(0, -6)–(0, -8); (1, -6)–(1, -8); (2, -6)–(2, -8); (3, -6)–(3, -8); (4, -6)–(4, -8); (5, -6)–(5, -7);

(0, -9)–(9, -9); (0, -10)–(9, -10); (0, -11)–(8, -11); (0, -12)–(4, -12); (0, -13)–(1, -13);
(0, -9)–(0, -13); (1, -9)–(1, -13); (2, -9)–(2, -12); (3, -9)–(3, -12); (4, -9)–(4, -12); (5, -9)–(5, -11); (6, -9)–(6, -11); (7, -9)–(7, -11); (8, -9)–(8, -11); (9, -9)–(9, -10);

The (a) BCT and (b) Young diagrams associated to the fixed point of Figure <ref>. Our (nonstandard) convention is to draw the Young diagrams aligned on the right with longer rows on top.

Note that these Young diagrams can also be realized by taking the vertices in the left side of each butterfly, realigning them at the top, and replacing them with boxes. Under this realization, butterfly surgeries and botched butterfly surgeries can be realized as surgery operations on Young diagrams. For $j\neq j'$, let $\site$ be a set of boxes of $\lambda^{(j)}$ with the property that if a box is in $\site$, then any box below it is also in $\site$. Suppose deleting $\site$ from $\lambda^{(j)}$ leaves a Young diagram with distinct parts, and stacking those boxes below $\lambda^{(j')}$ results in a Young diagram with distinct parts. Then, call the operation on the set of Young diagrams $\{\lambda^{(j)}\ |\ 1\leq j \leq m\}$ that deletes $\site$ from $\lambda^{(j)}$ and stacks it below $\lambda^{(j')}$ a “Young diagram surgery". If $\site$ does not contain any boxes in the rightmost column, then the Young diagram surgery corresponds to a butterfly surgery. Otherwise, if $\site$ contains at least one box of the rightmost column, then the Young diagram surgery corresponds to a botched butterfly surgery. The constraint (<ref>) is reformulated as

\begin{equation} \label{eqn:box_constraint}
\site\text{ contains at least }c_j - c_{j'} + 1\text{ boxes of the rightmost column.}
\end{equation}

This is simply a reformulation of Section <ref> and Section <ref>. By Lemma <ref> and Lemma <ref>, Young diagram surgeries satisfying the additional constraint (<ref>) whenever a box in the rightmost column is moved correspond to pencils of invariant curves. Given a box $\bx$ in a Young diagram $\lambda$, define its height $y(\bx)$ by $y(\bx) = 1 - i$, where $\bx$ belongs to the $i$th row of $\lambda$. The tangent weight of the invariant curve is given by comparing the heights of any box in $\site$ before and after the surgery. Namely, the exponent of $\hb$ is $-\Delta_\surgery y = y(\bx)-y(\bx')$, where $\bx$ is any box in $\site$ and $\bx'$ is the result of applying $\surgery$ to $\bx$. Treating the Young diagram as a subset of $\RR^2$ and each box as a closed square, the number of connected components of $\site$ is precisely the number of connected components of the corresponding (botched) butterfly surgery. In Figure <ref>, we reformulate Figure <ref> in the language of Young diagram surgeries.

Let $\brane$ be a separated brane diagram and $p\in\bowvar(\brane)^\torus$. Let $\surgery$ be a surgery on the corresponding Young diagrams that moves boxes from $\lambda^{(j)}$ to $\lambda^{(j')}$. If $j<j'$ and the site $\site$ contains at least one box of the rightmost column, then impose the additional constraint that $\site$ contains at least $c_j-c_{j'}+1$ boxes of the rightmost column, where $c=(c_1,...,c_m)$ is the margin vector of D5 brane charges. Then, there is a $k$-dimensional pencil $\pen_\surgery$ of $\torus$-invariant curves containing $p$ with tangent weight

\[
\frac{\ub_j}{\ub_{j'}}\hb^{-\Delta_\surgery y},
\]

where $k$ is the number of connected components of $\site$.

$w = \frac{\ub_1}{\ub_3}\hb$

[scale=0.31, xscale=-1]

(0, -2)–(10, -2); (0, -3)–(10, -3); (0, -4)–(4, -4); (0, -5)–(1, -5);
(0, -2)–(0, -5); (1, -2)–(1, -5); (2, -2)–(2, -4); (3, -2)–(3, -4); (4, -2)–(4, -4); (5, -2)–(5, -3); (6, -2)–(6, -3); (7, -2)–(7, -3); (8, -2)–(8, -3); (9, -2)–(9, -3); (10, -2)–(10, -3);

(0, -6)–(5, -6); (0, -7)–(5, -7); (0, -8)–(4, -8);
(0, -6)–(0, -8); (1, -6)–(1, -8); (2, -6)–(2, -8); (3, -6)–(3, -8); (4, -6)–(4, -8); (5, -6)–(5, -7);

(0, -9)–(9, -9); (0, -10)–(9, -10); (0, -11)–(8, -11); (0, -12)–(4, -12); (0, -13)–(1, -13);
(0, -9)–(0, -13); (1, -9)–(1, -13); (2, -9)–(2, -12); (3, -9)–(3, -12); (4, -9)–(4, -12); (5, -9)–(5, -11); (6, -9)–(6, -11); (7, -9)–(7, -11); (8, -9)–(8, -11); (9, -9)–(9, -10);

[fill=gray] (4, -1) rectangle (5, -2); [fill=gray] (5, -1) rectangle (6, -2); [fill=gray] (6, -1) rectangle (7, -2);

[fill=gray] (1, -2) rectangle (2, -3); [fill=gray] (2, -2) rectangle (3, -3); [fill=gray] (3, -2) rectangle (4, -3);
[fill=gray] (4, -2) rectangle (5, -3); [fill=gray] (5, -2) rectangle (6, -3);

[fill=gray] (0, -3) rectangle (1, -4); [fill=gray] (1, -3) rectangle (2, -4); [fill=gray] (2, -3) rectangle (3, -4);
[fill=gray] (3, -3) rectangle (4, -4); [fill=gray] (4, -3) rectangle (5, -4);

[fill=gray] (0, -4) rectangle (1, -5); [fill=gray] (1, -4) rectangle (2, -5);

[->, thick] (13,-6.5)–(11,-6.5);

[xshift = 14cm, yshift = -2cm]
(0, 0)–(10, 0); (0, -1)–(10, -1); (0, -2)–(7, -2); (0, -3)–(6, -3); (0, -4)–(5, -4); (0, -5)–(2, -5);
(0, 0)–(0, -5); (1, 0)–(1, -5); (2, 0)–(2, -5); (3, 0)–(3, -4); (4, 0)–(4, -4); (5, 0)–(5, -4); (6, 0)–(6, -3); (7, 0)–(7, -2); (8, 0)–(8, -1); (9, 0)–(9, -1); (10, 0)–(10, -1);

(0, -6)–(5, -6); (0, -7)–(5, -7); (0, -8)–(4, -8);
(0, -6)–(0, -8); (1, -6)–(1, -8); (2, -6)–(2, -8); (3, -6)–(3, -8); (4, -6)–(4, -8); (5, -6)–(5, -7);

(0, -9)–(9, -9); (0, -10)–(9, -10); (0, -11)–(8, -11); (0, -12)–(4, -12); (0, -13)–(1, -13);
(0, -9)–(0, -13); (1, -9)–(1, -13); (2, -9)–(2, -12); (3, -9)–(3, -12); (4, -9)–(4, -12); (5, -9)–(5, -11); (6, -9)–(6, -11); (7, -9)–(7, -11); (8, -9)–(8, -11); (9, -9)–(9, -10);

[fill=gray] (4, -1) rectangle (5, -2); [fill=gray] (5, -1) rectangle (6, -2); [fill=gray] (6, -1) rectangle (7, -2);

[fill=gray] (1, -2) rectangle (2, -3); [fill=gray] (2, -2) rectangle (3, -3); [fill=gray] (3, -2) rectangle (4, -3);
[fill=gray] (4, -2) rectangle (5, -3); [fill=gray] (5, -2) rectangle (6, -3);

[fill=gray] (0, -3) rectangle (1, -4); [fill=gray] (1, -3) rectangle (2, -4); [fill=gray] (2, -3) rectangle (3, -4);
[fill=gray] (3, -3) rectangle (4, -4); [fill=gray] (4, -3) rectangle (5, -4);

[fill=gray] (0, -4) rectangle (1, -5); [fill=gray] (1, -4) rectangle (2, -5);

$w = \frac{\ub_3}{\ub_2}$

[scale=0.31, xscale=-1]

(0, 0)–(10, 0); (0, -1)–(10, -1); (0, -2)–(7, -2); (0, -3)–(6, -3); (0, -4)–(5, -4); (0, -5)–(2, -5);
(0, 0)–(0, -5); (1, 0)–(1, -5); (2, 0)–(2, -5); (3, 0)–(3, -4); (4, 0)–(4, -4); (5, 0)–(5, -4); (6, 0)–(6, -3); (7, 0)–(7, -2); (8, 0)–(8, -1); (9, 0)–(9, -1); (10, 0)–(10, -1);

(0, -6)–(5, -6); (0, -7)–(5, -7); (0, -8)–(4, -8);
(0, -6)–(0, -8); (1, -6)–(1, -8); (2, -6)–(2, -8); (3, -6)–(3, -8); (4, -6)–(4, -8); (5, -6)–(5, -7);

(0, -11)–(5, -11); (0, -12)–(5, -12); (0, -13)–(4, -13);
(0, -11)–(0, -13); (1, -11)–(1, -13); (2, -11)–(2, -13); (3, -11)–(3, -13); (4, -11)–(4, -13); (5, -11)–(5, -12);

[fill=gray] (5, -9) rectangle (6, -10); [fill=gray] (6, -9) rectangle (7, -10); [fill=gray] (7, -9) rectangle (8, -10);
[fill=gray] (8, -9) rectangle (9, -10);

[fill=gray] (4, -10) rectangle (5, -11); [fill=gray] (5, -10) rectangle (6, -11); [fill=gray] (6, -10) rectangle (7, -11);
[fill=gray] (7, -10) rectangle (8, -11);

[fill=gray] (0, -11) rectangle (1, -12); [fill=gray] (1, -11) rectangle (2, -12); [fill=gray] (2, -11) rectangle (3, -12);
[fill=gray] (3, -11) rectangle (4, -12);

[fill=gray] (0, -12) rectangle (1, -13);

[->, thick] (13,-6.5)–(11,-6.5);

[xshift = 14cm, yshift = 0cm]
(0, 0)–(10, 0); (0, -1)–(10, -1); (0, -2)–(7, -2); (0, -3)–(6, -3); (0, -4)–(5, -4); (0, -5)–(2, -5);
(0, 0)–(0, -5); (1, 0)–(1, -5); (2, 0)–(2, -5); (3, 0)–(3, -4); (4, 0)–(4, -4); (5, 0)–(5, -4); (6, 0)–(6, -3); (7, 0)–(7, -2); (8, 0)–(8, -1); (9, 0)–(9, -1); (10, 0)–(10, -1);

(0, -6)–(5, -6); (0, -7)–(5, -7); (0, -8)–(4, -8);
(0, -6)–(0, -8); (1, -6)–(1, -8); (2, -6)–(2, -8); (3, -6)–(3, -8); (4, -6)–(4, -8); (5, -6)–(5, -7);

(0, -9)–(9, -9); (0, -10)–(9, -10); (0, -11)–(8, -11); (0, -12)–(4, -12); (0, -13)–(1, -13);
(0, -9)–(0, -13); (1, -9)–(1, -13); (2, -9)–(2, -12); (3, -9)–(3, -12); (4, -9)–(4, -12); (5, -9)–(5, -11); (6, -9)–(6, -11); (7, -9)–(7, -11); (8, -9)–(8, -11); (9, -9)–(9, -10);

[fill=gray] (5, -9) rectangle (6, -10); [fill=gray] (6, -9) rectangle (7, -10); [fill=gray] (7, -9) rectangle (8, -10);
[fill=gray] (8, -9) rectangle (9, -10);

[fill=gray] (4, -10) rectangle (5, -11); [fill=gray] (5, -10) rectangle (6, -11); [fill=gray] (6, -10) rectangle (7, -11);
[fill=gray] (7, -10) rectangle (8, -11);

[fill=gray] (0, -11) rectangle (1, -12); [fill=gray] (1, -11) rectangle (2, -12); [fill=gray] (2, -11) rectangle (3, -12);
[fill=gray] (3, -11) rectangle (4, -12);

[fill=gray] (0, -12) rectangle (1, -13);

The two botched butterfly surgeries of Figure <ref> reformulated as Young diagram surgeries. The sites are shaded gray. Both sites are connected, so exactly two curves are obtained (no infinite pencils). The tangent weights $w$ of the corresponding invariant curves are also shown.

Let $\surgery$ be a Young diagram surgery with connected site $\site$. Then, $\surgery$ is uniquely determined by its action on any box in $\site$.

Since Young diagram surgeries are rigid translations, the action of $\surgery$ on a box in $\site$ completely determines the action of $\surgery$ on any other box in $\site$. It is trivial to show that the action $\surgery$ on any box in $\site$ completely determines whether or not each adjacent box is in $\site$. The result follows from these two statements and the connectedness of $\site$.

Let $\surgery_1$ and $\surgery_2$ be two connected Young diagram surgeries with $\Delta_{\surgery_1}y=\Delta_{\surgery_2}y$. Then, their sites are either disjoint or equal.

§ BCT BLOCK SWAP MOVES

An elementary result on BCTs is that any two BCTs with the same margin vectors are related by a sequence of “swap moves". A swap move acts on a BCT $M$ by taking a $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ minor and replacing it with $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ or vice-versa. A similar type of combinatorial move will provide an alternative combinatorial code for compact $\torus$-invariant curves.

Swap moves are also used in <cit.>, where they are called “simple moves" and appear in a generalization of the Chevalley-Monk formula.

Let us first introduce some preliminary notions.

Let $M$ be the BCT of some $p\in\bowvar(\brane)^\torus$. Call a minor

\[
\overline M = \overline{M}(i_1, i_2, j_1, j_2) = \left(M_{ij}\right)^{i = i_1, i_1+1, ..., i_2}_{j = j_1, j_2}
\]

consisting of two columns and any number of consecutive rows of $M$ a “block". Furthermore, define

\[
\delta(\overline M) = \begin{cases}
\sum_{i = i_1}^{i_2} (M_{i j_1} - M_{i j_2}) & \text{if }j_1 < j_2, \\
\sum_{i = i_1}^{i_2} (M_{i j_2} - M_{i j_1}) & \text{if }j_1 > j_2.
\end{cases}
\]

Call a block $\overline M$ “matched" if $\delta(\overline M) = 0$.

The following result provides an algorithmic way of constructing Young diagram surgeries. It will be a key ingredient in the proof of our main result, Theorem <ref>.

Let $M$ be the BCT of some $p\in\bowvar(\brane)^\torus$ where $\brane$ is separated with $m$ D5 branes. Then, using Definition <ref>,

* Every 01-pair $(i, j_0, j_1)$ or 10-pair $(i, j_1, j_0)$ in $M$ corresponds to a Young diagram surgery $\surgery$ moving boxes from $\lambda^{(j_1)}$ to $\lambda^{(j_0)}$.

* If $\overline{M}(i, i', j_0, j_1)$ is matched for some $i' > i$, then the site $\site$ of $\surgery$ does not contain any boxes of the rightmost column of $\lambda^{(j_1)}$. If no such matched block exists, then $\site$ contains $|\delta(\overline{M}(i, n, j_0, j_1))|$ boxes of the rightmost column of $\lambda^{(j_1)}$.

* The relative displacement of the boxes during $\surgery$ is given by

\[
\Delta_\surgery(y) = \sum_{k = 1}^{i} (M_{k j_1} - M_{k j_0}) - 1 = s_{i j_1} - s_{i j_0} - 1.
\]

We will provide the proof for 10-pairs. The result for 01-pairs follows from a symmetric argument. Make the simplifying assumption that $M$ is $n\times 2$. The general case follows from applying this argument to every pair of columns in $M$. To simplify the notation, let $\overline{M}(i_1, i_2) = \overline{M}(i_1, i_2, 1, 2)$.

Suppose the $i_1$th row of $M$ is $(1, 0)$. Let $\overline M$ be the smallest matched block of the form $\overline{M}(i_1, i_2)$ where $i_1 < i_2$, if such a block exists. Otherwise, if no such matched block exists, let $\overline M = \overline{M}(i_1, n)$. Recall that each 1 in the $j$th column of $\overline M$ represents a part of the Young diagram $\lambda^{(j)}$. Let $\overline{\lambda}^{(j)}$ be the partition consisting of the parts of $\lambda^{(j)}$ associated with the 1's in the $j$th column of $\overline M$. Denote the number of parts of $\overline{\lambda}^{(j)}$ by $\overline{c}_j$. Clearly, $|\delta(\overline{M}(i_1, i + 1)) - \delta(\overline{M}(i_1, i))| \leq 1$ for all $i_1 \leq i < n$, and $\delta(\overline{M}(i_1, i_1)) = 1$. If it were the case that $\delta(\overline M) < 0$, then there would exist $i > i_1$ for which $\delta(\overline{M}(i_1, i)) = 0$. It follows that $\delta(\overline M) \geq 0$, and therefore $\overline{c}_1 \geq \overline{c}_2$.

Since the first row of $\overline M$ is $(1, 0)$, we have $\overline{\lambda}^{(1)}_1 > \overline{\lambda}^{(2)}_1$. We claim that $\overline{\lambda}^{(1)}_{\overline{i}+1} \geq \overline{\lambda}^{(2)}_{\overline{i}}$ for all $\overline{i}=1, ..., \overline{c}_2 - 1$. Suppose to the contrary that for some $\overline{i}$, we have $\overline{\lambda}^{(1)}_{\overline{i}+1} < \overline{\lambda}^{(2)}_{\overline{i}}$. Let the $(1, 0)$ row corresponding to $\overline{\lambda}^{(1)}_{\overline{i}+1}$ be row $i'+1$ of $M$. Then, we have
$\delta(\overline{M}(i_1, i' + 1)) \leq 1$. It follows that $\delta(\overline{M}(i_1, i')) \leq 0$. By the intermediate value argument above, there is $i_1 < i < i_2$ such that $\overline{M}(i_1, i)$ is matched. This contradicts the definition of $\overline M$.

These inequalities imply that there is a connected Young diagram surgery which moves $\overline{\lambda}^{(1)}_{\overline{i}} - \overline{\lambda}^{(2)}_{\overline{i}} > 0$ boxes from $\overline{\lambda}^{(1)}_{\overline{i}}$ to $\overline{\lambda}^{(2)}_{\overline{i}}$. If $\overline M$ is matched, then this surgery constitutes a surgery on $\lambda^{(1)}$ and $\lambda^{(2)}$. Otherwise, this surgery can be extended to a surgery on $\lambda^{(1)}$ and $\lambda^{(2)}$ by moving the bottom $\delta(\overline{M}) > 0 $ rows of boxes in $\overline{\lambda}^{(1)}$ completely below $\lambda^{(2)}$. The displacement $\Delta_\surgery y$ is easily seen to be $\sum_{k = 1}^{i_1} (M_{k 1} - M_{k 2}) - 1$.

Let $\overline M$ be a minimal matched block whose top row is a 01- or 10- pair. The proof of Proposition <ref> shows that there is a connected butterfly surgery transforming $M$ by exchanging the columns of $\overline M$. The displacement of the surgery is given by applying the formula in Proposition <ref> (3) to the top row of $\overline M$. Also note that among any set of consecutive minimal matched blocks, those whose top row is a 10- or 01-pair are associated with surgeries with disjoint sites and the same displacement. Conversely, any matched block can be decomposed uniquely into consecutive minimal matched blocks. It follows that every matched block with top row $(1, 0)$ or $(0, 1)$ is associated with a pencil of compact invariant curves.

Let $M$ be a BCT.  A “block swap move" $\psi$ on $M$ consists of swapping the columns of a matched block $\overline M$ within $M$, where the first and last rows of $\overline M$ are $(1, 0)$ or $(0, 1)$. Call $\psi$ indecomposable if $\overline M$ is a minimal matched block.

Note that if the top row of a minimal matched block is $(1, 0)$, then the bottom row must be $(0, 1)$ and vice-versa. Moreover, any block swap move can be uniquely decomposed into simultaneous indecomposable block swap moves whose associated minimal matched blocks are separated by $(1, 1)$ and $(0, 0)$ rows.

Let $p\in\bowvar(\brane)^\torus$, with $\brane$ separated. Let $M$ be the BCT of $p$.

* Every block swap move $\psi$ on $M$ corresponds to a pencil $\Gamma$ of compact invariant curves containing $p$.

* Suppose $\psi$ decomposes into $k$ simultaneous indecomposable block swap moves $\psi_1, ... \psi_k$. Then, we have $\dim(\Gamma) = k$.

* The BCT for any fixed point in $\Gamma$ can be obtained by performing some subset of the indecomposable block swap moves $\psi_i$.

* Suppose row $i$ is the topmost row affected by $\psi$. Let $(i, j_0)$ be the index of the 0 in row $i$ that $\psi$ swaps to 1. Similarly, let $(i, j_1)$ be the index of the 1 in row $i$ that $\psi$ swaps to 0. Then, the tangent weight of $\Gamma$ at $p$ is $\frac{\ub_{j_1}}{\ub_{j_0}}\hb^{1+s_{i j_0}-s_{i j_1}}$.

It follows from Theorem <ref> that all pencils of compact invariant curves in $\bowvar(\brane)$ are associated with block swap moves.

§ CLASSIFICATION OF INVARIANT CURVES

Let $\bowvar(\brane)$ be a separated bow variety and $p\in\bowvar(\brane)^\torus$. Define three types of invariant curves containing $p$:

I curves arising from connected butterfly surgeries (Section <ref>),

II curves arising from connected botched butterfly surgeries (Section <ref>),

III nonsurgery curves (Section <ref>).

A priori, the type of a compact curve may depend on which of the two fixed points you apply this definition to. We will show in Corollary <ref> that, in fact, type II and III curves are always noncompact. Because butterfly surgeries are reversible, a type I curve containing $p_1$ is also type I when viewed with respect to its other fixed point $p_2$.

The type I, type II, and type III curves conatining a fixed point $p$ form three disjoint sets. Moreover, no type II curve has the same tangent weight as a type III curve.

It is easy to see from Lemma <ref> and the constraint (<ref>) that a type III curve cannot have the same tangent weight at $p$ as a type II curve. Thus, no curve is both type II and type III. A type I, type II, or type III curve $\gamma$ is constructed as the closure of the orbit of some point $p_\gamma$ for which we give an explicit representation in $\MM$. If $\gamma$ is type I, then the ranks of the $B$ maps of this representative are the same as those of the butterfly diagram for $p$. If $\gamma$ is type II, then the ranks of some of the $B$ maps of this representative are greater than those of the butterfly diagram. Thus, no curve can be both type I and type II. Finally, if $\gamma$ is type I, then this representative has $b=0$, and if $\gamma$ is type III, this representative has $b\neq 0$. Thus, no curve can be both type I and type III.

Curves of type I, II, and III will form a basic set from which all pencils of invariant curves can be generated.

Let $\gamma_1, ..., \gamma_k$ be distinct invariant curves containing $p\in\bowvar(\brane)^\torus$, each of type I, type II, or type III, and all having the same tangent weight at $p$. Then, there is a $k$-dimensional pencil of invariant curves $\Gamma$ with $\gamma_1, ..., \gamma_k$ in its boundary. Moreover, every vector in $T_p\gamma_1 \oplus\cdots\oplus T_p\gamma_k$ is in $T_p\gamma$ for some $\gamma\in\Gamma$.

By definition, the type I and II curves are associated with Young diagram surgeries. Since the curves are distinct, the surgeries must have distinct sites. By Corollary <ref>, the sites of these surgeries are disjoint. The type III curves have distinct tangent weights, as can be seen from Lemma <ref>. Hence, at most one of the curves is type III. It follows from Proposition <ref> that if one of the curves is type III, then the rest are type I. Note that the construction of type I and type III curves involve disjoint portions of the butterfly diagram. Thus, each $\gamma_i$ is constructed by adding a set $E_i$ of new edges to disjoint portions of the butterfly diagram of $p$. We may form the desired pencil by adding the edges of $E_i$ with factor $z_i$ for all $i=1,...,k$ (see Section <ref> for the construction of the pencil in the case where all curves are type I).

We say that the pencil $\Gamma$ in Proposition <ref> is “spanned" by the curves $\gamma_1, ..., \gamma_k$.

This leads us to our main result.

Let $\brane$ be separated and $p\in\bowvar(\brane)^\torus$. All $\torus$-invariant pencils of curves containing $p$ are spanned by curves of type I, type II, and type III.

Due to Proposition <ref>, it suffices to construct $k$ distinct type I, II, or III $\torus$-invariant curves with tangent weight $w$ at $p$ for each tangent weight $w$ of multiplicity $k$ in $T_p\bowvar(\brane)$. For simplicity, let us reduce to the case where $m=2$, and the BCT $M$ of $p$ is an $n\times 2$ matrix. The general case is obtained by applying the same argument to each pair of columns in $M$.

From Theorem <ref>, the tangent weights at $p$ come in two varieties: those of the form $\frac{\ub_1}{\ub_2}\hb^r$ and those of the form $\frac{\ub_2}{\ub_1}\hb^r$. It is easy to see from Proposition <ref> and Lemma <ref> that the weights of the latter form are accounted for by type I and II curves. Hence, we turn our attention to the former.

We restrict our attention to weights of the form $\frac{\ub_1}{\ub_2}\hb^r$. There is one such weight associated with each 01-pair. First, we consider the case where $c_1 \leq c_2$. In this case, the extra constraint (<ref>) is automatically satisfied. If there are no 10-pairs in $M$, then by Lemma <ref>, the tangent weights under consideration are accounted for by type III curves. Induct on the number of 10-pairs. Assume that $M$ has both 10- and 01-pairs. We see that whenever a 10-pair and a 01-pair are only separated by 00- or 11-pairs, the tangent weight of the curve associated with the 10-pair via Proposition <ref> is equal to the $\frac{\ub_1}{\ub_2}\hb^r$ tangent weight associated with the 01-pair. Note also that the minimal matched block between such a 01-pair and 10-pair makes no net contribution to the tangent weight formula of Theorem <ref> or the displacement formula of Proposition <ref> (3). A 01-pair with a 10-pair separated by only 00- or 11-pairs always exists as long as there are both 01- and 10-pairs, so deleting the block between them and applying induction completes the argument.

Next, we consider the case $c_1 > c_2$. Let the row indices of the 10-pairs be $i_1, ..., i_{c_1}$. Define $\overline{M}(i_1, i_2) = \overline{M}(i_1, i_2, 1, 2)$ and $\delta_i = \delta(\overline{M}(i, n))$. Clearly, $|\delta_n| \leq 1$, $\delta_1 = c_1 - c_2$, and $|\delta_i - \delta_{i + 1}| \leq 1$. Since $\delta_i - \delta_{i + 1} = 1$ if and only if $M$ has a 10-pair in the $i$th row, we have

\[
\{1, ..., c_1 - c_2\} \subset \{\delta_{i_k}\ |\ k = 1, ..., c_1\}.
\]

Moreover, defining

\[
k_1 = \max\{k\ |\ \delta_{i_k} = c_1 - c_2\},\, k_2 = \max\{k\ |\ \delta_{i_k} = c_1 - c_2 - 1\}, ...,\, k_{c_1 - c_2} = \max\{k\ |\ \delta_{i_k} = 1\},
\]

it follows from an intermediate value argument that $k_1 < k_2 < \cdots < k_{c_1 - c_2}$. We collect two basic facts:

* If $\delta_{i} - \delta_{i'} = 0$ for $i < i'$, then $\delta(\overline{M}(i, i' - 1)) = 0$.

* If $\delta_{i_k} - \delta_{i_{k'}} = 1$ for $k < k'$,  then $\delta(\overline{M}(i_k + 1, i_{k'} -1)) = 0$.

If $i_{k_1} > 1$, applying (1) with $i = 1, i' = i_{k_1}$ tells us that $\overline{M}(1, i_{k_1} - 1)$ is matched. Applying (2) to consecutive indices in the ordered sequence $k_1, ..., k_{c_1 - c_2}$ tells us that the blocks lying between the 10-pairs in rows $i_{k_1}, ..., i_{k_{c_1- c_2}}$ are matched. If $i_{k_{c_1 - c_2}} < n$, the previous two sentences imply that $\overline{M}(i_{k_{c_1 - c_2}} + 1, n)$ is matched. Hence, rows $i_{k_1}, ..., i_{k_{c_2 - c_1}}$ separate $M$ into matched blocks.

Our next goal is to show that every 10-pair except the those in rows $i_{k_1}, ..., i_{k_{c_1-c_2}}$ are associated with an invariant curve. By Proposition <ref>, a 10-pair in row $i$ is associated with a curve if it is the top row of some matched block or if it satisfies the constraint $\delta_i \geq c_1 - c_2 + 1$ (constraint (<ref>)). Let $(i, 1, 2)$ be a 10-pair where $i\neq i_k$ for $k = 1, ..., c_1 - c_2$. Suppose $\delta_i \leq c_1 - c_2$. If $\delta_i < 1$, then there exists $i' > i$ such that $\delta(\overline{M}(i, i')) = 0$ by an intermediate value argument. In other words, $(i, 1, 2)$ is the top of a matched block. If $1 \leq \delta_i \leq c_1 - c_2$, then $\delta_i = \delta_{i_k}$ for some $k$. We have $i < i_k$ by definition of $i_k$, so $\overline{M}(i, i_k - 1)$ is matched by (1).
Applying the argument above for the $c_1 \leq c_2$ case to each matched block lying between rows $i_{k_1}, ..., i_{k_{c_1 - c_2}}$ completes the proof.

While not necessary for the proof, it is easy to show that the 10-pairs in rows $i_{k_1}, ..., i_{k_{c_1 - c_2}}$ correspond to Young diagram surgeries that move at least one block of the right column but fail constraint (<ref>). edit: new remark

At first glance, the existence of nonsurgery (type III) curves and the constraint (<ref>) might seem mysterious. The proof of Theorem <ref> elucidates their role. By Theorem <ref>, each 01-pair corresponds to two tangent weights, $w$ and $\hb w^{-1}$. Each 01-pair is also associated with a Young diagram surgery, by Proposition <ref>. Hence, it is natural to expect half of the tangent weights to be accounted for by these surgeries, and indeed this is the case. On the other hand, we might expect the remaining half of the tangent weights to be accounted for by surgeries associated with 10-pairs. If there are fewer 10- than 01-pairs, we get a deficit of weights, which is made up by nonsurgery curves. Otherwise, if there are more 10- than 01-pairs, constraint (<ref>) kicks in to correct the surplus.

Let $\brane$ be separated and $p\in\bowvar(\brane)$. Invariant curves of type I containing $p$ are compact, and those of type II or III are noncompact.

We have already shown in Section <ref> that type I curves are compact. Let $\gamma$ be a compact invariant curve containing $p_1$. Then, $\gamma$ contains another fixed point $p_2$. Recall that the tangent weights of $\gamma$ at $p_1$ and $p_2$ are reciprocols. First, assume that $T_{p_1}\gamma = \frac{\ub_j}{\ub_{j'}}\hb^r$, where $j<j'$. Suppose to the contrary that $\gamma$ is type II or III. In Section <ref> and Section <ref>, we constructed a point $p_{\gamma, 1}\in\bowvar(\brane)$ such that $\gamma = \overline{\torus.p_{\gamma, 1}}.$ From the form of the tangent weight, we see that $p_{\gamma, 1}$ is represented by a point of $\prequot$ with $b\neq 0$. Viewing $\gamma$ from the perspective of $p_2$, Theorem <ref> implies that $\gamma$ belongs to a pencil of curves spanned by type I, II, and III curves containing $p_2$. Thus, the surgery constructions with respect to $p_2$ give us a point $p_{\gamma, 2}\in\bowvar(\brane)$ such that $\gamma = \overline{\torus.p_{\gamma, 2}}$. In particular, $p_{\gamma, 1}$ and $p_{\gamma, 2}$ are in the same $\torus$-orbit. It follows that $p_{\gamma, 2}$ must be represented by an element of $\prequot$ with $b\neq 0$. However, examining the constructions of Section <ref>, Section <ref>, and Section <ref>, pencils of curves with tangent weight $\frac{\ub_{j'}}{\ub_j}\hb^{-r}$, where $j<j'$, are given by orbit closures of points represented by elements of $\prequot$ with $b=0$. Contradiction!

The case where $T_{p_1}\gamma = \frac{\ub_{j'}}{\ub_{j}}\hb^r$, where $j<j'$ is similar. For this case, we do not need to consider type III curves. If the curve is type II, then the rank of $B$ in the representative of $p_{\gamma,1}$ is higher than the rank of $B$ in the butterfly diagram for $p_1$. The rank of $B$ in the butterfly diagram for $p_1$ is the same as in the butterfly diagram for $p_2$. Hence, the rank of $B$ in the representative of $p_{\gamma,2}$ is higher than that of the butterfly diagram for $p_2$. The only invariant curves containing $p_2$ with this property also have $b\neq 0$. Contradiction!

§ EXAMPLES OF INVARIANT CURVES

We will apply the classification of Section <ref> to describe the invariant curves of the example bow varieties from [8] in terms of Young diagram surgeries.

§.§ Invariant Curves for $\bowvar(\ttt{\fs 1\fs 2\fs 3\fs 4\fs 5\bs 2\bs})$

Letting $\brane = \ttt{\fs 1\fs 2\fs 3\fs 4\fs 5\bs 2\bs}$, the bow variety $\bowvar(\brane)$ is Hanany-Witten isomorphic to the 3d mirror dual of $T^*\mathrm{Gr}(2,5)$. Its $\torus$-fixed points are in bijection with the $2$-element subsets of the set $\{1,2,3,4,5\}$, where the subset $\{k,l\}$ corresponds to the tie diagram with $U_2$ connected to both $V_k$ and $V_l$, and $U_1$ connected to the three remaining NS5 branes. This fixed point will be denoted $kl$. The invariant curves from Figure <ref> were computed via Young diagram surgeries.

\[
\begin{tikzpicture}[scale=1.6]

\draw[] (3.5,-.5) -- (-0.5,3.5);
\draw[] (3.5,1.5) -- (.5,4.5);
\draw[] (3,4) -- (1.5,5.5);

\draw [](3,0) to [out=40,in=-40] (3,4);
\draw [](3,2) to [out=40,in=-40] (3,6);

\draw[] (1.5,.5) -- (3,2);
\draw[] (.5,1.5) -- (3.5,4.5);
\draw[] (-.5,2.5) -- (3.5,6.5);
\draw[] (3,0) -- (2.5,-.5);
\draw[] (3,6) -- (2.5,6.5);

\node[] at (2.5,.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^{-2}$}};
\node[] at (2.6,-.2) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^{2}$}};
\node[] at (3.5,.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^{-1}$}};
\node[] at (3.44,-.2) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^3$}};

\node[] at (1.5,1.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^{-1}$}};
\node[] at (1.6,.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^{2}$}};
\node[] at (2.55,1.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^{-1}$}};
\node[] at (2.45,.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^2$}};

\node[] at (0.6,2.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}$}};
\node[] at (0.6,1.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^2$}};
\node[] at (1.14,2.33) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^{-1}$}};
\node[] at (1.2,1.6) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb$}};

\node[] at (2.9,2.4) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^{-1}$}};
\node[] at (2.8,1.55) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb$}};
\node[] at (3.4,2.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}$}};
\node[] at (3.45,1.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^2$}};

\node[] at (1.6,3.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}$}};
\node[] at (1.6,2.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb$}};
\node[] at (2.4,3.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}$}};
\node[] at (2.4,2.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb$}};

\node[] at (-.4,3.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb$}};
\node[] at (-.4,2.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^2$}};
\node[] at (.5,3.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^{-1}$}};
\node[] at (.4,2.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}$}};

\node[] at (.6,4.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb$}};
\node[] at (.6,3.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb$}};
\node[] at (1.2,4.35) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}$}};
\node[] at (1.25,3.6) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}$}};

\node[] at (2.85,4.3) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}$}};
\node[] at (2.75,3.55) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}$}};
\node[] at (3.45,4.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb$}};
\node[] at (3.4,3.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb$}};

\node[] at (1.6,5.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb$}};
\node[] at (1.6,4.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}$}};
\node[] at (2.45,5.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb$}};
\node[] at (2.4,4.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}$}};

\node[] at (2.6,6.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb$}};
\node[] at (2.6,5.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}\hb^{-1}$}};
\node[] at (3.45,6.2) {\scalebox{.6}[.6]{$\frac{\ub_1}{\ub_2}\hb^2$}};
\node[] at (3.4,5.8) {\scalebox{.6}[.6]{$\frac{\ub_2}{\ub_1}$}};

\draw [](2.2,1.2) to [out=53,in=-53] (2.2,2.8);
\draw [](2.1,1.12) to [out=76,in=-76] (2.1,2.88);
\draw [](2,1.15) to [out=90,in=-90] (2,2.85);
\draw [](1.9,1.12) to [out=104,in=-104] (1.9,2.88);
\draw [](1.8,1.2) to [out=127,in=-127] (1.8,2.8);

\draw [](2.2,3.2) to [out=53,in=-53] (2.2,4.8);
\draw [](2.1,3.12) to [out=76,in=-76] (2.1,4.88);
\draw [](2,3.15) to [out=90,in=-90] (2,4.85);
\draw [](1.9,3.12) to [out=104,in=-104] (1.9,4.88);
\draw [](1.8,3.2) to [out=127,in=-127] (1.8,4.8);

\draw [](2,1) to (2.35,0.5);
\draw [](2,1) to (2.2,0.5);
\draw [](2,1) to (2 ,0.5);
\draw [](2,1) to (1.8,0.5);
\draw [](2,1) to (1.65,0.5);

\draw [](2,5) to (2.35,5.5);
\draw [](2,5) to (2.2,5.5);
\draw [](2,5) to (2 ,5.5);
\draw [](2,5) to (1.8,5.5);
\draw [](2,5) to (1.65,5.5);

\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (3,0) {\footnotesize$45$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (2,1) {\footnotesize$35$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (3,2) {\footnotesize$34$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (1,2) {\footnotesize$25$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (2,3) {\footnotesize$24$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (3,4) {\footnotesize$23$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (0,3) {\footnotesize$15$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (1,4) {\footnotesize$14$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (2,5) {\footnotesize$13$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (3,6) {\footnotesize$12$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (3,0) {\footnotesize$45$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (2,1) {\footnotesize$35$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (3,2) {\footnotesize$34$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (1,2) {\footnotesize$25$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (2,3) {\footnotesize$24$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (3,4) {\footnotesize$23$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (0,3) {\footnotesize$15$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (1,4) {\footnotesize$14$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (2,5) {\footnotesize$13$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (3,6) {\footnotesize$12$};

\end{tikzpicture}
\]

Illustration of $\torus$-fixed points and invariant curves (with their $\torus$-weights) of $\bowvar(\ttt{\fs 1\fs 2\fs 3\fs 4\fs 5\bs 2\bs})$, which is the 3d mirror dual of $T^*\mathrm{Gr}(2,5)$. edit: hyphens

As an example, consider the fixed point $13$, whose BCT and Young diagrams appear in Figure <ref>. By Theorem <ref>, $T_{13}\bowvar(\brane) = \frac{\ub_1}{\ub_2}\hb + \frac{\ub_2}{\ub_1} + \frac{\ub_1}{\ub_2}\hb + \frac{\ub_2}{\ub_1}$. Using Young diagram surgeries, four invariant curves are constructed, depicted in Figure <ref>, one for each tangent weight.

Each pair of surgeries that share a tangent weight (columns of Figure <ref>) have disjoint sites, and thus span pencils. The pencil of weight $\frac{\ub_2}{\ub_1}$ is spanned by two type I curves, so all the curves of this pencil are compact. The other pencil has weight $\frac{\ub_1}{\ub_2} \hb$ and is spanned by a type I curve and a type II curve, and thus only one curve of the pencil is compact.

Note that $13$ also has one Young diagram surgery that is not depicted, but it is an example of a surgery that does not satisfy constraint <ref>, as $c_1-c_2+1=2$.

\begin{pmatrix}
\end{pmatrix}

(1, 0)–(5, 0); (1, -1)–(5, -1); (3, -2)–(5, -2); (4, -3)–(5, -3);
(1, 0)–(1, -1); (2, 0)–(2, -1); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(0, -5)–(5, -5); (0, -6)–(5, -6); (2, -7)–(5, -7); (0, -5)–(0, -6); (1, -5)–(1, -6); (2, -5)–(2, -7); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

The (a) BCT and (b) Young diagrams associated to the fixed point $13$ of Figure <ref>.

*(a) $w = \frac{\ub_1}{\ub_2}\hb$

(1, 0)–(5, 0); (1, -1)–(5, -1); (3, -2)–(5, -2); (4, -3)–(5, -3);
(1, 0)–(1, -1); (2, 0)–(2, -1); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(0, -5)–(5, -5); (0, -6)–(5, -6); (2, -7)–(5, -7); (0, -5)–(0, -6); (1, -5)–(1, -6); (2, -5)–(2, -7); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

[fill=gray] (1, -1) rectangle (2, 0);

[->, thick] (6, -3.5)–(8, -3.5);

[xshift = 9cm]
(2, 0)–(5, 0); (2, -1)–(5, -1); (3, -2)–(5, -2); (4, -3)–(5, -3);
(2, 0)–(2, -1); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(0, -5)–(5, -5); (0, -6)–(5, -6); (1, -7)–(5, -7); (0, -5)–(0, -6); (1, -5)–(1, -7); (2, -5)–(2, -7); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

[fill=gray] (1, -7) rectangle (2, -6);

*(b) $w = \frac{\ub_2}{\ub_1}$

(1, 0)–(5, 0); (1, -1)–(5, -1); (3, -2)–(5, -2); (4, -3)–(5, -3);
(1, 0)–(1, -1); (2, 0)–(2, -1); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(0, -5)–(5, -5); (0, -6)–(5, -6); (2, -7)–(5, -7); (0, -5)–(0, -6); (1, -5)–(1, -6); (2, -5)–(2, -7); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

[fill=gray] (0, -5) rectangle (1, -6);

[->, thick] (6, -3.5)–(8, -3.5);

[xshift = 9cm]
(0, 0)–(5, 0); (0, -1)–(5, -1); (3, -2)–(5, -2); (4, -3)–(5, -3);
(0, 0)–(0, -1); (1, 0)–(1, -1); (2, 0)–(2, -1); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(1, -5)–(5, -5); (1, -6)–(5, -6); (2, -7)–(5, -7); (1, -5)–(1, -6); (2, -5)–(2, -7); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

[fill=gray] (0, 0) rectangle (1, -1);

*(c) $w = \frac{\ub_1}{\ub_2}\hb$

(1, 0)–(5, 0); (1, -1)–(5, -1); (3, -2)–(5, -2); (4, -3)–(5, -3);
(1, 0)–(1, -1); (2, 0)–(2, -1); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(0, -5)–(5, -5); (0, -6)–(5, -6); (2, -7)–(5, -7); (0, -5)–(0, -6); (1, -5)–(1, -6); (2, -5)–(2, -7); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

[fill=gray] (3, -1) rectangle (4, -2); [fill=gray] (4, -1) rectangle (5, -2); [fill=gray] (4, -2) rectangle (5, -3);

[->, thick] (6, -3.5)–(8, -3.5);

[xshift = 9cm]
(1, 0)–(5, 0); (1, -1)–(5, -1);
(1, 0)–(1, -1); (2, 0)–(2, -1); (3, 0)–(3, -1); (4, 0)–(4, -1); (5, 0)–(5, -1);

(0, -3)–(5, -3); (0, -4)–(5, -4); (2, -5)–(5, -5); (3, -6)–(5, -6); (4, -7)–(5, -7); (0, -3)–(0, -4); (1, -3)–(1, -4); (2, -3)–(2, -5); (3, -3)–(3, -6); (4, -3)–(4, -7); (5, -3)–(5, -7);

[fill=gray] (4, -7) rectangle (5, -6); [fill=gray] (4, -6) rectangle (5, -5); [fill=gray] (3, -6) rectangle (4, -5);

*(d) $w = \frac{\ub_2}{\ub_1}$

(1, 0)–(5, 0); (1, -1)–(5, -1); (3, -2)–(5, -2); (4, -3)–(5, -3);
(1, 0)–(1, -1); (2, 0)–(2, -1); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(0, -5)–(5, -5); (0, -6)–(5, -6); (2, -7)–(5, -7); (0, -5)–(0, -6); (1, -5)–(1, -6); (2, -5)–(2, -7); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

[fill=gray] (2, -6) rectangle (3, -7);

[->, thick] (6, -3.5)–(8, -3.5);

[xshift = 9cm]
(1, 0)–(5, 0); (1, -1)–(5, -1); (2, -2)–(5, -2); (4, -3)–(5, -3);
(1, 0)–(1, -1); (2, 0)–(2, -2); (3, 0)–(3, -2); (4, 0)–(4, -3); (5, 0)–(5, -3);

(0, -5)–(5, -5); (0, -6)–(5, -6); (3, -7)–(5, -7); (0, -5)–(0, -6); (1, -5)–(1, -6); (2, -5)–(2, -6); (3, -5)–(3, -7); (4, -5)–(4, -7); (5, -5)–(5, -7);

[fill=gray] (2, -1) rectangle (3, -2);

Four Young diagram surgeries representing four invariant curves of Figure <ref> containing fixed point $13$. The sites are shaded gray. The tangent weights $w$ of the corresponding invariant curves are also shown. Each pair of curves with the same weight span one of the two 2-dimensional pencils seen in Figure <ref>.

§.§ Invariant Curves for $\bowvar(\ttt{\fs 2\fs 3\fs 5\bs 3\bs 2\bs})$

In [8], the fixed curves are depicted for the bow variety $\bowvar(\ttt{\bs 1\fs 2\fs 2\bs 2\bs 1\fs})$. This is Hanany-Witten isomorphic to the separated bow variety $\bowvar(\ttt{\fs 2\fs 3\fs 5\bs 3\bs 2\bs})$. This isomorphism is $\torus$-equivariant up to the reparametrization of $\torus$ described in Section <ref>. The following computation will be done using tangent weights of fixed points in $\bowvar(\ttt{\fs 2\fs 3\fs 5\bs 3\bs 2\bs})$, which differs  from [8] in the exponents of $\hb$. The invariant curves of this bow variety can be seen in Figure <ref>.

\[
\begin{tikzpicture}[scale=2.5]

\draw[] (.07,.07) -- (.93,.93);
\draw[] (.07,1.93) -- (.93,1.07);
\draw[] (1.07,0.93) -- (1.93,0.07);
\draw[] (1.07,1.07) -- (1.93,1.93);
\draw[] (0,.1) -- (0,1.9);
\draw[] (2,.1) -- (2,1.9);

\draw[] (-.07,.07) -- (-.5,.5);
\draw[] (-.07,-.07) -- (-.5,-.5);
\draw[] (0,-.1) -- (0,-.5);
\draw[] (.07,-.07) -- (.5,-.5);

\draw[] (2.07,.07) -- (2.5,.5);
\draw[] (2.07,-.07) -- (2.5,-.5);
\draw[] (1.93,-.07) -- (1.5,-.5);
\draw[] (2,-.1) -- (2,-.5);

\draw[] (-.07,2.07) -- (-.5,2.5);
\draw[] (-.07,1.93) -- (-.5,1.5);
\draw[] (0,2.1) -- (0,2.5);
\draw[] (.07,2.07) -- (.5,2.5);

\draw[] (2.07,2.07) -- (2.5,2.5);
\draw[] (2.07,1.93) -- (2.5,1.5);
\draw[] (1.93,2.07) -- (1.5,2.5);
\draw[] (2,2.1) -- (2,2.5);

\draw[] (1,1) -- (1,1.6); \draw[] (1,1) -- (1.1,1.6); \draw[] (1,1) -- (1.2,1.6);   \draw[] (1,1) -- (1.3,1.6);  \draw[] (1,1) -- (1.4,1.6);  \draw[] (1,1) -- (1.5,1.6);
\draw[] (1,1) -- (1,.4);\draw[] (1,1) -- (.9,.4);\draw[] (1,1) -- (.8,.4);\draw[] (1,1) -- (.7,.4);\draw[] (1,1) -- (.6,.4);\draw[] (1,1) -- (.5,.4);

\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (0,0) {\footnotesize $1$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (1,1) {$3$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (2,0) {$2$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (0,2) {$4$};
\node[fill,white,draw,circle,minimum size=.5cm,inner sep=0pt] at (2,2) {$5$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (0,0) {\footnotesize $1$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (1,1) {\footnotesize$3$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (2,0) {\footnotesize$2$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (0,2) {\footnotesize$4$};
\node[draw,circle,minimum size=.5cm,inner sep=0pt] at (2,2) {\footnotesize$5$};

\node at (1.38,1.22) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}\hb$}};
\node at (.9,1.4) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}\hb$}};
\node at (.67,1.22) {\scalebox{.6}[.6]{$\frac{{\ub}_1}{{\ub}_2}$}};
\node at (.65,.76) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2}$}};
\node at (1.07,.6) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2}$}};
\node at (1.4,.75) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_1}\hb$}};

\node at (2.38,2.22) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}\hb^2$}};
\node at (1.89,2.33) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}\hb$}};
\node at (1.58,2.27) {\scalebox{.6}[.6]{$\frac{{\ub}_1}{{\ub}_3}\hb$}};
\node at (1.65,1.77) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2}\hb$}};
\node at (2.075,1.64) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_1}$}};
\node at (2.39,1.73) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2}$}};

\node at (2.4,.23) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}\hb$}};
\node at (1.93,.4) {\scalebox{.6}[.6]{$\frac{{\ub}_1}{{\ub}_3}$}};
\node at (1.59,.19) {\scalebox{.6}[.6]{$\frac{{\ub}_1}{{\ub}_2}\hb^{-1}$}};
\node at (1.63,-.24) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_1}\hb$}};
\node at (2.12,-.4) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_1}\hb^2 $}};
\node at (2.38,-0.27) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2}$}};

\node at (.32,.2) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}$}};
\node at (-.07,.4) {\scalebox{.6}[.6]{$\frac{{\ub}_1}{{\ub}_3}$}};
\node at (-.36,.22) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}\hb$}};
\node at (-0.37,-.25) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2}\hb$}};
\node at (.06,-.4) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2} $}};
\node at (.41,-0.25) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_1}\hb$}};

\node at (.35,2.2) {\scalebox{.6}[.6]{$\frac{{\ub}_1}{{\ub}_3}\hb$}};
\node at (-.1,2.4) {\scalebox{.6}[.6]{$\frac{{\ub}_1}{{\ub}_2}\hb$}};
\node at (-.4,2.24) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_3}\hb$}};
\node at (-0.35,1.78) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_2}$}};
\node at (-.07,1.5) {\scalebox{.6}[.6]{$\frac{{\ub}_3}{{\ub}_1}$}};
\node at (.31,1.815) {\scalebox{.6}[.6]{$\frac{{\ub}_2}{{\ub}_1}$}};
\end{tikzpicture}
\]

Illustration of $\torus$-fixed points and invariant curves (with their $\torus$-weights) of $\bowvar(\ttt{\fs 2\fs 3\fs 5\bs 3\bs 2\bs})$.

Let us consider the fixed point labeled $3$, which has BCT and Young diagrams in Figure <ref>. By Theorem <ref>, $T_{3}\bowvar(\brane) = \frac{\ub_2}{\ub_3}\hb + \frac{\ub_3}{\ub_2} + \frac{\ub_1}{\ub_2} + \frac{\ub_2}{\ub_1}\hb + \frac{\ub_2}{\ub_3}\hb + \frac{\ub_3}{\ub_2}$. We can produce an invariant curve for each of these tangent weights.

\begin{pmatrix}
\end{pmatrix}

(1, 0)–(4, 0); (1, -1)–(4, -1); (3, -2)–(4, -2);
(1,0)–(1,-1); (2,0)–(2,-1); (3,0)–(3,-2); (4,0)–(4,-2);

(2,-3)–(4,-3); (2,-4)–(4,-4);
(2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

(1, -5)–(4, -5); (1, -6)–(4, -6); (3, -7)–(4, -7);
(1,-5)–(1,-6); (2,-5)–(2,-6); (3,-5)–(3,-7); (4,-5)–(4,-7);

The (a) BCT and (b) Young diagrams associated to the fixed point $3$ of Figure <ref>.

First we check for type III curves. By Lemma <ref>, there is one type III curve, since $c_3-c_2=1$. This curve has tangent weight $\frac{\ub_2}{\ub_3}\hb$. If there are other curves with this weight, they must all be type I and, together with the type III curve, will span a pencil.

We now seek to construct five more invariant curves that come from Young diagram surgeries. To do this, we will consider surgeries between each pair of Young diagrams separately. There are two surgeries between the top two Young diagrams, three surgeries between the bottom two Young diagrams, and zero surgeries between the top and bottom Young diagrams. These five surgeries are depicted in Figure <ref> and Figure <ref>.

With these six curves described, we now build pencils of curves. There is a pencil of curves with weight $\frac{\ub_3}{\ub_2}$, which is spanned by a type I curve and type II curve, as seen in Figure <ref>. There is also a pencil of curves with weight $\frac{\ub_2}{\ub_3}\hb$, spanned by a type I curve from Figure <ref> and a type III curve. No other curves share weights, so these are the only multidimensional pencils of invariant curves.

*(a) $w = \frac{\ub_1}{\ub_2}$

(1, 0)–(4, 0); (1, -1)–(4, -1); (3, -2)–(4, -2);
(1,0)–(1,-1); (2,0)–(2,-1); (3,0)–(3,-2); (4,0)–(4,-2);

(2,-3)–(4,-3); (2,-4)–(4,-4);
(2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

[fill=gray] (1, -1) rectangle (2, 0);

[->, thick] (5, -2)–(7, -2);

[xshift = 7cm]
(2, 0)–(4, 0); (2, -1)–(4, -1); (3, -2)–(4, -2);
(2,0)–(2,-1); (3,0)–(3,-2); (4,0)–(4,-2);

(1,-3)–(4,-3); (1,-4)–(4,-4);
(1,-3)–(1,-4); (2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

[fill=gray] (1, -3) rectangle (2, -4);

*(b) $w = \frac{\ub_2}{\ub_1}\hb$

(1, 0)–(4, 0); (1, -1)–(4, -1); (3, -2)–(4, -2);
(1,0)–(1,-1); (2,0)–(2,-1); (3,0)–(3,-2); (4,0)–(4,-2);

(2,-3)–(4,-3); (2,-4)–(4,-4);
(2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

[fill=gray] (2, -3) rectangle (3, -4);

[->, thick] (5, -2)–(7, -2);

[xshift = 7cm]
(1, 0)–(4, 0); (1, -1)–(4, -1); (2, -2)–(4, -2);
(1,0)–(1,-1); (2,0)–(2,-2); (3,0)–(3,-2); (4,0)–(4,-2);

(3,-3)–(4,-3); (3,-4)–(4,-4);
(3,-3)–(3,-4); (4,-3)–(4,-4);

[fill=gray] (3, -2) rectangle (2, -1);

The two Young diagram surgeries possible between the top two Young diagrams of Figure <ref>. The sites are shaded gray and $w$ is the tangent weight of the corresponding invariant curve.

*(a) $w = \frac{\ub_2}{\ub_3}\hb$

(2,-3)–(4,-3); (2,-4)–(4,-4);
(2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

(1, -5)–(4, -5); (1, -6)–(4, -6); (3, -7)–(4, -7);
(1,-5)–(1,-6); (2,-5)–(2,-6); (3,-5)–(3,-7); (4,-5)–(4,-7);

[fill=gray] (2, -4) rectangle (3, -3);

[->, thick] (5, -5)–(7, -5);

[xshift = 7cm]
(3,-3)–(4,-3); (3,-4)–(4,-4);
(3,-3)–(3,-4); (4,-3)–(4,-4);

(1, -5)–(4, -5); (1, -6)–(4, -6); (2, -7)–(4, -7);
(1,-5)–(1,-6); (2,-5)–(2,-7); (3,-5)–(3,-7); (4,-5)–(4,-7);

[fill=gray] (2, -6) rectangle (3, -7);

*(b) $w = \frac{\ub_3}{\ub_2}$

(2,-3)–(4,-3); (2,-4)–(4,-4);
(2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

(1, -5)–(4, -5); (1, -6)–(4, -6); (3, -7)–(4, -7);
(1,-5)–(1,-6); (2,-5)–(2,-6); (3,-5)–(3,-7); (4,-5)–(4,-7);

[fill=gray] (1, -5) rectangle (2, -6);

[->, thick] (5, -5)–(7, -5);

[xshift = 7cm]
(1,-3)–(4,-3); (1,-4)–(4,-4);
(1,-3)–(1,-4); (2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

(2, -5)–(4, -5); (2, -6)–(4, -6); (3, -7)–(4, -7);
(2,-5)–(2,-6); (3,-5)–(3,-7); (4,-5)–(4,-7);

[fill=gray] (2, -4) rectangle (1, -3);

*(c) $w = \frac{\ub_3}{\ub_2}$

(2,-3)–(4,-3); (2,-4)–(4,-4);
(2,-3)–(2,-4); (3,-3)–(3,-4); (4,-3)–(4,-4);

(1, -5)–(4, -5); (1, -6)–(4, -6); (3, -7)–(4, -7);
(1,-5)–(1,-6); (2,-5)–(2,-6); (3,-5)–(3,-7); (4,-5)–(4,-7);

[fill=gray] (3, -7) rectangle (4, -6);

[->, thick] (5, -5)–(7, -5);

[xshift = 7cm]
(2,-3)–(4,-3); (2,-4)–(4,-4); (3,-5)–(4,-5);
(2,-3)–(2,-4); (3,-3)–(3,-5); (4,-3)–(4,-5);

(1,-6)–(4,-6); (1,-7)–(4,-7);
(1,-6)–(1,-7); (2,-6)–(2,-7); (3,-6)–(3,-7); (4,-6)–(4,-7);

[fill=gray] (4, -5) rectangle (3, -4);

The three surgeries possible between the bottom two Young diagrams of Figure <ref>. The sites are shaded gray and the tangent weights are included. Note that surgeries (b) and (c) share a tangent weight and have disjoint sites, and thus span a pencil of invariant curves.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Oxford University Press

journaltitleBulletin of the London Mathematical Society
titleConvexity and Commuting Hamiltonians

titleEquivariant Cohomology in Algebraic Geometry

journaltitleJ. Amer. Math. Soc.
titleElliptic Stable Envelopes

booktitleEncyclopaedia of Mathematical Sciences: Invariant Theory and Algebraic Transformation groups
titleAlgebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action

howpublishedPreprint arXiv:2308.07300
titleBow varieties: Stable envelopes and their 3d mirror symmetry

journaltitleCommun. Math. Phys.
titleModuli Spaces of Instantons on the Taub-NUT Space

journaltitleAdvances in Theoretical and Mathematical Physics
titleInstantons on the Taub-NUT space

journaltitleCommun. Math. Phys.
titleInstantons on Gravitons

journaltitleAnnals of Mathematics
titleThe Topological Schur Lemma and Related Results

Springer Singapore

booktitleSchubert Calculus and Its Applications in Combinatorics and Representation Theory
titleCharacteristic Classes of Orbit Stratifications, the Axiomatic Approach

Berlin, Springer-Verlag.

journaltitleInventiones mathematicae
titleEquivariant cohomology, Koszul duality, and the localization theorem

noteIn preparation
titleBow Varieties as Symplectic Reductions of $T^*(GL_n/P)$

titleToroidal embeddings 1

Société mathématique de France

booktitleSur les groupes algébriques
seriesMémoires de la Société Mathématique de France
titleSlices étales

titleQuantum Groups and Quantum Cohomology

journaltitleDuke Mathematical Journal
titleQuiver varieties and Kac-Moody algebras

howpublishedPreprint arXiv:1810.04293
titleTowards geometric Satake correspondence for Kac-Moody algebras–Cherkis bow varieties and affine Lie algebras of type $ A$

journaltitleSelecta Mathematica
titleCherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

Springer International Publishing

booktitleSingularities and Their Interaction with Geometry and Low Dimensional Topology
title$\hbar$-Deformed Schubert Calculus in Equivariant Cohomology, K-Theory, and Elliptic Cohomology

journaltitleCommunications in Analysis and Geometry
titleBow varieties—geometry, combinatorics, characteristic classes

abstractWe consider a pair of quiver varieties
$(X;X^\\{\\prime\\})\\$ related by 3D mirror symmetry, where
$X =T^*\\{Gr\\}(k,n)\\$ is the cotangent bundle of the Grassmannian of
$k\\$-planes of
$n\\$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on
$X \\times X^\\{\\prime\\} \\$ (the mother function) whose restrictions to
$X\\$ and
$X^\\{\\prime\\} \\$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for
$X\\$ and
$X^\\{\\prime\\}\\$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.
journaltitleInternational Mathematics Research Notices
titleThree-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

SIGMA (Symmetry, Integrability
Geometry: Methods

journaltitleSymmetry, Integrability and Geometry: Methods and Applications
titleThree-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

journaltitleQuantum Topology
titlePartial flag varieties, stable envelopes and weight functions

journaltitleJournal of Geometry and Physics
titleTrigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

journaltitleSelecta Mathematica
titleElliptic and K-theoretic stable envelopes and Newton polytopes

The University of North Carolina at Chapel Hill

titleBow Varieties—Geometry, Combinatorics, Characteristic Classes

journaltitleAdvances in Mathematics
title3d mirror symmetry and quantum K-theory of hypertoric varieties

journaltitlePublications of the Research Institute for Mathematical Sciences
titleNahm's Equations, Quiver Varieties and Parabolic Sheaves

Providence, RI

American Mathematical Society

booktitleSnowbird lectures in algebraic geometry
titleAn introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson

howpublishedPreprint arXiv:2310.11235
titleChevalley–Monk Formulas for Bow Varieties

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Oxford University Press

journaltitleBulletin of the London Mathematical Society
titleConvexity and Commuting Hamiltonians

titleEquivariant Cohomology in Algebraic Geometry

journaltitleJ. Amer. Math. Soc.
titleElliptic Stable Envelopes

booktitleEncyclopaedia of Mathematical Sciences: Invariant Theory and Algebraic Transformation groups
titleAlgebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action

howpublishedPreprint arXiv:2308.07300
titleBow varieties: Stable envelopes and their 3d mirror symmetry

journaltitleCommun. Math. Phys.
titleModuli Spaces of Instantons on the Taub-NUT Space

journaltitleAdvances in Theoretical and Mathematical Physics
titleInstantons on the Taub-NUT space

journaltitleCommun. Math. Phys.
titleInstantons on Gravitons

journaltitleAnnals of Mathematics
titleThe Topological Schur Lemma and Related Results

Springer Singapore

booktitleSchubert Calculus and Its Applications in Combinatorics and Representation Theory
titleCharacteristic Classes of Orbit Stratifications, the Axiomatic Approach

Berlin, Springer-Verlag.

journaltitleInventiones mathematicae
titleEquivariant cohomology, Koszul duality, and the localization theorem

noteIn preparation
titleBow Varieties as Symplectic Reductions of $T^*(GL_n/P)$

titleToroidal embeddings 1

Société mathématique de France

booktitleSur les groupes algébriques
seriesMémoires de la Société Mathématique de France
titleSlices étales

titleQuantum Groups and Quantum Cohomology

journaltitleDuke Mathematical Journal
titleQuiver varieties and Kac-Moody algebras

howpublishedPreprint arXiv:1810.04293
titleTowards geometric Satake correspondence for Kac-Moody algebras–Cherkis bow varieties and affine Lie algebras of type $ A$

journaltitleSelecta Mathematica
titleCherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

Springer International Publishing

booktitleSingularities and Their Interaction with Geometry and Low Dimensional Topology
title$\hbar$-Deformed Schubert Calculus in Equivariant Cohomology, K-Theory, and Elliptic Cohomology

journaltitleCommunications in Analysis and Geometry
titleBow varieties—geometry, combinatorics, characteristic classes

abstractWe consider a pair of quiver varieties
$(X;X^\\{\\prime\\})\\$ related by 3D mirror symmetry, where
$X =T^*\\{Gr\\}(k,n)\\$ is the cotangent bundle of the Grassmannian of
$k\\$-planes of
$n\\$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on
$X \\times X^\\{\\prime\\} \\$ (the mother function) whose restrictions to
$X\\$ and
$X^\\{\\prime\\} \\$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for
$X\\$ and
$X^\\{\\prime\\}\\$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.
journaltitleInternational Mathematics Research Notices
titleThree-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

SIGMA (Symmetry, Integrability
Geometry: Methods

journaltitleSymmetry, Integrability and Geometry: Methods and Applications
titleThree-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

journaltitleQuantum Topology
titlePartial flag varieties, stable envelopes and weight functions

journaltitleJournal of Geometry and Physics
titleTrigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

journaltitleSelecta Mathematica
titleElliptic and K-theoretic stable envelopes and Newton polytopes

The University of North Carolina at Chapel Hill

titleBow Varieties—Geometry, Combinatorics, Characteristic Classes

journaltitleAdvances in Mathematics
title3d mirror symmetry and quantum K-theory of hypertoric varieties

journaltitlePublications of the Research Institute for Mathematical Sciences
titleNahm's Equations, Quiver Varieties and Parabolic Sheaves

Providence, RI

American Mathematical Society

booktitleSnowbird lectures in algebraic geometry
titleAn introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson

howpublishedPreprint arXiv:2310.11235
titleChevalley–Monk Formulas for Bow Varieties

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Oxford University Press

journaltitleBulletin of the London Mathematical Society
titleConvexity and Commuting Hamiltonians

titleEquivariant Cohomology in Algebraic Geometry

journaltitleJ. Amer. Math. Soc.
titleElliptic Stable Envelopes

booktitleEncyclopaedia of Mathematical Sciences: Invariant Theory and Algebraic Transformation groups
titleAlgebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action

howpublishedPreprint arXiv:2308.07300
titleBow varieties: Stable envelopes and their 3d mirror symmetry

journaltitleCommun. Math. Phys.
titleModuli Spaces of Instantons on the Taub-NUT Space

journaltitleAdvances in Theoretical and Mathematical Physics
titleInstantons on the Taub-NUT space

journaltitleCommun. Math. Phys.
titleInstantons on Gravitons

journaltitleAnnals of Mathematics
titleThe Topological Schur Lemma and Related Results

Springer Singapore

booktitleSchubert Calculus and Its Applications in Combinatorics and Representation Theory
titleCharacteristic Classes of Orbit Stratifications, the Axiomatic Approach

Berlin, Springer-Verlag.

journaltitleInventiones mathematicae
titleEquivariant cohomology, Koszul duality, and the localization theorem

noteIn preparation
titleBow Varieties as Symplectic Reductions of $T^*(GL_n/P)$

titleToroidal embeddings 1

Société mathématique de France

booktitleSur les groupes algébriques
seriesMémoires de la Société Mathématique de France
titleSlices étales

titleQuantum Groups and Quantum Cohomology

journaltitleDuke Mathematical Journal
titleQuiver varieties and Kac-Moody algebras

howpublishedPreprint arXiv:1810.04293
titleTowards geometric Satake correspondence for Kac-Moody algebras–Cherkis bow varieties and affine Lie algebras of type $ A$

journaltitleSelecta Mathematica
titleCherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

Springer International Publishing

booktitleSingularities and Their Interaction with Geometry and Low Dimensional Topology
title$\hbar$-Deformed Schubert Calculus in Equivariant Cohomology, K-Theory, and Elliptic Cohomology

journaltitleCommunications in Analysis and Geometry
titleBow varieties—geometry, combinatorics, characteristic classes

abstractWe consider a pair of quiver varieties
$(X;X^\\{\\prime\\})\\$ related by 3D mirror symmetry, where
$X =T^*\\{Gr\\}(k,n)\\$ is the cotangent bundle of the Grassmannian of
$k\\$-planes of
$n\\$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on
$X \\times X^\\{\\prime\\} \\$ (the mother function) whose restrictions to
$X\\$ and
$X^\\{\\prime\\} \\$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for
$X\\$ and
$X^\\{\\prime\\}\\$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.
journaltitleInternational Mathematics Research Notices
titleThree-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

SIGMA (Symmetry, Integrability
Geometry: Methods

journaltitleSymmetry, Integrability and Geometry: Methods and Applications
titleThree-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

journaltitleQuantum Topology
titlePartial flag varieties, stable envelopes and weight functions

journaltitleJournal of Geometry and Physics
titleTrigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

journaltitleSelecta Mathematica
titleElliptic and K-theoretic stable envelopes and Newton polytopes

The University of North Carolina at Chapel Hill

titleBow Varieties—Geometry, Combinatorics, Characteristic Classes

journaltitleAdvances in Mathematics
title3d mirror symmetry and quantum K-theory of hypertoric varieties

journaltitlePublications of the Research Institute for Mathematical Sciences
titleNahm's Equations, Quiver Varieties and Parabolic Sheaves

Providence, RI

American Mathematical Society

booktitleSnowbird lectures in algebraic geometry
titleAn introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson

howpublishedPreprint arXiv:2310.11235
titleChevalley–Monk Formulas for Bow Varieties