Tangent Weights and Invariant Curves in Type A Bow Varieties

This paper provides a complete classification of torus-invariant curves in Cherkis bow varieties of type A. We develop combinatorial codes for compact and noncompact invariant curves involving the butterfly diagrams, Young diagrams, and binary contingency tables. As a key intermediate step, we also...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2025
Автори:	Foster, Alexander, Shou, Yiyan
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Мова:	Англійська
Опубліковано:	Інститут математики НАН України 2025
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Цитувати:	Tangent Weights and Invariant Curves in Type A Bow Varieties. Alexander Foster and Yiyan Shou. SIGMA 21 (2025), 016, 33 pages

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author	Foster, Alexander Shou, Yiyan
author_facet	Foster, Alexander Shou, Yiyan
citation_txt	Tangent Weights and Invariant Curves in Type A Bow Varieties. Alexander Foster and Yiyan Shou. SIGMA 21 (2025), 016, 33 pages
collection	DSpace DC
container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	This paper provides a complete classification of torus-invariant curves in Cherkis bow varieties of type A. We develop combinatorial codes for compact and noncompact invariant curves involving the butterfly diagrams, Young diagrams, and binary contingency tables. As a key intermediate step, we also develop a novel tangent weight formula. Finally, we apply this new machinery to example bow varieties to demonstrate how to obtain their 1-skeletons (union of fixed points and invariant curves).
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 016, 33 pages Tangent Weights and Invariant Curves in Type A Bow Varieties Alexander FOSTER a and Yiyan SHOU b a) Department of Mathematics, University of North Carolina at Chapel Hill, NC, USA E-mail: aofoster@live.unc.edu b) Independent Researcher, Alexandria, VA, USA E-mail: yiyanshou@gmail.com Received December 08, 2023, in final form February 20, 2025; Published online March 09, 2025 https://doi.org/10.3842/SIGMA.2025.016 Abstract. This paper provides a complete classification of torus-invariant curves in Cherkis bow varieties of type A. We develop combinatorial codes for compact and noncompact invariant curves involving the butterfly diagrams, Young diagrams, and binary contingency tables. As a key intermediate step, we also develop a novel tangent weight formula. Finally, we apply this new machinery to example bow varieties to demonstrate how to obtain their 1- skeletons (union of fixed points and invariant curves). Key words: Cherkis bow varieties; invariant curves; butterfly surgery; Young diagrams 2020 Mathematics Subject Classification: 14H10; 05E14 1 Introduction For a broad class of complex algebraic varieties with a torus action T ⟳ Y , the fixed points and invariant curves under the torus action play an important role in the study of the equivariant cohomology of Y . Here, we define an invariant curve to be the closure of a 1-dimensional T -orbit. This paper extends the work of [17] on the case where Y is a Cherkis bow variety of type A by providing combinatorial codes for invariant curves. For simplicity, assume that Y has finitely many fixed points. Under certain conditions, the map ι∗ : H∗ T (Y ) → H∗ T ( Y T ) induced by the inclusion of the fixed point locus is injective. In other words, each equivariant cohomology class is uniquely determined by its fixed point restrictions. This is the case for partial flag varieties, and more generally, Nakajima quiver varieties of type A. As H∗ T (pt) = C[u1, . . . , um], where m = dim(T ), this allows us to describe equivariant cohomol- ogy classes as tuples of polynomials. This approach is taken in [16] to study ℏ-deformed Schubert classes. In the setting where the variety is equivariantly formal with respect to the T -action, we can describe the image of ι∗ in terms of the fixed points and invariant curves. For instance, when Y possesses finitely many invariant curves, the result of [9] describes the image of ι∗ in terms of simple matching conditions on the polynomials (see also [25]). These matching condi- tions are determined by the invariant curves and their tangent data. When there are infinitely many invariant curves, the Chang–Skjelbred lemma [4] provides a more complicated description (see also [2]). Unfortunately, not every Cherkis bow variety is equivariantly formal nor has injec- tive restriction map [10]. The appendix of [3] shows, however, that there is a natural subalgebra (generated by the stable envelope) of H∗ T (Y ) on which ι∗ is injective. Determining whether this subalgebra can be described in terms of fixed points and invariant curves is an open problem. Besides providing a concrete way to describe equivariant cohomology classes, the fixed points and invariant curves play an important role in the study of stable envelopes [1, 11]. The stable envelope is an axiomatically defined T -equivariant characteristic class with important connec- mailto:aofoster@live.unc.edu mailto:yiyanshou@gmail.com https://doi.org/10.3842/SIGMA.2025.016 2 A. Foster and Y. Shou tions to quantum integrable systems, quantum groups, and quantum cohomology. In certain settings, the axioms can be formulated as conditions on fixed point restrictions that are entirely determined by the invariant curves and their tangent data. This formulation is, in a sense, local to the fixed points, in contrast to the original global axioms of [1, 11]. See [20, 21, 22] for a discussion of local stable envelope axioms in the Schubert calculus setting and [17, Section 7] for the Cherkis bow variety setting. In the Schubert calculus setting, the stable envelope agrees with the Chern–Schwartz–MacPherson classes of Schubert cells [8]. While we are focusing on equivariant cohomology here, this discussion generalizes to K-theory and elliptic cohomology. In the elliptic cohomology setting, the stable envelope plays an important role in the study of 3d (N = 4) mirror symmetry [3, 18, 19, 24]. The work of [3] shows that Cherkis bow varieties [5, 6, 7, 15] form a natural pool of varieties where the phenomenon of mirror symmetry manifests itself in the combinatorics and elliptic stable envelope. Bow varieties are holomorphic symplectic manifolds that come equipped with a torus action and generalize Nakajima quiver varieties [13]. In [17], a combinatorial framework for studying the fixed points of bow varieties of type A is established. This paper completes the picture by providing a complete combinatorial description of the invariant curves. A construction for compact invariant curves involving so called “butterfly surgeries” was previously known to the second author (see [23]). However, whether that construction captured all compact invariant curves remained a conjecture. This work resolves that conjecture in the affirmative and extends the result by also capturing the noncompact invariant curves using modified “butterfly surgery” constructions. While the main accomplishment of this paper is the complete classification of invariant curves, several intermediate developments are required, some of which are interesting in their own right. In Section 2, we review the combinatorial framework of [17] for the study of type A Cherkis bow varieties. In Section 3, we prove a novel and noncancellative formula for the tangent weights at a fixed point of a bow variety. This formula can be used to efficiently calculate tangent weights by hand or by computer and plays a pivotal role in the proof of our main result. In Section 4, we review the essentials of torus invariant curves and give a self-contained presentation of the butterfly surgery construction of [23]. This section also describes two novel constructions for noncompact invariant curves. Finally, we reformulate these constructions in terms of surgery operations on Young diagrams, resulting in a description of invariant curves in terms of familiar combinatorics. Section 5 further reformulates the combinatorics of Section 4 in terms of binary contingency tables (BCTs). The results of this section provide a simple algorithm for constructing all invariant curves in a bow variety. Section 6 contains the main result, a classification of invariant curves containing a given fixed point. Curves are characterized as one of three types, each having its own unique properties. In the final section, Section 7, we apply the machinery developed in the previous sections to the example bow varieties of [17]. 2 Background We begin by recalling the essential pieces of the combinatorial framework of [17] for the study of bow varieties. We will require brane diagrams, tie diagrams, table-with-margins, butterfly dia- grams, the formula for the tangent bundle of a bow variety, formulas for fixed point restrictions, and the Hanany–Witten transition. 2.1 Brane diagrams and bow varieties The [17] framework is based on the [15] construction of bow varieties. Here, we provide a sim- plified presentation. A brane diagram D consists of a horizontal line divided into segments by n lines of slope 1 and m lines of slope −1. The lines must be arranged so as not to create a triple Tangent Weights and Invariant Curves in Type A Bow Varieties 3 V1 2 2 V2 2 4 3 3 4 U3 3 U4 U5 2 2 NS5 branes D5 branesD3 branes . . . Figure 1. The first example of a brane diagram given in [17]. (or greater) intersection. Additionally, each segment is decorated by a nonnegative integer mul- tiplicity where the infinite segments on the far left and right are automatically decorated with 0. An example of a brane diagram is shown in Figure 1. The lines of slope 1 are called “NS5 branes”, and we label them from left to right as V1, . . . , Vn. The lines of slope −1 are called “D5 branes”, and we label them from left to right as U1, . . . , Um. The NS5 and D5 branes are referred to collectively as “5-branes”. The segments along with their multiplicities are referred to as “D3 branes”. The segments are labeled as X0, . . . , Xn+m and the multiplicity of a D3 brane X is denoted by dX . In particular, dX0 = dXn+m = 0. We do not bother displaying X0 and Xn+m in graphical representations of D. We also color the NS5 branes red and the D5 branes blue. These colors are purely for visual clarity and have no mathematical meaning. To each brane diagram D, we associate a smooth, torus equivariant, symplectic holomorphic variety T ⟳ C(D). This variety will be constructed as a GIT quotient. Given any brane β (D5, NS5, or D3), denote the brane immediately to the left by β− and right by β+. First, a vector space will be associated to each 5-brane. For each D3 brane X, letWX = CdX , and for each D5 brane U , let CU = C. � To each D5 brane U , associate a “triangle part” MU = Hom(WU+ ,WU−)⊕Hom(WU+ ,CU )⊕Hom(CU ,WU−) ⊕ End(WU−)⊕ End(WU+), whose elements will be denoted by (AU , bU , aU , BU , B ′ U ), as shown in the diagram WU− WU+ CU . BU ◦ ◦ bU AU B′ U ◦ aU � To each NS5 brane V , associate the “two-way part” MV = Hom(WV + ,WV −)⊕Hom(WV − ,WV +), whose elements will be denoted by (CV , DV ), as shown in the diagram WV − WV + . DV CV ◦ 4 A. Foster and Y. Shou Finally, define M = ⊕ U D5 MU ⊕ ⊕ V NS5 MV , the sum of all triangle and two-way parts. Next, define the groups T = C× ℏ × ∏ U D5 C× U and G = ∏ X D3 GL(WX), where C× ℏ and C× U are copies of C×. The group G acts on each WX and hence on M in a natural way. The torus T acts on M by scaling CU by the C× U factor and scaling each circled map in the above diagrams by the C× ℏ factor. Let M0 be the subset of M satisfying the following conditions: (1) For each D5 brane U , we have BUAU −AUB ′ U + aUbU = 0. (2) For each D3 brane X, we have � B′ X− −BX+ = 0 if X is in between two D5 branes (\−X − \), � CX+DX+ −DX−CX− = 0 if X is in between two NS5 branes (/−X − /), � −DX−CX− −BX+ = 0 if X− is an NS5 brane and X+ is a D5 brane (/−X − \), � CX+DX+ +B′ X− = 0 if X− is a D5 brane and X+ is an NS5 brane (\−X − /). These conditions arise from setting a certain moment map equal to 0 (see [15] or [17] for details). Hence, we will refer to them as “0-momentum” conditions. Let W = ⊕ X D3WX , and define three additional “stability” conditions: (1) For all D5 branes U , the only BU+-invariant subspace S ⊂WU+ with AU (S) = 0, bU (S) = 0 is S = 0. (2) For all D5 branes U , the only BU−-invariant subspace S ⊂WU− with im(AU )+im(aU ) ⊂ S is S =WU− . (3) The only subspace S = ⊕ X SX ⊂ W invariant under all A, B, C, —D maps such that im(aU ) ⊂ S and AU induces an isomorphism WU+/SU+ →WU−/SU− for all D5 branes U is S = W. Stability conditions 1 and 2 are the S1 and S2 conditions of [15, Section 2.2], and stability condition 3 is the ν2 condition of [15, Section 2.4]. Let M̃s be the subset of M0 satisfying the three stability conditions. M̃s is G- and T-invariant. Define the “Cherkis bow variety” corresponding to brane diagram D as the quotient C(D) = M̃s/G. There is a residual action of T on C(D), along with a collection of T-equivariant “tau- tological” vector bundles ξX → C(D), associated with the WX vector spaces. The rank of ξX is dX . A formula for the T-equivariant K-theory class of the tangent bundle T C(D) is given in [17, Section 3.2]: � Let h be the trivial line bundle whose fibres are scaled by the C× ℏ factor of T. � For a D5 brane U define TU = Hom(ξU+ , ξU−)⊕ hHom(ξU+ ,CU )⊕Hom(CU , ξU−) ⊕ hEnd(ξU−)⊕ hEnd(ξU+). � For an NS5 brane V define TV = hHom(ξV + , ξV −)⊕Hom(ξV − , ξV +). Tangent Weights and Invariant Curves in Type A Bow Varieties 5 2 2 2 4 3 3 4 3 2 2 Figure 2. An example of a tie diagram whose underlying brane diagram is the one displayed in Figure 1. Then, as elements of K0 T(C(D)), we have T C(D) = (⊕ U D5 TU ) ⊕ ( ⊕ V NS5 TV ) ⊖ (⊕ U D5 hHom(ξU+ , ξU−) ) ⊖ (⊕ X D3 (1 + h) End(ξX) ) . (2.1) 2.2 Torus fixed points Paper [17] provides three combinatorial codes for the T-fixed points of a bow variety C(D): tie diagrams, butterfly diagrams, and table-with-margins (BCTs). We will be utilizing all three in what follows. Let us begin with tie diagrams. A “tie diagram” is obtained from a brane diagram by adding dashed lines, called “ties”, between the 5-branes such that � each tie joins 5-branes of different type (one NS5 and one D5 brane), � for each segment X, there are dX distinct ties covering X. We allow at most one tie between any two 5-branes. In Figure 2, we added ties to the brane diagram in Figure 1 to create a tie diagram. This tie diagram corresponds to one of the T-fixed points. Next, we will create a correspondence between tie diagrams and binary contingency tables (BCTs). An intermediate notion will be required. An integer called the “charge” will be associated to each 5-brane. For an NS5 brane V , let charge(V ) = (dV + − dV −) + #{D5 branes left of V }. For a D5 brane U , let charge(U) = (dU− − dU+) + #{NS5 branes right of U}. Define the “row margin” and “column margin”, respectively, by r = (charge(V1), . . . , charge(Vn)), c = (charge(U1), . . . , charge(Um)). A “binary contingency table” (BCT) with row and column margins r, c is a binary matrix with row sums equal to r and columns sums equal to c. Note that the existence of such a matrix implies that ∑ i ri = ∑ j cj . Given a tie diagram, construct a BCT M with margins r, c as follows: � if Vi is to the left of Uj , set Mij = 1 if there is a tie between Vi and Uj and Mij = 0 otherwise, � if Vi is to the right of Uj , set Mij = 1 if there is no tie between Vi and Uj and Mij = 0 otherwise. 6 A. Foster and Y. Shou 5 2 2 0 2 U1 U2 U3 U4 U5 2 3 2 1 1 2 V6 V5 V4 V3 V2 V1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 Figure 3. The table-with-margins corresponding to the tie diagram in Figure 2. Assuming the underlying brane diagram D is fixed, this gives a bijection between tie diagrams and BCTs with margins r, c. However, the map sending all tie diagrams (with arbitrary un- derlying brane diagram) to their corresponding BCT is not injective. An additional piece of data, called the “separating line”, is required to make this map bijective. A separating line is a path running from the top left to the bottom right of an (m+ 1)× (n+ 1) lattice making only downward and rightward moves. We draw the separating line over the BCT, as depicted in Fig- ure 3. The separating line plays the role of encoding the order of the 5-branes. Downward moves correspond to NS5 branes and rightward moves to D5 branes. A BCT along with a separating line is called a “table-with-margins”. Table-with-margins are in bijection with tie diagrams. Remark 2.1. Paper [15] provides a more general construction of bow varieties of affine type A. In this more general setting, BCTs are replaced by the “Maya diagrams” of [14]. The precise relationship between BCTs and Maya diagrams is discussed in an appendix of [17]. Finally, we will describe butterfly diagrams. These provide a depiction of a representative in the prequotient M̃s of a T-fixed point of C(D). Fix a tie diagram and a D5 brane U . For each segment X, let dUX be the number of distinct ties connected to U covering X. Place a column of dUX vertices below X with fixed spacing, and align them in the following way: � If the 5-brane between two consecutive columns is D5, align the columns at the bottom. � For columns to the right of U , if the 5-brane between two consecutive columns is NS5, align the columns at the top. � For columns to the left of U , if the 5-brane between two consecutive columns is NS5, align the columns so that the top vertex of the left column is one position lower than that of the right. Also, place a special “framing” vertex (indicated by an open circle in our diagrams) below U . We will then join these vertices with directed edges: � If a column is adjacent to a D5 brane, create downward (black) edges between each con- secutive pair of vertices in the column. � If the 5-brane between two adjacent columns is D5, create leftward (blue) edges between horizontally adjacent pairs of vertices. � If the 5-brane between two adjacent columns is NS5, create (magenta dotted) edges point- ing left one position and down one position, wherever possible, and rightward (red) edges between horizontally adjacent pairs of vertices. � If dUU− > 0, create an (green) edge from the framing vertex to the top vertex under U−. � If dUU+ > dUU− , create an (green) edge from vertex dUU+ −dUU− (counted from top to bottom) to the framing vertex. Tangent Weights and Invariant Curves in Type A Bow Varieties 7 2 2 2 4 3 3 4 3 2 2 Figure 4. The tie diagram from Figure 2 along with its corresponding butterfly diagram. The resulting directed graph is called a “butterfly”. Taking the disjoint union of the butterflies of all the D5 branes gives us a “butterfly diagram”. Figure 4 displays the butterfly diagram corresponding to the tie diagram in Figure 2. For each segment X, pick a basis for WX and identify the basis vectors with the vertices of the butterfly diagram below X. For each D5 brane U , identify CU with the framing vertex of the butterfly of U . Then, interpret the edges as A, −B, C, D, a, −b maps. Note that when X− and X+ are both D5 branes, there are two B maps acting on WX , one from the triangle part of X− and one from the triangle part of X+. Considering the 0-momentum condition, we take these maps to be equal, both represented by the black edges below X. The maps determined by the butterfly diagram constitute an element p̃ ∈ M̃s. Moreover, this element descends to a T-fixed point of C(D). The converse is also true, that is, every T-fixed point of C(D) can be represented by an element of M̃s determined by some butterfly diagram. This is the content of [17, Theorem 4.8]. 2.3 Fixed point restrictions Recall that a bow variety C(D) comes equipped with tautological bundles ξX , one for each segment. Let p ∈ C(D)T, and consider the corresponding butterfly diagram. Each non-framing vertex will be assigned a “height” relative to the butterfly it belongs to. Definition 2.2. Fix a butterfly and define the “height” of each of its vertices as follows: � If the tail of a green edge is the framing vertex, the head is a vertex of height 0. � If the head of a green edge is the framing vertex, the tail is a vertex of height 1. � The height decreases by 1 when we go from one vertex to another one immediately below it, and is constant across horizontal rows of vertices. Given any vertex v in a butterfly diagram, denote its height by y(v). Note that the height of v is measured with respect to the butterfly containing v. Let u be the trivial line bundle over p whose T-action is given by scaling by the CU factor. Similarly, let h be the trivial line bundle scaled by Cℏ. Decorate each vertex v of the U butterfly by uhy(v) ∈ K0 T(p). Figure 5 shows these decorations for the butterfly diagram of Figure 4. 8 A. Foster and Y. Shou u1 u1 u1 u1 u2h 2 u2h 2 u2h 2 u2h 2 u2h 2 u2h u2h u2u2 u2 h u2 h u3 h u3 h u3 h u3 u2 h2 u2 h2 u4 h u4 h u4 h u4 h u4 u4 Figure 5. The butterfly diagram of Figure 4 decorated by elements of K0 T(p). Then, we have that ξX \|p is equal the sum of the line bundles decorating the vertices of the butterfly diagram below X as elements of K0 T(p). For example, in Figure 5, we have ξX7 = u2h 2 + u3 + u3 h + u4 h . Of central importance in this work is the T-equivariant K-theory class of Tp C(D). To express this class in terms of u and h bundles, one simply takes the tangent bundle formula (2.1) and substitutes each tautological bundle with the formula for its fixed point restriction calculated as above. Then, one can apply bilinearity of Hom and the isomorphism Hom(α, β) ∼= β ⊗ α∗ to expand this expression into a sum of line bundles. In our running example, after a great deal of cancellations, we obtain Tp C(D) = u2h 4 u5 + u2h 4 u3 + u1h 3 u3 + u2h 3 u3 + u4h 2 u5 + u3h u5 + u4h u5 + u5h u3 + u3 u5 + u5 u4 + u5 u4h + u3 u2h2 + u3 u1h2 + u3 u2h3 + u5 u2h3 . (2.2) Each line bundle in this expansion is called a “tangent weight at p”. In other words, the tangent weights at p are the K-theoretic Chern roots of Tp C(D). 2.4 Hanany–Witten transition and separated bow varieties Let D contain a D5 brane U followed immediately by an NS5 brane V . Let D̃ agree with D except at U and V , where they differ as in the picture d1 d2 d3 U V X d1 d̃2 d3 ŨṼ X̃ , where d2 + d̃2 = d1 + d3 + 1. Then, there is a natural holomorphic symplectic isomorphism C(D) ∼=→ C(D̃) called a “Hanany–Witten transition”. In other words, Hanany–Witten transitions allow one to exchange two consecutive 5-branes of different type (cf. [17, Section 8]) at the expense of changing the multiplicity of the D3 brane in between. The Hanany–Witten transition Tangent Weights and Invariant Curves in Type A Bow Varieties 9 A B C D E HW transition on fixpoints B A D C ¬E Figure 6. The action of Hanany–Witten transition on tie diagrams. The E and ¬E symbols mean that the tie labeled E is present if and only if the tie labeled ¬E is not. 2 3 4 6 9 11 6 4 2 2 Figure 7. The tie diagram of Figure 2 put into separated form using Hanany–Witten transitions. is equivariant with respect to the reparametrization ũ = uh (the other factors of the torus are not affected). The Hanany–Witten transition induces a bijection of T-fixed points. This bijection can best be understood through transformations of tie diagrams and tables-with-margins. The action of the Hanany–Witten transition on tie diagrams is described by Figure 6. In words, switch U and V while keeping all ties attached. If there was a tie between U and V , remove it, and if there was no such tie, create one. The Hanany–Witten transition acts on tables-with-margins by perturbing the separating line while leaving the BCT fixed. By a sequence of Hanany–Witten transitions, one may transform any brane diagram into a brane diagram with all NS5 branes on the left and all D5 branes on the right. Such a brane diagram will be called “separated”, as will the associated bow variety. Figure 7 depicts the result of applying this procedure to separate the tie diagram of Figure 2. Separated bow varieties will be the focus of the remainder of this work. Any results derived can be translated to non- separated bow varieties by making a suitable reparametrization of T. In practical terms, this amounts to adjusting the exponents of h in the formulas. 3 A combinatorial formula for tangent weights Fix a separated brane diagram D and torus fixed point p ∈ C(D)T. We will give a combinatorial formula for the tangent weights at p (i.e., K-theoretic Chern roots of Tp C(D)) in terms of the associated table-with-margins. Unlike the approach used to derive (2.2), this formula will return the tangent weights without the need for any further cancellations or simplifications. Note that since the bow variety is separated, the separating line always runs along the left and bottom sides. We ignore it and consider only the BCT M . Again, since the bow variety is separated, we have Mij = { 1 if there is a tie between Vi and Uj , 0 otherwise. 10 A. Foster and Y. Shou Definition 3.1. For 0 ≤ i ≤ n, 1 ≤ j ≤ m, define sij = i∑ i′=1 Mi′j . Furthermore, define a “kl-pair” to be a tuple of indices (i, j, j′) such that Mij = k, Mij′ = l, and j < j′. Theorem 3.2. Given a separated brane diagram D and a fixed point p ∈ C(D)T, we have Tp C(D) = ∑ (i,j0,j1)01-pair ( uj0 uj1 hsij1−sij0 + uj1 uj0 hsij0−sij1+1 ) as elements of K0 T(p). Proof. To aid in our proof, let us establish some notation. For 1 ≤ j ≤ m and r ∈ N, let r ∗ uj = uj + uj h + uj h2 + · · ·+ uj hr−1 ∈ K0 T(p). Using this notation, we can rephrase the formula for the fixed point restriction of tautological bundles of Section 2.3 as ξk\|p =  m∑ j=1 hk−n(skj ∗ uj) if k ≤ n, m∑ j=k−n+1 snj ∗ uj if n ≤ k ≤ n+m− 1, 0 otherwise. (3.1) Let ξ (j) k be the subspace of ξk\|p corresponding to the Uj butterfly. Then, each of the summands in (3.1) corresponds to a ξ (j) k . Namely, we have ξ (j) k =  hk−n(skj ∗ uj) if k ≤ n, snj ∗ uj if n ≤ k ≤ n+ j − 1, 0 otherwise. (3.2) It will also be helpful to summarize the tangent bundle formula (2.1) using the diagram ξ1 · · · ξn−1 ξn ξn+1 · · · ξn+m−1 u1 u2 um−1 um 1 −1−h h 1 h 1 −1−h h −1 1−h −1+h h 1−h h 1−h −1+h h1 1 1 1 (3.3) where ξk = ξXk . Finally, divide the diagram (3.3) into four types of pieces Tangent Weights and Invariant Curves in Type A Bow Varieties 11 ξi−1 ξi −1 1 h −h P1i = Hom(ξi−1, ξi) + hHom(ξi, ξi−1) − End(ξi−1) − hEnd(ξi) for 1 ≤ i ≤ n. ξn+j−1 ξn+j1−h −1+h P2j = (1 − h)Hom(ξn+j , ξn+j−1) + (−1 + h) End(ξn+j) for 1 ≤ j ≤ m. ξn+j−1 ξn+j uj h1 P3j = Hom(uj , ξn+j−1)+hHom(ξn+j ,uj) for 1 ≤ j ≤ m, where ξn+m = 0 by convention. ξn −1+h P4 = (−1 + h) End(ξn). Inspecting the tangent bundle formula (2.1) and the fixed point restriction formula (3.1), we see that Tp C(D) = f(h) + ∑ j0 ̸=j1 uj0 uj1 gj0j1(h), where f, gj0j1 ∈ C [ h±1 ] for all 1 ≤ j0, j1 ≤ m. Fix j0 < j1 and define δ (j) i = ξ (j) i − ξ (j) i−1 and ζ (j) i = ξ (j) i − hξ (j) i−1. Then, the contribution of P1i to gj0j1 is given by P̂1i = −hHom ( ζ (j1) i , δ (j0) i ) . The only nonzero contribution of the P2j is P̂2j0 = (1− h)Hom ( ξj1n+j0 , ξ (j0) n+j0−1 ) . From (3.2), we have ξ (j1) n+j0 = ξ (j1) n and ξ (j0) n+j0−1 = ξ (j0) n . Thus, P̂2j0 cancels with the contribu- tion P̂4 of P4. Finally, the only nonzero contribution of the P3j is P̂3j0 = hHom ( ξ (j1) n+j0 ,uj0 ) . Summing everything together yields uj0 uj1 gj0j1 = P̂3j0 + n∑ i=1 P̂1i = n∑ i=1 P̂1i + m∑ j=1 ( P̂2j + P̂3j ) + P̂4 = hHom ( ξ (j1) n+j0 ,uj0 ) − n∑ i=1 hHom(ζ (j1) i , δ (j0) i ) = hHom ( ξ(j1)n ,uj0 ) − n∑ i=1 hHom(ζ (j1) i , δ (j0) i ). Note that ξ (j) n = ∑n i=1 h n−iζ (j) i . Substituting into the first term above results in uj0 uj1 gj0j1 = n∑ i=1 hHom ( ζ (j1) i ,hi−nu0 − δ (j0) i ) . 12 A. Foster and Y. Shou The following formulas can be derived easily from (3.2): δ (j) i = hi−n(uj + (Mij − 1)ujh −sij ), ζ (j) i = hi−n ( Mijujh −sij+1 ) . From these formulas, we can deduce gj0j1 = n∑ i=1 (1−Mij0)Mij1h sij1−sij0 = ∑ i (i,j0,j1) 01-pair hsij1−sij0 . From a symmetric calculation, one can obtain gj1j0 = ∑ i (i,j0,j1)01−pair hsij0−sij1+1. Alternatively, once can appeal to the selfduality of the tangent bundle: T C(D) = hT C(D)∨. This selfduality property follows from general considerations but also from the explicit construc- tion of a polarization bundle in [23, Section 4.4.2]. Finally, a similar calculation can be done for f P̂1i = − m∑ j=1 (Mijh sij +Mij(Mij − 1)), P̂2j = 0, P̂3j = snj−1∑ k=0 h−k, P̂4 = (−1 + h) m∑ j=1 End ( ξ(j)n ) . Since Mij(Mij − 1) = 0, we have n∑ i=1 P̂1i = − m∑ j=1 n∑ i=1 Mijh sij = − m∑ j=1 snj∑ k=1 hk, from which is easy to see that ∑ i P̂1i + ∑ j P̂3j = −P̂4. It follows that f = 0, completing the proof. ■ Suppose now that D is not separated and p ∈ C(D)T. By applying a sequence of Hanany– Witten transitions, the tie diagram for p can be transformed into a separated diagram D̃, representing a fixed point p̃ ∈ C(D̃)T̃ on a separated bow variety. This changes the separating line, but not the BCT. Hence, we can immediately apply Theorem 3.2 to obtain the tangent weights at p̃. As discussed in Section 2.4, these differ from the tangent weights at p only in the exponents of h. Starting from D̃ and going to D, each time we move a D5 brane U to the left, we must multiply u by h. Corollary 3.3. Let D be a brane diagram and p ∈ C(D)T. Let M be the corresponding BCT and σj be the number of entries in the j-th column of M lying below the separating line. Then, the tangent weights at p are given by Tp C(D) = ∑ (i,j0,j1)01-pair ( uj0 uj1 hsij1−sij0+σj0 −σj1 + uj1 uj0 hsij0−sij1+σj1 −σj0 +1 ) . Observe that the tangent weights come in pairs of the form w+hw−1. This is a manifestation of the fact that bow varieties admit polarization bundles [23, Section 4.4.2]. Theorem 3.2 combined with the dimension formula [17, equation (4)] gives us a roundabout proof of the following purely combinatorial result. Tangent Weights and Invariant Curves in Type A Bow Varieties 13 Corollary 3.4. Let M be a BCT with margins r, c. Then, the number of 01-pairs in M is 1 2 ( m∑ j=1 (cj(cj + 1) + cj−1(cj−1 + 1)) + 2 n∑ i=1 ri−1ri − 2 m∑ j=1 c2j − 2 n−1∑ i=1 r2i ) , where cj = ∑j j′=1 cj′, and ri = ∑i i′=1 ri′. In particular, this number depends only on the margins. 4 Butterfly surgeries and invariant curves In this section, we will develop a combinatorial representation of T-invariant curves involving surgery operations on pairs of butterflies (see Section 2.2). Our goal is to identify all T-invariant curves. We will accomplish this using an exhaustive approach. For each fixed point p and tangent weight w that appears with multiplicity k at p, we will produce a k-dimensional pencil of invariant curves with tangent weight w at p. 4.1 Torus invariant curves Let us begin by collecting some standard facts regarding invariant curves that will be used throughout what follows. These facts can be found in [9, Section 7] and [2, Section 7.2]. We address specifically the properties of invariant curves in bow varieties, though these properties are quite general and hold for a much larger family of spaces. Fix a bow variety C(D) with the corresponding action by the torus T. By a “T-invariant curve” γ, we mean the closure of a 1-dimensional T-orbit in C(D). The following are true of T-invariant curves: � A 1-dimensional T-orbit is isomorphic to C×, and its closure is obtained by adding up to two T-fixed points. � An invariant curve is compact if it contains two fixed points and noncompact otherwise. Compact invariant curves can be parametrized by equivariant embeddings CP1 → C(D), and noncompact invariant curves with one fixed point can be parametrized by equivariant embeddings C → C(D). � If γ is an invariant curve, and p ∈ γ is a fixed point, then Tpγ is a T-weight space of Tp C(D). If p1, p2 ∈ γ are two distinct fixed points, then the T-weights of Tp1γ and Tp2γ are reciprocols. � In contrast to [9], bow varieties may contain infinitely many invariant curves. In this case, the curves can be arranged into parametrized families or “pencils”. Pencils of invariant curves will be described in Section 4.2. An invariant curve is a special case of a pencil, where the dimension of the pencil is 1. The pencils of invariant curves have important implications for the study of the equivariant cohomology of C(D). When C(D) is equivariantly formal and has only finitely many fixed points and invariant curves (no multidimensional pencils), [9] shows that the map H∗ T(C(D)) → H∗ T ( C(D)T ) induced by the inclusion of the fixed point locus is an injective ring homomorphism whose image is characterized by simple matching conditions on pairs of fixed points joined by an invariant curve. When multidimensional pencils of invariant curves are present, however, these matching conditions are not sufficient, and must be supplemented by more general matching conditions. See [4] and [2, Section 7.4]. It also appears that not all bow varieties are equivariantly formal [10]. From now on, we will only be interested in invariant curves containing at least one fixed point. The following result will be pivotal in classifying all such curves. 14 A. Foster and Y. Shou Figure 8. Fixed point with the site of a butterfly surgery outlined. Proposition 4.1. Let C(D) be a bow variety and p ∈ C(D)T. Then, we have (1) All T-invariant curves containing p are smooth. (2) If γ1 and γ2 are T-invariant curves containing p, and Tpγ1 = Tpγ2 as vector subspaces of Tp C(D), then γ1 = γ2. Proof. Let p ∈ U ⊂ C(D) be an affine T-invariant neighborhood. By the Luna slice theo- rem [12, Appendix of Chapter 1], there exists a T-equivariant morphism ϕ : V → Tp C(D) such that p ∈ V ⊂ U is a T-invariant neighborhood, ϕ(p) = 0, and ϕ is étale. Due to equivariance, ϕ maps invariant curves containing p to invariant curves containing 0. It can be easily seen from the structure of the tangent weights provided by Corollary 3.3 that the only invariant curves in Tp C(D) containing 0 are lines. Étale morphisms induce isomorphisms on tangent spaces. This proves (1). Moreover, since our spaces are smooth, ϕ restricts to a diffeomorphism on some analytic neighborhood of p. This proves (2). ■ Our approach will be to construct a collection of invariant curves containing p such that every T-invariant tangent line at p appears as the tangent line to one of the curves. Then, Proposition 4.1 will guarantee that all curves containing p are accounted for. 4.2 Butterfly surgery A fixed point p1 ∈ C(D)T corresponds to a butterfly diagram, that is, a collection of butterflies, one centered on each D5 brane. We stack the butterflies on top of each other, so that the centers Tangent Weights and Invariant Curves in Type A Bow Varieties 15 of the butterflies listed from top to bottom go from left to right in D. Fix two butterflies β1 and β′1 with centers U and U ′, respectively. Suppose that there is a (possibly disconnected) subgraph s of β1, such that stacking s below β′1 and creating new edges between β′1 and s according to the rules of Section 2.2 results in a butterfly β′2. The subgraph s, called the site of the surgery, must be translated vertically, in a rigid fashion, without any lateral movement, rotation, or deformation, and we do not create new edges within s. Let β2 be obtained by deleting s and all adjacent edges from β1, and assume that β2 is also a butterfly. The operation of replacing β1 with β2 and β ′ 1 with β ′ 2 is called a “butterfly surgery”. It transforms the butterfly diagram for p1 into the butterfly diagram for another fixed point p2 ∈ C(D)T. We will use S to denote butterfly surgeries. Clearly, in a butterfly surgery, s must contain all edges of β1 with both endpoints within s. Moreover, the 0-momentum condition forces s to be A, A−1, B, C, D-invariant. By this, we mean that all B, C, D edges in β1 incident to s are directed into s, and s contains all incident A edges. An invariant curve arises from a butterfly surgery as follows. Consider the newly created edges in the butterfly diagram for p2. Each edge starts in β′1 and ends in the translated copy of s, which we will call s′. Create the corresponding edges starting in β′1 and ending in s ⊂ β1 in the butterfly diagram for p1, as in Figure 10. Identifying vertices with basis vectors of the WX spaces and interpreting the edges as linear maps, the resulting graph gives an element p̃γ ∈ M. The 0-momentum condition holds by construction, as the rules in Section 2.2 governing the vertices and edges of a butterfly directly reflect the 0-momentum conditions (see the proof of [17, Theorem 4.8]). The S1, S2, and ν stability conditions hold as well. Lemma 4.2. The point p̃γ ∈ M satisfies the stability conditions S1, S2, and ν. Proof. For each D5 brane U0, S1 and S2, place a condition on the AU0 , BU0 , aU0 , bU0 maps. Let s ∈ ker(AU0) ∩ ker(bU0). Inspecting the structure of p̃γ , s must be in the span of the vertices of the U0 butterfly above the source of the bU0 edge. However, sufficiently many applications of BU0 + will result in a vector no longer in the kernel of bU0 . This proves S1. For S2, observe that im(AU0) contains all vertices under U0 − except possibly for some in the U0 butterfly. Any vertex of the U0 butterfly can be reached from im(aU0) by sufficiently many applications of BU0 − . To verify the ν stability condition, observe that for any D5 brane U0, every vertex of the U0 butterfly can be reached from the target vertex of aU0 or from the highest vertex under U+ 0 by following a sequence of A, A−1, B, C, D edges. By following an A−1 edge, we simply mean following an A edge in the direction opposite to its orientation. ■ It follows that p̃γ ∈ M̃s descends to a point pγ ∈ C(D). Denote the T-orbit closure of pγ by γ. For a vertex v of the butterfly diagram, recall that its height (see Definition 2.2) is denoted by y(v). Now, let v ∈ s ⊂ β. Applying the surgery yields the corresponding vertex v′ ∈ s′ ⊂ β′. The change in height ∆Sy = y(v′)− y(v) is constant with respect to v ∈ s. Consider the action of T on pγ . By a suitable change of basis in WX spaces (the G-action), a representation of the resulting point can be obtained by multiplying the new edges by the weight h−∆Syu/u′ and leaving the other edges alone. By fixing u, h and taking u′ → ∞, we see that p1 ∈ γ. Next, set u′ = h = 1, so that the new edges are acted upon by u. Apply the G-action that multiplies each vertex in s by u−1. Since all edges in β1 adjacent to s point into s, those edges will be multiplied by u−1. Meanwhile, the factor on the new edges will cancel to 1, and all other edges will be unchanged. Taking u → ∞, the edges adjacent to s are killed, leaving behind β2 and β′2. Thus, p2 ∈ γ. There is a surjective map C× → T.pγ given by sending t ∈ C× to the point obtained by multiplying the new edges by t. It follows that γ is at most 1-dimensional. Since γ contains two distinct fixed points p1, p2, it must be that dim(γ) = 1. Therefore, γ is a compact invariant curve. Its tangent weight at p1 is precisely h−∆Syu/u′, and its tangent weight at p2 is h ∆Syu′/u. 16 A. Foster and Y. Shou Figure 9. New fixed point resulting from butterfly surgery. From general considerations, γ is isomorphic to CP1. We obtain an explicit parametrization by multiplying each of the new edges by t ∈ C. This gives us an isomorphism C → γ \ {p2}. Compactifying C with a point ∞ and mapping ∞ to p2 yields a parametrized curve γ : CP1 → C(D) (we are abusing notation and using γ to refer both to the parametrization and its image). Note that performing the inverse butterfly surgery yields the same invariant curve with a different parametrization, one with γ(0) = p2. Suppose s has k connected components s1, . . . , sk. By multiplying the new edges pointing into si by zi, where z = (z1, . . . , zk) ∈ (C \ {0})k, we obtain a k-parameter family of T-invariant curves containing p1 and p2. In the closure of their union, there are other fixed points formed by stacking some of the connected components but not others, as well as invariant curves between them. For each subset I ⊂ {1, . . . , k}, consider the fixed point pI to be obtained from the butterfly surgery with site the union of components si for i ∈ I. So p1 = p∅ and p{1,...,k} = p2. Now consider the hypercube ( CP1 )k endowed with the diagonal T-action with every tangent weight equal to h−∆Syu/u′ at (0, . . . , 0). We obtain a fixed point xI ∈ ( CP1 )k for each I ⊂ {1, . . . , k} where the i-th component of xI is ∞ if i ∈ I and 0 otherwise. Next we obtain a T-equivariant map ΓS : ( CP1 )k → C(D) that sends each xI to pI . We define this map first on the interior and on the facets containing x∅ by sending each z = (z1, . . . , zk) ∈ Ck to the butterfly diagram of p∅ with new edges ofS pointing into si multiplied by zi, providing all curves containing p∅ in the image of ΓS. We similarly define that for each z = (z1, . . . , zk) ∈ ( CP1 )k with ∞ in each component of index in I ⊂ {1, . . . , k}, ΓS maps z to the butterfly diagram of pI with all new edges of S that still can be drawn in the butterfly diagram of pI (the ones with site Tangent Weights and Invariant Curves in Type A Bow Varieties 17 Figure 10. Explicit construction of an invariant curve corresponding to butterfly surgery. The dashed lines are the new edges added to the butterfly diagram. Taking the T-orbit closure of the element of M̃s represented by this graph yields an invariant curve with tangent weight ( u1 u2 h )±1 at its fixed points. components si where zi is finite) multiplied by the appropriate finite z-values. Despite the fact that only the curves in ΓS ( (C \ {0})k ) are between p1 and p2, we will still call ΓS a pencil of invariant curves between p1 and p2. Consider the k individual butterfly surgeries corresponding to each of the k connected components of s. These butterfly surgeries produce invariant curves γ1, . . . , γk. We will show in Proposition 6.3 that the tangent spaces of these curves at p1 (and also at p2) are linearly independent. Moreover, we have ∀z ∈ Ck \ {0}, ∂ ∂t ∣∣∣∣ t=0 ΓS(tz) = k∑ i=1 ziγ ′ i(0). It follows that each invariant curve in ΓS has a distinct tangent space at p1 and is therefore distinct. Similar considerations can be made for each other fixed point pI . In other words, ΓS is injective. The results of this section are summarized in the following lemma. Lemma 4.3. Suppose a butterfly surgery S sends a T-fixed point p1 to the T-fixed point p2. Let the site s be moved from the U butterfly to the U ′ butterfly by S, where k is the number of connected components of s. Then, there is a k-dimensional pencil ΓS : ( CP1 )k → C(D), a T-equ- ivariant map where ΓS ( (C \ {0})k ) consists of compact invariant curves containing p1 and p2. Moreover, each curve in the pencil has tangent weight u u′h−∆Sy, at p1, where ∆Sy = y(v′) − y(v), for any v ∈ s and corresponding v′ ∈ s′. Here, y denotes the height of a vertex (see Definition 2.2). 18 A. Foster and Y. Shou z1 z2 Figure 11. Explicit 2-dimensional pencil of invariant curves with tangent weight ( u1 u2 )±1 at its fixed points. It will be shown that butterfly surgeries capture all of the compact invariant curves in C(D). To do so, our approach also requires us to construct noncompact invariant curves. 4.3 Botched butterfly surgery For the remainder of Section 4, we will assume that M is separated. Analogous results will follow for the nonseparated case by tracing the relevant constructions through Hanany–Witten transitions. See the proof of [15, Proposition 8.1] for the explicit construction of the Hanany– Witten isomorphism. We explicitly trace a butterfly through the Hanany–Witten transition in [23, Section 3.2.4]. First, let us define a surgery operation that closely resembles a butterfly surgery, but is not exactly a butterfly surgery. Let D be a separated brane diagram with n NS5 branes and m D5 branes. As in Section 3, index the branes from left to right. The NS5 branes are denoted V1, . . . , Vn, the D5 branes are denoted U1, . . . , Um, and the segments (D3 branes) are denoted X0, X1, . . . , Xn+m, where X0 andXn+m are the infinite left and right segments. We will divide D into a “left side” and a “right side”. The left side comprises X0, . . . , Xn and V1, . . . , Vn. The right side is the remaining portion of the diagram. The notion of left and right side extends naturally to butterflies and butterfly diagrams. The left side of a butterfly always has a certain inverted staircase-like shape. See Figure 12 for an example. Fix p ∈ C(D)T and consider its butterfly diagram. Note that the portion of the butterfly diagram below Xn and the right side will be the same for all fixed points. Consider a surgery operation S as in Section 4.2 on the left side of the butterfly diagram. That is, we take a B, C, D-invariant subgraph s of the left side of the Uj butterfly and vertically translate it to stack it Tangent Weights and Invariant Curves in Type A Bow Varieties 19 1 2 3 4 5 7 9 9 10 11 6 4 Figure 12. The butterfly diagram of a fixed point of a separated bow variety. The cyan dashed line indicates the boundary between the “left side” and “right side”. Two different botched butterfly surgeries are outlined in solid gray. The surgery with site in the U1 butterfly is subject to and satisfies constraint (4.1). The other surgery is not subject to this constraint. The additional edges and vertices added to s to form s are outlined in dotted gray. below the left side of the Uj′ butterfly, for some 1 ≤ j ̸= j′ ≤ m. Assume that the left side of the resulting diagram is the left side of some butterfly diagram (not necessarily associated to the same bow variety). Note that all butterfly surgeries meet these conditions. The key difference is that butterfly surgeries cannot alter the Xn column. If s includes at least one vertex in the Xn column, then call S a “botched butterfly surgery”. See Figure 12 for an example. In 20 A. Foster and Y. Shou Section 4.2, the B, C, D-invariance of s emerged from other constraints on s. Here, we must impose this property as an additional condition. We will produce a noncompact invariant curve from a botched butterfly surgery following the approach of Section 4.2. The right side of the butterfly diagram is organized in a grid formed by horizontal leftward edges, which represent A maps, and vertical downward edges, which represent −B maps (see Figure 12). Let s be the minimal A, A−1, B, C, D-invariant subgraph containing s. Let s′ be a copy of s translated according to S. Take the edges extending from the Uj′ butterfly to s′ imputed by the rules of Section 2.2, and create the corresponding edges between the Uj′ butterfly and s. As in Section 4.2, the resulting graph determines an element p̃γ ∈ M. If j > j′, it is easy to verify the 0-momentum condition. If j < j′, all momentum conditions will be satisfied except for BUjAUj − AUjB ′ Uj + aUjbUj = 0. Due to the structure of separated brane diagrams, the number of vertices in the Xn, Xn+1, . . . , Xn+k−1 column of the Uk butterfly is given by ck. Recall that c is the margin vector of D5 brane charges (see Section 2.2). Suppose s contains the bottom r vertices of the Xn column, where r ≥ cj − cj′ + 1. (4.1) Label the vertices of the Xn+j−1 column of the Uj butterfly from top to bottom as v1, . . . , vcj . Label the vertices of the Xn+j column of the Uj′ butterfly from top to bottom as v′1, . . . , v ′ cj′ . Add additional edges according to the following rule: � Create negative blue edges from v′k+cj′−cj+r to vk for 1 ≤ k ≤ cj − r. Recall that the blue edges represent A. � Create a negative green edge from v′cj′−cj+r to the Uj framing vertex. Recall that the green edge directed into the framing vertex represents −b (Section 2.2). This corrects the momentum, so that the 0-momentum condition is satisfied. This rule is depicted in Figure 14a. The point p̃γ is illustrated in Figure 13 for both of the surgeries in Figure 12. Using similar arguments to those in Section 4.2 (see Lemma 4.5), one can show that p̃γ ∈ M̃s, the orbit of its image pγ ∈ C(D) is 1-dimensional, and its closure γ is an invariant curve con- taining p. Additionally, if s has k connected components, then a k-dimensional pencil ΓS of invariant curves containing p can be constructed. These results are summarized by the following analog to Lemma 4.3. The question of compactness is postponed until Section 6. Lemma 4.4. Fix a separated bow variety C(D) and p ∈ C(D)T. Let S be a botched butterfly surgery moving the site s from the Uj butterfly to the Uj′ butterfly. If j < j′, impose the additional condition that s contains at least cj − cj′ +1 vertices of the Xn column, where c is the margin vector of D5 brane charges. Then, there exists a k-dimensional pencil ΓS of T-invariant curves containing p. Each curve has tangent weight u u′h−∆Sy, at p. 4.4 Nonsurgery curves We retain the conventions of Section 4.3 for this section. Fix 1 ≤ j < j′ ≤ m. We will construct max{0, cj′ − cj} many invariant curves. To construct the i-th curve, � create a green edge from v′i to the Uj framing vertex, � create blue edges from v′i+k to vk for 1 ≤ k ≤ cj . Figure 14b depicts these added edges. Using similar arguments to those in Section 4.2 (see Lemma 4.5), one can show that this results in a point of C(D) with a 1-dimensional orbit having p in its closure. The tangent weight of this curve at p can easily be determined by examining the added b edge and comparing the heights of the target and source. Tangent Weights and Invariant Curves in Type A Bow Varieties 21 −1 −1 Figure 13. Construction of the invariant curves corresponding to the two botched butterfly surgeries in Figure 12. Lemma 4.5. Suppose p̃γ ∈ M is obtained through a botched butterfly surgery or nonsurgery construction with respect to p ∈ C(D)T. Then, (1) p̃γ satisfies the S1, S2, and ν stability conditions. (2) pγ ̸= p, so that dim(T.pγ) > 0. 22 A. Foster and Y. Shou Uj -1 -1 -1 (a) botched surgery Uj (b) nonsurgery Figure 14. Local depictions of (a) the curve associated to a botched surgery moving a part of a higher butterfly to a lower butterfly and (b) a nonsurgery invariant curve. In (a), those visible vertices that are contained in s are outlined in gray. Proof. (1) S1 follows from the fact that the A maps are injective. S2 can be verified in the same way as for butterfly surgeries (see Lemma 4.2). For the ν condition, we have to modify the strategy for butterfly surgeries slightly. The novelty is in botched butterfly surgeries moving part of a higher butterfly to a lower butterfly and nonsurgery curves. In these cases, for some vertices w′ of the Uj′ butterfly under U+ j , there are two distinct AUj edges with source w′. Namely, Aγ Uj (w′) = AUj (w ′) + w, where w is a vertex of the Uj butterfly under U− j , and we use the superscript of γ to distinguish between the maps of p̃ and p̃γ . However, w can be obtained by repeatedly applying B− Uj to aUj (±1). Any vertex of the Uj′ butterfly can be obtained by applying a sequence of A, B, C, D maps to aUj′ (±1) and canceling out the components of AUj landing in the Uj butterfly, as described above. (2) Let p̃ ∈ M̃s be the point described by the butterfly diagram of p. The statement for bot- ched butterfly surgeries moving part of a higher butterfly to a lower butterfly and nonsur- gery curves follows immediately from the fact that bUj vanishes in p̃ but not in p̃γ . For butterfly surgeries moving part of a lower butterfly to a higher butterfly, let ℓ = d Uj′ U− j′ , the number of vertices of the Uj′ butterfly below U− j′ . So, B ℓaUj′ vanishes in p̃ but not in p̃γ . ■ Lemma 4.6. Let C(D) be a separated bow variety and let c = (c1, . . . , cm) denote the margin vector of D5 brane charges. Let p ∈ C(D)T, and 1 ≤ j < j′ ≤ m. Then, there are max{0, cj′−cj} invariant curves containing p with tangent weights uj uj′ h, uj uj′ h2, . . . , uj uj′ hmax{0,cj′−cj} at p. Tangent Weights and Invariant Curves in Type A Bow Varieties 23 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 1  (a) (b) Figure 15. The (a) BCT and (b) Young diagrams associated to the fixed point of Figure 12. Our (nonstandard) convention is to draw the Young diagrams aligned on the right with longer rows on top. 4.5 Young diagram surgery The combinatorics of Sections 4.2 and 4.3 can be reformulated using the more familiar lan- guage of partitions and Young diagrams. Again, we fix a separated bow variety C(D) and a fixed point p ∈ C(D)T. Let M be the associated BCT and c = (c1, . . . , cm) be the margin vector of D5 brane charges. For each D5 brane Uj , define a partition λ(j) with cj distinct parts λ (j) 1 > · · · > λ (j) ci > 0 by (1) letting i1 < i2 < · · · < icj be the indices for which M(ik, j) = 1 and (2) setting λ (j) k = n− ik + 1. In other words, the parts of the partition correspond to the vertical positions of the 1’s in the j-th column of M measured from the bottom. The Young diagrams associated with these partitions will be drawn in a nonstandard way with the boxes aligned on the right. Each row of boxes represents a part of the partition, and the longer rows will be drawn on top. See Figure 15 for an example. We will use λ(j) to denote the partition as well as its Young diagram. 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 ̸= j′, let s be a set of boxes of λ(j) with the property that if a box is in s, then any box below it is also in s. Suppose deleting s from λ(j) leaves a Young diagram with distinct parts, and stacking those boxes below λ(j ′) results in a Young diagram with distinct parts. Then, call the operation on the set of Young diagrams { λ(j) \| 1 ≤ j ≤ m } that deletes s from λ(j) and stacks it below λ(j ′) a “Young diagram surgery”. If s does not contain any boxes in the rightmost column, then the Young diagram surgery corresponds to a butterfly surgery. Otherwise, if s contains at least one box of the rightmost column, then the Young diagram surgery corresponds to a botched butterfly surgery. The constraint (4.1) is reformulated as s contains at least cj − cj′ + 1 boxes of the rightmost column. (4.2) This is simply a reformulation of Sections 4.2 and 4.3. By Lemmas 4.3 and 4.4, Young diagram surgeries satisfying the additional constraint (4.2) whenever a box in the rightmost column is moved correspond to pencils of invariant curves. Given a box bx in a Young diagram λ, define its height y(bx) by y(bx) = 1 − i, where bx belongs to the i-th row of λ. The tangent weight of the invariant curve is given by comparing the heights of any box in s before and after the surgery. Namely, the exponent of h is −∆Sy = y(bx)− y(bx′), where bx is any box in s and bx′ 24 A. Foster and Y. Shou (a) w = u1 u3 h (b) w = u3 u2 Figure 16. The two botched butterfly surgeries of Figure 12 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. is the result of applying S to bx. Treating the Young diagram as a subset of R2 and each box as a closed square, the number of connected components of s is precisely the number of connected components of the corresponding (botched) butterfly surgery. In Figure 16, we reformulate Figure 12 in the language of Young diagram surgeries. Lemma 4.7. Let D be a separated brane diagram and p ∈ C(D)T. Let S be a surgery on the corresponding Young diagrams that moves boxes from λ(j) to λ(j ′). If j < j′ and the site s contains at least one box of the rightmost column, then impose the additional constraint that s contains at least cj − cj′ +1 boxes of the rightmost column, where c = (c1, . . . , cm) is the margin vector of D5 brane charges. Then, there is a k-dimensional pencil ΓS of T-invariant curves containing p with tangent weight uj uj′ h−∆Sy, where k is the number of connected components of s. Lemma 4.8. Let S be a Young diagram surgery with connected site s. Then, S is uniquely determined by its action on any box in s. Proof. Since Young diagram surgeries are rigid translations, the action of S on a box in s completely determines the action of S on any other box in s. It is trivial to show that the action S on any box in s completely determines whether or not each adjacent box is in s. The result follows from these two statements and the connectedness of s. ■ Corollary 4.9. Let S1 and S2 be two connected Young diagram surgeries with ∆S1y = ∆S2y. Then, their sites are either disjoint or equal. 5 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 ( 1 0 0 1 ) minor and replacing it with ( 0 1 1 0 ) or vice-versa. A similar type of combinatorial move will provide an alternative combinatorial code for compact T-invariant curves. Remark 5.1. Swap moves are also used in [26], where they are called “simple moves” and appear in a generalization of the Chevalley–Monk formula. Let us first introduce some preliminary notions. Tangent Weights and Invariant Curves in Type A Bow Varieties 25 Definition 5.2. Let M be the BCT of some p ∈ C(D)T. Call a minor M =M(i1, i2, j1, j2) = (Mij) i=i1,i1+1,...,i2 j=j1,j2 consisting of two columns and any number of consecutive rows of M a “block”. Furthermore, define δ ( M ) =  i2∑ i=i1 (Mij1 −Mij2) if j1 < j2, i2∑ i=i1 (Mij2 −Mij1) if j1 > j2. Call a block M “matched” if δ ( 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 6.5. Proposition 5.3. Let M be the BCT of some p ∈ C(D)T where D is separated with m D5 branes. Then, using Definition 3.1, (1) Every 01-pair (i, j0, j1) or 10-pair (i, j1, j0) in M corresponds to a Young diagram surgery S moving boxes from λ(j1) to λ(j0). (2) In the context above, if M(i, i′, j0, j1) is matched for some i′ > i, then the site s of S does not contain any boxes of the rightmost column of λ(j1). If no such matched block exists, then s contains ∣∣δ(M(i, n, j0, j1) )∣∣ boxes of the rightmost column of λ(j1). (3) The relative displacement of the boxes during S is given by ∆S(y) = i∑ k=1 (Mkj1 −Mkj0)− 1 = sij1 − sij0 − 1. Proof. 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 × 2. The general case follows from applying this argument to every pair of columns in M . To simplify the notation, let M(i1, i2) = M(i1, i2, 1, 2). Suppose the i1-th row of M is (1, 0). Let M be the smallest matched block of the form M(i1, i2) where i1 < i2, if such a block exists. Otherwise, if no such matched block exists, let M = M(i1, n). Recall that each 1 in the j-th column of M represents a part of the Young diagram λ(j). Let λ (j) be the partition consisting of the parts of λ(j) associated with the 1’s in the j-th column of M . Denote the number of parts of λ (j) by cj . Clearly,∣∣δ(M(i1, i + 1) ) − δ ( M(i1, i) )∣∣ ≤ 1 for all i1 ≤ i < n, and δ ( M(i1, i1) ) = 1. If it were the case that δ ( M ) < 0, then there would exist i > i1 for which δ ( M(i1, i) ) = 0. It follows that δ ( M ) ≥ 0, and therefore c1 ≥ c2. Since the first row of M is (1, 0), we have λ (1) 1 > λ (2) 1 . We claim that λ (1) i+1 ≥ λ (2) i for all i = 1, . . . , c2 − 1. Suppose to the contrary that for some i, we have λ (1) i+1 < λ (2) i . 26 A. Foster and Y. Shou Let the (1, 0) row corresponding to λ (1) i+1 be row i′+1 of M . Then, we have δ ( M(i1, i ′+1) ) ≤ 1. It follows that δ ( M(i1, i ′) ) ≤ 0. By the intermediate value argument above, there is i1 < i < i2 such that M(i1, i) is matched. This contradicts the definition of M . These inequalities imply that there is a connected Young diagram surgery which moves λ (1) i − λ (2) i > 0 boxes from λ (1) i to λ (2) i . If M is matched, then this surgery constitutes a surgery on λ(1) and λ(2). Otherwise, this surgery can be extended to a surgery on λ(1) and λ(2) by moving the bottom δ ( M ) > 0 rows of boxes in λ (1) completely below λ(2). The displacement ∆Sy is easily seen to be ∑i1 k=1(Mk1 −Mk2)− 1. ■ Let M be a minimal matched block whose top row is a 01- or 10- pair. The proof of Propo- sition 5.3 shows that there is a connected butterfly surgery transforming M by exchanging the columns of M . The displacement of the surgery is given by applying the formula in Proposi- tion 5.3 (3) to the top row of 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. Definition 5.4. Let M be a BCT. A “block swap move” ψ on M consists of swapping the columns of a matched block M within M , where the first and last rows of M are (1, 0) or (0, 1). Call ψ indecomposable if 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. Proposition 5.5. Let p ∈ C(D)T, with D separated. Let M be the BCT of p. (1) Every block swap move ψ on M corresponds to a pencil Γ of compact invariant curves containing p. (2) Suppose ψ decomposes into k simultaneous indecomposable block swap moves ψ1, . . . , ψk. Then, we have dim(Γ) = k. (3) The BCT for any fixed point in Γ can be obtained by performing some subset of the inde- composable block swap moves ψi. (4) Suppose row i is the topmost row affected by ψ. Let (i, j0) be the index of the 0 in row i that ψ swaps to 1. Similarly, let (i, j1) be the index of the 1 in row i that ψ swaps to 0. Then, the tangent weight of Γ at p is uj1 uj0 h1+sij0−sij1 . It follows from Theorem 6.5 that all pencils of compact invariant curves in C(D) are associated with block swap moves. 6 Classification of invariant curves Definition 6.1. Let C(D) be a separated bow variety and p ∈ C(D)T. Define three types of invariant curves containing p: I) curves arising from connected butterfly surgeries (see Section 4.2), II) curves arising from connected botched butterfly surgeries (see Section 4.3), III) nonsurgery curves (see Section 4.4). Tangent Weights and Invariant Curves in Type A Bow Varieties 27 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 6.7 that, in fact, type II and III curves are always noncompact. Because butterfly surgeries are reversible, a type I curve containing p1 is also type I when viewed with respect to its other fixed point p2. Proposition 6.2. The type I, type II, and type III curves containing a fixed point p form three disjoint sets. Moreover, no type II curve has the same tangent weight as a type III curve. Proof. It is easy to see from Lemma 4.6 and the constraint (4.1) 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 γ is constructed as the closure of the orbit of some point pγ for which we give an explicit representation in p̃γ ∈ M̃s. Suppose γ is type II and has tangent weight hruj/u ′ j at p. If j > j′, then BℓaUj′ ̸= 0 for ℓ = d Uj′ U− j′ . On the other hand, for any type I curve, BℓaUj′ = 0 in p̃γ . If j < j′, then bUj′ ̸= 0. Any type I curve, however, has bUj′ = 0. It follows that no curve is both type I and type II. Finally, if γ is type I, then p̃γ has b = 0, and if γ is type III, p̃γ has b ̸= 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. Proposition 6.3. Let γ1, . . . , γk be distinct invariant curves containing p ∈ C(D)T, 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 Γ with γ1, . . . , γk in its boundary. Moreover, every vector in Tpγ1 ⊕ · · · ⊕ Tpγk is in Tpγ for some γ ∈ Γ. Proof. Since the tangent weights of the γi are all equal, Corollary 4.9 implies that the sites of the surgeries associated with any type I and II γi are disjoint. Since any botched butterfly surgery must move the bottom vertex of Xn, there is at most 1 type II γi. We can also see from Lemma 4.6 that the tangent weights of the type III curves are distinct. Therefore, there is at most 1 type III γi. From Proposition 6.2, type II and type III curves never share the same tangent weight. Hence, if one of the γi is type II, then none are type III, and vice-versa. In summary, there is at most 1 i for which γi is not type I. It follows from the above that each γi is constructed by adding a set Ei of new edges to disjoint portions of the butterfly diagram of p. Attaching a factor of t ∈ C to each edge in Ei results in a lift γ̃i : C → M̃s of γi in a neighborhood of p. Section 4.2 describes this construction for type I curves. According to [15, Section 2.5], there is a complex ⊕ X End(WX) M N,α β such that Tp C(D) ∼= kerβ/ imα. In particular, α is the differential of the G-action and ker(β) ∼= Tp̃M̃s, where p̃ ∈ M̃s is the representative of p given by the butterfly diagram. We will not describe this complex in detail, as it is not required for the proof. Note that γ̃′i(0) + imα spans Tpγi. Fix (z1, . . . , zk) ∈ Ck − {0}, and let s = ∑k i=1 zi(γ̃ ′ i(0) + imα). Let p̃γ ∈ M be obtained from p̃ by adding Ei with a factor of zi for all i. The arguments of Sections 4.2, 4.3, and 4.4 can be applied to show that p̃γ ∈ M̃s and dim(T.pγ) = 1, where pγ is the image of p̃γ in C(D). 28 A. Foster and Y. Shou Therefore, γ = T.pγ is a T-invariant curve. As before, we can construct a lift γ̃ = ∑k i=1 ziγ̃i of γ. Putting everything together, we have 0 ̸= Tpγ = C { γ̃′(0) + imα } = C { k∑ i=1 zi ( γ̃′i(0) + imα )} = C{s}. It follows that Tpγ1, . . . , Tpγk are linearly independent, and every vector in Tpγ1 ⊕ · · · ⊕ Tpγk is in Tpγ for some invariant curve γ. Allowing z1, . . . , zk to vary yields the desired pencil Γ of invariant curves. See Section 4.2 for the construction of Γ in the case where all γi are type I. ■ Definition 6.4. We say that the pencil Γ in Proposition 6.3 is “spanned” by the curves γ1, . . . , γk. This leads us to our main result. Theorem 6.5. Let D be separated and p ∈ C(D)T. All T-invariant pencils of curves containing p are spanned by curves of type I, type II, and type III. Proof. Due to Proposition 6.3, it suffices to construct k distinct type I, II, or III T-invariant curves with tangent weight w at p for each tangent weight w of multiplicity k in Tp C(D). For simplicity, let us reduce to the case where m = 2, and the BCT M of p is an n× 2 matrix. The general case is obtained by applying the same argument to each pair of columns in M . From Theorem 3.2, the tangent weights at p come in two varieties: those of the form u1 u2 hr and those of the form u2 u1 hr. It is easy to see from Proposition 5.3 and Lemma 4.7 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 u1 u2 hr. There is one such weight associ- ated with each 01-pair. First, we consider the case where c1 ≤ c2. In this case, the extra constraint (4.2) is automatically satisfied. If there are no 10-pairs in M , then by Lemma 4.6, 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 5.3 is equal to the u1 u2 hr 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 3.2 or the displacement formula of Proposition 5.3 (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 c1 > c2. Let the row indices of the 10-pairs be i1, . . . , ic1 . Define M(i1, i2) =M(i1, i2, 1, 2) and δi = δ ( M(i, n) ) . Clearly, \|δn\| ≤ 1, δ1 = c1−c2, and \|δi−δi+1\| ≤ 1. Since δi − δi+1 = 1 if and only if M has a 10-pair in the i-th row, we have {1, . . . , c1 − c2} ⊂ {δik \| k = 1, . . . , c1}. Moreover, defining k1 = max{k \| δik = c1 − c2}, k2 = max{k \| δik = c1 − c2 − 1}, . . . , kc1−c2 = max{k \| δik = 1}, it follows from an intermediate value argument that k1 < k2 < · · · < kc1−c2 . We collect two basic facts: (1) If δi − δi′ = 0 for i < i′, then δ ( M(i, i′ − 1) ) = 0. (2) If δik − δik′ = 1 for k < k′, then δ ( M(ik + 1, ik′ − 1) ) = 0. Tangent Weights and Invariant Curves in Type A Bow Varieties 29 If ik1 > 1, applying (1) with i = 1, i′ = ik1 tells us thatM(1, ik1 −1) is matched. Applying (2) to consecutive indices in the ordered sequence k1, . . . , kc1−c2 tells us that the blocks lying between the 10-pairs in rows ik1 , . . . , ikc1−c2 are matched. If ikc1−c2 < n, the previous two sentences imply that M(ikc1−c2 + 1, n) is matched. Hence, rows ik1 , . . . , ikc2−c1 separate M into matched blocks. Our next goal is to show that every 10-pair except the those in rows ik1 , . . . , ikc1−c2 are asso- ciated with an invariant curve. By Proposition 5.3, 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 δi ≥ c1 − c2 + 1 (con- straint (4.2)). Let (i, 1, 2) be a 10-pair where i ̸= ik for k = 1, . . . , c1 − c2. Suppose δi ≤ c1 − c2. If δi < 1, then there exists i′ > i such that δ ( M(i, i′) ) = 0 by an intermediate value argument. In other words, (i, 1, 2) is the top of a matched block. If 1 ≤ δi ≤ c1 − c2, then δi = δik for some k. We have i < ik by definition of ik, so M(i, ik − 1) is matched by (1). Applying the argument above for the c1 ≤ c2 case to each matched block lying between rows ik1 , . . . , ikc1−c2 completes the proof. ■ Remark 6.6. While not necessary for the proof, it is easy to show that the 10-pairs in rows ik1 , . . . , ikc1−c2 correspond to Young diagram surgeries that move at least one block of the right column but fail constraint (4.2). At first glance, the existence of nonsurgery (type III) curves and the constraint (4.2) might seem mysterious. The proof of Theorem 6.5 elucidates their role. By Theorem 3.2, each 01-pair corresponds to two tangent weights, w and hw−1. Each 01-pair is also associated with a Young diagram surgery, by Proposition 5.3. 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 (4.2) kicks in to correct the surplus. Corollary 6.7. Let D be separated and p ∈ C(D). Invariant curves of type I containing p are compact, and those of type II or III are noncompact. Proof. We have already shown in Section 4.2 that type I curves are compact. Let γ be a compact invariant curve containing p1. Then, γ contains another fixed point p2. Recall that the tangent weights of γ at p1 and p2 are reciprocols. First, assume that Tp1γ = uj uj′ hr, where j < j′. Suppose to the contrary that γ is type II or III. In Sections 4.3 and 4.4, we constructed a point pγ,1 ∈ C(D) such that γ = T.pγ,1. From the form of the tangent weight, we see that pγ,1 is represented by a point p̃γ,1 ∈ M̃s with b ̸= 0. Viewing γ from the perspective of p2, Theorem 6.5 implies that γ belongs to a pencil of curves spanned by type I, II, and III curves containing p2. Thus, the surgery constructions with respect to p2 give us a point pγ,2 ∈ C(D) such that γ = T.pγ,2. In particular, pγ,1 and pγ,2 are in the same T-orbit. It follows that pγ,2 must be represented by an element p̃γ,2 ∈ M̃s with b ̸= 0. However, examining the constructions of Sections 4.2, 4.3 and 4.4, pencils of curves with tangent weight uj′ uj h−r, where j < j′, are given by orbit closures of points represented by elements of M̃s with b = 0. Contradiction! The case where Tp1γ = uj uj′ hr, where j > j′ is similar. For this case, we do not need to consider type III curves. If the curve is type II, then b = 0 and BℓaUj′ ̸= 0 for ℓ = d Uj′ U− j′ in p̃γ,1. Hence, we also have b = 0 and BℓaUj′ ̸= 0 in p̃γ,2. The invariant curves containing p2 with tangent weight uj′ uj h−r all violate one of these two properties. Contradiction! ■ 7 Examples of invariant curves We will apply the classification of Section 6 to describe the invariant curves of the example bow varieties from [17] in terms of Young diagram surgeries. 30 A. Foster and Y. Shou u1 u2 h−2 u2 u1 h2 u1 u2 h−1 u2 u1 h3 u1 u2 h−1 u2 u1 h2 u1 u2 h−1 u2 u1 h2 u1 u2 u2 u1 h2 u1 u2 h−1 u2 u1 h u1 u2 h−1 u2 u1 h u1 u2 u2 u1 h2 u1 u2 u2 u1 h u1 u2 u2 u1 h u1 u2 h u2 u1 h2 u1 u2 h−1 u2 u1 u1 u2 h u2 u1 h u1 u2 u2 u1 u1 u2 u2 u1 u1 u2 h u2 u1 h u1 u2 h u2 u1 u1 u2 h u2 u1 u1 u2 h u2 u1 h−1 u1 u2 h2 u2 u1 45 35 3425 24 23 15 14 13 12 45 35 3425 24 23 15 14 13 12 Figure 17. Illustration of T-fixed points and invariant curves of C(/1/2/3/4/5\2\) (with their T- weights), which is the 3d mirror dual of T ∗Gr(2, 5). 7.1 Invariant curves for C(/1/2/3/4/5\2\) Letting D = /1/2/3/4/5\2\, the bow variety C(D) is Hanany–Witten isomorphic to the 3d mirror dual of T ∗Gr(2, 5). Its T-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 U2 connected to both Vk and Vl, and U1 connected to the three remaining NS5 branes. This fixed point will be denoted kl. The invariant curves from Figure 17 were computed via Young diagram surgeries. As an example, consider the fixed point 13, whose BCT and Young diagrams appear in Figure 18. By Theorem 3.2, T13 C(D) = u1 u2 h+ u2 u1 + u1 u2 h+ u2 u1 . Using Young diagram surgeries, four invariant curves are constructed, depicted in Figure 19, one for each tangent weight. Each pair of surgeries that share a tangent weight (columns of Figure 19) have disjoint sites, and thus span pencils. The pencil of weight u2 u1 is spanned by two type I curves, so all the curves of this pencil are compact. The other pencil has weight u1 u2 h 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, given by moving a single box in the rightmost column, but it is an example of a surgery that does not satisfy constraint (4.2), as c1 − c2 + 1 = 2. 7.2 Invariant curves for C(/2/3/5\3\2\) In [17], the fixed curves are depicted for the bow variety C(\1/2/2\2\1/). This is Hanany– Witten isomorphic to the separated bow variety C(/2/3/5\3\2\). This isomorphism is T- equivariant up to the reparametrization of T described in Section 2.4. The following computation will be done using tangent weights of fixed points in C(/2/3/5\3\2\), which differs from [17] in the exponents of h. The invariant curves of this bow variety can be seen in Figure 20. Tangent Weights and Invariant Curves in Type A Bow Varieties 31 0 1 1 0 0 1 1 0 1 0  (a) (b) Figure 18. The (a) BCT and (b) Young diagrams associated to the fixed point 13 of Figure 17. (a) w = u1 u2 h (b) w = u2 u1 (c) w = u1 u2 h (d) w = u2 u1 Figure 19. Four Young diagram surgeries representing four invariant curves of Figure 17 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 17. 1 3 2 4 5 1 3 2 4 5 u2 u3 h u2 u3 h u1 u2 u3 u2 u3 u2 u2 u1 h u2 u3 h2 u2 u3 hu1 u3 h u3 u2 h u3 u1 u3 u2 u2 u3 h u1 u3 u1 u2 h−1 u3 u1 h u2 u1 h2 u3 u2 u2 u3 u1 u3 u2 u3 h u3 u2 h u3 u2 u3 u1 h u1 u3 h u1 u2 h u2 u3 h u3 u2 u3 u1 u2 u1 Figure 20. Illustration of T-fixed points and invariant curves (with their T-weights) of C(/2/3/5\3\2\). Let us consider the fixed point labeled 3, which has BCT and Young diagrams in Figure 21. By Theorem 3.2, T3 C(D) = u2 u3 h + u3 u2 + u1 u2 + u2 u1 h + u2 u3 h + u3 u2 . We can produce an invariant curve for each of these tangent weights. 32 A. Foster and Y. Shou 1 0 1 0 1 0 1 0 1  (a) (b) Figure 21. The (a) BCT and (b) Young diagrams associated to the fixed point 3 of Figure 20. (a) w = u1 u2 (b) w = u2 u1 h Figure 22. The two Young diagram surgeries possible between the top two Young diagrams of Figure 21. The sites are shaded gray and w is the tangent weight of the corresponding invariant curve. (a) w = u2 u3 h (b) w = u3 u2 (c) w = u3 u2 Figure 23. The three surgeries possible between the bottom two Young diagrams of Figure 21. 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. First we check for type III curves. By Lemma 4.6, there is one type III curve, since c3−c2 = 1. This curve has tangent weight u2 u3 h. 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 Figures 22 and 23. With these six curves described, we now build pencils of curves. There is a pencil of curves with weight u3 u2 , which is spanned by a type I curve and type II curve, as seen in Figure 23. There is also a pencil of curves with weight u2 u3 h, spanned by a type I curve from Figure 23 and a type III curve. No other curves share weights, so these are the only multidimensional pencils of invariant curves. Acknowledgements We would like to thank R. Rimányi for helpful discussions on the topic. We would also like to thank the referees, whose comments have resulted in numerous corrections and improvements. References [1] Aganagic M., Okounkov A., Elliptic stable envelopes, J. Amer. Math. Soc. 34 (2021), 79–133, arXiv:1604.00423. [2] Anderson D., Fulton W., Equivariant cohomology in algebraic geometry, Cambridge Stud. Adv. Math., Vol. 210, Cambridge University Press, Cambridge, 2023. https://doi.org/10.1090/jams/954 https://arxiv.org/abs/1604.00423 https://doi.org/10.1017/9781009349994 Tangent Weights and Invariant Curves in Type A Bow Varieties 33 [3] Botta T.M., Rimányi R., Bow varieties: Stable envelopes and their 3d mirror symmetry, arXiv:2308.07300. [4] Chang T., Skjelbred T., The topological Schur lemma and related results, Ann. of Math. 100 (1974), 307– 321. [5] Cherkis S.A., Moduli spaces of instantons on the Taub-NUT space, Comm. Math. Phys. 290 (2009), 719– 736, arXiv:0805.1245. [6] Cherkis S.A., Instantons on the Taub-NUT space, Adv. Theor. Math. Phys. 14 (2010), 609–641, arXiv:0902.4724. [7] Cherkis S.A., Instantons on gravitons, Comm. Math. Phys. 306 (2011), 449–483, arXiv:1007.0044. [8] Fehér L.M., Rimányi R., Weber A., Characteristic classes of orbit stratifications, the axiomatic approach, in Schubert Calculus and Its Applications in Combinatorics and Representation Theory, Springer Proc. Math. Stat., Vol. 332, Springer, Singapore, 2020, 223–249, arXiv:1811.11467. [9] Goresky M., Kottwitz R., MacPherson R., Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83. [10] Ji Y., Bow varieties as symplectic reductions of T ∗(G/P ), arXiv:2312.04696. [11] Maulik D., Okounkov A., Quantum groups and quantum cohomology, Astérisque 408 (2019), ix+209, arXiv:1211.1287. [12] Mumford D., Fogarty J., Kirwan F., Geometric invariant theory, 3rd ed., Ergeb. Math. Grenzgeb., Vol. 34, Springer, Berlin, 1994. [13] Nakajima H., Quiver varieties and Kac–Moody algebras, Duke Math. J. 91 (1998), 515–560. [14] Nakajima H., Towards geometric Satake correspondence for Kac–Moody algebras, Cherkis bow varieties and affine Lie algebras of type A, Ann. Sci. Éc. Norm. Supér. 56 (2023), 1777–1824, arXiv:1810.04293. [15] Nakajima H., Takayama Y., Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A, Selecta Math. (N.S.) 23 (2017), 2553–2633, arXiv:1606.02002. [16] Rimányi R., ℏ-deformed schubert calculus in equivariant cohomology, K-theory, and elliptic cohomology, in Singularities and Their Interaction with Geometry and Low Dimensional Topology, Commun. Comput. Inf. Sci., Springer, Cham, 2021, 73–96, arXiv:1912.13089. [17] Rimányi R., Shou Y., Bow varieties – geometry, combinatorics, characteristic classes, Comm. Anal. Geom. 32 (2024), 507–575, arXiv:2012.07814. [18] Rimányi R., Smirnov A., Varchenko A., Zhou Z., Three-dimensional mirror self-symmetry of the cotangent bundle of the full flag variety, SIGMA 15 (2019), 093, 22 pages, arXiv:1906.00134. [19] Rimányi R., Smirnov A., Zhou Z., Varchenko A., Three-dimensional mirror symmetry and elliptic stable envelopes, Int. Math. Res. Not. 2022 (2022), 10016–10094, arXiv:1902.03677. [20] Rimányi R., Tarasov V., Varchenko A., Partial flag varieties, stable envelopes, and weight functions, Quan- tum Topol. 6 (2015), 333–364, arXiv:1212.6240. [21] Rimányi R., Tarasov V., Varchenko A., Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety, J. Geom. Phys. 94 (2015), 81–119, arXiv:1411.0478. [22] Rimányi R., Tarasov V., Varchenko A., Elliptic and K-theoretic stable envelopes and Newton polytopes, Selecta Math. (N.S.) 25 (2019), 16, 43 pages, arXiv:1705.09344. [23] Shou Y., Bow varieties – geometry, combinatorics, characteristic classes, Ph.D. Thesis, The University of North Carolina at Chapel Hill, 2021, arXiv:2012.07814. [24] Smirnov A., Zhou Z., 3d mirror symmetry and quantum K-theory of hypertoric varieties, Adv. Math. 395 (2022), 108081, 61 pages, arXiv:2006.00118. [25] Tymoczko J.S., An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson, in Snowbird Lectures in Algebraic Geometry, Contemp. Math., Vol. 388, American Mathemat- ical Society, Providence, RI, 2005, 169–188, arXiv:math.AG/0503369. [26] Wehrhan T., Chevalley–Monk formulas for bow varieties, arXiv:2310.11235. https://arxiv.org/abs/2308.07300 https://doi.org/10.2307/1971074 https://doi.org/10.1007/s00220-009-0863-8 https://arxiv.org/abs/0805.1245 https://doi.org/10.4310/atmp.2010.v14.n2.a7 https://arxiv.org/abs/0902.4724 https://doi.org/10.1007/s00220-011-1293-y https://arxiv.org/abs/1007.0044 https://doi.org/10.1007/978-981-15-7451-1 https://arxiv.org/abs/1811.11467 https://doi.org/10.1007/s002220050197 https://arxiv.org/abs/2312.04696 https://doi.org/10.24033/ast https://arxiv.org/abs/1211.1287 https://doi.org/10.1215/S0012-7094-98-09120-7 https://arxiv.org/abs/1810.04293 https://doi.org/10.1007/s00029-017-0341-7 https://arxiv.org/abs/1606.02002 https://doi.org/10.1007/978-3-030-61958-9_5 https://arxiv.org/abs/1912.13089 https://doi.org/10.4310/cag.241015012209 https://arxiv.org/abs/2012.07814 https://doi.org/10.3842/SIGMA.2019.093 https://arxiv.org/abs/1906.00134 https://doi.org/10.1093/imrn/rnaa389 https://arxiv.org/abs/1902.03677 https://doi.org/10.4171/QT/65 https://doi.org/10.4171/QT/65 https://arxiv.org/abs/1212.6240 https://doi.org/10.1016/j.geomphys.2015.04.002 https://arxiv.org/abs/1411.0478 https://doi.org/10.1007/s00029-019-0451-5 https://arxiv.org/abs/1705.09344 https://arxiv.org/abs/2012.07814 https://doi.org/10.1016/j.aim.2021.108081 https://arxiv.org/abs/2006.00118 https://doi.org/10.1090/conm/388/07264 https://doi.org/10.1090/conm/388/07264 https://arxiv.org/abs/math.AG/0503369 https://arxiv.org/abs/2310.11235 1 Introduction 2 Background 2.1 Brane diagrams and bow varieties 2.2 Torus fixed points 2.3 Fixed point restrictions 2.4 Hanany–Witten transition and separated bow varieties 3 A combinatorial formula for tangent weights 4 Butterfly surgeries and invariant curves 4.1 Torus invariant curves 4.2 Butterfly surgery 4.3 Botched butterfly surgery 4.4 Nonsurgery curves 4.5 Young diagram surgery 5 BCT block swap moves 6 Classification of invariant curves 7 Examples of invariant curves 7.1 Invariant curves ... 7.2 Invariant curves ... References
id	nasplib_isofts_kiev_ua-123456789-212875
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-21T18:31:00Z
publishDate	2025
publisher	Інститут математики НАН України
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spelling	Foster, Alexander Shou, Yiyan 2026-02-13T13:49:22Z 2025 Tangent Weights and Invariant Curves in Type A Bow Varieties. Alexander Foster and Yiyan Shou. SIGMA 21 (2025), 016, 33 pages 1815-0659 2020 Mathematics Subject Classification: 14H10; 05E14 arXiv:2310.04973 https://nasplib.isofts.kiev.ua/handle/123456789/212875 https://doi.org/10.3842/SIGMA.2025.016 This paper provides a complete classification of torus-invariant curves in Cherkis bow varieties of type A. We develop combinatorial codes for compact and noncompact invariant curves involving the butterfly diagrams, Young diagrams, and binary contingency tables. As a key intermediate step, we also develop a novel tangent weight formula. Finally, we apply this new machinery to example bow varieties to demonstrate how to obtain their 1-skeletons (union of fixed points and invariant curves). We would like to thank R. Rimányi for helpful discussions on the topic. We would also like to thank the referees, whose comments have resulted in numerous corrections and improvements. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tangent Weights and Invariant Curves in Type A Bow Varieties Article published earlier
spellingShingle	Tangent Weights and Invariant Curves in Type A Bow Varieties Foster, Alexander Shou, Yiyan
title	Tangent Weights and Invariant Curves in Type A Bow Varieties
title_full	Tangent Weights and Invariant Curves in Type A Bow Varieties
title_fullStr	Tangent Weights and Invariant Curves in Type A Bow Varieties
title_full_unstemmed	Tangent Weights and Invariant Curves in Type A Bow Varieties
title_short	Tangent Weights and Invariant Curves in Type A Bow Varieties
title_sort	tangent weights and invariant curves in type a bow varieties
url	https://nasplib.isofts.kiev.ua/handle/123456789/212875
work_keys_str_mv	AT fosteralexander tangentweightsandinvariantcurvesintypeabowvarieties AT shouyiyan tangentweightsandinvariantcurvesintypeabowvarieties

Tangent Weights and Invariant Curves in Type A Bow Varieties

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