JHEP09(2018)085
where the extra signs correspond to using a graded permutation operator. Let us also
mention that we will often use the notation
f
±
≡ f(u ± i/2), f
[+a]
≡ f(u + ia/2) (2.4)
for shifts of the spectr al parameter.
Multiplying several R-matrices together we obtain the monodromy matrix T (u) defin-
ing the spin chain, which acts in the tensor product of the Hilbert space H and an auxiliary
space C
m|n
,
T (u) = R
01
(u − θ
1
) ⊗ R
02
(u − θ
2
) ⊗ · · · ⊗ R
0L
(u − θ
L
) ⊗ g . (2.5)
We have introduc ed an extra twist matrix g which acts in the auxiliary space only and
corresponds to twisted boundary conditions. We take it to be diagonal,
g = diag (λ
1
, λ
2
, . . . , λ
m+n
) , (2.6)
and assume that the twists λ
i
are all distinct and in generic position. They serve as
regulators in the construction, and also ensure the r e is a 1-t o-1 correspondence between
spin chain states and solutions of the nested Bethe equations. We also introduce d the
parameters θ
L
which correspond t o inhomogeneities of the spin chain.
6
The definition of the tensor product of operators in (2.5) is nontrivial in the super-
symmetric case and involves extra signs reflecting the graded natur e of the ve c tor spaces
C
m|n
, see e.g. [
30]. Explicitly, the matrix elements of T (u) read
T
a
1
a
2
...a
L
j
a
′
1
a
′
2
...a
′
L
j
′
(u) = R
a
1
j
a
′
1
j
′′
(u − θ
1
)R
a
2
j
a
′
2
j
′′
(u − θ
2
) . . . R
a
L
j
a
′
L
j
′′
(u − θ
L
) g
j
′′
j
′
(2.7)
×(−1)
P
L
α=2
P
α−1
β=1
[a
′
β
]([a
α
]+[a
′
α
])
where th e indices a
1
, a
2
, . . . and a
′
1
, a
′
2
, . . . c or r es pond to individual C
m|n
factors of the
Hilbert space (
2.2) while j, j
′
label the auxiliar y space, and we assume summation over
repeated indices.
Although one could consider spin chains wit h arbitrary representations of su(m|n) in
the physical and the auxiliary spaces, we only discuss the c ase when both representations
are fundamental. Let us note, however, that other representations on the sites of the chain
can be obtained by fusion from the fundamental one, corresponding to a special choice of θ’s.
It is very useful to view T(u) as a matrix of size (m + n) × (m + n) whose elements T
i
j
act on the physical Hilbert space H. This matrix satisfies a graded version of the ce l ebr at e d
RTT relation which in components reads
7
R
ji
i
′
j
′
(u − v)T
i
′
i
′′
(u)T
j
′
j
′′
(v) (−1)
[j
′′
][j
′
]
= T
i
i
′
(v)T
j
j
′
(u)R
j
′
i
′
i
′′
j
′′
(u − v) (−1)
[i
′
][j]
. (2.8)
6
Although most of the checks we present later in this paper have been done assuming that all θ
k
are in
generic position, we expect our results to be valid for any choice of θ’s.
7
The notation we u s e is slightly different compared to [
49, 51], so that we have T
there
ij
= (−1)
[j]([i]+1)
T
i
j
.
Our notation ensures in particular that the Berezi n ia n has the standard form given below in (
3.4), while
the notation of [
49, 51] would lead to an extra sign in that expression .
– 4 –
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