JHEP02(2020)159
It is important to note that the pairing (2.8) is not unique. As a matter of fact, we
can embed it into the following formalism. Consider the exact sequence
0 O
M
Ω
1
M
··· Ω
m
M
0
∇
dW
∇
dW
∇
dW
(2.11)
called a twisted de Rham complex, where we introduced a differential
∇
dW
:= d + τdW ∧, (2.12)
which defines an integrable connection, or equivalently a flat line bundle, since ∇
2
dW
= 0.
As was the case before, we can define cohomology groups based on this complex
3
H
k
(M, ∇
dW
) :=
{ϕ
k
∈ Ω
k
M
|∇
dW
ϕ
k
= 0}
{∇
dW
ϕ
k−1
∈ Ω
k
M
|ϕ
k−1
∈ Ω
k−1
M
}
, (2.13)
which are spaces of ∇
dW
-closed modulo ∇
dW
-exact forms. We will call H
k
dW
:=H
k
(M,∇
dW
)
for short from now on. As in the case of (2.4), only k = m gives a non-trivial cohomol-
ogy [40] and hence we have
dim H
m
dW
= (−1)
m
χ(M). (2.14)
Let us introduce a dual twisted cohomology H
m
−dW
defined in the same way as H
m
dW
but
with W → −W . We will often refer to cohomology classes [ϕ
±
] ∈ H
m
±dW
as twisted cocycles
and specific representatives ϕ
±
as twisted forms. Duality of the two cohomologies is induced
by the intersection pairing
H
m
−dW
× H
m
dW
→ C (2.15)
defined by
hϕ
−
|ϕ
+
i
dW
:=
τ
2π
√
−1
m
Z
M
ϕ
−
∧ ϕ
c
+
(2.16)
and called an intersection number. The overall normalization is chosen for later conve-
nience. Since M is non-compact, for this definition to make sense one needs to use a
compactly-supported form ϕ
c
+
(i.e., one which vanishes in infinitesimal neighbourhoods of
the boundary divisor ∂M) in the same cohomology class as ϕ
+
for direct computations.
This not only makes the result depend on the potential W , but also means the above in-
tegral localizes on ∂M, as in the bulk of M we have ϕ
−
∧ ϕ
c
+
= ϕ
−
∧ ϕ
+
= 0 for two top
holomorphic forms, see, e.g., [5].
Intersection numbers are rational functions of α
i
’s and τ . Different ways of evaluating
them in practice were given in [4, 5, 27, 41–49]. For spaces M admitting a fiber bundle
decomposition (or, more precisely, such that the connection decomposes generically on the
fibers), the most efficient computation method is currently given by recursion relations [5].
When M = M
0,n
is the moduli space of genus-zero curves with n marked points, intersec-
tion numbers have an intrinsic interpretation as computing tree-level scattering amplitudes
3
Since M is finite-dimensional, one can equivalently think of our setup in terms of Batalin-Vilkovisky
(BV) formalism, see, e.g., [39]. More precisely, for a space MV
•
(M) := Γ(∧
•
T M ) of antisymmetric mul-
tivector fields on M (a counterpart of Ω
•
(M) = Γ(∧
•
T
∗
M)) and a BV differential ∂
dW
:= div + ι
dW
, the
corresponding BV complex (MV
•
(M), ∂
dW
) is isomorphic to (Ω
m−•
(M), ∇
dW
).
– 6 –
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