$SO(3)$-instantons on $L(p,q)\times \mathbf{R}$

Abstract:We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N = (0, 2) theories, we obtain a number of results, which include new 3d N = 2 theories T [M 3 ] associated with rational homology spheres and new results for Vafa-Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0, 2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines / walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d N = (0, 2) theories and 3d N = 2 theories, respectively.

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“…Let us note that in the case when M 3 is a lens space the "gluing kernel" K[M 3 ] can be explicitly computed using the results of [75,76].…”

Section: Gluing Along 3-manifoldsmentioning

confidence: 99%

Fivebranes and 4-Manifolds

Gadde

Gukov

Putrov

2016

Arbeitstagung Bonn 2013

173

390

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“…We finish with a short section on torus knots. was shown by Sasahira [36,Corollary 4.3] (see also Austin [6]) to equal…”

Section: Knot Homology: Explicit Calculationsmentioning

confidence: 94%

Link homology and equivariant gauge theory

Poudel

Saveliev

2017

Algebr. Geom. Topol.

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The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, torus knots, and Montesinos knots, as well as for several families of two-components links. IntroductionThis paper studies the Floer homology I * (Σ, L) of two-component links L ⊂ Σ in homology 3-spheres defined by Kronheimer and Mrowka [26] using singular SO (3) instantons. An important special case of this theory is the singular instanton knot Floer homology I ♮ (k) for knots k ⊂ S 3 obtained by applying I * (S 3 , L) to the link L which is a connected sum of k with the Hopf link. The Floer homology I * (Σ, L) has a relative Z/4 grading, which can be upgraded to an absolute Z/4 grading in the special case of I ♮ (k). [26] used I ♮ (k) and its close cousin I ♯ (k) to prove that the reduced Khovanov homology is an unknot-detector. Kronheimer and MrowkaThe definition of groups I * (Σ, L) uses singular gauge theory, which makes them difficult to compute. We propose a new approach to these computations which uses equivariant gauge theory in place of the singular one.Given a two-component link L in an integral homology sphere Σ, we pass 2010 Mathematics Subject Classification. 57M27, 57R58.Both authors were partially supported by NSF Grant 1065905. 1 to the double branched cover M → Σ with branch set L and observe that the singular connections on Σ used in the definition of I * (Σ, L) pull back to equivariant smooth connections on M . The generators of the Floer chain complex IC * (Σ, L), whose homology is I * (Σ, L), are then derived from the equivariant representations π 1 M → SO(3), and their Floer gradings are computed using equivariant rather than singular index theory † .As our first application of this approach, we determine the grading of the special generator in the Floer chain complex IC ♮ (k) of a knot k ⊂ S 3 , see Section 5. This fixes the absolute Z/4 grading on I ♮ (k) and confirms the conjecture of Hedden, Herald and Kirk [20].Theorem. For any knot k ⊂ S 3 , the grading of the special generator in the Floer chain complex IC ♮ (k) equals sign k mod 4.We also achieve significant simplifications in computing the Floer chain complexes IC ♮ (k) and IC * (Σ, L) for knots and links with simple double branched covers, such as torus and Montesinos knots and links, whose double branched covers are Seifert fibered manifolds. Explicit calculations for these knots and links are possible because the gauge theory on Seife...

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“…where we consider P = N − 1 ≈ N ∼ O(10). Since P ef f turns out to be large we can use in our evaluation of P ef f an approximate expression (45) for three different sets of integer parameters P, M, q, w, l. For a given G 2 manifold the values of P , M , and q are fixed whereas l and w can be chosen freely.…”

Section: A Coarse Tuningmentioning

confidence: 99%

“…Including the factor of φ −2/P into (45) gives a few percent correction. It is clear from (45) and the entries in the table that in order to get a large value of P ef f one has a tradeoff between the values of P and q. In particular, for P < ∼ 10 the value of q has to be very large, i.e.…”

Section: A Coarse Tuningmentioning

confidence: 99%

“…In four dimensions, two different vacua corresponding to different values of the gauge connection are separated by a domain wall which is effectively described by a gauge field instanton on Q × R, where R is a direction transverse to the wall [4], [40], [45]. Because gauge field configurations with a non-zero fractional instanton number carry a non-zero M2-brane charge (63), the domain wall corresponds to an M2-brane whose world-volume is R 2,1 ∈ Q × R 3,1 .…”

Section: Dynamical Neutralization Of the Vacuum Energymentioning

confidence: 99%

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Scanning the Fluxless G_2 Landscape

Bobkov¹

2009

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We show that there exists an exponentially large discretuum of vacua in G 2 -compactifications of M-theory without flux. In M-theory-inspired G 2 -MSSM, quantities relevant for particle physics remain virtually insensitive to large variations of the vacuum energy across the landscape. The purely non-perturbative vacua form a special subset of a more general class of vacua containing fractional Chern-Simons contributions. The cosmological constant can be dynamically neutralized via a chain of transitions interpolated by fractional gauge instantons describing spontaneous nucleation of M2-brane domain walls. Each transition is generically accompanied by a gauge symmetry breaking in some sector of the theory. In particular, the visible sector GUT symmetry breaking can likewise be triggered by a spontaneous nucleation of an M2-brane.

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$SO(3)$-instantons on $L(p,q)\times \mathbf{R}$

Cited by 19 publications

References 20 publications

Fivebranes and 4-Manifolds

Fivebranes and 4-Manifolds

Link homology and equivariant gauge theory

Scanning the Fluxless G_2 Landscape

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