#### Evaluation of the operatorial Q-system for non-compact super spin chains

HJE
Evaluation of the operatorial Q-system for non-compact super spin chains
Rouven Frassek 0 1 2
Christian Marboe 0 1 2 3
David Meidinger 0 1 2
Institut des Hautes E´tudes Scientifiques 0 1 2
Gauge Theory
0 IRIS Geb ̈aude , Zum Großen Windkanal 6, 12489 Berlin , Germany
1 College Green , Dublin 2 , Ireland
2 35 Route De Chartres , 91440 Bures-sur-Yvette , France
3 School of Mathematics, Trinity College Dublin
We present an approach to evaluate the full operatorial Q-system of all u(p, q|r + s)-invariant spin chains with representations of Jordan-Schwinger type. In particular, this includes the super spin chain of planar N = 4 super Yang-Mills theory at one loop in the presence of a diagonal twist. Our method is based on the oscillator construction of Q-operators. The Q-operators are built as traces over Lax operators which are degenerate solutions of the Yang-Baxter equation. For non-compact representations these Lax operators may contain multiple infinite sums that conceal the form of the resulting functions. We determine these infinite sums and calculate the matrix elements of the lowest level Q-operators. Transforming the Lax operators corresponding to the Q-operators into a representation involving only finite sums allows us to take the supertrace and to obtain the explicit form of the Q-operators in terms of finite matrices for a given magnon sector. Imposing the functional relations, we then bootstrap the other Q-operators from those of the lowest level. We exemplify this approach for non-compact spin −s spin chains and apply it to N = 4 at the one-loop level using the BMN vacuum as an example.
Bethe Ansatz; Lattice Integrable Models; Quantum Groups; Supersymmetric
1 Introduction 2
Q-operators for representations of oscillator type
Lax operators for Q-operators
Definition of the Q-operators
Representation in the quantum space
Non-compact Heisenberg spin chains
Non-compact R-operators and infinite sums
Comments on the Q-system
Ladder decomposition of R-operators
4.3 Integral representation for lowest level R-operators
Derivation of the R-operators for the spin −s models
Finite sum representation for lowest level R-operators
Towards higher level finite sum representations
Generating the operatorial Q-system
Lowest level Q-operators
Operatorial Q-system from functional relations
3
4
5
6
7
B
C
2.1
2.2
2.3
4.1
4.2
4.4
4.5
4.6
5.1
5.2
The BMN vacuum of fully twisted N = 4 SYM at leading order
Conclusions and outlook
A Formulas for the explicit evaluation of Q-systems
A.1 Matrix elements of lowest level R-operators
A.2 Calculating supertraces
A.3 Generalised Lerch transcendents
A.4 Formulas for discrete integrals
Highest level R-operators and their matrix elements
Normalisations of Q-operators and functional relations
– 1 –
Introduction
Non-compact spin chains are quantum integrable models that appear in certain limits of
four-dimensional quantum field theories [1–4]. In contrast to their compact counterparts
the physical or quantum space of non-compact spin chains is infinite-dimensional. Though
physically these spin chains are of very different nature, they can be uniformly described
algebraically in the framework of the quantum inverse scattering method, see e.g. [5].
Here a commuting family of transfer matrices is built from so called R-matrices which are
solutions to the Yang-Baxter equation which are studied systematically in the theory of the
Yangian and the universal R-matrix [
6, 7
]. For a given quantum space, transfer matrices
which arise from fusion in the auxiliary space, see e.g. [8] for an overview.
A distinguished role among the commuting operators is taken by the Q-operators [
9
].
They are related to the transfer matrices via the quantum Wronskian and satisfy so called
QQ-relations. For the case of interest see [10] and references therein where these relations
are discussed on the level of eigenvalues, i.e. Q-functions. In the pioneering work [11–13],
Q-operators were constructed as traces over an infinite-dimensional oscillator space. We
refer the reader to the more recent works [14–17] for a mathematical discussion of these
infinite-dimensional representations. Similarly, Q-operators for spin chains with diagonal
twist can be constructed in the framework of the quantum inverse scattering method.
The corresponding Lax operators for certain compact q-deformed higher rank spin chains
were written down explicitly in [18] and derived from the universal R-matrix in [19] , see
also [20–22] for earlier works. In the rational case the relevant Lax operators were obtained
in a series of papers [
23–27
] for gl(N |M ). These solutions allow for the definition of
Qoperators for more general representations and in particular as discussed in this article for
representations of the non-compact super algebras u(p, q|r+s) acting on the quantum space.
While for compact spin chains the Lax operators derived in [26] can straightforwardly
be used to evaluate the matrix elements of Q-operators, it is much more involved to extract
their matrix elements in the case of non-compact representations. The first obvious reason
is that the quantum space is infinite-dimensional, however noting that the Q-operators are
block diagonal this problem can be overcome by considering magnon blocks separately.
The other, more serious issue arises because the Lax operators relevant to construct
Qoperators were derived in the form of a Gauss decomposition. Besides its beauty, this form
is rather inconvenient for practical purposes. One has to sum over intermediate states to
compute explicit matrix elements, and in the case of non-compact representations there
are potentially infinitely many such states.
In this paper we overcome these difficulties which arise when evaluating Q-operators
for non-compact spin chains of Jordan-Schwinger type and present an efficient method to
determine the full operatorial Q-system1 for a fixed magnon block. Section 2 is a brief
1We use the term Q-system to denote the full set of Q-operators or Q-functions of a given integrable
first discuss non-compact spin −s chains in section 3. In this case there are two non-trivial
Lax operators, one involving an infinite sum, whose matrix elements we evaluate in full
detail. We generalise our approach to higher rank super spin chains in section 4. Here we
discuss which Q-operators involve infinite sums and provide a decomposition of the Lax
operators on which our approach is based. For the lowest level Lax operators we present an
integral formula which conveniently allows to evaluate them in terms of rational functions.
The evaluation of higher level Lax operators is discussed in section 4.6. In section 5, we
use the integral representation of the lowest level Lax operators to calculate the matrix
data. In section 6 we show how to apply these methods to the N = 4 SYM spin chain in
the presence of a full diagonal twist, and calculate the Q-functions of the BMN vacuum of
arbitrary length. We conclude our work in section 7 and speculate about the application
of Q-operators to the Quantum Spectral Curve of N = 4 SYM [
29, 30
]. We provide further
information on the operatorial Q-systems in the appendix. In particular, to facilitate the
application of our results, we collect all formulas which are needed for the calculation of
the Q-systems in appendix A. These include formulas for the matrix elements of the Lax
operators, and for the evaluation of the supertraces over the auxiliary Fock spaces.
2
Q-operators for representations of oscillator type
In this section we present the derivation of the Lax operators (R-operators) which allow to
construct the Q-operators of gl(N |M ) spin chains with representations realised via
JordanSchwinger oscillators as traces of monodromy matrices. The general construction reviewed
here was developed in a series of papers: the derivation of the R-operators follows the
bosonic case in [26] but incorporates the supersymmetric Lax matrices derived in [25]. For
the Lax operators of bosonic models, Schwinger oscillators were discussed in [
31
]. The
more general derivation of the Lax operators for generalised rectangular representations is
unpublished [
27
] while expressions for the resulting operators can be found in [
32
].
2.1
Lax operators for Q-operators
The study of supersymmetric rational spin chains goes back to Kulish [
33
] who introduced
the gl(N |M ) invariant Lax operators
N+M
a,b=1
L(z) = z +
X (−1)|b|eabEba ,
intertwining arbitrary representations of gl(N |M ) with the defining fundamental one. Here
the indices take the values a, b = 1, . . . , N + M while |a| denotes the grading |fermion| = 1
and |boson| = 0. The gl(N |M ) generators Eab satisfy the commutation relations
[Eab, Ecd] = δbcEad − (−1)(|a|+|b|)(|c|+|d|)δdaEcb ,
(2.1)
(2.2)
– 3 –
where we defined the graded commutator as [X, Y ] = XY − (−1)|X||Y |Y X. The generators
eab in (2.1) denote the defining fundamental generators of gl(N |M ) satisfying eabecd =
δbcead. In the following we restrict to the Schwinger oscillator realisation
Eab = χ¯aχb ,
where [χa, χ¯b] = δab.
The Lax operators for Q-operators with the defining representation of gl(N |M ) at each
spin chain site (the so-called quantum space) were derived in [25], and are given by
LI (z) =
(z − sI )δab − (−1)|b|ξ¯aa¯ξa¯b ξ¯a¯b .
−(−1)|b|ξa¯b
δa¯¯b
There are 2N+M such Lax operators labelled by the set I ⊆ {1, . . . , M + N }. The notation
here and in the rest of this article is as follows: we sum over repeated indices (appearing
two or more times); unbarred indices take values a, b ∈ I while barred ones take values in
its complement, a¯, ¯b ∈ I¯. The (N |M )×(N |M ) matrix in (2.4) is written in terms of the sub
blocks under this decomposition.2 The shift sI in the spectral parameter z is introduced
for convenience and reads
sI =
P
a¯∈I¯(−1)|a¯|
2
.
The oscillators (ξa¯a, ξ¯aa¯) satisfy the graded Heisenberg algebra
[ξa¯a, ξ¯b¯b] = ξa¯aξ¯b¯b − (−1)(|a|+|a¯|)(|b|+|¯b|)ξ¯b¯bξa¯a = δabδa¯¯b .
We can write down the defining Yang-Baxter equation for the R-operators which are the
building blocks for Q-operators when the sites of the quantum space are in a representation
space different from the fundamental representation. As in the bosonic case [26] this relation
is given by
L(x − y)LI (x)RI (y) = RI (y)LI (x)L(x − y) .
The form of R-operators was obtained in [
27
] and spelled out in [
32
]. The derivation
follows [26] and, as we will discuss in the following, simplifies significantly in the case
which we are interested in. As for gl(N ) one takes a factorised ansatz,
RI (z) = e(−1)|c|+|c||c¯|ξ¯cc¯Ecc¯ RI0(z) e−(−1)|d||d¯|+|d|+|d¯|ξd¯dEd¯d ,
and ends up with a difference equation for the middle part RI0(z). The solution to the
difference equation simplifies significantly for the choice of generators (2.3). For representations
of this type one finds that RI0(z) can be written in terms of a single Gamma function
RI0(z) = ρI (z) Γ z + 1 − sI −
X Ea¯a¯
a¯
!
.
2We remark that quantities labelled by the set I depend on the partition I ∪ I¯ = {1, · · · , N + M }. We
leave the dependence on this full set implicit.
– 4 –
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
Here ρI denotes a normalisation not fixed by the Yang-Baxter equation (2.7). As we will
see, a good choice for it is given by
ρI (z) =
1
which depends on the central charge C that can be expressed in terms of the number
operators Na = χ¯aχa as
We conclude that
C =
Na .
N+M
X
a=1
RI (z) = e(−1)|c|+|c||c¯|ξ¯cc¯χ¯cχc¯ Γ(z + 1 − sI − χ¯a¯χa¯) e−(−1)|d||d¯|+|d|+|d¯|ξd¯dχ¯d¯χd
solves the Yang-Baxter equation in (2.7). The normalisation (2.10) ensures that for the
empty set R∅(z) = 1 and renders RI (z) a polynomial in z for compact representations.
Finally we note that the middle part of (2.12) involves multiple Gamma functions for more
general representations, see [
26, 27, 32
]. However, for Jordan-Schwinger type
representations only one Gamma function appears, cf. [31].
2.2
Definition of the Q-operators
Using the Lax operators LI (z) defined in (2.4), the Q-operators for gl(N |M ) rational spin
chains were introduced in [25] for the defining fundamental representations at each site of
the quantum space. We are interested in more general representations of oscillators type,
cf. (2.3). However, as discussed in e.g. [26] for gl(N ) the construction of the Q-operators
and the functional relations among them should be independent of the quantum space.
Thus, following [25] we define the Q-operators as
QI (z) = eiz Pa∈I (−1)|a|φa sctr MI (z) .
Here the monodromy MI is built from the tensor product of the R-operators in (2.12) in
the space of oscillators (χ, χ¯) and multiplication in the auxiliary space of oscillators (ξ, ξ¯) as
The normalised supertrace sctr is defined by
MI (z) = R[I1](z) ⊗ R[I2](z) ⊗ . . . ⊗ R[IL](z) .
sctr X =
str e−i Pa,b(φa−φb)Nab X
str e−i Pa,b(φa−φb)Nab
,
where str denote the ordinary supertrace over the auxiliary Fock space spanned by
the states generated from acting with the operators ξ¯aa¯ on a Fock vacuum satisfying
ξa¯a|0i = 0. These states are labelled by the values of the number operators
Nab = ξ¯abξba ,
– 5 –
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
where no sum is implied over the indices a and b. The twist parameters φa which can be
interpreted as Aharonov-Bohm phases, cf. [25], break the gl(N |M ) invariance down to its
diagonal subalgebra. They are required for the convergence of the supertraces. Note that a
regularisation is needed to make some of the traces converge, even in the presence of twists;
one can use an iε prescription for the twists, such that Re(exp(−i Pa,b(φa − φb))) < 1,
see [24].
The Q-operators defined in (2.13) commute with each other, [QI (z), QI′ (z′)] = 0,
and, as a consequence of the Yang-Baxter equation (2.7), also with the transfer matrix
built from the Lax operators L realised via (2.3). For a discussion on how to obtain the
Hamiltonian from the Q-operators we refer the reader to [34]. Further it was argued in [25]
that depending on the grading the Q-operators satisfy either the bosonic QQ-relations
ΔabQI∪{a,b}(z)QI (z) = QI∪{a} z +
QI∪{b} z − 2
−QI∪{a} z − 2
QI∪{b} z +
1
2
1
where |a| = |b| or the fermionic QQ-relations
ΔabQI∪{a}(z)QI∪{b}(z) = QI∪{a,b}
z +
1
2
QI z − 2
1
−QI∪{a,b}
QI z +
,
where |a| 6= |b|. Here we defined the trigonometric prefactor
Δab = (−1)|a|2i sin
φa − φb
2
the partial order induced by the inclusion of indices, see for example [10].
The
relations (2.17) and (2.18) then constrain the operators on each quadrilateral of this diagram.
It is straightforward to compute the Q-operators for the empty set I = ∅ and the full set
I = {1, . . . , N + M } = ∅. Using the normalisation in (2.10) one finds
Q∅(z) = 1 ,
Q∅(z) =
Γ(z + 1)
Γ(z + 1 − C)
L
,
where we imposed the constraint3
N+M
a=1
X (−1)|a|φa = 0 .
This relation is needed for Q∅ to be a rational function of the spectral parameter.
2.3
Representation in the quantum space
So far we did not specify a representation in the quantum space. Parts of our derivations will
be independent of the concrete representation, but calculations of explicit matrix elements
of course require a concrete knowledge of the representation space. Here we focus on
3For gl(N |N ) one has the additional constraint P2aN=1 φa = 0.
– 6 –
1
1
z − 2
,
1
2
(2.17)
1
2
(2.18)
(2.19)
(2.20)
(2.21)
unitary highest- or lowest-weight representations of u(p, q|r + s) of oscillator type which
were first investigated in [35]. To specialise to a real form of the algebra, we have to
indicate, in addition to the grading |a| = 0, 1, which directions have opposite sign under
conjugation that can be realised via a particle-hole transformation. We indicate these using
the variables
ωa =
(+1 if oscillator a is not transformed
−1 if oscillator a is transformed
Then the generators Eab = χ¯aχb can be realised by the oscillators
(χa, χ¯a) = (b¯a, −ba)
(aa, a¯a)
(ca, c¯a)
(d¯a, da)
for |a| = 0 and ωa = +1
for |a| = 0 and ωa = −1
for |a| = 1 and ωa = +1
for |a| = 1 and ωa = −1
These oscillators act on a Fock space with a vacuum state satisfying aa|0i = ba|0i =
ca|0i = da|0i = 0, such that an orthonormal basis is given by
a† = a¯, b† = b¯, c† = c¯ and d† = d¯, the generators are those of u(p, q|r + s), satisfying
Ea†b = ωa1+|a|ωb1+|b|Eba .
Finally, the Fock space contains a series of representations labelled by the central charge
C = P
a Na where the number operators have to be expressed in terms of oscillators a, b,
c and d via (2.23). These representations are of highest or lowest weight type depending
on the order of the different types of oscillators.
3
Non-compact Heisenberg spin chains
In this section we provide the formulas necessary to evaluate Q-operators of non-compact
Heisenberg spin chains, which include for example the spin −1 model which is of interest
for QCD in the Regge limit [36]. These models constitute the simplest non-trivial case
which can be treated in the more general framework presented in section 4 and 5.
While the formula for the Lax operators (2.12) is extremely compact, it is rather
inconvenient for practical purposes where the matrix elements of the Lax operators and
Q-operators are of interest. In particular, for non-compact representations we encounter
infinite sums. To understand this problem we consider the R-operators (2.12) with |I| = 1
which for the case of gl(2) are given by
(2.24)
(2.25)
(3.1)
R{a}(z) = eξ¯aa¯Eaa¯ Γ(z + 12 − Ea¯a¯) e−ξa¯aEa¯a ,
Γ(z + 12 − C)
– 7 –
where a = 1, 2 and a¯ 6= a. For infinite-dimensional representations the Lax operator
contains two infinite sums emerging from the exponential functions. Using the algebraic
relations in (2.2) we note that the Lax operators can be rewritten as
+∞
X
n=−∞
(ξ¯aa¯Eaa¯)θ(+n)|n|M{a}(z; |n|)(−ξa¯aEa¯a)θ(−n)|n| ,
(3.2)
with θ(−m) = θ(0) = 0 and θ(m) = 1 for m ∈ N+. The middle part is given by an infinite
sum and only depends on Cartan elements
M{a}(z;|n|) =
1 Γ(z + 12 −Ea¯a¯)
|n|! Γ(z + 21 −C) 3F2 Ea¯a¯ −λ1,Ea¯a¯ −λ2 +1,−Naa¯;1+|n|,Ea¯a¯ +
1
2 −z;1 ,
(3.3)
with the gl(2) weights λ1 and λ2.4
We are interested in highest-weight state representations of the type discussed in
section 2.3. To describe non-compact spin chains with spin −s, where s is a positive half
integer, we take the Jordan-Schwinger realisation (2.3) and perform a particle hole
transformation on the oscillators of type 1:
( χ¯1, χ1) → (−b, b¯) ,
( χ¯2, χ2) → (a¯, a) .
For convenience we use a notation different from the rest of this article and label the states
in the spin −s representation as
|mis = |2s − 1 + m, mi ,
cf. (2.24). The highest-weight state |0is then satisfies
E12|0is = 0 ,
E11|0is = λ1|0is = −2s|0is ,
E22|0is = λ2|0is = 0 ,
(a)n := Γ(a + n)/Γ(a).5
and the other states of the representation can be generated from |0is by acting with the
operator E21. The central charge takes the value C|mis = −2s|mis.
Our goal is to obtain the matrix elements of the R-operators in (3.1). From (2.20) we
1
find that R∅(z) = 1 and R∅(z) = (z+1)2s , where the Pochhammer symbol is defined by
It is rather straightforward to obtain the matrix elements of R{2}. They are
polynomials in the spectral parameter z and can be obtained noting that the series representation
of the hypergeometric function in (3.3) truncates. One finds
sh m˜|R{2}(z)|mis =
4Note that in the rank 1 case the reformulation (3.2) with (3.3) is also valid for representations that
are not of Jordan-Schwinger type.
5In the following we will sometimes consider Pochhammer symbols where a can be a negative integer.
In this case we define the symbol using the identity Γ(a+n) = (−1)n Γ(Γ1(−1−a−a)n) , which follows from Euler’s
Γ(a)
reflection formula.
– 8 –
(3.4)
(3.5)
(3.6)
(3.7)
with the middle part which is diagonal in the auxiliary space
M{2}(z, N21, k, l) = (2s − 1 + l)! X
l
p=0
l (N21 + 1 + p − l)l−p(z + 12 + 2s)p
p
(2s − 1 + p)!(k + l − p)!
.
(3.8)
Here θ(−m) = θ(0) = 0 and θ(m) = 1 for m ∈ N+.
However, as already noted in [26] the operator R{1} yields infinite sums when
evaluated naively, since there are only raising operators acting on the states, cf. (3.1). This
makes it difficult to evaluate its matrix elements concretely in terms of rational functions.
Nevertheless, as we will show in section 4.4, using the integral representation of the
hypergeometric function and the Euler transformation the matrix elements can be obtained
(3.9)
(3.10)
(3.12)
from (3.2) and written as
sh m˜|R{1}(z)|mis =
s max(m, m˜)! max(2s − 1 + m, 2s − 1 + m˜)!
min(m, m˜)! min(2s − 1 + m, 2s − 1 + m˜)!
× (−ξ¯12) θ(m−m˜ )|m−m˜ |M{1}(z, N12, |m − m˜|, max(m, m˜))(−ξ21) θ(m˜ −m)|m˜ −m| ,
with the middle part taking the simple form
M{1}(z, N12, k, l) =
M{2}(z, N12, k, l − k)
(z − N12 − l + 12 )2l−k+2s
.
We see that also this non-polynomial R-operator can be written in a very compact form
and observe that the resulting expression is very similar to the polynomial R-operator. In
particular, both are simple rational functions of the spectral parameter and the auxiliary
oscillators. The only difference is the dependence on the auxiliary space number operators
in the denominator. This has important consequences for the analytic structure of the
resulting Q-operators.
The matrix elements of the corresponding Q-operators (2.13) can now be derived as
shm˜ |Q{a}(z)|mis = eizφa tbr shm˜ 1|R{a}(z)|m1is · · · sh m˜L|R{a}(z)|mLis .
(3.11)
The big advantage of first evaluating the R-operators in the quantum space and
subsequently taking the trace in the auxiliary space is that we can restrict to individual magnon
sectors with M = PL
i=1 mi = PiL=1 m˜i. For each such sector, the Q-operators can then be
realised as matrices of finite size. As we will show in section 5, the matrix elements of the
Q-operators corresponding to the R-operators with non-truncating sums are non-rational
functions and can be written in terms of the Lerch transcendent (Lerch zeta-function)
defined as
To give the reader an impression of the resulting Q-functions we consider the
con1
crete case of a spin chain with spin − 2 . For small length L and magnon number M the
Q-operators resulting from the monodromy construction can easily be diagonalised. The
eigenvalues and eigenvectors containing the twist parameters are rather involved. For the
Φℓτ (z) =
τ k
∞
X
k=0 (k + z)ℓ
.
– 9 –
case L = 2 and M = 0, 1, 2 one easily obtains explicit though rather lengthy expressions for
the eigenvalues and eigenvectors. Due to the constraint (2.21), there is only one
independent twist parameter with φ1 = −φ2. For small values of φ1 the eigenvalues corresponding
to the highest-weight states of the untwisted spin chain are given by:
M
0
1
2
Q{1}(z)
1
2iφ1 ψ′(−z − 2 ) + O(φ21)
1
2iφ1 × (−4) × 1 + (z + 1)ψ′(−z − 2 ) + O(φ21)
Q{2}(z)
1
(z + 1) + O(φ1)
2iφ1 × 9 × (z + 1) + (z2 + 2z + 1132 )ψ′(−z − 2 ) + O(φ21) (z2 + 2z + 1132 ) + O(φ1)
1
HJEP09(217)8
Here the non-polynomial Q-functions are expressed in terms of the Polygamma function
ψ′(z) = Φ12(z). We observe that for fixed M , the prefactors of these functions in Q 1 are
{ }
given by the functions Q{2}, which are known in closed form and given by Hahn
polynomials [37, 38]. Expanding the factor Δ12 = 2iφ1 + O(φ31) in the functional relation (2.17),
we see that the functions 2i1φ1 Q{1}(z) and Q 2 (z) satisfy the functional relations of the
{ }
untwisted spin chain, where the factor Δ12 is not present.
4
Non-compact R-operators and infinite sums
In section 2.1 we introduced the R-operators with Jordan-Schwinger oscillator
realisation (2.3) in the quantum space. As for more general representations in the quantum space
the R-operators naturally decompose into three factors, cf. (2.12). However, as discussed
for the case of non-compact spin chains with u(1, 1) symmetry in section 3, this
undoubtedly elegant expression has a drawback when considering non-compact representations.
The exponentials appearing on the right (left) in the R-operators (2.12) do not truncate
in the case with only creation (annihilation) operators in the quantum space. Thus one
has to sum over an infinite tower of states.6 This issue likewise appears for higher rank
algebras u(p, q|r + s) as introduced in section 2.3. Furthermore, in this case even the
Roperators with truncating sums are complicated, and naively expanding the exponentials
leads to a large number of crossterms which grows exponentially; most of these terms do
not contribute to a given matrix element.
We start with a survey of the Q-system of u(p, q|r+s) and discuss its analytic structure.
Afterwards we study a different representation of (2.12), that is simpler to evaluate in the
non-compact, but also in the compact, case. We obtain a convenient formula (4.9) to
compute the lowest order Lax operators which, as we will see in section 5, is sufficient to
determine the whole Q-system. Finally we speculate about generalisations beyond the first level.
4.1
Comments on the Q-system
The Q-system of u(p, q|r+s) contains a total number of 2p+q+r+s Q-operators QI built from
the operators RI with I ⊆ {1, . . . , p+q +r +s}. Their functional relations can conveniently
6Note that at this stage we do not consider the auxiliary space. Later on when evaluating Q-operators
we will take the trace over the infinite-dimensional Fock space, see (2.13).
,
c
d
(0, q, 0)
HJEP09(217)8
hole transformed.
transformed.
Projection of the Hasse diagram with Q-operators on the lattice points (i, j, k).
Rational Q-operators are marked with
and non-rational ones with
corresponding to truncating
and non-truncating R-operators respectively.
p
i × j ×
q
k
be depicted in a Hasse diagram spanned by a hypercube. For our purposes it is convenient to
consider a projection of the Hasse diagram onto an ordinary three-dimensional cube. This
projection is visualised in figure 1. In this diagram, each lattice point (i, j, k) is occupied by
r+s Q-operators, namely those Q-operators whose index set I includes i bosonic
indices which are not particle-hole transformed according to (2.23), j bosonic indices which
are transformed, and k fermionic indices. Here i = 1, . . . , p, j = 1, . . . , q and k = 1, . . . , r+s.
The total number of indices in the set I is referred to as the level k = |I| which can
be thought of as a diagonal slice of the cube. Each level contains p+q+r+s Q-operators.
Interestingly, for only 2r+s(2p + 2q) of them all exponentials in the R-operators (2.12)
truncate.
This can be seen from the action of the exponentials on the states defined
k
in (2.24). An R-operator RI has matrix elements with truncating sums if one or both of
the following conditions hold:
• I does not contain any indices corresponding to bosonic oscillators that are
particle• I contains all indices corresponding to bosonic oscillators that are not particle-hole
Else the matrix elements of the R-operator will involve infinite sums. Due to the nilpotency
of the fermionic oscillators, the fermionic degrees of freedom do not change the truncating
or non-truncating nature of the R-operators. It follows that the truncating R-operators
are located at the lattice sites (i, 0, k) and (p, j, k) where i = 1, . . . , p, j = 1, . . . , q and
k = 1, . . . , r + s. We denote the vertices with truncating ones by
and the non-truncating
ones by
in figure 1. At a given lattice point the latter ones contain j(p − i) pairs of
exponentials which do not truncate.
Truncating R-operators yield Q-operators whose matrix elements are rational functions
of the spectral parameter multiplied by an exponential function including the twist phases.
In the case of non-truncating R-operators the resulting matrix elements of the Q-operators
are written in terms of rational functions and the generalised Lerch transcendent, see
section 5 and appendix A.
In this section we introduce the decomposition of the R-operators (2.12) on which our
approach to evaluate Q-operators for non-compact spin chains is based. Using only algebraic
relations we reduce the number of infinite sums and make the actual challenge of evaluating
R-operators manifest. Since the factors in the exponentials of the R-operators (2.12) appear often in the
HJEP09(217)8
following derivations, we define abbreviations for them:
Yaa¯ = (−1)|a|+|a¯||a|ξ¯aa¯ χ¯aχa¯ ,
Xaa¯ = (−1)|a¯|+|a|+|a¯||a|ξa¯a χ¯a¯χa .
(4.1)
The main idea is to expand the exponentials and to combine terms with the same difference
in the powers of the matching factors Xaa¯ and Yaa¯ in the exponents. This can be done
+∞
X
n=−∞
k=0
k!(|n| + k)!
(Yaa¯)θ(+n)|n| X∞ (−1)kYaka¯Xaka¯ f (Na −k, Na¯ +k)(−Xaa¯)θ(−n)|n|
that can be derived using the oscillator algebra. We can furthermore express the factor
using the formula
eYaa¯ f (Na, Na¯)e−Xaa¯ =
Yaka¯Xaka¯ as
RI (z) =
∞
X
"
{naa¯}=−∞
a, a¯
MI (z, {N}, {n}) =
The purely diagonal part MI is then given by
Yaka¯Xaka¯ = (−1)|a|+|a¯|
Γ(1 + Naa¯)
Γ(1 + Na)
Γ(1 + Naa¯ − k) Γ(1 + Na − k)
Γ(Na¯ + k + (−1)|a¯|)
Applying (4.2) to the R-operators (2.12), we find that they can be written in a form which
features a minimal number of creation and annihilation operators,
#
#
#
(4.2)
(4.3)
(4.5)
#
Y(Yaa¯)θ(naa¯)|naa¯| MI (z, {N}, {n})
Y(−Xaa¯)θ(−naa¯)|naa¯| . (4.4)
"
a, a¯
∞
X
"
Y
"
Y
(−1)(|a|+|a¯|+1)kaa¯
Γ(1 + Naa¯)
{kaa¯}=0 a,a¯ Γ(kaa¯ + 1)Γ(|naa¯| + kaa¯ + 1) Γ(1 + Naa¯ − kaa¯)
Γ(1 + Na)
×
×
a Γ(1 + Na − P
a¯ kaa¯)
a¯
Γ(z + 1 − sI − P
a¯ Na¯ − Pa,a¯ kaa¯) .
I looks rather complicated, this representation is in fact quite convenient.
First note that for any matrix element of (4.4), the outer sums over the variables naa¯ are
always finite, and only serve to introduce enough creation and annihilation operators to
produce overlapping states. For the lowest and the highest level of the Q-system, only
a single term contributes for any matrix element, which is then effectively given by the
diagonal part. Compared to naively expressing each term in the exponentials by their
power series, this representation already removes half of the infinite sums.
So far our discussion was purely algebraic and we did not specify the spectrum of
the number operators. Assuming that these operators act on a Hilbert space as given
in (2.24), the spectrum of the Na is positive or negative integer valued depending on
whether the corresponding oscillators are particle-hole transformed, cf. (2.23). For compact
representations, all N are positive or zero and the sums over the variables kaa¯ in (4.5) are
finite; they are effectively truncated by the Gamma functions in the denominator. However,
for non-compact representations some N take negative integer values such that some of the
sums may not truncate. The evaluation of those sums is however simplified by the fact
that they only involve diagonal operators.
Integral representation for lowest level R-operators
In this subsection we focus on the R-operators of the lowest level R{a}. For these
Roperators only one term in the sum (4.4) contributes to any matrix element, which is then
directly given by the diagonal part M{a}. Furthermore, as will be discussed in section 5,
the corresponding Q-operators determine the full Q-system. Here we derive an integral
representation of the diagonal part (4.5), which can easily be evaluated and from which
rational finite sum expressions can readily be obtained, cf. section 4.4 and 4.5.
We first specialise the expression given in (4.4) and (4.5) to the lowest level, and write
the Lax operators as
Y(Yaa¯)θ(+na¯)|na¯| M{a}(z, {N}, {n})
Y(−Xaa¯)θ(−na¯)|na¯| , (4.6)
#
#
∞
X
"
{na¯}=−∞
a¯
"
a¯
with X and Y given in (4.1). Here the diagonal part reads
M{a}(z, {N}, {n}) =
∞
X
Y
"
{ka¯}=0 a¯
(−1)(|a|+|a¯|+1)ka¯ Γ(1 + Naa¯)
ka¯! (|na¯| + ka¯)! Γ(1 + Naa¯ − ka¯)
Γ(Na¯ + (−1)|a¯| + ka¯)
#
Γ(Na + 1)
× Γ(Na + 1 − P
To obtain the aforementioned integral representation, we evaluate all sums over the
variables ka¯ in the diagonal part (4.7). Since the intermediate formulas are quite lengthy, we
only sketch this derivation.
Consider the first sum, which we take to be over some index ¯b. It is straightforward to
see that this sum can be written as a product involving Gamma functions and the following
hypergeometric function:
3F2
P
a¯6=¯b ka¯ − Na
− Na¯b
N¯b + (−1)|¯b|
−z + 21 Pa¯(−1)|a¯| + P
a¯6=¯b(Na¯ + ka¯) + N¯
b
1 + |n¯b|
; (−1)|a|+|¯b| .
!
(4.8)
Since the other summation variables appear in the arguments of this hypergeometric
function, the other sums cannot be taken easily. To remedy this, and to disentangle the sums,
one can use an integral representation of the hypergeometric function. The type of
integral however depends on the spectrum of the operator Na. If the oscillator with index a is
bosonic and particle-hole transformed, ωa = −1, the first argument P
pergeometric function (4.8) takes positive integer values, and we can use the standard Euler
type integral, expressing the function 3F2 as an integral over the interval (0, 1) on the real
line involving the function 2F1. For all other cases, the Gamma function Γ(Na + 1 − P
in the denominator of (4.7) truncates the range of the summation variables such that the
argument P
values. For negative arguments, one can use an analytic continuation of the Euler integral
a¯6=¯b ka¯ − Na of the hypergeometric function (4.8) takes non-positive integer
a¯6=¯b ka¯ −Na of the
hyemploying the Pochhammer contour. For negative integers, this contour collapses into a
contour integral around the origin.
Using the appropriate integral formulas to rewrite the hypergeometric function (4.8),
one finds that all subsequent summations decouple, and can be performed easily using the
series representation of the hypergeometric function 2F1. We then arrive at the result
M{a}(z,{N},{n})
=
Z
dt t−Na−1(1−t)−z−1+C+ 12 Pa¯(−1)|a¯| Y
1
a¯ |na¯|!
2F1
1+|na¯|
Na¯ +(−1)|a¯| −Naa¯ ;(−1)|a¯|+|a|t ,
(4.9)
!
where C is the central charge defined in (2.11) and the integration is
Z
(−1)Na Z 1
dt = Γ(−Na) 0
Γ(1 + Na) I
2πi
dt
t=0
if |a| = 0 and ωa = −1
dt
ωa = 1 or |a| = 1
.
(4.10)
This means that for truncating R-operators, the integral just computes a residue, while for
the non-truncating ones, it is an integral over the interval (0, 1). Note that strictly speaking,
the integral is only convergent for appropriately chosen values of the spectral parameter
z; this however poses no problem, since the result for any matrix element is a rational
function which can be analytically continued to any value of the spectral parameter.
Further we note that while it might seem that we did not gain much by writing the
potentially infinite sums of the R-operator in terms of an integral, this integral is in fact
trivial to evaluate, by expanding the integrand and, depending on the case, either taking
a simple residue or evaluating the line integral in terms of a Beta function. It provides a
convenient way of treating the truncating and non-truncating R-operators in a joint way,
the essentially only difference being the contour of integration. In the next section we
show how the integral formula (4.9) is used to recover the matrix elements in the case of
the spin −s chains discussed in section 3. Subsequently, we demonstrate how the integral
formula can be rewritten in terms of finite sums in section 4.5.
4.4
Derivation of the R-operators for the spin −s models
The integral representation for R-operators of the lowest level, given in (4.6) together
with (4.9), can easily be evaluated in practise. To show that it also serves as a good
starting point to obtain representations in terms of finite sums, we now derive the
formulas (3.7), (3.8), (3.9) and (3.10) for the R-operators of the spin −s spin chains considered in
section 3. For these models, both oscillators are bosonic, |·| = (0, 0), and the first oscillator
is particle-hole transformed, ω = (−1, +1). The central charge is constrained to C = −2s,
such that the states are given by |mis = |2s − 1 + m, mi, cf. (3.5).
We begin with the truncating R-operator R{2}. The matrix elements sh m˜|R{2}(z)|mis
can be determined from (4.6) by noting that the summation variable n1 is fixed to be
n1 = m˜ − m, and that the diagonal part then acts on a state | mˆis with mˆ = min(m, m˜ ),
cf. (A.1) and (A.3). Using this it is straightforward to show that the general structure
of the Lax operator exactly matches (3.7). The diagonal part M 2 could in principle be
derived directly from expression (4.7); we nevertheless start from the generally applicable
{ }
formula (4.9) expressing it as a contour integral. This integral can be evaluated by plugging
in the series representations of the hypergeometric function and of the power of (1 − t),
= mˆ!
X
k=0
mˆ (z + 12 + 2s)mˆ −k(1 + N21 − k)k(2s + mˆ − k)k
,
(| m˜ − m| + k)!
"X∞ (z + 21 + 2s)ℓ ℓ
# "mˆ +2s−1 (2s + mˆ − k)k(1 + N21 − k)k k
X
#
1 − 2s − mˆ
− N21 ; t
1 + | m˜ − m|
k=0
(| m˜ − m| + k)!k!
which is the same as (3.8).
Next we turn to the non-truncating R-operator R{1}. For each matrix element, the
summation variable is fixed to n2 = m − m˜ , and the diagonal part acts on | mˆis, where now
mˆ = max(m, m˜ ). One finds that the form of the matrix elements in (3.9) is reproduced
by (4.6). The diagonal part (4.9) is then given by
shmˆ |M{1}(z)|mˆ is =
(−1)2s+mˆ
Z 1
(2s−1+mˆ )!| m˜−m|! 0
dt t2s−1+mˆ (1−t)−z− 12 −2s2F1
mˆ+1 −N12 ;t .
1+| m˜−m|
(4.12)
To write the matrix elements as finite sums, we have to apply the Euler transformation
2F1(n, b; m; z) = (1 − z)m−n−b2F1(m − n, m − b; m; z) to the hypergeometric function. Then
this function can be written as a finite sum and the integral can be evaluated using the
integral representation of the Beta function:
shmˆ |M{1}(z)|mˆ is =
(−1)2s+mˆ
Z 1
(2s−1+mˆ )!| m˜−m|! 0
=
k=0
dt t2s−1+mˆ (1−t)−z− 21 −2s−min(m,m˜ )+N12
−min(m, m˜) 1+|m˜ −m|+N12 ;t
1+| m˜−m|
dt t2s−1+mˆ +k(1−t)−z− 12 −2s−min(m,m˜ )+N12
×2F1
×
Z 1
0
t
(4.11)
k=0
×B 2s+ mˆ+k,−z +
1
2 −2s−min(m, m˜)+N12 . (4.13)
This expression is identical to (3.10), upon using B(x, y) = ΓΓ((xx)+Γ(yy)) .
In the next section we represent the generalisation of this finite sum formula for the
nontruncating R-operators of arbitrary non-compact spin chains of Jordan-Schwinger type.
Finite sum representation for lowest level R-operators
Evaluating the integral formula in (4.9) is a very efficient way to determine matrix elements
of truncating as well as non-truncating R-operators. It is however also possible to directly
derive finite sum expressions from the integral representation using the same ideas as in
the previous section. Here one has to treat the truncating and non-truncating R-operators
separately, corresponding to the two integration contours in (4.10). In the truncating case,
evaluating the residue returns the expression given in (4.7), which can be expressed in terms
of number operators for the particle-hole transformed oscillators (2.23); then all sums are
manifestly finite.
For the non-truncating R-operators R{a} with |a| = 0 and ωa = −1, one can evaluate
the integral as follows: first, we decompose the set into sets of indices corresponding to the
different types of oscillators I¯ = I¯a ∪ Ib ∪ Ic ∪ I¯d, cf. (2.23). Subsequently we apply the
¯ ¯
Euler transformation to the hypergeometric functions corresponding to the set I¯a and use
the series expansion of all such functions to perform the Beta integral. We find
M{a}(z,{N},{n})
=
∞
X
{ka¯}=0
(−1)1+Nba
Nba !
1
Q
a¯∈I¯ka¯!(|na¯|+ka¯)!
×
×
a¯∈I¯a
a¯∈I¯c
Y (|na¯|−Naa¯ )ka¯ (|na¯|+Naa¯ +1)ka¯
Y (−Nba¯ )ka¯ (−Naa¯)ka¯
Y (−1)ka¯ (Nca¯ −1)ka¯ (−Naa¯)ka¯
Y (−1)ka¯ (−Nda¯ )ka¯ (−Naa¯)ka¯
(4.14)
a¯∈I¯b
a¯∈I¯d
2
a¯∈I¯
a¯∈I¯a
×B
−z +C+
1 X(−1)|a¯| + X (Naa¯ +|na¯|−Naa¯ ),Nba +1+Xka¯ ,
a¯∈I¯
!
where we denoted the number operators of the particle-hole transformed oscillators as
Naa = a¯aaa et cetera. All the Pochhammer symbols involving these operators are of the
form (−m)k with m ≥ 0 which gives (−m)k = (−1)k Γ(Γm(m++1−1)k) , such that all sums truncate.
Note that the fact that |na¯ − Naa¯ | ≤ 0 can be seen from the structure of the outer sums
in (4.6), see also appendix A.1.
4.6
We have seen that the R-operators of the lowest level can conveniently be written either
using the integral representation (4.9) or as finite sums as in (4.7) and (4.14). Here we want
to discuss the generalisation of such representation to the remaining levels of the Q-system.
First note that the R-operators are almost symmetric under the exchange I ↔ I¯; it
is therefore possible to proceed with the highest level R-operators similarly as with the
lowest level ones. These results are summarised in appendix B. The intermediate levels can
be much more involved. However, we note that the difficulty of deriving representations
without infinite sums does not necessarily increase according to the level |I| of the operators,
but rather by the number of infinite sums, or more precisely by the number of indices a ∈ I
which correspond to bosonic and particle-hole transformed oscillators, |a| = 0 and ωa = −1.
If no such indices appear in the index set of RI , the formula (4.5) contains no infinite sum
to begin with, cf. section 4.1. Furthermore, for the case that there is exactly one such
index, one can apply the same strategy as was used for the lowest level. Let this index be
b; then one can perform all sums over the variables kba¯ in (4.5), and obtain a formula with
finite sums and an integral as in section 4.3. Writing the integral in terms of finite sums
as in section 4.5, one obtains a formula in terms of finite sums only.
The first case where more severe difficulties arise can best be discussed using a concrete
example. Consider a u(2, 2) invariant model, with oscillators a1, a2, b3 and b4. Then the
operator R{3,4} contains two particle-hole transformed indices. After performing similar
calculations as for the lowest level, one finds the following representation:
R{3,4}(z) = e−Pa,a¯ ξ¯aa¯aaba¯ Γ(z −Na1 −Na2 ) e−Pa,a¯ ξa¯aa¯ab¯a¯
Γ(z −C)
#
¯
−ξaa¯aaba¯
θ(+naa¯)|naa¯| M{3,4}(z,{N},{n}))
−ξa¯aa¯ab¯a¯
θ(−naa¯)|naa¯| ,
"
Y
a,a¯
(4.15)
#
(4.16)
Y
{naa¯}=−∞ a,a¯
×
×
Z 1
0
M{3,4}(z) can be written in terms of finite sums and an integral,
where the indices run over a ∈ I = {1, 2} and a¯ ∈ I¯ = {3, 4}. Here the diagonal part
M{3,4}(z,{N},{n}) =
∞
X
(−1)Nb3 +Nb4
(k13 +k23 +Nb3 )!
k13,k23=0 n14!n24!Nb3 !Nb4 ! k13!k23!(k13 +n13)!(k23 +n23)!
(n13 −Na1 )k13 (n23 −Na2 )k23 (1+n13 +N13)k13 (1+n23 +N23)k23
(n13 +n23 +N13 +N23 −Nb3 +1−z)k13+k23+Nb3
tNb4 (1−t)C−z 3F2(Na1 +1,1−n13 +Na1 ,−N14;1−k13 −n13 +Na1 ,1+n14;t)
× 3F2(Na2 +1,1−n23 +Na2 ,−N24;1−k23 −n23 +Na2 ,1+n24;t)
where the central charge is C = Na1 + Na2 − Nb3 − Nb4 − 2. It resembles the integral
formula (4.9), but this time involving generalised hypergeometric functions. Indeed it is
even possible to write the integral in terms of finite sums, using an analogue of the Euler
transformation which can be found in [39]. This identity is however rather involved and
not very explicit, and requires finding the zeros of an auxiliary polynomial. Note that the
formula for M{3,4}(z) follows from first using the result (4.14) for finite sum representations
of the lowest levels to make half of the sums finite. Then one applies the same strategy
to the remaining sums. The fact that the next step requires implicit identities of the type
just discussed for the hypergeometric functions renders it difficult to treat cases with more
infinite sums in this recursive fashion.
Nevertheless, as discussed in the next section, for the purpose of calculating
Qoperators explicitly there is no need to evaluate higher level R-operators as the whole
Q-system can be obtained from the lowest level.
5
Generating the operatorial Q-system
HJEP09(217)8
Above we focused on the calculation of matrix elements for lowest level R-operators. In
fact, as we will discuss now, the lowest level R-operators are sufficient to generate the entire
operatorial Q-system. Our strategy is to first combine the R-operator’s matrix elements
into matrix elements of the respective Q-operator by taking products and tracing out the
auxiliary Fock space. For each magnon block, the Q-operators can be represented explicitly
as matrices of finite size. Systematically solving the functional relations (2.17) and (2.18),
we determine all other Q-operators in the corresponding magnon block. To facilitate
concrete calculations, we collect all formulas necessary in this process in appendix A. This
allows to perform all calculations in computer algebra systems such as Mathematica.
5.1
Lowest level Q-operators
Using the matrix elements of the R-operators of the lowest level one can construct matrix
elements of the corresponding Q-operators via (2.13). Due to the remaining u(1)N+M
invariance which persists in the presence of diagonal twists, the Q-operators are block
diagonal. These blocks correspond to sectors with a fixed number of magnons; they are
labelled by the total excitation numbers PiL=1 m(ai), where a = 1, . . . , p + q + r + s, of the
oscillators of the representation of u(p, q|r + s) given in (2.23), see also (2.24), and the
number of sites L. For each such magnon block, the matrix elements can therefore be
combined into a matrix of finite size. This gives the operatorial form of Q-operators in a
subspace of the infinite-dimensional Hilbert space of non-compact models. For spin chains
of length L the matrix elements of the lowest level Q-operators can be expressed as
hm˜ (L)
| · · · hm˜ (1)
Q{a}(z) |m(1)
i · · · |m(L)
i
= (−1)Pi<j |m˜ (j)|(|m(i)|+|m˜ (i)|) eizφa sctr hm˜ (1)
|R{a}(z)|m(1)
i · · · hm˜ (L)
|R{a}(z)|m(L)i .
Here we denote the Graßmann degree of the state |m(i)i defined in (2.24) by |m(i)|. The
matrix elements of the R-operators hm˜ (i)
|R{a}(z)|m(i)i follow immediately from the
integral representation given in (4.6) and (4.9), or the finite sum for the non-truncating
these matrix elements still depend on the auxiliary space operators ξ¯aa¯ and ξa¯a.
R-operators given in (4.14). They can be found in full detail in appendix A.1. Of course,
To evaluate (5.1), one first commutes all the auxiliary space operators either to the
left or to the right, and combines them into number operators Naa¯. All terms containing
(5.1)
icn ¯cd¯, ❞t
ffrem tseI ❞t
❞t ✲
number of bosonic
indices in set I (a¯, b¯)
signals the need to solve the difference equation (5.2) to obtain the Q-operators on the grey node.
All Q-operators on the white nodes can then be obtained from the determinant formulas (5.5). The
lattice shown here is a projection of the one used in figure 1.
any off-diagonal terms, i.e. raising or lowering operators, can then be dropped since they
do not contribute to the supertrace. The normalised supertrace is then given in terms of
ordinary sums over these remaining diagonal terms, which however need to be regularised,
by giving the twist angles small imaginary parts, as discussed in section 2.2. Note that
the definition of the trace (2.15) factors into traces over the individual Fock spaces of the
different auxiliary space oscillators, str = Qa,b sctrab, where sctrab traces out the oscillator
c
(ξ¯ab, ξba). One finds that only a closed set of a few different types of sums can occur
when calculating the traces sctrab, including sums over rational functions and the Lerch
transcendent (3.12). Formulas for all these sums are collected in appendix A.2.
5.2
Operatorial Q-system from functional relations
Knowing the Q-operators with a single index as explicit matrices for a given magnon
block, one can produce explicit matrices for all operators of higher level by imposing the
QQ-relations (2.17) and (2.18).7 A naive way of solving the bosonic relation (2.17) however
involves a matrix inversion, which is problematic given that the Q-operators are expressed
in terms of special functions.
A more efficient strategy is to first calculate the Q-operators with one bosonic and one
fermionic index, Q{a,b} with |a| 6= |b|. To obtain these, we need to solve the first order
difference equation given by (2.18):
Q{a,b}(z) − Q{a,b}(z + 1) = −ΔabQ{a}
z +
{ }
z +
The formal solution to this equation can be written in terms of the discrete analogue of
integration, which we denote by Σ and define through Σ [f (z) − f (z + 1)] = f (z) + P.
7Using the approach we present here, recovering the known expression for Q∅ given in (2.20) constitutes
a non-trivial check of these relations, which we performed for specific examples.
Here P is periodic, P(z) = P(z + 1). The discrete integral can be written as a sum,
Σ[f (z)] = P∞
n=0 f (z + n), whenever this sum converges. For the Q-operators with one
bosonic and one fermionic index we can thus write
Q{a,b}(z) = −Δab Σ Q{a}
z +
Q b
{ }
z +
(5.3)
We describe the explicit realisation of this operation on the encountered basis of functions
in appendix A.4, where we also make it clear that all Q-operators are given in terms of
linear combinations of rational functions and generalised Lerch transcendents which are
likewise defined there. It is important to note that in contrast to the untwisted case, the
arbitrary periodic function P is fixed to be zero if we require the Q-operators obtained
from (5.3) to be identical to the monodromy construction, since P is incompatible with
the exponential scaling in terms of the twist phases.
Via the QQ-relations, it is possible to write all other Q-operators as determinants of
Q{a} and Q{a,b} with |a| 6= |b|,
Q{a1,...,am,b1,...,bn} = Q
Qim=1
Qjn=1 Δaibj
1≤i<j≤m Δaiaj Q
1≤i<j≤n Δbibj
(5.4)
×
(−1)(n−m)mǫk1,...,kn Qrm=1 Δarbkr
1
Q[{⋆a]r,bkr }
Qsn=−1m Q[{nb−kmm++s1}−2s] m < n
ǫk1,...,km Qrm=1 Q{akr ,br}
m = n ,
1
(−1)(n−m)nǫk1,...,km Qrn=1 Δakr br
Q[{⋆a]kr ,br}
Qsm=−1n Q[{mak−nn++s}1−2s] m > n
see e.g. [
10, 30, 40
]. Here |aj | = 0 and |bj | = 1, Q[n] = Q(z + n2 ), and ⋆ can take any value
in −|m − n|, −|m − n| + 2, . . . , |m − n| − 2, |m − n|. The prefactor is a consequence of the
normalisation we use for the Q-operators (2.13), cf. appendix C.
The procedure to construct all Q-operators in this way is shown in figure 2. As a
consequence of this construction, one finds that the Q-operators only develop poles at
z ∈ N or z ∈ N − 12 , depending on the number of indices.
6
The BMN vacuum of fully twisted N = 4 SYM at leading order
In this section we want to show how the Q-operator construction and the methods for their
evaluation can be applied to the N = 4 SYM spin chain at the one-loop level. To make
comparisons to other approaches easier, we also show how to convert our expressions to
the conventions commonly used in the literature on the Quantum Spectral Curve, see in
particular [41], where the twisted case is discussed. From our construction we obtain the
Q-operators for the theory with a full diagonal twist. This generalises the well-know γi and
β deformation [42–44] and includes twists of the space-time part of the symmetries, such
that the field theory is non-commutative.8 The results can be specialised to the γi and
β deformed cases, or to the untwisted theory by choosing the twist angles appropriately.
8See [45] for a discussion of the subtleties which arise when trying to deduce the precise non-commutative
field theory from the integrable spin chain description.
While this leads to divergent matrix elements in the Q-operators, their eigenstates and the
conserved charges such as the Hamiltonian which can be obtained from the Q-operators as
described in [34] remain finite.
To specialise our construction to N = 4 SYM at one-loop, we first restrict to the
singleton representation of u(2, 2|4) by choosing a grading and applying particle-hole
transformation as
(|a|)8a=1 = (0, 0, 1, 1, 1, 1, 0, 0) ,
(ωa)8a=1 = (+1, +1, −1, −1, −1, −1, −1, −1) ,
(6.1)
and requiring that the central charge vanishes, i.e. C = 0.
Comparing with (2.23),
this gives the representation of the fields of N = 4 SYM in terms of the oscillators
(a¯1, a¯2, d¯1, d¯2, d¯3, d¯4, b¯1, b¯2) typically used in the spin chain description of N = 4 SYM
at weak coupling and first investigated in [46]. With this choice, the representation has the
scalar field Z as the lowest-weight state:
|Zi = d¯1d¯2|0i = |0, 0, 1, 1, 0, 0, 0, 0i .
(6.2)
(6.3)
(6.4)
To facilitate the application of our results to N = 4 SYM and to make comparisons
with the literature easier, we note that our conventions can easily be transformed into
those typically employed by literature on the Quantum Spectral Curve of N = 4 SYM.
There, bosonic and fermionic indices are treated separately. To obtain Q-functions with
the expected asymptotics, we call the Q-operators of the lowest level
8
(Qa)a=1 = (Q∅|1, Q∅|2, Q1|∅,Q2|∅,Q3|∅,Q4|∅,Q∅|3,Q∅|4) .
We note that the eigenvalues of these operators correspond to the leading perturbative
contribution to the functions that appear in the Pµ and Qω systems of the Quantum
Spectral Curve, which govern the monodromy properties of the Q-system of N = 4 SYM
at any coupling [
29, 30
]. To obtain the twist variables which were used in [41], we set
(e−iφa )8a=1 = (τa)i=a = (y1, y2, x1, x2, x3, x4, y3, y4) .
8
Finally, the spectral parameter used in the QSC is related to the one used here by z+ 21 = iu,
and the Lerch transcendents are given in terms of so-called η functions, which in the twisted
case are defined by ηax(u) := P∞
xk
k=0 (u+ik)a . For the generalised Lerch transcendents see
appendix A.3. These conventions ensure that the Q-operators have poles at positions in
the spectral parameter plane which are expected from the Quantum Spectral Curve.
As a simple application of the formulas derived in this paper, and in order to give some
further examples of how they can be used in practise, we calculate the matrix elements
of the single-index Q-operators with the BMN vacuum tr Z
these states constitute their own “magnon blocks”, we directly obtain the corresponding
Q-functions in this case. We consider the matrix elements of the single-index R-operators
of the form hZ|R|Zi. These can be determined from the integral representation of the
diagonal part of the R-operators given in (4.9); equivalently one can use the finite sum
representation in (4.14) or (4.7). Further relevant formulas are given in appendix A.1
L of arbitrary length L. Since
where we describe the combinatorial structure arising from the oscillator algebra, see in
particular (A.5). For the matrix elements under consideration, there are in fact no
combinatorial factors and no signs. Since we look at matrix elements on the diagonal, there are
no auxiliary space operators, which means that mA = m˜A = mˆA in equation (A.5). Thus
we only have to evaluate the diagonal part given in (4.9), where all na¯ are zero.
We can now evaluate the integrals appearing in the diagonal part (4.9). The matrix
elements of the operators R{1}, . . . , R{6} are polynomials in the spectral parameter and
in the number operators in the auxiliary space, since all sums truncate. In this case the
integral in (4.9) is a contour integral which computes a residue, and can be evaluated by
using the series representations of hypergeometric functions 2F1. The operators R{7} and
R{8} have non-truncating sums and their matrix elements are rational functions of both the
spectral parameter as well as the auxiliary space operators. For them, the integral in (4.9)
has to be taken along the interval (0, 1); using 2F1
a a−b ; x
= (1−x)b and 2F1
0cb ; x
= 1,
one directly finds Beta integrals, which give these rational functions. Performing these
calculations one finds the following matrix elements:
(6.5)
(6.6)
(6.7)
(6.8)
hZ|R{1}|Zi = hZ|R{2}|Zi = hZ|R{3}|Zi = hZ|R{4}|Zi = 1 ,
hZ|R{5}|Zi = z +
hZ|R{6}|Zi = z +
hZ|R{7}|Zi =
hZ|R{8}|Zi =
1
2
1
2
+ N51 + N52 + N53 + N54 ,
+ N61 + N62 + N63 + N64 ,
1
1
z + 12 − N71 − N72 − N73 − N74
z + 12 − N81 − N82 − N83 − N84
,
.
We now calculate the actual Q-functions as
hZL|Q{a}(z)|ZLi = τa−(−1)|a|zsctr hZ|R{a}(z)|Zi
L ,
where the BMN vacuum state of length L is |ZLi = |Zi⊗L. All formulas that are needed
to evaluate the supertraces over the auxiliary Fock space are collected in appendix A.2,
and can directly be applied to the matrix elements under consideration. From (6.5) we
immediately see that
hZL|Q{a}(z)|ZLi = τa−(−1)|a|z ,
a = 1, 2, 3, 4 .
Using the multinomial theorem and the formula for the supertrace of polynomials in the
number operators given in (A.7) we find
hZL|Q{5}(z)|ZLi =
L
×τ5z Xzk
k=0
X
k0+k1+k2
+k3+k4=L−k
L
k,k0,k1,k2,k3,k4 2k0 τ5−τ1
τ5
τ2
τ5−τ2
Pk3
ℓ3=0 ℓ3
k3 τ5 ℓ3+1−δk3,0 Pk4
τ3
δk1,0−1
δk2,0−1
ℓ4=0 ℓ4
τ4
k4 τ5 ℓ4+1−δk4,0 #
1− ττ35 k3 1− ττ45 k4 ,
where we abbreviate the twist angles as τa = e−iφa . The Q-function hZ | { }
L Q 6 (z)|ZLi =
hZL|Q{5}(z)|ZLi|τ5→τ6 is obtained by a simple relabelling of the result for Q{5}. For the
non-rational Q-functions we can use (A.9) to first evaluate the fermionic traces; the first
bosonic trace generates Lerch transcendents according to (A.9). The last trace can then
be evaluated using (A.10), and the resulting expressions simplified via the identity (A.12).
For Q{7} we find
"
and
gives hZ | { }
L Q 8 (z)|ZLi
=
hZ |
L Q 7 (z)|Z
{ }
L
i|τ7→τ8 . Quite remarkably, the Q-functions for these most trivial states
of the theory are already rather complicated, due to the presence of the full twist. The
higher level Q-functions for the BMN vacuum can be generated from the ones given above
as described in section 5.2, using (5.3) and (5.5). We note that the calculations for excited
states are not much more difficult; the corresponding Q-operators for each magnon block
can likewise be evaluated using the formulas in appendix A.
7
Conclusions and outlook
In this article, we discussed the oscillator construction of the Baxter Q-operators of
integrable models for the case of non-compact super spin chains with representations of
Jordan-Schwinger form, focusing on the concrete evaluation of these operators. We
outlined the derivation of the Lax operators on which this construction is based, and defined
the Q-operators with their functional relations. For non-compact spin chains with
infinitedimensional state spaces, these Lax operators are given in terms of infinite sums which
hides the analytic properties of the resulting Q-system and complicates their evaluation.
We proposed a strategy to overcome these difficulties. For the Lax operators of the lowest
level, we derived a representation without infinite sums, which allows to compute explicit
matrix elements. Employing a small set of formulas for the normalised supertrace, it is
then possible to obtain the matrix elements of the corresponding Q-operators. Due to the
remaining symmetry, the Q-operators can be realised as finite matrices for each magnon
block, and the functional relations then allow to uniquely recover the entire Q-system
starting from the lowest level. For all the steps in this procedure we provided explicit formulas
which can directly be implemented in computer algebra systems for practical calculations.
Although our approach only relies on the Q-operators of the lowest level to determine
the whole Q-system, it would be desirable to find analogues of our integral formula (4.9)
also for the Lax operators of higher levels. Our initial studies in section 4.6 indicate that
this rather difficult task might require novel ideas. A promising route might be to derive
these formulas directly from the Yang-Baxter equation (2.7). Our approach naturally
incorporates compact spin chains with symmetric representations at each site of the quantum
space. It would furthermore be interesting to study whether it can be generalised to more
general representations and in particular to principal series representations. Furthermore,
it should be straightforward to apply our method to open spin chains, for which the study
of Baxter Q-operators was initiated only recently in [
47
]. The main motivation of our
work is to allow the application of Q-operators to concrete physical problems. Apart from
applications in high energy physics, we hope that our method can be applied in the context
of the ODE/IM correspondence [
48
] and the computation of correlation functions [
49–51
].
Currently our main focus lies on N = 4 SYM where a similar Q-system arises in the form
of the Quantum Spectral Curve. So far, the QSC of N = 4 SYM has only been investigated
on the eigenvalue level, and it is tempting to ask how it lifts to the operatorial level, see also
the discussion in [52]. The individual Q-functions are multivalued functions of the spectral
parameter with particular monodromies and asymptotic behaviour. These Q-functions
are believed to be eigenvalues of Q-operators, but the nature of the operatorial Q-system
remains a mystery. Our approach should be equivalent to the construction of the leading
perturbative contribution to this system. It is well-understood how to iteratively construct
perturbative corrections to the Q-functions [53, 54]. There is no immediate reason why
these methods should not lift to the operatorial level, even though the perturbative solution
of the QSC with general twists, where eigenstates possibly correspond to spin chain states
with non-zero momentum, has not yet been examined in detail in the literature. Thus a
systematic way of performing the untwisting would be of great practical value. On the
eigenvalue level, a rather general method was proposed in [41], but it has to be applied to
each state individually; a discussion on the operator level can be found in [23, 55] for the
case of the Heisenberg spin chain, see also [56]. Such computations would yield access to
perturbative information about the operatorial Q-system, which might give hints about its
deeper nature. Furthermore, diagonalisation of the Q-operators would immediately yield
higher loop corrections to the eigenstates of the dilatation operator. This information
might shed light on higher-point correlation functions about which many aspects are still
not fully understood and also on the emergence of the integrable system that underlies
N = 4 SYM from the field theory. We plan to address these questions in future work.
It would furthermore be interesting to apply our construction to N = 4 SYM in the
presence of defects [57, 58] where Q-operators have already been used to calculate one-point
functions, and to the integrable chiral field theories discovered recently [59].
We finally note that there are other approaches to the construction of Q-operators,
which superficially are quite distinct from the oscillator construction pursued in this work.
It would be interesting to see how results analogous to those presented here can be obtained
from the approach developed in [60, 61], and if the interesting Q-operator construction
in [62] can be generalised to non-compact spin chains.
Acknowledgments
We like to thank Matthias Staudacher, Dmytro Volin, Zengo Tsuboi, Gregory Korchemsky,
Gregor Richter, Leonard Zippelius, Ivan Kostov, Didina Serban, and Stijn van Tongeren
for interesting discussions. RF thanks Vasily Pestun for related discussions. We thank
the referees for useful remarks. Further we thank the IPhT, Saclay and “Mathematische
Physik von Raum, Zeit und Materie”, Humboldt University Berlin for hospitality. DM
received support from GK 1504 “Masse, Spektrum, Symmetrie”. RF is supported by the
IH E´S visitor program. The research leading to these results has received funding from the
People Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme FP7/2007- 2013/ under REA Grant Agreement No 317089 (GATIS).
A
A.1
Formulas for the explicit evaluation of Q-systems
Matrix elements of lowest level R-operators
Explicit matrix elements hm˜ |R{a}(z)|mi of the R-operators of the lowest level with the
states defined in (2.24) can be obtained using simple oscillator algebra from (4.6) with
the diagonal part M{a} given by the integral representation (4.9), or equivalently by the
finite sum formulas (4.14). First note that the values of the summation variables na¯, a¯ ∈ I¯
in (4.6) are fixed by the difference in occupation numbers
na¯ = −ωa¯( m˜a¯ − ma¯) ,
because each of the corresponding oscillators only appears in a single factor Xaa¯ and a
single factor Yaa¯ in (4.6). The powers of the oscillators with index a in the left and right
factors are
Nℓ =
X θ(na¯)|na¯|
a¯
and
Nr =
X θ(−na¯)|na¯| .
a¯
Then the occupation numbers of the state on which the diagonal part M{a} acts are given by
mˆa = m˜a − ωaNℓ = ma − ωaNr ,
mˆ a¯ =
(max(ma¯, m˜a¯) if ωa¯ = 1
min(ma¯, m˜a¯)
if ωa¯ = −1
.
(A.3)
(A.1)
(A.2)
(A.5)
For the different types of oscillators, given in (2.23), this sets the number operators in the
diagonal part M{a} to
Nc = −1 − b¯cbc → −1 − mˆc
a¯cac →
c¯ccc →
mˆc
mˆc
1 − d¯cdc →
1 − mˆc
|c| = 0 and ωc = +1
|c| = 0 and ωc = −1
|c| = 1 and ωc = +1
|c| = 1 and ωc = −1
.
(A.4)
Finally, collecting all combinatorial factors arising from the oscillator algebra, we can
write the matrix elements as
hm˜ |R{a}(z)|mi
= ξ¯θa(1−ω1(m˜ 1−m1))|−ω1(m˜ 1−m1)| ···ξaK
¯θ(−ωK(m˜ K−mK))|−ωK( m˜K−mK)|
×(−1)Pa¯ |na¯|caa¯
mˆa!
√ m˜a!ma!
−ωa Y
a¯
min(ma¯,m˜ a¯)!
s max(ma¯, m˜a¯)! M{a}(z,{ mˆ},{−ωa¯(m˜ a¯ −ma¯)})
×ξθK(ωaK(m˜ K−mK))|−ωK(m˜ K−mK)| ···ξθ1(aω1(m˜ 1−m1))|−ω1(m˜ 1−m1)| ,
where K = p + q + r + s for u(p, q|r + s) models and the sign is determined by
caa¯ = h(|a|+|a||a¯|)θ(na¯)+(1+|a||a¯|)θ(−na¯) +
i 12 h(|a|+1)(1−ωa)θ(na¯)+(|a¯|+1)(1−ωa¯)θ(−na¯)
i
+ |a¯|
X
1≤c<a¯
K
c=1
+ (|a|+|a¯|)X|c| m˜cθ(na¯)+mcθ(−na¯)
|c| (θ(nc)+m˜ c)θ(na¯)+(θ(−nc)+mc)θ(−na¯) ,
# "
+ |a|
X
1≤c<a
#
|c| m˜cθ(na¯)+mcθ(−na¯)+δca¯
where we set na = 0.
A.2
Calculating supertraces
For the Q-operators of the lowest level, the following formulas are sufficient to perform the
occurring sums. First, for polynomials in the number operators, we can use
sctrab Nkab =
Pkn=0 hnki ττa n+1−δk,0
τa−τb
τa
1− ττbab k
1−δk,0
bosonic
fermionic
,
where nk are the Eulerian numbers defined by
k
n
X(−1)j k + 1
j
(n − j + 1)k .
Here and in the following we abbreviate the twist angles via τa = exp(−iφa).
For the non-rational Q-operators, we also need the Lerch transcendent Φ defined
in (3.12), since the matrix elements of the R-operators are rational functions of the number
operators. Concretely one encounters traces of the form
If further traces have to be evaluated, summation formulas for the Lerch transcendent are
sctrab (Nab + r)ℓ = τbτ−bτa Pk
Nkab
τb−τa δk,0τb r1ℓ − τa (r+1)ℓ
1 1
m=0 mk (−r)k−mΦℓτ−a/mτb (r)
sctrab Φℓτ (Nab + r) = τττbb−−ττaa τ Φℓτ (r) − ττab Φℓτa/τb (r)
τb
τb−τa τbΦℓτ (r) − τaΦℓτ (r + 1)
and in the general case
sctrab NkabΦℓτ (Nab + r)
=
τb−τa
τb
1
τb−τa τbδk,0Φℓτ (r) − τaΦℓτ (r + 1)
k
δk,0+P1tk−=1τahτ/tτ−b1ik+1
τa/τb t
τ
Φℓτ (r) − τ
1 Psk=0 s
Pjs=0 sj (1 − r)s−j Φℓτ−a/jτb (r − 1)
1
− δs,0 (r−1)ℓ
k−s
k δs,k+Ptk=−1s ht−1i
1− τaτ/τb
)
bosonic
fermionic
.
bosonic
fermionic
,
τa/τb t
τ
k−s+1
bosonic
fermionic
#
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
HJEP09(217)8
It is evident that these types of supertraces are the only ones which can appear.
This implies that all matrix elements of the lowest level Q-operators are either rational
functions for truncating R-operators or can be written in terms of the Lerch transcendent
for non-truncating R-operators. We note that one can usually reduce the number of terms
containing this function using the identity
Φτk(z + 1) =
1
Φτk(z) − zk
1
.
A.3
Generalised Lerch transcendents
When solving difference equations in order to obtain the Q-operators with one bosonic
and one fermionic index, cf. section 5.2, we need to apply the discrete integration Σ. It
can be realised as Σ[f (z)] = P∞
k=0 f (z + k) when the sum is convergent. For non-rational
Q-operators, these sums lead to generalisations of the Lerch transcendent (3.12). The
treatment here is equivalent to that of η-functions given in [53, 63].
HJEP09(217)8
X
τ1k1 τ2k2 · · · τnkn
(z + k1)a1 (z + k2)a2 · · · (z + kn)an ,
where the number of parameters n is arbitrary. Note that while the variables τi will be
given in terms of the twist variables in concrete calculations, here their indices i are not
gl(N |M ) indices, but simply label these arbitrary arguments.
It is clear from the definition that the generalised Lerch transcendent satisfies the
following shift identity, generalising equation (A.12):
Φτa,,τa11,,......,,τann (z) =
τ1 · · · τn Φτa11,,......,,τann (z + 1) + τ τ1 · · · τnΦτa,,τa11,,......,,τann (z + 1) .
Importantly, Φ-functions satisfy so called stuffle-relations, e.g.
Φτa11 Φτa22 = Φτa11,,τa22 + Φτa11τ+2a2 + Φτa22,,τa11 .
These can be used to linearise all products of these functions. The Lerch transcendents are
related to η-functions used in the Quantum Spectral Curve literature by
Φτa11,,τa22,,......,,τann (z) = ia1+a2+...+an ηaτ11,,τa22,,......,,τann (iz) .
A.4
Formulas for discrete integrals
To generate the full Q-system, we need to apply the discrete integration Σ to the following
four classes of functions, which form a closed set.
Polynomials. In the case τ 6= 1, Σ(τ zza) is another polynomial with an overall
exponential factor of the form p(z) = τ z(caza + . . . + c0) satisfying the constraint p(z) − p(z + 1) =
τ zza. This constraint fixes p(z) completely. In the case τ = 1, the polynomial is of degree
a + 1 instead.
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
Shifted inverse powers. From the definition of the generalised Lerch transcendent one
finds
Terms of the form (zτ+mΦ)a .
z
Σ
z
(z + m)a
Note that
= τ zΦτa(z + m) .
Σ
τ zΦτa11,,τa22,,......,,τann (z + 1)
= τ zΦτa,,τa11,,τa22,,......,,τann (z) .
(z+m)a
To evaluate Σ
h τzΦτa11,,τa22,,......,,τann i, use (A.14) to align the shifts and then use (A.18).
Terms of the form τ zzaΦ. To evaluate products of monomials and generalised Lerch
transcendents, one can use the finite difference analogue of partial integration,
.
For Σ[τ zzaΦτa11,,τa22,,......,,τann ], set f = (τ1 · · · τn)zΦτa11,,τa22,,......,,τann and g − g[2] =
be used recursively until no terms of this type are present.
Highest level R-operators and their matrix elements
In section 4.3 we derived a representation of R-operators for the lowest level of the
Qsystem which allows to evaluate their matrix elements. Here we summarise similar results
for the R-operators of the highest non-trivial level, i.e. those where the index set I contains
all but one single index which we denote by a¯.
According to equation (4.4) we can write these R-operators as
R{a¯}(z) =
∞
X
Y(Yaa¯)θ(+na)|na| M
"
a
{a¯}(z, {N}, {n})
Y(−Xaa¯)θ(−na)|na| , (B.1)
#
where X and Y are given in (4.1). After performing an almost identical computation as
for the R-operators with a single index, the diagonal part can be written as
(A.17)
(A.18)
(A.19)
z za. This can
M
{a¯}(z, {N}, {n}) = − Γ(z + 1 − c − 21 (−1)|a¯|)
Γ(z + 1 + 12 (−1)|a¯|)
dt (−t)Na¯+(−1)|a¯|−1(1 − t)−z−1− 12 (−1)|a¯| Y
(B.2)
1
a |na|!
2F1
1 + |na|
−Na − Naa¯ ; (−1)|a¯|+|a|t .
Here the integration again depends on whether the oscillator with index a¯ is particle-hole
transformed or not,
1
Z
dt = Γ(Na¯ + (−1)|a¯|) 0
Z 1
dt
(−1)Na¯+1 Γ(1 − Na¯ − (−1)|a¯|) I
2πi
t=0
dt
else
if |a| = 0 and ωa = +1
(B.3)
We can easily obtain matrix elements hm˜ |R{a¯}(z)|mi from this representation. First
we note that occupation numbers m˜ and m fix the values of the summation variables na,
a ∈ I in (B.1) to na = ωa( m˜a − ma). The powers of the a¯ oscillators in the left and right
factors are Nℓ = P
a θ(na)|na| and Nr = Pa θ(−na)|na|, such that the occupation numbers
of the states on which the diagonal part acts are
mˆ a¯ = m˜a¯ + ωa¯Nℓ = ma¯ + ωa¯Nr
mˆa =
(min(ma, m˜a)
if ωa = 1
max(ma, m˜a) if ωa = −1
.
(B.4)
We can thus write the matrix elements as
HJEP09(217)8
= ξ¯θ1(a¯ω1(m˜ 1−m1))|ω1(m˜ 1−m1)|
· · · ξ¯θK(ωa¯K(m˜ K−mK))|ωK( m˜K−mK)|
× (−1)Pa |na|c′aa¯
√ m˜a¯!ma¯!
ωa¯ Y
min(ma, m˜a)!
s max(ma, m˜a)! M(z, { mˆ}, {ωa( m˜a − ma)})
× ξθa¯(K−ωK(m˜ K−mK))|ωK(m˜ K−mK)|
· · · ξθa¯(1−ω1(m˜ 1−m1)|ω1(m˜ 1−m1)| ,
where K = p + q + r + s for u(p, q|r + s) models and the sign follows from
c′aa¯ = h(|a|+|a||a¯|)θ(na)+(1+|a||a¯|)θ(−na) +
i 12 h(|a|+1)(1−ωa)θ(na)+(|a¯|+1)(1−ωa¯)θ(−na)
i
+ |a|
X
1≤c<a
K
c=1
+ (|a|+|a¯|)X|c| m˜cθ(na)+mcθ(−na)
|c| (θ(nc)+m˜ c)θ(na)+(θ(−nc)+mc)θ(−na) .
# "
X
|c| m˜cθ(na)+mcθ(−na)+δca
#
(B.5)
#
(B.6)
C
Normalisations of Q-operators and functional relations
There are different conventions for the functional relations of Q-operators and twisted
Qfunctions which have been used in the literature. These are related to the normalisation
of the Q-operators. To facilitate comparisons to other works, we summarise and compare
these conventions in this appendix. The twists will be parametrised by τa = e−iφa .
In this work, we use the normalisation which is typically employed in the literature
on the oscillator construction of Q-operators, see e.g. [25, 26]. The operators are defined
in (2.13), and the prefactor there can be written as Q
. Note that this
normalisation is compatible with indexing the operators by sets (which is natural for the oscillator
relations are given in (2.17) and (2.18), and involve the factors Δab = (−1)|a| √τbτ−aττab .
construction), since it does not impose an ordering of the gl(N |M ) indices. The functional
Other possible choices induce an ordering of the gl(N |M ) indices. This ordering can
be reflected by indexing the Q-operators with antisymmetric multi-indices; we will instead
label the Q-operators with sets, and keep track of the ordering in the functional relations.
a∈I τa−(−1)|a|z
One possibility, discussed on the level of Q-functions for example in [41], uses a
normalisation without exponential scaling factors:
Qˆ I (z) =
Y (τa − τb)(−1)|a|+|b|
sctr MI (z) ,
a,b∈I
a<b
which gives functional relations
QI∪{a,b}Qˆ I = τ Qˆ +
ˆ
I∪{a}Qˆ −
I∪{b} − τ˜Qˆ I−∪{a}Qˆ +
I∪{b} |a| = |b|
QI∪{a}Qˆ I∪{b} = τ Qˆ +
ˆ
I∪{a,b}Qˆ I− − τ˜Qˆ I−∪{a,b}Qˆ I−
|a| 6= |b|
,
(C.2)
where τ = τa and τ˜ = τb if a < b or τ = τb and τ˜ = τa if b < a, and we used the notation
f ± = f (z ± 21 ).
Another normalisation we want to discuss is given by
Qˇ I (z) =
Y τ (−1)|a|(z+sI ) Y (τa − τb)(−1)|a|+|b|
a
sctr MI (z) .
a∈I
a,b∈I
a<b
Here the shift sI is the one defined in (2.5). The functional relations for these operators
are identical to those of untwisted Q-functions:
QI∪{a,b}Qˇ I = Qˇ +
ˇ
I∪{a}Qˇ −
I∪{b} − Qˇ −
I∪{a}Qˇ +
I∪{b} |a| = |b|
QI∪{a}Qˇ I∪{b} = Qˇ +
ˇ
I∪{a,b}Qˇ I− − Qˇ −
I∪{a,b}Qˇ I−
|a| 6= |b|
.
(C.1)
(C.3)
(C.4)
In the first equation, we have to assume that a < b if |a| = |b| = 0, or b < a if |a| = |b| = 1;
otherwise, the left hand side changes its sign.
Open Access.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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