The sound of Knots and 44-manifolds -manifolds Sergei Gukov Mikhail Khovanov Piotr Sulkowski Knots ? = Knots ? = 4-manifolds 4_1 FIGURE EIGHT ALEXANDER JONES 1 SL(3) GENUS AMPHICHEIRAL HOMFLY-PT COLORED Why integer coefficients? 1 i [M.Khovanov] 2 1 0 -1 -2 -5 -3 -1 1 3 5 j Paul Dirac (1939): “Pure mathematics and physics are becoming ever more closely connected, though their methods remain different … It is difficult to predict what the result of all this will be. Possibly, the two subjects will ultimately unify … At present we are, of course, very far from this stage, even with regard to some of the most elementary questions. For example, only four-dimensional space is of importance in physics, while spaces with other numbers of dimensions are of about equal interest in mathematics.” Extra Dimensions Kaluza-Klein compactification 10d string theory on 10-n x Mn “effective” theory T[Mn ] in 10-n dimensions depends on topology and geometry of Mn 10 = 4 + 6 E.Calabi S.-T.Yau Geometry of extra dimensions determines the spectrum of 4d particles and their interactions 0 104 108 1012 Energy Scale (GeV) 1016 R.Dijkgraaf ‘03 • = spectrum of BPS particles in supersymmetric theories Strings in 6d 6d fivebrane theory on 6-n x Mn “effective” theory T[Mn ] in 6-n dimensions depends on topology and geometry of Mn 6=3+3 3-manifold M3 complex flat connections 3d N = 2 theory T[M 3] supersymmetric vacua 6=3+3 3d N = 2 theory 3-manifold M3 T[M 3] supersymmetric vacua complex flat connections spectrum of BPS states Knot homology = BPS Classifying Phases of Matter Quiver Chern-Simons theory vertex a U(1) Chern-Simons at level a a ai edge aj cf. [D.Belov, G.Moore] [A.Kapustin, N.Saulina] [J.Fuchs, C.Schweigert, A.Valentino] : Quiver Chern-Simons theory integrate out A 3d Kirby moves 3d Kirby moves A is Lagrange multiplier Integrating out A makes B pure gauge and removes all its Chern-Simons couplings Plumbing graphs Intersection form on : 6=2+4 4-manifold M4 2d N = (0,2) theory T[M 4] 6d fivebrane theory on 2 x M4 depends on topology and geometry of M4 6=2+4 4-manifold M4 very rich math 2d N = (0,2) theory T[M 4] very rich physics 6=2+4 4-manifold M4 0-handle adding 2-handles: ADE more general plumbing graphs, Kirby moves, … Vertex Operator Algebra Heisenberg algebra affine Kac-Moody VOAs associated with even positive lattices … line 26 /7 BMY c = 12c (abelian ) Unitarity constraints rm o f e d E Het at ns o i at ge r la he t e No N ine l r c = 0 (large N) Phases Phases [M.Stephanov] Can we quantitatively understand confinement and the mass gap? • Extensively tested in computer simulations • Paper-and-pencil computation? $1,000,000 Prize The answer may involve gravity! Confinement in 2d ‘t Hooft Solvable Gauge Theories • • • • 2d Yang-Mills: almost “topological” QED: confinement / screening 2d N = 0 QCD: one Regge trajectory QCD with a massive adjoint: higher Regge Solvable Gauge Theories • • • • • • • 2d Yang-Mills: almost “topological” QED: confinement / screening 2d N = 0 QCD: one Regge trajectory QCD with a massive adjoint: higher Regge 2d N = (1,1) SQCD: very similar 2d N = (0,2) SQCD: ? 2d N = (2,2) SQCD: ? : 2d N = (0,2) SQCD • New (0,2) SCFTs • Exactly solvable string phenomenology • Holographic dual of the Veneziano limit higher spin theory Building walls … … vs building bridges Interview with Sir Michael Atiyah on math, physics and fun What makes a mathematics problem fun for you? The main thing that interests me in mathematics always is the interconnection between different parts of mathematics, the fact that one problem may have half a dozen different ways of being looked at in different subjects, a bit of algebra, a bit of geometry, a bit of topology. It's this interaction and bridges that interest me.
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