Partial domain wall partition functions

We consider six-vertex model configurations on an (n × N) lattice, n ≤ N, that satisfy a variation on domain wall boundary conditions that we define and call partial domain wall boundary conditions. We obtain two expressions for the corresponding partial domain wall partition function, as an (N × N)-determinant and as an (n × n)-determinant. The latter was first obtained by I Kostov. We show that the two determinants are equal, as expected from the fact that they are partition functions of the same object, that each is a discrete KP τ-function, and, recalling that these determinants represent tree-level structure constants in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} = 4\;{\text{SYM}} $\end{document}, we show that introducing 1-loop corrections, as proposed by N Gromov and P Vieira, preserves the determinant structure.


JHEP07(2012)186
3 Domain wall partition function in the infinite-rapidity limit 10 3.1 One rapidity becomes infinite 1 Introduction The discovery of classical and quantum integrable structures on both sides of the anti-de Sitter/conformal field theory correspondence, AdS/CFT, in the late 1990's and early 2000's, culminating in [1,2] and in the intensive rapid developments that followed these seminal works, has been beneficial to all subjects involved. 1 On the one hand, integrability is widely considered to be a viable approach to proving the AdS/CFT correspondence. On the other, ideas and insights from AdS/CFT will continue to enrich the subject of integrability.

Partial domain wall boundary conditions
The purpose of this note is to study six-vertex model configurations on a rectangular lattice with n horizontal and N vertical lines, n N , as in figure 1, that satisfy (n×N ) partial domain wall boundary conditions, pDWBC's. In our conventions, 2 these are defined as follows.
1. All arrows on the left and right boundaries point inwards, 2. n u (n l ) arrows on the upper (lower) boundary, such that n u + n l = N − n, also point inwards, 3. The remaining n + N arrows on the upper and lower boundaries point outwards, 4. The locations of the inward-pointing arrows on the upper and lower boundaries, with n u and n l fixed, are summed over.

A brief history of partial domain wall configurations
The configurations considered in this work were first introduced in the work of Bogoliubov, Pronko and Zvonarev in their study of boundary correlation functions in the presence of DWBC's [7], and subsequent works [8][9][10][11]. However, in these works, they were camouflaged by the fact that they were paired with complementary configurations to produce (N ×N ) configurations with conventional DWBC's, and that only a subset of the positions that the inverted n u (n l ) arrows on the upper (lower) boundaries can take were included in the statistical sum, as one does in computations of boundary correlation functions. In [12,13], Escobedo, Gromov, Sever and Vieira studied 3-point functions of three gauge-invariant single-trace length-N i operators, O i (x i ), i ∈ {1, 2, 3}, that are composed of elementary scalars in SU (2) subsectors of N = 4 supersymmetric Yang-Mills theory. 3 Using the connection with quantum integrable models, Escobedo et al. obtained a sum expression for the studied structure constants. In [14], Gromov, Sever and Vieira considered the same structure constants as in [12,13] in the special case where O 1 and O 3 , in the conventions of [14], are BPS operators. That is, the rapidity variables {x 1 } and {x 3 } that characterize these two operators are taken to infinity. In this special case, the sum expression of [12] simplifies to a sum expression for a quantity that they refer to as A{x 2 }. arrows on the upper and lower boundaries point inwards, while all other arrows on the upper and lower boundaries point outwards. All these choices could have been reversed. 3 In [12,13], {Oi}, i ∈ {1, 2, 3} are chosen such that their lengths Li satisfy non-extremal length conditions, Li < Lj + L k for any distinct {i, j, k}, and further, they are characterized by rapidity variables {xi}, such that they are non-BPS ({xi} has finitely many elements that are finite, rather than infinite), and have well-defined conformal dimensions (the elements of {xi} satisfy Bethe equations).

JHEP07(2012)186
In [15], the sum expression of [12,13] was evaluated in determinant form. This determinant is essentially Slavnov's determinant expression for the scalar product of a Bethe eigenstate and a generic state in a periodic spin-1 2 XXX chain. In vertex model terms, Slavnov's determinant is the partition function of rational six-vertex configurations with 2n horizontal lines (auxiliary spaces), N vertical lines (quantum spaces), and boundary conditions that are specified in [15,16]. The limit used in [14] to produce A{x 2 } is achieved in six-vertex model terms by simply deleting the n horizontal lines that represent the Bethe eigenstate, as explained in the sequel. The resulting configurations are the partial domain wall configurations discussed in this note.

Outline of contents
In section 2, we recall basic definitions related to the six-vertex model. In section 3, we start from (N×N ) domain wall configurations and delete N − n horizontal lines to obtain (n×N ) partial domain wall configurations. For simplicity we consider the case n u = N − n and n l = 0. Once this case is understood, the general case is straightforward to obtain. The corresponding (n × N ) pDWPF is obtained starting from Izergin's (N × N ) determinant expression for the (N × N ) DWPF of the initial domain wall configuration, taking the rapidity variables of the lines that are deleted to infinity, and normalizing appropriately to obtain the pDWPF Z N×N in (N ×N ) determinant form.
In section 4, we start from the (2n×N ) configurations that describe the scalar product of an n-magnon Bethe eigenstate and an n-magnon generic state, on an N -site periodic spin-1 2 chain. We delete the n horizontal lines that describe the Bethe eigenstate to obtain (n×N ) partial domain wall configurations with n u = N − n and n l = 0. The corresponding (n × N ) pDWPF is obtained starting from Slavnov's (n × n) determinant expression for the scalar product, taking the rapidity variables of the lines that we deleted (which are the Bethe roots) to infinity, and normalizing appropriately to obtain the pDWPF Z n×n in (n×n) determinant form. Z n×n was first derived by Kostov [17][18][19]. Expanding Z n×n , one obtains the sum expression of Gromov et al. [12]. Starting from the trigonometric Slavnov scalar product, we also derive the trigonometric version of Z n×n .
Each determinant, Z N×N and Z n×n , is a function of the set {x} of cardinality n, associated with the n horizontal lines, and the set {y} of cardinality N , associated with the N vertical lines. In section 5, as an independent check of the correctness of our expressions for Z N×N and Z n×n , we show that they can be written as polynomials in each of their variables x i , with the same bound on their degree, and that they satisfy the same recursion relations and initial condition. This proves that they are equal, as expected from the fact that they are different expressions for the same partition function. In section 6, we recall basic facts regarding Casorati determinants (the discrete analogues of Wronskians) and discrete KP τ -functions, then we show that pDWPF's are discrete KP τ -functions in the {x} as well as in the {y} variables.
In section 7, we recall a mapping that Gromov and Vieira use in [20,21] to introduce 1-loop corrections into the 0-loop expressions of certain structure constants in N = 4 supersymmetric Yang-Mills theory, and show that the (n×N ) pDWPF remains a determinant under this mapping. In section 8, we include remarks on recent developments.

JHEP07(2012)186
1.4 Glossary of frequently used notation {x} ({y}) is a set of rapidity variables that do not satisfy Bethe equations and that flow along horizontal (vertical) lines. We always take {x} and {y} to be free variables. {b} is a set of rapidity variables that do satisfy the Bethe equations and that flow along horizontal lines. When a set {x} has cardinality N , we sometimes indicate this by writing {x} N . At times we also use the notation  [16], and the determinant expressions for these objects [5,22]. Finally, we define the partial domain wall configurations and the corresponding partial domain wall partition functions.

Lines, orientations and rapidity variables
Consider a square lattice with n horizontal lines and N vertical lines that intersect at (n×N ) points, n N . We order the horizontal lines from bottom to top and assign the i-th line an orientation from left to right and a rapidity variable x i . We order the vertical lines from left to right and assign the j-th line an orientation from bottom to top and a rapidity variable y j . See figure 1. The orientations that we assign to the lattice lines are matters of convention and are meant to make the vertices of the six-vertex model, that we introduce shortly, unambiguous.

Segments, arrows and vertices
Each lattice line is divided into segments by all other lines that are perpendicular to it. Bulk segments are attached to two intersection points. Boundary segments are attached to one intersection point only. Assign each segment an arrow that can point in either direction, and define the vertex v ij as the union of the intersection point of the i-th horizontal line and the j-th vertical line, the four line segments attached to this intersection point, and the arrows on these segments.

Weights, configurations and partition functions
Assign every vertex v ij a weight w ij that depends on the specific orientations of its arrows, and the rapidities x i and y j that flow through it. Any lattice configuration with a definite assignment of arrows is assigned a weight equal to the product of the weights of its vertices. The partition function of the lattice in figure 1 is the sum of the weights of all lattice configurations which respect the boundary conditions that we impose.

Six vertices that conserve arrow flow
Since every arrow can point in either direction, there are 2 4 = 16 possible (types of) vertices. We are interested in models with 'conservation of arrow flow'. That is, the only vertices with non-zero weights are those such that the number of arrows that point toward the intersection point of the vertex is equal to the number of arrows that point away from it. These are six such vertices shown in figure 2. The remaining vertices have zero weights.
In this work, we study the rational and the trigonometric six-vertex model. The former is a special case of the latter. For the rational six-vertex model, we use the weights For the trigonometric six-vertex model, we use the weights where [x] ≡ sinh(x). The weights in equations (2.1) and (2.2) satisfy the Yang-Baxter equations and unitarity. The parametrization in the trigonometric case is not unique. The reason for using the parametrization in equation (2.2) is explained in subsection 3.6.

Limiting form of the weights
From equations (2.1) and (2.2), as x → ∞, the rational weights become while the trigonometric weights become

The domain wall partition function, DWPF
This standard object is defined in six-vertex model terms as the partition function of the configurations in figure 3, [4,6]. It depends on two sets of variables {x} N = {x 1 , . . . , x N } and {y} N = {y 1 , . . . , y N }, and we denote it by Z({x} N |{y} N ).

Izergin's determinant
Following [5], in the rational parametrization of equation (2.1) the DWPF is given by In the trigonometric parametrization of equation (2.2), the DWPF is given by where we use the notation |x| = N k=1 x k .

The scalar product
This is another standard object that is defined in this work in six-vertex model terms 4 as the partition function of the configuration in figure 4, [16,22,23]. It depends on three sets of variables {x} n = {x 1 , . . . , x n }, {b} n = {b 1 , . . . , b n }, {y} N = {y 1 , . . . , y N }, where n N . We denote it by S({x} n , {b} n |{y} N ).

Slavnov's determinant
Following [22] we assume that one set of variables in figure 4, {b} n , obeys the Bethe equations. 5 In the rational and trigonometric parametrizations they are given by JHEP07(2012)186 respectively. Assuming that the Bethe equations hold, in the rational parametrization the scalar product has the determinant representation In the trigonometric parametrization, it is given by

The partial domain wall partition function, pDWPF
Let n be an integer satisfying 1 n N . Consider the partition function generated by deleting the top (N − n) rows from the lattice in figure 3, or the top n rows from the lattice in figure 4, and whose top boundary is summed over all arrow configurations. We denote these objects by Z 1 ({x} n |{y} N ) and Z 2 ({x} n |{y} N ) respectively, and represent them by the lattices in figure 5.
We emphasize that, unlike the usual domain wall configurations, the top boundary segments in figure 5 are not fixed to definite arrow configurations but are summed over just as the bulk segments.
As we will see in subsection 3.2, up to a numerical coefficient, Z 1 ({x} n |{y} N ) is the leading term in Z({x} N |{y} N ) as x N , . . . , x n+1 → ∞. In this limit, the contribution from JHEP07(2012)186 One can also calculate the pDWPF Z 2 ({x} n |{y} N ) as the leading term of the scalar product S({x} n , {b} n |{y} N ) as b n , . . . , b 1 → ∞. In this limit, the contribution from the top n rows of figure 4 becomes trivial, and we are left with the lattice shown in figure 5. This is discussed in subsection 4.1.
In this paper we calculate Z 1 ({x} n |{y} N ) and Z 2 ({x} n |{y} N ) by taking the two limits described above. The starting points for these calculations are, respectively, Izergin's determinant formula for Z({x} N |{y} N ) and Slavnov's determinant formula for S({x} n , {b} n |{y} N ). In the case of the rational six-vertex model, whose vertex weights are invariant under the reversal of all arrows, the two quantities Z 1 ({x} n |{y} N ) and Z 2 ({x} n |{y} N ) are in fact equal (which is easily verified by comparing the two lattices in figure 5). Therefore in the rational parametrization we obtain two different determinant expressions for the same object.

Deleting lines from opposite boundaries
By symmetry of Z({x} N |{y} N ) in the variables {x} N , we are free to distribute the rapidities x N , . . . , x n+1 over the horizontal lines of the lattice in any way we wish, prior to taking the limit x N , . . . , x n+1 → ∞.
For example, we can choose to place the variables x N , . . . , x m+1 on the lowest lines of the lattice and x m , . . . , x n+1 on the highest, where m is some integer satisfying n m N . In the limit x N , . . . , x n+1 → ∞, the bottom (N − m) and the top (m − n) rows become trivial, and we obtain the lattice shown in figure 6. The lattice sum in figure 6 is equal to the one on the left of figure 5, up to multiplication by an overall factor. In the rational six-vertex model, this factor is the binomial coefficient N −n N −m .

JHEP07(2012)186
3 Domain wall partition function in the infinite-rapidity limit In this section, we obtain a determinant expression for the pDWPF starting from Izergin's formula for the DWPF, equation (2.5), and taking appropriate limits. The (N × N ) determinant that we obtain is 'hybrid' in the sense that it contains n rows of the type in Izergin's formula, and (N − n) rows of Vandermonde determinant-type.

One rapidity becomes infinite
Consider the rational DWPF. Due to the domain wall boundary conditions, and the conservation of arrow flow, the top row of the lattice always contains precisely one c + vertex, while all remaining vertices in that row are of the type a + or b + . Using the asymptotic behaviour of the vertex weights in equation (2.3) and the definition of it is easy to see that Consider the lattice representation of the pDWPF Z 1 ({x} i |{y} N ), for some 1 i N . The top boundary of this lattice consists of a sum over N i possible arrow configurations. From now on, we make remarks which apply to the internal part of this lattice, assuming that the boundary is fixed to any one of these N i configurations. Consider the x i -row of vertices. 6 Any configuration that this row takes must contain at least one c + vertex and m pairs of {c + , c − } vertices, m = 0, 1, 2, · · · In the large x i limit, the leading contribution to Z 1 ({x} i |{y} N ) corresponds to m = 0. Taking multiple counting into consideration, we obtain Iterating this result through i = {N, . . . , n + 1} we obtain where the limits are to be taken sequentially, starting with x N . 6 All vertices through which the xi rapidity variable flows.

Limit of Izergin's determinant as one rapidity becomes infinite
Starting from the expression in equation (2.5) for Z 1 ({x} N |{y} N ), it is simple to take the limit specified in equation (3.2). Absorbing the factor where for all 1 i N we have defined the function and setȳ j = y j − 1 for convenience. In the limit, every entry of the final row goes to 1, hence is the i-th complete symmetric function in two variables y j ,ȳ j , given by the generating series Proof. Let P N −n denote the proposition that equation (3.9) is true. Based on equation (3.7) for the one-rapidity case, we see that P 1 is true. Let us assume that P N −n is true and show that this implies P N −n+1 . Multiplying equation (3.9) by x n , making the change of variables x n = 1/ǫ and taking the limit ǫ → 0 gives Consider the functions f (n) j (ǫ) in the n-th row of the above determinant, which is defined in equation (3.6). Since ǫ is small, they can be expanded in powers of ǫ using the definition of the elementary and complete symmetric functions where e k (y 1 , . . . , y N ) and h k (x 1 , . . . , x n−1 ) are elementary and complete symmetric functions, given respectively by the generating functions Using the series expression in equation (3.11) for the row of entries f since they are linear combinations of the lower (N − n) rows, and the first sum starts at k = (N − n). Taking the limit ǫ → 0, all higher order terms in this sum vanish.
Studying the k = (N − n) term in the series in equation (3.11), it is clear that many of its sub-terms do not contribute to the determinant either. In fact, only the sub-term corresponding to l = m = (N − n) survives, and this is identically h N −n (y j ,ȳ j ). Therefore we obtain which proves P N −n+1 . This completes the proof of equation (3.9) for all 0 n N − 1, by induction.

A 'partial Vandermonde' way to write the determinant
Using row operations to cancel all terms of lower order and extracting an overall factor of (N −n)! from the determinant in equation (3.9), we obtain Equation (3.14) is our (N × N ) determinant expression for the pDWPF. As previously mentioned, the top n rows are of Izergin determinant-type, whereas the lower (N − n) rows are of Vandermonde determinant-type.

Towards the trigonometric pDWPF
Using the asymptotic behaviour of the trigonometric weights given in equation (2.4), we can repeat the procedure of subsection 3.2 to derive the relation between trigonometric partial domain wall partition functions. Iterating this result through i = {N, . . . , n + 1}, we obtain We omit the details since they are similar to those in the rational case.

Slavnov scalar product in the infinite-rapidity limit
In this section we obtain an alternative expression for the pDWPF, by starting from Slavnov's formula for the scalar product (equation (2.8)) and taking appropriate limits. The resulting expression is an (n×n) determinant.

n rapidities become infinite
Using the six-vertex model representation of the scalar product, figure 4, it is possible calculate the partial domain wall partition function in the alternative way where the limits are sequentially, starting with b n . The argument which underlies equation (4.1) is the same as the one that underlies equation (3.4).

The infinite rapidity limit of Slavnov's determinant
To obtain an alternative determinant expression for the pDWPF, we start from Slavnov's determinant in equation (2.8) and perform the limits specified in equation (4.1). We do this using induction, by proving the following result.
Proof. Let equation (4.2) be a proposition, P n−m . We begin with the proof of P 1 . Using equation (2.8) for the scalar product, we have In the limit being considered, we have t which proves P 1 . Now we assume that P n−m is true and show that this implies P n−m+1 . Taking equation (4.2) as our starting point, we find that lim bn,...,bm→∞ where ǫ is small in view of the limit being taken. Then we can write where we abbreviate det(·) 1 i n by | · |, and where the first (m − 1) columns are as before, so we do not write them. Now we can move all terms in the prefactor which depend on b m inside the m-th column, and take the limit. This gives us lim bn,...,bm→∞

Kostov's determinant
where we have used t (0) i ≡ 1. Combining equations (4.1) and (4.12), we see that the partial domain wall partition function can be written as where we have simultaneously reversed the order of variables in the Vandermonde and the order of columns in the determinant. In contrast to the determinant in equation (3.14), which is (N ×N ), the determinant in equation (4.13) is (n×n). equation (4.13) is due to I Kostov [17][18][19].

Writing Z 2 ({x} n |{y} N ) as a sum over partitions
To conclude the section, we show that the determinant in equation (4.13) can be expanded as a certain sum, which is the precise form in which it appears in [14]. Let us define the functions Then we have

The trigonometric version of Kostov's determinant
For completeness, we give the trigonometric version of the pDWPF obtained taking limits of Slavnov's scalar product in equation (2.9). The starting point in the calculation is the relation which arises from the lattice version of the scalar product in figure 4 and that of the pDWPF on the right of figure 5, and the asymptotic behaviour of the weights in equation (2.4).

JHEP07(2012)186
Repeating the ideas already developed in this section and working from Slavnov's determinant in equation (2.9), equation (4.18) ultimately leads us to the expression for the trigonometric pDWPF.

Equivalence of determinants
In this section we show directly that the determinants in equations (3.14) and (4.13) are equal. We do this for completeness and as a check of our limit calculations, because on the surface it is not apparent that the two expressions coincide. We restrict our attention to the determinants obtained from the rational six-vertex model, because as we have already mentioned the two lattice sums in figure 5 are not equivalent in the trigonometric parametrization.
Our approach is to convert both equations (3.14) and (4.13) to polynomials, by multiplying them by an overall factor (precisely the factor present in the denominator of the b and c weights). We distinguish the resulting expressions by calling them Z N×N ({x} n |{y} N ) and Z n×n ({x} n |{y} N ), in reference to the size of the determinants in question, and show that both Z N×N ({x} n |{y} N ) and Z n×n ({x} n |{y} N ) satisfy a list of conditions which the pDWPF must itself obey. This proves that they are equal, since the conditions only admit a unique solution.

Set of properties which characterize the pDWPF
Let Z({x} n |{y} N ) be the partition function of either of the lattices in figure 5, but whose weights are given by This polynomial version of Z({x} n |{y} N ) is obtained from the rational version by multiplying by n i=1 N j=1 (x i − y j + 1). Following Korepin [4], one can show that A. Z({x} n |{y} N ) is a polynomial in x n of degree bounded by (2N − 1).

JHEP07(2012)186
D. Z(x 1 |{y} N ) is known explicitly for all N 1, and is given by The second equality in equation (5.4) evaluates the sum over l, and can be established by induction on N .
These four properties uniquely determine the functions Z({x} n |{y} N ), for all 1 n N . This is because, from A, Z({x} n |{y} N ) is a polynomial in x n , and from B and C, it is known at more points than its degree.

Z N×N ({x} n |{y} N ) satisfies the four properties
Consider the polynomial version of the pDWPF in equation (3.14), obtained by multiplying by n i=1 N j=1 (x i − y j + 1). Let us denote this by Z N×N ({x} n |{y} N ). Usingȳ = y − 1, it is given by . . . . . .
Combining results, we see that property C is satisfied. Finally, when n = 1, we have Laplace expanding this determinant along the first row and using the Vandermonde determinant identity, we obtain Comparing this polynomial in x 1 with the polynomial in equation (5.4) at the points x 1 = {y 1 ,ȳ 1 , . . . , y N ,ȳ N }, we find that they are equal, and property D is satisfied.

Z n×n ({x} n |{y} N ) satisfies the four properties
Consider the polynomial version of the pDWPF in equation (4.13), obtained by multiplying the expression in that equation by n i=1 N j=1 (x i − y j + 1). We denote this by Z n×n ({x} n |{y} N ). Usingx = x + 1,ȳ = y − 1, it is given by Clearly Z n×n ({x} n |{y} N ) is a polynomial in x n , and the highest possible degree it can obtain in this variable is N . So property A is satisfied. Since all {y} N dependence is in the products N k=1 (x i −ȳ k ) and N k=1 (x i − y k ), property B is satisfied. Setting x n = {y N ,ȳ N }, the entries of the final row of the determinant in equation (5.10) become

JHEP07(2012)186
Rearranging the entries of the top (n − 1) rows to write them as Extracting factors which are common to each row of the determinant, we obtain Subtracting (column j + 1)/y N from (column j) for all 1 j < n, it is easy to show that Substituting this into equation (5.13), we have verified that property C is satisfied. Finally, when n = 1, observe that Z n×n (x 1 |{y} N ) is identically equal to the right hand side of equation ( 1 , x 2 , x 3 , x

Useful identities for h i {x}
From equation (6.1), it follows that From equation (6.2), one obtains

Discrete derivatives
The discrete derivative ∆ m h i {x} of h i {x} with respect to x m ∈ {x} is defined using equation (6.2) as Note that by applying ∆ m to a degree i complete symmetric function, h i {x}, one obtains a complete symmetric function h i−1 {x} of degree i − 1, in the same set of variables {x}.

The discrete KP hierarchy
Discrete KP is an infinite hierarchy of integrable partial difference equations in an infinite set of continuous Miwa variables {x 1 , x 2 , . . . } with multiplicities {m 1 , m 2 , . . . }. Time evolution is obtained by changing the multiplicities of the Miwa variables. In this work, we take the number of non-zero Miwa variables to be finite, and set all continuous Miwa variables apart from {x 1 , . . . , x N } to zero. In this case, the discrete KP hierarchy can be written in bilinear form as the (n×n) determinant equations where 3 n N , and

JHEP07(2012)186
In words, if τ {x} has m i copies of the variable x i , then τ +i {x} has (m i + 1) copies of x i and the multiplicities of all other variables remain the same, while τ −i {x} has one more copy of each variable except x i . In the simpler notation τ +i {x} = τ {m 1 , . . . , (m i + 1), . . . , m N } (6.8) 1), . . . , m i , . . . , (m N + 1)} the simplest discrete KP bilinear difference equation is where {x i , x j , x k } ∈ {x} and {m i , m j , m k } ∈ {m} are any three continuous Miwa variables and their corresponding multiplicities.

Casoratian matrices and determinants
Ω is a Casoratian matrix if and only if its matrix elements ω ij satisfy where ∆ m is the discrete derivative with respect to any variable x m ∈ {x}. It is redundant to choose a specific variable x m , since ω ij {x} is symmetric in {x}. If Ω is a Casoratian matrix, then det Ω is a Casoratian determinant. Casoratian determinants are discrete analogues of Wronskian determinants.

Notation for column vectors and determinants
We introduce the column vector notation for the corresponding column vector where the multiplicity of the subset of variables x k 1 ,. . . , x kn is increased by 1. We introduce the determinant notation for the determinant with shifted multiplicities.

Identities for Casoratian determinants
Following [24], equations (6.11) and (6.12) can be used to perform column operations in the determinant expressions for τ [1] and τ [1,...,n] , to obtain the two identities 6.9 Casoratian determinants are discrete KP τ -functions Following [25], consider the (2N ×2N ) determinant which is identically zero. For notational clarity, we have used subscripts to label the position of columns of zeros. Laplace expanding the left hand side of equation (6.19) in (N ×N ) minors along the top (N ×2N ) block, we obtain

JHEP07(2012)186
Using the Vandermonde determinant identity with 1 x k · · · x n−2 k denoting the omission of the k-th row of the matrix, we see that equation (6.21) is the cofactor expansion of the determinant in equation (6.6) along its last column. Hence we conclude that Casoratian determinants satisfy the bilinear difference equations of discrete KP.
In this subsection we show that Z N×N ({x} n |{y} N ), in equation (5.5), is a Casoratian determinant. The discrete derivatives are taken with respect to any of the variables y j . From the above discussion, this is sufficient to show that Z N×N ({x} n |{y} N ) is a τ -function of discrete KP in {y} N .
The first step is to rearrange equation (5.5) by bringing the numerator of the prefactor P({x} n , {y} N ) inside the determinant. We do this by multiplying the j-th column of the determinant by n k=1 (x k − y j )(x k −ȳ j ), for all 1 j N . The j-th column of the resulting determinant has entries which are polynomial in y j . After a routine calculation, we obtain where the coefficients c ik {x} depend on the row of the matrix and are given by It remains to take the Vandermonde ∆{−y} N inside the determinant of equation (6.23). This is essentially the same as proving the Jacobi-Trudi identity for Schur functions, see [26]. The final result is Up to the Vandermonde factor in the denominator, which is a constant in {y} N , this is clearly a Casoratian determinant.
6.11 Z n×n ({x} n |{y} N ) is a discrete KP τ -function in {x} n We can repeat the above procedure to write Z n×n ({x} n |{y} N ) as a Casoratian determinant, whose discrete derivatives are with respect to any of the variables x i . Starting from JHEP07(2012)186 equation (5.10), we already have Z n×n ({x} n |{y} N ) as a determinant whose i-th row entries are polynomials in x i . Expanding these polynomials in powers of x i , we obtain where the coefficients d kj {y} depend on the column of the matrix and are given by Taking the Vandermonde ∆{x} n inside the determinant of equation (6.26), we have Hence Z n×n ({x} n |{y} N ) is a Casoratian determinant, and satisfies the discrete KP equations in {x} n .

The Gromov-Vieira polynomial version of partial domain wall partition functions
Following [14], partial domain wall configurations are (the essential part of) 3-point functions of tree-level single-trace operators in the SU(2) sector of SYM 4 , with two BPS and one non-BPS operators. In [20,21], Gromov and Vieira showed that 1-loop corrections can be introduced using the mapping discussed in this section. In the sequel, we show that the determinant form of these objects at tree-level is preserved under the GV mapping, thus the corresponding 1-loop corrected objects in SYM 4 can also be expressed as determinants.

Aim of this section
Our aim is to show that, up to O(g 2 ), the GV mapping acts on a Casoratian determinant Z N×N ({x} n |{y} N ) to return a new determinant. We show this by explicitly evaluating [Z N×N ({x} n |{y} N )] y .

Remarks on notation
Henceforth we reserve i and j for the row and column indices of a determinant, respectively, and assume that they range over all values 1 i, j N . For example, we write the determinant in equation (6.25) as  Acting on equation (7.10) with G 2 and using equation (7.8), we find The first two terms come from the sum N l=1 in equation (7.10), while the final term comes from the sum 1 l 1 <l 2 N . All other terms vanish under the action of G 2 , either because they have the wrong degree or give rise to a determinant with two equivalent rows.
Combining the first and third determinant in equation (7.13), which are the same up to the ordering of their rows, we obtain  where the column index j ranges over all values 1 j N , as usual. For the second part of the GV mapping in equation (7.1), we wish to calculate Putting together the results of the previous subsections, namely equations (7.12) and (7.14), we find that       The result in equation (7.19) is such a simple modification of the original expression, obtained by setting g 2 → 0, that we expect that higher derivative versions of the GV mapping will also preserve the determinant form of the pDWPF. 7 Since the GV mapping is an expansion around the homogeneous limit at which all variables y i = 0, we cannot consider the determinant in equation (7.19) to be a discrete KP τ -function in the {y} variables. On the other hand, according to the methods of section 7, the determinant in equation (7.19) is not in Casorati form, hence we cannot conclude that it is a discrete KP τ -function in the {x} variables.

Summary of results
Rational and trigonometric partial domain wall partition functions, pDWPF's, are partition functions of six-vertex model configurations on lattices with unequal numbers of horizontal lines L h and vertical lines L v . They can be regarded as less restrictive variations on Korepin's rational and trigonometric domain wall partition functions, DWPF's, which require L h = L v , but can be deduced from them, as well as from configurations that describe scalar products, by taking some of the rapidities to infinity.
In this work, we gave explicit derivations of the determinant expressions for pDWPF's as limits of Izergin's DWPF determinant, as well as of Slavnov's determinant for the scalar product of a Bethe eigenstate and a generic state, in the rational and trigonometric cases, 7 An earlier draft of this work contained an incorrect version of equation (7.3) that led to a more complicated version of equation (7.19). We thank D Serban for pointing this out.

JHEP07(2012)186
and studied some of their properties. The rational pDWPF was first derived from Slavnov's determinant by I Kostov [17]. We showed how the two determinants obtained as limits of Izergin's determinant and of Slavnov's determinant are different (one is (N ×N ) while the other is (n×n), where n < N ), but can be directly related, that they are KP τ -functions in each of two sets of variables, and that they remain determinants under the mapping of Gromov and Vieira. 8

Taking the free variables to infinity in Slavnov's determinant
In section 5, following Kostov [17], we derived pDWPF's from Slavnov's scalar products. We kept the free rapidity variables {x} finite, and took the rapidity variables that satisfy Bethe equations, {b}, to infinity. The result is finite and non-trivial.
If we would have kept the Bethe roots {b} finite and took {x} to infinity, the result would have been zero. The reason is that this limit corresponds to the scalar product of a Bethe eigenstate, labeled by {b}, and a descendant of the reference state (the result of the action of spin-lowering operators on the reference state, that lower the net spin but do not introduce Bethe roots [12]). Since the scalar product of the Bethe eigenstate |{b} and the reference state vanishes, the scalar product of |{b} with a descendant of the reference state also vanishes. In other words, a pDWPF with auxiliary space (horizontal line) rapidities that obey Bethe equations vanishes.

Asymptotics
In [14,[17][18][19], pDWPF's were studied in the thermodynamic limit L v → ∞, such that the ratios L h /L v and x i /L v , i ∈ {1, . . . , L h }, remain finite, where L v (L h ) is the number of vertical (horizontal) lattice lines, L h < L v , and {x} are the rapidities of the horizontal lines. 9 While, strictly speaking, the variables {x} are free, in applications, such as computations of 3-point functions of two BPS and one non-BPS operators in the scalar sector of SYM 4 , they are restricted to obey the Bethe equations of a spin chain of length L, such that L = L v . For that reason, Bethe Ansatz asymptotics apply, but the pDWPF is nonetheless non-vanishing. This is the set-up used in [14,18,19].
Following [14,18], in the above thermodynamic limit, the variables {x}, which are solutions of Bethe equations of a spin chain of length L > L v , L ∼ L v , condense on a set of contours Γ = k Γ k , with linear density ρ{x}, ρ ∼ O(1), x i ∼ O(L v ). In the homogeneous limit, y i = 0, i ∈ {1, . . . , L v }, the asymptotic pDWPF can be expressed as an exponential of a contour integral over a dilogarithm function z n n 2 (8.1) 8 It is likely that the determinant expression is preserved under the action of higher derivative versions of the GV mapping. We did not pursue this since, at this stage, the relation between the higher derivative versions and the inclusion of higher loop corrections to the 3-point functions is not clear. However, in [27], D Serban argued that this is indeed the case, at least in the limit Li → ∞, i ∈ {1, 2, 3}. That is, when all three operators are represented by asymptotically long spin chain states. 9 Because of the condition that xi/Lv, i ∈ {1, . . . , L h }, remains finite, this limit is also known as the 'Sutherland limit' [28,29].