Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew

We explore QCD$_4$ quark matter, the $\mu$-T (chemical potential-temperature) phase diagram, possible 't Hooft anomalies, and topological terms, via non-perturbative tools of cobordism theory and higher anomaly matching. We focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then analyze the continuous and discrete global symmetries and pay careful attention to finite group sectors. We input constraints from $T=CP$ or $CT$ time-reversal symmetries, implementing QCD on unorientable spacetimes and distinct topology. Examined phases include the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We introduce a possibly useful but only approximate higher anomaly, involving discrete 0-form axial and 1-form mixed chiral-flavor-locked center symmetries, matched by the above four QCD phases. We also enlist as much as possible, but without identifying all of, 't Hooft anomalies and topological terms relevant to Symmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) or SU(3) QCD$_d$-like matter theories in general in any spacetime dimensions $d=2,3,4,5$ via cobordism.


Contents
1 Introduction and Summary

Physics in QCD quark matter
We are made of atoms, which are made out of quarks, the particles 1 of Quantum Chromodynamics (QCD) vacuum, plus some electrons. The majority of our mass is from the mass of nuclei. While about the 2% of our mass is from Higgs condensate, the surprising significant 98% of our mass is from QCD chiral condensate. Meanwhile, we live in the chiral symmetry breaking (ChSB) phase of QCD vacuum. In order to investigate the nature of QCD matter and its vacuum structure, it is helpful to move out from this particular vacuum (ChSB) to other new foreign phases. In condensed matter language, we try to explore other unfamiliar foreign phases outside the familiar domestic phase, away from the ground state (i.e., vacuum) we live in, by tuning parameters in the QCD phase diagram (see recent selected reviews [1][2][3]). Namely, we should explore different new vacua or ground state structures and their excitation spectra.
In this work, we will look at some simplified ideal toy models. One model is that quarks are nearly massless and on bi-fundamentals of SU (3). Here the quarks are the Dirac spinors in 3+1 dimensional spacetime (we denoted as 3+1D or 4d). We will consider various types of curved spacetime manifolds with different topology and with Spin structure, Pin + , or Pin − or other twisted structures. 2 In physics language, quarks are 4d Dirac fermions, in the fundamental representation 3 of SU(3) color gauge group and fundamental representation of SU(3) flavor global symmetry group. We denote them as the representation (Rep) 3 c in SU(3 c ) V for color and 3 f in SU(3 f ) V for flavor, where subindex V stands for the vector symmetry. Follow the notations in [4], for the Euclidean path integral, we have the schematic partition function (1.1) 1 In particle physics, quarks are elementary particles. In condensed matter viewpoint, it may be beneficial to alternatively view the quarks as quasiparticles, quasi-excitations out of certain vacuum. 2 We consider the smooth differentiable manifolds with a metric g as spacetime -if the fermions/spinor can live on them, we require Spin structure; if we require time-reversal T = CP , or CT or other reflection symmetries, we require Pin + , Pin − or other semi-direct ( ) product or twisted structures between the spacetime tangent bundle T M and the gauge bundle EG of the gauge group G. See more in the main text and see an overview of our setting in [4].
This path integral describes an SU(N c ) = SU(3 c ) gauge theory with 1-form gauge field a and 2-form field strength F a for the (exact) color N c = 3 with three colors in fundamental: red (r), green (g), and blue (b). It also describes the (approximate) N f = 3-flavor in fundamental for fermions ψ q , where we choose the lightest bare quarks in nature: the up quark (u), the down quark (d), and the strange quark (s). At the massless limit m q = 0, the topological θ-term can be absorbed by axial U(1) A rotations of quarks, and we may set θ = 0. The fermion ψ q are quarks that carry quantum number (denoted the quark quantum number q) of color (c), flavor (f ), the spacetime self-rotational spin (s), whose quark-pairing can also carry the spatial momentum k, angular momentum L, and parity quantum number P , etc. 3 See Table 1 and 2 for some examples of quantum numbers for different quark-pairing condensates. In Eq. (1.1), we can tune the temperature T = β E = 1 L E proportional to the inverse size of the Euclidean time circle L E , and we can also tune the chemical potential µ, which changes the density of quark matter.
The standard lore from the pioneer studies of quark matter [1,2] teaches us that four dominant phases occur at different regions of QCD phase diagram drawn in the µ-T (chemical potential v.s. temperature) axes. Below we aim to revisit some of these four phases, and applying modern perspectives of symmetries and anomalies to constrain these quantum systems.
(I). Global Symmetries: For symmetries, we explore and exhaust both continuous and discrete global symmetries and pay special attention to finite group sectors.
(II). Higher Symmetries: For gauge theories, there are extended operators of lines and surfaces, etc. They can also carry quantum numbers thus also charged under the higher generalized global symmetries [5].
There are also corresponding symmetry generators as charge operators. We also need to pay attention to higher symmetries.
(III). Anomalies and Higher Anomalies: Given the global symmetry, there can be potential obstructions to couple the symmetry to background gauge field or to subsequently gauging the symmetry -the phenomena are known as the 't Hooft anomalies [6]. For higher symmetries, there are also associated higher 't Hooft anomalies. By anomalies, we mean to include both • Perturbative local anomalies calculable from perturbative Feynman diagram loop calculations, classified by the integer group Z classes (or the so-called free classes). Selective examples include: (1): Perturbative fermionic anomalies from chiral fermions with U(1) symmetry, originated from Adler-Bell-Jackiw (ABJ) anomalies [7,8] with Z classes. (2): Perturbative gravitational anomalies [9].
• Non-perturbative global anomalies, classified by finite groups such as Z N (or the so-called torsion classes). Some selective examples from QFT or gravity include: (1): An SU(2) anomaly of Witten in 4d or in 5d [10] with a Z 2 class, which is a gauge anomaly.
(IV). Symmetry Protected Topological states (SPTs)/ Symmetry Enriched Topologically ordered states (SETs) or Higher SPTs/SETs: Quantum systems (usually the quantum vacuum or the ground state) can be protected by global symmetry in a topological way. These are known as the interacting generalizations of topological insulators (TI) and topological superconductors (TSC) [18][19][20][21][22] known as the SPTs for interacting bosons and interacting fermions (see the overview [23][24][25]). Ref. [26][27][28] propose mathematical theories of cobordism classifying these SPTs and their low energy iTQFTs. In the context that we require to apply is the SU(N) and time-reversal symmetry generalization of SPTs studied in Ref. [4] suitable for Yang-Mills and QCD systems. In this work, we will apply a generalized cobordism theory including the higher-SPTs classifications (given by higher classifying spaces and higher symmetries) based on the computations and tools in [29].
By keeping in minds and utilizing the above modern concepts of quantum systems and QFTs beyond Ginzburg-Landau symmetry-breaking paradigm, here we revisit the standard lore of four QCD quark matter phases [1,2] and list down their global symmetries 4 in Fig. 1 (without time-reversal symmetry) and Fig. 2 (with certain time-reversal symmetries): 1. QGP (quark-gluon plasma/liquid) at high T: For general N c and N f , we have the symmetry: where the [SU(N c ) V ] color is gauged as a gauge group. 5 The U(1) V /Z Nc,V is the vector symmetry associated to the baryon number B conservation. The SU(N f ) L and SU(N f ) R are the left/righthanded Weyl spinor SU(N f ) flavor symmetries under the projection of P L/R = 1∓γ 5 2 . 6 We will mainly focus on N c = N f = 3.

ChSB (chiral symmetry breaking) at low T and at lower densities and low µ:
For general N c and N f , we have the symmetry: where the [SU(N c ) V ] color is gauged. Due to the chiral condensate ψ ψ = 0 in this vacum, the SU(N f ) L and SU(N f ) R flavor symmetries of the left/right-handed Weyl spinor are broken down to the diagonal vector subgroup SU(N f ) V . We will mainly focus on N c = N f = 3. In ChSB, by the spontaneously symmetry breaking (SSB) from QGP, we gain 8 pseudo-Goldstone bosons (as mesons: π 0 , π ± , K 0 ,K 0 , η, K ± ), while there is one massive η meson. If the ChSB further forms the nucleon superfluid, the U(1) V is SSB, and we gain 1 more Goldstone boson. 4 It is worthwhile to emphasize the old literature may happen to pay less attention to the finite group and discrete sectors. However, the finite group and discrete sectors are important for topological terms and non-perturbative global anomalies later we compute from the cobordism group. Thus, we aim to be as precise as possible writing down the global symmetries, see also [4]. 5 The [Gg] specifies that Gg is dynamically gauged. 6 It is worthwhile mentioning that the discrete axial symmetry Z2N f ,A that are not broken via the (ABJ) effect, still remains and sits inside:  Table 1: Pairing of quark-quark condensate for 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity). The L and R are for left/right-handed spinors.
Pairing function Wavefunction Parity P Spin or orbital where the [SU(2 c ) V ] color is gauged thus there is an SU(2) gauge theory. The 2SC pairing is shown in Table 1. The Z F 2,V is the fermion parity symmetry, which is a vector symmetry. The 2SC pairs 2-flavor u-d and 2-color r-g both into SU(2) singlets as a color superconductor.
although there is a part of the global symmetry containing the electromagnetic [U(1)]Q which is gauged not global. 7 Figure 1: We revisit the QCD 4 matter phase in the µ-T (chemical potential-temperature) phase diagram: The high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We do not attempt to address the nature of phase transitions in this figure, thus we make some of the phase boundaries blur. The¨Z 6,A or¨Z 4,A means that part of the discrete axial symmetry (A) is broken: In general, if the G sym group is broken, we denote it as¨G sym .
By including time reversal symmetries into the QCD system, we can choose any suitable outer automorphism of the color gauge or flavor global symmetry group as a Z 2 -time reversal symmetry, which is a Z 2 -reflection symmetry by putting the Euclidean QCD 4 path integral on an unorientable spacetime.
In fact, several recent works have attempted to study QCD matter phases based on the languages of higher symmetries and anomalies above. We should quickly overview some of these pioneer works: (1). Whether the color superconductivity can be topological in some way was questioned in Ref. [30]. What Ref. [30] concerns is the topological insulators (TI) / topological superconductors (TSC) in the free non-interacting quadratic mean-field Hamiltonian systems. Thus the classifications in Ref. [30] are only either 0 or Z classes for mean-field free fermion systems. The new input in our context is that we consider fully interacting systems and enlist possible SPTs for these QCD matter phases by cobordism group classifications.
(2). Ref. [31][32][33][34]] explores a related system of 4d adjoint quantum chromodynamics (QCD 4 ) with an SU(2) gauge group and two massless adjoint Weyl fermions. Higher symmetries and higher anomalies play an important role. Depend on the complex mass parameters of fermions, we can land onto different phases, and there are interesting quantum phase transitions between bulk phases. There are implications and constraints for 3+1D deconfined quantum critical points (dQCP), quantum spin liquids (QSL) or Here we choose a semi-direct product of The semi-direct product is more general, which also includes the direct product case of ×Z CT 4 as the CT symmetry.
fermionic liquids in condensed matter, and constraints on 3+1D ultraviolet-infrared (UV-IR) duality. See Ref. [34] for an overview of the proposed phases at quantum critical points.
(3). Ref. [35,36] explores quantum phase transitions between Landau ordering phase transitions but beyond the Landau paradigm, for example, due to the effects of topological θ-terms. Ref. [36] suggests that SU(2) QCD 4 with large odd number of flavors of quarks could be a direct second order phase transition between two phases of U(1) gauge theories as well as between a U(1) gauge theory and a trivial vacuum (e.g. a Landau symmetry-breaking gapped paramagnet). The gauge group is enhanced to be non-Abelian at and only at the transition. It is characterized as Gauge Enhanced Quantum Critical Points.
(5). Quark-hadron continuity outside the Ginzburg-Landau paradigm: Quark-hadron continuity [44] asserts that hadronic matter superfluid phase is continuously connected to color-superconductor without phase transitions when the µ increases. This proposal is based on Ginzburg-Landau theory where two sides of phases have the similar symmetry breaking patterns and the similar gapless and gapped energetic spectrum. Ref. [45] questions the quark-hadron continuity to be invalid, by suggesting there must be a phase transition due to the topological fractionalization of excitations are different. Ref. [46] re-analyzes the scenario based on higher-symmetry is not spontaneously broken, and found that the quark-hadron continuity is still plausible.
In the remained of this article, we like to point out a higher anomaly involving discrete 0-form axial chiral and 1-form mixed chiral-flavor-locked center symmetries that can be matched by the above four QCD phases. Then we will give a quick mathematical introduction of tools we used in Sec. 1.3. After then, we will list down various data and tables computed from cobordism theory, with a view toward the applications of QCD matter phases, to be studied in the future [47]. The QCD d matter symmetries, anomalies, and topological terms without time-reversal symmetry, classified by the cobordism theory, are studied in Sec. 2 via a cobordism theory. The QCD d matter symmetries, anomalies, and topological terms with time-reversal symmetry, classified by the cobordism theory, are studied in Sec. 3, in general in any spacetime dimensions d = 2, 3, 4, 5 via a cobordism theory.

Approximate higher anomaly constraint on the QCD phase diagram
In this section, we point out a higher 't Hooft anomaly involving discrete 0-form axial and 1-form mixed chiral-flavor-locked center symmetries can be matched by the above four QCD phases. Our approach is related but still somehow different from Ref. [37][38][39][40][41][42].
For simplicity, we will set N c = N f = N below. If N c = N f , we just need to replace the N below to their greatest common divisor gcd(N c , N f ) and make some moderate but a straightforward generalization.
First, under some assumptions that we will comment later, we hope to point out that there is an approximate 1-form electric-magnetic (e-m) global symmetry Z N cf , [1] that mixed between 1-form color and flavor (CF) center symmetry: Z N cf , [1] . To recall, focus on the kinematics of the UV path integral of QFT, is gauged, we still are left with part of the projective special unitary symmetry = PSU(N f ) V from the flavor symmetry. We can regarded it as a 1-form magnetic symmetry that can be coupled to the flavor symmetry background probed fields . This idea is implemented already in [38][39][40][41][42].
(•3) Now we combine the above two 1-form e and m symmetries into a diagonal 1-form symmetry, and name it as 1-form electric-magnetic (e-m) global symmetry Z N cf , [1] . We define this 1-form e-m color-flavor-locked global symmetry Z N cf , [1] as the diagonal symmetry that rotates oppositely the colorfundamental (W c ) and the flavor-fundamental (W f ) Wilson line along the 1d curve γ 1 , called where the a c is. The (W c W f ) have ends that can be opened by having bi-fundamental quark and antiquark at each of two ends. Let us call the 2-dimensional surface operator that is the 1-form Z N cf , [1] symmetry generator on 2d area Σ 2 as: Here we write down the electric and magnetic 2-surface operators based on the conventions and notations in [16]. So that even if the bi-fundamental quarks can open up the the color-fundamental and the flavorfundamental Wilson line, it will not be charged under the 1-form e-m Z N cf , [1] symmetry, because the obtained phase is exactly cancelled: when the 2d U surface and the 1d combined W c W f line have a nontrivial linking number in a 4d spacetime. 8 shows W c is charged under 1-form Z N cf , [1] with a charge exp(i 2π N ). Similarly, the 1-form e-m Z N cf , [1] symmetry still acts nontrivially on the flavor-fundamental Wilson line because this linking shows W f is charged under 1-form Z N cf , [1] symmetry with a charge exp(−i 2π N ). We propose that there is an approximate anomaly mixing between the discrete Z 2N f ,A axial (chiral) symmetry and the 1-form e-m color-flavor-locked Z N cf , [1] global symmetry. We call the 2-form Z N valued background field of 1form Z N cf , [1] as B (2) cf . We find that there is an anomaly captured by the Z 2N f ,A axial (chiral) symmetry transformation, such that the partition function Z gains a fractionalized term cf ) is not a 4d SPTs but a fractionalized 4d SPTs, not a counter term, thus cannot be absorbed into 4d, and should be regarded as a 5d higher-SPTs/higher-iTQFT -which in fact is an indicator of the 4d higher 't Hooft anomaly of the QCD 4 .
The caveat however is that this 4d higher 't Hooft anomaly of the QCD 4 is only approximate. The subtlety is that it is only a precise anomaly if we also "gauge" the ] gauge theory. The disadvantages of our approximate anomaly (1.12) are that the flavor Wilson lines are only probed but not dynamical objects, so we do not really have the 1-form symmetry unless we at least weakly gauge the flavor symmetry. 9 Thus another interpretation of our gauge theory and anomaly is indeed the bi-fundamental gauge theory in 3+1 dimensions studied in [50,51].
In comparison, other approaches in Ref. [37][38][39][40][41][42] also have their own disadvantages. For example, Ref. [39] derives a different kind of anomaly only under a certain twisted flavor boundary condition in 4d: and its dimensional reduction in 3d where B c and B f are color/flavor background fields respectively. Ref. [38] derives constraints that have limitations on the chemical potential and boundary conditions as well.
In Figure 3, we show how our approximate anomaly in (1.12) is still required and can be matched by the four QCD 4 matter phases. We use the triple data with three imputs Fig. 3 can be also denoted as N c-shift indicates it is dimensionally reduced from 1-form symmetry Z N c-shift , [1] to a 0-form symmetry. The 1-form color-shift symmetry Z N c-shift , [1] is introduced in Ref. [38].
Let us indicate how the higher 't Hooft anomalies in (1.12) and in (1.14) can be matched by breaking some of the global symmetries in the triple data (Z 2N f ,A ; Z N c-shift ). Our notations are that if the G sym is broken, we denote it as¨G sym , if the G sym is preserved, we would either indicate G sym remained, or simply omit the symbol as we did in the Fig. 3. 1. QGP (quark-gluon plasma/liquid) at high T: Fig. 3 QGP. (1.17) 2. ChSB (chiral symmetry breaking) at low T and at lower densities and low µ: The four phases can also be matched and cancelled by the anomaly (1.14), Z f ) introduced in Ref. [38]. We use the triple data (Z 2N f ,A ; Z N cf ) to label the discrete 0-form axial symmetry Z 2N f ,A , a dimensionally reduced color-shift Z N c-shift symmetry introduced in Ref. [38] , and the 1-form mixed chiral-flavor-locked center symmetry Z (1) N cf that we introduce.

2SC (2-color superconductivity) at low T and at intermediate densities and µ:
( Fig. 3 2SC. (1. 19) 4. CFL (3-color-flavor locking superconductivity) at low T and at high density and high µ: Fig. 3 CFL. (1.20) What we have shown above is that higher 't Hooft anomalies in (1.12) and in (1.14) can indeed be matched by four phases via breaking some of the global symmetries. We thus can constrain other possible QCD phases via the proposed approximate anomaly (1.12), based on higher 't Hooft anomaly matching and cancellation.

Mathematical Primer
In this article, we use spectral sequences (Adams spectral sequence, Atiyah-Hirzebruch spectral sequence, and Serre spectral sequence) to compute several cobordism groups which appear in QCD matter phases (QGP, ChSB, 2SC and CFL in Sec. 1.1). See [4,29,52] for a primer.
We aim to compute the cobordism group Ω G d for d ≤ 5 where G is the gauge group of QCD matter phases (QGP, ChSB, 2SC and CFL).
By the generalized Pontryagin-Thom isomorphism, which identifies the cobordism group Ω G d with the homotopy group of the Madsen-Tillmann spectrum M T G.
We have the Adams spectral sequence Here A p is the mod p Steenrod algebra, Y is any spectrum. For any finitely generated abelian group G, G ∧ p = lim n→∞ G/p n G is the p-completion of G. In particular, A 2 is generated by Steenrod squares Sq i .
In our cases, the cobordism groups only have 2-torsion and 3-torsion.
We will use Adams spectral sequence to compute the 2-torsion part of the cobordism groups (we consider Y = M T G in Adams spectral sequence, and we focus on p = 2).
We will use Serre spectral sequence and Atiyah-Hirzebruch spectral sequence to compute the 3-torsion part of the cobordism groups. In our cases, in order to compute the 3-torsion part of the cobordism groups, we need only to compute the cobordism group Ω SO d (BG ) for some group G .
We have the Atiyah-Hirzebruch spectral sequence Since we need to know the integral homology groups H p (BG , Z). In order to obtain this data, we compute the integral cohomology groups H p (BG , Z) using Serre spectral sequence (we find a fibration of which BG is the total space).
For example, if M T G = M Spin ∧ X where X is any spectrum, by Corollary 5.1.2 of [53], we have for t − s < 8. Here A 2 (1) is the subalgebra of A 2 generated by Sq 1 and Sq 2 .
So for the dimension d = t − s < 8, we have (1.26) The H * (X, Z 2 ) is an A 2 (1)-module whose internal degree t is given by the * .
Our computation of E 2 pages of A 2 (1)-modules is based on Lemma 11 of [29]. More precisely, we find a short exact sequence of A 2 (1)-modules 0 → L 1 → L 2 → L 3 → 0, then apply Lemma 11 of [29] to compute Ext s,t A 2 (1) (L 2 , Z 2 ) by the data of Ext s,t A 2 (1) (L 1 , Z 2 ) and Ext s,t A 2 (1) (L 3 , Z 2 ). Our strategy is choosing L 1 to be the direct sum of suspensions of Z 2 on which Sq 1 and Sq 2 act trivially, then we take L 3 to be the quotient of L 2 by L 1 . We can use this procedure again and again until Ext s,t A 2 (1) (L 3 , Z 2 ) is determined.

QCD Symmetries, Anomalies and Topological Terms Without Time-Reversal
Before we consider the cobordism theory and co/bordism group of the following four QCD matter phases, we need to convert our notations to involve both the internal symmetry and the spacetime symmetry. Here are the results of conversions for the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. QGP: . ChSB: (2)) ×(U(1)×U (1)).

Chiral symmetry breaking
For ChSB with a global symmetry: We have a fibration (2.5) Hence we have the Serre spectral sequence, see Figure 4.
There is another approach: we have a fibration (2.9) Hence we have the Serre spectral sequence, see Figure 5.
(2.12) In Figure 5, we find that the 3d Z 3 survives the spectral sequence, so in Figure 4, there is no nontrivial differential from (0,2) to (3,0). Since the differential is a derivation, we conclude that in Figure 4 the dashed arrow from (0,4) to (3,2) does not actually exist. So there is a Z 3 in 5d survives the spectral sequence, thus in Figure 5, the dashed arrow from (0,4) to (2,3) also does not actually exist.
By the Atiyah-Hirzebruch spectral sequence, we have  For t − s < 8, since there is no odd torsion, we have the Adams spectral sequence We have where c i is the Chern class of the SU(3) bundle.
By Thom isomorphism, where c 1 is the Chern class of the U(1) bundle and U is the Thom class.
The  So there is no 2-torsion in Ω Combine the 2-torsion and 3-torsion results, we have

Bordism group
d Ω Table 3: Bordism group. The notation e(A, B) denotes a group extension of A by B, that is, a group that fits into the following short exact sequence 0 → B → e(A, B) → A → 0.

3-Color-Flavor locking superconductivity
For t − s < 8, since there is no odd torsion, we have the Adams spectral sequence We have where c i is the Chern class of the SU(3) bundle.

2-Color Superconductivity:
For t − s < 8, since there is no odd torsion, we have the Adams spectral sequence By Thom isomorphism, we have where w i is the Stiefel-Whitney class of the SO(4) bundle and U is the Thom class.
We also have where c 1 is the first Chern class of the U(1) bundle.
The    Here e i is the Euler class, p i is the Pontryagin class, c i is the Chern class. Here w i = w i (SO(4)), p 1 = p 1 (SO(4)), e i = e i (SO(4)).η is the mod 2 index of 1d Dirac operator, Arf is the Arf invariant. The ? is an undetermined cobordism invariant which also appears in [52].

Quark Gluon Plasma/Liquid
Since the localization of Spin We have where c i is the Chern class of the SU(3) bundle.
By Thom isomorphism, where c 1 is the Chern class of the U(1) bundle and U is the Thom class.
The  So there is no 2-torsion in Ω We have a fibration (2.27) Hence we have the Serre spectral sequence, see Figure 15.
The dashed arrows are possible differentials.
Compared with the 2-torsion result, we find that the differential from (0,2) to (3,0) kills one Z since the 2d cobordism group contains only one Z, and the differential from (0,4) to (3,2) kills two Z since the 4d cobordism group contains four Z while one Z is from Ω SO 4 . and (2.34) Here ? is an undetermined 3-torsion group.
By the Atiyah-Hirzebruch spectral sequence, we have )).  Table 6: Bordism group. The notation e(A, B) denotes a group extension of A by B, that is, a group that fits into the following short exact sequence 0 → B → e(A, B) → A → 0. Here n = 1 or 2, while ? is an undetermined 3-torsion group.

QCD Symmetries, Anomalies and Topological Terms With Time-Reversal
Now we consider putting the QCD matters on the smooth differentiable and unorientable spacetime manifolds -if the fermions/spinor can live on them, we require Spin structure; if we require time-reversal T = CP , or CT or other reflection symmetries, we require Pin + , Pin − or other semi-direct ( ) product or twisted structures between the spacetime tangent bundle T M and the gauge bundle E G of the gauge group G. See more in the main text and see an overview of our setting in [4]. Follow Fig. 1 and Fig. 2, we can choose any suitable outer automorphism of the color gauge or flavor global symmetry group as possible time-reversal symmetries, which can be any reasonable Z 2 -reflection symmetry. This implies putting the Euclidean QCD 4 path integral on an unorientable spacetime. The most general case is a semi-direct product Z T 4 , which is all allowed total group made from the exact sequence: Here we will only focus on two cases: The direct product ×Z T 4 which implies the Pin + structure, where Z T 4 ⊃ Z F 2 . We may also denote such a Z T 4 := Z T F 4 to indicates it includes Z F 2 as a normal subgroup. The direct product ×Z T 2 × Z F 2 which implies the Pin − structure. For other possible time-reversal symmetries, we leave them in a future work [47].

Chiral symmetry breaking
Since the localization of Pin ± × Z 2 U(3) For t − s < 8, since there is no odd torsion, we have the Adams spectral sequence We have where c i is the Chern class of the SU(3) bundle.
By Thom isomorphism, where c 1 is the Chern class of the U(1) bundle and U is the Thom class.
Also by Thom isomorphism, where w 1 is the Stiefel-Whitney class of the O(1) bundle and V is the Thom class.

Bordism group
d Ω By Künneth formula, For t − s < 8, since there is no odd torsion, we have the Adams spectral sequence

2-Color Superconductivity:
We have the constraint w 2 = w 2 where w i is the Stiefel-Whitney class of the tangent bundle, w i is the Stiefel-Whitney class of the SO(4) bundle.
For t − s < 8, since there is no odd torsion, we have the Adams spectral sequence  Table 10: Bordism group. Here w i is the Stiefel-Whitney class of the tangent bundle, w i is the Stiefel-Whitney class of the SO(4) bundle, c 1 (c 1 ) is the Chern class of the U(1) bundle, η Spin(4) is similar to the η SU(2) defined in [4]. We have the constraint w 2 + w 2 1 = w 2 where w i is the Stiefel-Whitney class of the tangent bundle, w i is the Stiefel-Whitney class of the SO(4) bundle.
For t − s < 8, since there is no odd torsion, we have the Adams spectral sequence w 4 1 , w 4 , w 2 2 , w 2 1 c 1 , w 2 1 c 1 , w 2 c 1 , w 2 c 1 , c 1 c 1 mod 2, (w 3 1 + w 1 w 2 + w 3 )η 5 Z 3 2 w 2 w 3 , w 4η , (w 3 1 + w 1 w 2 + w 3 )Arf Table 11: Bordism group. Here w i is the Stiefel-Whitney class of the tangent bundle, w i is the Stiefel-Whitney class of the SO(4) bundle, c 1 (c 1 ) is the Chern class of the U(1) bundle,η is the mod 2 index of 1d Dirac operator, Arf is the Arf invariant. where c 1 is the Chern class of the U(1) bundle and U is the Thom class.

Quark Gluon Plasma/Liquid
Also by Thom isomorphism, where w 1 is the Stiefel-Whitney class of the O(1) bundle and V is the Thom class. The