Testing holography through lattice simulations

Transcript

1. Testing holography through lattice simulations Raghav G. Jha Syracuse University April 04, 2017 arXiv 1704.XXXXX (with Simon Catterall, David Schaich & Toby Wiseman) Quantum Gravity, String theory and Holography Kyoto, Japan
2. Outline 1 Motivation 2 Lattice supersymmetry : Problems and resolutions 3 Topological twist and lattice action for N = 4 SYM on A∗ 4 lattice 4 Holographic principle, 2d SYM and dual gravity predictions • Phase transition • Homogeneous D1 phase • Localized D0 phase Holography & Lattice Raghav G. Jha, Syracuse University 2
3. 1 Motivation 2 Lattice supersymmetry : Problems and resolutions 3 Topological twist and lattice action for N = 4 SYM on A∗ 4 lattice 4 Holographic principle, 2d SYM and dual gravity predictions • Phase transition • Homogeneous D1 phase • Localized D0 phase Holography & Lattice Raghav G. Jha, Syracuse University 3
4. Why lattice supersymmetry ? Lattice discretization provides non-perturbative, gauge-invariant regularization of gauge theories • Insights into confinement, symmetry breaking, conformal field theories , etc. • Gauge/gravity (AdS/CFT) duality −→ potential non-perturbative definition of string theory Lattice studies are important for non-perturbative λ and finite-N regime of gauge theories. Holography & Lattice Raghav G. Jha, Syracuse University 4
5. 1 Motivation 2 Lattice supersymmetry : Problems and resolutions 3 Topological twist and lattice action for N = 4 SYM on A∗ 4 lattice 4 Holographic principle, 2d SYM and dual gravity predictions • Phase transition • Homogeneous D1 phase • Localized D0 phase Holography & Lattice Raghav G. Jha, Syracuse University 5
6. Lattice SUSY : Problems and resolutions Problem Supersymmetry generalizes Poincaré symmetry, adding spinorial generators Q and Q to translations, rotations, boosts The algebra includes QQ + QQ = 2σµPµ , Pµ generates infinitesimal translations, which don’t exist on the lattice =⇒ supersymmetry explicitly broken at the classical level. Solution Preserve a subset of SUSY algebra exactly on the lattice. Possible for theories with Q ≥ 2D. For ex : N = 4 supersymmetric Yang–Mills (SYM). Methods are based on orbifold construction and twisiting. I will focus on the latter in this talk. Holography & Lattice Raghav G. Jha, Syracuse University 6
7. N = 4 SYM is a particularly interesting theory —Important for holographic dualities at large-N. —Arguably simplest non-trivial field theory in four dimensions Basic features: • SU(N) gauge theory with four fermions ΨI and six scalars ΦIJ, all massless and in adjoint rep. • Supersymmetric: 16 supercharges QI α and Q I ˙ α with I = 1, · · · , 4 Fields and Q’s transform under global SU(4) ≃ SO(6) R-symmetry • Conformal: β function is zero for any ’t Hooft coupling λ = g2 Y M N Holography & Lattice Raghav G. Jha, Syracuse University 7
8. 1 Motivation 2 Lattice supersymmetry : Problems and resolutions 3 Topological twist and lattice action for N = 4 SYM on A∗ 4 lattice 4 Holographic principle, 2d SYM and dual gravity predictions • Phase transition • Homogeneous D1 phase • Localized D0 phase Holography & Lattice Raghav G. Jha, Syracuse University 8
9. Topological twisting =⇒ Exact SUSY on lattice SO(4)tw ≡ diag [SO(4)euc × SO(4)R ] ; SO(4)R ⊂ SO(6)R The 16-real components of the spinors in N = 4 SYM fill up the Dirac-Kähler multiplet :    Q1 α Q2 α Q3 α Q4 α Q1 ˙ α Q2 ˙ α Q3 ˙ α Q4 ˙ α    = Q + Qµ γµ + Qµν γµ γν + Qµ γµ γ5 + Qγ5 −→ Q + γa Qa + γa γb Qab with a, b = 1, · · · , 5 Q’s transform with integer spin under the “twisted rotation group”. This twisting and repackaging gives us a nilpotent, scalar supercharge Q which can be exactly preserved on the lattice even at finite lattice spacing. Holography & Lattice Raghav G. Jha, Syracuse University 9
10. Twisted N = 4 SYM fields Useful to start in a 5d setup QI α and Q I ˙ α −→ Q, Qa and Qab −→ 1 + 5 + 10 ΨI and Ψ I −→ η, ψa and χab Aµ and ΦIJ −→ Aa and Aa Everything transforms with integer spin under SO(4)tw — no spinors. Then under dimensional reduction : Q, Qa and Qab −→ Q + Qµ γµ + Qµν γµ γν + Qµ γµ γ5 + Qγ5 Aa −→ (Aµ , ϕ) + i(Bµ , ϕ) where, a b runs from 1 ·· 5 and µ from 1 ·· 4 Holography & Lattice Raghav G. Jha, Syracuse University 10
11. Lattice action The lattice action consists of Q-exact and Q-closed terms. Slat = N 4λlat Q ∑ ( χab Fab + η[ ¯ Da , Da ] − 1 2 ηd ) − N 16λlat ∑ ϵabcde χab ¯ Dc χde • Covariant derivatives −→ finite difference operators eg. Da ψb → Ua (x)ψb (x + a) − ψb (x)Ua (x + b) and anti-symmetric field tensor defined as, Fab = [Da , Db ] Catterall et al. 1405.0644 Q acts as : QAa = ψa Qχab = − ¯ Fab Qη = d Qψa = 0 Q ¯ Aa = 0 Qd = 0 Holography & Lattice Raghav G. Jha, Syracuse University 11
12. 5d → 4d dimensional reduction gives A∗ 4 lattice —Can think of A∗ 4 lattice as 4d analog of 2d triangular lattice —Preserves S5 point group symmetry — Basis vectors are non-orthogonal S5 irreps precisely match onto irreps of SO(4)tw 5 = 4 ⊕ 1 : Ua −→ Aµ + iBµ , ϕ + iϕ ψa −→ ψµ , η 10 = 6 ⊕ 4 : χab −→ χµν , ψµ Holography & Lattice Raghav G. Jha, Syracuse University 12
13. 5d → 4d dimensional reduction gives A∗ 4 lattice —Can think of A∗ 4 lattice as 4d analog of 2d triangular lattice —Preserves S5 point group symmetry — Basis vectors are non-orthogonal S5 irreps precisely match onto irreps of SO(4)tw 5 = 4 ⊕ 1 : Ua −→ Aµ + iBµ , ϕ + iϕ ψa −→ ψµ , η 10 = 6 ⊕ 4 : χab −→ χµν , ψµ Holography & Lattice Raghav G. Jha, Syracuse University 12
14. Public code for maximal SYM Our parallel code is publicly developed and available at : github.com/daschaich/susy We have recently generalized this for arbitrary N to access holographic dualities. The computational costs scales as : N7/2 D. Schaich, RGJ et al. (In preparation, 2017). This code evolved from MILC lattice QCD code and was first presented in arXiv:1410.6971 (restricted to N ≤ 4). Holography & Lattice Raghav G. Jha, Syracuse University 13
15. 1 Motivation 2 Lattice supersymmetry : Problems and resolutions 3 Topological twist and lattice action for N = 4 SYM on A∗ 4 lattice 4 Holographic principle, 2d SYM and dual gravity predictions • Phase transition • Homogeneous D1 phase • Localized D0 phase Holography & Lattice Raghav G. Jha, Syracuse University 14
16. Holographic applications Original AdS/CFT correspondence 4D N = 4 U(N) super-Yang-Mills theory associated with N D3-branes, is dual to Type IIB string theory on AdS5 × S5 in the large N limit. More general holographic dualities in lower dimensions Maximally superymmetric YM in p + 1 dimensions dual to Dp-branes At low temperatures (not very low !) , and in the decoupling limit : dual description in terms of black holes in Type II A/B supergravity Holography & Lattice Raghav G. Jha, Syracuse University 15
17. Two-dimensional SYM (p=1) • Dimensional reduction : A∗ 4 −→ A∗ 2 giving a skewed torus with γ = 1/2 (γ = cos θ). • ’t Hooft coupling (λ) is dimensionful in two dimensions and we construct a dimensionless coupling given by ˆ λ = λβ2, where β = 1/T . • Extent of spatial and time circles can be written as dimensionless quantities ; rx = √ λR and rτ = √ λβ = 1/t , where t is the dimensionless temperature. In addition, we also have γ. • Much more interesting than 1-d QM case, phase transition between uniform D1 and localized D0 phase with spatial Wilson line being the order parameter. • Three interesting regimes : 1) rτ ≪ rx , 2) rτ ∼ rx > 1 and, 3 ) rτ ≫ rx Holography & Lattice Raghav G. Jha, Syracuse University 16
18. Regime of valid supergravity description To have a valid SUGRA description, we need : • Radius of curvature should be large in units of α′. This implies rτ ≫ 1. • String coupling should be small. We can combine both requirements to get a constraint on the effective dimensionless coupling we can probe for a well-defined SUGRA description (p < 3) 1 ≪ λp β3−p ≪ N 10−2p 7−p This can be written in terms of our dimensionless coupling for p=1 as, 1 ≪ rτ ≪ N 2 3 Holography & Lattice Raghav G. Jha, Syracuse University 17
19. IIA sugra IIB sugra rx 2 ccritrΤ , ccrit 2.29 G L transition Px 0 Deconfinement Px 0 Confinement PΤ 0 everywhere 0 1 rx 1 rΤ Figure 1: Different regions of the phase space for γ = 0. From : arXiv:1008.4964 Holography & Lattice Raghav G. Jha, Syracuse University 18
20. Phase transition at large-N Figure 2: Phase transition for α = rx /rτ = 4. With, χ = N2( ⟨ P2 x ⟩ − ⟨Px ⟩2) Holography & Lattice Raghav G. Jha, Syracuse University 19
21. rτ ≪ rx • High temperature (weak coupling) regime of the theory, well approximated by BQM (Bosonic Quantum mechanics). • Higher order phase transition(s) expected around : r3 x ∼ 2.09rτ (Note that this is r3 x ∼ 1.35rτ for γ = 0) • Bosonic action density ∼ t2. 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 (1/N2)(c - <s>)/(λ) t SU(6), 16nt4 SU(9), 16nt4 SU(12), 16nt4 SU(9), 24nt6 SU(12), 24nt6 HTE for α=4 Figure 3: High-temperature lattice results Holography & Lattice Raghav G. Jha, Syracuse University 20
22. 1 Motivation 2 Lattice supersymmetry : Problems and resolutions 3 Topological twist and lattice action for N = 4 SYM on A∗ 4 lattice 4 Holographic principle, 2d SYM and dual gravity predictions • Phase transition • Homogeneous D1 phase • Localized D0 phase Holography & Lattice Raghav G. Jha, Syracuse University 21
23. rτ ∼ rx > 1 • For the skewed torus, the gravity results are not yet available and our current lattice simulations predict phase transition along : r2 x = 3.21(5) rτ . • Earlier work (γ = 0) predicted the transition along r2 x ≈ 2.45rτ which has been confirmed from calculations on the gravity side two months back [1702.07718]. Holography & Lattice Raghav G. Jha, Syracuse University 22
24. D1 thermodynamics The bosonic action ‘density’ for the dual D1 branes is found to be independent of angle of skewness of the torus. In this homogeneous phase, it predicts, like the rectangular torus : sBos,homog N2λ = − 23π 5 2 34 t3 ≈ −1.728t3 for the SYM bosonic action density, with t = 1/rτ the dimensionless temperature. Our lattice simulations seem consistent with this prediction. Holography & Lattice Raghav G. Jha, Syracuse University 23
25. D1 thermodynamics - Results from lattice simulations 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Localized BH Homogenous BS (1/N2)(c - <s>)/λ t SU(9), 16nt8 SU(12), 16nt8 SU(16), 16nt8 SU(12), 24nt12 D1 prediction : 1.728 t3 Figure 4: The lattice results and dual gravity prediction for α = 2. Holography & Lattice Raghav G. Jha, Syracuse University 24
26. 1 Motivation 2 Lattice supersymmetry : Problems and resolutions 3 Topological twist and lattice action for N = 4 SYM on A∗ 4 lattice 4 Holographic principle, 2d SYM and dual gravity predictions • Phase transition • Homogeneous D1 phase • Localized D0 phase Holography & Lattice Raghav G. Jha, Syracuse University 25
27. rτ ≫ rx : Localised phase (D0 thermodynamics) For the localized D0 phase, the prediction for the bosonic action density depends on the skewness of the torus and follows a different temperature-dependence than the D1 prediction sBos,localized N2λ ≃ −2.469 t16 5 α2 5 (1 − γ2)7 5 . α is the aspect ratio defined as rx /rτ . Holography & Lattice Raghav G. Jha, Syracuse University 26
28. D0 thermodynamics - Results 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (1/N2)(c - <s>)/λ t N=9, 8nt16 N=12, 8nt16 D0 prediction : 3.6935 t3.2 α-2/5 Figure 5: The lattice results and dual gravity prediction for the localized phase, α = 1/2 Holography & Lattice Raghav G. Jha, Syracuse University 27
29. Brief recap and future work • Lattice supersymmetry is both enticing and challenging • N = 4 SYM is practical to study on the lattice thanks to exact preservation of susy subalgebra Q2 = 0 • We can now access relatively large-N to explore conjectured holographic dualities. • We hope to study 3d SYM with sixteen supercharges in the future for which lattice construction is again possible. • Computing the √ λ dependence of the static potential in N = 4 SYM is also being pursued. Holography & Lattice Raghav G. Jha, Syracuse University 28
30. Thank you ! Holography & Lattice Raghav G. Jha, Syracuse University 29
31. Thank you ! Funding and computing resources Holography & Lattice Raghav G. Jha, Syracuse University 29
32. The sign problem In lattice gauge theory we compute operator expectation values ⟨O⟩ = 1 Z ∫ [dU][dU] O e−SB[U,U] pf D[U, U] pf D = |pf D|eiα can be complex for lattice N = 4 SYM −→ Complicates interpretation of [ e−SB pf D ] as Boltzmann weight Sign problem: The complex phase produces a ’sign problem’ if ⟨ eiα ⟩ is consistent with zero Holography & Lattice Raghav G. Jha, Syracuse University 28
33. Illustration - sign (problem, but no problem) We have carefully analyzed the sign of the Pfaffian in this theory and find no evidence of it being a problem. • With periodic temporal fermion boundary conditions we have an obvious sign problem, ⟨ eiα ⟩ consistent with zero • With anti-periodic BCs and all else the same ⟨ eiα ⟩ ≈ 1 Figure 6: Holography & Lattice Raghav G. Jha, Syracuse University 28
34. Holography & Lattice Raghav G. Jha, Syracuse University 28
35. Truncated lattice theory A naïve truncation of U(N) supersymmetric theory to SU(N) does not work. • To maintain SU(N) gauge invariance it is necessary to keep the fermions in gl(N, C), explicitly breaking the lattice supersymmetry that relates Ua to ψa in the U(N) construction. • Solution : Represent the truncated gauge links as Ub = eigaAb to argue that the continuum supersymmetry relating Aa and ψa is approximately realized in the large-N limit even at non-zero lattice spacing since g → 0 in the decoupling limit. Holography & Lattice Raghav G. Jha, Syracuse University 28
36. Continuum vs. lattice coupling The non-orthogonal basis vectors of the A∗ 2 lattice leads to mismatch in ’￿￿t Hooft coupling between lattice and continuum. The target continuum 2d-SYM coupling (rτ,cont. ) differs from the lattice coupling as, rτ,lattice = √ 3 √ 3 rτ,cont. Holography & Lattice Raghav G. Jha, Syracuse University 28