Completeness of Wronskian Bethe Equations for Rational ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ Spin Chains

Abstract

We consider rational integrable supersymmetric ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities ${{\theta }_{\ell }}$, $\ell =1,\ldots ,L$ satisfy ${{\theta }_{\ell }}+\hbar \ne {{\theta }_{\ell '}}$ for $\ell <\ell '$. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of $\theta _\ell$. In particular, we consider the regime $\theta _{\ell +1}/\theta _{\ell }\gg 1$ for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.

Appendices

Wronskian Algebra: A Pedestrian Approach

1.1 Some facts from commutative algebra and algebraic geometry

1.1.1 Basic definitions.

The study of the polynomial equations can be done in an analytic (geometric) or in an algebraic way. In the analytic approach, to a set of m polynomial equations in n variables $P_{i}(x_1,\ldots ,x_n)=0$, $i=1,\ldots , m$ is assigned an algebraic variety ${\mathcal {A}}$—a set of points $x\equiv (x_1,\ldots ,x_n)\in {\mathbb {C}}^n$ where these equations hold. The algebraic approach attaches to the equations an ideal generated by $P_{\ell }$, ${{\mathcal {I}}}=\langle P_{\ell } \rangle$ which is the set of all possible polynomials in n variables $Q\in {\mathbb {C}}[x_1,\ldots , x_n]$ that can be written in the form $Q=\sum _{\ell }q_{\ell }P_{\ell }$ for some polynomials $q_{\ell }$.

The relation between the two approaches is established by Hilbert’s Nullstellensatz: if Q vanishes on ${\mathcal {A}}$ then $Q^{r}\in {{\mathcal {I}}}$ for some integer r. The ideal constructed by all polynomials that vanish on ${\mathcal {A}}$ is called the radical of ${{\mathcal {I}}}$ and is denoted by $\sqrt{{{\mathcal {I}}}}$.

The algebraic description is more abstract and is less used in physics but it allows one to more accurately formulate some of the properties of the Wronskian algebra. In particular, we can work over fields different to ${\mathbb {C}}$, e.g. the field of fractions ${\mathbb {C}}({{\theta }_{}})$.

The next concept is to consider functions on ${\mathcal {A}}$ formalised as the quotient ring

$$\begin{aligned} {{\mathcal {R}}}={\mathbb {C}}[x_1,\ldots ,x_n]/{{\mathcal {I}}}\,. \end{aligned}$$

(A.1)

If ${{\mathcal {I}}}=\sqrt{{{\mathcal {I}}}}$ then ${{\mathcal {R}}}$ is called the coordinate ring of ${\mathcal {A}}$. Note that in this paper not all ideals are equal to their radicals.

Finally, recall that a ring is said to be an integral domain if $ab=0$ implies $a=0$ or $b=0$. The corresponding ideal is then called prime ($ab\in {{\mathcal {I}}}$ implies $a\in {{\mathcal {I}}}$ or $b\in {{\mathcal {I}}}$).

1.1.2 Polynomial division.

Easiness of the study of polynomials in one variable exists mainly due to the unambiguous polynomial division procedure. Recall how it works: let P be a monic polynomial in x of degree b which is one (of those polynomials) that generates the ideal ${{\mathcal {I}}}$ in ${\mathbb {C}}[x]$. Let Q be any polynomial in x. If Q contains a monomial $c\, x^a$ with $a\geqslant b$, we represent Q as a combination $Q=c\, x^{a-b} P+(Q-c\, x^{a-b}P)$ in which the first term is divisible by P and the second term has no monomial of degree a. One performs the same procedure with $P'=Q-c\, x^{a-b}P$ and continue it recursively until no monomials divisible by $x^b$ remain. So one obtains a representation $Q=q\,P+r$ , where the degree of r is strictly smaller than b. Both q and r are fixed uniquely. Furthermore, one can guarantee the Bézout’s lemma, that is, one can find such $\alpha ,\beta \in {\mathbb {C}}[x]$ that $\alpha P_1+\beta P_2={\mathrm{GCD}}(P_1,P_2)$ for any polynomials $P_1,P_2$, and hence conclude that any ideal in one variable is principal, i.e. it is generated by a single polynomial—the GCD of polynomials $P_1,\ldots ,P_m$ that generate the ideal.

A practical application in our case would be: if a Bethe algebra is generated by a single operator ${{\hat{x}}}$ then this algebra is guaranteed to be isomorphic to a quotient ${\mathbb {C}}[x]/{{\mathcal {I}}}$ where ${{\mathcal {I}}}$ is the ideal generated by the minimal polynomial of ${{\hat{x}}}$. As we have L generating operators ${{\hat{c}}}_{\ell }$, things are not that simple.

1.1.3 Gröbner bases.

Many problems in systems with multiple variables arise from difficulties with the polynomial division. First, to even define a division algorithm one needs to introduce a total order on the set of monomials $x^d\equiv \prod \nolimits _{i=1}^n x_i^{d_i}$ that should be an order in which 1 is the smallest monomial and $a<b$ implies $a\,c<b\,c$ for any a, b, c. A diversity of monomial orders is available in contrast to only one option for the single-variable case. We shall use below only lexicographic orders which form a small subset of all possibilities.

After fixing a monomial order and denoting by $P_{i}$ the generators of the ideal ${{\mathcal {I}}}$, one can perform long polynomial division (exclusion of all monomials that are divisible by leading monomials of $P_{i}$) to represent any polynomial Q as

$$\begin{aligned} Q=\sum _{i} q_{i}\,P_{i}+r\,. \end{aligned}$$

(A.2)

Unfortunately, neither the procedure nor its result are unique if $P_{i}$ are arbitrary generators, so the division is essentially meaningless.

However, if $P_{i}$ form a special set called Gröbner basis then r is uniquely defined by the polynomial division.³⁹ Hence $Q\in {{\mathcal {I}}}$ iff $r=0$. $q_{i}$ are not unique though, but uniqueness of r suffices for the study of the quotient ring (A.1).

A set of polynomials $P_i$ forms a Gröbner basis of an ideal ${{\mathcal {I}}}$ if i) they generate ${{\mathcal {I}}}$, ii) the set is closed under computation of S-polynomials, see e.g. [118] for further explanations. If moreover, for all $i\ne i'$, $P_i$ does not contain monomials divisible by the leading monomial of $P_{i'}$ then such a Gröbner basis is called the reduced one and it is unique for the given choice of a monomial order. By a Gröbner basis we mean the reduced basis in the following.

1.1.4 Monomial basis.

Let us fix a Gröbner basis. The set of monomials that can arise in the remainders of polynomial divisions forms a basis in the quotient ring ${{\mathcal {R}}}$ considered as a vector space. This basis shall be called the monomial basis.

We can use the monomial basis to realise the regular representation of an algebra in terms of explicit matrices, see the example on page 22. Such a basis has an important advantage: all computations in it are performed in the original field, and so the coefficients of ${\check{x}}$ will belong to the same field, where ${\check{x}}$ is a matrix in the regular representation, see Sect. 3.2.

1.2 ${\mathbb {C}}(\chi )$-module and invariance of solutions multiplicity

Consider the ring of polynomials in 2L variables ${\mathbb {C}}[\chi ][c]\equiv {\mathbb {C}}[\chi _{1},\ldots \chi _{L}][c_1,\ldots ,c_L]$. We define the Wronskian algebra as ${{{\mathcal {W}}}_{\varLambda }}:={\mathbb {C}}[\chi ][c]/{{\mathcal {I}}}_\varLambda$, where ${{\mathcal {I}}}_{\varLambda }=\langle {{\,\mathrm{SW}\,}}_{\ell }(c)-\chi _{\ell }\rangle$ is the ideal generated by Wronskian relations. As we can simply exclude $\chi _{\ell }$ using equations $\chi _{\ell }={{\,\mathrm{SW}\,}}_{\ell }(c)$, ${{{\mathcal {W}}}_{\varLambda }}$ is isomorphic over ${\mathbb {C}}$ to ${\mathbb {C}}[c]$—the ring of polynomials in L variables. Hence, in particular, ${{{\mathcal {W}}}_{\varLambda }}$ is an integral domain and ${{\mathcal {I}}}_{\varLambda }$ is a prime ideal.

In the case of prime ideals, it is quite easy to promote rings to fields. In this subsection, we shall consider $\chi _{\ell }={{\,\mathrm{SW}\,}}_{\ell }(c)$ as an equation on $c_{\ell }$ in the field of fractions ${\mathbb {C}}(\chi )$ and ${{{\mathcal {W}}}_{\varLambda }}$ as a ring over ${\mathbb {C}}(\chi )$. “Easiness” of promotion lies in the following statement: any polynomial in variables $c_{\ell }$ and $\chi _{\ell }$ that belongs to ${{{\mathcal {W}}}_{\varLambda }}$ considered as an object in a ring over ${\mathbb {C}}(\chi )$ would also belong to ${{{\mathcal {W}}}_{\varLambda }}$ considered as an object in a ring over ${\mathbb {C}}[\chi ]$.

When we work over a field of fractions, we can compute a Gröbner basis. Simply, instead of conventional computation in ${\mathbb {C}}[c_1,\ldots ,c_L]/\langle {{\,\mathrm{SW}\,}}_\ell (c)-\bar{\chi }_{\ell }\rangle$ with numerical $\bar{\chi }_{\ell }\in {\mathbb {C}}$, we do a computation in ${\mathbb {C}}(\chi )[c_1,\ldots ,c_L]/\langle {{\,\mathrm{SW}\,}}_\ell (c)-\chi _{\ell }\rangle$ with symbolic $\chi _{\ell }\in {\mathbb {C}}(\chi )$. When the Gröbner basis is computed, we can construct the corresponding monomial basis and conclude what is the dimension of ${{{\mathcal {W}}}_{\varLambda }}$ (as a vector space over ${\mathbb {C}}(\chi )$) and hence what is the number of solutions of the Wronskian equations. Note that the solutions themselves would typically only exist in an algebraic closure of ${\mathbb {C}}(\chi )$. However, computation of the monomial basis can be performed directly in ${\mathbb {C}}(\chi )$ and this is the only thing needed.

Working over ${\mathbb {C}}(\chi )$ is equivalent to considering $\chi _{\ell }$ in generic position, when no accidental relations happen. When we specialise to a concrete numerical value $\bar{\chi }_{\ell }$ of $\chi _{\ell }$, we are interested whether the number of solutions changes. We can formulate (a bit stronger) question from the point of view of the Gröbner basis: does it remain a Gröbner basis upon specialisation?

Lemma A.1

Let the Gröbner basis of the ideal ${{\mathcal {I}}}_\varLambda =\langle {{\,\mathrm{SW}\,}}_{\ell }-\chi _{\ell }\rangle$ in ${\mathbb {C}}(\chi )[c]$ w.r.t. some monomial order < be given by polynomials

$$\begin{aligned} s_m=c^{m}+\sum _{m'<m}p_{mm'}(\chi )\,c^{m'}\,,\quad m\in M\,, \end{aligned}$$

(A.3)

where $m:=(m_1,\ldots ,m_L)$, $c^{m}:=\prod \nolimits _{\ell =1}^L c_{{\ell }}^{m_{\ell }}$, M is a set of tuples m, and $p_{mm'}\in {\mathbb {C}}(\chi )$.

Let $p_{mm'}$ be finite numbers when evaluated at $\chi _{\ell }=\bar{\chi }_{\ell }\in {\mathbb {C}}$. Then ${\bar{s}}_m=c^{m}+\sum \limits _{m'<m}p_{mm'}(\bar{\chi })c^{m'}, m\in M$, form the Gröbner basis of the ideal ${{\mathcal {I}}}_\varLambda (\bar{\chi })=\langle {{\,\mathrm{SW}\,}}_{\ell }-\bar{\chi }_{\ell }\rangle$ in ${\mathbb {C}}[c]$ for the same monomial order.

In other words, it is safe to specialise a Gröbner basis at those values of $\chi _{\ell }$ where denominators of $p_{mm'}$ do not vanish.

Proof

To verify the statement first we check that the declared set of ${{\bar{s}}}_{m}$ generates ${{\mathcal {I}}}_\varLambda (\bar{\chi })$. To this end, use long division in ${\mathbb {C}}(\chi )[c_1,\ldots ,c_L]$ to write ${{\,\mathrm{SW}\,}}_{\ell }-\chi _{\ell }=\sum _m q_m(\chi ) s_m$. From the algorithm of long division it is clear that $q_m(\chi )$ are not singular at $\chi =\bar{\chi }$ if $p_{mm'}(\chi )$ are not singular which is the case by the condition of the theorem. Hence ${{\,\mathrm{SW}\,}}_{\ell }-\chi _{\ell }=\sum _m q_m(\chi ) s_m$ can be evaluated and still holds at $\chi =\bar{\chi }$. To check that ${\bar{s}}_{m}$ form a Gröbner basis we need to e.g. compute S-polynomials but this is combinatorially the same exercise as for $s_m$ since the leading monomials are not affected by specialisation. $\square$

Wronskian equations can be obviously specialised at arbitrary point $\bar{\chi }$ and so the ring ${{{\mathcal {W}}}_{\varLambda }}(\bar{\chi })$ is always a well-defined object. Now we would like to show that, for a given $\bar{\chi }$, one can find a Gröbner basis that can be specialised at this point and its vicinity. This is the key point to prove the following theorem:

Theorem A.2

$d_{\varLambda }:= \dim _{{\mathbb {C}}}{{{\mathcal {W}}}_{\varLambda }}(\bar{\chi })$ does not depend on $\bar{\chi }$.

In other words, the number of solutions of Wronskian equations counted with multiplicities is always the same, even on the degeneration set ${{\mathcal {X}}}_{{\mathrm{crit}}}$.

Proof

We know that the theorem holds for all points $\bar{\chi }\notin {{\,\mathrm{SW}\,}}(D)$ since all the solutions of the Wronskian equations are distinct there and so the dimension of the quotient ring coincides with the number of solutions that we denote as $d_{\varLambda }$. We can path-connect any two regular points and the number of solutions cannot change along the path, see Sect. 3.1.

Take L linearly independent constant vectors $w_{\ell }=(w_{\ell 1},\ldots ,w_{\ell L})$ and define $x_{\ell }=\sum _{\ell '}w_{\ell \ell '}c_{\ell '}$. For almost any choice of $w_{\ell }$, the Gröbner basis of ${{\mathcal {I}}}_{\varLambda }$ in ${\mathbb {C}}(\chi )$ w.r.t. the monomial order $x_1<x_2<\ldots x_L$ should have the form

$$\begin{aligned}&x_{1}^{d_{\varLambda }}+a_{1}^{(d_{\varLambda }-1)}(\chi )x_{1}^{d_{\varLambda }-1}+\ldots a_{1}^{(0)}\,, \end{aligned}$$

(A.4a)

$$\begin{aligned}&x_{2}-\sum _{k=0}^{d_{\varLambda }-1}b_{2k}(\chi ) x_1^{k}\,,\nonumber \\ \ldots \nonumber \\&x_{L}-\sum _{k=0}^{d_{\varLambda }-1}b_{Lk}(\chi ) x_1^{k}\,. \end{aligned}$$

(A.4b)

Indeed, take a point $\bar{\chi }\notin {{\mathcal {X}}}_{{\mathrm{crit}}}$ for which the conditions of Lemma A.1 hold. At such a point, leading monomials of the Gröbner basis are the same before and after specialisation, and so we can judge about the Gröbner basis from its specialised version. Since $\bar{\chi }\notin {{\mathcal {X}}}_{{\mathrm{crit}}}$, $\check{x}_{\ell }$ (regular representation of $x_\ell$, written as a matrix in the monomial basis) should have $d_{\varLambda }$ distinct eigenvalues for almost any choice of $\omega _{\ell }$, and therefore the minimal polynomial equation it satisfies is of degree $d_{\varLambda }$ which is (A.4a). In the chosen lexicographic order this equation should belong to the Gröbner basis. Other variables $x_2,\ldots ,x_L$ should satisfy (A.4b) (i.e. they are uniquely fixed if $x_1$ is fixed) otherwise dimension of ${{{\mathcal {W}}}_{\varLambda }}(\bar{\chi })$ would exceed $d_{\varLambda }$.

By the properness of WBE $a_1^{(a)}(\chi )$ cannot have singularities, hence they are simply polynomials in $\chi _{\ell }$. Coefficients $b_{\ell k}$ however are rational functions of $\chi _{\ell }$ that can contain poles. Everywhere outside of these poles, the conditions of Lemma A.1 hold and we can perform specialisation asserting that the dimension of the specialised polynomial ring is $d_{\varLambda }$.

It remains to show that for any $\bar{\chi }\in {\mathbb {C}}^L$, one can choose $\omega _{\ell }$ such that $b_{\ell k}$ are not singular at $\bar{\chi }$. To this end, we can actually explicitly express $b_{\ell k}$ in terms of solutions of the Wronskian system. Let $x_\ell =x_{\ell }^{(i)}$ be the i-th solution. Then polynomials (A.4b) can be rewritten as

$$\begin{aligned} x_{\ell }-\sum _{k=0}^{d_{\varLambda }-1}b_{\ell k}(\chi ) x_1^{k}= \frac{\det \left| \begin{matrix} x_{\ell } &{} 1 &{} x_1 &{} x_1^2 &{}\ldots \\ x_\ell ^{(1)} &{} 1&{} x_1^{(1)} &{} (x_1^{(1)})^2 &{} \ldots \\ x_\ell ^{(2)} &{} 1&{} x_1^{(2)} &{} (x_1^{(2)})^2 &{} \ldots \\ x_\ell ^{(3)} &{} \ldots \\ \ldots \end{matrix}\right| }{\det \left| \begin{matrix} 1 &{} x_1^{(1)} &{} (x_1^{(1)})^2 &{} \ldots \\ 1 &{} (x_1^{(2)}) &{} (x_1^{(2)})^2 &{} \ldots \\ 1 &{} \ldots \end{matrix}\right| }\,. \end{aligned}$$

(A.5)

Indeed, equality of the above polynomials to zero implies $x_\ell =x_{\ell }^{(i)}$ precisely when $x_1=x_1^{(i)}$.

While $x_{\ell }^{(i)}$ belong to an algebraic closure of ${\mathbb {C}}(\chi )$, the above ratio of determinants is symmetric under permutations $x_{\ell }^{i}\rightarrow x_{\ell }^{\sigma (i)}$ and hence should be a polynomial in $x_1$ with coefficients in the base field, i.e. ${\mathbb {C}}(\chi )$. This follows for instance from $\sum _{i=1}^{d_{\varLambda }} f(x^{(i)})={\mathrm{Tr}}\, f({\check{x}})$ and basic combinatorial arguments. Of course, one can conclude the same from the fact that (A.4b) are obtained in the process of computation of the Gröbner basis.

At points $\bar{\chi }$ where all $x_{1}^{(i)}$ are distinct, the denominator of (A.5) is non-zero and hence $b_{\ell k}$ are non-singular. As discussed, for a given regular $\bar{\chi }$, we can adjust $\omega _{\ell }$ in a way that $x_1$ has non-degenerate solutions.

When $\bar{\chi }\in {{\,\mathrm{SW}\,}}(D)$ all $x_{\ell }$ degenerate. Then consider a one-parametric smooth path $\chi (t)$ in the space of parameters such that $\chi (t=0)=\bar{\chi }$ is the degeneration point of interest and $\bar{\chi }(t\ne 0)\notin {{\mathcal {X}}}_{{\mathrm{crit}}}$. Moreover, one chooses such a path that all $x_{\ell }^{(i)}$ are distinct along the path for sufficiently small t, except for the point $t=0$ itself.

The value of the ratio of determinants in (A.5) is not well-defined at $t=0$ but it can be computed as the limit $t\rightarrow 0$. Since this ratio is a rational function of $\chi _{\ell }$, the limit, if finite, should produce polynomials (A.4b) specialised at $t=0$.

To compute the limit, note that all $x_{\ell }$, for generic enough choice of $\omega _{\ell }$, satisfy one-variable equations $x_{\ell }^{d_{\varLambda }}+a_{\ell }^{(d_{\varLambda })}x_{\ell }^{d_{\varLambda }-1}+\ldots =0$, where $a_{\ell }^{(k)}$ are polynomials in $\chi$ and hence are well-defined even at $t=0$. Define $x_{\ell }^i(t)$ as solutions of these equations that coincide with solutions of Wronskian equations for $t\ne 0$; their $t=0$ value is then defined as the continuation $t\rightarrow 0$. If $\mu _{i\ell }$ is the degree of degeneration of solution $x_{\ell }^{(i)}$ at $t=0$ (i.e. $\mu _{i\ell }$ solutions of the one-variable equation on $x_{\ell }$ coincide at this point) then $x_{\ell }^{(i)}(t)$ is expanded in the Puiseux series

$$\begin{aligned} x_{\ell }^{(i)}(t)=x_{\ell }^{(i)}(0)+r_{\ell i,1}t^{1/\mu }+r_{\ell i,2}t^{2/\mu }+\ldots \,, \end{aligned}$$

(A.6)

where $\mu ={\mathrm{LCM}}(\mu _{11},\ldots ,\mu _{LL})$.

One should know finitely many terms in the series (A.6) to compute the determinants ratio in (A.5) in the limit $t\rightarrow 0$. We require that for these finitely many terms, for each k and i, if at least one $r_{\ell i,k}$ is non-zero then all $r_{\ell i, k}$, $\ell =1,\ldots , L$, are non-zero. It is sufficient to guarantee that the ratio is finite, while imposing such a requirement excludes a measure zero subspace from acceptable values of $\omega _{\ell }$. Recall that we already excluded the space of $\omega _{\ell }$ where the degree of the minimal polynomial of $\check{x}_{\ell }$ is less than $d_{\varLambda }$ which is of measure zero as well. The majority of $\omega _{\ell }$ are outside of the stated restrictions, and we can choose any valid option to guarantee the regularity of $b_{\ell k}$ at $\chi =\chi ^{(0)}$ and hence the possibility to specialise the Gröbner basis (A.4) at this point thus concluding that $\dim _{{{\mathbb {C}}}}{{{\mathcal {W}}}_{\varLambda }}(\bar{\chi })=d_{\varLambda }$. $\square$

The proposed proof shows that there is a close analogy between WBE and a polynomial equation in a single variable. Indeed, for any point $\bar{\chi }\in {{\mathcal {X}}}$, regular or not, we can choose a variable $x_1$ that satisfies (A.4a) and such that there is a neighbourhood ${{\mathcal {O}}}_{\bar{\chi }}$ where $b_{\ell ,k}$ are non-singular which allows one to compute all elements of ${{{\mathcal {W}}}_{\varLambda }}$ using (A.4b). So a single-variable equation (A.4a) contains all information about ${{{\mathcal {W}}}_{\varLambda }}$ in the selected neighbourhood.

1.3 Freeness of ${{{\mathcal {W}}}_{\varLambda }}$ and trivialisation of a vector bundle

For each ${{\mathcal {O}}}_{\bar{\chi }}$, we have a basis generated by powers of $x_1$. Two different bases constructed at $\bar{\chi }$ and $\bar{\chi }'$ are related by a transition matrix which is regular together with its inverse on the intersection of ${{\mathcal {O}}}_{\bar{\chi }}$ and ${{\mathcal {O}}}_{\bar{\chi }'}$. Hence we get a structure of a holomorphic bundle with fibers being $d_{\varLambda }$-dimensional vector spaces over the field ${\mathbb {C}}$ and with base ${{\mathcal {X}}}$. The existence of this holomorphic bundle is the same as saying that ${{{\mathcal {W}}}_{\varLambda }}$ is a projective ${\mathbb {C}}[\chi ]$-module. This is the so-called Serre-Swan correspondence [119].⁴⁰

Because the base ${{\mathcal {X}}}\simeq {\mathbb {C}}^L$ is contractible, this bundle must be topologically trivial, that is, we can find $d_{\varLambda }$ global holomorphic sections forming a basis of the fiber at each point. A much more complicated question, already asked by Serre [119], is whether we can choose these global sections to be polynomials of ${{{\mathcal {W}}}_{\varLambda }}$. A positive answer was given by the Quillen-Suslin theorem [80]. This theorem requires that ${{{\mathcal {W}}}_{\varLambda }}$ is a finitely generated ${\mathbb {C}}[\chi ]$-module, i.e. that there exist finitely many elements ${\tilde{{{\mathfrak {b}}}}}_1,\ldots ,{\tilde{{{\mathfrak {b}}}}}_{{{\tilde{d}}}}$ such that any element of ${{{\mathcal {W}}}_{\varLambda }}$ is their linear combination with coefficients from ${\mathbb {C}}[\chi ]$. This is easy to see to be the case. Take for instance the finite set of ${{\tilde{d}}}=L\times d_{\varLambda }$ monomials $x^n:=x_1^{n_1},x_2^{n_2},\ldots ,x_L^{n_L}$ with $0\leqslant n_i< d_{\varLambda }$, where $x_i$ are the ones from the proof of Theorem A.2. Due to properness, $x_i$ satisfy a degree-$d_{\varLambda }$ equations with polynomial coefficients, cf. (A.4a), and hence any higher powers of $x_i$ are expressible as linear combinations of the first $d_{\varLambda }$ powers.

The Quillen-Suslin theorem establishes that there are no non-trivial algebraic vector bundles over ${\mathbb {C}}^L$ or equivalently by the Serre-Swan correspondence, that any finitely generated projective ${\mathbb {C}}[\chi ]$-module is free, with a basis given by the aforementioned global sections. Applied to ${{{\mathcal {W}}}_{\varLambda }}$, this is precisely the statement that it is a free module over ${\mathbb {C}}[\chi ]$, see Sect. 3.3.

1.4 Non-symmetric functions

Most of the time we work with only symmetric combinations $\chi _{\ell }$ of inhomogeneities. However, the Baxter operators as explicit matrices acting on the spin chain have coefficients from ${\mathbb {C}}[\theta ]\equiv {\mathbb {C}}[{{\theta }_{1}},\ldots ,{{\theta }_{L}}]$. This prompts us to understand some properties of ${\mathbb {C}}[\theta ]$-modules as compared to ${\mathbb {C}}[\chi ]$-modules. Also, one can consider equations

$$\begin{aligned} \chi _{\ell }({{\theta }_{1}},\ldots ,{{\theta }_{L}})=\chi _{\ell }\end{aligned}$$

(A.7)

as a toy model for (2.27) with $c_{\ell }={{\theta }_{\ell }}$ and ${{\,\mathrm{SW}\,}}_{\ell }(c)=\chi _{\ell }$.

First, we demonstrate how to use the Gröbner basis techniques to conclude that the polynomial ring ${\mathbb {C}}[\theta ]$ is a free ${\mathbb {C}}[\chi ]$-module and count the number of solutions to (A.7). To this end denote the elementary symmetric polynomial of degree $\ell$ in k variables $\theta _{1},\ldots ,\theta _{k}$ as $\chi _{\ell }^{(k)}$. Being roots of $\prod \nolimits _{\ell =1}^k(u-\theta _{\ell })$, inhomogeneities satisfy the characteristic equations

$$\begin{aligned} s_k :=\theta _{k}^{k}+\sum _{\ell =1}^{k}(-1)^{n}\chi _{\ell }^{(k)}\theta _k^{k-\ell }=0\,,\quad k=1,\ldots ,L\,. \end{aligned}$$

(A.8)

Now note that the polynomials $\chi _{\ell }^{(k)}$ can be rewritten as polynomials in $\chi _{\ell '}\equiv \chi _{\ell '}^{(L)}$, with ${\ell '}\leqslant \ell$, and $\theta _{m}$, $m>k$. As a result, $s_k$ become

$$\begin{aligned} s_1&={{\theta }_{1}}+(-\chi _{1}+\sum _{i=2}^L\theta _i)\,, \end{aligned}$$

(A.9a)

$$\begin{aligned} s_2&=\theta _2^2+\theta _2(-\chi _{1}+\sum _{i=3}^L\theta _i)+(\chi _{2}-(\chi _{1}-\sum _{i=3}^L\theta _i)\sum _{i=3}^L\theta _i-\sum _{3\leqslant i<j\leqslant L}\theta _i\theta _j)\,, \end{aligned}$$

(A.9b)

$$\begin{aligned}&\ldots \,,\nonumber \\ s_L&=\theta _{L}^{L}+\sum _{\ell =1}^{L-1}(-1)^{n}\chi _{\ell }\,{{\theta }_{L}}^{L-\alpha }+(-1)^L \chi _{L}\,. \end{aligned}$$

(A.9c)

Now let’s make a small formalisation: treat ${{\theta }_{\ell }}$ and $\chi _{\ell }$ as independent variables and consider an ideal ${\mathcal {I}}=\langle s_1,\ldots ,s_L\rangle$ as an ideal in ${\mathbb {C}}[\chi ][{{\theta }_{}}]$. In the quotient ring ${\mathbb {C}}[\chi ][{{\theta }_{}}]/{{\mathcal {I}}}$, excluding $\chi _{\ell }$ in favour of ${{\theta }_{\ell }}$ is easy. However, we are interested in the opposite—to solve for ${{\theta }_{\ell }}$ in terms of $\chi _{\ell }$. We won’t do this explicitly because this requires an algebraic closure but compute a Gröbner basis instead.

Lemma A.3

The above-introduced polynomials $s_1,\ldots ,s_L$ form the Gröbner basis of the ideal ${\mathcal {I}}=\langle s_1,\ldots ,s_L\rangle$ w.r.t. a lexicographic order for which ${{\theta }_{1}}>\theta _2>\ldots>{{\theta }_{L}}>\chi _{\ell }$, for any $\ell$.

Proof

First, by definition, $s_k$ generate the ideal ${{\mathcal {I}}}$. Then, $s_k$ has $\theta _k^k$ as its leading monomial. Indeed, the other monomials are products of $\theta _{k}^{k'}$ with $k'<k$, powers of $\theta _{k'}$ with $k'>k$, and $\chi _{\ell }$ which hence are lexicographically smaller than $\theta _k^k$. Finally, as the leading monomials enjoy the property ${\mathrm{GCD}}(\theta _k^k,\theta _{k'}^{k'})=1$, the S-polynomials between $s_{k}$ and $s_{k'}$ do not produce new relations and so this set of ideal generators is indeed a Gröbner basis. $\square$

Conceptually the Gröbner basis tells us how to algorithmically find $\theta _{\alpha }$ from the values of their symmetric combinations $\chi _{\ell }$. First one needs to solve (A.9c) for fix $\theta _{L}$ (L solutions), then one needs to substitute the found value of $\theta _{L}$ to the equation $s_{L-1}=0$ and solve it for $\theta _{L-1}$ ($L-1$ solutions) etc.

Very similarly to the analysis of the Wronskian algebra, we note that the ring ${\mathbb {C}}[\chi ][{{\theta }_{}}]/{{\mathcal {I}}}$ is isomorphic (over ${\mathbb {C}}$) to ${\mathbb {C}}[{{\theta }_{}}]$, but it is also naturally endowed with the structure of a ${\mathbb {C}}[\chi ]$-module. The computation of the Gröbner basis above immediately implies that this module is free and of rank L!. Indeed, the corresponding monomial basis are given by monomials $\theta _2^{n_2}\theta _3^{n_3}\ldots \theta _L^{n_L}$, with $n_\ell <\ell$. Any relations between these monomials is impossible precisely because $s_k$ form a Gröbner basis and leading monomials of $s_k$ do not belong to the monomial basis. Of course we know that L! is an expected number, if to count with multiplicities: the equation $\theta ^L-\chi _{1}\theta ^{L-1}+\ldots +(-1)^L\chi _{L}=0$ has L solutions, and any permutation of solutions is allowed as well.

Finally, let us extend Theorem 4.2 to the case of non-symmetric polynomials in ${{\theta }_{\ell }}$. Consider first the following example

• Example: Let the generators of a Wronskian algebra ${{\mathcal {W}}}$ satisfy equations $c_1+c_2=\chi _{1}$, $c_1c_2=\chi _{2}$⁴¹, and (a hypothetical) Bethe algebra ${{\mathcal {B}}}$ is generated by $2\times 2$ diagonal matrices ${{\hat{c}}}_1=\theta _1\times {{\,\mathrm{{\mathbbm {1}}}\,}}_2$, ${{\hat{c}}}_2=\theta _2\times {{\,\mathrm{{\mathbbm {1}}}\,}}_2$. These Wronskian and Bethe algebras are isomorphic as ${\mathbb {C}}[\chi ]$-modules. Let us now consider the extension of the Wronskian algebra ${{\mathcal {W}}}^{\theta }\simeq {{\mathcal {W}}}\otimes _{{\mathbb {C}}[\chi ]}{\mathbb {C}}[{{\theta }_{}}]$, i.e. consider generators satisfying $c_1+c_2=\theta _1+\theta _2$, $c_1c_2=\theta _1\theta _2$ and treat this algebra as a ${\mathbb {C}}[\theta ]$-module. This is a rank-two ${\mathbb {C}}[\theta ]$-module. In contrast, the Bethe algebra considered as a ${\mathbb {C}}[\theta ]$-module is of rank one.

By Lemma 4.1 we actually know that generators of type ${{\theta }_{\ell }}\times {{\,\mathrm{{\mathbbm {1}}}\,}}$ cannot appear in polynomial combinations of $c_\ell$, and so the hypothetical Bethe algebra in the above example cannot exist.

More generally, we can show that all polynomial relations satisfied ${{\hat{c}}}_{\ell }$, even with non-symmetric coefficients, should follow from the Wronskian algebra in the following sense. We can add non-symmetric polynomials by hand to the Wronskian algebra by considering ${{\mathcal {W}}}_{\varLambda }^{\theta }\simeq {{{\mathcal {W}}}_{\varLambda }}\otimes _{{\mathbb {C}}[\chi ]}{\mathbb {C}}[{{\theta }_{}}]$. Likewise, non-symmetric polynomials (times the identity operator) are not elements of the Bethe algebra by Lemma 4.1, and hence appending them as extra generators is also realised as ${{\mathcal {B}}}_{\varLambda }^{\theta }\simeq {{{\mathcal {B}}}_{\varLambda }}\otimes _{{\mathbb {C}}[\chi ]}{\mathbb {C}}[{{\theta }_{}}]$. Isomorphism between ${{\mathcal {W}}}_{\varLambda }^{\theta }$ and ${{\mathcal {B}}}_{\varLambda }^{\theta }$ as ${\mathbb {C}}[{{\theta }_{}}]$-algebras is then obvious from the isomorphism between ${{{\mathcal {B}}}_{\varLambda }}$ and ${{{\mathcal {W}}}_{\varLambda }}$ as ${\mathbb {C}}[\chi ]$-algebras. We also note that ${{\mathcal {W}}}_{\varLambda }^{\theta }$ and ${{\mathcal {B}}}_{\varLambda }^{\theta }$ are free as ${\mathbb {C}}[{{\theta }_{}}]$-modules and ${\mathbb {C}}[\chi ]$-modules as follows e.g. from Lemma A.3.

Cyclicity of Representations

The goal of this appendix is to build all the formalism necessary for the proof of Theorems 4.5 and 4.6 . There are two reasons why proving isomorphism of the specialised map $\varphi _{{{\bar{\theta }_{}}}}$ (4.4) is problematic. First, setting ${{\theta }_{\ell }}$ to numerical values, which is done for the Bethe algebra, is more restrictive than setting their symmetric combinations $\chi _{\ell }$ to numerical values, which is done for the Wronskian algebra. Second, the specialisation procedure is actually native to the representation of an algebra, not to the algebra alone. Namely, we set to numerical values coefficients of a matrix which is more restrictive than setting to numerical values only the factors that multiply this matrix as a whole.

To overcome these difficulties, we want to “rigidify” the algebra isomorphism (4.2) by also proving isomorphism between certain representations of these algebras. As was already mentioned in Sect. 4.3, the only natural choice of a representation for the Wronskian algebra ${{{\mathcal {W}}}_{\varLambda }}$ is its regular representation. As for the Bethe algebra, it acts on the a priori unrelated physical space ${{\,\mathrm{End}\,}}(U_\varLambda )\otimes {\mathbb {C}}[{{\theta }_{}}]$. These two representations are not isomorphic. This is why we need to introduce an alternative Yangian representation dubbed symmetrised representation. Using a cyclic vector argument we prove that its weight subspaces ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$ are indeed isomorphic to ${{{\mathcal {W}}}_{\varLambda }}$ as representations of ${{{\mathcal {B}}}_{\varLambda }}\simeq {{{\mathcal {W}}}_{\varLambda }}$, which resolves the second difficulty. This symmetrised representation has the virtue to manifestly depend only on symmetric combinations of inhomogeneities. We show that, under some explicit restriction on ${{\bar{\theta }_{}}}$, its specialisation at a point ${\bar{\chi }}$ is isomorphic to the spin chain representation at a point ${{\bar{\theta }_{}}}$ which resolves the first difficulty.

The discussed approach was developed in [9] for ${\mathfrak {gl}_{{\mathsf {m}}}}$ spin chains. The below-presented generalisation to the supersymmetric case is conceptually very straightforward. The only difference, apart from the way we present the results, is in the proof of Lemma B.7 which is in line with the ideas of Theorem 4.2.

1.1 Symmetrised Yangian representation

Consider the Yangian spin chain representation at point ${{\bar{\theta }_{}}}=({{\bar{\theta }_{1}}},\ldots ,{{\bar{\theta }_{L}}})$ defined in Sect. 2.2. We note that the order of inhomogeneities in ${{\bar{\theta }_{}}}$ is often superfluous. Indeed, the operator $r_{\ell }({{\bar{\theta }_{}}})=({{\bar{\theta }_{\ell }}}-{{\bar{\theta }_{\ell +1}}}){\mathcal {P}}_{\ell ,\ell +1}+\hbar {{\,\mathrm{{\mathbbm {1}}}\,}}$ satisfies (4.1) evaluated at ${{\theta }_{}}={{\bar{\theta }_{}}}$. If ${{\bar{\theta }_{\ell }}}\ne {{\bar{\theta }_{\ell +1}}}\pm \hbar$, it is invertible and hence an intertwiner between the two representations that differ by the permutation of ${{\bar{\theta }_{\ell }}}$ and ${{\bar{\theta }_{\ell +1}}}$. From here we conclude that the isomorphism class of the representation at point ${{\bar{\theta }_{}}}$ is decided only by $\bar{\chi }_{\ell }$ if there is no $\ell ,\ell '$ such that ${{\bar{\theta }_{\ell }}}-{{\bar{\theta }_{\ell '}}}=\pm \hbar$.

More generally, the following facts hold for supersymmetric representations of Yangians:

Proposition B.1

If $({{\bar{\theta }_{1}}},\ldots ,{{\bar{\theta }_{L}}})$ satisfy ${{\bar{\theta }_{\ell }}}+\hbar \ne {{\bar{\theta }_{\ell '}}}$ for $\ell <\ell '$ then the spin chain representation of ${\mathrm{Y}}({\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}})$ at point ${{\bar{\theta }_{}}}$ is cyclic with cyclic vector $\mathbf{e}^+=\mathbf{e}_1\otimes \ldots \otimes \mathbf{e}_1$, where $\mathbf{e}_1$ is the highest-weight vector of the defining ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ representation⁴². $\square$

This statement follows from Theorem 5.2 of [120].

$\mathbf{e}^+$ is obviously a highest-weight vector of the Yangian representation, i.e. it satisfies the condition $T_{ij}\mathbf{e}^+=0$ for $i<j$. Its weight is given by

$$\begin{aligned} T_{ii}\,\mathbf{e}^+=Q_{\theta }(u+\hbar \,\delta _{i,1})\,\mathbf{e}^+\,. \end{aligned}$$

(B.1)

Proposition B.2

The spin chain representation at point ${{\bar{\theta }_{}}}$ is irreducible if there are no $\ell ,\ell '$ such that ${{\bar{\theta }_{\ell }}}-{{\bar{\theta }_{\ell '}}}=\hbar$.

For the $Y({\mathfrak {gl}_{{\mathsf {m}}}})$ case, this is a standard result appearing in the study of Kirillov-Reshetikhin modules [121]. For $Y(\mathfrak {gl}_{1|1})$ it was proven in [122], Theorem 5. For the $Y(\mathfrak {gl}_{{{\mathsf {m}}}|{{\mathsf {n}}}})$ case, it apparently follows from [120], Proposition 5.4. But as this was not stated explicitly we give an alternative argument for irreducibility in the style of statistical lattice models.

Proof

Let ${\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}}$ be the Hilbert space of the $\ell$-th node of the spin chain, consider also the auxiliary space ${\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}}$ with basis vectors $\mathbf{e}_{\alpha }^{{\mathrm{aux}}}$, $\alpha =1,2,\ldots ,{{\mathsf {m}}}+{{\mathsf {n}}}$. The dual basis vectors of $\mathbf{e}_{\alpha }^{{\mathrm{aux}}}$ shall be denoted $\mathbf{e}^{\alpha }_{{\mathrm{aux}}}$. Define the Lax matrix acting on the tensor product of the mentioned spaces as ${\mathcal {L}}(u-{{\theta }_{\ell }})=(u-{{\theta }_{\ell }}){{\,\mathrm{{\mathbbm {1}}}\,}}+\hbar \, P$, where P is the graded permutation. In the notations of (2.8), ${\mathcal {L}}(u-\theta _\ell ):={(u-\theta _{\ell })}\sum \nolimits _{\alpha ,\beta }ev_{\theta _\ell }(t_{\alpha \beta })\otimes (\mathbf{e}_\beta ^{\mathrm{aux}}\otimes \mathbf{e}_{{\mathrm{aux}}}^{\alpha })$. The key property we use is that the Lax matrix becomes, up to normalisation, the graded permutation if $u={{\theta }_{\ell }}$.

Introduce $V^{{\mathrm{aux}}}$—the tensor product of L auxiliary spaces spanned by $\mathbf{e}_{A}^{{\mathrm{aux}}}=\mathbf{e}_{\alpha _1}^{\mathrm{aux}}\otimes \ldots \otimes \mathbf{e}_{\alpha _L}^{{\mathrm{aux}}}$—and define $B=\sum \nolimits _{A}\mathbf{e}_{A}^{\mathrm{aux}}T_{1\alpha _L}({{\bar{\theta }_{L}}})\times \ldots T_{1\alpha _2}({{\bar{\theta }_{2}}})\times T_{1\alpha _1}({{\bar{\theta }_{1}}})$. Given the above-mentioned key property, B maps V to $\mathbf{e}^+\otimes V^{{\mathrm{aux}}}$ which is easiest to see by a graphical representation of how B acts:

(B.2)

Here each vertical direction corresponds to a node ${\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}}$ of the spin chain and each horizontal direction corresponds to a tensor factor ${\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}}$ of $V^{{\mathrm{aux}}}$. Intersections are the places where the Lax matrices should be applied (considered as maps from South-West to North-East spaces). Red crosses are the places where the corresponding Lax matrix becomes the permutation.

The map B is also invertible. Indeed, up to a non-zero factor it reduces to an ordered product of $\frac{L(L-1)}{2}$ Lax matrices, e.g. these are the three Lax matrices marked by the encircled numbers in the image above. Each of these Lax matrices is invertible since ${{\bar{\theta }_{\ell }}}-{{\bar{\theta }_{\ell '}}}\ne \hbar$.

Because B is invertible, for any $v\in V$ one can find a vector $v^*\in (V^{{\mathrm{aux}}})^*$ such that $(v^*,B)v=\mathbf{e}^+$. Then irreducibility follows from Proposition B.1. $\square$

Remark

We expect that a further refinement of argument (B.2) can be used to prove Propostion B.1 about cyclicity.

For finite-dimensional irreducible ${\mathrm{Y}}({\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}})$ representations, the highest-weight vector exists and is unique and the representation is fully determined, up to an isomorphism, by the vector’s weight [123]. Note that the weight of $\mathbf{e}^+$ depends only on symmetric combinations of inhomogeneities according to (B.1). Then we can consider the induced representation from $\mathbf{e}^+$ which is isomorphic to the spin chain one by the above-mentioned uniqueness but, in contrast to the spin chain realisation is manifestly invariant under permutations of inhomogeneities.

We shall now introduce a different permutation-invariant realisation which does not require the irreducibility argument and will be formulated for inhomogeneities being abstract variables.

Yangian centraliser. Consider the vector space ${\mathcal {V}}\simeq ({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}\otimes {\mathbb {C}}[{{\theta }_{1}},\ldots ,{{\theta }_{L}}]$ on which the Yangian representation $ev_{{{\theta }_{}}}$ (2.11) is realised. An interesting question is what is the centraliser of the Yangian action on ${\mathcal {V}}$.

Define operators $S_{\ell }$ acting on ${\mathcal {V}}$ by

$$\begin{aligned} S_{\ell }={{\mathcal {P}}_{\ell ,\ell +1}}\varPi _{\ell ,\ell +1}-\frac{\hbar }{{{\theta }_{\ell }}-{{\theta }_{\ell +1}}}\left( \varPi _{\ell ,\ell +1}-{{\,\mathrm{{\mathbbm {1}}}\,}}\right) \,, \end{aligned}$$

(B.3)

where ${\mathcal {P}}_{\ell ,\ell +1}$ is the graded permutation in $({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}$, and $\varPi _{\ell ,\ell +1}$ permutes variables ${{\theta }_{\ell }}$ and ${{\theta }_{\ell +1}}$. These permutations were already used in the proof of Lemma 4.1 on page 27.

Although $S_{\ell }$ contains ${{\theta }_{\ell }}$ in denominator, its action on polynomials in ${{\theta }_{\ell }}$ yields again polynomials and hence its action on ${\mathcal {V}}$ is well-defined.

Lemma B.3

For $\ell =1,\ldots ,L-1$, $S_{\ell }$ commutes with the ${\mathrm{Y}}({\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}})$ action, (i.e. $[S_\ell ,T_{\alpha \beta }]=0$) and they form a representation of the symmetric group ${\mathsf {S}}_L$ on ${\mathcal {V}}$.

Proof

The commutativity follows from $({{\theta }_{\ell }}-{{\theta }_{\ell +1}})S_{\ell }=-\varPi _{\ell ,\ell +1}r_{\ell }+\hbar {{\,\mathrm{{\mathbbm {1}}}\,}}$ and (4.1). Then, by explicit computation one checks $S_{\ell }^2={{\,\mathrm{{\mathbbm {1}}}\,}}$ and $(S_{\ell }S_{\ell +1})^3={{\,\mathrm{{\mathbbm {1}}}\,}}$—the defining relations of ${\mathsf {S}}_L$. $\square$

dAHA. In the limit $\hbar \rightarrow 0$, ${\mathsf {S}}_{L}$ becomes an explicit permutation defined on a graded space that commutes with the action of ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$. Hence $\hbar \ne 0$ should be considered as a generalisation of the Schur-Weyl duality to the case of the Yangian algebra. This statement was made mathematically precise for the bosonic ${\mathfrak {gl}_{{\mathsf {m}}}}$ case [124,125,126]: $(S_{\ell })_{1\leqslant \ell \leqslant L-1}$ together with $(\theta _{\ell }\times {{\,\mathrm{{\mathbbm {1}}}\,}})_{1\leqslant \ell \leqslant L}$ form a representation of ${\mathcal {H}}_L$, the degenerate affine Hecke algebra (dAHA) on L sites. Moreover, the dAHA and the Yangian form a dual pair: they are maximal mutual centralisers of one another when acting on ${\mathcal {V}}$. More formally, one can view ${\mathcal {V}}$ as the tensor product of ${\mathsf {S}}_{L}$-modules ${\mathcal {H}}_L\otimes _{{\mathsf {S}}_{L}}({\mathbb {C}}^{{\mathsf {m}}})^{\otimes L}$ where ${\mathsf {S}}_{L}$ acts on ${\mathcal {H}}_L$ as a subalgebra and on $({\mathbb {C}}^{{\mathsf {m}}})^{\otimes L}$ by permutation of tensor factors. This point of view is conceptually interesting because to generalise Schur-Weyl duality to supersymmetric Yangians, one does not need to change the defining relations of the dAHA ${\mathcal {H}}_L$ but simply to replace the usual action of ${\mathsf {S}}_{L}$ on $({\mathbb {C}}^{{\mathsf {n}}})^{\otimes L}$ by the graded action on $({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}$ as in (B.3). The full mathematical treatment (for the affine Hecke algebra⁴³) can be found in [127].

We are only going to use the slightly weaker statement of Lemma B.3 that $S_\ell$ commute with the Yangian action.

${\mathbb {C}}[\chi ]$-Yangian module. Define ${\mathcal {V}}^{\mathsf {S}}\subset {\mathcal {V}}$ as the subspace of $S_{\ell }$-invariant vectors. As $[S_{\ell },T_{AB}]=0$, the Yangian action is well-defined on ${\mathcal {V}}^{\mathsf {S}}$. Multiplication by symmetric polynomials is also well-defined on ${\mathcal {V}}^{\mathsf {S}}$ and, moreover, ${\mathbb {C}}[\chi ]\times {{\,\mathrm{{\mathbbm {1}}}\,}}$ belongs to $ev_{{{\theta }_{}}}({\mathrm{Y}}({\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}))$ due to (2.27). Hence we shall call this action of the Yangian on ${\mathcal {V}}^{\mathsf {S}}$ the symmetrised Yangian representation. Note that ${\mathcal {V}}^{\mathsf {S}}$ is also naturally a ${\mathbb {C}}[\chi ]$-module.

1.2 Symmetrised Bethe modules and their characters

Since ${\mathsf {S}}_L$ commutes with the global ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ action it is consistent to define ${\mathcal {V}}^{\mathsf {S}}_{\varLambda }={\mathcal {V}}^{\mathsf {S}}\cap ({V_{\varLambda }}\otimes {\mathbb {C}}[{{\theta }_{1}},\ldots ,{{\theta }_{L}}])$—the weight $\varLambda$ subspace of ${\mathcal {V}}^{\mathsf {S}}$—and ${\mathcal {V}}^{{\mathsf {S}}+}_{\varLambda }={\mathcal {V}}^{\mathsf {S}}\cap ({V_{\varLambda }^+}\otimes {\mathbb {C}}[{{\theta }_{1}},\ldots ,{{\theta }_{L}}])\subset {\mathcal {V}}^{\mathsf {S}}_{\varLambda }$—the subspace of ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ highest-weight vectors corresponding to the Young diagram ${\varLambda ^+}$. These spaces are also naturally ${\mathbb {C}}[\chi ]$-modules.

Characters. For an element v of ${\mathcal {V}}^{\mathsf {S}}$, define its degree as the maximal degree of the monomials in ${{\theta }_{\ell }}$’s which occur in v. Define ${\mathcal {F}}_k {\mathcal {V}}^{\mathsf {S}}$ as the space of all vectors of degree less or equal to k. Finally, define the character

$$\begin{aligned} {\mathrm{ch}}({\mathcal {V}}^{\mathsf {S}})=\sum _{k=0}^\infty \left( \dim {\mathcal {F}}_k/{\mathcal {F}}_{k-1}\right) t^k\,. \end{aligned}$$

(B.4)

Since the ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ action does not change the degree, we can also define in an analogous way ${\mathrm{ch}}({\mathcal {V}}^{\mathsf {S}}_{\varLambda })$ and ${\mathrm{ch}}({\mathcal {V}}^{{\mathsf {S}}+}_{\varLambda })$.

Proposition B.4

${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}}}$ is a free ${\mathbb {C}}[\chi ]$-module of rank $\left( {\begin{array}{c}L\\ \lambda _1\ldots \nu _{{{\mathsf {n}}}}\end{array}}\right)$ and its character is given by

$$\begin{aligned} {\mathrm{ch}}({\mathcal {V}}^{\mathsf {S}}_{\varLambda })=t^{\varUpsilon _\varLambda }\prod _{a=1}^{{{\mathsf {m}}}}\prod _{k=1}^{\lambda _a}\frac{1}{1-t^k}\prod _{i=1}^{{{\mathsf {n}}}}\prod _{k=1}^{\nu _i}\frac{1}{1-t^k}\,, \end{aligned}$$

(B.5)

where $\varUpsilon _{\varLambda }:=\sum \nolimits _{i=1}^{{{\mathsf {n}}}}\frac{\nu _i(\nu _i-1)}{2}$.

Proof

${\mathcal {V}}^{\mathsf {S}}_{\varLambda }$ is the image of ${V_{\varLambda }}\otimes {\mathbb {C}}[\theta ]$ by the projector $p:=\frac{1}{L!}\sum \nolimits _{\sigma \in {\mathsf {S}}_L}S_\sigma$, where the symmetry group acts with $S_\sigma$ on ${V_{\varLambda }}\otimes {\mathbb {C}}[\theta ]$ as generated from $S_\ell$ (B.3). Since the construction is polynomial in $\hbar$ and the $\hbar$-term of $S_\ell$ lowers degrees of polynomials the proposition is true iff it is true for $\hbar =0$. Hence we will consider only the $\hbar =0$ case. Every element $\sigma \in {\mathsf {S}}_L$ is then represented as $S_\sigma ={\mathcal {P}}_\sigma \varPi _\sigma$, where ${\mathcal {P}}_\sigma$ is the graded permutation acting on ${V_{\varLambda }}$ and $\varPi _\sigma$ is the ordinary permutation acting on ${\mathbb {C}}[\theta ]$.

Denote by $\mathbf{e}_\alpha$ the standard basis vectors of ${\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}}$ defined by ${\mathsf {E}}_{\beta \beta }\mathbf{e}_{\alpha }=\delta _{\alpha \beta }\mathbf{e}_{\alpha }$. The standard spin basis of ${V_{\varLambda }}$ is indexed by tuples $I=(i_1,\ldots ,i_L)$ such that $|\{k : i_k=a\}|=\lambda _a$ and $|\{k : i_k=i\}|=\nu _i$ corresponding to vectors $\mathbf{e}_I:=\otimes _{\ell =1}^L\mathbf{e}_{i_\ell }$.

To get ${\mathcal {V}}^{\mathsf {S}}_{\varLambda }$, it is enough to consider the image of ${{\mathsf {w}}}\otimes {\mathbb {C}}[\theta ]$ by p with

$$\begin{aligned} {{\mathsf {w}}}=\underbrace{\mathbf{e}_1\otimes \ldots \otimes \mathbf{e}_1}_{\lambda _1}\otimes \ldots \otimes \underbrace{\mathbf{e}_{{\mathsf {m}}}\otimes \ldots \otimes \mathbf{e}_{{\mathsf {m}}}}_{\lambda _{{\mathsf {m}}}}\otimes \underbrace{\mathbf{e}_{{{\mathsf {m}}}+1}\otimes \ldots \otimes \mathbf{e}_{{{\mathsf {m}}}+1}}_{\nu _1}\otimes \ldots \otimes \underbrace{\mathbf{e}_{{{\mathsf {m}}}+{{\mathsf {n}}}}\otimes \ldots \otimes \mathbf{e}_{{{\mathsf {m}}}+{{\mathsf {n}}}}}_{\nu _{{\mathsf {n}}}}\,. \end{aligned}$$

(B.6)

Indeed, for all I we can always find $\sigma \in {\mathsf {S}}_L$ such that ${\mathcal {P}}_\sigma \cdot \mathbf{e}_I=\mathbf{e}_{\sigma (I)}=\pm {{\mathsf {w}}}$.

Denote by ${\mathsf {S}}_\varLambda :=\prod _{a=1}^{{\mathsf {m}}}{\mathsf {S}}_{\lambda _a}\prod _{i=1}^{{\mathsf {n}}}{\mathsf {S}}_{\nu _i}$ the stabiliser of ${{\mathsf {w}}}$ and by $H:={\mathsf {S}}_L/{\mathsf {S}}_\varLambda$ the space of orbits with respect to the right group multiplication. Then the projection by p can be represented as

$$\begin{aligned} {{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}}}\simeq p({{\mathsf {w}}}\otimes {\mathbb {C}}[{{\theta }_{}}])=\frac{1}{L!}\sum _{[\sigma ]\in H} S_{[\sigma ]}\cdot \left( {{\mathsf {w}}}\otimes R_B\cdot C_F\cdot {\mathbb {C}}[\theta ]\right) \,, \end{aligned}$$

(B.7)

where $R_B:=\sum \nolimits _{\sigma \in \prod _{a=1}^{{\mathsf {m}}}{\mathsf {S}}_{\lambda _a}}\varPi _\sigma$ and $C_F:=\sum \nolimits _{\sigma \in \prod _{i=1}^{{\mathsf {n}}}{\mathsf {S}}_{\nu _i}}(-1)^{|\sigma |}\varPi _\sigma$.

To decide about linear independence in ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}}}$, it is enough to consider one term in the sum $\sum _{[\sigma ]\in H} S_{[\sigma ]}$, e.g. $[\sigma ]=[{{\,\mathrm{{\mathbbm {1}}}\,}}]$ since different terms would be proportional to ${\mathcal {P}}_{\sigma }{{\mathsf {w}}}$ which are linearly independent in ${V_{\varLambda }}$. Also, $R_BC_F$ commutes with symmetric polynomials and hence we conclude that ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}}}$ and $R_BC_F\cdot {\mathbb {C}}[{{\theta }_{}}]$ are isomorphic as ${\mathbb {C}}[\chi ]$-modules.

It is easy to describe $R_BC_F\cdot {\mathbb {C}}[{{\theta }_{}}]$: it is spanned by $Q_{\varLambda }\times {\mathrm{Sym}}_{\lambda _1}\times \ldots \times {\mathrm{Sym}}_{\nu _{{{\mathsf {m}}}}}$, where $Q_\varLambda :=\prod \nolimits _{i=1}^{{\mathsf {n}}}\prod \nolimits _{1\leqslant k<l\leqslant \nu _i}(\theta _{j_i+k}-\theta _{j_i+l})$ with $j_i:=\sum _{a=1}^{{\mathsf {m}}}\lambda _a+\sum _{s=1}^{i-1}\nu _s$, and ${\mathrm{Sym}}_{k}$ is the space of symmetric polynomials in k variables. $Q_{\varLambda }\times {\mathrm{Sym}}_{\lambda _1}\times \ldots \times {\mathrm{Sym}}_{\nu _{{{\mathsf {m}}}}}$ is free under the action of ${\mathbb {C}}[\chi ]$, see Appendix A.4, and its character is (B.5). Rank is computed from $\frac{{{\,\mathrm{ch}\,}}({{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}}})}{{{\,\mathrm{ch}\,}}({\mathbb {C}}[\chi ])}|_{t=1}$. $\square$

Corollary B.5

${\mathcal {V}}^{\mathsf {S}}=\bigoplus _\varLambda {\mathcal {V}}^{{\mathsf {S}}}_{\varLambda }$ is a free ${\mathbb {C}}[\chi ]$-module of rank $({{\mathsf {m}}}+{{\mathsf {n}}})^L$. $\square$

Proposition B.6

${\mathcal {V}}^{{\mathsf {S}}+}_\varLambda$ is a free ${\mathbb {C}}[\chi ]$-module of rank $\frac{L!}{\prod \limits _{(a,s)\in {\varLambda ^+}} h_{a,s}}=\dim {V_{\varLambda }^+}$, where $h_{a,s}$ is the hook length at box (a, s). Its character is given by

$$\begin{aligned} {\mathrm{ch}}({\mathcal {V}}^{{\mathsf {S}}+}_{\varLambda })=t^{\varUpsilon _{\varLambda }^+}\prod _{(a,s)\in {\varLambda ^+}}\frac{1}{1-t^{h_{a,s}}}\,, \end{aligned}$$

(B.8)

where $\varUpsilon _{\varLambda }^+=\sum _{s=1}^{\lambda _1}\frac{h_s(h_s-1)}{2}$ with $h_s$ being the height of the s-th column of ${\varLambda ^+}$.

Note that $\varUpsilon _{\varLambda }^+\geqslant \varUpsilon _{\varLambda }$ which is consistent with ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}\subset {{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}}}$.

Proof

As in the previous proof, it is enough to consider $\hbar =0$.

The space ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ can be constructed from $V\otimes {\mathbb {C}}[{{\theta }_{}}]$ as follows. Take the standard Young tableau ${{\mathcal {T}}}$ which is obtained by filling the shape ${\varLambda ^+}$ first by filling the boxes defining the weight $\lambda _1$, then $\lambda _2$, $\ldots$, then $\nu _{{{\mathsf {n}}}}$. For instance, for $[\lambda _1|\nu _1,\nu _2]=[4|2,2]$, . Take the normalised⁴⁴ Young symmetriser $S_{{\mathcal {T}}}\propto R_{{\mathcal {T}}}C_{{\mathcal {T}}}$, where $R_{{\mathcal {T}}}$ is the symmetrisation over rows and $C_{{\mathcal {T}}}$ is the antisymmetrisation over columns. Then ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ is the image by the projector p of $S_{R_{{\mathcal {T}}}C_{{\mathcal {T}}}} {{\mathsf {w}}}\otimes {\mathbb {C}}[{{\theta }_{}}]$. Singling out a special vector ${{\mathsf {w}}}$ and (correlated to it) tableau ${{\mathcal {T}}}$ is enough to construct ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ because p sums over all permutations. Using this feature of p again, and by repeating the same construction as (B.7), one gets

$$\begin{aligned} {{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}\simeq & {} p(S_{R_{{\mathcal {T}}}C_{{\mathcal {T}}}} {{\mathsf {w}}}\otimes {\mathbb {C}}[{{\theta }_{}}])=p( {{\mathsf {w}}}\otimes \varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}{\mathbb {C}}[{{\theta }_{}}]) \nonumber \\= & {} \frac{1}{L!}\sum _{[\sigma ]\in H} S_{[\sigma ]}\cdot \left( {{\mathsf {w}}}\otimes R_B C_F\varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}\cdot {\mathbb {C}}[\theta ]\right) \,, \end{aligned}$$

(B.9)

and so ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$, as a ${\mathbb {C}}[\chi ]$-module, is isomorphic to $R_B C_F\varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}\cdot {\mathbb {C}}[\theta ]$. We can omit $R_BC_F$ as it has zero kernel when acting on $\varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}\cdot {\mathbb {C}}[\theta ]$. Indeed, $\varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}R_B C_F\varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}=\varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}$. But the module $\varPi _{C_{{\mathcal {T}}}R_{{\mathcal {T}}}}\cdot {\mathbb {C}}[\theta ]$ is the standard application of (reversed) Young symmetriser to a polynomial ring which is well understood, see e.g. [104, 128,129,130]. It is a free ${\mathbb {C}}[\chi ]$-module with character given by (B.8)⁴⁵. Again, the rank is computed from $\frac{{{\,\mathrm{ch}\,}}({{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}})}{{{\,\mathrm{ch}\,}}({\mathbb {C}}[\chi ])}|_{t=1}$. $\square$

Young diagram dependence. One may wonder how comes that the character (B.8) and, in fact, the ${\mathbb {C}}[\chi ]$-isomorphism class of ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ do not depend on ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ but only on the Young diagram ${\varLambda ^+}$. To understand this property, let us extend the underlying symmetry algebra from ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ to $\mathfrak {gl}_{{{\mathsf {m}}}'|{{\mathsf {n}}}'}$, where ${{\mathsf {m}}}'={\mathrm{max}}({{\mathsf {m}}},{\mathrm{h}}_{{\varLambda ^+}})$ and ${{\mathsf {n}}}'={\mathrm{max}}({{\mathsf {n}}},{\underline{\lambda }}_1)$. In addition to $a=1,\ldots ,{{\mathsf {m}}}$, $i={{\hat{1}}},\ldots ,{\hat{{{\mathsf {n}}}}}$, introduce also ${\mathfrak {z}}$ to label the new indices that appeared due to the extension. In the order $a<i<{\mathfrak {z}}$, the highest-weight vectors do not involve $v_{\mathfrak {z}}$ and hence ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ for the extended system is isomorphic to the one of the ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ system. Now we perform a chain of fermionic duality transformations (odd Weyl reflections) to get any other order in the set $a,i,{\mathfrak {z}}$ (the order between separately bosonic and fermionic indices will be preserved though). The procedure is described for instance in [97]. It changes the highest-weight vectors, i.e. the actual embedding of $V_{\varLambda }^+$ inside V is modified, but this change is performed by acting with elements of the global $\mathfrak {gl}_{{{\mathsf {m}}}'|{{\mathsf {n}}}'}\subset Y(\mathfrak {gl}_{{{\mathsf {m}}}'|{{\mathsf {n}}}'})$ that commutes with the dAHA and in particular with the action of ${\mathbb {C}}[\chi ]$. Since the procedure is invertible it establishes a ${\mathbb {C}}[\chi ]$-isomorphism between two spaces ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ that differ by the choice of the order defining the highest-weight vector. The order is bijected to a Manhattan-type path (e.g. the one in the Example on page 42), and only those indices that belong to the Young diagram part of the path participate in the highest-weight vectors. This last observation allows us to choose ${{\mathsf {m}}}',{{\mathsf {n}}}'$ to be any pair such that $({{\mathsf {m}}}',{{\mathsf {n}}}')$ lies on the boundary of ${\varLambda ^+}$ (black/red dots of Fig. 1) or outside of ${\varLambda ^+}$.

The dependence of ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$, as a ${\mathbb {C}}[\chi ]$-module, on the Young diagram alone parallels results of Sect. 5 that show that the isomorphism class of the twist-less ${{{\mathcal {B}}}_{\varLambda }}$, as a ${\mathbb {C}}[\chi ]$-algebra, only depends on the Young diagram. This is of course not a coincidence because the twist-less ${{{\mathcal {B}}}_{\varLambda }}$ and ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ are isomorphic as ${\mathbb {C}}[\chi ]$-modules which follows from the results of the next subsection.

1.3 Cyclicity of symmetrised Bethe modules

We shall use the notation ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$ to cover both ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}}}$ and ${{\mathcal {V}}_{\varLambda }^{{\mathsf {S}}+}}$ in the discussion below.

We know that ${{{\mathcal {W}}}_{\varLambda }}$ and ${{{\mathcal {B}}}_{\varLambda }}$ algebras are isomorphic as ${\mathbb {C}}[\chi ]$-algebras. The goal of this subsection is to show that the regular representation ${{{\mathcal {W}}}_{\varLambda }}$ of the Wronskian algebra and the symmetrised Bethe module ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$ are isomorphic as ${\mathbb {C}}[\chi ]$-modules. To do so we need a map from ${{{\mathcal {W}}}_{\varLambda }}$ to ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$ commuting with the action of ${{{\mathcal {W}}}_{\varLambda }}\equiv {{{\mathcal {B}}}_{\varLambda }}$. A standard approach is to take a vector $\omega \in U_\varLambda \otimes {\mathbb {C}}[{{\theta }_{}}]$ and to consider the morphism of representations

$$\begin{aligned} \begin{array}{cccl} \psi _\omega : &{}{{{\mathcal {W}}}_{\varLambda }}&{} \longrightarrow &{} U_\varLambda \otimes {\mathbb {C}}[{{\theta }_{}}]\,, \\ ~&{} c_\ell &{} \longmapsto &{} {\hat{c}}_\ell \,\omega \, \end{array}\,. \end{aligned}$$

(B.10)

Of course this map has no reason to be an isomorphism. Nevertheless we can prove the following.

Lemma B.7

For any non-zero vector $\omega \in U_\varLambda \otimes {\mathbb {C}}[{{\theta }_{}}]$, $\psi _\omega$ is injective.

Proof

Take $P\in {{{\mathcal {W}}}_{\varLambda }}$ such that ${\hat{P}}\,\omega =0$. As before, use WBE to write P as a polynomial in $c_\ell$ only. Around a regular point $\theta$, we know from our previous results that the Bethe algebra can be fully diagonalised and that the spectrum of the operators ${\hat{c}}_\ell$ are exactly the solutions of the Bethe equations. Denote by $W(\theta )$ the corresponding $\theta$-dependent change of basis that diagonalises ${{\hat{c}}}_\ell$. At least one of the components of $W(\theta )\,\omega$ has to be non-zero at $\theta$ and hence in some L-dimensional ball ${\mathcal {O}}$ around $\theta$. Then ${\hat{P}}\omega =0$ implies that for one of the solutions $c_\ell (\theta )$, $P(c_\ell (\theta ))=0$ for $\theta \in {\mathcal {O}}$. Since $c_\ell$ is a local diffeomorphism, P vanishes on a L-dimensional ball and thus $P=0$. $\square$

We hence see that for all $\omega$, $\psi _\omega$ is an isomorphism on its image. Let us take a very precise $\omega$: the vector of the smallest degree⁴⁶ (as a polynomial in ${{\theta }_{\ell }}$) that belongs to ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$.

Lemma B.8

$\psi _{\omega }({{{\mathcal {W}}}_{\varLambda }})={{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$. Therefore $\psi _{\omega }$ is an isomorphism of ${\mathbb {C}}[\chi ]$-modules between ${{{\mathcal {W}}}_{\varLambda }}$ and ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$.

Proof

Injectivity is proven by Lemma B.7. The fact that $\psi _{\omega }$ preserves the action of ${\mathbb {C}}[\chi ]$ is obvious from the definition. Note also that $\psi _{\omega }$ just increases the degree by $\varUpsilon _{\varLambda }$ (by $\varUpsilon _{\varLambda }^+$ in the non-twisted case) and otherwise preserves the natural filtrations on ${{{\mathcal {W}}}_{\varLambda }}$ and ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$. Therefore surjectivity of $\psi _{\omega }$ follows from the comparison of the corresponding characters computed in Sects. 3.4 and B.2 . $\square$

1.4 Specialisations

Let us summarise what has been done so far. On one side, we have the Wronskian algebra ${{{\mathcal {W}}}_{\varLambda }}$ acting on itself via the regular representation. On the other side, we have the Bethe algebra ${{{\mathcal {B}}}_{\varLambda }}$ acting on the space ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$. Moreover the two couples $({{{\mathcal {W}}}_{\varLambda }}, {{{\mathcal {W}}}_{\varLambda }})$ and $({{{\mathcal {B}}}_{\varLambda }}, {{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}})$ are isomorphic via $(\varphi ,\psi _\omega )$.

Now consider the ideal ${{\mathcal {I}}}:=\langle {{\,\mathrm{SW}\,}}_1-\bar{\chi }_{1},\ldots {{\,\mathrm{SW}\,}}_L-\bar{\chi }_{L}\rangle$ of ${{{\mathcal {W}}}_{\varLambda }}$ and its image $\hat{{{\mathcal {I}}}}$ in ${{{\mathcal {B}}}_{\varLambda }}$ under $\varphi$. Automatically, the isomorphisms $(\varphi ,\psi _\omega )$ will induce isomorphisms between $({{{\mathcal {W}}}_{\varLambda }}/{{\mathcal {I}}}, {{{\mathcal {W}}}_{\varLambda }}/{{\mathcal {I}}})$ and $({{{\mathcal {B}}}_{\varLambda }}/\hat{{{\mathcal {I}}}}, {{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}/\psi _\omega ({{\mathcal {I}}}))$. Moreover, as shown in Lemma 4.4$\psi _\omega ({{\mathcal {I}}})={{\mathcal {J}}}\cdot {{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$, where ${{\mathcal {J}}}:=\langle \chi -\bar{\chi }\rangle \subset {\mathbb {C}}[\chi ]$. Denote ${{{\mathcal {B}}}_{\varLambda }}(\bar{\chi }):={{{\mathcal {B}}}_{\varLambda }}/\hat{{{\mathcal {I}}}}$ and recall that ${{{\mathcal {W}}}_{\varLambda }}(\bar{\chi }):={{{\mathcal {W}}}_{\varLambda }}/{{\mathcal {I}}}$.

We also have a third pair in this correspondence, namely ${{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$, the Bethe algebra evaluated at ${{\bar{\theta }_{}}}$, acting on $U_\varLambda$. Our final goal is to show that $\varphi _{{{\bar{\theta }_{}}}} : {{{\mathcal {W}}}_{\varLambda }}(\bar{\chi })\simeq {{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$. This is equivalent to showing that ${{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})\simeq {{{\mathcal {B}}}_{\varLambda }}(\bar{\chi })$. Let us emphasise that a priory ${{{\mathcal {B}}}_{\varLambda }}({\bar{\chi }})$ and ${{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$ are two different objects.

• Example: Consider ${{\mathcal {W}}}$ and ${{\mathcal {B}}}^{{\mathrm{bad}}}$ from the example on page 29. ${{\mathcal {B}}}^{{\mathrm{bad}}}$ acts on the space ${\mathcal {V}}={\mathbb {C}}^2\otimes {\mathbb {C}}[{{\theta }_{1}},\theta _2]$. ${\mathcal {V}}$ is a ${\mathbb {C}}[\chi _{1},\chi _{2}]$-module of rank four, we can take $\begin{pmatrix} 1\\ 0\end{pmatrix}$, ${{\theta }_{1}}\begin{pmatrix} 1\\ 0\end{pmatrix}$, $\begin{pmatrix} 0\\ 1\end{pmatrix}$, ${{\theta }_{2}}\begin{pmatrix} 0\\ 1\end{pmatrix}$ as its basis elements. In this basis

  $$\begin{aligned} {{\hat{c}}}_1^{{\mathrm{bad}}}= \begin{pmatrix} 0 &{} -\chi _{2} &{} 0 &{} 0 \\ 1 &{} \chi _{1} &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -\chi _{2} \\ 0 &{} 0 &{} 1 &{} \chi _{1} \end{pmatrix}\,. \end{aligned}$$

  (B.11)

  Take a vector $\omega =A\begin{pmatrix} 1\\ 0\end{pmatrix}+ B\begin{pmatrix} 0\\ 1\end{pmatrix}$, where A, B are some polynomials in $\chi _{1},\chi _{2}$. Then ${\mathcal {U}}^{\mathsf {S}}:=\psi _{\omega }({{\mathcal {W}}})$ is a ${\mathbb {C}}[\chi _{1},\chi _{2}]$-module of rank two spanned by $\xi _1:=\omega$, and $\xi _2:=A\,{{\theta }_{1}}\begin{pmatrix} 1\\ 0\end{pmatrix}+B\,{{\theta }_{2}}\begin{pmatrix} 0\\ 1\end{pmatrix}$. ${{\hat{c}}}_1^{{\mathrm{bad}}}\in {{\mathcal {B}}}^{{\mathrm{bad}}}$ acting on ${\mathcal {U}}^{\mathsf {S}}$ is $\left( \begin{matrix} 0&{}-\chi _{2}\\ 1&{}\chi _{1}\end{matrix}\right)$ in the basis $\xi _1,\xi _2$. Specialisation ${{\mathcal {B}}}^{{\mathrm{bad}}}(\bar{\chi })$ is two-dimensional for any ${\bar{\chi }}$ and is clearly ${\mathbb {C}}$-isomorphic to ${{\mathcal {W}}}(\bar{\chi })$, in particular ${{\hat{c}}}_1^{{\mathrm{bad}}}(\bar{\chi })=\left( \begin{matrix} 0&{}-\bar{\chi }_{2}\\ 1&{}\bar{\chi }_{1}\end{matrix}\right)$. Note that the statement is completely independent of the choice of $\omega$. It holds even if $A(\bar{\chi }_{1},\bar{\chi }_{2})=A(\bar{\chi }_{1},\bar{\chi }_{2})=0$ because $\xi _i\notin \psi _\omega ({{\mathcal {I}}}):=\langle \chi -{\bar{\chi }}\rangle {\mathcal {U}}^{\mathsf {S}}$, $i=1,2$.

We chose ${{\mathcal {B}}}^{{\mathrm{bad}}}$ in the example above to explicitly demonstrate that ${{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$ and ${{{\mathcal {B}}}_{\varLambda }}(\bar{\chi })$ can be in principle non-isomorphic.

Instead of showing isomorphism between ${{{\mathcal {B}}}_{\varLambda }}({\bar{\chi }})$ and ${{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$ directly, let us show isomorphism between ${{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}/{{\mathcal {J}}}\cdot {{\mathcal {U}}_{\varLambda }^{{\mathsf {S}}}}$ and $U_\varLambda$. Since these spaces both carry representations of the Bethe algebra, if one can find an isomorphism commuting with these actions, it would automatically imply ${{{\mathcal {B}}}_{\varLambda }}({\bar{\chi }})\equiv {{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$ as is argued in Sect. 4.3. The advantage of this strategy is that we can leverage Yangian representation theory to prove such an isomorphism.

To relate the symmetrised Yangian representation at point $\bar{\chi }$ and the spin chain Yangian representation at point ${{\bar{\theta }_{}}}$, recall that ${\mathcal {V}}^{{\mathsf {S}}}$ is a subspace of ${\mathcal {V}}:=({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}\otimes {\mathbb {C}}[{{\theta }_{1}},\ldots ,{{\theta }_{L}}]$ and so we can define a map $\mathrm {Ev}_{{{\bar{\theta }_{}}}} :{\mathcal {V}}^{{\mathsf {S}}} \rightarrow ({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}$ simply by evaluating all vectors at ${{\bar{\theta }_{}}}$. Since ${{\mathcal {J}}}\cdot {\mathcal {V}}^{{\mathsf {S}}}\subset \mathrm {Ker}~\mathrm {Ev}_{{{\bar{\theta }_{}}}}$, this induces a well-defined map

$$\begin{aligned} \mathrm {ev}_{{{\bar{\theta }_{}}}}:{\mathcal {V}}^{{\mathsf {S}}}(\bar{\chi })\rightarrow ({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}\,, \end{aligned}$$

(B.12)

where ${\mathcal {V}}^{{\mathsf {S}}}(\bar{\chi }):={\mathcal {V}}^{{\mathsf {S}}}/{{\mathcal {J}}}\cdot {\mathcal {V}}^{{\mathsf {S}}}$. Concretely this just means the following: take a class $[v]\in {\mathcal {V}}^{{\mathsf {S}}}(\bar{\chi })$, represent it by some $v\in {\mathcal {V}}$ and evaluate it at ${{\bar{\theta }_{}}}$.

Note that ${\mathcal {V}}^{{\mathsf {S}}}(\bar{\chi })$ is the space where the symmetrised Yangian representation at point $\bar{\chi }$ is realised and $({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}$ is the Hilbert space of the spin chain. We can realise on it the spin chain Yangian representation at point ${{\bar{\theta }_{}}}$.

We are ready to formulate the main conceptual result of this appendix which is Proposition 3.5 of [9] generalised to the supersymmetric case.

Theorem B.9

Let ${{\bar{\theta }_{}}}=({{\bar{\theta }_{1}}},\ldots ,{{\bar{\theta }_{L}}})$ be a solution of equations $\chi _{\ell }(\theta )=\bar{\chi _{\ell }}$ such that ${{\bar{\theta }_{\ell }}}+\hbar \ne {{\bar{\theta }_{\ell '}}}$ for $\ell <\ell '$. Then $\mathrm {ev}_{{{\bar{\theta }_{}}}}$ is an isomorphism of ${\mathrm{Y}}({\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}})$ representations.

Proof

Since $\mathrm {ev}_{{{\bar{\theta }_{}}}}$ commutes with the Yangian action, it defines a homomorphism from the symmetrised representation to the spin chain representation. Assume ${{\mathsf {m}}}\geqslant 1$ and consider the vector $\mathbf{e}^+:=\mathbf{e}_1^{\otimes L}\in {\mathcal {V}}^{{\mathsf {S}}}(\bar{\chi })$. Since $\mathrm {ev}_{{{\bar{\theta }_{}}}}:\mathbf{e}^+\mapsto \mathbf{e}^+$, and $\mathbf{e}^+$ is a cyclic vector of the spin chain module by Theorem B.1, $\mathrm {ev}_{{{\bar{\theta }_{}}}}$ is surjective. As ${\mathcal {V}}^{{\mathsf {S}}}({\bar{\chi }})$ and $({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}$ are of the same dimension by Corollary B.5, $\mathrm {ev}_{{{\bar{\theta }_{}}}}$ is an isomorphism.

If ${{\mathsf {m}}}=0$, one can check that ${\mathcal {V}}^{{\mathsf {S}}}$ contains the vector $\prod _{1\leqslant \ell <\ell '\leqslant L}({{\theta }_{\ell }}- {{\theta }_{\ell '}}+\hbar )\mathbf{e}^+$, whose image under $\mathrm {ev}_{{{\bar{\theta }_{}}}}$ is nonzero as long as ${{\bar{\theta }_{\ell }}}+\hbar \ne {{\bar{\theta }_{\ell '}}}$ for $\ell <\ell '$. The rest of the proof is the same. $\square$

1.5 An explicit case study

Finally, we provide a concrete comprehensive example to illustrate the above-discussed ideas.

Explicit Q-operators. Consider the $L=3$ ${\mathfrak {\mathfrak {gl}}_{2}}$ spin chain in the absence of twist. By convention, we use the following basis of $\left( {\mathbb {C}}^2\right) ^{\otimes 3}$

(B.13)

In this basis, the periodic (“twist-less”) limit of the Q-operators is:

$$\begin{aligned} Q_{\varnothing }&=1,&Q_1&=\begin{pmatrix} 1&{}\\ &{}M_{1}&{}\\ &{}&{}M_{1}&{}\\ &{}&{}&{}1 \end{pmatrix},&Q_{12}&=\prod _{i=1}^3 (u-\theta _i) \,, \end{aligned}$$

(B.14)

where $M_{1}$ is the following $3\times 3$ block matrix

$$\begin{aligned} M_{1}= & {} \underbrace{\frac{1}{3} \begin{pmatrix} 1&{}1&{}1\\ 1&{}1&{}1\\ 1&{}1&{}1 \end{pmatrix}}_{P_s}+\left( u-2\frac{{{\theta }_{1}}+\theta _2+\theta _3}{3}\right) \underbrace{\frac{1}{3} \begin{pmatrix} 2&{}-1&{}-1\\ -1&{}2&{}-1\\ -1&{}-1&{}2 \end{pmatrix}}_{P_h}\nonumber \\&\quad +\frac{1}{6} \,\times \,\underbrace{ \begin{pmatrix}2\theta _2+2\theta _3&{}-\hbar -2\theta _3&{}\hbar -2\theta _2\\ \hbar -2\theta _3&{}2{{\theta }_{1}}+2\theta _3&{}-\hbar -2{{\theta }_{1}}\\ -\hbar -2\theta _2&{}\hbar -2{{\theta }_{1}}&{}2{{\theta }_{1}}+2\theta _2 \end{pmatrix} }_{c_0}\,. \end{aligned}$$

(B.15)

Notice that the projector to the symmetric irrep

is $\left( {\begin{matrix} 1\\ {} &{}P_s\\ &{}&{}P_s\\ &{}&{}&{}1 \end{matrix}} \right)$, whereas the projector to hook irreps is $\left( {\begin{matrix} 0\\ {} &{}P_h\\ &{}&{}P_h\\ &{}&{}&{}0 \end{matrix}} \right)$. Also notice that $c_0 P_s=0$, i.e. $c_0$ only affects the hook irreps

In addition to these notations, use $\chi _1={{\theta }_{1}}+{{\theta }_{2}}+{{\theta }_{3}}$, $\chi _2={{\theta }_{1}}{{\theta }_{2}}+{{\theta }_{1}}{{\theta }_{3}}+{{\theta }_{2}}{{\theta }_{3}}$, $\chi _3={{\theta }_{1}}{{\theta }_{2}}{{\theta }_{3}}$, and get

$$\begin{aligned} Q_2= & {} \frac{-6 u^4+8 u^3 \chi _1+u^2\left( 3\hbar ^2-12 \chi _2\right) +u(-2\hbar ^2 \chi _1+24 \chi _3)}{24 \hbar } \left( {\begin{matrix} 1\\ {} &{}P_s\\ &{}&{}P_s\\ &{}&{}&{}1 \end{matrix}} \right) \nonumber \\&\quad +\left( -\frac{u^3}{2\hbar }+\frac{\hbar \chi _1}{12}-\frac{\chi _3}{\hbar }\right) \left( {\begin{matrix} 0\\ {} &{}P_h\\ &{}&{}P_h\\ &{}&{}&{}0 \end{matrix}} \right) +\left( \frac{u^2}{4\hbar }-\frac{\hbar }{12}\right) \left( {\begin{matrix} 0\\ {} &{}c_0\\ &{}&{}c_0\\ &{}&{}&{}0 \end{matrix}} \right) \,, \end{aligned}$$

(B.16)

$$\begin{aligned} {\mathbb {Q}}_{0,1}= & {} \left( 3\hbar u^2-2\hbar \chi _1\,u+\hbar \chi _2+\frac{\hbar ^3}{4}\right) \left( {\begin{matrix} 1\\ {} &{}P_s\\ &{}&{}P_s\\ &{}&{}&{}1 \end{matrix}} \right) \nonumber \\&\quad +3\hbar \, u \left( {\begin{matrix} 0\\ {} &{}P_h\\ &{}&{}P_h\\ &{}&{}&{}0 \end{matrix}} \right) -\frac{\hbar }{2} \left( {\begin{matrix} 0\\ {} &{}c_0\\ &{}&{}c_0\\ &{}&{}&{}0 \end{matrix}} \right) \,. \end{aligned}$$

(B.17)

Restriction to the hook irreps. If we restrict to the subspace , we obtain $2\times 2$ matrices written for instance in the basis⁴⁷

(B.18)

In this basis, $c_0$ becomes the $2\times 2$ matrix

$$\begin{aligned} c:=c_0= \begin{pmatrix} 2\chi _1-\sqrt{3} ({{\theta }_{1}}-\theta _3)&{}\chi _1-3\theta _2+\sqrt{3} \hbar \\ \chi _1-3\theta _2-\sqrt{3} \hbar &{}2\chi _1+\sqrt{3} ({{\theta }_{1}}-\theta _3) \end{pmatrix} \end{aligned}$$

(B.19)

and equations (B.15), (B.16) and (B.17) become respectively

(B.20)

(B.21)

(B.22)

Symmetrised modules. Let us now compute these operators in the symmetrised Yangian representation. This results in presenting a ${{\theta }_{}}$-dependent change of basis such that all the matrix coefficients of $c$, the only non-trivial operator of the Bethe algebra, are symmetric polynomials in ${{\theta }_{\ell }}$. We will explicitly compute this basis by using the proof of B.8.

Let us first consider the case $\hbar =0$. The normalised Young symmetriser for is given by

$$\begin{aligned} S_{{\mathcal {T}}}=\frac{1}{3} R_{{\mathcal {T}}}C_{{\mathcal {T}}}=\frac{1}{3}(1+(2~1~3)-(3~2~1)-(3~1~2))\,. \end{aligned}$$

(B.23)

We now have to compute $S_{{\mathcal {T}}}\cdot {\mathbb {C}}[{{\theta }_{}}]$ and moreover to find a ${\mathbb {C}}[\chi ]$-basis for it. Start by picking a ${\mathbb {C}}[\chi ]$-basis of ${\mathbb {C}}[{{\theta }_{}}]$, for example the Schubert polynomials (in the case of ${\mathsf {S}}_3$ they are given by $\{1, {{\theta }_{1}},{{\theta }_{1}}+{{\theta }_{2}},{{\theta }_{1}}^2,{{\theta }_{1}}{{\theta }_{2}},{{\theta }_{1}}^2{{\theta }_{2}}\}$)⁴⁸. Since $S_{{\mathcal {T}}}$ commutes with multiplication by symmetric polynomials we just have to compute its action on Schubert polynomials. Eliminating obvious redundancies we obtain only two ${\mathbb {C}}[\chi ]$-independent basis elements

$$\begin{aligned} \eta _1:={{\theta }_{1}}+{{\theta }_{2}}-2{{\theta }_{3}}\qquad \eta _2:=2{{\theta }_{1}}{{\theta }_{2}}-{{\theta }_{1}}{{\theta }_{3}}-{{\theta }_{2}}{{\theta }_{3}}\,. \end{aligned}$$

(B.24)

At this stage we can already check the character formula (B.8). Indeed

$$\begin{aligned} \frac{{{\,\mathrm{ch}\,}}({\mathcal {V}}^{{\mathsf {S}}+}_{(2,1)})}{{{\,\mathrm{ch}\,}}({\mathbb {C}}[\chi ])}=\frac{t(1-t)(1-t^2)(1-t^3)}{(1-t)^2(1-t^3)}=t(1+t)=t^{\deg \eta _1}+t^{\deg \eta _2}\,. \end{aligned}$$

(B.25)

To obtain a ${\mathbb {C}}[\chi ]$-basis of ${\mathcal {V}}^{{\mathsf {S}}+}_{(2,1)}$ it remains to compute $p({{\mathsf {w}}}\otimes \eta _1)$ and $p({{\mathsf {w}}}\otimes \eta _2)$ which can be done straightforwardly.

Now assume $\hbar \ne 0$. This case is more complicated because now we have to take $\hbar$ corrections into account. In particular now $p({{\mathsf {w}}}\otimes \eta _1), p({{\mathsf {w}}}\otimes \eta _2)\notin {\mathcal {V}}^{{\mathsf {S}}+}_{(2,1)}$. Nevertheless $p({{\mathsf {w}}}\otimes \eta _1), p({{\mathsf {w}}}\otimes \eta _2)\in {\mathcal {V}}^{{\mathsf {S}}}_{(2,1)}$ and we have to correct them by some vectors of lower degree such that they belong to ${\mathcal {V}}^{{\mathsf {S}}+}_{(2,1)}$. Since $\deg p({{\mathsf {w}}}\otimes \eta _1)=\deg \eta _1=1$ it can only be corrected by a vector of degree zero. There is only one such vector in ${\mathcal {V}}^{{\mathsf {S}}}_{(2,1)}$: the totally symmetric combination ${|{\uparrow \downarrow \downarrow }\rangle }+{|{\downarrow \uparrow \downarrow }\rangle }+{|{\downarrow \downarrow \uparrow }\rangle }$. Its coefficient can be uniquely fixed by requiring that the corrected vector is highest-weight. By a similar argument we can compute the $\hbar$ corrections to $p({{\mathsf {w}}}\otimes \eta _2)$. In the end we obtain the following ${\mathbb {C}}[\chi ]$-basis of ${\mathcal {V}}^{{\mathsf {S}}+}_{(2,1)}$

$$\begin{aligned} \begin{aligned} \xi _1:=&\frac{1}{6}(-2{{\theta }_{1}}+{{\theta }_{2}}+{{\theta }_{3}}-3\hbar ){|{\uparrow \downarrow \downarrow }\rangle }\\&~~+\frac{1}{6}({{\theta }_{1}}-2{{\theta }_{2}}+{{\theta }_{3}}){|{\downarrow \uparrow \downarrow }\rangle }\\&~~~~+\frac{1}{6}({{\theta }_{1}}+{{\theta }_{2}}-2{{\theta }_{3}}+3\hbar ){|{\downarrow \downarrow \uparrow }\rangle }\\ \xi _2:=&\frac{1}{18}(-3{{\theta }_{1}}{{\theta }_{2}}-3{{\theta }_{1}}{{\theta }_{3}}+6{{\theta }_{2}}{{\theta }_{3}}-\hbar ({{\theta }_{1}}+4{{\theta }_{2}}+4{{\theta }_{3}})){|{\uparrow \downarrow \downarrow }\rangle }\\&~~+\frac{1}{18}(-3{{\theta }_{1}}{{\theta }_{2}}+6{{\theta }_{1}}{{\theta }_{3}}-3{{\theta }_{2}}{{\theta }_{3}}-\hbar (4{{\theta }_{1}}+{{\theta }_{2}}-5{{\theta }_{3}})-3\hbar ^2){|{\downarrow \uparrow \downarrow }\rangle }\\&~~~~+\frac{1}{18}(6{{\theta }_{1}}{{\theta }_{2}}-3{{\theta }_{1}}{{\theta }_{3}}-3{{\theta }_{2}}{{\theta }_{3}}+\hbar (5{{\theta }_{1}}+5{{\theta }_{2}}-{{\theta }_{3}})+3\hbar ^2){|{\downarrow \downarrow \uparrow }\rangle } \end{aligned} \end{aligned}$$

(B.26)

that is, ${\mathcal {V}}^{{\mathsf {S}}+}_{(2,1)}={\mathbb {C}}[\chi ]\xi _1\oplus {\mathbb {C}}[\chi ]\xi _2$. Changing $c$ to this basis we finally obtain its symmetrised representative

$$\begin{aligned} c^{\mathsf {S}}:= \begin{pmatrix} -\hbar &{} -2\chi _2-\frac{2}{3}\hbar (\chi _1+\hbar )\\ 6 &{} 4\chi _1+\hbar \end{pmatrix}\,. \end{aligned}$$

(B.27)

A few remarks are in order. First, note that $c$ is homogeneous of degree 1 (if we consider $\deg \hbar =1$) whereas $c^{\mathsf {S}}$ is non homogeneous of degree 2. This has to do with the fact that $\xi _1$ and $\xi _2$ are ${{\theta }_{}}$-dependent and of different degrees. Second, one can check that $c$ and $c^{\mathsf {S}}$ have the same characteristic polynomial and therefore the same spectrum. However there is a crucial difference: if we set $\hbar =0$ and take ${{\bar{\theta }_{1}}}={{\bar{\theta }_{2}}}={{\bar{\theta }_{3}}}$ (a potentially “bad” point by Theorem B.9), $c$ becomes proportional to the identity whereas $c^{\mathsf {S}}$ does not, so they cannot be related by a change of basis. In particular the symmetrised Bethe algebra ${{{\mathcal {B}}}_{\varLambda }}(\bar{\chi })$ is still maximal, whereas the evaluated Bethe algebra ${{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$ is not. Again this has to do with the fact that the basis $(\xi _1,\xi _2)$ is ${{\theta }_{}}$-dependent: at this particular point it becomes degenerate as a basis of the physical vector space, but not as a basis of the quotient ${\mathcal {V}}^{{\mathsf {S}}+}_{(2,1)}(\bar{\chi })$. Actually, the determinant of the matrix of change of basis between (B.18) and $(\xi _1,\xi _2)$ is given by $\frac{-1}{4\sqrt{3}}({{\theta }_{1}}-{{\theta }_{2}}+\hbar )({{\theta }_{1}}-{{\theta }_{3}}+\hbar )({{\theta }_{2}}-{{\theta }_{3}}+\hbar )$ and it equates to zero precisely at the “bad” points of Theorem B.9. Note however that even at most “bad” points ${{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})$ and ${{{\mathcal {B}}}_{\varLambda }}(\bar{\chi })$ are still isomorphic and maximal.

This example shows that a ${\mathbb {C}}[\chi ]$-basis of the symmetrised Yangian representation is quite difficult to compute. As long as the hypothesis of Theorem B.9 is satisfied, the traditional physical frame is perfectly equivalent to the symmetrised one and can be used without trouble for all practical applications.

Q-operators Belong to the Bethe Algebra

In order to show that the Q functions/operators belong to the Bethe algebra we will show how to find them from ${\mathbb {T}}$ functions/operators. When $a=1$, the contraction with the Levi-Civita tensor in the Wronskian expression (2.15) reduces to $({{\mathsf {m}}}-1)!\sum \limits _b (-1)^bQ_b^{[{{\mathsf {m}}}-{{\mathsf {n}}}+s]}Q_{{\bar{b}}}^{[-s]}$, hence we can express the t-dependent sum (where t is a free parameter)

$$\begin{aligned} S(t):= \sum _b \left( \sum _{s\geqslant 1} Q_{\bar{b}}^{[-{{\mathsf {m}}}+{{\mathsf {n}}}-2s]} t^s\right) Q_{b} \end{aligned}$$

(C.1)

as an infinite linear combination of the ${\mathbb {T}}_{(s^1)}$: this linear combination looks like $\sum \nolimits _{s\geqslant 1} {\mathbb {T}}_{(s^1)}^{[-{{\mathsf {m}}}+{{\mathsf {n}}}-s]} t^s$, up to the first terms (when $s<a-{{\mathsf {m}}}+{{\mathsf {n}}}$) and up to factors that have no impact on the present argument and would make expressions extremely bulky. These are the factors $Q_{{{\bar{\varnothing }}}|{{\bar{\varnothing }}}}$, ${{\,\mathrm{Ber}\,}}G$ (which we set to 1), $({{\mathsf {m}}}-1)!$ and the proportionality⁴⁹ factor denoted by the symbol $\propto$ in (2.15).

This infinite sum (where s runs from 0 to $+\infty$) converges in the disk $|t|<\min \left| \frac{1}{x_b}\right|$ and is then analytically continued to . Indeed, if we denote $Q_{\bar{b}}= \left( \frac{1}{x_b}\right) ^{u/\hbar } \sum _{k=0}^{M_{\bar{b}}}c_{{{\bar{b}}}}^{(k)}u^{k}$ then for $|t|<\min \left| \frac{1}{x_b}\right|$ we have:

$$\begin{aligned}&\sum _{s\geqslant 1} Q_{{\bar{b}}}^{[-{{\mathsf {m}}}+{{\mathsf {n}}}-2s]} t^s=\nonumber \\&\quad \left( \frac{1}{x_b}\right) ^{\frac{-{{\mathsf {m}}}+{{\mathsf {n}}}}{2}}\sum _{j=0}^{M_{{\bar{b}}}} (-1)^j \left( \sum _{k=j}^{M_{{\bar{b}}}} {\left( {\begin{array}{c}k\\ j\end{array}}\right) } (u^{[-{{\mathsf {m}}}+{{\mathsf {n}}}]})^{k-j} c^{(k)}_{{\bar{b}}} \right) \underbrace{\sum _{s\geqslant 1} s^j \left( t \,x_b\right) ^s}_{\frac{ \sum _{k=0}^{j-1} A(j,k)\left( t\, x_b\right) ^{k+1} }{\left( 1-t\, x_b\right) ^{j+1}}\,,} \end{aligned}$$

(C.2)

where the combinatorial factors A(j, k) are the positive integers known as the Eulerian numbers, and where the sum $\sum \limits _{k=0}^{j-1} A(j,k)\left( t\, x_b\right) ^{k+1}$ should be replaced by 1 in the ill-defined case $j=0$.

If the eigenvalues are $x_b$ are pairwise distinct we deduce that

$$\begin{aligned} Q_b\propto \lim _{t\rightarrow \frac{1}{x_b}}\left( 1-t\, x_b\right) ^{M_{{\bar{b}}}+1} S(t)\,. \end{aligned}$$

(C.3)

This expression allows one to conclude that, at the level of representations, it belongs to the Bethe algebra. Indeed, the Bethe Algebra forms a linear subspace of the space of operators on the Hilbert space, which is finite-dimensional. It is hence topologically closed, so that the sum S(t) belongs to the Bethe algebra, not only when $|t|<\min \left| \frac{1}{x_b}\right|$ but even for arbitrary t by analytic continuation. Then, by taking the limit (C.3) $Q_b$ belongs to the Bethe Algebra.

In addition to the Q-operators $Q_b$ ($1\leqslant b\leqslant {{\mathsf {m}}}$), the same approach also allows producing the operators $Q_i$ (${{\hat{1}}}\leqslant i\leqslant {\hat{{{\mathsf {n}}}}}$) by focusing on a sum of the form $\sum _{a\geqslant 0} {\mathbb {T}}_{(1^a)}^{[-{{\mathsf {m}}}+{{\mathsf {n}}}+a]}t^a$. Hence it allows expressing all Q-operators using (2.21) and (2.19).

In [65], explicit computations of such infinite sums and of their limit were performed combinatorically at the level of the representation $ev_{{\theta }}$ (for twist with pairwise-distinct eigenvues, as in the above discussion). This explicit construction of the Q-operators shows that their matrix coefficients are indeed polynomial functions of u and of the inhomogeneities ${{\theta }_{\ell }}$, and that they are rational functions of the twist eigenvalues $z_\alpha$. It also shows that the degree $M_{A|I}$ of $q_{A|I}$ is indeed given by (2.25).

For comparison purposes, we note that the infinite sum which was computed in [65] is actually of the form $\sum _{s\geqslant 1} {\mathbb {T}}_{(s^1)}^{[+s]} t^s$, by contrast with the opposite shifts in the above discussion. Consequently the explicit combinatorial description gives an expression of $Q_{{\bar{b}}}$ instead of $Q_b$, and $Q_b$ was extracted after a few more steps—after ${{\mathsf {m}}}-1$ successive limits—and the whole Q-system is expressed explicitly.

Details About Structural Study of Bethe Equations

In this section we will adopt the analytic point of view of Sect. 3.1 on inhomogeneities. Namely we will think them as complex numbers that we are going to vary. Then $c_{\ell }$—the coefficients of (twisted) Baxter polynomials—turn out to be algebraic functions of inhomogeneities. The main purpose of this section is to prove properness of WBE—that all solutions $c_{\ell }$ are bounded if ${{\theta }_{\ell }}$ (and hence $\chi _{\ell }$) are bounded. A natural consequence of this analysis will be the behaviour of $c_{\ell }$ when ${{\theta }_{\ell }}$ tend to infinity (in the twist-less case) which we analyse in D.3 to provide the necessary technical results for Sect. 6.2.

1.1 Properness, twisted case

The feature that ensures that $c_{\alpha }$ are bounded at finite $\theta _{\alpha }$ is the fact that $c_\alpha$ are coefficients of polynomials in u and the quantization condition (2.26) is an equation on these polynomials.

Assume that there is a point $\bar{\chi }\in {{\mathcal {X}}}$ by approaching which some of $c_{\alpha }$ diverge (become unbounded). If $c_{\alpha }$ is a coefficient of a twisted polynomial Q(u) then divergence of $c_{\alpha }$ implies divergence of some of the roots of Q(u). Factorise Q(u) in the form $Q=Q^{\gg }Q^{\lesssim }$, where $Q^{\gg }$ is a polynomial containing all diverging roots, and $Q^{\lesssim }$ is the function containing the twist prefactor and all finite roots.

Consider first the equation $Q_{a|i}^+-Q_{a|i}^-=Q_{a|\varnothing }Q_{\varnothing |i}$. One can rewrite $Q_{a|i}^+-Q_{a|i}^-=Q^{\gg }_{a|i}((Q^{\lesssim }_{a|i})^+-(Q^{\lesssim }_{a|i})^-)+H$, where H is the function that ensures equality. By taking a point $\chi$ sufficiently close to the point $\bar{\chi }$, one can ensure that H is small in the following sense: there exists R such that all roots of $(Q^{\lesssim }_{a|i})^+-(Q^{\lesssim }_{a|i})^-$ lie inside the circle $|u|=R$, all roots of $Q^{\gg }$ lie outside it, and that the absolute value of H is smaller than the absolute value of $Q^{\gg }_{a|i}((Q^{\lesssim }_{a|i})^+-(Q^{\lesssim }_{a|i})^-)$ when $|u|=R$. Then by Rouché’s theorem, the number of zeros of $Q_{a|i}^+-Q_{a|i}^-$ inside and outside of the circle is the same as that of $Q^{\gg }_{a|i}((Q^{\lesssim }_{a|i})^+-(Q^{\lesssim }_{a|i})^-)$. Hence existence of large zeros of $Q_{a|i}$ imply existence of large zeros, in the same amount, in the product $Q_{a|\varnothing } Q_{\varnothing |i}$. Their distribution between $Q_{a|\varnothing }$ and $Q_{\varnothing |i}$ depends on the solution we consider.

Consider now the WBE (2.26) and recall that SW is explicitly the determinant (2.20). Applying the same logic, we write

$$\begin{aligned} {{\,\mathrm{SW}\,}}(C_{\varLambda })=\prod _{a=1}^{{{\mathsf {m}}}} Q_{a|\varnothing }^{\gg }\prod _{i=1}^{{{\mathsf {n}}}}Q_{\varnothing |i}^{\gg }\,{{\,\mathrm{SW}\,}}(Q^{\lesssim })+H\,, \end{aligned}$$

(D.1)

and then conclude using Rouché’s theorem that $Q_{\theta }={{\,\mathrm{SW}\,}}(Q)$ has large zeros, i.e. the point $\bar{\chi }$ cannot have $\bar{\chi }_{\ell }$ all finite. For the argument to work, one needs to ensure that ${{\,\mathrm{SW}\,}}(Q^{\lesssim })$ is not vanishing but it is straightforward as the presence of twist prefactors $z_{\alpha }^{u/\hbar }$ ensures that already the leading-u term in ${{\,\mathrm{SW}\,}}(Q^{\lesssim })$ is non-vanishing.

1.2 Properness, twist-less case

To study the twist-less case, we will focus on the bosonised parameterisation of the Q-system on a Young diagram (5.10). In particular, one has

$$\begin{aligned} Q_{\theta }={\mathbb {Q}}_{0,0}\propto W(B_1,\ldots ,B_{{{\mathsf {m}}}})\,, \end{aligned}$$

(D.2)

where ${{\mathsf {m}}}=h_{\varLambda ^+}$. Equation (D.2) contains in principle the full information since the ${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|0}}$ Q-system is a possible way to parameterise the Bethe algebra ${{{\mathcal {B}}}_{\varLambda }}$.

To prove properness, we would like to use an argument similar to that of (D.1), however cancellations in the Wronskian determinant make things more subtle.

• Example: Take $B_1=u$, $B_2=(u-\varLambda )(u+1)$, $B_3=(u-\varLambda )^3$. Let $\varLambda \rightarrow \infty$ if $\chi \rightarrow \bar{\chi }$. Formally there are four divergent roots when $\varLambda \rightarrow \infty$. However $W(B_1,B_2,B_3)=u^3+u(3\varLambda -\hbar ^2)-\varLambda ^2(\varLambda +3)\,$ which has three divergent roots.

The issue in the example comes (at least) from the fact that in the decomposition $Q=Q^{\gg }Q^{\lesssim }$, $B_1^{\lesssim }$ and $B_2^{\lesssim }$ are polynomials of the same degree (equal to one).

To continue, we do a couple of formalisations.

Parametric factorisation. Let $\varLambda$ be a parameter, and we intend to consider the $\varLambda \rightarrow \infty$ behaviour of Q-functions that are algebraic functions of $\varLambda$. Define a scale function $S=\varLambda ^{\beta }$ for some real $\beta$. We say that $Q=Q^{\gg S}Q^{\lesssim S}$ is the parametric factorisation of the polynomial Q at scale S if all roots of the monic polynomial $Q^{\gg S}$ are much larger than S, and all roots of the monic polynomial $Q^{\lesssim S}$ are comparable to or smaller than S. More precisely, for each ${u}$ that satisfies $Q^{\gg S}({u})=0$ one has $\lim \limits _{\varLambda \rightarrow \infty } S/{u}=0$, and for each ${u}$ that satisfies $Q^{\lesssim S}({u})=0$ the $\varLambda \rightarrow \infty$ limit of ${u}/S$ is finite.

Then the argument around (D.1) can be formalised by the following lemma.

Lemma D.1

For some polynomials $Q_1,\ldots ,Q_a$, let $Q_{12\ldots a}=W(Q_1,\ldots ,Q_a)$ and let $Q=Q^{\gg S}Q^{\lesssim S}$ be the parametric factorisation at scale S.

If degrees $\deg Q_1^{{\lesssim } S}$, $\deg Q_2^{{\lesssim } S}$, $\deg Q_3^{{\lesssim } S}$, $\dots$ are pairwise distinct, and degrees $\deg Q_1$, $\deg Q_2$, $\deg Q_3$, $\dots$ are also pairwise distinct then

$$\begin{aligned} \deg Q^{\gg S}_{12\ldots a}=\sum _{a'=1}^{a} \deg Q^{\gg S}_{a'}\,. \end{aligned}$$

(D.3)

Proof

Perform an equivalent of decomposition (D.1) and apply Rouché’s theorem. An important ingredient is that $W(u^{\deg Q_1},\dots , u^{\deg Q_a})$ (resp. $W(u^{\deg Q_1^{{\lesssim } S}},\dots , u^{\deg Q_a^{{\lesssim } S}})$) is not zero and hence provides the term of the highest degree of $W(Q_1,\ldots , Q_a)$ (resp. $W(Q_1^{{\lesssim } S},\ldots , Q_a^{{\lesssim } S})$) which is why the restriction on the degrees is imposed. $\square$

Now we recall that not all coefficients of $B_a$ bear physical information as they are subject to the symmetry transformation (2.35). We shall benefit from (2.35) to ensure that a parametric factorisation of $B_a$ satisfies the conditions of the above lemma.

Lemma D.2

Let $B_1,\ldots ,B_{{\mathsf {m}}}$ be monic polynomials with $\deg B_1<\ldots <\deg B_{{\mathsf {m}}}$. For any scale S, one can find a “rotation”

$$\begin{aligned} B_a&\rightarrow B_a+\sum _{b<a}h_{ab} B_b\,, \end{aligned}$$

(D.4)

where the $h_{ab}$’s are complex-valued functions⁵⁰ of $\varLambda$, such that, after the rotation, the degrees of ${B}_{1}^{{\lesssim } S}$, $B_2^{{\lesssim } S}$, $\ldots$, ${B}_{{{\mathsf {m}}}}^{{\lesssim } S}$ are pairwise distinct.

We note that the proof below is constructive and it provides an algorithm to find $h_{ab}$ explicitly.

Proof

Without loss of generality one can set $S=1$ in which case we denote the parametric factorisation as $Q=Q^{\gg }Q^{\lesssim }$. Indeed, we can always perform the rescaling $u\rightarrow u\,S$.

In the labelling of polynomials $B_a=u^{\lambda _a+{{\mathsf {m}}}-a}+\ldots +b_k^{(a)} u^k+\ldots + b_1^{(a)} u+b_0^{(a)}$, consider all $b_{k'}^{(a)}$ that have the largest exponent when $\varLambda \rightarrow \infty$ and choose $b_{(k)}^{(a)}$ with the largest k among them. For instance, in $u^3+\varLambda u^2+\varLambda ^3 u+2\varLambda ^3$, it is $b_{(1)}=\varLambda ^3$. Then $\deg B_a^{\lesssim }=k$.

If there exist such a, b, $b<a$ that $\deg B_a^{\lesssim }=\deg B_b^{\lesssim }=k$ then perform the transformation $B_a\rightarrow B_a-\frac{b_{(k)}^{(a)}}{b_{(k)}^{(b)}}\,B_b$. This transformation will affect the parametric factorisation of $B_a$. Two things can happen. First, $\deg B_a^{\lesssim }$ becomes smaller. Second, all terms with the largest exponent are cancelled out from $B_a$ in which case one gets a new (smaller) largest exponent and a new value for $\deg B_a^{\lesssim }$ (in principle arbitrarily large, only bounded by the degree of $B_a$).

We repeat recursively the procedure of comparison between all available pairs of a, b and terminate when $\deg B_a^{\lesssim }$ become pairwise distinct. The recursion will terminate in a finite number of steps and produce a meaningful result for the following reasons: there are finitely many polynomials of finite degree to operate with, the maximal exponents can only decrease in the procedure and they are bounded by zero from below, and $B_a$ cannot vanish entirely as $\deg B_a$ are pairwise distinct and so the leading monomial is never affected by the performed transformations. $\square$

• Example: For $S=1$ and the system in the Example on page 70, the rotation is done as follows. First, transformation $B_2\rightarrow B_2+\varLambda B_1=u^2+u-\varLambda$ drops the degree of $B_2^{\lesssim }$ to zero. Now both degrees of $B_2^{\lesssim }$ and $B_3^{\lesssim }$ are zero. We perform transformation $B_3\rightarrow B_3-\varLambda ^2 B_2$. This drops the maximal exponent in $B_3$ from three to two, and $\deg B_3^{\lesssim }$ computed with respect to the new maximal exponent is two. Now all $\deg B_a^{\lesssim }$ are pairwise distinct. In summary, we get the rotated values $B_1=u,B_2=u^2+u-\varLambda ,B_3=u^3-\varLambda (\varLambda +3)\,u^2+2\varLambda ^2\,u$. $B_2$ has two divergent roots, $B_3$ has one divergent root, $W(B_1,B_2,B_3)$ remains unchanged by the performed rotation and it has three divergent roots.

Now we are ready to prove properness as declared on page 21. Assume that there is a finite point $\bar{\chi }\in {{\mathcal {X}}}$ by approaching which some coefficients of $B_a$ diverge. We follow some path parameterised by $\varLambda$, and $\varLambda \rightarrow \infty$ corresponds to the approach of the point. Choose $S=1$ and perform the transformation (D.4) to get $\deg B_a^{\lesssim }$ pairwise distinct. If there are still divergent coefficients after this transformation, we get $\deg Q_{\theta }^{\gg }>0$ by Lemma D.1 and (D.2) and hence reach a contradiction. Thus all $B_a$ have finite coefficients. Compute ${\mathbb {Q}}_{a,s}$ following (5.10) and use the procedure in the proof of Lemma 5.3 to compute the set $C_{\varLambda }$ introduced after (2.35). $c_{\ell }$ appearing in this set are hence non-divergent when we approach $\bar{\chi }$ which is the properness in the sense of Sect. 3.

1.3 Labelling solutions with standard Young tableaux, technical details

1.3.1 All solutions approach (6.6).

The key step is to justify the formula (6.5). Consider the situation when $\theta _L=\varLambda$, $\varLambda \rightarrow \infty$, and all other ${{\theta }_{\ell }}$ are finite. In any scaling $S=\varLambda ^{\beta }$ with $0<\beta <1$, we rotate $B_a$ to a frame where $\deg B_a^{\lesssim }$ are pair-wise distinct. By Lemma D.1, there is precisely one $a=a_0$ for which precisely one root of $B_{a_0}$ diverges, and all roots of $B_{a\ne a_0}$ stay finite.

Recall that $\deg B_a=\lambda _a+{{\mathsf {m}}}-a$ and $\deg B_i=d_i$. The assignment rules are explained in Fig. 4. Since $B_a$ are in the frame with pair-wise distinct $\deg B_a^{\lesssim }$, if $a_0\ne {{\mathsf {m}}}$, it must be that $\lambda _{a_0}>\lambda _{a_0+1}$ (equality is impossible). Then, there exists $s_0$ such that $d_{s_0}=\deg B_{a_0}-1$ and so the box $(a_0,s_0)$ is a corner box of the Young diagram.

Finally, we consider (5.10) to decide which ${\mathbb {Q}}_{a,s}$ have a divergent root. If $a>a_0$ then ${\mathbb {Q}}_{a,s}$ does not depend on $B_{a_0}$ and hence has no divergent roots. If $s>s_0$ then all polynomials of degree from 0 to $\deg B_{a_0}-1$ appear in the Wronskian determinant. Then the polynomial structure of $B_{a_0}$ is irrelevant, we can replace it with the leading monomial and so ${\mathbb {Q}}_{a,s}$ cannot have divergent roots. Finally if $a\leqslant a_0$ and $s\leqslant s_0$ then $B_{a_0}$ is present in the Wronskian and there is no polynomial of degree $\deg B_{a_0}-1$ in the Wronskian, so the conditions of the Lemma D.1 are satisfied, hence ${\mathbb {Q}}_{a,s}$ has precisely one divergent root.

Now we can deduce that roots scale exactly as $\varLambda$. Indeed, for $\beta '\in ]\beta ,1[$, the rotation of Lemma D.2 may change but the ${\mathbb {Q}}$ functions are invariant under this triangular rotation. If the value of $a_0$ changes at scale $\varLambda ^{\beta '}$ compared to scale $\varLambda ^{\beta }$, then there would be another corner-box $(a_0',s_0')\ne (a_0,s_0)$ such that at scale $\varLambda ^{\beta '}$ the ${\mathbb {Q}}$-functions with diverging roots are the nodes with $a\leqslant a_0'$ and $s\leqslant s_0'$. This is impossible because all ${\mathbb {Q}}$ functions that diverge at scale $\varLambda ^{\beta '}$ also diverge at scale $\varLambda ^{\beta }$. Therefore $a_0$ is independent of $\beta \in ]0,1[$, the number of diverging roots is thus also independent of $\beta$ and the diverging roots scale exactly as $\varLambda$. Finally, we get (6.5), and also that ${\tilde{{\mathbb {Q}}}}_{a,s}$ introduced alongside (6.5) is a Q-system on the Young diagram with the box $(a_0,s_0)$ removed.

1.3.2 Unambiguous continuation of the limiting solution (6.6) for each SYT to finite inhomogeneities.

To discuss this question, we should not send inhomogeneities one after another to infinity, but do a more smooth realisation of the limit (6.4). Namely, for $\alpha _1$, $\alpha _2$, $\dots$, $\alpha _L\in {\mathbb {C}}^*$ and $\beta _L>\beta _{L-1}>\dots>\beta _1>0$, we parameterise ${{\theta }_{1}}=\alpha _1\,\varLambda ^{\beta _1}$, ${{\theta }_{2}}=\alpha _2\,\varLambda ^{\beta _2}$, $\dots$, ${{\theta }_{L}}=\alpha _L\,\varLambda ^{\beta _L}$. All roots and all coefficients of all Q-polynomials are algebraic functions⁵¹ of the parameter $\varLambda$ and hence have a large-$\varLambda$ behavior of the form $\alpha \,\varLambda ^\beta$ which allows applying scaling argumentation from previous sections.

If $\beta _\ell$ are spaced apart well, we can recover the same results as if inhomogeneities are sent to infinity one by one. But now, after we know that all solutions of the Q-system approach (6.6), a sharper judgement about possible $\beta _\ell$ can be made by observing that the leading order of the large $\varLambda$ expansion (6.6) is solved by⁵²

$$\begin{aligned} \forall a&\leqslant {{\mathsf {m}}},&B_a\sim u^{{{\mathsf {m}}}-a}\prod _{s=1}^{{\underline{\lambda }}_{a}}\left( u-({{\mathsf {m}}}-a+s){N_{a-1,s-1}^{({{\mathcal {T}}}_{a,s})}}{{\theta }_{{{\mathcal {T}}}_{a,s}}}\right) \,. \end{aligned}$$

(D.5)

Let us now parameterise the Q-system by $\kappa _1,\ldots ,\kappa _L$ as follows

$$\begin{aligned} \forall a&\leqslant {{\mathsf {m}}},&B_a=u^{{{\mathsf {m}}}-a}\prod _{s=1}^{{\underline{\lambda }}_{a}}\left( u-({{\mathsf {m}}}-a+s){N_{a-1,s-1}^{({{\mathcal {T}}}_{a,s})}}\kappa _{{{\mathcal {T}}}_{a,s}}\varLambda ^{\beta _{{{\mathcal {T}}}_{a,s}}}\right) \,. \end{aligned}$$

(D.6)

Recall that ${{\theta }_{\ell }}=\alpha _\ell \varLambda ^{\beta _\ell }$. Then $Q_\theta =W(B_1,\ldots ,B_{{\mathsf {m}}})$ realises a map from $\kappa _\ell$ to $\alpha _\ell$ which analytically depends on $1/\varLambda$ for $\beta _{\ell }$ being integers. When $1/\varLambda =0$ this map is simply an identity map with obviously non-zero Jacobian. Hence we can apply the analytic implicit function theorem to invert the map. By the theorem, for some neighbourhood of the point $1/\varLambda =0$, $\kappa _\ell$ are analytic functions of $\alpha _1,\ldots ,\alpha _L$ and $1/\varLambda$ and hence each limiting solution (6.6) can be continued to finite values of inhomogeneities.

Notations

Typical values of indices

a, b:

1 to ${{\mathsf {m}}}$

i, j:

${{\hat{1}}}$ to ${\hat{{{\mathsf {n}}}}}$ (hat is omitted sometimes)

$\alpha ,\beta$ :

from the set $\{1,\ldots ,{{\mathsf {m}}},{{\hat{1}}},\ldots ,{\hat{{{\mathsf {n}}}}}\}$

$\ell$ :

1 to L

Parameters

$z_\alpha$ :

twist eigenvalues, $z_a\equiv x_a$, $z_{{{\hat{i}}}}\equiv y_i$

${{\theta }_{\ell }}$ :

inhomogeneities (as variables)

${{\bar{\theta }_{\ell }}}$ :

inhomogeneities (fixed number)

$\chi _{\ell }$ :

elementary symmetric polynomials

$\bar{\chi }_{\ell }$ :

$=\chi _{\ell }({{\bar{\theta }_{1}}},\ldots ,{{\bar{\theta }_{L}}})$

Lie algebra

${\mathfrak {\mathfrak {gl}}_{{{\mathsf {m}}}|{{\mathsf {n}}}}}$ :

symmetry of the system (broken to Cartan in the twisted case)

${\mathsf {E}}_{\alpha \beta }$ :

abstract generators and defining representation

${{\mathcal {E}}}_{\alpha \beta }$ :

global spin chain action

${\varLambda ^+}=({\underline{\lambda }}_1,{\underline{\lambda }}_2,\ldots )$ :

Young diagram $\equiv$ integer partition (typically of L)

$({\underline{\lambda }}'_1,{\underline{\lambda }}'_2,\ldots )$ :

transposed partition, $h_{{\varLambda ^+}}:={\underline{\lambda }}'_1$.

${\varLambda }=[\lambda _1,\ldots ,\lambda _{{\mathsf {m}}}|\nu _1,\ldots ,\nu _{{\mathsf {n}}}{]}$ :

weight (eigenvalues of ${{\mathcal {E}}}_{\alpha \alpha }$)

$({\hat{\lambda }}_1\ldots ,{\hat{\lambda }}_{{{\mathsf {m}}}'}|{\hat{\nu }}_1,\ldots ,{\hat{\nu }}_{{{\mathsf {n}}}'})$ :

shifted weight (describes ${\varLambda ^+}$ with marked point)

Spin chain

V :

Hilbert space of the spin chain ($\simeq ({\mathbb {C}}^{{{\mathsf {m}}}|{{\mathsf {n}}}})^{\otimes L}$)

${V_{\varLambda }}$ :

subspace of V spanned by states of weight ${\varLambda }$

${V_{\varLambda }^+}$ :

subspace of V spanned by highest weight states of irreps ${\varLambda ^+}$

${U_{\varLambda }}$ :

either ${V_{\varLambda }}$ or ${V_{\varLambda }^+}$

$d_\varLambda$ :

dimension of ${U_{\varLambda }}$

Bethe and Wronskian algebras

${{\hat{c}}}_k^{(d)}$, ${{\hat{c}}}_\ell$:

operators acting on spin chain, coefficients in Baxter Q-operators, e.g. $Q_k=u^{M_k}(1+\frac{{{\hat{c}}}_{k}^{(1)}}{u}+\ldots )$

$c_k^{(d)}$, $c_\ell$:

abstract variables and/or eigenvalues of ${{\hat{c}}}_k^{(d)}$, ${{\hat{c}}}_\ell$

${{{\mathcal {B}}}_{\varLambda }}$ :

Bethe algebra restricted to ${U_{\varLambda }}$ (generated by ${{\hat{c}}}_\ell$), a ${\mathbb {C}}[\chi ]$-module

${{{{\mathcal {B}}}_{\varLambda }}({{\bar{\theta }_{}}})}$ :

specialised Bethe algebra (for spin chain representation at point ${{\bar{\theta }_{}}}$)

${{{\mathcal {B}}}_{\varLambda }}(\bar{\chi })$ :

specialised Bethe algebra (for symmetrised representation at point $\bar{\chi }$)

${{{\mathcal {W}}}_{\varLambda }}$ :

Wronskian algebra (generated by $c_\ell$ subject to Wronskian Bethe equations)

${{{\mathcal {W}}}_{\varLambda }}(\bar{\chi })$ :

specialised Wronskian algebra

Functional relations conventions

u :

spectral parameter

$\hbar$ :

Unit of discrete shift in e.g. Baxter equation, typically $\hbar =\pm i,\pm 1,\pm 2$

$f^{[n]}$ :

$f^{[n]}\equiv f(u+\frac{\hbar }{2}n)$, $f^\pm \equiv f^{[\pm 1]}$

$f\propto g$ :

f and g, as functions of u, are equal up to a normalisation