Gauge interactions and topological phases of matter

Progress of Theoretical and Experimental Physics, Sep 2016

We study the effects of strongly coupled gauge interactions on the properties of the topological phases of matter. In particular, we discuss fermionic systems with three spatial dimensions, protected by time-reversal symmetry. We first derive a sufficient condition for the introduction of a dynamical Yang–Mills field to preserve the topological phase of matter, and then show how the massless pions capture in the infrared the topological properties of the fermions in the ultraviolet. Finally, we use the S-duality of <mml:math display="inline"><mml:mrow><mml:mi mathvariant="script" class="MJX-tex-caligraphic">N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>N=2 supersymmetric <mml:math display="inline"><mml:mrow><mml:mtext>SU</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>SU(2) gauge theory with <mml:math display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math>Nf=4 flavors to show that the <mml:math display="inline"><mml:mrow><mml:mi>ν</mml:mi><mml:mo>=</mml:mo><mml:mn>16</mml:mn></mml:mrow></mml:math>ν=16 phase of Majorana fermions can be continuously connected to the trivial <mml:math display="inline"><mml:mrow><mml:mi>ν</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>ν=0 phase.

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Gauge interactions and topological phases of matter

Prog. Theor. Exp. Phys. Gauge interactions and topological phases of matter Yuji Tachikawa 0 Kazuya Yonekura 0 0 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1.1. Symmetry-protected topogical phases and classification of quantum field theories There are by now bewildering varieties of quantum field theories (QFTs) realized experimentally and/or constructed theoretically. One way to put them in order is to try to classify them. To start a classification, we need to decide which kinds of QFTs we treat, and what equivalence relation we use. The classification becomes more tractable when we treat the simpler QFTs under a coarser equivalence relation. For a simple classification, let us only treat those whose excitations are all gapped or equivalently massive. Then, the infrared limit is almost empty, in that there can only be a finite number of vacuum states on a given space. Let us further demand that there is in fact only a unique vacuum state, whatever the topology of the space, as long as it is compact without any boundary. Such theories are said to have no intrinsic topological order. We now fix a symmetry group G and the spacetime dimension d + 1, and then consider all possible (d + 1)-dimensional gapped QFTs with G symmetry without intrinsic topological order. Here, the symmetry G can be arbitrarily chosen to your liking: it can include a spacetime discrete symmetry such as time reversal T, and also an internal continuous symmetry such as SU(2). We also need to specify whether we consider bosonic or fermionic QFTs, in the sense that the theory under consideration detects the spin structure of the spacetime manifold or not. To make the classification more tractable, we use the coarsest equivalence relation among such QFTs. This is done by declaring that two QFTs are equivalent when they can be continuously deformed to each other without leaving the class of such gapped QFTs with symmetry G. An equivalence class is then called a symmetryprotected topological (SPT) phase protected by G. We can always consider a completely trivial theory, such that there is only one state in the Hilbert space with a trivial G action and the transition amplitude is always 1 in any situation. The equivalence class containing this trivial theory is the trivial SPT phase, while the other SPT phases are the topological phases of matter. In order to see if two QFTs X1 and X2 belong to the same SPT phase or not, it is useful to consider the setup where the region y < 0 is filled with the system X1 and the region y > 0 by X2, with a thin transition region between the two. We call this transition region the boundary. When X1 and X2 belong to the same SPT phase, we can choose the configuration on the total space such that the system is gapped everywhere, without any topological order even at the boundary. In contrast, when X1 and X2 are distinct SPT phases, something must happen at the boundary of X1 and X2: there might be a gapless mode, or a spontaneous symmetry breaking of G, or a gapped surface topological order at the boundary. 1.2. Fermionic SPT phases A fermionic SPT phase, when G is given by a global U(1) symmetry together with a few discrete symmetries, is called a topological insulator in the literature. This terminology is due to the fact that in a real insulator the excitation is gapped and the electromagnetic U(1) symmetry is unbroken. Note that the U(1) symmetry in this case is considered as a global symmetry. Similarly, a fermionic SPT phase, when G is just given by a few discrete symmetries, is called a topological superconductor, because in a superconductor the excitation is again gapped and the electromagnetic U(1) symmetry is broken and not there in the infrared. In general, these systems have complicated interactions. To make the classification even simpler, it is instructive to start by considering only free massive fermions. It was found that the possible choice of discrete symmetries can be summarized in a tenfold way, and the dependence on the number of spacetime dimensions follows a uniform pattern. The result is now known as the periodic table of free fermionic SPT phases [1–3].1 Let us recall one concrete case that is the focus of our paper: the (3+1)-dimensional free fermionic SPT phases, protected by the time-reversal symmetry T such that T2 = (−1)F , where F is the fermion number. They are topological superconductors, and this choice of protecting discrete symmetry is 1 See also, e.g., [4–7] for a sample of papers in hep-th on the SPT phases. known as class DIII. As we will consider only relativistic systems, we can freely replace the timereversal symmetry T with the CP symmetry CP, which in this case satisfies CP2 = (−1)F . We will mostly use this latter nomenclature, since this will be more familiar to readers of hep-th. The basic example is a single Majorana fermion. The CP invariance forces the coefficient m of the mass term to be real. In order for the system to be gapped, we need m = 0. So the systems can be classified into two disconnected pieces: those with m > 0 and those with m < 0. Therefore they represent two distinct SPT phases.2 One manifestation of the distinctness is when we have a space-dependent mass term: suppose we have a y-dependent mass term m(y) such that m(y) > 0 (y > ), m(y) < 0 (y < − ) for a small positive . Then there is necessarily one massless CP-invariant (2 + 1)-dimensional Majorana fermion at the boundary region y ∼ 0. More generally, when we have N Majorana fermions, the coefficient of the mass term is a real symmetric matrix Mij. Let us denote by ν(M ) the number of negative eigenvalues. Now, consider again a y-dependent mass term. Then the generic number of massless CP-invariant (2 + 1)-dimensional Majorana fermions in the middle is given by the difference of ν(M ) on y > 0 and ν(M ) on y < 0. From this analysis, we see that the (3+1)-dimensional free topological superconductors of class DIII are characterized by an integer ν, i.e. they are classified by Z. Weak perturbation cannot change this classification, but interaction effects when it is strong can and indeed do change this classification. The first example that was found was the (1+1)-dimensional fermionic SPT phase of class BDI. At the level of the free fermion, it is characterized by an integer ν ∈ Z mostly as above. But with suitable four-fermion interactions added, it has been shown that the ν=8 phase and the ν=0 phase can be continuously connected while gapped [8]. Stated differently, the classification collapses from Z to Z8 due to the interaction effects. We now have ample pieces of evidence [9–12] that in the (3+1)-dimensional fermionic SPT phase protected by the time-reversal symmetry T2 = (−1)F , the classification similarly collapses from Z to Z16 once we include the effects of interactions. There is an ongoing effort to classify interacting bosonic and fermionic SPT phases by finding the right mathematical language to describe them, see e.g. [13,14]. These analyses have confirmed the collapse of the free classification due to the interactions recalled above, and have shown that there can also be genuinely interacting SPT phases that cannot be continuously connected to free SPT phases. 1.3. Gauge interactions and SPT phases In this paper, we consider the effects of dynamical gauge fields on the properties of SPT phases. Before going further, it is useful to recall how the gauge fields have been used in their study in the literature. First, for an SPT phase protected by a symmetry G, it is extremely convenient to consider coupling the system to background gauge fields for the internal symmetry part of G, and to a non-trivial background metric for the spacetime symmetry part of G. In a sense, an SPT phase can be characterized 2 In continuum QFTs, there is no point in saying which of m > 0 or m < 0 is the trivial SPT phase, since this notion depends on the UV regularization. One way is to fix the sign of the mass term of the Pauli–Villars regulator fermion to be positive. Then the m > 0 case is the trivial phase and the m < 0 case is the ν=1 phase. A more invariant way to state the situation is that the m > 0 case and the m < 0 case differ by the ν=1 SPT phase. by the response of the system to these background fields. The U(1) Maxwell field is often utilized in this manner for SPT phases protected by U(1), and unoriented background manifolds are used for SPT phases protected by time-reversal symmetry. Secondly, the topological properties of strongly coupled confining gauge theories in the infrared have been studied, e.g. in [15–17]. The systems considered there do, however, have intrinsic topological order in the infrared, in the sense that there are multiple degenerate vacua on non-trivial spacetime manifolds, and they correspond to what are usually referred to as symmetry-enriched topological (SET) phases in the literature, and not to genuine SPT phases in the narrower technical sense. In this paper, we study of the effects of dynamical gauge interactions, which can be strongly coupled, on the SPT phases. The main questions we pose are twofold. The first question is when the introduction of dynamical gauge fields to a given system does not destroy the SPT phase of the original system. Suppose an SPT phase X protected by a symmetry G to be considered is realized by a QFT with an additional symmetry H . Then, we can introduce a dynamical gauge field for H . We denote the combined system with dynamical gauge field by X /H . Naively, when the gauge interactions become very strong and confine themselves, it should be possible to integrate them out. When there is no G × H mixed anomaly, the process of integrating out should not introduce any interaction that breaks G, and therefore it should not change the SPT phase. Our main claim in this regard is the following: Suppose the SPT phase X protected by a symmetry G with three spatial dimensions we would like to consider has an additional symmetry H to which we can couple a dynamical gauge field. When the group H is simple, connected, and simply connected, and furthermore the effective theta angle of H is zero, the system X /H after the introduction of the gauge field can be obtained as a continuous deformation of the original system X without closing the gap. In particular, X and X /H are in the same SPT phase protected by G. After making a general argument leading to this claim, we perform a detailed check in the case when the original SPT phase is a system of free Majorana fermions protected by time-reversal symmetry T with T2 = (−1)F . We will see that the SPT phase of the ultraviolet fermions is indeed captured by the non-linear sigma model of the massless pions in the infrared for certain flavor numbers of quarks. These analyses will be performed in Sects. 2 and 3. The second question is whether the knowledge of the dynamics of strongly coupled gauge theories we acquired in the last three decades is useful in shedding new light on the properties of standard SPT phases. For example, can it be used to show that the interaction effects should collapse the classification of the (3+1)-dimensional topological superconductor of type DIII from Z to Z16? We would like to answer this question in the affirmative. We know from the seminal work of Seiberg and Witten [18,19] two decades ago that N = 2 supersymmetric SU(2) gauge theory with Nf =4 flavors has the S-duality, meaning that when the coupling constant of the SU(2) gauge group is made extremely large, there is a dual description of the same system using a dual SU(2) gauge group and dual matter contents such that the coupling constant is weak. A simple counting shows that the hypermultiplets of this system consist of 16 Majorana fermions, and that the introduction of the SU(2) gauge field should not change the SPT property of these fermions, according to the criterion we mentioned above. We will show that this S-duality allows us to connect the ν=16 phase continuously to the ν=0 phase. This we will do using the following strategy. First, we add to the free ν=16 SPT phase SU(2) gauge interactions and other fields that do not change the SPT properties so that the system is the N = 2 SU(2) gauge theory with Nf =4 flavors softly broken to zero supersymmetry. Second, we increase the gauge coupling constant, and pass to the dual weakly coupled description. Third, we add various interaction terms in the dual description to show that it is in the ν=0 phase. We will detail the procedure in Sect. 4. 1.4. Remark on the types of symmetries Before moving on, it would be instructive to emphasize here that there are three different types of symmetries considered in this paper. The reader is advised to distinguish them to avoid possible confusion. ◦ Global symmetries protecting SPT phase: For example, a topological superconductor might be protected by T and a topological insulator by T U(1). We typically use a letter such as G to denote a symmetry protecting the SPT phases. In most of the paper, we take it to be just the time-reversal symmetry G = T, with the exception that in Sect. 2.5 we also discuss the case G = T × F for an additional internal symmetry F. ◦ Dynamical gauge symmetries: These are associated with dynamical gauge fields living inside the bulk material we are considering, and we integrate over these fields in the path integral. We typically use letters such as H (in this section) or G (in other sections) for dynamical gauge groups. Notice that all the physical states in the Hilbert space are singlets under the gauge group (on a compact space), and in that sense the gauge symmetry is not a symmetry of physical systems. ◦ Accidental symmetries: Sometimes, a system we consider happens to have more symmetries than G. We may call these accidental symmetries. For example, free fermion systems can have much larger symmetry than just T. We can easily break them explicitly by introducing some (possibly higher-dimensional) operators in the Lagrangian if we do not like them to exist. We typically use a letter such as F to denote them. In particular, we emphasize that the U(1) of electromagnetism in the case of topological insulators is not a dynamical gauge symmetry in our terminology, but should be considered as a part of the global symmetry protecting the SPT phase to which the background non-dynamical electromagnetic field is coupled. 1.5. Organization of the paper In the rest of the paper, we always consider relativistic systems with 3+1 spacetime dimensions, protected by the time-reversal symmetry T with T2 = (−1)F , or equivalently by the CP symmetry with CP2 = (−1)F . We refer to these systems simply as the SPT phases in this paper.3 The rest of the paper is organized as follows. In Sect. 2, we first describe a general argument that when the gauge group G is simple, simply connected, and connected, and when the effective theta angle is zero, then the original phase X and the system with the gauge field X /G are in the same 3 Two justifications of this abuse of terminology are as follows. First, it is simply too tedious to repeat the phrase “the topological superconductor protected by the time-reversal symmetry T with T2 = (−1)F .” Second, our discussions in this paper can be generalized to SPT phases other than topological superconductors. See Sect. 2.5 for a brief discussion on this point. SPT phase. We then check this statement by studying the effect of non-trivial gauge bundles on the η invariant produced by the fermion path integral. In Sect. 3, we study the low-energy gauge dynamics of non-supersymmetric gauge theories belonging to the class found in Sect. 2 which preserve the SPT properties. We will find that the σ -model of the massless pions in the infrared correctly reproduces the η invariant of the Majorana fermions in the ultraviolet. In Sect. 4, we use the S-duality of N = 2 supersymmetric SU(2) gauge theory with Nf =4 flavors to continuously connect the ν=16 phase to the ν=0 phase, thus explicitly implementing the collapse of the classification by interaction from Z to Z16. The basic idea is to note that the hypermultiplets in this N = 2 supersymmetric theory consist of 16 Majorana fermions, and that an extremely strongly coupled region of this theory can be analyzed in a dual weakly coupled frame. We conclude the paper with a short discussion in Sect. 5. We have a few appendices: In Appendix A, the rudimentary facts on CP and T transformations in 3+1 dimensions are summarized, paying due attention to various subtle signs important to us. Then in Appendix B, we discuss how the CP transformations are implemented in various concrete gauge theories. We discuss both nonsupersymmetric and supersymmetric examples. Appendix C summarizes the properties of Wess– Zumino–Witten terms. Appendix D describes the process of S-duality in N = 2 supersymmetric gauge theory in more detail. Finally, Appendix E is a complement to Sect. 4. Before proceeding, we would like to recommend that readers from the hep-th side of the community go through the excellent paper and lecture notes by E. Witten [20,21], which have heavily influenced this paper. Effects of gauge fields on SPT phases In this section, we propose and justify a sufficient condition for when the coupling of dynamical gauge fields to an SPT phase can be considered as a continuous deformation. We first give a general argument in Sect. 2.1, and provide a detailed analysis verifying the argument when the original SPT phase is given by free fermions in Sects. 2.2, 2.3, and 2.4. In Sect. 2.5 we discuss a simple application of our findings in this section to the structure of the interaction terms that can collapse a free-fermion classification. 2.1. General construction Suppose we are given a system X whose infrared (IR) limit realizes an SPT phase protected by a CP symmetry with CP2 = (−1)F . See Appendix A for details on the relation between CP and T. The theory X is by definition gapped. We denote by MX the mass scale of the gap.4 As can be easily seen from the explicit construction below, the discussion can be generalized to a more general global symmetry G protecting the SPT phase. Let us further assume that X has an additional continuous non-Abelian G flavor symmetry. Because of the mass gap of X , there is no ’t Hooft anomaly for this continuous symmetry. We can then couple a G gauge field to the original system X . We denote the combined system by X /G. We stress here again that the G here needs to be distinguished from the G discussed in the introduction, which is used for the definition of SPT phases. In this paper we are mainly concerned with the case G = CP (or equivalently G = T) unless otherwise stated, although many of our results can 4 Throughout the paper we use the natural units of high energy physics where = c = 1. In addition, to simplify our analysis, we demand that G is connected and simply connected: be generalized to other G. The dynamical gauge group G is, in contrast, a non-Abelian Lie group such as SU(N ). Let us first assume that the dynamical scale G of the gauge theory is far below the gap of the original system X , i.e. G MX . The Lagrangian of the system in the scale intermediate between G and MX is given by5 L = − 4g1e2ff FμAν FAμν + 6θ4eπff2 μνρσ FμAν FρAσ . This is the effective action of the gauge field, which is obtained after integrating out the degrees of freedom of X . We normalize the theta angle so that one BPST instanton of the gauge group gives amplitudes proportional to eiθ . The CP invariance of the combined system at this level requires that θeff is 0 or π . Depending on θeff , the following is believed to happen in a pure Yang–Mills theory. A pure Yang–Mills confines in the IR and has a mass gap. If θeff = 0, then there is a unique vacuum which preserves CP. However, if θeff = π , it is believed (see, e.g., [22–25]) that the CP is spontaneously broken and there are two vacua related by CP. The SPT phase classification assumes that the symmetry under consideration is not broken in the bulk. Therefore, we exclude the case θeff = π in the following analysis, and consider only the case when the theta angle is zero: These are again to keep the system in the standard framework of the SPT phases. For example, in the gauge group G, we could have included a discrete gauge group such as Zk , but such a gauge group gives topological degrees of freedom in the IR which contradicts the basic assumption of the SPT phases. Such cases are excluded by the condition π0(G) = 0. Even if G does not contain such a discrete gauge group, there is still a possibility that a discrete gauge group appears as a low-energy effective theory of the confining gauge group. Let us consider the case of SO(3) pure Yang–Mills as an example. In this case, the low-energy theory contains a Z2 gauge group [16,26] which can be detected by a Z2 one-form symmetry acting on the ’t Hooft loop operators. More generally, whenever G is not simply connected, the gauge theory has a one-form symmetry [17] and it is believed that we get non-trivial topological degrees of freedom in the IR. Therefore we impose the condition π1(G) = 0. We will give another but related reason for the condition π1(G) = 0 below. Our main claim can now be formulated as given below; the aim of the rest of the paper is to give substance to this claim: When the conditions (2.2) and (2.3) are satisfied, i.e. when G is (semi)simple, connected, and simply connected and the effective theta angle is zero, the system X /G after the introduction of the gauge field can be obtained as a continuous deformation of the original system X without closing the gap. In particular, X and X /G are in the same SPT phase. 5 Here we are neglecting possible discrete theta angles. They do not exist after imposing the condition (2.3). Let us first construct a continuous deformation explicitly. It is generally believed that when there is a field in a gauge theory such that all Wilson lines can be dynamically screened by pair creation of particles, the Higgs phase and the confined phase are continuously connected without any phase boundary. This observation goes back to Refs. [27,28]. More specifically, this folklore theorem stipulates the existence of a family Y (μ) of bosonic systems with flavor symmetry G parameterized by a mass parameter μ, with the following properties: When Y (μ) is considered alone, ◦ when μ > 0 all the bosons have masses of order μ and G is unbroken, and ◦ when μ < 0 the bosons have vacuum exptectation values (vevs) of order |μ| that break G completely, such that when we couple a dynamical G gauge field to this system, the resulting gauged system by Y (μ)/G is ◦ in the confined phase in the limit μ ◦ in the Higgsed phase in the limit μ with no phase boundary between the two limits. For example, when G = SU(N ) we can just take N − 1 copies of scalars in the fundamental representation. Similarly, when G = Sp(N ), we can take 2N copies of scalars in the fundamental representation.6 We now consider a combined system (X × Y (μ))/G, namely, the original system X together with the scalar system Y (μ) with a potential specified by a parameter μ, coupled to a single G gauge field. When μ < 0 with |μ| MX , the gauge group G is completely broken in an energy scale much higher than the gap MX of the system X . Then we have X × Y (μ) −μ−→−−−∞→ X . (2.4) G When μ > 0 with μ MX , the scalars in Y can be integrated out in a scale much higher than the gap MX of the system, and therefore we have X ×GY (μ) −μ−→−+−∞→ XG . (2.5) Now we see that X and X /G are continuously connected. However, for this assumption to be the case, the scalars in Y (μ) need to be able to screen all Wilson lines of the Lie algebra of G, because otherwise some Wilson line shows the area law in the confining phase, which can be distinguished from the Higgs phase. Thus we must impose the condition π1(G) = 0 so that all representations of the Lie algebra of G are actually allowed by the Lie group G. In the rest of the section, we would like to give further credence to the discussion above, by analyzing the case when X is a system of free massive fermions more explicitly. 2.2. Gauging free fermions L = −iψ σ μ(∂μ + ρ(TA)AAμ)ψ − 12 m [ψ ψ + c.c.] − 41g2 FμAν FAμν + 64θπ 2 μνρσ FμAν FρAσ , (2.6) 6 It would be interesting to construct such Y (μ) for other groups more explicitly. Here we consider their existence as part of the folklore theorem we rely on. where ψ are fermions, ρ(TA) are generators of the gauge group in a representation ρ, m is a mass parameter, g is the gauge coupling, and θ is the theta angle. We assume that the Majorana fermions are in a strictly real representation ρ of the gauge group G. We choose the CP transformation to commute with the gauge symmetry. This is possible because the representation ρ is strictly real. For more on our conventions, see Appendices A and B. Let us first recall the following simple fact about the chiral anomaly. By a change of variables ψ = eiαψ in the path integral, the parameters are changed as where tρ is an integer defined by tr[ρ(TA)ρ(TB)] = −tρ δAB, in a normalization that the adjoint representation has tadj = h∨, where h∨ is the dual coxeter number of G. When the mass parameter is positive and much larger than the dynamical scale of the theory, the IR effective theory is given by a pure Yang–Mills theory with the θ unchanged from the UV.7 Then, by the anomaly discussed above, we conclude that the θeff in the low-energy effective action in the general mass case is given by where arg(m) is the phase of m: arg(|m|) = 0 and arg(−|m|) = π . As recalled already, the system is believed to spontaneously break the CP invariance when θeff = π . We would like to retain the ability to change the sign of m from positive to negative, keeping the fact that θeff = 0. This requires that tρ ∈ 2Z . There is another way to see the condition tρ ∈ 2Z. Let us consider a fermion mass that depends on the space coordinate y := x3, given by m(y) > 0, (y > ) m(y) < 0, (y < − ) for a small positive number . In this situation, one manifestation of the non-trivial SPT phases is that localized gapless Majorana fermions appear at the boundary y ∼ 0. Now let us gauge the massless Majorana fermions at the boundary by a gauge group G in a representation ρ. In the three-dimensional theory, there is a parity anomaly. One manifestation of this anomaly is that under a gauge transformation, the fermion functional determinant changes sign as (−1)tρn, where n is an integer determined by the topology of the gauge transformation. This anomaly exists when tρ is an odd integer. To cancel this anomaly, we have to introduce a Chern–Simons term with half-integer Chern–Simons level.8 We have to distinguish two cases. If the gauge field is living solely on the three-dimensional boundary, the CP is explicitly broken when tρ is odd. If the gauge field lives in the four-dimensional bulk, the parity anomaly is cancelled by the anomaly inflow mechanism. This is because the theta angles on both sides of the boundary y = 0 are different: θeff = 0 on one side, and θeff = π on the 7 It is better to regard this statement as the definition of the phase of the fermion path integral. In the Pauli–Villars regularization, this means that we are taking the regulator mass parameter to be positive. 8 More precisely, we should use the language of the η invariants to state what is going on [29]. For our purposes here, using a somewhat naive language of half-integer Chern–Simons level already implies that we need tρ to be even, which is all we need at this point. other. However, in this case, the CP is spontaneously broken in the region with θeff = π as discussed above, and hence we cannot apply the SPT phase classification. Therefore, in any case, we have to impose the condition that tρ is an even integer. So far, we have discussed a necessary condition such that the SPT phase is not spoiled by the gauge interaction. In the previous section, we argued that if G is further assumed to be simple, connected, and simply connected, and if θeff = 0 is satisfied, the vacuum of the gauge theory is in the same SPT phase as the the original theory without the gauge field. In our free fermionic case, the condition on the theta angle imposes the condition tρ ∈ 2Z. We would like to make further checks of this conclusion by considering the partition function of these systems on various manifolds. Suppose that we have a theory which has a mass gap and no topological degrees of freedom in the sense that vacuum states in the Hilbert space are onedimensional in any manifold. The infrared limit of such a theory is called an invertible topological field theory. Now we consider the partition function of this theory on a manifold M , where we take its metric to be extremely large. The partition function then is given by a phase factor Z (M ) = eiϕ up to uninteresting contributions which can be continuously deformed to be absorbed by local gravitational counterterms. When Z (M ) are different as functions of the choice of the manifold M , the SPT phases are definitely different. We stress that this criterion does not require any detail of the UV theory. For example, there can be a strongly coupled gauge theory in the intermediate energy scale between the UV and the IR, as long as the IR theory is gapped and is described by an invertible field theory. In terms of the partition function, one consequence of our claim is then as follows: Suppose the group G is simple, connected, and simply connected and tρ is even. Then, in the lowenergy limit, the phase of the partition function of the theory of ν= dim ρ free massive Majorana fermions on a manifold M is the same as that of the gauge theory (2.6) with θ = 0. In the next two subsections we will establish the claim above, by first relating the phase arg Z (M ) to the properties of the η invariant, and then by studying the dependence of the η invariant on the dynamical gauge fields. 2.3. Partition function and the η invariant The aim of this subsection is to reduce the computation of the phase of the partition function of the gauge theory to a property of the eta invariant (2.23). The property (2.23) itself will be established in the next subsection. The partition function Z (M ) of the gauge theory is given as Note that this gauge field action is real and positive. Z (M ) = SG = Using the four-component Majorana fermion the Euclidean-space Lagrangian of fermion fields can be written as where C = diag( αβ , α˙ β˙) is the charge conjugation matrix acting on spinor indices, and D/ = Dμγ μ is the Dirac operator. This form is more appropriate when we consider the Lagrangian on unorientable manifolds. Now we study the fermion partition function Zψ , which is given by where Pf is the Pfaffian of the fermion functional space, and we have introduced the Pauli–Villars regulator with mass > 0. The analysis below is essentially the same as the one given in [20], except that we now have a gauge field A. We may define the Pfaffian in the following way. First, note that the charge conjugation matrix C has the property that where λ is an eigenvalue of iD/, and we have used the fact that the representation ρ(TA) is real. One can check that and C ∗ transform in the same way under CP, and hence they are sections of the same pin+ structure. Thus if is an eigenfunction, then C ∗ is also an eigenfunction of the same eigenvalue. Furthermore, these two eigenfunctions are guaranteed to be distinct because of the identity C(C ∗)∗ = − . We learned that the eigenvalues always come in pairs. We define the Pfaffian as the product of eigenvalues, where we take one eigenvalue from each pair ( , C ∗) of eigenfunctions. We get where the product is over all the pairs ( , C ∗) of the eigenfunctions of the Dirac operator. Taking the argument, we have − arg When the eigenvalue is much smaller than |m|, the phase of (λ + im)/λ is essentially π2 sign(m) sign(λ). Here the sum is taken over all the eigenmodes , not over pairs ( , C ∗) as we did above. When some λ is zero, we formally define sign(λ = 0) = +1 and then η is defined as above. Comparing the two expressions, we see that k≥1 where the correction terms k≥1 O(m−k ) go away in the limit |m| → ∞. Physically speaking, these correction terms correspond to higher-dimensional operators in the low-energy effective action of the gauge field after integrating out the massive fermion fields. We expect that these terms can be neglected if the mass |m| is much larger than the length scale of the manifold M and the dynamical scale of the gauge theory. So we assume that the mass is large and we neglect the correction terms. Now the path integral becomes Z (M ) = π [DA] exp −i 4 (1 − sign(m))η(M , A) + crr |Zψ (M , A)|e−SG , where crr represents the correction terms. We will show in the next subsection that if the condition (2.10) is satisfied, the η invariant η(M , A) mod 4Z is independent of the gauge field A, so we can write where n ∈ Z is an integer which is essentially the instanton number of the gauge field. The property (2.23) immediately follows, using the fact that tρ ∈ 2Z. We show (2.26) below by combining the Atiyah–Patodi–Singer theorem and the obstruction theory. The factor |Zψ (M , A)|e−SG is manifestly positive. Therefore, when the correction terms can be neglected, we get This is exactly the same as in the case of free Majorana fermions. 2.4. Topology of gauge bundles and the η invariant In this subsection, we show that the η(M , A) is given as 2.4.1. The index theorem The Atiyah–Patodi–Singer index theorem for a (5 = 4 + 1)-dimensional unoriented manifold N states [30,31] that the index (for pin+ structure) is given by9 M = Di \ Di ⊂ Md = n(i)≤d [−M2] is meant to reverse the pin+ structure of M2. 2.4.2. Some obstruction theory Next we need to understand the topology of gauge bundles, which can be understood by the obstruction theory. Let us first recall the notion of the CW-complex for a manifold M . We write the manifold M as where (i) Di ∩ Dj = ∅ for i = j inside M , (ii) each cell Di is homeomorphic to an open disk of dimension n(i), and (iii) the points in the closure Di but not in Di are contained in lower-dimensional IndD/5d(N , A5d) ∈ 2Z. One can also, show similarly to (2.17), that the five-dimensional index is an even number for Majorana For example, an n-dimensional sphere Sn has a CW complex Sn = D1 ∪ D2, where D1 is a zerodimensional point and D2 is homeomorphic to an n-dimensional open disk, such that all points on the boundary of D2 map to the single point D1. Let us define the d-dimensional skeleton of M as 9 In general, an index can be defined if we have a Z2 grading = ±1 and a self-adjoint elliptic operator iD/ which is odd under the Z2 grading, i.e., D/ = −D/ . For the five-dimensional pin+ structure with gamma matrices I (I = 1, . . . , 5), we use = 1 2 3 4 5 as the Z2 grading and iD/5d = 6 I DI , where 6 is an additional gamma matrix with ( 6)2 = 1 and 6 I + I 6 = 0 (I = 1, . . . , 5) so that the relation D/5d = −D/5d is satisfied. A reflection CR in a direction nˆI is defined as → nˆI I , which commutes with as it should so that the Z2-graded pin+ bundle is well defined. In a cylinder N = M × R we have iD/5d = 6 5(∂5 + iD/4d), where iD/4d = iγ μDμ and γ μ := −i 5 μ (μ = 1, 2, 3, 4). The eta invariant for the index problem is defined by using this iD/4d in the subspace = +1 (↔ 5 = γ1γ2γ3γ4). Because of the lack of a perturbative anomaly in five dimensions, the index gets contributions only from the boundary eta term as in (2.27). In this setup, we can also define a charge conjugation matrix C such that C I∗ = I C (I = 1, . . . , 5) and C 6∗ = 6C, and hence C ∗ is a section of the same bundle as with the same eigenvalue. For a more mathematical exposition, see [31]. Note that ηthere = ηhere/2. This Md is not necessarily a manifold, but it is a reasonably well-behaved topological space. Let us take a smaller n(i)-dimensional disk Di whose closure is contained inside Di, i.e., D i ⊂ Di. Then, Md is constructed by gluing the d-dimensional disks Di (n(i) = d) with the space Md := Md \ n(i)=d Di ∼ {pt}, Md ∼ Md−1, Di ∩ Md ∼ Sd−1. where ∼ means the homotopy equivalence.10 The gluing region Di ∩ Md is homotopy equivalent to a sphere Sd−1, Now we have done enough preparation to discuss the topology of the G bundle on M = M4. We will use the following facts about a simple, connected, simply connected Lie group G: π0(G) = 0, π1(G) = 0, π2(G) = 0, π3(G) = Z. Suppose inductively that the gauge bundle on Md−1 can be trivialized. Then, because of the homotopy equivalence, the bundle on Md is also trivial. The disks Di (n(i) = d) are homotopically trivial and hence the bundle on them can also be trivialized. We construct Md by gluing Md and Di (n(i) = d). If the bundle is trivialized on each Md and Di (n(i) = d), the gluing of the bundle is specified by an element of πd−1(G) for each i (n(i) = d). When d < 4, the homotopy group πd−1(G) is zero and hence the bundle on Md is again trivial. Thus the induction continues when d < 4. When d = 4, the element of π3(G) = Z associated to the gluing of Di can be thought of as the instanton number localized on the disk Di. It is clear that topologically we can gather all the instantons to a single four-dimensional disk (say D0) by continuous deformation, and define the total instanton number n. If the manifold is orientable, this is the end of the classification of G bundles. The G bundle on M is classified by the integer n ∈ Z which is the instanton number. However, if M is not orientable, there is one more twist to the story. Locally on the disk D0, we can define an orientation and distinguish instantons from anti-instantons. However, globally, if we move an instanton through a path along which the orientation flips sign, an instanton comes back as an anti-instanton. This process changes the instanton number from n to n − 2. Therefore, only the n mod 2 can be a topological invariant. Recall that n instanton amplitude is proportional to einθ . This phase factor is consistent with the mod 2 nature of n only if θ is 0 or π . This is precisely the same as the requirement of CP invariance in gauge theory. In fact, we can put the theory on an unorientable manifold if and only if the theory has a CP invariance. Another way to present what we have found here is as follows. The obstruction theory as described here defines an analogue of the characteristic class c2 of a unitary bundle for any simple, connected, 10 More precisely, the situation is as follows. Let X be a topological space and Y ⊂ X its subspace. Suppose that there exists a continuous one-parameter family of maps ft : X → X (0 ≤ t ≤ 1) such that f0 is the identity map, f1(X ) ⊂ Y , and f1(y) = y for y ∈ Y . If such an ft exists, Y is said to be a deformation retract of X . Now, if there is some vector bundle E on X , we can consider a one-parameter family of bundles Et = ft∗E on X such that E0 = E and E1 is a pull-back of a bundle E|Y on Y . Then the topology of the bundle E = E0 is classified by the topology of E|Y . In our situation, we are using the case (X , Y ) = (Di, {pt}) and (Md , Md−1). and simply connected gauge bundle, which we still denote by c2 by a slight abuse of the notation. This c2 is a class in H 4(M , Z). When M is orientable, this cohomology group is Z, and then c2 defines an integer-valued instanton number. When M is unorientable, however, this cohomology group itself is Z2, and then c2 only gives us the instanton number modulo 2. 2.4.3. Derivation Now we can show our crucial identity (2.26) by using the facts established above. We have gathered instantons on a single disk D0 inside the manifold M . Then, we can represent the manifold M as a connected sum M #S4, where M and S4 are connected by a tube. The M #S4 is the same as M as a manifold, but we can put all the instantons on S4. The connected sum M ∼= M #S4 is equivalent to the direct sum M + S4 in the cobordism group, and we can use the cobordism invariance to compute η as η(M , A) = η(M , Atrivial) + η(S4, An instantons) mod 4Z where we have used the fact that η mod 4Z in an oriented manifold is the same as the Atiyah–Singer index, which is given by 2tρ n in an n-instanton background in S4. Equivalently, one can also see the fact that η(S4, An instantons) = 2tρ n mod 4Z from (2.8) and (2.22). This establishes our claim. 2.5. Flavor symmetries and the η invariant Up to now, we have considered only the CP symmetry as the protecting symmetry defining the SPT phase. In this subsection, as an application of the analysis of the η invariant in the previous two subsections, we consider what happens when the theory possesses other global symmetries F. This subsection is slightly outside the main points of this paper, and can be skipped in the first reading. For simplicity we assume that F commutes with CP. Put differently, we are going to study the system as an SPT phase protected by CP × F. We assume that a gauge bundle has no effect on η, as discussed before. However, when we have a flavor symmetry F, we can introduce a background flavor gauge field for F. Such a background field defines a bundle which we denote as EF. The η(M , EF) in general has dependence on this flavor symmetry bundle. If F is a Lie group that is (semi-)simple, connected, and simply connected, the effect of EF can be classified in completely the same way as in the case of a gauge bundle. We have the relation η(M , EF) = ν η0(M ) + 2tFn mod 4Z for some parameter tF. If tF is odd, then the effect of a flavor bundle is non-trivial. However, F need not be (semi-)simple, connected, or simply connected. Rather than doing a systematic analysis, let us give a simple example to illustrate the point. Suppose that we have ν free Majorana fermions with the same mass parameter. Then the theory has O(ν) flavor symmetry. Let us suppose that we add interactions to this system, and the symmetry is explicitly broken down to a subgroup, say F = (Z2)ν , which acts on each Majorana fermion as (−1). In any unoriented manifold M of spacetime dimension d, there is an orientation line bundle E = ∧d TM . The transition function of this bundle can be taken to be ±1, and hence it is a Z2 bundle. Using this bundle, we can consider a flavor bundle given by EF = i=1 where si = 0 or 1. If we have a pin+ structure, then pin + := E ⊗ pin+ is another pin+ structure which is conjugate to the original one in the sense that all the eigenvalues of the Dirac operator on pin + have the opposite sign from those of pin+. From this fact, we can see that η under the above flavor bundle is given by i=1 The implication of this equation is as follows. For ν=16, η(M ) mod 4Z is trivial if fermions are not coupled to non-trivial bundles. However, once we introduce a flavor bundle EF, η(M , EF) becomes non-trivial. This means that the boundary theory of this SPT phase must be non-trivial, when we require that the interactions preserve the F symmetry. Now let us consider the case that we are not imposing F as a symmetry protecting SPT phases, but it is just an accidental symmetry. We denote this accidental symmetry as F. Then the above discussion gives us a simple necessary criterion for an interaction that collapses the free fermionic classification: The interaction term that collapses the free fermionic classification must be sufficiently generic that the accidental flavor symmetry F which remains unbroken by the interaction is small enough such that the quantity η(M , EF ) mod 4Z does not depend on the flavor symmetry bundle EF . For example, in the case of the (1 + 1)-dimensional system of class BDI with ν=8, the interactions which Fidkowski and Kitaev introduced [8] to gap the boundary mode breaks the symmetry from O(8) down to Spin(7). The bundle of Spin(7) in two dimensions is always trivial, so it has no effect on η. Here it is important that the unbroken group is Spin(7) instead of SO(7) = Spin(7)/Z2. Quantum chromodynamics as SPT phases In the previous section, we have seen that the SPT phases of the free Majorana fermions do not change even if we add the gauge interaction, as long as tρ ∈ 2Z and the gauge group is simple, connected, and simply connected. In this section, we study this statement from the viewpoint of the low-energy effective theory of Goldstone bosons after the color confinement in theories of quantum chromodynamics (QCD), with gauge groups SU(N ), Spin(N ), and Sp(N ). 3.1. The models We consider SU(N ) theory with Nf fundamental flavors ψ and ψ˜ in the representation ρ = Nf ⊗ (N ⊕ N), Spin(N ) theory with Nf fundamental flavors ψ in ρ = Nf ⊗ N, and Sp(N ) theory with 2Nf half-flavors ψ in ρ = 2Nf ⊗ 2N. Here we are considering Spin(N ) rather than SO(N ) to agree with our condition π1(G) = 0, but this difference is not so important as far as the Goldstone bosons are concerned. First, let us summarize what is believed to happen in these theories; see, e.g., [32] for a standard textbook. In the massless case, these gauge theories have flavor symmetry F0, which is F0 = SU(Nf )L × SU(Nf )R in SU(N ) theory,11 F0 = SU(Nf ) in Spin(N ) theory, and F0 = SU(2Nf ) in Sp(N ) theory. 11 There is also U(1) baryon symmetry, but it is irrelevant for the discussion below and we neglect it. If we add a mass, these flavor symmetries are broken down to a subgroup. Maximal possible flavor symmetries with massive fermions are F = SU(Nf ) in SU(N ) theory, F = SO(Nf ) in SO(N ) theory, and Sp(Nf ) in Sp(N ) theory. It was proved, under some technical assumptions, by Vafa and Witten [33] that these flavor symmetries F preserved by the mass term are not spontaneously broken. It is also believed that symmetries which are in F0 but not in F are all spontaneously broken when N Nf . Let us see each case in more detail. The fermion Lagrangians are given as follows: L = −iψ σ μDμψ − iψ˜ σ μDμψ˜ − m(ψ˜ ai ψia + ψ iaψ˜ ia), We took the mass matrix to be proportional to the unit matrix for SU(N ) and Spin(N ), and to J for Sp(N ). The action of CP is discussed in greater detail in Appendix B; see (B.5), (B.1), and (B.11) in particular. The parameter ν and the flavor symmetries in the massless case F0 and the massive case F are summarized in Table 1. When N is large enough and m is small enough, the low-energy dynamics is described by (pseudo-)Goldstone bosons associated with the spontaneous symmetry breaking from F0 to F. The condensate is given by (J −1)ik (J −1)abψk ψj = −2v3U i, a b j where v is the mass scale of the condensate, and U is the unitary matrix representing the Goldstone bosons. The CP transformation acts on the matrix U as The properties of the matrix U can be summarized as follows: ◦ For SU(N ), U takes values in [SU(Nf )L × SU(Nf )R]/SU(Nf ) SU(Nf ) and hence it is a special unitary matrix, U †U = 1 and det U = 1. ◦ For Spin(N ), U takes values in SU(Nf )/SO(Nf ) and it is a special unitary matrix, U †U = 1 and det U = 1, which is also symmetric U T = U . The fact that U takes values in SU(Nf )/SO(Nf ) may be seen by writing it as U = VV T , where V ∈ SU(Nf ) with gauge invariance V ∼ VW for W ∈ SO(Nf ). CP(U ) = U †. ◦ Finally, for Sp(N ), U = JU takes values in SU(2Nf )/Sp(Nf ) and it is a unitary matrix, U †U = 1, which is anti-symmetric, U T = −U and Pf (U ) = 1. The fact that U takes values in SU(2Nf )/Sp(Nf ) may be seen by writing it as U = VJV T , where V ∈ SU(2Nf ) with gauge invariance V ∼ VW for W ∈ Sp(Nf ). When the mass is zero, these Goldstone bosons are massless, but when the mass is turned on, they have a potential energy Vpotential = −mv3(tr U + tr U †). If m > 0, the vacuum is at U = 1. The number tρ is given uniformly by tρ = Nf . For the reasons discussed in the previous section we require tρ ∈ 2Z, so we restrict attention to the case of even Nf . Assuming this, when m < 0, the vacuum is at U = −1. Notice that the condition Nf ∈ 2Z is necessary from this point of view because det(−1) = (−1)Nf for SU(N ) and Spin(N ), and Pf (−J ) = (−1)Nf for Sp(N ). 3.2. Phases from the Goldstone boson effective action The Wess–Zumino–Witten (WZW) term in the low-energy theory of Goldstone bosons is crucial in reproducing the non-trivial value of arg Z(M ) on a manifold M obtained in the UV path integral argument. The basic properties of the WZW terms are reviewed in Appendix C. Let us now discuss concrete examples in which the low-energy effective action of a strongly coupled gauge theory gives a non-trivial phase arg Z(M ), reproducing the non-trivial SPT phase. The situation is as follows. When the mass m is non-zero, the Goldstone boson gets massive and there is a unique vacuum with unbroken CP symmetry, since we assume Nf ∈ 2Z. We want to compute the arg Z(M ) of theories with m > 0 and m < 0, and see whether they match the expectation from the UV path integral analysis. More precisely, we consider the difference12 of arg Z(M ) for m > 0 and m < 0, which can be computed as follows. Let us smoothly change the mass parameter m within some range I = [−m0, m0], where m0 > 0. Then we have I5(V −1dV ), I5 = 2π κ 5 = 2π κ ((2−π 1i))3 52!! tr(V −1dV )5, 12 The value of arg Z(M ) itself cannot be computed for the following reason. From the UV point of view, the value of arg Z(M ) depends on the sign of the Pauli–Villars mass parameter, but that information is missing in the Goldstone boson effective action. Also, some manifolds that give non-trivial values of arg Z(M ) such as RP4 cannot be a boundary of any five-dimensional manifold and hence there is no natural way to define the WZW term. This is one of the limitations on the low-energy effective theory of Goldstone bosons. Still, the difference of the phase is a perfectly well-defined quantity and can be computed using the WZW term. where V takes values in the coset space F0/F and is related to U as discussed above, and the value of κ is determined by the ’t Hooft anomaly for (F0)3. Note that this integral over I × M makes sense even on an unorientable manifold, since I5 receives an additional sign change when the orientation is reversed, due to the action of CP on V . In practice, we compute this integral by considering an oriented double cover of M , which we denote as M˜ . This M˜ is oriented, and M is obtained from M˜ as M = M˜ /Z2, where the Z2 action on the manifold is given by an orientation-reversing diffeomorphism that we denote as σ : M˜ → M˜ . Then we have where the factor of 1/2 comes from the fact that M is half of M˜ . On the double cover M˜ , the field V must be consistent with the fact that it must reduce to a configuration on M . Denoting the coordinates of I and M˜ as t and x respectively, the correct rule is that under the action of σ it behaves as where VCP is the CP action on the value of V . We also need to impose the condition that the values of V at the boundaries of I go to the vacuum expectation values V |t=m0 = Vvac,m>0, V |t=−m0 = Vvac,m<0. Let us further restrict our attention to the case M = RP4 and M˜ = S4. Because of the condition (3.14), we may think of I × S4 as S5 by shrinking S4 at the ends of I . The north pole and the south pole of S5 correspond to t = m0 and t = −m0, respectively. Realize S5 as a unit sphere in a flat six-dimensional space with coordinates X I (I = 1, . . . , 6), and let Xˆ I be the points on S5 with (Xˆ )2 = 1. Let Xˆ0 be the north pole. The action of σ is then given by This action fixes the north pole Xˆ = Xˆ0 and the south pole Xˆ = −Xˆ0. We now compute δ arg Z (M ) for some specific examples. 3.2.1. SU(N ) with Nf =4 In this case, we can take V = (V1, V2) ∈ SU(Nf ) × SU(Nf )R and U = V1V2−1. In terms of these variables, the WZW term is given by where we used the fact that κ = N . We consider the configuration (C.9), U = P+( · Xˆ0)( · Xˆ ), where I=1,...,6 are the 8 × 8 Gamma matrices and P+ is the projection to the positive chirality space. Because of this projection, the right-hand side of (3.17) can be regarded as a 4 × 4 matrix, suitable for Nf = 4. This configuration has exactly the desired properties: it has the vacuum values U (Xˆ0) = 1 and U (−Xˆ0) = −1 at the north and south poles, and it satisfies the condition (3.13), where UCP = U †. The computation of the WZW action is reviewed in (C.10), with the result where we have used ν=2Nf N = 8N . By using the fact that η(RP4) = −1/4, this result exactly reproduces the path integral computation. 3.2.2. Spin(N ) with Nf = 8 where we have used the fact that κ = N , 5(VV ) = 5(V ) + 5(V ) for a closed manifold, 5(V T ) = 5(V ), and hence 5(U ) = 2 5(V ). We want to consider a configuration like (3.17), but we cannot directly use it because U must satisfy the condition U T = U . Instead, we can consider the configuration 1 U = 2 where S and A are the symmetric and anti-symmetric parts of P+( · Xˆ0)( · Xˆ ), respectively. This is possible for Nf = 8. One can check that it satisfies the desired properties. Then we get 3.2.3. Sp(N ) with Nf = 2 where we have used κ = 2N and 5(U ) = 2 5(V ). The U must satisfy (JU )T = −JU . Actually, it turns out that we can just use (3.17) in this case. The six-dimensional Clifford algebra I has a charge conjugation matrix C which has the property that (C M )T = −C M and [C M , P+] = 0. Then, we can identify J as J = C · Xˆ0. This gives the desired property (JU )T = −JU . Therefore we get 4.4.3. Step 3: Pinning the adjoint vev In the following, we pick a positive constant m0, and we change the mass parameter m from m = m0 > 0 to m = −m0 < 0. We always keep m positive. To pick the point A as our vacuum when m = ±m0, we introduce a SUSY-breaking potential VN =0 = λ| Tr 2 − 8m02|2 + λ Tr[ , †]2 on the dual side. The vev of ∼ diag(2m0, −2m0) up to very small corrections in g. This vev breaks SU (2) to U (1). We would like to take λ and λ in (4.22) to be sufficiently large to pin the vev of , while keeping everything perturbative. Recall that we are using a convention common in N = 2 studies such that the kinetic term for is non-canonical, such that the Kähler potential is given by K = 1/g2 Tr( † ). After canonically normalizing the fields, the condition of perturbativity is given by g4λ, g4λ 1. To pin the vacuum at (4.23) we need λm40 g2m2 m20, so that the contribution of VN =0 to the total potential is much more significant than the SUSY-preserving ones, which include a term g2|m + · · · |2. Later we need to impose the condition g2m > 2m0 in (4.38), and hence we need λ g−2. In summary, we take λ, λ to be in the region g2 g4λ, g4λ 1. For simplicity we consider the formal limit in which λ, λ → ∞, g → 0 such that the condition just stated is satisfied. Now, the system has a minimum at the point A when m = ±m0. When m deviates from ±m0, the point of the massless dual quark on the u-plane at = (2m, −2m) is different from the point (4.23). We always keep (4.23) by the above potential. 4.4.4. Step 4: Determination of the vev of the squarks At this point the potential for the scalar components of the dual quarks Q, Q˜ is still given by that of the N = 2 theory. Using the standard formula for the supersymmetric Lagrangian and replacing by the vev given above, the potential of the scalar components of the dual quarks is now given by V = VF + VD, where VF is obtained from the superpotential W = i4=1(−Q˜ i Qi) + 2mQ˜ 1Q1 + 12 m Tr 2 as i=1 4m m0 − (Qi+Q˜ +i − Qi−Q˜ −i) i=1 i=1 j=2 + 4(m0 − m)2(|Q1+|2 + |Q˜ +1|2) + 4(m0 + m)2(|Q1−|2 + |Q˜ −1|2) and VD is the D-term potential given by i=1 i=1 (|Qi+|2 − |Qi−|2 − |Q˜ +i|2 + |Q˜ −i|2) where the ± on the quark fields are the SU (2) indices a = ±, or more explicitly we are using the notations Qi = (Qia) = (Qi+, Qi−)T and Q˜ i = (Q˜ ai) = (Q˜ +i, Q˜ i ). − When m = ±m0, we see that the vacuum is given by Q1 = (2√m m0, 0)T , Q1 = (0, 2√m m0)T , Q˜ 1 = (2√m m0, 0) Q˜ 1 = (0, −2√m m0) (m = m0 > 0), (m = −m0 < 0). The vevs of Qi and Q˜ i for i = 2, 3, 4 are all zero. 4.4.5. Step 5: Determination of the dual CP transformation At this point we have determined the vevs of all the fields on the dual side, and we can complete the determination of the CP transformation on the dual side, whose last step was left unfinished at the end of Sect. 4.2. The action of CP on the dual adjoint field is simply given by However, this CP0 has two problems. First, its square is given by (CP0)2 = (−1)F (−1)G, where (−1)G is the center of SU (2); (−1)G = −1 for dual quarks and (−1)G = +1 for other fields. But we want the relation CP2 = (−1)F so that the dual theory can be put on a pin+ manifold. Second, this CP0 is broken by the vevs of the dual quarks given above, because its action on Q1 and Q˜ 1 is given by CP0(Q1) = iQ˜ 1 and CP0(Q˜ 1) = −iQ1. These problems can be solved at the same time by introducing a gauge transformation CSU(2) ∈ SU (2) given by and then the vev = diag(2m, −2m) (4.23) is invariant. On the dual quarks, the action can be written as follows. First, we define CP0. which commutes with the SU (2) gauge group, as CP( ) = CP0 := CSU(2)RCSO(8)Pinv. (i = 2, 3, 4). Under this CP, the transformations of the fields are given as CSU(2) = diag(−i, i) CP :=CSU(2)CP0 = CSU(2)CSU(2)RCSO(8)Pinv. CP(Q˜ ±1) = ±Q1±, CP(Q˜ ±i) = ∓Qi± They preserve the vevs (4.27), (4.28) of the scalar components of the dual quarks. The fact that we need to mix the gauge transformation CSU(2) to the definition of CP implies that the symmetry group of the dual theory is not simply SU (2) × Pin+, but is more complicated. We will discuss more on this point in Sect. 4.6. 4.4.6. Step 6: Decoupling of unwanted scalars When m = ±m0, the only non-zero vev of the scalar components of the dual quarks are CP-even, and the vev is neutral under SO(6), which acts on Qi, Q˜ i for i = 2, 3, 4. The potential V (4.24) is at most quartic, and invariant under CP and SO(6). We can then safely add large mass terms to the scalars charged under SO(6), and to the scalars neutral under SO(6) but odd under CP, to remove them. More concretely, we proceed as follows. First we add mass terms, i=2 Vadd1 = M 2 As long as M is much larger than other mass scales, we can integrate out the SO(6) charged quarks Qi, Q˜ i for i = 2, 3, 4 and set them to zero. Next, we add a gauge- and CP-invariant potential )Q1 − (2m0 + )Q1 + (2m0 − = 16λ m02 |Q1+ − Q˜ +1|2 + |Q1− + Q˜ −1|2 , where we set = diag(2m0, −2m0). This gives masses to the CP-odd scalars Q1+ − Q˜ +1 and Q1− + Q˜ −1, and we can set them to zero if λ is large enough. The remaining scalars are complex fields z and w given by Then, the potential (4.24) simplifies to This is now far easier to analyze. Now we can find the potential minimum for general m. We require that parameters satisfy the relation Under this condition, when m = 0, the minimum is given by |w|2 = w = 0 z = 0 The phase of z or w is eaten by the U (1) gauge field by the Higgs mechanism and they become massive together. There is a unique vacuum with a mass gap. When m = 0, the minimum of V is realized by (z, w) satisfying the conditions |z|2 + |w|2 = (4g2m m0 − 8m02)/g2 and zw = 0. This potential itself leads to a first-order phase transition from (z = 0, w = 0) to (z = 0, w = 0). To avoid such a phase transition, we further deform the potential by adding SU (2)- and CP-symmetric terms given by Vadd3 = − 41 g 2|Q1τ AQ1 − Q˜ 1τ AQ˜ 1|2 = −4g 2|zw|2, Vadd5 = − (Q˜ 1J Q1 + c.c.) = −2 (zw + c.c.). Let us explain the roles of each of them: ◦ The Vadd3 term cancels the term 4g2|zw|2 by taking g → g, which makes the analysis of the potential a little easier (but this is not absolutely necessary). After turning on Vadd3, the potential minima are given by |z|2 + |w|2 = (4g2m m0 − 8m20)/g2. ◦ Then, by turning on Vadd4, we can fix the ratio of the absolute values of z and w, |z|/|w|, to whatever values we want. Thus we can smoothly connect the points (z = 0, w = 0) to (z = 0, w = 0) by smoothly changing the parameter μ. ◦ Finally, in the intermediate region where both z and w are non-zero, there remains a massless boson coming from the relative phase arg(zw). This is the Goldstone boson associated with the U(1) flavor symmetry acting on (Q1, Q˜ 1). (The overall phase is absorbed by the dual U (1) gauge field.) By turning on Vadd5 with small , this Goldstone boson is eliminated since this Vadd5 breaks the flavor U(1) symmetry. This completes the argument that we can continuously deform from m = m0 > 0 to m = −m0 < 0 without having massless bosons (i.e., scalars or gauge fields). In particular, the vevs of z and w do not break CP, so the CP is preserved during this continuous deformation. Let us note the following point, in relation to the criterion we found in Sect. 2.5. Before adding the term Vadd5 the theory has the flavor symmetry U(4), which is preserved by the mass m. After adding the Vadd5 the flavor symmetry is reduced to SU(4), which is the double cover of SO(6). Here we consider the double cover SU(4) instead of SO(6) because the quarks of the original electric theory are in the fundamental representation of SU(4). Then we can avoid the no-go argument given in Sect. 2.5 because this SU(4) is simple, connected, and simply connected and has tF = 2 in the notation of that section. Note that the value tF = 2 is realized in different ways in the electric theory and magnetic theory. In the electric theory, we have two copies of the (4 + 4)-dimensional representation of SU(4). In the magnetic theory, we have two copies of the 6-dimensional representation of SU(4), which is the vector of SO(6). In fact, if U(4) were preserved, in particular we would have (Z2)4 ⊂ U(4), where each Z2 acts on each flavor of quarks Qi, Q˜ i in the electric theory. Then, by completely the same argument as the one around (2.38), it would be impossible to make the boundary theory trivial. Thus the term Vadd5 is really crucial. 4.4.7. Step 7: Analysis of the fermions Now let us consider the fermion masses. In the process of continuous deformation, the scalar components of the dual quarks Qj, Q˜ j (j = 2, 3, 4) never get a vev. Then, the fermions contained in Qj, Q˜ j (j = 2, 3, 4) do not mix with other fermions and their masses are given by the vev = (2m0, −2m0), which is constant. Therefore these fermions do not become massless during the deformation. The remaining fermions are 3 + 3 + 2 + 2 = 10 fermions coming from the N = 1 gauginos (λ++, λ−−, λ0), and the fermions in (ψ ++, ψ −−, ψ 0) and in Q1, Q˜ 1 (q+, q−, q˜+, q˜−). This implies that we can realize at most the |ν| ≤ 10 SPT phase. Now, note that ν mod 16 is preserved by coupling to the gauge fields as we discussed in Sect. 2 because the partition function depends on ν mod 16. Thus the only logical possibility is ν=0. This simple argument guarantees our success. For those who are not satisfied by the above argument, the fermion mass matrix is treated explicitly in Appendix E. Up to now, we have taken the masses of quarks Qi, Q˜ i as m1 = m2 = m3 = m4 = m. However, we can also consider other masses such as m1 = m2 = ma, m3 = m4 = mb. Then we can change only mb from m0 to −m0 while fixing ma at ma = m0. This case corresponds to the case of the ν=8 SPT phase because we are only changing the masses of half the quarks. In this case, we should not be able to deform the parameters while preserving the mass gap and without breaking the CP. Let us look at what happens. The dual quark masses are given by m1 = ma + mb, m2 = ma − mb, m3 = m4 = 0. Let us change mb from positive to negative values while fixing ma. When mb > 0, the potential minimum is realized by giving a vev to Q˜ +1, Q1+. When mb < 0, the potential minimum is realized by a vev of Q˜ 2 , Q+. + 2 However, recall that the action of CP was given as in (4.33). From this, one can check that the vev of Q˜ +1, Q1+ does preserve the CP in the region mb > 0 but that the vev of Q˜ +2, Q2+ does not preserve the CP in the region mb < 0. This originates from the difference in the actions of CSO(8) on Q˜ +1, Q+ 1 and Q˜ +j, Qj+ (j = 2, 3, 4). This does not mean that we do not have any CP symmetry in the region mb < 0. We can take a gauge transformation (−1)G := diag(−1, −1) ∈ SU (2) and define a new CP as CP = (−1)GCP. Notice that this new CP also satisfies CP 2 = (−1)F . However, the definition of the unbroken CP changes when we pass through the region mb = 0. Let us consider a deformation process from mb = m0 to mb = −m0. We assume that the adjoint scalar vev = diag(2m0, −2m0) is fixed, but the vevs of other scalars can be arbitrary by introducing various terms in the potential. We can mix gauge transformations to CP, but to satisfy CP2 = (−1)F and keep the adjoint vev invariant, the only possible CP are (4.32) and (4.46). We need to consider two cases separately: ◦ During the continuous deformation, we go through a parameter region where both CP and CP = (−1)GCP are broken. ◦ During the continuous deformation, we go through a parameter region where both CP and CP = (−1)GCP are unbroken. In the first case, CP is broken in that parameter region and we cannot smoothly connect mb = m0 and mb = −m0. In the second case, the (−1)G ∈ SU (2) is unbroken in the parameter region where both CP and CP = (−1)GCP are unbroken. However, the (−1)G is broken at the initial and final points of the deformation mb = m0 and mb = −m0, and hence we have to encounter a phase transition during the continuous deformation at which (−1)G is recovered. Therefore, in either case, we cannot show that the SPT phases corresponding to mb = m0 and mb = −m0 are the same. Of course, this is as expected since ν = 8 is guaranteed to be non-trivial by the consideration of the phase of the partition function on RP4, but it is reassuring that we also obtain a consistent result from this analysis. 4.6. Pin structure in the dual theory We have established that the ν=16 phase and the ν=0 phase can be continuously connected on a flat space, as shown schematically in Fig. 2. During the discussion, we have found that we have to mix a gauge transformation CSU(2) = diag(−i, i) to define the CP action on the dual magnetic theory, as CP =CSU(2)CP0 = CSU(2)CSU(2)RCSO(8)Pinv. Let us consider putting the theory on an unoriented pin+ manifold. Then the above fact means that we have to mix gauge transformations in the definition of the pin+ structure. Let us see more precisely how it works. The reason we discuss it here is to make sure that there is no inconsistency analogous to the anomaly of the spin–charge relation discussed in [29]. For this purpose, we have to identify the symmetry group of the theory that is left unbroken by various mass and deformation parameters. The CP0 commutes with the gauge symmetry, but its square is (CP0)2 = (−1)F (−1)G and hence the unbroken group cannot be simply Pin+(3, 1) × SU (2). It also implies that if we naively forget the gauge group SU (2), the theory cannot be put on a pin+ manifold because the quarks would have the relation (CP0)2 = 1, meaning that we need a pin− structure, but a pin− structure cannot be put on a generic pin+ manifold.14 So let us see what is going on. Before introducing various deformations, the dual N = 2 theory with N = 2 preserving mass term has the symmetry group SU (2) × SU(2)R × SO(2) × SO(6) × Pin−(3, 1) at the level of its Lagrangian. Here, Pin−(3, 1) is defined by using Pinv which came from the (5 = 4 + 1)-dimensional Lagrangian. Let (−1)G ∈ SU (2), (−1)R ∈ SU(2)R, and (−1)F ∈ Pin−(3, 1) be the centers of the respective groups. One can check that the product (−1)G(−1)R(−1)F acts trivially on all the fields of the theory, so the symmetry group is actually × SO(2) × SO(6), where Z2 is generated by (−1)G(−1)R(−1)F . After introducing deformations, this is broken to a subgroup. To describe it, take (Z4)R ⊂ SU(2)R which is a Z4 subgroup generated by CSU(2)R; (−1)R is a unique order-2 element in this (Z4)R. We 14 A quick way to see this is as follows. Let M be a d-dimensional manifold and E = ∧d TM be its orientation line bundle. Let Ed+1 = TM ⊕ E and Fd+3 = TM ⊕ E ⊕ E ⊕ E be orientable bundles. Then the spin structures of Ed+1 and Fd+3 reduce to the pin− and pin+ structures of M , respectively. The Stiefel–Whitney class of TM is denoted as w = 1 + w1 + w2 + · · · . Then the Stiefel–Whitney classes of Ed+1 and Fd+3 are given by w(Ed+1) = (1 + w1 + w2 + · · · )(1 + w1) = 1 + (w2 + w12) + · · · and w(Fd+3) = (1 + w1 + w2 + · · · )(1 + w1)3 = 1+w2 +· · · , where we have used the fact that the line bundle E has the Stiefel–Whitney class 1+w1. Therefore, a pin− structure requires w2(Ed+1) = w2 + w2 1 = 0, while a pin+ structure requires w2(Fd+3) = w2 = 0. This means that a manifold with w2 1 = 0 can have at most only one of pin− or pin+, but not both. For example, the Stiefel–Whitney class of RPd is given by w = (1 + a)d+1, where a is the generator of H 1(RPd , Z2). Thus w2(Ed+1) = 21 (d + 1)(d + 2)a2 and w2(Fd+3) = 21 d(d + 1)a2. If we put d = 4, we can see that RP4 has pin+ but it cannot have pin−. define a homomorphism π1 : Pin− → Z2 such that elements of Pin − which reverse orientation map to the element (−1) ∈ Z2, and also define π2 : (Z4)R → Z2 such that π2(CSU(2)R) = (−1). Furthermore, we take a Z2 subgroup (Z2)S ⊂ SO(2). Then we define a new group P˜in(3, 1) as P˜in(3, 1) := {(c, d, e) ∈ (Z4)R × (Z2)S × Pin −; π2(c) = d = π1(e)}. This contains the elements (−1)R, (−1)F , and CP0 = CSU(2)RCSO(8)Pinv, and it is a double cover of both Pin−(3, 1) and Pin+(3, 1). Indeed, the projection to the third component just gives Pin−(3, 1), which is 2 : 1, and the Z2 quotient with respect to (−1)R(−1)F is Pin+(3, 1). Then the unbroken group after the deformations is SU (2) ×Z2 P˜in(3, 1) := times SO(6), which is uninteresting to us. Here, the Z2 quotient is taken with respect to the element (−1)G(−1)R(−1)F as before. This group (4.50) is the structure group of the magnetic theory when we put the theory on a non-trivial manifold. This is an analog of the Pinc group. There is a homomorphism from the group SU (2)×Z2 P˜in(3, 1) to Pin+(3, 1) such that the following diagram commutes: SU (2) ×Z2 P˜in(3, 1) This can be described as follows. Mathematically, we have a forgetful map from the left-hand side to P˜in(3, 1)/Z2, which equals Pin+(3, 1) as discussed above. More physically, consider a gaugeinvariant fermion operator of the theory. For example, we may take where λ is the gaugino. Then the transformations in SU (2) ×Z2 P˜in(3, 1) acts as transformations in Pin+(3, 1) on . Therefore, by looking at the transformation of , we can define the above homomorphism. If we are given a pin+ manifold M , the theory can be put on M as follows. The pin+ bundle is defined on M by a set of transition functions. We specify an uplift of transition functions from Pin+(3, 1) to SU (2) ×Z2 P˜in(3, 1), Pin +(3, 1) → SU (2) ×Z2 P˜in(3, 1), such that it is consistent with the homomorphism (4.51). There exists such an uplift of the bundle; for example, we can use (4.47) for this purpose. For the existence of this uplift, the division by Z2 = {1, (−1)G(−1)R(−1)F } in (4.50) is important, because the square of CP is given by CP2 = C2SU(2)C2SU(2)RPi2nv = (−1)G(−1)R, where we have used Pi2nv = 1. This is equal to (−1)F only if we impose (−1)G(−1)R(−1)F = 1. One consequence of the division by Z2 is that if there is a field which is a singlet under P˜in(3, 1), then that field must have gauge charge under SU (2)/Z2 SO(3). In particular, the SU (2) bundle does not exist in itself in general. Note that the uplift with the above property is not unique, but remember that SU (2) is a dynamical gauge group and hence we integrate over all the possible uplifts in the path integral. In this way we can consistently define the magnetic theory on a pin+ manifold. In this paper, we studied the effects of strongly coupled gauge interactions on the topological phases of matter. We mainly studied those SPT phases protected by the CP symmetry with CP2 = (−1)F , in 3+1 spacetime dimensions with relativistic symmetry. We first discussed under which conditions the introduction of (possibly strongly coupled) gauge interactions preserve the topological phases. The rule of thumb we found is that if the π0 and π1 of the gauge group are trivial and the effective theta angle is zero, the system with dynamical gauge field can be continuously connected with the system before the addition of the gauge field, and thus is in the same topological phase. We gave a general derivation of this statement, and then verified it in more detail in the case of Majorana fermion systems coupled to the gauge field. We then tested our statement by studying non-supersymmetric QCD with various groups in the infrared, using the WZW action of the pseudo-Goldstone modes. Next, we showed that knowledge of the strong-coupling dynamics allows us to directly show that the ν=16 phase of the topological superconductor can be continuously connected to the ν=0 phase, thus explicitly demonstrating the collapse of the free fermion classification Z to Z16 due to the interaction effects. The crucial input we used was the S-duality of N = 2 SU(2) gauge theory with Nf = 4 flavors. Clearly, what was given in this paper is just the tip of the iceberg of the connection between the rich field of topological phases of matter and strongly coupled gauge dynamics. A couple of further directions that immediately come to our mind: ◦ In this paper, we mainly discussed the gauged topological phases from the bulk point of view, and did not discuss the dynamics of the boundary theory much. We should definitely study them; it might or might not be helpful in finding the boundary theory explicitly. ◦ In this paper, we studied the behavior of the CP invariance under duality only in the case of N = 2 SU(2) theory with Nf = 4 flavors to the minimal extent necessary for our analysis. The result was much subtler than naively expected. It would be interesting to carry out a systematic analysis of the CP actions for other known dualities, to see if they tell us anything new about topological phases of matter. We would like to come back to some of these questions in the future. The authors thank Yu Nakayama for collaboration during the initial stage of this work. It was his journal club on the so-called Kitaev–Wen mechanism [38,39] that started this work, and his contribution was absolutely essential. The authors also thank Ayuki Kamada for discussions on the Wess–Zumino–Witten term. The work ofYT is partially supported by JSPS Grant-in-Aid for Scientific Research No. 25870159, and by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. The work of KY is supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. Basics of CP and T transformations In this appendix we review the basics of CP and T transformations. The discussions in this section are mainly to set up the notations and the conventions. Our conventions for the spinor fields follow those of Wess and Bagger [34]. Let us very briefly recall them. The double cover of the Lorentz group SO(1, 3) is given by SL(2, C), and hence representations of operators and fields under the Lorentz group can be specified by putting spinor indices of SL(2, C) like α = 1, 2 and α˙ = 1, 2 in the fundamental and anti-fundamental representations of SL(2, C), respectively. Using this notation, a general operator can be represented as Oα1···αpβ˙1···β˙q . Under complex conjugation, an index without dot becomes an index with dot and vice versa: (Oα1···αpβ˙1···β˙q )† = Oα˙1···α˙pβ1···βq . These spinor indices are related to the spacetime index μ = 0, 1, 2, 3 by using four 2 × 2 matrices (σ α˙ α)μ given as where τ i are the standard Pauli matrices. There are also totally anti-symmetric invariant tensors of SL(2, C) denoted as αβ , αβ , α˙ β˙, and α˙ β˙, which are explicitly given as 12 = − 21 = +1 and 12 = − 21 = −1. By using them, we can raise and lower indices; e.g., if we have an operator ψα, we define ψ α = αβ ψβ . The αβ is the inverse matrix to αβ , and hence we get ψα = αβ ψ β . We also use abbreviation: if we have, e.g., ηα and ξα, then ξ η := ξ αηα, ηξ := ηα˙ ξ α˙ , and ξ σ μη = ξ α˙ (σ α˙ α)μηα. More details can be found in Appendix A of Wess and Bagger [34]. A.1. CPT in four dimensions Any relativistic unitary theory is symmetric under the CPT transformation, and it is given in four dimensions as follows. For an arbitrary operator Oα1···αpβ˙1···β˙q it is defined as CPT(Oα1···αpβ˙1···β˙q (x)) = (−1)q · i(p+q)2 · (Oα1···αpβ˙1···β˙q )†(−x). For a derivation in the axiomatic framework, see, e.g., Sect. II.5 of [40]. Note that CPT2 = (−1)F . Using CPT invariance, T and CP are related and we can use either of them depending on our preference. We will use CP for definiteness. Here we note that Open Access funding: SCOAP3. T2 = 1 ↔ CP2 = 1, T2 = (−1)F ↔ CP2 = (−1)F . This follows from the fact that T(CPT) = (−1)F (CPT)T because of the factor of i(p+q)2 in (A.2), or more geometrically in Euclidean signature with gamma matrices γ μ, a reflection along a direction nˆμ is given by nˆ · γ up to phase for fermions, and (nˆ · γ )(nˆ · γ ) = −(nˆ · γ )(nˆ · γ ) if nˆ and nˆ are orthogonal. In this paper we will be concerned with the case CP2 = (−1)F . The CP reverses all the spatial coordinates x → −x. If we instead consider CR [20], which only reverses one coordinate, say x1 → −x1, we get CR2 = (−1)F CP2 in 3+1 spacetime dimensions, because of the difference by the 2π rotation on the x2 − x3 plane. The cases CR2 = 1 (↔ CP2 = (−1)F ) and CR2 = (−1)F (↔ CP2 = 1) correspond to the Pin+ and Pin− symmetries, respectively, when the CP is embedded in the double cover of the unoriented Lorentz group O(3, 1). A review of these Pin symmetries can be found, e.g., in [14,20]. A.2. Single Majorana fermion Suppose that we have a single Weyl fermion ψα which has two components α = 1, 2. The action of CP is dictated by the Lorentz symmetry up to a phase, In particular, we have CP2(ψ ) = −ψ . Of course, this minus sign can also be derived without any explicit computation. Note that CP commutes with the spatial SU(2) rotation, and also that CP sends the doublet ψ of SU(2) to the complex conjugate ψ . Such a transformation is possible only if the representation is strictly or pseudo-real, and the transformation squares to +1 or −1 depending on whether it is strictly or pseudo-real, respectively. As is well known, an irreducible half-integer spin representation of SU(2) is pseudo-real, meaning that CP acting on an irreducible fermion needs to square to −1. Now suppose that the fermion ψα has a real Majorana mass m, where we extracted a factor of i from the phase, for later convenience. By raising and lowering the indices, we get in the convention of Wess and Bagger [34], where the minus sign is due to the fact that αβ and αβ differ by a sign as a matrix. By taking complex conjugates, we also obtain For the Majorana mass to respect the CP, we need from which we conclude η = ±1. Suppose we perform a field redefinition is invariant. In this paper, when we speak of the sign of the mass term, we refer to this combination. where M ij is a symmetric matrix, M T = M , and the notation ∗ means taking complex conjugates of entries of the matrix. By imposing the condition that this mass term is invariant under the CP, we get where we have used matrix notation. By acting CP twice, we get or equivalently, ηM is Hermitian. Now let us perform a field redefinition where again we have used matrix notation. We would like to have (CP)2 = (−1)F , or equivalently, we require that CP is a part of the Pin+(3, 1) symmetry. Then ηη∗ = 1. Therefore η is a unitary symmetric matrix: ηη† = 1 and ηT = η. Recalling (A.15), we have that A.3. Multiple Majorana fermions Let us extend CP transformation to the case where there are multiple fermions ψi (i = 1, 2, . . .). We can define a CP by by a unitary matrix V . This effects the following changes to the various matrices introduced above: M = V T MV , η = (V )−1η(V T )−1, η M Therefore, the eigenvalues of ηM are invariant under this redefinition. We can define an invariant ν(ηM ) as the number of negative eigenvalues, As we saw in Sect. 1.2, this number classifies the free fermionic SPT phases. A.4. Gauge fields We are mainly interested in the case in which Majorana fermions are coupled to a gauge group G in some strictly real representation ρ. The kinetic term of the fermions is where TA (A = 1, . . . , dim G) are generators of the gauge group G and ρ(TA) are the representation matrices of the generators, which are real anti-symmetric matrices. By requiring the CP invariance of this kinetic term, we see that the CP transformation acts on the gauge fields as follows: CP(A0A) = A0 , CP(AiA) = −AiA (i = 1, 2, 3). A A.5. CP transformation and supersymmetry Let us extend the CP transformation to supersymmetric theories. First, supercharges must transform as CP(Qα) = iηQQα˙ . By a redefinition of the phase of Qα, we can always set ηQ = 1. Then we get where ηK is a phase factor, K (θ , θ , x) is the complex conjugate of K (θ , θ , x), and CP(t, x) = (t, −x). This transformation law is as it should be if we admit the existence of the superspace, since once the action of a group on the (super)space is given, the action of the group on functions on that (super)space is naturally given by the pull-back. For those who prefer to start from just the super-algebra acting on the Hilbert spaces of the quantum theory, the same transformation law (A.25) can also be derived as follows. In that approach, the general superfield K (θ , θ , x), whose lowest component operator is K (x), is defined by using the supercharges Q and Q as K (θ , θ , x) = ei(θQ+θQ)K (x)e−i(θQ+θQ). Then, the CP transformation is given by CP(K (θ , θ , x)) := ei(θCP(Q)+θCP(Q))CP(K (x))e−i(θCP(Q)+θCP(Q)). Here we have used the fact that the meaning of CP(O) acting on a quantum mechanical operator O is given by (CP)O(CP)†, with (CP) now interpreted as the operator which acts on the Hilbert space. Then (CP) in this operator sense commutes with θ and θ . We remark that the precise meaning of “CP” appearing in CP(θ ) and CP(θ ) is different from the one in the operator sense; CP(θ ) and CP(θ ) are just defined by (A.24). We take the transformation of the lowest component as CP(K (x)) = ηK K (CP(x)). Then, by using θ αCP(Qα) = CP(θ α˙ )Qα˙ and θ α˙ CP(Qα˙ ) = CP(θ α)Qα, we get (A.25). As a special case, consider a chiral superfield Correspondingly, we define so that we get θ αCP(Qα) = CP(θ α˙ )Qα˙ and θ α˙ CP(Qα˙ ) = CP(θ α)Qα. Generally, for a superfield K (θ , θ , x) whose lowest component is a scalar, the rule of CP transformation is Now let us discuss what action can preserve the CP invariance. Notice that combinations such as θ 2 = θ αθα and θ 2 = θ α˙ θ α˙ satisfy This observation makes superspace analysis quite straightforward. First, the kinetic terms of chiral fields, including gauge superfields V A, are given by where, as before, we consider a strictly real representation ρ, and ρ(TA) are anti-Hermitian. It is easy to see that this kinetic term is invariant if the phase factor of V is ηV = 1, i.e., Next, let us analyze the superpotential. For simplicity, let us use a “Majorana basis” for the chiral fields such that η = +1. Consider a superpotential of the form W = mij i j + yijk i j k + · · · The term in the action is then d2θ W + d2θ W . The condition for CP invariance is simply that all the parameters mij, yijk , etc. are real. The more general phase factor η can be treated in the same way as in Sect. A.3. CP in concrete gauge theories Now let us discuss CP invariance of some gauge theories. First, a remark on the terminology. What we call CP is usually called P in standard textbooks in the case of SU(N ) or Sp(N ) gauge theories. However, our interest is in the SPT phases related to free or interacting Majorana fermions, so our focus is centered around Majorana fermions rather than gauge fields. Therefore we stick to calling the symmetry CP rather than P. B.1. Non-supersymmetric theories Here we consider G = SO(N ), SU(N ), or Sp(N ) gauge theories with Nf flavors of fermion fields in the defining representation. B.1.1. SO(N ) theories The vector representation is a strictly real representation, so we can simply consider N × Nf Majorana fermions and then gauge them by SO(N ). By an appropriate redefinition of fields, the CP transformation is just given by (A.5) with η = 1. The Lagrangian is where i, j, . . . are flavor indices and mij is a real symmetric matrix. Dμ is the covariant derivative, and gauge indices are suppressed. In the notation of Sect. A.3, we write B.1.2. SU(N ) theories One flavor of quarks in the fundamental representation means that we introduce fermions in the representation N ⊕ N. Each of N and N is a complex representation, but their sum N ⊕ N is a strictly real representation and hence we can take the generators ρ(Ta) to be real in a certain basis. Therefore, we can define CP in the way discussed in Appendix A. However, as it is sometimes more convenient to use the usual complex basis, let us define CP directly there. We introduce fermions ψia and ψ˜ ai , where a = 1, . . . , N is a gauge index and i = 1, . . . , Nf is a flavor index. Notice that the CP transformation and the SU(N ) × SU(Nf ) symmetry commute, as can be seen in a Majorana basis. We therefore define where we have used the notation that, e.g., the complex conjugate of ψia is given by (ψia) = ψ ia by exchanging the upper and lower indices, which makes the group action of SU(N ) × SU(Nf ) more transparent. The Lagrangian is −iψ σ μDμψ − iψ˜ σ μDμψ˜ − (mjiψ˜ aj ψia + (m†)jiψ jaψ˜ ia). The CP invariance requires that the matrix m is Hermitian: m = m†. This is just a special case of the more general discussion in Sect. A.3. In fact, in the basis M = B.1.3. Sp(N ) theories The fundamental representation 2N of the Sp(N ) is pseudo-real. If we have an even number of copies of the fundamental representation, the total representation can be made to be strictly real. So we take the number of fundamental Weyl fermions as 2Nf ; of course, this is also required by cancellation of the global gauge anomaly. Let us denote the fields as ψia, where a = 1, . . . , 2N and i = 1, . . . , 2Nf . The CP commutes with Sp(N ) × Sp(Nf ) symmetry, and we define where J ab and Jij are anti-symmetric invariant tensors of Sp(N ) and Sp(Nf ), respectively. The Lagrangian is given by 1 a b −iψ σ μDμψ − 2 [mij(J −1)abψi ψj − (m∗)ij(J )abψ iaψ jb], where m is an anti-symmetric matrix mT = −m. The CP invariance requires In the notation of Sect. A.3, we have m∗ = JmJ T . B.2. Supersymmetric theories It is straightforward to extend the non-supersymmetric SO(N ), SU(N ), or Sp(N ) theories discussed above to N = 1 supersymmetric theories if we only consider chiral fields in the fundamental representation. Corresponding to the non-supersymmetric mass term we just consider the superpotential W = QiMijQj, where Qi denotes quark superfields in the fundamental representation of the gauge group. The CP is extended as described in Sect. A.5, and the condition for M is completely the same as in the non-supersymmetric case. When we extend the supersymmetry to N = 2, we have to introduce an adjoint chiral superfield . Quarks also have to form hypermultiplets of N = 2 SUSY. Consider hypermultiplets (Qia, Q˜ a) i in a complex representation. Strictly real and pseudo-real representations are just a special case of complex representation. The coupling of these hypermultiplets to the adjoint chiral field is given by where we have suppressed superspace coordinates, and † acts on the matrix = Hermitian conjugate. We can also add mass terms that are consistent with the CP as Wmass = Q˜ aimjiQja + 12 m Tr 2, where m is Hermitian. The first term is consistent with N = 2 SUSY, while the second term breaks N = 2 to N = 1. Appendix C. The Wess–Zumino–Witten term C.1. Definitions Here we summarize the properties of the Wess–Zumino–Witten term. It is not difficult to discuss it in arbitrary even dimensions d = 2n. For simplicity we assume in this subsection that manifolds are orientable. Suppose that we have a theory with global symmetry F0 that is spontaneously broken to F ⊂ F0. Also, suppose that F0 has a ’t Hooft anomaly represented by the anomaly polynomial where the trace is taken in some representation, κ is the ’t Hooft anomaly coefficient, and F = 1 2 Fμν dxμ∧dxν with Fμν = ∂μAν −∂ν Aμ+[Aμ, Aν ] is the field strength two-form of the background gauge field of the flavor symmetry. κ is defined such that the trace over all fermions under the symmetry F0, denoted by Trfermions, is given by Trfermions = κ tr. I2n+2 is a (2n + 2)-form, and we define the (2n + 1)-form I2n+1(A) as dI2n+1(A) = I2n+2. Now, let V be the Goldstone boson field that takes values in F0. We impose the gauge invariance V ∼ VW for W ∈ F so that V is the variable taking values in F0/F. In this case, the WZW term (including the background field A) is given by SWZW = (I2n+1(AV ) − I2n+1(A)) up to manifestly local terms in 2n dimensions, V = V −1AV + V −1dV . One can check the following: (1) This action only depends on the boundary value of the fields modulo 2π κ because I2n+1(A) is the Chern–Simons action and we are taking the difference of the Chern–Simons actions I2n+1(AV ) − I2n+1(A) which differ only by “the gauge transformation by V .” (2) This action reproduces the ’t Hooft anomaly under the F0 flavor transformation A → gAg−1 +gdg−1 and V → gV , because AV is invariant while the term I2n+1(A) gives the anomaly by the standard anomaly descent argument. (3) Under the gauge transformation V → VW with W ∈ F, AV changes as AV → W −1AV W + W −1dW and hence gives the anomaly from I2n+1(AV ) by the descent equation argument. This is zero (up to contributions that are cancelled by manifestly local counterterms in 2n dimensions) if the current of F is free from the ’t Hooft anomaly. Assuming that is the case, SWZW is invariant under the transformation W if we choose appropriate counterterms. This assumption is satisfied in the theories considered in this paper. More explicitly, I2n+1 is given by Instead of (C.5), one can also use the following definition of I2n+1, which differs from (C.5) by a total derivative. (This paragraph is outside the main line of argument and may be skipped.) Let us split the background gauge field as A = A + A , where A takes values in the Lie algebra of F, and A is orthogonal to the Lie algebra of F inside F0. Then we have I2n+2(A) − I2n+2(A ) = dI2n+1(A , A ), where Here, F is the field strength of A , and D is the covariant exterior derivative using A . If F is free from the ’t Hooft anomaly as assumed above, we get I2n+2(A ) = 0 and hence I2n+2(A) = dI2n+1(A , A ), so we can use I2n+1(A , A ) in the definition of the WZW term. The point is that I2n+1(A , A ) is manifestly invariant under the gauge transformation of F, and hence the WZW term is manifestly invariant under V → VW without any counterterm. C.2. Computations using Clifford algebras Let us compute the integral of 2n+1 defined above for a few specific configurations.15 First, we take the gamma matrices in 2n + 2 dimensions M which satisfy Their sizes are 2n+1 × 2n+1. Let = in+1 1 · · · 2n+2 be the chirality matrix. We consider a unit sphere S2n+1 embedded in R2n+2 with the coordinates X I (I = 1, . . . , 2n + 2). We denote the points on S2n+1 by Xˆ M with (Xˆ )2 = 1. We also take a specific point Xˆ0M on S2n+1. Then, we consider a configuration of the Goldstone field V on S2n+1 given by V = P+( · Xˆ0)( · Xˆ ), where P+ = (1 + )/2 is the chirality projection, · Xˆ = M Xˆ M , and it should be understood that we only take the block of the matrix which has positive chirality. Then V is a 2n × 2n matrix and is unitary. Let MN = 21 ( M N − M N ). Then one can check that V −1dV = P+ MN Xˆ M dXˆ N and (V −1dV )2 = −d(V −1dV ) = −P+ MN dXˆ M dXˆ N . By using these equations we get by a straightforward computation that 2n+1 = d 2n+1 = 1, 15 The argument here was reviewed in [41]. Finally, notice the following property of V as a function of Xˆ : If we take the Hermitian conjugate, we get V (Xˆ )† = V (−Xˆ + 2Xˆ0(Xˆ0 · Xˆ )). This means that the coordinates of the directions orthogonal to Xˆ0 flip sign by the Hermitian conjugate. We will use this property in Sect. 3.2. More details on S-duality In this appendix we give a more detailed description of the continuous deformation from small electric gauge coupling region g 1 to large coupling region g 1 which corresponds to small magnetic gauge coupling g 1. For this purpose, it is most convenient to use the class S description which is manifestly symmetric under S-dual. See, e.g., [36] and references therein for the background of this appendix. We consider A1 = SU(2) class S theory on a Riemann sphere with four regular punctures, which corresponds to the N = 2 SU(2) theory with Nf = 4 flavors. The SO(8) flavor symmetry has the subgroup (4.16) SU(2)1 × SU(2)2 × SU(2)3 × SU(2)4 ⊂ SO(4) × SO(4) ⊂ SO(8), and each puncture, which we denote pi (i = 1, 2, 3, 4), is associated with SU(2)i. For simplicity, we consider mass parameters (ma, ma, mb, mb) for the four flavor quarks. This corresponds to the case where the punctures p1 and p3 have the mass parameters ma and mb respectively, while the punctures p2 and p4 do not have any mass parameters. Let z be the coordinate of the Riemann sphere regarded as C ∪ {∞}. The Pinv comes from a Lorentz transformation of the six-dimensional N = (2, 0) theory, and hence it acts on z as an orientationreversing anti-holomorphic automorphism. Its action can be taken as z → z. The positions of the punctures should also be fixed by this action.16 So the punctures must be aligned on the S1 given by z = z. We take them to be where q ∈ R is a real parameter corresponding to the gauge coupling in a certain way. There are three possible regions, 0 < q < 1, q < 0, and q > 1, and we will argue that our setup corresponds to 0 < q < 1. This fixes the cyclic order of the punctures on S1 as p1, p2, p4, p3. The Seiberg–Witten curve of the system is given by where λ is the Seiberg–Witten differential. This is determined by requiring that λ has poles with residues ma and mb at p1 (z = ∞) and p3 (z = 0), respectively. From the above Pinv action, we can easily see that u transforms under Pinv (and hence CP) as Pinv(u) = u. 16 There is another logical possibility that two punctures are exchanged under Pinv, but in that case Pinv does not commute with SO(8). The singular points on the u-plane can be found as the positions where the curve degenerates. The possibilities are either that the polynomial m2z2 − uz + qm2b (i) has a degenerate root, or (ii) has a a zero at z = 1 or z = q. Then we find singular points as If q is a small positive value 0 < q 1, these positions are precisely as expected in the field theory shown in the left-hand side of Fig. 1. Another possible region of q, given by q < 0, does not reproduce our expectation. This is because if q is negative, the points A and B are pure imaginary and they are exchanged under CP. This case corresponds to the case with the theta angle given by θ = π . The region q > 1 is just equivalent to 0 < q < 1 by reparametrizations z → z−1, q → q−1, ma ↔ mb, and u → q−1u. Therefore, we can focus our attention on 0 < q < 1, and q 1 corresponds to the small electric coupling g 1. Then the region 0 < 1 − q 1 should correspond to the large electric coupling, or equivalently small magnetic coupling. Indeed, if we take ma = mb = m, and renormalize u as u = u − 21 (1 + √q)2m2, the singular points are located as (1 − √q)2 m2, (D.5) which reproduce the situation in the right-hand side of Fig. 1. In summary, just by changing q in the region 0 < q < 1 from q ∼ 0 to q ∼ 1, we can smoothly go from weakly coupled electric description to weakly coupled magnetic description. Finally, let us comment on what the S-duality is. The above analysis using the Riemann sphere is manifestly symmetric under the S-duality. However, we can make a change of the coordinate of the Riemann sphere from z to z = 1 − z. This exchanges the positions of the punctures p2 and p3, and also changes the parameter q to q = 1 − q. This coordinate change corresponds to the S-duality. Of course, the physics is independent of the coordinate system, and hence the theory has the S-duality. Under the S-duality which exchanges p2 and p3, the symmetry groups SU(2)2 and SU(2)3 are exchanged. This is the fact used in Sect. 4.2. Appendix E. Explicit analysis of the fermion mass matrix on the dual side Here, we analyze the mass matrix of the fermions explicitly, as a complement to Sect. 4.4. In this appendix we use the abbreviation that the scalar components of the quarks Q1 and Q˜ 1 are just denoted as Q and Q˜ , and their fermionic components are denoted as q and q˜. The N = 1 gauginos are denoted as λ and the fermions in are denoted as ψ . The mass terms involving gauginos λ are given by Notice that q˜+ and q˜− have the dual U (1) charges −1 and +1 respectively; one might want to write them as q˜± := q˜∓. The other part of mass terms are determined by the superpotential. By a and the matrix M is given by 0 ⎟⎟ , −z⎟⎠ w 4m0/g2 m w −z w −z 2(m − m0) 0 −w Z = It is not too hard to analyze this matrix analytically. First recall that when m = 0, only one of the z or w gets a vev. Then, the above mass matrix has a U(1) symmetry that comes from the diagonal part of U (1) × U(1)flavor, which is unbroken by the vev. 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Yuji Tachikawa, Kazuya Yonekura. Gauge interactions and topological phases of matter, Progress of Theoretical and Experimental Physics, 2016, DOI: 10.1093/ptep/ptw131