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\markboth{\hfill MARTIN A. GUEST\hfill} {\hfill INTRODUCTION TO
HOMOLOGICAL GEOMETRY: PART II\hfill}

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\centerline{\hfill INTRODUCTION TO HOMOLOGICAL GEOMETRY: PART
II\hfill}

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\centerline{MARTIN A. GUEST}

\renewcommand{\thefootnote}{} \footnote{The
author was partially supported by a Grant-in-Aid for Scientific
Research from the Ministry of Education.}

Quantum cohomology is a concrete manifestation of the deep
relations between topology and integrable systems which have been
suggested by quantum field theory. It is a generalization of
ordinary cohomology theory, and (thanks to the pioneering work of
many people) it can be defined in a purely mathematical way, but
it has two striking features. First, there is as yet no general
set of computational techniques analogous to the standard
machinery of algebraic topology. Many calculations have been
carried out for individual spaces, but few general methods of
calculation are known. Second, because of the link with integrable
systems, a fundamental role is played by various functions
(differential operators, connections, etc.), and the variables
involved in these functions are homology or cohomology classes. It
seems likely that this function-theoretic point of view will form
the basis of the desired machinery. In fact, major progress in
this direction has been initiated by A. Givental, who has invented
a new term for it, {\lq\lq}homological
geometry{\rq\rq}.

It is easy to explain the origin of these homological functions: a
cohomology theory amounts to a discrete collection of data which
expresses the possible intersections of the various homology
classes for a given manifold $M$, and the functions in question
serve as generating functions for this morass of combinatorial
information. For ordinary cohomology, the data is finite, but for
quantum cohomology it is usually not. Therefore the problem is one
of dealing expeditiously with this collection of numbers, which
includes, in particular, the {\lq\lq}structure constants{\rq\rq}
of the quantum cohomology ring. These numbers are called
Gromov-Witten invariants.

As the authors of the textbook [GKP] state (at the beginning
of chapter 7), {\lq\lq}The most powerful way to deal with
sequences of numbers, as far as anybody knows, is to manipulate
infinite series that generate those sequences{\rq\rq}.
Accordingly, generating functions form the basis of the
function-theoretic approach to quantum cohomology. It is
particularly gratifying when these generating functions,
originally formal and devoid of meaning, begin to take on a life
of their own. For example, it is a standard observation that
combinatorial identities can sometimes be expressed using
derivatives of generating functions. This is exactly what happens
in the case of quantum cohomology, and it leads to very
interesting differential equations which often have a geometrical
meaning.

These notes are a continuation of [Gu], where a very brief
introduction to the function-theoretic aspects of quantum
cohomology was given. Needless to say, neither [Gu] nor the
present article is written for experts. They are meant to be
readable by mathematicians approaching the subject for the first
time, preferably with some background knowledge of differential
geometry and algebraic topology. Much of our exposition is lifted
directly from [Gi1]-[Gi6] and [Co-Ka]; the only
novel aspect lies in what we have chosen to delete rather than
what we have added (though we have worked out some very detailed
examples in the appendices). Apart from the general goal of
explaining how differential equations arise in quantum cohomology,
one of our aims is to arrive at a point of contact with the
{\lq\lq}mirror phenomenon{\rq\rq}. Even with such an imprecise
goal, our discussion is very incomplete, and in particular we have
to admit that we have not followed up on the loose ends from
[Gu].

Fortunately there are a number of excellent sources for further
information. The most comprehensive and the most elementary is
the book [Co-Ka]. The sheer quantity of material as well as
the emphasis on algebraic geometry may be forbidding, but the
careful and helpful presentation makes it invaluable; in addition
there is a lot of new material (in particular new proofs of known
results, and new examples). Many of the geometric aspects of the
subject have been developed in a series of fundamental papers by
B. Dubrovin, and [Du2] in particular is now a classic
reference --- although here the tremendous breadth of the subject
matter is a barrier for the beginner. The papers
[Gi1]-[Gi6] of A. Givental provide the inspiration for
the whole subject of {\lq\lq}homological geometry{\rq\rq},
although the groundbreaking nature of the arguments makes them
hard to follow. Nevertheless, even for the beginner, Givental's
articles are recommended because of the wealth of motivation
provided. Other foundational papers are those of M. Kontsevich and
Y. Manin and (for the underlying mathematical physics) those of E.
Witten and C. Vafa. For a historical perspective and many more
references, [Co-Ka] should be consulted, as well as the
recent foundational book [Ma].

We shall generally use the notation of [Gu]; a brief review
follows. Let $M$ be a simply connected (and compact, connected)
K\"{a}hler manifold, of complex dimension $n$, whose nonzero
integral cohomology groups are of even degree and torsion-free. We
choose a basis $b_0,b_1,\dots,b_s$ of $H^\ast(M;{\mathbf Z})$,
such that $b_1,\dots,b_r$ form a basis of $H^2(M;{\mathbf Z})$.
The Poincar\'{e} dual basis of $H_\ast(M;{\mathbf Z})$ will be
denoted by $B_0,B_1,\dots,B_s$. The dual basis of
$H^\ast(M;{\mathbf Z})$ with respect to the intersection form $(\
,\ )$ will be denoted by $a_0,a_1,\dots,a_s$. Thus, we have
$(a_i,b_j) = \langle a_i,B_j\rangle = \langle b_i, A_j \rangle =
\delta_{ij}$. We shall choose $b_0=1$, the identity element of
$H^\ast(M;{\mathbf Z})$, so that $B_0$ is the fundamental homology
class of $M$; sometimes we write $B_0=M$.

For homology classes (or representative cycles of such classes ---
we blur the distinction) $X_1,\dots,X_i$, when $i\ge 3$, the
notation $\langle X_1 \vert \dots \vert X_i \rangle_D$ will denote
the {\lq\lq}usual{\rq\rq} genus 0 Gromov-Witten invariant obtained
using moduli spaces of {\lq\lq}stable rational curves with $i$
marked points{\rq\rq}. For the definition and properties of
$\langle X_1 \vert \dots \vert X_i\rangle_D$ we refer to
[Fu-Pa] and chapter 7 of [Co-Ka] (where the standard
notation $\langle I_{0,i,D}\rangle(x_1,\dots,x_i)_{0,D}$ is used).
Here, $D$ is an element of $\pi_2(M)$, so we may write
$D=\sum_{i=1}^r s_i A_i$, and we shall assume as in [Gu] that
the homotopy class $D$ contains holomorphic maps ${\mathbf C}
P^1\to M$ only when $s_i\ge 0$ for all $i$.

It is necessary to issue a warning at this point. In \S 7 of
[Gu], the notation $\langle X_1 \vert \dots \vert
X_i\rangle_D$ had a different meaning, namely the intersection
number $\vert{\rm Hol}_D^{X_1,p_1} \cap \dots \cap {\rm
Hol}_D^{X_i,p_i} \vert$, where the points $p_1,\dots,p_i$ are
fixed. To avoid confusion the latter will be denoted by $\langle
X_1 \vert \dots \vert X_i \rangle^{fix}_D$ in the present article.
For $i=3$, the two definitions agree. For $i\ge 4$, they are (in
the words of [Fu-Pa]) solutions to two different enumerative
problems, and they have somewhat different properties.

A general element of $H^\ast(M;{\mathbf C})$ will be denoted by
$\hat{t} = \sum_{i=0}^s t_i b_i$. Since elements of
$H^2(M;{\mathbf C})$ play a special role, we reserve the symbol
$t$ for a general element of $H^2(M;{\mathbf C})$, i.e.
$t=\sum_{i=1}^r t_i b_i$.

The {\lq\lq}large{\rq\rq} quantum product on the vector space
$H^\ast(M;{\mathbf C})$ is defined by

\[
\langle a\circ_{\hat{t}} b, C\rangle = \sum\limits_{D,k\ge 0}
\frac{1}{k!} \langle A\vert B \vert C \vert \hat{T}\leftarrow k
\rightarrow \hat{T} \rangle_D
\]
where (as the notation indicates) the Poincar\'{e} dual $\hat T$
of $\hat t$ appears $k$ times in the general term of the series.
The $D=0$ term is special, because $\langle A\vert B \vert C \vert
\hat T \leftarrow k \rightarrow \hat T \rangle_0$ is zero unless
$k=0$. Using this fact, and the {\lq\lq}divisor rule{\rq\rq}
$\langle A \vert B \vert C \vert T \leftarrow k \rightarrow T
\rangle_D= \langle A \vert B \vert C \rangle_D \langle t,
D\rangle^k$, we see that the {\lq\lq}small{\rq\rq} quantum product
$a{\circ_{t}} b$ is equal to the quantum product which was used in
[Gu]. We shall not be concerned with the question of
convergence of infinite series like this; we shall assume that the
series converges in a suitable region or simply treat it as a
formal series. Each of $\circ_{\hat{t}}$ and ${\circ_{t}}$ endows
$H^\ast(M;{\mathbf C})$ with the structure of a commutative
algebra (over ${\mathbf C}$) with identity element $b_0=1$.

The (large) quantum product on the vector space $H^\ast(M;{\mathbf
C})$ is determined by giving all quantum products of the basis
elements $b_i$; these in turn are determined by the following
function, which is called the Gromov-Witten potential:

\[
\Phi(\hat{t})= \sum\limits_{D,k\ge 3} \frac{1}{k!} \langle \hat{T}
\leftarrow k \rightarrow \hat{T} \rangle_D.
\]
This may be regarded as a generating function for the
Gromov-Witten invariants; it is rather unwieldy, of course, and
one of the main themes of the subject is the fact that there are
alternative expressions for it. Because of the linearity of
Gromov-Witten invariants, we have

\[
{\frac{\partial}{\partial t_i}} \langle \hat{T} \vert A \vert B
\vert \dots \rangle = \langle B_i \vert A \vert B \vert \dots
\rangle
\]
and hence

\[
\langle b_i \circ_{\hat{t}} b_j, B_k \rangle = \frac
{{\partial}^3} {{\partial} t_i {\partial} t_j {\partial} t_k }
\Phi = b_i b_j b_k \Phi
\]