CONFORMAL FIELD THEORIES

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1 CONFORMAL FIELD THEORIES Definition 0.1 (Segal, see for example [Hen]). A full conformal field theory is a symmetric monoidal functor { } 1 dimensional compact oriented smooth manifolds {Hilbert spaces}. conformal cobordisms (Here the monoidal structure on the left is given by disjoint union, while the monoidal structure on the right is given by the completed tensor product.) Definition 0.2. A chiral conformal field theory consists of the following data: To each 1-manifold S as above we associate a category C S. Furthermore, to each object λ C, we associate a Hilbert space H λ. To each complex cobordism Σ from between 1-manifolds S in and S out, we associate a functor f Σ : C in C out, depending on the complex structure of Σ in a conformal way. Furthermore, to each object λ C in we associate a linear map H λ H fσ(λ). As a model for chiral CFT, we will use vertex algebras. Given a vertex algebra V, the category C S 1 will be given by the category Rep(V ). Given a representation λ Rep(V ), the corresponding Hilbert space will be given by the vector space underlying the representation. And the linear maps should be constructed from chiral blocks in some way. Given a chiral CFT, we need to make a lot of choices in a compatible way in order to get a full CFT. The FRS theorem says that given a chiral CFT corresponding to a rational vertex algebra V, making all of these choices is equivalent to choosing a Frobenius algebra object A Rep(V ). (When V is rational, Rep(V ) is a modular tensor category, so it is reasonable to talk about Frobenius algebra objects.) We will define some of these terms in the rest of this talk, and give examples of vertex algebras. 1. Vertex algebras Definition 1.1. A vertex algebra (V, 0, T, Y (, z)) consists of the following data: the space of states: a graded complex vector space V = i 0 V i the vacuum vector: 0 V 0 the translation operator: T : V V a linear map of degree 1 Date: February

2 2 CONFORMAL FIELD THEORIES the vertex operators: Y (, z) : V End V [z, z 1 ] a linear map such that if we have A V i and write Y (A, t) = n Z A (n) z n 1, then the ( n 1)th coefficient A (n) End V is of degree n + i 1. These data are subject to the following conditions: The vacuum axiom: Y ( 0, t) = id V Furthermore, for any A V, A (n) 0 = 0 for n 0, and A ( 1) 0 = A. The translation axiom: [T, Y (A, z)] = z Y (A, z) T 0 = 0 A V The locality axiom: For any A, B V there exists N N such that (z w) N [Y (A, z), Y (B, w)] = 0 End V [z ±, w ± ] Remark 1.2. We can modify this definition in the super setting, resulting in vertex superalgebras Example: Heisenberg and lattice vertex algebras. Definition 1.3. The Heisenberg Lie algebra is the central extension ĥ of the commutative Lie algebra C((t)) 0 C1 ĥ C((t)) 0, with generators {b n = t n } and 1 satisfying the commutation relations [1, b n ] = 0, [b n, b m ] = nδ n, m 1. More generally, let L be a lattice (i.e. a finitely generated free abelian group), and suppose (, ) is a positive definite symmetric bilinear form on L. 1 Definition 1.4. The Heisenberg Lie algebra modelled on L is the central extension ĥ L 0 C1 ĥl L C C((t)) 0, defined by setting [A f(t), B g(t)] = (A, B)(Res f(t)g (t)dt), where Res h(t)dt = Res t=0 h(t)dt = h 1 for any h(t) = i Z h it i C((t)). The Lie algebra ĥl is generated by elements of the form α(n).= α t n (α L, n Z), and by 1, satisfying the relations [α(m), 1] = 0, [α(m), β(n)] = m(α, β)δ m, n 1. We define representations of ĥ and ĥl called Fock spaces by inducing onedimensional representations of commutative subalgebras. 1 If we assume L is even, then the construction we re about to describe will result in a vertex algebra; otherwise we will obtain a vertex superalgebra.

3 CONFORMAL FIELD THEORIES 3 For example, let C 0 be the one dimensional representation of ĥ+ C1 on which 1 acts as the identity and ĥ+ acts trivially. (By ĥ+ we mean the subalgebra generated by positive powers of t.) Then we define π 0.= U(ĥ) U(^h + C1) C 0. We can see that as a ĥ-representation, π 0 is isomorphic to the polynomial algebra in infinitely many variables C[b 1, b 2,...], where a generator b n of ĥ acts by multiplication by the variable b n for n < 0, by differentiation n b n for n > 0 and by zero for n = 0. Similarly, for any λ in the lattice L, we have a representation π λ which is generated by a single vector λ and satisfies Set 1 λ = λ ; α(n) λ = 0, n > 0; α(0) λ = (λ, α) λ. V L.= λ L π λ. Claim 1.5. (1) The vector space V L is a vertex (super)algebra, called the lattice vertex algebra associated to (L, (, )). (2) The component π 0 is a vertex subalgebra of V L, called the Heisenberg vertex algebra. We will sketch the construction of the data required to specify the vertex algebra structure. For the details, see Chapter 7 of [FLM88] and also Sections 4.2 and 4.4 of [FBZ01]. (1) The parity: We set the parity of π λ to be equal to (λ, λ) mod 2. (In particular, if (, ) takes values in 2Z, then V L will be purely even and hence an ordinary vertex algebra.) The grading: We define the degree of a vector of the form m λ = α 1 (n 1 )α 2 (n 1 ) α k (n k ) λ to be equal to (λ, λ)/2 k j=1 n j. The vacuum vector: we choose the vector 0, noting that it is indeed even and of degree 0. The translation operator: we define T : V L V L by setting T λ.= λ( 1) λ [T, α(n)].= nα(n 1) λ L α L, n Z Note that this determines the action of T on all of V L by induction on the length k of the monomial m from (1): T α k+1 (n k+1 )m λ = α k+1 (n k+1 )T m λ + [T, α k+1 (n k+1 )]m λ It is clear that T is even and of degree one.

4 4 CONFORMAL FIELD THEORIES (2) The vertex operators: For any α L we set α(z) = Y (α( 1) λ, z).= n Z α(n)z n 1 ; here α(n) is regarded as a linear operator on V L : note that it has degree n and is even. By the Strong Reconstruction Theorem (Theorem 3.6.1, [FBZ01]) it suffices to define the vertex operators V λ (z) = Y ( λ, z) for λ L; these together with (2) uniquely determine the remaining vertex operators. We define S λ : π μ π μ+λ by setting S λ ( μ ).= μ + λ and requiring that [S λ, α(n)] = whenever α L and n 0. Then we set ( V λ (z).= S λ z λ0 exp ) ( λ n n z n exp ) λ n n z n n<0 n>0 (Here z λ 0 acts on π μ by z λ,μ.) We omit the proof that this gives a vertex (super)algebra structure on V L. Remark 1.6. We can see that the coefficients of the vertex operators Y ( λ, z) are endomorphisms which act on the L-graded pieces of V L as follows: π μ π μ+λ. In particular, the coefficients of Y ( 0, z) restrict to give endomorphisms of π 0. It is for this reason that π 0 is a vertex subalgebra. The Heisenberg vertex algebra for the lattice L = Z is used to model the theory of a free boson. When we introduce the lattice L we study the compactified version of this theory Example: Kac-Moody vertex algebras. Let g be a simple Lie algebra of finite dimension d = dim g. Definition 1.7. The loop algebra of g is the Lie algebra with Lie bracket given by Lg = g C C((t)), [A f(t), B g(t)] = [A, B] f(t)g(t), A, B g, f(t), g(t) C((t)). Definition 1.8. The affine Kac-Moody Lie algebra associated to g is the central extension with bracket given by 0 CK ĝ Lg 0, [A f(t), B g(t)] = [A, B] f(t)g(t) (Res t=0 fdg)(a, B)K. As before, we consider a subalgebra of our Lie algebra and we induce a onedimensional represention of this subalgebra to ˆg. In this case, we note that g[t] CK is a Lie subalgebra, and for any k C we consider its one-dimensional representation C k defined by letting K act by the scalar k, and g[t] act trivially.

5 CONFORMAL FIELD THEORIES 5 Definition 1.9. The vacuum representation of level k is V k (g).= U(ĝ) U(g [t ] CK) C k. Claim We have a canonical structure of vertex algebra on V k (g). Fact If k Z +, then V k (g) is irreducible as a ĝ-module, with a proper submodule I k generated by (e αmax 1 ) k+1 v k. (Here α max is the maximal root of the Lie algebra g, and v k is the image of 1 1 in V k (g), and is the vacuum vector.) But L k (g).= V k (g)/i k is irreducible as a ĝ-module, and acquires the structure of a simple vertex algebra. When physicists talk about the WZW-model, they are talking about the theory corresponding to L k (g) Example: the Virasoro vertex algebra. Notation We let K = C((t)), O = C[t]. We denote by Der K = C((t)) t the Lie algebra of continuous derivations of K, and similarly Der O = C[t] t. Definition The Virasoro Lie algebra is the central extension with Lie bracket defined by 0 CC Vir Der K 0, [f(t) t, g(t) t ] = (fg f g) t 1 12 (Res t=0 fg dt)c. It has topological generators of the form L n = t n+1 t and C, and these satisfy the relations [C, L n ] = 0, [L n, L m ] = (n m)l n+m + n3 n (3) δ n, m. 12 (These are known as the Virasoro relations.) Fix a constant c C and consider the action of the Lie subalgebra Der O CC of Vir on C defined by letting C act by the scalar c and Der O act trivially. We denote this representation by C c, and we consider the induced representation U(Vir) U(Der O CC) C c. This has a structure of vertex algebra; we call it the Virasoro vertex algebra of central charge c, and denote it by Vir c. Fact The representation Vir c is reducible as a Vir-module precisely when c is of the form c(p, q) = 1 6(p q)2 pq for p, q > 1 coprime integers. In that case, we can form the irreducible quotient, which we denote by L c(p,q) ; it is a simple vertex algebra. When physicists talk about the minimal model they are referring to a theory corresponding to L c(p,q). 2. Conformal vertex algebras and other representations Definition 2.1. A vertex algebra is conformal of central charge c if it is equipped with a distinguished vector ω V 2 such that the modes L V n of Y (ω, z) = L V n z n 2 n Z are endomorphisms of V satisfying the Virasoro relations of central charge c.

6 6 CONFORMAL FIELD THEORIES Equivalently, there is a unique vertex algebra homomorphism Vir c V, which sends the vacuum vector v c of Vir c to the vacuum vector 0 of V, and which also maps L 2 v c to ω. Definition 2.2. This vector ω is called the conformal vector Examples. (1) The Virasoro vertex algebra of central charge c is conformal of central charge c, with conformal vector ω = L 2 v c. (2) The Heisenberg vertex algebra π 0 has a family of conformal vectors ω λ π 0, given by ω λ = 1 2 b2 1 + λb 2. Then ω λ is conformal of central charge c λ = 1 12λ 2. In particular, setting λ = 0 we obtain a conformal vector of central charge 1. (3) Let V k (g) be the Kac-Moody vertex algebra of level k, and assume that k h, where h is the dual Coxeter number of g. Then the Sugawara kd construction produces a conformal vector ω of central charge k+h. (Recall that d is the dimension of g.) Definition 2.3. Let V = (V, 0, T, Y (, z)) be a vertex algebra. A representation of V (or a V -module) consists of the following data: a graded space M = n Z M n such that M n = 0 for n sufficiently small; a translation operator T M : M M, linear of degree 1; vertex operators Y M (, z) : V End(M)[z, z 1 ]. We require these objects to satisfy some axioms making them compatible with the vertex algebra structure on V. Definition 2.4. Suppose in addition that V is conformal of central charge c with conformal vector ω. Then a representation M as above is called conformal if the mode L M 0 of ω, defined by setting Y M (ω, z) = n Z L M n z n 2, coincides with the gradation operator on M, up to a shift Examples. (1) If we have a homomorphism of vertex algebras V W, then W is a representation of V. Moreover, every representation of W automatically becomes a representation of V. (2) In particular, every conformal vertex algebra of central charge c is a conformal representation of Vir c. (3) From Remark 1.6, we can see that each Fock space π λ is a representation of the Heisenberg vertex algebra π 0. Definition 2.5. We say that a conformal vertex algebra V is rational if every V -module is completely reducible. Here are some consequences of V being rational: (1) Up to isomorphism, V has only finitely many simple modules M.

7 CONFORMAL FIELD THEORIES 7 (2) The graded components M n of each simple module M are all finite dimensional. (3) Each simple module M is conformal. (4) The category Rep V is a modular tensor category. 3. Conformal blocks Fix a rational vertex algebra V and a collection M 1, M 2,..., M n of representations of V. We have a trivial bundle on the moduli space M n,g of genus g curves with n marked points given by (M 1 M n ) * M n,g M n,g. Our goal is to identify a nice subbundle of this vector bundle by imposing conditions that come from V and depend on our choice of marked curve (X, x 1,..., x n ) M n,g. That is, for each choice (X, x 1,..., x n ), we need to specify a collection of linear forms M 1 M n C. Step 1. To begin, assume than n = 1, and fix a smooth projective curve X together with a single point x 1 X. Let s take M 1 to be the natural representation of V on itself. The first thing to do is use V to build a vector bundle on X. We proceed as follows. Define Aut X.= { (x, t x ) x X, t x : ˆO X,x } O = C[t]. (We say that t x is a formal coordinate on X at x.) This gives a principal Aut O - bundle on X. Then notice that conditions of V being conformal give in particular an integrable action of Der O on V, so that we can view V as a representation of Aut O. We define V to be the associated bundle on X: V.= Aut X AutO V X. This is a bundle with flat connection (the construction of the connection also comes from the action of the Virasoro on V ). Denote by V x the restriction of V to the formal disc D x around x. Note that a choice of formal coordinate t x at x gives an identification of V x with V. Recall that the definition of Y (, z) means that for every choice of A, v V and φ V * we obtain an element φ, Y (A, z)v of C((t)). Using our formal coordinate t x, we obtain a meromorphic End(V)-valued section of the bundle V * on the punctured disc D x, which we ll call Y x. That is, for any x X, A V x, and φ V * x, we obtain a section (4) φ, Y x A of V * on D x. We can show that the definition of Y x is independent of the choice of coordinate t x. Definition 3.1. We say that φ V * x is a conformal block if for every choice A V x the section (4) extends to give a regular section of V * on all of X x. We denote by C(V, X, x) the space of all conformal blocks.

8 8 CONFORMAL FIELD THEORIES Step 2. Now we generalise to allow for more insertion points {x 1,..., x n } and arbitrary representations M i Rep(V ) associated to each point x i. For simplicity, let s assume that for each i, L Mi 0 acts on M i with integral eigenvalues. Then as above, we construct vector bundles (with flat connection) M i = Aut X AutO M i X, and we define End(M i )-valued sections Y Mi,x i of V * on D x i. Definition 3.2. An element φ ( n i=1 M i,x i ) * is called a conformal block if for any fixed choices A i M i,xi, all of the sections of V * over the punctured discs given by D x i φ, A 1 (Y Mi,x i A i ) A n can be extended to the same regular section of V * over X {x 1,..., x n }. This gives a space C V (X, (x i ), (M i )) of conformal blocks associated to the data V, X, x 1,..., x n, and M 1,..., M n. Step 3. Now we assemble these spaces into a sheaf of conformal blocks. Recall that M 1 M n X n is a vector bundle with flat connection. By definition, C V (X, (x i ), (M i )) is a subspace of the fibre of (M 1 M n ) * at (x 1,..., x n ) X n. (Here by X n we mean the complement in X n to all of the diagonals.) Allowing (x i ) to vary over X n, we obtain a subsheaf of (M 1 M n ) *, preserved by the flat connection. Step 4. Given an element φ C V (X, (x i ), (M i )) we obtain a horizontal section φ y1,...,y n of the sheaf of conformal blocks in a neighbourhood of (x 1, ldots, x n ). Choosing local sections A i (y i ) of each M i near x i, we form a function around the point (x 1,..., x n ): φ y1,...,y n A 1 A n. Definition 3.3. This function is called a chiral correlation function. Question 3.4. How do chiral correlation functions behave near the diagonals? Here is one result. Suppose that all x i are near the same point x; let us choose a coordinate z at x, and denote by z i the corresponding coordinate at x i. Then φ, A 1 (z 1 ) A n (z n ) = φ, Y (A 1, z 1 )Y (A 2, z 2 ) Y (A n, z n ) 0 (as an element of C[t 1,..., t n ][(z i z 1 j )] i j ). 4. Free field realisation This is a technique for understanding the conformal blocks of something complicated in terms of the conformal blocks of something easier. Key observation 4.1. Let V W be a morphism of conformal vertex algebras. Let X be a smooth projective curve with n marked points {x 1,..., x n }, and let M 1,... M n be conformal representations of W. Recall from the first example of 2.2 that each M i automatically acquires the structure of a V -module.

9 CONFORMAL FIELD THEORIES 9 We have an embedding C W (X, (x i ), (M i )) C V (X, (x i ), (M i )) of the subspaces of ( n i=1 M i) *. Indeed, the conditions imposed by V are weaker than those imposed by W, since the action of V factors through the action of W. Moreover, the flat connections on the corresponding sheaves are determined only by the action of the Virasoro on the modules M i, and this action is the same whether we view M i as a W -module or as a V -module. It follows this embedding extends to an embedding of the sheaves of conformal blocks, compatible with the flat connections. Therefore, if we know the horizontal sections of the sheaf of conformal blocks corresponding to W, we obtain horizontal sections for V as well. So if we are interested in studying the conformal blocks of a vertex algebra V, we should try to find a map from V to some vertex algebra W that we understand better. The simplest non-trivial vertex algebras are given by the Heisenberg algebras, and indeed our main application of will be by constructing a map (5) V k (g) W, where W is a Heisenberg vertex algebra corresponding to some Heisenberg Lie algebra ĥ. Then horizontal sections of the sheaf of conformal blocks corresponding to W are understood in terms of the theory of free bosonic fields, which is not too bad. On the other hand, horizontal sections of the sheaf of conformal blocks associated to V k (g) correspond to solutions to the Knizhnik Zamolodchikov equations, and are important in the study of the WZW-model. Note that this kind of argument cannot be done at the level of the Lie algebras ĝ and ĥ, and we do need to work with the machinery of vertex algebras and conformal blocks. However, to produce a map V k (g) W, it suffices to give a map of Lie algebras g ĥ. Moreover, the Heisenberg Lie algebra is (roughly) the Lie algebra of differential operators on an affine space, and so to produce the desired map we just need to give an action of the Lie algebra g on the affine space by vector fields. We will not give the full construction, but it amounts to defining a formal loop space on the flag variety G/B, called the semi-infinite flag manifold. Roughly, we should think of the space of maps This is the space on which g will act. D = Spec C((t)) G/B. 5. W-algebras In this section, we will sketch the definition of W-algebras: these form a 1- parameter family of vertex algebras associated to a simple Lie algebra g. Step 1: Form the BRST complex. First recall that given a finite dimensional vector space U, U((t)) U * ((t))dt is a complete topological vector space. We associate to it a complete topological algebra Cl(U), called the Clifford algebra. The Clifford algebra has an irreducible Fock space representation, which we denote

10 10 CONFORMAL FIELD THEORIES by U. It has the structure of a vertex superalgebra, isomorphic to a tensor product of copies of the free fermionic vertex superalgebra. We introduce a second grading on the spaces Cl(U) and U, which we call the charge gradation. Now given a simple Lie algebra g with Cartan decomposition g = n + h fn, we form a complex by taking the tensor product Ck(g) = V k (g), n where the grading of the complex is given by the charge gradation. This complex aqcuires the structure of a vertex superalgebra. We define an element Q of Ck (g) (it is given by an explicit formula, but we haven t introduced any of the notation necessary, so we will not give the formula here); then the standard differential d st of charge 1 is given by the coefficient Q ( 0) of the vertex operator Y (Q, z). The resulting complex (Ck (g), d st) is the standard complex of semi-infinite cohomology of the Lie algebra n + ((t)) with coefficients in V k (g). Step 2: Twist by the Drinfeld Sokolov character. The Drinfeld Sokolov character χ is defined on e α n = e α t n (α a root of g, n Z) by { χ(e α 1 if α is simple and n = 1, n) = 0 otherwise. This definition extends to give a linear functional on n + ((t)), and it is a character: χ([x, y]) = 0. We denote also by χ the corresponding element of n * +((t))dt Cl(n + ). We have that χ 2 = 0, and [d st, χ] = 0, and so it follows that d.= d st + χ is a differential on C k (g). Definition 5.1. The resulting complex (Ck (g), d) is called the BRST complex of the quantum Drinfeld Sokolov reduction. Step 3: Take cohomology. We denote the cohomology of (Ck (g), d) by H k (g). It is a vertex superalgebra, and in particular Hk 0 (g) is a vertex algebra. Definition 5.2. We set W k (g).= Hk 0 (g); it is called the W-algebra associated to ĝ at level k. Example 5.3. When g = sl 2, the vertex algebra W k (sl 2 ) is generated by a single vector W 1, which can be chosen to be the conformal vector if k h = 2. It follows that W k (sl 2 ) is isomorphic to the Virasoro vertex algebra Vir c (k). (Here the central charge is given by c(k) = 1 6(k+1)2 k+2. When k = 2, W 2 (sl 2 ) is a commutative vertex algebra; it is isomorphic to C[S n ] n 2. We have a free field realisation for W-algebras as well. Recall from Remark 1.6 that the vertex operators of the vertex algebra corresponding to a lattice L give rise to maps π λ π λ+μ for λ, μ L. In our case, we let α 1, α d denote the simple roots of g; we let ν = k + h, and we denote the corresponding maps by V αi/ν(z) : π 0 π αi/ν.

11 CONFORMAL FIELD THEORIES 11 Definition 5.4. The operators V αi/ν(z)dz : π 0 π αi/ν are known as the screening operators. Recall that π 0 is the Heisenberg vertex algebra, in this case associated to the Heisenberg Lie algebra modelled on the Cartan h of g. Theorem 5.5. For k h, W k (g) is the subalgebra of π 0 given by the intersection of the kernels of the screening operators. References [FBZ01] Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves, volume 88 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, [FLM88] I. Frenkel, J. Lepowsky, and A. Meurman. Vertex operator algebras and the Monster, volume 134 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, [Hen] André Henriques. Three-tier CFTs from Frobenius algebras. arxiv:math/ v2.