VERTEX OPERATOR ALGEBRAS AND DIFFERENTIAL GEOMETRY

Abstract

Mathematical beauty is more than its own reward. The purpose of the series of the lectures is to introduce some applications of vertex operator algebras to differential geometry. These applications are in the same spirit of the applications of Grassmannian algebras to differentiable mani-folds that lead to exterior differential forms and the exterior differential operator, and the applications of Clifford algebras and spinor representations that lead the Dirac operators on spin Riemannian manifolds. In particular, we will explain the relationship with elliptic genera. The vertex operator algebra we exploit include the semi-infinite wedge product and the semi-infinite symmetry power of an infinite dimensional space. We believe such applications will shed some lights on the ge-ometry and topology of infinite dimensional manifolds that naturally arise in string theory. We will emphasize on supersymmetry and supersymmetric indices, which already appear in the classical setting. Supersymmetry already appears in the Hodge theory of Laplacian operators on Riemannian manifolds and Kähler manifolds. The cor-responding indices coincide with the Euler characteristic and Hirzeburch χ y genus respectively. Elliptic genera, which generalize classical genera, naturally appear in the infinite dimensional setting as one considers the supersymmetric indices of the associated superconformal vertex algebras. A very important notion in string theory is that of an N = 2 superconformal field theory (SCFT). Physicists showed that the primary chiral fields of an N = 2 SCFT form an algebra. The proof of this fact in physics literature share many common features of the Hodge theory of Kähler manifolds. See e.g. [14, 21]. A closely related notion is that of a topological vertex algebra of which one can consider the BRST cohomology. Given an N = 2 SCFT, there are two ways to twist it to obtain a toplogical vertex algebra. The BRST cohomology groups of these two toplogical vertex algebras correspond to the algebras of the primary chiral and anti-chiral fields respectively of the orignial N = 2 SCFT. Given a Calabi-Yau manifold M , it has been widely discussed in physics literature for many years that there is an N = 2 SCFT associated to it, with the two twists giving the so-called the A-theory and the B-theory respectively. See e.g. [1]. Malikov, Schechtman, and Vaintrob [17] have constructed for any Calabi-Yau manifold a sheaf of topological vertex algebras. Their theory corresponds to the A-theory. In [28] we give a different approach based on standard techniques in differential geometry. We use holomorphic vector bundles of N = 2 superconformal vertex algebras on a complex manfiold M , and the ¯ ∂ operator on such bundles. We show that the corresponding cohomology group has a natural structure of an N = 2 1 2 JIAN ZHOU superconofrmal vertex algebra, whose two twists provide the desired A theory and B theory. Vafa [20] suggested an approach to quantum cohomology based on vertex alge-bra constructed via semi-infinite forms on loop space. Recall that a closed string in a manifold M is a smooth map from S 1 to M . The configuration space of all closed string is just the free loop space LM . Earlier researches in algebraic topology mostly dealt with the ordinary cohomology of the loop spaces. However the coho-mology theory related to semi-infinite forms on the loop space seems to be more interesting. As is well-known in the theory of vertex algebras, the space of such forms has a natural structure of a vertex algebra being the Fock space of a natural infinite dimensional Clifford algebra. One also has to consider the semi-infinite symmetry product which also has a natural structure of a vertex algebra being the bosonic space of a natural infinite dimensional Heisenberg algebra. Superconformal structures naturally arise when the fermionic and bosonic parts are combined. We begin with the Hodge theory on Riemannian manifolds in 1. There is a underlying Lie superalgebra which we call the U (1) supersymmetry algebra. The corresponding supersymmetric index is exactly the Euler characteristic. We study an algebraic analogue in 2. More precisely we study differential operators on the space of differential forms with polynomial coefficients. By taking suitable metric we obtain a formal Hodge theory analogous to the Hodge theory on Riemannian manifold. We also compute the corresponding supersymmetric index. We then move onto the Hodge theory on Kähler manifolds in 4 and present some larger Lie superalgebras underlying it. A suitably defined supersymmetric index in this case gives the Hirzebruch χ y genus. An algebraic analogue is studied in 5. The next natural topological invariant to consider is the elliptic genus. This involves the constructions in [28] of an N = 1 superconformal vertex algebra asso-ciated to any Riemannian manifold, and an N = 2 superconformal vertex algebra associated to any complex manifold. First of all on the algebraic level one needs to take the number of variables to infinity in the examples studied in 2 and 5. More precisely, we will study exterior algebras with infinitely many generators in 7 and polynomial algebras with infinitely many generators in 8. This leads us naturally to vertex algebras whose definition is presented in 9. We recall some well-known constructions of vertex algebras in 10. We present some basics of N = 2 supercon-formal vertex algebras in 11. For applications to differential geometry, the reader can consult [28].

Full text

Page 1

VERTEX OPERATOR ALGEBRAS AND DIFFERENTIAL
GEOMETRY
JIAN ZHOU
Mathematical beauty is more than its own reward.
P.A.M. Dirac
The purpose of the series of the lectures is to introduce some applications of
vertex operator algebras to differential geometry. These applications are in the
same spirit of the applications of Grassmannian algebras to differentiable mani-
folds that lead to exterior differential forms and the exterior differential operator,
and the applications of Clifford algebras and spinor representations that lead the
Dirac operators on spin Riemannian manifolds. In particular, we will explain the
relationship with elliptic genera. The vertex operator algebra we exploit include
the semi-infinite wedge product and the semi-infinite symmetry power of an infinite
dimensional space. We believe such applications will shed some lights on the ge-
ometry and topology of infinite dimensional manifolds that naturally arise in string
theory.
We will emphasize on supersymmetry and supersymmetric indices, which already
appear in the classical setting. Supersymmetry already appears in the Hodge theory
of Laplacian operators on Riemannian manifolds and K¨ ahler manifolds. The cor-
responding indices coincide with the Euler characteristic and Hirzeburch χygenus
respectively. Elliptic genera, which generalize classical genera, naturally appear in
the infinite dimensional setting as one considers the supersymmetric indices of the
associated superconformal vertex algebras.
A very important notion in string theory is that of an N = 2 superconformal
field theory (SCFT). Physicists showed that the primary chiral fields of an N = 2
SCFT form an algebra. The proof of this fact in physics literature share many
common features of the Hodge theory of K¨ ahler manifolds. See e.g. [14, 21].
A closely related notion is that of a topological vertex algebra of which one
can consider the BRST cohomology. Given an N = 2 SCFT, there are two ways
to twist it to obtain a toplogical vertex algebra. The BRST cohomology groups of
these two toplogical vertex algebras correspond to the algebras of the primary chiral
and anti-chiral fields respectively of the orignial N = 2 SCFT. Given a Calabi-Yau
manifold M, it has been widely discussed in physics literature for many years that
there is an N = 2 SCFT associated to it, with the two twists giving the so-called
the A-theory and the B-theory respectively. See e.g. [1].
Malikov, Schechtman, and Vaintrob [17] have constructed for any Calabi-Yau
manifold a sheaf of topological vertex algebras. Their theory corresponds to the
A-theory. In [28] we give a different approach based on standard techniques in
differential geometry. We use holomorphic vector bundles of N = 2 superconformal
vertex algebras on a complex manfiold M, and the¯∂ operator on such bundles. We
show that the corresponding cohomology group has a natural structure of an N = 2
1

Page 2

2JIAN ZHOU
superconofrmal vertex algebra, whose two twists provide the desired A theory and
B theory.
Vafa [20] suggested an approach to quantum cohomology based on vertex alge-
bra constructed via semi-infinite forms on loop space. Recall that a closed string
in a manifold M is a smooth map from S1to M. The configuration space of all
closed string is just the free loop space LM. Earlier researches in algebraic topology
mostly dealt with the ordinary cohomology of the loop spaces. However the coho-
mology theory related to semi-infinite forms on the loop space seems to be more
interesting. As is well-known in the theory of vertex algebras, the space of such
forms has a natural structure of a vertex algebra being the Fock space of a natural
infinite dimensional Clifford algebra. One also has to consider the semi-infinite
symmetry product which also has a natural structure of a vertex algebra being the
bosonic space of a natural infinite dimensional Heisenberg algebra. Superconformal
structures naturally arise when the fermionic and bosonic parts are combined.
We begin with the Hodge theory on Riemannian manifolds in §1. There is a
underlying Lie superalgebra which we call the U(1) supersymmetry algebra. The
corresponding supersymmetric index is exactly the Euler characteristic. We study
an algebraic analogue in §2. More precisely we study differential operators on the
space of differential forms with polynomial coefficients. By taking suitable metric
we obtain a formal Hodge theory analogous to the Hodge theory on Riemannian
manifold. We also compute the corresponding supersymmetric index.
We then move onto the Hodge theory on K¨ ahler manifolds in §4 and present
some larger Lie superalgebras underlying it. A suitably defined supersymmetric
index in this case gives the Hirzebruch χygenus. An algebraic analogue is studied
in §5.
The next natural topological invariant to consider is the elliptic genus. This
involves the constructions in [28] of an N = 1 superconformal vertex algebra asso-
ciated to any Riemannian manifold, and an N = 2 superconformal vertex algebra
associated to any complex manifold. First of all on the algebraic level one needs to
take the number of variables to infinity in the examples studied in §2 and §5. More
precisely, we will study exterior algebras with infinitely many generators in §7 and
polynomial algebras with infinitely many generators in §8. This leads us naturally
to vertex algebras whose definition is presented in §9. We recall some well-known
constructions of vertex algebras in §10. We present some basics of N = 2 supercon-
formal vertex algebras in §11. For applications to differential geometry, the reader
can consult [28].
1. Hodge Theory for Riemannian Manifolds and N = 1
Supersymmetric Index
Throughout this section M is a compact oriented manifold of real dimension n.
1.1. De Rham cohomology and Euler characteristic. The space C∞(M) of
smooth functions on M is a commutative algebra with unit. In particular,
(f · g) · h = f · (g · h),
f · g = g · f,
1 · f = f · 1 = f,

Page 3

VERTEX OPERATOR ALGEBRAS AND DIFFERENTIAL GEOMETRY3
for f,g,h ∈ C∞(M). Let x1,...,xnbe local coordinates, then we regard a smooth
function locally as a formal power series:
f = fi1,...,ikxi1···xik.
We have
xi· xj= xj· xi.
On the other hand, the space of differential forms is a graded commutative
algebra with unit. In particular,
(α ∧ β) · γ = α ∧ (β · γ),
α · β = (−1)|α|·|β|β ∧ α,,
1 ∧ α = α · 1 = α,
for α,β,γ ∈ Ω∗(M). In local coordinates x1,...,xn, a smooth differential form can
be written as
α = αi1,...,ikdxi1∧ ··· ∧ dxik.
We have
dxi∧ dxj= −dxj∧ dxi.
The exterior differential operator
d : Ω∗(M) → Ω∗+1(M)
satisfies the following properties:
d(α ∧ β) = dα ∧ β + (−1)|α|α ∧ dβ,
d2= 0.
(2)
In local coordinates we have
(1)
dα = dxi∧
∂
∂xiα.
There are two kinds of operators involved here: {
i = 1,...,n}. Note
∂
∂xi : i = 1,...,n} and {dxi∧ :
∂
∂xi
(dxi∧)(dxj∧) = −(dxj∧)(dxi∧).
∂
∂xj=
∂
∂xj
∂
∂xi,
The second property (2) of d implies that Imd ⊂ kerd, hence one defines the de
Rham cohomology group
Hp(M) = kerd|Ωp(M)/Imd|Ωp−1(M).
The first property (1) implies that there is an induced structure of a graded com-
mutative algebra with unit on
H∗(M) = ⊕n
The de Rham theorem states that the de Rham cohomology of M is isomorphic
to its singular or simplicial cohomology, hence it is a topological invariant. In
particular, Hp(M) is finite dimensional for each p. The Euler characteristic of M
is defined by:
n
?
p=0Hp(M).
χ(M) =
p=0
(−1)pdimHp(M).

Page 4

4JIAN ZHOU
1.2. Laplace operator and Hodge theory. Suppose now M is endowed with a
Riemannian metric g. Then one can define the Hodge star operator
∗ : Ωp(M) → Ωn−p(M).
(3)
It satisfies the following property:
∗2|Ωp(M)= (−1)p(n−p).
One can define a metric on Ω∗(M) by:
?α,β? =
?
M
α ∧ ∗β.
Define d∗: Ωp(M) → Ωp−1(M) by
d∗= − ∗ d ∗ .
Then by (3) and the Stokes theorem it is straightforward to see that
?dα,β? = ?α,d∗β?.
It follows from (2) and (3) that
(d∗)2= 0.
The Laplace operator is defined by;
∆ = dd∗+ d∗d = (d + d∗)2.
It is easy to see that
[∆,d] = [∆,d∗] = 0,
?∆α,β? = ?α,∆β?.
(4)
(5)
A differential form α is said to be harmonic if
∆α = 0.
It is easy to see that α is harmonic if and only if
dα = d∗α = 0.
I.e.,
α ∈ kerd ∩ kerd∗= ker(d + d∗).
Denote by H∗(M) the space of harmonic forms on M.
Now ∆ is a self-adjoint elliptic operator, by standard theory there are the Hodge
decompositions
Ω∗(M) = H∗(M) ⊕ Imd ⊕ Imd∗= H∗(M) ⊕ Im(d + d∗).
One can see from the Hodge decomposition
(6)
H∗(M)∼= H∗(M),
(7)
i.e., every de Rham cohomology class is represented by a unique harmonic form.

Page 5

VERTEX OPERATOR ALGEBRAS AND DIFFERENTIAL GEOMETRY5
1.3. Heat operator and its trace. There is also an eigenspace decomposition:
Ω∗(M) = ⊕λΩ∗(M)λ,
(8)
where the sum is taken over all eigenvalues of ∆. The eigenvalues of ∆ has some
nice properties, for example, they form a discrete set that goes to infinity and every
eigenspace is finite-dimensional. Furthermore, an operator e−t∆: Ω∗(M) → Ω∗(M)
can be defined such that its action on the eigenspace Ω∗(M)λis multiplication by
e−tλ, and it is of trace class:
?
Set
tre−t∆=
λ
e−λtdimΩ∗(M)λ.
Ω0(M) = ⊕p evenΩp(M),
Then Q = d + d∗maps Ω0(M) to Ω1(M) and vice versa. Since Q commutes with
∆, it maps Ω∗(M)λto itself. By the Hodge decomposition it is easy to see that
Q|Ω∗(M)λis an isomorphism for λ ?= 0. Define an operator (−1)F: Ω∗(M) →
Ω∗(M) by
(−1)Fα = (−1)pα,
It follows that
Ω1(M) = ⊕p oddΩp(M).
α ∈ Ωp(M).
tr((−1)Fe−t∆|Ω∗(M)λ) = 0,λ ?= 0.
(9)
We need the fermionic number operator J : Ω∗(M) → Ω∗(M) defined by:
J(α) = pα,α ∈ Ωp(M).
It is clearly self-adjoint, and (−1)F= (−1)J.
In physics literature, eigenvectors of zero eigenvalues are often referred to as the
zero modes, eigenvectors of nonzero eigenvalues are the nonzero modes. Hence the
harmonic forms are the zero modes of the Laplacian operator. A straightforward
consequence of the isomorphism (7) is
χ(M)=
n
?
tr((−1)Fe−t∆|Ω∗(M)0) = tr((−1)Fe−t∆|Ω∗(M)),
where in the last identity we have used (9). This is the starting point of the proof
of the Gauss-Bonnet-Chern theorem by heat kernel method proposed by McKean
and Singer. Also note for λ ?= 0,
Ωp(M)λ= dΩp−1(M)λ⊕ d∗Ωp+1(M)λ,
and d induces an isomorphism:
p=0
(−1)pdimHp(M) =
n
?
p=0
(−1)ptr(e−t∆|Ωp(M)0)
=
dΩp−1(M)λ∼= d∗Ωp(M)λ.
Therefore one has [2]:
tryJe−t∆=
n
?
p=0
ypdimHp(M) + (1 + y)qe−t(y),