Adjoint QCD$_4$, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension

TL;DR

This paper investigates the construction of 4d topological quantum field theories for adjoint QCD$_4$, revealing obstructions to fully trivializing certain higher anomalies and discussing implications for quantum critical phenomena and dualities.

Contribution

It generalizes the symmetry-extension method to higher symmetries and proves specific anomalies cannot be fully trivialized, indicating constraints on TQFTs with certain symmetries.

Findings

Certain higher anomalies cannot be trivialized, indicating obstructions to specific TQFT constructions.

The symmetry-extension method saturates some anomalies but not others, revealing limitations.

Implications for deconfined quantum critical points and dualities in 4d quantum field theories.

Abstract

Recent work explores the candidate phases of the 4d adjoint quantum chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4d topological quantum field theories (TQFTs) to saturate the higher 't Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from high energy. In this work, we generalize the symmetry-extension method [arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We prove that the symmetry-extension method saturates certain anomalies, but also prove that neither $A \mathcal{P}_2(B_2)$ nor $\mathcal{P}_2(B_2)$ can be fully trivialized, with the background 1-form field $A$, Pontryagin square $\mathcal{P}_2$ and 2-form field $B_2$. Surprisingly, this indicates an obstruction to constructing a fully 1-form center and 0-form chiral symmetry (full discrete axial symmetry) preserving 4d TQFT with confinement, a no-go scenario via symmetry-extension for specific higher anomalies. We comment on the implications and constraints on deconfined quantum critical points (dQCP), quantum spin liquids (QSL) or quantum fermionic liquids in condensed matter, and ultraviolet-infrared (UV-IR) duality in 3+1 spacetime dimensions.

Tables

Table 1. Table 1: The Scenario 1 , 2 , 3 , and 4 are from the list summarized in Sec.  IV.1.2 . The “SSB” stands for “spontaneous symmetry breaking.” The (…) means that symmetry ( … ) … (\dots) leads to SSB. We find an obstruction for Scenario 3 based on the higher symmetry-extension construction of Ref.  Wang  et al. ( 2018a ) . We should note that, educated by Ref.  Cordova and Dumitrescu ( 2018 ) and summarized in Fig.  3 , the Scenario 1 is consistent with the supersymmetry (SUSY) breaking of 𝒩 = 2 𝒩 2 {\cal N}=2 SYM from the magnetic monopole point and dyon point (as 2 copies of ℂ ​ ℙ 1 ℂ superscript ℙ 1 \mathbb{CP}^{1} model). The Scenario 2 is consistent with the SUSY breaking of 𝒩 = 2 𝒩 2 {\cal N}=2 SYM from the generic point from u ≠ 0 𝑢 0 u\neq 0 , and u ≠ ± Λ 2 𝑢 plus-or-minus superscript Λ 2 u\neq\pm\Lambda^{2} . The Scenario 4 is consistent with the SUSY breaking of 𝒩 = 2 𝒩 2 {\cal N}=2 SYM from the u = 0 𝑢 0 u=0 with a self-dual coupling τ 𝜏 \tau .

Scenario	$\begin{array}{c}\text{Internal global}\\ \text{symmetry }𝔾\end{array}$	$\begin{array}{c}\text{Chiral}\\ \text{Symmetry}\end{array}$	$\begin{array}{c}\text{1-form }{\mathbb{Z}}_{2,\left[1\right]}^e\text{ Symmetry;}\\ \text{De-/Confinement}\end{array}$	$\begin{array}{c}\text{Anomaly}\\ \text{matched}\\ \text{with UV}\end{array}$	$\begin{array}{c}\text{Plausible}\\ \text{Candidates}\end{array}$
1. Ref. Cordova and Dumitrescu (2018)	$O\left(2\right)\times U{\left(1\right)}_{\left[1\right]}$	$\begin{array}{c}\text{SSB}\end{array}$	Enhanced and preserved; Confined.	Yes	$\begin{array}{c}\text{Yes}\\ \text{(favored by energetic?)}\end{array}$
2. Ref. Cordova and Dumitrescu (2018)	$\left(\frac{\mathrm{SU}\left(2\right)\times {\mathbb{Z}}_{4,A}}{{\mathbb{Z}}_2^F}\right)\times {\mathbb{Z}}_{2,\left[1\right]}^e\times \dots$	$\begin{array}{c}\text{SSB}\\ {\mathbb{Z}}_{8,A}\to {\mathbb{Z}}_{4,A}\end{array}$	$\begin{array}{c}\text{Preserved; Confined.}\\ \text{But + new deconfined }{\mathbb{Z}}_2\text{-TQFT}\\ \text{with emergent new }{\mathbb{Z}}_{2,\left[1\right]}\text{ SSB.}\end{array}$	Yes	Yes
3. Ref. Bi and Senthil (2018)	$\left(\frac{\mathrm{SU}\left(2\right)\times {\mathbb{Z}}_{8,A}}{{\mathbb{Z}}_2^F}\right)\times {\mathbb{Z}}_{2,\left[1\right]}^e$	Preserved	Preserved; Confined.	$\begin{array}{c}\text{Obstruction}\\ \text{of symmetric}\\ \text{TQFT}\end{array}$	$\begin{array}{c}\text{Obstruction.}\\ \text{Not compatible w/}\\ \text{symmetry-extension }\text{<cite>Wang <em>et al.</em> <span>(</span><a>2018a</a><span>)</span></cite>}\end{array}$
4. Ref. Cordova and Dumitrescu (2018)	$\left(\frac{\mathrm{SU}\left(2\right)\times {\mathbb{Z}}_{8,A}}{{\mathbb{Z}}_2^F}\right)\times \overline{)U{\left(1\right)}_{\left[1\right]}^e}\times \overline{)U{\left(1\right)}_{\left[1\right]}^m}$	Preserved	Enhanced but SSB; Deconfined.	Yes	Yes

Table 2. Table 2: Bordism group Ω i Spin × ℤ 2 ( SU ​ ( 2 ) × ℤ 2 ℤ 8 ) superscript subscript Ω 𝑖 subscript subscript ℤ 2 Spin subscript subscript ℤ 2 SU 2 subscript ℤ 8 \Omega_{i}^{{\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})} in dimensions i 𝑖 i . Here a 𝑎 a is the generator of H 1 ⁡ ( B ​ ℤ 4 , ℤ 4 ) superscript H 1 B subscript ℤ 4 subscript ℤ 4 \operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{4}) , b 𝑏 b is the generator of H 2 ⁡ ( B ​ ℤ 4 , ℤ 4 ) superscript H 2 B subscript ℤ 4 subscript ℤ 4 \operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{4}) . w i ′ superscript subscript 𝑤 𝑖 ′ w_{i}^{\prime} is the Stiefel-Whitney class of the universal SO ​ ( 3 ) SO 3 {\rm SO}(3) bundle.

$i$	${\Omega }_i^{\mathrm{Spin}{\times }_{{\mathbb{Z}}_2}\left(\mathrm{SU}\left(2\right){\times }_{{\mathbb{Z}}_2}{\mathbb{Z}}_8\right)}$	cobordism invariants
0	$\mathbb{Z}$
1	${\mathbb{Z}}_4$	$a$
2	$0$
3	${\mathbb{Z}}_4$	$ab$
4	${\mathbb{Z}}^2$
5	$\mathbb{Z}\times {\mathbb{Z}}_2\times {\mathbb{Z}}_4$	$w_2^{\prime }{w}_3^{\prime },a{b}^2$

Table 3. Table 3: Bordism group Ω i Spin × ℤ 2 ( SU ​ ( 2 ) × ℤ 2 ℤ 8 ) ​ ( B 2 ​ ℤ 2 ) superscript subscript Ω 𝑖 subscript subscript ℤ 2 Spin subscript subscript ℤ 2 SU 2 subscript ℤ 8 superscript B 2 subscript ℤ 2 \Omega_{i}^{{\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})}(\mathrm{B}^{2}\mathbb{Z}_{2}) in dimensions i 𝑖 i . Here a 𝑎 a is the generator of H 1 ⁡ ( B ​ ℤ 4 , ℤ 4 ) superscript H 1 B subscript ℤ 4 subscript ℤ 4 \operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{4}) , b 𝑏 b is the generator of H 2 ⁡ ( B ​ ℤ 4 , ℤ 4 ) superscript H 2 B subscript ℤ 4 subscript ℤ 4 \operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{4}) . a ~ = a mod 2 ~ 𝑎 modulo 𝑎 2 \tilde{a}=a\mod 2 , b ~ = b mod 2 ~ 𝑏 modulo 𝑏 2 \tilde{b}=b\mod 2 . w 2 = w 2 ​ ( T ​ M ) subscript 𝑤 2 subscript 𝑤 2 𝑇 𝑀 w_{2}=w_{2}(TM) is the Stiefel-Whitney class of the tangent bundle. Note that w 2 ​ x 3 = w 3 ​ x 2 subscript 𝑤 2 subscript 𝑥 3 subscript 𝑤 3 subscript 𝑥 2 w_{2}x_{3}=w_{3}x_{2} (see Wan and Wang ( 2018a ) ). w i ′ superscript subscript 𝑤 𝑖 ′ w_{i}^{\prime} is the Stiefel-Whitney class of the universal SO ​ ( 3 ) SO 3 {\rm SO}(3) bundle.

$i$	${\Omega }_i^{\mathrm{Spin}{\times }_{{\mathbb{Z}}_2}\left(\mathrm{SU}\left(2\right){\times }_{{\mathbb{Z}}_2}{\mathbb{Z}}_8\right)}\left(B^2{\mathbb{Z}}_2\right)$	cobordism invariants
0	$\mathbb{Z}$
1	${\mathbb{Z}}_4$	$a$
2	${\mathbb{Z}}_2$	$x_2$
3	${\mathbb{Z}}_2\times {\mathbb{Z}}_4$	$\tilde{a}{x}_2,ab$
4	${\mathbb{Z}}^2\times {\mathbb{Z}}_2\times {\mathbb{Z}}_4$	$\tilde{b}{x}_2,{𝒫}_2\left(x_2\right)$
5	$\mathbb{Z}\times {\mathbb{Z}}_2^3\times {\mathbb{Z}}_4^2$	$w_2^{\prime }{w}_3^{\prime },\tilde{a}\tilde{b}{x}_2,w_2{x}_3,a{b}^2,a{𝒫}_2\left(x_2\right)$

Table 4. Table 4: Bordism group Ω i Spin ​ ( B ​ ℤ 8 ) superscript subscript Ω 𝑖 Spin B subscript ℤ 8 \Omega_{i}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}) in dimensions i 𝑖 i . Here η ~ ~ 𝜂 \tilde{\eta} is the 1d eta invariant, Arf is the Arf invariant, 𝔓 𝔓 \mathfrak{P} is the Postnikov square. a 𝑎 a is the generator of H 1 ⁡ ( B ​ ℤ 8 , ℤ 8 ) superscript H 1 B subscript ℤ 8 subscript ℤ 8 \operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{8}) , b 𝑏 b is the generator of H 2 ⁡ ( B ​ ℤ 8 , ℤ 8 ) superscript H 2 B subscript ℤ 8 subscript ℤ 8 \operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{8}) . a ~ = a mod 2 ~ 𝑎 modulo 𝑎 2 \tilde{a}=a\mod 2 , b ~ = b mod 2 ~ 𝑏 modulo 𝑏 2 \tilde{b}=b\mod 2 .

$i$	${\Omega }_i^{\mathrm{Spin}}\left(B{\mathbb{Z}}_8\right)$	cobordism invariants
0	$\mathbb{Z}$
1	${\mathbb{Z}}_2\times {\mathbb{Z}}_8$	$\tilde{\eta }$, $a$
2	${\mathbb{Z}}_2^2$	$\tilde{a}\tilde{\eta }$, Arf
3	${\mathbb{Z}}_2\times {\mathbb{Z}}_{16}$	$\tilde{a}$Arf, $𝔓\left(a\right)$
4	$\mathbb{Z}$
5	${\mathbb{Z}}_8\times {\mathbb{Z}}_2$101010This bordism group and its cobordism invariants have already been calculated in Gilkey and Botvinnik (1996); Hsieh (2018). We are grateful to Chang-Tse Hsieh for informing us his result Hsieh (2018).

Table 5. Table 5: Bordism group Ω i Spin ​ ( B ​ ℤ 8 × B 2 ​ ℤ 2 ) superscript subscript Ω 𝑖 Spin B subscript ℤ 8 superscript B 2 subscript ℤ 2 \Omega_{i}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2}) in dimensions i 𝑖 i . Here η ~ ~ 𝜂 \tilde{\eta} is the 1d eta invariant, Arf is the Arf invariant, 𝔓 𝔓 \mathfrak{P} is the Postnikov square. a 𝑎 a is the generator of H 1 ⁡ ( B ​ ℤ 8 , ℤ 8 ) superscript H 1 B subscript ℤ 8 subscript ℤ 8 \operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{8}) , b 𝑏 b is the generator of H 2 ⁡ ( B ​ ℤ 8 , ℤ 8 ) superscript H 2 B subscript ℤ 8 subscript ℤ 8 \operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{8}) . a ~ = a mod 2 ~ 𝑎 modulo 𝑎 2 \tilde{a}=a\mod 2 , b ~ = b mod 2 ~ 𝑏 modulo 𝑏 2 \tilde{b}=b\mod 2 .

$i$	${\Omega }_i^{\mathrm{Spin}}\left(B{\mathbb{Z}}_8\times B^2{\mathbb{Z}}_2\right)$	cobordism invariants
0	$\mathbb{Z}$
1	${\mathbb{Z}}_2\times {\mathbb{Z}}_8$	$\tilde{\eta }$, $a$
2	${\mathbb{Z}}_2^3$	$\tilde{a}\tilde{\eta }$, Arf, $x_2$
3	${\mathbb{Z}}_2^2\times {\mathbb{Z}}_{16}$	$\tilde{a}$Arf, $x_3$, $𝔓\left(a\right)$
4	$\mathbb{Z}\times {\mathbb{Z}}_2^2$	$\tilde{b}{x}_2$, $\frac{{𝒫}_2\left(x_2\right)}{2}$
5	${\mathbb{Z}}_2^2\times {\mathbb{Z}}_8\times {\mathbb{Z}}_2$	$\tilde{a}\tilde{b}{x}_2$, $\tilde{a}\frac{{𝒫}_2\left(x_2\right)}{2}$, 111111We omit the cobordism invariants of ${\Omega }_5^{\mathrm{Spin}}\left(B{\mathbb{Z}}_8\right)={\mathbb{Z}}_8\times {\mathbb{Z}}_2$ which have already been calculated in Gilkey and Botvinnik (1996) and Hsieh (2018).

Equations

$$\text{Adjoint QCD4 at UV }\stackrel{\text{RG flow (?) to a long distance}}{\to }$$

$$\text{Massless 1 Dirac (2 Weyl) fermion + 4d TQFT at IR (?)}.$$

$$\text{Type I anomaly/topological invariant}:{\mathrm{e}}^{\mathrm{i}\frac{k\pi }{2}\int A\cup {\mathcal{P}}_2(B_2)}.$$

$$\begin{gathered} \text{Type II anomaly/topological invariant}: \\ {\mathrm{e}}^{\mathrm{i}\pi \int w_2(TM){\mathrm{Sq}}^1{B}_2}={\mathrm{e}}^{\mathrm{i}\pi \int w_3(TM)B_2}. \end{gathered}$$

$$Z_{\text{UV}}$$

$$S_{\text{UV}}$$

$$\int (\frac{\theta }{8{\pi }^2}\text{Tr}F\wedge F).$$

$$(\frac{G_{\text{spacetime}}\times {\mathbb{G}}_{\text{internal}}}{{\mathbb{Z}}_2^F})\equiv G_{\text{spacetime}}{\times }_{{\mathbb{Z}}_2^F}{\mathbb{G}}_{\text{internal}},$$

$$\displaystyle{{\rm Spin}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)}\times\mathbb{Z}_{2,[1]}^{e}}$$

$$\displaystyle{{\rm Pin}^{-}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)}\times\mathbb{Z}_{2,[1]}^{e}}$$

$$\Omega_{d}^{{\rm Spin}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)}\times\mathrm{B}\mathbb{Z}_{2,[1]}^{e}}(pt)\\ \equiv\Omega_{d}^{{\rm Spin}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)}}(\mathrm{B}^{2}\mathbb{Z}_{2,[1]}^{e}).$$

$$\Omega_{d}^{{\rm Pin}^{-}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)}\times\mathrm{B}\mathbb{Z}_{2,[1]}^{e}}(pt)\\ \equiv\Omega_{d}^{{\rm Pin}^{-}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)}}(\mathrm{B}^{2}\mathbb{Z}_{2,[1]}^{e}).$$

$${\mathrm{e}}^{\mathrm{i}\frac{k\pi }{2}\int A\cup {\mathcal{P}}_2(B_2)}$$

$$={\mathrm{e}}^{\mathrm{i}\frac{k\pi }{2}\int A\cup (B_2\cup B_2+B_2{\cup }_1\delta B_2)}$$

$$={\mathrm{e}}^{\mathrm{i}\frac{k\pi }{2}\int A\cup (B_2\cup B_2+B_2(2{\mathrm{Sq}}^1{B}_2))},$$

$$1\to K\to H\to G\to 1,$$

$$r^{\ast }{\nu }_d^G={\nu }_d^H=\delta {\mu }_{d-1}^H\in H^d(H,\mathrm{U}(1)).$$

$$\mathrm{B}K\to \mathrm{B}H\to \mathrm{B}G,$$

$$\mathrm{B}{K}_{[0]}\times {\mathrm{B}}^2{K}_{[1]}\to \mathrm{B}\mathbb{H}\to \mathrm{B}\mathbb{G},$$

$$\mathrm{B}{K}_{[0]}⋉{\mathrm{B}}^2{K}_{[1]}\to \mathrm{B}\mathbb{H}\to \mathrm{B}\mathbb{G}$$

$${\mathrm{e}}^{\mathrm{i}\pi \int w_2(TM){\mathrm{Sq}}^1{B}_2}={\mathrm{e}}^{\mathrm{i}\pi \int w_3(TM)B_2}.$$

$$\mathrm{B}{\mathbb{Z}}_2\to \mathrm{B}\mathrm{Spin}(d)\times {\mathrm{B}}^2{\mathbb{Z}}_2\to \mathrm{B}\mathrm{SO}(d)\times {\mathrm{B}}^2{\mathbb{Z}}_2.$$

$$\underset{\text{5d higher SPTs (4d higher anomaly)}}{\underbrace{{\mathrm{e}}^{\mathrm{i}\pi {\int }_{M^5}{w}_2(TM){\mathrm{Sq}}^1{B}_2}}}\cdot \underset{\text{locally a 4d Z2-TQFT with emergent fermions and spin-structure}}{\underbrace{\sum _{\genfrac{}{}{0ex}{}{a\in C^1((\partial M{)}^4,{\mathbb{Z}}_2),}{b\in C^2((\partial M{)}^4,{\mathbb{Z}}_2)}}\exp (\mathrm{i}2\pi {\int }_{(\partial M{)}^4}\frac{1}{2}(b\delta a)+\dots )}}.$$

$$\displaystyle\mathrm{B}\mathbb{Z}_{2}\to\mathrm{B}({{\rm Spin}\times\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)})\times\mathrm{B}^{2}\mathbb{Z}_{2,[1]}^{e}}\to\mathrm{B}({{\rm Spin}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)})\times\mathrm{B}^{2}\mathbb{Z}_{2,[1]}^{e}},$$

$${\mathrm{B}}^2{\mathbb{Z}}_2\to \mathrm{B}\mathrm{SO}(d)\times {\mathrm{B}}^2{\mathbb{Z}}_4\to \mathrm{B}\mathrm{SO}(d)\times {\mathrm{B}}^2{\mathbb{Z}}_2.$$

$$\underset{\text{5d higher SPTs (4d higher anomaly)}}{\underbrace{{\mathrm{e}}^{\mathrm{i}\pi {\int }_{M^5}{w}_2(TM){\mathrm{Sq}}^1{B}_2}}}\cdot \underset{\text{locally a 4d Z2-TQFT, on which the 1-form Z2,[1]e-symmetry acts projectively}}{\underbrace{\sum _{\genfrac{}{}{0ex}{}{a^{\prime }\in C^1((\partial M{)}^4,{\mathbb{Z}}_2),}{b^{\prime }\in C^2((\partial M{)}^4,{\mathbb{Z}}_2)}}\exp (\mathrm{i}2\pi {\int }_{(\partial M{)}^4}\frac{1}{2}(a^{\prime }\delta b^{\prime })+\dots )}}.$$

$$\displaystyle\mathrm{B}^{2}\mathbb{Z}_{2,[1]}\to\mathrm{B}({{\rm Spin}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)})\times\mathrm{B}^{2}\mathbb{Z}_{4,[1]}^{e}}\to\mathrm{B}({{\rm Spin}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)})\times\mathrm{B}^{2}\mathbb{Z}_{2,[1]}^{e}},$$

$${\mathrm{e}}^{\mathrm{i}\frac{k\pi }{2}\int A\cup {\mathcal{P}}_2(B_2)}={\mathrm{e}}^{\mathrm{i}\frac{k\pi }{2}\int A\cup (B_2\cup B_2+B_2{\cup }_1\delta B_2)}$$

$${\mathrm{e}}^{\mathrm{i}\pi \int A\cup {\mathcal{P}}_2(B_2)}$$

$$\mathrm{B}{\mathbb{Z}}_2\to \mathrm{B}(\mathrm{Spin}\times (\mathrm{SU}(2){\times }_{{\mathbb{Z}}_2}{\mathbb{Z}}_8))\times {\mathrm{B}}^2{\mathbb{Z}}_2\to$$

$$\mathrm{B}{\mathbb{Z}}_2\to \mathrm{B}\mathbb{G}\to \mathrm{B}(\mathrm{Spin}\times \mathrm{SU}(2)\times {\mathbb{Z}}_8)\times {\mathrm{B}}^2{\mathbb{Z}}_2\to {\mathrm{B}}^2{\mathbb{Z}}_2.$$

$$\mathrm{B}{\mathbb{Z}}_2\times {\mathrm{B}}^2{\mathbb{Z}}_2\to \mathrm{B}(\mathrm{Spin}\times (\mathrm{SU}(2){\times }_{{\mathbb{Z}}_2}{\mathbb{Z}}_8))\times {\mathrm{B}}^2{\mathbb{Z}}_4\to$$

Full text

1. I Introduction and Summary of Main Results

2. II Theory of Adjoint QCD4

3. II.1 Global Symmetries

4. II.2 Anomalies

5. III Higher Symmetry-Extension

6. III.1 Summary of Ordinary Symmetry-Extension

7. III.2 Higher Symmetry Generalization

8. III.3 Saturate Type II anomaly: Symmetric TQFTs

9. III.4 Saturate Type I anomaly: Obstruction

10. III.5 Saturate both Type I (for even class in $\mathbb{Z}_{4}$) and II anomalies

11. III.6 Other Examples

12. IV Conclusion

13. IV.1 The Fate of the Dynamics of QCD4

14. IV.1.1 Possible Fates of the Dynamics of Fundamental QCD4 with $N_{f}$ Dirac fermions

15. IV.1.2 Possible Fates of the Dynamics of Adjoint QCD4 with $N_{f}$ Weyl fermions

16. IV.2 Deconfined Quantum Criticality, Quantum Spin/Fermionic Liquids in 3+1 Dimensions and More Comments

17. V Acknowledgments

18. A Cobordism Theory and Higher Symmetry-Extension: Construction of Symmetric TQFTs

19. A.1 Pullback trivialization of $A\mathcal{P}_{2}(B_{2})$ in $\Omega_{d}^{{\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})}(\mathrm{B}^{2}\mathbb{Z}_{2})$

20. A.1.1 Computation

21. A.1.2 Manifold generator

22. A.2 Pullback trivialization

23. A.2.1 Computation

24. A.2.2 Further trivialization: first approach

25. A.2.3 Further trivialization: second approach

26. A.2.4 Further trivialization: third approach

27. A.2.5 Summary

28. B Proof: a counterexample

29. C Pullback trivialization of $\mathcal{P}_{2}(B_{2})$ in $\Omega_{4}^{{\rm SO}}(\mathrm{B}^{2}\mathbb{Z}_{2})$

I Introduction and Summary of Main Results

Recent work explores the candidate phases of the adjoint quantum chromodynamics in 4 dimensional spacetime (QCD4) with an SU(2) gauge group and two massless adjoint Weyl fermions (equivalently, two massless adjoint Majorana fermions, or one massless adjoint Dirac fermion) Anber and Poppitz (2018); Cordova and Dumitrescu (2018); Bi and Senthil (2018); Seiberg et al. (2018).111 In this case, we denote the adjoint Weyl fermion flavor $N_{f}=2$ and the gauge group $N_{c}=2$ for SU($N_{c}$). This adjoint QCD4 has a 1-form electric $\mathbb{Z}_{2}$ center global symmetry, which is a generalized global symmetry of higher differential form Gaiotto et al. (2015). This adjoint QCD4 has the SU(2) gauge theory coupling to the matter fields in the adjoint representation, thus it gains a 1-form electric $\mathbb{Z}_{2}$ center symmetry; while the usual fundamental QCD4 has the gauge theory coupling to the matter fields in the fundamental representation, which lacks the 1-form symmetry. We will soon learn that this 1-form symmetry plays a crucial rule to constrain the higher ’t Hooft anomaly-matching ’t Hooft et al. (1980) of the quantum phases of the adjoint QCD4. (See Sec. II for more detailed information regarding the global symmetries and ’t Hooft anomalies of this adjoint QCD4.)

Given the adjoint QCD4 at the high energy scale, it is known that this theory is weakly coupled thus asymptotically free at ultraviolet (UV free) when the number of Weyl fermion flavor $N_{f}\leq 5$. Viewing the adjoint QCD4 as a UV completion of a quantum field theory (QFT), we should ask what this QFT flows to under a renormalization-group (RG) flow from ultraviolet (UV) to the low energy at infrared (IR). Both Cordova-Dumitrescu Cordova and Dumitrescu (2018) and Bi-Senthil Bi and Senthil (2018) propose its low energy candidate phases at IR, saturating the higher ’t Hooft anomalies involving the 1-form symmetry.

In particular, Bi-Senthil Bi and Senthil (2018) suggests a fully-symmetric 4d TQFT to saturate higher ’t Hooft anomalies without breaking any UV global symmetries of the adjoint QCD4. Namely, an interesting RG flow from Bi-Senthil Bi and Senthil (2018) speculates that:

[TABLE]

The IR theory only involves a massless free 1 Dirac (or 2 Weyl) fermion, and a decoupled 4d TQFT. Since the massless 1 Dirac fermion has only the ordinary 0-form symmetry but no 1-form symmetry, so the massless fermion sector alone cannot saturate the higher anomaly of the adjoint QCD4. Thus the crucial and nontrivial check on Bi-Senthil Bi and Senthil (2018) proposal of this UV-IR duality eq. (1) relies on the explicit construction of the fully-symmetric 4d TQFT to saturate all higher ’t Hooft anomalies involving 1-form symmetry. One of the motivation of our present work is to rigorously verify the validity of this symmetric anomalous 4d TQFT.

In this work, we have two goals:

We generalize the symmetry-extension method of Wang-Wen-Witten Wang et al. (2018a) to higher symmetries. We formulate a higher group cohomology or a higher cobordism theory approach of LABEL:Wang2017loc170506728 to construct “symmetric anomalous TQFTs” that can live on the boundary of symmetry protected topological states (SPTs). The “symmetric anomalous TQFTs” is an abbreviation of “the TQFTs that saturates the (higher) ’t Hooft anomalies of a given global symmetry by preserving the global symmetry.” Previous works in condensed matter physics suggest that the long-range entangled anomalous topological order (whose effective low energy theory is a TQFT) can live on the boundary of a short-range entangled SPT state, see Senthil (2015) and References therein on this exotic phenomenon. The boundary of SPTs protected by symmetry group $G$ (called $G$-SPTs) has the ’t Hooft anomaly of symmetry $G$. Ref. Wang et al. (2018a) provides a systematic way to construct the symmetric anomalous TQFTs for a $G$-SPTs of a given symmetry $G$. In particular, among other results, Ref. Wang et al. (2018a) proves that:

“For any bosonic $G$-SPTs protected by a finite group $G$ (unitary or anti-unitary time-reversal symmetry) in a 2-dimensional spacetime (2d) or above ($\geq 2d$), there always exists a finite group $K$ bosonic gauge theory which is a TQFT, saturating the $G$-’t Hooft anomaly, that can live on the boundary of $G$-SPTs, based on the symmetry-extension method via a short exact sequence $1\to K\to H\to G\to 1$, where all $G,K$ and $H$ are finite groups of 0-form symmetry.”

In this article, we will explore the related phenomenon of Ref. Wang et al. (2018a) but we improve the formulation by replacing the 0-form $G$ symmetry to include generalized higher symmetries of Ref. Gaiotto et al. (2015). 2. 2.

We apply the above generalized higher symmetry-extension method from Ref. Wang et al. (2018a) either to construct the higher-symmetric anomalous TQFTs, for adjoint QCD4; or to show the invalidity of the TQFTs via a symmetry-extension method.

Specifically, we find an obstruction to construct certain symmetric 4d TQFTs via symmetry extension, for the mixed anomaly mixing between the discrete axial symmetry (here the 0-form $\mathbb{Z}_{2N_{c}N_{f}}$ = $\mathbb{Z}_{8}$ symmetry, with $N_{c}=N_{f}=2$) and the 1-form electric center symmetry (denoted as $\mathbb{Z}_{2,[1]}^{e}=\mathbb{Z}_{2,[1]}$). This higher anomaly is abbreviated as the Type I higher anomaly in Ref. Bi and Senthil (2018). The Type I anomaly in 4d has a $\mathbb{Z}_{4}$ class (below $k\in\mathbb{Z}_{4}$ class), one can explicitly write down the 5d topological (abbreviated as “topo.”) invariant Cordova and Dumitrescu (2018) which is a cobordism invariant (see mathematical details in Wan and Wang (2018a) and Sec. A),

[TABLE]

Here $A$ is the $\mathbb{Z}_{4}$-valued background 1-form gauge field coupling to the 0-form $\mathbb{Z}_{8}/\mathbb{Z}_{2}^{F}=\mathbb{Z}_{4}$ part of the axial global symmetry. The $\mathbb{Z}_{2}^{F}$ is the fermionic parity symmetry which is $(-1)^{N_{F}}$, assigning a minus to the state of system when there is an odd number of total number of fermions ${N_{F}}$. The $B_{2}$ is the $\mathbb{Z}_{2}$-valued background 2-form gauge field coupling to the 1-form $\mathbb{Z}_{2,[1]}^{e}$-symmetry. The $\cup$ is the cup product, and the $\mathcal{P}_{2}$ is the Pontryagin square, see more details in Sec. II. In Sec. A, we will prove the non-existence of anomalous symmetric 4d TQFTs (of finite groups or higher groups) for this 4d higher anomaly (or equivalently, 5d higher SPTs) of eq. (2), via the symmetry extension method. However, we clarify that our proof does not necessarily imply a no go theorem for the anomalous symmetric 4d TQFTs for Bi-Senthil Bi and Senthil (2018) in general, it could be due to the limitation of the symmetry extension Wang et al. (2018a) we used. Nevertheless, it is known that Wang et al. (2018a)’s method is general and systematic enough to construct symmetric TQFT for all bosonic anomalies of the ordinary 0-form finite group symmetries; thus the obstruction from Wang et al. (2018a) is severe and interesting by itself to be presented here. This proof indicates a no-go scenario for anomalous symmetric 4d TQFTs if we only limit the construction under the symmetry-extension construction of TQFTs.

In contrast, we find that the generalized symmetry-extension method can indeed construct another symmetric 4d TQFT saturating a different higher mixed anomaly, mixing between the background gravity (or the curved spacetime geometry) and the 1-form center symmetry (denoted as $\mathbb{Z}_{2,[1]}$). This higher anomaly is abbreviated as the Type II higher anomaly in Ref. Bi and Senthil (2018). We can explicitly write down the 5d topological (abbreviated as “topo.”) invariant Cordova and Dumitrescu (2018) as the following cobordism invariant (see mathematical details in Wan and Wang (2018a) and Sec. A),

[TABLE]

Here $w_{j}(TM)$ has the $w_{j}$ as the $j$-th Stiefel-Whitney (SW) class Milnor and Stasheff (1974), as the probed background spacetime $M$ connection over the spacetime tangent bundle $TM$. The $\mathrm{Sq}^{1}$ is the Steenrod operation. We demonstrate the explicit construction of the 4d symmetric anomalous TQFT for this 4d higher anomaly (or equivalently, 5d higher SPTs) of eq. (3) in Sec. III.

Physically, the above description concerns the physics side of the story, relating to quantum field theory, QCD or the strongly-correlated systems in condensed matter physics.

Mathematically, we ask the following questions (corresponding to the physics story above) and find an obstruction to a positive answer for a Bi-Senthil’s scenario Bi and Senthil (2018) via the symmetry-extension alone, generalizing the method of Wang et al. (2018a):

1. Question 1.

Can we trivialize the topological term $A\cup\mathcal{P}_{2}(B_{2})$ via extending the global symmetry by 0-form symmetry and 1-form symmetry? To answer this, we deal with the trivialization problem of the cobordism invariant $A\cup\mathcal{P}_{2}(B_{2})$ of the bordism group $\Omega_{5}^{{\rm Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}}(\mathrm{B}^{2}\mathbb{Z}_{2})$.222 In this work, we will use the term $d$d “cobordism invariant” to describe the $d$d topological term or $d$d (higher) SPTs. On a manifold with boundary, the boundary of such a cobordism invariant (or SPTs) has a ’t Hooft anomaly. We denote the bordism group $\Omega_{d}^{G}$, while we denote the cobordism group $\Omega^{d}_{G}$. We prove that the answer is negative. 2. Question 2.

We also solve the trivialization problem of the cobordism invariant $\mathcal{P}_{2}(B_{2})$ of the bordism group $\Omega_{4}^{{\rm SO}}(\mathrm{B}^{2}\mathbb{Z}_{2})$: Can we trivialize the topological term $\mathcal{P}_{2}(B_{2})$ via extending the global symmetry by 0-form symmetry and 1-form symmetry? We prove that the answer is also negative.

The plan of the article goes as follows.

In Sec. II, we detail the related global symmetries and higher anomalies relevant for our goal, following a remarkable Ref. Cordova and Dumitrescu (2018).

In Sec. III, we discuss the higher symmetry-extension generalization of Wang et al. (2018a), and successfully apply the method to construct a 4d symmetric anomalous TQFT for Type II anomaly eq. (3). But this method shows an obstruction for the Type I anomaly eq. (2).

We leave rigorous but more formal and mathematical details of the calculation in Appendices.

In Appendix A, we find a potential obstruction: The Type I anomaly eq. (2) cannot be saturate by a symmetric anomalous finite group/higher group TQFT, at least by a symmetry extension method.

In Appendix B, we give a counter example as the proof for the failure of the symmetry extension method applying to trivializing the 5d $A\cup\mathcal{P}_{2}(B_{2})$.

In Appendix C, we show a similar obstruction: The 4d $\mathcal{P}_{2}(B_{2})$ cannot be saturated by a symmetric anomalous finite group/higher group TQFT, at least by a symmetry extension method.

We note that the Appendix A, B, and C are more technical and mathematical demanding. For readers who are not familiar with the mathematical background for these three sections, one can either consult Wan and Wang (2018a) and Guo et al. (2018) (e.g. the Appendix of Guo et al. (2018)), or simply skip them and proceed to the conclusion Sec. IV which we summarize the physics interpretations of the above three sections.

We conclude in Sec. IV.

The mathematical details of our cobordism calculations can be found in a companion paper Wan and Wang (2018a).

II Theory of Adjoint QCD4

We have an SU(2) gauge theory coupled to 2 $\mathbf{3}$ ($N_{f}=2$ for the 2, and the $\mathbf{3}$ for the triplet) adjoint Weyl fermions in the adjoint representation of SU(2). The path integral (or partition function) of this adjoint QCD4, in the Minkowski signature, viewed as a UV QFT theory can be written as:

[TABLE]

The eq. (5) contains the first term as the Dirac Lagrangian, and the second term as the Yang-Mills Lagrangian. The $[{\cal D}\dots]$ is the path integral measure for the quantum fields. The $\bar{\sigma}^{\mu}\equiv(1,-\vec{\sigma})$ contains the standard Pauli sigma matrices $\vec{\sigma}$. Here the Weyl fermion $\psi^{jb}_{\alpha}$ has:

$\bullet$ the flavor index $j$ (of the classical U(2) flavor symmetry, or more precisely the $\frac{{\rm SU}(2)\times\mathbb{Z}_{8}}{\mathbb{Z}_{2}^{F}}$ flavor symmetry in a quantum theory, see later),

$\bullet$ the gauge index $b$ of the gauge SU(2) of adjoint triplet,

$\bullet$ the Lorentz index $\alpha$ of the Lorentz group.

The hermitian conjugation of fermion field is $\bar{\psi}^{b^{\prime}}_{j}$ $=\psi^{jb^{\prime}\dagger}$. With the Lorentz index, we have $\bar{\psi}^{b^{\prime}}_{\dot{\alpha}j}$ $=\psi^{jb^{\prime}\dagger}_{\alpha}$, following the standard supersymmetry notation.

Here are some other comments:

$\bullet$ The $g$ is the dimensionless Yang-Mills coupling, which is a running coupling in the quantum theory.

$\bullet$ The $F$ is the SU(2) gauge field $a$’s 2-form field strength. The $\star F$ is the $F$’s Hodge dual.

$\bullet$ One can consider the deformation of the theory as extra terms in the $\dots$, such as the mass deformation Seiberg et al. (2018), e.g. $({m_{ij}}{}\delta_{b^{\prime}b}\psi_{\alpha}^{ib^{\prime}}\epsilon^{\alpha\beta}\psi_{\beta}^{jb}+c.c.)$. In the classical theory, we can add the $\theta$-term,

[TABLE]

However, in the quantum theory, with the presence of the fermion fields $\psi$, we can rotate the $\theta$ away. If we have the mass term for the fermions, we can absorb the $\theta$-term into the complex fermion mass matrix in the mass deformation.

II.1 Global Symmetries

The global symmetries of the adjoint QCD4 eq. (4) has been analyzed systematically in Cordova and Dumitrescu (2018). Here we recap the results and will write the results suitable for the cobordism theory analysis later in Appendices A to C.

Flavor symmetry $\frac{{\rm SU}(2)\times\mathbb{Z}_{8}}{\mathbb{Z}_{2}^{F}}$: The classical flavor symmetry of 2 triplet Weyl fermions is the flavor ${\rm U}(2)=\frac{{\rm SU}(2)\times{\rm U}(1)_{\rm A}}{\mathbb{Z}_{2}^{F}}$. However, the axial symmetry ${\rm U}(1)_{\rm A}$ is broken down to a discrete axial symmetry $\mathbb{Z}_{2N_{c}N_{f},{\rm A}}$, which is $\mathbb{Z}_{2N_{c}N_{f},{\rm A}}=\mathbb{Z}_{8}$ here, due to the Adler-Bell-Jackiw (ABJ) anomaly.333For a clarification of different meanings of anomalies, such as the three different types of physics of anomalies: (1) Classical global symmetry is violated at the quantum theory: ABJ anomaly. (2) Quantum global symmetry is well-defined and preserved but with the ’t Hooft anomaly. (3) Dynamical gauge anomaly; the readers can consult, for example, the Section 1 Introduction of Wan and Wang (2018b) and References therein. It is a standard calculation of the $U(1)_{\rm A}$-axial symmetry is explicitly broken by the dynamical ${\rm SU}(N_{c})$-gauge instanton down to $\mathbb{Z}_{2N_{c}N_{f},{\rm A}}$-axial symmetry.

So the flavor symmetry is simply $\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}=\frac{{\rm SU}(2)\times\mathbb{Z}_{8}}{\mathbb{Z}_{2}^{F}}$ for the quantum theory. The ${\rm SU}(2)$ is also written as the ${\rm SU}(2)_{R}$ as the $R$-symmetry thanks to the standard convention in ${\cal N}=2$ supersymmetric Yang-Mills theory (SYM) Seiberg and Witten (1994). In the ${\cal N}=2$ SYM, the adjoint fermions are gauginos. 2. 2.

The 1-form center symmetry $\mathbb{Z}_{2,[1]}^{e}\equiv\mathbb{Z}_{2,[1]}$: The adjoint QCD has the matter in adjoint representation, so the SU($N_{c}$) (here SU(2)) fundamental Wilson line is charged under the 1-form electric center symmetry $\mathbb{Z}_{2,[1]}^{e}$ measured by a 2-surface “charge” operator. The “charged” fundamental Wilson line (spin-1/2 representation of SU(2)) has an odd $\mathbb{Z}_{2}$ charge. The odd half integer spin-$n$/2 representation of SU(2) has an odd $\mathbb{Z}_{2}$ charge of 1-form symmetry. Wilson lines of other integer spin-$n$ representation (e.g. the adjoint) of SU(2) has a trivial (namely even) $\mathbb{Z}_{2}$ charge of 1-form symmetry.

Importantly the 1-form center symmetry $\mathbb{Z}_{2,[1]}^{e}$ is preserved means that the electric Wilson loop ($e$-loop) is unbreakable, or called tension-ful Bi and Senthil (2018). Since the adjoint QCD has the 1-form center symmetry, we can use the 1-form center symmetry charged object to detect:

$\bullet$ Confinement: If 1-form symmetry is preserved, and all the Wilson loops (of all representations) obey the area law.

$\bullet$ Deconfinement: If 1-form symmetry is spontaneously broken, then the Wilson loops of odd half integer spin-$n$/2 representation (e.g. fundamental representation) obey the perimeter law. 3. 3.

Spacetime symmetry: In Lorentz signature, we have the Poincaré group symmetry which contains the Lorentz group. We also have the discrete $CPT$ symmetries. There is no charge conjugation $C$ for SU(2) gauge theory due to the lack of SU(2) outer automorphism. So there is only $T$ and $P$ symmetry interchangeably thanks to the $CPT$ theorem. If we focus on orientable spacetime for the adjoint QCD in $d$d, we can consider the ${\rm Spin}(d)$ spacetime symmetry, for the purpose of classifying the ’t Hooft anomalies through the cobordism theory Freed and Hopkins (2016); Wan and Wang (2018a). If we consider the non-orientable spacetime for the adjoint QCD in $d$d, we should consider the ${\rm Pin}^{-}(d)$ spacetime symmetry, for the purpose of classifying the ’t Hooft anomalies through a cobordism theory, See Ref. Freed and Hopkins (2016); Wan and Wang (2018a). This adjoint QCD is a fermionic theory, the spacetime symmetry ${G_{\text{spacetime}}}$ and the internal symmetry ${\mathbb{G}_{\text{internal}}}$ shares the fermionic parity $\mathbb{Z}_{2}^{F}$, so the precise way to write the full global symmetry would be:

[TABLE]

where the common ${\mathbb{Z}_{2}^{F}}$ is mod out, while the “$\times_{\mathbb{Z}_{2}^{F}}$” notation follows Freed and Hopkins (2016).

By combining the internal global symmetry (flavor and 1-form center symmetries) and the spacetime global symmetry above, the overall global symmetry can be written as:

[TABLE]

[TABLE]

Below we follow Ref. Wan and Wang (2018a), which generalizes a theorem in a remarkable work of Freed-Hopkins Freed and Hopkins (2016). Freed-Hopkins Freed and Hopkins (2016) formulates a cobordism theory — whose cobordism group, of the ordinary 0-form global symmetries, classifies a class of symmetric invertible TQFTs, which is relevant to the SPT classification. Ref. Wan and Wang (2018a) generalizes Freed and Hopkins (2016) to a cobordism theory of the higher global symmetries (e.g. including 0-form global symmetries and 1-form global symmetries) and computes some examples of such cobordism groups.

In terms of bordism group notation, which later will be helpful for identifying all the (higher) ’t Hooft anomalies and the SPT classes via the computations of Wan and Wang (2018a), we write their corresponding bordism groups $\Omega_{d}$ as:444 Here $\mathrm{B}G$ means the classifying space of $G$, and $pt$ means the point.

• •

Bordism group for eq. (8):

[TABLE]

• •

Bordism group for eq. (9):

[TABLE]

For adjoint QCD4 in 4d, the higher ’t Hooft anomalies are classified by the dimension $d=5$ for the above bordism groups.555On the other hand, if we aim to know the 4d SPTs compatible with the symmetry of adjoint QCD4, then we need to consider the dimension $d=4$ for the above bordism groups. This research direction is pursued by LABEL:2017arXiv171111587GPW for the related SU($N_{c}$) Yang-Mills gauge theories. See more details in LABEL:Wan2018zql1812.11968.

II.2 Anomalies

Now consider the $d=5$ bordism groups above in eq. (10) and eq. (11), we like to match their selective 5d cobordism invariants to the anomalies captured by the 4d adjoint QCD4.

Cordova-Dumitrescu Cordova and Dumitrescu (2018) have captured several anomalies, which we now overview:

The SU(2) Witten anomaly Witten (1982a) for the flavor SU(2)R sector, due to that there is an odd number of SU(2)R flavor doublet. The appearance of SU(2) Witten anomaly also indicates the IR fate of this adjoint QCD4 is gapless instead of fully gapped. 2. 2.

The $(\mathbb{Z}_{8,{\rm A}})^{3}$ anomaly captured by a perturbative anomaly (i.e., a triangle 1-loop Feynman diagram in 4d). 3. 3.

The $(\mathbb{Z}_{8,{\rm A}})$-(gravity)2 anomaly captured by a perturbative anomaly (i.e., a triangle 1-loop Feynman diagram in 4d). The gravity part is due to the diffeomorphism of the background geometry. 4. 4.

The $(\mathbb{Z}_{8,{\rm A}})$-(SU(2)R)2 anomaly captured by a perturbative anomaly (i.e., a triangle 1-loop Feynman diagram in 4d).

Ref. Cordova and Dumitrescu (2018) explains the two interesting mixed ’t Hooft higher anomalies involving 1-form symmetry, the Type I eq. (2) and Type II eq. (3) anomalies earlier.

Type I higher anomaly: mixing between the 1-form electric center symmetry ($\mathbb{Z}_{2,[1]}^{e}$) and the 0-form discrete axial symmetry ($\mathbb{Z}_{2N_{c}N_{f}}$ = $\mathbb{Z}_{8}$). We can write eq. (2) as

[TABLE]

see Wan and Wang (2018a) for introducing the cup products, higher cup products and the Steenrod square Sq. 2. 6.

Type II higher anomaly: mixing between the 1-form center symmetry (denoted as $\mathbb{Z}_{2,[1]}$) and the background gravity (or the curved spacetime geometry) in eq. (3).

The UV theory as an adjoint QCD4 has all of the above ’t Hooft anomalies, captured also by a particular 5d cobordism invariant, in eq. (10) and eq. (11).

Following our Introduction, in the next Sec. III, we formulate the higher symmetry-extension generalizing Wang et al. (2018a), and successfully construct a 4d symmetric anomalous TQFT for Type II anomaly eq. (3). But we will soon show an obstruction to construct symmetric TQFT for the Type I anomaly eq. (2).

III Higher Symmetry-Extension

III.1 Summary of Ordinary Symmetry-Extension

Ref. Wang et al. (2018a) sets up the symmetry-extension problem as follows. Consider the $d$d SPTs protected by an internal symmetry group $G$, whose boundary theory has $(d-1)$d ’t Hooft anomaly in $G$. There are three different ways to phrase the question asked by Wang et al. (2018a), but their underlying meanings are the same:

1. Q1.

Condensed matter statement: Can we find a total group $H$ such that $G$ is its quotient group, and such that the $G$-SPTs becomes a trivial gapped vacua in $H$? More precisely, there is a local unitary transformation preserving the symmetry $H$ (but breaking the symmetry $G$), such that when the $G$-SPTs is viewed as an $H$-SPTs, it can be deformed to a trivial gapped insulator in $H$ via a local unitary transformation, without breaking $H$ and without any phase transition.666This procedure has been demonstrated explicitly in a many body quantum system recently in Ref. Prakash et al. (2018), which constructs an explicit path in the enlarged $H$-symmetric quantum Hilbert space. 2. Q2.

QFT or high energy particle physics statement: Given a $(d-1)$d ’t Hooft anomaly in $G$, can we find an enlarged group $H$, with a total group $H$ having $G$ as its quotient group, such that the ’t Hooft anomaly in $G$ becomes anomaly-free in $H$? (i.e., the $G$-anomaly becomes trivial in $H$.) 3. Q3.

Mathematical and algebraic topology statement: Given a $d$d topological term of a group $G$, here the topological term can be:

• •

the $d$d cocycle for a $d$-th cohomology group $\operatorname{H}^{d}(\mathrm{B}G,{\rm U}(1))$ in a group cohomology theory.

• •

the $d$d co/bordism invariant for a $d$-th cobordism group $\Omega^{d}_{-}(\mathrm{B}G,{\rm U}(1))$ or bordism group $\Omega_{d}^{-}(\mathrm{B}G)$ or bordism group, in a cobordism theory;777Here the $-$ can be chosen as co/bordism with different structures such as special/orthogonal ${\rm SO}$/${\rm O}$, spin/pin, or ${\rm Spin}$/${\rm Pin}^{\pm}$ structures.

can we find an extended group $H$ with $G$ its quotient group, via a short exact sequence

[TABLE]

such that the topological term of a group $G$ can be pulled back to a trivial topological term of a group $H$?

Suppose the above answer is positive, and suppose that $G$, $H$ and $K$ are finite groups, then Ref. Wang et al. (2018a) shows, valid for both the lattice Hamiltonian and the path integral construction, that the $G$-SPTs in $d$d can allow:

$\bullet$ $H$-symmetry extended gapped boundary in any spacetime dimension $d\geq 2$,

$\bullet$ $G$-symmetry preserving and topological $K$-gauge theory gapped boundary: Topological emergent $K$-gauge theory with preserving global symmetry $G$ on a bulk $d\geq 3$.

Ref. Wang et al. (2018a) addresses the above questions Q1, Q2 and Q3, by proving that at least for a finite group $G$ (with $G$ a unitary symmetry group or anti-unitary symmetry group involving time-reversal symmetry), by the following positive answers, with the always-existences on the validity of the symmetric gapped boundary construction:

1. A1.

For any bosonic SPT state with a finite onsite symmetry group $G$, including both unitary and anti-unitary symmetry, there always exists an $H$-symmetry-extended (or $G$-symmetry-preserving) gapped boundary via a nontrivial group extension by a finite $K$, given the bulk spacetime dimension $d\geq 2$. 2. A2.

For any $G$-anomaly in $(d-1)$d given by a cocycle $\nu_{d}^{G}\in\operatorname{H}^{d}(\mathrm{B}G,{\rm U}(1))$ of group cohomology of a finite group $G$, there always exists a pull back to a finite group $H$ via a certain group extension $1\rightarrow K\rightarrow H\overset{r}{\rightarrow}G\rightarrow 1$, extended by a finite $K$, such that $G$-anomaly becomes $H$-anomaly free, given the dimension $d\geq 2$ 3. A3.

For any $G$-cocycle $\nu_{d}^{G}\in\operatorname{H}^{d}(\mathrm{B}G,{\rm U}(1))$ of a finite group $G$, there always exists a pull back to a finite group $H$ via a certain short exact sequence of a group extension $1\rightarrow K\rightarrow H\overset{r}{\rightarrow}G\rightarrow 1$ by a finite $K$, such that

[TABLE]

Here $r$ is the pullback operation, and $\delta$ is the coboundary operation. Namely, a $G$-cocycle becomes a $H$-coboundary, which splits to a one-lower dimensional $H$-cochains $\mu^{H}_{d-1}$, given the dimension $d\geq 2$.

The proof of Wang et al. (2018a) has also been verified later by Tachikawa (2017). The related constructions similar to Wang et al. (2018a) are explored also in specific cases or from different perspectives in Kapustin and Thorngren (2014); Cheng et al. (2017).

III.2 Higher Symmetry Generalization

Now we generalizes the approach in Wang et al. (2018a). The short exact sequence of a group extension $1\rightarrow K\rightarrow H\overset{r}{\rightarrow}G\rightarrow 1$ extended by a finite $K$ given in Wang et al. (2018a) also implies an induced fiber sequence from the fibration

[TABLE]

where all $G,K$ and $H$ are finite groups of 0-form symmetry such that the $G$-SPTs protected by a finite group $G$ becomes trivial $H$-SPTs by pulling pack $G$ to $H$, under the above criteria A1, A2 and A3.

We consider the higher symmetry-extension problem. A simpler example is

[TABLE]

where $K_{[0]}$ is an extension from a normal 0-form symmetry $K_{[0]}$, while $K_{[1]}$ is an extension from a less familiar and more exotic 1-form symmetry $K_{[1]}$. However, our goal is more ambitious to check a more general fibration

[TABLE]

where $\mathbb{G}$ and $\mathbb{H}$ are 2-groups, $K_{[0]}$ and $K_{[1]}$ are finite abelian groups of 0-form symmetry and 1-form symmetry respectively such that the higher-$\mathbb{G}$-SPTs protected by a 2-group $\mathbb{G}$ becomes the trivial higher-$\mathbb{H}$-SPTs by pulling back $\mathbb{G}$ to $\mathbb{H}$.888For the related physics topics on higher group symmetries and higher SPTs, the readers can find from the recent developments Cordova et al. (2018); Benini et al. (2018); Delcamp and Tiwari (2018); Zhu et al. (2018); Wen (2018); Delcamp and Tiwari (2019) and References therein.

Here ${\mathrm{B}K_{[0]}\ltimes\mathrm{B}^{2}K_{[1]}}$ is the total space $\mathrm{B}\mathbb{K}$ of the fibration

[TABLE]

Similar to questions in Q1, Q2 and Q3 of Sec. III.1, we ask a set of generalized questions:

1. Q4.

Condensed matter statement: Can we find a total 2-group $\mathbb{H}$ as a total space such that $\mathrm{B}\mathbb{G}$ is $\mathrm{B}\mathbb{H}$’s orbit (or base space), and such that the $\mathbb{G}$-SPTs becomes a trivial gapped vacua in $\mathbb{H}$? More precisely, there is a local unitary transformation preserving the symmetry $\mathbb{H}$ (but breaking the symmetry $\mathbb{G}$), such that when the $\mathbb{G}$-SPTs is viewed as an $\mathbb{H}$-SPTs, it can be deformed to a trivial gapped insulator in $\mathbb{H}$ via a local unitary transformation (note that the locality also need to be generalized to higher dimensional extended object such as a line instead of just a point, due to the 2-group structure), without breaking $\mathbb{H}$ and without any phase transition in the enlarged $\mathbb{H}$-symmetric quantum Hilbert space. 2. Q5.

QFT or high energy particle physics statement: Given a $(d-1)$d ’t Hooft anomaly in a higher group $\mathbb{G}$, can we find an enlarged group $\mathbb{H}$, with a total group $\mathbb{H}$ obeying eq. (15), such that the ’t Hooft anomaly in $\mathbb{G}$ becomes anomaly-free in $\mathbb{H}$? (i.e., the $\mathbb{G}$-anomaly becomes trivial in $\mathbb{H}$.) 3. Q6.

Mathematical and algebraic topology statement: Given a $d$d topological term of a higher group $\mathbb{G}$, here the topological term can be:

• •

the $d$d cocycle for a $d$-th cohomology group $\operatorname{H}^{d}(\mathrm{B}\mathbb{G},{\rm U}(1))$ in a higher group cohomology theory.

• •

the $d$d co/bordism invariant for a $d$-th cobordism group $\Omega^{d}_{-}(\mathrm{B}\mathbb{G},{\rm U}(1))$ or bordism group $\Omega_{d}^{-}(\mathrm{B}\mathbb{G})$ or bordism group, in a cobordism theory;999Here the “$-$” follows the earlier footnote 7.

can we find an extended group $\mathbb{H}$ obeying eq. (15) such that the topological term of a group $\mathbb{G}$ can be pulled back to a trivial topological term of a group $\mathbb{H}$?

In the next two subsections, we implement the strategy eq. (15) by asking the questions in Q4, Q5 and Q6, for the two examples: Type I anomaly/topo. invariant in eq. (2), and Type II anomaly/topo. invariant in eq. (3).

We relegate more formal and mathematical details of the calculation of the above two subsections into Appendices A, B, and C.

III.3 Saturate Type II anomaly: Symmetric TQFTs

We first try to do higher symmetry extension to trivialize 4d Type II higher anomaly (given by a 5d topological invariant) eq. (3)

[TABLE]

We have found that eq. (3) is a topological invariant in $d=5$, for:

$\bullet$ $\operatorname{H}^{d}(\mathrm{B}^{2}\mathbb{Z}_{2},{\rm U}(1))$ group cohomology of a higher classifying space finite group, as well as

$\bullet$ $\Omega_{d}^{{\rm SO}}(\mathrm{B}^{2}\mathbb{Z}_{2})$ cobordism group of a higher classifying space finite group. Below we can either use the group cohomology or the cobordism group viewpoint to understand the trivialization of 4d Type II higher anomaly.

The first way to trivialize this 4d Type II higher anomaly is extending the spacetime symmetry from special orthogonal group ${\rm SO}(d)={\rm Spin}(d)/\mathbb{Z}_{2}^{F}$ to ${\rm Spin}(d)$:

[TABLE]

This extension works since $w_{2}(TM)=0$ vanishes on Spin manifold. Thus, eq. (3) is trivialized once we pull back eq. (3) into $\mathrm{B}{\rm Spin}(d)\times\mathrm{B}^{2}\mathbb{Z}_{2}$. According to the interpretation in Sec. III.1 and Ref. Wang et al. (2018a), the fibration $\mathrm{B}\mathbb{Z}_{2}$ contains an emergent 0-form global symmetry which is anomaly-free and can be dynamically gauged. Indeed, the natural way to interpret the eq. (21) as the generalized construction of Wang et al. (2018a) is that there is an emergent 1-form $\mathbb{Z}_{2}$ gauge theory (dynamically gauged from emergent 0-form global symmetry $\mathrm{B}\mathbb{Z}_{2}$), such that the $\mathbb{Z}_{2}$ gauge theory has additional emergent fermionic particle excitations due to the emergent spin structure (the Spin$(d)$ in the total space in eq. (21)). In terms of the full 4d symmetric TQFT saturating the higher ’t Hooft anomaly (coupling to the 5d higher SPTs), we can write the involved QFT sectors into a partition function, which looks like the following locally:

[TABLE]

Here $a$ is the $\mathbb{Z}_{2}$-valued 1-form gauge field (the standard notation as the 1-cochain in $C^{1}$), $b$ is the $\mathbb{Z}_{2}$-valued 2-form gauge field (the standard notation as the 2-cochain in $C^{2}$), the $\delta$ is the coboundary operator here $\delta=2\mathrm{Sq}^{1}$, and we use the cup product $\cup$. See also our previous explanations around eq. (3) for notations. The $\dots$ are additional coupling terms between dynamical gauge fields and background fields. The $\dots$ also include additional sectors from the UV adjoint QCD4 from eq. (4), in order to saturate the other anomalies. Note that the similar emergent dynamical spin structure with $\mathbb{Z}_{2}$ gauge field has been studied in LABEL:Wang:2018qoyWWW. The important thing is that the 1-form gauge field $a$ can be regarded as the difference between two spin-structures, while the gauge field $a$ becomes dynamical.

Moreover, we can write the extension of eq. (21) in terms of the full symmetry eq. (8):

[TABLE]

while the physical interpretation remains the same as eq. (21) and eq. (22). 2. 2.

The second way to trivialize this 4d Type II higher anomaly is extending the 1-form symmetry:

[TABLE]

This way works since $B$ is pulled back to $\tilde{B}\in\operatorname{H}^{2}(\mathrm{B}^{2}\mathbb{Z}_{4},\mathbb{Z}_{2})$, and $\mathrm{Sq}^{1}\tilde{B}=0$ (see Appendix A.2.4).

According to the interpretation in Sec. III.1 and Ref. Wang et al. (2018a), the fibration $\mathrm{B}^{2}\mathbb{Z}_{2}$ is associated to an emergent 1-form global symmetry $\mathbb{Z}_{2,[1]}$ which is anomaly-free and can be dynamically gauged. Indeed, the natural way to interpret the eq. (24) as the generalized construction of Wang et al. (2018a) is that there is an emergent 2-form $\mathbb{Z}_{2}$ gauge theory (dynamically gauged from emergent 1-form global symmetry $\mathrm{B}\mathbb{Z}_{2}$) with a 2-form gauge field $b^{\prime}$. The original 1-form $\mathbb{Z}_{2,[1]}^{e}$-symmetry acts projectively on the emergent 2-form $\mathbb{Z}_{2}$ gauge theory, but the extended 1-form $\mathbb{Z}_{4,[1]}^{e}$-symmetry acts on it faithfully.

We can write the involved QFT sectors into a following partition function, which looks like the following locally:

[TABLE]

Here $b^{\prime}$ is the $\mathbb{Z}_{2}$-valued 2-form gauge field (the standard notation as the 2-cochain in $C^{2}$), $a^{\prime}$ is the $\mathbb{Z}_{2}$-valued 1-form gauge field (the standard notation as the 1-cochain in $C^{1}$), while other notations are explained around eq. (3) and eq. (25). The $\dots$ are additional coupling terms between dynamical gauge fields and background fields. The $\dots$ also include additional sectors from the UV adjoint QCD4 from eq. (4), in order to saturate the other anomalies. We can also write the extension of eq. (24) in terms of the full symmetry eq. (8):

[TABLE]

while the physical interpretation remains the same as eq. (24).

III.4 Saturate Type I anomaly: Obstruction

We now try to do higher symmetry extension to trivialize 4d Type I higher anomaly (given by a 5d topological invariant of higher SPTs) eq. (2)

[TABLE]

Below we show that

When $k=2\in\mathbb{Z}_{4}$, the Type I anomaly eq. (2) can be trivialized, thanks to the fact that we can rewrite eq. (2) as

[TABLE]

where we have used the fact that $\mathrm{Sq}^{1}\tilde{A}=0$ where $\tilde{A}=A\mod 2$, and the Wu formula. See also useful information in Wan and Wang (2018a).

So when $k=2\in\mathbb{Z}_{4}$, if we extend the global symmetry by

[TABLE]

then the Type I anomaly eq. (2) vanishes. This extension works since $w_{1}(TM)=w_{2}(TM)=0$ vanishes on Spin manifolds. Thus, eq. (2) is trivialized once we pull back eq. (2) into $\mathrm{B}({\rm Spin}\times({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))\times\mathrm{B}^{2}\mathbb{Z}_{2}$. 2. 2.

When $k=1,3$ $\in\mathbb{Z}_{4}$, or $k$ odd, the Type I anomaly eq. (2) cannot be trivialized by extensions.

We have tried three approaches, which we relegate the details in Appendix A.2 while we summarize the physics story and implication here.

• •

The first approach (Appendix A.2.2) is a breaking case since we set $B$ to be zero. Physically this means that in order to saturate the ’t Hooft anomaly, we can break 1-form $\mathbb{Z}_{2}$-symmetry to nothing. In comparison, this 1-form $\mathbb{Z}_{2}$-symmetry breaking is a different scenario from Cordova and Dumitrescu (2018); Bi and Senthil (2018).

• •

In the second approach (details and notations explained in Appendix A.2.3), we define $\mathbb{G}$ to be a group which sits in a homotopy pullback square

[TABLE]

Hence we have a fiber sequence

[TABLE]

In this case, $B_{2}=B$ is identified with $\beta_{(2,8)}A$ where $A\in\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{8})$, and $\mathrm{Sq}^{1}B=0$, but $\tilde{A}\cup\frac{B^{2}}{2}$ is still not trivialized. This case is also a breaking case, since $B$ is locked with $A$. In physics, the locking between two probed background fields means that the global symmetry between two sectors are locked together, thus which results in global symmetry breaking.

Physically this means that in order to saturate the ’t Hooft anomaly, we still need to break symmetry in some way.

• •

In the third approach, we extend both the 0-form symmetry and the 1-form symmetry:

[TABLE]

But in this case, $\tilde{A}\cup\frac{B^{2}}{2}$ is still not, and cannot be, trivialized.

In summary, we finally conclude that when $k$ is odd, $k=1,3\in\mathbb{Z}_{4}$, the Type I anomaly eq. (2) cannot be trivialized by extensions and give a proof in Appendix B. In comparison, Ref. Bi and Senthil (2018) proposes a full symmetry-preserving TQFT different all of our scenarios above, which contradicts to our proof in Appendix B.

III.5 Saturate both Type I (for even class in $\mathbb{Z}_{4}$) and II anomalies

When $k=2$, such that the Type I anomaly survives as only a $\mathbb{Z}_{2}$ subclass (even $k$) in the original $k\in\mathbb{Z}_{4}$ class (of $kA\cup\mathcal{P}_{2}(B_{2})$), however, we can actually trivialize the $\mathbb{Z}_{2}$ subclass of Type I anomaly and the full Type II anomaly together via the fibration:

[TABLE]

The above is achieved by combining both eq. (21) and eq. (28) into eq. (23). Since we only care $k=2$, this also means that the $\mathbb{Z}_{8,{\rm A}}$-symmetry only needs to be survived as a $\mathbb{Z}_{4,{\rm A}}$-symmetry. Physically this means that $\mathbb{Z}_{8,{\rm A}}$-symmetry can be spontaneously broken down to a $\mathbb{Z}_{4,{\rm A}}$-symmetry. Thus the eq. (36) really implies a fibration of a smaller symmetry (e.g. a smaller classifying space) as:

[TABLE]

For such a 4d TQFT preserving a $(\frac{{\rm SU}(2)\times\mathbb{Z}_{4,{\rm A}}}{\mathbb{Z}_{2}^{F}})$-chiral symmetry and 1-form $\mathbb{Z}_{2,[1]}^{e}$-symmetry (from UV adjoint QCD4), saturating the higher ’t Hooft anomaly (coupling to the 5d higher SPTs), we can write the involved QFT sectors into a partition function, which looks like the following locally:

[TABLE]

again the 1-form gauge field $a$ can be regarded as the difference between two spin-structures; the 1-form emergent dynamical $\mathbb{Z}_{2}$ gauge field $a$ is associated to a dynamical spin structure (similar to a situation in LABEL:Wang:2018qoyWWW).

We note that the $\dots$ terms can involve additional ’t Hooft anomaly cancellation for the UV’s adjoint QCD4, such as the gapless sector proposed in Anber and Poppitz (2018); Cordova and Dumitrescu (2018); Bi and Senthil (2018); Seiberg et al. (2018). Besides, the $\dots$ terms also involve the coupling terms between dynamical gauge fields and background fields, so that the full partition function can be made gauge-invariant. Although eq. (37) already suggests a formation definition of TQFTs (based on the extension construction of bulk-boundary coupled TQFTs, see Wang et al. (2018a) and related constructions in Wang et al. (2018); Guo et al. (2018)), it may be worthwhile to formulate a cochain or continuum TQFT description following Wang et al. (2018); Guo et al. (2018) — which we leave them for future work. It may also be worthwhile to give a continuum 4d TQFT formulation for the higher-form gauge theory analogous to Dijkgraaf-Witten Dijkgraaf and Witten (1990) like gauge theory, similar to the continuum TQFT formulation given in Putrov et al. (2017); Delcamp and Tiwari (2018).

III.6 Other Examples

In our companion work Wan and Wang (2018b), we consider similar trivialization problem for 5d topological invariants of 4d Yang-Mills SU(N) gauge theory (in particular at $\theta=\pi$) anomaly. The successful trivialization by the pulling back via a finite group extension suggests that the low energy fate of these 4d Yang-Mills SU(N) gauge theories can also be a fully symmetry-preserving TQFT with the full ’t Hooft anomaly matched.

The fully symmetry-preserving TQFT means that there is no 0-form global symmetry (e.g. time-reversal symmetry) breaking, nor the 1-form center symmetry breaking (thus it is in a confined phase in the usual definition, where the charged Wilson loop is tension-ful and has an area law); but there is an additional emergent discrete gauge theory as a TQFT associated to the finite group extension.

In recent work, LABEL:Wan2019oyrWWZ1904.00994 studies the related fully symmetry-preserving TQFTs of various 4d Yang-Mills SU(2) gauge theories where its SU(2)-fundamental Wilson lines can be Kramers singlet/doublet and bosonic/fermionic. LABEL:Wan2019oyrWWZ1904.00994 uses the higher-symmetry extension method to suggest where the 4d UV Yang-Mills SU(2) gauge theories can flow to candidate fully symmetry-preserving TQFTs as new reasonable IR fates.

We find that many other examples of 5d topological invariants of 4d Yang-Mills anomaly can be trivialized by extending the 0-form symmetry and 1-form symmetry. Hence, the higher symmetry-extension generalization LABEL:Wang2017loc170506728 is powerful enough to trivialize a lot of other higher bosonic types of anomalies thus to construct exotic fully-symmetric anomalous TQFTs, although it gives an obstruction to saturate the Type I anomaly at an odd $k$ while preserving the full symmetry.

IV Conclusion

We conclude by summarizing the implications of the higher-symmetry extension construction of TQFTs on the low energy dynamics of QCD4. Then we comment about the constraints on the deconfined quantum critical phenomena, or so called the deconfined quantum critical point (dQCP) Senthil et al. (2004), in 3+1 spacetime dimensions Bi and Senthil (2018).

IV.1 The Fate of the Dynamics of QCD4

IV.1.1 Possible Fates of the Dynamics of Fundamental QCD4 with $N_{f}$ Dirac fermions

First, we recall the possible fates of the dynamics of QCD4 with $N_{f}$ Dirac fermions in fundamental representations of SU($N_{c}$). The conventional wisdom teaches us that the phase structure of dynamics of QCD4 via tuning $N_{f}$ (with a fix $N_{c}$), shown in Fig. 1, is that:

$\bullet$ At lower $N_{f}$, there should be a confinement (IR confinement) and chiral symmetry breaking (IR ChSB).

$\bullet$ At larger $N_{f}$, there is a range of $N_{f}$, such that at IR, the QFT flows to an interacting conformal field theory (“CFT”), this is known as the range of conformal window phenomena studied by Bank-Zaks Banks and Zaks (1982) and others.

$\bullet$ Let $N_{f}^{\text{asym.free}}=\frac{11}{2}N_{c}$, when $N_{f}<N_{f}^{\text{asym.free}}$, the UV theory is weak coupling known as the asymptotic freedom (or UV free) Gross and Wilczek (1973); Politzer (1973). When $N_{f}>N_{f}^{\text{asym.free}}$, the UV theory becomes strongly coupled while the coupling $g$ flows weak at IR, at least perturbatively.

IV.1.2 Possible Fates of the Dynamics of Adjoint QCD4 with $N_{f}$ Weyl fermions

Now we organize the possible fates of the dynamics of QCD4 with $N_{f}$ Weyl fermions in adjoint representations of SU($N_{c}$). The possible phase structure of dynamics of QCD4 via tuning $N_{f}$ (with a fix $N_{c}$) is shown in Fig. 2. We remark that the candidate adjoint phases are summarized very elegantly in Cordova and Dumitrescu (2018), we recap into a concise Fig. 2, while also list down the related Scenario 1, 2, 3, and 4, from Ref. Cordova and Dumitrescu (2018); Bi and Senthil (2018), and from the list summarized in Sec. IV.1.2.

The conventional wisdom teaches us that the phase structure of dynamics of adjoint QCD4 via tuning $N_{f}$ (with a fix $N_{c}$), shown in Fig. 2, is that:

• •

At $N_{f}=0$, it is a pure SU($N_{c}$) Yang-Mills gauge theory (say SU(2)), potentially with a $\theta$-term eq. (6). At $\theta=0$, the phase is a trivially gapped confined phase (IR confinement) with no SPT state. However, at $\theta=\pi$, the phase has mixed higher anomalies Gaiotto et al. (2017) and potentially newly found higher ’t Hooft anomalies Wan and Wang (2018b).

• •

At $N_{f}=1$, it is a pure ${\cal N}=1$ supersymmetric Yang-Mills gauge theory (SYM) Intriligator and Seiberg (1996). Moreover, there are $N_{c}$ supersymmetric breaking vacua due to gaugino condensation Witten (1982b), which breaks $\mathbb{Z}_{2N_{c}}$ down to $\mathbb{Z}_{2}$ (simply $\mathbb{Z}_{2}^{F}$). This ${\cal N}=1$ SYM phase is also known to be confined through monopole condensation, by embedding into a ${\cal N}=2$ SYM theory with $N_{c}=2$ Seiberg and Witten (1994).

• •

At lower $N_{f}$, there should be a confinement (IR confinement) and chiral symmetry breaking (IR ChSB).

• •

At larger $N_{f}$, one expects again a range of $N_{f}$ with a range of conformal window phenomena of Bank-Zaks Banks and Zaks (1982).

To proceed further, we recall that the UV internal global symmetry is \big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,{\rm A}}}{\mathbb{Z}_{2}^{F}}\big{)}\times\mathbb{Z}_{2,[1]}^{e}. Now we organize a list of possible fates of the dynamics of adjoint QCD4 with $N_{f}$ Weyl fermions proposed from Cordova and Dumitrescu (2018); Bi and Senthil (2018). There are four scenarios, summarized in Table 1 and below:

The $N_{c}$ copies of (or more specifically here $N_{c}=2$) of 4d $\mathbb{CP}^{1}$ sigma model at low energy with spontaneous symmetry breaking Goldstone modes, proposed by Cordova and Dumitrescu (2018). Its global symmetry:

[TABLE]

In summary, the scenario 1 has:

[TABLE]

To digest better about the target space of $\mathbb{CP}^{1}$ sigma model, here we can consider the breaking of the 0-form symmetry group $G$ as the total space $E$ breaking to a smaller fiber $F$ (a subgroup or a normal subgroup, as the fiber or the stabilizer), where the order parameter parametrizes the base manifold $B$ (the base space or the orbit). In short, we formally and mathematically write:

[TABLE]

Then we obtain a relation for the scenario 1:

[TABLE]

or more precisely a relation:

[TABLE]

The $\mathbb{CP}^{1}\rtimes\frac{\mathbb{Z}_{8,{\rm A}}}{{\mathbb{Z}_{2}}\times\mathbb{Z}_{2}^{F}}$ has two copies of $\mathbb{CP}^{1}$ as the target space, parametrizing the order parameter of the base manifold $B$ (the base space or the orbit). 2. 2.

A free massless Dirac fermion (equivalently, two massless Weyl fermions, or two massless Majorana fermions) and a $\mathbb{Z}_{2}$ discrete gauge theory as a 4d TQFT with a $\mathbb{Z}_{4}$ symmetry (spontaneously broken from the $\mathbb{Z}_{8}$ symmetry), proposed by Cordova and Dumitrescu (2018). The IR symmetry is

[TABLE]

In summary, the scenario 2 has:

[TABLE]

However, as explained in Cordova and Dumitrescu (2018), there is an additional emergent new deconfined $\mathbb{Z}_{2}$-TQFT with emergent new $\mathbb{Z}_{2,[1]}$ symmetries spontaneously broken. 3. 3.

A free massless Dirac fermion (equivalently, two massless Weyl fermions, or two massless Majorana fermions) and a 4d TQFT preserving the full $\mathbb{Z}_{8}$ symmetry, proposed by Bi and Senthil (2018). The two massless Weyl fermions actually have a U(2) continuous global symmetry. The IR symmetry we focus is:

[TABLE]

In summary, the scenario 3 proposed that:

[TABLE] 4. 4.

A 4d U(1) gauge theory in Coulomb phase with a $\mathbb{Z}_{2N_{c}N_{f}}$ = $\mathbb{Z}_{8}$ symmetry, proposed by Cordova and Dumitrescu (2018). The IR symmetry we focus is:

[TABLE]

The (…) means a spontaneous symmetry breaking of $\dots$, thus for 1-form symmetry breaking here, it leads to a deconfinement of U(1) gauge theory. In summary, the scenario 4 proposed that:

[TABLE] 5. 5.

Note that there is another scenario from Ref. Anber and Poppitz (2018) proposing only a free massless Dirac fermion at IR (equivalently, two massless Weyl fermions, or two massless Majorana fermions), and two vacua (two degenerate ground states) due to $\mathbb{Z}_{8,{\rm A}}\to\mathbb{Z}_{4,{\rm A}}$, without any 1-form symmetry. This scenario is certainly incomplete due to the lack of matching the higher ’t Hooft anomalies of 1-form symmetry. As Ref. Anber and Poppitz (2018) also notices later, the more complete scenario is adding a TQFT sector, following the Scenario 2.

IV.2

Deconfined Quantum Criticality, Quantum Spin/Fermionic Liquids in 3+1 Dimensions and More Comments

In this work, we obtain a higher-symmetry extension generalization of Ref. Wang et al. (2018a)’s method to construct symmetric anomalous TQFT saturating higher ’t Hooft anomalies. We have obtained a symmetric anomalous TQFT, valid for Scenario 2 from Cordova-Dumitrescu (Ref. Cordova and Dumitrescu (2018)), see eq. (37) and eq. (38). However, we are unable to obtain a symmetric anomalous TQFT proposed by Scenario 3 motivated by Bi-Senthil (Ref. Bi and Senthil (2018)) based on a symmetry-extension construction.

It is worthwhile to digest the exotic and interesting physics of Scenario 3 better. The Scenario 3 is motivated by the deconfined quantum criticality in 3+1 dimensions. It is proposed that a critical theory can be realized as a phase transition between two conventional Landau-Ginzburg symmetry-breaking orders Senthil et al. (2004), or a phase transition between two different SPT orders (see Bi and Senthil (2018) and References therein). The adjoint QCD4 is a UV description (UV side of eq. (1)) of the phase transition, while the IR description is currently unclear (IR side of eq. (1)).

The novelty of Scenario 3 is that the gapless sector is a free CFT as two free Weyl fermions (a single free Dirac fermion). So the hope is that the possible UV-IR duality eq. (1) in 3+1D is between a strongly coupled and interacting UV gauge theory and a free non-interacting massless IR theory, up to a gapped fully-symmetric TQFT sector to saturate the higher ’t Hooft anomalies.

Our present work shows an obstruction for Scenario 3 from a symmetry-extension construction alone. The implications of our finding are follows:

1. I.

We should remind the readers that the symmetry-extension construction is fairly general enough to saturate a large class of higher ’t Hooft anomalies of bosonic systems. Although the adjoint QCD4 is a fermionic system (the UV completion requires fermionic degrees of freedom, where there are gauge-invariant fermionic operators), the Type I and II anomalies, eq. (2) and eq. (3), are bosonic anomalies in nature. 2. II.

Despite the fact that fully-symmetric TQFT under Scenario 3 cannot be obtained via our symmetry-extension construction, we may still be able to use the symmetry-extension construction to derive other symmetric anomalous TQFTs, suitable to propose new candidate phases of other deconfined quantum criticality (dQCP), in 3+1 and other dimensions.

We should also notice that the recent numerical attempts Del Debbio et al. (2009); Athenodorou et al. (2015) suggest that the adjoint QCD4 with SU(2) gauge group and $N_{f}$ number of adjoint Weyl fermions may have IR dynamics as follows:

• •

At $N_{f}=2$, (as 1 adjoint Dirac fermion), according to Athenodorou et al. (2015), the IR theory may be very close to the onset of the conformal window, instead of the conventional confining behavior. In addition, the anomalous dimension of the fermionic condensate is reported to be close to 1. The numerical data seems to suggest the IR theory can be an interacting CFT (more exotic), instead of a free CFT (all the proposed scenarios so far, discussed in Table 1).

• •

At $N_{f}=4$, (as 2 adjoint Dirac fermion), Ref. Del Debbio et al. (2009) discusses the candidate IR theory. Ref. Del Debbio et al. (2009) points out the theory is gapless (or massless), while future endeavor is required to distinguish whether it shows the confinement or the conformal behavior.

To unambiguously determine the IR dynamics, apart from the given numerical inputs Del Debbio et al. (2009); Athenodorou et al. (2015), we note that further lattice studies are still necessary.

In addition, the adjoint QCD4 system has the $\mathbb{Z}_{2}^{F}$ fermionic parity symmetry un-gauged; thus $\mathbb{Z}_{2}^{F}$ remained an honest global symmetry. This system can be viewed as the emergent QFT of a 3+1 spacetime dimensional Quantum Fermionic Liquids — a fermionic analog of the familiar bosonic Quantum Spin Liquids (QSL). In condensed matter, the “spin” in QSL implies the “isospin” which is bosonic in nature. While QSL does not require manifolds with spin structures to be realized, the adjoint QCD4 requires certain manifolds endorsed with analogs of spin structure given in eq. (8) and eq. (9).

Finally, we remark that many anomalies discussed in Sec. II.2, following Cordova and Dumitrescu (2018), are non-perturbative global anomalies instead of perturbative anomalies. The non-perturbative anomalies have classifications from finite groups (e.g. $\mathbb{Z}_{n}$ classes), instead of a $\mathbb{Z}$ classification. Examples include the old and the new SU(2) anomalies Witten (1982a); Wang et al. (2018b), and also the recent higher ’t Hooft anomalies of SU(N) YM gauge theory, see Gaiotto et al. (2017) and Wan and Wang (2018b), and References therein. For these non-perturbative global anomalies, we can saturate certain ’t Hooft anomalies of ordinary or higher global symmetries by symmetry-preserving TQFTs or so-called the long-ranged entangled topological order sectors, via our higher symmetry-extension approach, see a companion work along this direction Wan and Wang (2018b).

V Acknowledgments

The authors are listed in the alphabetical order by the standard convention. JW thanks the participants of Developments in Quantum Field Theory and Condensed Matter Physics (November 5-7, 2018) at Simons Center for Geometry and Physics at SUNY Stony Brook University for giving valuable feedback where this work is publicly reported Wang . We thank Edward Witten for helpful comments and sharing his proof from another perspective. We are grateful to Chang-Tse Hsieh for Email correspondences on Ref. Gilkey and Botvinnik (1996), and informing his related result in Ref. Hsieh (2018), such that we make a correction in our Table 4 and 5 after the Phys. Rev. D publication Wan and Wang (2019). JW especially thanks Clay Cordova, also thanks Kantaro Ohmori, Nathan Seiberg, and Shu-Heng Shao for illuminating conversations. Part of this short result emerges from an unpublished work in Seiberg et al. (2018). ZW acknowledges support from NSFC grants 11431010 and 11571329. JW acknowledges the Corning Glass Works Foundation Fellowship and NSF Grant PHY-1606531. This work is also supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics” and Center for Mathematical Sciences and Applications at Harvard University.

Appendix A Cobordism Theory and Higher Symmetry-Extension: Construction of

Symmetric TQFTs

By the result of the page 251 of Ref. Hatcher (2002), the cohomology ring of the infinite Lens space $\mathrm{B}\mathbb{Z}_{2^{n}}=S^{\infty}/\mathbb{Z}_{2^{n}}$ with coefficients $\mathbb{Z}_{2^{n}}$ is the polynomial ring generated by $a$ and $b$ over $\mathbb{Z}_{2^{n}}$ quotient by the relation $a^{2}=2^{n-1}b$:

[TABLE]

where $a\in\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{2^{n}},\mathbb{Z}_{2^{n}})$, $b\in\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{2^{n}},\mathbb{Z}_{2^{n}})$.

[TABLE]

where $\tilde{a}=a\mod 2$, $\tilde{b}=b\mod 2$, there is a $(2,2^{n})$-Bockstein $\beta_{(2,2^{n})}$ with $\beta_{(2,2^{n})}(a)=\tilde{b}$. Here $\operatorname{H}^{*}$ is the cohomology ring, $\Lambda_{\mathbb{Z}_{2}}$ denotes the exterior algebra over $\mathbb{Z}_{2}$, $\otimes$ is the tensor product. The $(2,2^{n})$-Bockstein homomorphism $\beta_{(2,2^{n})}:\operatorname{H}^{*}(-,\mathbb{Z}_{2^{n}})\to\operatorname{H}^{*+1}(-,\mathbb{Z}_{2})$ is associated to the extension $\mathbb{Z}_{2}\to\mathbb{Z}_{2^{n+1}}\to\mathbb{Z}_{2^{n}}$.

Notice that the notation $a\in\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{2^{n}},\mathbb{Z}_{2^{n}})$ and $b\in\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{2^{n}},\mathbb{Z}_{2^{n}})$ will be abused later, since we will encounter the cases $n=2$ and $n=3$. We will use the uniform notation and explain wherever they appear.

A.1 Pullback trivialization of $A\mathcal{P}_{2}(B_{2})$ in $\Omega_{d}^{{\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})}(\mathrm{B}^{2}\mathbb{Z}_{2})$

Follow the mathematical conventional notation, we will also denote the 5d topological term

[TABLE]

in Appendix A and after. The $a$ here is a background probed field, which should not be confused with the SU(2) dynamical gauge field.

A.1.1 Computation

${\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})\equiv({\rm Spin}\times({\rm SU}(2)\times\mathbb{Z}_{8})/\mathbb{Z}_{2})/\mathbb{Z}_{2}$ where the quotient is with respect to the diagonal center $\mathbb{Z}_{2}$ subgroup.

Since the computation involves no odd torsion, we can use the Adams spectral sequence

[TABLE]

Here $\operatorname{Ext}$ is the Ext functor, $\mathcal{A}_{2}$ is the mod 2 Steenrod algebra, more precisely, $\operatorname{Ext}_{\mathcal{A}_{2}}^{s,t}$ is the internal degree $t$ part of the $s$-th derived functor of $\operatorname{Hom}_{\mathcal{A}_{2}}^{*}$. $MT({\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))$ is the Madsen-Tillmann spectrum of the group ${\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})$, the bordism group $\Omega_{d}^{{\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})}(\mathrm{B}^{2}\mathbb{Z}_{2})=\pi_{d}(MT({\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))\wedge(\mathrm{B}^{2}\mathbb{Z}_{2})_{+})$ is the stable homotopy group of the spectrum $MT({\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))\wedge(\mathrm{B}^{2}\mathbb{Z}_{2})_{+}$, here $\wedge$ is the smash product, $X_{+}$ is the disjoint union of the space $X$ and a point. “$\Rightarrow$” means “convergent to”. For more detail, see Wan and Wang (2018a).

Similarly as the discussion in Campbell (2017); Guo et al. (2018), we know

[TABLE]

where $2\xi$ is twice the sign representation, $(\mathrm{B}\mathbb{Z}_{4})^{2\xi}$ is the Thom space $\text{Thom}(\mathrm{B}\mathbb{Z}_{4};2\xi)$, $M{\rm Spin}$ is the Thom spectrum of the group ${\rm Spin}$, $M{\rm SO}(3)$ is the Thom spectrum of the group ${\rm SO}(3)$, $\Sigma$ is the suspension.

Note that $(\mathrm{B}\mathbb{Z}_{4})^{2\xi}=\Sigma^{-2}M\mathbb{Z}_{4}$.

We have a homotopy pullback square

[TABLE]

where $\tilde{b}$ is the generator of $\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{2})$, $w_{2}=w_{2}(TM)$ is the Stiefel-Whitney class of the tangent bundle $TM$, $w_{2}^{\prime}=w_{2}^{\prime}({\rm SO}(3))$ is the Stiefel-Whitney class of the universal ${\rm SO}(3)$ bundle.

Hence we have the constraint

[TABLE]

Since $\operatorname{H}^{*}(M{\rm Spin},\mathbb{Z}_{2})=\mathcal{A}_{2}\otimes_{\mathcal{A}_{2}(1)}\{\mathbb{Z}_{2}\oplus M\}$ where $\mathcal{A}_{2}(1)$ is the sub-algebra of $\mathcal{A}_{2}$ generated by $\mathrm{Sq}^{1}$ and $\mathrm{Sq}^{2}$, and $M$ is a graded $\mathcal{A}_{2}(1)$-module with the degree $i$ homogeneous part $M_{i}=0$ for $i<8$.

For $t-s<8$, we can identify the $E_{2}$ page with

[TABLE]

$\operatorname{H}^{*+3}(M{\rm SO}(3),\mathbb{Z}_{2})=\mathbb{Z}_{2}[w_{2}^{\prime},w_{3}^{\prime}]U$ where $U$ is the Thom class and $w_{i}^{\prime}$ is the Stiefel-Whitney class of the universal ${\rm SO}(3)$ bundle.

The $\mathcal{A}_{2}(1)$-module structure of $\operatorname{H}^{*+3}(M{\rm SO}(3),\mathbb{Z}_{2})$ is shown in Figure 4.

$\operatorname{H}^{*+2}(M\mathbb{Z}_{4},\mathbb{Z}_{2})=(\mathbb{Z}_{2}[\tilde{b}]\otimes\Lambda_{\mathbb{Z}_{2}}(\tilde{a}))U$ where $U$ is the Thom class, $\tilde{a}$ is the generator of $\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{2})$, $\tilde{b}$ is the generator of $\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{2})$.

The $\mathcal{A}_{2}(1)$-module structure of $\operatorname{H}^{*+2}(M\mathbb{Z}_{4},\mathbb{Z}_{2})$ is shown in Figure 5.

$\operatorname{H}^{*}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})=\mathbb{Z}_{2}[x_{2},x_{3},x_{5},\dots]$ where $x_{2}$ is the generator of $\operatorname{H}^{2}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$, $x_{3}=\mathrm{Sq}^{1}x_{2}$, $x_{5}=\mathrm{Sq}^{2}x_{3}$, etc.

The $\mathcal{A}_{2}(1)$-module structure of $\operatorname{H}^{*}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$ is shown in Figure 6.

The $\mathcal{A}_{2}(1)$-module structure of $\operatorname{H}^{*+3}(M{\rm SO}(3),\mathbb{Z}_{2})\otimes\operatorname{H}^{*+2}(M\mathbb{Z}_{4},\mathbb{Z}_{2})\otimes\operatorname{H}^{*}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$ is shown in Figure 7.

There is a differential $d_{2}$ corresponds to the $(2,4)$-Bockstein May and Milgram (1981) as indicated in Figure 5. There is also a differential $d_{2}$ maps $x_{2}x_{3}+x_{5}$ to $x_{2}^{2}h_{0}^{2}$ since $\beta_{(2,4)}(\mathcal{P}_{2}(x_{2}))=x_{2}x_{3}+x_{5}$ Wan and Wang (2018a). Since $\beta_{(2,4)}(a\mathcal{P}_{2}(x_{2}))=\tilde{b}x_{2}^{2}+\tilde{a}(x_{2}x_{3}+x_{5})$, there is a differential $d_{2}$ maps $\tilde{b}x_{2}^{2}+\tilde{a}(x_{2}x_{3}+x_{5})$ to $\tilde{a}x_{2}^{2}h_{0}^{2}$.

Note that the $\mathcal{A}_{2}(1)$-module structure of $\operatorname{H}^{*+3}(M{\rm SO}(3),\mathbb{Z}_{2})\otimes\operatorname{H}^{*+2}(M\mathbb{Z}_{4},\mathbb{Z}_{2})$ is contained in that of $\operatorname{H}^{*+3}(M{\rm SO}(3),\mathbb{Z}_{2})\otimes\operatorname{H}^{*+2}(M\mathbb{Z}_{4},\mathbb{Z}_{2})\otimes\operatorname{H}^{*}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$, we draw the $E_{2}$ page for it individually in Figure 8. The rest part is shown in Figure 9.

See Table 2 for the bordism group data.

See Table 3 for the bordism group data.

A.1.2 Manifold generator

Now we determine the manifold generator of the $\mathbb{Z}_{4}$-valued invariant $a\cup\mathcal{P}_{2}(x_{2})$.

[TABLE]

Here bordism is an equivalence relation. $(M,f,g)$ and $(M^{\prime},f^{\prime},g^{\prime})$ are bordant if there exists a 6-manifold ${\cal M}$ and maps $F:{\cal M}\to\mathrm{B}({\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))$, $G:{\cal M}\to\mathrm{B}^{2}\mathbb{Z}_{2}$ such that the boundary of ${\cal M}$ is the disjoint union of $M$ and $M^{\prime}$ and the induced ${\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})$ structures on $M$ and $M^{\prime}$ from that determined by $F$ on ${\cal M}$ coincide with those determined by $f$ and $f^{\prime}$ respectively, and $G|_{M}=g$, $G|_{M^{\prime}}=g^{\prime}$.

We have the homotopy pullback square (71).

In order to give a map $f:M\to\mathrm{B}({\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))$, we need only give maps $f_{1}:M\to\mathrm{B}{\rm SO}$, $f_{2}:M\to\mathrm{B}{\rm SO}(3)$ and $f_{3}:M\to\mathrm{B}\mathbb{Z}_{4}$ with $f_{1}^{*}(w_{2})=f_{2}^{*}(w_{2}^{\prime})+f_{3}^{*}(\tilde{b})$.

The bordism invariant $a\cup\mathcal{P}_{2}(x_{2})$ is actually $f_{3}^{*}(a)\cup\mathcal{P}_{2}(g^{*}(x_{2}))=f_{3}\cup\mathcal{P}_{2}(g)$.

Now let $M$ be the Lens space $S^{5}/\mathbb{Z}_{4}$, $M$ is orientable but not spin.

Take $f_{1}=TM$ (since $M$ is orientable, the tangent bundle $TM$ determines a map $M\to\mathrm{B}{\rm SO}$), $f_{2}=0$, $f_{3}$ be the generator of $\operatorname{H}^{1}(M,\mathbb{Z}_{4})$.

By the cell structure of the Lens space, $f_{3}$ induces a chain map between the cellular chain complexes of $M$ and $\mathrm{B}\mathbb{Z}_{4}$, we draw the chain map below degree 2:

[TABLE]

So $f_{3}^{*}(\tilde{b})$ is nonzero, since $f_{1}^{*}(w_{2})$ is also nonzero, the cohomology group $\operatorname{H}^{2}(M,\mathbb{Z}_{2})$ is $\mathbb{Z}_{2}$, we have a commutative diagram

[TABLE]

So we get a map $f:M\to\mathrm{B}({\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))$.

Take $g=w_{2}(TM)$, and

[TABLE]

The partition function is

[TABLE]

So $(M,f,g)$ is the manifold generator of the $\mathbb{Z}_{4}$-valued invariant $f_{3}\cup\mathcal{P}_{2}(g)$.

A.2 Pullback trivialization

Consider the pullback of $\mathrm{B}({\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8}))$ to $\mathrm{B}{\rm Spin}\times\mathrm{B}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})$:

[TABLE]

Since $w_{2}=0$ in Spin, so $w_{2}x_{3}=w_{3}x_{2}$ is trivialized.

Furthermore, consider the pullback of $\mathrm{B}{\rm Spin}\times\mathrm{B}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})$ to $\mathrm{B}{\rm Spin}\times\mathrm{B}{\rm SU}(2)\times\mathrm{B}\mathbb{Z}_{8}$:

[TABLE]

To simplify the computation, we only compute $\Omega_{5}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2})$ which is a subgroup of $\Omega_{5}^{{\rm Spin}}(\mathrm{B}{\rm SU}(2)\times\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2})$.

Note that $\mathcal{P}_{2}(x_{2})=x_{2}^{2}=\mathrm{Sq}^{2}(x_{2})=(w_{2}(TM)+w_{1}(TM)^{2})x_{2}=0\mod 2$ on Spin manifolds where we have used the Wu formula, so $\mathcal{P}_{2}(x_{2})$ can be divided by 2.

A.2.1 Computation

We have the Adams spectral sequence

[TABLE]

For $t-s<8$,

[TABLE]

The $\mathcal{A}_{2}(1)$-module structure of $\operatorname{H}^{*}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{2})\otimes\operatorname{H}^{*}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$ is shown in Figure 10.

Note that the $\mathcal{A}_{2}(1)$-module structure of $\operatorname{H}^{*}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{2})$ is contained in that of $\operatorname{H}^{*}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{2})\otimes\operatorname{H}^{*}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$, we draw the $E_{2}$ page for it individually in Figure 11. The rest part is shown in Figure 12.

There is a differential $d_{3}$ corresponding to the $(2,8)$-Bockstein May and Milgram (1981) as indicated in Figure 10 and a differential $d_{2}$ corresponding to the $(2,4)$-Bockstein $\beta_{(2,4)}(\mathcal{P}_{2}(x_{2}))=x_{2}x_{3}+x_{5}$.

See Table 4 for the bordism group data.

See Table 5 for the bordism group data.

One $\mathbb{Z}_{2}$-valued bordism invariant of $\Omega_{5}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2})$ is $\tilde{a}\cup\frac{\mathcal{P}_{2}(x_{2})}{2}$. Here $\tilde{a}$ is the generator of $\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{2})$, $x_{2}$ is the generator of $\operatorname{H}^{2}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$.

A.2.2 Further trivialization: first approach

Define $\mathbb{G}$ to be a group which sits in a homotopy pullback square

[TABLE]

where $j_{1}^{*}(w_{2})=j_{2}^{*}(w_{2}^{\prime}+\tilde{b})$, $\tilde{b}$ is the generator of $\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{4},\mathbb{Z}_{2})$, $w_{2}=w_{2}(TM)$ is the Stiefel-Whitney class of the tangent bundle $TM$, $w_{2}^{\prime}=w_{2}^{\prime}({\rm SO}(3))$ is the Stiefel-Whitney class of the universal ${\rm SO}(3)$ bundle.

In general, if we have a homotopy pullback square

[TABLE]

then there is a fiber sequence

[TABLE]

where $\Omega Z$ is the loop space of $Z$.

So there is a fiber sequence

[TABLE]

where the last map is $(u,v)\mapsto j_{1}^{*}(w_{2})(u)-x_{2}(v)=j_{2}^{*}(w_{2}^{\prime}+\tilde{b})(u)-x_{2}(v)$.

Then we define $\mathbb{G}^{\prime}$ to be a group which sits in a homotopy pullback square

[TABLE]

Since $w_{2}$ is identified with $x_{2}$ in $\mathrm{B}\mathbb{G}$, it is trivialized in $\mathrm{B}\mathbb{G}^{\prime}$ because $x_{2}=w_{2}=0$ due to the spin structure, so $a\cup\mathcal{P}_{2}(x_{2})$ is clearly trivialized by being pulled back to $\Omega_{5}^{\mathbb{G}^{\prime}}$.

Although our starting point was the symmetry-extension, this turns out to be a symmetry breaking case in disguise.

A.2.3 Further trivialization: second approach

Define $\mathbb{G}$ to be a group which sits in a homotopy pullback square

[TABLE]

In general, if we have a homotopy pullback square

[TABLE]

then there is a fiber sequence

[TABLE]

where $\Omega Z$ is the loop space of $Z$.

So there is a fiber sequence

[TABLE]

where the last map is $(u,v)\mapsto j_{3}^{*}(\tilde{b})(u)-x_{2}(v)$.

Since $\mathcal{P}_{2}(x_{2})=x_{2}\underset{{\scriptscriptstyle 0}}{\cup}x_{2}+x_{2}\underset{{\scriptscriptstyle 1}}{\cup}\delta x_{2}$, $\delta x_{2}=2\mathrm{Sq}^{1}x_{2}$, $x_{2}\underset{{\scriptscriptstyle 0}}{\cup}x_{2}=x_{2}^{2}$, so $\frac{\mathcal{P}_{2}(x_{2})}{2}=\frac{x_{2}^{2}}{2}+x_{2}\underset{{\scriptscriptstyle 1}}{\cup}\mathrm{Sq}^{1}x_{2}$.

Since $x_{2}$ is identified with $\tilde{b}=\beta_{(2,8)}a$ in $\mathrm{B}\mathbb{G}$ where $a\in\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{8},\mathbb{Z}_{8})$ and $\mathrm{Sq}^{1}\beta_{(2,8)}=0$ Wan and Wang (2018a), so $\tilde{a}\cup(x_{2}\underset{{\scriptscriptstyle 1}}{\cup}\mathrm{Sq}^{1}x_{2})$ is trivialized in $\Omega_{5}^{\mathbb{G}}$.

Note that $\tilde{a}\cup\frac{x_{2}^{2}}{2}$ is still not trivialized.

This is also a symmetry breaking case, since $x_{2}$ is locked with $a$. In physics, the locking between two probed background fields means that the global symmetry between two sectors are locked together, thus which results in global symmetry breaking.

A.2.4 Further trivialization: third approach

Consider the pullback of $\mathrm{B}^{2}\mathbb{Z}_{2}$ to $\mathrm{B}^{2}\mathbb{Z}_{4}$:

[TABLE]

Since $\frac{\mathcal{P}_{2}(x_{2})}{2}=\frac{x_{2}^{2}}{2}+x_{2}\underset{{\scriptscriptstyle 1}}{\cup}\mathrm{Sq}^{1}x_{2}$, $x_{2}\in\operatorname{H}^{2}(\mathrm{B}^{2}\mathbb{Z}_{2},\mathbb{Z}_{2})$ is pulled back to $\tilde{x}_{2}\in\operatorname{H}^{2}(\mathrm{B}^{2}\mathbb{Z}_{4},\mathbb{Z}_{2})$ and the following diagram

[TABLE]

is commutative by the naturality of Bockstein homomorphism, we have $\mathrm{Sq}^{1}\tilde{x}_{2}=0$, so $\tilde{a}\cup(x_{2}\underset{{\scriptscriptstyle 1}}{\cup}\mathrm{Sq}^{1}x_{2})$ is trivialized in $\Omega_{5}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{4})$.

Note that $\tilde{a}\cup\frac{x_{2}^{2}}{2}$ is still not trivialized.

A.2.5 Summary

The term $\tilde{a}\cup\frac{x_{2}^{2}}{2}$ cannot be trivialized.

Consider $M=S^{1}\times S^{2}\times S^{2}$, the partition function is

[TABLE]

Since

[TABLE]

where the two generators of $\operatorname{H}^{2}(S^{2}\times S^{2},\mathbb{Z})$ are $a,b$, the generator of $\operatorname{H}^{4}(S^{2}\times S^{2},\mathbb{Z})$ is $ab$.

No matter how to pullback, when $x_{2}=a+b\mod 2$, $(-1)^{k\int_{S^{2}\times S^{2}}\frac{x_{2}^{2}}{2}}=(-1)^{k}$ can be nontrivial.

This conclusion will be stated more formally and proved in the next appendix.

In this appendix, we compute the bordism group $\Omega_{5}^{{\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})}(\mathrm{B}^{2}\mathbb{Z}_{2})$ and find a bordism invariant $a\cup\mathcal{P}_{2}(x_{2})$ of it. Then we find the manifold generator of $a\cup\mathcal{P}_{2}(x_{2})$, and consider the pullback trivialization problem of $a\cup\mathcal{P}_{2}(x_{2})$. We first compute the bordism group $\Omega_{5}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2})$, and find that $a\cup\mathcal{P}_{2}(x_{2})$ becomes $\tilde{a}\cup\frac{\mathcal{P}_{2}(x_{2})}{2}$ in $\Omega_{5}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2})$. Moreover, we find that the summand $\tilde{a}\cup(x_{2}\underset{{\scriptscriptstyle 1}}{\cup}\mathrm{Sq}^{1}x_{2})$ of $\tilde{a}\cup\frac{\mathcal{P}_{2}(x_{2})}{2}$ can be trivialized (note that $\mathcal{P}_{2}(x_{2})=x_{2}^{2}+2x_{2}\underset{{\scriptscriptstyle 1}}{\cup}\mathrm{Sq}^{1}x_{2}$), but $\tilde{a}\cup\frac{x_{2}^{2}}{2}$ cannot be trivialized. We conclude that $a\cup\mathcal{P}_{2}(x_{2})$ cannot be trivialized via extending the global symmetry by 0-form symmetry and 1-form symmetry.

Appendix B Proof: a counterexample

By direct computation, we find that $a\cup\mathcal{P}_{2}(x_{2})$ is a bordism invariant of $\Omega_{5}^{{\rm Spin}\times_{\mathbb{Z}_{2}}({\rm SU}(2)\times_{\mathbb{Z}_{2}}\mathbb{Z}_{8})}(\mathrm{B}^{2}\mathbb{Z}_{2})$.

We consider the trivialization problem: Can we trivialize the topological term $a\cup\mathcal{P}_{2}(x_{2})$ via extending the global symmetry by 0-form $K_{[0]}$ symmetry and 1-form $K_{[1]}$ symmetry?

We can reformulate it mathematically: Can we find find finite abelian groups $K_{[0]}$ and $K_{[1]}$ such that

[TABLE]

is a fibration and $(fg)^{*}(a\cup\mathcal{P}_{2}(x_{2}))=0$ for any 5-manifold $M$ and any map $g:M\to\mathrm{B}\mathbb{G}$?

There is a group homomorphism:

[TABLE]

So the trivialization problem is asking whether we can find $\mathbb{G}$ and $f$ such that $\phi^{*}(a\cup\mathcal{P}_{2}(x_{2}))=0$ for any $(M,g)\in\Omega_{5}^{\mathbb{G}}$.

By direct computation, we find that $a\cup\mathcal{P}_{2}(x_{2})$ becomes $\tilde{a}\cup\frac{\mathcal{P}_{2}(x_{2})}{2}$ in $\Omega_{5}^{{\rm Spin}}(\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2})$.

Our main result is

Claim 1: We cannot find finite abelian groups $K_{[0]}$ and $K_{[1]}$ such that

[TABLE]

is a fibration and $(fg)^{*}(a\cup\mathcal{P}_{2}(x_{2}))=0$ for any 5-manifold $M$ and any map $g:M\to\mathrm{B}\mathbb{G}$.

Claim 2: We cannot find finite abelian groups $K_{[0]}$ and $K_{[1]}$ such that

[TABLE]

is a fibration and $(fg)^{*}(\tilde{a}\cup\frac{\mathcal{P}_{2}(x_{2})}{2})=0$ for any 5-manifold $M$ and any map $g:M\to\mathrm{B}\mathbb{G}$.

Clearly Claim 2 implies Claim 1 since if we can find abelian groups $K_{[0]}$ and $K_{[1]}$ such that

[TABLE]

is a fibration and $(fg)^{*}(a\cup\mathcal{P}_{2}(x_{2}))=0$ for any 5-manifold $M$ and any map $g:M\to\mathrm{B}\mathbb{G}$, then we can define $\mathbb{G}^{\prime}$ which sits in a homotopy pullback square

[TABLE]

Then

[TABLE]

is a fibration and $(f^{\prime}g^{\prime})^{*}(\tilde{a}\cup\frac{\mathcal{P}_{2}(x_{2})}{2})=0$ for any 5-manifold $M$ and any map $g^{\prime}:M\to\mathrm{B}\mathbb{G}^{\prime}$.

Since $\operatorname{H}_{i}(\mathrm{B}{\rm Spin},\mathbb{Z})=0$ for $i=1,2,3$, $\operatorname{H}^{2}(\mathrm{B}{\rm Spin},K_{[0]})=\operatorname{H}^{3}(\mathrm{B}{\rm Spin},K_{[1]})=0$ by the universal coefficient theorem, similarly we have $\operatorname{H}^{2}(\mathrm{B}{\rm SU}(2),K_{[0]})=\operatorname{H}^{3}(\mathrm{B}{\rm SU}(2),K_{[1]})=0$, so in order to prove claim 2, we need only prove

Claim 3: We cannot find finite abelian groups $K_{[0]}$ and $K_{[1]}$ such that

[TABLE]

is a fibration and $(fg)^{*}(\tilde{a}\cup\frac{\mathcal{P}_{2}(x_{2})}{2})=0$ for any Spin 5-manifold $M$ and any map $g:M\to\mathrm{B}\mathbb{G}$.

We prove Claim 3 by finding a counterexample.

For $M=S^{1}\times S^{2}\times S^{2}$, let $a,b$ be the generators of $\operatorname{H}^{2}(S^{2}\times S^{2},\mathbb{Z}_{2})$, $c$ be the generator of $\operatorname{H}^{1}(S^{1},\mathbb{Z}_{8})$, let $h:M\to\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2}$ be given by $(c,a+b)$. The lifting problem

[TABLE]

has a solution, but $(c\mod 2)\cup\frac{\mathcal{P}_{2}(a+b)}{2}\neq 0$.

In general, if $F\to E\stackrel{{\scriptstyle p}}{{\to}}B\stackrel{{\scriptstyle q}}{{\to}}\Sigma F$ is a fiber sequence, then $[M,F]\to[M,E]\stackrel{{\scriptstyle p_{*}}}{{\to}}[M,B]\stackrel{{\scriptstyle q_{*}}}{{\to}}[M,\Sigma F]$ is an exact sequence of abelian groups, so the lifting problem has a solution if and only if $q_{*}(h)=0$ where $q:\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2}\to{\mathrm{B}^{2}K_{[0]}\ltimes\mathrm{B}^{3}K_{[1]}}$. So we need prove that $q\circ h=0$.

Again apply the exact sequence $[M,\Sigma F]\to[M,\Sigma E]\to[M,\Sigma B]$ to $M=S^{1}\times S^{2}\times S^{2}$ and the fibration

[TABLE]

we get that if the image of $q\circ h$ in $[M,\Sigma B]$ is zero, then $q\circ h$ is the image of some map in $[M,\Sigma F]$.

We can write the composition $q^{\prime}$ of the map ${\mathrm{B}^{2}K_{[0]}\ltimes\mathrm{B}^{3}K_{[1]}}\to\mathrm{B}^{2}K_{[0]}$ and $q$ as

[TABLE]

where $q_{1}\in\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{8},K_{[0]})$, $q_{2}\in\operatorname{H}^{2}(\mathrm{B}^{2}\mathbb{Z}_{2},K_{[0]})$. We assume that $q_{2}=0$ to ensure that the 1-form symmetry (here the 1-form $\mathbb{Z}_{2}^{e}$-symmetry) is not broken (write $f:\mathrm{B}\mathbb{G}\to\mathrm{B}\mathbb{Z}_{8}\times\mathrm{B}^{2}\mathbb{Z}_{2}$ as $f=(f_{1},f_{2})$, then $q_{1}\circ f_{1}+q_{2}\circ f_{2}=0$, if $q_{2}$ is nonzero, then $f_{1}$ and $f_{2}$ are locked, hence the 1-form symmetry is broken).

We see that $q^{\prime}\circ h=q_{1}\circ c+q_{2}\circ(a+b)$, since $q_{1}\circ c=0$ and $q_{2}\circ(a+b)=0$ for $M=S^{1}\times S^{2}\times S^{2}$, so $q\circ h$ is the image of some map in $[M,\mathrm{B}^{3}K_{[1]}]$. So $q\circ h=q_{3}\circ c+q_{4}\circ(a+b)$ where $q_{3}\in\operatorname{H}^{3}(\mathrm{B}\mathbb{Z}_{8},K_{[1]})$, $q_{4}\in\operatorname{H}^{3}(\mathrm{B}^{2}\mathbb{Z}_{2},K_{[1]})$. Since $q_{3}\circ c=0$ and $q_{4}\circ(a+b)=0$ for $M=S^{1}\times S^{2}\times S^{2}$, we have proven that $q\circ h=0$.

In this appendix, we give a proof of the conclusion in the previous appendix. This answers the first question (Question 1) in Sec. I.

Appendix C Pullback trivialization of $\mathcal{P}_{2}(B_{2})$ in $\Omega_{4}^{{\rm SO}}(\mathrm{B}^{2}\mathbb{Z}_{2})$

There is a group homomorphism:

[TABLE]

We want to extend the 1-form $\mathbb{Z}_{2}$ symmetry by 0-form $K_{[0]}$ symmetry and 1-form $K_{[1]}$ symmetry such that $\rho^{*}\mathcal{P}_{2}(\tilde{g})=\mathcal{P}_{2}(fg)=0$ for any $(M,g)\in\Omega_{4}^{{\rm SO}}(X)$ where $(M,\tilde{g})\in\Omega_{4}^{{\rm SO}}(\mathrm{B}^{2}\mathbb{Z}_{2})$.

We consider the trivialization problem: Does there exist a fibration $f:X\to\mathrm{B}^{2}\mathbb{Z}_{2}$ with fiber ${\mathrm{B}K_{[0]}\ltimes\mathrm{B}^{2}K_{[1]}}$ where $K_{[0]}$ and $K_{[1]}$ are finite abelian groups such that $\mathcal{P}_{2}(fg)=0$ for any oriented 4-manifold $M$ and any map $g:M\to X$?

The answer to this problem is negative, for $M=S^{2}\times S^{2}$, let $a,b$ be the generators of $\operatorname{H}^{2}(S^{2}\times S^{2},\mathbb{Z}_{2})$. The lifting problem

[TABLE]

always has a solution, but $\mathcal{P}_{2}(fg)=\mathcal{P}_{2}(a+b)$ is nontrivial. Similarly as before, we need only prove that the composition map $S^{2}\times S^{2}\stackrel{{\scriptstyle a+b}}{{\to}}\mathrm{B}^{2}\mathbb{Z}_{2}\stackrel{{\scriptstyle{q}}}{{\to}}{\mathrm{B}^{2}K_{[0]}\ltimes\mathrm{B}^{3}K_{[1]}}$ is zero.

This can be proven similarly as before.

So $\mathcal{P}_{2}(x_{2})$ cannot be trivialized.

In this appendix, we consider the pullback trivialization problem of $\mathcal{P}_{2}(x_{2})$, we give a similar proof that $\mathcal{P}_{2}(x_{2})$ also cannot be trivialized via extending the global symmetry by 0-form symmetry and 1-form symmetry. This answers the second question (Question 2) in Sec. I.