AdS 6 /CFT 5 in Type IIB

Transcription

1 AdS 6 /CFT 5 in Type IIB Part II: Dualities, tests and applications Christoph Uhlemann UCLA Strings, Branes and Gauge Theories APCTP, July 2018 arxiv: , , , , , , , , with E. D Hoker, M. Gutperle, A. Karch, C. Marasinou, A. Trivella, O. Varela, O. Bergman, D. Rodríguez-Gómez, M. Fluder

2 Introduction Higher-dimensional CFTs are interesting: mysterious, non-lagrangian interesting implications for d 4 string/m-theory connections... 5d SCFTs: often fixed points for asymptotically safe gauge theories, large classes engineered in Type IIB via 5-brane webs 1

3 Recap Any planar 5-brane junction realizes 5d SCFT on intersection point (p 1, q 1 ) (p 2, q 2 ) p i, q i Z pi = q i = 0 (p 3, q 3 ) Characterized entirely by external 5-brane charges. May or may not have gauge theory deformations. 2

4 Recap Supergravity solutions for fully localized 5-brane intersections, constructed directly as near-horizon limit: (p L, q L ) (p 1, q 1 )... (p 3, q 3 ) (p 2, q 2 ) 3

5 Recap Supergravity solutions for fully localized 5-brane intersections, constructed directly as near-horizon limit: Z L + q L + ip L Z 1 + q 1 + ip r L r 3 s 1 s 2 r 1 s 3... s L 2 r 2 Σ AdS 6 S 2 Σ with Σ = disc 5-branes emerge at poles on Σ parametrized entirely by residues =conserved 5-brane charges AdS susies = F (4) Z 3 + q 3 + ip 3 Z 2 + q 2 + ip 2 aturally fit to string theory picture. Large : all (p i, q i ) large. 3

6 Recap Claim: poles groups of 5-branes that are unconstrained by s-rule. Constrained junctions realized by bringing 7-branes into web: S5 Σ k T k D5 D7 S5 Supergravity: Σ = disc with puncture(s), [p, q] 7-brane monodromy Position of puncture on Σ choice of face in which 7-brane is placed with no 5-branes attached(?) 4

7 AdS 6 /CFT 5 in Type IIB Goal of this talk: convincing case for proposed identifications and dualities, first lessons on 5d SCFTs 5

8 AdS 6 /CFT 5 in Type IIB Outline Testing AdS 6 /CFT 5 with stringy operators Holographic S 5 partition functions and EE, # d.o.f. QFT partition functions: Precision test of AdS 6 /CFT 5 (Massive) spin-2 fluctuations 6

9 Testing AdS 6 /CFT 5 with strings

10 Testing AdS 6 /CFT 5 with strings 5-brane picture: gauge invariant operators from strings and string junctions connecting external 5-branes 1 (p 1, q 1 ) 2 (p 2, q 2 ) 1 (p 1, q 1 ) 2 (p 2, q 2 ) 3 (p 3, q 3 ) 3 (p 3, q 3 ) Supergravity: operators with = O() probe strings or string junctions in supergravity background (1 F(S 5 )). Strategy: study gauge theory deformations, identify stringy BPS operators, extrapolate to SCFT, compare to supergravity 7

11 Stringy operators in +,M theories M [] SU() M 1 [] SU() 2 U(1) M B U(1)M 1 I SU() 2 SU(M) 2 U(1) 8

12 Stringy operators in +,M theories M [] SU() M 1 [] SU() 2 U(1) M B U(1)M 1 I SU() 2 SU(M) 2 U(1) M i j = (x (1) x (M) ) i j (,, 1, 1) = 3 2 M Q = 1 2 M B (k) = det(x (k) ) (1, 1, M, M) = 3 2 Q = 1 2 8

13 Stringy operators in +,M theories M B (k) [] SU() M 1 [] M i j SU() 2 U(1) M B U(1)M 1 I SU() 2 SU(M) 2 U(1) M i j = (x (1) x (M) ) i j (,, 1, 1) = 3 2 M Q = 1 2 M B (k) = det(x (k) ) (1, 1, M, M) = 3 2 Q = 1 2 M i j F1 between D5, B(k) D1 between S5. S-dual quiver deformation realizes all D1 operators, subset of F1 operators. 8

14 Stringy operators in +,M solution M S5 D5 Σ D5 M S5 9

15 Stringy operators in +,M solution M S5 D5 Σ D5 M S5 global AdS 6 /CFT 5 on R S 4 AdS 6 t dilation: = H 9

16 Stringy operators in +,M solution M S5 D5 Σ D5 M S5 global AdS 6 /CFT 5 on R S 4 AdS 6 t dilation: = H Action and EOM for (p, q) string along t and Σ: S (p,q) = 2T d 2 ξf 6 ρ w qmq ( )( )( p 1 χ p qmq = e 2φ q χ χ 2 + e q) 4φ 0 = w w w w + ( w w w w ) ln ( f 2 6 ρ 2 q T Mq ) 9

17 Stringy operators in +,M solution M S5 D5 Σ D5 F1 D1 global AdS 6 /CFT 5 on R S 4 M S5 AdS 6 t dilation: = H F1 between D5 poles, D1 between S5 poles: F1 = 3 2 M D1 = 3 2 Solve EOM, scaling dimensions match field theory exactly 9

18 Stringy operators in +,M solution R-charge of string states from coupling to SU(2) bulk gauge field fluctuation dual to R-symmetry current (S 2 isometries). Relevant part from KK ansatz + SU(2) gauge invariance: dx µ dx µ + K µ I AI δc (2) = dc f I A I +... K I S 2 Killing vector fields df I = S 2K I 10

19 Stringy operators in +,M solution R-charge of string states from coupling to SU(2) bulk gauge field fluctuation dual to R-symmetry current (S 2 isometries). Relevant part from KK ansatz + SU(2) gauge invariance: dx µ dx µ + K µ I AI δc (2) = dc f I A I +... K I S 2 Killing vector fields df I = S 2K I (p, q) strings couple to δc (2) through WZ term Q (p,q) = T (p Re(dC) q Im(dC)) Σ (p,q) = Q F1 = 1 2 M Q D1 =

20 Stringy operators in T [Benini,Benvenuti,Tachikawa 09] ext. 5-branes can be connected with D1-F1-(1, 1) triple string junction Q = 3 2 ( 1) Q = 1 ( 1) 2 (a) (,, ) of SU() 3 global symmetry 4d version in [Gaiotto,Maldacena] 11

21 Stringy operators in T [Benini,Benvenuti,Tachikawa 09] ext. 5-branes can be connected with D1-F1-(1, 1) triple string junction Q = 3 2 ( 1) Q = 1 ( 1) 2 (a) (,, ) of SU() 3 global symmetry 4d version in [Gaiotto,Maldacena] Triple junction in supergravity: S5 z F1 = ξ z D1 = e πi/3 ξ z (1,1) = e 2πi/3 ξ = 3 2 Q = 1 2 D1 (1, 1) Σ F1 D5 Agrees with T operator at large. (1, 1) 11

22 Stringy operators in +,M,k M M M k T k T /2 M (a) M (b) (c) [] () M k 1 () ( k) ( 2k) (k) y [k] 12

23 Stringy operators in +,M,k M M M k T k T /2 M (a) M (b) (c) [] () M k 1 () ( k) ( 2k) (k) O ã = [x b 1 x M y] ã k b y [k] = 3 (M ) 2 k + 1 Q = 1 3 O (i) = det x i = 3 2 Q =

24 Stringy operators in +,M,k M M S5 k T k D5 D7 Σ M (a) M (b) S5 [] () M k 1 () ( k) ( 2k) (k) O ã = [x b 1 x M y] ã k b y [k] = 3 (M ) 2 k + 1 Q = 1 3 O (i) = det x i = 3 2 Q = 1 2 D1, F1 in supergravity agree on and Q at large, M 12

25 Spectrum of high-dimension operators Similar stories for Y, X,M, + M 2 M (a) (a) (a) Precise quantitative agreement of field theory analyses and supergravity results on scaling dimensions at large Confirms brane junction interpretation for solutions without and for solutions with monodromy. 13

26 Holographic S 5 partition functions and entanglement entropies, # d.o.f.

27 Entanglement entropy vs. free energy First steps in AdS/CFT: Holographic interpretation consistent? Warped product, poles, renormalization? 14

28 Entanglement entropy vs. free energy First steps in AdS/CFT: Holographic interpretation consistent? Warped product, poles, renormalization? S EE (disc) finite? = F(S 5 ) 8d min. surface wrapping S 2 and Σ on-shell action, C (4) = 0 total derivative [Okuda,Trancanelli 08] 14

29 Entanglement entropy vs. free energy First steps in AdS/CFT: Holographic interpretation consistent? Warped product, poles, renormalization? S EE (disc) finite? = F(S 5 ) 8d min. surface wrapping S 2 and Σ on-shell action, C (4) = 0 total derivative [Okuda,Trancanelli 08] translates to non-trivial identities on Σ, both sensitive to entire geometry, including poles on Σ 14

30 Entanglement entropy vs. free energy First steps in AdS/CFT: Holographic interpretation consistent? Warped product, poles, renormalization? S EE (disc) finite? = F(S 5 ) 8d min. surface wrapping S 2 and Σ on-shell action, C (4) = 0 total derivative [Okuda,Trancanelli 08] translates to non-trivial identities on Σ, both sensitive to entire geometry, including poles on Σ Computations free from conceptual subtleties o divergences from poles, no finite contributions. 14

31 Entanglement entropy vs. free energy EE factorizes into geometric and theory-dependent part S EE = 1 Area(γ 4 ) 8Vol S 2 d 2 w κ 2 G 4G 3 S 5 partition function from renormalized on-shell action S E IIB = 5Vol S 2 3G Vol AdS6,ren rl l k,m n Σ Z [lk] Z [mn] dx ln x r k r l r k ln x r m 1 r m r n x r n 15

32 Entanglement entropy vs. free energy EE factorizes into geometric and theory-dependent part S EE = 1 Area(γ 4 ) 8Vol S 2 d 2 w κ 2 G 4G 3 S 5 partition function from renormalized on-shell action S E IIB = 5Vol S 2 3G Vol AdS6,ren rl l k,m n Σ Z [lk] Z [mn] dx ln x r k r l r k ln x r m 1 r m r n x r n Related as expected, S EE (disc) finite = F(S 5 ) 15

33 General scaling of # d.o.f. In general F(S 5 ) depends on all 5-brane charges in junction 1 (p 1, q 1 ) 2 (p 2, q 2 ) Z l ± l (q l + ip l ) 3 (p 3, q 3 ) Simple behavior under homogeneous rescaling of all charges: i α i i = F(S 5 ) α 4 F(S 5 ) Compare α 5/2 for USp() theory from D4/D8/O8 [Jafferis,Pufu] 16

34 Examples 5d T theory w/ gauge theory deformation SU( 1) SU(2) 2 [Benini,Benvenuti,Tachikawa 09;Bergman,Zafrir 14] F sugra (S 5 ) = 27 8π 2 ζ(3) 4 17

35 Examples 5d T theory w/ gauge theory deformation SU( 1) SU(2) 2 [Benini,Benvenuti,Tachikawa 09;Bergman,Zafrir 14] F sugra (S 5 ) = 27 8π 2 ζ(3) 4 S5/D5 intersection: [Aharony,Hanany,Kol 97] M SU() M 1 F sugra (S 5 ) = π 2 ζ(3) 2 M 2 M 17

36 Field theory partition functions and precision tests of AdS 6 /CFT 5

37 Field theory partition functions Strategy: study gauge theory deformation to obtain SCFT results. 5d SYM partition function on (squashed) S 5 [Källen et al.; H.-C. Kim, S. Kim;Imamura;Lockhart,Vafa] Expected large- simplifications: instantons exponentially suppressed, saddle point approximation exact 18

38 Field theory partition functions Strategy: study gauge theory deformation to obtain SCFT results. 5d SYM partition function on (squashed) S 5 [Källen et al.; H.-C. Kim, S. Kim;Imamura;Lockhart,Vafa] Expected large- simplifications: instantons exponentially suppressed, saddle point approximation exact Remaining challenge for zero instanton partition functions: Z 0 = i,j dλ (j) i exp ( F) Gauge group node becomes effectively continuous parameter at large. Scaling of λ (j) i not necessarily homogeneous. 18

39 Field theory partition functions umerical evaluation [Herzog,Klebanov,Pufu,Tesileanu]: Replace saddle point eq. by set of particles w/ coordinates λ (j) i in potential F F λ (j) i = 0 F (t) = dλ dt λ (j) i (j) i (t) Equilibrium configurations are solutions to saddle point equation. Approximate solutions from late-time behavior of λ (j) i (t). 19

40 Field theory partition functions umerical evaluation [Herzog,Klebanov,Pufu,Tesileanu]: Replace saddle point eq. by set of particles w/ coordinates λ (j) i in potential F F λ (j) i = 0 F (t) = dλ dt λ (j) i (j) i (t) Equilibrium configurations are solutions to saddle point equation. Approximate solutions from late-time behavior of λ (j) i (t). Zero instanton saddle point approximation to F(S 5 ), conformal central charge C T from squashing deformations [Chang,Fluder,Lin,Wang] 19

41 Field theory partition functions Explicit numerical results for S 5 partition functions for T : ,M : 2, M 30 Computationally more expensive conformal central charges: T : ,M : 2, M 15 20

42 Field theory partition functions Explicit numerical results for S 5 partition functions for T : ,M : 2, M 30 Computationally more expensive conformal central charges: T : ,M : 2, M 15 +,M consistency check: SCFT partition functions obtained from quiver and S-dual quiver should agree M vs. M Relative error 0 for large, M. Below 1% for, M 16 20

43 Field theory partition functions Results for T : T -F S C T T umerical results as dots, degree-4 polynomial from least-squares fit as dashed line. F S 5 on the left, C T on the right. Clear quartic scaling, as predicted holographically. 21

44 Precision test of AdS 6 /CFT 5 Comparison to supergravity: T -F S 5 T -F S Supergravity: F S 5 = 27 8π 2 ζ(3) 4 C T = 2160 π 4 ζ(3) 4 Coefficients of leading terms, extracted via fit of numerical results to degree-4 polynomial, agree within 1 o / oo 22

45 Field theory partition functions Results for +,M : for fixed clear quadratic scaling with M, and for fixed M clear quadratic scaling with,m -F S M = 30 M = 25 M = 20 M = 15 M = 10 M = ,M C T M = 15 M = 13 M = 11 M = 9 M = 7 M = 5 M = 3 Supergravity: F S 5 = π 2 ζ(3) 2 M 2 C T = 7560 π 4 ζ(3) 2 M 2 Coefficients of leading terms, extracted via fit of numerical results to degree-2 polynomial in M, agree within 2 o / oo 23

46 Massive spin-2 excitations

47 Massive spin-2 excitations Identifying fluctuations non-trivial: non-trivial background, weak symmetry constraints large set of coupled PDE s on Σ. Spin-2 fluctuations exclusively from metric decoupling trivial. Used for warped AdS 4 fluctuations by [Bachas,Estes 11]. 5d SCFTs should have conserved T µν massless AdS 6 graviton. Decoupling of states on external 5-branes? 24

48 Massive spin-2 excitations Metric perturbation on unit-radius AdS 6 : ds 2 = f6 2 ( ds 2 AdS6 + δg ) + ĝ ab dy a dy b With h [tt] µν transverse-traceless δg = h [tt] µν φ lm Y lm dx µ dx ν AdS6 h [tt] µν = (M 2 2)h [tt] µν Type IIB EOM reduce to: 6 a ( G 2 η ab b φ lm ) l(l + 1) ( 9κ 2 G + 6 G 2) φ lm + M 2 κ 2 Gφ lm = 0 25

49 Massive spin-2 excitations Two families of universal, regular and normalizable solutions: (i) φ lm = G l M 2 = ( 5) = 5 + 3l (ii) φ (1) lm + iφ(2) lm = ( ) Gl A + Ā = 6 + 3l M 2 3l(3l + 5) for regular solutions, saturated by (i). Massless graviton for M = l = 0 5d conserved T µν. 26

50 Massive spin-2 excitations Two families of universal, regular and normalizable solutions: (i) φ lm = G l M 2 = ( 5) = 5 + 3l (ii) φ (1) lm + iφ(2) lm = ( ) Gl A + Ā = 6 + 3l M 2 3l(3l + 5) for regular solutions, saturated by (i). Massless graviton for M = l = 0 5d conserved T µν. Match Q 4 descendants in B 2, A 4 multiplets with scalar primaries = 3 + 3l and prim. A 4 = 4 + 3l [Cordova,Dumitrescu,Intriligator] prim. B 2 Universal part in spectrum of dual SCFTs. Similarities to massive IIA [Passias,Richmond 18], different in details and interpretation. 26

51 Summary & Outlook

52 Summary Holographic interpretation of solutions appears consistent: poles/external 5-branes decouple, 5d conserved T µν Quantitative match of stringy operators in supergravity solutions to string theory picture and field theory predictions. Field theory partition functions and conformal central charges match supergravity predictions for T and +,M. First lessons: 4 d.o.f., universal = O(1) operators, stringy spectrum for theories with no suitable gauge theory deformations. 27

53 Outlook Compelling picture for AdS 6 /CFT 5 in Type IIB: large classes of solutions, coherent string theory interpretation, match to SCFTs. Full fluctuation spectrum = O(1) operators. (Consistent) KK reduction to 6d gauged supergravity? on-universal parts? More stringy operators and op. s corresponding to e.g. D3 branes. Correlation functions, defects, Wilson lines, more tests,... Classification of large- SCFTs? 28

54 Outlook Compelling picture for AdS 6 /CFT 5 in Type IIB: large classes of solutions, coherent string theory interpretation, match to SCFTs. Full fluctuation spectrum = O(1) operators. (Consistent) KK reduction to 6d gauged supergravity? on-universal parts? More stringy operators and op. s corresponding to e.g. D3 branes. Correlation functions, defects, Wilson lines, more tests,... Classification of large- SCFTs? Thank you! 28