HS Gauge Fields in Frame-Like Formulation

∇²−(s − 2)(s + d − 3) − s l²_AdS



ψµ1···µs = 0 (4.31)

∇^µψ_µµ2···µs = 0 ψ_µµ^µ _3···µs = 0,

which are the Fierz-Pauli equations on a curved space. To completely fix the gauge, the gauge parameter µ1···µs−1 should satisfy the equations



∇²−(s − 1)(s + d − 3) l²_AdS



_µ1···µs−1 = 0 (4.32)

∇^µµµ2···µ_s−1 = 0

^µ_{µµ3···µs−1} = 0.

Fronsdal equations are linear and are based on a metric-like formulation. Next we will discuss HS gauge fields in the frame-like formulation. This is relevant for introducing interactions, as shown by Vasilliev.

4.2.2 HS Gauge Fields in Frame-Like Formulation

Vasiliev’s non-linear HS theory is based on the frame-like formulation and gen-eralizes the vielbein approach to gravity using the differential forms language.

In gravity we can introduce respectively the vielbein and spin connection

e^a_µ w^ab_µ. (4.33)

The vielbein is related to the metric by

gµν = η_abe^a_µe^b_ν. (4.34) Here the spin connection w_µ^abcorrespond to a gauge field with the local Lorentz rotations acting as gauge symmetries. The vielbein e^a_µand spin connection w_µ^ab can be combined as components of the one-forms,

e^a= e^a_µdx^µ w^ab = w^ab_µdx^µ. (4.35)

These forms obey the Cartan structure equations

de^a+ w^a_b ∧ e^b= T^a (4.36)

dw^ab+ w^a_c∧ w^cd= R^ab R^ab= 1

2R^ab_µνdx^µ∧ dx^ν,

where T^a is the torsion two-form and R^ab is the Riemann tensor two-form.

When T^a= 0 we have

R_µνρσ = R^ab_µνe_ρae_σb. (4.37) Since w^ab is a gauge field of local Lorentz transformations and ^a transforms as a vector under Lorentz transformations, the Lie algebra is of the form

[Mab, Mcd] = i(ηbcMad− η_bdMca− η_acMbd+ ηadMcb) (4.38) [M_ab, Pc] = i(η_bcPa− η_acP_b)

[P_a, P_b] = i l_AdS² M_ab,

where P_a corresponds to the generators of local translations. This is the AdS algebra SO(d, 2). We can build a one form W as follows

W = −i(e^aP_a+1

2w^abM_ab), (4.39)

which is intepreted as the gauge field of the Lie algebra in (4.38). The curvature of W is

dW + W ∧ W = − i(de^a+ w^a_b ∧ e^b)Pa+ 1

2(dw^ab+ w_c^a∧ w^cb+ 1

l²_AdSe^a∧ e^b)Mab) (4.40)

= − i(T^aPa+1

2(R^ab+ 1

l²_AdSe^a∧ e^b)M_ab).

For a flat connection on W the above equation becomes

dW + W ∧ W = 0. (4.41)

The solution to this equation is

T^a= 0 and R^ab = − 1

l²_AdSe^a∧ e^b ⇒ R_µνρσ = − 1

l_AdS² (g_µρg_νσ− g_µσg_νρ).

(4.42) This shows that the flat connection is related to a maximally symmetric back-ground. The gauge transformation for W is

δW = d + [W, ] (4.43)

 = −i(^aPa+1

2^abMab),

where ^a and ^ab are the gauge parameters for local translations and local Lorentz transformations respectively. The generators P_a, M_ab are combined into ^d(d+1)₂ generators TAB, where A, B = 0, · · · , d and W = −iw^ABTAB [39].

Generalizing to higher spins, the vielbein is given by the one-form

e^{a1,··· ,as−1} = e^a_µ^{1···as−1}dx^µ, (4.44) where e^{a1···as−1} is totally symmetric and traceless in the indices a1· · · a_s−1,

η_abe^{aba3···as−1} = 0. (4.45) We will use the Young tableau notation [n1, n2, n3, · · · , nk] where ni is the number of boxes in the i^th row. The tensor e^{a1···as−1} is in a reducible SO(d) representation and it decomposes as follows

[1, 0, 0, · · · ] ⊗ [s − 1, 0, · · · ] = [s, 0, · · · ] + [s − 2, 0, · · · ] + [s − 1, 1, 0, · · · ] (4.46) where the first two representation on the RHS represent a symmetric and double traceless HS field, which is the Fronsdal field (ψµ1···µ_s). The third representation on the RHS is a hook Young diagram representation which corresponds to gauge redundancies, so the corresponding gauge field has to be of the form w^aµ^{1···as−1,b}. This is the general spin connection in HS field theory.

Unlike the s = 2 case, solving the analog of (4.41) in HS theory does not give

a unique spin connection w^aµ^{1···as−1,b}. Rather a tower of HS spin connections are required to fix this gauge redundancy. Therefore HS fields in the framelike description have the following vielbein and spin connections [39],

e^a_µ^1···as (4.47)

w^a_µ^{1···as,b1···bt} t = 1, 2, · · · , s − 1.

The HS analog of (4.41), which will allow the HS spin connection to be solved in terms of an HS vielbein is given by the equations

R^{a1···as−1} =0 (4.48)

R^{a1···as−1,b}=0 ...

R^{a1,···as−1,b1···bs−1} =e_(0)c∧ e₍₀₎

dC^{a1···as−1c,b1···bs−1d}

where R^{a1···as−1,b1···bs−1} are curvature two-forms that generalize (4.36) and C^{a1···as−1c,b1···bs−1d}is the HS generalized Weyl tensor. It is built out of s deriva-tives of the Fronsdal field, and e^a₍₀₎is the tree level vielbein field from the fluc-tuation e^a_µ= e^a_(0)µ+ ˆe^a_µ. Just as in the s = 2 case, e^{a1···as−1} and w^{a1···as−1,b1···bt} can be combined into a gauge field w^{A1···As−1,B1···Bs−1} where A, B = 0, · · · , d.

This gauge field is now in the representation [s − 1, s − 1, 0, · · · ] of the AdS algebra. Each gauge field can be associated to a generator TA1···A_s−1,B1···B_s−1

in the same representation. These HS fields can be combined into one field

W = −i^X

s

w^{A1···As−1,B1···Bs−1}T_A1···A_s−1,B1···B_s−1. (4.49)

The generators TA1···A_s−1,B1···B_s−1 are the same generators from the CFT HS algebra constructed in terms of Killing vectors. These generators on the AdS side form the gauge algebra. To linearize (4.49), expand around the AdS background to linear order in the fluctuations w = w0+ w1, to obtain

W = −iw₀^ABTAB− i^X

s

w₁^{A1···As−1,B1···Bs−1}TA1···As−1,B1···Bs−1. (4.50)

The linearized gauge transformation becomes

δw₁ = d + [w₀, ] (4.51)

where w₀ = w₀^ABT_AB and  = ^P_s^{A1···As−1,B1···Bs−1}T_A1···As−1,B1···Bs−1. The linearized curvature in (4.48) becomes

dW + W ∧ W = dw1+ [w0, w1]. (4.52) The Vasiliev HS gauge theory is non-linear. Its non-linear gauge symmetry is given by

δW = d + [W, ]. (4.53)

This is the extent to which we will discuss the HS gauge theory. If we are ever to reproduce the full non-linear structure of this theory from CFT, it is clear that we need a good understanding of the free CFT. This is a major motivation for this PhD.

We have seen that the free scalar field CFT has a tower of higher spins primary operators packaged in a Gegenbauer polynomial. These higher spin primary operators are constructed using bilinears (2 copies of) of the scalar fields. This does not exhaust the spectrum of primaries of the CFT. Indeed, as we will see in the next chapter, even the spectrum of a single free scalar field theory is much richer than this.

Chapter 5

Primary Fields in Free Scalar Conformal Field Theory in 4-dimensions

This long chapter will contain the work of the papers that were published in [19][40]. This work contains novel results for the free conformal field theory of a scalar field in 4 dimensions (CFT4). Using representation theory a general generating function for the number of primary operators constructed from n copies of the free scalar field is derived. The generating function yields the correct counting for the primary fields. The counting is then specialised to counting primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4, 2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of algebraic and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.

5.1 Introduction

In [41] we showed that free scalar four dimensional conformal field theory can be formulated as an infinite dimensional associative algebra, admitting a decomposition into linear representations of SO(4,2), and equipped with a bilinear product satisfying a non-degeneracy condition. This algebraic

struc-ture gives a formulation of the CFT4 as a two dimensional topological field theory (TFT2) with SO(4,2) invariance, where crossing symmetry is expressed as associativity of the algebra. TFT2 structure had previously been identified as a unifying structure in the study of combinatorics and correlators in BPS sectors of N = 4 SYM, quiver gauge theories, matrix models, tensor models, and in Feynman graph combinatorics [42, 43, 44, 45]. The theme of TFT2 as a powerful unifying structure for QFT combinatorics was also developed in [41] in the context of counting primary fields. In this chapter we return to a systematic study of primaries in free field theories in four dimensions.

We consider scalar, vector and matrix models. Another motivation for the detailed construction of primary fields in four dimensional scalar QFT is that free field calculations have been found to be useful in calculating the anoma-lous dimensions of operators at the Wilson-Fischer fixed point in the epsilon expansion [18, 46, 47, 48, 49].

We start by developing some explicit formulae for the counting of primary fields, using characters of representations of so(4, 2). This makes extensive use of previous literature on the subject, notably [17]. This is followed by considering the problem of constructing the primary fields. A useful remark is that the algebraic problem of finding composite fields of the form

(∂ · · · ∂φ)(∂ · · · ∂φ) · · · (∂ · · · ∂φ) (5.1) where there are n φ fields involved, can be conveniently rephrased in terms of a question about multi-variable polynomial functions of 4n variables : Ψ(x_µ) where µ runs over the space-time coordinates and I runs from 1 to n. This re-lies on a function space realisation of the conformal algebra. We explain how this function space realisation arises naturally in radial quantization. The question of constructing primaries, when phrased in terms of the functions Ψ(xµ) can viewed as a many-body quantum mechanics problem, where F is a many-body wavefunction of n particles moving on R⁴. These many-body wavefunctions have to obey three simple conditions :

• They have to obey Laplace’s equation in each of the variables x^I_µ for I = 1 · · · n.

• They have to be invariant under the simultaneous translation x^I_µ→ x^I_µ+ aµ, for µ = 1 . . . 4.

• They have to be invariant under permutations x^I_µ→ x^σ(I)_µ for any permuta-tion σ ∈ Sn.

An infinite class of solutions of the Laplacian condition are obtained by choos-ing a complex structure to identify R⁴ = C² so that x_µ → (z, w, ¯z, ¯w) and considering holomorphic functions of z, w. These primaries correspond to holomorphic polynomial functions on

(C²)ⁿ/(C²× S_n) (5.2)

which can also be written as

(Cⁿ/C × Cⁿ/C)/Sn. (5.3)

The modding out by C² is the condition of invariance under the shift while the S_n invariance comes from the permutation symmetry. A special class of these primary fields correspond to functions of z only i.e functions on

(Cⁿ)/(C × S_n). (5.4)

These primaries were constructed in [50] using an oscillator realization of the conformal algebra, which is close to the differential realization used here. An extensive study of the representation of so(4, 2) on function spaces with em-phasis on relations to quarternions, is developed in [51].

The association of primaries to functions on the orbifold has several inter-esting consequences. Since the holomorphic polynomial functions form a ring, and a class of primaries are in 1 − 1 correspondence with these functions, we are finding a ring structure on this subspace of primary operators. This ring structure is different from the algebra structure related to the operator prod-uct expansion. The interplay between this prodprod-uct and the OPE would be an interesting subject for future study. The Hilbert series of the polynomial ring (5.2) has a very interesting palindromy property which we prove. The proof relies on an interesting algebraic structure based on symmetric groups in the problem. For fixed number of primaries n, this is

∞

M

k,l=0

C(S_n) ⊗ C(S_n) ⊗ C(S_k) ⊗ C(S_l) (5.5)

where C(S_n) is the group algebra of the symmetric group S_n. As recently discussed in the context of Hilbert series for moduli spaces of supersymmetric vacua of gauge theories [52, 53], the palindromy property of Hilbert series is indicative that the ring being enumerated is Calabi-Yau. The precise math-ematical statement is due to Stanley [54]. We show that the orbifold (5.2) indeed admits a unique non-singular nowhere-vanishing top-dimensional holo-morphic form, which is inherited from the covering space.

Our work involves an interesting interplay between representations of so(4, 2) and representations of symmetric groups. Let V₊ be the lowest weight repre-sentation corresponding to local operators built from derivatives acting on the field φ. The construction of primaries built from derivatives acting on n copies of φ, amounts to finding explicit formulae for the lowest weight states of irre-ducible representations in the symmetrized tensor product Symⁿ(V+). If we consider the primaries which arise at dimension n + k, of the class associated to the geometry (5.4) this can be mapped to a problem about multiplicities of Sn× S_k irreps in V_H^×k where VH is the n − 1 dimensional representation of Sn. A formula for these multiplicities, derived in [50], is found to be useful in the study of the geometry of (5.2). The connection between representation theory of symmetric groups and that of non-compact groups has also been discussed in [55] in the context of higher spin theories.

We extend this approch to primary fields to the case of vector fields in four dimensions. The underlying orbifold geometry for holomorphic primaries in this case is

(C²ⁿ/C × C²ⁿ/C)/S_n[S₂]. (5.6) The group Sn[S2] is the group of Sn which is generated by the n pairwise permutations (1, 2), (3, 4), · · · , (2n − 1, 2n) along with the n! permutations of these pairs. It is called the wreath product of S_n with S₂. We establish the palindromy property of the Hilbert series in this case. For the case of primary

fields in the free theory of matrices in four dimensions, we again find the underlying orbifold geometry

((C²)ⁿ× S_n)/(C²× S_n) = (Cⁿ/C × Cⁿ/C × S_n)/S_n (5.7) with a palindromic Hilbert series.

The chapter is organised as follows. In section 2 we describe a realisation of the conformal algebra so(4, 2) in terms of differential operators acting on polynomial functions of spacetime coordinates x_µ in R⁴. This is related, by a duality which we explain, to the standard realization of the conformal algebra in terms of derivatives acting on a scalar field. In section 3 we obtain a number of useful general formulae for the counting of primary fields. The first step is to start from the character of the irrep V+ of so(4, 2) which contains all the local operators consisting of derivatives acting on a single scalar field. This is a function of variables s, x, y which keep track of dimension, left spin and right spin i.e eigenvalues of D (the scaling operator) and JL, JR ( the Cartan generators for the two SU (2)’s in SO(4) = SU (2) × SU (2)). We then derive a generating function for the Cauchy identity. We describe a specialisation of these formulae relevant to what we call extremal primaries. These include the leading twist primaries studied in the context of deep inelastic scattering in QCD. Taylor expansion of the generating function leads to explicit results for n = 3, 4 which take the form of rational functions of s, x, y. In section 4 we describe the construction of the primary fields using the polynomials rep-resentations. A new counting formula for the extremal primaries is obtained by exploiting the permutation group algebras C(Sn) ⊗ C(Sn) ⊗ C(Sk) ⊗ C(Sl) in the problem of building primaries from n fields φ and corresponding to polynomials of degree k in one holomorphic variable and degree l in the other.

This is shown to be consistent with the derivation based on the Taylor ex-pansion method of the previous section. These primary fields form a ring and the counting is recognised as a Hilbert series, which encodes aspects of the generators and relations of the ring. This is a ring of functions of an orbifold which we identify. The counting formula based on S_n× S_k × S_l symmetry for the extremal sector is shown to have a palindromy property indicative of a Calabi-Yau nature of the orbifold. As further support for the Calabi-Yau nature of the orbifold, we construct the explicit top-dimensional holomorphic form. In section 5 we extend the results on counting and construction of

pri-maries, and the underlying Calabi-Yau orbifold geometries, to the case of a four dimensional vector model. In section 6 we develop the story for the case of free four dimensional 1-matrix theory.

5.2 Representation of so(4, 2) on multi-variable

In document Algebraic structures in the counting and construction of primary operators in free conformal field theory (Page 54-64)