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K-Decompositions and 3d Gauge Theories

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Tudor Dimofte, Maxime Gabella, Alexander B. Goncharov

High Energy Physics - Theory · Submitted 2 Jan 2013 · 118 references

Pith's verdict: A serious, mostly honest extension of the K=2 construction to general K: the K2-isotropy theorem is real, and the load-bearing gap is stated openly rather than buried.

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CitationAlexander B. Goncharov, Maxime Gabella, Tudor Dimofte. "K-Decompositions and 3d Gauge Theories." arXiv:1301.0192 (2013). https://pith.science/paper/XJOPH3IG
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The paper

The paper claims

The paper argues that the flat $PGL(K, C)$ connections on a 3-manifold's boundary that extend into the interior form a Lagrangian subvariety, and proves a stronger multiplicative version of that statement by cutting each tetrahedron into $K

The reading

The paper asks whether the space of flat $PGL(K, C)$ connections on the boundary of a 3-manifold that extend into the interior sits inside the boundary phase space as a Lagrangian — a half-dimensional subspace on which the symplectic form vanishes. It answers a stronger, arithmetic version of that question: the multiplicative refinement of the symplectic form, its $K_2$ class, should restrict to zero. The strategy is to cut each ideal tetrahedron of a triangulation into $K(K^2-1)/6$ octahedra, attach to each octahedron the same three cross-ratio parameters that Thurston's gluing equations use for hyperbolic tetrahedra, and glue them back with monomial relations at every internal lattice point. Using a map from configurations of decorated flags to the Bloch complex, the authors show the resulting subvariety is $K_2$-isotropic whenever the boundary has no torus or annulus components. The same gluing data is then read as a recipe for a 3d $N=2$ abelian Chern-Simons-matter theory, conjecturally the one produced by wrapping $K$ M5-branes on the manifold.

What carries the argument

The load-bearing device is the hypersimplicial $K$-decomposition: intersecting the simplex $sum x_i = K$ with the integer hyperplanes $x_i = s$ cuts an $n$-simplex into hypersimplices, of which the 3d octahedron $Delta^{1,1}$ is the basic building block. Each octahedron is given the elementary Lagrangian pair $L_diamond = {z'' + z^{-1} = 1} subset P_diamond = {z z' z'' = -1} simeq (C^*)^2$, and the whole manifold's moduli space is conjectured to be the holomorphic symplectic quotient of the product of these by the gluing monomials. The proof of isotropy runs through a chain map $alpha_bullet$ from the complex of configurations of decorated flags to the Bloch complex, built by sending each hypersimplex to a configuration of vectors in a Grassmannian; commutativity of the right square converts the boundary $K_2$ class into $delta$ of a Bloch element, which is zero in $K_2$ by construction.

Extended reading notes

Core claim

The central claim is that the moduli space $L_K(M)$ of framed flat $PGL(K, C)$ connections on $M$ that extend to the boundary can be built locally, octahedron by octahedron, and that the result is a $K_2$-Lagrangian in the framed unipotent boundary phase space $X^{un}_K(M)$. Concretely, the hypersimplicial $K$-decomposition replaces each tetrahedron by $K(K^2-1)/6$ octahedra; each octahedron carries a triple $(z, z', z'')$ with $z z' z'' = -1$ and the Plücker-type relation $z'' + z^{-1} = 1$, exactly the relation that defines the elementary $K=2$ tetrahedron Lagrangian. Products of these parameters at internal lattice points give gluing monomials $c_k$, and products at boundary lattice points reproduce the cluster cross-ratio and triple-ratio coordinates. Theorem 4.2 then shows that, for manifolds whose boundary consists of big boundary with holes filled by small discs, the $K_2$ class $W$ of the symplectic form restricts to $L_K(M)$ as $delta$ applied to the Bloch-group element $sum_a {z_a}_2$ built from octahedron parameters, hence vanishes in $K_2$; a dimension count upgrades isotropy to the full Lagrangian property for convex polyhedra. The authors state the general symplectic-quotient statement as Conjecture 4.1 and explicitly do not prove it in full generality.

Load-bearing premise

The construction rests on the Symplectic Gluing Conjecture: that the boundary phase space really is the holomorphic symplectic quotient of the product of octahedron phase spaces by the gluing monomials, and that $L_K(M)$ is the image of the product octahedron Lagrangian under that quotient — a statement the authors formulate and explicitly do not prove in general.

Editorial extensions

If this is right

  • If the gluing picture holds, quantizing the pair $L_K(M) subset X^{un}_K(M)$ yields difference operators that annihilate $PGL(K, C)$ Chern-Simons wavefunctions on any triangulated 3-manifold, generalizing the $K=2$ construction.
  • The Bloch-group element built from octahedron parameters gives a $PGL(K, C)$ volume whose variation is computed entirely from boundary data, extending the Neumann-Zagier variation formula beyond $K=2$.
  • Because a 2-3 Pachner move decomposes into elementary moves on individual octahedra, triangulation independence of the 3d theory $T_K[M]$ reduces to the infrared duality between $N_f = 1$ SQED and the XYZ model.
  • The number of degrees of freedom of $T_K[M]$ scales as the number of octahedra, $K(K^2-1)/6$, giving cubic growth in $K$ that matches holographic expectations for $K$ coincident M5-branes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The construction suggests a purely combinatorial route to $K$-dependent knot invariants: for a knot complement the Lagrangian is a curve generalizing the $A$-polynomial, and its quantization should produce recursion relations for colored HOMFLY polynomials, a step the paper points at but does not carry out.
  • Since the octahedron relation $z'' + z^{-1} = 1$ is the same five-term identity that governs cluster mutations, one could try to repackage the entire construction as a single cluster transformation on a four-dimensional cluster variety, with $K$ labelling hypersimplices rather than indexing a rank.
  • A natural place to look for failure is the paper's own remark that some triangulations give an empty $tilde{L}_K(M, t^{3d})$ — for instance when a tetrahedron is glued to itself along two faces — which hints that the construction quietly depends on a positivity or angle-structure condition on the triangulation, not just on the topology of $M$.
  • If the symplectic quotient is only a partial reduction, as the paper's Proposition 5.1 allows, then $L_K(M)$ may be isotropic without being Lagrangian for manifolds with torus boundaries, which would put the knot-complement applications on a weaker footing than the polyhedron cases.
Original abstract

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.83 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.63 · compares with prior results 0.85 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 This paper combines several new constructions in mathematics and physics. contribution 0.73 0.75 0.02 asserted 0.98
s1 Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. contribution 0.99 0.88 0.03 asserted 0.97
s2 We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. contribution 0.99 0.94 0.04 asserted 0.99
s3 Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. contribution 0.63 0.41 0.19 asserted 0.82
s4 We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. method 0.73 0.94 0.10 asserted 1.00
s5 Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). result 0.93 0.94 0.19 asserted 1.00
s6 In some cases this implies that L_K(M) is K_2-Lagrangian. interpretation 0.79 0.85 0.10 hedged possibility 0.73
s7 For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. method 0.74 0.86 0.09 asserted 1.00
s8 Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. contribution 0.98 0.94 0.14 asserted 0.60
s9 This extends known constructions for K=2. background 0.43 0.33 0.04 asserted 0.99
s10 Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. contribution 0.40 0.48 0.06 asserted 1.00
s11 Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. result 0.75 0.72 0.07 asserted 1.00
s12 In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K. result 1.00 0.93 0.08 hedged possibility 0.50

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The 2–3 move: decomposing a bipyramid into two or three tetrahedra. The gluing of (1.1) was done in such a way that, generically, the definition of T2[M] would be independent of the choice of triangulation. Topologically, any two ideal triangu￾lations can be related by local 2–3 Pachner moves as depicted in
Figure 2. Stacking 1 6K(K2 − 1) octahedra to form an AK−1 tetrahedron (for K = 4). In [12] it was shown that an open part of XK(∂M) is covered by affine charts XK(∂M, t) of specific type, the cluster Poisson coordinate charts, labeled by 2d ideal triangulations t of ∂M. Here we similarly glue together LeK(M) from affine varieties LeK(M, t3d), labeled by 3d ideal triangulations t3d. To define LeK(M, t3d), we use an important tool: the hypersimplicial K-decomposition of an ideal triangulation, or K-decomposition for short. Given an ideal triangulation t3d = {∆i} N i=1 of M, we further divide…
Figure 3. (a) truncated tetrahedra, glued together to form 3-manifolds that have either (b) big boundaries tiled by hexagonal faces with holes filled in by small vertex triangles; or (c) small torus boundaries tiled exclusively by vertex triangles. Let us truncate a tetrahedron by cutting off its vertices. The resulting truncated tetra￾hedron has two kinds of faces: (small) triangles replacing the original vertices, and (big) hexagons. Let us glue truncated tetrahedra into an oriented manifold with boundary by al￾lowing pairs of hexagonal faces to be glued, but not triangular ones (
Figure 4. A topological surface with lamination and marked points is equivalent to a big boundary with small annuli and small discs connecting its holes. Here, two loops of a lamina￾tion are replaced by two annuli (depicted as long tubes); and two marked points are replaced by discs. The remaining big boundary consists of two 3-punctured spheres. There is also a nice physical interpretation of the admissibility conditions. We ultimately want to study 3-manifolds M on which the 6d (2, 0) theory can be compactified. The big boundaries we have just described are asymptotic regions of M in six…
Figure 5. Parallel transport of flags to the common point p. Given a framed P GL(K, C)-vector bundle V with a flat connection on a 2d surface C with punctures, and given an ideal triangulation of C, i.e. a triangulation with vertices at the punctures, any ideal tri￾angle t of the triangulation gives rise to a configuration of three flags (A, B, C) in a K-dimensional complex vector space VK. Namely, consider the restriction of the flat bundle V to the triangle t. Then the flat sections defining the framing provide flat sections of the as￾sociated flag bundle VB at each punctured disc on C in…
Figure 6. Forming ∆1,1 . We now describe a construction that plays a crucial role in this paper, namely the hypersimplical K-decomposition of an ideal triangulation in dimension n. Let p and q be two non-negative integers satisfying p + q = n − 1. An n-dimensional hypersimplex ∆p,q is defined [78] as the intersection of the n + 1-dimensional cube 0 ≤ xi ≤ 1 with the hyperplane Pn i=0 xi = q + 1. Combinatorially, it is isomorphic to the convex hull of the centers of p-dimensional faces of an n-dimensional simplex. The hypersimplices ∆p,0 and ∆0,q are just simplices. The simplest hypersimplex…
Figure 7. 2-simplex with lattice points (a, b, c), a + b + c = 3. shaded white, and labeled by the triples (a, b, c) with a + b + c = K − 1; K 2  of them are “upside-down” ∆1,0 -triangles, shaded black and labeled by the triples (a, b, c) with a + b + c = K − 2. 3. When n = 3 we get a K-decomposition of a tetrahedron. It consists of K+2 3  upright ∆0,2 -tetrahedra, K+1 3  octahedra ∆1,1 , and K 3  upside-down ∆2,0 -tetrahedra. It in￾duces a K-triangulation of each of the four faces of the tetrahedron. The lattice points pabcd are labeled by quadruples of non-negative integers that sum to…
Figure 8. The 3-decomposition of the tetrahedron. The upright ∆0,2 -tetrahedra (white) correspond to lines, the octahedra (black) to planes, and the upside-down ∆2,0 -tetrahedron (red) to a 3-space. Adjacent hypersimplices obey incidence relations, which can be represented as a hypergraph, shown on the right. 4. When n = 4 we get a K-decomposition of a four-dimensional simplex into hypersimplices of type ∆0,3 , ∆1,2 , ∆2,1 , ∆3,0 . 2.6 Configuration of flags and hypersimplices Consider a generic configuration of n+1 flags (F • 0 , . . . , F• n ) in a K-dimensional complex vector spac
Figure 9. 3-triangulation of a triangle, with triples of integers corresponding to points pabc with a+b+c = 3, lines Labc with a+b+c = 2 (black dots), and planes Pabc with a+b+c = 1 (white dots). The blue connectors between black and white dots stand for incidence relations between lines and planes. Next, every internal lattice point pabc of the K-triangulation, which is labeled by three strictly positive integers that add to K, gives rise to a 3-space Va−1,b−1,c−1 = A a−1 ∩ B b−1 ∩ C c−1 . This space contains all three planes and all six lines on the black and white triangles sur￾rounding i
Figure 10. Two triangles glued along an edge. The points p0bc0 on the edge are shown in magenta, together with the incidence relations with the neighboring planes (white dots). The lines (black dots) on adjacent white triangles are identified. Internal lattice points are shown in yellow. The K − 1 lattice points pabcd on the edge have labels (0, b, c, 0) with b, c > 0 and b + c = K. They belong to the K-decomposition of both adjacent triangles. The lines on the white triangles adjacent to the edge are identified. The four lines surrounding p0bc0 have labels L1,b−1,c−1,0, L0,b,c−1,0,…
Figure 11. Unipotent holonomy constraints around a puncture. coordinates (−x•) around the first loop and six coordinates (−x•) around the second loop must both equal one. (The origin of these constraints is explained in [12]; we will re-derive the constraints in Section 3.3.) Altogether, we find that cross-ratios and triple-ratios give dK(g, h) = −(K2 − 1)χ(C) − (K − 1)h (3.6) independent coordinates for the moduli space X un K (C). This is precisely the expected complex dimension. The coordinates are C ∗ -valued and cover an open patch. Notice that the cross￾ratios and triple-ratios never eq
Figure 12. Poisson structure on a 4-triangulated triangle. The coordinates x and x 0 at the lattice points p and p 0 have the Poisson bracket {x, x0} = +xx0 since there is one positively oriented edge from p to p 0 . After projection to the traditional moduli space LocK(C), the symplectic and Poisson structures agree with the standard Goldman symplectic form on the non-singular part of the moduli space. The latter is expressed simply in terms of complex flat connections A as Ω = R C Tr (δA ∧ δA). 3.2 The Lagrangian pair assigned to a 3d triangulation of M Having described coordinates for t
Figure 13. Coordinates for framed P GL(2, C)-connections on an ideal tetrahedron. with Poisson brackets {z, z0 } = zz0 , {z 0 , z00} = z 0 z 00 , {z 00, z} = z 00z . (3.14) Now let’s fill in the bulk of the tetrahedron. Since the bulk is contractible, any bundle with a flat connection on the bulk is trivial. So the choice of invariant flags A, B, C, D becomes absolutely crucial — the flags are unrestricted, and they carry the only nontrivial degrees of freedom of framed flat connections in the bulk. After filling in the tetrahedron we can parallel-transport all flags to a common point p of…
Figure 14. Left: 3-decomposition of a tetrahedron. Right: Cross-ratios for an octahedron. and triple-ratio coordinates for the lattice points on these faces and on the edge, just as in
Figure 15. Ten vectors parametrizing the ten lines on the K = 3 tetrahedron. We have used a GL(3, C) transformation to put the vectors in the given form; otherwise they are completely generic. We also show the corresponding octahedron parameters (z, w, x, y), on their corresponding vertices. Here zz0 z 00 = −1 and z 00+z −1−1 = 0, so that z 0 = (1−z) −1 , z00 = 1 − z −1 , and similarly for w, x, y. is equal to the product xw0 z 00 . Finally, let us look at the internal lattice points of the K-decomposition of a tetrahedron: points pabcd with none of a, b, c, d vanishing. (This first o
Figure 16. Lattice point on a pair of glued faces. In Section 3.2.2 we found that at every internal point in the K-decomposition of a tetrahedron the product of octahedron pa￾rameters is one. We also saw that the product of parameters on a boundary point is equal to the cross-ratio or triple-ratio coordi￾nate in the boundary phase space. Both of these facts generalize. Now the product at every internal point in the K-decomposition of a 3-manifold — including points on faces and edges of glued tetrahedra — is one; while products at external points are equal to big-boundary coordinates. We…
Figure 17. Hexagonal bipyramid obtained by gluing six tetrahedra along a common edge. Next, consider points on an internal edge E in M shared by N tetrahedra. We picture this arrangement as an N-gonal bipyramid, as in
Figure 18. Left: Configuration of octahedra around a point on a trivalent internal edge. Right: Point on an external edge. vertices to define octahedron parameters throughout the bipyramid.10 Each of the K − 1 lattice points on E is touched by N octahedron vertices. The product of the N octahedron cross-ratios at each lattice point involves numerators and denominators that cancel in pairs, and so automatically equals one. For example, a situation with N = 3 is shown on the left of
Figure 19. 3d snakes in the (K −1)-decomposition of a tetra￾hedron, from the top vertex to the opposite face (here for K = 4). To define a 3d snake, we begin with an oriented path on the edges of the (K − 1)-decomposition of a tetrahe￾dron, which starts at any of the tetrahedron vertices and ends at a lattice point on the opposite face, consisting of K − 1 segments (the small edges it traverses). We orient the segments toward the opposite face. We decorate each segment of the path by a fin. To define a fin observe that a path segment is an edge of a unique ∆0,2 tetrahedron; the fin is a face…
Figure 20. Correspondence between the K and (K − 1)-decompositions, for K = 3. (It is much more natural to draw snakes on the latter.) The K lines Labcd that label the vertices of a 3d snake are generic since the original configuration of four flags is generic. Let us associate a basis (v1, . . . , vK) to the snake so that the vector vi lies in the line at the i-th vertex of the snake (numbered from tail to head), and the vector vi ± vi+1 belongs to the line at the vertex to which the i-th fin points to. Here we take a sum if the fin points to the right and a difference if the fin points to…
Figure 21. Using the i-th segment of a snake to select part of a projective basis. Here there are three lines L1, L2, L3 associated with the vertices of a small black tetrahedron. The indicated vectors must belong to these lines. within a tetrahedron. They are shown in
Figure 22. The four fundamental snake moves. In each case we go from the orange snake to the green snake. Move IV is shown for K = 4. Move I can only occur at the head of a snake, adjacent to the face where it ends. It moves the head across a small black tetrahedron, keeping the fins on the same tetrahedron face, and acts as FK−1, sending   v1 . . . vK−1 vK   7→ FK−1   v1 . . . vK−1 vK   =   v1 . . . vK−1 vK−1 + vK   . (3.37) – 47 –
Figure 23. Moving a snake from one edge to another, for K = 3. The single most important thing to observe about snakes on a tetrahedron is that the projective bases associated with them are well-defined — i.e. the P GL(K, C) holonomy inside 11This transformation is perhaps the most nontrivial to derive. It suffices to apply it to snakes on a K = 3 tetrahedron, in which case the parametrizations of
Figure 24. Moving a snake around a black tetrahedron to obtain coordinate constraints. On one hand, if we derive both octahedron parameters and projective bases by using a configuration of four flags, then the bases are automatically well defined and the relations are automatically satisfied. Conversely, if we simply place a triple of unconstrained coordinates zabcd, z0 abcd, z00 abcd ∈ C ∗ − {1} on every octahedron and define the P GL(K, C) transformations from one snake to another using the rules of
Figure 25. The fifth snake move: jumping between glued tetrahedra, fins along the common face. We simply need one more elementary move to transport a snake from one tetrahedron to another, across a common face (
Figure 26. Moving a snake around a lattice point of the K-triangulation that lies on a pair of glued faces. The four images on the left (orange snakes) are from one tetrahedron, and the images on the right (green snakes) are from a second. common glued face (corresponding to a white triangle in the (K−1)-triangulation of the face). It is easy to see that corresponding holonomy matrix is diag(1, . . . , 1, z00w 0xy0v 00t, . . . , z00w 0xy0v 00t), with K − i nontrivial entries. For a trivial holonomy the product z 00w 0xy0v 00t of the six octa￾hedron parameters that touch p must equal one —…
Figure 27. The associahedron, and canonical snakes corresponding to three of its 24 vertices
Figure 28. Connecting glued associahedra. In combination with loops inside tetrahedra, the face and edge loops just discussed are enough to generate all the homotopically trivial transformations of snakes. Thus, we find that imposing all the standard octahedron gluing constraints allows to define a flat P GL(K)-connection on the space of snakes, which induces a flat P GL(K)-connection on the 3-manifold M. To be more explicit about the last step, we may “doubly trun￾cate” every big tetrahedron ∆ in the triangulation of M (
Figure 29. Computing the holonomy around a cycle of the torus boundary. – 53 –
Figure 30. The two elementary 2–3 moves for octahedra, corresponding to cobordisms through ∆1,2 hypersimplices (left) and ∆2,1 hypersimplices (right). The proof of the invariance of the motivic volume map under the 2–3 moves suggests that the effect of the Pachner 2–3 move on the framed flat P GL(K)-connections should be presented as a composition of the elementary moves of two types, parametrized by the set of all ∆1,2 - and ∆2,1 -hypersimplices in the K-decomposition of a 4-simplex. This includes • the K+2 4  elementary moves via cobordisms through the ∆1,2 -hypersimplices; and • the K+1 4…
Figure 31. The sequence of elementary 2–3 moves for K = 3 (labels are for future reference). We apply the 2 → 3 move on the left of
Figure 32. The sequence of elementary 2–3 moves for K = 5. “underlined” or “decorated.” The decoration serves (in part) to encode how octahedra occur – 58 –
Figure 33. K-decompositions of an ideal tetrahedron, for K = 2, 3, 4. Z Z! Z!! Z Z! Z!!
Figure 34. La￾beling of vertices. Each octahedron is oriented and its faces can be canonically assigned two distinct colors, according to whether they align with big faces of a tetrahedron (black) or with the truncations of vertices (white). The colors alternate on neighboring faces. We label the six vertices of every octahedron with formal complex parameters Zi , Z0 i , Z00 i , equal on opposite vertices and occurring in a clockwise order around every black face. (Thus there are three distinct ways to label any octahedron, related by a cyclic permutation Zi → Z 0 i → Z 00 i → Zi .) Notice…
Figure 35. Four parallel sets of slices in the K = 4 decomposition. A B C A C B
Figure 36. Continuing slices from one tetrahedron to another; the sets of slices on the right here are associated with vertices A, B, and C, respectively. of ∂M acquires a family of K − 1 global slices throughout M centered around that small boundary and having the same topology. For example, a small torus boundary comes with K − 1 toroidal slices in the K-decomposition. 5.2 Coordinates on the product phase space We now want to revisit the symplectic gluing of elementary Lagrangian pairs for octahedra to form the Lagrangian pair for an admissible 3-manifold M. We will assume that the boundary…
Figure 37. Open (γ1) and closed (γ2) paths on a slice in M. Recall that there are K − 1 global slices asso￾ciated with every small boundary component of M, which extend from one tetrahedron to the next. We always view the slices from above, that is, from the point of view of the small boundary. On any one of these slices, we consider oriented paths made of segments that begin at the midpoint of a side of a small black or white triangle, curve to the left or right inside the triangle, and end at the midpoint of another side. Two examples are shown in Fig￾ure 37. The entire path may either be cl
Figure 38. Path homo￾topy. We will call these functions path-coordinates. For example, the open path γ1 in
Figure 39. Gluing functions from loops on a slice of M. Every function Ck is the path-coordinate (minus 2πi) for a small closed loop surrounding a lattice point on a slice of M. The three types of vertices described above correspond to the three types of closed loops shown in
Figure 40. Homotopy through a vertex p. When homotoping a path through a lattice point, as in Fig￾ure 40, its path-coordinate simply picks up a contribution ±C for the gluing function assigned to that point. A nice application of this fact is to illustrate the K − 1 linear relations among the Ck coming from every small torus boundary of M. Consider any of the K − 1 global slices of M associated with a small torus boundary — each having the topology of a torus. Choose two small adjacent triangles on this slice, as in
Figure 41. Using path homotopy to obtain relations among Ck’s. – 80 –
Figure 42. The big boundary functions Xj as path-coordinates on slices. The functions Xj can be interpreted as path-coordinates for open paths on the slices associated with small disc boundaries of M. This is illustrated in
Figure 43. Longitude and meridian paths on the second slice of a torus boundary. projections to the actual torus boundary are in the homology classes of the A and B cycles, respectively. An example of such paths is shown in
Figure 44. Arrows representing the Poisson bracket on big boundaries. commute. Third, the big boundary coordinates commute with torus coordinates, {Xj , U(ν) a } = {Xj , V (ν) a } = 0 , (5.8c) and the torus coordinates satisfy  V (ν) a , U(ν 0 ) a 0
Figure 45. Loops surrounding a hole on the big boundary (left), and a contractible path on the 2nd slice parallel to the hole that demonstrates the constraint on eight Xj functions (right). For every hole in the big boundary of M (filled in by a small disc) we expect K − 1 central relations among the Xj . The hole in the big boundary is filled in by a small disc, and in the octahedral decomposition of M there are K − 1 slices parallel to this disc. The path￾coordinate for a path on the periphery of such a slice, as on the right of
Figure 46. From a combinatorial perspective, they are absolutely identical. ! W1 W2 W3 X1 X2 P1 P2 Z2 Z1 X1 X2 P1 P2 ! 2–3 2–3 W1 W3 W2 X1 X2 P1 P2 Z2 Z1 X1 X2 P1 P2 Z1 Z2
Figure 47. The sequence of moves on octahedra that realizes a full 2–3 move for K = 3. – 89 –
Figure 48. Transforming paths during the 2–3 move 5.5.2 Composing elementary moves In order to perform a full 2–3 move on ideal tetrahedra that have K-decompositions, we iterate the elementary 2–3 moves. We recall in
Figure 49. Longitudinal polarization on mirror-symmetric clumps of octahedra. In the longitudinal polarization, the cluster of two octahedra is very simple. We take Πlong = (X+, X−; P+, P−) := (Z1, Z2; −Z 0 1 , Z00 2 ). Since the final positions X± match the initial positions Z1,2, the theory is just a product of two octahedron theories, a.k.a. an N = 4 19The Lagrangian LSUSY is the supersymmetric parameter space of a theory on C × S 1 . See [1, 53, 87] and Appendix A.6 for a brief review. – 94 –
Figure 50. Equatorial polarization on mirror-symmetric clumps of octahedra. Explicitly, on the 2-side, we set (X+, X−; P+, P−) = (Z 0 1 + Z 00 2 , Z00 1 + Z 0 2 ;Z 0 2 , Z0 1 ), or   X+ X− P+ P−   =   −1 0 −1 1 0 −1 1 −1 0 −1 0 −1 −1 0 −1 0     Z1 Z2 Z 00 1 Z 00 2   − iπ   −1 −1 −1 −1   . (6.11) This results in N = 2 SQED, T N =2 SQED :    Dynamical U(1) theory with two chirals q, qe; flavor symmetry U(1)+ × U(1)− corresponding to (axial ± topological) charges: q qe v+ v− G 1 −1 0 0 X+ −1 0 1 0 X− −1 0 0 1 R 2 0 0 0 CS levels: G…
Figure 51. Octahedra and fi￾nal polarization for K = 3. The four octahedra in a K = 3 tetrahedron can be as￾signed triples of parameters (z, w, s, t) as in
Figure 52. Octahedra, final polarization, and operators for K = 4. The octahedron in the middle of the back edge is drawn separately; note that its parameter is not part of the polarization. On the right we show the three slices around the top vertex. As dictated by this polarization, the first four U(1)i flavor symmetries rotate the first four φi ’s; while the for 5 ≤ i ≤ 9 the field φi has charge +1 and φ10 has charge −1. Moreover, the first nine fields have R-charge zero and φ10 has R-charge 2. The background Chern-Simons levels can be determined after a precise choice of momenta Pi in Π…
Figure 53. The central cluster of the K = 5 tetra￾hedron. It is instructive to build the tetrahedron and the correspond￾ing gauge theory in pieces. We start with a cluster of four octa￾hedra, which have a small upright tetrahedron (∆0,2 simplex) in their center (
Figure 54. Assembling the SL(5) tetrahedron from a central cluster and 16 peripheral octahedra. Notice that the four products of chirals φi1φi2φi3 are associated with vertices on the groups of peripheral octahedra that become identified with the four distinguished vertices of the central cluster during the gluing. The four superpotential couplings combine the operators on the two sides of the gluing. 6.4 The K = 3 bipyramid We now come to the theory of the bipyramid for K = 3. We expect that by splitting the bipyramid into ideal tetrahedra two different ways we will find two different…
Figure 55. The equatorial (left) and tropical (right) polarizations for the K = 3 bipyramid. We only specify the seven position coordinates, indicating them by dark-blue dots on the K-triangulation of the boundary. The light-blue dots are lattice points/coordinates that commute with the positions (and so are assigned operators as well). flavor symmetries. In any polarization, we will always have mirror symmetry T3[bip-2] ' T3[bip-3] . (6.30) We begin by describing the theories on the two sides, and then explain how the mirror symmetry works. 6.4.1 Two tetrahedra z8 z6 z7 z5 z z4 3 z2 z1 φ1 φ2…
Figure 56. Decomposition into eight octahedra, with Πtrop. Let us focus on the tropical polarization, since it makes the two-tetrahedron theory especially simple. The decomposition of the bipyramid into two tetrahedra, and subsequently into eight octahedra, is shown in
Figure 57. Decomposi￾tion into 12 octahedra, with Πtrop. The decomposition of the bipyramid into three tetrahedra, and subsequently into twelve octahedra, is shown in
Figure 58. The core of the SL(3) bypyramid, built from two copies of N = 4 SQED. In order to complete the bipyramid, we must add a “halo” of six octahedra to the core of
Figure 59. Assembling the SL(3) bipyramid from a core and a halo. For the tropical polarization on the final boundary, we should represent the halo as three copies of N = 2 SQED (cf. (6.12) and
Figure 60. The first two steps in going from two SL(3) tetrahedra to three. We work in the tropical polarization, and start with the theory T3[bip-2] (6.31) built from two tetrahedra and eight octahedra. Consider the three pairs of chirals (φ3, φ6), (φ4, φ7), (φ5, φ8) corresponding to the three pairs of octahedra that touch across the big internal face. They represent three N = 4 hypers, coupled by an N = 2 superpotential W = φ3φ4φ5φ6φ7φ8 . (6.36) The first set of three 2 → 3 moves dualizes the hypers to three copies of N = 4 SQED (
Figure 61. The final step in going from two SL(3) tetrahedra to three: forming the core. poles of the bipyramid, and think of each pair (r, φ1) and (r, φ2) as an N = 4 hyper. We want to dualize these pairs to N = 4 SQED. This is subtle: the chirals r, r0 are not gauge-invariant, and we are effectively assuming that it is ok to interchange an order of limits (of gaugings) as we flow to the IR. With this assumption, the result of the duality is the core theory of (6.34): two copies of N = 4 SQED, for which a combination of axial and topological symmetries — identified with U(1)d — has been…
Figure 62. Construction of the trefoil knot complement from two SL(3) tetrahedra. Octa￾hedron edges with matching colors are glued together. The complement of the trefoil knot can be decomposed into two tetrahedra, as shown in the
Figure 63. First and second slices of the torus boundary for the trefoil knot. The paths µa and νa represent the meridian and longitude, while the small circular paths drawn in green correspond to the gluing constraints. In order to have canonical Poisson brackets, we redefine the longitude coordinates as in section 5.3.2: Ve1 := 1 3 (2V1 + V2 + C1 − C2 + C6) , (7.20a) Ve2 := 1 3 (V1 + 2V2 − C1 + C2 − C6) . (7.20b) One can check that now {Vea, Ub} = δab. The division by K = 3 here (i.e. by the determinant of the Cartan matrix) is a convenient way of lifting from P GL(3, C) representations
Figure 64. Construction of the figure-eight knot complement from two SL(3) tetrahedra. Using the K = 3 decomposition into tetrahedra and octahedra shown in
Figure 65. First and second slices of the torus boundary for the figure-eight knot. The coordinates ua and va for the meridian and longitude paths drawn in
Figure 66. The wheel and crown arrangements that can occur on slices. The approach will be to look at how the octahedron parameters contributing to X can appear on the slice S. There are two basic cases: either X contains a sum of parameters in a crown-shaped or wheel-shaped arrangement of small triangles on S. These two arrangements are shown in
Figure 67. Slices in the neighborhood of a point p (blue dot), with octahedron parameters appearing in X marked by red dots. If p is on a glued face shared by two tetrahedra, then there are five relevant families of slices — one centered around each vertex of the big bipyramid that contains p. In the three families centered around the vertices of the big glued face, the slices containing p and those above and below p have X parameters, in wheels or crowns (
Figure 68. Slices in the neighborhood of an external point p (blue dot). Finally, suppose that p is on a big boundary of M. Then the slices containing X param- – 133 –
Figure 69. Uniformiza￾tion of paths inside a tetrahedron. From Section B.1 we know that {U γ , Uλ} only depends on the homotopy class of γ and λ (since closed paths commute with all gluing constraints Ck). Therefore, we may “uniformize” the paths, so that 1) they are smooth (have no bounces); 2) when￾ever they turn counterclockwise (left) inside a big tetrahedron they stay close to the edge they wind around; and 3) whenever they turn clockwise (right) they stay as far away from the edge as possible (
Figure 70. Paths turning together within a tetrahedron. γ and the octahedron parameters contributing to it are drawn in red, while possible options for λ are in green. Thus, let us first consider slices of a tetrahedron ∆ in which the paths turn the same way, modulo orientation (which may be equal or opposite). The path γ, on the a-th slice, either turns left or right. If it turns left (making a small turn) then the octahedron parameter (call it Zi) contributing to this segment of γ also appears on the (a + 1)-st slice. This is shown on the left of
Figure 71. Paths converging and then diverging without a net crossing. If the projections of γ and λ to the common small boundary do not have a net crossing during the run, there are two possible cases to analyze, depicted in
Figure 72. Paths converging and then diverging with a crossing. hγ, λi = ∓1 on the left (right). left (resp., right) of
Figure 73. Paths entering and exiting a tetrahedron on the same pair of faces. K − a and K − a + 1 with respect to the bottom. Any possible paths λ at worst pick up Zi itself, and so commute with this segment of γ. Alternatively, if γ turns to the right, then its octahedron parameters appear on the (K − 1)-st (i.e. the last) slice with respect to the bottom vertex. They appear in two horizontal rows — the position of these rows depending on a. Again, all possible paths λ at worst pick up the same octahedron parameters as in γ, so that the contribution to {U γ , Uλ} vanishes. Now we consider…
Figure 74. Paths entering and exiting a single tetrahedron on four distinct faces. First, it is possible for ∆0 and ∆1 to coincide — i.e. γ and λ both enter and exit a single tetrahedron separately. This is depicted in
Figure 75. Paths entering and exiting a pair of glued tetrahedra on four distinct faces: I top slice K − a K − a +1 K − a +2 K − 1 bottom slice E top slice bottom slices a a ··· ··· −1+1 ··· ··· −1 +1 −1 +1 ··· ··· ··· ··· ··· II +1 −1 +1 −1 −1 +1 +1 − −1 1 −1 +1 +1
Figure 76. Paths entering and exiting a pair of glued tetrahedra on four distinct faces: II In each of these three cases, there are two options for orienting γ. It is then shown how its octahedron parameters appear on various slices with respect to the bottom vertex of E – 139 –
Figure 77. Paths entering and exiting a pair of glued tetrahedra on four distinct faces: III (the arrangements of +’s and −’s follow by combining the arrangements in Figures 73 and 74). Now the potential λ paths do pick up parameters that don’t commute with those in U γ . However, the non-commuting contributions always cancel in pairs — either within ∆0 and ∆1 independently, or between ∆0 and ∆1. We let the figures speak for themselves. This finally exhausts all possible ways in which the two paths γ and λ may come close enough to share an octahedron — and thus potentially have nontrivial…

Figures reproduced from arXiv:1301.0192 by the authors.

References
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68 later papers cite it

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Uses its method or tool 2 of 13 papers with this label · 1 by the same authors

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  • Followed this paper's framework in three dimensions, together with another work's framework in two dimensions, to construct coordinates for spaces of framed flat connections on framed 3-manifolds and their boundaries.

    RG Domain Walls and Hybrid Triangulations (2013) (same authors) · 0.93 · paraphrased: the licence does not allow quoting

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68 arXiv papers cite this in 194 sentences

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Papers citing it · 126 sentences without a confident label

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Uses its method or tool 13 papers
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  • Under this change of coordinates five of our found six roots of shaded A-polynomials coincide with roots constructed in [ref] , and one extra root is spurious: [equation] This observation delivers a support to our assumptions about a behavior of huge representations of $U_q()$ leading to quasi-classical equations [ref].

    Shading A-polynomials via huge representations of $U_q(\mathfrak{su}_N)$ · Trefoil knot · 0.85

  • [Some additional subtleties and restrictions arise, especially as compared to [ref] and Appendix A of [ref] , because we focus only on boundaries of framed 3-manifolds rather than their interiors.]

    A Spectral Perspective on Neumann-Zagier · Framed flat connections on boundaries · 0.96

  • Interestingly, this simple description is reminiscent of the type of theories obtained by compactifying the 6d $(2,0)$ $𝔞_2$ theory on a three-manifold that is a bipyramid, as proposed in [ref] .

    Half-BPS Boundaries and the RG-Wall of $\mathcal{N}=2$ $SU(N)$ SYM · The $[]$ theory · 0.87

Cites it as background 33 papers
  • Although the dictionary between the dynamics of $T[M_3]$ and topological properties of $M_3$ is incredibly rich [ref] and only partially explored, there are two very fundamental relations between $M_3$ and $T[M_3]$.

    A 3d-3d appetizer · Introduction · 0.94

  • Let $M$ be an oriented 3-manifold with nonempty framed boundary, in the sense of [ref] .

    Complex Chern-Simons theory at level k via the 3d-3d correspondence · Topological data · 0.81

  • The program to systematically describe these theories and their properties began in a series of papers [ref] , inspired by earlier physical studies [ref] , and has since been extended and clarified in [ref] , among other works.

    3d Superconformal Theories from Three-Manifolds · Introduction · 0.96

What it cites · 89 sentences without a confident label

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  • The main physical goal of the present paper is to extend the construction of [ref] to all $𝔤=A_K-1$ Lie algebras, obtaining theories $T_K[M, Π]$ that correspond, in principle, to a stack of $K$ M5-branes wrapping $M$.

    3-Manifolds and 3d Indices · Introduction · 1.00

  • The main physical goal of the present paper is to extend the construction of [ref] to all $𝔤=A_K-1$ Lie algebras, obtaining theories $T_K[M, Π]$ that correspond, in principle, to a stack of $K$ M5-branes wrapping $M$.

    Gauge Theories Labelled by Three-Manifolds · Introduction · 1.00

  • The discussion here is essentially identical to that in [ref] , with $K=2$ tetrahedra replaced by octahedra.

    Quantum Riemann Surfaces in Chern-Simons Theory · 2–3 moves and path coordinates · 0.94

Uses the method or tool of 6 papers
  • The ellipsoid partition function can be obtained from data $(g,σ)$ using a symplectic-reduction approach [ref] ( [ref] ) that is still not fully rigorous; it is also expected to be equivalent to the “state integral” invariants of [ref] .

    Quantum Riemann Surfaces in Chern-Simons Theory · Quantum partition functions · 0.91

  • The ellipsoid partition function can be obtained from data $(g,σ)$ using a symplectic-reduction approach [ref] ( [ref] ) that is still not fully rigorous; it is also expected to be equivalent to the “state integral” invariants of [ref] .

    A TQFT from quantum Teichm\"uller theory · Quantum partition functions · 0.91

  • The ellipsoid partition function can be obtained from data $(g,σ)$ using a symplectic-reduction approach [ref] ( [ref] ) that is still not fully rigorous; it is also expected to be equivalent to the “state integral” invariants of [ref] .

    The quantum content of the gluing equations · Quantum partition functions · 0.91

Compares with 2 papers
  • Major differences between [ref] and the present work include our understanding of both bulk and boundary moduli spaces $_K(M)⊂_K^un(M)$ as Lagrangian pairs obtained from symplectic gluing (which necessitates the introduction of “big” boundary); our results on variations of the volume and its motivic avatar; and the eponymous geometric $K$-decompositions underlying all our constructions.

    Gluing equations for PGL(n,C)-representations of 3-manifolds · Mathematics perspective · 1.00

  • [This is the gauge-theory equivalent of Chern-Simons constructions in [ref] .]

    The quantum content of the gluing equations · The gauge theory · 0.81

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Cites as background 20 papers

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