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M2-Branes Ending on M5-Branes

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M2-Branes Ending on M5-Branes M2-branes ending on M5-branes Vasilis Niarchos Crete Center for Theoretical Physics, University of Crete 7th Crete Regional Meeting on String Theory, 27/06/2013 based on recent work with K. Siampos 1302.0854, ``The black M2-M5 ring intersection spins’‘ Proceedings Corfu Summer School, 2012 1206.2935, ``Entropy of the self-dual string soliton’’, JHEP 1207 (2012) 134 1205.1535, ``M2-M5 blackfold funnels’’, JHEP 1206 (2012) 175 and older work with R. Emparan, T. Harmark and N. A. Obers ➣ blackfold theory 1106.4428, ``Blackfolds in Supergravity and String Theory’’, JHEP 1108 (2011) 154 0912.2352, ``New Horizons for Black Holes and Branes’’, JHEP 1004 (2010) 046 0910.1601, ``Essentials of Blackfold Dynamics’’, JHEP 1003 (2010) 063 0902.0427, ``World-Volume Effective Theory for Higher-Dimensional Black Holes’’, PRL 102 (2009)191301 0708.2181, ``The Phase Structure of Higher-Dimensional Black Rings and Black Holes’‘ + M.J. Rodriguez JHEP 0710 (2007) 110 2 Important lessons about the fundamentals of string/M-theory (and QFT) are obtained by studying the low-energy theories on D-branes and M-branes. Most notably in M-theory, recent progress has clarified the low-energy QFT on N M2-brane and the N3/2 dof that it exhibits. Bagger-Lambert ’06, Gustavsson ’07, ABJM ’08 Drukker-Marino-Putrov ’10 Our understanding of the M5-brane theory is more rudimentary, but efforts to identify analogous properties, e.g. the N3 scaling of the massless dof, is underway. Douglas ’10 Lambert,Papageorgakis,Schmidt-Sommerfeld ’10 Hosomichi-Seong-Terashima ’12 Kim-Kim ’12 Kallen-Minahan-Nedelin-Zabzine ’12 ... 3 It is believed that the M5 theory is a theory of strings. Townsend ’95, Strominger M2-branes can end on M5-branes ’95, Becker-Becker ’96 ... just like F-strings can end on D-branes in string theory. The above intersection is 1/4-BPS (preserves 8 supercharges). 4 The IR dynamics is controlled by a (1+1)-dim (4,4) SCFT that lives in the intersection. We would like to identify this theory, or key features of this theory e.g. how does the central charge of this CFT scale with N2 , N5 ? Technically, we can approach this question in two different ways: • from a microscopic analysis of the M5/M2 brane physics • from a supergravity analysis of the corresponding black brane intersection (e.g. near-extremal black brane thermodynamics gives 3 2 for M2-branes S N 2 T ⇠ for M5-branes S N 3 T 5 ) e.g. Klebanov-Tseytlin ’96 ⇠ 5 Both approaches are technically complicated. I will describe work in the SUGRA approach. 6 M5 point of view: the Howe-Lambert-West solution Descriptions of the intersection are possible either from the M2 or M5 brane point of view. I will not say much about the M2 point of view, as it will be less relevant for the approach of this talk. Basu-Harvey ’04, (ABJM model with boundary conditions...) Berman et al. ’06, ’09 ... From the M5 point of view the string intersection appears as a solitonic solution of the M5 brane worldvolume theory. 7 Howe, Sezgin, West, Bandos, Lechner, Nurmagambetov, Pasti, Sorokin, Tonin, Aganagic, Park, Poperscu, Schwarz ’96, ’97 The Howe-Lambert-West solution: N5=1, N2>0. The abelian worldvolume theory on a single M5 brane is known. It is a theory of a self-dual 3-form field strength and 5 transverse scalars (plus their fermion partners). Key point: this theory is the leading term in a long-wavelength derivative expansion (analogous to the DBI theory for D-branes). 8 Howe-Lambert-West ’97 The 1/4-BPS self-dual string soliton solution Generalizations (M-theory analog of BIon solution) Gauntlett-Lambert-West ’98 Callan-Maldacena ’97 x6 = z 1 2Qsd 6 x direction suppressed z(σ)= ,z:= x 3 σ2 S H = H = dz σ (3) ⇤6 (3) ⇤4 3 An S spike describes N2 M2-branes ending on N5=1 M5-branes. 9 The solution (and the associated derivative expansion) breaks down at some radius, but a `miracle’ happens: at the tip of the spike one recovers the tension of the orthogonal M2 branes. The leading order solution works much better than naively expected. We will re-encounter and extend this feature below. 10 The non-abelian M5-brane worldvolume theory remains a mostly open subject. Lambert,Papageorgakis ’10 Saemann-Wolf ’11, ’12 Ho-Huang-Matsuo ’11 Bonetti-Grimm-Hohenegger ’12 Samtleben-Sezgin-Wimmer-Wulff ’11, ’12 Chu-Ko ’12 ... This obstructs a similar analysis at generic N2, N5 , but recent progress has been possible. Chu-Ko-Vanichchapongjaroen ’12 Chu-Vanichchapongjaroen ’13 Managed to reproduce one of our supergravity predictions... 11 Known supergravity solutions Supergravity allows us to examine the system in the limit N2, N5 >>1. Brane intersections in supergravity is a subject with a long history and impressive achievements. Nevertheless, in many cases it is technically challenging to find explicit solutions that describe fully localized intersections. ☞ The Einstein equations imply complicated systems of PDEs. The challenge is greater as we reduce the amount of supersymmetry, or if we have no supersymmetry at all, e.g. for non-extremal solutions. 12 In the case of the 1/4-BPS orthogonal M2-M5 intersection we are looking for a solution with SO(1,1) x SO(4) x SO(4) symmetry. A partially localized solution with metric element 2 1/3 2/3 1 2 1 2 1 2 2 5 2 ds = H H (H H )− ( dt +(dx ) )+H− ((dx ) + ...+(dx ) )+ 2 5 2 5 − 5 h 1 6 2 7 2 10 2 H− (dx ) +(dx ) + +(dx ) , 2 ··· i 2 H =0, H 2 + 2 H =0 r(789(10)) 5 5r(2345) r(789(10)) 2 ⇣ ⌘ Duff-Ferrara-Khuri-Rahmfeld ’95 is known (delocalized along the 6-direction). Tseytlin ’96, Gauntlett ’97 Youm ’99, Smith ’02 13 Recent progress towards a fully localized solution ☞ Lunin: 0704.3442 (near-horizon), 0706.3396 (asymptotically flat) derives the appropriate PDEs, solutions are not explicit ☞ D’Hoker-Estes-Gutperle-Krym (near-horizon) 0810.1484, 0806.0605, 0810.4647, 0906.0596 & follow-up: Estes-Feldman-Krym 1209.1845, Berdichevsky-Dahan 1304.4389 explicit families of solutions, but so far unclear which solution describes the orthogonal M2-M5,... 14 I will describe a different treatment of the fully localized solution in SUGRA that • works within a long-wavelength expansion scheme (and is thus technically & conceptually closer to the non-gravitational M5 worldvolume description) • gives immediate intuitive information, and • extends easily to more complicated (less symmetric) configurations that are well beyond the reach of current exact solution generating techniques. analogous description of BIon (F1-D3) solution: Grignani, Harmark, Marini, Obers, Orselli ’10,’11 15 Part of a more general still-developing programme in (super)gravity to: ☞ systematically capture the dynamics of (black) brane solutions with long- wavelength expansion schemes around exact solutions ☞ develop the gravitational analog of DBI-theory for supergravity solutions 16 Blackfold theory basics Emparan, Harmark, VN, Obers Blackfolds provide a general effective (long-wavelength) worldvolume description of black brane dynamics They describe how a black p-brane fluctuates, spins and bends Local patch of Lorentz transformed black p-brane r0 flat black p-brane a thin curved blackfold 17 The fluid/gravity correspondence illustrates nicely the idea. Bhattacharyya-Hubeny- • a spin-off of the AdS/CFT correspondence Minwalla-Rangamani ’07,... • it describes temperature and velocity fluctuations of AdS black branes 1 L2 in the long-wavelength approximation λ AdS T ⇠ r0 in terms of a relativistic conformal fluid. T µ⌫ =0 rµ • the fluid lives on a time-like surface in the asymptotic region of the black hole spacetime (AdS boundary) • there is a constructive perturbative procedure that maps uniquely the solutions of the fluid equations to regular bulk spacetimes 18 Blackfolds add co-dimension to the fluid-gravity correspondence ➠ a mix of fluid dynamics + `DBI’. Long-wavelength fluctuations of a black brane are captured by an effective hydrodynamic theory that describes a fluid on an dynamical worldvolume. The effective degrees of freedom are the slowly-varying temperature, fluid velocity, worldvolume bending scalars, etc. The equations of motion of the effective degrees of freedom are conservation equations for the stress-energy tensor, charge currents... 19 matched asymptotic expansions Outstanding issues Equations of Perturbative expansion of Effective Hydrodynamic Theory (super)gravity equations ☞ Effective Hydro theory has been constructed only at few leading orders. Emparan-Harmark-VN-Rodriguez-Obers ’07 Camps, Emparan ’12 ☞ In progress: effective hydro theory at all orders, general map between hydro solutions and regular bulk spacetimes. We will focus on the leading order of this expansion scheme. 0th order blackfold solution ➠ 1st order thermodynamics 20 M2-M5 blackfold funnels We want to describe a spiky deformation of the planar M5 black brane (with dissolved M2 brane charge) Izquierdo-Lambert- Papadopoulos-Townsend ’95 ➠ how the planar black M2-M5 bound state deforms Russo-Tseytlin ’96 1 F4 = dC3 + D− ?dC6 21 The parameters of the planar M2-M5 bound state are promoted to slowly- varying functions of the effective 5-brane worldvolume coordinates Effective degrees of freedom a a a a a a ˆ r0(ˆσ ) ,↵(ˆσ ) ,✓(ˆσ ) ,u(ˆσ ) ,X?(ˆσ ) , V(3) unit 3-volume form local temperature M2, M5 charges local transverse boosts scalars controls embedding of M2 charge in 5-brane wv 22 Leading order blackfold equations i Kab : extrinsic curvature `hydrodynamic’ part tensor `DBI’ part intrinsic equations extrinsic equations i ab ab Kab T =0 DaT =0 d J =0,J= Vˆ ⇤ 3 3 Q2 (3) d J =0,J= Vˆ ⇤ 6 6 Q5 (6) local stress-energy tensor, constitutive relations,..
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