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

3. 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/g s, and become light, producing the KK tower of the 11d theory. d s 2 11 = e ¡ 2 Á = 3 d s 2 10 + e 4 Á = 3 ( d x 11 + A ) 2

4. 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? R F 0 A D 0

5. 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 N 3/2. 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 N 3/2. Is that the generic behavior? Is that the generic behavior? g s » ¸ = N i n A d S 5 an d g s » ¸ 5 = 4 = N i n A d S 4

6. 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 e ¡ 2 Á ¡ R MN + 2 r M r N Á ¡ 1 4 H M PQ H NPQ ¢ = P k = 0 ; 2 ; 4 T F k MN T F k MN = 1 2 ( k ¡ 1 ) ! F M M 2 ::: M k F NM 2 ::: M k ¡ 1 4 k! F M 1 ::: M k F M 1 ::: M k g MN 1 4 ( P k = 2 ; 4 F 2 0 ; k ¡ 1 + P k = 0 ; 2 ; 4 F 2 ? ; k ) F k = e 0 ^ F 0 ; k ¡ 1 + F ? ; k

7. 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 F 0 ≠ 0, then the rhs. Therefore. Typically, the lhs is order 1/R 2, thus ( ¿ ` ¡ 2 s ) Z C a e ¡ B X k F k = n a ( 2 ¼ ` s ) a ¡ 1 > 1 = ` 2 s e Á ¿ 1 e Á » < ` s = R.

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 F 0 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. The N=6 CSM theory of N M2 branes in C 4 /Z k 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 ( C I ) ¤ ; ( Ã I ) ¤ i n ( ¹ N ; N ) t h e i rcon j uga t es C I ; Ã I i n ( N ; ¹ N ) ma tt er ¯ e ld s A ¹ ; ~ A ¹ gauge ¯ e ld s C I = ( A a ; B ¤ _ a ) : W = 2 ¼ k ² a b ² _ a _ b ( A a B _ a A b B _ b )

12. Dual geometry The gauge theory coupling is 1/k. Fixing, the usual ‘t Hooft limit is a string theory. One obtains IIA on AdS 4 × CP 3 with N units of F 4 and k units of F 2 in CP 3 For, one finds small curvature and a large dilaton. Lifts to M-theory on ¸ = N = k, N ! 1 g IIA » ¸ 1 = 4 k R 2 s t r = 2 5 = 2 ¼ p ¸ N À k 5 A d S 4 £ S 7 = Z k

13. 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, n 0 -k) cannot be cancelled with the matter fields. This motivates the idea that the total CS level should be related to the F 0 flux. [Gaiotto Tomasiello, Fujita Li Ryu Takayanagii] U ( N 1 ) k £ U ( N 2 ) ¡ k + n 0

14. The light U(1) on the moduli space has a level n 0 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] k CS ( A 1 ) + ( n 0 ¡ k ) CS ( A 2 ) + j X j 2 ( A 1 ¡ A 2 ) 2

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

16. Large N limit In the ‘t Hooft limit, these solutions are small deformations of the AdS 4 x CP 3 N=6 IIA supergravity solution. What about the large N limit for fixed levels? When n 0 = 0, this results in strong coupling, and a lift to M-theory. We now know that this is impossible for n 0 ≠ 0.

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

18. Parameters and fluxes The four parameters of the sugra solution are related to the quantized fluxes, R e ¡ B F k = n k ( 2 ¼ ` s ) k ¡ 1 ` ´ R A d S =( 2 ¼ ` s )

19. A new regime These relations can be inverted explicitly. Take n 4 = 0, n 2 = k, n 6 =N W h en N ¿ k 3 n 2 0 ¾ ! 2, t h e F u b i n i - S t u d yme t r i c, an d` » N 1 = 4 k 1 = 4, g s » N 1 = 4 k 5 = 4 d e f orma t i ono f t h e N = 6 so l u t i on. W h en N À k 3 n 2 0 ¾ ! 1, t h enear l y- K a hl erme t r i c, an d` » N 1 = 6 n 1 = 6 0, g s » 1 N 1 = 6 n 5 = 6 0 anewwea kl ycoup l e d reg i me !

20. 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 n 0 =1, n 2 = 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. 1 2 ¼ ` s R F 2 ^ A D 2 ( n D 2 n 2 + n D 0 n 0 ) R R A » k 2 » N 2 = 3 L 5 g s n D 2 = n 0, n D 0 = ¡ n 2. L g s p k 2 + L 4.

21. Field theory interpretation Define the ‘t Hooft couplings, where n 4 =0 for simplicity. In these variables, the transition occurs for To have better control over the CFT, we turn to the N=2 case. ¸ 1 = N k 1 ; ¸ 2 = N ¡ k 2 ; ¸ § = ¸ 1 § ¸ 2 N » n 3 2 n 2 0 ) ¸ ¡ » ¸ 2 +

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. 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. R S 2 F a = 2 ¼n ¾ = n = 2 [Borokhov Kapustin Wu] ¾ a = k ¡ 1 ¹

24. Anomalous dimension N=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. ¡ 1 2 P f erm i ons j q e j Q F

25. 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: ¾ i = 1 2 d i ag ( w 1 i ;:::; w N i i ). n i = P w a i ( k i w 1 i ;:::; k i w N i i ) P k i n i = 0 w 2 = ( 1 ;::: k 1 :::; 1 ; 0 ;::: ) AA y ¡ B y B = k 1 2 ¼ w 1, BB y ¡ A y A = ¡ k 2 2 ¼ w 2. w 1 A = A w 2 ; w 2 B = B w 1 w 1 = ( 1 ;::: k 2 :::; 1 ; 0 ;::: ) an d

26. 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 ( n 1 ¡ n 2 ) 2 ¡ ( N 1 ¡ N 2 )( n 1 ¡ n 2 ) k 1 k 2 2 + ( k 2 ¡ k 1 ) 2 ¡ ( k 2 ¡ k 1 )( N 1 ¡ N 2 )

27. N=2 solution The internal metric is SO(4) invariant. It has the form of T 1,1 fibered over an interval. One S 2 shrinks at each end. Depends on 4 parameters, L, g s, b,, where 0 is the undeformed solution. Related to the four quantized fluxes. d s 2 6 = e 2 B 1 ( t ) 4 d s 2 S 2 1 + e 2 B 2 ( t ) 4 d s 2 S 2 2 + 1 8 ² 2 ( t ) d t 2 + 1 64 ¡ 2 ( t )( d a + A 2 ¡ A 1 ) 2 Ã 1 2 [ 0 ; p 3 ]

28. N=2 solution The solution can be reduced to three first order differential equations. W h erew i = 4 e 2 B i ¡ 2 A, C t ; à = cos 2 ( 2 t ) cos 2 ( 2 à ) ¡ 1, e 2 A i s t h ewarp f ac t or o f t h e A d S me t r i c, an dà appears i n t h esp i nors.

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. Two phases again Take n 4 = 0, n 2 = k, n 6 =N W h en N À k 3 n 2 0, Ã 1 ! p 3, acon i f o ld s i ngu l ar i t yappears, an d` » N 1 = 6 n 1 = 6 0, g s » 1 N 1 = 6 n 5 = 6 0. W h en N ¿ k 3 n 2 0, Ã 1 ! 0, ge t F u b i n i - S t u d yme t r i c, an d` » N 1 = 4 k 1 = 4, g s » N 1 = 4 k 5 = 4.

31. Match of light D2 branes A D2 brane wrapping the diagonal S 2 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. m D 2 L = n 0 L ( 2 ¼ ) 2 g s ` 3 s R p d e t ( g + F ¡ B ) = ¡ n 2 2 2 ¡ n 0 n 4 ¢ F ( Ã 1 )

32. A new weakly coupled string regime In the massive IIA solution dual to U(N) k × U(N) -k+n 0, 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. R s t r » ³ N n 0 ´ 1 = 6 g s » 1 ( N n 5 0 ) 1 = 6 1 G N » N 5 = 3 n 1 = 3 0

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.