symmetries in the infrared. When this assumption is not valid, a modification of this
procedure [9] is applied as we demonstrate in some of our examples.
On these general grounds, we expect the (0, 2) Landau-Ginzburg theory to flow to a
heterotic SCFT with the left-moving spectrum governed by Virasoro symmetry and right-
moving spectrum governed by N = 2 super-Virasoro symmetry. The partition function of
the theory on the torus i.e.
Z = Tr q
L
0
¯q
¯
L
0
(1.1)
is invariant under the modular transformations
1
τ → (aτ + b)/(cτ + d) for a, b, c, d ∈ Z
and q = e
2πiτ
. This condition puts a strong constraint on the spectrum apart from the
symmetries. In the class of examples we study, the above considerations turn out to be
strong enough to determine the low energy theory completely.
2 Search for solvable LG models
Before we start the search for solvable Landau-Ginzburg model, a quick introduction to
their Lagrangian is in order. A (0, 2) Landau-Ginzburg model is constructed using p chiral
superfields Φ
i
and q Fermi superfields Ψ
a
. The chiral multiplet consists of a complex scalar
φ and a complex right-moving fermion λ and the Fermi multiplet consists of a single complex
left-moving fermion ψ. The supersymmetry allows for two types of interaction terms, the
J-type and the E-type. The J-type interaction is analogous to the superpotential term.
Most compactly, it is presented as the integral over half the superspace
Z
dθ
+
X
a
Ψ
a
J
a
(Φ
i
) + c.c. (2.1)
where J
a
are holomorphic functions of Φ
i
. For brevity, we will drop c.c. from now on. The
E-type interaction is induced somewhat unconventionally as supersymmetry variation of
the Fermi field i.e. by requiring
¯
D
+
Ψ
a
= E
a
(Φ
i
) instead of
¯
D
+
Ψ
a
= 0. The N = (0, 2)
supersymmetry requires
P
a
E
a
J
a
= 0. In terms of component fields, these interactions get
spelled out as follows
L = . . . −
X
a
|J
a
|
2
+ |E
a
|
2
−
X
a
X
i
ψ
a
¯
λ
i
∂J
a
∂φ
i
+
¯
ψ
a
λ
i
∂E
a
∂φ
i
+ c.c.
. (2.2)
The E
a
and the J
a
-type interactions are interchanged by conjugating the Fermi multiplet
Ψ
a
. This means the action of supersymmetry on
¯
Ψ is given by
¯
D
+
¯
Ψ
a
= J
a
. In this
paper, we will set all the E-terms to zero. The cohomology of the free supercharge
¯
D
(0)
+
is
generated by Φ
i
and
¯
Ψ
a
. Quantum mechanically, after integrating out the auxiliary fields
at tree level, the supercharge gets the correction
¯
D
(1)
+
= J
a
δ
δ
¯
Ψ
a
. (2.3)
1
By modular invariance we mean invariance up to an overall simple factor due to ’t Hooft and gravi-
tational anomalies. Also, we will consider the partition function with anti-periodic boundary conditions
for the fermions along both cycles of the torus. In this sector the partition function is expected to have
invariance under the subgroup Γ
0
(2) i.e. the subgroup of SL(2, Z) that is generated by S and T
2
elements.
– 2 –
