Novikov ring Quantum cohomology

various choices of coefficient ring quantum cohomology of x possible. ring chosen encodes information second homology of x. allows quantum cup product, defined below, record information pseudoholomorphic curves in x. example, let

h

2


(
x
)
=

h

2


(
x
,

z

)

/


t
o
r
s
i
o
n



{\displaystyle h_{2}(x)=h_{2}(x,\mathbf {z} )/\mathrm {torsion} }

be second homology modulo torsion. let r commutative ring unit , Λ ring of formal power series of form

λ
=

∑

a
∈

h

2


(
x
)



λ

a



e

a


,


{\displaystyle \lambda =\sum _{a\in h_{2}(x)}\lambda _{a}e^{a},}

where

the coefficients




λ

a




{\displaystyle \lambda _{a}}

come r,
the




e

a




{\displaystyle e^{a}}

formal variables subject relation




e

a



e

b


=

e

a
+
b




{\displaystyle e^{a}e^{b}=e^{a+b}}

,
for every real number c, finitely many ω(a) less or equal c have nonzero coefficients




λ

a




{\displaystyle \lambda _{a}}

.

the variable




e

a




{\displaystyle e^{a}}

considered of degree



2

c

1


(
a
)


{\displaystyle 2c_{1}(a)}

,




c

1




{\displaystyle c_{1}}

first chern class of tangent bundle tx, regarded complex vector bundle choosing complex structure compatible ω. Λ graded ring, called novikov ring ω. (alternative definitions common.)