1 / 33

Massive type IIA string theory cannot be strongly coupled

Massive type IIA string theory cannot be strongly coupled. Daniel L. Jafferis Institute for Advanced Study. 16 November, 2010 Rutgers University. Based on work with Aharony, Tomasiello, and Zaffaroni. Motivations. What is the fate of massive IIA at strong coupling?

sophie
Download Presentation

Massive type IIA string theory cannot be strongly coupled

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Massive type IIA string theory cannot be strongly coupled Daniel L. Jafferis Institute for Advanced Study 16 November, 2010 Rutgers University Based on work with Aharony, Tomasiello, and Zaffaroni

  2. Motivations • What is the fate of massive IIA at strong coupling? • What is the dual description of 3d CFTs at large N and fixed coupling? • Explore the N=1, 2 massive IIA AdS × CP3 solutions and their dual CFTs.

  3. 2 = = Á Á 2 2 3 2 4 3 ¡ ( ) d d d A + + s e s e x = 1 1 1 1 1 0 IIA string theory at strong coupling • The strong coupling limit of IIA string theory is M-theory, so this regime is again described by supergravity. • D0 branes have a mass 1/gs, and become light, producing the KK tower of the 11d theory.

  4. R F A D 0 0 Massive IIA at strong coupling • Would seem to be a lacuna in the web of string dualities. • The D0 branes have tadpoles, • There is no free “massive” parameter in 11d supergravity. • A more fundamental question: are there any strongly coupled solutions of IIA supergravity?

  5. = 5 4 = = ¸ d d ¸ d N A S N A S i i g n a n g n » » 5 4 s s Behavior of 3d CFT at large N • In the ‘t Hooft limit, one always finds a weakly coupled string dual. • In 3d, it is natural to consider taking N large with k fixed. • In the N=6 theory, this results in light disorder operators corresponding to the light D0 branes of IIA at strong coupling. There is an M-theory sugra description with entropy N3/2. • Is that the generic behavior?

  6. F 1 M M 1 M M k T F F F F k k 2 1 ¡ : : : : : : g = F M N M M M M M N Á ¡ ¢ 1 P Q 2 ¡ M N ( ) P k ! k ! 0 Á 2 1 4 k R r r H H T ¡ k k 2 1 2 + ¡ F F F : : : : : : + ^ e = M N M N M N P Q 1 e 2 2 = k P k ? P k ( ) 0 1 k M N ¡ 0 2 4 F F 4 + = ; ; ; ; k ? k k k 0 1 2 4 ¡ 0 2 4 4 = = ; ; ; ; ; A bound on the dilaton • In string frame, the Einstein equations are this is exact up to 2 derivative order even when the coupling is large. • The 00 component can be written using frame indices as where

  7. 2 ¡ ( ) ` 2 ¿ Z = ` 1 > X B s 1 ¡ ¡ Á ( ) a ` F 2 s 1 e n ¼ ¿ = k e a s Á = ` R C < k e a s . » Massive IIA solutions cannot be strongly coupled and weakly curved • This equation is satisfied at every point in spacetime. All of the terms in parentheses on the left side must be small , otherwise the 2 derivative sugra action cannot be trusted. • The fluxes , on a compact a-cycle are quantized. • Thus is F0 ≠ 0, then the rhs . • Therefore . Typically, the lhs is order 1/R2, thus

  8. In strongly curved backgrounds? • In a generic background with string scale curvature, the notion of 0-form flux is not even defined. • No signs of strong coupling in known massive IIA AdS solutions. • UV completion of Sagai-Sugimoto still unknown, but the region between the D8 branes is not both weakly curved and at large coupling.

  9. In some special cases, one might make sense out of a strongly curved, strongly coupled region in a massive IIA solution: • If it were a part of a weakly curved solution, probably it will be small (string scale). • If there were enough supersymmetry, it might be related by duality to a better description. For example T-dualizing to a background without F0 flux. [Hull,…]

  10. Massive IIA AdS duals of large N 3d CFTs • To gain further insight into this result, will look at AdS vacua of massive IIA. • This results in interesting statements about the dual field theories. • We will find that the string coupling never grows large. • At large N for fixed couplings, the behavior will be completely different than the massless case.

  11. ~ I I ¹ ¹ ( ) ( ( ) ( ) ) Ã Ã ¯ l d h ¯ l d ¤ ¤ C A C A N N N N i i i j t t t t n g a u n g e e m s a e r e e r c o s n u g a e s I I ¹ ¹ ; ; ; ; ; I ( ) C A B ¤ = _ a ; : a 2 ( ) ¼ W A B A B ² ² = _ _ b b _ b b _ a a a k a The N=6 CSM theory of N M2 branes in C4/Zk • U(N)k x U(N)-k CSM with a pair of bifundamental hypermultiplets • Field content: • SU(2) x SU(2) global symmetry, which does not commute with SO(3)R, combining to form SU(4)R

  12. p = 2 5 2 ¸ R 2 ¼ 7 = = d Z A S S t £ s r = = ¸ k N N 1 4 k 4 ¸ 1 = ! , g » I I A 5 k k N À Dual geometry • The gauge theory coupling is 1/k. Fixing , the usual ‘t Hooft limit is a string theory. • One obtains IIA on AdS4 × CP3 with N units of F4 and k units of F2 in CP3 • For , one finds small curvature and a large dilaton. Lifts to M-theory on

  13. ( ) ( ) U N U N £ k k 1 2 ¡ + n 0 Massive IIA • Consider deforming the N=6 CSM theory by the addition of a level a CS term for the second gauge group. • In this theory the monopole operators corresponding to D0 branes develop a tadpole, since the induced electric charge (k, n0-k) cannot be cancelled with the matter fields. • This motivates the idea that the total CS level should be related to the F0 flux. [Gaiotto Tomasiello, Fujita Li Ryu Takayanagii]

  14. 2 2 ( ) ( ) ( ) j j ( ) k k C S A C S A X A A + ¡ + ¡ n 1 0 2 1 2 • The light U(1) on the moduli space has a level n0 Chern-Simons term, matching the coupling of the D2 worldvolume to the Romans mass. • For such deformations of N=6 CSM, there are field theories with N = 3,2,1,0 differing by the breaking of the SU(4) into flavor and R-symmetry. [Tomasiello; Gaiotto Tomasiello]

  15. R R k F F N N ¡ k k F n n = = ¡ = = 2 R 2 2 4 4 2 1 1 n F N 2 = C P = C P 0 0 1 2 n = = 6 6 1 3 C P 2 2 2 d d d + s s s = d d 3 N A S 1 2 3 C P N 1 2 3 w a r p e = = 4 ; ; ; ; ; Review of massive AdS4 solutions • The dual geometries are topologically the same as the N=6 solution, but are now warped. • Metric on CP3 has SO(5), SO(4), SO(3) isometry in the N = 1,2,3 cases. Last solution only known perturbatively.

  16. Large N limit • In the ‘t Hooft limit, these solutions are small deformations of the AdS4 x CP3N=6 IIA supergravity solution. • What about the large N limit for fixed levels? When n0 = 0, this results in strong coupling, and a lift to M-theory. • We now know that this is impossible for n0 ≠ 0.

  17. p ³ ´ ( ) ( = ) q 2 2 5 ¡ ¡ k k 1 1 i i j j 2 ¾ ¾ 2 2 2 R 5 ( ) d d d R A ¯ B J R + + ¡ + s x ² x s = = = d A S 3 4 S C P N 8 2 1 ( ) 2 + 2 2 1 + ¾ = ¾ ¾ ; 2 [ ] 2 2 ¾ 5 ; N=1 detailed analysis • The SO(5) invariant metric on CP3 is given by where the space is regarded as an S2 bundle over S4. • The parameter , where 2 is the Fubini-Study metric.

  18. = ( ) ` ` R 2 1 ( ) ´ f ¼ 0 1 d A S ¾ b s 0 1 0 1 0 1 0 1 0 0 0 l n n g 0 s 0 k B 1 R ¡ ¡ ( ) ` F 2 e n ¼ = k k B C s l ( ) B C B C B C f b B C ¾ b 1 0 0 2 n n B C B C B C 2 g 2 B C s B C B C B C B C ´ = B C B C B C 3 B C l 1 b 2 ( ) b b f B C B C B C 1 0 n n ¾ B C 4 4 4 2 B C B C B C g B C s @ A @ A @ A @ A 1 1 b 3 2 5 b b b 1 l ( ) n n f 6 6 ¾ 6 2 6 g s Parameters and fluxes • The four parameters of the sugra solution are related to the quantized fluxes,

  19. 3 3 k k h h W W N N ¿ À e e n n 2 2 n n 0 0 h h b l h d l F K S i i i i 1 2 t t t t t ¾ ¾ e e n e u a r n y - - a u e y r m m e e r r c c ! ! , , , , = = = 1 1 6 4 1 4 N N N 1 d d ` ` a a n n g g » » » » = = = = s s 1 1 4 6 5 4 5 6 , , k k = 1 6 N n n 0 0 d f f h l N i i 6 t t t e o r m a o n o e s o u o n = k l l d ! i . a n e w w e a y c o u p e r e g m e A new regime • These relations can be inverted explicitly. • Take n4 = 0, n2 = k, n6=N

  20. 5 = L p L 1 2 2 3 R R ( ) 2 4 k k N F L A A + + ^ n n n n » » D D D 2 2 2 0 2 0 ¡ ` R 2 . n n n n = = g ¼ g D D 2 0 0 2 s s s , . Particle-like probe branes • In the massive IIA solutions, D0 branes have a tadpole. Just as in the massless case, so do D2 branes, • A D2/D0 bound state has a total worldvolume tadpole . . Take • Consider n0=1, n2 = k for simplicity. Then the mass of the D-brane is • In the first phase, the D0s dominate the mass while in the second phase, the D2 dominates the mass • D4 branes always exist, and have mass in AdS units, which is order N in both phases, as expected for a baryon.

  21. N N ¸ ¸ ¸ ¸ ¸ § = = = 1 2 1 2 § k k ; ; ¡ 1 2 3 2 n ¸ ¸ N 2 » ) » ¡ 2 + n 0 Field theory interpretation • Define the ‘t Hooft couplings, where n4=0 for simplicity. • In these variables, the transition occurs for • To have better control over the CFT, we turn to the N=2 case.

  22. Light disorder operators in the CFT dual? • There are clearly no light D-branes in this limit of the N=1 solution. • One expects that the monopole operators of the CFT will get large quantum corrections to their dimensions. • However, in the N=2 case, they are protected.

  23. 1 ¡ = k 2 ¾ ¾ n ¹ = = a R F 2 ¼ n = 2 a S Monopoles operators • There are monopole operators in YM-CS-matter theories, which we follow to the IR CSM. • In radial quantization, it is a classical background with magnetic flux , and constant scalar, . Of course, in the CSM limit, • It is crucial that the fields in μ are not charged under a. • This operator creates a vortex. [Borokhov Kapustin Wu]

  24. 1 P j j Q ¡ q F f i e 2 e r m o n s Anomalous dimensionN=2 case • We work in the UV to compute the 1-loop correction to the charge of a monopole operator under some flavor (or R–symmetry, or gauged) U(1). One finds • This is an addition to the usual, mesonic charge of the operator.

  25. N k k y y y y 1 ( ) k k A A B B i B B A A 1 2 ¡ ¡ ¡ N w w w w 1 = = 1 i i 1 2 ( ) d i i i i ; : : : ; 2 , 2 ¼ ¼ ¾ a g w w = i i i ; : : : ; . 2 P A A B B a w w w w = = n w 1 2 2 1 = i . ; i P k 0 n = i i ( ) 1 1 0 ( ) d 1 1 0 w = k 2 w a n = k ; : : : : : : ; ; ; : : : 1 1 ; : : : : : : ; ; ; : : : 2 Monopoles in the massive duals • Take Then • Sits in an irrep with weight . • Gauge invariant combinations with the matter fields require that • In our case, take • There are solutions to the equations:

  26. k k 2 ( ) ( ) ( ) k k k k N N 1 2 + ¡ ¡ ¡ ¡ 2 1 2 1 1 2 2 2 ( ) ( ) ( ) N N ¡ ¡ ¡ ¡ n n n n 1 2 1 2 1 2 Dimensions • There are two adjoint fermions with R-charge +1 in the vector multiplets, and four bi-fundamental fermions with R-charge -1/2. • This results in a quantum correction to the R-charge of the monopole • Combines with the matter dimension to give

  27. p [ ] Ã 0 3 2 1 ( ) ( ) B B 2 2 t t ; 1 1 1 2 e e 2 2 2 2 2 2 2 ( ) ( ) ( ) d d d d d ¡ A A t t t + + + + ¡ s s s ² a = 2 1 2 2 6 S S 4 4 8 6 4 1 2 N=2 solution • The internal metric is SO(4) invariant. • It has the form of T1,1 fibered over an interval. One S2 shrinks at each end. • Depends on 4 parameters, L, gs, b, , where 0 is the undeformed solution. • Related to the four quantized fluxes.

  28. B A A 2 2 2 2 2 ¡ ( ) ( ) h à h f W C i 4 2 2 1 t t t i ¡ e r e w e c o s c o s e s e w a r p a c o r = = à i t , , ; 2 ( ) ( ) ( ) à C i 4 2 2 t + + s n w w c o s w w f h d d à h A S i i i à t t t 1 2 1 2 t o e m e r c a n a p p e a r s n e s p n o r s 0 à ; , . = ( ) ( ) ( ) 2 à C i 4 2 2 t + + s n w w c o s w w à 1 2 1 2 t ; 2 ( ( ) ) à C i 4 2 2 2 ¡ ¡ w w w w s n w à 1 1 2 2 1 t 0 ; w = 1 ( ) ( ) ( ) 2 à C i 4 2 2 t + + s n w w c o s w w à 1 2 1 2 t ; 2 ( ( ) ) à C i 4 2 2 2 ¡ ¡ w w w w s n w à 2 1 2 1 2 t 0 ; w = 2 ( ) ( ) ( ) 2 à C i 4 2 2 t + + s n w w c o s w w à 1 2 1 2 t ; N=2 solution • The solution can be reduced to three first order differential equations.

  29. The shape of the solution • At each end of the interval one sphere shrinks. The size of the other at that point is plotted versus the deformation parameter. Develops a conifold singularity!

  30. 3 3 k k p h h à à W W N N 0 3 ¿ À e e n n ! ! 1 1 2 2 , , , , n n 0 0 f b l d d l F S i i i i i i t t t t g a e c o n u o n - s n u g u y a r m e y r a p c p e a r s , , = = = 1 1 6 4 1 4 N N N 1 d d ` ` a a n n g g » » » » = = = = s s 1 1 4 6 5 4 5 6 , , . . k k = 1 6 N n n 0 0 Two phases again Take n4 = 0, n2 = k, n6=N

  31. 2 p L ¡ ¢ R n ( ) ( ) n d à L F B F 0 2 t + ¡ ¡ m e g n n = = D 2 0 4 1 ( ) 2 3 ` 2 2 ¼ g s s Match of light D2 branes • A D2 brane wrapping the diagonal S2 can be supersymmetric. The tadpole is cancelled by appropriate worldvolume flux. • The mass can be calculated to give • Remarkably, computing numerically, F is a constant, equal to 1. • Precisely matches the field theory.

  32. 1 g » = 5 s ( ) 1 6 N = n 1 6 0 ³ ´ N = R 1 3 = 1 5 3 » N t s r n n » 0 0 G N A new weakly coupled string regime • In the massive IIA solution dual to U(N)k × U(N)-k+n0, we found • This is in spite of the fact that the N=2 theory has light monopole operators. • It would be interesting to understand the general behavior.

  33. Conclusions • There are no strongly coupled solutions of massive IIA supergravity. Regions of strong curvature still need to be fully understood. • Conifold singularities seen to arise in AdS backgrounds. • The emergence of weakly coupled strings in a new regime of field theories.

More Related