Topological strings, D-model, and knot contact homology

Aganagic, Mina; Ekholm, Tobias; Ng, Lenhard; Vafa, Cumrun

doi:10.4310/atmp.2014.v18.n4.a3

Cited by 41 publications

(11 citation statements)

References 57 publications

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“…To summarize, by taking the limit (2.71)-(2.72) in the physical system (2.68), we obtain a relation between the bottom row of the HOMFLY-PT homology of K and the homology of In fact, the geometry and topology of M 3 = L K is closely related to that of the knot complement S 3 \ K, see [59]. Namely, for a knot (= a link with one component), both…”

Section: -Manifolds and The "Bottom Row"mentioning

confidence: 94%

BPS spectra and 3-manifold invariants

Gukov

Pei

Putrov

et al. 2020

J. Knot Theory Ramifications

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132

331

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We provide a physical definition of new homological invariants H a (M 3 ) of 3manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M 3 times a 2-disk, D 2 , whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d N = 2 theory T [M 3 ]: D 2 × S 1 half-index, S 2 × S 1 superconformal index, and S 2 × S 1 topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern-Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of M 3 . The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.C. Categorification of the Turaev-Viro invariants 733 It is in fact (Tor H1(M3, Z)) * /Z2 that is canonically identified with components of abelian flat connections.However, as the distinction between Tor H1(M3, Z) and its dual is only important in section 2.2, we will use the same set of labels {a, b, . . .} for elements in both groups. 4 A related conjecture was made in [28]. However it did not include the S-transform, which is crucial for restoring integrality and categorification. 5 The constant positive integer c depends only on M3 and in a certain sense measures its "complexity". In many simple examples c = 0, and the reader is welcome to ignore 2 −c factor which arises from some technical subtleties. Its physical origin will be explained later in the paper. 6 Later in the text we will sometimes use slightly redefined quantities, Za(q) → q ∆ Za(q), where ∆ is a common, a independent rational number.7 Recall that Tor H1(M3, Z), as a finitely generated abelian group, can be decomposed into Tor H1(M3, Z) = i Zp i . We ask for a fairly weak condition that Z2 doesn't appear in this decomposition. In other words, M3 is a Z2-homology sphere. Equivalently, there is a unique Spin structure on M3, so that there is no ambiguity in specifying Nahm-pole boundary condition for N = 4 SU (2) SYM on M3 × R+ [18]. The general case, in principle, could also be worked out. We leave it as an exercise to an interested reader.8 The presence of 1/2 factors that produce 2 −c overall factor in (2.7) can be interpreted as presence of factors ∼ = C[x] with deg q x = 0 in Ha[M3]. The q-graded Euler charecteristic of C[x] is naively divergent: 1+1+1+. . ., but its zeta-regularization gives 1/2.

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Section: -Manifolds and The "Bottom Row"mentioning

confidence: 94%

BPS spectra and 3-manifold invariants

Gukov

Pei

Putrov

et al. 2020

J. Knot Theory Ramifications

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132

331

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“…The first major piece of the desired structure came with the construction of Khovanov-Rozansky homology [5][6][7] that belongs to the lower left corner in Figure 1. This corner is by far the most developed element of the sought after 2d-4d TQFT on D ⊂ M 4 , and even that only for M 4 = R 4 and D = R × K. Its physical interpretation, proposed in [8], led to many new predictions and connections between various areas, which include knot contact homology [9], gauge theory [10,11], and algebras of interfaces [12,13], just to name a few. (A more complete account of these connections can be found, e.g., in [13,14].…”

Section: Numbersmentioning

confidence: 99%

“…The supersymmetric completions of momentum and winding operators play an important role in mirror symmetry [49] and in various extensions of the sine-Liouville theory. 9 One of the main goals of this paper is to relate Seiberg-Witten invariants of M 4 (and their generalizations) to correlation functions of the operators V λ in the half-twisted theory T [M 4 ]. In the absence of a background charge, all such correlators are subject to a "neutrality condition" which states that correlation functions vanish unless the total charge λ of the operators is equal to zero.…”

Section: Half-twisted Model With Target Spacementioning

confidence: 99%

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Vertex Algebras and 4-Manifold Invariants

Dedushenko¹,

Gukov²,

Putrov³

2018

Geometry and Physics: Volume I

112

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We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d N = (0, 2) theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations. 5 Non-abelian generalizations 51 5.1 Coulomb-branch index and new 4-manifold invariants 54 A Curvature of the canonical connection on the universal bundle 57 A.1 SU (N f ) invariance and the moment map 59 B Topological twist of SQED 60 1.1 Unorthodox invariants of smooth 4-manifolds Searching for new invariants of smooth 4-manifolds, that potentially could go beyond the Seiberg-Witten and Donaldson invariants, it was proposed in [1] to consider a twodimensional quantum field theory T [M 4 , G] as a rather unusual invariant of smooth structures on a 4-manifold M 4 . Specifically, T [M 4, G] is a 2d N = (0, 2) superconformal theory that, apart from M 4 , also depends on a choice of a root system G and is invariant under the Kirby moves. Luckily, conformal field theories in two dimensions exhibit rich mathematical structure which, on the one hand, is rich enough to (potentially) describe the wild world of smooth 4-manifolds and, on the other hand, is rigorous enough to hope for a precise mathematical definition of the invariant T [M 4 , G]. In fact, for many practical purposes and applications in this paper, a mathematically inclined reader can think of T [M 4 , G] as a functor that assigns a vertex operator algebra (VOA) to a smooth 4-manifold M 4 (and a "gauge" group G).Composing it with other functors that assign various quantities to 2d conformal theories, one can obtain more conventional invariants of smooth 4-manifolds:Here, Z can be any invariant of a 2d conformal theory with N = (0, 2) supersymmetry, e.g., its elliptic genus, chiral ring, moduli space of marginal couplings, or central charge. Since 2d theory T [M 4 ; G] is systematically determined by M 4 and invariant under the Kirby moves, all such invariants lead to various 4-manifold invariants; some are simple and some are quite powerful. In particular, it was conjectured in [1] that chiral ring of the theory T [M 4; G] for G = SU (2) or, equivalently, its Q + -cohomology knows about Donaldson invariants of M 4 . One of the main goals in this paper is to present some evidence to this conjecture and to build a bridge between VOA G [M 4 ] and more traditional 4-manifold invariants. Conjecturally, at least for manifolds with b + 2 > 1, one can trade SU (2) Donaldson invariants for Seiberg-Witten invariants defined in a simpler gauge theory, with gauge group G = U (1), which would be too trivial if not for an extra ingredient, the additional spinor fields. We wish to study how these invariants are realized in 2d theory T [M 4 ; G] with G = SU (2) and G = U (1), respectively. In particular, we shall see that in 2d realizati...

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“…For example quantum A-polynomials for knots reduce to classical Apolynomial algebraic equations, which on one hand encode information about S n -colored knot polynomials for large n, and on the other hand capture classical BPS invariants for knots [29]. In case of multiple variables -which arise for example for knots colored by non-symmetric representations, or for links whose components are independently colored -higher-dimensional quantum and classical varieties can be considered, such as those discussed in [47,48]. Note that via the knots-quivers correspondence, quantum A-polynomials for knots at the same time provide difference equations for generating series of quivers associated to knots, in this case with all variables x i identified with a single variable x, as discussed in [16].…”

Section: Quantum Curves and A-polynomials For Quiversmentioning

confidence: 99%

Topological strings, strips and quivers

Panfil

Sułkowski

2019

J. High Energ. Phys.

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We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

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Topological strings, D-model, and knot contact homology

Cited by 41 publications

References 57 publications

BPS spectra and 3-manifold invariants

BPS spectra and 3-manifold invariants

Vertex Algebras and 4-Manifold Invariants

Topological strings, strips and quivers

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