauli matrix acting on
the $i,j$ indices)

$\displaystyle T_{\bm{a}_{1}}:$ $\displaystyle
n\to\left(\begin{array}[]{ccc}I_{3}&0&0\\\
0&\exp\left(i\frac{2\pi}{3}\sigma_{y}^{4,5}\right)&0\\\
0&0&\exp\left(-i\frac{2\pi}{3}\sigma_{y}^{6,7}\right)\end{array}\right)n\left(\begin{array}[]{ccc}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{array}\right),$ (86) $\displaystyle T_{\bm{a}_{2}}:$
$\displaystyle n\to\left(\begin{array}[]{ccc}I_{3}&0&0\\\
0&\exp\left(i\frac{2\pi}{3}\sigma_{y}^{4,5}\right)&0\\\
0&0&\exp\left(-i\frac{2\pi}{3}\sigma_{y}^{6,7}\right)\end{array}\right)n\left(\begin{array}[]{ccc}-1&0&0\\\
0&1&0\\\ 0&0&-1\end{array}\right),$ (93) $\displaystyle
T_{-\bm{a}_{1}-\bm{a}_{2}}:$ $\displaystyle
n\to\left(\begin{array}[]{ccc}I_{3}&0&0\\\
0&\exp\left(i\frac{2\pi}{3}\sigma_{y}^{4,5}\right)&0\\\
0&0&\exp\left(-i\frac{2\pi}{3}\sigma_{y}^{6,7}\right)\end{array}\right)n\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&-1\end{array}\right).$ (100)

The $C_{6}=C_{2}\times C_{3}$ rotation is implemented as

$C_{6}:n\to\left(\begin{array}[]{ccccc}I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&-1&0&0\\\ 0&0&0&1&0\\\
0&0&0&0&-1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right).$ (101)

Finally, the reflection $\mathcal{R}_{y}$ (preserving $\bm{a}_{1}$ but
exchanging $\bm{a}_{2}$ with $-\bm{a}_{1}-\bm{a}_{2}$) acts as

$\mathcal{R}_{y}:n\to\left(\begin{array}[]{ccccc}I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\
0&0&0&0&-1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&1\\\ \end{array}\right).$ (102)

The symmetry actions are chosen so that all components of the $n_{ji}$ field
are nontrivial under some symmetry action. This makes the state somewhat
stable since the most fundamental fields are not allowed by symmetry as
perturbations. One can check that the conserved currents of the $SO(7)\times
SO(3)$ symmetry are also forbidden, which is important since these operators
have scaling dimension $2$ and are relevant. Whether the theory is actually a
stable phase depends on the (yet unknown) relevance or irrelevance of
composite operators like $n_{ji}n_{j^{\prime}i^{\prime}}$.

One can check that the symmetry actions are consistent with the group algebra
and indeed gives a homomorphism. One can also check the anomaly-matching
conditions as follows. First, we pull back the SW classes to the physical
symmetries:

$\displaystyle w_{1}^{O(7)}$ $\displaystyle=$ $\displaystyle t,$
$\displaystyle w_{2}^{O(7)}$ $\displaystyle=$ $\displaystyle
w_{2}^{SO(3)}+t^{2}+c^{2}+r^{2}+rc,$ $\displaystyle w_{4}^{O(7)}$
$\displaystyle=$
$\displaystyle(w_{2}^{SO(3)}+t^{2})(c^{2}+r^{2}+rc)+tcr(c+r),$ $\displaystyle
w_{1}^{O(3)}$ $\displaystyle=$ $\displaystyle r,$ $\displaystyle w_{2}^{O(3)}$
$\displaystyle=$ $\displaystyle xy+xr.$ (103)

Notice that the restriction $w_{1}^{TM}=w_{1}^{O(7)}+w_{1}^{O(3)}=r+t$ is
satisfied. Now plugging these into the IR anomaly Eq. (35), we obtain

$\displaystyle S_{\rm bulk}$ $\displaystyle=$ $\displaystyle
i\pi\int_{X_{4}}((w_{2}^{SO(3)}+t^{2})(xy+c^{2}+xr+rc)$ (104)
$\displaystyle+(c^{2}+r^{2}+rc)(r^{2}+xy+xr)+r^{4}$
$\displaystyle+tcr(c+r)+r^{2}(xy+xr)).$

We now examine the relations among these terms. Again, we set $ry=cx=cy=0$,
since the crystalline symmetries involved in each product do not commute. So
the second and third lines of the above anomaly become

$\displaystyle c^{2}r^{2}+cr^{3}+tcr(c+r)$ (105) $\displaystyle=$
$\displaystyle(r+t)cr(c+r)$ $\displaystyle=$ $\displaystyle w_{1}^{TM}cr(c+r)$
$\displaystyle=$ $\displaystyle{\rm Sq}^{1}(c^{2}r+cr^{2})$ $\displaystyle=$
$\displaystyle 0.$

So only the first line of Eq. (104) remains, which is exactly what is required
as discussed in Sec. VII.2.

How do we think of this $N=7$ SL theory on lattice? We can interpret the
$n_{ji}$ field as a collection of fluctuating order parameters, whose nature
is decided by their symmetry properties. The first three rows of the field
$n_{ji}$ ($1\leqslant j\leqslant 3$), being a triplet in $SO^{s}(3)$,
describes magnetic fluctuations at the three $\bm{M}$ points ($(\pi,0)$,
$(0,\pi)$ and $(\pi,\pi)$) in the Brillouin zone. A possible pattern of
symmetry-breaking order is

$\langle n\rangle\sim\left(\begin{array}[]{c}O_{3\times 3}\\\ 0_{4\times
3}\end{array}\right),$ (106)

where $O_{3\times 3}$ is a $3\times 3$ orthogonal matrix and $0_{4\times 3}$
is a zero-matrix. This describes a non-coplanar magnetic order, with the
expectation of the spin operator $S_{i}$ ($i=1,2,3$) on the site
$\bm{r}=n\bm{a}_{1}+m\bm{a}_{2}$ ($n,m\in\mathbb{Z}$)

$\langle S_{i}\rangle\sim
O_{i1}\cos[(n+m)\pi]+O_{i2}\cos(n\pi)+O_{i3}\cos(m\pi),$ (107)

where the three vectors $O_{i1},O_{i2},O_{i3}$ are by construction
orthonormal. This non-coplanar magnetic order is also known as the tetrahedral
order on triangular lattice.

The theory can also form a VBS order by condensing $n_{ji}$ with $4\leqslant
j\leqslant 7$. By Eq. (86) this VBS order has momentum $\bm{K}+\bm{M}$, which
is the same as the commonly studied $12$-site VBS on triangular lattice.

We therefore conclude that the SL(7) theory can naturally arise as a
competition (or intertwinement) between the tetrahedral magnetic order and the
$12$-site VBS order on triangular lattice. The tetrahedral order is known,
numerically, to arise in the $J_{1}-J_{2}-J_{\chi}$ model Gong  _et al._
(2017), where $J_{1}$ and $J_{2}$ are the nearest and next-nearest neighbor
Heisenberg couplings, respectively, and $J_{\chi}$ is the spin chirality
$\bm{S}_{i}\cdot(\bm{S}_{j}\times\bm{S}_{k})$. This model, however, explicitly
breaks the time-reversal and reflection symmetries due to the chirality term.
Since time-reversal breaking perturbations are relevant for Dirac spin liquids
(SL(6)), it may also be relevant for SL(7). Therefore, to search for SL(7), it
may be useful to find a time-reversal invariant lattice Hamiltonian that
realizes the tetrahedral order, and study the effect of various perturbations
on top of it. We also note that the tetrahedral order may be realized in
higher-spin systems with additional $(\bm{S}_{i}\cdot\bm{S}_{j})^{2}$ coupling
Yu and Kivelson (2020). So it is also interesting to explore whether SL
physics can arise in those systems. A smoking-gun signature for the SL(7)
state is that both the non-coplanar magnetic and VBS order parameters (i.e.,
the $21$ matrix elements of $n$) are critical with identical critical
exponents, which is a consequence of the emergent $SO(7)\times SO(3)$ global
symmetry. Similar physics has been numerically confirmed for the SL(5) (i.e.,
DQCP) Nahum  _et al._ (2015a).

####  VII.4.2 Kagome lattice

We now consider a Kagome lattice with spin-$1/2$ per site. The Kagome lattice
has the same lattice symmetries as the triangular. There are three spin-$1/2$
moments in each unit cell, so the LSM anomaly involving translation symmetries
is identical to the triangular case. The only essential difference with the
triangular lattice is the lack of spin moment at the $C_{6}$ rotation center.
So instead of Eq. (57), the full LSM anomaly on Kagome is

$i\pi\int_{X^{4}}(w_{2}^{SO(3)}+t^{2})(xy+rx).$ (108)

Figure 6: Kagome lattice and the relevant symmetries. Each filled red circle
represents an odd number of spin-1/2’s. In this case the rotation center of
$C_{6}$ does not host any spin. Again, the dashed line is the reflection axis
of $R_{y}$.

We now describe a symmetry embedding (a lattice realization) of the SL(7)
theory on Kagome lattice. The spin rotation and time-reversal act on the first
three rows of $n$, in the same way as the triangular realization following Eq.
(78) and (79). The translation symmetries act as (see Fig. 6)

$\displaystyle T_{\bm{a}_{1}}:$ $\displaystyle n\to
n\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right),$ (112)
$\displaystyle T_{\bm{a}_{2}}:$ $\displaystyle n\to
n\left(\begin{array}[]{ccc}-1&0&0\\\ 0&1&0\\\ 0&0&-1\end{array}\right),$ (116)
$\displaystyle T_{-\bm{a}_{1}-\bm{a}_{2}}:$ $\displaystyle n\to
n\left(\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\ 0&0&-1\end{array}\right).$ (120)

The $C_{6}$ rotation acts as

$C_{6}:n\to\left(\begin{array}[]{ccccc}-I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\
0&0&0&0&1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right).$ (121)

Finally, the $\mathcal{R}_{y}$ reflection acts the same way as the triangular
case:

$\mathcal{R}_{y}:n\to\left(\begin{array}[]{ccccc}I_{3}&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\
0&0&0&0&-1\end{array}\right)n\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&1\\\ \end{array}\right).$ (122)

We can now go through the anomaly calculation in a similar way as our earlier
examples. We shall omit the intermediate steps here and simply state that the
final result is indeed the required anomaly in Eq. (108).

Similar to the triangular case, we can again interpret this SL(7) theory as a
result of competition between non-coplanar magnetic and VBS orders, now both
at momenta $\bm{M}$ from Eq. (112). The non-coplanar magnetic order also has a
$C_{6}$ angular momentum $l=1$. This is known as the cuboctahedral order (more
precisely, the cuboc1 order in the language of Ref. Messio  _et al._ (2011)).
This order has been numerically found for the $J_{1}-J_{2}-J_{3}$ Heisenberg
model in certain regimes Gong  _et al._ (2015). Our results motivate further
exploration of the phase diagram near the cuboctahedral order, and see if the
SL(7) state could be realized. Again, just like the realization on the
triangular lattice, a smoking-gun signature of the SL(7) state is that the
non-coplanar magnetic and VBS order parameters are critical and have identical
critical exponent, due to the emergent $SO(7)\times SO(3)$ symmetry.

##  VIII Discussion

In this paper, based on a nonlinear sigma model defined on a Stiefel manifold,
$SO(N)/SO(4)$, supplemented with a Wess-Zumino-Witten (WZW) term, we have put
forward the theory of Stiefel liquids, which are a family of critical quantum
liquids that have many extraordinary properties. For example, they have a
large emergent symmetry, a cascade structure, and nontrivial quantum
anomalies. Some of these Stiefel liquids are argued to be dual to the well
known deconfined quantum critical point and $U(1)$ Dirac spin liquid, and
others are conjectured to be non-Lagrangian, i.e., its corresponding RG fixed
point cannot be described by any weakly-coupled mean-field theory at any
scale, which, in particular, means that these states are beyond parton (mean-
field) construction widely used in the study of exotic quantum phases and
phase transitions.

We make some comments on why the “non-Lagrangian” conjecture for the
$N\geqslant 7$ Stiefel liquids may be reasonable. The most obvious gauge
theory candidates for such WZW fixed points are some kinds of QCD3 with
gapless Dirac fermions coupled to some gauge fields. However, as we review in
Appendix D, typical QCD3 correspond to WZW theories defined on Grassmannian
manifolds $G(2N)/G(N)\times G(N)$, where $G$ can be $U,SU,USp,SO$. In fact,
this is also why Stiefel liquids with $N=5,6$ do have gauge theory
descriptions, since the corresponding Stiefel manifolds in these two cases
also happen to be some kinds of Grassmannian:
$SO(5)/SO(4)=USp(4)/(USp(2)\times USp(2))$ and $SO(6)/SO(4)=SU(4)/(SU(2)\times
SU(2))$. For $SO(N\geqslant 7)/SO(4)$, we do not have such identification, so
the corresponding Stiefel liquids are not captured by some simple QCD3. We
also note the special role played by the $SO(N-4)$ symmetry in Stiefel
liquids. For $N=6$, this $SO(2)$ symmetry is realized in the gauge theory as
the flux conservation symmetry of the dynamical $U(1)$ gauge field. It is not
clear how this $SO(2)$ flux conservation symmetry could be generalized to
higher $SO(N-4)$ in different gauge theories. Besides these constraints from
symmetries, the intricate anomaly structures of the Stiefel liquids discussed
in Sec. VI also impose further nontrivial constraints on its possible
renormalizable-Lagrangian description. One possibility is that the $N\geqslant
7$ Stiefel liquids can be realized by gauge theories with significantly lower
symmetries in the UV Lagrangians, and the full IR symmetries emerge through
some nontrivial dynamics. This scenario will be hard to rule out, and if true,
it will likely shed new light on the dynamics of $(2+1)$-d gauge theories.

We mention that the fixed points of some quantum loop models were also
proposed to be non-Lagrangian Freedman  _et al._ (2004, 2005); Dai and Nahum
(2020). The nature of such loop quantum criticality appears to be very
different from those studied in this paper. For example, they are not Lorentz
invariant.

Note that although the most commonly used parton approach uses canonical
bosonic/fermionic partons to construct a (non-interacting) mean field, there
are also some parton approaches that use other types of partonic DOFs, in
particular, ones that are subject to some constraints and are strongly
fluctuating at all scales (see, e.g., Ref. Xu and Sachdev (2010)). One may
wonder if the latter constrained-parton-based approach can lead to a
construction to the non-Lagrangian Stiefel liquids. Although it is not ruled
out, we believe this approach is difficult, and even if it can be achieved,
novel ideas are still needed to make it work. This is because: i) As far as we
know, all states constructed with constrained partons ultimately fall into the
paradigm of mean field plus weak fluctuations, but these (conjectured) non-
Lagrangian states are beyond this paradigm. ii) More technically, in such an
approach, one often (if not always) encounters Dirac fermions coupled to sigma
fields (i.e., bosonic fields subject to some constraints), but these sigma
fields live in certain Grassmannian manifold, which is difficult to be
converted into a Stiefel manifold relevant here, unless some nontrivial
mathematical facts can be used, in a way similar to the case of Stiefel
liquids with $N=6$.

In Sec. VII, we have proposed an approach based on the hypothesis of
emergibility, which is complementary to the conventional parton approach, to
study quantum phases and phase transitions. This approach is benchmarked with
some known examples, and then applied to predict that spin-1/2 triangular and
Kagome lattices can host one of the non-Lagrangian Stiefel liquids. This
approach can also predict some detailed properties of such lattice
realizations of these Stiefel liquids, such as the quantum numbers of various
critical order parameters with identical scaling exponents.

It may be useful to comment on the parton approach and compare it with our
anomaly-based approach. The parton approach is explicit, concrete, and
relatively easy to manipulate, and it has led to tremendous success and deep
insights in the study of strongly-correlated quantum matter. However, this
parton approach also has some drawbacks. More specifically, there are two
common treatments of a parton construction, leading to a projected wave
function and an effective gauge theory, respectively. The wave-function-based
treatment starts with an enlarged Hilbert space of the partons, and performs a
projection of a valid ground state of these partons, in order to return to the
physical Hilbert space. Although the projected wave function is indeed in the
physical Hilbert space, a priori, it may not describe any ground state of a
local Hamiltonian in the physical Hilbert space, and it is unclear what
universal properties it exhibits – these have to be checked case by case, say,
using numerical calculations. On the other hand, from the effective gauge
theory, it is more analytically tractable to deduce what ground state it
describes and what universal properties it possesses. However, a priori, it is
unclear whether such an effective gauge theory can really emerge from the
physical Hilbert space, although it is often assumed so without any rigorous
analytically controlled justification151515We note that sometimes such an
effective gauge theory is phrased in terms of a lattice gauge theory, where
the partons hop on the lattice sites and are coupled to the gauge fields
living on the lattice links. In such an interpretation, the physical Hilbert
space is often manifestly just a gauge-fluctuation-free subspace of the
Hilbert space of the lattice gauge theory, where certain hard gauge
constraints are imposed.. Our anomaly-based approach may be more abstract by
the contemporary standards, but it is closer to the intrinsic characterization
of the universal many-body physics discussed in the introduction, and it
directly hinges on the emergibility: states that fail to satisfy the anomaly-
matching condition are necessarily not emergible, although at this stage we
cannot rigorously prove that states that do satisfy the anomaly-matching
condition must be emergible. We also note that the parton approach is in fact
also essentially an attempt to verify anomaly-matching between the IR theory
and the microscopic setup, but through an explicit mean-field-like
construction.

Let us also comment on the significance of non-Lagrangian, or intrinsically
non-renormalizable, theories specifically in condensed matter physics. Since
most condensed matter systems do have some natural UV cutoffs, the concept of
renormalizability should not play a fundamental role in condensed matter
physics. This means that we should have a theoretical framework that can
handle both Lagrangian and non-Lagrangian theories – there should be no
intrinsic difference between the two. However, the fact is that not only we do
not have many tools to analyze non-Lagrangian theories, we did not even have
many serious examples of non-Lagrangian theory prior to this work. From this
perspective, what is really surprising is how difficult it was to find such
non-Lagrangian examples – this is perhaps rooted in our heavy reliance on
perturbative quantum field theories in the past. The examples found here may
also require and inspire us to develop new tools to analyze strongly
correlated systems in more intrinsic manners. One example is the problem of
“emergibility”, which was the focus of the later half of our paper: the
intrinsic non-renormalizability, or lack of mean-field construction, forced us
to further develop the anomaly-matching approach which may become useful for
future works on strongly correlated systems in general. Therefore, even though
renormalizability per se is not of fundamental importance in condensed matter
physics, being able to go beyond renormalizable theories is. Our work on
(likely) non-Lagrangian quantum criticality represents a step toward this
ambitious goal.

We finish this paper with some interesting open questions that we leave for
future work.

  1. 1.

Although we have given a derivation of the effective theory of some of the
Stiefel liquids based on the gauge theoretic descriptions of Dirac spin
liquids, it is desirable to give a more explicit derivation. For example, it
is desirable to explicitly derive the expression of Skyrmion current in Eq.
(23) in terms of the $\mathcal{P}$ field. Also, it is nice to show how the WZW
term on the Grassmannian manifold reduces to that on the Stiefel manifold.

  2. 2.

The quantum anomalies of the Stiefel liquids labeled with an even $N$ have not
been fully pinned down. It is interesting to finish this anomaly analysis. For
the purpose of studying Stiefel liquids, using their cascade structure may be
sufficient to fully determine their anomalies, just as what we have done in
this paper. However, we note that deriving the full quantum anomalies directly
based on the WZW action is also an intriguing theoretical challenge.

  3. 3.

Although we have argued that the Stiefel liquids labeled by $N\geqslant 6$ can
flow to a conformally invariant fixed point under RG, we have not been able to
establish this with a controlled analysis. It is well motivated to find ways
to systematically study the IR dynamics of the Stiefel liquids. For example,
it is natural to ask if the Stiefel liquid fixed points predicted in this work
can be found in numerical conformal bootstrap. The large emergent symmetry
group $SO(N)\times SO(N-4)$ and the likely irrelevance of singlet operators
may provide some reasonable starting point for such investigation. Also, given
that the Stiefel liquids are described by a matrix model, it is interesting to
explore if they have any holographic dual.

  4. 4.

The conjecture that the Stiefel liquids with $N>6$ are non-Lagrangian has not
been proved. It is important to prove or falsify it. We expect that the method
to prove or falsify it will necessarily bring in useful general insights.
Also, even if it is falsified, these Stiefel liquids are still interesting.
Relatedly, although a wave function is not strictly necessary for the
intrinsic characterization of the universal physics of a many-body system, it
may be interesting and useful to find a wave function for these Stiefel
liquids.

  5. 5.

It is natural to ask whether similar WZW models on target manifolds other than
the Stiefel can lead to interesting fixed points. The most natural manifolds
are the Grassmannians, which have been studied Bi  _et al._ (2016) and related
to various gauge theories Komargodski and Seiberg (2018) (see also Appendix
D). It will be interesting to either better understand the Grassmannian
theories, or to contemplate on theories based on other types of manifolds.

  6. 6.

The hypothesis of emergibility has not been proved rigorously, and it is
important to prove or falsify it. If it can be proved, or at least further
justified, it can be applied to other cases to study other exotic quantum
phases and phase transitions. We expect this approach to give rise to many
more interesting results and novel insights in the future. If it will be
falsified, it is still very useful to find the correct general rules that
govern the emergibility of a given low-energy theory for a system.

  7. 7.

Perhaps the most important question is whether some of the critical Stiefel
liquids can be realized in real materials. We have suggested that the $N=7$
Stiefel liquid may arise near certain non-coplanar magnetic orders.
Numerically those non-coplanar orders can arise in some relatively simple
lattice spin models Gong  _et al._ (2015); Yu and Kivelson (2020); Gong  _et
al._ (2017). It will be fascinating to explore further in the phase diagram of
those systems and see if the Stiefel liquid can indeed be found. This may
provide valuable guidance towards ultimate experimental realizations.

  8. 8.

We have seen that the theories of the deconfined quantum critical point and
the $U(1)$ Dirac spin liquid can be formulated in terms of local DOFs. It may
be interesting to see if other exotic non-quasiparticle critical quantum
liquids can also be formulated in a similar way, and such a new formulation
may bring in new insights. For example, can the theory of a Fermi surface
coupled to a $U(1)$ gauge field in $(2+1)$-d be formulated purely in terms of
local DOFs? It is likely that such a formulation will explicitly involve the
infinitely many collective excitations around the Fermi surface, and the ideas
from Ref. Mross and Senthil (2011) may be useful.

###### Acknowledgements.

We thank Maissam Barkeshli, Zhen Bi, Vladimir Calvera, Davide Gaiotto, Meng
Guo, Chao-Ming Jian, Theo Johnson-Freyd, Steve Kivelson, John McGreevy, Subir
Sachdev, Cenke Xu, Weicheng Ye and Yi-Zhuang You for illuminating discussions.
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Industry Canada and
by the Province of Ontario through the Ministry of Colleges and Universities.

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##  Appendix A More on the proposed WZW action

In the main text a WZW action for the $(2+1)$-d system of our interest is
proposed in Eq. (6). In this appendix we present more details on its
mathematical aspects.

For any even integer $d\geqslant 0$, we can define a WZW term for a $d+1$
(spacetime) dimensional system on a Stiefel manifold $V_{N,N-(d+2)}\equiv
SO(N)/SO(d+2)$, where $N\geqslant d+3$. An element on this Stiefel manifold
can be parameterized by an $N$-by-$\left(N-(d+2)\right)$ matrix, $n$, such
that its columns are orthonormal, i.e., $n^{T}n=I_{N-(d+2)}$. The
corresponding WZW action is

$\displaystyle S_{\rm
WZW}^{(N,d)}[n]=\frac{2\pi}{\Omega_{d+2}}\int_{0}^{1}du\int
d^{d+1}x\sum_{k_{1},k_{2},\cdots,k_{\frac{d+2}{2}}=1}^{N-(d+2)}\det(\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})})$
(123)

where $\Omega_{d+2}$ is the volume of $S^{d+2}$ with unit radius, and the
$N$-by-$N$ matrix $\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})}$ is
given by

$\displaystyle\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})}=(n,\partial_{x_{1}}n_{k_{1}},\partial_{x_{2}}n_{k_{1}},\partial_{x_{3}}n_{k_{2}},\partial_{x_{4}}n_{k_{2}},\cdots,\partial_{x_{d+1}}n_{k_{\frac{d+2}{2}}},\partial_{u}n_{k_{\frac{d+2}{2}}})$
(124)

where $x_{1,2,\cdots,x_{d+1}}$ is the coordinate of the physical spacetime,
$n_{k_{i}}$ is the $k_{i}$th column of $n$ (note that the repeated subscripts
$k_{i}$’s are not summed over in the right hand side of (124)). That is, the
first $N-(d+2)$ columns of
$\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})}$ is just $n$, and its last
$d+2$ columns are derivatives of the columns of $n$ arranged in the above way.
More explicitly,

$\displaystyle\begin{split}\det&(\tilde{n}_{(k_{1},k_{2},\cdots,k_{\frac{d+2}{2}})})=\frac{1}{(N-(d+2))!}\epsilon^{i_{1}i_{2}\cdots
i_{N-(d+2)}}\epsilon^{j_{1}j_{2}\cdots
j_{N}}n_{j_{1}i_{1}}n_{j_{2}i_{2}}\cdots n_{j_{N-(d+2)}i_{N-(d+2)}}\\\
&\cdot\partial_{x_{1}}n_{j_{N-(d+2)+1}k_{1}}\partial_{x_{2}}n_{j_{N-(d+2)+2}k_{1}}\partial_{x_{3}}n_{j_{N-(d+2)+3}k_{2}}\partial_{x_{4}}n_{j_{N-(d+2)+4}k_{2}}\cdots\partial_{x_{d+1}}n_{j_{N-1}k_{\frac{d+2}{2}}}\partial_{u}n_{j_{N}k_{\frac{d+2}{2}}}\end{split}$
(125)

where the $\epsilon$’s are the fully anti-symmetric symbols with rank
$N-(d+2)$ and $N$, respectively.

It is straightforward to see that the WZW term presented in the main text is
precisely the special case of the above one with $d=2$, and in the main text
we denote $V_{N,N-4}$ by $V_{N}$. Also, it is clear that such a term can be
defined only if $d$ is even, and it is interesting to compare this observation
with the fact that the form of the homotopy groups of the Stiefel manifold
$V_{N,N-(d+2)}$ is qualitatively different for even $d$ and odd $d$. That is,
the first nontrivial homotopy group of $V_{N,N-(d+2)}$ is

$\displaystyle\pi_{d+2}V_{N,N-(d+2)}=\left\\{\begin{array}[]{lr}\mathbb{Z},&d+2\
{\rm even\ or\ }N=d+3\\\ \mathbb{Z}_{2},&d+2\ {\rm odd\ and\
}N>d+3\end{array}\right.$ (128)

The validity of the above WZW term requires it to be the integral of the
pullback of a closed $(d+2)$-form on $V_{N,N-(d+2)}$. The $(d+2)$-form on
$V_{N,N-(d+2)}$ that this WZW term is associated with is

$\displaystyle\begin{split}\omega&=\frac{1}{(N-(d+2))!}\epsilon^{i_{1}i_{2}\cdots
i_{N-(d+2)}}\epsilon^{j_{1}j_{2}\cdots
j_{N}}n_{j_{1}i_{1}}n_{j_{2}i_{2}}\cdots n_{j_{N-(d+2)}i_{N-(d+2)}}\\\
&\quad\cdot dn_{j_{N-(d+2)+1}k_{1}}\wedge dn_{j_{N-(d+2)+2}k_{1}}\wedge
dn_{j_{N-(d+2)+3}k_{2}}\wedge dn_{j_{N-(d+2)+4}k_{2}}\cdots\wedge
dn_{j_{N-1}k_{\frac{d+2}{2}}}\wedge dn_{j_{N}k_{\frac{d+2}{2}}}\end{split}$
(129)

where the repeated subscripts $k_{i}$’s are summed over. It can be shown that
the above form is indeed closed 161616We thank Vladimir Calvera for giving a
mathematical proof to the closedness..

It remains to fix the normalization factor in front of the WZW term. We start
with two observations:

  1. 1.

For $N=d+3$ Eq. (123) is the familiar WZW term on $S^{d+2}$ with the correct
normalization factor.

  2. 2.

For $N>d+3$, if we fix the first column of $n$ to a constant, say
$n_{1}=(1,0,0...)^{T}$, the proposed WZW term for $V_{N,N-(d+2)}$ becomes that
for $V_{N-1,(N-1)-(d+2)}$.

Mathematically, fixing the first column of $n$ describes an inclusion map
$i:V_{N-1,N-d-3}\to V_{N,N-d-2}$:

$\displaystyle n_{(N-1)\times(N-d-3)}\to\left(\begin{array}[]{cc}1&0\\\
0&n_{(N-1)\times(N-d-3)}\end{array}\right).$ (132)

This map then induces a homomorphism between the homotopy groups
$\pi_{d+2}(V_{N-1,N-d-3})\to\pi_{d+2}(V_{N,N-d-2})$. Based on the two
observations made above, the normalization factor in Eq. (123) will be
justified if the homomorphism
$\pi_{d+2}(V_{N-1,N-d-3})\to\pi_{d+2}(V_{N,N-d-2})$ induced by $i$ is an
isomorphism. The last statement can be proved using the long exact sequence of
homotopy groups associated with the fibration

$\displaystyle V_{N-1,N-d-3}\to V_{N,N-d-2}\to S^{N-1}.$ (133)

A pictorial consequence of the above argument is that a “generator” of
$\pi_{d+2}(V_{N,N-(d+2)})$ is given by fixing the entries of the first
$N-(d+2)-1$ columns of $n$ to be $n_{ji}=\delta_{ji}$, and letting the last
column, which now lives on $S^{d+2}$ with unit radius, wrap around the
$S^{d+2}$ once.

Some properties of the WZW term for the case with $d=2$ are discussed in Sec.
IV.1, and with minor modifications many of them also apply appropriately to
the case with a general even $d$.

##  Appendix B A gauge theory description of SL(N=5,k)

In this appendix, we show that SL(N=5,k) has a natural gauge-theoretic
description, i.e., a QCD3 theory with $N_{f}=2$ flavor of fermions interacting
with a $USp(2k)$ gauge field. A special case of $k=1$, i.e., SL(5) or the
DQCP, was already discussed in Ref. Wang  _et al._ (2017).

The global symmetry of the $N_{f}=2$ $USp(2k)$ QCD3 theory is
$USp(2N_{f})=USp(4)\cong SO(5)$, which is identical to SL(5,k). The fermions
are in the fundamental representation of the $USp(2k)$ gauge group and
$USp(4)$ global symmetry (i.e., spinor representation of $SO(5)$). The gauge
invariant operators with lowest scaling dimensions shall be the fermion mass
terms, which are $SO(5)$ vector and $SO(5)$ singlet. The $SO(5)$ vector mass
can be identified as the order parameter field $n$ of the NLSM. To see the
relation more explicitly, we can couple the $SO(5)$ vector mass to a $SO(5)$
bosonic vector $n_{i}$, and then integrate out fermions. This will yield an
$SO(5)$ NLSM, and the original $USp(2k)$ gauge field is expected to just
confine since it does not couple to any low-energy degrees of freedom.
Moreover, due to the Abanov-Wiegmann mechanism Abanov and Wiegmann (2000),
integrating out fermions also yields a level-$k$ WZW term of the $SO(5)$
vector $n_{i}$. The level-$k$ comes from the fact that there are $k$ copies of
fermions as the gauge group is $USp(2k)$. Therefore, we have derived that the
$N_{f}=2$ $USp(2k)$ QCD3 theory is dual to SL(N=5,k).

From the gauge-theoretic description of SL(N=5,k), we can gain some intuition
about the properties of SL with a general $(N,k)$:

  1. 1.

Stability. It is clear that the larger $k$ is, it is more likely that the QCD3
theory will confine. Naturally, we expect that this feature will also hold for
$N>5$: for a given $N$ there exists a critical $k_{c}$, such that the SL can
flow into a critical phase when $k\leqslant k_{c}$.

  2. 2.

Neighboring topological order. By turning on a time-reversal-breaking singlet
mass, the SL(N=5,k) will become the $USp(2k)_{\mp 1}^{s}$ 171717The
superscript $s$ refers to the fact that the gauge field is a spin gauge field.
TQFT, which is dual to the $USp(2)_{\pm k}=SU(2)_{\pm k}$ TQFT. $SU(2)_{1}$ is
just the semion topological order discussed in the main text. In Sec. VI.5, we
have argued that SL(N,1) with a general $N\geqslant 5$ can flow to the
$SU(2)_{\pm 1}$ TQFT under a time-reversal-breaking deformation. So it is
possible that the SL(N>5,k) will also flow to the $SU(2)_{\pm k}$ TQFT under
appropriate time-reversal-breaking perturbation.

  3. 3.

Higgs descendent. As shown in Sec. IV B of Ref. Wang  _et al._ (2017), for the
case with $k=1$, adding a flavor-singlet Higgs field can break the gauge
structure to $U(k)$ with $k=1$, and the resulting theory is equivalent to a
QED3 coupled to 4 gapless Dirac fermions, with one of the six monopole
operators added to the Lagrangian, i.e., schematically we have $USp(2)+{\rm
Higgs}=U(1)+{\rm monopole}$. This is how the DQCP is related to the $U(1)$
DSL. The argument there can actually be generalized to any $k$ to show that
$USp(2k)+{\rm Higgs}=U(k)+{\rm monopole}$. This is nicely compatible with the
cascade structure of the SLs, and our results that SL(5,k) and SL(6,k) can
also be described by $USp(2k)$ gauge theory and $U(k)$ gauge theory,
respectively.

  4. 4.

Microscopic realization. For a spin-$k/2$ system, there is a natural parton
construction for the SL(N=5,k).

We first fractionalize spin operators into partons,

$S^{i}=-\frac{1}{4}\textrm{Tr}(X^{\dagger}X\sigma^{i}),$ (134)

with $X$ being a $2k\times 2$ matrix,

$X=\left(\begin{matrix}\psi_{1}^{\dagger}&\cdots&\psi^{\dagger}_{k}&\psi^{\dagger}_{k+1}&\cdots&\psi^{\dagger}_{2k}\\\
\psi_{k+1}&\cdots&\psi_{2k}&-\psi_{1}&\cdots&-\psi_{k}\end{matrix}\right)^{T}.$
(135)

Here $\psi_{i}^{\dagger}$ is the fermion creation operator. $X$ satisfies the
reality condition $X^{*}=\Omega^{c}X\Omega^{s}$ with

$\Omega^{c}=\left(\begin{matrix}0&I_{k}\\\
-I_{k}&0\end{matrix}\right),\quad\Omega^{S}=\left(\begin{matrix}0&1\\\
-1&0\end{matrix}\right).$ (136)

Here $I_{k}$ is a $k\times k$ identity matrix. So the spin operators can be
written as
$S^{i}=-\frac{1}{4}\textrm{Tr}(\Omega^{s}X^{T}\Omega^{c}X\sigma^{i})$. It is
apparent that this parton decomposition has a $USp(2k)$ gauge invariance,
namely, the spin operators are invariant under a $USp(2k)$ (left) rotation of
$X$, $RX$, as $R^{T}\Omega^{c}R=\Omega^{c}$. On the other hand, the $SO(3)$
spin rotation acts as the $USp(2)\cong SU(2)$ (right) rotation of $X$. The
local contraint of the parton construction is,

$X\Omega^{s}X^{T}=-\Omega^{c},$ (137)

or equivalently, $\psi_{i}^{\dagger}\psi_{i}=\psi_{i+k}^{\dagger}\psi_{i+k}$
for $i=1,\cdots,k$. One can further show that the spin operators defined in
Eq. (134) satisfy $[S^{i},S^{j}]=i\varepsilon^{ijk}S^{k}$ and
$\sum_{i}(S^{i})^{2}=\frac{k}{2}\left(\frac{k}{2}+1\right)$.

Therefore, the above parton construction has an emergent $USp(2k)$ gauge
structure, and the fermions $(\psi_{1},\cdots,\psi_{k})$ form a $USp(2k)$
fundamental. Putting $\psi_{i}$ fermions into a band structure with two Dirac
cones, we will get the $N_{f}=2$ $USp(2k)$ QCD3, or equivalently, SL(5,k).

So it is natural to look for SL(5,k) in a spin-$k/2$ system. Furthermore, as
proposed in Sec. V, SL(6,k) is dual to the $N_{f}=2$ $U(k)$ QCD3, so it is
also possible that SL(6,k) can emerge in a spin-$k/2$ system. This is
discussed in detail recently Calvera and Wang (2020). These observations lead
us to further conjecturing that it is also true SL(N>6,k) can also emerge in a
spin-$k/2$ system. Indeed, in Sec. VII.4, we have argued that SL(7) can emerge
in a spin-1/2 system.

##  Appendix C Full quantum anomaly of the DQCP

The quantum anomaly of the DQCP, or equivalently, SL(5), was partially
analyzed in Ref. Wang  _et al._ (2017), and an anomaly associated with the
$SO(5)$ symmetry was found, which is described by a $(3+1)$-d topological
response function, $i\pi\int_{M}w_{4}^{SO(5)}$, where $M$ is the closed
manifold in which the $(3+1)$-d bulk corresponding to the DQCP lives, and
$w_{4}^{SO(5)}$ is the fourth Stiefel-Whitney (SW) class of the $SO(5)$ gauge
bundle that couples to this bulk.

Besides the $SO(5)$ symmetry, the DQCP also enjoys a time reversal symmetry,
$\mathcal{T}$, which also contributes to the anomaly. To understand the full
anomaly associated with both the $SO(5)$ and $\mathcal{T}$ symmetries, it is
useful to enlarge the $SO(5)$ symmetry to $O(5)$ by including the improper
$Z_{2}$ rotation. Then the action of the time reversal symmetry is a
combination of this improper $Z_{2}$ rotation and a flip of the time
coordinate. In this appendix, we show that the full anomaly of the DQCP is
described by a $(3+1)$-d topological response function

$\displaystyle S=i\pi\int_{M}w_{4}^{O(5)}$ (138)

with a constraint $w_{1}^{O(5)}=w_{1}^{TM}\ ({\rm mod\ }2)$, where $TM$
denotes the tangent bundle of $M$. This constraint simply indicates that the
improper $Z_{2}$ rotation of the $O(5)$ symmetry is accompanied with a flip of
the time coordinate. For notational brevity, in the following we will suppress
the superscript “$TM$” of a SW class of $TM$.

To obtain the above result, let us look at the most general form that the
anomaly can take:

$\displaystyle
S=i\pi\int_{M}\left[a_{1}w_{4}^{O(5)}+a_{2}[w_{2}^{O(5)}]^{2}+a_{3}w_{1}^{2}w_{2}^{O(5)}+a_{4}w_{1}^{4}+a_{5}w_{2}^{2}\right]$
(139)

where we have used the (mod 2) relations $w_{1}^{O(5)}=w_{1}$,
$w_{1}w_{3}=w_{1}^{4}+w_{2}^{2}+w_{4}=0$, $w_{1}w_{2}=0$,
$[w_{2}^{O(5)}]^{2}=(w_{2}+w_{1}^{2})w_{2}^{O(5)}$, ${\rm
Sq}^{1}(w_{3}^{O(5)})=w_{1}w_{3}^{O(5)}$, ${\rm Sq}^{1}\cdot{\rm
Sq}^{1}(w_{2}^{O(5)})=0$, and $w_{3}^{O(5)}={\rm
Sq}^{1}(w_{2}^{O(5)})+w_{1}^{O(5)}w_{2}^{O(5)}$ to remove some terms.

The above topological response function must satisfy the following known
properties of the DQCP Vishwanath and Senthil (2013); Kapustin (2014); Wang
_et al._ (2017); Bi  _et al._ (2015), which help us to deduce the values of
the $a$’s unambiguously:

  1. 1.

As mentioned above, if only the $SO(5)$ symmetry is considered, the anomaly is
described by $i\pi\int_{M}w_{4}^{SO(5)}$.

In this case, $w_{4}^{O(5)}=w_{4}^{SO(5)}$, $w_{1}=0$, and
$w_{2}^{O(5)}=w_{2}^{SO(5)}$. So Eq. (139) becomes
$S=i\pi\int_{M}\left[a_{1}w_{4}^{SO(5)}+a_{2}[w_{2}^{SO(5)}]^{2}+a_{5}w_{2}^{2}\right]$,
which implies that $a_{1}=1$ and $a_{2}=a_{5}=0$.

  2. 2.

Ignoring the $SO(5)$ symmetry and implementing $\mathcal{T}$ by
$n\rightarrow-n$, there is an anomaly associated with $\mathcal{T}$, described
by $i\pi\int_{M}w_{1}^{4}$. The corresponding bulk is known as $eTmT$ Wang and
Senthil (2013).

In this case, $w_{4}^{O(5)}=w_{1}^{4}$ and $w_{2}^{O(5)}=0$. So Eq. (139)
becomes $S=i\pi\int_{M}(1+a_{4})w_{1}^{4}$, which implies that $a_{4}=0$.

  3. 3.

Suppose the full symmetry is broken to $SO(2)\times\mathcal{T}$, where the
$SO(2)$ rotates the first 2 components of $n$ and $\mathcal{T}$ flips its last
3 components, the anomaly is described by
$S_{3}=i\pi\int_{M}w_{1}^{2}w_{2}^{SO(2)}$. The corresponding bulk is known as
$eCmT$ Wang and Senthil (2013); Metlitski  _et al._ (2013).

In this case, $w_{4}^{O(5)}=w_{1}^{2}w_{2}^{SO(2)}$ and
$w_{2}^{O(5)}=w_{2}^{SO(2)}+w_{1}^{2}$. So Eq. (139) becomes
$S=i\pi\int_{M}\left[(1+a_{3})w_{1}^{2}w_{2}^{SO(2)}+a_{3}w_{1}^{4}\right]$,
which implies that $a_{3}=0$.

In summary, the above three conditions imply that $a_{1}=1$ and
$a_{2}=a_{3}=a_{4}=a_{5}=0$. So the full anomaly of the DQCP is described by
Eq. (138).

##  Appendix D WZW models on the Grassmannian manifold
$\frac{G(2N)}{G(N)\times G(N)}$

The Grassmannian manifolds $\frac{G(2N)}{G(N)\times G(N)}$ (with
$G=U,SU,SO,USp$) also have $\pi_{4}(\frac{G(2N)}{G(N)\times G(N)})=\mathbb{Z}$
and $\pi_{3}(\frac{G(2N)}{G(N)\times G(N)})=0$, so one can define $2+1$-d WZW
models on these Grassmannians. Using the argument in Sec. IV.4 we can obtain a
similar phase diagram (at least for large $N$). Namely, as one tunes the
coupling constant of NLSM, there are three fixed points: 1) an attractive
fixed point of a spontaneous-symmetry-breaking phase, with the ground state
manifold being $\frac{G(2N)}{G(N)\times G(N)}$; 2) a repulsive fixed point of
order-disorder transition; 3) an attractive fixed point of a critical quantum
liquid. The last attractive fixed point is the Grassmannian version of our
proposed SLs. Interestingly, the Grassmannian WZW models have simple
candidates of renormalizable Lagrangian descriptions, i.e., Dirac fermions
coupled to non-Abelian gauge fields Komargodski and Seiberg (2018). More
concretely, we have

  1. 1.

The QCD3 theory with $N_{f}=2N$ Dirac fermions coupled to a $SU(k)$ gauge
field is a UV completion of the $\frac{U(2N)}{U(N)\times U(N)}$ NLSM model
with a level $k$ WZW term.

  2. 2.

The QCD3 theory with $N_{f}=2N$ Majorana fermions coupled to a $SO(k)$ gauge
field is a UV completion of the $\frac{SO(2N)}{SO(N)\times SO(N)}$ NLSM model
with a level $k$ WZW term.

  3. 3.

The QCD3 theory with $N_{f}=2N$ Dirac fermions coupled to a $USp(2k)$ gauge
field is a UV completion of the $\frac{USp(4N)}{USp(2N)\times USp(2N)}$ NLSM
model with a level $k$ WZW term.

  4. 4.

The QCD3 theory with $N_{f}=2N$ Dirac fermions coupled to a $U(k)$ gauge field
is a UV completion of the $\frac{SU(2N)}{SU(N)\times SU(N)}$ NLSM model with a
level $k$ WZW term.181818This one is relatively new and will be discussed more
carefully elsewhere.

One may expect to see this correspondence by using the trick that appears
several times in the paper. We first couple the color-singlet fermion mass of
the QCD3 theory to a bosonic field that lives on the Grassmannian
$\frac{G(2N)}{G(N)\times G(N)}$, and then integrate out fermions. This will
give a NLSM of the bosonic field and will also generate a WZW term Abanov and
Wiegmann (2000). At last, the gauge field will confine by itself without doing
anything to the Grassmannian $\frac{G(2N)}{G(N)\times G(N)}$ WZW models. The
level $k$ (instead of 1) comes from the color multiplicity of the gauge field.
It is worth mentioning if one couples the fermion mass to a field living on
$\frac{G(2N)}{G(2N-M)\times G(M)}$ with $M\neq N$, integrating out Dirac
fermions will generate a Chern-Simons term for the gauge field as well. In
this case the gauge field will not confine, and one ends up with the
$\frac{G(2N)}{G(2N-M)\times G(M)}$ WZW model coupled to a Chern-Simons gauge
field.

We can also compare the global symmetry of the gauge theories and the
Grassmannian WZW models. The simplest one is the last case, where both the
gauge theory and the Grassmannian WZW models have an explicit $USp(4N)$ global
symmetry. For the first case, the $SU(k)$ gauge theory has an explicit
$SU(2N)\times U(1)$ symmetry, and the $U(1)$ symmetry is carried by the baryon
operator. The $\frac{U(2N)}{U(N)\times U(N)}$ WZW model has an explicit
$SU(2N)$ symmetry, which acts directly on the NLSM field. The nontrivial part
is the $U(1)$ symmetry, which comes from the topological property of the
manifold $\pi_{2}(\frac{U(2N)}{U(N)\times U(N)})=\mathbb{Z}$. The operator
charged under this topological $U(1)$ symmetry is the Skyrmion creation
operator, which is fermionic (or bosonic) if the level $k$ of the WZW term is
odd (or even) Komargodski and Seiberg (2018). This nicely matches the
statistics of the baryon operator of the $SU(k)$ gauge theory. Similarly, for
the second case one can also match the global symmetry by using
$\pi_{2}(\frac{SO(2N)}{SO(N)\times SO(N)})=\mathbb{Z}_{2}$ for $N>2$.

The identification of Grassmannian WZW models as the QCD3 theories further
corroborates the existence of SLs as critical quantum liquids, as discussed in
Sec. IV.4.

##  Appendix E Explicit homomorphism between the $su(4)$ and $so(6)$
generators

To be self-contained, in this appendix we present the explicit homomorphism
between the $su(4)$ and $so(6)$ generators that is used in this paper.

Recall that the we write the $su(4)$ generators as
$\sigma_{ab}\equiv\frac{1}{2}\sigma_{a}\otimes\sigma_{b}$, with $a,\
b=0,1,2,3$ but $a$ and $b$ not simultaneously zero. Here $\sigma_{0}=I_{2}$
and $\sigma_{1,2,3}$ are the standard Pauli matrices. The correspondence
between the $su(4)$ and $so(6)$ generators are given as follows

$\displaystyle\begin{split}&\sigma_{01}\leftrightarrow T_{16},\
\sigma_{02}\leftrightarrow T_{62},\ \sigma_{03}\leftrightarrow T_{12},\\\
\sigma_{10}\leftrightarrow T_{54},\ &\sigma_{11}\leftrightarrow T_{32},\
\sigma_{12}\leftrightarrow T_{31},\ \sigma_{13}\leftrightarrow T_{63},\\\
\sigma_{20}\leftrightarrow T_{53},\ &\sigma_{21}\leftrightarrow T_{24},\
\sigma_{22}\leftrightarrow T_{14},\ \sigma_{23}\leftrightarrow T_{46},\\\
\sigma_{30}\leftrightarrow T_{34},\ &\sigma_{31}\leftrightarrow T_{25},\
\sigma_{32}\leftrightarrow T_{51},\ \sigma_{33}\leftrightarrow
T_{56}\end{split}$ (140)

where $(T_{ij})_{kl}=i(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$ is a
6-by-6 matrix, which generates rotations on the $(i,j)$-plane. One can
explicitly check that the above correspondence is indeed a homomorphism
between the $su(4)$ and $so(6)$ algebras.

##  Appendix F $I^{(N)}$ anomalies of SL(N)

In this appendix, we present the details of the monopole-based approach to the
anomalies associated with the $I^{(N)}$ symmetry, where $I^{(N)}=(SO(N)\times
SO(N-4))/Z_{2}$ for even $N$ and $I^{(N)}=SO(N)\times SO(N-4)$ for odd $N$.
Our strategy is to consider $(3+1)$-d bosonic $I^{(N)}$-SPTs that are also
compatible with the discrete $\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$
symmetries, gauge the $I^{(N)}$ symmetry, and use the statistics and quantum
numbers of the fundamental $I^{(N)}$-monopoles of the resulting gauge theory
to characterize the SPT we start with. This also gives us a characterization
and classification of the $I^{(N)}$-anomalies of the $(2+1)$-d theories. Note
that this is not a full classification of the anomalies associated with both
$I^{(N)}$ and the discrete symmetries, and such a full classification is
expected to be a more refined version of the one presented here.

The main results are as follows.

  1. 1.

If $N=2\ ({\rm mod\ }4)$, the SPTs or anomalies are classified into a
$\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{4}$ structure. The fundamental monopoles
of the root states are given by (148).

  2. 2.

If $N=0\ ({\rm mod\ }4)$, the SPTs or anomalies are classified into a
$\mathbb{Z}_{2}^{3}\times\mathbb{Z}_{4}$ structure, i.e., it has one more
$\mathbb{Z}_{2}$ factor compared to the case with $N=2\ ({\rm mod\ }4)$. In
addition to fundamental monopoles of the types given in (148), this additional
$\mathbb{Z}_{2}$ factor corresponds to one more possible type of the
fundamental monopole, given in (151).

  3. 3.

If $N$ is odd, the SPTs or anomalies are classified into a $Z_{2}^{5}$
structure. The fundamental monopoles of root states are listed in Table 5.

  4. 4.

In all these cases, the statistics and quantum numbers of the $SO(N)$ and
$SO(N-4)$ monopoles of the root states are derived, given by Table 2 for even
$N$ and Table 5 for odd $N$.

  5. 5.

In all these cases, we identify anomalies corresponding to theories that are
compatible with the cascade structure of the SLs, as discussed in Sec. IV.3.
In particular, two conditions need to be satisfied:

    1. (a)

If the symmetry is broken to $SO(5)$, we can consider the $SO(5)$ monopole of
the resulting theory. An $SO(5)$ monopole breaks the $SO(5)$ symmetry to
$SO(2)\times SO(3)$. For a SL, the $SO(5)$ monopole of the resulting theory
should carry no charge under the $SO(2)$ but a spinor representation under
$SO(3)$.

    2. (b)

If the symmetry of a SL is broken to $(SO(4)\times SO(N-4))/Z_{2}$, the
resulting theory should have no anomaly.

For even $N$, we find that only root 3 in (148) and its inverse satisfy both
conditions. For odd $N$, there is a single anomaly class that satisfies both
conditions, as discussed at the end of Appendix F.2.

###  F.1 The case with an even $N$

We start the discussion with the case with an even $N$, as in this case
results not captured by Eq. (35) may arise.

As in the main text, we write the $SO(N)$ and $SO(N-4)$ gauge fields as
$A^{SO(N)}=A_{a}^{L}T_{a}^{L}$ and $A^{SO(N-4)}=A_{a}^{R}T_{a}^{R}$,
respectively, where $\\{T^{L}\\}$ and $\\{T^{L}\\}$ form the generators of
$SO(N)$ and $SO(N-4)$, respectively. For even $N$, the field configuration of
a fundamental monopole will be taken as

$\displaystyle A_{12}^{L}=A_{34}^{L}=A_{56}^{L}=\cdots
A_{N-1,N}^{L}=A_{12}^{R}=A_{34}^{R}=A_{56}^{R}=\cdots
A_{N-5,N-4}^{R}=\frac{A_{U(1)}}{2}$ (141)

where $A_{ij}^{L}$ ($A_{ij}^{R}$) is the gauge field corresponding to the
generator associated with rotations on the $(i,j)$-plane of $SO(N)$
($SO(N-4)$) symmetry, and $A_{U(1)}$ is the field configuration of a unit
monopole in a $U(1)$ gauge theory, which can taken to be of the form in Ref.
Wu and Yang (1975). Namely, this monopole is obtained by embedding many
half-$U(1)$-monopoles into the maximal Abelian group of $I^{(N)}$. The
configuration of such a monopole breaks the gauge symmetry from $I^{(N)}$ to
$(SO(2)^{N-2})/Z_{2}$. So it is convenient to denote a general excitation in
this $I^{(N)}$ gauge theory by the following excitation matrix:

$\displaystyle\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}=\left(\begin{array}[]{cccc|cccc}q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
m_{12}^{L}&m_{34}^{L}&\cdots&m_{N-1,N}^{L}&m_{12}^{R}&m_{34}^{R}&\cdots&m_{N-5,N-4}^{R}\end{array}\right)_{s}$
(146)

where the first (second) row represents the electric (magnetic) charges of
this excitation under $A_{ij}^{L,R}$, $s=0\ ({\rm mod\ }2)$ ($s=1\ ({\rm mod\
}2)$) represents that this excitation is a boson (fermion), and the vertical
line separates the charges related to the original $SO(N)$ and $SO(N-4)$
subgroups of $I^{(N)}$. The above fundamental monopole has
$\bm{m}=(\frac{1}{2},\frac{1}{2},\cdots,\frac{1}{2})$, and its $\bm{q}$ and
$s$ will characterize the corresponding SPT. Because the statistics of any
excitation can be unambiguously determined by its $\bm{q}$ and the statistics
of the fundamental monopole, later we will sometimes suppress the subscript
related to the statistics of this excitation.

In such a theory, the structures of the possible excitations are constrained
by the following conditions.

  1. 1.

The pure gauge charges are built up with bosons in the bifundamental
representation of $I^{(N)}$. That is, if $\bm{m}=0$, then all entries of
$\bm{q}$ are integers that add up to an even integer. An example of the
elementary pure gauge charge has $\bm{m}=0$ and $\bm{q}=(1,0,0,\cdots,0,0,1)$.

  2. 2.

The Dirac quantization condition for two excitations
$\left(\begin{array}[]{c}\bm{q}_{1}\\\ \bm{m}_{1}\end{array}\right)$ and
$\left(\begin{array}[]{c}\bm{q}_{2}\\\ \bm{m}_{2}\end{array}\right)$:
$\bm{q}_{1}\cdot\bm{m}_{2}-\bm{q}_{2}\cdot\bm{m}_{1}\in\mathbb{Z}$.

  3. 3.

If an excitation exists, its $\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$
partners also exist. We take the actions of these discrete symmetries on the
excitation given by (146) to be

$\displaystyle\begin{split}&\mathcal{C}:\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}\rightarrow\left(\begin{array}[]{cccc|cccc}-q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&-q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
-m_{12}^{L}&m_{34}^{L}&\cdots&m_{N-1,N}^{L}&-m_{12}^{R}&m_{34}^{R}&\cdots&m_{N-5,N-4}^{R}\end{array}\right)_{s}\\\
&\mathcal{R}:\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}\rightarrow\left(\begin{array}[]{cccc|cccc}-q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
m_{12}^{L}&-m_{34}^{L}&\cdots&-m_{N-1,N}^{L}&-m_{12}^{R}&-m_{34}^{R}&\cdots&-m_{N-5,N-4}^{R}\end{array}\right)_{s}\\\
&\mathcal{T}:\left(\begin{array}[]{c}\bm{q}\\\
\bm{m}\end{array}\right)_{s}\rightarrow\left(\begin{array}[]{cccc|cccc}q_{12}^{L}&q_{34}^{L}&\cdots&q_{N-1,N}^{L}&-q_{12}^{R}&q_{34}^{R}&\cdots&q_{N-5,N-4}^{R}\\\
-m_{12}^{L}&-m_{34}^{L}&\cdots&-m_{N-1,N}^{L}&m_{12}^{R}&-m_{34}^{R}&\cdots&-m_{N-5,N-4}^{R}\end{array}\right)_{s}\end{split}$
(147)

  4. 4.

The remaining $(SO(2)^{N-2})/Z_{2}$ has a normalizer subgroup in $I^{(N)}$. If
an excitation exists, its partners under the actions of the normalizer
subgroup also exist.

These are necessary conditions for a theory to be consistent, and we believe
they are also sufficient.

The above conditions impose strong constraints on the possible $\bm{q}$ of a
fundamental monopole. First, there is an element in the normalizer subgroup
associated with the remaining $(SO(2)^{N-2})/Z_{2}$, whose action is to
exchange the first two columns of the excitation matrix. Applying this
operation to the fundamental monopole yields a normalizer-partner of it. The
bound state of this normalizer-partner and the anti-particle of the
fundamental monopole has all entries in the excitation matrix being $0$,
except that the first two entries in the first row are
$\pm(q_{12}^{L}-q_{34}^{L})$. Because this is a pure gauge charge,
$q_{12}^{L}=q_{34}^{L}\ ({\rm mod\ }1)$. Similarly, it is easy to see that the
first $N/2$ entries in $\bm{q}$ of the fundamental monopoles are all equal mod
1, and the last $(N-4)/2$ entries in $\bm{q}$ are also all equal mod 1.
Second, it is always possible to attach the fundamental monopole with some
pure gauge charge, such that all its $\bm{q}$-entries are in the interval
$(-1,1]$. These two observations imply that we can always write a fundamental
monopole as
$\left(\begin{array}[]{cccc|cccc}q^{L}&q^{L}&\cdots&q^{L}&q^{R}&q^{R}&\cdots&q^{R}\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{s}$
if $N=2\ ({\rm mod\ }4)$, while if $N=0\ ({\rm mod\ }4)$, besides this
possibility, there is one more possible type:
$\left(\begin{array}[]{cccc|cccc}q^{L}&q^{L}&\cdots&q^{L}&q^{R}&q^{R}&\cdots&q^{R}+1\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{s}$.

####  F.1.1 The case with $N=2\ ({\rm mod\ }4)$

Now let us focus on the case with $N=2\ ({\rm mod\ }4)$. The case with $N=6$
needs some special treatment, so we defer the discussion on it for a moment.
If $N>6$, the 4-particle bound state of the fundamental monopole and its
$\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$ parnters is a pure gauge charge
with excitation matrix
$\left(\begin{array}[]{ccccc|ccccc}0&4q^{L}&4q^{L}&\cdots&4q^{L}&0&4q^{R}&4q^{R}&\cdots&4q^{R}\\\
0&0&0&\cdots&0&0&0&0&\cdots&0\end{array}\right)$. This implies that
$4q^{L,R}\in\mathbb{Z}$. The Dirac quantization condition on the fundamental
monopole and its $\mathcal{T}$-partner implies
$\frac{Nq^{L}+(N-8)q^{R}}{2}\in\mathbb{Z}$. So $q^{L}+q^{R}\in\mathbb{Z}$. Now
it is straightforward to see that there are only two elementary possibilities
of $(q^{L},q^{R})$, i.e., $(q^{L},q^{R})=(0,1)$ and
$(q^{L},q^{R})=(\frac{1}{4},-\frac{1}{4})$, and these possibilities are
elementary in the sense that all other possibilities can be obtained from them
by forming bound states of them and/or attaching pure gauge charges, i.e.,
they can be taken as the monopoles of the root states. So far we have not
considered the statistics of the fundamental monopole, and it can actually be
either bosonic or fermionic. These results suggest a
$\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{4}$ classification of the fundamental
monopoles, and the roots can be taken to be

$\displaystyle\begin{split}&{\rm root\
}1:\left(\begin{array}[]{cccc|cccc}0&0&\cdots&0&0&0&\cdots&0\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{f}\\\
&{\rm root\ }2:\left(\begin{array}[]{cccc|cccc}0&0&\cdots&0&0&0&\cdots&1\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}\\\
&{\rm root\
}3:\left(\begin{array}[]{cccc|cccc}\frac{1}{4}&\frac{1}{4}&\cdots&\frac{1}{4}&-\frac{1}{4}&-\frac{1}{4}&\cdots&-\frac{1}{4}\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}\end{split}$
(148)

Notice that in writing root 2, we have attached pure gauge charge to it. One
can check that these roots satisfy all 4 conditions listed at the beginning of
this subsection.

The above results also apply to the case with $N=6$, but the argument needs to
be slightly modified. For $N=6$, these 4 excitations exist:
$\left(\begin{array}[]{ccc|c}-q^{L}&-q^{L}&-q^{L}&q^{R}\\\
\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}\end{array}\right)_{s}$,
$\left(\begin{array}[]{ccc|c}q^{L}&-q^{L}&q^{L}&q^{R}\\\
-\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\end{array}\right)_{s}$,
$\left(\begin{array}[]{ccc|c}-q^{L}&-q^{L}&q^{L}&q^{R}\\\
-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{array}\right)_{s}$, and
$\left(\begin{array}[]{ccc|c}q^{L}&-q^{L}&-q^{L}&q^{R}\\\
\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\end{array}\right)$. The
4-particle bound state of these 4 excitations is
$\left(\begin{array}[]{ccc|c}0&-4q^{L}&0&4q^{R}\\\
0&0&0&0\end{array}\right)_{b}$, which means in this case we also have
$4q^{L,R}\in\mathbb{Z}$. Furthermore, the Dirac quantization condition for a
fundamental monopole and its $\mathcal{T}$-partner gives
$-3q^{L}+q^{R}\in\mathbb{Z}$. These conditions still suggest a
$Z_{2}^{2}\times Z_{4}$ classification, and the roots can still be taken as
the ones in (148).

It is useful to derive the structure of the $SO(N)$ and $SO(N-4)$ monopoles
from these fundamental monopoles. An $SO(N)$ monopole can be viewed as the
2-particle bound state of the fundamental monopole and its $\mathcal{R}$
partner:
$\left(\begin{array}[]{cccc|ccc}0&2q^{L}&\cdots&2q^{L}&2q^{R}&\cdots&2q^{R}\\\
1&0&\cdots&0&0&\cdots&0\end{array}\right)_{q_{L}+q_{R}}$. The $SO(N-4)$
monopole can be viewed as the 2-particle bound state of the fundamental
monopole and its $\mathcal{T}$ partner:
$\left(\begin{array}[]{ccc|cccc}2q^{L}&\cdots&2q^{L}&0&2q^{R}&\cdots&2q^{R}\\\
0&\cdots&0&1&0&\cdots&0\end{array}\right)_{q_{L}-3q_{R}}$. In physical terms,
the properties of these monopoles are summarized in Table 2.

|  $SO(N)$ monopole |  $SO(N-4)$ monopole
---|---|---
root 1 | (singlet, singlet, boson) | (singlet, singlet, boson)
root 2 | (singlet, singlet, fermion) | (singlet, singlet, fermion)
root 3 | (spinor, spinor, boson) | (spinor, spinor, fermion)
root 4 with $N=0\ ({\rm mod\ }8)$ | (singlet, singlet, fermion) | (singlet, vector, boson)
root 4 with $N=4\ ({\rm mod\ }8)$ | (singlet, singlet, boson) | (singlet, vector, fermion)
Table 2: Properties of the $SO(N)$ and $SO(N-4)$ monopoles of the root states
for even $N$. The first three roots apply to all even $N$, and root 4 only
applies to the case with $N$ an integral multiple of 4. The $SO(N)$ monopole
breaks the $I^{(N)}$ symmetry to $(SO(2)\times SO(N-2)\times SO(N-4))/Z_{2}$,
and it always has no charge under the $SO(2)$. Its three corresponding entries
represent its representation under the $SO(N-2)$, its representation under the
$SO(N-4)$, and its statistics, respectively. The $SO(N-4)$ monopole breaks the
$I^{(N)}$ symmetry to $(SO(N)\times SO(N-6)\times SO(2))/Z_{2}$, and it always
has no charge under the $SO(2)$. Its three corresponding entries represent its
representation under the $SO(N)$, its representation under the $SO(N-6)$, and
its statistics, respectively. For the case with $N=6$, the second entry does
not exist for its $SO(N-4)$ monopole. Notice these properties are determined
up to attaching pure gauge charges.

Let us check which anomalies correspond to states that satisfy the two
conditions listed at the beginning of this appendix. In order for the first
condition to be satisfied, according to Table 2, the state must contains the
anomaly corresponding to root 3 or its inverse.

It is a bit more complicated to check the second condition, but it actually
suffices to check a weaker condition: if the remaining $SO(N-4)$ symmetry is
further broken to $SO(2)^{\frac{N}{2}-2}$, the system is anomaly-free. To this
end, let us condense
$\left(\begin{array}[]{cccccc|cccc}1&1&\cdots&1&0&0&-1&-1&\cdots&-1\\\
0&0&\cdots&0&0&0&0&0&\cdots&0\end{array}\right)_{b}$. This condensate breaks
the $I^{(N)}$ symmetry into $(SO(2)^{\frac{N}{2}-2}\times SO(4))/Z_{2}$, and
the gauge fields corresponding to the remaining $\frac{N}{2}-2$ $SO(2)$
symmetries can be taken as
$A_{12}^{\prime}=\frac{1}{2}(A_{12}^{L}+A_{12}^{R}),A_{34}^{\prime}=\frac{1}{2}(A_{34}^{L}+A_{34}^{R}),\cdots,A_{N-5,N-4}^{\prime}=\frac{1}{2}(A_{N-5,N-4}^{L}+A_{N-5,N-4}^{R})$,
and for the $SO(4)\backsimeq\frac{SU(2)\times SU(2)}{Z_{2}}$, we denote the
gauge fields corresponding to these two $SU(2)$ subgroups by $B_{1,2}$, which
together form the $SO(4)$ gauge field.

The fundamental monopole remains deconfined in this condensate and becomes the
fundamental monopole of the resulting theory. In terms of the remaining
symmetries, it should be written as
$\left(\begin{array}[]{cccc|cc}q^{L}+q^{R}&q^{L}+q^{R}&\cdots&q^{L}+q^{R}&q^{L}&q^{L}\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{array}\right)_{s}$.
By adding to the theory a proper theta-term of the $SO(4)$ gauge field,
$\theta\epsilon_{\mu\nu\lambda\rho}{\rm
Tr}(\partial^{\mu}B_{1}^{\nu}\partial^{\nu}B_{1}^{\rho}-\partial^{\mu}B_{2}^{\nu}\partial^{\lambda}B_{2}^{\rho})$,
which preserves all remaining symmetries (including the remaining
$\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$ symmetries), this fundamental
monopole can be converted to
$\left(\begin{array}[]{cccc|cc}q^{L}+q^{R}&q^{L}+q^{R}&\cdots&q^{L}+q^{R}&0&0\\\
\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{array}\right)_{s}$.
In order for the theory to be anomaly-free, this monopole should be a boson
with trivial projective quantum numbers. This means $s=0\ ({\rm mod\ }2)$ and
$q^{L}+q^{R}=0\ ({\rm mod\ }2)$. Therefore, only root 3 and multiple copies of
it satisfy this condition.

To satisfy both conditions, the only possibilities are root 3 and its inverse.
Because the SLs with $(N,\pm 1)$ satisfy both conditions, we conclude that the
anomalies of these SLs are precisely the same as root 3 and its inverse.
Notice that from this analysis we cannot determine which of $(N,\pm 1)$
corresponds to root 3, and which corresponds to its inverse.

Now that we have identified the SLs with $(N,\pm 1)$ as states that realize
the anomalies of root 3 and its inverse, it may also be worth mentioning which
states realize the anomalies of the other 2 roots.

The $I^{(N)}$-anomaly of root 1 can be realized by a $Z_{2}$ topological order
(TO). Denote the $Z_{2}$ charge and flux by $e$ and $m$, respectively, the
symmetry actions on these topological sectors are given in Table 3.

| $e$ | $m$
---|---|---
$I^{(N)}$ | (singlet, vector) | (singlet, vector)
$\mathcal{C}$ | $e$ | $m$
$\mathcal{R}$ | $e$ | $m$
$\mathcal{T}$ | $e$ | $m$
Table 3: Projective symmetry actions on the topological sectors of the $Z_{2}$
TO that realizes the $I^{(N)}$ anomaly of root 1. In the row corresponding to
$I^{(N)}$, the two entries represent the representations of this excitation
under the $SO(N)$ and $SO(N-4)$ subgroups of $I^{(N)}$, respectively. When
$N=6$, the vector representation of the $SO(2)\subset I^{(6)}$ means charge 1
under $SO(2)$. Their partners under $\mathcal{C}$, $\mathcal{R}$ and
$\mathcal{T}$ are shown as above. Notice one can further specify data like
$T^{2}$ for $e$ and $m$, but it is unnecessary for our purpose.

To see that this $Z_{2}$ TO realizes the $I^{(N)}$-anomaly of root 1, consider
threading a fundamental monopole through the system, which leaves a flux.
Because of the above symmetry assignment, when an $e$ or $m$ circles around
this flux, it acquires a phase factor $-1$, no matter how far it is away from
the flux. Since this monopole threading process is local for the $(2+1)$-d
system, this $-1$ phase factor has to be cancelled by requiring that the flux
also trap an anyon that has $-1$ mutual braiding with both $e$ and $m$. This
anyon is $\epsilon$, the fermionic bound state of $e$ and $m$. Furthermore,
time reversal symmetry ensures that no polarization charge is induced around
this flux. Then the composite of this flux and $\epsilon$ is a fermion in the
trivial representation of $I^{(N)}$. Therefore, the fundamental monopole of
this $Z_{2}$ TO has precisely the structure as in root 1.

This $Z_{2}$ TO can be explicitly constructed using the layer construction in
Ref. Wang and Senthil (2013), and it is an analog of $eCmC$, the surface state
of a $(3+1)$-d bosonic topological insulator protected by $U(1)$ charge
conservation and time reversal Vishwanath and Senthil (2013); Wang and Senthil
(2013).

The $I^{(N)}$-anomaly of root 2 can be realized by a $Z_{4}\times Z_{2}$ TO.
Denote the $Z_{4}$ charge and flux by $e_{1}$ and $m_{1}$, and the $Z_{2}$
charge and flux by $e_{2}$ and $m_{2}$, respectively. We take the convention
such that the mutual statistics between $e_{1}$ and $m_{1}$ is $i$ if $N=6\
({\rm mod\ }8)$, and $-i$ if $N=2\ ({\rm mod\ }8)$. The symmetry actions on
these topological sectors are given in Table 4.

| $e_{1}$ | $m_{1}$ | $e_{2}$ | $m_{2}$ | $\epsilon_{2}\equiv e_{2}m_{2}$
---|---|---|---|---|---
$I^{(N)}$ | (singlet, fund.) | (singlet, anti-fund.) | (anti-fund., anti-fund.) | (fund., fund.) | (singlet, singlet)
$\mathcal{C}$ | $e_{1}^{-1}$ | $m_{1}^{-1}$ | $e_{2}^{-1}$ | $m_{2}^{-1}$ | $\epsilon_{2}$
$\mathcal{R}$ | $e_{1}$ | $m_{1}^{-1}e_{1}^{-2}\epsilon_{2}$ | $e_{2}^{-1}e_{1}^{-2}$ | $m_{2}^{-1}e_{1}^{2}$ | $\epsilon_{2}$
$\mathcal{T}$ | $e_{1}^{-1}$ | $m_{1}e_{1}^{2}\epsilon_{2}$ | $e_{2}e_{1}^{2}$ | $m_{2}e_{1}^{-2}$ | $\epsilon_{2}$
Table 4: Projective symmetry actions on the topological sectors of the
$Z_{4}\times Z_{2}$ TO that realizes the $I^{(N)}$ anomaly of root 2. In the
row corresponding to $I^{(N)}$, the two entries in each parenthesis represent
the representations of this excitation under the associated spin group of the
correponding $SO(N)$ and $SO(N-4)$ subgroups of $I^{(N)}$, where “fund.”
(“anti-fund.”) represents fundamental (anti-fundamental) representation of the
relevant spin group (when $N=6$, the fund. (anti-fund.) in the second entry in
the parenthesis means charge $1/2$ ($-1/2$) under $SO(2)\subset I^{(6)}$).
Their partners under $\mathcal{C}$, $\mathcal{R}$ and $\mathcal{T}$ are shown
as above. Notice one can further specify data like $T^{2}$ for various anyons,
but it is unnecessary for our purpose.

To see that this theory realizes the $I^{(N)}$-anomaly of root 2, we can again
consider threading a fundamental monopole through the system and use a similar
argument as before. One can check that the flux left by the fundamental
monopole will trap an anyon $e_{1}m_{1}^{-1}$, and this fundamental monopole
is indeed a boson in the vector representation under $SO(N-4)$, so we conclude
that this theory realizes the $I^{(6)}$-anomaly of root 2.

We believe this $Z_{4}\times Z_{2}$ TO with the above symmetry implementation
is a consistent theory, although we do not have an explicit construction for
it.

####  F.1.2 The case with $N=0\ ({\rm mod\ }4)$

Next we turn to the case with $N=0\ ({\rm mod\ }4)$. From similar analysis as
before, now the constraints we obtain are $4q^{L,R}\in\mathbb{Z}$ and
$2(q^{L}+q^{R})\in\mathbb{Z}$. The three roots in (148) still satisfy these
constraints, but we find one additional root 191919This root is absent in the
case with $N=2\ ({\rm mod\ }4)$ because $q_{L}+q_{R}\in\mathbb{Z}$ in that
case, which is violated here.:

$\displaystyle{\rm root\
}4:\left(\begin{array}[]{ccc|cccc}0&\cdots&0&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\\\
\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}\end{array}\right)_{b}$
(151)

Similar as before, we can also derive the properties of the $SO(N)$ and
$SO(N-4)$ monopoles for this root, and the results are given in Table 2. From
this Table, we observe that there are not only differences in the anomalies
for the cases with $N=2\ ({\rm mod\ }4)$ and $N=0\ ({\rm mod\ }4)$, but also
differences in the anomalies for the cases with $N=0\ ({\rm mod\ }8)$ and
$N=4\ ({\rm mod\ }8)$. It is known that the spinor representations of $SO(N)$
in these three cases are different, i.e., they are complex, real, and
pseudoreal for $N=2\ ({\rm mod\ }4)$, $N=0\ ({\rm mod\ }8)$ and $N=4\ ({\rm
mod\ }8)$, respectively Zee (2016). Our analysis suggests a connection between
these two results.

Lastly, analogous arguments as before indicate that only root 3 and its
inverse satisfy both conditions discussed at the beginning of this appendix.
Because SLs with $(N,\pm 1)$ also satisfy those two conditions, we conclude
that these SLs have the same anomaly as root 3 and its inverse.

###  F.2 The case with an odd $N$

Now we turn to the case with odd $N$, so $I^{(N)}=SO(N)\times SO(N-4)$. Notice
this analysis also applies to the case with an even $N$ if we add to the
system bosonic DOF in the vector representation of $SO(N)$.

In this case, the fundamental monopoles are simply the usual $SO(N)$ and
$SO(N-4)$ monopoles, so these two monopoles will characterize the $I^{(N)}$
SPTs and anomalies.

An $SO(N)$ monopole breaks $I^{(N)}$ into $SO(2)\times SO(N-2)\times SO(N-4)$.
This monopole can be denoted as
$\left(\begin{array}[]{cc|c}q_{N}&r_{N}^{L}&r_{N}^{R}\\\
1&0&0\end{array}\right)_{s_{N}}$, where $q_{N}$ represents the fractional
charge under the remaining $SO(2)$, $r_{N}^{L}$ represents the projective
quantum number under the remaining $SO(N-2)$, $r_{N}^{R}$ represents the
projective quantum number under $SO(N-4)$, and $s_{N}$ is the statistics of
this monopole. Similarly, an $SO(N-4)$ monopole can be denoted by
$\left(\begin{array}[]{c|cc}r_{N-4}^{L}&q_{N-4}&r_{N-4}^{R}\\\
0&1&0\end{array}\right)_{s_{N-4}}$. Just as in the usual $(3+1)$-d bosonic
topological insulator, time reversal symmetry and the bosonic statistics of
the pure gauge charge require that $q_{N}=q_{N-4}=0$ Vishwanath and Senthil
(2013). Furthermore, the Dirac quantization condition requires that whenever
the $SO(N)$ monopole carries a spinor representation under $SO(N-4)$, the
$SO(N-4)$ monopole must also carry a spinor representation under $SO(N)$, and
vice versa. These are all the constraints, and we get a $\mathbb{Z}_{2}^{5}$
classification of the $I^{(N)}$ SPTs or anomalies, and the structures of the
monopoles of the 5 root states are in Table 5.

|  $SO(N)$ monopole |  $SO(N-4)$ monopole | topological response function
---|---|---|---
root 1 | (singlet, singlet, fermion) | (singlet, singlet, boson) | $(w_{2}^{SO(N)})^{2}$
root 2 | (singlet, singlet, boson) | (singlet, singlet, fermion) | $(w_{2}^{SO(N-4)})^{2}$
root 3 | (spinor, singlet, boson) | (singlet, singlet, boson) | $w_{4}^{SO(N)}$
root 4 | (singlet, singlet, boson) | (singlet, spinor, boson) | $w_{4}^{SO(N-4)}$
root 5 | (singlet, spinor, boson) | (spinor, singlet, boson) | $w_{2}^{SO(N)}w_{2}^{SO(N-4)}$
Table 5: Properties of the $SO(N)$ and $SO(N-4)$ monopoles of the root states
for odd $N$. The $SO(N)$ monopole breaks the $I^{(N)}$ symmetry to
$SO(2)\times SO(N-2)\times SO(N-4)$, and it always has no fractional charge
under the $SO(2)$. Its three corresponding entries represent its
representation under the $SO(N-2)$, its representation under the $SO(N-4)$,
and its statistics, respectively. The $SO(N-4)$ monopole breaks the $I^{(N)}$
symmetry to $SO(N)\times SO(N-6)\times SO(2)$, and it always has no charge
under the $SO(2)$. Its three corresponding entries represent its
representation under the $SO(N)$, its representation under the $SO(N-6)$, and
its statistics, respectively. The last column lists the topological response
function corresponding to these root states.

Using similar arguments as before, one can show that only one of the
$2^{5}=32$ anomaly classes satisfies both conditions discussed at the
beginning of this appendix, which is (root 2)$\otimes$(root 3)$\otimes$(root
4)$\otimes$(root 5), i.e., a composed system made of root 2, root 3, root 4
and root 5. This anomaly class is identified with the one for the SLs, and
this result agrees with Eq. (35).

##  Appendix G Explicit calculations of the $U(1)$ DSL

In this appendix, we explicitly derive the properties of the fundamental
$I^{(6)}$-monopole in a $U(1)$ DSL, whose effective theory is given by Eq.
(4). We will see that this fundamental $I^{(6)}$-monopole has precisely the
structure of root 3 in Eq. (44), which further strengthens our proposal that
the $U(1)$ DSL and SL(6) are equivalent. Our results also agree with a more
formal calculation in Ref. Calvera and Wang (2021).

Recall that we can take the Dirac fermions in a DSL to be in either the
fundamental or anti-fundamental representation of $SU(4)$, and take the
monopole of $a$ to have either charge $1$ or $-1$ under $U(1)_{\rm top}$, so
there are 4 different choices of the symmetry implementation. To be general,
we will consider the 4 cases together by introducing parameters $\zeta$ and
$\xi$, such that $\zeta=1$ ($\zeta=-1$) if the Dirac fermions are in the
fundamental (anti-fundamental) representation of $SU(4)$, and $\xi=\pm 1$ if
the monopole of $a$ carries charge $\pm 1$ under $U(1)_{\rm top}$202020Since
we can redefine the theory through a charge conjugation
$\tilde{\psi}=\psi^{\dagger}$, $\tilde{a}=-a$, the two signs $\zeta$ and $\xi$
can be flipped simultaneously without physical effect. Only the product
$\zeta\xi$ will eventually matter in the following discussions..

Next we will calculate the $q^{L}$, $q^{R}$ and $s$ of the fundamental
$I^{(6)}$-monopole for a DSL. In particular, we will thread a flux
corresponding to the fundamental $I^{(6)}$-monopole in Eq. (38), which has a
$\pi$-flux for $A_{12}$, $A_{34}$, $A_{56}$ and $A_{\rm top}$.

It is useful to denote the 4 flavors of Dirac fermions by $\psi_{\uparrow+}$,
$\psi_{\uparrow-}$, $\psi_{\downarrow+}$, and $\psi_{\downarrow-}$,
respectively. This notation is motivated by the lattice realizations of a DSL,
where $\uparrow$ and $\downarrow$ represent two physical spins, and $+$ and
$-$ represent the two valleys. According to the homomorphism between $su(4)$
and $so(6)$ in Appendix E, the charges under $(A_{12},A_{34},A_{56})$ carried
by $\psi_{\uparrow+}$, $\psi_{\uparrow-}$, $\psi_{\downarrow+}$ and
$\psi_{\downarrow-}$ are respectively $\zeta(1/2,1/2,1/2)$,
$\zeta(1/2,-1/2,-1/2)$, $\zeta(-1/2,1/2,-1/2)$ and $\zeta(-1/2,-1/2,1/2)$.
Note that the Dirac fermions are neutral under $A_{\rm top}$.

When the flux specified above is thread, $\psi_{\uparrow+}$ sees a total
$3\zeta\pi/2$ flux, while the other 3 flavors of Dirac fermions see a total
$-\zeta\pi/2$ flux. To construct a gauge invariant state corresponding to the
local fundamental $I^{(6)}$-monopole, one can consider a state where the
internal gauge field, $a$, has a flux of $\zeta\pi/2$, such that at the end
$\psi_{\uparrow+}$ sees a $2\zeta\pi$ flux and contributes a zero mode in this
flux background, and the other 3 flavors see no flux and contribute no zero
mode. Because there is a single zero mode in the background of a flux with
magnitude $2\pi$, no matter it is occupied or not, we get a bosonic state, so
$s=b$ for the fundamental $I^{(6)}$ monopole in Eq. (38). To determine
$q^{L,R}$ for this fundamental monopole, we need to determine whether this
zero mode is occupied or not.

The usual way to do this is to demand that the zero modes are half-filled.
However, that works only if the theory has a symmetry that preserves the flux
but flips the charge. In the present case, there is no such a symmetry, so a
different approach should be taken. To proceed, we will regularize the DSL as
follows. First, to preserve the $SU(4)$ flavor symmetry, for each flavor we
add to the system a gapped Dirac fermion, which contributes to the effective
action a term $-\pi\eta/2$ when combined with the original gapless Dirac
fermion with the same flavor, where $\eta$ is the $\eta$-invariant of the
Dirac operator corresponding to each flavor of gapless and gapped Dirac
fermions Witten (2016). These $\eta$-invariants will generally break the
$\mathcal{T}$ symmetry. To maintain the $\mathcal{T}$ symmetry, next we put
the system on the boundary of a $(3+1)$-d bulk that contributes an appropriate
theta-term to the bulk partition function, such that the combined partition
function of the boundary and bulk is $\mathcal{T}$ invariant. This particular
regularization of the theory should suffice to yield $q^{L,R}$ of the
fundamental monopole in Eq. (38), up to attaching local DOF.

With this regularization, the effective action212121More precisely the
partition function is $Z=|\det(\not{D})|\exp(iS_{\rm{eff}})$ in Euclidean
signature Witten (2016). is

$\displaystyle S_{\rm eff}=$
$\displaystyle\sum_{i=1}^{4}\left[-\frac{\pi}{2}\eta(a_{i})+\frac{1}{2}\frac{1}{4\pi}\int
d^{3}xa_{i}da_{i}\right]+\frac{\xi}{2\pi}\int d^{3}xA_{\rm top}da$ (152)

with

$\displaystyle\begin{split}&a_{1}=a+\frac{\zeta(A_{12}+A_{34}+A_{56})}{2}\\\
&a_{2}=a+\frac{\zeta(A_{12}-A_{34}-A_{56})}{2}\\\
&a_{3}=a+\frac{\zeta(-A_{12}+A_{34}-A_{56})}{2}\\\
&a_{4}=a+\frac{\zeta(-A_{12}-A_{34}+A_{56})}{2}\end{split}$ (153)

and $i=1,2,3,4$ correspond to contributions from $\uparrow+$, $\uparrow-$,
$\downarrow+$, $\downarrow-$, respectively. Notice that the second term in the
above effective action comes from reducing the theta-term of the $(3+1)$-d
bulk to the boundary. Furthermore, the coefficient of the CS term of the
dynamical gauge field $a$ is $1/(2\pi)$, which is well defined in $(2+1)$-d
and implies that the $(3+1)$-d bulk does not really need a dynamical gauge
field, consistent with the general expectation that a theory with a ’t Hooft
anomaly can live on the boundary of a short-range entangled bulk (see, e.g.,
Ref. Ning  _et al._ (2020) for examples of theories that have more severe
anomalies than a ’t Hooft anomaly and thus can only live on the boundary of a
long-range entangled bulk). Also notice that all dependence on the metric of
the spacetime manifold of the system is suppressed, which will not affect our
following analysis.

This effective action can be used to read off the resulting charges under
various gauge fields when the flux is thread:

$\displaystyle\begin{split}&Q_{a}=N_{\psi_{\uparrow+}}+\frac{B_{a_{1}}+B_{a_{2}}+B_{a_{3}}+B_{a_{4}}}{4\pi}+\frac{\xi
B_{\rm top}}{2\pi}=N_{\psi_{\uparrow+}}+\frac{B_{a}}{\pi}+\frac{\xi B_{\rm
top}}{2\pi}\\\
&Q_{12}=\zeta\left(\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{a_{1}}+B_{a_{2}}-B_{a_{3}}-B_{a_{4}}}{8\pi}\right)=\zeta\cdot\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{12}}{4\pi}\\\
&Q_{34}=\zeta\left(\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{a_{1}}-B_{a_{2}}+B_{a_{3}}-B_{a_{4}}}{8\pi}\right)=\zeta\cdot\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{34}}{4\pi}\\\
&Q_{56}=\zeta\left(\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{a_{1}}-B_{a_{2}}-B_{a_{3}}+B_{a_{4}}}{8\pi}\right)=\zeta\cdot\frac{N_{\psi_{\uparrow+}}}{2}+\frac{B_{56}}{4\pi}\\\
&Q_{\rm top}=\frac{\xi B_{a}}{2\pi}\end{split}$ (154)

where $Q_{(\cdot)}$ and $B_{(\cdot)}$ represent the charge and flux under the
corresponding gauge field, respectively, and $N_{\psi_{\uparrow+}}$ determines
whether the zero mode contributed by $\psi_{\uparrow+}$ is occupied, i.e., if
the flux seen by the fermion is positive (negative), then
$N_{\psi_{\uparrow+}}=0$ means it is occupied (unoccupied).

According to the previous discussion, now we have $B_{12}=B_{34}=B_{56}=B_{\rm
top}=\pi$ and $B_{a}=\zeta\pi/2$. To be gauge invariant, $Q_{a}=0$, which
means $N_{\psi_{\uparrow+}}=-(\zeta+\xi)/2$. Substituting this into the rest
of the equations yields $Q_{12}=Q_{34}=Q_{56}=-\frac{\zeta\xi}{4}$ and $Q_{\rm
top}=\frac{\zeta\xi}{4}$. That is, if $(\zeta,\xi)=(1,-1)$ or
$(\zeta,\xi)=(-1,1)$, $q^{L}=-q^{R}=1/4$, corresponding to root 3 in Eq. (44).
If $(\zeta,\xi)=(1,1)$ or $(\zeta,\xi)=(-1,-1)$, $q^{L}=-q^{R}=-1/4$,
corresponding to the inverse of root 3 in Eq. (44). Therefore, the $I^{(N)}$
anomaly of the $U(1)$ DSL is indeed identical to that of the SL(6,±1), which
even further strengthens our proposal that they are dual.

##  Appendix H More on the LSM constraints

In the main text, we have used some physical arguments to propose that Eq.
(57) describes the complete set of LSM constraints for various lattice
systems. In this appendix, we extract some LSM contraints from Eq. (57) that
were not used in deriving Eq. (57). We will also give an alternative
expression for the LSM anomaly on a square lattice.

For convenience, we copy Eq. (57):

$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})[xy+c^{2}+r(x+c)]$ (155)

where $w_{2}^{SO(3)}$ is the second SW class of the $SO(3)$ gauge field
corresponding to the spin rotational symmetry, $t$ is the gauge field
corresponding to the time reversal symmetry, $x$ and $y$ are the gauge field
corresponding to translation along $T_{1}$ and $T_{2}$, respectively, $c$ is
the gauge field corresponding to $C_{2}$ site-centered lattice rotation, and
$r$ is the gauge field corresponding to the reflection symmetry $R_{y}$. There
is a constraint $r+t=w_{1}^{TM}\ ({\rm mod\ }2)$. This anomaly polynomial
should be viewed as a topological response function of the system under the
various gauge fields.

Physically, we expect that there will be LSM anomalies associated with
reflection symmetry $R_{x}$. To read them off, we need to design a
configuration of the gauge field corresponding to $R_{x}$ in Eq. (155).
Although the gauge field corresponding to $R_{x}$ does not explicitly appear
in Eq. (155), because $R_{x}$ is a combination of $C_{2}$ and $R_{y}$, we can
still have a gauge connection of $R_{x}$ by writing $c=c_{0}+r$. This means
that whenever there is a gauge connection corresponding to $R_{y}$, a gauge
connection corresponding to $C_{2}$ is also induced. Therefore, now $r$
actually represents a gauge connection corresponding to $R_{x}$, and $c_{0}$
is the gauge connection for pure $C_{2}$ rotation. Substituting $c=c_{0}+r$
into Eq. (155) yields

$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})[xy+c_{0}^{2}+r(x+c_{0})]$ (156)

The physical meaning of this new anomaly polynomial can be understood by
looking at various sub-symmetries of the system. For example, ignoring
translation symmetries, i.e., setting $x=y=0$, it becomes $S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})(c_{0}^{2}+rc_{0})$. The first term
in the second parenthesis, $c_{0}^{2}$, physically means that there is an LSM
anomaly if there is an odd number of spin-1/2’s at the $C_{2}$ center, just as
Eq. (54). The other term, $rc_{0}$, represents an LSM anomaly associated with
$R_{x}$ and $C_{2}$, if there is a $C_{2}$ center at the $R_{x}$-invariant
line and this $C_{2}$ center hosts an odd number of spin-1/2’s.

As another example, we can also ignore the $C_{2}$ symmetry by setting
$c_{0}=0$, and consider translations. On a triangular lattice, the
$R_{x}$-invariant line has a translation symmetry generated by
$T_{1}T_{2}^{2}$. The gauge field corresponding to such a translation symmetry
can be obtained by writing $y=y_{0}+2x$. Similar as above, now $x$ represents
the gauge field corresponding to $T_{1}T_{2}^{2}$, and $y_{0}$ represents the
gauge field corresponding to $T_{2}$. Substituting $c_{0}=0$ and $y=y_{0}+2x$
into Eq. (155) yields $S_{\rm
LSM}=i\pi\int_{X^{4}}(w_{2}^{SO(3)}+t^{2})(xy_{0}+rx)$. Now the first term in
the second parenthesis, $xy_{0}$, represents an LSM anomaly associated with
having an odd number of spin-1/2’s in each unit cell corresponding to the
translations $T_{2}$ and $T_{1}T_{2}^{2}$. The second term, $rx$, represents
an LSM anomaly associated with $R_{x}$ and $T_{1}T_{2}^{2}$, if there is an
odd number of spin-1/2’s in each unit cell of $T_{1}T_{2}^{2}$.

Similarly, we can also obtain the LSM anomaly associated with $R_{x}$ and
$R_{y}$. To do so, we can set $x=y=0$ and $r=c+r_{0}$ in Eq. (155). The first
of these two conditions amounts to ignoring translation symmetries, while the
second condition means that now $c$ is really a gauge field corresponding to
the $R_{x}$ symmetry, and $r_{0}$ is the gauge field corresponding to the
$R_{y}$ symmetry. Substituting these conditions into Eq. (155) yields $S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})cr_{0}$. This represents an LSM
anomaly if there is an odd number of spin-1/2’s at the intersecting point of
the reflection axes of $R_{x}$ and $R_{y}$. This LSM anomaly of course encodes
the one associated with the $C_{2}$ center. To see it formally, one way is to
further restrict $c=r_{0}$, which turns both $c$ and $r_{0}$ into the gauge
field corresponding to the $C_{2}$ rotation. Then this anomaly polynomial
becomes Eq. (54).

The above discussion motivates us to write the LSM anomaly on a square lattice
in terms of gauge fields corresponds to $T_{1,2}$, $R_{x,y}$, $SO(3)$ and
$\mathcal{T}$ symmetries. Denote the gauge fields corresponding to $R_{x,y}$
by $r_{x,y}$. Using an argument similar to that in the main text, the LSM
anomaly on a square lattice can be written as

$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})(x+r_{x})(y+r_{y})$ (157)

with a constraint $t+r_{x}+r_{y}=w_{1}^{TM}\ ({\rm mod\ }2)$. Note that this
expression is manifestly $C_{4}$ rotationally invariant.

To reproduce the Eq. (155), we want to write this anomaly in terms of gauge
fields of $R_{y}$, $C_{2}$, $T_{1,2}$, $SO(3)$ and $\mathcal{T}$. So we let
$r_{x}=c$ and $r_{y}=c+r$, then $c$ is the gauge field for $C_{2}$ and $r$ is
the gauge field for $R_{y}$. Then the above LSM anomaly precisely recover Eq.
(155), after using that $cx=cy=0$. It is interesting to note that in order to
derive Eq. (157) from Eq. (155), one needs to first replace the latter by

$\displaystyle S_{\rm
LSM}=i\pi\int_{X_{4}}(w_{2}^{SO(3)}+t^{2})[xy+c^{2}+r(x+c)+cy]$ (158)

Because $cy=0$, this expression should be equivalent to Eq. (155). Then by
setting $c=r_{x}$ and $r=r_{y}+r_{x}$ and using that $r_{x}x=r_{y}y=0$, this
anomaly becomes Eq. (157).

All the above results are consistent with the physical expectations. The
method employed above can also be readily applied to other situations to
extract other LSM anomalies.

##  Appendix I Anomaly matching of the $U(1)$ DSL on a triangular lattice

In this appendix we show that the $U(1)$ DSL on a triangular lattice indeed
has the correct LSM anomaly.

The symmetries we will focus on are $SO^{s}(3)$ spin rotation, time reversal
$\mathcal{T}$, translations $T_{\bm{a}_{1},\bm{a}_{2}}$, site-centered $C_{2}$
rotation, and reflection $R_{y}$ that keeps $\bm{a}_{1}$ invariant. Their
actions on the $U(1)$ DSL on a triangular lattice are Song  _et al._ (2020,
2019):

$\displaystyle\begin{split}&SO^{s}(3):\
n\rightarrow\left(\begin{array}[]{cc}I_{3}&\\\
&SO^{s}(3)\end{array}\right)n,\\\ &\mathcal{T}:\
n\rightarrow\left(\begin{array}[]{cc}I_{3}&\\\ &-I_{3}\end{array}\right)n,\\\
&T_{\bm{a}_{1}}:n\rightarrow\left(\begin{array}[]{cccc}-1&&&\\\ &1&&\\\
&&-1&\\\
&&&I_{3}\end{array}\right)n\exp\left(i\frac{2\pi}{3}\sigma_{y}\right),\\\
&T_{\bm{a}_{2}}:n\rightarrow\left(\begin{array}[]{cccc}1&&&\\\ &-1&&\\\
&&-1&\\\
&&&I_{3}\end{array}\right)n\exp\left(i\frac{2\pi}{3}\sigma_{y}\right),\\\
&C_{2}:n\rightarrow\left(\begin{array}[]{cc}I_{3}&\\\
&-I_{3}\end{array}\right)n\sigma_{z},\\\
&R_{y}:n\rightarrow\left(\begin{array}[]{cccc}&&-1&\\\ &1&&\\\ -1&&&\\\
&&&I_{3}\end{array}\right)n\end{split}$ (159)

From these symmetry actions, we get

$\displaystyle\begin{split}&w_{1}^{O(6)}=t+r+c,\ w_{1}^{O(2)}=c,\\\
&w_{2}^{O(6)}=xy+xr+rt+rc+c^{2}+w_{2}^{SO^{s}(3)}+t^{2},\ w_{2}^{O(2)}=0,\\\
&w_{4}^{O(6)}=(w_{2}^{SO^{s}(3)}+t^{2})(xy+xr+rc)+rc^{2}(t+c)\end{split}$
(160)

with the meanings of these symbols identical as those in the main text.
Substituting these expressions into Eq. (35) and performing some algebraic
manipulations yield

$\displaystyle S_{\rm
bulk}=i\pi\int_{M}[xy+c^{2}+r(x+c)]\left(w_{2}^{SO^{s}(3)}+t^{2}\right)=S_{\rm
LSM}$ (161)

which shows that the $U(1)$ DSL on a triangular lattice indeed has the correct
LSM anomaly.