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Bott Periodicity and Realizations of Chiral Symmetry in Arbitrary Dimensions ∗ Richard Dejonghe, Kimberly Frey, Tom Imbo

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Bott Periodicity and Realizations of Chiral Symmetry in Arbitrary Dimensions ∗ Richard Dejonghe, Kimberly Frey, Tom Imbo Physics Letters B 718 (2012) 603–609 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Bott periodicity and realizations of chiral symmetry in arbitrary dimensions ∗ Richard DeJonghe, Kimberly Frey, Tom Imbo Department of Physics, University of Illinois at Chicago, 845 W. Taylor St., Chicago, IL 60607, United States article info abstract Article history: We compute the chiral symmetries of the Lagrangian for confining “vector-like” gauge theories with Received 28 August 2012 massless fermions in d-dimensional Minkowski space and, under a few reasonable assumptions, Received in revised form 18 October 2012 determine the form of the quadratic fermion condensates which arise through spontaneous breaking Accepted 19 October 2012 of these symmetries. We find that for each type (complex, real, or pseudoreal) of representation of the Available online 22 October 2012 gauge group carried by the fermions, the chiral symmetries of the Lagrangian, as well as the residual Editor: B. Grinstein symmetries after dynamical breaking, exactly follow the pattern of Bott periodicity as the dimension Keywords: changes. The consequences of this for the topological features of the low-energy effective theory are Chiral symmetry breaking considered. Extra dimensions © 2012 Elsevier B.V. All rights reserved. Vector-like gauge theories Bott periodicity 1. Introduction arbitrary representation of G carried by the fermions) under cer- tain assumptions. Different patterns of “chiral” symmetry breaking have been found in 2 + 1 dimensions for fermions in a complex Spontaneous breaking of chiral symmetry is a central non- representation of any gauge group [4–7], in the fundamental (pseu- perturbative feature of QCD. It not only explains the large effec- doreal) representation in an SU 2 gauge theory [8–10], and in the tive mass of quarks bound within hadrons, but also allows one to ( ) adjoint (real) representation in an SU N gauge theory [9–11]. understand pions as Goldstone bosons of this broken symmetry. ( c) Our analysis is consistent with the aforementioned results and Given the importance of chiral symmetry breaking in QCD, we will goes well beyond them. Under reasonable assumptions, we not investigate this phenomenon in a wider class of theories — namely, only determine the form of the relevant condensates generated confining vector-like gauge theories in Minkowski space of arbi- by dynamical breaking for arbitrary d and G,butfindthatfor trary dimension d — in an attempt to obtain a broader perspective each type (complex, real, or pseudoreal) of representation of G on the nature of chiral symmetry breaking. (What we mean by a carried by the fermions, the chiral symmetries of the massless La- “vector-like” theory for d oddwillbemadeclearbelow.)Ford > 4 grangian, as well as the residual symmetries after dynamical break- certain aspects of these models1 may be relevant to higher dimen- ing, exactly follow the pattern of Bott periodicity as the dimension sional extensions of the Standard Model, while some models with changes. The consequences of this for the topological features of d < 4 might be relevant in condensed matter physics. the low-energy effective theory of the Goldstone boson degrees of Various results are known for d 4. In 3+ 1 dimensions, Peskin freedom are considered, including an analysis of the interpretation [1,2] and Preskill [3] have worked out the patterns of spontaneous of baryons as topological solitons. chiral symmetry breaking (for an arbitrary gauge group G and an 2. Chiral symmetries of the Lagrangian * Corresponding author. E-mail address: [email protected] (T. Imbo). We consider a confining gauge theory in d-dimensional Min- 1 Although gauge theories in d > 4 are not perturbatively renormalizable, we may kowski space with compact gauge group G, where the gauge fields consider such models as effective theories arising from an appropriate UV comple- are coupled to N flavors of massless fermions which all transform tion. However, since these gauge theories are naively free in the infrared, we assume the presence of additional degrees of freedom (or other dynamical modifications) under a single irreducible unitary representation r of G.Wede- i,a which conspire to render the theory confining, but which do not otherwise play a note the fermion fields by ψ , where i = 1,...,N is the flavor role in determining the patterns of chiral symmetry breaking. More robustly, similar index and a = 1,...,dim r is the “color” index. The spinor index is chiral symmetry breaking patterns will be obtained for any strongly coupled vector- suppressed. like field theory of massless fermions (in any dimension) transforming irreducibly i,a under an internal symmetry group G, and for which an appropriate G-invariant con- When d iseven,wetakeeachψ to be a Dirac spinor. Here i,a i,a densate forms. Using gauge symmetries simply makes our analysis more concrete. ψ decomposes uniquely into left-handed (ψL ) and right-handed 0370-2693/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2012.10.043 604 R. DeJonghe et al. / Physics Letters B 718 (2012) 603–609 i,a [ ]= ∈ 2 = (ψR ) Weyl spinors, corresponding to the two inequivalent irre- (i.e. J, ρ0(g) 0forallg G0) and also satisfies J I (re- ducible representations of the group Spin(1, d − 1).Whend is odd, spectively, J 2 =−I), where I is the identity operator. (Such a J however, there is only a single irreducible representation (up to is unique up to a phase.) If ρ0 is neither real nor pseudoreal it is equivalence) of Spin(1, d − 1); i.e. a single type of Weyl spinor. called complex.Ifρ0 is irreducible, then it falls into exactly one Hence, in order for there to be “left-handed” and “right-handed” of these three categories. Given a representation ρ1 of some other spinors (and therefore a notion of chirality), we must consider each group G1, we can form the outer tensor product representation i,a ψ to consist of two copies of this unique Weyl spinor, where ρ0 ⊗ ρ1 of G0 × G1. Now assume ρ0 and ρ1 (and hence ρ0 ⊗ ρ1) the parity transformation is defined such that the “left-handed” are irreducible. Then, if either ρ0 or ρ1 is complex, ρ0 ⊗ ρ1 will and “right-handed” spinors are interchanged [4,5,12]. As a conse- be as well. If, instead, ρ0 and ρ1 are either both real or both pseu- quence, when d is odd we may alternatively think of the theory as doreal, ρ0 ⊗ ρ1 will be real. In all other cases, ρ0 ⊗ ρ1 will be having 2N identical “Weyl flavors”. In what follows, we will sup- pseudoreal. For a more detailed discussion see [13]. press flavor and color indices, and simply denote the fermion fields by ψ. 2.1. Symmetries of L for d odd A gauge theory (in any dimension) will be called vector-like if it has the fermion content just described, and a Lagrangian which For d odd, we have that the irreducible representation of treats “left-handed” and “right-handed” spinors democratically. In Spin(1, d − 1) (which we denote by s)isrealford = 1, 3(mod8) particular, we take the Lagrangian density to be and pseudoreal for d = 5, 7 (mod 8). In the cases in which r (i.e. the irreducible representation of the gauge group G)is ¯ μν ≡ ⊗ L = ψD/ψ + Tr F Fμν, (1) also real or pseudoreal, the product representation ρ s r of Spin(1, d − 1) × G is either real or pseudoreal. As such, there ex- = μ + c c μ where D/ iγ (∂μ igtr Aμ),withg the coupling constant, γ the ists the antilinear equivariant map J on the carrier space of ρ 2 ¯ = † 0 c described above. Additionally, Schur’s lemma guarantees that the appropriate Dirac matrices, ψ ψ γ , and tr the generators of the Lie algebra of G in the representation associated with r. scalars are the only linear maps which commute with ρ(g) for all We also take the fermion fields to be Grassmann-valued. g ∈ Spin(1, d − 1) × G.Byextendingρ to include the trivial rep- 6 We define the chiral symmetries to be the global symmetries of resentation on “flavor” space, we have that the most general real L which can be represented as real linear transformations3 Z act- linear transformation commuting with ρ is an operator of the form ing on the indices of ψ such that Z commutes with all gauge X + YJ, where J has been extended to act on “flavor” space simply 7 and spinor transformations.4 This definition reproduces the stan- as complex conjugation, and X, Y are (standard) linear operators dard notion of chiral symmetry for vector-like theories in 3 + 1 which act non-trivially only on “flavor” space. dimensions [1]. The reason we consider real linear transformations One can show that L remains invariant under the transforma- (and not just the linear ones) is that these are more natural when tion ψ → (X + YJ)ψ if and only if ψ is not in a complex representation of Spin(1, d − 1) × G.For † 0 † 0 0 example, when ψ carries a representation ρ in which the group X γ DX/ − (YJ) γ D/(YJ) = γ D/ (2) elements are all represented by real matrices, the real and imagi- and nary parts of ψ do not mix under ρ; hence, these can be thought of as independent fields. However, the natural transformations on T X†γ 0D/(YJ)K = X†γ 0D/(YJ)K , (3) the carrier space of ρ which allow the real and imaginary parts of ψ to transform independently are not linear, but merely real where K denotes the complex conjugation operator on the carrier linear.
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