CFT4 as SO(4,2)-invariant TFT2

We show that correlators of local operators in four dimensional free scalar field theory can be expressed in terms of amplitudes in a two dimensional topological field theory (TFT2). We describe the state space of the TFT2, which has $SO(4,2)$ as a global symmetry, and includes both positive and negative energy representations. Invariant amplitudes in the TFT2 correspond to surfaces interpolating from multiple circles to the vacuum. They are constructed from SO(4,2) invariant linear maps from the tensor product of the state spaces to complex numbers. When appropriate states labeled by 4D-spacetime coordinates are inserted at the circles, the TFT2 amplitudes become correlators of the four-dimensional CFT4. The TFT2 structure includes an associative algebra, related to crossing in the 4D-CFT, with a non-degenerate pairing related to the CFT inner product in the CFT4. In the free-field case, the TFT2/CFT4 correspondence can largely be understood as realization of free quantum field theory as a categorified form of classical invariant theory for appropriate SO(4,2) representations. We discuss the prospects of going beyond free fields in this framework.


Introduction
In this paper we develop a new perspective on correlators of four dimensional conformal field theory using two dimensional topological field theory. We are primarily concerned with the correlators of local operators in CFT 4 on R 3,1 (or the Wick-rotated R 4 ), where the space of states forms representations of the conformal group SO (4,2). The explicit calculations in the paper will start from the simplest CFT 4 , namely the free scalar field. We will outline how elements of the discussion generalize in various related free theories, and we will also see that many key elements in the discussion would be sensible for interacting theories. The symmetry SO(4, 2) plays the role of a global symmetry in the TFT 2 .
The space of local operators of the CFT 4 will determine the state space of the TFT 2 . In a TFT 2 , we associate a state space W to a circle, and tensor products W ⊗k to a disjoint union of k circles. In G-invariant TFT 2 , the state space is a linear representation of G.
Here the state space W is a linear representation of G = SO (4,2). TFT 2 associates, to an interpolating surface (cobordism) from k circles to the vacuum, an SO(4, 2) invariant multi-linear map from W ⊗k to complex numbers. For the case of the free scalar field, we will specify this map explicitly. There are two basic ingredients that go into this map. One is the fact that the 2-point function of the elementary field ϕ of scalar field theory can be viewed as a generator for the matrix elements of the SO(4, 2) invariant map from V + ⊗ V − → C. Here V + is the basic positive energy (lowest weight) representation of SO(4, 2) formed by ϕ and its derivatives, while V − is the dual negative energy (highest weight) representation. The other ingredient that goes in the construction of the map from W ⊗k → C is the combinatorics of Wick contractions. Using this invariant map, we can compute the k-point correlation functions of arbitrary composite operators. For the general background on TFT 2 , including careful definition of orientations, of ingoing versus outgoing boundaries and of cobordisms and their equivalences, we found [1] to be a very useful reference. Section 2 starts with the motivations from AdS/CFT leading to this work. It has recently been found that the combinatoric part of extremal correlators, notably three-point functions, of half-BPS operators can be expressed in terms of 2d TFTs built from lattice gauge theories where permutation groups play the role of gauge groups [2,3,4,5,6]. These belong to the class of theories considered by Dijkgraaf-Witten [7], and form examples of TFTs obeying the axioms stated by Atiyah [8]. This naturally raises the question of whether the full spacetime dependent correlators, not just the combinatoric part, can be understood using an appropriate TFT 2 . Since the space-time dependences of three-point correlators are completely determined by the invariance of the theory under the conformal group SO(4, 2), the right TFT 2 has to have a global SO (4,2) invariance. The notion of TFT 2 with a global G-invariance has been explained in [9]. One subtlety we have to deal with is that standard TFT 2 's have finite dimensional state spaces. This requirement of finite dimensionality follows from the way the axioms are set up, and physically relates to the fact that the TFT 2 's have amplitudes corresponding to surfaces of arbitrary genus. To allow infinite dimensional state spaces, we restrict to genus zero surfaces, so we have a genus zero TFT 2 , which involve a genus zero subset of the equations and algebraic structures entering finite TFT 2 's. We describe this genus zero subset of equations and corresponding geometry of two dimensional cobordisms. Key among these properties are the existence of a product, corresponding to the 3-holed sphere with two ingoing and one outgoing boundary. This product is associative.
Another key property is the existence of a non-degenerate pairing W ⊗ W → C, which has to be SO (4,2) invariant. The student of perturbative QFT is familiar with the fact that zero-dimensional Gaussian integrals provide a brilliant toy model to learn about QFT. Here the Gaussian integration model is used to provide what is arguably the simplest example of TFT 2 , having infinite dimensional state space, associativity and non-degeneracy.
Section 3 describes how the basic two-point function of the elementary scalar in free scalar quantum field theory is understood in terms of SO (4,2) invariants. Let V + be the irreducible representation of SO (4,2), with states of positive scaling dimensions (positive energy in radial quantization), consisting of the field ϕ and its derivatives, with the equations of motion set to zero. V − is the conjugate representation. There is no SO(4, 2) invariant in the tensor product V + ⊗ V + , but there is an SO(4, 2) invariant bilinear mapη : V + ⊗ V − → C. η is extended to an invariant map fromη : V ⊗ V → C, where V = V + ⊕ V − . This bilinear invariant plays a crucial role, so we study some of its properties. We describe the invariant explicitly by exploiting SU (2) × SU (2) and SL(2) subalgebras of SO (4,2). The SO (4,2) invariance is neatly expressed as a set of partial differential equations obeyed by the generator of matrix elements ofη. Some additional representation theoretic constructions are described, notably a map ρ : V + → V − , which is related to an automorphism of the so(4, 2) Lie algebra. By usingη, ρ, we can construct an inner product g on V + and on V − . The positivity of g is related to unitarity of the CFT. The invariance ofη is important in giving a TFT 2 interpretation of the 2-point function in terms of an SO(4, 2) invariant map. These ideas can be understood in a simple toy model. Consider the spin half representation of SU (2), denoted V1 2 . A problem in classical invariant theory is to count the number of times the trivial (one-dimensional) representation C appears in the Clebsch-Gordan decomposition of various tensor powers. If we take V 1 2 ⊗ V 1 2 it appears once. A more refined question is to describe the form of this state, which is We are using the usual notation for SU (2) reps where states in an irrep are labeled by |j, m > with J 3 = m and quadratic Casimir equal to j(j + 1). Now if we let x 2J 3 1 ⊗ x 2J 3 2 act on this invariant state, we get back the state with a factor.
The equations (1.1) (1.2) are equivalent ways to describe the form of the invariant state in In the application to the TFT 2 construction of CFT 4 correlators, these x 1 , x 2 are replaced by 2 spacetime coordinates, and we are describing the invariant representation C in V + ⊗ V − . The precise description of the invariant state in this way is a refinement of the counting of invariants. In this sense, this is a categorification of invariant theory and the TFT 2 construction of free field correlators involves a categorification of invariant theory for certain representations of SO(4, 2). Section 4 describes the state space W of the TFT 2 corresponding to the free field CFT 4 . Loosely speaking W contains states corresponding to all the composite operators in free field theory. The slight surprise is that it is V = V + ⊕ V − , rather than V + or V − alone which enters the construction of W. This is related to the fact mentioned above that there is an SO(4, 2) invariant in V + ⊗ V − but not in V + ⊗ V + , so a construction of CFT 4 correlators from SO(4, 2) invariants in TFT 2 has to involve both V + and V − in the construction of the TFT 2 state space. We have The n = 0 subspace W 0 = C. The n-fold symmetric product arises because of the bosonic statistics of the free scalar. TFT 2 involves assigning SO(4, 2) invariant maps to interpolating surfaces (cobordisms) from disjoint unions of circles to the vacuum. We describe such an invariant map from W ⊗k → C for any k ≥ 0. It is constructed from the basic invariantη, several copies of which are tensored according to Wick contraction combinatorics of QFT.
We identify the basic field as a linear superposition of states, labeled by position x ∈ R 4 , living in V + and V − , with x related to x by inversion. Using tensor products of this field, we have states corresponding to composite fields living in W n for all n. Choosing coordinates x 1 , x 2 · · · x k for the composite fields thus defined, using tensor products and applying the invariant map, we get arbitrary correlators of composite fields at non-coincident points.
In section 5 we focus attention on the 2-point functions of arbitrary composite operators, viewed from the TFT 2 perspective. This is the amplitude for two circles going to the vacuum, denoted η : W ⊗ W → C. We show that the non-degeneracy equation is satisfied, i.e. there is an inverseη of η. This equation corresponds to the fact that we can glue a cylinder with two incoming boundaries to one with two outgoing boundaries, along one boundary from each, to give a cylinder with one in and one out boundary (see Figure 2). There is no gluing along two boundaries to produce a torus, which would give infinity because of the infinite dimensionality of the state spaces. This restriction is clear at the level of equations, but subtle at the level of rigorous category theoretic axiomatics. These subtleties are discussed in Section 9. The approach we take in the bulk of the paper is to define TFT 2 in terms of this restricted set of genus zero equations.
In section 6, we discuss 3-point functions and the operator product expansion. The relation between the two is provided by the inverse of η discussed in Section 5. The amplitude for 3 circles to vacuum is the 3-point function. The amplitude for 2-circles to one circle is the OPE. The amplitude for one circle to two is the co-product.
In section 7, we discuss crossing and associativity. We explicitly prove the crossing property of the 4-point amplitude of TFT 2 . This is related, using the non-degeneracy condition, to associativity, and also to what is sometimes called the Frobenius equation or the nabladelta equation.
Section 8 looks at a basic problem in free scalar field theory, which is the enumeration of primary fields according to SO(4, 2) representation and multiplicity. We find that TFT 2 's with infinite dimensional state spaces, of the kind described in Section 2, play a role in the counting and lead to explicit new results for the case of three primary fields. This shows that the notion of genus zero TFT 2 's, with infinite dimensional state spaces which we have identified, is integral to the architecture of CFTs -not just to the whole CFT, but also to how the whole CFT is assembled from simpler parts. We hope to return to this theme by considering the construction of primary fields in the future. This would be another application of the counting to construction philosophy which finds various applications in the study of BPS states, integrability of giant graviton fluctuations and quiver combinatorics [10,11,6]. Section 9 discusses outstanding problems and future directions. In particular, Section 9.4 considers generalized free fields, where the irrep V + is replaced by a more general irrep of SO (4,2). In this more general set-up, we can still construct an SO(4, 2) invariant TFT 2 as defined in section 2. However, there is an additional condition related to having a local stress tensor that is not satisfied in the case of generalized free fields. We outline how this additional stress tensor condition can be expressed in terms of the SO(4, 2)-invariant TFT 2 data.
The TFT 2 construction we have developed with SO(4, 2) can be repeated after replacing SO(4, 2) with other groups. If we consider SO(d, 2), we can relate CFTs in d dimensions to TFT 2 . We can also consider a compact group G. This will give G-invariant TFT 2 's. A unique invariant mapη : V ⊗V → C can be defined for any finite dimensional representation V of G which contains, with unit multiplicity, the trivial irrep in the tensor product decomposition V ⊗ V . V can be an irreducible representation if that irrep is self-dual, or it can be a direct sum of some irreducible representation V + with its dual V − . We can define the state space The amplitudes C A 1 ,··· ,A k can be defined using tensor products of the elementaryη as in Section 4. Since the proofs of non-degeneracy in Section 5 and of associativity in Section 7 are purely combinatoric, they will continue to hold in this more general set-up. It would be interesting to investigate applications of this general construction and to find path integral constructions which give rise to these TFT 2 's.
It is worth noting here that connections between 4D quantum field theories and two dimensional topological field theories, in diverse incarnations, have been a fruitful area of research. Superconformal indices of 4D theories have been related to 2D TFT [12]. In such applications the 2D surface has a physical origin as the surface that a 6D theory has to be compactified on to arrive at the 4D theory. Another way to related 4D QFT to 2D TFT is to twist the 4D theory so it becomes topological and then consider the 4D theory on a product of Riemann surfaces [13]. It is instructive to compare the present construction with these precedents. We have a 4D theory, and we are looking at local correlators, with non-trivial spacetime dependences. There is no dimensional reduction and the 2D surface arises as a geometrical device to encode, via its cobordism equivalences, the crossing and non-degeneracy properties of the 4D conformal field theory. The spacetime coordinates of local operators have become labels of states in the state space W associated boundary circles of the 2D surface. This is somewhat like string theory where spacetime momenta become labels of vertex operators inserted at points on the worldsheet. The observables do not depend on worldsheet metric, because we integrate out the worldsheet metrics, hence the topological nature. In this sense, our construction has some analogies to the twistor string proposal for N = 4 SYM [14].

Motivations and strategy
The approach to correlators of CFT 4 we develop here, is motivated by studies of extremal correlators of half-BPS operators in N = 4 super Yang Mills theory with U (N ) gauge group. The half-BPS states correspond to multi-traces of an N × N complex matrix Z. For every positive integer n, these are gauge invariant observables which can be constructed using The correlators can be written as [2,3,4,5,6] T 1 is the conjugacy class of the permutation σ 1 . T 2 is the conjugacy class of the permutation σ 2 . C σ 3 is the number of cycles in the permutation σ 3 . The combinatoric part of the correlator is constructed from a quantity Z TFT 2 (Sn) (T 1 , T 2 , T 3 ) which is a function of 3 conjugacy classes This 2D topological field theory is an example from the class of TFT 2 's associated with finite groups H (here S n ), which were first discussed by Dijkgraaf and Witten [7]. For closed Riemann surfaces, this sums over homomorphisms from the fundamental group of the surface to the group H. For manifolds with boundary, we sum over homomorphisms subject to a condition that the boundary group elements are restricted to some conjugacy classes. In the above case, we have the partition function on a 3-holed sphere, with T 1 , T 2 , T 3 being the three specified conjugacy classes at the boundaries.
Given that the combinatoric part has an elegant TFT 2 description, it is natural to ask if the same is true for the space-time dependent part of the correlator 2.2. This is known to be determined by the conformal group SO(4, 2). The simplest set-up to investigate this question is to consider ordinary (non-matrix) free scalar field theory. The main result of this paper is to describe this as an SO(4, 2)-invariant TFT 2 . We return to matrix scalar field theories briefly in Section 9.3 and outline how the SO(4, 2) and S n appear in the TFT 2 description in that case.
A TFT 2 associates a vector space W to a circle and tensor products of W to disjoint unions of circles. Cobordisms are associated to homomorphisms between tensor products of the vector spaces. From a physical point of view, once we have chosen a basis, there is discrete data : structure constants C ABC and a bilinear pairing η AB , which obey some consistency conditions. These consistency conditions correspond to equivalences between different ways to construct cobordisms. They include, importantly, a non-degeneracy condition and an associativity condition. A variation on the above is associated to theories with global symmetry group G. Then W is a representation of G. The homomorphisms ρ k 1 ,k 2 : W ⊗k 1 → W ⊗k 2 are G-equivariant, that is for any g ∈ G the homomorphism and g action commute ρ k 1 ,k 2 •g = g •ρ k 1 ,k 2 . So the vector space of states W, whose basis states are labeled by A, B, · · · , form a representation of a group G (or its Lie algebra, when G is a Lie group). The data η AB and C ABC , are equivariant maps to the trivial representation C, equivalently they are G-invariant maps. This notion of TFT 2 with global symmetry group G is mentioned in [9] prior to developing TFT 2 with local G-symmetry, where the geometrical category involves circles with G-bundles and the cobordisms involve surfaces equipped with G-bundles.

Genus zero restriction and infinite dimensional state spaces
The standard axiomatic approach to TFT2 [1] requires the state spaces to be finite dimensional and includes finite amplitudes for surfaces of arbitrary genus. The first observation is that there is a well-defined subset, which we may call the genus zero subset of the TFT 2 equations, which do not involve summing over states in a (stringy) loop. These genus zero equations consist of a rich algebraic system including an associative product, non-degeneracy, unit, co-unit and co-product. These equations allow solutions involving infinite dimensional state spaces. So the η AB , C ABC are infinite dimensional arrays of numbers, with A, B, · · · running over an infinite discrete set of states. We will first write down some of these genus zero equations, and then show that there are simple non-trivial solutions, which we will call toy model solutions. The first toy model can be viewed as quantum field theory reduced to zero dimension and consists of Gaussian integration. A second toy model is related to SU (2) tensor product multiplicities, which has applications in counting primary fields in CFT 4 , as we will see in Section 8.
In subsequent sections we will show how these equations -with an appropriate choice of state space -provide a realization of free scalar field CFT 4 as a TFT 2 . The equations obeyed by these structure constants have geometrical analogues in terms of equivalences of cobordisms (see [1] for the geometrical definitions and the equations, and a physics presentation in [15]). We write the key genus zero equations.
• Non-degeneracy : The 2-point function η AB has an inverseη AB η AB corresponds to the cobordism from vacuum to two circles, while the identity on the RHS corresponds to the cylinder. Corresponding to (2.4) is the relation between cobordisms in Figure 2. In the case of finite dimensional state spaces, we also have η ABη BA = Dimension, which is closely related to non-degeneracy. This is a genus one cobordism from vacuum to vacuum which is excluded from our genus zero subset of equations.
• G invariance : There invariance under a group G, which can be finite or a Lie group.
In case of a Lie group, the invariance can be expressed in terms of the Lie algebra. For an element L in the Lie algebra of G, we have In our application to CFT 4 , G is SO (4,2). In the toy model of Section 2.5, G is trivial.
• Using the inverseη of η, the 3-to-vacuum amplitude C ABC can be related to a 2-to-1 amplitude, which is the structure constant of an algebra.
In the applications to free field theory, this structure constant will be related to the operator product expansion, while C ABC will be related to 3-point correlators. As a relation between cobordisms, this is shown in Figure 3. Note that we might imagine associating one-dimensional pictures to such data, i.e. in this case a trivalent graph, and describing the equations in terms of relations between graphs. However a trivalent graph is not a manifold. Indeed in one dimension, cobordisms exist from one set of points to another only if the numbers of points are both even or both odd [1]. Here we keep as closely as possible to the standard topological field theory framework of cobordisms, hence two dimensions are naturally selected as the right geometrization of the equations.
• Usingη , we can also relate the 3-to-vacuum amplitude C ABC to a 1-to-2 amplitude, which is called a co-product.
The figure corresponding to this is Figure 3.
• Symmetry Relations • Associativity : This corresponds to the fact that the two different gluings shown in Figure 5 give equivalent cobordisms.
• Crossing : Using the non-degeneracy equation, this is equivalent to associativity, which we elaborate on in Section 2.4. We also show there the equivalence to the Frobenius relation.
• G-invariance conditions forη AB and C C AB follow from the previous equations (2.5) • In the context of the TFT 2 /CFT 4 construction in Section 3, we will use an automorphism ρ of G to define an inner product g on the state space. The relation is of the form g(·, ·) = η(ρ(·), ·). We can impose a unitarity constraint on the TFT 2 by requiring positivity of this inner product. • Higher point correlators can be constructed from three-point correlators, e.g There is a similar construction for n-point correlators.
We will show that all these equations have realizations in the context of discrete data underlying correlators of CFT 4 . These same equations are also realized by simpler toy-models. One of them is Gaussian integration. Another is related to SU (2) fusions and has applications in the counting of primary fields. Both of these toy models have infinite dimensional state spaces. TFT 2 defined by these equations thus contains the discrete structure of CFT 4 as well as the related toy models. We will take these equations for C ABC , η AB as our definition of TFT 2 -they are essentially genus zero restrictions of standard TFT 2 equations. We have not given an axiomatic definition of the kind that exists for the case of usual TFT 2 (corresponding to Frobenius algebras) where all the possible gluings of the basic η, C,η are allowed, higher genus surfaces are included and state spaces are constrained to be finite dimensional. Finding the right axiomatic framework for the equations presented here is an interesting problem, which we discuss in section 9.1.

Equations related to the properties of the vacuum state
As mentioned in the introduction and described in detail in section 4, the state space W in the case of the TFT 2 construction of CFT 4 is graded by the degree of the symmetric tensors The pairing η : W ⊗ W → C is diagonal in the grading in the sense that η(W n , W m ) ∝ δ n,m (2.14) The case n = m = 1 is the case of the basic pairing, denotedη. The case n = m = 2 and higher corresponds to sums over all possible Wick contractions between composite fields, which are quadratic in ϕ etc. The explicit formulae for n ≥ 1 are in later sections. To describe the degree zero or vacuum sector introduce the state e (0) so that W 0 = Ce (0) . That is, a general state in W 0 is ae (0) for some complex number a. This is the one-dimensional representation of SO(4, 2). We define Then bilinearity requires that η(a 1 e (0) , a 2 e (0) ) = a 1 a 2 (2.16) Let us define the co-unit. This a homomorphism : W → C. It corresponds to the cobordism from circle to vacuum. We define (ae (0) + e (1) + e (2) + · · · ) = a (2.17) Figure 4: Relating correlator to co-product It just picks up the coefficient of the vector in the trivial representation. Since this is a projector to the trivial irrep of SO(4, 2), it is SO(4, 2) equivariant. If we denote e A the general basis vectors of W (say a basis that diagonalizes the CFT inner product -which can be constructed by group theory) and we denote A = 0 the vector e (0) , then we have Then we have the unit, which is a map˜ : C → W. Pictorially it is the map from vacuum to circle.˜ We define˜ A as the coefficient of the A'th basis vector in˜ (1). Then we can writẽ This means that the S 2 partition function of the TFT 2 , which is obtained by gluing the vacuum-to-circle amplitude, with the circle-to-vacuum amplitude is 1.
The definition of C ABD for general degree states is given later in terms of Wick-contractions. Letting D be a degree zero state amounts to only having contractions between A, B. It follows that we will have When both B, D are degree zero states, With these definitions, the equations corresponding to capping off an incoming circle or an outgoing circle hold.
If we have 3-circles going to vacuum, and cap off one circle, then we get just the amplitude for 2 circles going to vacuum.
The figure for this equation is Figure 6. Then the equation which is consistent with the definition (2.25). We can write this as The cobordism equivalence corresponding to this equation is Figure 7.

Associativity and crossing equations
Here we use the non-degeneracy property (2.4) to show that associativity, crossing and Frobenius equations are equivalent. The Frobenius relation is given in Figure 8.
The crossing equation is Using the η to lower indices The symmetry of C follows because we have a CFT of bosons and is an algebraic property of the Wick contraction map acting on W ⊗ W ⊗ W. We can do the same steps on the RHS and arrive at Now raise the D index on both side usingη and we get the associativity equation is We also have the Frobenius equation [1] ( sometimes called the nabla-delta equals deltanabla relation), which can demonstrated using the existence of the inverseη . Start from the nabla-delta equation and lower the C, D indices to obtain The RHS can be rearranged using the symmetry of C ... as follows This proves that the crossing equation implies both associativity and the equality of nabladelta and delta-nabla. This last equation is called the Frobenius relation. These manipulations use the inverse of η, without ever encountering η ABη AB which diverges.

Toy Model : Gaussian Integration
Our toy model employs the ring of polynomials in one variable. The states of the model are defined by It is straight forward to introduce an inner product on this space of states This inner product will play the role of the bilinear pairing of the TFT. Carrying out the integral above, we find Since g nm is clearly invertible, we have proved that this model has a non-degeneracy equation. Now, define the TFT structure constants by Again, carrying out the integral we find The structure constants C c ab give a product and we can get a co-product from C bc The integral expression for C abc prove that the structure constants are symmetric. The above structure constants and bilinear pairing will define a TFT 2 provided the crossing equation A simple generalization of the above model, is to consider a Gaussian Hermitian matrix model. The natural analog of the observables above are parametrized by [n, σ], where n is a positive integer and σ is a permutation in S n φ [n,σ] =: Again, a natural inner product on this set of states (dX is the usual U (N ) invariant measure for Hermitian matrices) gives us the bilinear pairing of the TFT. Carrying out the integral above, we find T 1 is the conjugacy class of the permutation σ. T 2 is the conjugacy class of the permutation τ . C σ 3 is the number of cycles in the permutation σ 3 . This model again has a non-degeneracy equation. Note however, the pairing (2.46) is not diagonal in this permutation basis. This will obscure the associativity of the model, although it could be verified by explicit computations. A simpler description is obtained by changing basis with a Fourier transform on the symmetric group Above R n is a Young diagram with n boxes and χ R (σ) is a character of the symmetric group. Our states are now the Schur polynomials for which the pairing is diagonal with f R the product of the factors of Young diagram R [4]. The TFT structure constants are now The crossing equation will hold because, again, the two expressions appearing above are two ways of calculating the integral needed to evaluate the right hand side. The product takes the form We have defined a starproduct labeled by k. In this Fourier basis of Young diagrams, the k = 0 product is given by Littlewood-Richardson coefficients. In the original permutation basis this is the outer product which takes σ 1 ∈ S n 1 , σ 2 ∈ S n 2 to give (σ 1 • σ 2 ) ∈ S n 1 × S n 2 ∈ S n 1 +n 2 . For the case, n 1 = n 2 and k = n 1 = n 2 , the product is the ordinary product of permutations. The intermediate cases correspond to products where one permutation acts on a set of n 1 integers, the second on a subset of n 2 integers, where the sets overlap over k elements. This structure is readily derived using diagrammatic tensor space techniques [16,17].

2.6
Zero area YM2 : An example of infinite dimensional state spaces and restricted amplitudes In the above discussions and in most of the paper, we focus on examples where TFT 2 with infinite dimensional state spaces arise, with a restriction to genus zero. There are also examples where infinite dimensional state spaces arise, and infinities are avoided by restricting to surfaces of genus greater than one. The partition function of YM2 on a surface of genus G with area A [18], with g 2 Y M set to 1, is where the sum is over irreps R of the gauge group, DimR is the dimension of the representation R, C 2 (R) is the quadratic Casimir. In the zero area limit, we have This diverges for G = 0, 1 for a Lie group, e.g. SU (N ), since there are infinitely many irreps with arbitrarily large dimensions. However the partition function is well defined for G > 1. The A = 0 limit is interesting as a topological limit, from the point of view of the moduli space of flat SU (N ) connections [19] and also from the large N expansion and gauge-string duality [20,21].

Basic CFT 4 2-point function as an invariant map in TFT 2
In free massless scalar field theory in four dimensions, all the correlators of composite local operators inserted at distinct points, can be obtained from the basic 2-point function of the elementary field In this section we will show that this quantity encodes the way the one-dimensional representation C appears inside the tensor product where the lowest energy state has dimension D = 1, corresponding to the state ϕ of CFT 4 , and the other states correspond to derivatives of ϕ, or equivalently to strings of P µ acting on the lowest weight state. We use energy/dimension/weight interchangeably in this paper, since we are working with radial quantization. Its conjugate is V − which is the representation with highest weight (energy) having D = −1 and where states are generated by K µ .
We start by describing the properties of the bilinear SO(4, 2) invariant mapη : V + ⊗V − → C. A Lie algebra element L ∈ so(4, 2) acts on the tensor product as The complex number field is the one-dimensional representation where the Lie algebra acts as zero. An equivariant mapη to C obeys The LHS is zero because C is the trivial irrep, and the expanding the RHS giveŝ This equivariance property exactly fits the definition of what is required in TFT 2 with SO(4, 2) as global symmetry.
The requirement that this map is SO(4, 2) invariant fixes it up to an overall constant. Equivalently the decomposition of the tensor product V + ⊗ V − in terms of irreducible representations contains a unique copy of the trivial one-dimensional representation C, where the Lie algebra elements act as zero. The states in V + are of the form where the C I are symmetric traceless tensors. We will introduce (Euclidean) spacetime coordinates x µ to keep track of these states. We can think of x µ as a way to describe the states in V + via a continuous variable as opposed to a discrete variable. Analogously for V − we have states obtained by acting with K µ and the variable x is the continuous variable. We will show that the invariant mapη, described in this spacetime basis is the 2-point function F (x 1 , x 2 ). Further, in the spacetime basis, there are a simple set of differential equations expressing the SO(4, 2) invariance ofη. Finally, usingη and a map ρ : V + → V − we are able to define an inner product on V + and on V − . This is a map from V ± × V ± to C which is sesquilinear. The inner product obtained in this way is the usual one which is used, for example, to study the bounds unitarity places on operator dimensions [22].

Flat space quantization and radial quantization
Start from the so(4, 2) algebra (we use g = diag(−, −, +, ..., +) ) The indices M, N, · · · run over {−1, 0, 1, · · · d}. Note that the structure constants are all real and the generators are all antihermittian S † M N = −S M N . We can write this algebra in two different ways, which make different subgroups manifest. For useful background material see [23] and section 2.1.2 of [24].
This rewriting is relevant for quantization of the theory on R 4 or R 3,1 . Each equal time slice is a copy of 3-dimensional Euclidean space, R 3 . Identify From the anti-hermiticity of the S M N s we find i.e. all of the generators have pure imaginary eigenvalues. The algebra obeyed by these generators is Notice that the M µν generate SO(3, 1) and D generates SO(1, 1).

Manifest so(4) × so(2) subgroup
This rewriting is relevant for the radial quantization. Equal "time slices" are three-spheres, with p ∈ {1, · · · , d}. Notice that from the anti-hermitticity of the S M N 's we find Thus, D will have real eigenvalues and M pq will have purely imaginary eigenvalues. The algebra obeyed by these generators is Clearly then, M pq generate the SO(4) subgroup, while D generates the SO(2) subgroup.

Invariant pairing
We will use the writing of so(4, 2) which makes the so(4) × so(2) subalgebra manifest. We want to consider two different so(4, 2) representations V + and V − . V + is built on the lowest weight state v + which obeys The remaining states in this irrep are constructed by acting with traceless combinations of P p s on v + . A convenient way to describe this is to write where the tensor C I p 1 ···p l is symmetric traceless in the p 1 , · · · , p l indices. The index I runs over the states in the SO(4) irrep ( l 2 , l 2 ). We will often trade I for SU (2) × SU (2) state labels (m L , m R ).
The remaining states in this irrep are constructed by acting with traceless combinations of K p 's on v − . The representations that we consider most of the time are relevant for the description of a free massless bosonic scalar field in 4 dimensions, in which case we set d = 1.
For the remainder of this section, we will set d = 1.
The invariant pairing isη : On the right hand side we have traded I for (m L , m R ) and J for (m L , m R ). In the next section we will prove that the requirement of so(4, 2) invariance determines f l,m L ,m R ;l m L ,m R up to an overall constant. A convenient way of summarizing the action ofη is in terms of the tensors which are themselves nicely summarized as To evaluate (3.18), we can use the so(4, 2) invariance of the pairing to shift (say) P p 's from the left slot to the right slot. The action of P p on the state in the right slot is then easily computed by using the so(4, 2) algebra as well as the fact that P p annihilates v − . In the next section, this logic will be applied also to SU (2) × SU (2) and SL(2) sub-algebras, to derive an explicit formula for f l,m L ,m R ;l m L ,m R . A straight forward computation now gives For a closely related discussion, see section 3 of [25].

Description of pairing in terms of SU (2) × SU (2) and SL(2) subalgebras
The requirement of so(4, 2) invariance determines f l,m L ,m R ;l m L ,m R up to an overall constant. This is most easily demonstrated by requiring thatη is invariant under sl(2) and su(2)×su(2) subalgebras. We will prove that theη obtained in this way enjoys the full so(4, 2) invariance.
The complete set of states of V + can be obtained by acting with elements of the sl(2) and su(2) × su(2) subalgebras on v + , and the complete set of states of V − can be obtained by acting with elements of the sl(2) and su(2) × su(2) algebras on v − . This can be seen by considering the SO(4, 2) character in equation (B.5) for V + . The coefficient of s q+1 in this character is χ q 2 (x)χ q 2 (y). This implies that the states of scaling dimension q + 1 fill out a multiplet of spin (j L , j R ) = ( q 2 , q 2 ). This complete multiplet can be generated by applying SU (2) × SU (2) rotations to (H + ) q v + , which shows that we do indeed generate the complete set of states in V + by acting with the sl(2) and su(2) × su(2) subalgebras on v + . A similar argument shows that we generate the complete set of states of dimension −q − 1 in V − by acting with elements of su(2) × su (2) Invariance under sl(2) and su(2) × su(2) leads to We will demonstrate how invariance under sl(2) fixes f l (SL (2)). The demonstrations for f m L (SU (2)) and f m R (SU (2)) are very similar.
The subalgebra that we study is described in detail in Appendix A. The two su(2) algebras have raising and lowering operators given by J R ± and J L ± , while the raising and lowering operators of sl(2) are H ± . In terms of these generators, the pairingη iŝ To demonstrate how sl(2) invariance fixes f sl (2) d,l , consider the positive discrete series irrep V + of sl(2) and the negative discrete series irrepṼ − Note thatṼ ± are subspaces of the irreducible SO(4, 2) representations V ± that we introduced above. Invariance of the pairingη(H l which shows it vanishes unless l = l , so that Then we have This is solved by A very similar argument requiring invariance under su (2), shows that

so(4, 2) invariance of the pairing
To obtain the pairingη we have required invariance under the sl(2) and su(2) × su (2) algebras. In this section we will show that the pairing we have obtained enjoys the bigger so(4, 2) invariance. The requirement of so(4, 2) invariance translates into a set of partial differential equations for F (x, y ). The demonstration then amounts to showing the F (x, y ) does indeed obey these partial differential equations.
Recall that To start, we will consider invariance under dilatations. Towards this end, note that and Thus, the statement of invariance under dilatations is equivalent to the differential equation It is straightforward to check that the function given in (3.20) obeys this equation.
Next, some algebra shows that It is also clear that Consequently, the statement is equivalent to the differential equation Again, this equation is obeyed by (3.20).
Finally, note that Consequently, the statement is equivalent to the differential equation This equation is again obeyed by (3.20), which completes the demonstration of so(4, 2) invariance.

Invariant pairing to Inner product via twist map ρ
In this section we define an inner product g : V ± × V ± → C. Since we have the invariant map η : V + × V − → C, if we introduce a map ρ : V + → V − we can construct an inner product by composing ρ andη. We will choose the map ρ so that the inner product obtained is the usual inner product of the CFT used to test unitarity. The positivity of this inner product is what puts constraints on dimensions of fields. For example, a scalar should not have dimension lower than 1, as first proved by Mack [22].η is the building block of SO(4, 2) invariant maps, which can be used to construct correlators. The ρ map gives the relation betweenη and g, and is related to an automorphism.
It is also useful to note that This explains why this inner product has the usual hermiticity property. Finally we observe that the map α(L a ) ≡ −L † a used above is an automorphism of the Lie algebra 4 State space, amplitudes, and correlators in the TFT 2 In section 3 we have made use of two representations V + and V − . This has allowed us to describe the two point function of the basic field of free scalar field theory in four dimensions in terms of the invariant irrep C of SO(4, 2) appearing in the tensor product V + ⊗ V − . In this section we will consider a larger state space W, which will allow us to extend our discussion to arbitrary correlation functions in CFT 4 .
The state space W is a vector space V + is again the lowest weight representation obtained by acting with some L + a in the enveloping algebra of so(4, 2) on the lowest weight state v + . The plus in L + a indicates that this is the subalgebra generated by the P µ 's. V − is a lowest weight representation obtained by acting with L − a (products of K µ 's) on the highest weight state v − . Our field is represented by the following sum of states in where a primed coordinate is always related to the unprimed coordinate by inversion This is a key equation translating between fields in CFT 4 and the states in TFT 2 . The explicit (x ) 2 multiplying the second term is needed to ensure that both terms have the same scaling dimension. If v + has dimension d we would have We will demonstrate that the TFT 2 correctly computes arbitrary CFT 4 correlation functions. Our demonstration will build up from the basic two point function in a series of steps. Let us denote by < ϕ(x 1 )ϕ(x 2 )...ϕ(x 2k ) > CF T 4 the usual 4D CFT free field correlator. We are using the hatted notation for the usual quantum field to distinguish from the sum of states in V + ⊕ V − which we use as the foundational equation in our TFT 2 approach (4.3). We want to show that We have already described the two point function as the SO(4, 2) invariant. To extend this idea to arbitrary correlation functions, we need a bilinear so(4, 2)-invariant map The basic building block for this map will bê which is the mapη, described in section 3. It is used to definê by symmetry asη We may writeη by the direct sumη =η +− ⊕η −+ (4.13) It is useful to think of thisη as a block off-diagonal matrix. In section 3 we have given a formula forη a + b − , in the basis where a + = l, p L , p R and b − = l , q L , q R , aŝ Next, suppose we have a 4-point function. The TFT 2 computes this correlator by mapping the tensor product to a number. This is accomplished with the map defined by η =η 12η34 +η 13η24 +η 14η23 . We are using the products of the basic map (4.13) on V ⊗ V to produce a map on V ⊗ V ⊗ V ⊗ V . The details of how we do this are fixed so that we reproduce the combinatorics of Wick's theorem. This action gives The 1 2 for eachη ij comes from multiplying the two factors of 1 √ 2 appearing in (4.3). The factor of 2 comes from adding theη(Φ + (x 1 ), Φ − (x 2 )) = G(x 1 , x 2 ) andη(Φ − (x 1 ), Φ + (x 2 )) = G(x 1 , x 2 ) -where Φ + (x) and Φ − (x) live in V + and V − . This is in perfect agreement with the usual free field computation In general, to construct the map η : V ⊗ 2k → C we sum over (2k)! 2 k k! pairings which can be parametrized by choosing i 1 < i 2 · · · < i k and i 1 < j 1 , i 2 < j 2 , · · · , i k < j k . The pairings are These are determined by permutations σ with n cycles of length 2, which form a conjugacy class denoted as [2 k ] of S 2k , so we may write {i σ l , j σ l }, which is a set of k pairs (i, j) uniquely determined by σ. Here we start with the state (4.20) and then act with the SO(4, 2) invariant map The 1 2 comes from the normalization factors and the two from theη(Φ + , Φ − ) andη(Φ − , Φ + ). This is the correct CFT 4 correlator.
Consider next correlators of descendents of ϕ(x). Our TFT 2 proposal is to insert the derivatives of the basic Φ from (4.3) where I ρ µ appearing in is the local Lorentz transformation for an inversion.
In CFT 4 , correlators of descendents follow by taking the appropriate derivatives of the Green's functions. The TFT 2 applies the map (4.21) after the derivatives of Φ(x) have been taken. The equality of the TFT 2 and CFT 4 computations follows because taking derivatives of (4.20) and then applying (4.21) is the same as computing the pairings and then doing the derivatives. These commute because things like x µ 1 1 · · · x µs 1 P µ 1 · · · P µs v + as live in V + ⊗ C[x 1 ], the tensor product of the SO(4, 2) irrep with the function space C[x 1 ]. The different orders of taking derivatives commute becauseη acts on the V + ⊗ V − whereas the derivatives act on the C[x 1 ] ⊗ C[x 2 ] and operators acting on different tensor factors commute.
As an example consider iP µ e iP ·y v + , 2x ρ I ρ ν (x ) 2 ρ(e iP ·x v + ) + 1 2η iP µ e iP ·y v + , (x ) 4 I ρ ν ρ(iP ρ e iP ·x v + ) Lets discuss the evaluation of the first term on the RHS. Use the so(4, 2) invariance ofη to move the P µ from the first slot to the second slot. After evaluating the ρ map in the second slot, we have to evaluate P µ acting on e −iK·x v − . This can be done using the so(4, 2) algebra, as explained in section 3.2.2. Evaluating all four terms in this way, it is now simple to find which confirms the argument given earlier that taking derivatives with respect to the spacetime coordinates commutes with the evaluation ofη, so that the CFT 4 correlator for scalars with derivatives is reproduced by the TFT 2 construction.
To complete the discussion, consider correlators involving powers of the elementary field.
The starting point to get this on the TFT 2 side is We need n 1 + n 2 + · · · + n k ≡ 2M to be even for a non-zero correlator. To this we apply a sum of products ofη's, schematically written as η ij (4.29) The sum is over all pairings of 2M objects, avoiding cases where (i, j) belong to the same subset of n 1 or n 2 etc. integers. To write a more explicit version of (4.29) note that the pairings we are summing over correspond to permutations in the conjugacy class [2 M ] in the symmetric group S 2M , but are not in the subgroup S n 1 × S n 2 × · · · × S n k ≡ S n . Hence the invariant map is For eachη factor there is a 1 2 × 2G(x i , x j ) as before. In this way we again see that the TFT 2 correctly computes the CFT 4 correlator One can also consider applying derivatives in each of the x 1 , · · · , x k . Again the TFT 2 proposal is to apply the derivatives to the state (4.28) which gives, schematically which is equal to by using the argument given above.

Non-degeneracy of the Invariant Pairing
We can explicitly describe the pairing using SU (2) × SU (2) and SL(2) sub-algebras.
Using the states 3) The f factors are given earlier in Section 3.2.1.
For the space W given in (4.1) we define η if v 1 , v 2 belong to Sym(V ⊗n 1 ) and Sym(V ⊗n 2 ) for n 1 = n 2 . Define e a 1 • · · · • e an = 1 n! τ ∈Sn e a τ (1) ⊗ · · · ⊗ e a τ (n) = P sym e a 1 ⊗ · · · ⊗ •e an (5.7) The projector P sym = 1 n! σ∈Sn σ is the projector for the symmetric part. For fixed n, define η a 1 ,...,an;b 1 ,··· ,bn ≡ η(e a 1 • e a 2 · · · • e an , e b 1 • · · · • e bn ) (5.8) by the equation There is an inverse,η, which obeys where these capital indices run over the states in W. Choose a basis running over the degree n of the symmetric tensor product Sym(V ⊗n ), and for each n, we run over states v a 1 •· · · v an . We defineη for v 1 ∈ Sym(V ⊗n 1 ) ⊂ W, v 2 ∈ Sym(V ⊗n 2 ) ⊂ W with n 1 = n 2 . And for v 1 , v 2 in the same symmetric power, With these definitions both η,η are block-diagonal, with blocks labeled by the degree of the symmetric tensors.
The non-trivial part of the check of the inverse property, in each block, involves showing This expresses the fact the LHS is non-zero only when the symmetric part of e a 1 ⊗e a 2 · · ·⊗e an is identical to the symmetric part of e c 1 ⊗ · · · ⊗ e cn . The proof is a simple calculation.
In the last step we use the invariance of the sum over S n , under group multiplication. The final answer on the right is the matrix element of the identity operator on symmetric tensors.
There is a physical way to understand the non-degeneracy we have just described. Let us define the vector space V + T spanned by states of the form This corresponds to local operators ∂ µ 1 · · · ∂ µs ϕ in CFT. Similarly we have the dual representation V − T spanned by The mapη can be defined to act on vector space V + T ⊗ V − T . Using the SO(4, 2) algebra as well as the SO(4, 2) invariance of η, it is straight forward to demonstrate that This shows that P µ P µ v + is a degenerate state for the invariant pairing. It is also a degenerate state for the inner product g defined in terms of η in Section 3.3. This is a purely representation theoretic fact related to the possibility of imposing the equation of motion ∂ µ ∂ µ φ = 0 in a way consistent with SO(4, 2). The P µ P µ v + state, together with all of its descendants, are null. We can quotient V + T by the null states to get V + : the space V + is the vector space made from states obtained by acting with symmetric traceless products of P 's, i.e products of the form C I µ 1 ···µp P µ 1 · · · P µp . This corresponds to the derivatives of the elementary scalar ϕ, with the equation of motion imposed. The 2-point function F (x 1 , x 2 ) derived from the path integral obeys the property ∂ x µ 1 ∂ x µ 1 F = 0, related to the fact that equations of motion are satisfied by the quantum field, inside correlators, as operator equations. From the representation theory point of view, when we work with V + the equation of motion has already been imposed. We then take V = V + ⊕ V − to get a self-dual object which has a non-degenerate SO(4, 2) pairing. In this section we considered W = ∞ n=0 Sym(V ⊗n ) and demonstrated that the non-degenerate bilinear pairing on V extends to this space. From the physical point of view, this is not surprising. We know that once we have accounted for the equations of motion on the basic field ϕ, the construction of composite fields has no further source of null states, than the ones that come from the equations of motion on each elementary field. It is nevertheless useful to exhibit this directly as a fact in representation theory, since the non-degeneracy is a crucial ingredient in the SO(4, 2)-invariant TFT 2 .

Three-point functions and OPE
In Section 4 we have described a state space W (with discrete basis e A ) and shown how to construct a multi-linear map from the tensor product W ⊗k to the complex numbers C using tensor products of the basic invariant mapη according to Wick combinatorics. Insertions of states in W corresponding to composite scalar field operators, labeled by positions x 1 , x 2 , · · · , x k , gives the k-point functions of free 4D scalar field theory. In particular there are C ABC which give the 3-point function. In TFT 2 we can form C C AB = C ABDη DC which gives a product structure to W, i.e. a map W ⊗ W → W.
Given any state e A ⊗ e B in W ⊗ W we have a state C D AB e D in W. This state has the property that the 3-point function of e A ⊗ e B with any e C is just the two-point function η(C D AB e D , e C ). Indeed This is the TFT2 expression of a familiar construction in CFT, the operator product expansion (OPE). For any two local operators at spacetime positions x 1 and x 2 , the OPE expresses their product as a sum of local operators at x where x may be x 1 , x 2 or even the midpoint x 1 +x 2 2 , according the convention adopted. In conformal field theories this expansion is a convergent expansion [25] and it provides a powerful approach for understanding the correlation functions in the theory. The three point function for operators located at positions x 1 , x 2 and x 3 is reduced to the computation of a sum of two point functions, after the OPE is used to take the product of two of the operators in the correlator.
To make this link between product in TFT 2 and operator product in CFT 4 explicit, recall that the basic field ϕ(x) corresponds to the following state in the state space W of the TFT 2 . The composite field ϕ 2 (x) of QFT corresponds to the tensor product Φ(x) ⊗ Φ(x) ∈ Sym(V ⊗2 ). More generally any local operator corresponds to an expansion of the form for e A running over a discrete basis in W. Given the way the C ABC are constructed from tensor products ofη (equations (4.21), (4.30)) we see that we may write the operator product of O 1 (x 1 ) and O 2 (x 2 ) in the TFT 2 language as a sum of the form (1 +η +η ⊗η + · · · ) acting on We are organizing the product by the number of Wick contractions between the first two operators. The first term in the sum corresponds to the 3-point function for an O 3 (x 3 ) that receives only contribution from terms having no Wick contractions between the first two operators. This is followed by terms with one or more Wick contractions. Hence the first term is the ordinary tensor product, while the subsequent terms involve the application of the invariant mapη which acts on V ± and V ∓ factors from O 1 and O 2 . The symmetric role played by positive and negative energy representations in the TFT 2 construction of CFT 4 naturally leads to the correct form of the OPE.
Let us make the discussion even more concrete by taking O 1 and O 2 to correspond to the quadratic composite field ϕ 2 . Here we will have (1 +η 13 +η 14 +η 23 +η 24 +η 13 ⊗η 24 +η 13 ⊗η 24 ) (Φ( In this expression we have not chosen whether to expand the RHS around x 1 or x 2 or the mid-point. That is a subsequent choice that can be made and the above state in W expanded accordingly.

CFT crossing and associativity
The OPE is associative. Since the OPE can be used to construct correlation functions, associativity of the OPE implies relations among the CFT correlators, namely crossing symmetry. It is known that crossing symmetry implies strict constraints on operator dimensions and OPE coefficients (or equivalently, on the C C AB of TFT 2 ). In this section we give a combinatoric proof of crossing in the TFT 2 framework. The reader will expect this to work since the TFT 2 framework has already been shown to reproduce correlators of the free CFT 4 . It is however useful to give an explicit proof without appealing to the path integral of the free CFT 4 . It allows us to see that it continues to work for generalizations where SO(4, 2) is replaced by any G and V by any space with a unique invariant in V ⊗ V .
We can parametrize the sum over contractions by decomposing the sets S a as For a pair of sets U, V of same cardinality |U | = |V |, we define the contractor where Sym(V ) is the symmetric group of all permutations of the set V . So the correlator is computed by applying the map The sum runs over all possible decompositions (7.3).
Although we have written C S,T in terms of sums over permutations above, we can also write it without permutations but using lists and give another proof of 7.7. Let L S be a list constructed from the set S. A set does not know about any ordering. Let |S| be the cardinality of S. Then |S|! is the number of lists constructed from S, with list size |S|. Let L S 0 be a fixed list, with some fixed chosen ordering of the elements. We can write By definition, the delta function on the lists pairs the first from L S 0 with the first from L T , second with second etc. In what follows the delta on the cardinalities is often suppressed. Let us define U = (S 1 \ S 12 ) ∪ (S 2 \ S 21 ) and V = (S 3 \ S 34 ) ∪ (S 4 \ S 43 ). We can write the LHS of (7.7) as Now a pairing δ(L U 0 , L V ) determines • a subset S 13 of (S 1 \ S 12 ) and a subset S 31 of S 3 \ S 34 which are contracted with each other.
• the order in L U 0 picks an ordered list L S 13 0 from S 13 . The sum over L V includes a sum over all ordered lists L S 31 in S 31 .
• Similarly S 14 , S 41 are determined as in Figure 9. L S 14 0 and L S 41 are also determined.
We conclude that which is the desired identity (7.7). See Figure 9 for a graphical representation of the discussion.
Computing the correlator by using the OPE in the (14)(23) channel, we first choose subsets S 14 , S 41 and S 32 , S 23 and then do the corresponding contractions. Then we do the contractions between (S 1 \ S 14 ) ∪ (S 2 \ S 23 ) and (S 3 \ S 32 ) ∪ (S 4 \ S 41 ). For this channel we are applying the map The equality between (7.6) and (7.11) follows simply by swapping the orders of summation.

Counting of Primaries
A rather basic question about free scalar field theory is to enumerate the SO(4, 2) irreducible representations appearing among the composite fields made out of n = 2, 3, · · · fundamental fields. For example, we need these multiplicities to compute the spectrum of primary operators in the CFT 4 . This question amounts to decomposing, into irreducible representations, the tensor product Sym(V ⊗n + ), where V + = D [1,0,0] in the notation of [27]. The three integer labels in D [d,j L ,j R ] are the dimension and two Lorentz spins.
We have seen simple examples of TFT 2 in Section 2, arising from 1-variable Gaussian integration, as well as from matrix integration. In this section we will introduce a simple TFT 2 that organizes the counting of primaries. Our results give a formula for the multiplicities of irreducible representations of SO(4, 2) in the tensor product V ⊗n + . Some key results that are needed to reproduce the results of this section have been collected in Appendix B.

Results on tensor products
It is known that [28] where A d,j L ,j R is another class of irreducible representation of SO(4, 2), see [27] for more details. For n = 3, we have, using results in [27], or alternatively by manipulating characters, For n = 4, we get This is an easy application of the previously derived formula for V ⊗3 + along with equation (4.7) of [27]. For general n, we have

TFT 2 for counting primaries
Consider an algebra of polynomials in variables X j where j ∈ {0, 1/2, 1, · · · }. The structure constants are given by the fusion rules of SU (2) This is the large k limit of the fusion rule algebra in the SU (2) WZW model. Call this algebra A su . Introduce the pairing < X j 1 , X j 2 >= δ j 1 j 2 (8.6) Given the above structure constants and the pairing, we have associativity equations such as Now, take a second copy of this algebra, generated by Y j . Introduce the algebra of power series in s with rational coefficients, called Q[s]. A pairing on Q[s] is given by In summary, we have an algebra A su ⊗ A su ⊗ Q[s] which is associative and has a nondegenerate pairing. Thus, we have defined a TFT 2 of the kind we described in Section 2. Let us call this the SU (2) × SU (2)-fusion-TFT 2 . Computations of the SU (2) fusion multiplicities can be easily programmed in Mathematica.
We will now explain how this TFT 2 can be used to compute the multiplicities of the primaries in the CFT 4 . The character χ V + (s, x, y) is obtained by computing tr V + X with In computing this trace, the null state associated with the equation of motion for the free massless boson, together with all of its descendents, are subtracted. The character of V ⊗n is then χ V + (s n , x n , y n ). Since the fundamental field of our CFT is a boson, taking a product of n copies gives Sym(V ⊗n + ). The character χ Sym(V ⊗n + ) (s, x, y) can be computed as the trace of a symmetrizer acting on V ⊗n which leads to In the last line, C i (σ) is the number of cycles of length i in the permutation σ. Using the formula (B.4) from the Appendix B, we can express the characters χ V + (s i , x i , y i ) in terms of χ j L (x)χ j R (y), which can be replaced by X j L Y j R . The result is then simplified using the product (8.5) and written as a sum of characters of SO(4, 2) irreducible representations. We will illustrate the procedure for n = 3 in the next subsection.
Finally, the characters useful for understanding the tensor product decomposition are where P (s, x, y) = 1

Symmetrized tensor products and primaries in free field theory
In this section we will focus on the case n = 3. A simple application of (8.9) gives The first term above can be simplified using (8.2). For the second and third terms we use the formula (B.4) in Appendix B.2 for χ V + (s n , x n , y n ). In view of (8.10), to identify the sum of characters we obtain we need to have an overall factor of P (s, x, y) appearing. One way to achieve this is to introduce the inverse of P (s, x, y) An overall factor of P (s, x, y) can now be arranged by including a factor of Q. For more details see Appendix B.2. With this motivation, consider the following state of the TFT χ n = Qχ n (8.14) where It is straight forward to compute χ n in Mathematica, with the result The coefficients of C n,k,k 1 ,k 2 give the multiplicities of SO(4, 2) irreducible representations in V ⊗n . By computing these numbers explicitly we could for example, derive the results of section 8.1.
We now specialize to the case n = 3. For fixed k, at least one of k 1 or k 2 in (8.15) is equal to k 2 . Also, χ n is symmetric under exchange of X and Y . With these properties in mind, it is useful to introduce the operation S defined by In terms of S we can write where µ is the residue of q modulo 6, ν is the residue of l modulo 6 and where we have used the Kronecker delta δ(x) which is 1 if x = 0 and zero otherwise.
9 Discussion and Future directions 9.1 Genus zero TFT 2 : Equations in search of Axioms We have given a set of genus zero equations which form part of a truncation of the usual finite dimensional TFT 2 equations. These include a product, co-product, unit, co-unit, associativity and crossing. This construction is realized in ordinary Gaussian integrals, and also as an SO(4, 2)-covariant TFT 2 which reproduces the correlators and OPE of 4D CFT. It is a genus zero system because of the infinite dimensionality of the state space. A torus amplitude would diverge.
An obvious open problem is to find the axiomatic framework which accommodates this truncated TFT 2 system of equations and which therefore admits solutions in terms of infinite dimensional state spaces. The usual definitions of TFT 2 and Frobenius algebras force the state spaces to be finite dimensional and naturally include amplitudes for all genera.
Infinite dimensionality of the state spaces W implies some accompanying subtleties. We avoided these issues by focusing attention on a discrete basis, which can be chosen to have nice properties, such as diagonalising the CFT 4 inner product. We gave the equations in terms of arrays of numbers C ABC , η AB ,η AB with indices labeling this basis set. Going beyond this somewhat basis dependent approach, requires being more careful about specifying what sorts of linear combinations we have in mind when we talk of the vector space W. As a first step let us consider defining as the vector space W of finite linear combinations of the specified basis set. Since the basis set has an infinite number of elements, this is an infinite dimensional space. For W ⊗ W, we also consider finite linear combinations F AB e A ⊗ e B . Acting on these η is obviously well defined. For generic infinite sums, it would not be well defined. Similarly we can work with F A 1 ···An e A 1 ⊗ e A 2 · · · ⊗ e An , with F having only finitely many nonzero entries, as our definition of W ⊗n . Then the map from W ⊗n → C is well defined. Now consider the amplitude for vacuum going to two circles. Using the discrete dataη AB we can buildη AB e A ⊗ e B (9.1) But this is an infinite linear combination. So we are not getting a map from C → W ⊗ W, as we might have expected. The infinite dimensionality has thus entered in a crucial way here. Indeed this is expected. If the amplitude for vacuum to 2 circles exists, and 2 circles to vacuum exists, then if we stick with a category theory framework , we can compose these two and get a torus, which looks like it cannot be anything but infinite (hence ill-defined) in the case of infinite dimensional state spaces.
So we conclude that we cannot make sense of the amplitude for the vacuum going to 2 circles in terms of a map C → W ⊗ W. However, we found good uses ofη AB in our system of equations, e.g. in relating crossing to associativity. How can we modify the axiomatic framework so as to associate some sort of algebraic data to the picture of vacuum going to 2 circles ?
One sensible possibility would be to associate specified finite dimensional subspaces of W to one of the two outgoing circles corresponding toη AB . We could think of this as placing a subspace defect on one of the circles. The effect is to restrict one of the indices ofη AB , to run over a specified subset of the basis vectors in W. If we do that to one of the outgoing circles, the torus amplitude is no longer infinite, but depends on the dimension of the chosen subspace. In general, for an amplitude C B 1 ,··· ,Bm A 1 ,A 2 ,···An corresponding to n incoming circles and m outgoing circles, we would restrict m − 1 of the outgoing circles with subspace defects.
There are generalizations of TFT 2 as a functor from the geometrical category of cobordisms to a more general target category which replaces the category of vector spaces (e.g. chain complexes) [9]. The target category we need here would be the one where an object is not a vector space, but rather a vector space along with some class of its subspaces. In our applications the vector space would be W and we would also allow its subspaces. So a circle can be associated to W or to some subspace thereof.
Subspace defects can be considered even in TFTs with finite dimensional state spaces. They may have interesting combinatoric applications. For the case of 2D TFTs based on lattice constructions with a gauge group G, natural observables to consider, with some similarity to subspace defects, include those where the holonomy of a specified boundary is in a subgroup H ⊂ G. Amplitudes including such observables have applications in counting Feynman graphs and BPS states of quiver theories [29,6].
We believe there is an interesting story with the space of finite linear combinations, with the inclusion of subspace defects. Even if that were solved, there would remain another layer of subtleties involved in giving a complete axiomatic framework for CFT 4 , based on the TFT 2 picture developed here. This is because states of interest in computing correlators, namely the Φ(x) (eq. 4.3) used to construct CFT 4 correlators, are actually infinite linear combinations! Careful treatment of these issues gets into the subject of topological vector spaces. This can be a subtle subject, e.g. how to define tensor products, see for example [30]. The right definition of "tensor product" in our case, would have to take into account the fact that if we insert Φ(x) at one circle and again the same Φ(x) at another circle ( with the same x ), then the amplitude is not well-defined. As pragmatic physicists, we can be content with a description of the TFT 2 in terms of genus-restricted version of the usual TFT 2 equations, and this is the point of view we have taken in most of the paper. However, solving the problem of finding the right axiomatic framework for the current TFT 2 formulation of CFT 4 would give an interesting mathematical perspective on AdS/CFT.

Correlators and Clebsch-Gordan coefficients
We have written can be used to generate the matrix elements of this homomorphism, i.e. the Clebsch-Gordan coefficients, which are often of interest as special functions. As far as we are aware, this QFT connection has not been used to develop explicit formulae for SO(4, 2) Clebsch-Gordan coefficients in the discrete basis. This would be an interesting exercise.

Further examples of CFT 4 /TFT 2
Let us recap, from section 2.5, that for the case of the Gaussian integration model, we have a state space At each level we have a 1-dimensional space, isomorphic to C. W n corresponds to φ n . The product is given by Gaussian integration. For the free scalar field in four dimensions, we had W n = Sym(V ⊗n ), where V = V + ⊕ V − . For Gaussian Hermitian Matrix model, let us work at large N , we have W 0 = C. At level n, we have a vector space of dimension p(n), the number of partitions of N . The basis vectors can be taken to be the multi-traces The numbers p i for i ∈ {1, · · · , n} determine a partition of n, i.e. n = i ip i . We may write The product is obtained from matrix integration. If can be written neatly in terms of permutations in S n . We won't make that explicit here.
For the 4D CFT of a free hermitian matrix field Φ, in the large N limit, we have for W n , a direct sum over partitions p. For each p The Cyc i (V ) is defined as the projection of V ⊗i to the subspace invariant under cyclic permutations of the i factors. Sym p i (W ) is the subspace of W ⊗p i invariant under symmetric group S p i .
If we want to do this at finite N , we can use the technology from [31]. The primaries in a scalar field theory are obtained from ⊕ ∞ n=0 SymV ⊗n + , where V + = D [1,0,0] . For the matrix scalar theory, we need to consider (V + ⊗ V N ⊗V N ) ⊗n and project to invariants of U (timesN ) and S n . We can use the following decompositions In the first line we are decomposing V ⊗n + in terms of irreps of SO(4, 2) and the commuting S n . The vector space V Λ,Λ 1 is the space of multiplicities for given irreps of SO(4, 2) and S n . U (N ) invariance forces R = S. S n invariants requires that V R ⊗ V R ⊗ V Λ 1 contains the trivial irrep of S n . The multiplicity C(R, R, Λ) of this trivial irrep. appears in the construction of the operators.
Let us outline some problems related to the above models and obvious, but interesting, generalizations : • For the free scalar or hermitian matrix CFT 4 , write the products C C AB and corresponding associativity equations explicitly.
• For the extremal correlators of the complex one-matrix model, the description of the state space will follow much the same steps as above for the matrix scalar field. This is relevant to the half-BPS sector in AdS/CFT. The explicit formulae for the invariant maps leading to correlators will need to take into account that the correlators involve insertion of multiple holomorphic operators and a single anti-holomorphic operators.
• Develop the CFT 4 /TFT 2 of free Maxwell Gauge fields, including TFT 2 construction of the correlators of E and B-fields. Extend this to free non-abelian gauge fields and free fermionic fields. These steps would provide the foundation for approaching perturbative N = 4 super-Yang Mills theory as a TFT 2 . We give more discussion of going beyond free fields in Section 9.5.
• Conversely, the TFT 2 construction we have developed with SO(4, 2) can be modified by replacing SO(4, 2) with a compact group G. Any finite dimensional irreducible representation V of G which contains the trivial irrep in the tensor product decomposition V ⊗ V , will allow the definition of an invariant mapη : V ⊗ V → C. Then we can define the state space Sym(V ⊗n ) (9.9) and the amplitudes C A 1 ,··· ,A k can be defined using tensor products of the elementaryη. This will give G-invariant TFT 2 's. Do these constructions have a realization in terms of a path integral ?

Generalized free fields and the stress tensor condition
Our reformulation of the free scalar field CFT 4 in terms of TFT 2 has started from the representation V + = V ∆=1,j L =0,j R =0 and the dual V − = (V + ) * , has constructed V = V + ⊕V − , and then built the state space W = ∞ n=0 Sym(V ⊗n ). There is a basic SO(4, 2) invariant mapη : V ⊗V → C which leads to the 2-point function. Multilinear SO(4, 2) invariant maps corresponding to the cobordism between k circles and the vacuum in TFT 2 , are constructed from W ⊗k → C, using tensor products of the basic invariantη. These include, for the case k = 2, a bilinear invariant pairing η : W ⊗ W → C with η AB in some convenient discrete basis e A of W. For the amplitude for 3-circles to vacuum, we have C ABC .
Given this reformulation, it is very natural to ask if all SO(4, 2) invariant TFT 2 's lead to some CFT 4 . A key result of this section is to construct examples of SO(4, 2) invariant TFT 2 's that do not correspond to a CFT 4 . Our discussion was motivated by recent activity aimed at solving the constraints imposed by crossing symmetry [32,33,34]. By understanding the crossing symmetry constraints one may hope to understand and completely characterize the space of all possible CFTs. In this section we will consider the problem of solving the constraints imposed by crossing, in the framework of our novel approach to CFT 4 .
The construction of new solutions might begin by replacing V + ⊕ V − above with V = V ∆,j L ,j R ⊕ V * ∆,j L ,j R . Since any discrete series lowest weight irrep (having positive energy) has a dual highest weight irrep (having negative energy) such that V ∆,j L ,j R ⊗ V * ∆,j L ,j R contains a unique copy of the identity representation, we know that we will have a uniqueη ab as matrix elements of V ⊗ V → C. Aside from this replacement, the construction of W, η, C proceeds as before. This is a generalized free field construction, which, as we will explain, gives an SO(4, 2) covariant TFT 2 . Further conditions can be imposed, such as unitarity and existence of a stress tensor, which can be reasonably considered necessary for these to be proper CFT 4 .
We will want to place additional constraints on this solution. For example we can ask for a unitary TFT 2 . We have explained how to define an inner product g : V ⊗ V → C, which is sesquilinear and hermitian. This extends to an inner product on W which coincides with the CFT inner product. If we require that this inner product on the TFT 2 state space is non-degenerate with strictly positive eigenvalues, then we have the usual restriction of unitarity in CFT 4 , which leads to constraints on dimensions of fields. For example, for a scalar (where j L , j R = 0) we have ∆ ≥ 1.
Another natural constraint is to require that the theory contains a stress tensor. For the free scalar field which has ∆ = 1, Sym(V ⊗2 ) contains a local stress tensor, for which V ∆,j L ,j R = V 4,1,1 .
An interesting case to consider is where the basic free field is itself the representation V 4,1,1 . Now V has the right SO(4, 2) transformation properties to be a stress tensor. There is a T µν which is treated as an elementary field in this generalized free field construction. We want to know whether we can make sense of in our TFT 2 approach to this generalized free field system. To define P µ along the lines of the first equation, we would need to define some sum over states in the representation V = V + ⊕ V − . This would give P µ as a state in W. So P µ = P A µ e A where e A is a set of basis vectors for W. There is a corresponding operator ( P µ ) C B ≡ P A µ C C AB in the endomorphisms of W. The state O is a general state in W, i.e. O = O A e A and there is a corresponding operator ( O) C B ≡ O A C C AB . The LHS can be written as The RHS of the second equation is another well defined state in W, so we can consider The question is whether there is a good definition of P µ , i.e. whether the description as a sum above can be made precise enough to prove that the equality holds for arbitrary O.
Since we do not have a Lagrangian for this prospective new generalized free field CFT, we have to formulate the requirement of existence of a good stress tensor along the algebraic lines outlined here. Although this has been instructive as an exercise in using the new TFT language, the above construction does not work, as we now explain. The argument we employ appeared recently in the context of CFT d in [35]. Our structure constants C ABC are symmetric. By looking at the dimensions of fields, it is clear that the energy momentum tensor does not appear in the product of the field with itself. Equivalently, the field will not appear in the product of the energy momentum tensor with the field which contradicts (9.18). This contradiction rules out generalized free fields in CFT d unless they have a dimension ∆ = d−2 2 . Although the generalized free field theories are not CFT's, our discussion makes it clear what they are: they are SO(d, 2) covariant TFT 2 's.

Beyond free fields
We highlight here some key ingredients in the connection between CFT 4 and TFT 2 we have developed, which will continue to hold beyond the set-up of free fields that we have described explicitly here.
• The description of the state space in terms of discrete series representations of SO (4,2).
The description of the two-point function of operators in terms of an invariant pairing between positive and negative energy representations. This is necessitated simply by the fact that there is no SO(4, 2) invariant in the tensor product of two positive energy representations.
• The relation of the bilinear invariant pairing η AB to a sesquilinear positive definite inner product, via a map ρ between positive and negative energy representations.
• The non-degeneracy of the pairing, i.e existence ofη AB , and its relation to nondegeneracy of the Zamolodchikov metric.
• The OPE and C C AB = C ABDη DC .
• Relation between crossing and associativity.
One element which needs to be worked out in a case by case basis, is the explicit construction of the correlators in terms of representation theoretic data as we have done for the free field case. Here we have developed the construction by expressing the usual Wick contraction rules of free field theory in terms of applications of a tensor product of elementary invariant pairings.
A very interesting problem is to develop this beyond free fields to perturbative CFTs, e.g. N = 4 supersymmetric Yang Mills theory. Elucidating the role of regularization and renormalization in this TFT 2 framework will be a fascinating problem. There is good reason to believe that this problem will have an elegant solution. This expectation comes from the fact that one-loop N = 4 SYM has been formulated using representation theory of P SU (2, 2|4) via the formalism of the one-loop dilatation operator [36,37] and this approach is also known to generalize to higher loops [38]. The key elements of the link to TFT 2 have been aspects of SO(4, 2) representation theory. In the case of N = 4 SYM we would expect a P SU (2, 2|4)-invariant TFT 2 .
Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. We thank Chong Sun Chu, David Garner, Hally Ingram, Dimitri Polyakov, Douglas Smith, Gabriele Travaglini, Rodolfo Russo for discussions.
A sl(2) and su(2) × su(2) subalgebras of so (4,2) In this Appendix we will describe the sl(2) and su(2) × su(2) subalgebras of so(4, 2) that were used in section 3.2.1 to construct the invariant pairingη. In terms of the generators of section 3.1.2, introduce which contains useful information about the states that appear in V + .