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Some Comments on Equivariant Gromov-Witten Theory and 3D Gravity (A Talk in Two Parts) This Theory Has Appeared in Various Interesting Contexts

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Some Comments on Equivariant Gromov-Witten Theory and 3D Gravity (A Talk in Two Parts) This Theory Has Appeared in Various Interesting Contexts Some comments on equivariant Gromov-Witten theory and 3d gravity (a talk in two parts) This theory has appeared in various interesting contexts. For instance, it is a candidate dual to (chiral?) supergravity with deep negative cosmologicalShamit constant. Kachru Witten (Stanford & SLAC) Sunday, April 12, 15 Tuesday, August 11, 15 Much of the talk will be review of things well known to various parts of the audience. To the extent anything very new appears, it is based on two recent collaborative papers: * arXiv : 1507.00004 with Benjamin, Dyer, Fitzpatrick * arXiv : 1508.02047 with Cheng, Duncan, Harrison Some slightly older work with other (also wonderful) collaborators makes brief appearances. Tuesday, August 11, 15 Today, I’d like to talk about two distinct topics. Each is related to the general subjects of this symposium, but neither has been central to any of the talks we’ve heard yet. 1. Extremal CFTs and quantum gravity ...where we see how special chiral CFTs with sparse spectrum may be important in gravity... II. Equivariant Gromov-Witten & K3 ...where we see how certain enumerative invariants of K3 may be related to subjects we’ve heard about... Tuesday, August 11, 15 I. Extremal CFTs and quantum gravity A fundamental role in our understanding of quantum gravity is played by the holographic correspondence between conformal field theories and AdS gravity. In the basic dictionary between these subjects conformal symmetry AdS isometries $ primary field of dimension ∆ bulk quantum field of mass m(∆) $ ...... Tuesday, August 11, 15 I For compact target manifold, discrete string spectrum ! discrete symmetry groups I For one-loop partition function, worldsheet is torus any trace I For compact target manifold,! discretefunction string spectrumshould be invariant under ! discreteSL symmetry(2, Z) groups I For one-loop partition function, worldsheet is torus any trace ! function should be invariant under SL(2, Z) observables, which of course are also the ones we will be studying here, because of their phenomenological importance, and because, since the B model is soluble classically, the relation given byB. mirror The symmetryelliptic betweengenus observables and Mathieu of the A (moonshineX)modelandobservables observables, which of course are also the ones we will be studying here, because of their of the B(Y )modelisparticularlyuseful. phenomenological importance, and because, since the B model is soluble classically, the What is moonshine? The theory describingOutline strings propagating on a K3 relation given byB. mirror The symmetryelliptic betweengenusMonstrous moonshine observables and Mathieu of the A (moonshineX)modelandobservables There are famous examplesConway moonshine where the correspondence can surface is a 2d non-linearConclusions sigma model with target a of2. the PreliminariesbeB(Y made)modelisparticularlyuseful. very precise. One, relevant to this conference, Appearance in stringWhat is moonshine? theory The theoryK3 describing surface,Outline i.e.strings a theory propagating of maps on a K3 To begin with, we recall theMonstrous standard moonshine supersymmetric nonlinear sigma model in two relates the sigmaConway model moonshine with target Hilb(K3) to Conclusions 2.dimensions. Preliminariessurface2 It governsis a 2d maps non-linearΦ : Σ → X ,sigma with Σ modelbeing a Riemannwith target surface anda X a superstringsI In string theory wecompactified me consider 2d on AdS3xS3xK3. Riemannian manifoldAppearanceK3 of metricsurface, in stringg.Ifwepicklocalcoordinates theory i.e. a theory of mapsz, z on Σ and φI on X, To begin with, wesigma recallφ models,: ⌃ the maps standardK from3 supersymmetric nonlinear sigma model in two then Φ can be describedworldsheet locally to! target via manifold functions φI (z, z¯). Let K and K be the canonical dimensions.2 It governs maps Φ : Σ → X, with Σ being a Riemann surface and X a and anti-canonicalI In string line theory bundles we me of considerΣ (the 2d bundles of one formsoftypes(1, 0) and (0, 1), Riemannian manifold of metric g.Ifwepicklocalcoordinatesz, z on Σ and φI on X, sigmaφ models,: ⌃ mapsK from31/2 respectively), and let K1/2 and K be square roots of these. Let TX be the complexified then Φ canThis be described2dworldsheet field locally totheory! target via manifold functions has someφI (z, z¯basic). Let Kfieldsand K φ be (along the canonical with I 1/2 ∗ tangent bundle of X. The fermi fields of the model are ψ+,asectionofK ⊗ Φ (TX), and anti-canonicalsuperpartners), line bundles1/2 of Σ (theand bundles an action of one formschematicallysoftypes(1, 0) like: and (0, 1), and ψI ,asectionofK ⊗ Φ∗1(/TX2 ). The Lagrangian is3 respectively),This− 2d and letfieldK1/ 2theoryand K hasbe square some roots basic of these. fields Let TX φ be(along the complexified with tangent bundle of X. The fermi fields1 of the model arei ψI ,asectionofK1/2 ⊗ Φ∗(TX), L =2t d2z g (Φ)∂ φI ∂ φJ + g + ψI D ψJ I superpartners),1/2 ∗ andIJ anz actionz¯ schematically3IJ − z − like: and ψ ,asectionofK !⊗ Φ ("TX2 ).Sarah The M. Harrison Lagrangianc =12Moonshine2 is − (2.1) i I J 1 I J K L 2 1+ gIJψ+IDzψJ+ + i RIJKLI ψ+ψJ+ψ− ψ− . L =2t d z gIJ2(Φ)∂zφ ∂z¯φ + 4gIJψ−Dzψ− # ! "2 Sarah M. Harrison c =12Moonshine2 (2.1) Monday, August 10, 15 ¯ Here t is a coupling constant, andi RIJKLI isJ the Riemann1 tensorI J K of XL . Dz¯ is the ∂ operator + gIJψ Dzψ + RIJKLψ ψ ψ ψ . on K1/2 ⊗ Φ∗(TX)constructedusingthepullbackoftheLevi-Civitaconnect2 + + 4 + + − −# ion on TX. InTuesday, formulas August 11, 15 (using a local holomorphic trivialization of K1/2), HereMonday,t is August a coupling 10, 15 constant, and RIJKL is the Riemann tensor of X. Dz¯ is the ∂¯ operator 1/2 ∗ J on K ⊗ Φ (TX)constructedusingthepullbackoftheLevi-CivitaconnectI ∂ I ∂φ I K ion on TX. Dz¯ψ = ψ + Γ ψ , (2.2) In formulas (using a local holomorphic+ trivialization∂z + ∂z of KJK1/2+), I J 1/2 ∗ with ΓJK the affine connectionI of X.Similarly∂ I ∂φDzIis theK ∂ operator on K ⊗ Φ (TX). Dz¯ψ+ = ψ+ + ΓJKψ+ , (2.2) The supersymmetries of the model∂z are generated∂z by infinitesimal transformations 1/2 with ΓI the affine connection ofI X.SimilarlyI D Iis the ∂ operator on K ⊗ Φ∗(TX). JK δΦ = i%−ψ+ + i%+ψz − The supersymmetries of theI model are generatedI K byI infinitesiM mal transformations δψ+ = −%−∂zφ − i%+ψ− ΓKMψ+ (2.3) I I I δΦ =I i%−ψ + i%I+ψ K I M δψ− = −%++∂zφ −−i%−ψ+ ΓKMψ− , I I K I M δψ+ = −%−∂zφ − i%+ψ− ΓKMψ+ (2.3) 2 This discussion will be at the classical level, and we will notworryabouttheanomaliesthat I I K I M arise and spoil some assertionsδψ− if= the−% target+∂zφ space− i%− isψ+ notΓKM a Caψlabi-Yau− , manifold. 3 2 2 Here d z is the measure −idz ∧ dz¯.Thusifa and b are one forms, a ∧ b = This2 discussion will be at the classical level, and we will notworryabouttheanomaliesthat i d z (azbz¯ − az¯bz). The Hodge ! operator is defined by !dz = idz, !dz¯ = −idz¯. $ arise and spoil some assertions if the target space is not a Calabi-Yau manifold. 3$ 2 Here d z is the measure −idz ∧ dz¯.Thusif4 a and b are one forms, a ∧ b = 2 i d z (azbz¯ − az¯bz). The Hodge ! operator is defined by !dz = idz, !dz¯ = −idz¯. $ $ 4 Here is a natural question: given a desired spectrum of bulk AdS fields (“set of elementary particles”), does there exist a dual CFT which defines it non-perturbatively? This question is far beyond our reach. But we can explore simple examples. Perhaps the simplest is pure 3d quantum gravity. Do there exist conformal field theory duals to pure 3d (super)gravity? Tuesday, August 11, 15 You see that for eigenstates of the “Hamiltonian” it becomes obviousThe spectrum that thisof pure is just: 3d gravity would be just the graviton multipletThese transformations and the associated are multiparticleassociated with the TheseβH transformations are associatedgroup SL(2,Z). with the Tr e− , β 2⇡R states. ⇠ group SL(2,Z). Now instead ofWe a canparticle use the moving Thisspectrum means in imaginaryto that read the off partition time,the partition function function we were This meanscomputing, ofthat the the conjectural partition that summarizes functiondual CFT: wethe were spectrum of string computing,consider that a string: summarizesstates, should the behave spectrum well of under string SL(2,Z): states, should behave well under SL(2,Z): n 2⇡i⌧ Z = n cnq q = e n 2⇡i⌧ Z = cnq q = e n P cn = # of states at mass level n P cn = # of states at mass level n more properly, There are now two parameters that fix the shapea⌧+b of should behave as Z( c⌧+d )moreZ properly,(⌧) a⌧+b should⇠ behave as a modular function the torus, instead of justZ (ac ⌧radius.+d ) Z(⌧) ⇠ a modular function or form.... or form.... Monday, August 10, 15 Tuesday, August 11, 15 Thursday, May 15, 14 Monday, August 10, 15 Thursday, May 15, 14 Monday, August 10, 15 Thursday, May 15, 14 factorization, and mainly consider only the holomorphic sector of the theory. The full partition function is the product of the function we determine and a similar antiholomor- phic function. These partition functions have been studied before [31,32], with different motivation. 3.1. The Bosonic Case We begin with the bosonic case. What are the physical states ofpuregravityina spacetime asymptotic at infinity to AdS3? Since there are no gravitational waves in the theory, the onlystatethatisobvious at first sight is the vacuum, corresponding in the classical limit to Anti de Sitter space. In a conformal field theory with central charge c =24k,thegroundstateenergyisL0 = c/24 = k.ThecontributionofthegroundstateΩ to the partition function Z(q)= − − | " L0 k Tr q is therefore q− . Of course, there is more to the theory than just the ground state.
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