# T-system on T-hook: Grassmannian solution and twisted Quantum Spectral Curve

Vladimir Kazakov, Sébastien Leurent, Dmytro Volin

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Vladimir Kazakov, Sébastien Leurent, Dmytro Volin. T-system on T-hook: Grassmannian solution and twisted Quantum Spectral Curve. Journal of High Energy Physics, 2016, 12, pp.044. 10.1007/JHEP12(2016)044. hal-01522682

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# T-system on T-hook: Grassmannian solution and twisted Quantum Spectral Curve

Vladimir Kazakov,ª,ᵇ Sébastien Leurentᶜ and Dmytro Volinᵈ,ᵉ

ªLPT, École Normale Superieure,
24, rue Lhomond 75005 Paris, France

ᵇUniversité Paris-VI,
Place Jussieu, 75005 Paris, France

ᶜInstitut de Mathématiques de Bourgogne, UMR 5584 du CNRS, Univ. Bourgogne Franche-Comté,
9 avenue Alain Savary, 21000 Dijon, France

ᵈSchool of Mathematics, Trinity College Dublin,
College Green, Dublin 2, Ireland

ᵉNordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

E-mail: kazakov@lpt.ens.fr, sebastien.leurent@normalesup.org,
vel145@gmail.com

**ABSTRACT:** We propose an efficient grassmannian formalism for solution of bi-linear finite-difference Hirota equation (T-system) on T-shaped lattices related to the space of highest weight representations of gl(K₁, K₂|M) superalgebra. The formalism is inspired by the quantum fusion procedure known from the integrable spin chains and is based on exterior forms of Baxter-like Q-functions. We find a few new interesting relations among the exterior forms of Q-functions and reproduce, using our new formalism, the Wronskian determinant solutions of Hirota equations known in the literature. Then we generalize this construction to the twisted Q-functions and demonstrate the subtleties of untwisting procedure on the examples of rational quantum spin chains with twisted boundary conditions. Using these observations, we generalize the recently discovered, in our paper with N. Gromov, AdS/CFT Quantum Spectral Curve for exact planar spectrum of AdS/CFT duality to the case of arbitrary Cartan twisting of AdS₅×S⁵ string sigma model. Finally, we successfully probe this formalism by reproducing the energy of gamma-twisted BMN vacuum at single-wrapping orders of weak coupling expansion.

**KEYWORDS:** AdS-CFT Correspondence, Bethe Ansatz, Integrable Field Theories, Super-symmetric gauge theory

ARXIV ePRINT: 1510.02100
# Contents

1 Introduction 1

2 Algebraic properties of Q-system and solution of Hirota equations 5

2.1 Hirota equation in historical perspective 6

2.2 Notations 9

2.3 QQ-relations and flags of $C^N$ 11

2.4 Hodge duality map 13

2.5 Symmetry transformations on Q-systems 13

2.5.1 Rescalings (gauge transformations) 14

2.5.2 Rotations 14

2.6 Solution of Hirota equation on a strip 15

2.6.1 Gauge symmetry of the Hirota equation 16

2.6.2 Proof A: existence of solutions to Hirota equation 17

2.6.3 Proof B: uniqueness of the solution to Hirota equation 17

2.7 On finite-difference (Baxter) equation and Bäcklund transforms 19

2.8 Solution of Hirota equation on an L- or T-shaped lattice 23

2.8.1 Bijection between supersymmetric and non-supersymmetric Q-systems 23

2.8.2 QQ-relations with a grading 25

2.8.3 Expression for T-functions in a T-hook 28

2.8.4 Symmetries 29

2.8.5 Bäcklund flow in supersymmetric case 30

3 Polynomiality and twist 32

3.1 Polynomiality and spin chains 32

3.2 Twisted spin chains and Q-system 35

3.2.1 Twist as an exponential prefactor 35

3.2.2 Twist as a holomorphic connection 36

3.3 Dependence on twist and the untwisting limit: illustration on examples 38

3.3.1 $su(2)$: untwisting should be supplemented with a rotation 39

3.3.2 $su(2)$: analytic continuation in twist meets representation theory 42

3.3.3 $gl(1|1)$: lowest weight depends on a nesting path 44

3.3.4 $su(2|1)$: states in short representations involve zero Q-functions 45

3.3.5 $sl(2)$: non-compactness leads to Stokes phenomena 51

3.4 Dependence on twist and the untwisting limit: general picture 55

4 Twisted Quantum Spectral Curve 58

4.1 Twisting of Quantum Spectral Curve 60

4.2 Twisted $P_\mu$ and $Q_\omega$ systems 63

4.3 Asymptotics of Q-functions for full and partial twistings 64

4.4 Particular cases of twisting 67
4.4.1 Leading asymptotics for fully twisted case 67

4.4.2 $\gamma$-deformation 68

4.4.3 $\beta$-deformation 69

5 BMN vacuum in gamma-deformed case, weak coupling expansion 70

5.1 Input data, notations, and symmetries 70

5.2 Asymptotic $P\mu$-system 73

5.3 Asymptotic Q-system 76

5.4 Asymptotic T-system and energy 79

6 Conclusion 81

A Further details and proofs 83

A.1 Derivation of (2.21) via Plücker identities 83

A.2 Q-functions for the semi-infinite strip 83

A.3 Index splitting QQ-relations 84

A.4 Example of a non-Wronskian solution to the Hirota equation 87

A.5 Derivation of the Baxter equation in the form (2.62) 88

A.6 Proofs of [derivative] QQ-relations in a supersymmetric Q-system 88

A.7 Proof of the Wronskian solution of Hirota on the T-hook 91

B More details of the zero-twist limit in spin chains 94

B.1 Large Bethe roots and zeros of Laguerre polynomials 94

B.2 Construction of the rotation 95

B.3 Rational spin chain's Q-operators 97

B.4 L-hook reduction in the case of short multiplets 98

C Twisted asymptotics and weight of the representation 101

D Leading QSC asymptotics for some particular cases of twisting 105

D.1 Computer implementation of leading asymptotics of QSC Q-functions 106

D.1.1 Sagemath implementation 107

D.1.2 Mathematica implementation 108

D.2 Fully untwisted case 109

D.3 The case $x_1 = y_4$ 109

D.4 The case $x_1 = x_2$ 110

D.5 Twisted $sl_2$-sector 110

E Links to TBA: mirror Q- T- and Y-functions and the energy 111

E.1 Formulae for T-functions of mirror T-hook 112

E.2 Derivation of TBA formula for energy from QSC 112
# 1 Introduction

In 1931, Hans Bethe analysed the very first example of a quantum integrable model — Heisenberg SU(2) XXX spin chain — and showed that it can be reduced to algebraic equations which now bear his name [1]. The roots of these equations, called Bethe roots, enter the observable quantities only through their symmetric combinations. This is one of many reasons to work with the Baxter Q-polynomial — a polynomial with zeros at Bethe roots, $Q(u) = \prod_{k=1}^{L}(u - u_k)$. Later, several different techniques have been developed to determine $Q(u)$. For instance, instead of the Bethe equations one can use the Baxter equation

$$ \phi \left( u + \frac{i}{2} \right) Q(u + i) + \phi \left( u - \frac{i}{2} \right) Q(u - i) = T(u) Q(u), \quad \phi(u) = u^L, \quad (1.1) $$

and search for such solutions that $Q(u)$ and $T(u)$ are both polynomials.

Another reformulation of the same problem is to demand the Wronskian identity

$$ W \equiv \frac{1}{\phi(u)} \left| \begin{array}{cc} Q_1(u + \frac{i}{2}) & Q_1(u - \frac{i}{2}) \\ Q_2(u + \frac{i}{2}) & Q_2(u - \frac{i}{2}) \end{array} \right| = 1 \quad (1.2) $$

to be satisfied. Indeed, it is easy to show that for any two solutions $Q_1(u)$, $Q_2(u)$ of the Baxter equation the Wronskian combination $W$ is an $i$-periodic function. We can further normalize the solutions so as to put $W = 1$, resulting in (1.2). Then it is enough to demand that both $Q_1$ and $Q_2$ solving (1.2) are polynomials to get solutions of (1.1) describing the physical spectrum of the spin chain. On this example we see that there are actually two Q-functions appearing.

The present paper introduces an interpretation of the Wronskian condition (1.2) in a natural geometric way. Consider $\mathbb{C}^2$ and denote by $\zeta_1$ and $\zeta_2$ two basis vectors in it. A one-form is a (u-dependant) linear combination of the basis vectors, i.e. a function of the form

$$ Q_{(1)}(u) \equiv Q_1(u)\zeta_1 + Q_2(u)\zeta_2, \quad (1.3) $$

where $Q_1$ and $Q_2$ are a priori arbitrary functions. Any one-form is associated to a $u$-dependent embedding $V_{(1)}(u) : \lambda \mapsto \lambda Q_{(1)}(u)$ of the complex line $\mathbb{C}$ into $\mathbb{C}^2$. We will also denote its image as $V_{(1)}(u) \equiv \{\lambda Q_{(1)}(u) | \lambda \in \mathbb{C}\} \subset \mathbb{C}^2$, which can be characterized as the set of points $\mathbf{x}$ satisfying<fnref n="1" /> $Q_{(1)} \wedge \mathbf{x} = 0$. The coordinates $Q_i$ of the form $Q_{(1)}$ are then called the *Plücker coordinates* of the line $V_{(1)}$.

The Wronskian condition (1.2) can be written as

$$ Q_{(1)} \left( u + \frac{i}{2} \right) \wedge Q_{(1)} \left( u - \frac{i}{2} \right) = \phi(u) \zeta_1 \wedge \zeta_2. \quad (1.4) $$

First, it implies that the lines $V_{(1)}(u + \frac{i}{2})$ and $V_{(1)}(u - \frac{i}{2})$ are not collinear — except when $u$ is a zero of $\phi$. Second, it describes an XXX spin chain if we demand that the embedding is polynomial (i.e. realized by Plücker coordinates being polynomial functions of $u$) and, as a consequence, $\phi(u)$ in (1.4) is a polynomial which we denote as $\phi(u) = \prod_{k=1}^{L}(u - \theta_k)$. Homogeneous spin chains considered in (1.1), correspond to $\theta_j = 0, \forall j$.

<footnote n="1">¹We denote by the wedge symbol an arbitrary bilinear antisymmetric product such that $\zeta_1 \wedge \zeta_2 \neq 0$. Consequently, one has $\mathbf{x} \wedge \mathbf{y} = \det(\mathbf{x}, \mathbf{y}) \zeta_1 \wedge \zeta_2$.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1. Q-functions define a fibration of grassmannians over the Riemann surface of the spectral parameter $u$. Relation between grassmannians of different rank is restricted by (1.5).</figcaption></figure>

We are ready to establish the following map: to each polynomial embedding $V_{(1)}(u)$, such that $V_{(1)}(u + \frac{i}{2}) \cap V_{(1)}(u - \frac{i}{2}) = \{0\}$ should correspond an eigenstate of the SU(2) XXX spin chain of length $L$ in the fundamental representation with inhomogeneities $\theta_1, \theta_2, \dots, \theta_L$. The correspondence is established after factoring out elementary symmetry transformations as it will be described in the text: on the one hand, eigenstates belonging to the same SU(2) multiplet correspond to the same embedding. And on the other hand, embeddings related by global $GL_2$ transformations should not be distinguished — they correspond to the same eigenstate.

In this way, we reformulated the solution of XXX spin chain in a geometric fashion. This point of view can be generalised to integrable systems with a higher rank symmetry algebras of gl type as follows. Let $V_{(n)}(u)$ be a $u$-dependent $n$-dimensional linear subspace of $\mathbb{C}^N$, i.e. we associate to $u \in \mathbb{C}$ an element $V_{(n)}(u)$ of the Grassmannian $\mathbf{G}_N^n$ — the set of all $n$-dimensional linear subspaces of $\mathbb{C}^N$. Consider a collection $V_{(0)}(u), V_{(1)}(u), \dots, V_{(N)}(u)$ of linear subspaces of increasing dimensions, and demand the property

$$V_{(n)}(u + i/2) \cap V_{(n)}(u - i/2) = V_{(n-1)}(u), \quad \forall n \in \{1, 2, \dots, N-1\} \quad (1.5)$$

to hold for any $u$ save a discrete number of points, see figure 1. In principle, equation (1.5) is equivalent to the existence of $N$ linearly independent functions $Q_i$, as we will see in section 2. However, the geometric formulation (1.5) provides us with a useful intuition and prepares us to introduce a richer set of functions: the Q-system.

We will advocate in this article that solving equation (1.5) supplemented with appropriate analytic constraints is equivalent to finding the spectrum of certain integrable models. For the case of compact rational spin chains equation (1.5) is an analog of fusion procedure and the analytic constraints are reduced to the demand that Q-functions, which are defined as Plücker coordinates for $V_{(n)}$, are polynomials in $u$. However, this example is not unique. Equation (1.5) appears to be generic and applies to many quantum integrable systems, including (1+1)-dimensional QFT's, with gl(N) symmetry or gl(k|N-k) super-symmetry, or even for non-compact (super)algebras su(K₁, K₂|M). It is closely re-
lated to the fact that the transfer-matrices and their eigenvalues, such as the T-function of eq. (1.1), satisfy the so-called Hirota bi-linear finite-difference equation (2.1) which, as we will see later, can be solved in terms of Wronskian expressions through a finite number of Q-functions. The Q-functions are not obliged to be polynomials, as it is the case in integrable non-compact spin chains and (1+1)-dimensional QFT's. Moreover, there are situations when an approach similar to the coordinate or algebraic Bethe ansatz is not known, and yet the equation (1.5) holds.

Moreover, the equation (1.5) is also central to the spectral problem of integrable two-
dimensional quantum field theories, and in particular sigma-models. It even allows for
a concise and efficient description for exact spectrum of energies (anomalous dimensions)
of AdS<sub>5</sub>/CFT<sub>4</sub> duality. It is because the quantum spectral curve (QSC) of the model,
describing the dynamics of quantum conservation laws, is most adequately formulated in
terms of the Q-system based on the equation (1.5) and the psu(2,2|4) superconformal
symmetry algebra [2, 3] related to it.

Since (1.5) is such a generic equation, expected to appear in virtually all quantum
integrable models, its properties deserve to be studied in detail, which is one of the main
goals of this paper.

One should always bear in mind that Q-functions are a way to introduce a *coordi-nate* system, hence they are not uniquely defined. For instance, we can replace $Q_2 \to Q_2 + \text{const} \times Q_1$ without any consequence for the Wronskian condition (1.2), and the pos-sible linear transformations are not exhausted by this example. In addition, the overall rescaling of all Q-functions by any function of $u$ does not affect the embeddings $V_{(n)}$. In section 2 we will construct the T-functions as determinants of Q-functions; $T(u)$ in the Baxter equation (1.1) is one of them: $T(u)\zeta_1 \wedge \zeta_2 = Q_{(1)}(u+i) \wedge Q_{(1)}(u-i)$. T-functions should be thought as certain volume elements in $\mathbb{C}^N$, i.e. they are represented by a full form. They are invariant under rotation of the basis but still transform under rescalings. The fully invariant objects are Y-functions which are certain ratios of T-functions. Al-though the description in terms of Y's is a more invariant way to parameterise the system, the description in terms of $V_n(u)$ has an important advantage since usually the analytic properties of Q-functions, directly related to T- and Y-functions by Wronskian solutions, are significantly simpler than the ones of T's or Y's.

In this article we discuss the following applications of the proposed approach. In section 2.6 we show how the Hirota equation (T-system) for integrable systems with gl(N) type of symmetry is solved in terms of Q-functions and also discuss how the Wronskian-type formulation (1.5) is related to higher-rank Baxter equations. This is a quite well established topic in the literature, in particular its geometric interpretation can be easily spotted from discussion in [4]. We include it into the paper as a simple example which contains the guiding lines useful for the further generalizations to supergroups and noncompact representations.

Then, in section 2.8, we generalise the gl(N) solution and show how to get from our
formalism the generic Wronskian solution of Hirota equation with the boundary condi-
tions of the “T-hook” type, describing the weight space of highest weight non-compact
representations appearing in integrable models with su(K₁, K₂|M) symmetry. Note that
the T-hook itself was first proposed as a formulation of AdS$_5$/CFT$_4$ Y-system [5] with superconformal psu(2, 2|4) symmetry. The generic symmetry algebra su(K$_1$, K$_2$|M) also includes two interesting particular cases: the compact supersymmetric algebra su(K|M) and the non-compact one su(K$_1$, K$_2$), the latter should be relevant for Toda-like systems. We emphasize here a remarkable fact that the supersymmetric generalization still relies on the same equation (1.5), with N = K$_1$ + K$_2$ + M. However, a convenient way to properly treat it is to choose a subspace C$^M$ in C$^N$ and work with Q-functions in specially re-labeled Grassmannian coordinates obtained by a Hodge-duality transformation in C$^M$. We propose a nice interpretation of this re-labeling as a specific rotation of (K + N)-dimensional hypercube representing the so called Hasse diagram of bosonic u(K + N) algebra.<fnref n="2" />

The Wronskian solution of Hirota equations on “T-hook” was given for the most in-
teresting case of psu(2, 2|4) symmetry by Gromov, two of the authors, and Tsuboi in [6],
and then it was presented for the generic case in the work of Tsuboi [7]. We believe that
the formalism of exterior forms developed here presents these results in a much more con-
cise and geometrically transparent way. We also establish several interesting new relations
among the Q-functions, especially elegantly written in terms of the exterior forms. Some
of them have been extensively used in the study of the Q-system emerging in AdS/CFT
integrability case [2, 3].

In section 3 we discuss how the construction can be amended to include the case of
integrable spin chains with twisted boundary conditions. It happens in a very natural way:
one should gauge the global rotational GL(N) symmetry w.r.t. the space of spectral param-
eter, making it local and hence introducing a new object: a holomorphic connection A. The
non-local relation (1.5) is modified by inserting a parallel transport P exp $\left[ \int_{u-\frac{1}{2}}^{u+\frac{1}{2}} A(u')du' \right]$
of the plane $V_{(n)}(u - i/2)$, so that the intersection in (1.5) naturally happens at the same
point (see figure 2). This parallel transport precisely realizes the twisting.

The new properties emerging in the twisted case are thoroughly studied, mainly on the
examples of rational spin chains. Especial attention is paid to the untwisting limit which
is singular and quite non-trivial. In particular, we give a detailed description how relation
between the asymptotics at infinity and the representation theory depends on the presence
or absence of particular twists.

The Wronskian solution of Hirota equation (2.1) in the case of super-conformal algebra su(2, 2|4) and the grassmannian structure of the underlying Q-system have played an important role in the discovery of the most advanced version of equations for the exact spectrum of anomalous dimensions in planar $N = 4$ SYM theory — the quantum spectral curve (QSC) [2, 3]. In fact, many of the relations discussed and re-derived in the present paper using the formalism of exterior forms have been already present in [3] in the coordinate form. As an interesting generalization of the QSC, we will present in section 4 its twisted version, in the presence of all (3+3) angles describing the gamma deformation and a non-commutativity deformation of the original $N = 4$ SYM theory [8–10]. The corresponding $\mathbf{P} - \mu$ and $\mathbf{Q} - \omega$ equations of [2, 3], as well as all Plücker QQ-relations,

<footnote n="2">²Each of all 2$^{M+N}$ multi-index Q-functions corresponds to a vertex of the hypercube and different Q-functions are related by Plücker relations on each 2-dimensional edge.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Deformation of the fibration by introducing a connection. This connection “rotates” the spaces $V_{(n)}$ via the parallel transport from point $u \pm \frac{i}{2}$ to point $u$ where the equation (1.5) can be used.</figcaption></figure>

will be essentially unchanged and the whole difference with the untwisted case will reside in the large $u$ asymptotics of Q-functions with respect to the spectral parameter $u$, which are modified due to the presence of twists by certain exponential factors. This is the only change in the analytic properties of QSC due to the twisting. The algebraic part of the twisted QSC formulation will be simply a particular $(2|4|2)$ case of the twisted version of the general $(K_1|M|K_2)$ Q-system presented in this paper. One can also note that a similar twisted version of the QSC was already introduced in [11] for a specific choice of boundary conditions introduced in [12, 13] at the level of the TBA (see also [14] which appeared simultaneously to the present paper).

Finally, in section 5 we probe our conjectures for twisted QSC on an interesting case of $\gamma$-deformed BMN vacuum of this AdS/CFT duality. For a particular case, $\beta$-deformation, the Y-system and T-system for the twisted case were formulated and tested in [15, 16] (see also [17, 18] at the level of the S-matrix). We reproduce by our method the one-wrapping terms in the energy of this state, known by direct solution of TBA equations [19, 20], which was also known by the direct perturbation theory computation [21]. A potential advantage of our method is the possibility to find the next corrections to this state on a regular basis, by the methods similar to [22, 23] as well as application of the efficient numerical procedure of [24], but this is beyond the scope of the current paper.

## 2 Algebraic properties of Q-system and solution of Hirota equations

In this section we show how the Q-system is used to solve Hirota equations on $(K_1|M|K_2)$ T-hooks. We also establish notations and algebraic properties of the Q-system. Although this solution was already presented in the literature [7], we take a look on it from a different, more geometric point of view, and we believe it will be a useful contribution to the subject as the technicality of the involved formulae is significantly reduced and the solution itself is made more transparent.
## 2.1 Hirota equation in historical perspective

The bi-linear discrete Hirota equation, sometimes also called the Hirota-Miwa equation [25–27]

$$T_{a,s}\left(u + \frac{\dot{i}}{2}\right) T_{a,s}\left(u - \frac{\dot{i}}{2}\right) = T_{a+1,s}(u) T_{a-1,s}(u) + T_{a,s+1}(u) T_{a,s-1}(u) \quad (2.1)$$

appears in numerous quantum and classical integrable systems. In these notations, typically used in the context of quantum integrable spin chains and sigma-models, $T_{a,s}(u)$ are complex-valued functions of two integer indices $a$ and $s$ parameterising a $\mathbb{Z}^2$ lattice, and of a parameter $u \in \mathbb{C}$ usually called spectral parameter. Although the parameter $u$ enters the equation only with discrete shifts and hence can be treated as another discrete variable, the analytic dependence of $T_{a,s}$ on $u$ is an important piece of information used to specify the physical model. We will exploit this analytic dependence starting from section 3.

In integrable quantum spin chains with gl(N) symmetry, $T_{a,s}$ appears to be the transfer matrix in the representation $s^a$, with the $a \times s$ rectangular Young diagram, as shown in figure 3b, while $u$ plays the role of the spectral parameter. Equation (2.1) describes the fusion procedure among these transfer-matrices [4, 28–31]. The statement generalises to supersymmetric case [32–35] and, with a particular modification of (2.1), to other semi-simple Lie algebras, see [4, 27, 30, 31] and the references therein. In integrable 2d CFT’s at finite size or finite temperature, and in particular in 2d sigma-models, this Hirota equation first appeared in relation to quantum KdV [36] and more recently it was successfully used for the finite size analysis, including excited states, for the SU(N) × SU(N) principal chiral field (PCF) and some related models [37, 38]. It was also proposed as a version of the AdS/CFT Y-system [39] appearing in the spectral problem of the planar $\mathcal{N} = 4$ SYM theory and it was successfully exploited there for extracting many non-trivial results at arbitrary strength of the ’t Hooft coupling and in various physically interesting limits [40]. The finite-difference Hirota equation (2.1) is also related in different way to the classical integrability, besides the standard classical limit $\hbar \to 0$ of the original quantum system. It can be obtained from the canonical Hirota equation for $\tau$-function of classical integrable hierarchies of PDE’s by introduction of discrete Miwa variables [26]. And in particular, a generating series of transfer-matrices of gl(N) quantum Heisenberg spin chains can be interpreted as a $\tau$-function of the mKP hierarchy [41].

As was shown in the past, Hirota equation admits general and exact solutions for specific boundary conditions on the $\mathbb{Z}^2$ lattice. In particular, if one demands $T_{a<0,s} = 0$ then all T-functions can be expressed explicitly in terms of $T_{0,s}$ and $T_{1,s}$ by

$$T_{a,s}(u) = \frac{\det_{1 \le i,j \le a} T_{1,s+i-j} \left( u + i \frac{1+a-i-j}{2} \right)}{\prod_{k=1}^{a-1} T_{0,s} \left( u + i \frac{a+1-2k}{2} \right)}, \quad (2.2)$$

which is a particular case of the Cherednik-Bazhanov-Reshetikhin (CBR) determinant [34, 38, 42, 43] formulae. This determinant relation is a generic solution of the Hirota equation in the sense it can be proven recursively in $a$ assuming $T_{a \ge 0,s} \ne 0$; if $T_{a,s} = 0$ for some positive
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>(b) Identification of a node of the strip to a rectangular Young diagram.</figcaption></figure>

**Figure 3.** The Young diagrams of compact representations of gl(N) group are confined to a half-strip, depicted on figure (a), of width N on infinite representational (a, s)-lattice. The vertices within this strip are in one-to-one correspondence with rectangular Young tableux of size a × s, as depicted on figure (b), as well as with corresponding characters or T-functions.

a then (2.2) may be violated, however in practice this affects only T's which do not have an explicit physical interpretation, and we choose to define these T's such that (2.2) holds.

If we impose a more severe restriction on T's and demand them to be non-zero only in the black nodes of figure 3a (i.e. for s = 0 or a ≥ 0 or s > 0, N ≥ a ≥ 0) then we get the Hirota equation appearing in integrable models with gl(N) symmetry and related to the compact representations of the latter. For such boundaries, we can recognise in CBR determinants a quantum generalisation of standard Gambelli-Jacobi-Trudi formulae for characters of gl(N) irreps. The analog of (2.2) looks especially simple

$$ \chi_{a,s}(G) = \det(\chi_{1,s+j-k}(G))_{1\le j,k\le a}, \qquad (2.3) $$

where $G = \{x_1, \dots, x_N\}$ is a Cartan subgroup element. This character satisfies the simplified Hirota equation<fnref n="3" />

$$ \chi_{a,s}(G)\chi_{a,s}(G) = \chi_{a+s,s}(G)\chi_{a-1,s}(G) + \chi_{a,s+1}(G)\chi_{a,s-1}(G); \qquad (2.4) $$

it can be derived directly from (2.5) due to the Jacobi relation for determinants (see e.g. the appendix of [34]).

In the case of characters, we know that there exists a more explicit, Weyl formula expressing the character as a determinant involving the Cartan elements:

$$ \chi_{a,s}(G) = \frac{\det_{1 \le j,k \le N} x_k^{N-j+s \Theta_{a,j}}}{\det_{1 \le j,k \le N} x_k^{N-j}}, \qquad \text{where } \Theta_{i,j} \equiv \begin{cases} 1 & \text{if } i \ge j \\ 0 & \text{if } i < j \end{cases}. \qquad (2.5) $$

It is clear that it should be possible to generalize the Weyl formula from characters to T-functions. Such a quantum generalization was known since quite a while [4] in terms of

<footnote n="3">³It is sometimes called the Q-system in the mathematical literature. We will avoid this in order to confuse it with the Baxter's Q-functions $Q_I(u)$ which we use all over the paper. We rather call the collection of these Q-functions as the Q-system.</footnote>
<table><caption>**Table 1.** Expression of the GL(N) characters and their generalization to T-functions. Representations are labeled by Young diagrams $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_{|\lambda|})$, and $|\lambda'|$ denotes $\lambda_1$. Characters $\chi_\lambda(g)$ are written in terms of the eigenvalues $(x_1, x_2, \dots, x_N)$ of a group element $g$. The CBR formula and Wronskian expression of T-functions are written in this table under specific gauge constraint. In other gauges they hold up to division by $T_{(0)}$ or $Q_0$, cf. (2.2) and (2.36), the normalisation is clarified in section 2.6.1.</caption><thead><tr><th>Characters of the GL(N) group</th><th>T-functions on the GL(N) strip</th></tr></thead><tbody><tr><td>2<sup>nd</sup> Weyl formula<br>χ<sub>λ</sub>(G) = det<sub>1≤j,k≤|λ|</sub> χ<sub>(λ<sub>k</sub>+j−k)</sub>(G)</td><td>Cherednik-Bazhanov-Reshetikhin formula<br>T<sub>λ</sub>(u) = det<sub>1≤j,k≤|λ|</sub> T<sub>(λ<sub>k</sub>+j−k)</sub> (u + i λ<sub>k+1+|λ|-|λ'|-j−k</sub> / 2)</td></tr><tr><td>1<sup>st</sup> Weyl formula<br>χ<sub>λ</sub>(G) = det<sub>1≤j,k≤N</sub> x<sub>k</sub><sup>λ<sub>j</sub>+N−j</sup> / det<sub>1≤j,k≤N</sub> x<sub>k</sub><sup>N−j</sup></td><td>Wronskian expression<br>T<sub>λ</sub>(u) = det<sub>1≤j,k≤N</sub> Q<sub>k</sub>(u + i λ<sub>j-2j+1+|λ|-|λ'|-j</sub> / 2)</td></tr></tbody></table>

the Wronskian-type determinant:

$$T_{a,s}(u) = \det_{1 \le j,k \le N} Q_k \left( u + i \frac{a+1+s(2\Theta_{a,j}-1)}{2} - j \right). \qquad (2.6)$$

It gives, up to rescaling of T-functions, the general solution of Hirota equation for a half-strip boundary conditions of figure 3 in terms of N independent Q-functions $Q_1(u), \dots, Q_N(u)$. More precisely, it applies for the semi-infinite rectangular domain $s \ge 0, N \ge a \ge 0$; the rest of non-zero T-functions, corresponding to the black nodes of figure 3, $a = 0, s < 0$ and $s = 0, a > N$ are easily restored.<fnref n="4" />

The parallels between character formulae and T-functions (or, when meaningful, transfer matrices) extend beyond the rectangular representations $s^a$, the equivalent formulae for arbitrary finite-dimensional representations of gl(N) algebra are summarised in table 1.

The Gambelli-Jacobi-Trudi-type formulae (2.3) and their quantum counterpart (2.2) remain unchanged if one generalises the symmetry to the case of superalgebras of gl (or rather sl) type, including the non-compact cases. They are used, however, under different boundary conditions outlined in figure 4.

The super-analogues of Weyl-type formulae are not obtained by a straightforward generalisation, yet they are also known. For the compact case $su(K|M)$ the determinant expressions for characters were established in [44]. In the non-compact case $su(K_1, K_2|M)$ certain expression for characters were given in [45] and their determinant version for the case of rectangular representations<fnref n="5" /> was elaborated in [46]. The generalization to the quantum case was first presented for finite-dimensional irreps of $su(K|M)$ in [35], then for $su(2, 2|4)$ in [6] (this is the most interesting case for physics as it is realised in the context of AdS/CFT integrability, see a review [40] for introduction into the subject) and finally

<footnote n="4">⁴Indeed, the Hirota equation gives $T_{0,s-1} = T_{0,s}^+ T_{0,s}^- / T_{0,s+1}$, and $T_{a+1,0} = T_{a,0}^+ T_{a,0}^- / T_{a-1,0}$ for $a > 0$, allowing to iteratively restore the boundary T-functions.</footnote>

<footnote n="5">⁵Determinant character formulae for non-rectangular highest-weight representation were not published explicitly to our knowledge.</footnote>
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4. Fat hook and T-hook, for supersymmetric symmetry groups.</figcaption></figure>

generalized to any $su(K_1, K_2|M)$ in [7]. In the case of rectangular $s^a$ irrep, the formulae of [7] give the generic (up to a gauge transformation, as explained below) Wronskian solution of Hirota equation (2.1) within the $(K_1|M|K_2)$-hook presented in figure 4b (which was also called T-hook due to its shape). The so-called fat hook of the figure 4a, which we also call L-hook, is a particular case $K_2 = 0$ of $su(K_1, K_2|M)$ corresponding to the compact representations of $su(K|M)$.

In [7], the Weyl-type solution of Hirota equation is presented in terms of an explicit finite determinant and it summarises the whole progress achieved in this field. However, the corresponding expressions are extremely bulky which somewhat obscures their nice geometric and algebraic properties. The main aim of this section is to present a more concise and more intuitive formalism, based on the exterior forms of Baxter-type Q-functions. It will clarify the Grassmannian nature of Wronskian solutions for T-functions on supergroups and allow simple and general proofs for these formulae. We will re-derive several relations already proven in [7] in this new language and present some new useful relations.

## 2.2 Notations

The Wronskian solution of Hirota equation (2.1) with boundary conditions shown in figures 3a and 4 will be written in subsequent sections in terms of a set of Q-functions $Q_{b_1b_2...}$ which are labeled by several indices $b_k$ and which are antisymmetric under permutations of these indices. There exist relations between the Q-functions, and there are two equivalent ways to formulate them: either as an algebraic statement — the “QQ-relations” — or as a geometric statement — in terms of the intersection property (1.5).

Algebraically, the QQ-relations read (in the non-super-symmetric case of section 2.6) [47–52]

$$Q_A Q_{Ab} = Q_{Ab}^{+} Q_{Ac}^{-} - Q_{Ab}^{-} Q_{Ac}^{+} \qquad (2.7)$$

All other QQ-relations derived below ultimately follow from (2.7), hence we will pause for a while to accurately introduce the notational conventions related to (2.7) and to Q-system in general.
<figure><img src="image_8.png" /><figcaption>Figure 5. Hasse diagram for g(3) Q-functions.</figcaption></figure>

The Q-functions are functions of the spectral parameter $u$. This dependence is typically assumed implicitly, and the shifts of $u$ are denoted following the convention

$$f^{[\pm n]} = f\left(u \pm n \frac{i}{2}\right), \qquad f^{\pm} = f\left(u \pm \frac{i}{2}\right), \qquad i \equiv \sqrt{-1}. \qquad (2.8)$$

The indices $b, c$ in (2.7) take value in the “bosonic” set $\mathcal{B} = \{1, 2, \dots, N\}$. The multi-index $A$ of the bosonic set can for instance contain one single index $a \in \mathcal{B}$, or no index at all (it is then denoted as $A = \emptyset$), or all indices (which is denoted as $A = \mathcal{B} = \emptyset$), etc. The multi-index 2, 1 is different from the multi-index 1, 2 (one has $Q_{2,1} = -Q_{1,2}$), and we will say that the multi-index $A = a_1 a_2 \dots a_n$ is sorted if $\forall k < n, a_k < a_{k+1}$. The sum over all sorted multi-indices of length $n$ is denoted by $\sum_{|A|=n}$.

For a multi-index $A$, $\{A\}$ means the associated set (for instance $\{2, 1\} = \{1, 2\}$), and we denote by $\bar{A}$ the sorted multi-index obeying $\{\bar{A}\} = \mathcal{B}\{A\}$ (for instance $\overline{1, 3} = \{2, 4, 5, \dots, N\}$).

There are $2^N$ different Q-functions corresponding to the different subsets of $\mathcal{B}$. They can be arranged as a Hasse diagram forming an N-dimensional hypercube, see figure 5. Each facet of the Hasse diagram is associated with a QQ-relation: for instance the bottom facet in figure 5 is associated to the relation

$$Q_2 Q_{123} = Q_{12}^{+} Q_{23}^{-} - Q_{12}^{-} Q_{23}^{+}, \qquad (2.9)$$

which is the case$^6 A = 2, b = 1, c = 3$ in (2.7).

Given a basis of $N$ independent elements $\zeta_1, \zeta_2, \dots, \zeta_N$ and an associative antisymmetric bilinear product “$\wedge$”, we also introduce the $n$-form

$$Q_{(n)} = \sum_{|A|=n} Q_A \zeta_A, \quad \text{where } \zeta_{b_1 b_2 \dots b_n} \equiv \zeta_{b_1} \wedge \zeta_{b_2} \wedge \dots \wedge \zeta_{b_n}, \quad (\text{and } \zeta_{\emptyset} = 1). \qquad (2.10)$$

With explicit indices, (2.10) reads: $Q_{(n)} = \sum_{1 \le b_1 < b_2 < \dots < b_n \le N} Q_{b_1 b_2 \dots b_n} \zeta_{b_1} \wedge \zeta_{b_2} \wedge \dots \wedge \zeta_{b_n}$.

We also introduce the Hodge dual $\star \omega$ of an arbitrary $n$-form $\omega$ as the linear transformation such that

$$\star \zeta_A = \epsilon_{A\bar{A}} \zeta_{\bar{A}}, \qquad (2.11)$$

<footnote>$^6$More precisely, (2.7) gives the relation $Q_2 Q_{213} = Q_{21}^{+} Q_{23}^{-} - Q_{21}^{-} Q_{23}^{+}$, which is equivalent due to the antisymmetry.</footnote>
where $\epsilon_{b_1b_2...b_N}$ is the completely antisymmetric tensor with the sign choice $\epsilon_{12...N} = 1$. For instance this definition gives $\star\zeta_{13} = -\zeta_{2,4,5,...,N}$.

The Hodge-dual Q-functions are denoted using the super-script labelling:

$$Q^A \equiv \epsilon^{\bar{A}A} Q_{\bar{A}}, \quad \text{so that} \quad \star Q_{(n)} = \sum_{|A|=|\bar{B}|-n} Q^A \zeta_A. \qquad (2.12)$$

The sign convention for the completely antisymmetric tensor $\epsilon^{b_1b_2...b_N}$ is also $\epsilon^{12...N} = 1$. We will interchangeably use upper- and lower-indexed $\epsilon$ to emphasise the covariance in relations.

Note that the inverse Hodge-dual operation given by

$$Q_A = \epsilon_{A\bar{A}} Q^{\bar{A}} \qquad (2.13)$$

has certain difference in signs compared to (2.12).

**Plücker identity.** Throughout this text, we will frequently use Plücker identities. The simplest one is

$$\begin{aligned} & \star (\zeta_{b_1} \wedge \zeta_{b_2} \wedge \cdots \wedge \zeta_{b_N}) \star (\zeta_{c_1} \wedge \zeta_{c_2} \wedge \cdots \wedge \zeta_{c_N}) = \\ & = \sum_{a=1}^{n} \star (\zeta_{b_1} \wedge \cdots \wedge \zeta_{b_{N-1}} \wedge \zeta_{c_a}) \star (\zeta_{c_1} \wedge \cdots \wedge \zeta_{c_{a-1}} \wedge \zeta_{b_N} \wedge \zeta_{c_{a+1}} \wedge \cdots \wedge \zeta_{c_N}), \quad (2.14) \end{aligned}$$

where the Hodge operation “★” simply transforms each product $\zeta_{b_1} \wedge \zeta_{b_2} \wedge \cdots \wedge \zeta_{b_N}$ into the number $\epsilon_{b_1b_2...b_N}$.

More generally, one has

$$\begin{aligned} & \star (\mathbf{x}_1 \wedge \mathbf{x}_2 \wedge \cdots \wedge \mathbf{x}_N) \star (\mathbf{y}_1 \wedge \mathbf{y}_2 \wedge \cdots \wedge \mathbf{y}_N) = \\ & = \sum_{a=1}^{N} \star (\mathbf{x}_1 \wedge \cdots \wedge \mathbf{x}_{N-1} \wedge \mathbf{y}_a) \star (\mathbf{y}_1 \wedge \cdots \wedge \mathbf{y}_{a-1} \wedge \mathbf{x}_N \wedge \mathbf{y}_{a+1} \wedge \cdots \wedge \mathbf{y}_N), \quad (2.15) \end{aligned}$$

where $\mathbf{x}_i = \sum_{j=1}^{N} x_{i,j} \zeta_j$ and $\mathbf{y}_i = \sum_{j=1}^{N} y_{i,j} \zeta_j$ are arbitrary sets of vectors.

**Asymptotics.** The asymptotic behavior of functions at large $u$ will have some importance later on in this article. We will then use the notation $f \sim g$ to say that $\lim_{|u| \to \infty} \frac{f}{g} = 1$ and $f \sim g$ to say that there exists $\alpha \in \mathbb{C}^\times$ such that $\lim_{|u| \to \infty} \frac{f}{g} = \alpha$.

## 2.3 QQ-relations and flags of $\mathbb{C}^N$

The geometric counterpart of the algebraic relation (2.7) is the intersection condition (1.5). Our nearest goal is to justify this statement.

The functions $Q_A$ with $|A| = n$ should be thought as Plücker coordinates of the hyperplane $V_{(n)}$; they define $V_{(n)}$ as the collection of points $\mathbf{x}$ that satisfy $Q_{(n)} \wedge \mathbf{x} = 0$. Note that for a generic $n$-form $\omega_n$ the condition $\omega_n \wedge \mathbf{x} = 0$ does not define an $n$-dimensional hyperplane (for instance if $\omega = \zeta_1 \wedge \zeta_2 + \zeta_3 \wedge \zeta_4$, the condition is satisfied only by $\mathbf{x} = 0$).
However, as it will become clear in this subsection, the relation (2.7) insures that the $Q_A$ are indeed the Plücker coordinates of $n$-dimensional hyperplanes.

To derive (2.7) from the intersection condition (1.5) we note that the latter can be equivalently reformulated as the following union property

$$ \forall n \in \{1, 2, \dots, N-1\}, \qquad V_{(n)}^{+} + V_{(n)}^{-} = V_{(n+1)}, \qquad (2.16) $$

which implies, in particular, that the sequence $\{0\} \equiv V_{(0)} \subset V_{(1)}^{+} \subset \dots \subset V_{(N)}^{[+N]} \equiv \mathbb{C}^N$ is a maximal flag of $\mathbb{C}^N$. The union property should hold for almost all values of the spectral parameter save a discrete set of points.

Since $V_{(1)}$ is a line there exists a one-form

$$ Q_{(1)} = \sum_{a=1}^{N} Q_a \zeta_a \qquad \text{such that} \qquad V_{(1)} = \{ \mathbf{x} \in \mathbb{C}^N ; Q_{(1)} \wedge \mathbf{x} = 0 \}. \qquad (2.17) $$

This defines $Q_{(1)}$ up to a normalisation, i.e. up to the transformation $Q_{(1)}(u) \mapsto f(u)Q_{(1)}(u)$, where $f$ is a $\mathbb{C}$-valued function of $u$. Next, one can immediately see from (2.16) that

$$ V_{(n)} = V_{(1)}^{[n-1]} + V_{(1)}^{[n-3]} + \dots + V_{(1)}^{[-n+1]} \qquad (2.18) $$

$$ = \{ \mathbf{x} \in \mathbb{C}^N ; Q_{(1)}^{[n-1]} \wedge Q_{(1)}^{[n-3]} \wedge \dots \wedge Q_{(1)}^{[-n+1]} \wedge \mathbf{x} = 0 \}. \qquad (2.19) $$

We can therefore define the forms $Q_{(n)}$ by the relation

$$ Q_{(n)} = f_n Q_{(1)}^{[n-1]} \wedge Q_{(1)}^{[n-3]} \wedge \dots \wedge Q_{(1)}^{[-n+1]} \qquad \text{if } n > 1, \qquad (2.20) $$

where $f_n(u)$ is a normalisation freedom that we will have to fix.

The definition (2.20) enforces the coordinates $Q_A$ to obey the relation

$$ Q_A Q_{Ab} \frac{f_{|A|+1}^{+} f_{|A|+1}^{-}}{f_{|A|} f_{|A|+2}} = Q_{Ab}^{+} Q_{Ac}^{-} - Q_{Ab}^{-} Q_{Ac}^{+}, \qquad (2.21) $$

a proof is given in appendix A.1, and it is based on a simple application of the Plücker identity (2.15).

The equation (2.7) corresponds to a particular choice of normalisation such that $f_{|A|+1}^{+} f_{|A|+1}^{-} = f_{|A|} f_{|A|+2}$, i.e. $f_n = \frac{g^{[n-1]}}{g^{[1-n]}}$ for some function $g$. Note that (2.7) can be modified if one decides to use a different prescription for $f_n$; equation (2.21) is an invariant version of (2.7). Still, we stick to the normalisation choice of (2.7) in this paper, this is also a common choice in the literature.

Plugging the expression $f_n = \frac{g^{[n-1]}}{g^{[1-n]}}$ into (2.20) and using $g^-/g^+ = f_0 \equiv Q_0$, we finally get

$$ Q_{(n)} = \frac{Q_{(1)}^{[n-1]} \wedge Q_{(1)}^{[n-3]} \wedge \dots \wedge Q_{(1)}^{[1-n]}}{\prod_{1 \le k \le n-1} Q_{\emptyset}^{[n-2k]}} \qquad (2.22) $$
or equivalently, when written in terms of coordinates,

$$Q_{b_1 b_2 \dots b_n} = \frac{\det_{1 \le j,k \le n} Q_{b_j}^{[n+1-2k]}}{\prod_{1 \le k \le n-1} Q_{\emptyset}^{[n-2k]}}. \qquad (2.23)$$

It is easy to see that the above expression is the general solution to QQ-relation (2.7),<fnref n="7" /> which proves that the geometric statement (1.5) is equivalent to QQ-relation (2.7).

In what precedes, we defined the Q-system by a very simple 3-terms bilinear relation (2.7). It implies many other, in general multilinear, equations relating Q-functions. Equation (2.23) is one example of such a relation and a few other relations are given throughout the text and in appendix A.3.

## 2.4 Hodge duality map

Whereas the form $Q_{(n)}$ defines a plane $V_{(n)}$ of dimension $n$ in $\mathbb{C}^N$, it can be also used to define a plane of dimension $N-n$ in the dual space. It is easy to see that the intersection condition (1.5) and the union condition (2.16) exchange their roles in the dual space and, hence, we can devise a Q-system for the dual geometric construction which, quite naturally, is simply given by Hodge-dual Q-functions (2.12). In practice, this means that Q-functions with upper indices obey exactly the same algebraic relations as the Q-functions with lower indices. For instance, one can derive

$$Q^{b_1 b_2 \dots b_n} = \frac{\det_{1 \le j,k \le n} (Q^{b_j})^{[n+1-2k]}}{\prod_{1 \le k \le n-1} Q_{\emptyset}^{[n-2k]}}, \qquad (2.24)$$

etc.

Note that, technically speaking, Hodge duality is not a symmetry of a given Q-system, in the sense that it relates Q-functions with different set of indices. We can think about it as a map, a natural way to construct another collection of Q-functions obeying (2.7) — i.e. another Q-system — differing from the original one by a relabelling of the Q-functions.

## 2.5 Symmetry transformations on Q-systems

In this section we discuss other symmetries of the equation (2.7). Like the Hodge transformation, they map a given set of Q-functions (Q-system) to another Q-system. By contrast with the Hodge transformation, which maps the spaces $V_{(n)}$ to the dual space, the transformations we will consider essentially leave the spaces $V_{(n)}$ invariant.

We have seen that the QQ-relations is a way to rewrite the geometric intersection property in a coordinate form. But any coordinatisation is sensible to a choice of basis,

<footnote n="7">⁷The statement is true if there is no $A$ such that $Q_A = 0$. For instance, if $N = 4$, and $Q_\emptyset = 1$, $Q_{(1)} = \sum_{i=1}^4 \zeta_i$, $Q_{(2)} = 0 = Q_{(3)}$, $Q_{(4)} = \zeta_1 \wedge \zeta_2 \wedge \zeta_3 \wedge \zeta_4$, then the QQ-relations (2.7) hold, whereas (2.22) do not hold. A singular situation with $Q_A = 0$ may appear in practical applications, we observed it in cases related to short representations of supersymmetric algebra, see section 3.3.4. In the situations we encountered, (2.22) holds even if $Q_A = 0$ for some $A$.</footnote>
hence there exist transformations which change a basis but do not affect the relation (2.7) itself. These basis-changing transformations of Q-system are of two types: rescalings and rotations.

## 2.5.1 Rescalings (gauge transformations)

Plücker coordinates are projective: rescaling them does not change the point in Gaussian that they define. Hence the transformation $Q_A \to g_{|A|}Q_A$ is a symmetry of the QQ-relation (2.7). As we saw in the last section, this rescaling, defined by arbitrary $N+1$ functions $g_0(u), g_1(u), \dots, g_N(u)$, modifies $f_i$ in (2.21). As we agreed to work in the normalisation compatible with (2.7), only 2 out of $N+1$ functions remain independent. We can summarize the admissible rescalings that preserve (2.7) in a compact form as

$$Q_A \mapsto g_{(+)}^{[|A|]} g_{(-)}^{-[|A|]} Q_A, \qquad (2.25)$$

where $g_{(\pm)}$ are certain combinations of $g_i$.

These rescaling transformations are also known as gauge symmetries of the Q-system. Indeed, they are local transformations because $g_{(\pm)}$ depend on $u$.

## 2.5.2 Rotations

One can also rotate<fnref n="8" /> the basis frame, that is to choose different basis vectors $\zeta_1, \dots, \zeta_N$. However we cannot rotate the frames independently at different values of the spectral parameter as the QQ-relations are non-local. Therefore, the following transformation

$$\tilde{Q}_b \mapsto \sum_{c \in \mathcal{B}} h_{bc} Q_c \qquad (2.26a)$$

of single-indexed Q-functions together with the transformation

$$\tilde{Q}_{b_1 b_2 \dots b_n} = \sum_{c_1, c_2, \dots, c_n \in \mathcal{B}} h_{b_1 c_1}^{[n-1]} h_{b_2 c_2}^{[n-1]} \dots h_{b_n c_n}^{[n-1]} Q_{c_1 c_2 \dots c_n} \qquad (2.26b)$$

of multi-indexed Q-functions is a symmetry of the QQ-relation (2.7) if $h_{bc}$ are i-periodic functions of $u$:

$$h_{bc}^{+} = h_{bc}^{-}. \qquad (2.27)$$

The transformations (2.26) will be called H-transformations [53] or simply rotations.

Note that the case $h_{bc} = h \delta_{bc}$ can be viewed as a particular case of the rescaling symmetry with $g_{(+)}g_{(-)} = 1$ and $g_{(+)}^{+}g_{(-)}^{-} = h$. Hence one may restrict to the case of unimodular H-transformations:

$$\det_{1 \le b, c \le N} h_{bc} = 1. \qquad (2.28)$$

In contrast to two local rescaling symmetries, rotations should be thought as a global symmetry. Indeed, periodic functions, e.g. (2.27), in the case of finite-difference equations play the same role as constants in the case of differential equations. Eventually, we will gauge the rotational symmetry, in order to formulate a twisted Q-system in section 3. But until then, this symmetry will remain global.

<footnote n="8">⁸In this article we allowed a freedom of speech to call any linear non-degenerate transformation as rotation. There is no metric to preserve, hence it does not lead to confusion.</footnote>
<figure><img src="image_9.png" /><figcaption>Figure 6. Infinite horizontal strip.</figcaption></figure>

## 2.6 Solution of Hirota equation on a strip

This section is devoted to solving the Hirota equation (2.1) on a strip. One case of our interest is the semi-infinite strip of figure 3 which corresponds to compact representations of gl(N). We remind that in this figure T-functions are identically zero outside the nodes denoted by black dots. The solution for these boundary conditions had been already written in [4] and then analysed in a handful of follow-up works. We revise this case as a warm-up for our subsequent studies of T-systems related to non-compact supergroups.

The semi-infinite strip should be thought as a special reduction<fnref n="9" /> of an infinite horizontal strip shown in figure 6, i.e. related to the solution such that $T_{a,s}$ is identically zero outside the band $0 \le a \le N$. We write down the generic solution for this case as well. It was already successfully used in [37, 38, 54] for the study of TBA and physical Y-system for the spectrum of principal chiral field (PCF) model at finite space circle.

On the infinite horizontal strip of figure 6, the generic solution to the Hirota equation is given by<fnref n="10" />

$$ T_{a,s} = \star (Q_{(a)}^{[+s]} \wedge P_{(N-a)}^{-[s]}) \quad \text{when } 0 \le a \le N \quad \text{and } T_{a,s} = 0 \text{ otherwise.} \quad (2.29) $$

By letters $P$ and $Q$ we denote two independent sets of Q-functions, each of them expressed through (2.22).<fnref n="11" />

On the semi-infinite strip of figure 3a, a solution to the Hirota equation is given by:

$$ T_{a,s} = \star (Q_{(a)}^{[+s]} \wedge Q_{(N-a)}^{-[s-N]}) \quad \text{when } s \ge 0 \text{ and } 0 \le a \le N \quad (2.30) $$

In components, the last relation becomes

$$ T_{a,s} = (-1)^{a(N-a)} \sum_{|A|=a} Q_A^{[+s]} (Q^A)^{-[s-N]}. \quad (2.31) $$

The solution (2.30) has to be supplemented with $T_{0,s} = T_{0,s+1}^+ T_{0,s+1}^- / T_{0,s+2} = \star Q_0^{[+s]} Q_{(N)}^{-[s-N]}$ for $s < 0$ and $T_{a,0} = T_{a-1,0}^+ T_{a-1,0}^- / T_{a-2,0} = \star Q_0^{-[a]} Q_{(N)}^{[a-N]}$ for $a > N$.

<footnote n="9">⁹Up to minor adjustments, namely the question is about the vertical line $s = 0$ in figure 3. We can replace this line, for the same solution of Hirota equation, by the horizontal line $a = N$ and demand that T-functions are non-zero on this horizontal line instead.</footnote>

<footnote n="10">¹⁰The only purpose of Hodge $\star$-operation is to convert (N)-forms to (0)-forms, as $\star(\zeta_1 \wedge \zeta_2 \wedge \cdots \wedge \zeta_N) = 1$.</footnote>

<footnote n="11">¹¹This means in particular that $P_{(n)} = \frac{P_{(1)}^{[n-1]} \wedge P_{(1)}^{[n-3]} \wedge \cdots \wedge P_{(1)}^{[1-n]}}{\prod_{1 \le k \le n-1} P_0^{[n-2k]}}$.</footnote>
We will now discuss what are the symmetry transformations of Hirota equation and of formulae (2.29) and (2.30), then we will give a proof that (2.29) and (2.30) are indeed the generic solution of the Hirota equation on the corresponding strips.

## 2.6.1 Gauge symmetry of the Hirota equation

Hirota equation, for any “shape” of non-zero T-functions, is invariant under the transformation

$$ T_{a,s}(u) \mapsto g_{(++)}^{[+a+s]} g_{(+-)}^{[+a-s]} g_{(-+)}^{-[a+s]} g_{(--)}^{-[a-s]} T_{a,s}(u), \quad (2.32) $$

where $g_{(++)}$, $g_{(+-)}$, $g_{(-+)}$ and $g_{(--)}$ are four arbitrary functions of the spectral parameter $u$. This transformation is usually called the gauge transformation.

One can reformulate the Hirota equation (2.1) as a Y-system:

$$ \frac{Y_{a,s}(u + \frac{i}{2})Y_{a,s}(u - \frac{i}{2})}{Y_{a+1,s}(u)Y_{a-1,s}(u)} = \frac{(1 + Y_{a,s+1}(u))(1 + Y_{a,s-1}(u))}{(1 + Y_{a+1,s}(u))(1 + Y_{a-1,s}(u))}, \quad (2.33) $$

using the Y-functions defined as $Y_{a,s} = \frac{T_{a,s}^+ T_{a,s}^-}{T_{a+1,s} T_{a-1,s}}$. The Y-functions are obviously invariant under the gauge transformation (2.32). Typically, physically relevant quantities can be expressed only through the gauge-invariant functions.

If the gauge functions $g_{(\pm\pm)}$ are i-periodic, i.e. if they obey $g^+ = g^-$, then the gauge transformation is the multiplication of $T_{a,s}(u)$ with a single i-periodic function. Such transformation will be called a normalization. For instance,<fnref n="12" /> the prefactor $(-1)^{a(N-a)}$ in (2.31) can be removed by an appropriate normalisation.

As T-functions are determinants of Q-functions, unimodular rotations of the Q-basis have no effect on T-functions. By contrast, the rescaling gauge transformation of Q-system precisely generates gauge transformations of the T-functions. Indeed, one can spot from (2.29) that the rescaling

$$ Q_A \mapsto g_1^{[A]} g_2^{-[A]} Q_A, \qquad P_A \mapsto g_3^{[A]} g_4^{-[A]} P_A \quad (2.34a) $$

induces the following gauge transformation<fnref n="13" />

$$ T_{a,s} \mapsto g_1^{[a+s]} g_3^{[N-a-s]} g_2^{-[a+s]} g_4^{[a-s-N]} T_{a,s}. \quad (2.34b) $$

In a more restrictive case of (2.30), the rescaling of Q-functions generates only two gauge transformations:

$$ Q_A \mapsto g_1^{[A]} g_2^{-[A]} Q_A, \qquad T_{a,s} \mapsto g_1^{[a+s]} g_1^{-[a-s]} g_2^{-[a+s]} g_2^{[a-s-2N]} T_{a,s}. \quad (2.35) $$

In fact, the solution (2.30) is written in a specific so-called Wronskian gauge in which $T_{1,0} = T_{0,-1}$ and $T_{N+1,0} = T_{N,1}$. In arbitrary gauge, the semi-infinite strip solution should be written as

$$ T_{a,s} = \frac{f_1^{[a+s]} f_2^{[a-s-N]}}{f_1^{-[a-s]} f_2^{-[a-s-N]}} \star (Q_{(a)}^{[+s]} \wedge Q_{(N-a)}^{-[s-N]}) \quad \text{when } s \ge 0 \text{ and } 0 \le a \le N, \quad (2.36) $$

<footnote n="12">¹²For instance $(-1)^{a(N-a)}$ can be written as $((-1)^{(N-1)iu})^{[a]} /((-1)^{(N-1)iu})^{[-a]}$. One should note that in this example of normalization, the functions $g_{(\pm\pm)}$ are not all periodic, but their product is.</footnote>

<footnote n="13">¹³This transformation clearly matches (2.32) up to relabeling the functions $g$ and their shifts.</footnote>
where $f_1$ and $f_2$ are two additional arbitrary functions of the spectral parameter $u$. Hence, obviously, we speak about (2.30) as a general solution modulo gauge symmetry.

One can use (2.35) to set, for instance, $Q_\emptyset = Q_\emptyset = 1$. We note that if $Q_\emptyset = 1$ then the expression (2.30) becomes a determinant $T_{a,s}(u) = \det_{1 \le j,k \le N} Q_k(u + i \frac{2s\Theta_{a,j} - 2j + 1 + a - s}{2})$ which coincides (for rectangular Young diagrams) with the determinant expression written in table 1.

## 2.6.2 Proof A: existence of solutions to Hirota equation

Let us first prove that (2.29) provides a solution to the Hirota equation (2.1). Since the Hirota equation is invariant under the gauge transformations (2.34), it is sufficient to prove that it is satisfied when $P_\emptyset = Q_\emptyset = 1$.

We can start by writing

$$T_{a,s}^{-} T_{a,s}^{+} = \star(\mathbf{x}_{1} \wedge \mathbf{x}_{2} \wedge \cdots \wedge \mathbf{x}_{N}) \star(\mathbf{y}_{1} \wedge \mathbf{y}_{2} \wedge \cdots \wedge \mathbf{y}_{N}), \quad (2.37)$$

where

$$(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_a) = (Q_{(1)}^{[a+s-2]}, Q_{(1)}^{[a+s-4]}, \dots, Q_{(1)}^{[-a+s]}), \quad (2.38)$$

$$(\mathbf{x}_{a+1}, \mathbf{x}_{a+2}, \dots, \mathbf{x}_N) = (P_{(1)}^{[-s+N-a-2]}, P_{(1)}^{[-s+N-a-4]}, \dots, P_{(1)}^{[-s-N+a]}), \quad (2.39)$$

$$(\mathbf{y}_1, \mathbf{y}_2, \dots, \mathbf{y}_a) = (Q_{(1)}^{[a+s]}, Q_{(1)}^{[a+s-2]}, \dots, Q_{(1)}^{[-a+s+2]}), \quad (2.40)$$

$$(\mathbf{y}_{a+1}, \mathbf{y}_{a+2}, \dots, \mathbf{y}_N) = (P_{(1)}^{[-s+N-a]}, P_{(1)}^{[-s+N-a-2]}, \dots, P_{(1)}^{[-s-N+a+2]}). \quad (2.41)$$

We can use (2.15), and notice that $N-2$ terms of the sum in the r.h.s. vanish because they contain a factor $\mathbf{x}_k \wedge \mathbf{y}_{k+1}$ (which is zero if $k \neq a$). This gives

$$T_{a,s}^{-} T_{a,s}^{+} = \star(\mathbf{x}_1 \wedge \cdots \wedge \mathbf{x}_{N-1} \wedge \mathbf{y}_1) \star(\mathbf{x}_N \wedge \mathbf{y}_2 \wedge \cdots \wedge \mathbf{y}_N) \\ + \star(\mathbf{x}_1 \wedge \cdots \wedge \mathbf{x}_{N-1} \wedge \mathbf{y}_{a+1}) \star(\mathbf{y}_1 \wedge \cdots \wedge \mathbf{y}_a \wedge \mathbf{x}_N \wedge \mathbf{y}_{a+2} \wedge \cdots \wedge \mathbf{y}_N) \quad (2.42)$$

$$\text{i.e.} \qquad T_{a,s}^{-} T_{a,s}^{+} = T_{a+1,s} T_{a-1,s} + T_{a,s-1} T_{a,s+1}, \quad (2.43)$$

which proves that the Hirota equation (2.1) is then satisfied for $0 < a < N$. Also, the Hirota equation reduces to $T_{a,s}^{+} T_{a,s}^{-} = T_{a,s+1} T_{a,s-1}$ (resp $0 = a = N$) if $a = 0$ or $a = N$ (resp $a < 0$ or $a > N$), so that it is clearly satisfied at the boundaries of the strip as well.

It is also clear that the T-functions given by (2.30) obey the Hirota equation for $s > 0$, since they are a particular case of (2.29). At the line $s = 0$, the Hirota equation reduces (if $a > 0$) to $T_{a,0}^{+} T_{a,0}^{-} = T_{a+1,0} T_{a-1,0}$, and it indeed holds because $T_{a,0} = Q_\emptyset^{[a-N]}$. Similarly it holds on the line $a = 0$ (arbitrary $s$), explicit formulae are given after (2.31). Now it is immediate to see that it holds if we put $T_{a,s} = 0$ outside the black dots of figure 3.

## 2.6.3 Proof B: uniqueness of the solution to Hirota equation

We showed above that if T-functions are expressed by the Wronskian ansatz (2.29) (resp (2.30)) then they obey the Hirota equation. We now focus on the opposite question: given a solution of the Hirota equation, does there exist Q-functions such that (2.29) (resp (2.30)) holds?
The answer is generically yes, as one can convince oneself by a simple counting argument: if the functions $T_{a,s}$ are non-zero within the infinite strip of figure 6 then a solution of the Hirota equation is characterized by the $2N + 2$ independent functions $T_{a,0}$ and $T_{a,1}$ (where $0 \le a \le N$), whereas the T-functions written in (2.29) are characterized by the $2N + 2$ independent function $Q_0, Q_1, Q_2, \dots, Q_N, P_0, P_1, P_2, \dots, P_N$. Similarly in the case of the semi-infinite strip of figure 3a, the solution of the Hirota equation is characterized by the $N + 3$ independent functions $T_{0,0}, T_{1,0}$ and $T_{a,1}$, whereas the T-functions written in (2.36) are characterized by the $N + 3$ independent function $f_1, f_2, Q_0, Q_1, Q_2, \dots, Q_N$.

In this subsection, we however provide a constructive proof that Q-functions exist for a generic solution of Hirota equation. We will focus on the case of the infinite strip, whereas the generalization to the semi-infinite strip is done in appendix A.2.

Furthermore, there exist degenerate solutions of the Hirota equation, for which some T-functions are identically zero, which cannot be expressed in terms of Q-functions by the Wronskian expression (2.29). An example of this is given in appendix A.4.

**Construction of the Q-functions.** Let us first notice that if $T_{a,s}$ is given by (2.29), then the single-indexed Q-functions are solutions of the following finite-difference “Baxter equation” [4] (see explanations below):

$$ \left( \sum_{r=0}^{N} Q_{(1)}^{[+2r]} \psi_r \right) \wedge \left( \sum_{r=0}^{N} T_{1,s_0+r}^{[-s_0+r]} \psi_r \right) \wedge \left( \sum_{r=0}^{N} T_{1,r+s_0+1}^{[-s_0-1+r]} \psi_r \right) \wedge \cdots \wedge \left( \sum_{r=0}^{N} T_{1,r+s_0+N-1}^{[-s_0-N+1+r]} \psi_r \right) = 0 \quad (2.44) $$

for any $s_0 \in \mathbb{Z}$, where $\psi_0, \psi_1, \dots, \psi_N$ are a set of variables such that the antisymmetric product $\psi_0 \wedge \psi_1 \wedge \cdots \wedge \psi_N$ does not vanish. For instance, if $N = 2$, this equation takes the form

$$ \forall s_0 \in \mathbb{N}, \quad \forall i \in \{1, 2\}, \quad \left| \begin{array}{ccc} Q_i & Q_i^{[+2]} & Q_i^{[+4]} \\ T_{1,s_0}^{[-s_0]} & T_{1,s_0+1}^{[-s_0-1]} & T_{1,s_0+2}^{[-s_0-2]} \\ T_{1,s_0+1}^{[-s_0-1]} & T_{1,s_0+2}^{[-s_0-2]} & T_{1,s_0+3}^{[-s_0-3]} \end{array} \right| = 0. \quad (2.45) $$

The equation (2.44) is a consequence of (2.29): indeed, (2.29) implies that $\forall k \in \{0, 1, \dots, N-1\}$, $\sum_{r=0}^N T_{1,r+s_0+k}^{[-s_0-k+r]} \psi_r = \sum_{a=1}^N \alpha_{a,k} (\sum_{r=0}^N Q_a^{[+2r]} \psi_r)$, where $\alpha_{a,k} = (P^a)^{[-2s_0-2k]}$. This implies that all the $N+1$ vectors in the wedge product (2.44) are linear combinations of $N$ vectors $\sum_{r=0}^N Q_i^{[+2r]} \psi_r$, hence the vanishing of the l.h.s. of (2.44).

Let us now show that this Baxter equation (2.44) can be used to define the Q-functions for a generic solution of Hirota equation, and express the T-functions by the relation (2.29). To this end, we assume that for a given value of $s_0$ the Baxter equation (2.44) has $N$ independent solutions $Q_1, Q_2, \dots, Q_N$. We also assume that for this value of $s_0$, the vectors $\vec{T}_k \equiv \sum_{r=0}^N T_{1,s_0+r+k}^{[r-s_0-k]} \psi_r$ (where $k = 0, 1, \dots, N-1$) are independent.<fnref n="14" /> Then the equation (2.44) states that $N$ independent vectors $\vec{Q}_a \equiv \sum_{r=0}^N Q_a^{[+2r]} \psi_r$ belong to the $N$-dimensional space spanned by the vectors $\vec{T}_k$, which implies that the $\vec{T}_k$ are linear

<footnote n="14">¹⁴While we use the forms notation (2.10) for combinations of the basis elements $\zeta_A$, we use the arrow for combinations of the variables $\psi_k$.</footnote>
combinations of them, i.e. there exists functions $\alpha_{i,k}(u)$ such that $\vec{T}_k = \sum_{i=1}^N \alpha_{a,k} \vec{Q}_a$ (for $k = 0, 1, \dots, N-1$), i.e. such that

$$ T_{1,r+s_0+k}^{[-s_0-k+r]} = \sum_{a=1}^{N} \alpha_{a,k} Q_a^{[+2r]}, \quad k = 0, 1, \dots, N-1, \quad r = 0, \dots, N. \quad (2.46) $$

One can see that the coefficients $\alpha_{a,k}$ are not independent: for any $k \ge 1$ (and any $r = 0, \dots, N-1$), if we plug the condition $T_{1,r+s_0+k}^{[-s_0-k+r]} = (T_{1,r+1+s_0+k-1}^{[-s_0-(k-1)+r+1]})^{[-2]}$ into (2.46), we obtain $\sum_{a=1}^N (\alpha_{a,k} - \alpha_{a,k-1}^{[-2]}) Q_a^{[+2r]} = 0$. Hence the independence of $Q_1, Q_2, \dots, Q_N$ implies that $\alpha_{i,k} = \alpha_{i,k-1}^{[-2]}$, i.e.

$$ \alpha_{i,k} = \alpha_{i,0}^{[-2k]}. \quad (2.47) $$

We therefore define $Q_\emptyset, P^\emptyset$ and the functions $P^a$ (where $1 \le a \le N$) as follows:<fnref n="15" />

$$ T_{0,s} = Q_{\emptyset}^{[+s]} (P^{\emptyset})^{-[s]} \qquad P^a = \alpha_{a,0}^{[+2s_0]}. \quad (2.48) $$

One defines the Q- and P-functions for arbitrary multi-indices by (2.23) and by applying (2.24) for the functions P.

Then the functions $\tilde{T}_{a,s} = \star(Q_{(a)}^{[+s]} \wedge P_{(N-a)}^{-[s]})$ provide a solution to the Hirota equation, which coincides with $T_{a,s}$ when $a=0$ and when $a=1$ and $s=s_0, s_0+1, \dots, s_0+2N-1$. It is then easy to see that one can iteratively show that $\tilde{T}_{a,s} = T_{a,s}$ using the Hirota equation (assuming that $T_{a,s}$ is generic, i.e. $T_{a,s} \ne 0$ for all $a, s$ inside the infinite strip). This concludes the proof that, with the functions P and Q defined above, $T_{a,s}$ is given by the relation (2.29).

## 2.7 On finite-difference (Baxter) equation and Bäcklund transforms

In the previous sections, we reproduced the previously-known generic solution [4] of Hirota equation, using fact that this solution is of a Wronskian type. There exists also an interpretation of the Q-functions from a Bäcklund flow [4, 55–57]. Here we remind the main points of this construction, as it gives an interesting point of view on the Wronskian solution. In particular, we will relate it to the known method of “variation of constants”, a standard trick used in the resolution of differential or difference equation.

In the proof for wronskian relation $T \to Q$ above, the existence of finite-difference equation (2.44) (Baxter equation) played the decisive role. Let aside for a while the goal of solving Hirota equation and discuss some generic finite-difference equation of the N-th order:

$$ \sum_{n=0}^{N} c_n Q^{[2n]} = 0. \quad (2.49) $$

<footnote n="15">¹⁵The existence of two functions $Q_\emptyset$ and $P^\emptyset$ such that $T_{0,s} = Q_\emptyset^{[+s]}(P^\emptyset)^{[+s]}$ is a consequence of the Hirota equation at $a=0$, which reads $T_{0,s}^+ T_{0,s}^- = T_{0,s+1} T_{0,s-1}$.</footnote>
If $Q_1, Q_2, \dots, Q_N$ are $N$ independent solutions, the equation can be also rewritten as

$$ \left| \begin{array}{cccc} Q & Q^{[+2]} & \dots & Q^{[2N]} \\ Q_1 & Q_1^{[+2]} & \dots & Q_1^{[2N]} \\ \vdots & \vdots & \ddots & \vdots \\ Q_N & Q_N^{[+2]} & \dots & Q_N^{[2N]} \end{array} \right| = 0. \qquad (2.50) $$

Suppose we know one solution of (2.49), say $Q_1$. What simplification in the search for other solutions could we make? The standard trick (known as “variation of the constant”) is to write the ansatz $Q = \Psi Q_1$ and to derive the equation on $\Psi$. After simple manipulations, this new equation can be written as an equation of degree $N-1$ for the function

$$ W = Q_1 Q_1^{[2]} (\Psi - \Psi^{[2]}), \qquad (2.51) $$

where we introduced the prefactor $Q_1 Q_1^{[2]}$ for further convenience. The message is clear: we reduced the problem of solving a degree-$N$ equation to solving the equation of degree $N-1$ plus solving the linear equation (2.51). The linear equation can be always solved, at least in terms of a semi-infinite sum$^{16}$ — the analog of integration in the case of differential equations.

Notably for us, the determinant representation for the equation on $W$ is

$$ \left| \begin{array}{cccc} W & W^{[+2]} & \dots & W^{[2N-2]} \\ Q_{12}^{+} & Q_{12}^{[3]} & \dots & Q_{12}^{[2N-1]} \\ \vdots & \vdots & \ddots & \vdots \\ Q_{1N}^{+} & Q_{1N}^{[3]} & \dots & Q_{1N}^{[2N-1]} \end{array} \right| = 0, \qquad (2.52) $$

which implies that $N-1$ solutions for $W$ are $Q_{12}^{+}, Q_{13}^{+}, \dots, Q_{1N}^{+}$, where $Q_{ab}$ are precisely the Q-functions that we discuss in this paper (in the gauge $Q_\emptyset = 1$, i.e. $Q_{ab} = Q_a^+ Q_b^- - Q_a^- Q_b^+$).

Obviously, the argument is repeated recursively. If we happened to find one solution for $W$, say $Q_{12}^{+}$, we can further reduce the degree of equation by one and get the equation which is solved by $Q_{123}^{[2]}, Q_{124}^{[2]}, \dots, Q_{12N}^{[2]}$ etc.

Hence, resolution of any degree-$N$ finite-difference equation is inherently linked to the construction of a Q-system. A part of this construction is to determine $N$ Q-functions in the set $Q_{b_1}, Q_{b_1 b_2}, \dots, Q_{b_1 \dots b_N}$. The Bäcklund flow precisely realises this goal, but now for Q-functions of the specific finite-difference equation (2.44).

A Bäcklund transform (one step in the Bäcklund flow) is introduced as follows. For any Wronskian solution $T_{a,s}$ (2.30) of the Hirota equation on the semi-infinite gl($N$)-strip of figure 3a, one can notice that for any $b \in \mathcal{B}$, the function

$$ F_{a,s} = \star (Q_{(a)}^{[+s]} \wedge Q_{(N-1-a)}^{[-s-N+1]} \wedge \zeta_b) \quad \text{when } s \ge 0 \text{ and } 0 \le a \le N-1 \qquad (2.53) $$

<footnote>$^{16}$For instance, solving equations like (2.51) is routinely performed in the perturbative computation of the AdS/CFT spectrum [22].</footnote>
is a solution of the Hirota equation on the GL(N − 1)-strip of figure 3a, which obeys the Lax pair condition [4]

$$
T_{a+1,s} F_{a,s}^{+} - T_{a,s}^{+} F_{a+1,s} = T_{a+1,s-1}^{+} F_{a,s+1} \quad (2.54a)
$$

$$
T_{a,s+1} F_{a,s}^{-} - T_{a,s}^{-} F_{a,s+1} = T_{a+1,s} F_{a-1,s+1}^{-} \quad (2.54\text{b})
$$

If $T$ obeys the Hirota equation and $F$ obeys (2.54), then $F$ is called the Bäcklund transform of $T$, and it automatically obeys the Hirota equation. Moreover, one can impose the following gauge conditions on $T$, and see that they automatically propagate to $F$ (due to (2.54)):

$$
\left\{
\begin{array}{l}
T_{0,s} = T_{0,0}^{[-s]} \\
T_{a,0} = T_{0,0}^{[+a]} \\
T_{N,s} = T_{N,0}^{[-s]}
\end{array}
\right.
\quad \Rightarrow \quad
\left\{
\begin{array}{l}
F_{0,s} = F_{0,0}^{[-s]} \\
F_{a,0} = F_{0,0}^{[+a]} \\
F_{N-1,s} = F_{N-1,0}^{[-s]}
\end{array}
\right.
\qquad (2.55)
$$

One can then iterate this procedure: a Bäcklund transform of $F$ is a solution of Hirota equation on the gl(N − 2)-strip. The simplest example of a sequence of Bäcklund transformations is given by characters , i.e. for the case when $T_{a,s}(u) = \chi_{a,s}(G)$ for some $G \in \text{GL}(N)$. We can denote by $G^{(b_1,b_2,\dots,b_n)} \in \text{GL}(n)$ a matrix with eigenvalues $x_{b_1}, x_{b_2}, \dots, x_{b_n}$ (where $x_1, x_2, \dots, x_N$ are the eigenvalues of $G$), and for any multi-index $A \subset \{1, \dots, N\}$ denoting the nesting path, we set

$$
T_{a,s}^{(A)} = \chi_{a,s} \left( G^{(A)} \right) \left( \prod_{j \in \{A\}} x_j \right)^{-\frac{i u - |A| - \frac{s-a-1}{2}}{2}} . \quad (2.56)
$$

Then each function $T^{(A)}$ is a Bäcklund transform of $T^{(Ab)}$ (for any $b \notin \{A\}$). These successive Bäcklund transforms, labeled by a multi-index $A \subset \{1, \dots, N\}$ can be represented by Hasse diagram (see figure 7) [35].

From this example, as well as from the boundary condition in (a, s) space, we see that
each Bäcklund transform can be viewed as a decrease by one of the rank of the symmetry
group, as one might already guess from the “variation of constants” method described above
which decreases the degree of the finite-difference equation by one at each step as well.

Since the Bäcklund transform of T-functions fits into the same Hasse diagram as for
these characters, one can define Q-functions as

$$
Q_A(u) = T_{0,0}^{(A)} \left( u + \frac{i}{2} |A| \right) . \qquad (2.57)
$$

Let us now call nesting path a sequence of Bäcklund transforms from $T^{(\bar{\emptyset})}$ to $T^{(\emptyset)}$ (e.g. such as the green sequence of arrows on figure 7). Each nesting path is associated to a sequence of multi-indices $A_0, A_1, \dots, A_N$, where

$$
\emptyset \equiv \{A_0\} \subset \{A_1\} \subset \cdots \subset \{A_N\} \equiv \mathcal{B}, \qquad (2.58)
$$
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Hasse diagram for gl(3) T-functions. $T^{(123)}$ is the original T-function out of which a sequence of Bäcklund transforms is constructed. Each arrow corresponds to a Bäcklund transformation, reducing by one the number $|A|$ of indices labeling the function $T^{(A)}$. One should note that different sequences of arrows having the same starting and ending point (e.g. the different types of wavy arrows) correspond to sequences of Bäcklund transformations resulting in the same T-functions. The double arrows form a nesting path, i.e. a sequence of arrows from $T^{(\emptyset)}$ to $T^{(\emptyset)}$.</figcaption></figure>

such that $|A_n| = n$: for instance the green nesting path of figure 7 is associated to $A_0 = \emptyset$, $A_1 = 3$, $A_2 = 13$, $A_3 = 123$. Then one can show [4, 55] from (2.54) that

$$ \sum_{s \ge 0} T_{1,s}^{[+s]} e^{i s \partial_u} = Q_{\emptyset}^{[1-N]} W_{A_N;A_{N-1}} W_{A_{N-1};A_{N-2}} \cdots W_{A_1;A_0} Q_{\emptyset}^{-} \quad (2.59) $$

$$ \text{with } W_{I;J} = \left( 1 - \frac{Q_I^{[+3-|I|]}}{Q_I^{[+1-|I|]}} \frac{Q_J^{[-1-|J|]}}{Q_J^{[+1-|J|]}} e^{i \partial_u} \right)^{-1}, \quad (2.60) $$

where $e^{i\partial_u} f(u) = f(u + i) e^{i\partial_u}$ and where $(1 - f e^{i\partial_u})^{-1} = 1 + f e^{i\partial_u} + f e^{i\partial_u} f e^{i\partial_u} + \dots$.

One can then show (see [6, 58]) that the QQ-relation (2.7) arises<fnref n="17" /> from the constraint $W_{Aab;Aa}W_{Aa;A} = W_{Aab;Ab}W_{Ab;A}$, i.e. the statement that two sequences of Bäcklund transformations having the same starting point and the same endpoint in the Hasse diagram (e.g. the red and blue arrows in figure 7) should give rise to the same T-functions. Moreover, one can show that each function $T^{(A)}$ is then given by

$$ T_{a,s}^{(A)} = \star (Q_{(a)}^{[+s]} \wedge Q_{(|A|-|a|)}^{[-s-|A|]} \wedge \zeta_{\bar{A}}) \epsilon^{A\bar{A}} \quad \text{when } s \ge 0 \text{ and } 0 \le a \le |A|. \quad (2.61) $$

Another interesting remark is that Bäcklund flow suggests a different way to generate the Baxter equation:

$$ [ \mathcal{O}^N \cdot Q(u + i s/2) ]|_{a=N} = 0, \quad (2.62) $$

where $\mathcal{O}(u; a, s) = T_{a,s-1} e^{i \partial_u - \partial_a} \frac{1}{T_{a,s-1}} - T_{a,s}^+ e^{-\partial_s - \partial_a} \frac{1}{T_{a,s}^+}$. We present the proof in appendix A.5. Note that equations (2.62) and (2.44) do not coincide literally. We need to extensively exploit the Hirota equation to show their equivalence.

Although the Bäcklund flow was introduced for the case of Hirota equation on semi-infinite strip, the logic survives if we consider the case of the infinite strip of figure 6. For instance, (2.62) holds in either of cases.

<footnote n="17">¹⁷More precisely, this procedure gives the QQ-relation up to an i-periodic constant factor which can be viewed as an irrelevant normalisation, analogous to the factors $f_{|A|}$ in (2.21).</footnote>
## 2.8 Solution of Hirota equation on an L- or T-shaped lattice

### 2.8.1 Bijection between supersymmetric and non-supersymmetric Q-systems

In this section we will describe quite a remarkable fact: one does not need to change the geometric description to accommodate the Q-system for integrable systems with $gl(K|M)$ supersymmetry. One can still use the same Q-system as was used for $gl(K+M)$ case. I.e. one still considers $u$-dependent hyperplanes of $\mathbb{C}^N$ and imposes the same intersection property (1.5), however one needs to introduce a different set of coordinates to parameterise it.

Consider a decomposition

$$ \mathbb{C}^N = \mathbb{C}^K \oplus \mathbb{C}^M \qquad (2.63) $$

and choose coordinate vectors $\xi_1, \dots, \xi_N$ of $\mathbb{C}^N$ such that first $K$ of them span $\mathbb{C}^K$ and the latter span $\mathbb{C}^M$. Correspondingly, we introduce a set of “bosonic” indices $B = \{1, 2, \dots, K\}$ and a set of “fermionic” ones $\mathcal{F} = \{K+1, K+2, \dots, N\}$. Since we will see that in most setups, there is no risk of confusion<fnref n="18" /> between “bosonic” and “fermionic” indices, one may also label the latter as $\mathcal{F} = \{1, 2, \dots, M\}$.

The Q-functions which were used in previous paragraphs will be denoted here by small $q$ to avoid a clash with notations introduced below. The labelling of $q$'s is done according to the decomposition (2.63), i.e. $q_{AI}$, where $A$ is a multi-index from $B$ and $I$ is a multi-index from $\mathcal{F}$, denote the components of the (p;q)-form

$$ q_{(p;q)} = \sum_{|A|=p, |I|=q} q_{AI} \xi_{AI}, \qquad (2.64) $$

where $\xi_{b_1 b_2 \dots b_p i_1 i_2 \dots i_q} \equiv \xi_{b_1} \wedge \xi_{b_2} \wedge \dots \wedge \xi_{b_p} \wedge \xi_{i_1} \wedge \dots \wedge \xi_{i_q}$. The sum of $(n-k;k)$-forms

$$ q_{(n)} = \sum_{k=0}^{n} q_{(n-k;k)} \qquad (2.65) $$

is nothing but the $n$-form (2.10) which defines the hyperplane $V_{(n)}$ obeying (1.5).

We define the Q-functions $Q_{A|I}$, which form what will be called the supersymmetric $gl(K|M)$ Q-system, by a simple relation<fnref n="19" />

$$ Q_{A|I} \equiv \epsilon^{\bar{I}I} q_{A\bar{I}}, \qquad \text{where} \quad \{\bar{I}\} = \mathcal{F} \setminus \{I\}, \qquad (2.66) $$

i.e., it is a simple relabeling of the purely bosonic Q-functions. In geometric terms, the supersymmetric Q-system is obtained by the partial Hodge transformation along $\mathbb{C}^M$ direction of $\mathbb{C}^N$ of the non-supersymmetric Q-system.

This partial Hodge transformation can also be viewed as a rotation of the Hasse diagram, see section 3.1 and figure 9.

The supersymmetric Q-system of $gl(K|M)$-type was introduced in [56](see also [33, 35]). In [6], it was observed that the inverse of relation (2.66) can be used to map from $gl(K|M)$ to

<footnote n="18">¹⁸We should still view the two symbols $1 \in \mathcal{B}$ and $1 \in \mathcal{F}$ as two distinct objects, but the context will allow to know which of them is referred to when we use the symbol 1.</footnote>

<footnote n="19">¹⁹No summation over $\bar{I}$.</footnote>
gl(N) system and it was named “bosonisation” (or “fermionisation”) trick. We extensively rely on this mapping in various places of the paper.

As should be clear from (2.66), $Q_{a_1...a_p|i_1...i_q}$ is antisymmetric under a permutation of bosonic a-indices and under a permutation of fermionic i-indices. Correspondingly, the graded Q-forms are defined by

$$Q_{(n|p)} \equiv \sum_{|A|=n, |I|=p} Q_{A|I} \xi_A \wedge \zeta_I, \qquad (2.67)$$

where $\zeta$'s are some anti-commuting variables independent of $\xi$ ($\zeta$'s and $\xi'$s are defined to anti-commute between them as well).

For the following discussions, it would be convenient to introduce the Hodge duality map. It is induced from (2.11) which can be written more explicitly as $\star\zeta_{AI} = \epsilon_{AI\bar{A}\bar{I}} \xi_{\bar{A}\bar{I}} \equiv (-1)^{|I||\bar{A}|} \epsilon_{A\bar{A}} \epsilon_{II'} \xi_{\bar{A}\bar{I}}$. One deduces that the hodge-dual Q-functions should be defined by:

$$Q^{A|I} \equiv (-1)^{|A||\bar{I}|} \epsilon^{\bar{A}A} \epsilon^{II'} Q_{\bar{A}|\bar{I}}. \qquad (2.68)$$

Finally, for the sake of notational simplicity, we will also sometimes denote $Q_A \equiv Q_{A|\emptyset}$ and $Q_\emptyset \equiv Q_{\emptyset|\emptyset}$ (and the same for Q-functions with upper indices).

The bijection between supersymmetric and non-supersymmetric Q-systems is quite a remarkable property; we spend the remainder of this subsection discussing it.

One thing to note is a possibility to rewrite the Hodge transformation as a Grassmannian Fourier transform. Namely, introduce the sums

$$q[u; \xi_1, \dots, \xi_N] \equiv \sum_{n=0}^{N} q_{(n)} = \sum_{n=0}^{N-M} \sum_{m=0}^{M} \sum_{|A|=n, |I|=m} q_{AI} \xi_A \xi_I, \qquad (2.69a)$$

$$Q[u; \xi_1, \dots, \xi_{N-M}, \zeta_1, \dots, \zeta_M] \equiv \sum_{n=0}^{N-M} \sum_{m=0}^{M} (-1)^{M-m+1} \sum_{|A|=n, |I|=m} Q_{A|I} \xi_A \zeta_I, \qquad (2.69b)$$

Then they are related by the Grassmann integral

$$Q = \int \prod_{i \in \mathcal{F}} d\xi_i q e^{\xi_i \zeta_i}. \qquad (2.70)$$

This representation suggests adopting a Dirac sea point of view on the bijection transformation. Whereas the description in terms of $q$'s corresponds to “excitations” of the “bare vacuum”, description in terms of $Q$'s corresponds to “excitations” of the “sea” created by filling the bare vacuum with all the excitations from the set $\mathcal{F}$. Such an interpretation has a close relation to the Grassmannian construction in the works of Jimbo and Miwa [26].

A similar relation exists between the characters of gl(K|M) and gl(N) algebras. The characters of compact representations are, correspondingly, the Schur symmetric polynomials $s_\lambda(x)$ and Schur supersymmetric polynomials $s_\lambda(x|y)$, where $\lambda$ is a Young diagram, see e.g. [59]. Schur polynomials form a ring

$$s_{\lambda} s_{\mu} = \sum_{\nu} c_{\lambda \mu}^{\nu} s_{\nu}, \qquad (2.71)$$
where $c_{\lambda\mu}^{\nu}$ are the Littlewood-Richardson coefficients which are the same for the ordinary and supersymmetric cases. Hence, in the limiting case of $M \to \infty$ and $K \to \infty$ when none of $s_{\lambda}$ is zero due to the bound on a group rank, the rings of ordinary and supersymmetric Schur polynomials are isomorphic. It is not difficult to construct the isomorphism mapping explicitly. We can do this by exploiting the 2nd Weyl formula from table 1, the reader may also focus on the most important case of rectangular representations when the Weyl formula reduces to (2.3) and can be derived directly from the simplified Hirota equation (2.4). The 2nd Weyl formula expresses all the characters through $\chi_{1,s}$ — the characters for the representation $\lambda = (s, 0, 0, \dots)$. On the other hand, the generating function for $\chi_{1,s}(x|y)$ of $gl(K|M)$ is known:

$$ \frac{\prod_{i \in \mathcal{F}} (1 + \epsilon y_i)}{\prod_{a \in \mathcal{B}} (1 - \epsilon x_a)} = \sum_{s=0}^{\infty} \epsilon^s \chi_{1,s}(x|y). \qquad (2.72) $$

Hence the map between supersymmetric and non-supersymmetric characters is induced by the replacements

$$ 1 + \epsilon y \leftrightarrow \frac{1}{1 - \epsilon x} \qquad (2.73) $$

in the generating function. The mapping becomes an isomorphism in the limit when the numbers of y's and x's are infinite. This relation can be thought of as a statement (equivalent to the partial Hodge transformation) that “adding a fermionic index” is the same as “removing a bosonic index”. Indeed, adding a fermionic index or removing a bosonic index is realised by multiplication of the generating function by a factor $1 + \epsilon\alpha$ (where $\alpha$ is either $-y$ or $x$). See also section 2.8.5 for a motivation of this principle in terms of Bäcklund transforms.

### 2.8.2 QQ-relations with a grading

As explained in the previous section, the supersymmetric Q-system is obtained by a simple relabelling of ordinary Q-functions: we just use $Q_{A|I}$ instead of $q_{A\bar{I}}$. Therefore, all the QQ-relations in the supersymmetric basis would be, eventually, an algebraic consequence of (2.7). Nevertheless, despite the simplicity of (2.7), the emergent algebraic structure turns out to be very rich.

First of all, the original QQ-relation (2.7) splits into three equations due to possibility of multiplying Q-functions with different gradings [50, 52, 56, 57, 60, 61]

$$ Q_{A|I}Q_{Aa|I} = Q_{Aa|I}^{+}Q_{Ab|I}^{-} - Q_{Aa|I}^{-}Q_{Ab|I}^{+}, \qquad (2.74a) $$

$$ Q_{Aa|I}Q_{A|Ii} = Q_{Aa|Ii}^{+}Q_{A|I}^{-} - Q_{A|I}^{+}Q_{Aa|Ii}^{-}, \qquad (2.74b) $$

$$ Q_{A|I}Q_{A|Iij} = Q_{A|Ii}^{+}Q_{A|Ij}^{-} - Q_{A|Ii}^{-}Q_{A|Ij}^{+}, \qquad (2.74c) $$

It is easy to see how they correspond to (2.7), especially if to note the general rule that adding a fermionic index in $Q_{A|I}$ is equivalent to removing this index from $q_{A\bar{I}}$.

Now, we present a handful of algebraic relations which all follow from (2.74). Their derivation is given in appendix A.6.
Firstly, we have the obvious relations

$$Q_{(n|0)} = \frac{Q_{(1|0)}^{[n-1]} \wedge Q_{(1|0)}^{[n-3]} \wedge \cdots \wedge Q_{(1|0)}^{[1-n]}}{\prod_{1 \le k \le n-1} Q_{\emptyset}^{[n-2k]}}, \quad Q_{(0|p)} = \frac{Q_{(0|1)}^{[p-1]} \wedge Q_{(0|1)}^{[p-3]} \wedge \cdots \wedge Q_{(0|1)}^{[1-p]}}{\prod_{1 \le k \le p-1} Q_{\emptyset}^{[p-2k]}}, \quad (2.75)$$

which are identical to the relations of section 2.6 because they do not mix “bosonic” and “fermionic” indices. Secondly, the following expressions give all Q-functions in terms of $Q_\emptyset$, $Q_{(1|0)}$, $Q_{(0|1)}$ and $Q_{(1|1)}$:$^{20}$

$$Q_{(n|n)} = \frac{(-1)^{\frac{n(n-1)}{2}} Q_{(1|1)}^{n}}{n! Q_{\emptyset|\emptyset}^{n-1}} \equiv \frac{(-1)^{\frac{n(n-1)}{2}} Q_{(1|1)} \wedge Q_{(1|1)} \wedge \cdots \wedge Q_{(1|1)}}{n! Q_{\emptyset|\emptyset}^{n-1}} \quad (n \text{ times}) \quad (2.76)$$

$$Q_{(n|p)} = \frac{Q_{(p|p)}^{[t]}}{Q_{\emptyset}^{[t]}} \wedge Q_{(n-p|0)} \qquad \text{where } n \ge p \\ \text{and } t \in \{n-p, n-p-2, \dots, -n+p\} \quad (2.77)$$

$$Q_{(n|p)} = (-1)^{n(p+1)} \frac{Q_{(n|n)}^{[t]}}{Q_{\emptyset}^{[t]}} \wedge Q_{(0|p-n)} \qquad \text{where } n \le p \\ \text{and } t \in \{p-n, p-n-2, \dots, -p+n\}. \quad (2.78)$$

These expressions were already implicitly incorporated into *sparce determinants* of [35], and rewritten in terms of forms in [3] for the *psu(2, 2|4)* case without proofs.

These relations can be recast into equations for the components $Q_{A|I}$ of these forms: the relation (2.76) becomes

$$Q_{A|I} = \frac{\det Q_{a|i}}{\prod_{i \in I} (Q_{\emptyset})^{n-1}} \quad \text{or more explicitly} \quad Q_{b_1, b_2, \dots, b_n | f_1, f_2, \dots, f_n} = \frac{\det Q_{b_i | f_j}}{(Q_{\emptyset})^{n-1}}, \quad (2.79)$$

whereas the equation (2.77) (resp (2.78)) states that if $|A| = n$, and $|I| = p$ with $n \ge p$ (resp $n \le p$), then for any $t \in \{|n-p|, |n-p|-2, \dots, -|n-p|\}$ we have

$$Q_{A|I} = \frac{(-1)^{p(n+1)}}{Q_{\emptyset}^{[t]}} \sum_{\substack{|B|=|I| \\ |C|=|A|-|I|}} Q_{B|I}^{[t]} Q_{C|\emptyset}^{\delta_A^{BC}}, \quad (2.80)$$

$$\text{resp. } Q_{A|I} = \frac{(-1)^{n(p+1)}}{Q_{\emptyset}^{[t]}} \sum_{\substack{|J|=|A| \\ |K|=|I|-|A|}} Q_{A|J}^{[t]} Q_{\emptyset|K}^{\delta_I^{JK}}, \quad (2.81)$$

$$\text{where } \delta_{i_1 i_2 \dots i_n}^{j_1 j_2 \dots j_n} = \det_{1 \le a, b \le n} \delta_{ia}^{j_b}. \quad (2.82)$$

Note that the role of e.g. $\delta_A^{BC}$ in (2.80) is to anti-symmetrise the index BC.

Another interesting class of relations is obtained by using both Q-functions and their Hodge duals:

<footnote>$^{20}$One should note that $Q_\emptyset$, $Q_{(1|0)}$, $Q_{(0|1)}$ and $Q_{(1|1)}$ are not independent: they are related by (2.74b), which states that $Q_{(1|0)} \wedge Q_{(0|1)} = Q_{(1|1)}^+ Q_\emptyset^- - Q_{(1|1)} Q_\emptyset^+$.</footnote>
$$Q_{\emptyset|J} = \frac{(-1)^{n|J|}}{Q_{\emptyset|\emptyset}^{[t]}} \sum_{|A|=n} Q^{A|\emptyset} Q_{A|J}^{[t]} \quad \text{when } n = |\mathcal{B}| - |\mathcal{F}| + |J| \ge 0 \\ \text{and } t \in \{-n, -n+2, \dots, n-2, n\}, \qquad (2.83)$$

$$Q_{A|\emptyset} = \frac{(-1)^{n|A|}}{Q_{\emptyset|\emptyset}^{[t]}} \sum_{|J|=n} Q^{\emptyset|J} Q_{A|J}^{[t]} \quad \text{when } n = |\mathcal{F}| - |\mathcal{B}| + |A| \ge 0 \\ \text{and } t \in \{-n, -n+2, \dots, n-2, n\}, \qquad (2.84)$$

$$\sum_{|A|=|I|} Q_{A|I} Q^{A|J} = (-1)^{|I|(|\mathcal{F}|+1)} \sum_{|L|=|J|-|I|} \delta_{LI}^{J} Q^{\emptyset|L} Q_{\emptyset|\emptyset} \quad \text{when } |J| - |I| = |\mathcal{F}| - |\mathcal{B}| \ge 0, \qquad (2.85)$$

$$\sum_{a \in \mathcal{B}} Q_a^{[+t]} Q^a = \begin{cases} (-1)^{|\mathcal{B}|+|\mathcal{F}|+1} (Q^\emptyset)^\pm Q_\emptyset^{[t \mp 1]} & \text{if } t = \pm n \text{ where } n = |\mathcal{B}| - |\mathcal{F}| > 0 \\ 0 & \text{if } t \in \{n-2, n-4, \dots, -n+2\}, \end{cases} \qquad (2.86)$$

$$\sum_{a \in \mathcal{B}} Q_a Q^a = (Q^\emptyset)^{-} Q_\emptyset^{+} - (Q^\emptyset)^{+} Q_\emptyset^{-} \quad \text{if } |\mathcal{B}| = |\mathcal{F}|. \qquad (2.87)$$

Note that equation (2.84) is obtained from (2.83) by interchanging bosonic and fermionic indices. Obviously the same can be done for any other relation. For instance it follows from (2.86) that

$$\sum_{i \in \mathcal{F}} Q_{\emptyset|i}^{[\pm n]} Q^{\emptyset|i} = (-1)^{|\mathcal{B}|+|\mathcal{F}|+1} (Q^\emptyset)^{\pm} Q_\emptyset^{[\pm n \mp 1]} \quad \text{when } n = |\mathcal{F}| - |\mathcal{B}| > 0. \qquad (2.88)$$

Similarly, one can take the Hodge dual of each relation, i.e. perform the substitutions

$$Q_{A|I} \mapsto Q^{A|I}, \qquad Q^{A|I} \mapsto (-1)^{(|A|+|I|)(|\bar{A}|+|\bar{I}|)} Q_{A|I}, \qquad (2.89)$$

which leave the QQ-relations invariant, and are compatible with the sign in (2.68). For instance (2.83) becomes

$$Q^{\emptyset|J} = \frac{(-1)^{n|\mathcal{F}|}}{(Q^{\emptyset|\emptyset}^{[t]})[t]} \sum_{|A|=n} Q_{A|\emptyset}(Q^{A|J})^{[t]} \quad \text{when } n = |\mathcal{B}| - |\mathcal{F}| + |J| \ge 0 \\ \text{and } t \in \{-n, -n+2, \dots, n-2, n\}, \qquad (2.90)$$

where the sign $(-1)^{n|\mathcal{F}|}$ is obtained by simplifying the expression $(-1)^{n|J|+|A||\bar{A}|}$ obtained from the substitution (2.89).

**Examples.** It turned out [3] that in the case of AdS/CFT (where T-hook in figure 8 has $K_1 = K_2 = M_1 = M_2 = 2$; and $Q_{\emptyset|\emptyset} = Q_{\bar{\emptyset}|\bar{\emptyset}} = 1$), the above-listed relations are very useful. We give below some of them specified to this particular case:

* Setting $|J| = 1$ or $|A| = 1$ in (2.83), (2.84), one gets two interesting relations

$$Q_{\emptyset|i} = - \sum_a Q^{a|\emptyset} Q_{a|i}^{\pm}, \qquad Q_{a|\emptyset} = - \sum_i Q^{\emptyset|i} Q_{a|i}^{\pm}, \qquad (2.91)$$

which correspond to (4.14a1)-(4.14b1) in [3]. The other relations (4.14) are obtained by Hodge duality (2.89).

* Setting $|I| = |J| = 1$ in (2.85), one gets $\sum_a Q^{a|i} Q_{a|j} = -\delta_j^i$, which is the relation (4.15a) in [3].
<figure><img src="image_11.png" /><figcaption>Figure 8. The right, left, and upper strip of the T-hook are delimited by the diagonals (dashed, gray). The figure corresponds to $K_1 = 3$, $K_2 = 2$, $M_1 = 1$ and $M_2 = 2$.</figcaption></figure>

* In the AdS/CFT case, the r.h.s. of (2.87) vanishes, giving the relation (4.16a) in [3].

We did not describe all possible relations in this section. For instance, another interesting class worth mentioning involves equations of finite difference type of order $\ge 2$. Such kind of Baxter-type relations were exploited for instance in [53]. Very recently, the fourth-order equation having $Q_{\emptyset|i}$ as four solutions played the decisive role in the derivation of the BFKL equation from the AdS/CFT integrability [62].

We see that the algebra of Q-functions is indeed very rich. We should think about these relations as an opportunity for discovering short-cuts through the Q-system that link the physically most-improtant Q-functions for practical problems to solve. For each particular problem or calculation, one should look for a specific, most convenient subset of these relations.

### 2.8.3 Expression for T-functions in a T-hook

At the level of Q-functions, we have seen that it was necessary to introduce two different sets of indices, which we called “bosonic” and “fermionic”, and which are distinguished in the QQ-relations (2.74). If we denote by $K$ (resp $M$) the number $|\mathcal{B}|$ (resp $|\mathcal{F}|$) of bosonic (resp fermionic) labels, then the Q-functions are related to the algebra $\mathfrak{gl}(K|M)$.

At the level of T-functions which obey the Hirota equation (2.1) on a generic T-hook figure 8, one should also specify a real form: it is $\mathrm{su}(K_1, K_2|M)$ in the most general case, with $K_1 + K_2 = K$). As a consequence, the set $\mathcal{B}$ of bosonic indices should be split into a union of two non-intersecting sets $\mathcal{B}_1$ and $\mathcal{B}_2$:

$$ \mathcal{B} = \mathcal{B}_1 \sqcup \mathcal{B}_2, \quad \text{where } |\mathcal{B}_1| = K_1, \quad |\mathcal{B}_2| = K_2. \qquad (2.92) $$

With these two sets, we introduce *graded* and *ungraded* forms in the same way as in, respectively, (2.67) and (2.64):

$$ Q_{(n_1,n_2|p)} \equiv \sum_{|R|=n_1, |L|=n_2, |I|=p} Q_{RL|I} \xi_R \wedge \xi_L \wedge \zeta_I, \qquad (2.93) $$

$$ q \equiv \sum_{|R|=n_1, |L|=n_2, |I|=p} q_{RLI} \xi_R \wedge \xi_L \wedge \xi_I, \quad R \subset \mathcal{B}_1, L \subset \mathcal{B}_2, I \subset \mathcal{F}. \qquad (2.94) $$
With these notations, the Hirota equation on the $(K_1|M_1 + M_2|K_2)$ T-hook has the following solution

$$ T_{a,s} = \varepsilon_r(a, s) \star (Q_{(a,0|0)}^{[\tilde{s}]} \wedge Q_{(K_1-a,K_2|M)}^{[-\tilde{s}]}), \quad \text{if } \tilde{s} \ge \tilde{a}, \quad (2.95a) $$

$$ T_{a,s} = \varepsilon_u(a, s) \star (Q_{(K_1,0|M_1-s)}^{[+\tilde{a}]} \wedge Q_{(0,K_2|M_2+s)}^{[-\tilde{a}]}), \quad \text{if } \tilde{a} \ge |\tilde{s}|, \quad (2.95b) $$

$$ T_{a,s} = \varepsilon_l(a, s) \star (Q_{(K_1,K_2-a|M)}^{[-\tilde{s}]} \wedge Q_{(0,a|0)}^{[+\tilde{s}]}) \quad \text{if } \tilde{s} \le -\tilde{a}, \quad (2.95c) $$

where $M_1 + M_2 = M$ and the choice of $M_1$ and $M_2$ is arbitrary and defines the origin of the T-hook, as in figure 4b. In (2.95), we used the notations

$$ \tilde{s} = s - s_0, \quad \tilde{a} = a - a_0, \quad s_0 = \frac{-K_1 + K_2 + M_1 - M_2}{2}, \quad a_0 = \frac{K - M}{2}, \\ \varepsilon_r(a, s) = \dot{\mathbb{I}}^{M(a-s)}(-1)^a K_2, \quad \varepsilon_l(a, s) = \dot{\mathbb{I}}^{M(a-s)}(-1)^{a(K_1+M)}, \quad (2.96) \\ \text{and} \\ \varepsilon_u(a, s) = \dot{\mathbb{I}}^{M(a-s)}(-1)^{(a+K)(M_2+s)+K_2(K_1+M)}. $$

The practical meaning of these notations is: $(s_0, a_0)$ is the coordinate of the intersection of the diagonals on figure 8, and $(\tilde{s}, \tilde{a})$ are the coordinates, with respect to this point, of an arbitrary node on the T-hook.

The proof that (2.95) indeed solve the Hirota equation is given in appendix A.7. There we use, in particular, a possibility to represent the solution in terms of the bosonised functions (2.94).

The semi-infinite strip of figure 3a is the case $K_1 = N$, $K_2 = M_1 = M_2 = 0$ of the T-hook. In this case, the above expressions of T-functions match the expressions of section 2.6 up to an overall redefinition of the (shift of the) Q-functions:

$$ Q_{(n)} \mapsto Q_{(n)}^{[3N/2]}, \quad (2.97) $$

which obviously leaves all QQ-relations invariant.

Other interesting special cases of the Wronskian solution (2.95) include: the compact real form $\mathrm{su}(M|K_2)$ corresponding to $B_1 = \emptyset$ and L-hook shape of non-zero T-functions shown in figure 4a; the compact real form $\mathrm{su}(K_1, |M)$ corresponding to $B_2 = \emptyset$ and a mirror-reflected L-hook;<fnref n="21" /> and, finally the non-compact and non-supersymmetric case $\mathrm{su}(K_1, K_2)$ which corresponds to $\mathcal{F} = \emptyset$ and the “slim-hook” shape first discussed in [46] (see e.g. figure 1b in [63]). The slim-hook is solved using purely bosonic Q-system constructed on $\mathbb{C}^{K_1+K_2}$. We expect that Hirota equation on such a hook will appear in affine Toda integrable models.

### 2.8.4 Symmetries

Similarly to its bosonic version, the graded Q-system has rotational and rescaling symmetry.

<footnote n="21">²¹For real forms we use notations of [63], a more detailed exposition is planned in [64]. Although the real form $\mathrm{su}(K_1, |M)$ is isomorphic to $\mathrm{su}(K_1|M)$ and hence comma is usually not written, we should distinguish the case with comma and without when $\mathrm{su}(K_1, |M)$ and $\mathrm{su}(M|K_2)$ are simultaneously subalgebras of a bigger non-compact algebra $\mathrm{su}(K, |M|K)$.</footnote>
**Gauge transformations.** It is suitable to parameterise two available rescalings (gauge transformations) by

$$Q_{A|I} \mapsto g_1^{[|A|-|I|]} g_2^{-[|A|+|I|]} Q_{A|I} \qquad (2.98)$$

which replaces (2.25).

This transformation generates the following two gauge transformation of T-functions:

$$T_{a,s} \mapsto g_1^{[a+s-s_0]} g_2^{-[a+s-s_0]} g_1^{-[a-s+s_0+K-M]} g_2^{[a-s+s_0-K+M]} T_{a,s} \qquad (2.99)$$

Another two gauge degrees of freedom of T-functions (cf. (2.32)) are actually fixed for what concerns the solution (2.95). This solution was specially written to satisfy the Wronskian gauge:

$$T_{K_1+1,M_1} = T_{K_1,M_1+1} \quad \text{and} \quad T_{K_2+1,-M_2} = T_{K_2,-M_2-1}, \qquad (2.100)$$

which immediately implies, by virtue of Hirota equation,

$$T_{K_1+n,M_1} = T_{K_1,M_1+n} \quad \text{and} \quad T_{K_2+n,-M_2} = T_{K_2,-M_2-n}, \quad n \ge 0. \qquad (2.101)$$

and reflects the fact that the corresponding characters are equal: $\chi_{K_1+n,M_1} = \chi_{K_1,M_1+n}$, $\chi_{K_2+n,-M_2} = \chi_{K_2,-M_2-n}$.

The signs $\epsilon_r$, $\epsilon_u$, and $\epsilon_l$ in (2.95) were chosen, in particular, to ensure the Wronskian gauge condition (2.100). Hence, as in the case of semi-infinite strip, we understand that (2.95) is a general solution modulo two gauge transformations.

**Rotations.** The graded $gl(N - M|M)$ Q-system is algebraically equivalent to its bosonised version and hence it is in principle invariant under $GL(N)$ H-transformations originating from (2.26). However, the obvious explicit rotations are only a subgroup $GL(N - M) \times GL(M)$ which leaves invariant the decomposition (2.63). All other rotations, implicitly there, would not preserve the T-functions and hence should not be considered.

Furthermore, the T-functions of T-hook are invariant only under unimodular rotations from $GL(K_1) \times GL(K_2) \times GL(M)$ which preserve the grading of the forms (2.93). It is important to realise that prior to constructing a T-hook, one has to agree how to decompose indices into sets $B_1, B_2$, and $F$ and then stick to the bases which respect such a decomposition. Also, it is possible to exchange the role of bosonic and fermionic indices and, in particular, decompose into sets $B, F_1, F_2$. The choice of a basis and decomposition into sets depends on a real form one wishes to associate to T-hook and how this real form is related to analytic properties of Q-functions. From the same $GL(K|M)$-system we can construct different T-hooks. It is an additional question to justify which of the T-hooks (maybe several) are physically meaningful in a given explicit problem and what is their physical interpretation.

### 2.8.5 Bäcklund flow in supersymmetric case

One can also introduce Q-functions from a sequence of Bäcklund transformations. It was demonstrated already for the bosonic case in section 2.7, and we saw that QQ-relations can be interpreted as the fact that different paths on the Hasse diagram (see figure 7) correspond to the same transformation.
This approach can be generalized to the super-symmetric case, i.e. for L-hook [56] and T-hook [65]. The relation to Wronskian determinants was shown in [7]. We remind the arguments for the L-hook case only. Consider the Lax pair condition (2.54) near the internal boundary of the hook, namely set $(a, s) = (K - 1, M)$ in (2.54a) and $(a, s) = (K, M - 1)$) in (2.54b). One can see that if $T$ obeys the Hirota equation on a $(K|M)$ fat hook, then $F$ can obey it on a $(K - 1|M)$ or a $(K|M + 1)$ fat hook. The transformation from a $(K|M)$ to a $(K - 1|M)$ fat hook is the exact analog of the Bäcklund transformation of section 2.7, and corresponds to the removal of a “bosonic” index from the Q- and T-functions. By contrast, it is the inverse of the transformation from a $(K|M)$ to a $(K|M + 1)$ fat hook which can be regarded as a Bäcklund transformation removing a “fermionic” index; in this case, the function $T$ of (2.54) is the Bäcklund transform of the function $F$. Hence we see that the same transformation “adds a fermionic index” or “removes a bosonic index”, justifying the partial Hodge transformation (2.66), and the analogous observation (2.73) at the level of characters.

Furthermore, one finds out from the linear system (2.54) and the definition (2.57) that the generating series (2.59) can be generalized to the L-hook. To this end, we encode a nesting path as a sequence of labels

$$ (A_0|I_0 \equiv \emptyset|\emptyset) \subset A_1|I_1 \subset A_2|I_2 \subset \cdots \subset (A_{K+M}|I_{K+M} \equiv \mathcal{B}|\mathcal{F}), \quad (2.102) $$

which are included into each other and obey $|A_n| + |I_n| = n$. Each step $n$ of the nesting path is a Bäcklund transform which can be either associated to a “bosonic” index (then $|A_{n+1}| = |A_n| + 1$ and $I_{n+1} = I_n$) or a “fermionic” index (then $A_{n+1} = A_n$ and $|I_{n+1}| = |I_n| + 1$). Then, the generalization of the generating series (2.59) is

$$ (-i)^M \sum_{s \ge 0} T_{1,s}^{[+s+(M-K)/2]} e^{i s \partial_u} = Q_{\emptyset}^{[1-K+M]} W_{K+M} W_{K+M-1} \cdots W_1 Q_{\emptyset}^{-}, \quad (2.103) $$

$$ \text{with } W_n = \begin{cases} \left( 1 - \frac{Q_{A_n|I_n}^{[3-|A_n|+|I_n|]}}{Q_{A_n|I_n}^{[+1-|A_n|+|I_n|]}} \frac{Q_{A_{n-1}|I_{n-1}}^{[-1-|A_{n-1}|+|I_{n-1}|]}}{Q_{A_{n-1}|I_{n-1}}^{[+1-|A_{n-1}|+|I_{n-1}|]}} e^{i \partial_u} \right)^{-1}, & \text{if } |A_n| = |A_{n-1}| + 1 \\ 1 - \frac{Q_{A_n|I_n}^{[-1-|A_n|+|I_n|]}}{Q_{A_n|I_n}^{[+1-|A_n|+|I_n|]}} \frac{Q_{A_{n-1}|I_{n-1}}^{[3-|A_{n-1}|+|I_{n-1}|]}}{Q_{A_{n-1}|I_{n-1}}^{[+1-|A_{n-1}|+|I_{n-1}|]}} e^{i \partial_u}, & \text{if } |I_n| = |I_{n-1}| + 1 \end{cases} \quad (2.104) $$

As an illustration (which will be used in the next section), the coefficient of $e^{i \partial_u}$ gives

$$ T_{1,1} = i^M Q_{\emptyset}^{[s_0]} \sum_{n=1}^{K+M} \varepsilon_n \frac{Q_{A_n|I_n}^{[+2\varepsilon_n-|A_n|+|I_n|-s_0]}}{Q_{A_n|I_n}^{[-|A_n|+|I_n|-s_0]}} \frac{Q_{A_{n-1}|I_{n-1}}^{[-2\varepsilon_n-|A_{n-1}|+|I_{n-1}|-s_0]}}{Q_{A_{n-1}|I_{n-1}}^{[-|A_{n-1}|+|I_{n-1}|-s_0]}} Q_{\emptyset}^{-[s_0]} \quad (2.105) $$

$$ \text{where } \varepsilon_n = (-1)^{|I_n|-|I_{n-1}|} \quad \text{and} \quad s_0 = \frac{M-K}{2}. $$

The QQ-relations (2.74) can be easily deduced¹⁷ from this generating series [6] (see also [58]). All T-functions can be expressed from (2.103) and (2.2), and the result which comes out coincides with the Wronskian expressions (2.95).
## 3 Polynomiality and twist

In the previous sections, in our study of general Wronskian solutions of Hirota functional equations with particular “hook” boundary conditions, as well as the QQ-relations, we had no need to precise the analyticity properties of the functions of spectral parameter $u$. If we now try to do it, generically it will impose severe restrictions on the analyticity properties of the whole ensemble of these functions. For instance if they are assumed to be polynomial or meromorphic functions, or having a given set of singularities we will have rather strong restrictions on the type and position of the singularities and zeros due to the Hirota and QQ functional relations. It turns out that the analyticity of the T- and Q-functions is an extremely important ingredient to characterise a given physical model. In this section, we discuss a well-known example of rational spin chains which correspond, in the case of compact representations, to polynomial T-functions, with polynomial Q-functions. In particular, we discuss the effect of the twist on polynomiality conditions. In the section 4, we will consider the AdS/CFT Q-system which corresponds to multivalued analytic Q-functions with specific monodromy properties.

### 3.1 Polynomiality and spin chains

The spectra of periodic rational spin chains in compact representations of $su(K|M)$ with integer fermionic Dynkin label are encoded in the polynomial solutions of the QQ-relations, with certain constraints on the polynomials that precise the details of the spin chain considered (length, representation, inhomogeneities). There are various ways to establish the correspondence between the spectrum of a spin chain and the QQ-relations, probably the most direct one is to construct the Q-operators acting in the quantum space of the spin chain (several constructions are available in the literature [41, 52, 66–68]) and identify Q-functions with the eigenvalues of these operators.

In appendix C, we list the required constraints on the polynomials for a generic case.<fnref n="22" /> In this section, we discuss one of the most simple and probably the most important examples — a homogeneous rational spin chain of length $L$ in the defining representation. For this spin chain one imposes

$$Q_{\emptyset} = 1, \qquad Q_{\emptyset}^{[s_0]} = T_{0,0} = u^L, \qquad (3.1)$$

where $s_0 = \frac{M-K}{2}$.

It is remarkable that, algebraically, the Q-system is the same for all symmetry algebras $su(K|M)$ with given value of $K + M$. The difference appears only in how the constraints (3.1) appear on the Hasse diagram. This phenomenon is illustrated in figure 9, where we see that the “bosonization trick” (2.66) amounts to a rotation of the Hasse diagram.

Note also how the Hodge duality map acts. It flips the Hasse diagram (upside-down), hence the boundary conditions (3.1) change to $Q_{\emptyset}^{[s_0]} = u^L$, $Q_{\emptyset} = 1$. These new boundary conditions correspond to the conjugation of the defining representation. It is not difficult

<footnote n="22">²²The discussion of the appendix applies beyond the polynomial case and includes any highest-weight type representations.</footnote>
<figure><img src="image_12.png" /></figure>

(a) Hasse diagram for an $su(3)$ spin chain's Q-functions: the Q-functions of figure 5 are not written explicitly, only the conditions $Q_{\emptyset} = 1$ and $Q_{\emptyset}^{[s_0]} = u^L$ are made manifest (the shift $[s_0]$ is omitted).

<figure><img src="image_13.png" /></figure>

(b) Hasse diagram for an $su(2|1)$ spin chain's Q-functions: two different equivalent orientations of the Hasse diagram are presented. In the notations of (2.66), the diagram to the left corresponds to functions $q_{AI}$ while the diagram to the right corresponds to $Q_{A|I}$.

<figure><img src="image_14.png" /></figure>

(c) Notation for the orientation of Hasse diagram: the dashed red lines indicate how to write the QQ-relation for a given facet.

<caption>**Figure 9.** Boundary conditions and orientation of the Hasse diagrams.</caption>

to guess then that the Hodge duality performs an outer automorphism $E_{ij} \mapsto -E_{ji}$ from the point of view of representation theory.

Although there are no other constraints than (3.1) on the structure of the polynomial Q-functions, the QQ-relations themselves strongly constrain possible polynomials, and one ends up with only a discrete set of possibilities. All of them can be found by solving Bethe equations for super-symmetric rational spin chains [69, 70] which are a set of algebraic equations for the roots of the polynomials.

The QQ-relations directly imply the Bethe equations as follows [56]: if $Q_{Aa|I}$ has a zero at position $u = \theta$, then equation (2.74a) implies that $\theta$ is also a zero of $Q_{A|I}^{+}Q_{Aab|I}^{+} - Q_{Aa|I}^{++}Q_{Ab|I}$ and of $Q_{A|I}^{-}Q_{Aab|I}^{-} + Q_{Aa|I}^{-+}Q_{Ab|I}$. Hence it is a zero of the linear combination $Q_{A|I}^{+}Q_{Aab|I}^{+}Q_{Aa|I}^{-} + Q_{A|I}^{-}Q_{Aab|I}^{-}Q_{Aa|I}^{++}$, and we get the equation

$$-1 = \frac{Q_{A|I}(\theta + \frac{\dot{i}}{2})Q_{Aa|I}(\theta - \dot{i})Q_{Aab|I}(\theta + \frac{\dot{i}}{2})}{Q_{A|I}(\theta - \frac{\dot{i}}{2})Q_{Aa|I}(\theta + \dot{i})Q_{Aab|I}(\theta - \frac{\dot{i}}{2})}, \quad \text{where } Q_{Aa|I}(\theta) = 0. \quad (3.2)$$

This equation involves the Q-functions corresponding to two successive “bosonic” Bäcklund transformations along the nesting path (2.102).

If one has two subsequent “fermionic” Bäcklund transformations, we get analogously

$$-1 = \frac{Q_{A|I}(\theta + \frac{\dot{i}}{2})Q_{A|Ii}(\theta - \dot{i})Q_{A|Iij}(\theta + \frac{\dot{i}}{2})}{Q_{A|I}(\theta - \frac{\dot{i}}{2})Q_{A|Ii}(\theta + \dot{i})Q_{A|Iij}(\theta - \frac{\dot{i}}{2})}, \quad \text{where } Q_{A|Ii}(\theta) = 0. \quad (3.3)$$
Finally, one also derives

$$1 = \frac{Q_{Aa|Ii}(\theta + \frac{i}{2})Q_{A|I}(\theta - \frac{i}{2})}{Q_{A|I}(\theta + \frac{i}{2})Q_{Aa|Ii}(\theta - \frac{i}{2})}, \quad \text{where } Q_{Aa|I}(\theta) = 0, \quad \text{or } Q_{A|Ii}(\theta) = 0, \quad (3.4)$$

if a “fermionic” Bäcklund transformation is followed by a “bosonic” one (case $Q_{Aa|I}(\theta) = 0$) or if a “bosonic” Bäcklund transformation is followed by a “fermionic” one (case $Q_{A|Ii}(\theta) = 0$).

There is a special case when all there terms of (2.74a) (or (2.74b), or (2.74c)) become zero at some $u = \theta$. Such zero can be an “exceptional” root of Bethe equations which was accidentally trapped into a singular point, we can resolve this singularity by introducing a twist, see (3.7). Another possibility, which is not realised for defining representation but is possible for other cases, is that such zero is not demanded to be a solution of Bethe equation; instead, it belongs to a source term thus specifying the type of a spin chain, see appendix C for further details.

When exceptional Bethe roots are properly accounted, a solution of the Bethe equations allows to restore the Q-functions and vice versa, hence the Q-system and the Bethe equations encode the same information. This is another way to see that the polynomial ansatz with boundary conditions of type (3.1) indeed corresponds to a rational spin chain, as there is a handful of ways to derive Bethe equations, including those not relying on construction of Q-operators or even T-operators.

Each eigenspace of the spin chain which forms an irrep of $su(K|M)$ symmetry algebra corresponds to a solution of the QQ-relations. For our particular example of the homogeneous spin chain in the defining representation, the commuting family of operators that act diagonally on the discussed eigenspaces includes the operator with only nearest-neighbour interactions of the spin chain sites: $\mathcal{H} = \sum_{i=1}^{L} P_{i,i+1}$, where $P$ is a permutation operator. It is usually interpreted, up to an addition or multiplication by a constant, as the physical Hamiltonian of the system. For an eigenstate characterized by a given solution of the QQ-relations, the eigenvalue (energy) is given by

$$E = i \partial_u \log T_{1,1}|_{u=0} = \frac{i \partial_u T_{1,1}}{T_{1,1}|_{u=0}}. \quad (3.5)$$

In the expression (2.105) for $T_{1,1}$, we see that due to the factor $Q_{\emptyset}^{[s_0]} = u^L$, the terms $n \neq K + M$ do not contribute to (3.5) if $L \ge 2$. Thus, we have

$$E = i \partial_u \log \left( (u \pm i)^L \frac{Q^\mp}{Q^\pm} \right)|_{u=0} = \pm \left( L - \sum_k \frac{1}{\theta_k^2 + \frac{1}{4}} \right), \quad (3.6)$$

where $\pm$ denotes $\epsilon_{K+M}$, the grading of the first Bäcklund transform of the nesting path, and $Q = Q_{A_{K+M-1}|I_{K+M-1}}^{[s_0]}$, while $\theta_k$ denote the roots of $Q \propto \prod_k (u - \theta_k)$.

Formulae of type (3.6) is an extra information one needs to introduce, apart from finding a solution of a QQ-relations, for computing the spectrum of rational spin chains. By contrast, in the case of the AdS/CFT integrable system, the Hamiltonian is part of the symmetry algebra charges that define the large-$u$ asymptotic of Q-functions (4.5). One can derive
the formulae like (3.6), at least in the asymptotic Bethe Ansatz limit [3], but now as a non-trivial consequence of analytic properties of Q-functions rather than an independent input.

## 3.2 Twisted spin chains and Q-system

Spin chains can be deformed by the introduction of a “twist”, which changes the periodicity condition.<fnref n="23" /> For rational spin chains, this twist $G$ can be chosen diagonal without a loss of generality, and we denote its eigenvalues as $x_1, \dots, x_K, y_1, \dots, y_M$.

It is known that in the presence of a twist, the Bethe equations of the rational spin chain are deformed and become

$$-\frac{x_b}{x_a} = \frac{Q_{A|I}(\theta + \frac{i}{2}) Q_{Aa|I}(\theta - \dot{i}) Q_{Aab|I}(\theta + \frac{i}{2})}{Q_{A|I}(\theta - \frac{i}{2}) Q_{Aa|I}(\theta + \dot{i}) Q_{Aab|I}(\theta - \frac{i}{2})}, \quad \text{where } Q_{Aa|I}(\theta) = 0, \qquad (3.7a)$$

$$\frac{x_a}{y_i} = \frac{Q_{Aa|Ii}(\theta + \frac{i}{2}) Q_{A|I}(\theta - \dot{i}) Q_{A|Iij}(\theta + \frac{i}{2})}{Q_{A|I}(\theta + \dot{i}) Q_{Aa|Ii}(\theta - \frac{i}{2})}, \quad \text{where } Q_{Aa|I}(\theta) = 0, \text{ or } Q_{A|Ii}(\theta) = 0, \qquad (3.7b)$$

$$-\frac{y_j}{y_i} = \frac{Q_{A|I}(\theta + \frac{i}{2}) Q_{A|Ii}(\theta - \dot{i}) Q_{A|Iij}(\theta + \frac{i}{2})}{Q_{A|I}(\theta - \frac{i}{2}) Q_{A|Ii}(\theta + \dot{i}) Q_{A|Iij}(\theta - \frac{i}{2})}, \quad \text{where } Q_{A|Ii}(\theta) = 0, \qquad (3.7c)$$

which constrains the roots of the polynomials $Q_{A|I}$. Hence these polynomial Q-functions do not obey the same QQ-relations (2.74) as in the absence of twist. There exist two equivalent ways to describe this situation: one can either add an exponential prefactor which breaks the polynomiality of Q-functions, or deform the QQ-relations.

### 3.2.1 Twist as an exponential prefactor

One possibility is to consider Q-functions which are not polynomials anymore. More precisely, $Q_{A|I}$ is the product of the exponential prefactor $(\prod_{a \in A} x_a)^{-i u}$ and of a polynomial function denoted by the letter $Q$:

$$Q_{A|I} \propto \left( \frac{\prod_{a \in A} x_a}{\prod_{i \in I} y_i} \right)^{-i u} Q_{A|I}, \qquad (3.8)$$

then it is immediate to see that (3.2)–(3.4) for $Q$ becomes (3.7a)–(3.7c) for $Q$, whereas it is a bit less trivial to see that (3.6) is not modified<fnref n="24" /> (in particular, $\partial_u \log \frac{Q^+}{Q^-}$ is invariant under multiplication of $Q$ by $x^{-iu}$).

In (3.8), the symbol “$\propto$” denotes an arbitrary normalization for the polynomial $Q_{A|I}$ (for instance the coefficient of the leading power can be set to one). This normalization is not very relevant, as it cancels out in (3.2)–(3.4) and (3.6).

<footnote n="23">²³More explicitly the Hamiltonian becomes $\mathcal{H} = \sum_{i=1}^{L-1} P_{i,i+1} + (P_{1,L} \cdot G^{-1} \otimes I^{\otimes L-2} \otimes G)$.</footnote>

<footnote n="24">²⁴In principle, one could expect that in (3.6), the factor $(u \pm i)^L$ (from $Q_0$) becomes $(\text{sdet } u)^{-iu}(u \pm i)^L$, if $\text{sdet } G \neq 1$. This is not the case as the expression (3.5) holds in a gauge where $T_{1,1}$ is polynomial, i.e. one has to divide the expression (2.105) by $(\text{sdet } u)^{-iu}$.</footnote>
In this setup, the simplest character solution of the QQ-relations (2.74) when the twist has pairwise-distinct eigenvalues is the following:

$$Q_{b_1, b_2, \dots, b_n | f_1, f_2, \dots, f_p}^{(\chi)} = \frac{\prod_{i=1}^{n} x_{b_i}^{-i u + \frac{p-n-1}{2}} \prod_{1 \le i < j \le n} (x_{b_i} - x_{b_j}) \prod_{1 \le i < j \le p} (y_{f_j} - y_{f_i})}{\prod_{j=1}^{n} y_{f_j}^{-i u + \frac{p-n+1}{2}} \prod_{i=1}^{n} \prod_{j=1}^{p} (x_{b_i} - y_{f_j})} . \quad (3.9)$$

It is obtained by solving the QQ-relation when $Q_{b|\emptyset} = x_b^{-iu}$, $Q_{\emptyset|f} = y_f^{iu}$. The corresponding T-function, obtained by plugging (3.9) into (2.95), is related to the characters $\chi_{a,s}(G)$ as follows:

$$T_{a,s}^{(\chi)} = \dot{i}^M (a+s+a s) \chi_{a,s}(G) Q_{\emptyset}^{[a-s+s_0]} . \quad (3.10)$$

Note that if the twist $G = \text{diag}(x_1, \dots, x_K, y_1, \dots, y_M)$ belongs to $\text{SL}(K|M)$, i.e. if $\prod_{a \in B} x_a = \prod_{i \in \mathcal{F}} y_a$, then the factor $Q_{\emptyset}^{[a-s+s_0]}$ is just a $u$-independent normalization (in particular, it is equal to the Vandermonde determinant in the bosonic case and the Cauchy double alternant in the $\text{SL}(M|M)$ case).

In more general situation, one can see that the T-functions are polynomial functions of $u$ if $G \in \text{SL}(K|M)^{25}$. By contrast, unlike the untwisted case, their Bäcklund transforms are in general not polynomial functions of $u$.

### 3.2.2 Twist as a holomorphic connection

A more geometric approach consists in adding to the fiber bundle described in section 2.3 a holomorphic $\text{GL}(N)$ connection $A$. In other words, one gauges the global rotational $\text{GL}(N)$ symmetry making it local.

In this setup, we slightly deform the definition (2.19) of Plücker coordinates of $V_{(n)}$: we now introduce the coordinates of $V_{(n)}$ as forms $Q_{(n)}$ such that

$$V_{(1)} = \left\{ \mathbf{x} \in \mathbb{C}^N ; \mathcal{Q}_{(1)} \wedge \mathrm{P} e^{\int_u^0 A(v)dv} \mathbf{x} = 0 \right\}, \quad (3.11)$$

$$V_{(n)} = \left\{ \mathbf{x} \in \mathbb{C}^N ; \mathcal{Q}_{(n)} \wedge \mathrm{P} e^{\int_{u-i}^0 \frac{A(v)}{2} dv} \mathbf{x} = 0 \right\}, \quad (3.12)$$

where the path-ordered integral $\mathrm{P} e^{\int_u^v A(v)dv}$ is the parallel transport from spectral parameter $u$ to $v$, and the shift $\dot{i}^{\frac{n-1}{2}}$ was introduced arbitrarily in (3.12) to simplify upcoming expressions. To obtain (3.11)–(3.12), one naturally chooses (as a generalization of (2.22)):

$$\frac{\mathcal{Q}_{(n)}}{\mathcal{Q}_{\emptyset}^{[-n]}} = \left( \mathrm{P} e^{\int_{u+i}^{u-i} \frac{n-1}{2} A(v)dv} \frac{\mathcal{Q}_{(1)}^{[n-1]}}{\mathcal{Q}_{\emptyset}^{[n-2]}} \right) \wedge \left( \mathrm{P} e^{\int_{u+i}^{u-i} \frac{n-1}{2} A(v)dv} \frac{\mathcal{Q}_{(1)}^{[n-3]}}{\mathcal{Q}_{\emptyset}^{[n-4]}} \right) \wedge \dots \wedge \frac{\mathcal{Q}_{(1)}^{[1-n]}}{\mathcal{Q}_{\emptyset}^{[-n]}} . \quad (3.13)$$

While the $A=0$ case corresponds to the non-twisted case of the previous sections, the twist corresponds to constant $A$: indeed, if $A$ is a constant diagonal matrix and we

<footnote>$^{25}$If $G \notin \text{SL}(K|M)$, then the T-functions are the product of $(\text{sdet } g)^{-i u}$ and of a polynomial, so that they are polynomial up to a gauge.</footnote>
denote diag(x₁, ..., xₖ, y₁, ..., yₘ) = G = e^A then we get (at the price of repeating the bosonization trick in the supersymmetric case)

$$Q_{A|I}Q_{A,a|I} = x_a Q_{A,a|I}^{+}Q_{A,b|I}^{-} - x_b Q_{A,a|I}^{-}Q_{A,b|I}^{+}, \quad (3.14a)$$

$$Q_{A|I,i}Q_{A,a|I} = x_a Q_{A,a|I,i}^{+}Q_{A|I}^{-} - y_i Q_{A,a|I,i}^{-}Q_{A|I}^{+}, \quad (3.14b)$$

$$Q_{A|I,i,j}Q_{A|I} = y_i Q_{A|I,j}^{+}Q_{A|I,i}^{-} - y_j Q_{A|I,j}^{-}Q_{A|I,i}^{+}. \quad (3.14c)$$

These relations imply the Bethe equations (3.7).

Obviously, this approach is equivalent to the approach of section 3.2.1, and the twisted QQ-relations (3.14) are equivalent to the standard QQ-relations (2.74) up to the change of variables

$$Q_{A|I} = Q_{A|I} \left( \frac{\prod_{a \in A} x_a}{\prod_{i \in I} y_i} \right)^{\frac{i u + \frac{|A|-|I|}{2}}{2}}. \quad (3.15)$$

This change of variable corresponds to

$$Q_{(n|p)} = P e^{\int_0^{u+i \frac{p-n+1}{2}} A(v)dv} Q_{(n|p)}, \quad (3.16)$$

which is a simple parallel transport to the origin.

One easily checks (for instance by use of the mapping (3.15) to the non-twisted case (2.75)–(2.78)) that the QQ-relations (3.14) are solved in a gauge where $Q_\emptyset = 1$,<fnref n="26" /> similarly to the untwisted case, by

$$Q_{(n|0)} = (G^{n-1} Q_{(1|0)}^{[n-1]}) \wedge (G^{n-2} Q_{(1|0)}^{[n-3]}) \wedge \dots \wedge Q_{(1|0)}^{[1-n]}, \quad (3.17a)$$

$$Q_{(0|p)} = Q_{(0|1)}^{[p-1]} \wedge (G Q_{(0|1)}^{[p-3]}) \wedge (G^2 Q_{(0|1)}^{[p-5]}) \wedge \dots \wedge (G^{p-1} Q_{(0|1)}^{[1-p]}), \quad (3.17b)$$

$$Q_{\emptyset|i}Q_{a|\emptyset} = x_a Q_{a|i}^{+} - y_i Q_{a|i}^{-} \quad (3.17c)$$

$$Q_{(n|n)} = \frac{(-1)^{\frac{n(n-1)}{2}}}{n!} Q_{(1|1)}^{n} \equiv \frac{(-1)^{\frac{n(n-1)}{2}}}{n!} Q_{(1|1)} \wedge Q_{(1|1)} \wedge \dots \wedge Q_{(1|1)} \quad (3.17d)$$

$$Q_{(n|p)} = Q_{(p|p)}^{[p-n]} \wedge Q_{(n-p|0)} \quad \text{where } n \ge p, \quad (3.17e)$$

$$Q_{(n|p)} = (-1)^{n(p+1)} Q_{(n|n)}^{[p-n]} \wedge Q_{(0|p-n)} \quad \text{where } n \le p, \quad (3.17f)$$

where the twist appears only in the non-local relations (3.17a)–(3.17c). Similarly, one can write the T-functions in terms of Q-functions instead of Q-functions as in (2.95). To this end, one should just substitute (3.15) into (2.95). We do not repeat here these expressions, which become slightly less compact than in the non-twisted case (2.95) (see [35] for similar formulae).

Quite curiously, the holomorphic connection point of view allows constructing Q-systems with arbitrary value of A, not only a constant one which we consider in this article. The non-constant value of A produces a Q-system which we cannot identify with systems studied in the literature. It would be indeed very interesting to explore this new case.

<footnote n="26">²⁶In a gauge where $Q_\emptyset$ is not equal to one, the relations (3.17) still hold up to a denominator, as in (2.75)–(2.78). This denominator can be restored by substituting $Q_{(n|p)} \rightsquigarrow Q_{(n|p)}/Q_\emptyset^{[n-p]}$ into the relations (3.17).</footnote>
**Remark: Bäcklund flow.** If we use these twisted QQ-relations, we can also understand the Q-functions in terms of Bäcklund transformations [56, 57], as in section 2.7, with the slight difference that the Lax Pair (2.54) has to be replaced with

$$T_{a+1,s}F_{a,s}^{+} - T_{a,s}^{+}F_{a+1,s} = g_{\alpha} T_{a+1,s-1}^{+}F_{a,s+1} \quad (3.18a)$$

$$T_{a,s+1}F_{a,s}^{-} - T_{a,s}^{-}F_{a,s+1} = g_{\alpha} T_{a+1,s}F_{a-1,s+1}^{-} \quad (3.18b)$$

In this expression, $g_\alpha$ denotes an eigenvalue of the twist (either $x_\alpha$ if $\alpha$ is a bosonic index or $y_\alpha$ otherwise) and its index $\alpha$ is the index which is removed by the Bäcklund transform, in the notations of figure 7.

### 3.3 Dependence on twist and the untwisting limit: illustration on examples

The dependence of a Q-system on twist can be quite non-trivial. For instance, the behaviour of the Q-functions is singular when two eigenvalues of the twist matrix tend to become equal. We can see it already on the example of the 1-st Weyl formula in the table 1 where both the numerator and denominator become zero in this limit. If we focus on the QQ- relation (3.14a), then we see that if the Q-functions are polynomial (consider the case of compact spin chains), then their degrees obey

$$\deg \mathcal{Q}_{A,a|I} + \deg \mathcal{Q}_{A,b|I} = \begin{cases} \deg \mathcal{Q}_{A|I} + \deg \mathcal{Q}_{A,a|I} & \text{if } x_a \neq x_b \\ \deg \mathcal{Q}_{A|I} + \deg \mathcal{Q}_{A,a,b|I} + 1 & \text{if } x_a = x_b. \end{cases} \quad (3.19)$$

For instance, in the case of the $su(2)$ Heisenberg spin chain of length $L$, we have $\deg \mathcal{Q}_{12} = L$ and $\deg \mathcal{Q}_{0} = 0$, which means that the degree of the polynomial $Q_1Q_2$ increases by one in the limit $x_1 - x_2 \to 0$. This seemingly harmless change in the degree leads, as we shall see, to a significant reorganisation of the Q-system.

Let us consider a more general picture now. The space of all possible diagonal twists is the projective space $CP^{N-1}$ parameterised by $[x_1 : x_2 : \dots : x_K : y_1 : y_2 \dots : y_M]$. We can study how a Q-system changes upon analytic continuation in this space. We then face several different effects when performing such a study:

* Untwisting limit. The limiting points on hyperplanes $x_a = x_b$, $x_a = y_i$, or $y_i = y_j$ are quite singular as we explained above. This type of limit receives the most of attention in this section, a special emphasis is put on the fully untwisted case when the twist matrix $G$ becomes the identity. In general, the result of the limit $G \to \mathbb{I}$ may depend on how the identity is approached. We discuss only the limit $G = \mathbb{I} + \epsilon g_0$ with $\epsilon \to 0$ and assume $g_0$ being in generic position.

* Degeneration of solutions. Singular points on $CP^{N-1}$ of other type live on hyperplanes $x_a = 0$ or $y_i = 0$. Space of solutions to QQ-relations degenerates there, and analytic continuation around such hyperplanes has a non-trivial monodromy.

* Borel ambiguities. Generic points on $CP^{N-1}$ also have certain interest. We included an example of a non-compact rational spin chain into this section. The definition of the associated Q-system for such a chain suffers from Borel-type ambiguities, with position of Borel singularities being dependent on the value of the twist.
* Relation to representation theory of $gl(K|M)$. In the presence of generic twist, Cartan sub-algebra of $gl(K|M)$ is the only remaining symmetry<fnref n="27" /> of a spin chain. However, the full $gl(K|M)$ symmetry is restored in the untwisting limit; this is another way to see why this limit is singular. Certain properties of irreducible representations (irreps) find their counterpart in analytic structure of Q-systems.

In this section, we will discuss several explicit examples based on small-rank algebras to illustrate the mentioned effects, the gained experience is then summarised in section 3.4. The generalisation from the explicit examples to an arbitrary rank is also done, but only for the question of untwisting limit and for the case of finite-dimensional irreps. We hope to study other phenomena beyond small-rank cases in future works.

We explore the above-mentioned properties purely assuming existence of a Q-system with certain analytic properties (mostly polynomiality), without questioning its origin. This approach is conceptually important because there are situations, e.g. the AdS/CFT, where the existence of a Q-system is known although there is presently no operatorial construction, beyond the leading order in the perturbation theory, of a Hamiltonian and T- (hence Q-)operators. However, we should note that for the spin chains discussed in this section, all analytic properties follow from operatorial constructions [66–68, 71, 72]. We explicitly demonstrate this link in subsection 3.3.4.

In the discussion of irreps, we use the following notations (with more details given in appendix C). The vector of an irrep is characterised by its fundamental weight<fnref n="28" />

$$[\lambda_1, \lambda_2, \dots, \lambda_K; \nu_1, \nu_2, \dots, \nu_M], \qquad (3.20)$$

where $\lambda$'s and $\nu$'s are eigenvalues of the corresponding Cartan generators. In physical jargon, $\lambda_a$ is called the “number of spin d.o.f.”, or “number of spins” in short<fnref n="29" /> in direction a; and $\nu_i$ is the “number of spins” in direction $\hat{i}$. Indeed, in the case of spin chains with sites in the defining representation, the weight of any eigenstate comprises non-negative integers which sum up to the number of sites, $\sum_a \lambda_a + \sum_i \nu_i = L$, each site is being thought of as a spin degree of freedom.

An irrep can be labelled by the weight of its lowest weight vector. We emphasise that the definition of the latter is not universal as it depends on a total order imposed on the set of indices $\{1, \dots, K, \hat{1}, \dots, \hat{M}\}$, see appendix C. The corresponding ambiguity finds its counterpart in the untwisting limit of a Q-system. However, quite expectedly, more invariant objects — T-functions — do not depend on the choice of order.

### 3.3.1 $su(2)$: untwisting should be supplemented with a rotation

The simplest example to commence with is the $su(2)$ XXX spin chain in the defining representation. The twist-related effects were studied probably the most on this example,

<footnote n="27">²⁷In the sense that Q- and T-operators commute with symmetry generators, see appendix B.3. Also note that the Cartan sub-algebra is realised in a standard way, by generators $E_{kk}$, if the twist is dingonal.</footnote>

<footnote n="28">²⁸Hats in $\nu_i$ may be omitted when expression is unambiguous.</footnote>

<footnote n="29">²⁹This terminology must not be confused with the name spin which is related to the value of Casimir operator.</footnote>
quite a detailed and illuminating analysis was presented in [66], including explicit examples of construction of Q-operators. We will partially repeat the known statements, but also complement this discussion, in the next subsection, with novel observations about analytic dependence of the Q-functions on the twist. In particular, we remark that the famous umbrella-shaped configurations of Bethe roots are well-approximated by zeros of Laguerre polynomials.

In this example, the only non-trivial QQ-relation is $Q_1^+Q_2^- - Q_1^-Q_2^+ = Q_0Q_{12}$. It is explicitly realised as<fnref n="30" />

$$Q_1^+Q_2^- - zQ_2^+Q_1^- \propto u^L, \qquad (3.21)$$

where $Q_a$ are polynomials defined modulo an overall normalisation, and where we defined $z \equiv x_2/x_1$.

According to (3.21), the degree of the polynomial $Q_1Q_2$ should be $L$ if $z \neq 1$ and $L+1$ if $z = 1$. But any limit of a degree $L$ polynomial cannot have a higher degree than $L!$ Hence something non-trivial should happen in the limit $z \to 1$. Let us find the explicit solutions of (3.21) for $L = 2$ to clarify the situation. One can do this study analytically, but numerical solution already suffices to demonstrate the effect. The Hilbert space is 4-dimensional, hence one should find 4 solutions. Moreover, one expects appearance of spin-1 and spin-0 irreps at the point $z = 1$. We find for $z = e^{-\frac{2\pi i}{100}}$ which is sufficiently close to 1:

* Solutions describing the triplet when $z \to 1$:

$$M_1=0: \begin{aligned} Q_1 &= 1, \\ Q_2 &= 1 + 6.28 \times 10^{-2} u + 1.97 \times 10^{-3} u^2, \end{aligned}$$

$$M_1=1: \begin{aligned} Q_1 &= 1 - 3.14 \times 10^{-2} u, \\ Q_2 &= 1 + 3.14 \times 10^{-2} u, \end{aligned}$$

$$M_1=2: \begin{aligned} Q_1 &= 1 - 6.28 \times 10^{-2} u + 1.97 \times 10^{-3} u^2, \\ Q_2 &= 1. \end{aligned}$$

* Solution describing the singlet when $z \to 1$:

$$M_1=1: \begin{aligned} Q_1 &= u + 7.85 \times 10^{-3}, \\ Q_2 &= u - 7.85 \times 10^{-3}, \end{aligned}$$

where $M_1$ is the degree of $Q_1$.

The small in magnitude numbers will become identically zero in the limit $z = 1$. We see that the degree of $Q_1Q_2$ actually drops, or it remains the same at most. Furthermore, both Q-functions approach the same value, $\lim_{z \to 1} Q_1 = \lim_{z \to 1} Q_2 = Q$, so that

$$\lim_{z \to 1} [Q_1^+Q_2^- - zQ_2^+Q_1^-] = 0, \qquad (3.22)$$

hence there is no way to normalise this Wronskian combination to $u^L$ in (3.21) without going to its subleading in $(z - 1)$ terms.

<footnote n="30">³⁰We drop the offset $[s_0]$ from the further discussion.</footnote>
The $\mathrm{su}(K)$ generalisation of the observed phenomenon is the following one: in the untwisting limit, all polynomials $Q_A$ with the same value of the number of indices $|A|$ tend to the same $Q$-functions which can be denoted as $Q_{\leftarrow|A|}$:

$$Q_{\leftarrow|A|} \propto \lim_{G \to \mathbb{I}} Q_A. \qquad (3.23)$$

We understand $G \to \mathbb{I}$ as $G = \mathbb{I} + \epsilon g_0$ with $\epsilon \to 0$; the equality (3.23) should hold for all but finite number of the limiting directions on the group given by $g_0$. Furthermore we understand that $Q_A$ is normalised to be neither infinite nor zero in the $G \to \mathbb{I}$ limit. That is we consider some value $u = u_0$ at which $Q_A(u_0) \neq 0$ for $G$ sufficiently close to the identity matrix (e.g. $u = 0$ for triplet solution in the example above), and normalise $Q_A(u_0) = 1$. For example, in this convention $Q_A = 1 + \frac{u}{z-1}$ would produce $\propto u$ in the untwisting limit $z \to 1$.

Equation (3.23) means that, when one takes the direct untwisting limit, one formally obtains only $N + 1$ non-equal $Q$-functions $Q_{\leftarrow|A|}$, with $|A| = 0, 1, 2, \dots, N$, out of $2^N$ $Q$-functions of the twisted system.

The functions $Q_{\leftarrow|A|}$ are quite special. First, we can obtain all these functions from $Q$-functions on a nesting path (2.58), hence the corresponding Bethe equations (3.2) would be well defined. Second, the energy of a state can be still computed using (3.6), with $Q = Q_{\leftarrow K-1}$. Hence, in principle, the emerging functions $Q_{\leftarrow|A|}$ contain all necessary information. Moreover, it is known (see appendix C) that

$$Q_{\leftarrow|A|} \sim u^{\sum_{b=1}^{|A|} \lambda_b}, \quad u \to \infty, \qquad (3.24)$$

where $[\lambda_1 \lambda_2 \dots \lambda_N]$ is the lowest weight of an irreducible multiplet associated to the (remnant of) $Q$-system in the full untwisting point.

On the other hand, it is quite dissatisfying that other $2^N - (N+1)$ $Q$-functions seem to be lost in the untwisting procedure. The art of obtaining these other $Q$-functions is to take the untwisting limit simultaneously with rotating the $Q$-system, e.g. to consider combinations of the type

$$\frac{Q_1 - Q_2}{x_1 - x_2} = \frac{x_1^{-i u} Q_1 - x_2^{-i u} Q_2}{x_1 - x_2}. \qquad (3.25)$$

In the limit $x_1 = x_2$ for $\mathrm{su}(2)$ case, this sample combination becomes a polynomial, and its degree can be larger than the degree of $Q_1$ or $Q_2$, due to the expansion

$$x^{-i u} = 1 - i u \log x + \dots \qquad (3.26)$$

Then, if we define $Q_\emptyset = Q_{\leftarrow 0} = 1$, $Q_1 = Q_{\leftarrow 1}$, $Q_{12} = Q_{\leftarrow 2} = u^L$ to be $Q$-functions for $z=1$ case, one still has to introduce $Q_2$ which would be given by $Q_2 \propto \lim_{z \to 1} \frac{x_1^{-i u} Q_1 - x_2^{-i u} Q_2}{x_1 - x_2}$. It is quite clear that $Q$-functions defined in such a way satisfy the desired $QQ$-relation $Q_1^+ Q_2^- - Q_2^+ Q_1^- = Q_\emptyset Q_{12}$.

In section 3.3.4 we explicit another example of implementation of rotation of the type (3.25), and we describe a general strategy of defining the untwisting limit alongside with rotation in appendix B.2.
<figure><img src="image_15.png" /></figure>

(a) Zeros of $Q_1$ on example of $L = 34, s = 15, m = 10$ and twist $z = e^{-2\pi i/3}$. Crosses show exact location of Bethe roots and circles — their approximation using (3.28). $Q_1$ degenerates into a degree-2 polynomial $Q = u^2 - \frac{1}{4} \left(\cot \frac{\pi}{L-1}\right)^2$ in the untwisting limit, if one follows the shortest path from $z = e^{-2\pi i/3}$ to $z = 1$.

<figure><img src="image_16.png" /></figure>

(b) Zeros of polynomial $Q_1^+Q_2^-$, an example with $M_1 = 3, M_2 = 5$, and $z = e^{-4\pi}$. Circles denote zeros of $Q_1^+$ and diamonds — zeros of $Q_2^-$. The monodromy around $z = 0$ results roughly in rotation of the zeros by the angle $\frac{2\pi}{L}$ around $u = 0$, thus producing new $Q_a$.

<caption>Figure 10. Sample patterns of Bethe roots.</caption>

### 3.3.2 su(2): analytic continuation in twist meets representation theory

One can pose a question: what should be known about twisted Q-functions to predict the Q-system emerging in the untwisting limit? For instance, can we predict the values of $\lambda_b$ in the large-u behaviour (3.24)? The first, naive expectation is that if we know the large-u behaviour of twisted Q-functions, e.g. in the $su(2)$ case

$$Q_1 \sim x_1^{-iu} u^{M_1}, \quad Q_2 \sim x_2^{-iu} u^{M_2}, \qquad (3.27)$$

with $M_1 + M_2 = L$ then we can deduce the large-u behaviour in the limit $z = 1$. This expectation is wrong as we can see from our explicit numerical example. Generically, both powers $M_1$ and $M_2$ will drop in the direct untwisting limit (without rotations), and one cannot predict by what amount without a more detailed information about the structure of Q-functions.

One however understands that degree of a polynomial drops if some of its zeros go to $\infty$. We can find approximate analytic solution describing the structure of these large zeros. As is derived in appendix B.1, all twisted Q-functions $Q_1, Q_2$ which tend to a polynomial $Q$ of degree $\frac{L}{2} - s$, have the following structure when $z \to 1$:

$$ \begin{aligned} Q_1(u) &\approx \mathbb{Q}(u) L_m^{(-2s-1)}(-iu \log z), \\ Q_2(u) &\approx \mathbb{Q}(u) L_{2s-m}^{(-2s-1)}(+iu \log z), \end{aligned} \qquad (3.28) $$

for $0 \le m \le 2s$, and where $L_m^{(\alpha)}(x)$ are associated Laguerre polynomials, see an example in figure 10a.

From the point of view of representation theory, solution $Q$ with $\deg Q = \frac{L}{2} - s$ corresponds to a spin-$s$ multiplet. The multiplet consists of $2s + 1$ states and we can
observe these states as coming from precisely $2s+1$ solutions (3.28) of the twisted Q-system. Note that $M_1 \equiv \deg Q_1 = \frac{L}{2} - s + m$ and $M_2 \equiv \deg Q_2 = \frac{L}{2} + s - m$ define the weight of an eigenstate:

$$M_a \text{ is the eigenvalue of the Cartan generator } E_{aa} \qquad (3.29)$$

in the twisted case. The eigenstate remains of the same weight at any value of twist, even at point $z=1$. However, the relation between the weight and the powers $M_1 = \deg Q_1$ and $M_2 = \deg Q_2$ does not work at the point $z=1$, due to the above-discussed power drop and degeneration effects.<fnref n="31" /> Moreover, the powers $M_a$ are even not uniquely defined when $z=1$. Indeed, for generic twist, one sees that for every $a$, the relation between degrees and charges associates $Q_a$ with the Cartan generator $E_{aa}$ and with the eigenvalue $\chi_a$ of the twist — the labelling of Q-functions is then unambiguously identified to the labeling of eigenstates of the twist, cf. (3.8). By contrast, in the $G \to \mathbb{I}$ limit, different labelings of Q-function are possible since one can always rotate them. Similarly to the labelling of Q-functions, their asymptotic behavior is ambiguous in the $G \to \mathbb{I}$ limit, since it is not rotation-invariant.

The result about $2s+1$ being the number of solutions (3.28) is obtained solely by analytic analysis of the QQ-relation (3.21), but it is, de-facto, in agreement with representation theory of $\mathfrak{su}(2)$. It would be interesting to generalise this analysis to higher ranks and to derive in this way a rich set of representation theory properties solely from analytic structure imposed by QQ-relations.

Approximation (3.28) is valid only in proximity to $z=1$. If we are far from this point, can we still predict what would happen with Q-functions in the untwisting limit? In fact, the result of the untwisting procedure depends on a path which connects a point $z \neq 1$ to $z=1$. Hence, twisted Q-functions are not assigned a-priory to some particular multiplet at $z=1$. There is a non-trivial monodromy around $z=0, \infty$ which allows to jump from one multiplet to another.

Consider for instance the vicinity of $z=0$. At the leading order of small-$z$ expansion one has

$$Q_1^+ Q_2^- \propto u^L, \quad \text{hence} \quad Q_1 = (u^-)^{M_1}, \quad Q_2 = (u^+)^{M_2}, \qquad (3.30)$$

i.e. all solutions with given weight $[M_1 M_2]$ degenerate to the solution (3.30). It is then well-expected that these solutions will mix when one performs an analytic continuation around the point $z=0$.

At the subleading order, the QQ-relation can be written as

$$Q_1^+ Q_2^- = (1-z) u^L + z Q_2^+ Q_1^- = (1-z) u^L + z(u+\hat{i})^{M_2} (u-\hat{i})^{M_1} + o(z). \qquad (3.31)$$

The polynomial on the r.h.s. has roots at positions

$$u_k = \frac{1}{\sqrt{z}} e^{i\pi \left(\frac{M_2+1}{L} - \frac{1}{2}\right)} e^{\frac{2\pi i k}{L}} + \mathcal{O}\left(\frac{1}{\sqrt{z}}\right)^2, \quad k = 0, 1, \dots, L-1, \qquad (3.32)$$

an example is shown in figure 10b.

<footnote n="31">³¹When $z=1$, a link to the representation theory is realised by (3.24).</footnote>
We have to assign these $L$ distinct zeros to either $Q_1^+$ or $Q_2^-$, and there are $\binom{L}{M_1}$ ways of doing this. Since $M_1$ can range from 0 to $L$, we conclude that there are precisely $2^L$ solutions of the QQ-relations (3.21), which is precisely the dimension of the Hilbert space of the length-$L$ XXX Heisenberg spin chain. We emphasise that this enumeration result was obtained solely by analysing (3.21), no connection with XXX spin chain was exploited. Historically, enumeration of solutions to Bethe equations was a non-trivial issue [1] which relied on the string hypothesis about the patterns of Bethe roots. This hypothesis is known to be, strictly speaking, wrong. In the above-proposed approach, enumeration becomes indeed simple, at a suitable value of the twist parameter $z = 0$, and it does not require any assumptions. We can look on this result also from the other side: in operatorial derivation of Q-system, we do know that Q-functions — the eigenvalues of Q-operators — are polynomials in $u$ and that QQ-relations are satisfied. However, it requires an extra analysis to show that all polynomial solutions of the QQ-relations are indeed eigenvalues of Q-operators. The obtained enumeration result is a way to resolve this issue.

Analytic continuation produces a cyclic permutation of the Bethe roots (3.32) and hence induces nontrivial monodromy on solutions of the Q-system. For instance, the singlet state from our numerical example is exchanged with a vector in the triplet state that has the same weight, upon the analytic continuation.

We saw that analytic continuation in twist is a useful tool allowing one to better control combinatorial and group-theoretical aspects of the Q-system. It can potentially have other interesting applications, one of them is analysing the above-mentioned string hypothesis, see appendix A of [63].

### 3.3.3 gl(1|1): lowest weight depends on a nesting path

We consider the gl case, assuming that $x/y \neq 1$, otherwise we won't be able to introduce a non-trivial twist. In higher-rank generalisations one can restrict to sl case only.

For a spin chain of length $L$, one has $Q_{\emptyset|\emptyset} = 1$ and $Q_{1|1} = u^L$, thus the only non-trivial QQ-relation $Q_{1|1}^+ Q_{\emptyset|\emptyset} - Q_{1|1}^- Q_{\emptyset|\emptyset} = Q_{1|\emptyset} Q_{\emptyset|1}$ becomes explicitly

$$ \left(u + \frac{i\dot{\epsilon}}{2}\right)^L - z \left(u - \frac{i\dot{\epsilon}}{2}\right)^L \propto Q_{1|\emptyset} Q_{\emptyset|1}, \qquad (3.33) $$

where $z \equiv y/x$.

Since the l.h.s. is a polynomial of degree $L$, we can distribute zeros of this polynomial between $Q_{1|\emptyset}$ and $Q_{\emptyset|1}$ in $2^L$ ways, thus correctly reproducing the dimension of the Hilbert space.

As before, asymptotics of Q-functions encodes the weight of the state $[\lambda; \nu]$, with $\deg Q_{1|\emptyset} = \lambda$, $\deg Q_{\emptyset|1} = \nu$.

In the untwisting limit, there is precisely one Bethe root going to infinity, and either from $Q_{1|\emptyset}$ or from $Q_{\emptyset|1}$. Hence, if we define the non-twisted Q-functions by

$$ Q_{\leftarrow 0|0} = 1, \quad Q_{\leftarrow 1|0} \propto \lim_{z \to 1} Q_{1|\emptyset}, \quad Q_{\leftarrow 0|1} \propto \lim_{z \to 1} Q_{\emptyset|1}, \quad Q_{\leftarrow 1|1} = u^L, \qquad (3.34) $$
they can originate from two different twisted Q-systems. And, indeed, all irreps of gl(1|1) with non-trivial action of sl(1|1) sub-algebra are 2-dimensional, hence the solution Q corresponds to the eigenstates in a certain 2-dimensional representation.

The question is how to label this representation. The representation consists of two vectors, with weights [deg $Q_{\leftarrow 1|0} + 1$; deg $Q_{\leftarrow 0|1}$] and [deg $Q_{\leftarrow 1|0}$; deg $Q_{\leftarrow 0|1} + 1$], and we have to pick one of them for the purpose of labelling. In contrast to non-supersymmetric case, there is no distinguished choice of what should be called lowest weight and we can choose either of options.

The situation becomes clearer with generalisation to higher ranks. In general, one can expect that untwisting without rotations of an $su(K|M)$ Q-system generates a set of $(K+1) \times (M+1)$ functions $Q$ described in [56]

$$Q_{\leftarrow k|m} \propto \lim_{G \to \mathbb{I}} Q_{A|J}, \quad \text{for } k = |A|, m = |J|, \qquad (3.35)$$

where the limit is understood in the same sense as in (3.23).

In supersymmetric algebras, lowest weights are not invariant objects.<fnref n="32" /> We, however, can associate the unique notion of lowest weight to the choice of the nesting path (2.102). It is done as follows: for certain $K+M+1$ functions $Q$ which are the untwisting limit (3.35) of $Q_{A|J}$ along certain path (2.102), one parameterises their degree as

$$\deg Q_{\leftarrow k|m} = \sum_{a=1}^{k} \lambda_a + \sum_{i=1}^{m} \nu_i. \qquad (3.36)$$

On the other hand, each nesting path is in the obvious one-to-one correspondence with the ordering in the set $\{1, \dots, K, \hat{1}, \dots, \hat{M}\}$, see appendix C. Hence we can say that the choice of the nesting path also defines the notion of the lowest weight. Then one can show that the weight $[\lambda_1, \dots, \lambda_K; \nu_1, \dots, \nu_M]$ is the lowest-weight vector defined by the nesting path used to choose the $K+M+1$ functions $Q$ in (3.36).

Specifying to the gl(1|1) example: if the nesting path is $(\emptyset|\emptyset) \subset (1|\emptyset) \subset (1|1)$, then $[\deg Q_{1|0}, \deg Q_{0|1} + 1]$ would be our lowest weight and if the nesting path is $(\emptyset|\emptyset) \subset (\emptyset|1) \subset (1|1)$, then $[\deg Q_{1|0} + 1, \deg Q_{0|1}]$ would be our lowest weight.

The formula (3.36) applies without any subtleties if the multiplet in question is long. If the multiplet is short, and there are plenty of them in $su(K|M)$ spin chains, we need to provide a further analysis.

### 3.3.4 $su(2|1)$: states in short representations involve zero Q-functions

The $K+M+1$ functions $Q_{\leftarrow}$ obtained from $K+M+1$ functions $Q_{A|J}$ along certain nesting path contain, in principle, all the information about the untwisted Q-system. In this sense, the situation is exactly the same as with $K+1$ functions $Q_{\leftarrow}$ in $su(K)$ case. However, the direct untwisting limit generates more than $K+M+1$ distinct functions, see (3.35). In this

<footnote n="32">³²To be more strict, we can change the lowest weight vector in non-supersymmetric case as well, by choosing a different Borel decomposition, but its weight would be just the same after we apply the automorphism $E_{ab} \mapsto E_{\sigma(a)\sigma(b)}$, where $\sigma$ is a permutation such that $E_{\sigma(a)\sigma(b)}$|lowest weight⟩ = 0 for $a > b$. This “cure” by an automorphism cannot be done in supersymmetric case.</footnote>
respect, the situation differs from the $\mathfrak{su}(K)$ case where other Q-functions are accessible only if the untwisting is supplemented with a rotation. It is a priory not obvious that Q-functions generated by (3.35) will be consistent with QQ-relations. One can see that [56]

$$ \mathcal{Q}_{\leftarrow k+1|m+1}^{+} \mathcal{Q}_{\leftarrow k|m}^{-} - \mathcal{Q}_{\leftarrow k+1|m+1}^{-} \mathcal{Q}_{\leftarrow k|m}^{+} \propto \mathcal{Q}_{\leftarrow k+1|m} \mathcal{Q}_{\leftarrow k|m+1}, \quad (3.37) $$

but sometimes it is possible to satisfy this equation only when the coefficient of proportionality is zero, very similarly to untwisting of (3.21).

The simplest example is the relation $Q_{a|i}^{+} - Q_{a|i}^{-} = Q_{a|0}Q_{0|i}$. If the polynomial $Q_{a|i} = (\frac{x_a}{y_i})^{\dot{i}u} Q_{a|i}$ becomes a constant in the untwisting limit then $\lim_{x_a, y_i \to 1} Q_{a|0}Q_{0|i} = 0$, and one should assign, for consistency, $Q_{\leftarrow a|0} = 0$ or $Q_{\leftarrow 0|i} = 0$, instead of a non-zero answer stemming from (3.35). We also can spot from (3.36) that $\deg Q_{a|i} = 0$ means $\lambda_a + \nu_i = 0$ which is a shortening condition for representations of $\mathfrak{su}(K|M)$.

We illustrate this issue on a very explicit and relatively simple case of the $\mathfrak{su}(2|1)$ spin chain with two sites ($L=2$) in the defining representation. We will further strengthen our claim that Q-systems can involve zero Q-functions and yet describe physical states by explicitly realising all Q-functions as eigenvalues of the Q-operators. This example is also rich enough to illustrate certain other twist-dependent effects introduced in previous sections.

First, we list below all possible polynomial solutions of the $\mathfrak{su}(2|1)$ QQ-relations without twist with the boundary conditions $Q_{\emptyset|0} = 1$ and $Q_{12|1} = u^2$. These solutions can be quickly found by brute force. There is one solution with non-zero Q-functions:

$$ \begin{aligned} Q_{\emptyset} &= 1, & Q_{1|\emptyset} &= 1, & Q_{2|\emptyset} &= \dot{i}u + \text{cst}, & Q_{\emptyset|1} &= -2, \\ Q_{12|\emptyset} &= 1, & Q_{1|1} &= 2\dot{i}u, & Q_{2|1} &= -u^2 - \frac{1}{4} + 2\dot{i}\text{cst}u, & Q_{12|1} &= u^2, \end{aligned} \quad (3.38) $$

where $\text{cst}$ denotes an irrelevant constant which originates from the GL(2) H-symmetry rotating bosonic indices.

There are also two solutions which contain zero Q-functions:

$$ \begin{aligned} Q_{\emptyset} &= 1, & Q_{1|\emptyset} &= 0, & Q_{2|\emptyset} &= -u^2, & Q_{\emptyset|1} &= R, \\ Q_{12|\emptyset} &= 0, & Q_{1|1} &= 1, & Q_{2|1} &= \Psi^{+}(u^2 R), & Q_{12|1} &= u^2, \end{aligned} \quad (3.39a) $$

$$ \begin{aligned} Q_{\emptyset} &= 1, & Q_{1|\emptyset} &= R, & Q_{2|\emptyset} &= -u^2 & Q_{\emptyset|1} &= 0, \\ Q_{12|\emptyset} &= - \begin{vmatrix} R^+ & R^- \\ (u^+)^2 & (u^-)^2 \end{vmatrix}, & Q_{1|1} &= 1, & Q_{2|1} &= 0, & Q_{12|1} &= u^2, \end{aligned} \quad (3.39b) $$

where $R$ is an arbitrary polynomial and $\Psi(u^2 R)$ is a polynomial that satisfies $\Psi(u^2 R) - \Psi^{++}(u^2 R) = u^2 R$.

Second, we assign the irreps in the Hilbert space to the presented solutions. The Hilbert space is 9-dimensional and it decomposes into two irreps of $\mathfrak{su}(2|1)$:
super-symmetrisation of two defining representations

<table><thead><tr><th>state</th><th>weight</th><th>1 &lt; 2 &lt; 1̂</th><th>1 &lt; 1̂ &lt; 2</th><th>1̂ &lt; 1 &lt; 2</th></tr></thead><tbody><tr><td>|↑↑⟩</td><td>[20;0]</td><td>HW</td><td>HW</td><td></td></tr><tr><td>|↑↓⟩ + |↓↑⟩</td><td>[11;0]</td><td></td><td></td><td></td></tr><tr><td>|↓↓⟩</td><td>[02;0]</td><td></td><td>LW</td><td>LW</td></tr><tr><td>|↑θ⟩ + |θ↑⟩</td><td>[10;1]</td><td></td><td></td><td>HW</td></tr><tr><td>|↓θ⟩ + |θ↓⟩</td><td>[01;1]</td><td>LW</td><td></td><td></td></tr></tbody></table>

and super-antisymmetrisation

<table><thead><tr><th>state</th><th>weight</th><th>1 &lt; 2 &lt; 1̂</th><th>1 &lt; 1̂ &lt; 2</th><th>1̂ &lt; 1 &lt; 2</th></tr></thead><tbody><tr><td>|↑↓⟩ - |↓↑⟩</td><td>[11;0]</td><td>HW</td><td></td><td>LW</td></tr><tr><td>|↑θ⟩ - |θ↑⟩</td><td>[10;1]</td><td></td><td>HW</td><td></td></tr><tr><td>|↓θ⟩ - |θ↓⟩</td><td>[01;1]</td><td></td><td>LW</td><td></td></tr><tr><td>|θθ⟩</td><td>[00;2]</td><td>LW</td><td></td><td>HW</td></tr></tbody></table>

The choice of the ordering $1 < 2 < 1̂$, $1 < 1̂ < 2$, or $1̂ < 1 < 2$ is in one-to-one correspondence with the preferred choice of the nesting path. For instance $1 < 1̂ < 2$ corresponds to $(\emptyset|\emptyset) \subset (1|\emptyset) \subset (1|1) \subset (12|1)$.

We can use (3.36) to identify Q-systems with corresponding irreps. We see that the four-dimensional representation corresponds to the Q-system (3.38). A less obvious claim is that both Q-systems (3.39) correspond to the five-dimensional representation. To perform identification of weights, we should choose a nesting path which avoids zero Q-functions: (3.39a) is used with the ordering $1̂ < 1 < 2$ while (3.39b) is used with the orderings $1 < 2 < 1̂$ or $1 < 1̂ < 2$. The reader can check correctness of (3.36) when we choose $R \propto 1$.

The five-dimensional representation is an example of short, or atypical representation. Such a representation is characterised by a property that some of states are annihilated by more than a half of fermionic generators. Hence, these states can be highest- or lowest-weight ones for more than one index ordering. The practical output that we rely on is a possibility to realise condition $\lambda_a + \nu_i = 0$ for a highest or lowest weight if one choose an appropriate ordering for which $a$ and $i$ are the neighbours in this ordering sequence. In compact rational spin chains all weights are non-negative integers. Hence $\lambda_a + \nu_i = 0$ implies $\lambda_a = \nu_i = 0$. Then condition of being lowest weight implies $\lambda_b = 0$ for $b \le a$ and $\nu_j = 0$ for $j \le i$. This significantly restricts the possible Q-systems describing short representations, it also explains why we chose $R \propto 1$ above.<fnref n="33" />

The choice $R \propto 1$ seems to be natural for the purpose of correct weight counting. And it also stems from the operatorial construction given below. However, quite remarkably, such invariant quantities as T-functions or energy do not depend on the choice of $R$. We can even

<footnote n="33">³³This last argument is based on the representation theory. It would be nice to observe it solely from analytic properties of a Q-system, we however do not perform this analysis here.</footnote>
put $R=0$ and get that the two solutions (3.39a) and (3.39b) coincide! To say more, many T-functions computed for the states in short representations are identically 0, the non-zero ones live on a smaller L-hook. The observed phenomena are present in supersymmetric Q- and T-systems of any rank. In fact, in the case of character solution (3.9), we can recognise in these effects one of the defining properties of supersymmetric Schur polynomials. We discuss this question in detail in appendix B.4.

Finally, we support the observations made above by explicit analysis of operators $\hat{Q}$ acting on the Hilbert space. These operators are constructed according to the procedure of [68], and the presence of twist is essential, see appendix B.3 for details. The explicit expressions obtained in the basis $|\uparrow\uparrow\rangle, |\uparrow\downarrow\rangle, |\uparrow\theta\rangle, |\downarrow\uparrow\rangle, |\downarrow\downarrow\rangle, |\downarrow\theta\rangle, |\theta\uparrow\rangle, |\theta\downarrow\rangle, |\theta\theta\rangle$ are

$$ \hat{Q}_{\emptyset} = 1, \qquad \hat{Q}_{12|1} = u^2 \frac{(x_1 - x_2)}{(x_1 - y)(x_2 - y)}, \qquad (3.40a) $$

$$ \hat{Q}_{1|\emptyset} = \begin{pmatrix} u^2 - \frac{x_1(x_2-y)}{(x_1-x_2)(x_1-y)} & (2iu + \frac{x_1+x_2}{x_1-x_2}) & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & u + \frac{i x_1}{x_2-x_1} + \frac{i x_1}{x_1-y} & 0 & -\frac{i x_2}{x_1-x_2} & 0 & 0 & 0 & 0 \\ 0 & 0 & u + \frac{i x_1}{x_2-x_1} + \frac{i x_1}{x_1-y} & 0 & 0 & 0 & -\frac{i y}{x_1-y} & 0 \\ 0 & -\frac{i x_1}{x_1-x_2} & 0 & u + \frac{i x_1}{x_2-x_1} + \frac{i x_1}{x_1-y} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -\frac{i x_1}{x_1-y} & 0 & 0 & 0 & u + \frac{i x_1}{x_2-x_1} + \frac{i x_1}{x_1-y} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \qquad (3.40b) $$

$$ \hat{Q}_{2|\emptyset} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & u + \frac{i x_2}{x_1-x_2} + \frac{i x_2}{x_2-y} & 0 & \frac{i x_2}{x_1-x_2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{i x_1}{x_1-x_2} & 0 & u + \frac{i x_2}{x_1-x_2} + \frac{i x_2}{x_2-y} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & u^2 - \frac{x_2(x_1-y)}{(x_2-x_1)(x_2-y)} & (2iu + \frac{x_2+x_1}{x_2-x_1}) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & u + \frac{i x_2}{x_1-x_2} + \frac{i x_2}{x_2-y} & -\frac{i y}{x_2-y} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{i x_2}{x_2-y} & u + \frac{i x_2}{x_1-x_2} + \frac{i x_2}{x_2-y} & 0 \end{pmatrix}, \qquad (3.40c) $$

$$ \hat{Q}_{\emptyset|1} = \begin{pmatrix} \frac{1}{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & u + \frac{i x_2}{x_2-y} + \frac{i y}{x_1-y} & 0 & 0 & \frac{i y}{x_1-y} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & u + \frac{i x_2}{x_2-y} + \frac{i y}{x_1-y} & 0 & \frac{i y}{x_2-y} & 0 \\ 0 & \frac{i x_1}{x_1-y} & 0 & 0 & 0 & u + \frac{i x_2}{x_2-y} + \frac{i y}{x_1-y} & 0 & 0 \\ 0 & 0 & 0 & \frac{i x_2}{x_2-y} & 0 & 0 & u + \frac{i x_2}{x_2-y} + \frac{i y}{x_1-y} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & u^2 + 2iu \frac{x_1 x_2 - y^2}{(x_1 - y)(x_2 - y)} - \frac{x_1 x_2 + y^2}{(x_1 - y)(x_2 - y)} \end{pmatrix}. \qquad (3.40d) $$

The presented 5 operators mutually commute. The $Q$-functions are their eigenvalues. The operators $\hat{Q}_{1|1}$, $\hat{Q}_{2|1}$, and $\hat{Q}_{2|1}$ were not shown explicitly. They are also polynomials in $u$ and rational functions in twist variables and they can be easily restored using the $QQ$-relations.

**Eigenstate $|\uparrow\uparrow\rangle$.** The most intriguing is to look on a state which becomes a member of atypical representation in the untwisting limit. We will concentrate on $|\uparrow\uparrow\rangle$, it is already an eigenstate of $\hat{Q}$-operators. After the change of variables (3.15), we obtain the following eigenvalues:
$Q_0 = 1,$

$$
Q_{1|0} = x_1^{-iu-3/4} \left( u^2 - \frac{x_1(x_2-y)}{(x_1-x_2)(x_1-y)} \left( 2iu + \frac{x_1+x_2}{x_1-x_2} \right) \right), \quad Q_{2|0} = x_2^{-iu-3/4}, \quad Q_{0|1} = y^{i(u-1/4)},
$$

$$
Q_{12|0} = (x_1 x_2)^{-i u - 5/4} \left( u^2 + i u \frac{x_1 + y}{x_1 - y} - \frac{1}{4} \right) (x_1 - x_2),
$$

$$
Q_{2|1} = \left(\frac{x_2}{y}\right)^{-iu-1/4} \frac{1}{x_2 - y}, \quad Q_{1|1} = \left(\frac{x_1}{y}\right)^{-iu-1/4} \frac{1}{x_1 - y} \left( u^2 - i u \frac{x_1 + x_2}{x_1 - x_2} - \frac{x_1^2 + 6x_1x_2 + x_2^2}{4(x_1 - x_2)^2} \right),
$$

$$
Q_{12|1} = \left(\frac{x_1 x_2}{y}\right)^{-iu-3/4} u^2 \frac{(x_1 - x_2)}{(x_1 - y)(x_2 - y)}. \qquad (3.41)
$$

For instance, one can compute the energy<sup>34</sup> of this state which turns out to be 2, as can
be seen in (3.6) where $r = 0$ ($Q_1$ has no roots).

If we perform a straightforward untwisting limit in the style of (3.23) and (3.35) we will
find that all Q-functions are proportional to identity, except for $\mathcal{Q}_{\leftarrow 2|0} \propto u$ and $\mathcal{Q}_{\leftarrow 2|1} \propto u^2$.
Such set of Q-functions does not satisfy the relation (3.37).

A sober way to proceed is to perform a rotation which will produce Q-functions that
have an explicitly regular limit $G \to \mathbb{I}$ (for almost any direction) but, however, may become
also zero. Then we guarantee that QQ-relations would survive the limit.

On one hand, one can use the rotation (2.26) with the matrix

$$
h = \begin{pmatrix}
\frac{-(x_1-x_2)^2(x_1-y)}{2\alpha} & 0 & 0 \\
(x_1-y) \left( \frac{-1}{x_1-x_2} - \frac{5}{4} - \frac{41}{32}(x_1-x_2) \right) & -\frac{2(x_2-y)}{(x_1-x_2)^3} & 0 \\
0 & 0 & \alpha
\end{pmatrix}, \quad (3.42a)
$$

where $\alpha$ is an arbitrary (but non-zero) constant. More precisely $\alpha$ is independent of $u$,
and it may be a function of $G$ such that $\epsilon \ll \alpha$ when $G = \mathbb{I} + \epsilon g_0 \to \mathbb{I}$. Then, one finds
that the Q-functions obtained after the rotation have a $G \to \mathbb{I}$ limit given by (3.39a), with
$R = \lim_{G \to \mathbb{I}} \alpha$. The choice $R=0$ can be obtained in several ways, for instance one can put
$\alpha = \sqrt{x_2 - y}$.

On the other hand, one can use the rotation (2.26) with the matrix

$$
h = \begin{pmatrix}
-\alpha \frac{(x_1 - x_2)^2 (x_1 - y)}{2(x_2 - y)} & 0 & 0 \\
(x_1 - y) \left( \frac{-1}{x_1 - x_2} - \frac{5}{4} - \frac{41}{32}(x_1 - x_2) \right) & -2 \frac{x_2 - y}{(x_1 - x_2)^3} & 0 \\
0 & 0 & \frac{x_2 - y}{\alpha}
\end{pmatrix}, \quad (3.42b)
$$

(with the same condition on $\alpha$ as before), and produce Q-functions with a $G \to \mathbb{I}$ limit given by (3.39b), with $R = \lim_{G \to \mathbb{I}} \alpha$. It is manifest that the two rotations differ by slight normalisations only, and the choice $\alpha = \sqrt{x_2 - y}$ makes them coincide.

The procedure to construct these rotations is the following one: the diagonal entries
are designed to make Q-functions along a chosen nesting path regular and non-zero in

<footnote n="34"><sup>34</sup>We remind that the present convention for the Hamiltonian is $\mathcal{H} = \sum_{i=1}^{L-1} \mathcal{P}_{i,i+1} + (\mathcal{P}_{1,L} \cdot G^{-1} \otimes \mathbb{I}^{\otimes L-2} \otimes G)$.</footnote>
the untwisting limit,<fnref n="35" /> i.e. we perform the limit of type (3.35) on the chosen $K+M+1$ Q-functions. The off-diagonal entries are introduced to reproduce all Q-functions of the untwisted Q-system, they execute the idea (3.25). If we are interested only in Q-functions of type $Q_{\leftarrow}$ then we can skip constructing off-diagonal terms. Note that in the presence of the off-diagonal terms, e.g. rotated $Q_{2|\emptyset}$ is not a product of polynomial and exponential prefactor. Such a mixture allows to get polynomials of higher degree in the untwisted limit, the off-diagonal terms are fine-tuned to achieve this goal. A generic algorithm to construct rotations is explained in appendix B.2.

**Eigenstate $|\downarrow\downarrow\rangle$.** This one also has the energy equal to 2. It is analysed in full analogy to $|\uparrow\uparrow\rangle$. Two rotational matrices yielding (3.39) in the untwisting limit are

$$h = \begin{pmatrix} \frac{x_1 - y}{\alpha} & 0 & 0 \\ (x_1 - y) \left( \frac{1/16}{x_1 - x_2} - \frac{3/2}{(x_1 - x_2)^2} + \frac{2}{(x_1 - x_2)^3} \right) & \frac{x_2 - y}{x_1 - x_2} & 0 \\ 0 & 0 & \alpha \end{pmatrix} \quad (3.43a)$$

and

$$h = \begin{pmatrix} \alpha & 0 & 0 \\ (x_1 - y) \left( \frac{1/16}{x_1 - x_2} - \frac{3/2}{(x_1 - x_2)^2} + \frac{2}{(x_1 - x_2)^3} \right) & \frac{x_2 - y}{2(x_1 - x_2)} & 0 \\ 0 & 0 & \frac{x_1 - y}{\alpha} \end{pmatrix} . \quad (3.43b)$$

**Eigenstate $|\theta\theta\rangle$.** The energy of this state is $-2$. One can use the rotation

$$h = \begin{pmatrix} 1 & 0 & 0 \\ \frac{-1}{x_1 - x_2} & \frac{1}{x_1 - x_2} & 0 \\ 0 & 0 & (x_1 - y)(x_2 - y) \end{pmatrix} \quad (3.44)$$

to reproduce (3.38) in the untwisting limit.

As this eigenstate is not cluttered with effects related to atypical representation, it is the simplest example to observe how off-diagonal elements of the rotation matrix allow to increase the degree of a polynomial. Indeed $Q_{1|\emptyset} = Q_{1|\emptyset} = 1$ on this state, however $Q_{2|\emptyset} = \mathbf{i}u + \text{cst}$ in (3.38).

**Other eigenstates.** The remaining 6 eigenstates are obtained by diagonalizing three $2 \times 2$ blocks in matrices (3.40). These states and their energies read as follows

$$\begin{pmatrix} \text{State} & \text{twisted energy} \\ \sqrt{x_1} |\uparrow\downarrow\rangle \pm \sqrt{x_2} |\downarrow\uparrow\rangle & \pm \frac{x_1+x_2}{\sqrt{x_1 x_2}} \\ \sqrt{x_1} |\uparrow\theta\rangle \pm \sqrt{y} |\theta\uparrow\rangle & \pm \frac{x_1+y}{\sqrt{x_1 y}} \\ \sqrt{x_2} |\downarrow\theta\rangle \pm \sqrt{y} |\theta\downarrow\rangle & \pm \frac{x_2+y}{\sqrt{x_2 y}} \end{pmatrix}, \quad (3.45)$$

<footnote n="35">³⁵Example: if we choose the nesting path $(\emptyset|\emptyset) \subset (\emptyset|1) \subset (1|1) \subset (12|1)$, then we obtain the rotation (3.42a) with $\alpha = 1$, whereas if we chose the nesting path $(\emptyset|\emptyset) \subset (1|\emptyset) \subset (1|1) \subset (12|1)$, then we obtain the rotation (3.42b) with $\alpha = 1$.</footnote>
and the reader can straightforwardly construct rotations which provide a smooth $G \to \mathbb{I}$ limit, following the lines of appendix B.2.

These states are examples demonstrating a non-trivial monodromy around co-dimension one hyperplanes $x_1 = 0$ etc, where the twist matrix $G$ becomes degenerate, cf. (3.30). Going around the degeneration points changes the branch of the corresponding square root. On the branch were $\sqrt{1} = 1$, the sign “+” in (3.45) corresponds to the states which become a part of the atypical (five-dimensional) representation in the untwisting limit (hence Q-functions have the limit (3.39)), whereas sign “-” corresponds to the states which become a part of the typical (four-dimensional) representation (i.e. Q-functions have the limit (3.38)).

### 3.3.5 sl(2): non-compactness leads to Stokes phenomena

Finally, we will consider the XXX spin chain in a non-compact representation. Such a spin chain is not described by entirely polynomial Q-functions, but it is still based on rational R-matrix, hence it is natural to consider it in the same section. Understanding certain features of such a system is quite important for further study of the AdS/CFT integrability which is also based on a non-compact algebra.

The non-compactness is distinguished by appearance of a certain singularity, a pole in the rational case: $Q_{12} = u^{-L}$, so the QQ-relation to solve is

$$Q_1^+ Q_2^- - z Q_1^- Q_2^+ \propto u^{-L}. \qquad (3.46)$$

One further demands that $Q_1$ will be a polynomial. Let us denote its degree $\deg Q_1 = M_1$. On the other hand, $Q_2$ cannot be a polynomial as $Q_2 \sim u^{-L-M_1}$ at large $u$.

Similarly to the compact case, the degree of the polynomial function $Q_1$ can drop in untwisting limit, and we can find the following analytic approximating solution

$$Q_1(u) \approx \mathbb{Q}(u) L_m^{(-2s-1)}(-i u \log z), \quad \text{with } \deg \mathbb{Q} = -\frac{L}{2} - s. \qquad (3.47)$$

The main difference with (3.28) is the change in sign of $s$, so now one has $-2s-1 \ge L-1 \ge 0$. As a consequence, we have no upper bound on the value of $m$, i.e. $m \in \mathbb{Z}_{\ge 0}$. This labelling by $m$ enumerates the eigenstates of an infinite-dimensional lowest-weight representation of sl(2), with spin $s$. Another consequence of $-2s-1 \ge 0$ is that all zeros of the associated Laguerre polynomial are real. The polynomial $\mathbb{Q}(u)$ satisfies untwisted Bethe equations, it is known to have real Bethe roots as well.

Unlike the compact case, the limits of $Q_1$ and $Q_2$ are two independent functions which we denote as $\mathbb{Q} \propto \lim_{z \to 1} Q_1$ and $\mathbb{Q}' \propto \lim_{z \to 1} Q_2$. The degree$^{36}$ of $\mathbb{Q}'$, $\deg \mathbb{Q}' = -\frac{L}{2} + s + 1$ is negative but it is larger than that of $Q_2$. It is indeed possible that the degree of a non-polynomial function increases in certain limits, a simple example is $\lim_{z \to 1} \frac{u^{M_2}}{1+(z-1)u}$.

Another interesting question to discuss is the interplay between dominant and sub-dominant solutions. Think of $Q_1, Q_2$, the Q-functions of the untwisted Q-system, as of two solutions of the Baxter equation (1.1). In the compact su(2) case, the function $Q_1 = \mathbb{Q}_{\leftarrow 1}$

<footnote>$^{36}$We define degree of a non-polynomial function the value of its exponent when $u \to \infty$.</footnote>
is obtained by the direct untwisting limit (3.23), whereas the function $Q_2$ is obtained with the help of a rotation. Apart from this difference related to untwisting limit, $Q_1$ can be also singled out as the sub-dominant solution of the Baxter equation at large $u$. $Q_2$ is a dominant solution and hence it is not defined uniquely: any combination of the type $Q_2 + \text{const } Q_1$ would still qualify as a dominant solution. The transformation $Q_2 \mapsto Q_2 + \text{const } Q_1$ is a residual H-rotation which respects the ordering in degree of Q-functions.

In the non-compact $sl(2)$ case, the situation appears to be contr-intuitive if one uses ordering in degree as a way to select “distinguished” Q-functions. The two Q-functions $Q_1 \equiv Q$ and $Q_2 \equiv Q'$ also satisfy the Baxter equation (1.1), but now with $\phi = u^{-L}$. $Q_1$ seems now to be a dominant one, nevertheless $Q_1$ is defined uniquely because this is the only polynomial solution. At first glance, $Q_2$ is sub-dominant and hence it should be defined uniquely. Alas, it is not. We are going to investigate this subtlety.

As we shall see, the subtlety is also present in the twisted case: the function $Q_1$ is uniquely determined from the fact that it is a polynomial, whereas it is less elementary to give a unique prescription for $Q_2$.

We will reconstruct $Q_2$ from the fact that it satisfies (3.46). We introduce an operator $\Psi_z$ which satisfies the property

$$ \Psi_z(f) - z \Psi_z^{++}(f) = f. \qquad (3.48) $$

Then, the most general expression for $Q_2$ can be written in the form

$$ Q_2 = Q_1 \Psi_z^+ \left( \frac{1}{u^L Q_1^+ Q_1^-} \right) + \mathcal{P}(u) Q_1 z^{\ddagger u}, \qquad (3.49) $$

where $\mathcal{P}(u)$ is an $\ddagger$-periodic function.

$\Psi_z$ is not defined by (3.48) uniquely. We further narrow the ambiguities in its definition by the requirement that $\Psi_z(f)$ is regular if $f$ is regular and that the large-$u$ asymptotic expansion of $\Psi_z(f)$ is related to the large-$u$ expansion of $f$ by

$$ \Psi_z(f) \approx \frac{1}{1-z e^{\ddagger \partial_u}} f = \left( \frac{1}{1-z} + \frac{\ddagger z}{(1-z)^2} \partial_u + \dots \right) f. \qquad (3.50) $$

For instance, these constraints on $\Psi_z(f)$ imply that $\Psi_z(f)$ is a polynomial if $f$ is a polynomial. Already from (3.50) we see that $z = 1$ is quite special. Large-$u$ expansion should be re-summed, we refer to [22] for a discussion of properties of $\Psi \equiv \Psi_{z=1}$ in the case $z = 1$.

As we seek for $Q_2$ with power-like asymptotics, the term with $z^{\ddagger u}$ factor should be discarded. Furthermore, one can always write a solution using the following ansatz (cf. [22])

$$ Q_2 = Q_1 \sum_{k=1}^{L} c_k \Psi_z^+ \left( \frac{1}{u^k} \right) + R(u), \qquad (3.51) $$

where coefficients $c_k$ are defined by the small-$u$ expansion $\frac{1}{u^L Q_1^+ Q_1^-} = \sum_{k=1}^{L} \frac{c_k}{u^k} + \mathcal{O}(u)^0$, and where $R(u)$ is a polynomial of degree $L-1$ which is uniquely fixed by requirement to cancel all positive powers in $u$ in the large-$u$ expansion of (3.51).
Hence we can focus on studying $\Psi_z(\frac{1}{u^k})$ which coincides, after an appropriate rescaling of parameters, with the Lerch transcendent. We can immediately write down its integral representation by rewriting a symbolic expression $\frac{1}{1-z e^{i \partial_u} \frac{1}{u^k}}$ as a Laplace integral

$$ \Psi_z \left( \frac{1}{u^k} \right) = \frac{1}{\Gamma(k)} \int_0^{\infty \times e^{i \phi}} \frac{t^{k-1} e^{-ut}}{1 - z e^{-i t}} dt. \qquad (3.52) $$

The announced ambiguity in construction of $Q_2$ can be explicitly seen here: the direction of integration is chosen to make the integral convergent, it is correlated with the direction in which (3.50) is chosen to hold and the analytic function defined by the integral *does depend* on the choice of direction.

A different perspective on this effect is to note that the expansion (3.50) produces asymptotic non-convergent series. Their Borel resummation leads to (3.52) which has Borel ambiguities at the poles

$$ t_n = -i \log z + 2\pi n, \quad n \in \mathbb{Z}. \qquad (3.53) $$

The resulting ambiguity in the definition of $Q_2$ is $\delta_n Q_2 \propto Q_1(u) z^{i u} e^{-2\pi n u}$, which has the form of the second term in (3.49) that we attempted to discard. But, as $n \in \mathbb{Z}$, one can always find Borel ambiguities which are exponentially suppressed in $u$. Due to this exponential suppression they are sub-dominant compared to the first term in (3.49) and we cannot discard such terms based on the large-$u$ behaviour argument. Hence $Q_2$ cannot be defined uniquely.

We observed here a qualitative distinction between differential and finite-difference equations: in the case of differential equations, the dominance of a solution is decided by analysing its large-$u$ asymptotic. In the case of finite-difference equations, solutions can be summed with periodic coefficients, not only constants, to produce a new solution; and it can happen that periodic coefficients themselves decide the dominance of different terms in the sum. This effect becomes visible in a non-compact case. Indeed, we can forbid non-constant periodic functions as coefficients in the compact case, by requiring polynomiality of the solutions. However, in the non-compact case, we cannot forbid periodic functions completely. Although we can require power-like behaviour when $u \to \infty$, at least in certain directions, periodic functions will still appear as subdominant terms due to Borel ambiguities.

For any value of twist $z$, except the cases $0 \le z < 1$ and $z > 1$, there are two distinguished choices for $\Psi_z$.

The first one, upper half-plane analytic (UHPA), is denoted by $\Psi_z^\uparrow$ and is defined as solution having large-$u$ expansion (3.50) valid in the largest possible cone containing $u \to +i \infty$. It can be defined by integration over $t$ in (3.52) from 0 to $-i \infty$ for $\Im m(u) > 0$ and then by analytic continuation. For $|z| \le 1$, this solution can be represented as a convergent series$^{37}$

$$ \Psi_z^\uparrow \left( \frac{1}{u^k} \right) = \sum_{s=0}^{\infty} \frac{z^s}{(u + i s)^k}. \qquad (3.54) $$

<footnote>$^{37}$For $z = 1$, the sum is marginally divergent when $k = 1$. We define such a sum assuming the same regularisation as for the case of digamma function, see [22].</footnote>
<figure><img src="image_17.png" /></figure>

(a) Borel complex plane. The position of poles are denoted by crosses. Integration contour A defines the UHPA solution. Integration contour B defines another, non-equivalent solution.

<figure><img src="image_18.png" /></figure>

(b) Sectors in u-plane where expansion (3.50) is applicable. Two sectors are shown, the ones corresponding to $\Psi_z$ defined by integration contours A and B.

**Figure 11.** Illustration of Borel ambiguities (left) and emergence of Stokes sectors (right). A case with $|z| > 1$.

Note that in the case of $|z| \le 1$ the sector of applicability of (3.50) (Stokes sector) is any direction save $u \to -i\infty$. The sector becomes smaller if $|z| > 1$ but it still includes both $u \to +\infty$ and $u \to -\infty$ directions, except when $z > 1$; example is shown in figure 11. For $z > 1$ the UHPA solution is not defined uniquely as there are two solutions which have Stokes sector of equal size.

The second one, the lower half-plane analytic (LHPA), is denoted by $\Psi_z^\downarrow$ and is defined as solution having large-u expansion (3.50) valid in the largest possible sector containing $u \to -i\infty$. Correspondingly, the integration in (3.52) is from 0 to $+i\infty$ for $\Im(m(u)) < 0$, and $\Psi_z^\downarrow$ is defined by analytic continuation if $\Im(m(u)) > 0$. The corresponding series

$$ \Psi_z^\downarrow \left( \frac{1}{u^k} \right) = - \sum_{s=1}^{\infty} \frac{z^{-s}}{(u - i s)^k} \qquad (3.55) $$

is convergent for $|z| \ge 1$. LHPA is not defined uniquely for $0 \le z < 1$.

The difference between two solutions can be found explicitly by computing the emerging integral by residues:

$$ \Psi_z^\uparrow \left( \frac{1}{u^k} \right) - \Psi_z^\downarrow \left( \frac{1}{u^k} \right) = \frac{(-\partial_u)^{k-1}}{\Gamma(k)} \int_{+i\infty}^{-i\infty} \frac{e^{-ut}}{1-z e^{it}} dt = \pi \frac{(-\partial_u)^{k-1}}{\Gamma(k)} \frac{(-z)^{i u}}{\sinh(\pi u)}. \qquad (3.56) $$

We see that this difference indeed produces the term of a type $P(u)z^{i u}Q_1(u)$ in (3.49), and this term is exponentially suppressed in $u$ in the region where large-u asymptotic expansion is applicable simultaneously for both UHPA and LHPA solutions.

The distinction between UPHA and LHPA Q-functions is paramount for the case of AdS/CFT quantum spectral curve, one could even understand the Riemann-Hilbert conditions of the spectral curve as a way to build UPHA system from LHPA and vice-versa [3]. However, we see now that the phenomenon appears already in rational non-compact spin chains.
Although the presence of twist is not an absolute requirement for the presented analysis, the twisted case illuminates and enriches the emergent Stokes effects. First, quite convenient series (3.54) and (3.55) cease to converge simultaneously if $|z| \neq 1$ and one start to look for a more universal integral definition (3.52) which clearly suffers from Borel ambiguities. Second, the Borel poles depend on twist, and the definition of UPHA and LHPA, which relies on position of these poles, is not smooth in $z$. Consider for instance UPHA solutions. Crossing the $z > 1$ line in $z$-plane requires to pick up a Borel pole in $t$-plane thus generating solution which is no longer an UPHA. We hence perceive the line $1 < z < \infty$ as a branch cut. Its branch points are of infinite degree. Indeed, the discontinuity $\Psi_{z+i0}^{\uparrow} - \Psi_{z-i0}^{\uparrow}$ across the cut involves $\log z$.

## 3.4 Dependence on twist and the untwisting limit: general picture

Back in section 2.3, we understood that Q-system realises a maximal flag in $\mathbb{C}^N$ whose $u$-dependence is constrained by the intersection property (1.5). Since then, we observed two conceptually different ways to parameterise this flag using $N$ Q-functions (in the gauge $Q_\emptyset = 1$). We list the essential properties of these two parameterisations (su($N$) case is kept in mind, but most of the statements can be generalised to supersymmetric and noncompact cases):

### Covariant parameterisation:

- Uses functions

$$[Q_1(u) : Q_2(u) : \dots : Q_N(u)]. \qquad (3.57)$$

It can be geometrically thought as a map $\Sigma \to \mathbb{CP}^{N-1}$, where $\Sigma$ is the space of spectral parameter $u$.

- Is not invariant (but co-variant) under H-symmetry transformations. Hence, a choice of a particular basis (3.57) fixes the H-symmetry freedom.

- All other Q-functions are restored straightforwardly and uniquely by the determinant relation (2.23).

- The physical constraint $Q_{\emptyset} = u^L$ (see figure 9) is a highly non-local equation in the $u$-plane.

### Nested parameterisation:

- Uses Q-functions along a nesting path, e.g.

$$Q_{\leftarrow 1} = Q_1, \quad Q_{\leftarrow 2} = Q_{12}, \quad \dots, \quad Q_{\leftarrow N} = Q_{12\dots N} \equiv Q_{\emptyset}. \qquad (3.58)$$

- Is invariant under the action of Borel subgroup of the H-symmetry transformations (lower-triangular matrices with respect to the chosen ordering).$^{38}$ Hence, H-symmetry is only partially broken by the choice of (3.58).

<footnote>$^{38}$More accurately, diagonal H-matrices still affect the overall normalisation of Q-functions. The overall normalisation is, however, not essential for our discussion.</footnote>
- All other Q-functions are restored by a systematic usage of the QQ-relation (2.7); the explicit computation requires to solve linear first-order finite-difference equations, and the Q-functions are found not uniquely but modulo the Borel subgroup of H-rotations.

- The constraint $Q_{\emptyset} = u^L$ is imposed naturally.

The nesting path parameterisation behaves smoothly in the untwisting limit in the sense that its limit can be used to parameterise an untwisted Q-system. In fact, a safe way to realise the untwisting limit of the whole Q-system is to choose the set of nested Q-functions

$$Q_{\leftarrow|A|} \equiv Q_A, \quad A \text{ belongs to a chosen nesting path,} \qquad (3.59)$$

normalise them to be non-singular and non-zero in the untwisting limit, take the limit only of these functions,<fnref n="39" />

$$\mathbb{Q}_{\leftarrow|A|} = \lim_{G \to \mathbb{I}} Q_{\leftarrow|A|}, \qquad (3.60)$$

and then restore all other Q-functions using QQ-relations. Note that $\mathbb{Q}_{\leftarrow|A|}$ are always non-zero by construction, and one can construct a meaningful Q-system from any nested set of non-zero Q-functions.

On the other hand, the covariant parameterisation is quite singular: the set functions (3.57) degenerates upon direct untwisting limit and we cannot use it anymore to parameterise a Q-system. The problem with covariant parameterization is that it defines a meaningful Q-system only if the basis Q-functions have a non-vanishing determinant: $Q_{(1)}^{[1-N]} \wedge Q_{(1)}^{[3-N]} \wedge \dots Q_{(1)}^{[N-1]} \neq 0$. The latter property can be violated when certain limits are taken, in particular this happens to be the case in the untwisting limit.

We can give a symmetry argument why the covariant parameterisation behaves badly in the untwisting limit. Imposing the gauge-fixing condition that the twist is diagonal essentially brakes the H-symmetry. In the untwisting limit, the gauge-fixing condition is no longer required, hence the (global) H-symmetry is restored. Any choice of a covariant basis would spontaneously brake the restored symmetry, but there is no preferred way to do this choice by performing the untwisting limit. So, instead of having a meaningful limit, the covariant parameterisation degenerates when $G \to \mathbb{I}$.

We can choose, by hands, how to brake the symmetry, by performing twist-dependent H-rotations, e.g. (3.20), and in this way to define a smooth limit of a covariant basis. It can be summarised by a formula

$$\mathbb{Q}_a = \lim_{G \to \mathbb{I}} (h_a^{a'} Q_{a'}), \qquad (3.61)$$

where $h$ is a twist-dependent rotation matrix which should be fine-tuned to remove degeneracy from the limiting system. Finding $h$ is equivalent in complexity to solving QQ-relations.

Nested parameterisation is invariant enough under H-transformation to have a smooth limit. It is, however, not fully invariant, and the choice of a nesting path plays a certain role which we will now discuss.

<footnote n="39">³⁹We use font $\mathbb{Q}$ to label a Q-system without twist.</footnote>
We need to precise how the Q-functions are labeled. In the diagonal twist gauge, we follow the following rule: the function $Q_A$ with index $A$ is the one with the exponential prefactor $\prod_{a \in A} x_a^{-i\frac{u}{2}}$ in the large-$u$ asymptotics. In the non-twisted case, the labelling is more subtle, but there still exists a natural choice. First, one rotates the basis such that all one-indexed Q-functions have a distinct degree, and then, among all Q-functions $Q_A$ with $|A| = n$, we assign the label $Q_{12...n}$ to the polynomial of the lowest degree.

In principle, there are $N!$ choices of nesting paths, or, equivalently of the total order in the set $\{1, 2, \dots, N\}$. In the twisted case all of the choices enter indeed on the same footing. However, in the non-twisted case, the order $1 < 2 < \dots < N$ in the above-introduced labelling scheme, used as an example in (3.58), is distinguished for three reasons:

- the Q-functions of the distinguished nesting can be uniquely defined as sub-dominant solutions of the corresponding Baxter equations.

- these nested Q-functions emerge from any nested set in the untwisting limit. I.e. if $Q_{\leftarrow n} = \lim_{G \to \mathbb{I}} Q_{\leftarrow n}$ then $Q_{\leftarrow n} = Q_{12...n}$ independently of how $Q_{\leftarrow n} = Q_{a_1...a_n}$ was chosen.

- The large-$u$ behaviour correctly reproduces the weights of the representation according to (3.24); note that the statement (3.24) is formulated following the distinguished nested path.

The present discussion can be generalised to $su(K_1, K_2|M)$ case. The full H-symmetry is broken to $GL(K_1) \times GL(K_2) \times GL(M)$ by analytic and boundary requirements on the solution. However the full H-symmetry is not broken on the level of QQ-relations, and it re-emerges partially producing some interesting ambiguities.

The Q-functions of the twisted Q-system have a natural labelling $Q_{A_1,A_2|J}$, where $A_1 \subset \mathcal{B} = \{1, 2, \dots, K_1\}$, $A_2 \subset \mathcal{B} = \{\hat{1}, \hat{2}, \dots, \hat{K_2}\}$, and $J \subset \mathcal{F} = \{\hat{1}, \hat{2}, \dots, \hat{M}\}$, imposed by the prefactor $\prod_{a \in A_1} \prod_{b \in A_2} \prod_{j \in J} \left(\frac{x_a x_b}{y_j}\right)^{-i\frac{u}{2}}$ in the large-$u$ asymptotic. Due to possible Borel ambiguities, which are remnants of broken H-symmetry, we have to further precise the maximal Stokes sector where one demands such an asymptotic behaviour. The covariant basis, comprised from one-indexed functions $Q_{a,\emptyset|\emptyset}$, $Q_{\emptyset,b|\emptyset}$, $Q_{\emptyset,\emptyset|j}$, definitely suffers from the Borel ambiguities. However, it is possible to choose a nesting path where all Q-functions are rational functions times the exponential prefactor and hence defined uniquely. Nesting paths with order in which $a < b$ if $a \in \mathcal{B}_1$ and $b \in \mathcal{B}_2$ have such a property.<fnref n="40" />

By analogy, one can define $Q_{\leftarrow k_1, k_2|m} \propto \lim_{G \to \mathbb{I}} Q_{A_1, A_2|J}$, where $k_i = |A_i|$ and $j = |J|$. It is expected that due to the degeneracy effect, the function $Q_{\leftarrow k_1, k_2|m}$ does not depend on the choice of sets $A_1, A_2, J$, but only on the number of indices involved.<fnref n="41" /> We then

<footnote n="40">⁴⁰The statement was made by inspecting the structure of Bethe equations for arbitrary highest-weight representation written in appendix C. Intrinsic Q-system study has still to be performed; it might show that a weaker constraint on the order suffices.</footnote>

<footnote n="41">⁴¹This is a conjecture which we believe should be true in a general position. The main argument is that each QQ-relation (2.7) has a potentially singular dependence only on one twist ratio $x_a/x_b$, $y_i/y_j$, or $x_a/y_j$. Hence, in general position we can essentially analyse each QQ-relation separately, and hence apply our detailed knowledge of rank-1 examples.</footnote>
constrain the distinguished orderings in the untwisted Q-system by demanding

$$ \mathbb{Q}_{1...k_1,1...k_2|1...m} = \mathbb{Q}_{\leftarrow k_1,k_2|m}. \qquad (3.62) $$

I.e $(K_1 + K_2 + M)!$ different nesting paths degenerate into $\frac{(K_1+K_2+M)!}{K_1!K_2!M!}$ paths by taking the untwisting limit. We further prefer to constrain to the paths which contain only rational Q-functions, hence we limit ourselves to $\frac{K_1!K_2!}{(K_1+K_2)!}(K_1+K_2+M)!$ possibilities in the twisted case and $\frac{(K_1+K_2+M)!}{(K_1+K_2)!M!}$ possibilities in the untwisted case.

Defining all Q-functions with distinguished order using (3.62) may lead to contradictions in QQ-relations, e.g. (3.37) might be violated. Hence we follow the above-outlined strategy: to choose one unique path, perform the limit for Q-functions along this path and then restore the other Q-functions using QQ-relations. Note that the strategy with the use of rotation matrix (3.61) also requires to choose a nesting path to decide which functions to only regularise/make non-zero, and which to rotate. The result of untwisting can indeed depend on the nesting choice, cf. (3.39a) vs. (3.39b) (this is another place where the broken H-symmetry re-emerges). However, the ambiguity proves to be unphysical as explained in appendix B.4. The untwisted result should also comply with weights expected from representation theory, according to (3.36), with $k = k_1 + k_2$. To get the agreement, the lowest weight should be defined by the same total order that defines the nesting path.

## 4 Twisted Quantum Spectral Curve

The AdS$_5$/CFT$_4$ system represents the most emblematic example of AdS/CFT duality between the Green-Schwarz-Metsaev-Tseytlin superstring sigma model on $AdS_5 \times S^5$ background on the string side of the duality, and $N = 4$ SYM theory on the CFT side. It was realized that, at least in the planar sector, this system is integrable [40]. Recently, the integrability equations, originally discovered as the AdS/CFT Y-system [5], and later brought into a TBA form [39, 73-75] were recast by the present authors together with N.Gromov into a concise and elegant finite system of Riemann-Hilbert equations — the Quantum Spectral Curve (QSC) [2, 3]. The QSC approach has shown its efficiency and universality in the recent papers [22-24, 62, 76] and has been then successfully applied to the AdS$_4$/CFT$_3$ duality [77-79].

In this section, we will generalise the AdS$_5$/CFT$_4$ QSC construction to the case of arbitrary diagonal twist which corresponds, in several subcases, to a handful of integrable modifications of $N = 4$ SYM (see e.g. [10]). In e.g. rational spin chains, the twist is a particular deformation of the spin chain boundary conditions. We saw in section 3 that such a deformation amounts to the introduction of a constant connection $A$. In the AdS$_5$/CFT$_4$ case, a description on the level of spin chains or other explicit physical model is not available at arbitrary coupling. Hence the twist of QSC is understood solely as the introduction of a constant connection $A$, we denote the eigenvalues of $e^A$ as $x_a$ and $y_i$. The use of the terminology “twist” is justified at weak and strong coupling where the introduced twist can be indeed given its more standard physical meaning.

As we discussed in section 3, there is a mapping between the twist parameters and the charges. For spin chain, the twist matrix $G = e^A$ does not only execute H-rotations, but
also it can be thought as an element of the Cartan subgroup of the symmetry group. Such a mapping also manifests itself in the large-u asymptotics: a multi-index function $Q_{A|I}$ has an asymptotics $Q_{A|I} \sim \prod_a (x_a^i u u^{\lambda_a})^{\alpha_a} \prod_i (y_i^{-i u u^{\nu_i}})^{\beta_i} u^n$, where $[\lambda; \nu]$ is the weight of state (3.20), and the numbers $\alpha_a, \beta_i, n$ depend on the indices $A|I$ but do not depend on the state considered.

By analogy, we think about twist parameters as group elements also in the case of the AdS$_5$/CFT$_4$. There are actually six twist parameters:

$$G = \{y_1, \dots, y_4 | x_1, \dots, x_4\} \in \text{PSU}(2, 2|4), \quad \text{where} \quad x_1 x_2 x_3 x_4 = 1, \quad y_1 y_2 y_3 y_4 = 1, \quad (4.1)$$

where the two constraints are due to super-unimodularity and projectivity of $\text{PSU}(2, 2|4)$ group.

Explicitly the charges $\lambda_a$ and $\nu_i$ associated correspondingly to $x_a$ and $1/y_i$ read as follows:

$$\lambda_1 = \frac{J_1 + J_2 - J_3}{2}, \quad \lambda_2 = \frac{J_1 - J_2 + J_3}{2}, \quad \lambda_3 = \frac{-J_1 + J_2 + J_3}{2}, \quad \lambda_4 = \frac{-J_1 - J_2 - J_3}{2}, \quad (4.2a)$$

$$\nu_1 = \frac{-\Delta + S_1 + S_2}{2}, \quad \nu_2 = \frac{-\Delta - S_1 - S_2}{2}, \quad \nu_3 = \frac{+\Delta + S_1 - S_2}{2}, \quad \nu_4 = \frac{+\Delta - S_1 + S_2}{2}. \quad (4.2b)$$

To avoid confusion, let us stress that twists do not define the value of charges. We just say that twists and (exponentiation of charges) are elements of the same group.

One introduces a freedom of speech and says that we twist a given symmetry if the value of the twist in the direction of the corresponding Cartan elements is different from one. For instance, we say that x-twists realise the twisting of $\text{su}(4) \simeq \text{so}(6)$ R-symmetry and that y-twists twist the $\text{su}(2, 2) \simeq \text{so}(2, 4)$ conformal symmetry of $\mathcal{N} = 4$ SYM. Note that a twisted model is generically no longer invariant under the original symmetry transformations, only invariance under the Cartan subalgebra action remains. In particular, unless $x_a = y_i$ for some $a, i$, the supersymmetry is fully broken in the presence of twist.

For what concerns twisting the charges, one can make a curious remark. It was noticed in [51], in the approximation of the twisted ABA equations,<fnref n="42" /> that

$$\frac{x_1 x_2}{y_1 y_2} = \frac{y_3 y_4}{x_3 x_4} = e^{i \mathcal{P}}, \quad (4.3)$$

where $\mathcal{P}$ is the total momentum of a state. The identities (4.3) are also true beyond the ABA approximation The first one is a trivial consequence of (4.1) and the second one is a dynamical consequence of the QSC equations. On the other hand, the combination $\frac{x_1 x_2}{y_1 y_2}$ twists the charge $E = \Delta - J_1$ which is nothing but the energy of a state. Hence the total momentum $\mathcal{P}$ plays a role of twist for the AdS time $\tau_{\text{AdS}}$.

On the CFT side of duality, e.g. from the point of view of asymptotic Bethe ansatz, the deformed theory represents a non-local spin chain with twisted periodic boundary

<footnote n="42">⁴²The dictionary between our twist notations and the angles of [51] in the sl(2) favored grading is: $y_1 = e^{i\phi_1}$, $x_1 = e^{i\phi_2}$, $x_2 = e^{i\phi_3}$, $y_2 = e^{i\phi_4}$, $y_3 = e^{i\phi_5}$, $x_3 = e^{i\phi_6}$, $x_4 = e^{i\phi_7}$, $y_4 = e^{i\phi_8}$. In the SU(2) favored grading, one has to exchange $x_j \leftrightarrow y_j$.</footnote>
conditions. The corresponding SYM action is not known for a general twisting. However, for some particular twistings such SYM theories were conjectured and successfully tested. Such is the case of the so-called $\gamma$-deformation, when y-twists are absent and x-twists are pure phase factors. The corresponding $\gamma$-deformed, $N=0$ SYM action has three exactly marginal deformations of scalar-scalar and fermion-scalar interactions: the commutators of scalar fields with themselves and with the fermions should be replaced by the deformed q-commutators [10]. It is believed that this $\gamma$-deformed theory is non-conformal anymore at finite $N_c$ but it still preserves its conformality at $N_c = \infty$.43 On the string side, the $\gamma$-deformed coset has been known already since long [82] and the corresponding string sigma model appears to be integrable. This AdS/CFT-correspondence was successfully tested by comparing the results of Lüscher correction from TBA on the string side [19, 20] with the leading order weak coupling correction on the Yang-Mills side [21, 80] for the BMN vacuum. We will reproduce this result in the next section for testing the twisted QSC.

For a particular one-parametric case of gamma twisting, the $\beta$-deformation (when all $\gamma_j = \beta$) the corresponding CFT dual is identified with a particular case of Leigh-Strassler $N=1$ SYM theory [83]. The $\beta$-deformation of the $AdS_5 \times S^5$ string sigma model was proposed in [84, 85] and it is related to Lunin-Maldacena background [8] on the string side of duality.

The y-twists describe the deformations of $AdS_5$. They presumably correspond to the introduction of a non-commutativity of space-time coordinates in the dual deformed SYM theory [10]. Generically, three y-twists (the condition (4.1) always imposed) break the conformal group $SU(2,2)$ to the remaining $U(1) \times U(1) \times \mathbb{R}$ subgroup, where the line $\mathbb{R}$ corresponds to the action of the dilatation operator $\mathcal{D}$ related to the non-compact translational isometry of the conformal group.

In this section, we first introduce the most general twisting of QSC, and then show how one can reduce the number of twists and consider some examples of partial twisting which correspond to preserving certain, generically non-abelian subalgebras of the full $\text{psu}(2,2|4)$. The untwisting procedure is far from trivial (it is already rather subtle for the spin chains, as we have seen in the previous section) and it can drastically modify the analyticity conditions of QSC system.

In the next section, we will demonstrate how one can work with twisted QSC by reproducing the known result for the anomalous dimension of the BMN vacuum at single wrapping orders.

## 4.1 Twisting of Quantum Spectral Curve

Now we will give the twisted version of QSC formulation of the AdS/CFT spectral problem. We will proceed with the generic twisting keeping all 6 independent twist variables

<footnote n="43">43 At finite $N_c$ the $\gamma$-deformed theory is non-conformal, and even at infinite $N_c$ certain operators of small length, such as $\text{tr}Z^2$, have a divergent dimension and demand the addition of double trace counterterms in the action, leading to the running coupling [80, 81]. The theory then runs into the $\beta$-deformed $N=1$ SYM. This is always the case at finite $N_c$. However, in the 't Hooft limit $N_c \to \infty$ we expect that these short operators can be decoupled from OPE's and most of the correlators will take the conformal form. In that sense, both conformality and integrability of $\gamma$-deformed SYM theory are restored in the 't Hooft limit.</footnote>
as arbitrary complex numbers. Recall [3] that the QSC is essentially characterized by certain of Riemann-Hilbert type conditions imposed on a set of $2^8 = 256$ Baxter's Q-functions $Q_{A|I}(u)$, which depend on the spectral parameter $u$ and on two sets of indices, $A \subset \{1, 2, 3, 4\}$, $I \subset \{1, 2, 3, 4\}$, and the dependence is antisymmetric with respect to permutations of the indices inside each of these sets $A$ and $I$. These Q-functions obey the QQ-relations (2.74). In the AdS/CFT context, and in a specific, most natural gauge, the following constraints hold:

$$Q_{\emptyset|\emptyset} = 1, \quad Q_{\bar{\emptyset}|\bar{\emptyset}} := Q_{1234|1234} = 1. \qquad (4.4)$$

The first condition is a simple normalization, whereas the second one is non-trivial and should be interpreted as following from the quantum analog of unimodularity [6]. Indeed, $\frac{Q_{\bar{\emptyset}|\bar{\emptyset}}^+}{Q_{\bar{\emptyset}|\bar{\emptyset}}}$ becomes a quantum determinant in the case of rational spin chain with the same

symmetry. $Q_{\bar{\emptyset}|\bar{\emptyset}}$ = const is the only solution of $\frac{Q_{\bar{\emptyset}|\bar{\emptyset}}^+}{Q_{\bar{\emptyset}|\bar{\emptyset}}}$ = 1 consistent with analytic properties required below and the projectivity constraint (4.1). Normalisation (4.4) can be achieved by a constant in $u$ basis rotation $H$, with $\det H \neq 1$.

We notice that the twisting of spin chains, as it was done in the previous section by (3.8), only modifies the analyticity of Q-functions, in particular, adding the exponential asymptotics at $u \to \infty$ without changing the Baxter relations and QQ relations. We will follow this inspiration in the case of the AdS/CFT QSC and assume that the QQ-relations and the Riemann-Hilbert relations remain intact after twisting. Hence we will modify the $u \to \infty$ asymptotics of Q-functions by exponential factors defined by twists. The consequent modification of analytic properties will be however greatly constrained by the structure of QSC equations.

In what follows, we use the notations of [3]: namely, the Q-functions $P_a$ (resp $Q_j$) denote the functions $Q_{a|\emptyset}$ (resp $Q_{\emptyset|j}$) in a specific gauge, discussed in detail in [3] and used through the whole present section. The functions $P^a$ and $Q^j$ are their Hodge dual: $P^a = Q^a|\emptyset = (-1)^a Q_{\bar{a}|1234}$ and $Q^j = Q^{\emptyset|j} = (-1)^j Q_{1234|\bar{j}}$ — where $\bar{n}$ denotes the sorted multi-index which forms the complement of $n$ in $\{1, 2, 3, 4\}$ (for instance $\bar{3} = 124$). The Riemann-Hilbert relations, in a particular form of $P\mu$ and $Q\omega$ systems will be detailed further in section 4.2.

A natural ansatz for the large $u$ asymptotics of QSC, generalizing the formulae of the paper [3] (section 3.2.3) to the twisted case, when all 6 twists are turned on, is

$$ \begin{aligned} \mathbf{P}_a &\approx A_a x_a^{\frac{i}{2}u} u^{-\lambda_a}, & \mathbf{P}^a &\approx A^a x_a^{-\frac{i}{2}u} u^{\lambda_a}, & a = 1, 2, 3, 4, \\ \mathbf{Q}_j &\approx B_j y_j^{-\frac{i}{2}u} u^{-\nu_j}, & \mathbf{Q}^j &\approx B^j y_j^{\frac{i}{2}u} u^{\nu_j}, & j = 1, 2, 3, 4, \end{aligned} \qquad (4.5) $$

where $A_a, A^a, B_j$ and $B^j$ are constant prefactors and the powers of $u$ are given, for the generic twisting, by equation (4.2).

This ansatz will be further justified in section 4.3, where we will show that in general, setting a part of twists to zero (or to one, in terms of $x$ and $y$ variables) leads to certain shifts by integers in powers of certain Q-functions with respect to the fully