Conformal field theory in the sense of Segal, modified for a supersymmetric context

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1 Conformal field theory in the sense of Segal, modified for a supersymmetric context Paul S Green January 27, Introduction In these notes, we will review and propose some revisions to the definition of a conformal field theory first articulated by Graeme Segal in a set of notes which have had considerable informal circulation and influence since 1989, although they were published only in 2004 in a festschrift volume for Professor Segal. The proposed revisions are in several directions. The explicit introduction of a supersymmetric version of Segal s definition. The observation that a Riemann surface with exactly one incoming and one outgoing boundary component with a marked point on each admits a canonical parametrization of the boundary which gives the set of all such surfaces the natural structure of a holomorphic semigroup. This semigroup and its supersymmetric generalization will be seen to play an important role in the theory. An important role for the Krichever-Novikov algebras and their supersymmetric generalizations. 2 Review and reformulation of Segal s definition of a CFT The version of Segal s definition that we give here is closer to his reformulation in the preface to the published version of his notes than it is to the 1

2 original notes; however it is not identical with either. A conformal field theory in the sense of Segal begins with a functor H from the category of compact closed connected one-manifolds and orientationpreserving diffeomorphisms to the category of Hilbert spaces and unitary isomorphisms, satisfying: (a) H( C) = H(C), where C denotes C with the opposite orientation. (b) If C and C are disjoint then H(C C ) = H(C) H(C ). (c) It is useful to adopt the convention that H(φ) = C, where φ denotes the empty one-manifold. The other part of the structure assigns to each Riemann surface X an element A(X) H( (X)). Here, X is given the orientation determined by its complex structure and (X) denotes the oriented boundary of X. In particular, if X is closed, then A(X) is a complex number. A satisfies: (d) If f : X X is a holomorphic equivalence, then A(X ) = H(f X)(A(X)). Before stating the next condition, we recall that if V and W are Hilbert spaces, then the Hilbert tensor product V H W is naturally isomorphic to the space linear functions from W to V of Hilbert-Schmidt type. We will say a Hilbert-Schmidt operator A from W to V is of trace class if A U is of trace class for some (and therefore for any) unitary equivalence U : V W. Note that the trace can be computed in the tensor product formulation from the formula tr(v w) =< v, U(w) >. 1 This allows us to extend the definition of trace class to a Hilbert tensor product of any number of factors. (e) If C and C are components of X, and ˆX is the result of gluing C to C by the orientation reversing diffeomorphism f, so that X = ˆX C C, then A(X) is in the domain of the map from H( X) to H( ˆX) induced by the pairing from H(C) H(C ) to C that takes α β to < β, H(f)(α) >, and its image is A( ˆX). Here, <, > is the Hermitian inner product in H(C ). 1 It follows from this observation that elements of V H W that are not of trace class are not expressible as finite sums of decomposables. The converse is not true. 2

3 We observe that the functor H induces an action of Diff(C) C on H(C), where Diff(C) denotes the Lie algebra of smooth vector fields on C. A consequence of the properties already listed, that will play a crucial role in the sequel, is (f) Let χ be a holomorphic vector field on X, then H(χ X)(A(X)) = Incoming and Outgoing Boundary Components So far, we have taken the boundary of X to be directed as the topological boundary of S, with the orientation determined by the complex structure. With that orientation, each component of the boundary is, by convention, outgoing. We may choose to reverse the orientation of some boundary components and, accordingly, regard them as incoming. In that case, we can regard A(X) as a linear transformation of trace class from H( in X) to H( out X), where in X and out X denote respectively the unions of the incoming and outgoing boundary components. 3 SuperRiemann Surfaces 3.1 Definitions and basic properties For our purposes, a super-riemann surface is a Riemann surface X, with or without boundary, together with a holomorphic line bundle S and a holomorphic pairing, {, } : S S T X. Such structures exist and are well known to be parametrized by H 1 (X, Z 2 ) if X is closed. To see that this remains the case if X has a boundary, let {X, S, {, }} and {X, S, {, } } be super-riemann surfaces with the same underlying Riemann surface X. Fix a base point x on X and let C be any curve in X beginning and ending at x. Let s and s be points on the fibres over x of S and S respectively such that so that {s, s} = {s, s }. Since any complex line bundle over a circle is trivial, S C has a nowhere vanishing section σ taking the value s at x 0. Let C be parametrized by x(t) with 0 t 1 so that x(0) = x(1) = x. Then there is a unique path σ (t) in the total space of S with σ (0) = s and {σ (t), σ (t)} = {σ(x(t)), σ(x(t))}. Then σ (1) = ±s, and the sign is independent of the choice of σ. This construction defines an element of Hom(π 1 (X, x ), Z 2 ) = H 1 (X, Z 2 ), and is precisely the obstruction to the equivalence of the two structures. 3

4 We define a super-circle to be a circle, together with a real square root of its (trivial) tangent bundle. Such a square root is either trivial or the Möbius bundle. We denote the corresponding supercircles as T-circles or M-circles. In either case, the circle is directed by the squares of spinors. If {X, S, σ} is a super-riemann surface and C is a smooth directed simple closed curve in X then S C has a real sub-bundle consisting of all s in the fibers of S such that σ(s s) is tangent to C in the given direction. This gives C the structure of a super-circle. Lemma 1 Let {X, S, σ} be a super-riemann surface and let C bound in X. Then the induced super-circle structure on C makes it an M-circle. Proof: Without loss of generality, we may assume C is the boundary of X. It follows that every spin structure on X extends uniquely to the closed Riemann surface X C D, where D is a standard disk sewn in along C. We will continue to use S and σ to denote the extended structure. Let z be a coordinate on D and let χ be a nowhere vanishing holomorphic section of S D. iz z is a holomorphic vector field on D that vanishes only at z = 0 and is tangent to C Then z z = f(z)σ(χ χ), where f is holomorphic and non-vanishing on D except for a simple zero at z = 0. It follows that f does not have a single-valued square root on C; hence the induced super-circle structure makes C an M-circle. Let {S, σ} and {S, σ } be super-riemann structures on X and let C be a smooth simple closed curve in X. We say the two structures are C- distinguished if C is a T-circle with respect to one and an M-circle with respect to the other. It is evident from the foregoing discussion that this is the case if and only if the cohomology obstruction to the equivalence of the two structures does not annihilate C. Corollary 1 Let {X, S, σ} be a super-riemann structures. Then the number of T-circles among the boundary components of X is even. Proof: Let ˆX be X with all boundary components capped off. Then if {S, σ} extends to ˆX, all the boundary components of X are M-circles since they all bound in ˆX. Otherwise, the cohomology obstruction to the equivalence of {S, σ} with a structure that does extend to ˆX annihilates the boundary of X. The corollary follows. We end this subsection by pointing out that the automorphism group for a connected super-circle or super-riemann surface is a central extension 4

5 by Z 2 of a subgroup of the automorphism group of the underlying circle or Riemann surface. The central Z 2 is generated by multiplication by 1 in the spinor bundle. In the sequel, we will confine ourselves to super-riemann surfaces all of whose boundary components are M-circles. 3.2 The super-lie algebra of a super-riemann surface We begin by recalling that a super-lie algebra L consists of a Lie algebra L 0, an L 0 -module L 1 and a symmetric pairing from L 1 L 1 to L 0, with respect to which the elements of L 0 act as derivations. We will emphasize this point of view by writing the action of L 0 on L 1 simply as adjacency with the element of L 0 on the left. Finally, it is required that the triple product from L 1 to L 0 given by {α, β}γ satisfies a Jacobi identity, which is equivalent to the requirement that {α, α}α = 0. We define the super-lie algebra associated with either a super-circle or a super-riemann surface. Let X = {X, S, {, }} (resp C = {C, S, {, }}) be a super-riemann surface (resp super-circle). We define the super-lie algebra L(X) (resp L(C))associated with it by setting L 0 to be the Lie algebra of holomorphic vector fields on X (resp smooth vector fields on C) and L 1 to be the complex (resp real) vector space of holomorphic (resp smooth) sections of S. The pairing {, } is the fiberwise bi-linear pairing already defined as part of the super-riemann surface (resp super-circle) structure. What remains to be defined is the action of L 0 on L 1. We use sigma for a generic section of S in either context and recall that we must have {σ, σ}σ = 0. We complete the definition by setting (f{σ, σ})(gσ) = (f{σ, σ}g 1 ({σ, σ}f)g)σ, 2 where f and g are holomorphic (resp smooth real-valued) functions. This definition, while not fiberwise, is local and is independent of the choice of σ. Lemma 2 L(X) and L(C) as defined above are super Lie algebras over C and R respectively. Moreover, if X is a super-riemann surface and C is an smooth directed simple closed curve in X with the induced super-circle structure, then there is a natural restriction homomorphism of super-algebras from L(X) to L(C) C. 5

6 3.3 Representations of super-lie algebras Let L be a super-lie algebra and let V = V 0 V 1 be a Z 2 -graded vector space. A representation ρ of L on V consists of representations of L 0 on V 0 and V 1 together with pairings L 1 V 0 V 1 and L 1 V 1 V 0 such that The elements of L 0 act as derivations of the pairings involving L 1. 2ρ(α) 2 = ρ({α, α}) for α L 1. We will be particularly interested in the case in which L admits a complex conjugation, V is a complex Hilbert space in which V 1 and V 0 are orthogonal, and ρ(α) = iρ(α) for α L 1. ρ(χ) = ρ(χ) for χ L 0. We remark that a tensor product V V of Z 2 -graded vector spaces is graded by setting (V V ) 0 = V 0 V 0+V 1 V 1 and (V V ) 1 = V 1 V 0+V 0 V Proposed definition of a super-conformal field theory The definition of a conformal field theory, given above, can easily be modified to define a super conformal field theory. In the first place, the functor H becomes a functor from the category of disjoint unions of super-circles and super-diffeomorphisms to the category of Z 2 -graded Hilbert spaces, and the action of Diff(C) on H(C) is extended to an action of L(C). The only change to (a)-(c) is that the tensor product becomes a graded tensor product. We require A(X) H( X) 0, where X is now a super-riemann surface, require f in (e) to be a superdiffeomorphism, and extend (f) to all elements of L(X). 4 Krichever-Novikov algebras and supersymmetric generalizations We recall that the Krichever-Novikov algebra KN(X, F ) of a closed Riemann surface X with respect to a distinguished finite subset F is the Lie Algebra 6

7 of meromorphic fields holomorphic on the complement of the distinguished subset. There is an immediate and straightforward extension to the case of a closed super-riemannian manifold, again with a distinguished finite subset. Let X be a closed Riemann surface and F a finite subset. Let the bundle S define a spin structure on X. We will define the super-krichever-novikov algebra SKN(X, F ) by setting SKN 0 (X, F ) = KN(X, F ) while SKN 1 (X, F ) consists of meromorphic sections η of S. 5 The semigroup of twice bounded Riemann surfaces One of the principal difficulties in working with Segal s definition of a conformal field theory is the relative intractability of the infinite dimensional diffeomorphism group of the circle. A partial way around this difficulty is provided by the fact that a Riemann surface with two boundary curves and a preferred point on each admits a canonical parametrization, which makes the set of such Riemann surfaces into a holomorphic semigroup G, that acts on H(S 1 ) where H is the Hilbert space valued functor of a conformal field theory and S 1 is the unit circle in C. More explicitly, let X be a Riemann surface with two boundary components. We direct one of them as incoming and the other as outgoingwe also choose points x in and x out on in (X) and out (X) respectively. We now observe that there is a unique harmonic function t on X such that t in (X) = 0, t out (X) is constant, and dt = out(x) in dt = 2π. (X) The one-form dt, together with the choice of x in and x out define canonical diffeomorphisms of each boundary component of X with the standard unit circle S 1 in C. We use θ as a parameter on each circle, taking θ = 0 at the marked point and dθ = dt. Using these identifications, one can interpret A(X) as a trace-class operator on H = H(S 1 ), and it follows that a conformal field theory induces a representation of the semigroup G by trace-class operators on a Hilbert space. Moreover, observation (f) of Section 2 tells us that KN( ˆX, 0, ) partially intertwines the actions of Diff(S 1 ) with G on H, where ˆX is the closed Riemann surface obtained from X by using the canonical parametrizations to sew disks to both boundary components, and 0 and are the respective centers of the disks sewn to the incoming and outgoing boundary components. 7

8 An important sub-semigroup of G is the punctured open unit disk D.For any conformal field theory, H decomposes into orthogonal subspaces on which z D operates as multiplication by z p z q where p and q are non-negative real numbers whose difference is an integer. We will concern ourselves mainly with chiral theories, for which only the case q = 0 occurs. We observe also that G = g G g where G g denotes the component of G consisting of Riemann surfaces of genus g. G g is a complex manifold of dimension 3g + 1, the composition is additive with respect to genus, and the composition G g G g G g+g is holomorphic. In particular, G 0 = D. We can define an analogous semigroup of superriemann surfaces, recalling that we are requiring the boundary components to be M-circles. In addition to the marked point on each boundary component, we must choose a square root in S of θ at each marked point. With these choices made, we obtain a semigroup G s. Moreover, G s g is a 2 g+1 -fold covers of G g. In particular, G s 0 = D s is the non-trivial two-fold cover of D. 5.1 The Hyperelliptic Semigroup We recall that a hyperelliptic Riemann surface is one that can be realized as a branched double cover of the Riemann sphere. We will consider specifically double covers of the form {(z, w) w 2 = zp(z), 0 < z 0 z z } where p(z) is a polynomial of degree 2g without multiple roots, and all the roots of p(z) have absolute values strictly between z 0 and z. the Riemann surface thus defined has genus g, and two boundary components respectively double-covered by the circles z = z 0 and z = z. In this case, we can set dt = 1 log z 2 z 0, and we choose the marked points on the two boundary components to lie over z 0 and z respectively. The deck transformation reduces this from a fourfold choice to a twofold choice. This construction defines a sub-semigroup S = S g of G with S 0 = G 0 = D. One of the reasons that the hyperelliptic semigroup is of interest is the form of the Krichever-Novikov algebra of a hyperelliptic curve. We are interested in meromorphic vector fields on a hyperelliptic Riemann surface S with poles only at one or both of the points w = z = 0 and w = z =. Such vector fields have the form z k w a (2w z + p (z) w ), where k is any integer and a is 0 or 1. Vector fields for which a = 1 are invariant under the deck transformation, while those for which a = 0 are reversed by the deck transformation. Only among the latter are the squares of spinors, which we take to be sections of a preferred spinor bundle corresponding to the divisor 8

9 (1 g)p where P is the point w = z = and g is the genus of the hyperelliptic curve in question. In particular, a spinor field corresponding to the divisor (1 g)p is a square root of 2w z + p (z) w. 9