Dimers for Type D Relativistic Toda Model

The integrability of relativistic Toda lattice (RTL) is characterized by the $R$-matrix $R_{a_{i},a_{j}}:V_{a_{i}}\otimes V_{a_{j}}\to V_{a_{i}}\otimes V_{a_{j}}$, which satisfies the Yang-Baxter equation:

The $2\times 2$ Lax matrix is a special case of this $R$-matrix with the choice of:

defined for the $n$-th particle. $\mathcal{H}_{n}$ is the Hilbert space of a particle, and $V_{\rm aux}=\mathbb{C}^{2}$ is called the auxiliary space. On each of the lattice site we define a $2\times 2$ Lax operator as a $GL_{2}$-valued function Kuznetsov:1994ur ; Iorgov:2007ks ; deVega:1993xi

where $\mathtt{p}_{n}=-\hbar\partial_{\mathtt{q}_{n}}$ and $\mathtt{q}_{n}$ are the canonically conjugated momentum and coordinate of the $n$-th particle. The $R$-matrix is given by

The commutation relations between two matrix components of the Lax operator is governed by the Yang-Baxter RLL-relation (train track relation)

which can be verified true by direct computation.

The monodromy matrix $\mathbf{T}(x)$ of the type A relativistic Toda lattice (with periodic boundary conditions) is an ordered product of the Lax matrices across $N$ particles:

It is evident that the monodromy matrix $\mathbf{T}(x)$ satisfies the same Yang-Baxter equation as the Lax operator:

The spectral curve of the integrable system is defined by introducing the spectral parameter $Y$:

E. Sklyanin points out that the monodromy matrix for type BCD relativistic Toda lattices (RTLs) can be obtained by introducing a reflection matrix as a boundary condition sklyanin1988boundary ; kuznetsov1992infinite . The transfer matrix of an RTL with boundary conditions is given by:

where $\mathbf{t}(x)\in\text{End}\left(\bigotimes_{n=1}^{N}\mathcal{E}\otimes V_{\text{aux}}\right)$ is the monodromy matrix of the type A RTL, and $K_{\pm}(x)\in\text{End}(V_{\text{aux}})$ are the reflection matrices obeying the reflection equations:

and

Given a simple solution $K_{\pm}(x)$ to the reflection equation, one can verify that:

satisfy the same reflection equation if $\mathbf{T}_{\pm}(x)$ satisfy the Yang-Baxter equation (11). The transfer matrix (trace of monodromy matrix) is the generating function of the integral of motion:

The reflection matrices $K_{\pm}$ that satisfy the reflection equation are given by:

These reflection matrices lead to the following (first) Hamiltonian:

The boundary reflection matrices $K_{\pm}$ for the RTL of type D, as discussed in Kuznetsov:1994ur , are given by:

The monodromy matrix $\mathbf{T}(x)$ of $\hat{D}_{N}$ RTL is defined by incorporating both the reflection matrices and the Lax operators. It is given by:

where the $\tilde{L}_{n}(x)$ operators are defined as:

These $\tilde{L}_{n}(x)$ operators are related to the original Lax operators $L_{n}(x)$ by

The construction of $\mathbf{T}(x)$ involves alternating beteween the reflection matrices $K_{+}(x)$ and $K_{-}(x)$ with the Lax operators and their inverses, reflecting the boundary conditions and the periodic structure of the $\hat{D}_{N}$ RTL.

The spectral curve of classical integrable system is defined by the characteristic polynomial of the monodromy matrix $\mathbf{T}(x)$. For RTL, this is given by ¹¹1The determinant is replaced by q-determinant when quantizing the integrable system.

To find the spectral curve, we first compute the determinant of the monodromy matrix $\mathbf{T}(x)$. It is given by the product of the determinants of its individual building blocks:

Next, we scale $Y\to Y{g}^{2N-4}(X-X^{-1})^{2}$ to simplify the expression

where $T(X)$ denotes the trace term related to $\mathbf{T}(X)$. This equation represents the spectral curve associated with the relativistic Toda lattice in the given parameterization.

The Type D relativistic Toda provides insights to the description of Seiberg-Witten ansatz for $SO(2N)$ pure gauge theory Martinec:1995by . For 5d $\mathcal{N}=1$ $SO(2N)$ pure super Yang-Mills (SYM) theory, the Seiberg-Witten curve is given by

To visualize the corresponding toric diagram $\mathcal{T}$, one can refer to the D-brane and O-plane construction. The toric diagram encapsulates the geometric data relevant for the gauge theory. For an illustration of this toric diagram in the context of $SO(8)$, see Fig. 1.

The relation between the toric diagram and dimer graph works in both directions. A dimer graph $\Gamma=(B,W,E)$ can be constructed based on a given toric diagram $\mathcal{T}$ Eager:2011dp ; Hanany:2005ve . In some cases multiple dimer graphs can be constructed based on a single toric diagram Lee:2023wbf .

We use the following two facts to construct the dimer model for type D RTL:

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  Seiberg-Witten curve of supersymmetric Yang-Mills theory of gauge group $G$ coincides with spectral curve of RTL of Lie algebra $\mathfrak{g}=\text{Lie}(G)$.

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  $SO(2N)$ gauge theory shares the same toric diagram with $SU(2N)$ gauge theory with 8 fundamental matters fine-tuned and can be embedded into $SU(2N)$ theory with folding to reduce the degree of freedom.

The dimer graph associated to the toric diagram for $SU(2N)+8F$ can be constructed by introducing impurity in the $Y^{2N-4,0}$ model (corresponding to the $A_{2N-4}$ RTL). This is indicated by the slopes of the four long edges of the toric diagram. The $D_{N}$ dimer can then be obtained through folding, which eliminates half of the degrees of freedom.

The 8 short vertical lines in the toric diagram $\mathcal{T}$ introduce impurity to the square dimer in the following way: When $N\geq 4$, we pick $4$ among $2N-4$ squares in the unit cell the $n_{i}$-th square, $i=1,2,3,4$, $n_{i}\in\{1,\dots,2N-4\}$, to introduce impurity. Two pairs of black and white nodes, which we will denote by $\tilde{\mathtt{w}}_{n_{i}}^{(1,2)}$ and $\tilde{\mathtt{b}}_{n_{i}}^{(1,2)}$ are added in the middle of the $n_{i}$-th square in the order white-black-white-black starting from the left of the unit cell. The vertical edges of $n_{i}$-th squares are removed and replaced by

• •

  Horizontal lines connecting the newly introduced black and white nodes, which also extended to outside of the the unit cell.

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  Two vertical lines connecting the newly introduced black nodes $\mathtt{b}_{n_{i}}^{(l)}$, $l=1,2$ to the square’s white nodes $\mathtt{w}_{n_{i}-\frac{1}{2}+\frac{(-1)^{n_{i}}}{2}}$.

• •

  Two slide edges connecting $\mathtt{w}_{n_{i}}^{l}$, $l=1,2$, to $\mathtt{b}_{n_{i}}$ and $\mathtt{b}_{n_{i}-1}$.

See Fig. 4 for illustration.

Alternatively, one can place double impurity to a single square in the $A_{2N-4}$ dimer. Four pairs of white and black nodes are added to the middle of the square. See Fig. 5 for illustration.

The final dimer graph will have exactly $2N-4+4\times 2=2N+4$ pairs of black and white nodes distributed on exactly $2N$ rows.

The dimer graph for $D_{4}$ system can be constructed by introducing impurities into the $A_{4}$ dimer graph. The impurity can be introduced in the form of 4 single impurities or as 2 double impurities. Figure 6 illustrates these two different bipartite graphs. More general cases can also be considered, which we will discuss in future work.

The Kastelyne matrix $K_{4,D}$ of the double impurity bipartite graph in Fig. 6 is:

In particular, when setting all parameter (beside $X$ and $Y$) to 1 gives

We choose the reference perfect matching such that each dimer covers $\mathtt{w}_{i}\to\mathtt{b}_{i}$ with all arrows going rightward. The number of 1-loops is the coefficient of $-X^{3}$ in (32): 36. These 1-loops are distributed across the 8 horizontal lines:

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  Lines with $\mathtt{w}_{1}$ and $\mathtt{w}_{7}$: 7 in total;

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  Lines with $\mathtt{w}_{3}$ and $\mathtt{w}_{9}$: 7 in total;

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  Lines with $\mathtt{w}_{5}$ and $\mathtt{w}_{b}$: 3 in total;

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  Lines with $\mathtt{w}_{6}$ and $\mathtt{w}_{c}$: 1 in total;

See Fig. 7 for illustration of the 1-loops. Most of the 1-loops can be built from straight lines $d_{j}$, $j=1,2,3,4$, and square loops $S_{n}$, $n=1,\dots,7$. See Fig. 8 for an illustration.

The double impurity dimer is constructed from the ${A}_{7}+8F$ toric diagram Fig.1. The masses $m_{f}$, $f=1,\dots,8$ are fine-tuned to

The first Hamiltonian for the double impurities – the dimer graph on the left of Fig. 6 — is the sum over all 1-loops:

The seventh Hamiltonian is the sum over all 7-loops:

To fold from $SU(8)+8F$ to $SO(8)$, we need to eliminate half of the degree of freedom to make $H_{1}=H_{7}$. This can be done by setting