Adjoint QCD$_4$, Symmetry-Enriched TQFT and Higher Symmetry-Extension

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 criticality and ultraviolet-infrared (UV-IR) duality in 3+1 spacetime dimensions.


I. INTRODUCTION AND SUMMARY OF MAIN RESULTS
Recent work explores the candidate phases of the adjoint quantum chromodynamics in 4 dimensional spacetime (QCD 4 ) with an SU(2) gauge group and two massless adjoint Weyl fermions (equivalently, two massless adjoint Majorana fermions, or one massless adjoint Dirac fermion) [1][2][3][4]. 1 This adjoint QCD 4 has a 1-form electric Z 2 center global symmetry, which is a generalized global symmetry of higher differential form [5]. This adjoint QCD 4 has the SU(2) gauge theory coupling to the matter fields in the adjoint representation, thus it gains a 1-form electric Z 2 center symmetry; while the usual fundamental QCD 4 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 [6] of the quantum phases of the adjoint QCD 4 . (See Sec. II for more detailed information regarding the global symmetries and 't Hooft anomalies of this adjoint QCD 4 .) Given the adjoint QCD 4 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 ≤ 5. Viewing the adjoint QCD 4 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 [2] and Bi-Senthil [3] propose its low energy candidate phases at IR, saturating the higher 't Hooft anomalies involving the 1-form symmetry.
In particular, Bi-Senthil [3] suggests a fully-symmetric 4d TQFT to saturate higher 't Hooft anomalies without breaking any UV global symmetries of the adjoint QCD 4 . Namely, an interesting RG flow from Bi-Senthil [3] speculates that: Adjoint QCD 4 at UV 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 0form symmetry but no 1-form symmetry, so the massless fermion sector alone cannot saturate the higher anomaly of the adjoint QCD 4 . Thus the crucial and nontrivial check on Bi-Senthil [3] 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: 1. We generalize the symmetry-extension method of Wang-Wen-Witten [7] to higher symmetries. We formulate a higher group cohomology or a higher cobordism theory approach of Ref. [7] 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 shortrange entangled SPT state, see [8] 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. [7] 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. [7] 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 (≥ 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 → K → H → G → 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. [7] but we improve the formulation by replacing the 0-form G symmetry to include generalized higher symmetries of Ref. [5].
2. We apply the above generalized higher symmetryextension method from Ref. [7] either to construct the higher-symmetric anomalous TQFTs, for adjoint QCD 4 ; 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 Z 2NcN f = Z 8 symmetry, with N c = N f = 2) and the 1-form electric center symmetry (denoted as Z e 2, [1] = Z 2, [1] ). This higher anomaly is abbreviated as the Type I higher anomaly in Ref. [3]. The Type I anomaly in 4d has a Z 4 class (below k ∈ Z 4 class), one can explicitly write down the 5d topological (abbreviated as "topo.") invariant [2] which is a cobordism invariant (see mathematical details in [9] and Sec. A), Type I anomaly/topo. invariant : e i kπ 2 A∪P2(B2) . (2) Here A is the Z 4 -valued background 1-form gauge field coupling to the 0-form Z 8 /Z F 2 = Z 4 part of the axial global symmetry. The Z F 2 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 Z 2 -valued background 2-form gauge field coupling to the 1-form Z e 2,[1] -symmetry. The ∪ is the cup product, and the 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 [3] in general, it could be due to the limitation of the symmetry extension [7] we used. Nevertheless, it is known that [7]'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 [7] 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 symmetryextension 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 Z 2, [1] ). This higher anomaly is abbreviated as the Type II higher anomaly in Ref. [3]. We can explicitly write down the 5d topological (abbreviated as "topo.") invariant [2] as the following cobordism invariant (see mathematical details in [9] and Sec. A), Type II anomaly/topo. invariant : Here w j (T M ) has the w j as the j-th Stiefel-Whitney (SW) class [10], as the probed background spacetime M connection over the spacetime tangent bundle T M . The 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 [3] via the symmetry-extension alone, generalizing the method of [7]: Question 1. Can we trivialize the topological term A ∪ 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 ∪ P 2 (B 2 ) of the bordism group Ω Spin× Z 2 Z8 5 (B 2 Z 2 ). 2 We prove that the answer is negative. Question 2. We also solve the trivialization problem of the cobordism invariant P 2 (B 2 ) of the bordism group Ω SO 4 (B 2 Z 2 ): Can we trivialize the topological term 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. [2].
In Sec. III, we discuss the higher symmetry-extension generalization of [7], 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 ∪ P 2 (B 2 ).
In Appendix C, we show a similar obstruction: The 4d 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 [9] and [11] (e.g. the Appendix of [11]), 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 [9].

II. THEORY OF ADJOINT QCD4
We have an SU(2) gauge theory coupled to 2 3 (N f = 2 for the 2, and the 3 for the triplet) adjoint Weyl fermions in the adjoint representation of SU (2). The path integral (or partition function) of this adjoint QCD 4 , in the Minkowski signature, viewed as a UV QFT theory can be written as: 2 In this work, we will use the term dd "cobordism invariant" to describe the dd topological term or dd (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 Ω G d , while we denote the cobordism group Ω d G .
The eq. (5) contains the first term as the Dirac Lagrangian, and the second term as the Yang-Mills Lagrangian. The [D .
. . ] is the path integral measure for the quantum fields. Theσ µ ≡ (1, − σ) contains the standard Pauli sigma matrices σ. Here the Weyl fermion ψ jb α has: • the flavor index j (of the classical U(2) flavor symmetry, or more precisely the SU(2)×Z8 Z F 2 flavor symmetry in a quantum theory, see later), • the gauge index b of the gauge SU(2) of adjoint triplet, • the Lorentz index α of the Lorentz group. The hermitian conjugation of fermion field isψ b j = ψ jb † . With the Lorentz index, we haveψ b αj = ψ jb † α , following the standard supersymmetry notation.
Here are some other comments: • The g is the dimensionless Yang-Mills coupling, which is a running coupling in the quantum theory.
• The F is the SU(2) gauge field a's 2-form field strength. The F is the F 's Hodge dual.
• One can consider the deformation of the theory as extra terms in the . . . , such as the mass deformation [4], e.g. (m ij δ b b ψ ib α αβ ψ jb β + c.c.). In the classical theory, we can add the θ-term, However, in the quantum theory, with the presence of the fermion fields ψ, we can rotate the θ away. If we have the mass term for the fermions, we can absorb the θ-term into the complex fermion mass matrix in the mass deformation.

A. Global Symmetries
The global symmetries of the adjoint QCD 4 eq. (4) has been analyzed systematically in [2]. Here we recap the results and will write the results suitable for the cobordism theory analysis later in Appendices A to C. However, the axial symmetry U(1) A is broken down to a discrete axial symmetry Z 2NcN f ,A , which is Z 2NcN f ,A = Z 8 here, due to the Adler-Bell-Jackiw (ABJ) anomaly. 3 It is a standard calculation of the U (1) A -axial symmetry is explicitly broken by the dynamical SU(N c )-gauge instanton down to Z 2NcN f ,A -axial symmetry.
So the flavor symmetry is simply for the quantum theory. The SU(2) is also written as the SU(2) R as the R-symmetry thanks to the standard convention in N = 2 supersymmetric Yang-Mills theory (SYM) [13]. In the N = 2 SYM, the adjoint fermions are gauginos.
2. The 1-form center symmetry Z e 2, [1] ≡ 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 Z e 2, [1] measured by a 2-surface "charge" operator. The "charged" fundamental Wilson line (spin-1/2 representation of SU(2)) has an odd Z 2 charge. The odd half integer spin-n/2 representation of SU(2) has an odd 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) Z 2 charge of 1-form symmetry.
Importantly the 1-form center symmetry Z e 2, [1] is preserved means that the electric Wilson loop (eloop) is unbreakable, or called tension-ful [3]. Since the adjoint QCD has the 1-form center symmetry, we can use the 1-form center symmetry charged object to detect: • Confinement: If 1-form symmetry is preserved, and all the Wilson loops (of all representations) obey the area law.
• 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. Spacetime symmetry: In Lorentz signature, we have the Poincaré group symmetry which contains the Lorentz group. We also have the discrete CP T 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 CP T theorem. If we focus on orientable spacetime for the adjoint QCD in dd, we can consider the Spin(d) spacetime symmetry, for the purpose of classifying the 't Hooft anomalies through the cobordism theory [9,14]. If we consider the non-orientable spacetime for the adjoint QCD in dd, we should consider the Pin − (d) spacetime symmetry, for the purpose of classifying the 't Hooft anomalies through a cobordism theory, See Ref. [9,14]. This adjoint QCD is a fermionic theory, the spacetime symmetry G spacetime and the internal symmetry G internal shares the fermionic parity Z F 2 , so the precise way to write the full global symmetry would be: where the common Z F 2 is mod out, while the "× Z F 2 " notation follows [14].
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: Below we follow Ref. [9], which generalizes a theorem in a remarkable work of Freed-Hopkins [14]. Freed-Hopkins [14] 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. [9] generalizes [14] 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 [9], we write their corresponding bordism groups Ω d as: 4 • Bordism group for eq. (8): [1] ). (10) 4 Here BG means the classifying space of G, and pt means the point.

B. 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 QCD 4 .
Cordova-Dumitrescu [2] have captured several anomalies, which we now overview: 1. The SU(2) Witten anomaly [15] 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 QCD 4 is gapless instead of fully gapped.
3. The (Z 8,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.
5. Type I higher anomaly: mixing between the 1-form electric center symmetry (Z e 2, [1] ) and the 0-form discrete axial symmetry (Z 2NcN f = Z 8 ). We can write eq. (2) as see [9] for introducing the cup products, higher cup products and the Steenrod square Sq.
The UV theory as an adjoint QCD 4 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 [7], 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).

A. Summary of Ordinary Symmetry-Extension
Ref. [7] sets up the symmetry-extension problem as follows. Consider the dd 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 [7], but their underlying meanings are the same: 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. 6 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.) Q3. Mathematical and algebraic topology statement: Given a dd topological term of a group G, here the topological term can be: • the dd cocycle for a d-th cohomology group H d (BG, U(1)) in a group cohomology theory.
• the dd co/bordism invariant for a d-th cobordism group Ω d − (BG, U(1)) or bordism group Ω − d (BG) or bordism group, in a cobordism theory; 7 can we find an extended group H with G its quotient group, via a short exact sequence 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. [7] shows, valid for both the lattice Hamiltonian and the path integral construction, that the G-SPTs in dd can allow: • H-symmetry extended gapped boundary in any spacetime dimension d ≥ 2, • G-symmetry preserving and topological K-gauge theory gapped boundary: Topological emergent K-gauge theory with preserving global symmetry G on a bulk d ≥ 3.
Ref. [7] 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: 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 ≥ 2.
A2. For any G-anomaly in (d − 1)d given by a cocycle ν G d ∈ H d (BG, 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)) 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 Here r is the pullback operation, and δ is the coboundary operation. Namely, a G-cocycle becomes a H-coboundary, which splits to a one-lower dimensional H-cochains µ H d−1 , given the dimension d ≥ 2.
The proof of [7] has also been verified later by [17]. The related constructions similar to [7] are explored also in specific cases or from different perspectives in [18,19].

B. Higher Symmetry Generalization
Now we generalizes the approach in [7]. The short exact sequence of a group extension 1 → K → H r → G → 1 extended by a finite K given in [7] also implies an induced fiber sequence from the fibration 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. Our goal is to find a fibration where G and 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 G-SPTs protected by a 2-group G becomes trivial higher H-SPTs by pulling back G to H. 8 Similar to questions in Q1, Q2 and Q3 of Sec. III A, we ask a set of generalized questions: Q4. Condensed matter statement: Can we find a total 2-group H as a total space such that BG is BH's orbit (or base space), 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 (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 H and without any phase transition in the enlarged Hsymmetric quantum Hilbert space.
Q5. QFT or high energy particle physics statement: Given a (d−1)d 't Hooft anomaly in a higher group G, can we find an enlarged group H, with a total group H obeying eq. (15), such that the 't Hooft anomaly in G becomes anomaly-free in H? (i.e., the G-anomaly becomes trivial in H.) Q6. Mathematical and algebraic topology statement: Given a dd topological term of a higher group G, here the topological term can be: • the dd cocycle for a d-th cohomology group H d (BG, U(1)) in a higher group cohomology theory.
• the dd co/bordism invariant for a d-th cobordism group Ω d − (BG, U(1)) or bordism group Ω − d (BG) or bordism group, in a cobordism theory; 9 can we find an extended group H obeying eq. (15) such that the topological term of a group G can be pulled back to a trivial topological term of a group 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.

C. 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) e i π w2(T M )Sq 1 B2 = e i π w3(T M )B2 .
We have found that eq. (3) is a topological invariant in d = 5, for: • H d (B 2 Z 2 , U(1)) group cohomology of a higher classifying space finite group, as well as • Ω SO d (B 2 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.
1. The first way to trivialize this 4d Type II higher anomaly is extending the spacetime symmetry from special orthogonal group SO(d) = Spin(d)/Z F 2 to Spin(d): Here the "−" follows the earlier footnote 7.
This extension works since w 2 (T M ) = 0 vanishes on Spin manifold. Thus, eq. (3) is trivialized once we pull back eq. (3) into BSpin(d) × B 2 Z 2 . According to the interpretation in Sec. III A and Ref. [7], the fibration BZ 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. (16) as the generalized construction of [7] is that there is an emergent 1-form Z 2 gauge theory (dy-namically gauged from emergent 0-form global symmetry BZ 2 ), such that the 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. (16)). 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: 5d higher SPTs (4d higher anomaly) Here a is the Z 2 -valued 1-form gauge field (the standard notation as the 1-cochain in C 1 ), b is the Z 2 -valued 2-form gauge field (the standard notation as the 2-cochain in C 2 ), the δ is the coboundary operator here δ = 2Sq 1 , and we use the cup product ∪. See also our previous explanations around eq. (3) for notations. The . . . are additional coupling terms between dynamical gauge fields and background fields. The . . . also include additional sectors from the UV adjoint QCD 4 from eq. (4), in order to saturate the other anomalies. Note that the similar emergent dynamical spin structure with Z 2 gauge field has been studied in Ref. [26]. 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. (16) in terms of the full symmetry eq. (8): while the physical interpretation remains the same as eq. (16) and eq. (17).
2. The second way to trivialize this 4d Type II higher anomaly is extending the 1-form symmetry: This way works since B is pulled back toB ∈ H 2 (B 2 Z 4 , Z 2 ), and Sq 1B = 0 (see Appendix A 2 d).
According to the interpretation in Sec. III A and Ref. [7], the fibration B 2 Z 2 is associated to an emergent 1-form global symmetry Z 2, [1] which is anomaly-free and can be dynamically gauged. Indeed, the natural way to interpret the eq. (19) as the generalized construction of [7] is that there is an emergent 2-form Z 2 gauge theory (dynamically gauged from emergent 1form global symmetry BZ 2 ) with a 2-form gauge field b . The original 1-form Z e 2,[1] -symmetry acts projectively on the emergent 2-form Z 2 gauge theory, but the extended 1-form Z e 4,[1] -symmetry acts on it faithfully.
We can write the involved QFT sectors into a following partition function, which looks like the following locally: Here b is the Z 2 -valued 2-form gauge field (the standard notation as the 2-cochain in C 2 ), a is the 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. (20). The . . . are additional coupling terms between dynamical gauge fields and background fields.
The . . . also include additional sectors from the UV adjoint QCD 4 from eq. (4), in order to saturate the other anomalies. We can also write the extension of eq. (19) in terms of the full symmetry eq. (8): while the physical interpretation remains the same as eq. (19).
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 b) 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 Z 2 -symmetry to nothing. In comparison, this 1-form Z 2 -symmetry breaking is a different scenario from [2,3]. • In the second approach (details and notations explained in Appendix A 2 c), we define G to be a group which sits in a homotopy pullback square Hence we have a fiber sequence 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: But in this case,Ã ∪ B 2 2 is still not, and cannot be, trivialized.
In summary, we finally conclude that when k is odd, k = 1, 3 ∈ Z 4 , the Type I anomaly eq. (2) cannot be trivialized by extensions and give a proof in Appendix B. In comparison, Ref. [3] proposes a full symmetry-preserving TQFT different all of our scenarios above, which contradicts to our proof in Appendix B. E. Saturate both Type I (for even class in Z4) and II anomalies When k = 2, such that the Type I anomaly survives as only a Z 2 subclass (even k) in the original k ∈ Z 4 class (of kA ∪ P 2 (B 2 )), however, we can actually trivialize the Z 2 subclass of Type I anomaly and the full Type II anomaly together via the fibration: The above is achieved by combining both eq. (16) and eq. (23) into eq. (18). Since we only care k = 2, this also means that the Z 8,A -symmetry only needs to be survived as a Z 4,A -symmetry. Physically this means that Z 8,A -symmetry can be spontaneously broken down to a Z 4,A -symmetry. Thus the eq. (27) really implies a fibration of a smaller symmetry (e.g. a smaller classifying space) as: For such a 4d TQFT preserving a ( )-chiral symmetry and 1-form Z e 2,[1] -symmetry (from UV adjoint QCD 4 ), 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: again the 1-form gauge field a can be regarded as the difference between two spin-structures; the 1-form emergent dynamical Z 2 gauge field a is associated to a dynamical spin structure (similar to a situation in Ref. [26]). We note that the . . . terms can involve additional 't Hooft anomaly cancellation for the UV's adjoint QCD 4 , such as the gapless sector proposed in [1][2][3][4]. Besides, the . . . 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. (28) already suggests a formation definition of TQFTs (based on the extension construction of bulk-boundary coupled TQFTs, see [7] and related constructions in [27,28]), it may be worthwhile to formulate a cochain or continuum TQFT description following [27,28] -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 [29] like gauge theory, similar to the continuum TQFT formulation given in [22,30].

F. Other Examples
In our companion work [12], we consider similar trivialization problem for 5d topological invariants of 4d Yang-Mills SU(N) gauge theory (in particular at θ = π) anomaly.
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 Ref. [7] 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 QCD 4 . Then we comment about the constraints on the deconfined quantum critical phenomena, or so called the deconfined quantum critical point (dQCP) [31], in 3+1 spacetime dimensions [3].  Fig. 1, is that: • At lower N f , there should be a confinement (IR confinement) and chiral symmetry breaking (IR ChSB).
• 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 [32] and others.
• Let N asym.free f = 11 2 N c , when N f < N asym.free f , the UV theory is weak coupling known as the asymptotic freedom (or UV free) [33,34]. When N f > N asym.free f , the UV theory becomes strongly coupled while the coupling g flows weak at IR, at least perturbatively.

Possible Fates of the Dynamics of Adjoint QCD4 with N f Weyl fermions
Now we organize the possible fates of the dynamics of QCD 4 with N f Weyl fermions in adjoint representations of SU(N c ). The possible phase structure of dynamics of QCD 4 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 [2], we recap into a concise Fig. 2, while also list down the related Scenario 1, 2, 3, and 4, from Ref. [2,3], and from the list summarized in Sec. IV A 2.  The conventional wisdom teaches us that the phase structure of dynamics of adjoint QCD 4 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 θ-term eq. (6). At θ = 0, the phase is a trivially gapped confined phase (IR confinement) with no SPT state. However, at θ = π, the phase has mixed higher anomalies [35] and potentially newly found higher 't Hooft anomalies [12].
• At N f = 1, it is a pure N = 1 supersymmetric Yang-Mills gauge theory (SYM) [36]. Moreover, there are N c supersymmetric breaking vacua due to gaugino condensation [37], which breaks Z 2Nc down to Z 2 (simply Z F 2 ). This N = 1 SYM phase is also known to be confined through monopole condensation, by embedding into a N = 2 SYM theory with N c = 2 [13].
• 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 [32].
To proceed further, we recall that the UV internal global symmetry is [1] . Now we organize a list of possible fates of the dynamics of adjoint QCD 4 with N f Weyl fermions proposed from [2,3]. There are four scenarios, summarized in Table I and [13], there is a moduli space of the supersymmetric vacua, labeled by the expectation value of the complex u = Re u + i Im u ≡ Tr(φ 2 ) ∈ C for the N = 2 chiral operator. The τ ≡ τ U(1) ≡ θ 2π + 2π i e 2 = i is the special point at the self-dual value of the coupling τ . The coupling τ is for the Coulomb phase of U(1) gauge theory (the Coulomb branch for the moduli space of vacua). Even though the SYM is supersymmetric, Ref. [2] enumerates the possible vacua by supersymmetry-breaking deformations. Let us relate the supersymmetric vacua to the adjoint QCD4 vacua listed in the Scenarios [2]: The vacua of magnetic monopole point (u = Λ 2 ) and dyon point (u = −Λ 2 ) is related by the broken symmetry Z8,A generator, which gives rise a Scenario 1.
A generic vacua of u = 0, and u = ±Λ 2 on the plane is related to a Scenario 2.
The vacua of the u = 0 with a self-dual coupling τ is related to a Scenario 4.
1. The N c copies of (or more specifically here N c = 2) of 4d CP 1 sigma model at low energy with spontaneous symmetry breaking Goldstone modes, proposed by [2]. Its global symmetry: [1] .
In summary, the scenario 1 has: "chiral symmetry breaking, and confinement." (31) To digest better about the target space of 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: Then we obtain a relation for the scenario 1: or more precisely a relation: The CP 1 Z8,A Z2×Z F 2 has two copies of CP 1 as the target space, parametrizing the order parameter of the base manifold B (the base space or the orbit).

A free massless Dirac fermion (equivalently, two
massless Weyl fermions, or two massless Majorana fermions) and a Z 2 discrete gauge theory as a 4d TQFT with a Z 4 symmetry (spontaneously broken from the Z 8 symmetry), proposed by [2]. The IR symmetry is [1] .

A free massless Dirac fermion (equivalently, two
massless Weyl fermions, or two massless Majorana fermions) and a 4d TQFT preserving the full Z 8 symmetry, proposed by [3]. The two massless Weyl fermions actually have a U(2) continuous global symmetry. The IR symmetry we focus is: [1] .
In summary, the scenario 3 proposed that: "chiral symmetry fully preserved, and confinement." 4. A 4d U(1) gauge theory in Coloumb phase with a Z 2NcN f = Z 8 symmetry, proposed by [2]. The IR symmetry we focus is: The¨(. . . ) means a spontaneous symmetry breaking of . . . , thus for 1-form symmetry breaking here, it leads to a deconfinement of U(1) gauge theory. In summary, the scenario 4 proposed that: "chiral symmetry preserved, and deconfinement." (40) 5. Note that there is another scenario from Ref. [1] 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 Z 8,A → Z 4,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. [1] also notices later, the more complete scenario is adding a TQFT sector, following the Scenario 2.  TABLE I. The Scenario 1, 2, 3, and 4 are from the list summarized in Sec. IV A 2. The "SSB" stands for "spontaneous symmetry breaking." The¨(. . . ) means that symmetry (. . . ) leads to SSB. We find an obstruction for Scenario 3 based on the higher symmetry-extension construction of Ref. [7]. We should note that, educated by Ref. [2] and summarized in Fig. 3, the Scenario 1 is consistent with the supersymmetry (SUSY) breaking of N = 2 SYM from the magnetic monopole point and dyon point (as 2 copies of CP 1 model). The Scenario 2 is consistent with the SUSY breaking of N = 2 SYM from the generic point from u = 0, and u = ±Λ 2 . The Scenario 4 is consistent with the SUSY breaking of N = 2 SYM from the u = 0 with a self-dual coupling τ.

B. Deconfined Quantum Criticality in 3+1 Dimensions and More Comments
In this work, we obtain a higher-symmetry extension generalization of Ref. [7]'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. [2]), see eq. (28) and eq. (29). However, we are unable to obtain a symmetric anomalous TQFT proposed by Scenario 3 motivated by Bi-Senthil (Ref. [3]) based on a symmetryextension 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 [31], or a phase transition between two different SPT orders (see [3] and References therein). The adjoint QCD 4 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: I. We should remind the readers that the symmetryextension construction is fairly general enough to saturate a large class of higher 't Hooft anoma-lies of bosonic systems. Although the adjoint QCD 4 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.
II. Despite the fact that fully-symmetric TQFT under Scenario 3 cannot be obtained via our symmetryextension 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 [38,39] suggest that the adjoint QCD 4 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 [39], 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 I).
• At N f = 4, (as 2 adjoint Dirac fermion), Ref. [38] discusses the candidate IR theory. Ref. [38] 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 [38,39], we note that further lattice studies are still necessary.
Finally, we remark that many anomalies discussed in Sec. II B, following [2], are non-perturbative global anomalies instead of perturbative anomalies. The nonperturbative anomalies have classifications from finite groups (e.g. Z n classes), instead of a Z classification. Examples include the old and the new SU(2) anomalies [15,26], and also the recent higher 't Hooft anomalies of SU(N) YM gauge theory, see [35] and [12], and References therein. For these non-perturbative global anomalies, we can saturate certain 't Hooft anomalies of ordinary or higher global symmetries by symmetrypreserving TQFTs or so-called the long-ranged entangled topological order sectors, via our higher symmetryextension approach, see a companion work along this direction [12].

V. ACKNOWLEDGMENTS
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. We thank Edward Witten for helpful comments and sharing his proof from another perspective. 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 [4]. 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.
Since the computation involves no odd torsion, we can use the Adams spectral sequence Here Ext is the Ext functor, A 2 is the mod 2 Steenrod algebra, more precisely, Ext s,t A2 is the internal degree t part of the s-th derived functor of Hom * A2 . M T (Spin × Z2 (SU(2) × Z2 Z 8 )) is the Madsen-Tillmann spectrum of the group Spin × Z2 (SU(2) × Z2 Z 8 ), the bordism group Ω here ∧ is the smash product, X + is the disjoint union of the space X and a point. "⇒" means "convergent to". For more detail, see [9].
Similarly as the discussion in [11,41], we know where 2ξ is twice the sign representation, (BZ 4 ) 2ξ is the Thom space Thom(BZ 4 ; 2ξ), M Spin is the Thom spectrum of the group Spin, M SO (3) is the Thom spectrum of the group SO(3), Σ is the suspension. Note that (BZ 4 ) 2ξ = Σ −2 M Z 4 . We have a homotopy pullback square whereb is the generator of H 2 (BZ 4 , Z 2 ), w 2 = w 2 (T M ) is the Stiefel-Whitney class of the tangent bundle T M , w 2 = w 2 (SO(3)) is the Stiefel-Whitney class of the universal SO(3) bundle.
Hence we have the constraint is the subalgebra of A 2 generated by Sq 1 and Sq 2 and M is a M is a graded 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 H * +3 (M SO(3), Z 2 ) = Z 2 [w 2 , w 3 ]U where U is the Thom class and w i is the Stiefel-Whitney class of the universal SO(3) bundle.
The Figure 7.
i Ω Here a is the generator of H 1 (BZ4, Z4), b is the generator of H 2 (BZ4, Z4).ã = a mod 2,b = b mod 2. w2 = w2(T M ) is the Stiefel-Whitney class of the tangent bundle. Note that w2x3 = w3x2 (see [9]). w i is the Stiefel-Whitney class of the universal SO(3) bundle.

b. Manifold generator
Now we determine the manifold generator of the Z 4valued invariant a ∪ P 2 (x 2 ).
Here bordism is an equivalence relation. The bordism invariant a ∪ P 2 (x 2 ) is actually f * 3 (a) ∪ P 2 (g * (x 2 )) = f 3 ∪ P 2 (g). Now let M be the Lens space S 5 /Z 4 , M is orientable but not spin.
Take f 1 = T M (since M is orientable, the tangent bundle T M determines a map M → BSO), f 2 = 0, f 3 be the generator of H 1 (M, Z 4 ).
By the cell structure of the Lens space, f 3 induces a chain map between the cellular chain complexes of M and BZ 4 , we draw the chain map below degree 2: So f * 3 (b) is nonzero, since f * 1 (w 2 ) is also nonzero, the cohomology group H 2 (M, Z 2 ) is Z 2 , we have a commutative diagram (A11) So we get a map f : M → B(Spin × Z2 (SU(2) × Z2 Z 8 )). Take g = w 2 (T M ). So (M, f, g) is the manifold generator of the Z 4 -valued invariant f 3 ∪ P 2 (g).

a. Computation
We have the Adams spectral sequence i Ω Spin Here we proposed that the 5d cobordism invariant of Z 16 is the Postnikov square. A caveat is that we have not yet been able to fully verify the Postnikov square is the only possibility, although partial evidences suggest this to be true. The readers need to be cautious using this particular result. The verification is left for future work.  (BZ8, Z8), b is the generator of H 2 (BZ8, Z8).ã = a mod 2,b = b mod 2.  In general, if we have a homotopy pullback square then there is a fiber sequence where ΩZ is the loop space of Z. So there is a fiber sequence Since x 2 is identified withb = β (2,8) a in BG where a ∈ H 1 (BZ 8 , Z 8 ) and Sq 1 β (2,8) is trivialized in Ω G 5 . Note thatã ∪ 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.