The Dirac-Ramond operator and vertex algebras

The Dirac-Ramond operator and vertex algebras Westfälische Wilhelms-Universität Münster cvoigt@math.uni-muenster.de http://wwwmath.uni-muenster.de/reine/u/cvoigt/ Vanderbilt May 11, 2011

Kasparov theory

Kasparov theory Definition Let A and B be (separable, ungraded) C -algebras. A Kasparov A-B-module (E, φ, F ) consists of a (countably generated) graded Hilbert B-module E = E 0 E 1 a graded -homomorphism φ : A L(E) an odd operator F L(E) such that [F, φ(a)], φ(a)(f 2 1), φ(a)(f F ) are contained in K(E) for all a A.

Kasparov theory

Kasparov theory Definition (Kasparov) Let A and B be C -algebras. The group KK(A, B) is the set of homotopy classes of Kasparov A-B-modules.

Kasparov theory Definition (Kasparov) Let A and B be C -algebras. The group KK(A, B) is the set of homotopy classes of Kasparov A-B-modules. KK(A, B) is a bifunctor on the category of C -algebras with values in abelian groups, covariant in B and contravariant in A.

Kasparov theory Every -homomorphism f : A B determines an element [f ] = (B, f, 0) KK(A, B). Theorem (Kasparov) There exists an associative bilinear product KK(A, B) KK(B, C) KK(A, C), (x, y) x B y generalizing the composition of -homomorphisms.

Bott periodicity

Bott periodicity The Dirac-dual Dirac method of Atiyah-Kasparov consists in defining

Bott periodicity The Dirac-dual Dirac method of Atiyah-Kasparov consists in defining the Dirac element α KK(C 0 (R 2n ), C) the dual Dirac (Bott) element β KK(C, C 0 (R 2n ))

Bott periodicity The Dirac-dual Dirac method of Atiyah-Kasparov consists in defining the Dirac element α KK(C 0 (R 2n ), C) the dual Dirac (Bott) element β KK(C, C 0 (R 2n )) and proving Theorem (Bott periodicity) The elements α and β are mutually inverse. Hence KK(A C 0 (R 2n ), B) = KK(A, B) = KK(A, B C 0 (R 2n )) for all A and B.

The Dirac element

The Dirac element The Dirac element α = (H, φ, F ) in KK(C 0 (R 2n ), C) is given by

The Dirac element The Dirac element α = (H, φ, F ) in KK(C 0 (R 2n ), C) is given by the Hilbert space H = L 2 (R 2n ) S where S = S + S is the spinor representation of the Clifford algebra Cl 2n. the -homomorphism φ : C 0 (R 2n ) L(H) is the operator F L(H) is φ(f )(h σ) = fh σ. F = where D is the Dirac operator D 1 + D 2 D = 2n j=1 c(e j ) x j.

The dual-dirac element

The dual-dirac element The dual Dirac element β = (E, ψ, b) in KK(C, C 0 (R 2n )) is given by

The dual-dirac element The dual Dirac element β = (E, ψ, b) in KK(C, C 0 (R 2n )) is given by the Hilbert C 0 (R 2n )-module E = C 0 (R 2n ) S where again S = S + S are the spinors. the -homomorphism ψ : C L(E) determined by ψ(1) = id. the operator b L(E) = C b (R 2n, Cl 2n ) given by b(x) = for x = (x 1,..., x 2n ) R 2n. 2n j=1 x j 1 + x 2

Proof of Bott periodicity

Proof of Bott periodicity We have to show that α and β are mutually inverse.

Proof of Bott periodicity We have to show that α and β are mutually inverse. The Kasparov product β α KK(C, C) is represented by the Kasparov C-C-module given by

Proof of Bott periodicity We have to show that α and β are mutually inverse. The Kasparov product β α KK(C, C) is represented by the Kasparov C-C-module given by the graded Hilbert space L 2 (Λ (R 2n )) the left action of C by scalar multiplication an operator B L(L 2 (Λ (R 2n ))) essentially given by harmonic oscillators. One computes ind(b) = 1, and this implies β α = [id].

The Dirac element for general manifolds If M is a (even-dimensional complete Riemannian spin) manifold one obtains the Dirac element [D] = (H, φ, F ) in KK(C 0 (M), C). Here H = L 2 (M, S) are the L 2 -spinors on M, the -homomorphism φ : C 0 (M) L(H) is given by pointwise multiplication, the operator F L(H) is F = D 1 + D 2 where D is the Dirac operator on M.

Supersymmetric mechanics

Supersymmetric mechanics Consider the supersymmetric mechanical system given by the Lagrange density L(X ) = 1 2 g µν(x ) τ X µ DX ν dθdτ.

Supersymmetric mechanics Consider the supersymmetric mechanical system given by the Lagrange density L(X ) = 1 2 g µν(x ) τ X µ DX ν dθdτ. Here X : R 1 1 M describes a superparticle moving in the complete Riemannian (even-dimensional spin) manifold (M, g). The Lagrangian is supersymmetric, the generator of supersymmetry is given by the odd vector field satisfying Q 2 = τ. Q = θ + θ τ

Supersymmetric mechanics The Euler-Lagrange equation is τ DX = 0 and the solutions form a symplectic supermanifold which can be identified with the odd vector bundle π (ΠTM) over TM. Here π : TM M denotes the bundle projection. The algebra C (π (ΠTM)) of classical observables contains in particular the functions on M.

Quantization More precisely, canonical quantization yields

Quantization More precisely, canonical quantization yields the graded Hilbert space H = L 2 (M, S) of L 2 -spinors on M,

Quantization More precisely, canonical quantization yields the graded Hilbert space H = L 2 (M, S) of L 2 -spinors on M, the action of the observables in C 0 (M) on H by pointwise multiplication,

Quantization More precisely, canonical quantization yields the graded Hilbert space H = L 2 (M, S) of L 2 -spinors on M, the action of the observables in C 0 (M) on H by pointwise multiplication, the Dirac operator D acting on H as quantization of the supercharge Q,

Supersymmetric σ-models In physics, a σ-model is a field theory in which the elementary fields are maps from spacetime to a target space, often a Riemannian manifold.

Supersymmetric σ-models In physics, a σ-model is a field theory in which the elementary fields are maps from spacetime to a target space, often a Riemannian manifold. The above mechanical system can be viewed as a 0 + 1-dimensional supersymmetric σ-model with target space M; the underlying spacetime R 1 1 has zero space dimensions plus one (even) time dimension,

Supersymmetric σ-models There is a 1 + 1-dimensional supersymmetric σ-model with target M given by the Lagrangian where now X : R 2 1 M. L(X ) = g µν (X )(DX µ )( z X ν )dθdzdz Here z, z are (independent) coordinates on R 2 and θ is a Grassmann variable as before.

Loop spaces and the Dirac-Ramond operator Heuristically, if M is a string manifold, the analogue of the Dirac cycle in Kasparov theory should encode the Dirac-Ramond operator on the loop space LM = C (S 1, M). Formally, this is the operator Q = 2π 0 ( dσψ µ (σ) i acting on L 2 -spinors over LM. D DX µ (σ) + g µν(x ) X ν ) σ

Loop spaces and the Dirac-Ramond operator Witten considered the (equivariant) index of Q by formally applying the equivariant index theorem with respect to the canonical S 1 -action on LM. This is the starting point for his interpretation of the elliptic genus. All index theoretic information of Q is captured on the normal bundle N M.

The loop space LR 2n

The loop space LR 2n We expand an element v in LR 2n = C (S 1, R 2n ) into Fourier modes, v(t) = v 0 + v k e 2πitk + v k e 2πitk with v 0 R 2n and v k C 2n. k=1

The loop space LR 2n We expand an element v in LR 2n = C (S 1, R 2n ) into Fourier modes, v(t) = v 0 + v k e 2πitk + v k e 2πitk k=1 with v 0 R 2n and v k C 2n. This yields LR 2n R 2n C 2n q k k=1

The loop space LR 2n We expand an element v in LR 2n = C (S 1, R 2n ) into Fourier modes, v(t) = v 0 + v k e 2πitk + v k e 2πitk k=1 with v 0 R 2n and v k C 2n. This yields LR 2n R 2n C 2n q k k=1 A suitable algebra of smooth functions on this manifold is C c (R 2n ) S q k (C 2n ) S q l (C 2n ) k=1 l=1

Spinors for LR 2n

Spinors for LR 2n Write g = so(2n) and let ĝ be the associated affine algebra. The spinor representation of ĝ acts on the (Ramond) Fock space S = S Λ q m(c 2n ) m=1 where S = S + S are the complex spinors for R 2n. The Hilbert space H of L 2 -spinors on LR 2n is a completion of C c (R 2n ) S q k (C 2n ) S Λ q m(c 2n ) S q l (C 2n ) k=1 m=1 l=1

The Dirac-Ramond operator for LR 2n

The Dirac-Ramond operator for LR 2n Formally, the Dirac-Ramond operator Q acting on H is Q = D + G 0

The Dirac-Ramond operator for LR 2n Formally, the Dirac-Ramond operator Q acting on H is where Q = D + G 0 the horizontal part D is the Dirac operator on R 2n coupled to the (trivial) infinite-dimensional vector bundle S q k (C 2n ) k=1 Λ q m(c 2n ) S q l (C 2n ) m=1 l=1

Further structure on the Hilbert space H

Further structure on the Hilbert space H On H there are natural commuting representations of the Virasoro algebra and the super-virasoro algebra.

Further structure on the Hilbert space H On H there are natural commuting representations of the Virasoro algebra and the super-virasoro algebra. The Virasoro algebra Vir is the Lie algebra with generators L k for k Z and C and relations for all k, l Z. [L k, L l ] = (k l)l k+l + δ k, l k 12 (k2 1)C [L k, C] = 0

Further structure on the Hilbert space H More precisely, there are (unbounded) operators L k, L k and G k for k Z acting in H satisfying the relations [L k, L l ] = (k l)l k+l + δ k, l k 12 (k2 1)c k [L k, L l ] = (k l)l k+l + δ k, l 12 (k2 1)c ( k [L k, G l ] = 2 l [L k, L l ] = 0, [G k, L l ] = 0 ) G k+l, [G k, G l ] = 2L k+l + c 3 with central charges c = 2n, c = 3n. ( k 2 1 ) δ k, l 4

Further structure on the Hilbert space H In conformal field theory, one works with the generating series L(z) = L k z k 2 k Z where z is a formal variable. L(z) = L k z k 2 k Z G(z) = G k z k 1 2 +Z k 3/2

Further structure on the Hilbert space H In conformal field theory, one works with the generating series L(z) = k Z where z is a formal variable. L k z k 2 L(z) = L k z k 2 k Z G(z) = G k z k 1 2 +Z k 3/2 These expressions can be viewed as operator-valued distributions on C \ {0} and are basic examples of vertex operators.

Vertex operators and vertex algebras

Vertex operators and vertex algebras Definition Let V be a vector space. A vertex operator on V is a series a(z) = n Z a (n) z n 1 End(V )[[z, z 1 ]] such that for all v V one has a (n) (v) = 0 for n 0. Vertex operators cannot be multiplied in a simple fashion. Vertex algebras axiomatize the algebra of vertex operators. They provide a rigorous mathematical framework for conformal field theory.

Vertex algebras

Vertex algebras Definition A vertex-c -algebra consists of a commutative C -algebra V(0) with unit Ω (vacuum) and a Hilbert V (0) -module V = n N 0 V (n) Write F(V) V for the algebraic direct sum of the spaces V (n). an antilinear involution : V V which preserves conformal weights and coincides on V(0) with the given involution. an element ω V(2) (conformal vector) satisfying ω = ω. a locally bounded linear map Y : F(V ) EndV(0) (F(V))[[z, z 1 ]] which maps v to the vertex operator Y (v, z) = v (n) z n 1 n= such that the following axioms are satisfied:

Vertex algebras (Vacuum axiom) One has Y (Ω, z) = id and Y (v, z)(ω) z=0 = v for all v F(V ). (Virasoro axiom) The Fourier coefficients Ln of Y (ω, z) satisfy n [L n, L m] = (n m)l n+m + δ n, m 12 (n2 1)c where c is the central charge of V. Moreover L 0 (v) = nv for v V (n). (Translation covariance) For all v F(V) one has [L 1, Y (v, z)] = z Y (v, z). (Locality) For all u, v V there exists N >> 0 such that (w z) N [Y (u, w), Y (v, z)] = 0. (Unitarity) For all v, u, w F(V) one has Y (v, z) u, w = u, Y (v, z)w where Y (v, z) = Y (e L 1 z ( z 2 ) L 0 (v ), z 1 ).

A (first attempt of a) definition of elliptic cycles

A (first attempt of a) definition of elliptic cycles Let A and B be separable C -algebras. An elliptic A-B-cycle consists of a N = 1 2 SUSY conformal vertex C -algebra V with dim(v (0) ) < a Z 2 -twisted Hilbert B-vertex module E over V a graded -homomorphism φ : A L(E) an odd homogeneous operator F L(E)

A (first attempt of a) definition of elliptic cycles

A (first attempt of a) definition of elliptic cycles These data are required to satisfy the following conditions. The operators [F, φ(a)], (F 2 1)φ(a), (F F )φ(a) are locally compact for all a A. The representation φ commutes strictly with all vertex operators. F commutes up to locally compact operators with the action of all holomorphic vertex operators. (Minimality) The vertex module E is cyclic. (Modular invariance) The character of (V, E) is modular invariant.

The corresponding homology theory Definition Let A and B be separable C -algebras. We write e(a, B) for the group of homotopy classes of elliptic A-B-cycles.

The corresponding homology theory Definition Let A and B be separable C -algebras. We write e(a, B) for the group of homotopy classes of elliptic A-B-cycles. Theorem There exists an associative bilinear product e(a, B) e(b, C) e(a, C) generalizing the composition of -homomorphisms. There is a natural ring homomorphism T : e (C, C) M (Z) where M (Z) is the ring of weakly meromorphic integral modular forms.

Modular forms

Modular forms SL 2 (Z) acts on the upper half plane H = {τ C Im(τ) > 0} by ( ) a b (τ) = aτ + b c d cτ + d.

Modular forms SL 2 (Z) acts on the upper half plane H = {τ C Im(τ) > 0} by ( ) a b (τ) = aτ + b c d cτ + d. Definition Let k Z. A (weakly meromorphic) modular form of weight k is a holomorphic function f : H C such that (( ) ) a b f (τ) = (cτ + d) k f (τ) c d and f is meromorphic at i, f (q) = n= N a n q n, q = e 2πiτ.

Modular forms A modular form is integral if all its Fourier coefficients are integers.

Modular forms A modular form is integral if all its Fourier coefficients are integers. The ring of (weakly meromorphic) integral modular forms is M (Z) = Z[E 4, E 6,, 1 ]/(E4 3 E6 2 1728 ) where E 4, E 6 are (normalized) Eisenstein series and = q (1 q k ) = η(q) 24 is the modular discriminant. k=1

An analogue of Bott periodicity for e-theory

An analogue of Bott periodicity for e-theory The Dirac-Ramond operator for R n defines an element α e(c 0 (R n ), C) iff the dimension n is a multiple of 24.

An analogue of Bott periodicity for e-theory The Dirac-Ramond operator for R n defines an element α e(c 0 (R n ), C) iff the dimension n is a multiple of 24. Theorem The element α e(c 0 (R n ), C) is invertible, and there are natural isomorphisms for all A, B. e(c 0 (R 24 ) A, B) = e(a, B) = e(a, C 0 (R 24 ) B) Under the map e (C, C) M (Z), the Dirac-Ramond element α for R 24 corresponds to 1.