Cauchy conformal fields in dimensions d>2
aa r X i v : . [ h e p - t h ] O c t RUNHETC-2015-10
Cauchy conformal fields in dimensions d > Daniel Friedan , ∗ and Christoph A. Keller , † NHETC and Department of Physics and AstronomyRutgers, The State University of New JerseyPiscataway, New Jersey 08854-8019, USA The Science Institute, The University of Iceland, Reykjavik, Iceland Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
Abstract
Holomorphic fields play an important role in 2d conformal field theory. We generalizethem to d > d = 3 and 4. ∗ [email protected] † [email protected] ontents d > M in terms of Casimir invariants . . . . . . . . . . . . . . . . . 12 V s for d = 2 n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 V s for d = 2 n + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 Summary of the Cauchy fields . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 The spin V s Cauchy fields as “free fields” . . . . . . . . . . . . . . . . . . . 18 ∂ µ ∂ µ φ = 0 . . . . . . . . . . . . . . . . . . . . . 205.3 Constrain the modes using the radial Cauchy equation . . . . . . . . . . . 215.4 Summary of the mode expansions . . . . . . . . . . . . . . . . . . . . . . . 25 φ k , | k | ≥ ∆ are all non-zero . . . . . . . . . . . . . . . . . . . . 287.2 The spin of φ † ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3 Invariant (anti-)commutators of the modes . . . . . . . . . . . . . . . . . . 30 A The irreducible representations of so ( d ) M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38A.3 Example: d = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 B Classification of Cauchy fields 39
B.1 Assuming unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.2 The non-unitary cases: d = 3 and d = 4 . . . . . . . . . . . . . . . . . . . . 40B.3 The non-unitary case: d > C First-order over-determination conditions for V = V s Introduction d > – the Cauchy condition Holomorphic quantum fields play a major role in two dimensional conformal field theory.Their defining property is the Cauchy-Riemann equation( ∂ x + i∂ y ) φ ( z ) = 0 . (1.1)Equation (1.1) defines a Cauchy problem. Namely, if we know the field φ on some contour,say for instance x = const , or the circle | x + iy | = const , then (1.1) uniquely determines φ everywhere. That φ depends holomorphically on the single complex variable z = x + iy is equivalent to the Cauchy-Riemann equation (1.1). A 2d holomorphic field has a modeexpansion φ n = Z C dz f n ( z ) φ ( z ) , φ ( z ) = X n f ∗ n ( z ) φ n , Z C f m ( z ) f ∗ n ( z ) = δ m,n (1.2)where the smearing functions f n ( z ) are a complete set of functions holomorphic in someneighborhood of the contour C (for example, a circle in the 2d space-time). The Cauchyproperty allows us to deform the contour without changing the result, within the regionwhere φ ( z ) and f n ( z ) are non-singular. By appropriately deforming the contour, wecan thus obtain the commutation relations of the modes from just the singular termsin the operator products of the holomorphic fields. The representation theory of thealgebra of the modes becomes a powerful tool for analyzing the quantum field theory.In a complementary vein, the global properties of the Cauchy-Riemann equation putstringent constraints on the correlation functions of holomorphic fields, making possibleexact solutions.Holomorphic quantum fields were first constructed and studied in string theory [1–4].Early uses of contour deformation techniques can be found in [5, 6]. P. Goddard wrote“People gradually realized through this time (1971-73) that one could more profitably usethe analytic properties of the fields in this way than think in terms of distributions onthe unit circle” [7]. Holomorphic fields and the contour deformation technique were laterrediscovered in [8].In this paper our goal is to generalize the notion of holomorphic conformal fields tohigher dimensions, and to attempt a classification of such fields. Our hope – unrealized –was that there might be as rich a variety of such fields as in two dimensions. To mimic thetwo dimensional case, we search for fields φ that satisfy a first order differential equationwhich has the Cauchy property — the property of uniquely determining φ everywhereonce φ is known on some codimension 1 surface S . This Cauchy property generalizesto d >
2, along with the techniques of mode expansion and contour deformation. Theproperty of depending on a single complex variable on the other hand does not seem togeneralize to conformal fields in d > j µ ( x ) satisfies the first order equation ∂ µ j µ = 0, but4nowing j µ on the hyperplane x d = 0 is not enough to determine j µ on nearby hyper-planes. We are thus looking for fields which satisfy a special type of first order differentialequation.An example of a field that does have the Cauchy property is a self-dual two form φ µν ( x ) in d = 4 dimensions. It satisfies the self-duality condition φ µν = 12 ǫ µν µ ′ ν ′ φ µ ′ ν ′ (1.3)and the first order differential equation ∂ µ φ µν = 0 . (1.4)Using the self-duality condition to eliminate φ ij in favor of φ kd , the first order equationbecomes ∂ d φ id = − ǫ ij k ∂ j φ kd . (1.5)If we know φ on a surface x d = const , then we can integrate this first order equation toget φ everywhere. Self-dual 2-forms in 4d are thus Cauchy fields. Our goal is to identifythe other fields of this type in all dimensions d > Differential equations for φ correspond to shortened representations of the conformal alge-bra so ( d, φ ( x ) cor-responds to an irreducible lowest weight representation of the conformal algebra so ( d, D . The lowestweight states | φ i form an irreducible representation V of the euclidean rotation Lie al-gebra so ( d ). The field φ ( x ) is a vector-valued field of spin V and conformal weight (orscaling dimension) ∆. The conformal transformation properties of φ ( x ) are completelydetermined by the data ( V, ∆).The field φ ( x ) satisfies a first order differential equation iff the states | φ i are annihilatedby some linear combination of the translation generators P µ . The P µ act as raisingoperators in the conformal algebra, adding 1 to the weight. If a linear combination ofthe P µ annihilates | φ i , this means that there is a null subspace of states at level 1 in therepresentation. The representation is said to be a level 1 short representation . In d = 2for instance, the Cauchy-Riemann differential equation (1.1) comes from the null statecondition ¯ L − | φ i = 0. For general d , classifying the primary fields that satisfy first orderdifferential equations is equivalent to classifying the level 1 short representations of theconformal algebra.The inner product matrix of the conformal representation at level 1 is easily computedin terms of the commutation relations of the conformal algebra. There is a null stateon level 1 iff the determinant of the level 1 inner product matrix equals 0. This is apolynomial equation on the conformal weight ∆ with coefficients that depend on the so ( d )representation V . This type of argument is of course very familiar from the derivation of5nitarity bounds. A necessary condition for the conformal representation to be unitary isthat the inner product matrix at level 1 be non negative. For a given so ( d ) representation V , the inner product matrix is positive definite for large values of ∆. As ∆ decreases, iteventually reaches a value where one or more null states appear. That is the unitaritybound on ∆. Below that value of ∆, the level 1 inner product matrix has at least onenegative eigenvalue. Thus, if we insist on unitarity, the only level 1 short representationsthat can occur are at the level 1 unitarity bound. If we do not require unitarity, morepossibilities open up, as there are in general several values of ∆ where null states appearon level 1. The conserved spin 1 current discussed above does have null states at level 1,which leads to its conservation equation, but it does not have the Cauchy property. Thecentral question of this work is thus: which level 1 null states give Cauchy differentialequations? Let us summarize our results. For unitary fields, we find a complete answer to thisquestion.In even dimensions d = 2 n , a unitary Cauchy field is a conformal field whose spin isan so ( d ) representation V s with highest weight λ s , V s = V λ s , λ s = ( | s | , . . . , | s | , s ) , s ∈ Z , (1.6)and whose conformal weight is∆ s = (cid:26) n − | s | s = 00 s = 0 . (1.7)The case s = 0 is the trivial case φ = 1, the identity operator, which of course satisfiesthe first order equation ∂ µ φ = 0.In odd dimensions d = 2 n + 1, the unitary Cauchy fields are the primary fields ( V s , ∆ s ) V s = V λ s , λ s = ( s, . . . , s ) , with s = 0 or s = 12 , (1.8)∆ = 0 , ∆ / = n − . (1.9)The case s = 0 is again the identity operator.We prove that these lists exhaust all unitary Cauchy fields in d > ∂ µ ∂ µ φ ( x ) = 0 (1.10)consist exactly of the free massless scalar field and the fields ( V s , ∆ s ). For this reasonthe fields in representations V s were called “free” in [9]. This was a misnomer, since theKlein-Gordon equation did not immediately imply that the correlation functions factorize6s free-field correlation functions into products of 2-point functions according to the Wickcontraction rule. For example, non-abelian current algebras in d = 2, and, more generally, W -algebras, satisfy the Klein-Gordon equation (1.10) but are certainly not free. Here we prove that all unitary Cauchy fields in d > d >
2, that the commutators of the modes are multiplesof the identity operator, which establishes that indeed the unitary Cauchy fields all havefree-field correlation functions. We find that only the massless spinor and the self-dual n -form field can have a local energy-momentum tensor, in accord with the Weinberg-Wittentheorem [11].It would have been very nice to find non-free Cauchy fields that could be used toconstruct non-trivial conformal field theories. In a sense however the negative result is notunexpected, as it is related to results on conserved higher spin currents in quantum fieldtheory in d >
2. Note that short representations lead to conserved currents: suppose wehave a field φ a ( x ) in a representation V of so ( d ), with a being an index for V , and supposethat it satisfies a first-order differential equation A µab ∂ µ φ a ( x ) = 0. Then J µb = A µab φ a ( x )is a conserved current. J will be a higher spin conserved current for all but the smallestrepresentations V . Note that this is somewhat different from the usual construction ofhigher spin currents as bilinears, since here J is linear in the underlying fields. Nonetheless,the Coleman-Mandula theorem [12] states that higher spin conserved currents in d > d = 3, in [14]for symmetric traceless fields in d = 4, and in [15] for 4-, 5- and 6-pt functions of theenergy-momentum tensor in d = 4. For higher dimensions there has been work [16] onthe construction of higher spin algebras, arguing that for d = 3 and d > d . We show that there always exist enoughsmearing functions to capture all the modes of the field on a surface of codimension 1.Our results for non-unitary conformal Cauchy fields are incomplete. We classify allpossible spins and scaling dimensions for d = 3 and 4. For d >
4, we find a restricted listof possible spins and scaling dimensions, but we do not prove that all the possibilities onthe list do in fact have the Cauchy property. We derive the mode expansions only for thenon-unitary cases where the spin is one of the so ( d ) representations V s but the scalingdimension is not the unitary value. We do not show that the non-unitary theories are For scalar fields with ∂ µ ∂ µ φ ( x ) = 0 in d > d >
2, whose properties could be investigated by a straightforward applicationof the methods used here.
We consider euclidean CFTs in the radial quantization. The conformal operator algebrais so ( d, d + 2)( d + 1) / P µ of translations, thegenerators L µν of so ( d ), the euclidean rotations, the generator D of dilations, and thegenerators K µ of the special conformal transformations. The commutation relations are[ P µ , P ν ] = 0 , [ K µ , K ν ] = 0 , [ K µ , P ν ] = 2 δ µν D − L µν [ D, P µ ] = P µ , [ D, L µν ] = 0 , [ D, K µ ] = − K µ (2.1)[ L µν , P σ ] = δ νσ P µ − δ µσ P ν , [ L µν , K σ ] = δ νσ K µ − δ µσ K ν [ L µν , L ρσ ] = δ νρ L µσ − δ νσ L µρ + δ µσ L νρ − δ µρ L νσ The adjointness relations are P † µ = K µ , D † = D , L † µν = − L µν . (2.2)Our generators differ by a factor of i from the usual ones in the physics literature.The conformal generators implement the conformal vector fields:generator vector field P µ ∂ µ D x µ ∂ µ L µν ( δ σµ x ν − δ σν x µ ) ∂ σ K µ (2 x µ x σ − x δ σµ ) ∂ σ . (2.3)The ground state is annihilated by all the conformal generators, so the correlation func-tions are invariant under the complex conformal algebra so ( d + 2 , C ). The conformalsymmetries of euclidean space form the real subalgebra so ( d + 1 , R : x µ x − x µ .Writing X [ v ] for the operator generator implementing the conformal vector field v , theadjoint is X [ v ] † = − X [ Rv ], thus the adjointness relations of (2.2) above. Extending X [ v ] to complex vector fields v , the vector fields satisfying Rv = ¯ v have skew-adjointgenerators. The usual Minkowski space quantization is constructed with respect to thereflection in the hyperplane x d = 0, R Mink : ( ~x, x d ) ( ~x, − x d ). So, in the Minkowskispace quantization, P † d = P d and P † i = − P i (recall that our generators differ from theusual ones by a factor of i ). The two reflections, R and R Mink are conjugate to each other8n the euclidean conformal group, so the two Lie algebras of skew-adjoint generators arenot the same, but they are isomorphic to each other, both isomorphic to so ( d, φ i ( x ) with scaling dimen-sions ∆ i , satisfying[ P µ , φ i ( x )] = ∂ µ φ i ( x ) , [ D, φ i ( x )] = ( x µ ∂ µ + ∆ i ) φ i ( x ) . (2.4)The radial quantization gives a space of states in one-to-one correspondence with the scal-ing fields. The operator-state correspondence maps the scaling field φ i ( x ) to an eigenstateof the dilation generator D , φ i ( x ) ↔ | φ i i = φ i (0) | i , D | φ i i = ∆ i | φ i i , (2.5)where | i is the ground state, corresponding to the identity field 1. The ground state | i is annihilated by all the conformal generators. Correlation functions are given by groundstate expectation values of radially ordered products of fields.The generators of the conformal algebra act as operators on the state space. Theeigenvalue of D is the conformal weight. The generators P µ raise the weight by 1, and thegenerators K µ lower the weight by 1. The conformal lowest weight states are the statesthat are killed by the lowering operators K µ , K µ | φ i = 0 , D | φ i = ∆ | φ i . (2.6)The full space of states is generated from the lowest weight states by the action of theraising operators P µ . The so ( d ) generators L µν commute with D , so they take lowestweight states to lowest weight states. The space of lowest weight states thus decomposesinto a sum of irreducible representations of so ( d ). We write | φ i for the finite dimensionalvector space of lowest weight states of weight ∆ in an irreducible so ( d ) representation V .The raising operators P µ acting on | φ i generate an irreducible lowest weight representationof the conformal algebra, with lowest weight ∆.The conformal primary fields φ ( x ) are in one-to-one correspondence with the conformallowest weight states | φ i and are labeled by the same data ( V, ∆). The representation V is called the spin of the field φ ( x ). Writing φ a ( x ) for the component fields of therepresentation V , the generators L µν act on the lowest weight states by matrices M µν on V , L µν | φ b i = M µν ab | φ a i . (2.7)The matrices M µν satisfy the same commutation relations as the L µν ,[ M µν , M ρσ ] = δ νρ M µσ − δ νσ M µρ + δ µσ M νρ − δ µρ M νσ . (2.8)There is a unique hermitian inner product on the representation V such that M † µν = − M µν . (2.9)The inner product on the entire conformal representation is the unique inner productdetermined by the adjointness relations (2.2).9he action of the conformal generators on the conformal fields can be derived fromthe operator state correspondence, the action of the conformal generators on the lowestweight states, and the translation covariance φ ( x ) = e x µ P µ φ (0) e − x µ P µ , φ ( x ) | i = e x µ P µ | φ i . (2.10)The results are[ P µ , φ ( x )] = ∂ µ φ ( x ) , (2.11)[ D, φ ( x )] = x µ ∂ µ φ ( x ) + ∆ φ ( x ) , (2.12)[ L µν , φ b ( x )] = ( x ν ∂ µ − x µ ∂ ν ) φ b ( x ) + M µν ab φ a ( x ) , (2.13)[ K µ , φ b ( x )] = (2 x µ x σ − x δ σµ ) ∂ σ φ b ( x ) + 2 x σ (∆ δ µσ δ ab − M µσab ) φ a ( x ) . (2.14) Let us now study the irreducible lowest weight representations of the conformal algebra inmore detail. Let φ ( x ) be a conformal field of dimension ∆ in the so ( d ) representation V .As we argued above, the full representation is obtained by acting with raising operators P µ . For a generic representation, we expect all those descendent states to be linearlyindependent. Such a representation is called long . We are interested in short or degenerate representations, where there are linear relations among some of the descendent states.In mathematical language, the Verma module is the vector space spanned by all theformal products P µ · · · P µ N | φ a i symmetric in the indices µ i . We have a linear map fromthe Verma module to the physical state space, taking the formal product to the physicalproduct of operators acting on the lowesst weight state. The kernel of this linear map isthe space of null states — the linear relations in the Verma module.The weight space D = ∆ + N is called level N . Let us consider level 1, spanned bythe states P µ | φ a i . Suppose the conformal representation is degenerate on level 1. Thenthere are identities of the form A µba P µ | φ b i = 0 . (2.15)This is equivalent to A µba ∂ µ φ b (0) | i = 0 (2.16)which, by a standard quantum field theory argument from translation invariance of cor-relation functions, is equivalent to A µba ∂ µ φ b ( x ) = 0 , (2.17)a first order differential equation with constant coefficients.Conversely, if φ ( x ) satisfies a first order differential equation with constant coefficients,the differential equation can alway be written as a set of equations in the form (2.17),which implies (2.16) and (2.15). So the conformal representation is degenerate on level 1.In fact, we do not need the condition of constant coefficients. If φ ( x ) is a conformalfield, then a first order differential equation with non-constant coefficients is equivalent10o a first order equation with constant coeffients. Suppose φ satisfies A µba ( x ) ∂ µ φ b ( x ) = 0 . (2.18)For each x , conjugate with translation operators to get A µba ( x ) ∂ µ φ b (0) | i = 0 . (2.19)This is equivalent to performing the radial quantization with x as origin. Equation (2.19)asserts that a certain subspace N ( x ) of level 1 states is null. Let N be the span of all the N ( x ) for all x , N = ⊕ x N ( x ) ⊂ C ⊗ V . (2.20)All the constraints on φ from the original differential equation (2.18) are expressed in thefact that N is a null subspace. The fact that N is a null subspace is expressed as well bythe differential equation with constant coefficients P N µν ∂ µ φ ( x ) = 0 , (2.21)where P N is the projection on the subspace N . So the original first order differentialequation on φ ( x ) with non-constant coefficients is equivalent to a differential equationwith constant coefficients.Therefore we can say that the conformal field φ ( x ) satisfies a first order differentialequation iff the conformal representation is degenerate on level 1.To see when level 1 is degenerate, we make the standard calculation of the matrix ofinner products of the level 1 states, h φ b ′ | P † ν ′ P ν | φ b i = h φ b ′ | K ν ′ P ν | φ b i = h φ b ′ | [ K ν ′ , P ν ] | φ b i (2.22)= h φ b ′ | (2 δ ν ′ ν D − L ν ′ ν ) | φ b i (2.23)= h φ b ′ | (2 δ ν ′ ν ∆ δ ab − M ν ′ ν ab ) | φ a i (2.24)The formal states P µ | φ a i of the Verma module form the so ( d ) representation C d ⊗ V ,where C d is the fundamental representation. Define the self-adjoint matrix ˆ M on C d ⊗ V ,ˆ M µaνb = M µνab . (2.25)Then the inner product on level 1 states given by equation (2.24) is the hermitian quadraticform on C d ⊗ V corresponding to the self-adjoint matrix 2(∆ − ˆ M ). Therefore the rep-resentation is degenerate on level 1 iff ∆ is an eigenvalue of ˆ M [17]. The null vectors arethen the eigenspace ˆ M = ∆.A primary field φ ( x ) thus satisfies a first order differential equation iff ∆ is an eigen-value of the self-adjoint matrix ˆ M acting on C d ⊗ V .The conformal representation is unitary on level 1 iff ∆ − ˆ M ≥ M . The largest eigenvalue of ˆ M is thelevel 1 unitarity bound , the lower bound on ∆ imposed by unitarity.11or example, consider a scalar field — a field in the trivial representation of so ( d ).The trivial representation has ˆ M = 0, so the level 1 unitarity bound is at ∆ = 0. Atthe unitarity bound, at ∆ = 0, the level 1 matrix of inner products is identically zero, soall the states on level 1 are null, so the field satisfies the first order differential equation ∂ µ φ ( x ) = 0, which implies that φ = 1, the identity field.For scalar fields, there is an additional unitarity condition at level 2,∆(∆ − ( d − / ≥ . (2.26)This of course allows the ∆ = 0 identity field – the ground state – but it forces anynon-trivial scalar field to have dimension at least that of the free scalar field, ( d − / V , the level 1 unitaritycondition is necessary and sufficient. This was established first for d = 3 [18] and d = 4 [19]and, finally, for any dimension [20]. ˆ M in terms of Casimir invariants Since finding the eigenvalues of the matrix ˆ M is the central issue, let us rewrite it interms of basic Lie algebra theoretic objects. Normalize the quadratic Casimir invariantof a representation V of so ( d ) as C d ( V ) = − M µν M µν . (2.27)The matrix ˆ M on C d ⊗ V is then simplyˆ M = ⊗ C d ( V ) + C d ( C d ) ⊗ − C d ( C d ⊗ V ) . (2.28)To see this, note that the fundamental representation, which is given in equation (2.1), isgenerated by the matrices M Fµν ρσ = δ ρµ δ νσ − δ ρν δ µσ (2.29)so M Fµν ρσ M µν ab = 2 M ρσab = 2 ˆ M ρaσb (2.30)so ˆ M = 12 M Fµν ⊗ M µν = ⊗ C d ( V ) + C d ( C d ) ⊗ − C d ( C d ⊗ V ) . (2.31)So the eigenvalues of ˆ M are gotten by decomposing C d ⊗ V into irreducible componentsand finding their quadratic Casimir invariants. The classical representation theory neededfor this is collected in the appendices. We postpone calculating until we have found howto tell which so ( d ) representations V give degenerate conformal representations that givefirst order differential equations with the Cauchy property.12 First order differential equations and the Cauchy property
A conformal field φ ( x ) of spin V satisfies a first order differential equation iff ∆ is an eigen-value of ˆ M . We want to know for what so ( d ) representations V and scaling dimensions∆ does this first order differential equation have the Cauchy property.The central object for that analysis is ˆ P ∆ , the projection matrix acting on C d ⊗ V that projects on the eigenspace ( C d ⊗ V ) ∆ where ˆ M = ∆,ˆ P ∆ : C d ⊗ V → ( C d ⊗ V ) ∆ . (3.1)To simplify notations, we will often just write ˆ P for ˆ P ∆ . The matrix elements of ˆ P areˆ P µrνs . The eigenspace ( C d ⊗ V ) ∆ is the space of null states on level 1, soˆ P µaνb P µ | φ a i = 0 . (3.2)Suppressing the indices for V , write ˆ P µν for the matrix on V with matrix elements ˆ P µaνb .Then the null state conditions are writtenˆ P µν P µ | φ i = 0 , (3.3)equivalent to the differential equationˆ P µν ∂ µ φ ( x ) = 0 . (3.4) The Cauchy property is the condition on the differential equation that the values of φ ( x )on a codimension 1 submanifold completely determine φ ( x ) everywhere. For a rotationallyinvariant first order equation with constant coefficients, this is simply the condition that ∂ d φ ( x ) is completely determined by the ∂ i φ ( x ), which is an algebraic condition.Suppose φ ( x ) and ˜ φ ( x ) have the same spatial derivatives at x , ∂ i φ ( x ) = ∂ i ˜ φ ( x ). Weneed the differential equation to imply that ∂ d φ ( x ) = ∂ d ˜ φ ( x ). Writing δφ = ˜ φ − φ , weneed ∂ i δφ ( x ) = 0 = ⇒ ∂ d δφ ( x ) = 0 . (3.5)Both φ ( x ) and ˜ φ ( x ) satisfy the differential equation (3.4) and the equation is linear, so δφ ( x ) also satisfies it, 0 = ˆ P µν ∂ µ δφ ( x ) . (3.6)Writing the indices for V explicitly, this isˆ P daνb ∂ d δφ a ( x ) + ˆ P iaνb ∂ i δφ a ( x ) = 0 . (3.7)But the spatial derivatives are zero, so the differential equation givesˆ P daνb ∂ d δφ a ( x ) = 0 . (3.8)13e need this to imply that ∂ d δφ r ( x ) = 0.Define ˆ P d to be the matrix with matrix elements ˆ P daνb . It is a linear map from V tothe ˆ M = ∆ eigenspace,ˆ P d : V → ( C d ⊗ V ) ∆ , ˆ P d v = ˆ P (ˆ e d ⊗ v ) , (3.9)where ˆ e d is the unit vector in the d -direction in C d .We need ˆ P d ∂ d δφ ( x ) = 0 = ⇒ ∂ d δφ ( x ) = 0 . (3.10)But ∂ d δφ ( x ) can be any vector in V . So this is simply the condition ∀ v ∈ V , ˆ P d v = 0 = ⇒ v = 0 , (3.11)which is the condition that ˆ P d is injective. So we have shown that the Cauchy propertyis equivalent to the condition that ˆ P d is injective, A1 The first order differential equation satisfied by φ ( x ) has the Cauchy property iffthe matrix ˆ P d : V → ( C d ⊗ V ) ∆ is injective.Now define ˆ P dd to be the matrix with matrix elements ˆ P dadb . It is a linear map from V to V , ˆ P dd : V → V , ˆ P dd v = Proj C ˆ e d ⊗ V ( ˆ P d v ) . (3.12)We will argue that A1 is equivalent to A2 The first order differential equation satisfied by φ ( x ) has the Cauchy property iffthe matrix ˆ P dd : V → V is invertible.To see the equivalence, first note that if ˆ P d is not injective, then there is a non-zero vector v with ˆ P d v = 0, which implies ˆ P dd v = 0 , so ˆ P dd cannot be invertible. Now suppose ˆ P dd isnot invertible. We will use the invariant inner-products on V and on the tensor productspace C d ⊗ V . If ˆ P dd is not invertible, there must be a non-zero v ∈ V such that ˆ P dd v = 0,so ( v, ˆ P dd v ) = 0 . (3.13)But ( v, ˆ P dd v ) = (ˆ e d ⊗ v, ˆ P ˆ e d ⊗ v ), so(ˆ e d ⊗ v, ˆ P ˆ e d ⊗ v ) = 0 . (3.14)ˆ P is a self-adjoint projection, so( ˆ P ˆ e d ⊗ v, ˆ P ˆ e d ⊗ v ) = (ˆ e d ⊗ v, ˆ P ˆ e d ⊗ v ) = (ˆ e d ⊗ v, ˆ P ˆ e d ⊗ v ) = 0 . (3.15)The invariant inner-product is positive definite, so we haveˆ P d v = ˆ P ˆ e d ⊗ v = 0 , (3.16)14o we have shown that ˆ P dd not invertible implies ˆ P d not injective. So the algebraic Cauchyconditions are equivalent.The symbol of the first order differential operator ˆ P µν ∂ µ in the differential equation(3.4) is the map from unit co-vectors to matrices,ˆ n µ ˆ n µ ˆ P µaνb . (3.17)By rotational invariance, we might as well choose ˆ n in the d -direction, in which case thesymbol is the matrix ˆ P d . So the symbol for any value of ˆ n µ is conjugate to ˆ P d by somerotation. So our Cauchy condition is exactly the condition that the symbol of the firstorder differential operator is injective.A differential operator is said to be elliptic when its symbol is invertible. A differentialoperator is said to be overdetermined elliptic when its symbol is injective but not invertible.(Sometimes the label elliptic is used for both.)It seems to us that the natural generalization of ¯ ∂ to conformal fields in d > overdetermined refers to the fact that not all initial data on the codimension1 submanifold is possible. The differential equation imposes linear constraints on thevalue of φ ( x ) on the submanifold. We discuss those constraints in section 8.3 below.As an illustration, let us explain in more detail how the first order equation satisfiedby a conserved current, ∂ µ j µ ( x ) = 0, fails to satisfy the conditions A1 and A2 . The firstorder differential equation (3.4) is ˆ P µανβ ∂ µ j α ( x ) = 0 (3.18)with the projection ˆ P being ˆ P µανβ = δ µα δ νβ . (3.19)Then ˆ P dανβ = δ dα δ νβ , ( ˆ P v ) νβ = ˆ P dανβ v α = v d δ νβ . (3.20)If v d = 0 then ˆ P d v = 0. So ˆ P d is not injective. The matrix ˆ P dd is P dαdβ = δ dα δ dβ (3.21)which is the projection matrix on the d -direction, which is obviously not invertible. From our discussion in the previous section, we define a field φ to be a Cauchy field ifit satisfies a first order equation with injective symbol. This property then determines φ everywhere uniquely once we know its value on some codimension 1 submanifold.15e argued in section 2.2 that short representations lead to differential equations.However, as we illustrated with the conserved current, not every short representation leadsto a differential equation which satisfies the Cauchy condition. Cauchy representationsare thus a proper subset of short representations, and we want to classify them.In this section, we present all the unitary Cauchy representations. We show here thatall of them have the Cauchy property. The proof that there are no other unitary Cauchyrepresentations is given in appendix B.1. The most obvious example is the trivial representation. If V is the trivial so ( d ) represen-tation, M µν = 0, then ∆ = 0 is the level 1 degeneracy condition. If ∆ = 0, then all oflevel 1 is null. The first order differential equation is ∂ µ φ ( x ) = 0, whose solution is theidentity field φ ( x ) = 1. In the following we will thus only consider non-scalar (non-trivial) so ( d ) representations V . V s for d = 2 n We saw at the beginning that the (anti-)self-dual n -forms Λ n ± are Cauchy fields in evendimensions d = 2 n . Let us now describe a larger set of examples in d = 2 n , namely therepresentations (1.6), V s = V λ s , λ s = ( | s | , . . . , | s | , s ) , s = 0 . (4.1)In particular, V ± are the chiral spinors, V ± are the (anti-)self-dual n -forms.For these examples we can make the rather abstract discussion in (3.2) much moreconcrete by giving very explicit expressions for the projectors ( ˆ P ∆ ) dd . First note from(A.12) that the tensor product C d ⊗ V s decomposes into exactly two components, C d ⊗ V s = V λ s + ǫ ⊕ V λ s ∓ ǫ n , for s = ±| s | , (4.2)so that ˆ M has two eigenspaces. Its eigenvalues can be calculated using its expression interms of quadratic Casimirs (2.28) and formula (A.11) for the quadratic Casimirs,ˆ M = ( −| s | on V λ s + ǫ n − | s | on V λ s ∓ ǫ n . (4.3)Since ˆ M has two distinct eigenvalues, and since the sum of the eigenvalues is n −
1, theprojection on the subspace ˆ M = ∆ is simplyˆ P ∆ = ˆ M − ( n − − ∆)∆ − ( n − − ∆) . (4.4)Since ˆ M dd = 0 we have ( ˆ P ∆ ) dd = − ( n − − ∆)∆ − ( n − − ∆) (4.5)16hich is a non-zero multiple of the identity matrix. So ˆ P dd is invertible. Therefore ( V s , ∆)is a Cauchy representation for ∆ either of the two eigenvalues of ˆ M .The largest eigenvalue of ˆ M is n − | s | , which is therefore the level 1 unitaritybound. In fact, since we are considering s = 0 only, it is the exact unitarity bound. So V = V s , ∆ = n − | s | gives a unitary Cauchy field, while V = V s , ∆ = −| s | gives anon-unitary Cauchy field. The Cauchy differential equation (3.4) is h ˆ M µν − ( m − ∆) δ µν i ∂ µ φ ( x ) = 0 , m = 12 ( d − . (4.6)For the unitary field, ∆ = | s | + m , the differential equation is (cid:16) ˆ M µν + | s | δ µν (cid:17) ∂ µ φ ( x ) = 0 . (4.7) V s for d = 2 n + 1The Cauchy fields in odd dimensions are even more restricted. From (A.12) we see thatfor s > / C d ⊗ V s = ( V λ s ⊕ V λ s + ǫ ⊕ V λ s − ǫ n , for s > / ,V λ s ⊕ V λ s + ǫ , for s = 1 / . (4.8)Only for s = 1 / s = 1 / s > / s = 1 /
2, the eigenvalues of ˆ M are ( d −
1) and − s . The unitarity bound is∆ = ( d − h ˆ M µν − ( m − ∆) δ µν i ∂ µ φ ( x ) = 0 , m = 12 ( d − . (4.9)For the unitary field, ∆ = m + | s | , the differential equation is (cid:16) ˆ M µν + | s | δ µν (cid:17) ∂ µ φ ( x ) = 0 . (4.10)It can be shown that this equation is equivalent to the massless Dirac equation. We now have the classification of the non-trivial Cauchy conformal fields in unitary con-formal field theories. Write m = ( d − d , the non-trivial Cauchy conformal fields are the conformal pri-mary fields of spin V and conformal weight ∆ in the set (cid:26) ( V, ∆) = ( V s , | s | + m ) : s ∈ Z , s = 0 (cid:27) (4.11)17n odd dimensions d = 2 n , the only non-trivial Cauchy conformal field is the masslessspinor field, V = V s , s = , ∆ = | s | + m .For both d even and for d odd, the Cauchy differential equation is h ˆ M µν − ( m − ∆) δ µν i ∂ µ φ ( x ) = 0 , m = 12 ( d − . (4.12)For the unitary cases, ∆ = | s | + m , (cid:16) ˆ M µν + | s | δ µν (cid:17) ∂ µ φ ( x ) = 0 . (4.13) V s Cauchy fields as “free fields”
Conformal fields in the representations V s were previously studied in somewhat differentcontext [9], as a subset of the so-called “free field” conformal representations — theconformal fields φ ( x ) satisfying the massless Klein-Gordon equation ∂ µ ∂ µ φ ( x ) = 0 . (4.14)Indeed, applying ∂ ν to equation (4.12) gives ∂ µ ∂ µ φ ( x ) = 0. Conversely, if P ν P ν | φ i = 0,then the identity [ K µ , P ν P ν ] | φ i = 0 gives exactly the non-trivial first order differentialequation (4.6), except for the case of a scalar field with the scaling dimension of thefree massless scalar field. A conformal field with non-zero spin satisfying equation (4.6)necessarily belongs to an so ( d ) representation V such that ˆ M has at most two eigenvalues.So the “free field” representations are the representations V s plus the representation ofthe free massless scalar field.The label “free field” for conformal fields satisfying ∂ µ ∂ µ φ ( x ) = 0 was a misnomer.Additional work is needed to show that φ ( x ) is actually a free field, i.e. a field whosecorrelation functions are gaussian, given in terms of its two-point function by Wick con-tractions. For d = 2, this is obviously not the case, as shown by the many non-freeexamples. In the following two sections we will develop the tools to show that for d > In this and the following section we prove two negative results about the Cauchy conformalfields in unitary theories in dimensions d >
2: First, they are free fields — their correlationfunctions are the products of the 2-point function following the free-field Wick contractionrules. Second, they have a local stress-energy tensor only for | s | ≤ φ ( x ), expanding in spherical harmonicson the ( d − so ( d )representations to show that the singular part of the operator product expansioncontains only the identity field.3. We then point out that the commutator depends only on the singular part of theoperator product expansion. It follows that the commutator is a multiple of theidentity, so that the correlation functions are indeed given by free-field Wick con-tractions.4. We finally point out that, for | s | >
1, there is no field in the operator productexpansion with the spin and scaling dimension of the stress-energy tensor.This is perhaps not the most elegant proof, but it has the virtue of efficiency. We devi-ate from the spirit of d = 2 holomorphic fields, since we do not calculate the commutatorsby contour deformation. In fact we have not yet established that contour deformation isvalid. We return to the question of deformability in section 8. It would be more elegantto use those techniques: construct the field modes by smearing φ ( x ) over a codimension1surface against smearing functions that satisfy the dual Cauchy equation, then calculatethe commutator of the field modes by contour deformation. This is a priori a more pow-erful technique, because the commutator so derived would depend only on the part of theoperator product expansion with singularity at least O ( r − d ), not on the entire singularpart.As it happens, our cruder approach is powerful enough to establish the two results. Wecan easily enforce a subset of the constraints implied by the Cauchy differential equation torestrict the modes enough to control the singular part of the operator product expansion.We do not need the precise list of field modes — some of the modes on our list mightbe zero. We will return to this question in section 7.1 to show that they are all in factnon-vanishing. For the moment however our list of the possible modes is small enough togive our two negative results on the unitary Cauchy fields. We describe the operator representation of a Cauchy field in the radial quantization byexpanding the field in spherical harmonics. Suppose φ ( x ) = φ ( r ˆ x ) is a Cauchy conformalfield with spin V s and conformal weight ∆.The rotation Lie algebra so ( d ) acts on the field by the generators M t otµν φ ( x ) = ( M o rbµν + M s pinµν ) φ ( x ) (5.1)where, as in (2.13), the M s pinµν generate the spin representation V s and the orbital gener-ators M o rbµν = − ( x µ ∂ ν − x ν ∂ µ ) (5.2)generate the rotations acting on the unit sphere, on functions of ˆ x . The functions onthe unit sphere decompose into irreducible representations of the M o rbµν , the spherical19armonics, L ( S d − ) = ∞ ⊕ l =0 S l , (5.3)where S l = V ( l, ,..., (5.4)is the irreducible representation of so ( d ) on traceless symmetric l -tensors.Expand the field φ in spherical harmonics φ ( r ˆ x ) = ∞ X l =0 Y l (ˆ x ) φ l ( r ) , Y l ∈ S l , (5.5) Y l (ˆ x ) φ l ( r ) = 1 l ! ˆ x µ · · · ˆ x µ k φ l ( r ) µ ··· µ k . (5.6)The mode φ l ( r ) is a traceless symmetric l -tensor with values in V s . Note that we aresuppressing the V s index in our labeling of the field φ ( x ) and its modes φ l ( x ).The field mode φ l ( r ) lies in a sub-representation of S l ⊗ V s . The next steps are to usethe Cauchy equation to put constraints on this sub-representation. ∂ µ ∂ µ φ = 0Again, applying ∂ ν to equation (4.12) gives ∂ µ ∂ µ φ = 0. So φ ( x ) is a harmonic function,with the usual expansion in spherical harmonics.To be explicit, the laplacian in polar coordinates is ∂ µ ∂ µ = r − h ( r∂ r ) + ( d − r∂ r − C d, o rb i (5.7)where C d, o rb = − M o rbµν M µν o rb (5.8)is the quadratic Casimir operator of the orbital representation. Applying the laplacian tothe mode expansion (5.5), using the quadratic Casimir invariants C d ( S l ) = 12 l ( l + d −
2) (5.9)gives ( r∂ r − l )( r∂ r + d − l ) φ l ( r ) = 0 . (5.10)The general solution is φ l ( r ) = r l φ + l + r − l − m φ − l . (5.11)Recall that we defined m = d − .We can read off the scaling dimensions directly, or use (2.4),dim( φ + l ) = ∆ + l , dim( φ − l ) = ∆ − l − m . (5.12)20ow we follow the usual convention of labeling the field modes by minus the scalingdimension, φ k = φ + − k − ∆ = φ + | k +∆ − m |− m : k + ∆ ≤ , < k + ∆ < m ,φ − k − m +∆ = φ −| k +∆ − m |− m : 2 m ≤ k + ∆ , (5.13) k + ∆ ∈ Z , φ k ∈ S | k +∆ − m |− m ⊗ V s , dim( φ k ) = − k , [ D, φ k ] = − kφ k . (5.14)The mode expansion is now φ ( x ) = X k +∆ ∈ Z | k +∆ − m |≥ m r − k − ∆ Y | k +∆ − m |− m (ˆ x ) φ k . (5.15)Next note that the correlation functions must be regular as r →
0, so φ ( x ) | i must beregular, where | i is the ground state of the radial quantization. Therefore, φ k | i = 0 , k > − ∆ (5.16)and the operator state correspondence is | φ i = φ (0) | i = φ − ∆ | i . (5.17)We will thus identify the φ k with k ≥ m − ∆ with annihilation modes, and the φ k with k ≤ − ∆ with creation modes. So far, we have only used the fact that any field satisfying ∂ µ ∂ µ φ = 0 has a discreteexpansion in harmonic functions to get the mode expansion (5.15). The mode φ k forms asubrepresentation of S | k +∆ − m |− m ⊗ V s . The next step is to show that the radial componentof the first order differential equation (4.12), x ν h ˆ M µν − ( m − ∆) δ µν i ∂ µ φ ( x ) = 0 , (5.18)fixes the quadratic Casimir invariant of φ k . Then we use the known decomposition of thetensor product S | k +∆ − m |− m ⊗ V s into irreducible representations to find that the Casimirinvariant of φ k either does not occur in the decomposition, or identifies a unique irre-ducible. Thus certain of the modes φ k must vanish identically, and each of the rest of the φ k are in specific irreducible so ( d ) representations.The representation of so ( d ) on φ ( x ) is the tensor product of the spin and orbitalrepresentations, as given in equation (5.1). Using equation (5.2) for the orbital generators,12 M o rbµν M µν s pin φ s ( x ) = x ν ∂ µ M µν rs φ r ( x ) = x ν ˆ M µν rs ∂ µ φ r ( x ) . (5.19)21he so ( d ) quadratic Casimir operators are C d, o rb = − M o rbµν M µν o rb , C d, s pin = − M s pinµν M µν s pin , C d, t ot = − M t otµν M µν t ot , (5.20)so 12 M o rbµν M µν s pin = C d, o rb + C d, s pin − C d, t ot , (5.21)so equation (5.18), the radial component of the Cauchy differential equation, can bewritten h C d, o rb + C d, s pin − C d, t ot − ( m − ∆) r∂ r i φ ( x ) = 0 (5.22)Substituting the mode expansion (5.15) and taking account of the independence of theterms in the expansion, we get h C d, o rb + C d, s pin − C d, t ot + ( m − ∆)( k + ∆) i φ k = 0 , (5.23)which, since φ k lies in S | k +∆ − m |− m ⊗ V s , fixes the quadratic Casimir invariant of φ k to be C d, t ot φ k = (cid:2) C d ( S | k +∆ − m |− m ) + C d ( V s ) + ( m − ∆)( k + ∆) (cid:3) φ k . (5.24)Using C d ( S l ) φ k = 12 l ( l + d − φ k , (5.25)with l = | k + ∆ − m | − m , (5.26)we get a simple formula for the quadratic Casimir of φ k , C d, t ot φ k = (cid:20) C d ( V s ) + 12 ( − ∆ + k ) (cid:21) φ k . (5.27)The second step is to compare these Casimir values with the Casimirs of the irreduciblerepresentations that occur in the decomposition of the tensor product S | k +∆ − m |− m ⊗ V s .That decomposition is given in [21], Props 9.4 and 9.5. For d even, S l ⊗ V ( | s | ,..., | s | , ±| s | ) = | s | ⊕ s ′ =max( −| s | , − l + | s | ) V ( l + s ′ , | s | ,..., | s | , ± s ′ ) . (5.28)For d odd, the decomposition of the tensor product is, for s = 0 and s = 1 / S l ⊗ V ( | s | ,..., | s | , | s | ) = | s | ⊕ s ′ =max( −| s | , − l + | s | ) V ( l + s ′ , | s | ,..., | s | , | s ′ | ) . (5.29)For d even, the Casimir invariants of the individual components are C d ( V ( l + s ′ , | s | ,..., | s | , ± s ′ ) ) = C d ( V s ) + 12 (cid:2) ( l + s ′ )( l + s ′ + d −
2) + s ′ − | s | ( | s | + d − − s (cid:3) . (5.30)22he Casimir of the component increases monotonically in s ′ , given the inequalities on s ′ in the decomposition (5.28). Therefore the Casimir of φ k can match the Casimir of atmost one component, so φ k will lie in an irreducible representation, or will vanish.The Casimir of the component labelled by s ′ agrees with the Casimir of φ k given inequation (5.27) iff s ′ satisfies the quadratic equation( l + s ′ )( l + s ′ + d −
2) + s ′ − | s | ( | s | + d − − s = − ∆ + k , (5.31)which can be re-arranged as (cid:20) s ′ + 12 ( l + m ) (cid:21) −
14 ( k − ∆ + m ) = ( | s | + m − ∆)(∆ + | s | ) . (5.32)The rhs vanishes, because ∆ has one of the two values: | s | + m in the unitary case, −| s | in the non-unitary case. Since l = | k + ∆ − m | − m , the two roots are s ′ = − | k + ∆ − m | ±
12 ( k − ∆ + m ) . (5.33)We also need s ′ to satisfy the inequalities dictated by the decomposition (5.28), − | s | ≤ s ′ , − l + | s | ≤ s ′ , s ′ ≤ | s | . (5.34)For both the annihilation modes with k + ∆ − m ≥ m and the creation modes with k + ∆ − m ≤ − m , we calculate the two roots s ′ , then check for which of the two valuesof ∆ and for which values of k each root satisfies all the inequalities (assuming d > m > k + ∆ − m ≥ m s ′ = − k none s ′ = m − ∆ ∆ = m + | s | , k ≥ ∆ k + ∆ − m ≤ − m s ′ = − m + ∆ ∆ = m + | s | , k ≤ − ∆ s ′ = k ∆ = −| s | , ∆ ≤ k ≤ − ∆In the unitary case ∆ = m + | s | , we see that the modes φ k with m − | s | ≤ k < ∆ havebeen eliminated, because their Casimirs do not occur in the decomposition of the tensorproduct. The mode expansion in the unitary case is φ ( x ) = X k +∆ ∈ Z , | k |≥ ∆ r − k − ∆ Y l (ˆ x ) φ k . (5.35)where ∆ = m + | s | , l = | k + ∆ − m | − m = ( k − ∆ + 2 | s | , k ≥ ∆ − k − ∆ , k ≤ − ∆ . (5.36)23nd the mode φ k is in the irreducible representation φ k ∈ V ( | k |− ∆+ | s | , | s | ,..., | s | , − ǫs ) ǫ = sgn( k ) . (5.37)At this point, it remains possible that some of these modes are identically zero, since wehave only enforced the radial component of the first order differential equation. When wecalculate the singular operator product expansion, we will find that all these modes arenon-zero.For the non-unitary case, ∆ = −| s | , only a finite number of modes can be non-zero, φ ( x ) = X k −| s |∈ Z , | k |≤| s | r − k + | s | Y | s |− k (ˆ x ) φ k . (5.38) φ k ∈ V ( | s | , | s | ,..., | s | ,ǫ ′ k ) , ǫ ′ = sgn( s ) . (5.39)Note that in the non-unitary case, the expansion in r is a polynomial. We will return tothe consequences of this observation in the next section.Finally, let us discuss the modes in odd dimension d . In this case, there is only onenon-trivial Cauchy representation, V ( s,...,s ) with s = 1 /
2. The decomposition (5.29) is S l ⊗ V ( s,...,s ) = ⊕ s ′ = ± / V ( l + s ′ ,s,...,s ) , s = 12 , l ≥ . (5.40)The quadratic Casimir invariants of the components are C d ( V ( l + s ′ ,s,...,s ) ) = C d ( V ( s,...,s ) ) + 12 [( l + s ′ )( l + s ′ + d − − s ( s + d − . (5.41)Matching to the Casimirs of the φ k , equation (5.27), gives the condition( l + s ′ )( l + s ′ + d − − s ( s + d −
2) = − ∆ + k , (5.42)which can be re-written, since s ′ = ± , s ′ | k + ∆ − m | = ( m − ∆)( k + ∆ − m ) + m (cid:18) m + 12 − ∆ (cid:19) . (5.43)For the unitary case, ∆ = ( d −
1) = m + , this is s ′ (cid:12)(cid:12)(cid:12)(cid:12) k + 12 (cid:12)(cid:12)(cid:12)(cid:12) = − (cid:18) k + 12 (cid:19) , (5.44)The conformal dimension ∆ = m + is an integer, so the weights k of the modes areintegers, so the solutions are : s ′ = for k ≤ − s ′ = − for k ≥ l > d odd, in the unitary case, is obtained by combining with theresults of the harmonic expansion, equation (5.15), φ ( x ) = X | k |≥ ∆ r − k − ∆ Y l (ˆ x ) φ k , (5.45)24 = m + 12 , l = | k + ∆ − m | − m = ( k − ∆ + 1 , k ≥ ∆ − k − ∆ , k ≤ − ∆ , (5.46) φ k ∈ V ( l − ǫs,s,...,s ) = V ( | k |− ∆+ s,s,...,s ) , ǫ = sgn( k ) . (5.47)For the non-unitary case, ∆ = − , the matching equation (5.43) is s ′ (cid:12)(cid:12)(cid:12)(cid:12) k − − m (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) m + 12 (cid:19) k − . (5.48)The only solutions are k = , s ′ = with l = 0 and k = − , s ′ = − with l = 1. Themode expansion is thus φ ( x ) = Y (ˆ x ) φ / + r Y (ˆ x ) φ − / (5.49) φ ± / ∈ V ( s,s,...,s ) . (5.50) We have established that the non-trivial unitary Cauchy fields are the fields with spin V s ,for d odd: s = 12 , (5.51)for d = 2 n even: s ∈ (cid:26) ± , ± , ± , . . . (cid:27) , (5.52)and scaling dimension ∆ = | s | + m , m = 12 ( d − , (5.53)and with mode expansion φ ( x ) = X | k |≥ ∆ r − k − ∆ Y | k +∆ − m |− m (ˆ x ) φ k . (5.54)for d odd: φ k ∈ V ( | k + s |− m − ǫs,s,...,s ) ǫ = sgn( k ) , (5.55)for d = 2 n even: φ k ∈ V ( | k |− ∆+ | s | , | s | ,..., | s | , − ǫs ) ǫ = sgn( k ) . (5.56) Using the mode expansions, we show now that the commutators of the modes of unitaryCauchy fields are multiples of the identity operator. Therefore the correlation functionscan be calculated by Wick contractions, so the unitary Cauchy fields are free fields.In a first step we will show that the commutator of two Cauchy fields is completelydetermined by the singular part of the operator product expansion of the two fields. Then25e show that the singular part of the operator product expansion of two Cauchy fieldscontains only the identity operator.We also show that the operator product of two Cauchy fields contains a field withthe spin and scaling dimension of the stress-energy tensor iff d is even and s = ± / s = ± d is odd and s = 1 /
2. So the only Cauchy fields with local stress-energy tensorsare the massless spinor fields in any dimension and the free (anti-)self-dual n -form fieldsin even dimensions d = 2 n . The first step is to show that the commutator of a mode φ k of a Cauchy field φ ( x ) withany field ψ ( x ) is completely determined by the singular part of the operator productexpansion of the two fields.Let us extract the mode φ k of the Cauchy field φ ( x ) by smearing over the sphere ofradius r with the appropriate vector-valued spherical harmonic, Z S d − r d Ω F k (ˆ x ) φ ( x ) = r − k − ∆ φ k . (6.1)Suppose ψ ( y ) is any local field. Let R ( φ ( x ) ψ ( y )) be the radially ordered operator product.The commutator of φ k with ψ ( y ) can be calculated by evaluating (6.1) at radius r = | y | + ǫ and at radius r = | y | − ǫ , Z S d − | y | + ǫ − Z S d − | y |− ǫ d Ω F k (ˆ x ) R ( φ ( x ) ψ ( y )) = ( | y | + ǫ ) − k − ∆ φ k ψ ( y ) − ( | y | − ǫ ) − k − ∆ ψ ( y ) φ k = | y | − k − ∆ [ φ k , ψ ( y )] + O ( ǫ ) . (6.2)If the OPE of φ ( x ) ψ ( y ) is non-singular, then R ( φ ( x ) ψ ( y )) is bounded in the integrals onthe left-hand side, so that the left-hand side is O ( ǫ ). Sending ǫ →
0, we see that thecommutator [ φ k , ψ ( y )] vanishes. Therefore the commutator only depends on the singularpart of the OPE.In fact, the deformability property of Cauchy fields as described in section 8 will allowus to deform the integration domain on the lhs of (6.2) to a small sphere centered at y . We then see that the commutator depends only on the singular part of the OPE of φ ( x ) ψ ( y ) that is at least as singular as | x − y | − ( d − . For the present purpose we do notneed this stronger result. The analysis in section 5 showed that there are no modes φ k in the range − ∆ < k < ∆.This has drastic consequences for the operator product expansions of Cauchy fields.Suppose φ ( x ) and φ ( x ) are Cauchy fields. We can suppose, without loss of generality,that ∆ ≥ ∆ (exchanging φ ↔ φ if necessary).26he conformal highest weight state corresponding to φ ( x ) is φ (0) | i = φ , − ∆ | i (6.3)The operator product φ ( x ) φ (0) is given by φ ( x ) φ (0) | i = X | k |≥ ∆ r − k − ∆ Y | k +∆ − m |− m (ˆ x ) φ ,k φ , − ∆ | i . (6.4)The singular part is( φ ( x ) φ (0)) sing | i = X k ≥ ∆ r − k − ∆ Y k +∆ − m (ˆ x ) φ ,k φ , − ∆ | i . (6.5)The state | φ ′ k − ∆ i = φ ,k φ , − ∆ | i (6.6)has conformal weight − k + ∆ . But k ≥ ∆ ≥ ∆ . By unitarity, there are no states withconformal weight <
0. Therefore | φ ′ k − ∆ i = 0 unless k = ∆ = ∆ , in which case | φ ′ k − ∆ i has weight 0, so must be proportional to the ground state, φ ,k φ , − ∆ | i = δ k, ∆ C | i , ∆ = ∆ . (6.7)Therefore, using ∆ = | s | + m ,( φ ( x ) φ (0)) sing = ( , ∆ = ∆ ,r − Y | s | (ˆ x ) C , ∆ = ∆ . (6.8)From the argument in section 6.1 it thus follows that all commutators between Cauchyfields of different scaling dimensions must vanish, and all commutators of Cauchy fieldsof the same scaling dimension must be proportional to the identity operator.We have a decomposition of any Cauchy field φ ( x ) into creation and annihilation oper-ators, so we can evaluate any correlation function by applying Wick’s theorem, commutingdestruction operators to the right and creation operators to the left. It thus follows thatall correlation functions of unitary Cauchy fields factorize into two point functions. Allunitary Cauchy fields are thus indeed free fields. Suppose φ ( x ) is a Cauchy field of dimension ∆. If there is a local energy momentumtensor T µν ( x ), it must have dimension d . The operator product T µν ( x ) φ (0) must containthe field φ (0) with a canonical non-zero coefficient. Therefore h φ † ( ∞ ) T µν ( x ) φ (0) i 6 = 0 . (6.9)Therefore, by a global conformal transformation h T µν ( ∞ ) φ † ( x ) φ (0) i 6 = 0 . (6.10)27herefore the operator product of φ † ( x ) φ (0) must contain a dimension d field. But thelowest dimension field occurring in the operator product φ † ( x ) φ (0), besides the identity,has dimension 2∆. So there cannot be a local energy momentum tensor T µν ( x ) unless2∆ ≤ d . Since ∆ = | s | + ( d − /
2, this is | s | ≤ | s | > j > Now that we know that every unitary Cauchy conformal field is free, we can finish theanalysis of the mode expansion. φ k , | k | ≥ ∆ are all non-zero Suppose φ ( x ) is a unitary Cauchy field of spin V s and scaling dimension ∆ = | s | + n − φ k , | k | ≥ ∆ are non-zero by calculating the commutators ofthe modes recursively using the raising operators P µ , and seeing that the commutatorsare non-zero.Let φ † ( x ) be the adjoint field, so h φ † ( x ) φ (0) i 6 = 0. The scaling dimension of φ † ( x ) isalso ∆. The spin of φ † ( x ) must be one of V ± s . Which one will be determined later.Write the mode expansions (5.35) in the form φ ( x ) = X k +∆ ∈ Z , | k |≥ ∆ r − k − ∆ φ k (ˆ x ) , φ † ( x ) = X k +∆ ∈ Z , | k |≥ ∆ r − k − ∆ φ † k (ˆ x ) . (7.1)The (anti-)commutators of the modes are multiples of the identity, so[ φ † k (ˆ x ) , φ k (ˆ x )] ∓ = C k (ˆ x ) . (7.2)The commutator is used for s ∈ Z , the anti-commutator for s ∈ + Z .The commutators of P µ with the modes are obtained by calculating[ x µ P µ , φ ( x )] = x µ ∂ µ φ ( x ) (7.3) X k +∆ ∈ Z , | k |≥ ∆ r − k − ∆ x µ [ P µ , φ k (ˆ x )] = X k +∆ ∈ Z , | k |≥ ∆ r − k − ∆ ( − k − ∆) φ k (ˆ x ) (7.4)which gives [ˆ x µ P µ , φ k +1 (ˆ x )] = ( − k − ∆) φ k (ˆ x ) . (7.5)28hen we derive the recursion relation:0 = h ˆ x µ P µ , [ φ † k +1 (ˆ x ) , φ − k (ˆ x )] ∓ i (7.6)= h [ˆ x µ P µ , φ † k +1 (ˆ x )] , φ − k (ˆ x ) i ∓ + h φ † k +1 (ˆ x ) , [ˆ x µ P µ , φ − k (ˆ x )] i ∓ (7.7)= h ( − k − ∆) φ † k (ˆ x ) , φ − k (ˆ x ) i ∓ + h φ † k +1 (ˆ x ) , ( k + 1 − ∆) φ − k − (ˆ x ) i ∓ (7.8)which is ( k + 1 − ∆) C k +1 (ˆ x ) = ( k + ∆) C k (ˆ x ) (7.9)which we solve to get C k (ˆ x ) = (cid:18) k + ∆ − − (cid:19) C ∆ (ˆ x ) k ≥ ∆ (7.10) C k (ˆ x ) = (cid:18) − k + ∆ − − (cid:19) C − ∆ (ˆ x ) k ≤ − ∆ . (7.11)The second equation is equivalent to the first under φ ↔ φ † , so we only need to consider k ≥ ∆.The mode φ − ∆ has l = 0, so φ − ∆ (ˆ x ) is independent of ˆ x , so the two-point function is h φ † ( x ) φ (0) i = h || x | − φ † ∆ (ˆ x ) φ − ∆ (ˆ x ) | i = | x | − C ∆ (ˆ x ) (7.12)therefore C ∆ (ˆ x ) is not identically zero, therefore, by the recursion relation (7.10), none ofthe C k (ˆ x ), k ≥ ∆ are identically zero, therefore none of the modes φ k , φ †− k , k ≥ ∆, areidentically zero. The same is true exchanging φ ↔ φ † . So all the modes φ k , φ † k , | k | ≥ ∆,are non-zero. φ † ( x )The two point function is h φ † ( x ) φ (0) i = h | φ † ∆ φ − ∆ | i (7.13)with φ − ∆ ∈ V s , (7.14)so φ † ∆ must be in V ∗ s , the dual space to V s . The representation V s is unitary, V s = V † s = ¯ V ∗ s ,so the dual space is the same as the complex conjugate, V ∗ s = ¯ V s . For d = 2 n even, all ofthe V s are generated by tensor products of V ± with itself, and V ± is the chiral spinorrepresentation. For d odd, V is the spinor representation. So, from the reality propertiesof spinors, φ † ∆ ∈ V ∗ s = ¯ V s = V s d = 2 n, n even V − s d = 2 n, n odd V s d odd . (7.15)29omparing to the representation of φ † ∆ as given by (5.55) and (5.56), we find that thespin of the adjoint field φ † ( x ) must be V ¯ s = V − s d = 2 n, n even V s d = 2 n, n odd V s d odd . (7.16) The mode φ − k lies in a representation given by (5.55) or (5.56), and the mode φ † k lies inthe dual representation. Using upper indices α, β for the representation of φ − k , and lowerindices α, β for the dual representation, the (anti-)commutator of the modes takes theform [ φ † k,β , φ α − k ] ∓ = c k δ αβ (7.17)where c k is a number.The terms in the mode expansions (7.1) take the form φ − k (ˆ x ) = Y l ( s, − k ; ˆ x ) α φ α − k , φ † k (ˆ x ) = Y l (¯ s, k ; ˆ x ) β φ † k,β , (7.18)where Y l ( s, − k ; ˆ x ) α is the vector spherical harmonic expressing the Clebsch-Gordan coef-ficients between S l , V s , and the representation of φ − k , only the last of which is labeled byan explicit index, α , and similarly for Y l (¯ s, k ; ˆ x ) β .Equation (7.2) becomes C k (ˆ x ) = c k Y l (¯ s, k ; ˆ x ) α Y l ( s, − k ; ˆ x ) α . (7.19)In particular, the starting point of the recursion formula is C ∆ (ˆ x ) = c ∆ Y | s | (¯ s, ∆; ˆ x ) α Y ( s, − ∆; ˆ x ) α = c ∆ Y | s | (¯ s, ∆; ˆ x ) , (7.20)where Y | s | (¯ s, ∆; ˆ x ) is the Clebsch-Gordan for S | s | , V ¯ s , and V s . The 2-point function is h φ † ( x ) φ (0) i = h || x | − φ † ∆ (ˆ x ) φ − ∆ (ˆ x ) | i = | x | − C ∆ (ˆ x ) = | x | − c ∆ Y | s | (¯ s, ∆; ˆ x ) . (7.21)The result (7.10) of the recursion gives us c k Y l (¯ s, k ; ˆ x ) α Y l ( s, − k ; ˆ x ) α = (cid:18) k + ∆ − − (cid:19) c ∆ Y | s | (¯ s, ∆; ˆ x ) , k ≥ ∆ , (7.22)which determines the numbers c k , after some group-theoretic work which we refrain fromdoing. 30 Deformability
Suppose φ ( x ) is a Cauchy field in the Cauchy conformal representaton ( V, ∆), not neces-sarily unitary. We want to be able to use the same deformation of contour arguments for φ ( x ) as for a holomorphic field in 2 dimensions. In particular, we would like to calculatea commutator [ φ k , ψ ( y )] by deforming the integrals for the mode φ k to an integral over asmall sphere centered at y , so that the commutator can be extracted from the operatorproduct expansion of φ ( x ) ψ ( y ).A deformable integral over codimension 1 surfaces in space-time is given by a conservedcurrent. So we want to represent each mode of φ ( x ) by smearing over a codimension 1surface S by a vector-valued current f µ ( x ), by φ [ f, S ] = Z S d d − x ˆ n µ h f µ ( x ) , φ ( x ) i , f µ ( x ) ∈ V (8.1)where the inner-product on the rhs is the invariant hermitian inner-product on V . Wecan deform S to any cobounding S ′ by Stokes’ theorem, if ∂ µ h f µ ( x ) , φ ( x ) i = 0 . (8.2)Since φ ( x ) satisfies a linear first order differential equation that contains only a linearcombination of derivatives of φ , f µ ( x ) must separately satisfy ∂ µ f µ ( x ) = 0 (8.3)and h f µ ( x ) , ∂ µ φ ( x ) i = 0 . (8.4)The Cauchy differential equation (3.4) can be written ∂ ν φ = (1 − ˆ P ) µν ∂ µ φ (8.5)where ˆ P is the self-adjoint projection on the null subspace in level 1 of the Verma module.So equation (8.4) is equivalent to (1 − ˆ P ) µν f ν ( x ) = 0 . (8.6)The smearing currents f µ ( x ) satisfying (8.3) and (8.6) are exactly those defining a mode φ [ f, S ] that can be evaluated on any deformation S ′ of S . We want to show now that there are enough smearing currents satisfying (8.3) and (8.6)to capture all the modes of φ ( x ). This means that the contour deformation technique canbe used to calculate with the modes. 31or simplicity, we take the codimension 1 surface S to be R d − . The Cauchy differentialequation is first order, so it will certainly be possible to covariantize. The complexifiedtangent space of space-time is C d . Decompose it into the tangential and normal subspaces, C d = C d − ⊕ C , ∂ µ = ( ~∂, ∂ d ) . (8.7)The projection matrix ˆ P : C d ⊗ V → C d ⊗ V (8.8)that enters in the Cauchy first order differential equation (3.4) and in equation (8.6) forthe smearing current decomposes into a block matrixˆ P = ←→ P ~P~P † ˆ P dd ! . (8.9)ˆ P is a self-adjoint projector, so the operators ˆ P dd and ←→ P are self-adjoint, but by themselvesthey are not projectors. The projector condition ˆ P = ˆ P is ←→ P + ~P ~P † = ←→ P or ~P ~P † = ←→ P (1 − ←→ P ) , (8.10) ←→ P ~P + ~P ˆ P dd = ~P or ~P ˆ P dd = (1 − ←→ P ) ~P , (8.11) ~P †←→ P + ˆ P dd ~P † = ~P † or ~P †←→ P = (1 − ˆ P dd ) ~P † , (8.12) ~P † ~P + ( ˆ P dd ) = ˆ P dd or ~P † ~P = ˆ P dd (1 − ˆ P dd ) . (8.13)Decomposing f into components ( ~f , f d ) and using the block matrix decomposition (8.9)of ˆ P , equation (8.6) becomes ←→ P ~f ( x ) + ~P f d ( x ) = ~f ( x ) (8.14) ~P † ~f ( x ) + ˆ P dd f d ( x ) = f d ( x ) . (8.15)Our approach is to pick f d ( x ), and then use the first equation (8.14) to determine ~f ,solving (cid:16) − ←→ P (cid:17) ~f ( x ) = ~P f d ( x ) , (8.16)to express ~f in terms of f d . This has a solution because the projector condition (8.10)implies that the image of ~P is orthogonal to the eigenspace ←→ P = 1. We can thus definethe inverse of 1 − ←→ P to be zero on the eigenspace ←→ P = 1 to obtain ~f ( x ) = (cid:0) − ←→ P (cid:1) − ~P f d ( x ) + ~f ( x ) , ←→ P ~f ( x ) = ~f ( x ) , (8.17)for some arbitrary ~f ( x ) in the eigenspace ←→ P = 1. The spatial piece ~f of the current isthus determined by f d up to the component of eigenvalue 1 under ←→ P . Using the projectorconditions (8.10-8.13), it is straightforward to check that (8.15) is then automatically32atisfied. This shows that (8.6) imposes no further constraints on our choice of f d . Theonly remaining condition on f µ ( x ) is (8.3), ∂ d f d ( x ) = − ~∂ · ~f ( x ) . (8.18)It follows that ∂ d f d as given by (8.18) and ∂ d ~f as given by (8.17) can be integrated forarbitrary initial data f d ( x ) on S . The corresponding mode is φ [ f, S ] = Z S d d − ~x h f d ( x ) , φ ( x ) i . (8.19)Since f d ( x ) is arbitrary on S , there are enough deformable smearing currents to captureevery mode of φ ( x ). In principle, for V = V s , we could now construct explicit smearingfunctions f µk ( x ) such that φ [ f k , S d − ] = φ k (8.20)for the modes φ k defined in section 5, but we will refrain from doing so here. Equation (8.1) suggests a duality between the Cauchy fields φ ( x ) and the smearing cur-rents f µ ( x ). But not every initial data φ ( x ) on S can be integrated to a solution of theCauchy equation. We will see below that the Cauchy equation is over-determined. Sothere must be an equivalence relation – a gauge symmetry – on the smearing currents.In fact, equations (8.3) and (8.6) on f µ ( x ) have the gauge symmetries f µ ( x ) → f µ ( x ) + δf µ ( x ) (8.21) δf µ ( x ) = ∂ σ g µσ ( x ) , g µσ ( x ) = − g σµ ( x ) , (1 − ˆ P ) µν g νσ ( x ) = 0 . (8.22)The mode φ [ f ] is gauge-invariant because (8.22) implies φ [ f + δf, S ] − φ [ f, S ] = Z S d d − x ˆ n µ h ∂ σ g µσ ( x ) , φ ( x ) i (8.23)= Z S d d − x ˆ n µ h g σµ ( x ) , ∂ σ φ ( x ) i (8.24)= 0 . (8.25)We would want to show that the solutions of equations (8.3) and (8.6) on f µ ( x ) modulothe gauge transformations (8.22) are exactly dual to solutions of the Cauchy differentialequation (3.4), with the pairing given by (8.1). We do not know how to do that. Itrequires describing the space of smearing functions f d ( x ) modulo gauge transformations,and the space of initial data φ ( x ) satisfying the overdetermination conditions. We suspectthat [22] is relevant.Here, we will only take a first look at the overdetermination conditions. Again takingthe codimension 1 surface S to be R d − , we ask for the conditions that φ ( x ) must satisfy33n S in order to extend to a solution of the Cauchy equation (3.4) off of S . We deriveonly the first such over-determination condition, which is a first order differential equationon S . Using the block decomposition (8.9) of ˆ P , the Cauchy differential equation (3.4)becomes ←→ P ~∂φ + ~P ∂ d φ = 0 (8.26) ~P † ~∂φ + ˆ P dd ∂ d φ = 0 . (8.27)The Cauchy condition is equivalent to the invertibility of ˆ P dd , so (8.27) can be written ∂ d φ = − (cid:0) ˆ P dd (cid:1) − ~P † ~∂φ , (8.28)which gives the normal derivative in terms of the data on S . This is the Cauchy property.Now we substitute for ∂ d φ in equation (8.26), getting ←→ P ~∂φ = 0 , (8.29)where ←→ P = ←→ P − ~P (cid:0) ˆ P dd (cid:1) − ~P † . (8.30)From the identities (8.10)–(8.13) it follows that (cid:0) ←→ P (cid:1) = ←→ P , ←→ P ←→ P = ←→ P ←→ P = ←→ P (8.31)so ←→ P is the projection on the eigenspace ←→ P = 1.The first-order differential equation (8.29) is the first over-determination condition. Itcontains no derivatives in the normal direction, so it is a differential equation on S . TheCauchy differential equation (3.4) is equivalent to the combination of (8.28) and (8.29).In Appendix C we work out the first-order over-determination condition explicitly for theunitary Cauchy representations V = V s .The covariant form of the first-order over-determination condition is the differentialequation on S , ˆ P ( x ) µν ∂ µ φ ( x ) = 0 , (8.32)where ˆ P ( x ) is the projection on the eigenspace with eigenvalue 1 of the operator ←→ P ( x ) = [ P ( T x S ) ⊗
1] ˆ P [ P ( T x S ) ⊗
1] (8.33)on C d ⊗ V , where P ( T x S ) is the projection on the complexified tangent space to S at x ,which is a subspace of C d .The first order over-determination condition (8.29) is a necessary condition on theinitial data φ ( x ) on S , but it is not necessarily a sufficient condition for integrating theCauchy equation off S . Equation (8.28) can be integrated to determine φ ( x ) uniquely onany nearby surface S ′ . But further integration requires φ ( x ) on S ′ to continue to satisfy(8.29). We need the integrability condition ∂ d ←→ P ~∂φ = 0 . (8.34)34et us write (8.28) as( ∂ d + A d ) φ = 0 , A d = A jd ∂ j = − (cid:0) ˆ P dd (cid:1) − (cid:0) ~P † (cid:1) j ∂ i (8.35)and (8.29) as A i φ = 0 , A i = ←→ P ki ∂ k φ . (8.36)The integrability condition (8.34) is0 = ∂ d A i φ = A i ∂ d φ = A i ( − A d ) φ = [ A d , A i ] φ − A d A i φ = [ A d , A i ] φ , (8.37)since we already have the first-order condition A i φ = 0. So we need the second-orderover-determination condition[ A d , A i ] φ = [ A jd , A ki ] ∂ j ∂ k φ = 0 . (8.38)If the second-order condition over-determination condition follows from the first-ordercondition, then we are done. If not, then we have to impose the second-order condition inaddition to the first-order condition, and then check the integrability of the second-ordercondition, and so on.In the unitary case, whatever the complete set of over-determination conditions, weknow that, when S is the unit sphere, all the solutions are the φ k (ˆ x ), because these arethe only possible modes and all are non-zero.The structure of the over-determination conditions is determined by the pattern ofhighest weight vectors in the Verma module. The null space ( P d + A jd P j ) φ generates asubmodule of the Verma module – all null states, all perpendicular to the entire Vermamodule. Complementary to this null submodule is the submodule with basis { P i · · · P i N | φ i} . (8.39)This is the so ( d − ,
2) Verma module generated by the so ( d ) representation V consideredas a representation of so ( d − A ki P k | φ i on level 1. The second-order condition corresponds to a nullspace on level 2. The second-order condition is independent if the corresponding level 2null subspace is not contained in the sub-module generated by the level 1 null space. Sospecifying the over-determination conditions is equivalent to finding the minimal set ofgenerators for the full null sub-module of the so ( d − ,
2) Verma module.
Acknowledgments
We thank Matt Buican, Thomas Dumitrescu, Simeon Hellerman, Siddhartha Sahi,Scott Thomas, and Sasha Zhiboedov for useful discussions. Peter Goddard helped uswith the history of 2d holomorphic fields. Siddhartha Sahi pointed us to the proof in [21]of the branching rule we used in section 5.3. Nolan Wallach pointed us to the generalunitarity bound in [20], and to [22], and suggested a possible argument to finish theclassification of the non-unitary Cauchy fields (section B.3). Nicolas Boulanger, Dmitry35onomarev, Evgeny Skvortsov and Massimo Taronna gave useful explanation about thestatus of the conformal Coleman-Mandula theorem. We are grateful to them for theirhelp.DF and CAK were supported by the Rutgers New High Energy Theory Center. CAKthanks the Aspen Center for Physics and the Harvard University High Energy TheoryGroup for hospitality, where part of this work was completed. CAK was supported in partby National Science Foundation Grant No. PHYS-1066293, by the Swiss National ScienceFoundation through the NCCR SwissMAP, and by U.S. DOE Grants No. doe-sc0010008and DOE-ARRA-SC0003883.
Appendices
A The irreducible representations of so ( d ) A.1 The highest weights representations
In this appendix we collect some basic results in the representation theory of so ( d ), takenfrom e.g. [23]. The irreducible representations of so ( d ) are written V λ , where λ is thehighest weight. For d = 2 n , the highest weights of the irreducible representations are λ = ( λ , . . . , λ n ) , λ ≥ λ ≥ · · · ≥ λ n − ≥ | λ n | . (A.1)For d = 2 n + 1, the highest weights of the irreducible representations λ = ( λ , . . . , λ n ) , λ ≥ λ ≥ · · · ≥ λ n − ≥ λ n ≥ . (A.2)In both cases, the λ i are all integers (for the vector representations) or all half-integers(for the spinor representations). 36ome notable representations are given by:for all d : trivial representation V = V (0 , ··· , (A.3)fundamental representation C d = V (1 , ,..., (A.4)symmetric traceless l -tensors S l = V ( l, ,..., (A.5)for d = 2 n : p -forms, 0 ≤ p ≤ n − p = V (1 ,..., |{z} p , ,..., |{z} n − p ) (A.6)(anti-)self-dual n -forms Λ n ± = V (1 , ,..., , ± (A.7)chiral spinors S ± = V ( , , ··· , , ± ) (A.8)for d = 2 n + 1: p -forms, 0 ≤ p ≤ n Λ p = V (1 ,..., |{z} p , ,..., |{z} n − p ) (A.9)spinors S = V ( , , ··· , , ) (A.10)The quadratic Casimir invariant — with the normalization given in equation (2.27) — is,for all d , C d ( V λ ) = 12 n X i =1 λ i ( λ i + d − i ) (A.11)Decomposing the tensor product C d ⊗ V λ into irreducibles gives C d ⊗ V λ = ⊕ λ ′ = λ ± ǫ k V λ ′ d = 2 n ⊕ λ ′ = λ,λ ± ǫ k V λ ′ d = 2 n + 1 (A.12)where ǫ k = (0 , . . . , , , , . . . ) is the weight that has 1 in the k -th position and 0 elsewhere,and the sums include all the λ ′ = λ ± ǫ k that are highest weights for so ( d ).Finally we will need the branching rules of an so ( d ) irreducible V λ decomposing into so ( d −
1) irreducibles V d − µ , V λ = ⊕ µ V d − µ (A.13)where the sum ranges over all µ that are highest weights for so ( d −
1) satisfying λ − µ ∈ Z and λ ≥ µ ≥ λ ≥ µ ≥ · · · ≥ λ n − ≥ µ n − ≥ | λ n | , d = 2 n ,λ ≥ µ ≥ λ ≥ µ ≥ · · · ≥ λ n − ≥ µ n − ≥ λ n ≥ | µ n | , d = 2 n + 1 . (A.14)37 .2 The eigenvalues of ˆ M In what follows we need to know the eigenvalues ˆ M λ ′ ,λ of ˆ M acting on the irreduciblecomponents V λ ′ ⊂ C d ⊗ V λ ,ˆ M λ ′ ,λ = C d ( V λ ) + C d ( C d ) − C d ( V λ ′ ) . (A.15)For the λ ′ of interest, the ones occurring in (A.12), this evaluates toˆ M λ + ǫ k ,λ = k − − λ k (A.16)ˆ M λ − ǫ k ,λ = λ k + d − − k (A.17)ˆ M λ,λ = 12 ( d −
1) (A.18)Recall that the number ˆ M λ ′ ,λ occurs as an eigenvalue only if λ ′ is actually a highest weight.To check unitarity, it is important to identify the largest eigenvalue of ˆ M . For d = 2 n ,the numbers ˆ M λ ′ ,λ satisfy the inequalitiesfor λ n > M λ + ǫ ,λ < · · · < ˆ M λ + ǫ n − ,λ < ˆ M λ + ǫ n ,λ < ˆ M λ − ǫ n ,λ < ˆ M λ − ǫ n − ,λ < · · · < ˆ M λ − ǫ ,λ , (A.19)for λ n = 0:ˆ M λ + ǫ ,λ < · · · < ˆ M λ + ǫ n − ,λ < ˆ M λ + ǫ n ,λ = ˆ M λ − ǫ n ,λ < ˆ M λ − ǫ n − ,λ < · · · < ˆ M λ − ǫ ,λ , (A.20)for λ n < M λ + ǫ ,λ < · · · < ˆ M λ + ǫ n − ,λ < ˆ M λ − ǫ n ,λ < ˆ M λ + ǫ n ,λ < ˆ M λ − ǫ n − ,λ < · · · < ˆ M λ − ǫ ,λ . (A.21)For d = 2 n + 1, the ˆ M λ ′ ,λ satisfyfor λ n >
0: ˆ M λ + ǫ ,λ < · · · < ˆ M λ + ǫ n ,λ < ˆ M λ,λ < ˆ M λ − ǫ n ,λ < · · · < ˆ M λ − ǫ ,λ , (A.22)for λ n = 0: ˆ M λ + ǫ ,λ < · · · < ˆ M λ + ǫ n ,λ < ˆ M λ,λ = ˆ M λ − ǫ n ,λ < · · · < ˆ M λ − ǫ ,λ . (A.23) A.3 Example: d = 4Let us give the more familiar expressions for d = 4 here. The general formula becomes C d ( V λ ) = 12 λ ( λ + 2) + 12 λ . (A.24)We usually write so (4) = su (2) L × su (2) R . The irreducible representations of so (4) are thetensor products of su (2) L,R irreducibles j L , j R . The corresponding so (4) highest weightis λ = j L + j R , λ = j L − j R , (A.25)giving C d ( V λ ) = j L ( j L + 1) + j R ( j R + 1) . (A.26)38 Classification of Cauchy fields
Next, we want to classify the Cauchy conformal fields. That is, we want to determinewhich representations V λ and which scaling dimensions ∆ satisfy the algebraic condition A1 , which is equivalent to the Cauchy property. Condition A1 is the condition that thematrix ˆ P d : V λ → ( C d ⊗ V λ ) ∆ be injective. Since we picked out the direction d , ˆ P d is notfully so ( d )-invariant, but it is so ( d − so ( d )representations into so ( d −
1) representations.We can obtain necessary conditions for ˆ P d to be injective by using Schur’s Lemma.Any non-trivial irreducible so ( d −
1) representation V µ that occurs in V λ must also occurin ( C d ⊗ V λ ) ∆ . Otherwise the so ( d −
1) invariant map ˆ P d must map it to the trivialrepresentation, which means that ˆ P d cannot be injective. Therefore, a necessary conditionfor ˆ P d to be injective is C1 Every inequivalent irreducible so ( d − V µ that occurs in V λ mustalso occur in ( C d ⊗ V λ ) ∆ .If we demand that the representation V λ lead to a unitary representation of the fullconformal group, it is necessary that ∆ is the largest eigenvalue of ˆ M . C2 For ( V λ , ∆) to be a unitary conformal representation, ∆ must be the largest eigen-value of ˆ M .Let us first discuss condition C1 . For λ non-trivial, the branching rules given in (A.14)imply that the only λ ′ which satisfy this necessary condition are d = 2 n : A : λ ′ = λ + ǫ B + : λ ′ = λ − ǫ n , λ n > , B − : λ ′ = λ + ǫ n , λ n < ,d = 2 n + 1 : A + : λ ′ = λ + ǫ C + : λ ′ = λ , (B.1)To see this, note that for instance if λ ′ = λ − ǫ j , j < n , then µ = λ is not in λ ′ ; for λ ′ = λ + ǫ j , µ = ( λ , . . . , λ j , λ j , . . . ) is in λ but not in λ ′ . Similar arguments eliminate theother cases. For λ = 0, the only λ ′ is C d , which satisfies the necessary condition. B.1 Assuming unitarity
Let us now show that the unitary Cauchy fields with spin V s listed in section 4.4 are theonly unitary Cauchy fields that satsify condition C2 . This completes the classification ofunitary Cauchy fields. 39irst note from (A.19) – (A.23) that unless λ = 0, λ ′ = λ + ǫ never leads to thelargest eigenvalue. In the following always assume that λ = 0.Next, for d = 2 n consider the case λ n > λ ′ = λ − ǫ n . From (A.19) we see that λ ′ canonly be the largest eigenvalue if none of the λ − ǫ j are representations of so ( d ). This isonly the case if λ = ( | s | , . . . , | s | , | s | ). Similarly, for λ n < λ ′ = λ + ǫ n , we see from (A.21)that λ ′ = λ + ǫ n is only the largest eigenvalue if λ = ( | s | , . . . , | s | , −| s | ). This establishesthe claim for even d .For odd d , λ ′ = λ has to lead to the largest eigenvalue. From (A.22) no representationswith λ − ǫ j can appear, so that λ = ( | s | , . . . , | s | , | s | ). Moreover λ − ǫ n must not appeareither, which is only the case if s = 1 /
2. This establishes the claim for odd d . B.2 The non-unitary cases: d = 3 and d = 4Let us now investigate the injectivity condition C1 if we do not require unitarity. Wedid not develop the general theory, but the special cases d = 3 and d = 4 are relativelystraightforward to work out.For d = 3, the problem reduces to decomposing so (3) into so (2) representations, whichis decomposing su (2) into u (1) representations. Since the latter are 1 dimensional it isenough to check that P dλ,λ ′ is non-vanishing on each u (1) representation µ that occursin λ , which is equivalent to the non-vanishing of the relevant Clebsch-Gordan coefficientbetween the fundamental C of so (3), λ , and λ ′ . In terms of Wigner 3- j symbols thecondition is thus (cid:18) λ λ ′ µ − µ (cid:19) = 0 , | µ | ≤ λ , λ − µ ∈ Z , (B.2)where by assumption λ ′ occurs in the fusion of λ with C . There is then just one additionalselection rule on the 3- j symbols, namely (cid:18) λ λ ′ µ − µ (cid:19) = 0 iff 1 + λ + λ ′ ∈ Z + 1 and µ = 0 . (B.3)This never affects the case A , but if λ = λ ′ is integer, then P dλ,λ ′ vanishes on µ = 0.Condition C1 is thus satisfied for d = 3 A : λ ′ = λ + ǫ injective C : λ ′ = λ injective iff λ ∈ + Z . (B.4)For d = 4, we show that the map P dλ,λ ′ is injective in all three of the non-unitary cases A , B + , B − .Use so (4) = sl (2) ⊕ sl (2). The representation ( λ , λ ) of so (4) is the representation( j ) ⊗ ( k ) of sl (2) ⊕ sl (2), with λ = j + k , λ = j − k . (B.5)40he fundamental representation ( ) of sl (2) is C with invariant anti-symmetric tensor ǫ ab , a, b = 1 ,
2. Take as basis for the sl (2) representation ( j ) the symmetric tensors ofrank 2 j on C . For the second sl (2), take ( C ) ∗ as the fundamental representation. Thisgives a basis for λ as tensors on C , t a ··· a j b ··· b k (B.6)symmetric separately in the a i and in the b i . The representation C d has basis t ab . The so (3) invariant is δ ab .The case A is j ′ = j + , k ′ = k + 1 /
2. The projection ˆ P is thus given by the mapˆ P : C d ⊗ V λ → ( C d ⊗ V λ ) λ ′ , ( t ab , t a ··· a j b ··· b k ) Sym a Sym b (cid:0) t a ··· a j b ··· b k t ab (cid:1) (B.7)The map P dλ,λ ′ is t a ··· a j b ··· b k Sym a Sym b (cid:16) t a ··· a j b ··· b k δ a j +1 b k +1 (cid:17) (B.8)which is pretty clearly injective. To see this, note that the irreducible components under so ( d −
1) have basis t a ··· a j b ··· b k = Sym a Sym b (cid:16) w a ··· a j − r b ··· b k − r δ a j − r +1 b k − r +1 · · · δ a j b k (cid:17) , w a ··· a j − r − cb ··· b k − r − c = 0 . (B.9)In this basis, P dλ,λ ′ is just the identity on each so ( d −
1) component of λ .The case B + is j > k > j ′ = j − , k ′ = k + 1 /
2. The map P dλ,λ ′ is t a ··· a j b ··· b k Sym b (cid:0) t a ··· a j b ··· b k ǫ a j b k +1 (cid:1) . (B.10)Again look at the action of P dλ,λ ′ on the so ( d −
1) component of λ with basis elementsgiven in (B.9), t a ··· a j b ··· b k Sym a Sym b (cid:16) w a ··· a j − r − a j b ··· b k − r ǫ a j b k − r +1 δ a j − r b k − r +2 · · · δ a j − b k +1 (cid:17) . (B.11)We only have to show that this is non-zero. Take the tensor w to have only one non-zerocomponent, w ··· ··· = 1. Then w ··· ··· ǫ = 0. So P dλ,λ ′ is injective.The case B − is the same as B + with j ↔ k .So, for d = 4, in all three of the non-unitary cases A , B + , B − , the map P dλ,λ ′ isinjective. B.3 The non-unitary case: d > V λ an irreducible representation of so ( d ), and V λ ′ one of the irreduciblecomponents of C d ⊗ V λ listed in (B.1), is the map P dλ,λ ′ : V λ → V λ ′ injective, where P dλ,λ ′ ( v ) = P λ,λ ′ (ˆ e d ⊗ v ) (B.12)where P λ,λ ′ : C d ⊗ V λ → V λ ′ (B.13)is the projection on the component V λ ′ . We cannot answer the question, but we convey asuggestion from N. Wallach: 41n the case of λ + ǫ mapping into λ ⊗ ǫ , P dλ,λ ′ is injective since it is justCartan multiplication (which is multiplication in an integral domain). In theeven dimensional case B ± , P dλ,λ ′ is adjoint to Cartan multiplication (whichis multiplication in an integral domain). That is we look at λ mapping into( λ + ǫ ) ⊗ ǫ , by realizing λ + ǫ in λ ⊗ ǫ and contracting. [24] C First-order over-determination conditions for V = V s As an illustration, let us now work out the first-order over-determination conditions ex-plicitly for the Cauchy representations V s . Take d = 2 n and V = V s , s = 0.Let us first work out the eigenvalues of P dd and ←→ P . Since they are self-adjoint, we candecompose C ⊗ V into eigenspaces of ˆ P dd and decompose C d − ⊗ V into eigenspaces of ←→ P C ⊗ V = ⊕ λ V λ , ˆ P dd (cid:12)(cid:12)(cid:12) V λ = λ (C.1) C d − ⊗ V = ⊕ λ W λ , ←→ P (cid:12)(cid:12)(cid:12) W λ = λ , (C.2)but since they are not projectors, they can have eigenvalues different from λ = 0 , v λ ∈ V λ and w λ ∈ W λ , ~P † ~P v λ = λ (1 − λ ) v λ (C.3)( ˆ P dd + λ − ~P † w λ = 0 (C.4)( ←→ P + λ − ~P v λ = 0 (C.5) ~P ~P † w λ = λ (1 − λ ) w λ . (C.6)The Cauchy condition Ker ˆ P dd = 0 means that V = 0, so that (C.3) and (C.6) respectivelyimply that Ker ~P = V , Im ~P † = X λ =0 , V λ (C.7)Ker ~P † = W ⊕ W , Im ~P = X λ =0 , W λ (C.8)whereas the second and third equation are equivalent to ~P V λ = W − λ , ~P † W λ = V − λ , λ = 0 , . (C.9)Let us now evaluate these expression for Cauchy representations V s . The projection ˆ P isˆ P = ˆ M − ( n − − ∆)∆ − ( n − − ∆) , (C.10)42ith ∆ = n − | s | in the unitary case and ∆ = −| s | in the non-unitary case. Sinceˆ M dd = 0, ˆ P dd has only one eigenvalue,ˆ P dd = − ( n − − ∆)∆ − ( n − − ∆) . (C.11)From (C.9) it follows that ←→ P has at most eigenvalues λ = 1 − − ( n − − ∆)∆ − ( n − − ∆) = ∆2∆ − ( n −
1) (C.12)and λ = 0 and 1. Having established that, let us work out the eigenspaces of ←→ P explicitly.Using ←→ P = ˆ M d − − ( n − − ∆)∆ − ( n − − ∆) , (C.13)the strategy is of course to use the so ( d −
1) invariance of ˆ M d − to express it in terms of so ( d −
1) Casimirs. Decomposing V s into representations of so ( d − V s = V µ | s | , µ | s | = ( | s | , . . . , | s | ) (C.14) C d − ⊗ V s = V µ | s | ⊕ V µ | s | + ǫ ⊕ V µ | s | − ǫ n − (C.15)where the right-most summand does not occur for | s | = 1 /
2. The quadratic Casimirs are C d − ( V µ ) = 12 n − X i =1 µ i ( µ i + d − − i ) (C.16) C d − ( C d − ) = 12 ( d −
2) = n − C d − ( V µ | s | ) = 12 ( n − | s | ( | s | + n −
1) (C.18) C d − ( V µ | s | + ǫ ) = C d − ( V µ | s | ) + | s | + n − C d − ( V µ | s | − ǫ n − ) = C d − ( V µ | s | ) − | s | (C.20)Writing ˆ M d − asˆ M d − = ⊗ C d − ( V µ | s | ) + C d − ( C d − ) ⊗ − C d − ( C d − ⊗ V µ | s | ) , (C.21)its eigenspaces and eigenvalues are n − V µ | s | (C.22) − | s | on V µ | s | + ǫ (C.23) n − | s | on V µ | s | − ǫ n − , (C.24)43o that the eigenvalues of ←→ P are ∆2∆ − ( n −
1) on V µ | s | (C.25)∆ − | s | − n + 12∆ − ( n −
1) on V µ | s | + ǫ (C.26)∆ + | s | − ( n −
1) on V µ | s | − ǫ n − (C.27)For the unitary case, ∆ = | s | + n −
1, these eigenvalues of ←→ P are∆2∆ − ( n −
1) on V µ | s | (C.28)0 on V µ | s | + ǫ (C.29)1 on V µ | s | − ǫ n − (C.30)For the non-unitary case, ∆ = −| s | ,∆2∆ − ( n −
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