Scalar Field Theories with Polynomial Shift Symmetries
SScalar Field Theories with Polynomial Shift Symmetries
Tom Griffin a , Kevin T. Grosvenor b,c , Petr Hoˇrava b,c and Ziqi Yan b,c a Blackett Laboratory, Department of PhysicsImperial College, London, SW7 2AZ, UK b Berkeley Center for Theoretical Physics and Department of PhysicsUniversity of California, Berkeley, CA, 94720-7300, USA c Theoretical Physics Group, Lawrence Berkeley National LaboratoryBerkeley, CA 94720-8162, USA
Abstract:
We continue our study of naturalness in nonrelativistic QFTs of the Lifshitztype, focusing on scalar fields that can play the role of Nambu-Goldstone (NG) modesassociated with spontaneous symmetry breaking. Such systems allow for an extension of theconstant shift symmetry to a shift by a polynomial of degree P in spatial coordinates. These“polynomial shift symmetries” in turn protect the technical naturalness of modes with ahigher-order dispersion relation, and lead to a refinement of the proposed classification ofinfrared Gaussian fixed points available to describe NG modes in nonrelativistic theories.Generic interactions in such theories break the polynomial shift symmetry explicitly to theconstant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomialshift symmetry of degree P , what are the lowest-dimension operators that preserve thissymmetry, and deform the theory into a self-interacting scalar field theory with the shiftsymmetry of degree P ? To answer this (essentially cohomological) question, we develop anew graph-theoretical technique, and use it to prove several classification theorems. First,in the special case of P = 1 (essentially equivalent to Galileons), we reproduce the knownGalileon N -point invariants, and find their novel interpretation in terms of graph theory, asan equal-weight sum over all labeled trees with N vertices. Then we extend the classificationto P > a r X i v : . [ h e p - t h ] A ug ontents
1. Introduction: Landscapes of Naturalness 12. Multicritical Nambu-Goldstone Bosons and Polynomial Shift Symmetries 4
3. Galileon Invariants 14
4. Beyond the Galileons 17
P, N, ∆) = (1 , ,
2) 184.1.2. (
P, N, ∆) = (3 , ,
4) 184.2. Introduction to the Graphical Representation 194.2.1. (
P, N, ∆) = (1 , ,
2) 194.2.2. (
P, N, ∆) = (3 , ,
4) 204.3. New Invariants via the Graphical Approach 214.3.1. The Minimal Invariant: (
P, N, ∆) = (2 , ,
5) 224.3.2. The Minimal Invariant: (
P, N, ∆) = (3 , ,
6) 234.3.3. Medusas and Total Derivative Relations 254.4. Superposition of Graphs 264.4.1. Quadratic Shift ( P =2) 274.4.2. Cubic Shift ( P =3) 294.4.3. Quartic Shift ( P =4) 304.4.4. Quintic Shift ( P =5) 31
5. Conclusions and Outlook 32A. Glossary of Graph Theory 33 – i – . Theorems and Proofs 35
B.1. The Graphical Representation 35B.1.1. Types of Graphs and Vector Spaces 37B.1.2. Maps 37B.1.3. Relations 38B.1.4. The Consistency Equation and Associations 39B.2. Building Blocks for Consistency Equations 40B.2.1. The Loopless Realization of L / R × -Relations 45B.2.4. A Basis for R × loopless P -invariant and an Exact Invariant 59B.4.2. Superposition of Minimal Loopless 1-invariants 60B.5. Unlabeled Invariants 63 C. Coset Construction 64
1. Introduction: Landscapes of Naturalness
Some of the most fundamental questions of modern theoretical physics can be formulated aspuzzles of naturalness [1]. Why is the observed cosmological constant so small compared tothe Planck scale? Why is the observed Higgs mass so small compared to any high particle-physics scale (be it the quantum gravity scale, or the scale of grand unification, or someother scale of new physics)? In both cases, the expected quantum corrections estimated inthe framework of relativistic effective field theory (EFT) predict natural values at a muchhigher scale, many orders of magnitude larger than the observed ones. The principle ofnaturalness is rooted in the time-honored physical principles of causality and the hierarchyof energy scales from the microscopic to the macroscopic. It is conceivable that somepuzzles of naturalness may only have environmental explanations, based on the landscapeof many vacua in the multiverse. However, before we give up naturalness as our guidingprinciple, it is important to investigate more systematically the “landscape of naturalness”:To map out the various quantum systems and scenarios in which technical naturalness doeshold, identifying possible surprises and new pieces of the puzzle that might help restorethe power of naturalness in fundamental physics.One area in which naturalness has not yet been fully explored is nonrelativistic grav-ity theory [2, 3]. This approach to quantum gravity has attracted a lot of attention in– 1 –ecent years, largely because of its improved quantum behavior at short distances, novelphenomenology at long distances [4], its connection to the nonperturbative Causal Dy-namical Triangulations approach to quantum gravity [5–7], as well as for its applicationsto holography and the AdS/CFT correspondence of nonrelativistic systems [8–10]. Thisarea of research in quantum gravity is still developing rapidly, with new surprises alreadyencountered and other ones presumably still awaiting discovery. Mapping out the quan-tum structure of nonrelativistic gravity theories, and in particular investigating the role ofnaturalness, represents an intriguing and largely outstanding challenge.Before embarking on a systematic study of the quantum properties of nonrelativis-tic gravity, one can probe some of the new conceptual features of quantum field theories(QFTs) with Lifshitz-type symmetries in simpler systems, without gauge symmetry, dy-namical gravity and fluctuating spacetime geometry. In [11], we considered one of the ubiq-uitous themes of modern physics: The phenomenon of spontaneous symmetry breaking, inthe simplest case of global, continous internal symmetries. According to Goldstone’s the-orem, spontaneous breaking of such symmetries implies the existence of a gapless Nambu-Goldstone (NG) mode in the system. For Lorentz invariant systems, the relativistic versionof Goldstone’s theorem is stronger, and we know more: There is a one-to-one correspon-dence between broken symmetry generators and the NG modes, whose gaplessness impliesthat they all share the same dispersion relation ω = ck . On the other hand, nonrelativisticsystems are phenomenologically known to exhibit a more complex pattern: Sometimes,the number of NG modes is smaller than the number of broken symmetry generators,and sometimes they disperse quadratically instead of linearly. This rich phenomenologyopens up the question of a full classification of possible NG modes. A natural and elegantapproach to this problem has been pursued in [12]: In order to classify NG modes, oneclassifies the low-energy EFTs available to control their dynamics.In the case of systems with nonrelativistic Lifshitz symmetries, this approach suggestsa classification of NG modes into two categories [12]: Type A, and Type B NG modes.Each Type A mode is associated with a single broken symmetry generator, and each TypeB mode with a pair. Upon closer inspection [11], it turns out that even such simpleexamples of Lifshitz-type QFTs exhibit rich and surprising features, often contrasting orcontradicting the intuition developed in relativistic QFTs. Our analysis of naturalness inthe patterns of spontaneous symmetry breaking in systems with Lifshitz symmetries hasrevealed a refined hierarchy of the Type A and B universality classes of NG dynamics, withrich low-energy phenomenology dominated by multicritical NG modes whose dispersion isof higher degree in momentum. These results shed some new, and perhaps surprising,light on the concept of naturalness in nonrelativistic quantum field theory. However, asusual, the naturalness of the multicritical dispersion relation turns out to be protected bya symmetry. This new kind of symmetry is generated by the shifts of the NG fields by apolynomial in spatial coordinates. In this paper, we continue our investigation of scalar field theories with such polynomial This symmetry is a natural generalization of two types of symmetries well-known in the literature: Thefamous constant shift symmetry observed in systems with relativistic NG modes, and the shift linear in thespacetime coordinates known from the relativistic theory of Galileons [13]. – 2 –hift symmetries of degree P . The paper is organized into several relatively self-containedblocks. In §
2, we review the physics background and discuss the structure of multicrit-ical symmetry breaking in nonrelativistic systems of the Lifshitz type, summarizing andexpanding on the findings of [11, 14]. We discuss the Goldstone theorem in the nonrela-tivistic regime, and give the refined classification of Nambu-Goldstone modes for systemswith Lifshitz symmetries into two hierarchies of multicritical fixed points of Type A n andB n , with n = 1 , , . . . . We present the nonrelativistic analog of the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem, and discuss its implications for the dynamics of themulticritical NG modes. Throughout, we stress the role played by the polynomial shiftsymmetry, as an approximate symmetry restored at the Gaussian infrared fixed points. Inmany general examples of multicritical symmetry breaking, the polynomial shift symmetryis broken by the self-interactions of the NG modes. It is then natural to ask: What ifwe impose the polynomial shift symmetry as an exact symmetry? What is the lowest-dimension operator that can be added to the action while preserving the symmetry? Thisis the task we address in the remainder of the paper.The classification of Lagrangians invariant under the polynomial shift of degree P upto a total derivative (which we will refer to as “ P -invariants” for short) is essentially a coho-mological problem. In §
3, we consider the polynomial-shift invariants in the simplest caseof linear shifts (i.e., “1-invariants”). In order to prepare for the general case of
P >
1, wedevelop a novel technique, based on graph theory. Having rephrased the defining relationfor the invariants into the language of graphs, we can address the classification problemusing the abstract mathematical machinery of graph theory. The basic ingredients of thistechnique are explained as needed in § §
4. However, we relegate all the technicalitiesof the graphical technique into a self-contained Appendix B (preceded by Appendix A, inwhich we offer a glossary of the basic terms from graph theory). Appendix B is rathermathematical in nature, as it contains a systematic exposition of all our definitions, theo-rems and proofs that we found useful in the process of generating the invariants discussedin the body of the paper. The good news is that Appendix B is not required for the un-derstanding of the results presented in § §
4: Once the invariants have been foundusing the techniques in Appendix B, their actual invariance can be checked by explicitcalculation (for example, on a computer). In this sense, the bulk of the paper ( § § N -point 1-invariants discussed in § § N -point 1-invariants in the language ofgraphs. They are simply given by the equal-weight sum over all labeled trees with N vertices!In § P -invariantswith P >
1, organized in the order of their scaling dimension. We find several series ofinvariants; some of them we prove to be the unique and most relevant (or, more accurately,least irrelevant) N -point P -invariants, while others represent hierarchies of P -invariants of– 3 –igher dimensions. We also show how to construct higher P -invariants from superposingseveral graphs that represent invariants of lower P . Appendix C contains a brief discus-sion of the connection between our invariant Lagrangians and the Chevalley-Eilenberg Liealgebra cohomology theory. In §
2. Multicritical Nambu-Goldstone Bosons and Polynomial Shift Symmetries
Some surprising features of naturalness in the regime of nonrelativistic field theories areillustrated by considering one of the classic problems in physics: The classification ofNG modes associated with possible patterns of spontaneous breaking of continuous globalinternal symmetries. In this section, we summarize and explain the results found in [11, 14],which lead to a refinement in the classification of NG modes in systems with Lifshitzsymmetries, characterized by a multicritical behavior which is technically natural, andprotected by a symmetry.
For clarity and simplicity, as in [11], in this paper we focus on systems on the flat space-time with Lifshitz spacetime symmetries. We define this spacetime to be M = R D +1 with a preferred foliation F by fixed spatial slices R D , and equipped with a flat metric.Such a spacetime with the preferred foliation F would for example appear as a ground-state solution of nonrelativistic gravity [2] whose gauge symmetry is given by the group offoliation-preserving spacetime diffeomorphisms, Diff( M, F ) (or a nonrelativistic extensionthereof [15]). It is useful to parametrize M by coordinates ( t, x = { x i , i = 1 , . . . D } ), suchthat the leaves of F are the leaves of constant t , and the metric has the canonical form g ij ( t, x ) = δ ij , N ( t, x ) = 1 , N i ( t, x ) = 0 (2.1)(here g ij is the spatial metric on the leaves of F , N is the lapse function, and N i the shiftvector).The isometries of this spacetime are, by definition, those elements of Diff( M, F ) thatpreserve this flat metric [16]. Explicitly, the connected component of this isometry groupis generated by infinitesimal spatial rotations and spacetime translations, δt = b, δx i = ω ij x j + b i , ω ij = − ω ji . (2.2)At fixed points of the renormalization group, systems with Lifshitz isometries develop anadditional scaling symmetry, generated by δx i = λx i , δt = zλt. (2.3)The dynamical critical exponent z is an important observable associated with the fixedpoint, and characterizes the degree of anisotropy between space and time at the fixedpoint. – 4 –he connected component of the group of isometries of our spacetime M with the flatmetric (2.1) is generated by (2.2), and we will refer to it as the “Lifshitz symmetry” group. The full isometry group of this spacetime has four disconnected components, which can beobtained by combining the Lifshitz symmetry group generated by (2.2) with two discretesymmetries: The time-reversal symmetry T , and a discrete symmetry P that reverses theorientation of space. In this paper, we shall be interested in systems that are invariantunder the Lifshitz symmetry group. Note that this mandatory Lifshitz symmetry does notcontain either the discrete symmetries T and P , or the anisotropic scaling symmetry (2.3). We are interested in the patterns of spontaneous symmetry breaking of global continu-ous internal symmetries in the flat spacetime with the Lifshitz symmetries, as definedin the previous paragraph. Our analysis gives an example of phenomena that are novelto Goldstone’s theorem in nonrelativistic settings, and can in principle be generalized tononrelativistic systems with even less symmetry.An elegant strategy has been proposed in [12]: In order to classify Nambu-Goldstonemodes, we can classify the corresponding EFTs available to describe their low-energy dy-namics. In this EFT approach, we organize the terms in the effective action by their in-creasing dimension. Such dimensions are defined close enough to the infrared fixed point.However, until we identify the infrared fixed point, we don’t a priori know the value of thedynamical critical exponent, and hence the relative dimension of the time and space deriva-tives – it is then natural to count the time derivatives and spatial derivatives separately.Consider first the “potential terms” in the action, i.e., terms with no time derivatives. Thegeneral statement of Goldstone’s theorem implies that non-derivative terms will be absent,and the spatial rotational symmetry further implies that (for
D >
1) all derivatives willappear in pairs contracted with the flat spatial metric. Hence, we can write the general“potential term” in the action as S eff , V = (cid:90) dt d x (cid:26) g IJ ( π ) ∂ i π I ∂ i π J + . . . (cid:27) (2.4)where g IJ ( π ) is the most general metric on the vacuum manifold which is compatiblewith all the global symmetries, and . . . stand for all the terms of higher order in spatialderivatives.If the system is also invariant under the primitive version T of time reversal, definedas the transformation that acts trivially on fields, T : (cid:40) t → − t,π I → π I , (2.5) It would be natural to refer to M with the flat metric (2.1) as the “Lifshitz spacetime”. Unfortunately,this term already has another widely accepted meaning in the holography literature, where it denotes thecurved spacetime geometry in one dimension higher, whose isometries realize the Lifshitz symmetries (2.2)plus the Lifshitz scaling symmetry (2.3) for some fixed value of z [17]. – 5 –he time derivatives will similarly have to appear in pairs, and the kinetic term will begiven by S eff , K = (cid:90) dt d x (cid:26) h IJ ( π ) ˙ π I ˙ π J + . . . (cid:27) , (2.6)where again h IJ is a general metric on the vacuum manifold compatible with all symmetries,but not necessarily equal to the g IJ that appeared in (2.4); and . . . are higher-derivativeterms.However, invariance under T is not mandated by the Lifshitz symmetry. If it is absent,the Lifshitz symmetries allow a new, more relevant kinetic term, (cid:101) S eff , K = (cid:90) dt d x (cid:8) Ω I ( π ) ˙ π I + . . . (cid:9) , (2.7)assuming one can define the suitable object Ω I ( π ) on the vacuum manifold so that all thesymmetry requirements are satisfied, and Ω I ( π ) ˙ π I is not a total derivative. Since Ω I ( π )plays the role of the canonical momentum conjugate to π I , if such Ω-terms are present inthe action, they induce a natural canonical pairing on an even-dimensional subset of thecoordinates on the vacuum manifold.In specific dimensions, new terms in the effective action that are odd under spatialparity P may exist. For example, in D = 2 spatial dimensions, we can add new terms tothe “potential” part of the action, of the form (cid:101) S eff , V = (cid:90) dt d x (cid:26)
12 Ω IJ ( π ) ε ij ∂ i π I ∂ j π J + . . . (cid:27) , (2.8)where Ω IJ is any two-form on the vacuum manifold that respects all the symmetries. Inthe interest of simplicity, we wish to forbid such terms, and will do so by imposing the P invariance of the action, focusing on the symmetry breaking patterns that respect spatialparity. This condition can of course be easily relaxed, without changing our conclusionssignificantly.This structure of low-energy effective theories suggests the following classification ofNG modes, into two general types: • Type A: One NG mode per broken symmetry generator (not paired by Ω I ). Thelow-energy dispersion relation is linear, ω ∝ k . • Type B: One NG mode per each pair of broken symmetry generators (paired by Ω I ).The low-energy dispersion relation is quadratic, ω ∝ k .In general, Type A and Type B NG modes may coexist in one system. Some examplesfrom condensed matter theory can be found in [12].Based on the intuition developed in the context of relativistic quantum field theory,one might be tempted to conclude that everything else would be fine tuning, as quantumcorrections would be likely to generate large terms of the form (2.4) in the effective actionif we attempted to tune them to zero. For example, if Ω I ( π ) suitable for (2.7) exist, one can take Ω IJ = ∂ [ I Ω J ] . – 6 – .3. Naturalness of Slow Nambu-Goldstone Modes Our careful study of a number of explicit examples revealed [11] that the naive intuitionabout fine-tuning summarized in the previous paragraph is incorrect. It turns out that theleading spatial-derivative term in (2.4) can be naturally small (or even zero), as we illus-trated in [11] by explicit calculations of loop corrections in a series of examples. The leadingcontribution to S eff , V then comes at fourth order in spatial derivatives; schematically, S eff , V = (cid:90) dt d x (cid:26) g IJ ( π ) ∂ π I ∂ π J + . . . (cid:27) , (2.9)where the “ . . . ” stand for all other terms of order four and higher in ∂ i . The dispersionrelation of this NG mode at the Gaussian infrared fixed point is then ω ∝ k (or ω ∝ k ), if the kinetic term is of Type A as in (2.6) (or of Type B as in (2.7)). The reasonwhy this behavior does not require fine tuning is simple [11]: As we approach this newGaussian infrared fixed point, the theory develops a new enhanced symmetry. Specifically,the symmetry that protects the Type A NG modes with the quadratic dispersion relation isa generalization of the constant shift symmetry of conventional NG modes: The generatorsof the new symmetry act by shifting each field component π I by a quadratic polynomialin spatial coordinates, δπ I ( t, x ) = a Iij x i x j + a Ii x i + a I . (2.10)The leading, quadratic part of this symmetry forbids the term (2.4) allowing only terms offourth order in ∂ i and higher to appear in the action in the free-field limit. The subleadinglinear and constant terms have been included in (2.10) because they would be generatedanyway by the action of spatial translations and rotations, which are a part of the assumedLifshitz symmetry of the system. Similarly, quadratic shift symmetries can also be extendedto the Gaussian limit of Type B NG modes. At the Gaussian fixed point, Ω I ( π ) of (2.7)reduces to a linear function of π , such that (cid:101) S eff , K is invariant under the quadratic shift upto a total derivative, and the extra shift symmetry yields Type B NG modes with a quarticdispersion relation.This construction can obviously be iterated. The quadratic shift symmetry (2.10) canbe promoted to a polynomial shift symmetry by polynomials of degree P = 2 , , . . . , leadingto a natural protection of higher-order dispersion relations ω ∝ k n (or ω ∝ k n ) for TypeA (or Type B) NG modes. Since the polynomial shift symmetries act on the fields π I ( t, x ) separately component bycomponent, from now on we shall focus on just one field component, and rename it φ ( t, x ).The generators of the polynomial shift symmetry of degree P act on φ by δ P φ = a i ...i P x i · · · x i P + . . . + a i x i + a. (2.11)The multicritical Gaussian fixed point with dynamical exponent z = n is described by S n = (cid:90) dt d x (cid:26)
12 ˙ φ − ζ n ( ∂ i . . . ∂ i n φ ) ( ∂ i . . . ∂ i n φ ) (cid:27) . (2.12)– 7 –n fact, it is a one-parameter family of fixed points, parametrized by the real positivecoupling ζ n . (Sometimes it is convenient to absorb ζ n into the rescaling of space, and wewill often do so when there is no competition between different fixed points.)The action S n is invariant under polynomial shift symmetries (2.11) of degree P ≤ n −
1: It is strictly invariant under the symmetries of degree
P < n , and invariant up toa total derivative for degrees n ≤ P ≤ n − time coordinates. The requirement of relativistic invariance is presumablythe main reason that has precluded the generalization of the Galileon symmetries past thelinear shift: The higher polynomial shift symmetries in spacetime coordinates would leadto actions dominated by higher time derivatives, endangering perturbative unitarity.So far, we considered shifts by generic polynomials of degree P , whose coefficients a i ...i (cid:96) are arbitrary symmetric real tensors of rank (cid:96) for (cid:96) = 0 , . . . , P . We note here in passingthat for degrees P ≥
2, the polynomial shift symmetries allow an interesting refinement.To illustrate this feature, we use the example of the quadratic shift, δ φ = a ij x i x j + a i x i + a . (2.13)The coefficient a ij of the quadratic part is a general symmetric 2-tensor. It can be decom-posed into its traceless part (cid:101) a ij and the trace part a ii , a ij = (cid:101) a ij + 1 D a kk δ ij . (2.14)Since this decomposition is compatible with the spacetime Lifshitz symmetries (2.2), onecan restrict the symmetry group to be generated by a strictly smaller invariant subalgebrain the original algebra generated by a ij . For example, setting the traceless part (cid:101) a ij ofthe quadratic shift symmetry to zero reduces the number of independent generators from( D + 2)( D + 1) / D + 2, but it is still sufficient to prevent ∂ i φ ∂ i φ from being an invariantunder the smaller symmetry. This intriguing pattern extends to P >
2, leading to intricatehierarchies of polynomial shift symmetries whose coefficients a i ...i (cid:96) have been restrictedby various invariant conditions. As another example, invariance under the traceless parthas been studied in [18]. In the interest of simplicity, we concentrate in the rest of thispaper on the maximal case of polynomial shift symmetries with arbitrary unrestricted realcoefficients a i ...i (cid:96) .The invariance of the action under each polynomial shift leads to a conserved Noethercurrent. Each such current then implies a set of Ward identities on the correlation functionsand the effective action. Take, for example, the case of n = 2 in (2.12): The currents for theinfinitesimal shift by a general function a ( x ) of the spatial coordinates x i are collectivelygiven by J t = a ( x ) ˙ φ, J i = a ( x ) ∂ i ∂ φ − ∂ j a ( x ) ∂ i ∂ j φ + ∂ i ∂ j a ( x ) ∂ j φ − ∂ i ∂ a ( x ) φ, (2.15)– 8 –nd their conservation requires˙ J t + ∂ i J i ≡ a ( x ) (cid:110) ¨ φ + ( ∂ ) φ (cid:111) − ( ∂ ) a ( x ) φ = 0 . (2.16)The term in the curly brackets is zero on shell, and the current conservation thus reducesto the condition ( ∂ ) a ( x ) φ = 0, which is certainly satisfied by a polynomial of degreethree, a ( x ) = a ijk x i x j x k + a ij x i x j + a i x i + a. (2.17)Note that if we start instead with the equivalent form of the classical action (cid:101) S = (cid:90) dt d x (cid:26)
12 ˙ φ −
12 ( ∂ i ∂ i φ ) (cid:27) , (2.18)the Noether currents will be related, as expected, by (cid:101) J t = J t , (cid:101) J i = a ( x ) ∂ i ∂ φ − ∂ i a ( x ) ∂ φ + ∂ a ( x ) ∂ i φ − ∂ i ∂ a ( x ) φ (2.19)= J i + ∂ j [ ∂ i a ( x ) ∂ j φ − ∂ j a ( x ) ∂ i φ ] . From these conserved currents, one can formally define the charges Q [ a ] = (cid:90) Σ d x J t . (2.20)However, for infinite spatial slices Σ = R D , such charges are all zero on the entire Hilbertspace of states generated by the normalizable excitations of the fields φ . This behavior isquite analogous to the standard case of NG modes invariant under the constant shifts, andit simply indicates that the polynomial shift symmetry is being spontaneously broken bythe vacuum. In its original form, Goldstone’s theorem guarantees the existence of a gapless mode whena global continuous internal symmetry is spontaneously broken. However, in the absenceof Lorentz symmetry, it does not predict the number of such modes, or their low-energydispersion relation.The classification of the effective field theories which are available to describe the low-energy limit of the Nambu-Goldstone mode dynamics leads to a natural refinement of theGoldstone theorem in the nonrelativistic regime. In the specific case of spacetimes withLifshitz symmetry, we get two hierarchies of NG modes: • Type A: One NG mode per broken symmetry generator (not paired by Ω I ) Thelow-energy dispersion relation is ω ∝ k n , where n = 1 , , , . . . . • Type B: One NG mode per each pair of broken symmetry generators (paired by Ω I ).The low-energy dispersion relation is ω ∝ k n , where n = 1 , , , . . . .– 9 –t is natural to label the members of these two hierarchies by the value of the dynamicalcritical exponent of their corresponding Gaussian fixed point. From now on, we will referto these multicritical universality classes of Nambu-Goldstone modes as “Type A n ” and“Type B n ”, respectively.The following few comments may be useful:(1) While Type B NG modes represent a true infinite hierarchy of consistent fixedpoints, the Type A NG modes hit against the nonrelativistic analog of the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem: At the critical value of n = D , theydevelop infrared singularities and cease to exist as well-defined quantum fields. Wecomment on this behavior further in § T invariance, while Type B break T . (This does not mean that asuitable time reversal invariance cannot be defined on Type B modes, but it wouldhave to extend T of (2.5) to act nontrivially on the fields.)(3) Our classification shows the existence of A n and B n hierarchies of NG modes de-scribed by Gaussian fixed points, and therefore represents a refinement of the classifi-cations studied in the literature so far. However, it does not pretend to completeness:We find it plausible that nontrivial fixed points (and fixed points at non-integer val-ues of n ) suitable for describing NG modes may also exist. In this sense, the fullclassification of all possible types of nonrelativistic NG dynamics – even under theassumption of Lifshitz symmetries – still remains a fascinating open question.(4) For simplicity, we worked under the assumption of spacetime Lifshitz symmetry.Obviously, this simplifying restriction can be removed, and the classification of mul-ticritical NG modes in principle extended to cases whereby some of the the spacetimesymmetries are further broken by additional features of the system – such as spatialanisotropy, layers, an underlying lattice structure, etc. We also expect that the clas-sification can be naturally extended to Nambu-Goldstone fermions associated withspontaneous breaking of symmetries associated with supergroups. Such generaliza-tions, however, are beyond the scope of this paper. In this section, we consider the Type A n and Type B n hierarchies of NG modes, and theirinfrared behavior. For simplicity, we will focus on theories that consist of Type A n (orType B n ) NG modes with a fixed n , and leave the generalizations to interacting systemsthat mix different types of NG modes for future studies.In relativistic systems, all NG bosons – if they exist – are Type A . However, whetheror not the corresponding symmetry is spontanously broken famously depends on the space-time dimension. This phenomenon is controlled by the celebrated theorem discovered in-dependently in condensed matter by Mermin and Wagner [19] and by Hohenberg [20], andin high-energy physics by Coleman [21]; we therefore refer to it, in the alphabetical order,as the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem.– 10 –his famous CHMW theorem states that no spontaneous breaking of global continuousinternal symmetries is possible in 1 + 1 spacetime dimensions. The proof is beautifullysimple: 1 + 1 represents the “lower critical dimension” of the massless scalar field φ , definedas the dimension where φ is formally dimensionless at the Gaussian fixed point. Quantummechanically, this means that its propagator is logarithmically divergent, and we need toregulate it by introducing an infrared regulator µ IR : (cid:104) φ ( x ) φ (0) (cid:105) = (cid:90) d k (2 π ) e ik · x k + µ ≈ − π log( µ IR | x | ) + const . + O ( µ IR | x | ) . (2.21)The asymptotic expansion in (2.21), valid in the regime µ IR | x | (cid:28)
1, clearly shows thatas we try to take µ IR → µ IR . We can still construct various composite operators out of derivatives andexponentials of φ , yielding consistent and finite renormalized correlation functions in the µ IR → φ itself does not exist as a quantum object. And since thecandidate NG mode φ does not exist, the corresponding symmetry could never have beenbroken in the first place, which concludes the proof.Going back to the general class of Type A n NG modes, we find an intriguing nonrela-tivistic analog of the CHMW theorem. The dimension of φ ( t, x ) at the A n Gaussian fixedpoint in D + 1 dimensions – measured in the units of spatial momentum – is[ φ ( t, x )] A n = D − n . (2.22)The Type A n field φ is at its lower critical dimension when D = n . Its propagator alsorequires an infrared regulator. There are many ways how to introduce µ IR in this case, forexample by (cid:104) φ ( t, x ) φ (0) (cid:105) = (cid:90) dω d D k (2 π ) D +1 e i k · x − iωt ω + k D + µ D IR , (2.23)or by (cid:104) φ ( t, x ) φ (0) (cid:105) = (cid:90) dω d D k (2 π ) D +1 e i k · x − iωt ω + ( k + µ ) D . (2.24)Either way, as we try to take µ IR →
0, the asymptotics of the propagator again behaveslogarithmically, both in space (cid:104) φ ( t, x ) φ (0) (cid:105) ≈ − π ) D/ Γ( D/
2) log( µ IR | x | ) + . . . for | x | D (cid:29) t (2.25)and in time, (cid:104) φ ( t, x ) φ (0) (cid:105) ≈ − π ) D/ D Γ( D/
2) log( µ D IR t ) + . . . for | x | D (cid:28) t. (2.26)Most importantly, the propagator remains sensitive to the infrared regulator µ IR . Con-sequently, we obtain the nonrelativistic, multicritical version of the CHMW theorem forType A NG modes and their associated symmetry breaking:– 11 – he Type A n would-be NG mode φ ( t, x ) at its lower critical dimension D = n ex-hibits a propagator which is logarithmically sensitive to the infrared regulator µ IR ,and therefore φ ( t, x ) does not exist as a quantum mechanical object. Consequently,no spontaneous symmetry breaking with Type A n NG modes is possible in D = n dimensions. By extension, this also invalidates all Type A n would-be NG modes with n > D : Theirpropagator grows polynomially at long distances, destabilizing the would-be condensateand disallowing the associated symmetry breaking pattern.In contrast, in the Type B n case (and assuming that all the NG field components areassigned the same dimension), we have[ φ ( t, x )] B n = D , and the lower critical dimension is D = 0. Hence, in all dimensions D >
0, the Type B n NG modes are free of infrared divergences and well-defined quantum mechanically for all n = 1 , , . . . , and the Type B nonrelativistic, multicritical CHMW theorem is limited tothe following statement: The Type B n symmetry breaking is possible in any D > and for any n = 1 , , . . . . In the special cases for Type A and Type B NG modes, the multicritical CHMWtheorems stated above reproduce the results reported in [22].
The conclusions of the nonrelativistic CHMW theorem appear rather unfavorable for TypeA n NG modes with n ≥ D . However, unlike in the relativistic case of n = 1 in 1 + 1dimensions, the nonrelativistic systems offer an intriguing way out [14], as we now illustratefor the case of the lower critical dimension D = n , with D > φ is logarithmically sensitive to theinfrared regulator. However, all is not lost – unlike in the relativistic case, the systemcan now provide its own natural infrared regulator, and flow under the influence of someof the relevant terms to another infrared fixed point of Type A n (cid:48) , with a lower value of n (cid:48) < n . And we know that this phenomenon can be arranged to happen hierarchically,in a pattern protected by the hierarchical breaking of the polynomial symmetries. Thus,we can break the polynomial shift symmetry at a high energy scale µ only partially, to apolynomial symmetry of a lower degree which is then broken at a lower energy scale µ (cid:48) .This process can continue until at some low scale µ (cid:48)(cid:48) the symmetry is broken all the wayto the constant shift and n (cid:48)(cid:48) = 1. This process of consecutive partial symmetry breakingopens up a hierarchy of energy scales µ (cid:29) µ (cid:48) (cid:29) . . . (cid:29) µ (cid:48)(cid:48) , (2.27) In the case of preudo-NG modes, even the constant shift symmetry can be broken explicitly at somescale. – 12 –ver which the propagator for φ exhibits a cascading behavior: First it appears logarithmicand the formation of a condensate seems precluded, and then it undergoes a series ofcrossovers to lower values of z < D until in the far infrared the condensate is no longerdestabilized by infrared fluctuations. The separation between two consecutive scales µ and µ (cid:48) can be kept large, as a result of the symmetry that is given by a larger-degree polynomialat scale µ than at scale µ (cid:48) . All in all, whether or not the original continuous global internalsymmetry (for which the field is the NG mode) is spontaneously broken is now a questionabout the competition of various scales in the system. We have established a new infinite sequence of symmetries in scalar field theories, andhave shown that they can protect the smallness of quantum mechanical corrections totheir low-energy dispersion relations near the Gaussian fixed points. The symmetries areexact at the infrared Gaussian fixed point, and turning on interactions typically breaksthem explicitly – as we have seen in the series of examples in [11]. Yet, the polynomialshift symmetry at the Gaussian fixed point is useful for the interacting theory as well: Itcontrols the interaction terms, allowing them to be naturally small, parametrized by theamount ε of the explicit polynomial symmetry breaking near the fixed point.Generally, this explicit breaking by interactions breaks the polynomial shift symmetriesof NG modes all the way to the constant shift, which remains mandated by the originalform of the Goldstone theorem (guaranteeing the existence of gapless modes). However,one can now turn the argument around, and ask the following question: Starting at a givenType A n or B n fixed point, what are the lowest-dimension scalar composite operators thatinvolve N fields φ and respect the polynomial shift symmetry of degree P exactly, up toa total derivative? Such operators can be added to the action, and for N = 3 , , . . . theyrepresent self-interactions of the system, invariant under the polynomial shift of degree P .More generally, one can attempt to classify all independent composite operators invariantunder the polynomial shift symmetry of degree P , organized in the order of their increasingdimensions.These are the questions on which we focus in the rest of this paper. In order toprovide some answers, we will first translate this classification problem into a more precisemathematical language, and then we will develop techniques – largely based on abstractgraph theory – that lead us to systematic answers. For some low values of the degree P of the polynomial symmetry and of the number N of fields involved, we can even find themost relevant invariants and prove their uniqueness. Strictly speaking, moving away from the Gaussian fixed point by turning on the self-interactions gen-erally yields additional corrections to the constant shift symmetries, if the underlying symmetry group ofthe interacting theory is non-Abelian. Such non-Abelian corrections vanish at the Gaussian fixed point,and each NG component effectively becomes an Abelian field with its own constant shift symmetry. Inthis paper, we will concentrate solely on the simplest Abelian case, with one Type A NG field φ and thesymmetry group U (1). – 13 – . Galileon Invariants Consider a quantum field theory of a single scalar field φ ( t, x ) in D spatial dimensionsand one time dimension. Consider the transformation of the field which is linear in spatialcoordinates: δφ = a i x i + a , where a i and a are arbitrary real coefficients. Other thanthe split between time and space and the exclusion of the time coordinate from the linearshift transformation, this is the same as the theory of the Galileon [13].The goal is to find Lagrangian terms which are invariant (up to a total derivative)under this linear shift transformation. We will classify the Lagrangian terms by theirnumbers of fields N and derivatives 2∆. Imposing spatial rotation invariance requiresthat spatial derivatives be contracted in pairs by the flat metric δ ij . Thus ∆ counts thenumber of contracted pairs of derivatives. It is easy to find Lagrangian terms which areexactly invariant (i.e., not just up to a total derivative): Let ∆ ≥ N and let at least twospatial derivatives act on every φ . For the linear shift case, all terms with at least twiceas many derivatives as there are fields are equal to exact invariants, up to total derivatives(Theorem 4). However, it is possible for a term to have fewer derivatives than this and stillbe invariant up to a non-vanishing total derivative. For fixed N , the terms with the lowest∆ are more relevant in the sense of the renormalization group. Therefore, we will focuson invariant terms with the lowest number of derivatives, which we refer to as minimalinvariants .These minimal invariants have already been classified for the case of the linear shift.There is a unique (up to total derivatives and an overall constant prefactor) N -pointminimal invariant, which contains 2( N −
1) derivatives (i.e., ∆ = N − N = 5. L = φ, (3.1a) L = ∂ i φ ∂ i φ, (3.1b) L = 3 ∂ i φ ∂ j φ ∂ i ∂ j φ, (3.1c) L = 12 ∂ i φ ∂ i ∂ j φ∂ j ∂ k φ ∂ k φ + 4 ∂ i φ ∂ j φ ∂ k φ ∂ i ∂ j ∂ k φ, (3.1d) L = 60 ∂ i φ ∂ i ∂ j φ ∂ j ∂ k φ ∂ k ∂ (cid:96) φ ∂ (cid:96) φ + 60 ∂ i φ ∂ i ∂ j φ ∂ j ∂ k ∂ (cid:96) φ ∂ k φ ∂ (cid:96) φ + 5 ∂ i φ ∂ j φ ∂ k φ ∂ (cid:96) φ ∂ i ∂ j ∂ j ∂ k ∂ (cid:96) φ. (3.1e)These are not identical to the usual expressions (e.g., in [13]), but one can easily check thatthey are equivalent. We can represent the terms in (3.1) as formal linear combinations of graphs. In thesegraphs, φ is represented by a • -vertex. An edge joining two vertices represents a pair ofcontracted derivatives, one derivative acting on each of the φ ’s representing the endpoints– 14 –f the edge. The graphical representations of the above terms are given below: L = • , (3.2a) L = , (3.2b) L = 3 , (3.2c) L = 12 + 4 , (3.2d) L = 60 + 60 + 5 . (3.2e)The structure of the graph (i.e., the connectivity of the vertices) is what distinguishesgraphs; the placement of the vertices is immaterial. This reflects the fact that the order ofthe φ ’s in the algebraic expressions is immaterial and the only thing that matters is whichcontracted pairs of derivatives act on which pairs of φ ’s. Therefore, for example, the graphsbelow all represent the same algebraic expression. (3.3)Similarly, the four graphs below represent the same algebraic expression. (3.4)A more nontrivial example is given by the following twelve graphs, which all represent thesame algebraic expression. (3.5)The graphs in the second line above appear to have intersecting edges. However, sincethere is no • -vertex at the would-be intersection, these edges do not actually intersect. There are three times as many graphs in (3.5) as there are in (3.4). It so happens thatthe coefficient with which the first graph in (3.5) appears in L (3.2d) is also three timesthe coefficient with which the first graph in (3.4) appears in L . This suggests thatthe coefficient with which a graph appears in a minimal term is precisely the number of– 15 –raphs with the exact same structure (i.e., isomorphic), just with various vertices and edgespermuted.One simple way to state this is to actually label the vertices in the graphs. If thevertices were labeled, and thus distinguished from each other, then all of the graphs ineach one of (3.3), (3.4) and (3.5) would actually be distinct graphs. Of course, this meansthat the corresponding algebraic expressions have φ ’s similarly labeled, but this labeling isfiducial and may be removed afterwards. Note the simplicity that this labeled conventionintroduces: L is the sum of all of the graphs in (3.4) and (3.5) with unit coefficients.The graphs in (3.4) and (3.5) have an elegant and unified interpretation in graphtheory. These graphs are called trees . A tree is a graph which is connected (i.e., cannotbe split into two or more separate graphs without cutting an edge), and contains no loops(edges joining a vertex to itself) or cycles (edges joining vertices in a closed cyclic manner).One can check that there are exactly 16 trees with four vertices and they are given by (3.4)and (3.5). Cayley’s formula, a well-known result in graph theory, says that the number oftrees with N vertices is N N − .For N = 3, the 3 − = 3 trees are in (3.3), and we indeed find that L is the sum ofall three graphs with unit coefficients. The same can be said for L and L . Therefore,the minimal terms for N = 1 , , N = 5 case, itwould strongly suggest that this may hold for all N .There are 5 = 125 trees for N = 5. They can be divided into three sets such that thetrees in each set are isomorphic to one of the three graphs appearing in L (3.2e). Thereare 60 graphs which are isomorphic to the first graph appearing in L ; 12 of these arelisted below and the rest are given by the five rotations acting on each of these 12 graphs:(3.6)There are 60 graphs which are isomorphic to the second graph appearing in L ; 12 ofthese are listed below and the rest are given by their rotations: (3.7)Finally, there are five graphs which are isomorphic to the third graph appearing in L ,which are simply the five rotations acting on that graph. Therefore, L is indeed thesum with unit coefficients of all trees with five vertices!Thus, we arrive at the main result of this section (proven in Appendix B.3):– 16 – he unique minimal N -point linear shift-invariant Lagrangian term is representedgraphically as a sum with unit coefficients of all labeled trees with N vertices.
4. Beyond the Galileons
Now, we extend the linear shift transformation to polynomials of higher degree. We willneed to develop the graphical approach further in order to tackle this problem and numer-ous technicalities will arise. However, a rather elegant and beautiful description of thesepolynomial shift invariants will emerge.Consider the problem of determining all possible terms in a Lagrangian that are in-variant under the polynomial shift symmetry: φ ( t, x i ) → φ ( t, x i ) + δ P φ, δ P φ = a i ··· i P x i · · · x i P + · · · + a i x i + a. (4.1)The a ’s are arbitrary real coefficients that parametrize the symmetry transformation, andare symmetric in any pair of indices. P = 0 , , , . . . corresponds to constant shift, linearshift, quadratic shift, and so on. Obviously, if a term is invariant under a polynomial shiftof order P , then it is also invariant under a polynomial shift of order P (cid:48) with 0 ≤ P (cid:48) ≤ P .We will call a term with N fields and 2∆ derivatives an ( N, ∆) term. We are interestedin interaction terms, for which N ≥
3. As previously mentioned, terms with the lowestpossible value of ∆ are of greatest interest. It is straightforward to write down invariantterms with ∆ ≥ N ( P + 1) since, if each φ has more than P derivatives acting on it, thenthe term is exactly invariant. Are there any invariant terms with lower values of ∆? If so,then these invariant terms will be more relevant than the exact invariants.To be invariant, a term must transform into a total derivative under the polynomialshift symmetry. In other words, for a specific P and given ( N, ∆), we are searching forterms L such that δ P L = ∂ i ( L i ) . (4.2)Here L is a linear combination of terms with N φ ’s and 2∆ ∂ ’s, and L i is a linear combi-nation of terms with N − φ ’s. Such L ’s are called P-invariants .How might we determine such invariant terms in general? For a given ( N, ∆), the mostbrute-force method for determining invariant terms can be described as follows. First, writedown all possible terms in the Lagrangian with a given ( N, ∆) and ensure that they areindependent up to integration by parts. Next, take the variation of all these terms underthe polynomial shift. There may exist linear combinations of these variations which areequal to a total derivative, which we call total derivative relations . If we use these totalderivative relations to maximally reduce the number of variation terms, then the required P -invariants form the kernel of the map from the independent Lagrangian terms to theindependent variation terms (Corollary 6). Let us consider some examples of this brute-force procedure in action. – 17 – .1. Brute-force Examples4.1.1. ( P, N, ∆) = (1 , , L = ∂ i φ ∂ j φ ∂ i ∂ j φ, L = φ ∂ i ∂ j φ ∂ i ∂ j φ. The variation under the linear shift symmetry (for P = 1) of these terms is given by δ ( L ) = 2 L × a , δ ( L ) = L × b , where L × a = a i ∂ j φ ∂ i ∂ j φ and L × b = ( a k x k + a ) ∂ i ∂ j φ ∂ i ∂ j φ . There is only one total derivativethat can be formed from these terms, namely ∂ i ( a i ∂ j φ ∂ j φ ) = 2 L × a . Therefore, there is a single invariant term for (
P, N, ∆) = (1 , , L = ∂ i φ ∂ j φ ∂ i ∂ j φ. ( P, N, ∆) = (3 , , L = ∂ i ∂ j φ ∂ k ∂ l φ ∂ i ∂ j ∂ k ∂ l φ, L = ∂ i ∂ j φ ∂ i ∂ k ∂ l φ ∂ j ∂ k ∂ l φ,L = ∂ i φ ∂ j ∂ k ∂ l φ ∂ i ∂ j ∂ k ∂ l φ, L = φ ∂ i ∂ j ∂ k ∂ l φ ∂ i ∂ j ∂ k ∂ l φ. The variation under the cubic shift symmetry (for P = 3) of these terms is given by δ ( L ) = 2 L × a , δ ( L ) = L × b + 2 L × c ,δ ( L ) = L × d + L × e , δ ( L ) = L × f . where L × a = (6 a ijm x m + 2 a ij ) ∂ k ∂ l φ ∂ i ∂ j ∂ k ∂ l φL × b = (6 a ijm x m + 2 a ij ) ∂ i ∂ k ∂ l φ ∂ j ∂ k ∂ l φL × c = 6 a ikl ∂ i ∂ j φ ∂ j ∂ k ∂ l φL × d = (3 a imn x m x n + 2 a im x m + a i ) ∂ i φ ∂ j ∂ k ∂ l φ ∂ i ∂ j ∂ k ∂ l φL × e = 6 a jkl ∂ i φ ∂ i ∂ j ∂ k ∂ l φL × f = ( a mnp x m x n x p + a mn x m x n + a m x m + a ) ∂ i ∂ j ∂ k ∂ l φ ∂ i ∂ j ∂ k ∂ l φ. There are three independent total derivatives that can be formed out of these: ∂ i [2(6 a ijm x m + 2 a ij ) ∂ k ∂ l φ ∂ j ∂ k ∂ l φ − a ijj ∂ k ∂ l φ ∂ k ∂ l φ ] = 2( L × a + L × b ) ,∂ i [6 a ijk ∂ j ∂ l φ ∂ k ∂ l φ ] = 2 L × c ,∂ i [6 a ijk ∂ j ∂ k ∂ (cid:96) φ ∂ (cid:96) φ ] = L × c + L × e . – 18 –t is a non-trivial exercise to find and verify this, and a more systematic way of finding thetotal derivative relations will be introduced later.Applying these relations, one finds a single invariant for ( P, N, ∆) = (3 , , L + 2 L = ∂ i ∂ j φ ∂ k ∂ l φ ∂ i ∂ j ∂ k ∂ l φ + 2 ∂ i ∂ j φ ∂ i ∂ k ∂ l φ ∂ j ∂ k ∂ l φ. Note that δ ( L + 2 L ) = 2( L × a + L × b ) + 2(2 L × c ), which is a total derivative. It is clear that even for these simple examples, the calculations quickly become unwieldy,and it becomes increasingly difficult to classify all of the total derivative relations. Atthis point we will rewrite these results in a graphical notation which will make it easierto keep track of the contractions of indices in the partial derivatives. Full details aboutthis graphical approach can be found in Appendix B, but we will summarize them here.In addition to the • -vertex and edges we introduced in §
3, we represent δ P φ by a ⊗ (a × -vertex). Note that there are at most P edges incident to a × -vertex since P + 1 derivativesacting on δ P φ yields zero, whereas an arbitrary number of edges can be incident to a • -vertex. Moreover, we introduce another vertex, called a (cid:63) -vertex, which will be used torepresent terms that are total derivatives. We require that a (cid:63) -vertex always be incidentto exactly one edge, and that this edge be incident to a • -vertex or × -vertex. This edgerepresents a derivative acting on the entire term as a whole, and the index of that derivativeis contracted with the index of another derivative acting on the φ or δ P φ of the • - or × -vertex, respectively, to which the (cid:63) -vertex is adjacent. Therefore, directly from thedefinition, any graph with a (cid:63) -vertex represents a total derivative term. The expansion ofthis derivative using the Leibniz rule is graphically represented by the summation of thegraphs formed by removing the (cid:63) -vertex and attaching the edge that was incident to the (cid:63) -vertex to each remaining vertex. This operation is denoted by the derivative map ρ . Thesymbols N ( • ), N ( × ) and N ( (cid:63) ) represent the numbers of each type of vertex. Note that N = N ( • ) + N ( × ) does not include N ( (cid:63) ) since (cid:63) -vertices represent neither φ nor δ P φ .We define three special types of graphs: A plain-graph is a graph in which all verticesare • -vertices. A × -graph is a plain-graph with one • -vertex replaced with a × -vertex. A (cid:63) -graph is a graph with one × -vertex and at least one (cid:63) -vertex.Note that the variation δ P of a plain-graph under the polynomial shift symmetry isgiven by summing over all graphs that have exactly one • -vertex in the original graph re-placed with a × -vertex. To illustrate the graphical approach, we rewrite the examples fromsections 4.1.1 and 4.1.2 using this new graphical notation. Since the algebraic expressionshave unlabeled φ ’s, the graphs in this section will be unlabeled. ( P, N, ∆) = (1 , , L = L =– 19 –he variation under the linear shift symmetry (for P = 1) is given by δ (cid:32) × (cid:33) = 2 × δ (cid:32) × (cid:33) = × The only independent total derivative that can be formed out of these terms is ρ (cid:32) (cid:70) × (cid:33) = 2 × As before, there is a single invariant for (
P, N, ∆) = (1 , ,
2) given by L . In this case, thegraphical version of (4.2) is given by δ (cid:32) × (cid:33) = 2 × = ρ (cid:32) (cid:70) × (cid:33) ( P, N, ∆) = (3 , , L = L = L = L =The variation under the cubic shift symmetry (for P = 3) is given by δ (cid:32) × (cid:33) = 2 × δ (cid:32) × (cid:33) = × + 2 × δ (cid:32) × (cid:33) = × + × δ (cid:32) × (cid:33) = × The independent total derivatives that can be formed out of these terms are ρ (cid:32) (cid:70) × − (cid:70) × (cid:33) = 2 × + 2 × ρ (cid:32) (cid:70) × (cid:33) = 2 × ρ (cid:32) (cid:70) × (cid:33) = × + × Once again, there is a single invariant for (
P, N, ∆) = (3 ,
4) given by L + 2 L . Thegraphical version of (4.2) is given by δ (cid:32) × + 2 × (cid:33) = 2 × + 2 × + 4 × = ρ (cid:32) (cid:70) × − (cid:70) × + 2 (cid:70) × (cid:33) (4.3)– 20 –o far all we have done is rewrite our results in a new notation. But the graphical notationis more than just a succinct visual way of expressing the invariant terms. The followingsection illustrates the virtue of this approach. As shown in Appendix B, the graphical approach allows us to prove many general theorems.In particular, we have the following useful outcomes:1. Without loss of generality, we can limit our search for invariants to graphs with veryspecific properties (Appendix B.2).2. There is a simple procedure for obtaining all the independent total derivative relationsbetween the variation terms for each P , N and ∆ (Theorem 1).3. The graphical method allows a complete classification of 1-invariants (Theorem 4).4. The graphical method allows many higher P -invariant terms to be constructed fromlower P invariants (Appendix B.4).We will expound upon the above points by presenting explicit examples. These examplesare generalizable and their invariance is proven in Appendix B. However, the reader canalso check by brute force that the terms we present are indeed invariant. We will nowsummarize points 1 and 2 and will return to point 4 in § § ∂ i ∂ i acting on a single φ and one can always integrate by partsto move one of the ∂ i ’s to act on the remaining φ ’s. We can also restrict to plain-graphswith vertices of degree no less that ( P + 1) (Proposition 7). This represents a significantsimplification from the previous procedure ( § § L and L are immediately discarded.Taking the variation of these terms yields × -graphs and we need to determine thetotal derivative relations between them. Since all plain-graphs we are considering areloopless, any × -graphs involved in these total derivative relations are also loopless. Thetotal derivative relations that we need to consider can be obtained with the use of graphscalled Medusas (Definition 10). A Medusa is a loopless (cid:63) -graph with all of its (cid:63) -verticesadjacent to the × -vertex and such that the degree of the × -vertex is given by:deg( × ) = P + 1 − N ( (cid:63) ) , (4.4)where, again, N ( (cid:63) ) is the number of (cid:63) -vertices. Note that because deg( × ) ≥ N ( (cid:63) ) for aMedusa, (4.4) implies that N ( (cid:63) ) ≤ ( P + 1) ≤ deg( × ) for any Medusa. Furthermore, weneed only consider Medusas with × -vertex and • -vertices of degree no less than ( P + 1)(Proposition 8). From each of these Medusas, we obtain a total derivative relation, con-taining only loopless × -graphs, by applying the map ρ and then omitting all looped graphs(Proposition 4). This map is denoted as ρ (0) in Definition 13. In § N and P , we call an invariant consisting of graphs containingthe lowest possible value of ∆ a minimal invariant (Definition 16). Minimal invariants areof particular interest in a QFT, since they are the most relevant N -point interactions. In § § N = 4 and P = 2 , ( P, N, ∆) = (2 , , N = 4. For P = 2, any Medusamust have N ( (cid:63) ) ≤ ( P + 1) = , and thus there is exactly one (cid:63) -vertex in a P = 2 Medusa.Furthermore, we need only consider Medusas in which each vertex has degree at least 2,since ( P + 1) = . Therefore, the counting implies that we need only consider P = 2Medusas with ∆ ≥ N + 1. In particular, when N = 4, the minimal ∆ is 5 (representingterms with 10 derivatives). In the following we show that there is exactly one 2-invariantwith ∆ = 5. The relevant Medusas are: M = (cid:70) × M = (cid:70) × The resulting loopless total derivative relations are: ρ (0) ( M ) = 2 × + × ≡ L × a + L × e ρ (0) ( M ) = × + × + × ≡ L × b + L × c + L × d (4.5)Note that, when acting on these P = 2 Medusas, ρ and ρ (0) are in fact the same.On the other hand, the invariants must be constructed out of plain-graphs containingvertices of degree no less than ( P + 1) = . The only possibilities are: L = L = L = L = L =The invariant cannot be constructed out of L since δ ( L ) is absent from the total deriva-tive relations (4.5). Hence, we need only consider the variations of L , L , L and L .We can now determine the 2-invariants. The total derivative relations allow us toidentify L × d ∼ − L × b − L × c and L × e ∼ − L × a . Up to total derivatives, δ L L L L = L × a L × b L × c L × d L × e ∼ − − − L × a L × b L × c (4.6)– 22 –he invariants form the nullspace of the transpose of the final 4 × , , , L + L + L + L , i.e.,+ + + (4.7)Therefore, (4.7) gives the only independent minimal 2-invariant for N = 4. ( P, N, ∆) = (3 , , P = 3, N = 4, for which each vertex degree must be at least ( P + 1) = 2. Again we would like to find the minimal invariant in this case. By countingalone, it is possible to write down Medusas with ∆ = 5. In fact, a 3-invariant with N = 4and ∆ = 5 would also be 2-invariant. The only possible 2-invariant with ( N, ∆) = (4 ,
5) is(4.7). However, this is not a 3-invariant because it is impossible for some graphs containedin it to appear in a 3-invariant. For example, by replacing a degree-2 vertex in L in(4.7) with a × -vertex, a × -graph Γ × is produced; Γ × and the only P = 3 Medusa M thatgenerates Γ × are given below:Γ × = × M = (cid:70)(cid:70) × But M contains a • -vertex of degree lower than 2, and therefore (4.7) cannot be 3-invariant.This sets a lower bound for ∆: ∆ ≥
6. For ∆ = 6, the Medusas are: M = (cid:70) × M = (cid:70) × M = (cid:70) × M = (cid:70) × M = (cid:70) × M = (cid:70) × M = (cid:70)(cid:70) × M = (cid:70)(cid:70) × These give the following total derivative relations: ρ (0) ( M ) = × + × + × ≡ L × a + L × b + L × c ρ (0) ( M ) = × + 2 × ≡ L × d + 2 L × e ρ (0) ( M ) = × + 2 × ≡ L × f + 2 L × g ρ (0) ( M ) = × + × + × ≡ L × a + L × h + L × i ρ (0) ( M ) = 2 × + × ≡ L × e + L × j – 23 – (0) ( M ) = × + × + × ≡ L × b + L × k + L × h ρ (0) ( M ) = 2 × + × + 4 × + 2 × ≡ L × (cid:96) + L × m + 4 L × n + 2 L × o ρ (0) ( M ) = 2 × + × + 2 × + 4 × ≡ L × p + L × q + 2 L × r + 4 L × s Then the invariants must be made up of plain-graphs whose variations are contained inthe total derivative relations above. In other words, the invariants are made from: L = L = L = L = L = L = L = L = L = L =We can now determine the 3-invaraints: δ L L L L L L L L L L = L × a L × b L × c L × d L × e L × f L × g L × h L × i L × j L × k L × (cid:96) L × m L × n L × o L × p L × q L × r L × s – 24 – − − − − − − − − − − − L × a L × b L × e L × f L × h L × l L × n L × o L × p L × r L × s The nullspace of the transpose of this matrix is spanned by (3 , , , , , , , , , N = 4.Note that L does not appear in the invariant. Indeed, it can be discarded immediately,since the unique Medusa associated with L is (cid:70)(cid:70) (cid:70) × This Medusa has an empty vertex, which violates the lower bound on vertex degree.
We have seen that Medusas play a central role in the search for P -invariants. It is thusworthwhile to discuss the key features of Medusas and to demonstrate how a total derivativerelation consisting of loopless × -graphs is constructed from a Medusa. Given any Medusa, ρ (0) ( M ) is in fact a total derivative relation, as can be seen from the following construction.For fixed P and N , consider a Medusa M that contains N ( (cid:63) ) (cid:63) -vertices. Then, bydefinition, it has a × -vertex of degree deg( × ) = P + 1 − N ( (cid:63) ). Since the maximal degreeof a × -vertex is P , graphs in ρ ( M ) contain at most N ( (cid:63) ) − (cid:63) -graph Γ ( (cid:96) ) from M by deleting (cid:96) ≤ N ( (cid:63) ) − (cid:63) -vertices in M and then adding (cid:96) loops to the × -vertex.By this definition, M = Γ (0) . In the algebraic expression represented by Γ ( (cid:96) ) , the N ( (cid:63) ) − (cid:96)(cid:63) -vertices stand for N ( (cid:63) ) − (cid:96) partial derivatives acting on the whole term. Distributing N ( (cid:63) ) − (cid:96) − ∂ ’s over all φ ’s in this algebraic expression will result in a linear combination– 25 –f total derivative terms. In the graphical representation, this is equivalent to acting ρ on Γ ( (cid:96) ) but keeping fixed exactly one (cid:63) -vertex and its incident edge. Setting to zero allcoefficients of graphs in the resulting linear combination, except for the ones containingexactly (cid:96) loops, generates a linear combination L ( (cid:96) ) of (cid:63) -graphs, each containing exactlyone (cid:63) -vertex. By construction, ρ (0) ( M ) = ρ N ( (cid:63) ) − (cid:88) α =0 ( − α L ( α ) . (4.9)The algebraic form of the RHS of (4.9) is explicitly a total derivative relation. For arigorous treatment of the above discussion, refer to Proposition 4 in Appendix B.2.3.As a simple example, we consider the Medusa M referred to in § M = (cid:70)(cid:70) × ⇒ ρ (0) ( M ) = ρ (cid:32) (cid:70) × + 2 (cid:70) × − (cid:70) × (cid:33) For a second example, we consider (
P, N, ∆) = (5 , ,
6) and the Medusa M = (cid:70)(cid:70) (cid:70) × By (4.9), we obtain ρ (0) ( M ) = ρ (cid:32) (cid:70) × + 2 (cid:70) × − (cid:70) × + (cid:70) × (cid:33) . This Medusa is involved in a 5-invariant that we will construct in § In §
3, we discovered an intriguing construction of the minimal 1-invariant for given N ,which is a sum with equal coefficients of all possible trees with N vertices. A close studyof the P -invariants with P > § superposition of equal-weight tree summations and exact invariants. Recall that an exact invariant is a linear combination that is invariant exactly,instead of up to a total derivative. In the graphical representation, a linear combinationof graphs is an exact P E -invariant if and only if all vertices are of degree larger than P E (Corollary 4). Each graph in an exact invariant is itself exactly invariant.Next we illustrate “the superposition of linear combinations” by explicit examples.When appropriate, we will consider labeled graphs and only remove the labels at the end.Consider two labeled graphs, Γ = Γ =– 26 –he superposition of Γ and Γ is defined to be the graph formed by taking all edges in Γ and adding them to Γ , i.e., Γ ∪ Γ =The superposition of two linear combinations, L A = (cid:80) k A i =1 a i Γ Ai and L B = (cid:80) k B i =1 b i Γ Bi , ofplain graphs Γ Ai , Γ Bj with the same N , is defined as L A ∪ L B ≡ k A (cid:88) i =1 k B (cid:88) j =1 a i b j Γ Ai ∪ Γ Bj . In the following we present numerous examples of invariants constructed by superposingequal-weight tree summations and exact invariants for various P ’s. In fact, Theorem 7 ofAppendix B.4 states: For fixed N , the superposition of an exact P E -invariant with the superposition of Q minimal loopless 1-invariants results in a P -invariant, provided P E + 2 Q ≥ P . We conjecture that the above result captures all P -invariants, up to total derivatives. Sincewe have classified all exact invariants and all 1-invariants (Theorem 4), it is straightforwardto construct the P -invariants in the above statement for any specific case. We now proceedto construct the minimal P -invariants for some important cases. P =2) N = : A 2-invariant can be constructed by superposing an equal-weight treesummation with an exact 0-invariant. In the labeled representation, all possible trees for N = 3 are given by (3.3),The sum of all three N = 3 trees with unit coefficients gives a 1-invariant, L . On theother hand, up to total derivatives, an exact 0-invariant with ∆ = 2 is isomorphic toΓ =Then Γ ∪ L contains the three superposed graphs as follows: Note that this theorem also applies when P E <
0, where we take an exact P E -invariant for P E < P E -invariant for any P E <
0, and superposing this graph on any other is equivalent to notsuperposing anything at all. – 27 – a) All 16 trees for N = 4. (b) Superposition of (4.11) and the 16 trees. Figure 1 : The most relevant 2-invariant for N = 4 from superposition of graphs.Summing over all superposed graphs with unit coefficients gives a 2-invariant for N = 3(after identifying isomorphic graphs), δ (cid:18) + 2 (cid:19) = ρ (0) (cid:32) (cid:70) × (cid:33) . (4.10)Note that for P = 2 and N = 3, we need only consider Medusas with at least four edges,since that the × -vertex and • -vertices have degree no less than 2. The Medusa in (4.10)is the only such Medusa with ∆ = 4. Therefore, this is the only independent minimal2-invariant. Note that the 2-invariant given in (4.10) and the 3-invariant in (4.3) happento be the same.In fact, we can prove a general minimality statement for N = 3. Consider a Medusawith ∆ = P + 1 for odd P , and ∆ = P + 2 for even P . The • -vertices of this Medusa havedegree at least ∆. For odd P this already saturates the lower bound for the degree of a • -vertex; no edge joining the two • -vertices can be removed and thus ∆ cannot be loweredfurther. For even P , one • -vertex saturates the lower bound on vertex degree and the other • -vertex has an excess of exactly one edge. Nevertheless, the same conclusion holds. N = : In § N = 4. It has the structure of a superposition of the sum with unit coefficients of all N = 4 trees (Figure 1a) and an exact 0-invariant: (4.11)The superposition of this 0-invariant with the trees in Figure 1a is given in Figure 1b. Thesum of all graphs in Figure 1b with unit coefficients gives the 2-invariant in (4.7) (with anoverall prefactor of 4). The trees are arranged in order of their Pr¨ufer sequences [23]. – 28 – .4.2. Cubic Shift ( P =3) N = : For P = 3, the only independent minimal invariant for N = 3 is given in(4.3), which can be written as a superposition of two equal-weight tree summations:As we already pointed out, this 3-invariant happens to be the minimal 2-invariant as well.We can produce more 3-invariants by superposing an exact 1-invariant on the equal-weight sum of N = 3 trees. This would have two more derivatives compared to the minimalterm above. For example, there are two independent exact 1-invariants for N = 3 with 3edges:which yield the following two 3-invariants:+ 2 N = : In § N = 4. It has the structure of a superposition of two sums with unit coefficients of alltrees in Figure 1a. As mentioned in §
3, there are two isomorphism classes of N = 4 trees: T A = T B = (4.12)Superposing these graphs on the N = 4 trees produces the graphs in Figure 2. If T and T (cid:48) are isomorphic trees, then superposing T on the trees in Figure 1a produces 16 graphswhich are isomorphic to the 16 graphs formed by superposing T (cid:48) on the same trees. Thereare four trees in the isomorphism class of T A and twelve for T B . Therefore, we just haveto give the 16 graphs in Figure 2a weight 4 and the 16 graphs in Figure 2b weight 12 andthen add them all up. The result is (4.8) with an overall prefactor of 4. Again, we havealready shown that this is the unique minimal 3-invariant for N = 4.As in the N = 3 case, we can produce non-minimal 3-invariants by superposing anexact 1-invariant on the equal-weight sum of N = 4 trees. For example, there are four– 29 – a) Superposition of T A and Figure 1a. (b) Superposition of T B and Figure 1a. Figure 2 : Superposition of graphs in (4.12) on trees in Figure 1a.independent exact 1-invariants for N = 4 with the lowest number of edges: P =4) N = 3 Case : As argued earlier, ∆ min = 6 in this case. There are two Medusas with thefewest edges such that the × -vertex and the • -vertices have degree no less than 3 (notethat ( P + 1) = in this case): (cid:70) (cid:70) × (cid:70) (cid:70) × There is exactly one minimal 4-invariant in this case, which is constructed by superposingtwo equal-weight sums of trees with an exact 0-invariant:+ 6 + 2Note that the sum of the coefficients is 9, as it should be, since there are three N = 3 trees,and thus there are nine superpositions of two N = 3 trees.Examples of non-minimal invariants can be constructed by superposing an equal-weightsum of trees with an exact 2-invariant, or two equal-weight sums of trees with an exact1-invariant.The proofs of uniqueness and minimality for the remaining N = 4 examples are lengthyand involve many more Medusas than the previous examples, but the process is the same.Therefore, we will simply write the invariants and state that they are unique and minimal.– 30 – =4 Case : We construct the minimal 4-invariant by superposing two copies of equal-weight sums of trees with an exact 0-invariant. There is one independent exact 0-invariant:Superposing this on the superposition of two copies of equal-weight sums of trees yields48 + 40 + 16 + 32+ 20 + 16 + 16 + 16+ 16 + 8 + 8 + 4+ 4 + 4 + 4 + 4Note that the sum of the coefficients is 256 = 16 . P =5) N = : In this case, ∆ min = 6. There are three Medusas with the fewest edges suchthat the × -vertex and the • -vertices have degree no less than 3 (note that ( P + 1) = 3): (cid:70) (cid:70) (cid:70) × (cid:70)(cid:70) (cid:70) × (cid:70) (cid:70) × There is exactly one minimal 5-invariant in this case, which is constructed by superposingthree equal-weight sums of trees:3 + 18 + 6Note that the sum of the coefficients is 27 = 3 . Also, note that this is proportional tothe unique independent minimal 4-invariant found in the previous section, which was thesuperposition of two equal-weight sums of trees and an exact 0-invariant.– 31 – = : The superposition of three equal-weight sums of N = 4 trees yields4 + 12 + 108 + 432 + 288+ 72 + 216 + 216 + 36 + 72+ 72 + 144 + 144 + 612 + 144+216 + 72 + 72 + 432 + 72+ 72 + 72 + 216 + 192 + 108Note that the sum of the coefficients is 4096 = 16 .To facilitate the check of the invariance of the above linear combination, denoted as L , we provide the linear combination of Medusas L M such that δ ( L ) = ρ (0) ( L M ):12 (cid:70)(cid:70) (cid:70) × + 72 (cid:70)(cid:70) (cid:70) × + 24 (cid:70)(cid:70) (cid:70) × + 72 (cid:70)(cid:70) × + 72 (cid:70)(cid:70) × + 144 (cid:70)(cid:70) × +144 (cid:70)(cid:70) × + 216 (cid:70)(cid:70) × + 72 (cid:70) × + 72 (cid:70) × + 72 (cid:70) × + 288 (cid:70) × +432 (cid:70) × + 108 (cid:70) × + 216 (cid:70) × + 144 (cid:70) × + 216 (cid:70) ×
5. Conclusions and Outlook
In this paper, we studied nonrelativistic scalar field theories with polynomial shift symme-tries. In the free-field limit, such field theories arise in the context of Goldstone’s theorem,where they lead to the hierarchies of possible universality classes of Nambu-Goldstonemodes, as reviewed in §
2. Our main focus in § § interacting effectivefield theories which respect the polynomial shift symmetries of degree P = 1 , , . . . . In or-der to find such theories, one needs to identify possible Lagrangian terms invariant underthe polynomial shift up to total derivatives, and organize them by their scaling dimension,starting from the most relevant. As we showed in § § P , number N of fields, the number 2∆– 32 –f spatial derivatives, and as a function of the spatial dimension D , in a way that is muchmore efficient than any “brute force” technique. Secondly, and perhaps more importantly,the graphical technique reveals some previously hidden structure even in those invariantsalready known in the literature. For example, the known Galileon N -point invariants aregiven by the equal-weight sums of all labeled trees with N vertices! This hidden simplic-ity of the Galileon invariants is a feature previously unsuspected in the literature, and itsmathematical explanation deserves further study. In addition, we also discovered patternsthat allow the construction of higher polynomials from the superposition of graphs rep-resenting a collection of invariants of a lower degree – again a surprising result, revealingglimpses of intriguing connections among the a priori unrelated spaces of invariants acrossthe various values of P , N and ∆.Throughout this paper, we focused for simplicity on the unrestricted polynomial shiftsymmetries of degree P , whose coefficients a i ...i (cid:96) are general real symmetric tensors of rank (cid:96) = 0 , . . . , P . As we pointed out in § P ≥
2, this maximal polynomial shift symmetryalgebra allows various subalgebras, obtained by imposing additional conditions on thestructure of a i ...i (cid:96) ’s. While this refinement does not significantly impact the classification ofGaussian fixed points, reducing the symmetry to one of the subalgebras inside the maximalpolynomial shift symmetry can lead to new N -point invariants, beyond the ones presentedin this paper. It is possible to extend our graphical technique to the various reducedpolynomial shift symmetries, and to study the refinement of the structure of polynomialshift invariants associated with the reduced symmetries.Our main motivation for the study of scalar field theories with polynomial shift sym-metries has originated from our desire to map out phenomena in which technical natu-ralness plays a crucial role, in general classes of field theories with or without relativisticsymmetries. The refined classification of the universality classes of NG modes and thenonrelativistic refinement of Goldstone’s theorem have provided an example of scenarioswhere our naive relativistic intuition about technical naturalness may be misleading, andnew interesting phenomena can emerge. We anticipate that other surprises of natural-ness are still hidden not only in the landscape of quantum field theories, but also in thelandscape of nonrelativistic theories of quantum gravity. Acknowledgments
We wish to thank Hitoshi Murayama and Haruki Watanabe for useful discussions. Thiswork has been supported by NSF Grant PHY-1214644 and by Berkeley Center for Theo-retical Physics.
A. Glossary of Graph Theory
In this section, we list the standard terminologies in graph theory to which we will refer.(These essentially coincide with the ones in [23].)
Graph A graph Γ is an ordered pair ( V (Γ) , E (Γ)) consisting of a set V (Γ) of vertices anda set E (Γ), disjoint from V (Γ), of edges , together with an incident function Ψ Γ that– 33 –ssociates with each edge of Γ an ordered pair of (not necessarily distinct) vertices ofΓ. If e is an edge and u and v are vertices such that Ψ Γ ( e ) = { u, v } , then e is said to join u and v . Isomorphism
Two graphs Γ A and Γ B are isomorphic if there exist a pair of bijections f : V (Γ A ) → V (Γ B ) and φ : E (Γ A ) → E (Γ B ) such that Ψ Γ A ( e ) = { u, v } if and onlyif Ψ Γ B ( φ ( e )) = { f ( u ) , f ( v ) } . Identical Graphs
Two graphs are identical , written Γ A = Γ B , if V ( G ) = V ( H ), E ( G ) = E ( H ) and Ψ G = Ψ H . Labeled Graph
A graph in which the vertices are labeled but the edges are not, is calleda labeled graph . This will be the notion of graphs that we will refer to most frequently.
Unlabeled Graph An unlabeled graph is a representative of an equivalence class of iso-morphic graphs. Finite Graph
A graph is finite if both of its vertex set and edge set are finite.
Null Graph
The graph with no vertices (and hence no edges) is the null graph . Incident
The ends of an edge are said to be incident to the edge, and vice versa . Adjacent
Two vertices which are incident to a common edge are adjacent . Loop A loop is an edge that joins a vertex to itself. Cycle A cycle on two or more vertices is a graph in which the vertices can be arrangedin a cyclic sequence such that two vertices are joined by exactly one edge if they areconsecutive in the sequence, and are nonadjacent otherwise. A cycle on one vertexis a graph consisting of a single vertex with a loop. Loopless Graph A loopless graph contains no loops. Note that a loopless graph may stillcontain cycles on two or more vertices. Vertex Degree
The degree of a vertex v , denoted by deg( v ), in a graph Γ is the numberof edges of Γ incident to v , with each loop counting as two edges. Empty Vertex
A vertex of degree 0 is called an empty vertex . Leaf
A vertex of degree 1 is called a leaf . Edge Deletion
The edge deletion of an edge e in a graph Γ is defined by deleting from Γthe edge e but leaving the vertices and the remaining edges intact. Vertex Deletion
The vertex deletion of a vertex v in a graph Γ is defined by deletingfrom Γ the vertex v together with all the edges incident to v . The resulting graph isdenoted by Γ − v . – 34 – onnected Graph A graph is connected if, for every partition of its vertex set into twononempty sets X and Y , there is an edge with one end in X and one end in Y . Connected Component A connected component of a graph Γ is a connected subgraphΓ (cid:48) of Γ such that any vertex v in Γ (cid:48) satisfies the following condition: all edges incidentto v in Γ are also contained in Γ (cid:48) . Tree A tree is a connected graph that contains no cycles. In particular, note that a treehas no empty vertices if it contains more than one vertex. Cayley’s Formula
The number of labeled trees on N vertices is N N − . B. Theorems and Proofs
B.1. The Graphical Representation
Consider the polynomial shift symmetry applied to a real scalar field φ , φ ( t, x i ) → φ ( t, x i ) + δ P φ, δ P φ = a i ··· i P x i · · · x i P + · · · + a i x i + a. (B.1)The polynomial ends at P th order in the spatial coordinate x i with P = 0 , , , . . . , respec-tively corresponding to constant shift, linear shift, quadratic shift, and so on. The a ’s arearbitrary real coefficients that parametrize the symmetry transformation, and are symmet-ric in any pair of indices. In the algebraic language, for a specific P , we are searching fora Lagrangian that is invariant under the polynomial shift up to a total derivative. Let L be a term in the Lagrangian with N φ ’s and 2∆ spatial derivatives. Then, δ P ( L ) = ∂ i ( L i ) , (B.2)where L i is an expression containing N − φ ’s and an index i , which is not contracted.Such L ’s are called P-invariants . We will mainly focus on interaction terms, i.e., N ≥ P -invariants using a graphical representation. The ingredi-ents of the graphical representation are:1. • -vertices, denoted in a graph by • .2. × -vertices, denoted in a graph by ⊗ .3. (cid:63) -vertices, denoted in a graph by (cid:70) .4. Edges, denoted in a graph by a line, that join the above vertices.In this context, a graph contains up to three types of vertices. This means that thesegraphs carry an additional structure regarding vertex type, compared to the conventionaldefinition of a graph in Appendix A.We construct graphs using the following rules:1. The maximal degree of a × -vertex is P . Any graph containing a × -vertex of degreegreater than P is identified with the null graph.– 35 – (a) ∂ i ( ∂ j φ∂ i ∂ j φ ) (b) ∂ j φ∂ i ∂ j φ∂ i ∂ φ Figure 3 : Examples for the graphical representation of algebraic expressions.2. There is at most one × -vertex in a graph.3. A (cid:63) -vertex is always a leaf (i.e., it has degree one).4. Two (cid:63) -vertices are not allowed to be adjacent to each other.We now describe what these graph ingredients represent. A • -vertex represents a φ anda × -vertex represents δ P φ . A pair of derivatives with contracted indices, each one actingon a certain φ or δ P φ , is represented by an edge joining the relevant • - and × -vertices.Note that Rule 1, which requires that there be at most P edges incident to the × -vertex,is justified since P + 1 derivatives acting on δ P φ gives zero.A graph with (cid:63) -vertices will represent terms which are total derivatives. By Rules 3and 4, a (cid:63) -vertex must always have exactly one edge incident to it, and this edge is incidentto a • -vertex or × -vertex. This edge represents a derivative acting on the entire term as awhole, and the index of that derivative is contracted with the index of another derivativeacting on the φ or δ P φ of the • - or × -vertex, respectively, to which the (cid:63) -vertex is adjacent.Therefore, any graph with a (cid:63) -vertex represents a total derivative term.Since the Lagrangian terms that these graphs represent have a finite number of φ ’s and ∂ ’s, we will consider only finite graphs. In addition, by the definition of graphs, all verticesand edges are automatically labeled, due to the fact that all elements in a set are distinctfrom each other. Therefore, a graph represents an algebraic expression in which each φ and ∂ carries a label. It will be convenient to keep the labels on φ , but it is unnecessaryto label the derivatives. This motivates the definition given in Appendix A for “labeled”graphs. In the rest of Appendix B, unless otherwise stated, a graph is understood to be alabeled graph.The desired algebraic expressions in which all φ ’s are identical can be recovered by iden-tifying all isomorphic graphs (for examples, refer to § P -invariantsalready capture all of the unlabeled ones (Appendix B.5).Note that not all algebraic expressions are captured in the graphical representationdescribed above. For example, ∂ ( ∂ j φ ∂ j φ ) cannot be represented by a graph, since two (cid:63) -vertices are forbidden to be adjacent to each other by Rule 4. However, this algebraicexpression can be written as 2 ∂ i ( ∂ i ∂ j φ ∂ j φ ), which is graphically represented in Figure 3a,disregarding the coefficient 2. Another peculiar example is ∂ i ( ∂ j φ ∂ j φ ) ∂ i ∂ φ , which is equalto 4( ∂ i ∂ j φ )( ∂ j φ )( ∂ i ∂ φ ). Although the graphical representation for the former expressionis beyond the current framework, the graphical representation for the latter one is givenin Figure 3b. One could generalize the graphical representation to include all possiblealgebraic expressions. However, for our purposes, the present framework will suffice.– 36 – .1.1. Types of Graphs and Vector Spaces We classify graphs by different combinations of vertices:
Definition 1.
1. A plain-graph is a graph in which all vertices are • ’s.2. A (cid:63) -ed plain-graph is a graph with vertex set consisting of only • -vertices and at leastone (cid:63) -vertex.3. A × -graph is a plain-graph with one • -vertex replaced with a × -vertex.4. A (cid:63) -graph is a graph with one × -vertex and at least one (cid:63) -vertex. We define sets of graphs and the real vector spaces that they generate:
Definition 2. G N, ∆ is the set of plain-graphs with N • -vertices and ∆ edges.2. G × N, ∆ is the set of × -graphs with N − • -vertices, one × -vertex and ∆ edges.3. G (cid:63)N, ∆ is the set of (cid:63) -graphs with N − • -vertices, one × -vertex, at least one (cid:63) -vertexand ∆ edges.In the above graphs, we choose the labels of the • - and × -vertices to go from v to v N andthe labels of the (cid:63) -vertices to go from v (cid:63) to v (cid:63)N ( (cid:63) ) , where N ( (cid:63) ) is the number of (cid:63) -vertices.Let L N, ∆ , L × N, ∆ and L (cid:63)N, ∆ be the real vector spaces of formal linear combinations generatedby G N, ∆ , G × N, ∆ and G (cid:63)N, ∆ , respectively. The zero vector in any of these vector spaces is thenull graph. Note that N = N ( • ) + N ( × ), where N ( • ) is the number of • -vertices and N ( × ) is thenumber of × -vertices. N does not include N ( (cid:63) ) since (cid:63) -vertices represent neither φ nor δ P φ .These sets of graphs are finite and therefore the vector spaces of formal linear combinationsare finite-dimensional.By Definition 2, graphs in a linear combination L ∈ L N, ∆ ( L × N, ∆ or L (cid:63)N, ∆ ) share thesame number of N and ∆. In most of the following discussion, N and ∆ are fixed. Wewill therefore omit theses subscripts as long as no confusion arises. However, the numberof (cid:63) -vertices N ( (cid:63) ) is not fixed in a generic linear combination of (cid:63) -graphs. B.1.2. Maps
We now define some maps between the sets and vector spaces in Definition 2. This willmodel the operations that act on the algebraic expressions represented by the graphs.Firstly, the variation under the polynomial shift δ P of an algebraic term, expressed bya graph Γ, is represented graphically by summing over all graphs that have one • -vertexin Γ replaced with a × -vertex. – 37 – efinition 3 (Variation Map) . Given a plain-graph Γ ∈ G , with V (Γ) = ( v , . . . , v N ) , themap δ P : G → L × is defined by δ P (Γ) = (cid:80) Ni =1 Γ × i , where Γ × i is a graph given by replacing v i with a × -vertex. This map extends to L → L × by distributing δ P over the formal sum. Note that Γ × i is the null graph if v i has degree greater than P . We will omit the subscript P in δ P as long as no confusion arises. It is also necessary to define a map that operatesin the reverse direction: Definition 4.
The map v : G × → G is defined by replacing the × -vertex with a • -vertex. In the algebraic expressions, a total derivative term looks like ∂ i L i , and the ∂ i can bedistributed over L i as usual, by applying the Leibniz rule. This feature will be capturedby the graphical representation in the following definition. Definition 5 (Derivative Map) . For a given (cid:63) -graph Γ (cid:63) ∈ G (cid:63) , the derivative map ρ : G (cid:63) →L × is defined using the following construction:1. For the (cid:63) -graph Γ (cid:63) , denote the • -vertices by v , . . . , v N − , the × -vertex by v N andthe (cid:63) -vertices by v (cid:63) , . . . , v (cid:63)k , k = N ( (cid:63) ) . Take any (cid:63) -vertex v (cid:63)i in Γ (cid:63) . For each j ∈ { , . . . , N } , form a graph Γ j by deleting v (cid:63) in Γ (cid:63) and then adding an edgejoining v j and the vertex that was adjacent to v (cid:63) in Γ (cid:63) .2. Apply the above procedure to each of the Γ j to form Γ j j by removing the next v (cid:63) .Iterate this procedure until all (cid:63) -vertices have been removed, forming the × -graph Γ j ...j k .3. Define ρ (Γ (cid:63) ) ≡ (cid:80) Nj ,...,j k =1 Γ × j ...j k .The domain of this map can be extended to L (cid:63) by distributing ρ over the formal sum. Thederivative map ρ can be similarly defined on (cid:63) -ed plain-graphs. Furthermore, we take ρ tobe the identity map when it acts on × -graphs. Note that the above definition is well-defined since ρ is independent of the order in whichthe (cid:63) -vertices are deleted. B.1.3. Relations
There are many linear combinations of plain- and × -graphs representing terms that canbe written as a total derivative. To take this feature into account, we define two notionsof relations for plain- and × -graphs, respectively. Definition 6 (Relations) . If a linear combination of plain-graphs L ∈ L can be written as ρ ( L (cid:48) ) , where L (cid:48) is a sum of (cid:63) -ed plain-graphs, then L is called a plain-relation . If a linearcombination of × -graphs L × ∈ L × can be written as ρ ( L (cid:63) ) , with L (cid:63) a sum of (cid:63) -graphs,then L × is called a × - relation . We shall denote the set of all plain-relations by R and the set of all × -relations by R × . R and R × have a natural vector space structure and are subspaces of L and L × , respectively.– 38 – .1.4. The Consistency Equation and Associations Recall that P -invariants are defined algebraically by equation (B.2), δ P ( L ) = ∂ i ( L i ). Thisequation is written in the graphical representation as δ P ( L ) = ρ ( L (cid:63) ) , (B.3)for L ∈ L and L (cid:63) ∈ L (cid:63) . We call (B.3) the consistency equation . Searching for P -invariantsis equivalent to constructing all consistency equations. Note that, by Definition 6, theconsistency equation implies that δ P ( L ) is a × -relation and so we make the followingdefinition: Definition 7 ( P -Invariant) . L ∈ L is a P -invariant if δ P ( L ) ∈ R × . Furthermore, there is a simple class of P -invariants, which we call exact P -invariants .These represent terms which are exactly invariant under the polynomial shift symmetry(B.1), not just up to a total derivative. Definition 8 (Exact P -Invariant) . L ∈ L is an exact P -invariant if δ P ( L ) = 0 . The following notion, called “association between graphs”, will turn out to be indis-pensable in constructing consistency equations.
Definition 9 (Associations) . The associations between pairs of plain- and × -graphs, plain-and plain-graphs, (cid:63) - and × -graphs, (cid:63) - and (cid:63) -graphs and plain- and (cid:63) -graphs are defined asfollows:1. Γ ∈ G and Γ × ∈ G × are associated with each other if Γ × is contained in δ P (Γ) , or,equivalently, v (Γ × ) = Γ .2. Any two graphs Γ , Γ ∈ G are associated with each other if either they are associatedwith the same Γ × ∈ G × , or Γ is identical to Γ .3. Γ (cid:63) ∈ G (cid:63) and Γ × ∈ G × are associated with each other if Γ × is contained in ρ (Γ (cid:63) ) .4. Any two graphs Γ (cid:63) , Γ (cid:63) ∈ G (cid:63) are associated with each other if either they are associatedwith the same Γ × ∈ G × , or Γ (cid:63) and Γ (cid:63) are identical to each other.5. Any two graphs Γ ∈ G and Γ (cid:63) ∈ G (cid:63) are associated with each other if they are associatedwith the same Γ × ∈ G × . It turns out that the associations between only plain-graphs and × -graphs have a simplestructure. Note that for any × -graph Γ × ∈ G × , v (Γ × ) uniquely defines the associatedplain-graph. Hence, Proposition 1. A × -graph is associated with exactly one plain-graph. The corollaries below directly follow:
Corollary 1.
For L ∈ L and Γ × a × -graph in δ ( L ) , L contains the plain-graph v (Γ × ) . – 39 – orollary 2. Any two associated plain-graphs are identical to each other.Proof.
If two distinct plain-graphs are associated with each other, then they are associatedwith a common × -graph, which violates Proposition 1. Therefore, only identical plain-graphs are associated with each other. Corollary 3.
For L ∈ L and a plain-graph Γ in L , δ ( L ) contains all × -graphs in δ (Γ) .Proof. Without loss of generality, suppose Γ appears in L with unit coefficient (otherwise,simply divide L by the coefficient of Γ). Let Γ × be a × -graph in δ (Γ). By Proposition 1,Γ is the only plain-graph associated with Γ × . Therefore, Γ × cannot drop out of δ (Γ + L (cid:48) )for any L (cid:48) ∈ L that does not contain Γ. Applying this statement to L (cid:48) = L − Γ proves thatΓ × must appear in δ ( L ).This now allows us to find all exact P -invariants in a simple manner: Corollary 4. L ∈ L is an exact P -invariant if and only if all vertices in all graphscontained in L have degree at least P + 1 .Proof. If there is a vertex v in some plain-graph Γ in L of degree lower than P + 1, then δ P (Γ) contains the × -graph where v is replaced with a × -vertex. But by Corollary 3, thismeans that δ P ( L ) also contains this × -graph, which contradicts δ P ( L ) = 0.All of the above definitions and conclusions make sense when extended to P <
0. Eventhough
P <
P <
0. Since any vertex has a non-negative degree,Corollary 4 implies:
Corollary 5. If P < , any L ∈ L is an exact P -invariant. Associations between (cid:63) -graphs and × -graphs also have a simple and useful property: Lemma 1.
Suppose that a × -graph Γ × ∈ G × is associated with a (cid:63) -graph Γ (cid:63) ∈ G (cid:63) thatcontains a single (cid:63) -vertex. Then Γ × appears in ρ (Γ (cid:63) ) with coefficient 1. In general, a × -graph can be associated with more than one (cid:63) -graph. Figure 4 presents asimple example with ( P, N, ∆) = (2 , , P -invariant. In the next section we will develop techniquesto deal with this difficulty. B.2. Building Blocks for Consistency Equations
In this section, we introduce the building blocks with which we will construct the con-sistency equation (B.3), δ P ( L ) = ρ ( L (cid:63) ). Any polynomial shift-invariant can be gen-erated using these building blocks. We show that we can constrain L to contain onlyloopless plain-graphs, and all other invariants are equal to these ones up to total deriva-tives. Consequently, δ P ( L ), and thus ρ ( L ∗ ), contains only loopless × -graphs. Therefore, ρ ( L (cid:63) ) = ρ (0) ( L (cid:63) ), where ρ (0) acts in the same way as ρ but omits any looped graphs (Defi-nition 13). In fact, we can restrict L (cid:63) to be a linear combination L M of a particular type of– 40 – (cid:32) × (cid:70)(cid:70) (cid:33) = 2 × + × + × = ρ (cid:32) × (cid:70) + × (cid:70) (cid:33) Figure 4 : Two different linear combinations of (cid:63) -graphs result in an identical × -relationfor P = 2. In particular, the × -graph with a coefficient 2 is associated with all three (cid:63) -graphs in the figure. (cid:63) -graph, such that ρ (0) ( L (cid:63) ) = ρ (0) ( L M ). These (cid:63) -graphs will be called Medusas (Definition10). In Appendix B.2.5 we determine a lower bound on the degree of a vertex in any graphthat appears in the consistency equation. B.2.1. The Loopless Realization of L / R There are usually many alternative expressions for a single P -invariant algebraic term,which are equal to each other up to total derivatives. In the graphical language, thegraphs representing these equivalent expressions are related by plain-relations. Therefore,we are interested in the space of linear combinations of plain-graphs modding out plain-relations, i.e., the quotient space L / R . We need to find subset of graphs, B ⊂ G , whosespan is isomorphic to L / R . In other words, every element of L can be written as a linearcombination of graphs in B and plain-relations. Furthermore, this means that there are noplain-relations between elements in the set B . The following proposition shows that theset of loopless plain-graphs realizes the set B . Proposition 2 (Loopless Basis) . The span of loopless plain-graphs is isomorphic to L / R .Proof. Denote the span of all loopless plain-graphs by L loopless . If ∂ i ∂ i acts on a single φ ,then one can always integrate by parts to move one of the ∂ i ’s to act on the remaining φ ’s.In the graphical language, this means that any graph with loops can always be written asa linear combination of loopless graphs up to a plain-relation. This proves L / R ⊂ L loopless .Now, we show that there are no plain-relations between the loopless plain-graphs.Suppose there exists a linear combination of loopless plain-graphs L that is a plain-relation.That is, there exists a linear combination L (cid:48) of (cid:63) -ed plain-graphs such that L = ρ ( L (cid:48) ). LetΓ (cid:48) be a (cid:63) -ed plain-graph in L (cid:48) . Then, ρ (Γ (cid:48) ) is a linear combination of plain-graphs eachof which has a number of loops no greater than the number of (cid:63) -vertices in Γ (cid:48) . There willbe exactly one graph, Γ f.l. , which is fully-looped (with the number of loops equal to thenumber of (cid:63) -vertices in Γ (cid:48) ), produced when all the original (cid:63) -vertices (and the edges incidentto them) are replaced with loops. Furthermore, Γ f.l. uniquely determines Γ (cid:48) by replacingeach loop in Γ f.l. with an edge incident to an extra (cid:63) -vertex. Choose the (cid:63) -ed plain-graphappearing in L (cid:48) with the largest number of (cid:63) -vertices (the maximally (cid:63) -ed plain-graphs).This maximum exists since the number of the edges incident to the (cid:63) -vertices is boundedabove by the number of edges ∆. The fully-looped graphs formed from these maximally (cid:63) -ed plain-graphs cannot cancel each other (by uniqueness) and cannot be canceled by– 41 –ny other graphs that are not fully-looped (by maximality). Therefore ρ ( L (cid:48) ) is a linearcombination containing looped graphs, which contradicts the initial assumption that L only consists of loopless graphs. This proves L loopless ⊂ L / R .Therefore, L loopless ∼ = L / R .Henceforth, we can restrict our search for P -invariants to L loopless . Note that if L ∈L loopless , then all the graphs in δ ( L ) are also loopless, so it is sufficient to consider onlyloopless × -graphs. We can also restrict to loopless × -relations, R × loopless ⊂ R × , which isthe vector space consisting of × -relations that are linear combinations of loopless graphs.We summarize this discussion in the following corollary: Corollary 6.
The P -invariants that are independent up to total derivatives are representedby L ∈ L loopless with δ ( L ) ∈ R × loopless . Equivalently, they span the kernel of the map: q ◦ δ : L loopless → L × loopless / R × loopless , where q is the quotient map q : L × loopless → L × loopless / R × loopless . B.2.2. Medusas and Spiders
Corollary 6 motivates us to look for a basis of R × loopless , the × -relations that are linear com-binations of loopless graphs. To classify all such loopless × -relations, we are led to studythe linear combinations of (cid:63) -graphs that give rise to these relations under the derivativemap ρ . We will realize a convenient choice for the basis of R × loopless , which will tremen-dously simplify our calculations: It turns out that the basis of R × loopless is in one-to-onecorrespondence with a particular subset of loopless (cid:63) -graphs, which we now define. Definition 10 (Medusa) . A Medusa is a loopless (cid:63) -graph with all (cid:63) -vertices adjacent tothe × -vertex, such that the degree of the × -vertex deg( × ) and the number of (cid:63) -vertices N ( (cid:63) ) satisfy deg( × ) = P + 1 − N ( (cid:63) ) . We denote the set of Medusas by M N, ∆ . We should point out that applying ρ to a Medusa does not necessarily generate a × -relationin R × loopless ; it will sometimes produce × -graphs with loops. In order to form a loopless × -relation, these looped × -graphs must be canceled by contributions from other (cid:63) -graphs.In the proof of the one-to-one correspondence between the basis of R × loopless and thesubset of Medusas, we will frequently refer to the following definitions. Definition 11 (Primary (cid:63) -Graphs) . A primary (cid:63) -graph is a (cid:63) -graph that contains exactlyone (cid:63) -vertex. Definition 12 (Spider) . A spider is a primary (cid:63) -graph with the (cid:63) -vertex adjacent to the × -vertex and deg( × ) = P . Since deg( × ) ≥ N ( (cid:63) ) for a Medusa, we have ( P + 1) ≤ deg( × ) ≤ P . In particular, if P = 1 or 2, then deg( × ) = P and N ( (cid:63) ) = 1 (i.e., a Medusa is a loopless spider for P = 1 or2). Spiders will play an important role in sorting out all independent loopless × -relationsin Appendix B.2.4, and loopless spiders will lead us to the classification of 1-invariants inAppendix B.3. – 42 –n B.2.3 and B.2.4 we will construct R × loopless by spiders and Medusas. We will needto keep track of graphs containing specific numbers of loops, and the following refinementof the derivative map ρ (Definition 5) will allow us to formulate the graphical operationsalgebraically. Definition 13.
The map ρ ( (cid:96) ) : L (cid:63) → L × is defined such that, for L (cid:63) ∈ L (cid:63) , ρ ( (cid:96) ) ( L (cid:63) ) isequal to ρ ( L (cid:63) ) , with the coefficient of any graph that does not contain (cid:96) loops set to zero. Note that ρ = (cid:80) ∞ i =0 ρ ( i ) . If Γ (cid:63) ∈ G (cid:63) contains a loop, then ρ (0) (Γ (cid:63) ) is identically zero. InTheorem 1 we will show that the map ρ (0) defines the one-to-one correspondence between M and the preferred basis of R × loopless .It is also useful to introduce the operation of “undoing” loops. Definition 14.
Given a Γ ∈ G × ∪ G (cid:63) which contains (cid:96) loops, labeled from 1 to (cid:96) , the map θ i : G × ∪ G (cid:63) → G (cid:63) is defined such that θ i (Γ) is a (cid:63) -graph constructed by deleting the i th loop from Γ , adding an extra (cid:63) -vertex v (cid:63) and adding an edge joining v (cid:63) to the vertex atwhich the i th deleted loop ended. Define θ : G × ∪ G (cid:63) → G (cid:63) by θ (Γ) ≡ θ ◦ · · · ◦ θ (cid:96) (Γ) . Thismap extends to L × ∪ L (cid:63) → L (cid:63) by distributing θ over the formal sum. The map θ can besimilarly defined on (cid:63) -ed plain-graphs. We will need to distinguish different types of loops:
Definition 15.
A loop at the × -vertex is called a × -loop , and a loop at a • -vertex is a • -loop . A graph that contains (cid:96) loops is called (cid:96) -looped . We can now prove a key formula:
Proposition 3.
If an (cid:96) -looped (cid:63) -graph Γ (cid:63) contains a loop at vertex v A , then ρ ( (cid:96) − ◦ θ A (Γ (cid:63) ) = ρ ( (cid:96) − ◦ θ A ◦ ρ ( (cid:96) ) (Γ (cid:63) ) + ρ ( (cid:96) − (cid:16) L ( (cid:96) − (cid:17) , (B.4) where θ A undoes a loop at v A , and L ( (cid:96) − is a linear combination of ( (cid:96) − -looped spiders.Every graph in a nonzero L ( (cid:96) − has one fewer × -loop than Γ (cid:63) . L ( (cid:96) − is nonzero if andonly if the following three conditions are satisfied: ( a ) v A is the × -vertex; ( b ) There is a (cid:63) -vertex that is not adjacent to the × -vertex; ( c ) deg( × ) ≥ P +1 − N • ( (cid:63) ) , where N • ( (cid:63) ) is the number of (cid:63) -vertices in Γ (cid:63) that are adjacentto • -vertices.Proof. Denote the vertices in Γ (cid:63) adjacent to (cid:63) -vertices by v i , i = 1 , . . . , k , and the × -vertexby v × . Note that v × and v A may coincide with each other and with some of the v i ’s. Forman ( (cid:96) − (cid:63) by deleting all (cid:63) -vertices and deleting a loop at v A . Wenow show that all graphs in ρ ( (cid:96) − ◦ θ A (Γ (cid:63) ) and ρ ( (cid:96) − ◦ θ A ◦ ρ ( (cid:96) ) (Γ (cid:63) ) in (B.4) contain Γ asa subgraph: – 43 – ρ ( (cid:96) − ◦ θ A (Γ (cid:63) ): Define Γ v β ...v βk to be a graph formed from Γ by adding k edgesjoining v i and v β i , respectively. Then ρ ( (cid:96) − ◦ θ A (Γ (cid:63) ) = (cid:88) v α (cid:54) = v A (cid:88) v β (cid:54) = v . . . (cid:88) v βk (cid:54) = v k Γ v α v β ...v βk , (B.5)where Γ v α v β ...v βk is the graph Γ v β ...v βk with an extra edge joining v A and v α . Notethat the graphs formed from Γ by adding edges joining v A (or v i ) to itself are notincluded in this sum, because these graphs are not ( (cid:96) − • ρ ( (cid:96) − ◦ θ A ◦ ρ ( (cid:96) ) (Γ (cid:63) ): Define Γ A to be the graph Γ with a loop added at v A . Then ρ ( (cid:96) ) (Γ (cid:63) ) = (cid:80) v βi (cid:54) = v i (cid:101) Γ v β ...v βk , where (cid:101) Γ v β ...v βk is the graph formed from Γ A by adding k edges joining v i and v β i , respectively. v β i (cid:54) = v i in the sum because only (cid:96) -loopedgraphs are included. Applying ρ ( (cid:96) − ◦ θ A gives: ρ ( (cid:96) − ◦ θ A ◦ ρ ( (cid:96) ) (Γ (cid:63) ) = (cid:88) v α (cid:54) = v A (cid:88) v β (cid:54) = v . . . (cid:88) v βk (cid:54) = v k (cid:101) Γ v α v β ...v βk , (B.6)where (cid:101) Γ v α v β ...v βk is formed from (cid:101) Γ v β ...v βk by deleting a loop at v A and adding anedge joining v A and v α . v α (cid:54) = v A in the sum because only ( (cid:96) − v α v β ...v βk and (cid:101) Γ v α v β ...v βk for v α (cid:54) = v A and v β i (cid:54) = v i . At first itmight seem that Γ v α v β ...v βk = (cid:101) Γ v α v β ...v βk , since both ultimately involve taking Γ and adding anedge joining v A and v α , and edges joining v i and v β i . However, there is a subtlety involved:Recall that graphs containing a × -vertex of degree larger than P are identified with thenull graph. Therefore, Γ v α v β ...v βk = (cid:101) Γ v α v β ...v βk , provided neither side is null or both sides arenull; when one side of this equation represents the null graph and the other does not, thenthis equation will not hold. This violation happens only if there is a difference in deg( × )of the graphs formed during the construction of Γ v α v β ...v βk and (cid:101) Γ v α v β ...v βk . Note that we addthe same k edges (joining v i and v β i ) to both Γ and Γ A to form the intermediate graphs,Γ v β ...v βk and (cid:101) Γ v β ...v βk , respectively. Thus the difference between the latter two graphs isthe same as the difference between Γ and Γ A : There is an extra loop at v A in Γ A comparedto Γ, which will only be deleted after the edges have been added. Hence, the violation ofthe equality, Γ v α v β ...v βk = (cid:101) Γ v α v β ...v βk , happens only if v A = v × . This is condition (a).From now on, we assume v A = v × . In addition, there must exist at least one v β i = v × in order for the violation to occur, since otherwise deg( × ) in any graph we are consideringnever exceeds the one in Γ (cid:63) , and thus no graph in (B.5) and (B.6) is null. Hence for at leastone v β i , v × = v β i (cid:54) = v i , which implies condition (b). From now on we assume v × = v β i (cid:54) = v i ,for at least one v β i , and denote the number of v β i equal to v × by b , where 1 ≤ b ≤ N • ( (cid:63) ).We know that deg( × ) in (cid:101) Γ v β ...v βk is 2 higher than deg( × ) in Γ v β ...v βk , due to the oneextra loop at v A in Γ A .Therefore, if Γ v α v β ...v βk vanishes, then (cid:101) Γ v α v β ...v βk should also vanish, since (cid:101) Γ v β ...v βk al-ready has a higher deg( × ). Furthermore, deg( × ) in (cid:101) Γ v β ...v βk is 1 higher than deg( × ) in– 44 – v α v β ...v βk , and thus Γ v α v β ...v βk (cid:54) = (cid:101) Γ v α v β ...v βk if and only if deg( × ) = P in Γ v α v βa ...v βk . In this case,Γ v α v β ...v βk does not vanish, but (cid:101) Γ v β ...v βk contains a × -vertex of degree P + 1 and is identifiedwith the null graph, which means (cid:101) Γ v α v β ...v βk is also null. So the equality is violated if andonly if deg( × ) = P + 1 in (cid:101) Γ v β ...v βk . We want to write this condition in terms of deg( × ) inΓ (cid:63) . Note that deg( × ) = P + 1 in (cid:101) Γ v β ...v βk if and only if deg( × ) = P + 1 − b − ( k − N • ( (cid:63) ))in Γ A , and deg( × ) = P − − b − ( k − N • ( (cid:63) )) in Γ. Finally this implies deg( × ) = P + 1 − b in Γ (cid:63) . Since 1 ≤ b ≤ N • ( (cid:63) ), we have that P ≥ deg( × ) ≥ P + 1 − N • ( (cid:63) ), which is condition(c). Moreover, (cid:88) v α (cid:54) = v A Γ v α v β ...v βk = ρ (Γ spider ) , where Γ spider is an ( (cid:96) − v β ...v βk by adding a (cid:63) -vertex andthen adding an edge that joins this (cid:63) -vertex and the × -vertex. By construction, Γ spider hasone fewer × -loop than Γ (cid:63) . Such spiders form the desired L ( (cid:96) − in (B.4). B.2.3. Constructing Loopless × -Relations Constructing a basis for R × loopless requires a thorough examination of (cid:63) -graphs. The nextlemma shows that any × -relation can be written as a derivative map acting on a linear com-bination of primary (cid:63) -graphs, which allows us to restrict to primary (cid:63) -graphs in classifyingall loopless × -relations. Lemma 2.
For any L × ∈ R × , there exists L (cid:63) ∈ L (cid:63) that contains only primary (cid:63) -graphs,satisfying L × = ρ ( L (cid:63) ) .Proof. Since L × is a × -relation, there exists (cid:101) L (cid:63) = (cid:80) i b i Γ (cid:63)i ∈ L (cid:63) such that ρ ( (cid:101) L (cid:63) ) = L × .Starting with Γ (cid:63)i , one can follow steps 1 and 2 in Definition 5 to construct a series of (cid:63) -graphs, (Γ (cid:63)i ) j , (Γ (cid:63)i ) j j , . . . (Γ (cid:63)i ) j ...j k − , with j α = 0 , . . . , n − α = 1 , . . . , k . Byconstruction, (Γ (cid:63)i ) j ...j k − contains exactly one (cid:63) -vertex (which makes it a primary (cid:63) -graph),and ρ (Γ (cid:63)i ) = ρ (cid:16)(cid:80) n − j ,...,j k − =0 (Γ (cid:63)i ) j ...j k − (cid:17) . Therefore, L × = (cid:88) i b i · ρ (Γ (cid:63)i ) = ρ (cid:88) i N − (cid:88) j ,...,j k − =0 b i (Γ (cid:63)i ) j ...j k − . This linear combination of primary (cid:63) -graphs (Γ (cid:63)i ) j ...j k − defines the desired L (cid:63) . Opera-tionally, such L (cid:63) is constructed from (cid:101) L (cid:63) by removing the (cid:63) -vertices one by one, as per thesteps in the definition of ρ , until only one (cid:63) -vertex remains.The following proposition presents a general construction for loopless × -relations. In thenext section we will prove that this procedure generates all elements in R × loopless . Proposition 4.
Let L (cid:63) ∈ L (cid:63) be a linear combination of (cid:96) -looped primary (cid:63) -graphs suchthat all × -graphs in ρ ( L (cid:63) ) are (cid:96) -looped. There exists an L (cid:63)(cid:96) ∈ L (cid:63) , such that ( a ) L (cid:63)(cid:96) − L (cid:63) contains no graph with more than (cid:96) − loops; – 45 – b ) ρ ( L (cid:63)(cid:96) ) = ( − (cid:96) ρ (0) ◦ θ ( L (cid:63) +L spider ) ∈ R × loopless . L spider is a linear combination of spiders.Proof. Since each graph in ρ ( L (cid:63) ) is (cid:96) -looped, ρ ( L (cid:63) ) = ρ ( (cid:96) ) ( L (cid:63) ). Let ρ ( (cid:96) ) ( L (cid:63) ) ≡ (cid:80) ki =1 b i Γ × i ,where each Γ × i contains (cid:96) loops. For each Γ × i , label the loops from 1 to (cid:96) . By Definition14, θ undoes the 1 st loop, and θ (Γ × i ) defines a primary (cid:63) -graph with (cid:96) − × i . By Lemma 1, Γ × i drops out of ρ ( L (cid:63) − b i · θ (Γ × i )). Moreover, all × -graphs in ρ ◦ θ (Γ × i ), except for Γ × i , are ( (cid:96) − L (cid:63) ≡ L (cid:63) − k (cid:88) i =1 b i · θ (Γ × i ) = L (cid:63) − θ ◦ ρ ( (cid:96) ) ( L (cid:63) )defines an L (cid:63) that satisfies ρ ( L (cid:63) ) = ρ ( (cid:96) − ( L (cid:63) ). Repeat this procedure for L (cid:63) and the 2 nd loop, in place of L (cid:63) and the 1 st loop, obtaining L (cid:63) ≡ L (cid:63) − θ ◦ ρ ( (cid:96) − ( L (cid:63) ) = L (cid:63) − θ ◦ ρ ( (cid:96) ) ( L (cid:63) ) + θ ◦ ρ ( (cid:96) − ◦ θ ◦ ρ ( (cid:96) ) ( L (cid:63) ) . The second equality holds because ρ ( (cid:96) − ( L (cid:63) ) = 0. Furthermore, ρ ( L (cid:63) ) = ρ ( (cid:96) − ( L (cid:63) ).Iterating this (cid:96) times, we will reach a linear combination of primary (cid:63) -graphs L (cid:63)(cid:96) ≡ L (cid:63) + (cid:96) (cid:88) i =1 ( − i X (cid:63)i , X (cid:63)i ≡ θ i ◦ ρ ( (cid:96) − i +1) ◦ · · · ◦ θ ◦ ρ ( (cid:96) − ◦ θ ◦ ρ ( (cid:96) ) ( L (cid:63) ) . (B.7)Here ρ ( L (cid:63)(cid:96) ) = ρ (0) ( L (cid:63)(cid:96) ) ∈ R × loopless . Moreover, X (cid:63)i only contains ( (cid:96) − i )-looped graphs. Thismeans that graphs in L (cid:63)(cid:96) − L (cid:63) = (cid:80) (cid:96)i =1 ( − i X (cid:63)i contain at most (cid:96) − L (cid:63)(cid:96) satisfies condition (a) of the proposition.To prove that L (cid:63)(cid:96) also satisfies condition (b), take L ( i )spider to stand for “any linearcombination of i -looped spiders” and, for 0 < k ≤ (cid:96) , define Z (cid:63)(cid:96) − k ≡ θ k ◦ . . . ◦ θ ( L (cid:63) ) − k (cid:88) α =2 θ k ◦ . . . ◦ θ α (cid:16) L ( (cid:96) − α +1)spider (cid:17) − L ( (cid:96) − k )spider which contains only ( (cid:96) − k )-looped graphs. Define Z (cid:63)(cid:96) ≡ L (cid:63) . Therefore, for 0 ≤ k ≤ (cid:96) ,applying Proposition 3, ρ ( (cid:96) − k − ◦ θ k +1 ◦ ρ ( (cid:96) − k ) (cid:0) Z (cid:63)(cid:96) − k (cid:1) = ρ ( (cid:96) − k − (cid:104) θ k +1 (cid:0) Z (cid:63)(cid:96) − k (cid:1) − L ( (cid:96) − k − (cid:105) = ρ ( (cid:96) − k − (cid:34) θ k +1 ◦ · · · ◦ θ ( L (cid:63) ) − k +1 (cid:88) α =2 θ k +1 ◦ . . . ◦ θ α (cid:16) L ( (cid:96) − α +1)spider (cid:17) − L ( (cid:96) − k − (cid:35) , i.e., ρ ( (cid:96) − k − ◦ θ k +1 ◦ ρ ( (cid:96) − k ) (cid:0) Z (cid:63)(cid:96) − k (cid:1) = ρ ( (cid:96) − k − (cid:0) Z (cid:63)(cid:96) − k − (cid:1) . (B.8)– 46 –ote that ρ (0) ( L (cid:63) ) = 0 and ρ (0) ( X (cid:63)i ) = 0 for i = 1 , . . . , (cid:96) −
1. Then, by (B.7) and (B.8), ρ (0) ( L (cid:63)(cid:96) ) = ( − (cid:96) ρ (0) ( X (cid:63)(cid:96) )= ( − (cid:96) ρ (0) ◦ θ (cid:96) ◦ ρ (1) ◦ · · · ◦ θ ◦ ρ ( (cid:96) − ◦ θ ◦ ρ ( (cid:96) ) ( Z (cid:63)(cid:96) )= ( − (cid:96) ρ (0) ◦ θ (cid:96) ◦ ρ (1) ◦ · · · ◦ θ ◦ ρ ( (cid:96) − ( Z (cid:63)(cid:96) − ) = . . . = ( − (cid:96) ρ (0) ( Z (cid:63) )= ( − (cid:96) ρ (0) (cid:32) θ (cid:96) ◦ . . . ◦ θ ( L (cid:63) ) − (cid:96) (cid:88) α =2 θ (cid:96) ◦ . . . ◦ θ α (cid:16) L ( (cid:96) − α +1)spider (cid:17) − L (0)spider (cid:33) = ( − (cid:96) ρ (0) ◦ θ (cid:32) L (cid:63) − (cid:96) +1 (cid:88) α =2 L ( (cid:96) − α +1)spider (cid:33) . Hence, ρ ( L (cid:63)(cid:96) ) = ρ (0) ( L (cid:63)(cid:96) ) = ( − (cid:96) ρ (0) ◦ θ ( L (cid:63) + L spider ) ∈ R × loopless . This gives condition (b), with L spider = − (cid:80) (cid:96) +1 α =2 L ( (cid:96) − α +1)spider . Corollary 7.
Given L s , a linear combination of spiders, ρ (0) ◦ θ ( L s ) ∈ R × loopless .Proof. It is enough to show that this corollary is true for one spider S . We claim thatProposition 4 holds for L (cid:63) = S and where L spider is null if we order the loops such thatloops 1 to (cid:96) × are × -loops and (cid:96) × + 1 to (cid:96) are • -loops and the loops are removed in thisorder. Define Z (cid:63)(cid:96) − k ≡ θ k ◦ . . . ◦ θ ( S ) and Z (cid:63)(cid:96) ≡ S . As in the proof of Proposition 4, weare done if we can prove (B.8), but with this new definition of Z (cid:63)(cid:96) − k (i.e., when L spider isalways taken to be zero).For k = 0, Z (cid:63)(cid:96) = S , a spider, which has no (cid:63) -vertex adjacent to a • -vertex in violationof Proposition 3( b ). Thus, ρ ( (cid:96) − ◦ θ ◦ ρ ( (cid:96) ) ( Z (cid:63)(cid:96) ) = ρ ( (cid:96) − ◦ θ ( Z (cid:63)(cid:96) ) = ρ ( (cid:96) − ( Z (cid:63)(cid:96) − ). This holdsregardless of which loop is chosen to be undone first. However, if the first loop is a × -loop,then Z (cid:63)(cid:96) − = θ ( S ) will continue to violate Proposition 3( b ). Therefore, if all of the × -loopsare undone first, then ρ ( (cid:96) − k − ◦ θ k +1 ◦ ρ (cid:96) − k ( Z (cid:63)(cid:96) − k ) = ρ ( (cid:96) − k − ( Z (cid:63)(cid:96) − k − ) holds for 0 ≤ k ≤ (cid:96) × .Now, there are no longer any × -loops. Whenever there are • -loops, Z (cid:63)(cid:96) − (cid:96) × will violateProposition 3( a ). Thus, ρ ( (cid:96) − k − ◦ θ k +1 ◦ ρ (cid:96) − k ( Z (cid:63)(cid:96) − k ) = ρ ( (cid:96) − k − ( Z (cid:63)(cid:96) − k − ) continues to holdall the way until k = (cid:96) . B.2.4. A Basis for R × loopless To find a basis for R × loopless , we first show that any loopless × -relation can be written as ρ (0) ◦ θ acting on a linear combination of spiders. We start with the following lemmas: Lemma 3.
Let L × ∈ R × loopless satisfy L × = ρ ( L (cid:63) ) , with L (cid:63) a linear combination of primary (cid:63) -graphs. For any Γ (cid:63)A in L (cid:63) that is associated with a looped × -graph Γ × , there exists another (cid:63) -graph Γ (cid:63)B (cid:54) = Γ (cid:63)A in L (cid:63) , such that Γ × is not contained in ρ (Γ (cid:63)A − Γ (cid:63)B ) .Proof. Suppose Γ (cid:63)A is the only (cid:63) -graph in L (cid:63) that is associated with Γ (cid:63)A . Assume that thecoefficient of Γ (cid:63)A in L (cid:63) is b A (cid:54) = 0. Therefore, none of the (cid:63) -graphs in L (cid:63) − b A Γ (cid:63)A is associatedwith Γ × , and thus Γ × is not contained in ρ ( L (cid:63) − b A Γ (cid:63)A ). Hence, the looped × -graph Γ × – 47 –ppears in ρ ( L (cid:63) ) = ρ ( L (cid:63) − b A Γ (cid:63)A ) + b A · ρ (Γ (cid:63)A ) with coefficient b A (cid:54) = 0, which contradictsthe fact that ρ ( L (cid:63) ) ∈ R × loopless .The above argument shows that there exists a Γ (cid:63)B (cid:54) = Γ (cid:63)A in L (cid:63) that is associated withΓ × . By Lemma 1, Γ × appears in both ρ (Γ (cid:63)A ) and ρ (Γ (cid:63)B ) with coefficient 1. Then, ρ (Γ (cid:63)A − Γ (cid:63)B )does not contain Γ × .Specifically, if Γ (cid:63)A is (cid:96) -looped and Γ × is ( (cid:96) + 1)-looped, then Γ (cid:63)B is also (cid:96) -looped. Lemma 4. If Γ (cid:63)A , Γ (cid:63)B ∈ G (cid:63) are (cid:96) -looped primary (cid:63) -graphs that are associated with the same ( (cid:96) + 1) -looped × -graph, then θ (Γ (cid:63)A − Γ (cid:63)B ) = 0 .Proof. Since Γ (cid:63)A and Γ (cid:63)B are both associated with the same ( (cid:96) + 1)-looped × -graph, Γ × , θ (Γ (cid:63)A ) = θ (Γ (cid:63)B ) = θ (Γ × ). Therefore, θ (Γ (cid:63)A − Γ (cid:63)B ) = 0. Proposition 5.
For any loopless × -relation L × ∈ R × loopless , there exists a linear combina-tion of spiders L s , such that L × = ρ (0) ◦ θ ( L s ) .Proof. In the following, we take L spider to stand for “any linear combination of spiders”.Since L × is a × -relation, there exists L (cid:63) ∈ L (cid:63) , consisting of primary (cid:63) -graphs, such that L × = ρ ( L (cid:63) ). Take the set, H (cid:96) = { Γ (cid:63) , . . . , Γ (cid:63)H (cid:96) } , of (cid:63) -graphs in L (cid:63) that contain the highestnumber, (cid:96) , of loops. Let b i(cid:96) be the coefficient of Γ (cid:63)i in L (cid:63) . Therefore, L (cid:63) − (cid:80) H (cid:96) i =1 b i(cid:96) Γ (cid:63)i contains no graphs with more than (cid:96) − H (cid:96) in order from Γ (cid:63) to Γ (cid:63)H (cid:96) :1. Define β (1) (cid:96) ≡ b (1) (cid:96) . Apply to Γ (cid:63) the construction outlined in Proposition 4:(a) If Γ (cid:63) is a spider: By Corollary 7, Proposition 4(b) becomes that there exists alinear combination of primary (cid:63) -graphs L (1) (cid:96) , such that ρ (cid:0) L (1) (cid:96) (cid:1) = ρ (0) ◦ θ (Γ (cid:63) ) = ρ (0) ◦ θ (L spider ) ∈ R × loopless . By Proposition 4(a), graphs in L (1) (cid:96) − Γ (cid:63) contain at most (cid:96) − (cid:63) is not a spider: ρ (Γ (cid:63) ) contains an ( (cid:96) + 1)-looped × -graph Γ × . By Lemma3, there exists an (cid:96) -looped (cid:63) -graph Γ (cid:63)j ∈ H , Γ (cid:63)j (cid:54) = Γ (cid:63) , that is associated with Γ × and Γ × is not in ρ (Γ (cid:63) − Γ (cid:63) ). By Lemma 4, θ (Γ (cid:63) − Γ (cid:63)j ) = 0. By Proposition 4, ρ (cid:0) L (1) (cid:96) (cid:1) = ρ (0) ◦ θ (L spider ) ∈ R × loopless , where graphs in L (1) (cid:96) − (Γ (cid:63) − Γ (cid:63)j ) contain at most (cid:96) − β ( i ) (cid:96) , for i >
1, as the coefficient (which may be zero) of Γ (cid:63)i in L (cid:63) − β (1) (cid:96) L (1) (cid:96) .3. If β (2) (cid:96) = 0, skip this step; if not, repeat step 1 for Γ (cid:63) , resulting in an L (2) (cid:96) with ρ (cid:0) L (2) (cid:96) (cid:1) = ρ (0) ◦ θ (L spider ) ∈ R × loopless . Redefine β ( i ) (cid:96) , for i >
2, to be the coefficient of Γ (cid:63)i in (cid:0) L (cid:63) − β (1) (cid:96) L (1) (cid:96) (cid:1) − β (2) (cid:96) L (2) (cid:96) .– 48 –. Repeat step 3 for Γ (3) (cid:96) , . . . , Γ ( H (cid:96) ) (cid:96) in sequence. This will eventually generate a linearcombination of (cid:63) -graphs, L (cid:63) − (cid:80) H (cid:96) i =1 β ( i ) (cid:96) L ( i ) (cid:96) , where ρ (cid:32) H (cid:96) (cid:88) i =1 β ( i ) (cid:96) L ( i ) (cid:96) (cid:33) = ρ (0) ◦ θ (L spider ) ∈ R × loopless , and all graphs in L (cid:63) − (cid:80) H (cid:96) i =1 β ( i ) (cid:96) L ( i ) (cid:96) contain at most (cid:96) − L (cid:63) − (cid:80) H (cid:96) i =1 β ( i ) (cid:96) L ( i ) (cid:96) . We will obtain L (cid:63) − H (cid:96) (cid:88) i =1 β ( i ) (cid:96) L ( i ) (cid:96) − H (cid:96) − (cid:88) i =1 β ( i ) (cid:96) − L ( i ) (cid:96) − , ρ H (cid:96) − (cid:88) i =1 β i(cid:96) L ( i ) (cid:96) − = ρ (0) ◦ θ (L spider ) ∈ R × loopless . In the first expression graphs contain at most (cid:96) − (cid:96) times to get (cid:101) L (cid:63) ≡ L (cid:63) − (cid:96) (cid:88) α =1 H α (cid:88) i =1 β iα L ( i ) α , which contains only loopless primary (cid:63) -graphs. In addition, ρ (cid:32) (cid:96) (cid:88) α =1 H α (cid:88) i =1 β iα L ( i ) α (cid:33) = ρ (0) ◦ θ (L spider ) ∈ R × loopless . Therefore, ρ ( (cid:101) L (cid:63) ) = L × − ρ (cid:32) (cid:96) (cid:88) α =1 H α (cid:88) i =1 β iα L ( i ) α (cid:33) ∈ R × loopless . Next we show that (cid:101) L (cid:63) is a linear combination of spiders. Suppose there exists a (cid:63) -graph Γ (cid:63)A in (cid:101) L (cid:63) that is not a spider. Then, by Lemma 3, there should exist another (cid:63) -graphΓ (cid:63)B (cid:54) = Γ (cid:63)A in (cid:101) L (cid:63) that is associated with the 1-looped × -graph Γ × in ρ (Γ (cid:63)A ). But since Γ × is associated with a unique loopless (cid:63) -graph (resulting from undoing the loop), this isimpossible. Therefore, graphs in (cid:101) L (cid:63) are loopless spiders, and thus ρ ( (cid:101) L (cid:63) ) = ρ (0) ◦ θ ( (cid:101) L (cid:63) ).Hence, L × = ρ (cid:32)(cid:101) L (cid:63) + (cid:96) (cid:88) α =1 H α (cid:88) i =1 β iα L ( i ) α (cid:33) = ρ (0) ◦ θ (L spider ) . The L spider within the last pair of parentheses is the desired L s .Thus we have shown that any loopless × -relation can be written as ρ (0) ◦ θ ( L s ). In fact,we can go further and show that it is equal to ρ (0) ( L M ), where L M is a linear combinationof Medusas. We start with the following Lemma: Lemma 5.
Given a spider S that contains (cid:96) × × -loops, there exist two linear combinationsof spiders Y (cid:63) , with graphs containing (cid:96) × × -loops but no • -loop, and W (cid:63) , with graphscontaining fewer than (cid:96) × × -loops, such that ρ (0) ◦ θ ( Y (cid:63) ) = ρ (0) ◦ θ (cid:0) S + W (cid:63) (cid:1) . – 49 – roof. Suppose S contains (cid:96) • • -loops, labeled from 1 to (cid:96) • . Denote the total number ofloops in S to be (cid:96) = (cid:96) × + (cid:96) • . Label the × -loops in S from (cid:96) • + 1 to (cid:96) . We will follow theproof of Proposition 4 to undo these • -loops. However, we want to keep the (cid:63) -vertex in S untouched. Therefore, define ρ s to be the usual derivative map except that it keeps theoriginal (cid:63) -vertex in S untouched; we can grade ρ s by number of loops in analogy with ρ .Apply θ to S to undo the 1 st • -loop. Define Y (cid:63) ≡ S − ρ s ◦ θ ( S ) and note that S drops out of Y (cid:63) . Furthermore, since S is the only (cid:96) -looped graph in ρ s ◦ θ ( S ), Y (cid:63) = S − ρ s ◦ θ ( S ) = − ρ ( (cid:96) − s ◦ θ ( S ) . Repeat this procedure for all graphs in Y (cid:63) and the 2 nd • -loop, in place of S and the 1 st • -loop, obtaining Y (cid:63) ≡ Y (cid:63) − ρ s ◦ θ ( Y (cid:63) ), which is a linear combination of ( (cid:96) − Y (cid:63) are ( (cid:96) − Y (cid:63) = ( − ρ ( (cid:96) − s ◦ θ ◦ ρ ( (cid:96) − s ◦ θ ( S ) . Iterate this (cid:96) • times, resulting in Y (cid:63)(cid:96) • = ( − (cid:96) • ρ ( (cid:96) − (cid:96) • ) s ◦ θ (cid:96) • ◦ · · · ρ ( (cid:96) − s ◦ θ ◦ ρ ( (cid:96) − s ◦ θ ( S ) . Graphs in Y (cid:63)(cid:96) • contain no • -loops. As in the derivation of (B.2.3) in Proposition 4, Y (cid:63)(cid:96) • =( − (cid:96) • ρ ( (cid:96) − (cid:96) • ) s (cid:32) θ (cid:96) • ◦ · · · ◦ θ ( S ) − (cid:96) • (cid:88) α =2 θ (cid:96) • ◦ · · · ◦ θ α (cid:16) L ( (cid:96) − α +1)spider (cid:17) − L ( (cid:96) − (cid:96) • )spider (cid:33) (B.9)By Proposition 3, each graph in L ( (cid:96) − α +1)spider and L ( (cid:96) − (cid:96) • )spider contains fewer than (cid:96) × × -loops.Take Y (cid:63) = ( − (cid:96) • Y (cid:63)(cid:96) • . Then, by Proposition 3, ρ (0) ◦ θ ( Y (cid:63) ) = ρ (0) ◦ θ (cid:96) ◦ · · · ◦ θ (cid:96) • +1 ( Y (cid:63) ) = ρ (0) ◦ θ ( S + W (cid:63) ) , where W (cid:63) is a linear combination of spiders containing fewer than (cid:96) × × -loops.The next proposition finally allows us to relate spiders and Medusas. Proposition 6.
For any linear combination of spiders L s ∈ L (cid:63) , there exists a linearcombination of Medusas L M , such that ρ (0) ◦ θ ( L s ) = ρ (0) ( L M ) .Proof. Take the set, H (cid:96) × = (cid:110) S ( (cid:96) × )1 , . . . , S ( (cid:96) × ) H (cid:96) × (cid:111) , of spiders in L s that contain the highest number, (cid:96) × , of × -loops. Denote the coefficientof S ( (cid:96) × ) i in L s as b ( (cid:96) × ) i . Therefore, graphs in L s − (cid:80) H (cid:96) × i =1 b ( (cid:96) × ) i S ( (cid:96) × ) i contain at most (cid:96) × − × -loops. Applying Lemma 5 to each S ( (cid:96) × ) i yields two linear combinations of spiders Y ( (cid:96) × ) ,comprised of graphs containing (cid:96) × × -loops but no • -loop, and W ( (cid:96) × ) , comprised of graphscontaining fewer than (cid:96) × × -loops, such that H (cid:96) × (cid:88) i =1 ρ (0) ◦ θ (cid:16) b ( (cid:96) × ) i S ( (cid:96) × ) i (cid:17) = ρ (0) ◦ θ (cid:16) Y ( (cid:96) × ) − W ( (cid:96) × ) (cid:17) . (B.10)– 50 –urthermore, L s − H (cid:96) × (cid:88) i =1 b ( (cid:96) × ) i S ( (cid:96) × ) i + W ( (cid:96) × ) (B.11)is a linear combination of spiders that contain at most (cid:96) × − × -loops. Replace L s with(B.11), and the above procedure applies, resulting in a linear combination of spiders thatcontain at most (cid:96) × − × -loops. Iterating (cid:96) × times, we will obtain L (0) s ≡ L s − (cid:96) × (cid:88) α =1 (cid:32) H α (cid:88) i =1 b ( α ) i S ( α ) i + W ( α ) (cid:33) , (B.12)By construction, L (0) s is a linear combination of spiders containing no × -loop. Moreover, ρ (0) ◦ θ (cid:16) Y ( α ) (cid:17) = ρ (0) ◦ θ (cid:32) H α (cid:88) i =1 b ( α ) i S ( α ) i + W ( α ) (cid:33) , (B.13)in analogy with (B.10). Finally, repeat the same procedure one last time with L (0) s in(B.12) in place of L s . Since there are no longer any × -loops in L (0) s , no W ( α ) ’s will arise.We obtain L (0) s = H (cid:88) i =1 b (0) i S (0) i , ρ (0) ◦ θ (cid:16) Y (0) (cid:17) = ρ (0) ◦ θ (cid:32) H (cid:88) i =1 b (0) i S (0) i (cid:33) . (B.14)Combining (B.12), (B.13) and (B.14), we obtain ρ (0) ◦ θ ( L s ) = ρ (0) ◦ θ (cid:96) × (cid:88) α =0 Y ( α ) . Since the Y ( α ) ’s are linear combinations of spiders with no • -loops and α × -loops, θ ( Y ( α ) ),is a linear combination of loopless (cid:63) -graphs with deg( × ) = P − α and N ( (cid:63) ) = α + 1. Thismeans deg( × ) + N ( (cid:63) ) = P + 1 in these loopless (cid:63) -graphs. By Definition 10, such graphsare Medusas. Therefore, L M = θ (cid:96) × (cid:88) α =0 Y ( α ) gives the desired linear combination of Medusas in ρ (0) ◦ θ ( L s ) = ρ (0) ( L M ). Theorem 1. ρ (0) ( M ) forms a basis of R × loopless .Proof. Proposition 5 states that any L × ∈ R × loopless can be written as ρ (0) ◦ θ ( L s ), with L s a linear combination of spiders. Proposition 6 states that there exists a linear combinationof Medusas L M , such that ρ (0) ◦ θ ( L s ) = ρ (0) ( L M ). Therefore, L × = ρ (0) ( L M ). This provesthe completeness.Suppose that there exists a linear combination of Medusas L M = (cid:80) ki =1 α i M i (cid:54) = 0,such that ρ (0) ( L M ) = 0. Note that if two Medusas M A and M B have × -vertices of different– 51 –egrees, then ρ (0) ( M A ) and ρ (0) ( M B ) do not have × -graphs in common. Therefore, withoutloss of generality, we can assume that the × -vertices in all M i have the same degree. Bythe definition of Medusas, P = deg( × ) + N ( (cid:63) ) −
1, and this implies that they also have thesame number of (cid:63) -vertices. Furthermore, we can assume that, after deleting the × -vertex,all M i ’s are identical since this must be the case in order for the ρ (0) ( M i ) to cancel eachother. Therefore, the only differences between the M i are in the edges incident to the × -vertex.For any • -vertex, v , take the set H = { (cid:102) M , . . . , (cid:102) M H } of distinct Medusas in L M , suchthat each (cid:102) M i contains the highest number, E , of edges that join v and the × -vertex amongall Medusas in L M . For each (cid:102) M i , form a specific graph, Γ × i , in ρ (0) ( (cid:102) M i ) by deleting all N ( (cid:63) ) (cid:63) -vertices and adding N ( (cid:63) ) edges joining the × -vertex and v . Now, Γ × i contains E + N ( (cid:63) )edges joining v and the × -vertex. By construction, the Γ × i contain the highest number ofedges joining v and the × -vertex, among all × -graphs in ρ (0) ( L M ). Therefore, the Γ × i canonly be canceled among ρ (0) ( (cid:102) M i )’s. Thus in order for Γ × i to be canceled, we need Γ × i = Γ × j for some i (cid:54) = j (note that this Γ × i appears in ρ (0) ( (cid:102) M i ) with unit coefficient). But this means (cid:102) M i = (cid:102) M j which is a contradiction since the (cid:102) M i are distinct. This proves independence.We can use this result to systematically find any P -invariant. When P = 0 or P = 1, asimple classification is possible for any N . The case P = 1 is studied in detail in AppendixB.3 and P = 0 is dealt with in the following corollary. Corollary 8.
Any loopless -invariant is an exact -invariant.Proof. There are no Medusas in P = 0 (since Medusas require N ( (cid:63) ) ≥ N ( (cid:63) ) ≤ deg( × ) = P + 1 − N ( (cid:63) ) ≤ P ). Therefore R × loopless = { } .Since we have already identified all exact invariants in Corollary 4, this classifies all loopless0-invariants. B.2.5. Lower Bound on Vertex Degree
Within the loopless basis, we will set a lower bound on the degree of any • - or × -vertex inany graph appearing in a consistency equation for a P -invariant. Proposition 7.
The vertices of a plain-graph Γ in a P -invariant L ∈ L loopless are of degreeno less than ( P + 1) .Proof. Let v be a vertex in Γ of degree less than ( P + 1). Let Γ × in δ (Γ) be the termgiven by replacing v with a × -vertex. By Corollary 3, Γ × is in δ ( L ). Since L is P -invariant,by Theorem 1, δ ( L ) = ρ (0) ( (cid:80) i α i M i ) for M i ∈ M and thus, Γ × is in ρ (0) ( M j ) for some j . Since M j is a Medusa, the × -vertex in M j is of degree no less than ( P + 1). Thiscontradicts the fact that the × -vertex in Γ × is of degree less than ( P + 1). Proposition 8. If L ∈ L loopless is a P -invariant and L M is a linear combination ofMedusas, such that δL = ρ (0) ( L M ) , then the vertices of any Medusa in L M are of de-gree no less than ( P + 1) . – 52 – roof. Suppose that a Medusa M in L M has a vertex, v , of degree lower than ( P +1). Foran interaction term, there exists at least one other vertex v in M to which the (cid:63) -verticescan be joined such that v is neither v nor the × -vertex. The resulting graph in ρ (0) ( M )is a × -graph Γ × with vertex v of degree lower than ( P + 1), and thus, by Proposition 7,does not appear in δ ( L ). Therefore, there must exist another Medusa M in L M such that M (cid:54) = M and Γ × is absent from ρ (0) ( M − M ). Therefore, M must produce Γ × afterdeleting the (cid:63) -vertices and adding the same number, N ( (cid:63) ), of edges joining the × -vertexand the other vertices. Since M (cid:54) = M at least one of these other vertices is not v , so thedegree of v is larger in M than in M . Now, form another × -graph, Γ × , in ρ (0) ( M ) bydeleting the (cid:63) -vertices in M and adding the same number of edges joining the × -vertexand v . Γ × again contains v with degree lower than ( P + 1). This × -graph must becanceled by introducing a third Medusa M . This procedure can be iterated to obtain aninfinite sequence of Medusas M , M , M , . . . in L M , with the number of edges incident to v in M i monotonically increasing with i . But this is impossible since the Medusas have afixed finite number of edges and thus we have a contradiction. B.3. Linear Shift Symmetry, Trees and Galileons
For P = 1, Medusas have a very limited configuration: A Medusa M has two subgraphs,which are disconnected from each other, one of which is (cid:70) × , and the other of which isa loopless plain-graph. This strongly restricts the possible associations between graphs. Inaddition, for P = 1, ρ ( M ) = ρ (0) ( M ) for any Medusa M . Proposition 9.
For P = 1 , a loopless × -graph is associated with at most one Medusa.Proof. If the × -vertex in the loopless × -graph has degree zero, then it cannot be associatedwith a Medusa. If the × -vertex in the loopless × -graph has degree one, then the loopless × -graph takes the form of a × -vertex and an edge joining this × -vertex to a vertex ina loopless plain-graph, Γ. The unique Medusa associated with this × -graph is given bydeleting the edge incident to the × -vertex and adding a (cid:63) -vertex together with an edgejoining the (cid:63) -vertex and the × -vertex.Following the same logic as in the proofs of Corollary 2 and 3, we obtain: Corollary 9.
For P = 1 , any two associated Medusas are identical to each other. Corollary 10.
For P = 1 , if M is a Medusa in a sum of Medusas L M , then ρ ( L M ) contains all graphs in ρ ( M ) . Corollary 11.
For P = 1 , if a Medusa M is associated with a plain-graph in a 1-invariant, L ∈ L loopless , then δ ( L ) contains all graphs in ρ ( M ) .Proof. If M is associated with Γ in the 1-invariant L , then there exists Γ × shared by δ (Γ)and ρ ( M ). Corollary 3 implies Γ × is in δ ( L ). Theorem 1 implies δ ( L ) = ρ ( L M ) where L M is a sum of Medusas. Therefore, Γ × is in ρ ( L M ). Proposition 9 implies M is in L M .Corollary 10 implies ρ ( L M ) contains ρ ( M ) and thus δ ( L ) contains ρ ( M ).– 53 – .3.1. Minimal InvariantsDefinition 16. A nonzero P -invariant L N, ∆ is minimal if there is no nonzero P -invariant L (cid:48) N, ∆ (cid:48) for any ∆ (cid:48) < ∆ . For a given P and N , let ∆ min denote the minimum ∆ for whicha P -invariant exists. Now, we prove that a minimal 1-invariant is a sum of trees. We start with the followinglemma.
Lemma 6 (Leaf Shuffling) . If a graph Γ A that contains a leaf v appears in a 1-invariant L ,then any graph Γ B that contains v as a leaf and satisfies Γ B − v = Γ A − v is also containedin L .Proof. Depicted below is a series of graphs, which are all associated with each other: v Γ A ⇒ × v Γ × A ⇒ (cid:70) × vM ⇒ × v Γ × B ⇒ v Γ B where the circles denote subgraphs. In particular, both Γ A and Γ B are associated with theMedusa M . By Corollary 11, δL contains all graphs in ρ ( M ). Furthermore, by Corollary1, L contains both Γ A and Γ B .We call the procedure that relates Γ B to Γ A described in the above lemma leaf shuffling .The corollaries below follow immediately. Corollary 12.
If a plain-graph Γ A is contained in a 1-invariant L , and Γ B is formed from Γ A by shuffling leaves, then Γ B is also contained in L . Corollary 13.
If a plain-graph Γ in a loopless 1-invariant L contains more than oneconnected component, then none of these connected components is a tree.Proof. Note that there are at least two leaves in a tree, if the tree is not an empty vertex.If in Γ there is a connected component T that is a tree, then we can shuffle all leaves in T to be joined to other connected components, while turning T into another tree withat least one fewer vertex. This procedure can be iterated until all but one vertex in T are shuffled to be joined to other connected components, which turns Γ into a graph (cid:101) Γcontaining an empty vertex. By Corollary 12, since L contains Γ, it also contains (cid:101) Γ, whichviolates the lower bound on vertex degree. Therefore, such Γ cannot appear in loopless1-invariants.
Proposition 10. If L is a nonzero minimal N -point loopless 1-invariant, then it containsall trees with N vertices.Proof. If L contains no plain-graphs with leaves, then the vertices in L have degree at least2 (note that an empty vertex is disallowed by Proposition 7). Then the number of edges∆ satisfies ∆ ≥ N. (B.15)– 54 –therwise, consider a plain-graph Γ contains a leaf v and appears in L . If there existany other leaves in Γ, shuffle them to be adjacent to v . Iterate for the resulting graphsuntil reaching a plain-graph Γ in which all leaves are adjacent to v . By Corollary 12, Γ is also contained in L . Denote the subgraph of Γ consisting of all leaves in Γ and v by T , which is a particular type of tree usually called a star . Define Γ (cid:48) as a subgraph of Γ that is formed by deleting from Γ all vertices in T .If Γ (cid:48) is not null, by Corollary 13, T cannot be disconnected from Γ (cid:48) , since otherwise T would be a tree disconnected from at least one other connected component in Γ . Moreover,since v is a leaf in Γ, Γ (cid:48) is joined to T by exactly one edge incident to v . Define N T as thenumber of vertices in T and N (cid:48) as the number of vertices in Γ (cid:48) , then N = N T + N (cid:48) . Notethat there is no leaf in Γ (cid:48) , and thus the vertices in Γ (cid:48) have degree at least 2. Therefore,∆ ≥ ( N T −
1) + N (cid:48) + 1 = N. (B.16)If Γ (cid:48) is null, then Γ = T and the original graph Γ is a tree. In this case, ∆ = N − L contains T . The Galileon invariants presented in § min = N − N vertices can be turned into a star by shuffling leaves. Since L contains the star with N vertices, it contains all trees with N vertices.Next, we prove the uniqueness of the minimal term. Proposition 11.
The minimal N -point loopless 1-invariant with ∆ = N − is unique upto proportionality.Proof. Let L and L be minimal N -point loopless 1-invariants with ∆ = N −
1. Let T bea tree in L and L , which exists by Proposition 10. Rescale L and L so that T appearsin each with unit coefficient. Then, T is not in L − L . However, L − L is a minimal N -point loopless 1-invariant. Therefore, Proposition 10 implies that L − L vanishes.Finally, we prove the existence of the minimal N -point loopless 1-invariant with ∆ = N − Theorem 2.
Any minimal N -point loopless -invariant is proportional to the sum withunit coefficients of all trees with N vertices.Proof. Let T N be a general tree with N • -vertices. Let T × N be a general tree with ( N − • -vertices and one × vertex. Let T (cid:63)N have two connected components, one of which is (cid:70) × and the other is a tree T N − . Let L , L × and L (cid:63) be the sum with unit coefficients of all T N , T × N and T (cid:63)N , respectively. Replacing a • -vertex in T N with a × produces a unique T × N ,since vertices are labeled. Therefore, δ ( L ) is simply L × . Similarly, ρ ( L (cid:63) ) = L × . Therefore, δ ( L ) = ρ ( L (cid:63) ) and L is 1-invariant, which is unique by virtue of Proposition 11.It is shown in [13] that the Galileon-like term in spacetime dimension d = D + 1 ≥ N : (cid:15) i ··· i N − k N ··· k D (cid:15) j ··· j N − k N ··· k D ∂ i φ ∂ j φ ∂ i ∂ j φ · · · ∂ i N − ∂ j N − φ, (B.17)is invariant up to a total derivative. By Theorem 2, the sum with unit coefficients ofall trees with N vertices is proportional to (B.17), up to a total derivative. In fact, theconstant of proportionality is 1. – 55 – .3.2. Non-minimal Invariants So far we have found the unique minimal 1-invariant for each N , that is with ∆ = N − > N −
1) is equal to anexact 1-invariant up to a plain-relation (a total derivative). We begin with the followingdefinition:
Definition 17 (Frame) . For Γ ∈ G N, ∆ , delete all edges incident to leaves; iterate thisprocedure until reaching a graph Γ f that contains no leaves. We call Γ f the frame of Γ . Note that, by definition, after deleting from Γ all edges that appear in Γ f , each of theconnected components in the resulting graph is a tree. We also define: Definition 18 (Frame Invariant) . A loopless 1-invariant L = (cid:80) ki =1 α i Γ i is a frame invari-ant if the frames of Γ i for all i = 1 , . . . , k are identical. For convenience, we define a map f on frame invariants, such that f ( L ) is the frame whichis common to all of the graphs contained in L .By definition, the nonempty vertices in any frame have degree greater than 1. Nosuch vertex can be turned into a × -vertex or be adjacent to a (cid:63) -vertex in a (cid:63) -graph associ-ated with Γ. This means that terms with different frames cannot lead to cancellations inthe consistency equation. Therefore, a consistency equation naturally splits into multipleconsistency equations, each one a frame invariant. This is summarized in the lemma below: Lemma 7.
Any loopless 1-invariant is a linear combination of frame invariants.
The proofs presented in Proposition 10 and 11 and Theorem 2 are directly applicable toframe invariants, from which we conclude:
Proposition 12.
Up to proportionality, any loopless frame invariant L is equal to the sumwith unit coefficients of all plain-graphs that satisfy the following conditions: ( a ) The plain-graph has a frame f ( L ) . ( b ) If there is more than one connected component in the plain graph, then none of themis a tree.
Note that condition (b) follows directly from Corollary 13. Proposition 12 effectively pro-vides an equivalent definition for frame invariants. Classifying all non-minimal 1-invariantsis thus reduced to classifying all non-minimal frame invariants. Furthermore, by Proposi-tion 2, we can restrict our search to loopless frame invariants. In the following we will showthat any loopless frame invariant is equal to an exact invariant up to total derivatives. Westart with a useful lemma:
Lemma 8.
Let Γ ( (cid:96) ) ∈ G be an (cid:96) -looped exact 1-invariant, such that all vertices with loopshave degree 2. Then ρ (0) ◦ θ (Γ ( (cid:96) ) ) is a loopless 1-invariant and is equal to Γ ( (cid:96) ) up to aplain-relation. – 56 – roof. Label the loops in Γ ( (cid:96) ) from 1 to (cid:96) . Note that ρ ◦ θ (Γ ( (cid:96) ) ) = Γ ( (cid:96) ) + ρ ( (cid:96) − ◦ θ (Γ ( (cid:96) ) ).Since θ (Γ ( (cid:96) ) ) is a (cid:63) -ed plain graph, ρ ◦ θ (Γ ( (cid:96) ) ) is a plain-relation. Hence ρ ( (cid:96) − ◦ θ (Γ ( (cid:96) ) )is 1-invariant. Define L ≡ ρ ( (cid:96) − ◦ θ (Γ ( (cid:96) ) ) = ρ ◦ θ (Γ ( (cid:96) ) ) − Γ ( (cid:96) ) ,L is by definition an exact 1-invariant up to a plain-relation ρ ◦ θ (Γ (cid:96) ). Furthermore, allgraphs in L are ( (cid:96) − ρ ◦ θ ( L ) = L + ρ ( (cid:96) − ◦ θ ( L ). Define L ≡ ρ ( (cid:96) − ◦ θ ( L ) = ρ ◦ θ ( L ) − L , which is also an exact 1-invariant up to plain-relations and consists of graphs that are( (cid:96) − (cid:96) times, we obtain L (cid:96) = ρ (0) ◦ θ (cid:96) − ◦ · · · ◦ ρ ( (cid:96) − ◦ θ ◦ ρ ( (cid:96) − ◦ θ (Γ ( (cid:96) ) ) = ρ (0) ◦ θ (Γ ( (cid:96) ) ) , which is an exact 1-invariant up to plain-relations. Theorem 3.
A non-minimal frame invariant is an exact invariant up to a plain-relation.Proof.
It is sufficient to consider any loopless non-minimal frame invariant L ( k ) ∈ L N, ∆ ,with k the number of empty vertices in f ( L ( k ) ). Note that since L ( k ) is non-minimal, it isnot a tree and therefore k < N . We prove the theorem by induction on k .1. k = 0: In this case, f ( L (0) ) = L (0) . By construction, a vertex in a graph in L (0) is ofdegree no less than 2, and thus L (0) is already exactly invariant.2. If any L ( k ) with k < α is an exact invariant plus a plain-relation: Consider anyloopless non-minimal frame invariant L ( α ) . Form an (cid:96) -looped exact 1-invariant Γ ( α ) from f ( L ( α ) ) by adding a loop to each empty vertex. By Lemma 8, ρ (0) ◦ θ (Γ ( α ) )is a loopless 1-invariant, equal to Γ ( α ) up to plain-relations. Using Lemma 7, ρ (0) ◦ θ (Γ ( α ) ) = (cid:80) i F i , where each F i is a frame invariant with a distinct frame. Provided α < n , there is exactly one F i , say (cid:101) F , with f ( (cid:101) F ) = f ( L ( α ) ), and all other F i havefewer than α empty vertices in f ( F i ). Since all other F i ’s have fewer than α emptyvertices in f ( F i ), they are exact 1-invariants up to a plain-relation, by the inductionhypothesis. But this means that (cid:101) F is also an exact 1-invariant up to plain-relations.By Proposition 12, since (cid:101) F and L ( α ) share the same frame, (cid:101) F is proportional to L ( α ) and thus L ( α ) is also an exact 1-invariant up to plain-relations.By induction, L ( k ) is exactly 1-invariant up to plain-relations for any 0 ≤ k < N .From the above discussion, we can conclude: The set of all exact 1-invariants with alllooped vertices of degree 2 generates all non-minimal 1-invariants, up to plain-relations.An example of these graphs for N = 3 and ∆ = 4 is shown in Figure 5.We end our discussion of the linear shift symmetry with a summary of the full classi-fication of 1-invariants: – 57 – igure 5 : All 1-invariant terms up to total derivatives, with N = 3, ∆ = 4 and P = 1. Theorem 4 (Classification of 1-invariants) . The sum with unit coefficients of all treeswith n vertices is the unique 1-invariant with ∆ = N − up to proportionality and plain-relations ) . The set of all graphs consisting of N vertices, with all vertices of degree higherthan 1 and any looped vertex of degree 2, generate all 1-invariants with ∆ > N − up toplain-relations ) . There are no invariants with ∆ < N − . B.4. Invariants from Superpositions
In this section, we describe a method of combining invariants to form other invariants.Therefore, we will need to keep track of the degree of the polynomial shifts under whichthe variation of various terms are taken. It is important to recall at this point that thevariation map δ P depends crucially on P . Therefore, all the different types of graphsdepend on P as well. Until now, this dependence on P has been kept implicit. We willnow make it explicit by referring to graphs as P -graphs .The method of combining invariants involves the notion of superposition, defined be-low, which combines graphs with different values of P . Definition 19 (Superposition of Graphs) . Given a P A -graph Γ A and a P B -graph Γ B ,which each have the same value of N , the superposition of Γ A and Γ B is a P -graph formedby applying the the following procedure:1. If there is a × -vertex in Γ B , replace the vertex in Γ A that has the same label as the × -vertex in Γ B with a × -vertex.2. Add any (cid:63) -vertices in Γ B to Γ A .3. Take all edges in Γ B and add them to Γ A , joining the same vertices as they do in Γ B .4. Identify the resulting graph as a null graph if deg ( × ) is higher than P or there aretwo × -vertices.The resulting graph is denoted by Γ A ∪ Γ B . Note that Γ A ∪ Γ B = Γ B ∪ Γ A . Note that the above definition of superposition depends on P . We will refer to such a superposition as a P -superposition . Definition 20 (Superposition of Linear Combinations) . Given the linear combinations L A = (cid:80) k A i =1 a i Γ Ai and L B = (cid:80) k B i =1 b i Γ Bi , where Γ Ai , Γ Bj are graphs with the same n , thesuperposition of L A and L B is defined as L A ∪ L B ≡ k A (cid:88) i =1 k B (cid:88) j =1 a i b j Γ Ai ∪ Γ Bj . – 58 – .4.1. Superposition of a P -invariant and an Exact Invariant This section involves the construction of new invariants by taking the superposition of a P -invariant with an exact invariant. Lemma 9.
1. Given a P -graph Γ ∈ L N, ∆ and a P E -graph Γ E ∈ L N, ∆ E , where Γ E is an exact P E -invariant, Γ E ∪ δ P (Γ) = δ P + P E +1 (Γ E ∪ Γ) , (B.18) where ∪ denotes ( P + P E + 1) -superposition.2. Given a P -graph Γ (cid:63) ∈ L (cid:63)N, ∆ and a P E -graph Γ E ∈ L N, ∆ E , where Γ E is an exact P E -invariant, Γ E ∪ ρ (Γ (cid:63) ) = ρ (Γ E ∪ Γ (cid:63) ) , (B.19) where ∪ denotes ( P + P E + 1) -superposition.Proof.
1. Operationally, a graph in the LHS of (B.18) is given by substituting one • -vertex, v , in Γ with a × -vertex and then adding the edges in Γ E to the result. Meanwhile,the RHS is given by adding the edges in Γ E to Γ first before substituting v by a × -vertex. Thus, (B.18) is violated only when a graph vanishes from one side and notthe other. A graph vanishes from the RHS if and only if the degree of the vertex v in Γ E ∪ Γ is greater than P + P E + 1. If this condition holds, then the graph alsovanishes from the LHS by the rules of ( P + P E + 1)-superposition. A graph couldalso possibly vanish from the LHS if deg( v ) > P in Γ. However, if deg( v ) > P in Γ,then deg( v ) > P + P E + 1 in Γ E ∪ Γ since the degree of a vertex in Γ E is at least P E + 1 (by Corollary 4). Therefore, the conditions for the vanishing of a graph fromeither side of (B.18) are identical and thus the equation holds.2. Once again, (B.19) is violated only when a graph vanishes from one side and not theother. A graph vanishes from the RHS if and only if the degree of the × -vertex in ρ (Γ E ∪ Γ (cid:63) ) is greater than P + P E + 1. If this condition holds, then the graph alsovanishes from the LHS by the rules of ( P + P E + 1)-superposition. A graph on theLHS could also possibly vanish if deg( × ) > P for a graph Γ × in ρ (Γ (cid:63) ). However,then deg( × ) > P + P E + 1 in Γ E ∪ Γ × , since the degree of a vertex in Γ E is at least P E + 1 (by Corollary 4). Therefore, the conditions for the vanishing of a graph fromeither side of (B.19) are identical and thus the equation holds.Now we apply Lemma 9 to prove the main result: Theorem 5.
For fixed N , the superposition of a P -invariant and an exact P E -invariantis a ( P + P E + 1) -invariant. – 59 – roof. Denote the P -invariant by L = (cid:80) ki =1 b i Γ i and the exact P E -invariant by L E = (cid:80) k E i =1 a i Γ Ei . By Corollary 4, all vertices in Γ Ei have degree greater than P E . Since L is a P -invariant, there exists a linear combination of (cid:63) -graphs, L (cid:63) = (cid:80) k (cid:63) i =1 c i Γ (cid:63)i , such that thefolowing consistency equation holds: δ P ( L ) = ρ ( L (cid:63) ) . (B.20)Define (cid:101) L ≡ L E ∪ L = (cid:80) i,j a i b j Γ Ei ∪ Γ j . Then, using Statement 1 of Lemma 9: δ P + P E +1 ( (cid:101) L ) = (cid:88) i,j a i b j δ P + P E +1 (Γ Ei ∪ Γ j )= (cid:88) i,j a i b j Γ Ei ∪ δ P (Γ j ) = (cid:88) i a i Γ Ei ∪ δ P ( L ) (B.21)Furthermore, define (cid:101) L (cid:63) ≡ (cid:80) i,j a i c j Γ Ei ∪ Γ (cid:63)j using ( P + P E + 1)-superposition. Using State-ment 2 of Lemma 9: ρ ( (cid:101) L (cid:63) ) = (cid:88) i,j a i c j ρ (Γ Ei ∪ Γ (cid:63)j ) = (cid:88) i,j a i c j Γ Ei ∪ ρ (Γ (cid:63)j ) = (cid:88) i a i Γ Ei ∪ ρ ( L (cid:63) ) (B.22)Combining (B.20), (B.21) and (B.22) we have that δ P + P E +1 ( (cid:101) L ) = ρ ( (cid:101) L (cid:63) ) and therefore (cid:101) L ≡ L E ∪ L is a ( P + P E + 1)-invariant. B.4.2. Superposition of Minimal Loopless 1-invariants
In this section we show that the superposition of Q minimal loopless 1-invariants results ina (2 Q − (cid:63) -graphs called “hyper-Medusas”, which we now define: Definition 21 (Hyper-Medusa) . A hyper-Medusa is a loopless (cid:63) -graph with all (cid:63) -verticesadjacent to the × -vertex, such that the degree of the × -vertex deg( × ) and the number of (cid:63) -vertices N ( (cid:63) ) satisfy deg( × ) ≥ P + 1 − N ( (cid:63) ) . Lemma 10.
Given M h a hyper-Medusa, there exists a linear combination of Medusas L M that satisfies ρ (0) ( M h ) = ρ (0) ( L M ) .Proof. Within the action of ρ (0) , we can delete deg( × ) + N ( (cid:63) ) − P + 1 ≥ (cid:63) -vertices andthen add the same number of edges in M h , yielding a linear combination of (cid:63) -graphs withexactly P + 1 − deg( × ) (cid:63) -vertices. These resulting graphs are Medusas.The following definition will allow us to construct the desired hyper-Medusas: Definition 22.
Take Γ to be any × -graph or (cid:63) -graph and for each i = 2 , ..., Q , take T i tobe any tree, such that Γ and T i have the same value of n . Label the × -vertex in Γ by v × and define T × i to be the graph formed from T i by replacing the vertex that is labeled by v × with a × -vertex. If v × is a leaf in T i , then define (cid:101) T i to be the unique P = 1 Medusa thatis associated with T × i , otherwise (cid:101) T i = T × i . Then we define: χ (Γ ∪ T ∪ · · · ∪ T Q ) ≡ Γ ∪ (cid:101) T ∪ · · · ∪ (cid:101) T Q . – 60 – heorem 6. For fixed N , the superposition of Q minimal loopless 1-invariants is a (2 Q − -invariant.Proof. By Theorem 2, any minimal N -point loopless 1-invariant is equal to the sum withunit coefficients of all trees with N vertices, up to proportionality. Therefore, denote the Q copies of the minimal loopless 1-invariants by L ( c ) n = (cid:80) N N − α c =1 T ( c ) α c , for c = 1 , . . . , Q . Weadd an additional structure to all graphs in this proof: We color all edges in all graphs in L ( c ) N by a distinct color ( c ). Throughout this proof, two graphs are equal if and only if theyare the same graph and, in addition, their edges are the same colors. Taking into accountthis coloring, all of the plain-graphs in L ≡ (cid:83) Qc =1 L ( c ) N now have unit coefficients, and thenumber of these plain-graphs is (cid:0) N N − (cid:1) Q . Moreover, L is the sum over α , . . . , α Q of allsuch T (1) α ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q ’s with unit coefficients. By Theorem 2, there is a unique linearcombination of Medusas (cid:80) β c M ( c ) β c satisfying δ (cid:16) L ( c ) N (cid:17) = δ N N − (cid:88) α c =1 T ( c ) α c = ρ (0) N ( N − N − (cid:88) β c =1 M ( c ) β c , where each T ( c ) α c is a distinct tree and each M ( c ) β c is a distinct P = 1 Medusa, consisting ofa subgraph tree and a disconnected subgraph (cid:70) × . In the following, we take the limit P → ∞ , so that no graph vanishes. Note that we have (cid:88) α δ ∞ (cid:16) T (1) α ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:17) = (cid:88) α (cid:16) δ ∞ ( T (1) α ) (cid:17) ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q = (cid:88) β ρ (0) (cid:16) M (1) β (cid:17) ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q + (cid:88) α ( δ ∞ − δ )( T (1) α ) ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q . (B.23)Define X L ≡ (cid:80) β ,α ,...,α Q M (1) β ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q and X R to be the sum with unitcoefficients of all distinct graphs contained in (cid:80) β ,α ,...,α Q χ ( M (1) β ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q ). Inthe following, we show that ρ (0) ( X L ) = ρ (0) ( X R ) . (B.24)Since T ( c ) α c , α c = 1 , . . . , N N − , and M ( c ) β c , β c = 1 , . . . , N ( N − N − , are all distinct fromeach other, all elements in X L and X R have unit coefficient. Therefore, it will suffice toshow that any graph in ρ (0) ( X R ) is also in ρ (0) ( X L ), and vice versa. RHS contains LHS:
Let Γ × be a × -graph in ρ (0) ( X L ). Then, Γ × is contained in ρ (0) (cid:0) M (1) β ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:1) for some β , α , . . . , α Q . The (cid:63) -graph, M (1) β ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q induces a unique χ (cid:0) M (1) β ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:1) , such that all × -graphs in ρ (0) (cid:0) M (1) β ∪ T (2) α ∪· · · ∪ T ( Q ) α Q (cid:1) (including Γ × ) are contained in ρ (0) ◦ χ (cid:0) M (1) β ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:1) . Therefore,Γ × is in ρ (0) ( X R ) and ρ (0) ( X R ) contains ρ (0) ( X L ). LHS contains RHS:
Let Γ × be a × -graph in ρ (0) ( X R ). Then, Γ × is contained in ρ (0) ◦ χ (cid:0) M (1) β ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:1) for some β , α , . . . , α Q . In particular, Γ × is contained in some– 61 – (0) (cid:0) M (1) β (cid:48) ∪ T (2) α (cid:48) ∪ · · · ∪ T ( Q ) α (cid:48) Q (cid:1) with T (2) α (cid:48) ∪ · · · ∪ T ( Q ) α (cid:48) Q in ρ (0) ◦ χ (cid:0) T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:1) . Since M (1) β (cid:48) ∪ T (2) α (cid:48) ∪ · · · ∪ T ( Q ) α (cid:48) Q is in X L , Γ × is in ρ (0) ( X L ) and ρ (0) ( X L ) contains ρ (0) ( X R ).From (B.24) we obtain (cid:88) β ,...,α Q ρ (0) (cid:16) M (1) β (cid:17) ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q = ρ (0) ( X R ) . (B.25)Similarly, (cid:88) α ,...,α Q ( δ ∞ − δ ) T (1) α ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q = ρ (0) (cid:16) (cid:101) X R (cid:17) , (B.26)with (cid:101) X R given by the sum with unit coefficients of all graphs contained in (cid:88) α ,...,α Q χ (cid:16) ( δ ∞ − δ ) T (1) α ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:17) . Therefore, by (B.23), (B.25) and (B.26), we conclude that δ ∞ ( L ) = (cid:88) α ,...,α Q δ ∞ (cid:16) T (1) α ∪ T (2) α ∪ · · · ∪ T ( Q ) α Q (cid:17) = ρ (0) (cid:16) X R + (cid:101) X R (cid:17) . (B.27)Finally, switch back to P = 2 Q −
1. Then graphs with deg( × ) > Q − P = 2 Q − δ Q − ( L ) = ρ (0) (cid:16) X R + (cid:101) X R (cid:17) . (B.28)Next we show that any graph, Γ, in X R + (cid:101) X R is a hyper-Medusa. By construction, Γresults from the superposition of graphs with either a × -vertex of degree 1 joined to a (cid:63) -vertex or a × -vertex of degree larger than 1. Therefore, deg( × ) in Γ satisfies deg( × ) ≥ N ( (cid:63) ) + 2 ( Q − N ( (cid:63) )). So, with P = 2 Q −
1, deg( × ) ≥ P + 1 − N ( (cid:63) ) and Γ is a hyper-Medusa. Hence, by Lemma 10, there exists a linear combination, L M , of Medusas, suchthat ρ (0) (cid:16) X R + (cid:101) X R (cid:17) = ρ (0) ( L M ). Therefore, combined with (B.28), we obtain δ Q − ( L ) = ρ (0) ( L M ), which proves that L is a (2 Q − P -invariants with a summary of all invariants that we found: Theorem 7.
For fixed N , the superposition of any exact P E -invariant with the superposi-tion of Q minimal loopless 1-invariants results in a P -invariant, provided P E + 2 Q ≥ P . We conjecture that the above theorem captures all P -invariants, up to total derivatives.Since we have classified all exact invariants and all 1-invariants, it is straightforward toconstruct the P -invariants in the above theorem for any specific case. This theorem applies even for P E <
0. Recall that, by Corollary 5, an exact P -invariant for P < P -invariant for any P < – 62 –ote that we have classified exact invariants and 1-invariants using the two parameters, N (number of vertices) and ∆ (number of edges). Finite connected graphs can always beembedded on a Riemann surface of some genus, in which case Euler’s theorem relates N , ∆and the number of faces F of the embedding to the genus g of the surface. Therefore, onecould also use the parameters N and F instead to classify invariants . Any finite graphthat can be embedded into a 2-sphere can also be embedded into a plane, and is known asa planar graph. In particular, this is true for any graph with N ≤ . One can check this statement for all of the examples in § § P = 0 (which is a trivial case), all superpositions involve trees, so that all superposedgraphs are connected. For example, all of the graphs in Figure 2 have three faces whenembedded into a plane. Indeed, when interpreted as Feynman diagrams, these graphshave three “loops”. In general, the superposition of Q minimal loopless 1-invariants yieldsgraphs have F “loops” as Feynman diagrams, where F is given by F = ( Q − N − P -invariant for P ≤ Q −
1. For example,the superposition of three minimal loopless 1-invariants with N = 4 produces a 5-invariantwith 6 faces. B.5. Unlabeled Invariants
So far we have been dealing entirely with labeled graphs, which represent algebraic termswhere each φ is given a distinct label. But we are primarily interested in invariants whereall φ ’s are the same. These are represented by unlabeled graphs, that is, where isomorphicgraphs are identified with each other. One may wonder whether or not the labeled P -invariants capture all of the unlabeled ones. The following proposition addresses thisquestion, and shows that our restriction to labeled P -invariants still allows us to find allunlabeled P -invariants. Proposition 13.
Given an unlabeled P -invariant L unlab , there exists a labeled P -invariant L lab , such that L lab reduces to an integer multiple of L unlab once the labels are removed.Proof. Define L × unlab = δ ( L unlab ), where δ ( L unlab ) contains δ (Γ unlab ) for all Γ unlab in L unlab .Label L unlab (i.e., label the vertices from 1 to N ) to form L lab . This labeling is fiducialsince we will eventually sum over all possible labelings. Do the same for L × unlab to form L × lab .Define L × lab (cid:48) = δ ( L lab ), which is a labeling of L × unlab , possibly distinct from L × lab . However,if Γ × lab in L × lab and Γ × lab (cid:48) in L × lab (cid:48) reduce to the same Γ × unlab in L × unlab once the labels areremoved, then Γ × lab and Γ × lab (cid:48) are simply related by a permutation. Therefore, (cid:88) σ ∈ S N σ ◦ δ ( L lab ) = (cid:88) σ ∈ S N σ ( L × lab ) = (cid:88) σ ∈ S N σ ( L × lab (cid:48) ) , (B.29)where S N is the group of permutations on the N vertices. In principle, ∆ and F could also be used, but this seems less natural. Thanks to Kurt Hinterbichler for bringing this issue to our attention at the 2014 BCTP Tahoe Summit. – 63 –ince L unlab is P -invariant, there exists L (cid:63) unlab such that δ ( L unlab ) = ρ ( L (cid:63) unlab ). Label L (cid:63) unlab to form L (cid:63) lab and define L × lab (cid:48)(cid:48) = ρ ( L (cid:63) lab ). Generically, there can be cancellationsbetween isomorphic graphs in L × lab (cid:48)(cid:48) , once the labels are removed, since one × -graph canbe associated with more than one (cid:63) -graph. Therefore, L × lab (cid:48)(cid:48) is not necessarily a labelingof L × unlab . Nevertheless, if α Γ × unlab appears in L × unlab with α (cid:54) = 0, then all of the graphs,Γ × , . . . , Γ × k in L × lab (cid:48)(cid:48) which are isomorphic to Γ × unlab appear in L × lab (cid:48)(cid:48) as a linear combination (cid:80) ki =1 α i Γ × i with (cid:80) ki =1 α i = α . Conversely, if (cid:80) ki =1 α i Γ × i appears in L × lab (cid:48)(cid:48) , but the graph,Γ × unlab , to which Γ × i reduces once the labels are removed, does not appear in L × unlab , then (cid:80) ki =1 α i = 0. Therefore, (cid:88) σ ∈ S N k (cid:88) i =1 α i · σ (Γ × i ) = k (cid:88) i =1 α k (cid:88) σ ∈ S N σ (Γ × lab ) = α (cid:88) σ ∈ S N σ (Γ × lab ) , which implies (cid:88) σ ∈ S N σ ◦ ρ ( L (cid:63) lab ) = (cid:88) σ ∈ S N σ ( L × lab (cid:48)(cid:48) ) = (cid:88) σ ∈ S N σ ( L × lab ) . (B.30)Combining Eqs. (B.29) and (B.30) with the facts that σ ◦ δ = δ ◦ σ and σ ◦ ρ = ρ ◦ σ foreach σ ∈ S N , yields the desired labeled consistency equation: δ (cid:18) (cid:88) σ ∈ S N σ ( L lab ) (cid:19) = ρ (cid:18) (cid:88) σ ∈ S N σ ( L (cid:63) lab ) (cid:19) . (cid:80) σ ∈ S N σ ( L lab ) is P -invariant and reduces to N ! L unlab once the labels are removed. C. Coset Construction
The standard technique for finding terms which are invariant under a nonlinear realizationof a symmetry is to use a coset construction [24–27]. In this appendix we explore theconnection between our invariant Lagrangians and this construction. Although we findthat the coset construction can reproduce some of the invariants that we have discovered,using this method is computationally difficult when compared to the graphical methodintroduced in this paper.We first review the coset construction applied to the polynomial shift symmetry aspresented in [18, 28]. In dimension D , consider the polynomial shift transformations of theGoldstone fields defined in (B.1), accompanied with space-time translations and spatialrotations. Denote the corresponding generators by Z, Z i , . . . , Z i ...i P for polynomial shifts, P i for spacial translations, P for temporal translations, and J ij for spatial rotations. Notethat there are no boost symmetries. For the nonlinear realization of space-time symmetry,the translation generators are treated as the broken generators [26, 27]. The Goldstonefields transform as δ P i φ = ∂ i φ, δ Z φ = 1 , δ Z i ...ik φ = 1 k ! x i . . . x i k , k = 1 , . . . , P. – 64 –he commutators between the operators can be readily calculated,[ P i , Z ] = 0 , (cid:2) P i , Z i ...i k (cid:3) = − i (cid:88) j k δ ji Z i ... ˆ ...i k , (cid:2) Z i ...i k , Z j ...j (cid:96) (cid:3) = 0 , where ˆ means that the index j is omitted.The commutators above given by the generators P , P i , J ij , Z and Z i ...i k , k = 1 , . . . , P define the Lie algebra of a Lie group G , and P and J ij correspond to the unbroken normalsubgroup H . Take left invariant differential N -forms on G/H to be N -cochains, and takethe coboundary operator d ( k ) to be the exterior derivative of differential forms. Denote thegroup of N -cocycles by Z k = Ker d ( k ) , and the group of k -coboundaries by B k = Im d ( k − .The Chevalley-Eilenberg cohomology group E k ( G/H ) is defined to be E k ( G/H ) = Z k / B k , which is isomorphic to the Lie algebra cohomology H k ( G/H ; Z ). (See [29] for details.)We associate the generator Z with the Goldstone field φ . To each generator Z i ...i k , k = 1 , . . . , P we associate a symmetric k -tensor field φ i ...i k . Indices can be lowered orraised by Kronecker delta symbols. The coset space is parametrized by g = exp (cid:0) i P i x i (cid:1) exp (cid:32) iZφ + i P (cid:88) k =1 Z i ...i k φ i ...i k (cid:33) . The Maurer-Cartan form is − ig − dg = P i dx i + Z ( dφ + φ i dx i ) + P − (cid:88) n =1 Z i ··· i n (cid:0) dφ i ··· i n + φ i ··· i n i dx i (cid:1) + Z i ...i P dφ i ...i P . Therefore, the basis dual to the generators is ω i P = dx i , ω i ··· i P = dφ i ··· i P ,ω = dφ + φ i dx i , ω i ··· i k = dφ i ··· i k + φ i ··· i k i dx i , k = 1 , . . . , P − . (C.1)Moreover, dω i P = 0 , dω i ··· i P = 0 ,dω = dφ i ∧ dx i , dω i ··· i k = dφ i ··· i k i ∧ dx i , k = 1 , . . . , P − . The inverse Higgs constraints [30, 31] imply the vanishing of ω and ω i ··· i k ( k < P ) in (C.1): φ i ··· i k = ( − k ∂ i · · · ∂ i k φ, k = 1 , . . . , P. (C.2)Having reviewed the coset construction, we now present examples for P = 2 and 3 (the P = 1 scenario is essentially the same as the Galileon case [28]). P =2 Case : For N = 3, the cohomology group is trivial for D <
2. Therefore, let us startwith the simplest nontrivial case, D = 2. We are looking for a closed form involving the– 65 –edge of three ω ’s, which are not ω P . There is one independent cohomology element, withthe lowest number of indices on ω ’s:Ω = (cid:15) ij ω ab ∧ ω ia ∧ ω jb = d (cid:0) (cid:15) ij φ ab dφ ia ∧ dφ jb (cid:1) ≡ dβ . These expressions can be extended to D ≥ D +1 = (cid:15) ijs ...s D ω ab ∧ ω ia ∧ ω jb ∧ ω s P ∧ · · · ∧ ω s D P ,β D = (cid:15) ijs ...s D φ ab dφ ia ∧ dφ jb ∧ dx s ∧ · · · ∧ dx s D . Taking the pullback of β D to the spacetime manifold and then applying the inverse Higgsconstraints in (C.2) gives a term proportional to (cid:15) ijs ...s D (cid:15) k(cid:96)s ...s D ∂ a ∂ b φ ∂ i ∂ k ∂ a φ ∂ j ∂ (cid:96) ∂ b φ, which is already contained in [18]. One can verify that this is equivalent up to integration byparts and overall prefactor to the invariant in (4.10). That term was found to be invariantfor P = 3, and is therefore also invariant for P = 2.Next, consider N = 4. The simplest nontrivial case is D = 3. We seek a closed 4-formgiven as the wedge of four ω ’s, which are not ω P . There is one independent cohomologyelement, with the lowest number of indices on ω ’s:Ω (cid:48) = (cid:15) ijk ω a ∧ ω ia ∧ ω jb ∧ ω kb = d (cid:20) (cid:15) ijk φ ac (cid:18) φ ac dφ jb + φ bc dφ ja (cid:19) ∧ dφ kb ∧ dx i (cid:21) ≡ dβ (cid:48) These expressions can be extended to D ≥ (cid:48) D +1 = (cid:15) ijks ...s D ω a ∧ ω ia ∧ ω jb ∧ ω kb ∧ ω s P ∧ · · · ∧ ω s D P ,β (cid:48) D = (cid:15) ijks ...s D φ ac (cid:18) φ ac dφ jb + φ bc dφ ja (cid:19) ∧ dφ kb ∧ dx i ∧ dx s ∧ · · · ∧ dx s D . Taking the pullback of β (cid:48) D to the spacetime manifold and then applying the inverse Higgsconstraints in (C.2) gives a term proportional to (cid:15) ijs ...s D (cid:15) k(cid:96)s ...s D ∂ ( a ∂ b φ ∂ c ) ∂ j ∂ (cid:96) φ ∂ a ∂ b φ ∂ c ∂ i ∂ k φ. One can verify that this is equivalent up to integration by parts and an overall prefactorto the invariant in (4.7). P =3 Case : Let us focus on N = 3. Again, we start with D = 2. There is one independentcohomology element:Ω = (cid:15) ij (cid:0) ω ab ∧ ω ia ∧ ω jb + 2 ω a ∧ ω ib ∧ ω jab − ω a ∧ ω ia ∧ ω jbb − ω ∧ ω iab ∧ ω jab + ω ∧ ω iaa ∧ ω jbb (cid:1) . It is quite a challenge to determine the potential, β , for this Ω . One can appreciate thepower of the graphical method at this point: We have already determined that there is– 66 –ne independent 3-invariant with N = 3. Therefore, we can immediately conclude withoutcalculation that the pullback of β must be proportional to the invariant in (4.3) up to totalderivatives. Again, Ω can be generalized to D ≥ ω P ’s on the end:Ω D +1 = (cid:15) ijs ...s D (cid:0) ω ab ∧ ω ia ∧ ω jb + 2 ω a ∧ ω ib ∧ ω jab − ω a ∧ ω ia ∧ ω jbb − ω ∧ ω iab ∧ ω jab + ω ∧ ω iaa ∧ ω jbb (cid:1) ∧ ω s P ∧ . . . ∧ ω s D P . References [1] G. ’t Hooft,
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