Conformal Field Theory In Four And Six Dimensions Edward Witten∗ Institute for Advanced Study Princeton, NJ 08540 USA February 7, 2008 1 Introduction In these lectures, I will be considering conformal field theory (CFT) mainly in four and six dimensions, occasionally recalling facts about two dimensions. The notion of conformal field theory is familiar to physicists. From a math- ematical point of view, we can keep in mind Graeme Segal’s definition [1] of conformal field theory. Instead of just summarizing the definition here, I will review how physicists actually study examples of quantum field theory, as this will make clear the motivation for the definition. When possible (and we will later consider examples in which this is not possible), physicists make models of quantum field theory using path inte- arXiv:0712.0157v2 [math.RT] 7 Feb 2008 grals. This means first of all that, for any n-manifold Mn, we are given a space of fields on Mn; let us call the fields Φ. The fields might be, for example, real-valued functions, or gauge fields (connections on a G-bundle over Mn for some fixed Lie group G), or p-forms on Mn for some fixed p, or they might be maps Φ : Mn → W for some fixed manifold W . Then we are given a local action functional I(Φ). “Local” means that the Euler-Lagrange equations for a critical point of I are partial differential equations. If we are constructing a quantum field theory that is not required to be conformally ∗Supported in part by NSF Grant PHY-0070928. Lectures at the conference on Topol- ogy, Geometry, and Quantum Field Theory, Oxford University (July, 2002). 1 invariant, I may be defined using a metric on Mn. For conformal field theory, I should be defined using only a conformal structure. For a closed Mn, the partition function Z(Mn) is defined, formally, as the integral over all Φ of e−I(Φ): Z(Mn)= DΦ exp(−I(Φ)). (1) Z If Mn has a boundary Mn−1, the integral depends on the boundary conditions. If we let ϕ denote the restriction of Φ to Mn−1, then it formally makes sense to consider a path integral on a manifold with boundary in which we integrate over all Φ for some fixed ϕ. This defines a function Ψ(ϕ)= DΦ exp(−I(Φ)). (2) Φ| =ϕ Z Mn−1 We interpret the function Ψ(ϕ) as a vector in a Hilbert space H(Mn−1) of L2 functions of ϕ. From this starting point, one can motivate the sort of axioms for quantum field theory that Segal considered. I will not go into details, as we will not need them in the present lectures. In fact, to keep things simple, we will mainly consider closed manifolds Mn and the partition function Z(Mn). Before getting to the specific examples that we will consider, I will start with a general survey of conformal field theory in various dimensions. Two- dimensional conformal field theory plays an important role in string theory and statistical mechanics and is also relatively familiar mathematically. 1 For example, rational conformal field theory is studied in detail using complex geometry. More general conformal field theories underlie, for example, mirror symmetry. Three and four-dimensional conformal field theory is also important for physics. Three-dimensional conformal field theory is used to describe sec- ond order phase transitions in equilibrium statistical mechanics, and a four- dimensional conformal field theory could conceivably play a role in models of elementary particle physics. Physicists used to think that four was the maximum dimension for non- trivial (or non-Gaussian) unitary conformal field theory. Initially, therefore, little note was taken of a result by Nahm [2] which implies that six is the maximum possible dimension in the supersymmetric case. (A different result 1In counting dimensions, we include time, so a two-dimensional theory, if formulated in Lorentz signature, is a theory in a world of one space and one time dimension. In these lectures, we will mostly work with Euclidean signature. 2 proved in the same paper – eleven is the maximal possible dimension for supergravity – had a large impact right away.) Nahm’s result follows from an algebraic argument and I will explain what it says in section 3. String theorists have been quite surprised in the last few years to learn that the higher dimensional superconformal field theories whose existence is suggested by Nahm’s theorem apparently do exist. Explaining this, or at least giving a few hints, is the goal of these lectures. One of the surprises is that the new theories suggested by Nahm’s theorem are theories for which there is apparently no Lagrangian – at least none that can be constructed using classical variables of any known sort. Yet these new theories are intimately connected with fascinating mathematics and physics of more conventional theories in four dimensions. In section 2, we warm up with some conventional and less conventional linear theories. Starting with the example of abelian gauge theory in four dimensions, I will describe some free or in a sense linear conformal field theories that can be constructed in arbitrary even dimensions. The cases of dimension 4k and 4k + 2 are rather different, as we will see. The most interesting linear theory in 4k + 2 dimensions is a self-dual theory that does not have a Lagrangian, yet it exists quantum mechanically and its existence is related to subtle modular behavior of the linear theories in 4k dimensions. In section 3, I will focus on certain nonlinear examples in four and six dimensions and the relations between them. These examples will be super- symmetric. The importance for us of supersymmetry is that it gives severe constraints that have made it possible to get some insight about highly non- linear theories. After reviewing Nahm’s theorem, I will say a word or two about supersymmetric gauge theories in four dimensions that are conformally invariant at the quantum level, and then about how some of them are appar- ently related to nonlinear superconformal field theories in six dimensions. 2 Gauge Theory And Its Higher Cousins First let us review abelian gauge theory, with gauge group U(1). (For general references on some of the following discussion of abelian gauge fields and self-dual p-forms, see [3].) The connection A is locally a one-form. Under a gauge transformation, it transforms by A → A + dǫ, with ǫ a zero-form. The curvature F = dA is invariant. 3 For the action, we take 1 iθ F F I(A)= 2 F ∧∗F + ∧ . (3) 2e ZM 2 ZM 2π 2π Precisely in four dimensions, the Hodge ∗ operator on two-forms is confor- mally invariant and so I(A) is conformally invariant. If M is closed, the 2 second term in I(A) is a topological invariant, i(θ/2) M c1(L) . In general, 2 c1(L) is integral, and on a spin manifold it is actually even.R So the integrand exp(−I(A)) of the partition function is always invariant to θ → θ +4π, while on a spin manifold it is invariant to θ → θ +2π. In general, even when M is not closed, this is a symmetry of the theory (but in case M has a boundary, the discussion becomes a little more elaborate). Now let us look at the partition function Z(M)= L DA exp(−I(A)), where we understand the sum over all possible connectionsP R A as including a sum over the line bundle L on which A is a connection. We can describe the ′ L path integral rather explicitly, using the decomposition A = A + Ah , where ′ L A is a connection on a trivial line bundle O, and Ah is (any) connection on L ′ L L of harmonic curvature Fh . The action is I(A) = I(A )+ I(Ah ), and the path integral is ′ ′ L DA exp(−I(A)) = DA exp(−I(A )) exp(−I(Ah )). (4) XL Z Z XL L ′ Here, note that Ah depends on L, but A does not. Let us look first at the second factor in eqn. 4, the sum over L. On the lattice H2(M; Z), there is a natural, generally indefinite quadratic form given, for x an integral harmonic two-form, by (x, x)= M x∧x. There is also a positive-definite but metric-dependent form hx, xi = RM x∧∗x, with ∗ being the Hodge star operator. The indefinite form (x, x) hasR signature (b2,+, b2,−), where b2,± are the dimensions of the spaces of self-dual and anti-self-dual harmonic two-forms. L Setting x = Fh /2π, the sum over line bundles becomes 4π2 θ exp − e2 hx, xi + i 2 (x, x) . (5) 2 Z x∈HX(M; ) θ 4πi If I set τ = 2π + e2 , then this function has modular properties with respect to τ. It is the non-holomorphic theta function of C. L. Siegel, which in the mid-1980’s was introduced in string theory by K. S. Narain to understand 4 toroidal compactification of the heterotic string. The Siegel-Narain function has a simple transformation law under the full modular group SL(2, Z) if M is spin, in which case (x, x)/2 is integer-valued. In general, it has modular properties for a subgroup Γ0(2) of SL(2, Z). In any case, it transforms as a modular function with holomorphic and anti-holomorphic weights (b2,+, b2,−). The other factor in eqn. 4, namely the integral over A′, DA′ exp(−I(A′)), is essentially a Gaussian integral that can be defined by zetaR functions.