Membranes and the Emergence of Geometry in MSYM and ABJM

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1 in MSYM and ABJM Department of Mathematics University of Surrey Guildford, GU2 7XH, UK Mathematical Physics Seminar 30 October 2012

2 Outline 1 Motivation 2 3 4

3 Outline 1 Motivation 2 3 4

4 Gauge theory / gravity correspondences QFTs in flat space with large local symmetry (gauge) groups, typically with O(N 2 ) degrees of freedom, at strong coupling λ are dual to weakly coupled theories of quantum gravity (string theories). This leads naturally to the following question: How are the geometry and topology (both of spacetime and membranes) encoded in long, gauge-invariant operators?

5 Outline 1 Motivation 2 3 4

6 Maximally supersymmetric N = 4 SYM theory in 3+1 dimensional flat space with an SU(N) gauge group is dual to Type IIB superstring theory on AdS 5 S 5 MSYM: 3 complex scalars Xb a, Y b a, Z b a gauge fields, spinors. in the adjoint of SU(N), IIB on AdS 5 S 5 : background metric: ds 2 = R 2 {dsads dss 2 } 5 self-dual 5-form: F 5 = 4R 4 {vol(ads 5 ) + vol(s 5 )} = dc 4 [Maldacena: hep-th/ ]

7 Dictionary between local operators and string theory states: Pictures by Dino Giovannoni (UCT) and from [DG, JM, AP: [hep-th] ]

8 How does the non-perturbative nature of membranes manifest in the gauge theory? Let us consider the simple 1 2 -BPS operators with O( N) fields which are dual to strings. The single trace and multi-trace operators: O n (Z) = [tr(z)] ν 1 [ tr(z 2 ) ] ν2 [tr(z n )] νn form an approximately orthonormal basis with respect to the free-field two-point correlator in MSYM to leading order in 1 N. Here the number of traces counts the number of strings in the dual description!

9 NB! Here O n(z) has an S n symmetry associated with interchanging Zs Notice that we can rewrite this generic multi-trace operator as O n(z) = [tr(z)] ν 1 [ tr(z 2 ) ] ν2 [tr(z n )] νn = tr{σz n } = Z a 1 a σ(1) Z an a σ(n) which can be labelled by the conjugacy class of σ S n. Interchanging Z s is equivalent to conjugating σ τ 1 στ and doesn t change the operator! The ν k in the original expression give cycle structure of this class. Hence each distinct O n(z) is associated with a partition of n: λ = (λ 1,..., λ n) with ν k = λ k λ k+1 (for 1 k n 1) and ν n = λ n and hence a Young diagram associated with an irrep R of S n

10 But, if we consider operators dual to membranes, the length of the operator becomes of O(N) and the orthogonality of single and multi-trace operators breaks down! In computing the free-field correlators, counting arguments give combinatoric factors which overpower the 1 N expansion. Single/multi-trace operators mix and we need a new orthonormal basis given, in the 1 2-BPS sector, by Schur polynomial operators χ R (Z) σ S n χ R (σ) tr{σ Z n } labeled by an irrep R [ Corkey, Jevicki & Ramgoolam: hep-th/ ]. The coefficients χ R (σ) in the expansion over our previous operators tr{σ Z n } are simply the characters of elements in S n ( # elements in the class). Orthogonality of Schurs follows from the orthogonality theorem for characters.

11 What membranes embedded into AdS 5 S 5 are dual to 1 2 -BPS Schur polynomial χ R(Z) operators? R = AdS giant graviton R = sphere giant graviton R = collection of two concentric AdS sphere giant gravitons R = collection of two concentric sphere giant gravitons the # rows/columns with O(N) boxes counts the # membranes!

12 The Schur polynomial labeled by the totally anti-symmetric representation R of S n (a Young diagram with one column) χ. (Z) On subdet (Z) = ɛ a1...a na n+1...a N ɛ b1...bnan+1...a N Z a1 b 1 Z an b n is proportional to a subdeterminant with maximum size n = N. [Balasubramanian et. al.: hep-th/ ] A natural interpretation of this maximum length from the string theory point of view is that the sphere giant graviton cannot grow to be bigger than the compact S 5 space.

13 The sphere giant graviton takes the shape of an S 3 S 5 and is 1 2-BPS. Being a D3-brane, it is supported by its coupling to the potential C 4 and its motion in the 5-sphere space S 5. [McGreevy, Susskind & Toumbas: hep-th/ ] [Grisaru, Myers & Tafjord: hep-th/ ]

14 Analogy: Electric dipole separated by a spring moving in a magnetic field It can be seen as a bound state of a large number of Kaluza-Klein gravitons which become polarized in the F 5 = dc 4 field through a version of the Myers effect [ Myers: hep-th/ ].

15 Giant gravitons are energetically degenerate with point gravitons... in fact the sphere giant effectively looks like a graviton after the compactification of the compact S 5 space. The fluctuation spectrum indicates stability, but there is no dependence in the spectrum on the size r 0 of the giant graviton.

16 Giant gravitons in AdS 5 S 5 with non-trivial geometries Less supersymmetric giant gravitons can be constructed as the intersection of S 5 C 3 with holomorphic surfaces C in C 3, then boosted into motion. A general holomorphic surface in C 3 parametrized by (z 1, z 2, z 3 ) is f (z 1, z 2, z 3 ) = C, with special cases: 1 2 z1 = C = 1 (r 0) 2 spherical S 3 S 5 giant graviton χ R (Z) 1 4 z1z2 = C non-spherical giant graviton χ R(XZ) 1 8 z1z2z3 = C non-spherical giant graviton χ R(XYZ) [ Mikhailov: hep-th/ ]

17 The holomorphic sector of MSYM theory How do we construct an exactly orthonormal operator basis (from the scalars X, Y, Z) for the holomorphic sector of MSYM theory? Single and multi-trace operators in this sector can be written as O n (X, Y, Z) = tr{σ X n 1 Y n 2 Z n 3 } with σ S n and n = n 1 + n 2 + n 3 the length of the operator. But these operators have only an S n1 S n2 S n3 S n symmetry! Conjugating σ τ 1 σ τ only leaves the operator invariant if τ S n1 S n2 S n3 is restricted to this subgroup of S n. These single and multi-trace operators in the holomorphic sector are therefore labeled by restricted conjugacy classes of S n.

18 It is now possible to construct a complete, orthonormal basis of restricted Schur polynomial operators: χ R,{r} (X, Y, Z) σ S n χ R,{r} (σ) tr{σ X n 1 Y n 2 Z n 3 } [ de Mello Koch et al: [hep-th] ] which are labeled by R, {r} = R, (r 1, r 2, r 3), (α, β, γ) with irrep R of S n irrep (r 1, r 2, r 3) of S n1 S n2 S n3 subduced by the irrep R of S n. (Here R is reducible when restricted to elements of the subgroup S n1 S n2 S n3.) additional multiplicity labels (α, β, γ), only necessary when (r 1, r 2, r 3) is subduced by R more than once. Here χ R,(r1,r 2,r 3 ),(α,β,γ)(σ) are the restricted characters of σ S n, obtained by tracing over all the blocks within the large diagonal (r 1, r 2, r 3) block of Γ R (σ).

19 Outline 1 Motivation 2 3 4

20 The ABJM model is an N = 6 Super Chern Simons-matter theory in 2+1 dimensions with a U(N) U(N) gauge group and is dual to Type IIA superstring theory on AdS 4 CP 3 ABJM: 4 complex scalars (A 1 ) a α, (A 2 ) a α, (B 1 ) a α, (B 2 ) a α in bifundamental reps of U(N) U(N), gauge fields, spinors. IIA on AdS 4 CP 3 : background metric: ds 2 = R 2 {ds 2 AdS ds 2 CP 3 } form fields: F 4 = 3 R 3 {vol(ads 4 )} = dc 3 F 6 = F 4 = dc 5 F 2 = k da = dc 1 F 8 = F 2 = dc 7 [Aharony, Bergman, Jafferis & Maldacena: hep-th/ ]

21 Schur polynomials in ABJM Schur polynomials operators in ABJM must be constructed out of composite scalars: (φ 11 ) a b = (A 1 B 1 )a b = (A 1 ) a α(b 1 ) b α (φ 21 ) a b = (A 2 B 1 )a b = (A 2 ) a α(b 1 ) b α (φ 12 ) a b = (A 1 B 2 )a b = (A 1 ) a α(b 2 ) b α (φ 22 ) a b = (A 2 B 2 )a b = (A 2 ) a α(b 2 ) b α to ensure gauge invariance. For example, Schur polynomials constructed out of φ 11 = A 1 B 1 take the form χ R (φ 11 ) = χ R (A 1 B 1 ) σ S n χ R (σ) tr{σ (A 1 B 1 ) n }.

22 The Schur polynomial labelled by the totally anti-symmetric representation R of S n can again be written as a subdeterminant: χ. (A 1 B 1 ) Osubdet n (A 1 B 1 ) = ɛa 1...a na n+1...a N ɛ b 1...b na n+1...a N (A 1 B 1 )a 1 b (A 1 1 B 1 )an b n which factorizes at maximum size into the product of two full determinants ON subdet (A 1 B 1 ) = (det A 1) (det B 1 ) These are ABJM dibaryons, which are dual to two D4-branes wrapped on different non-trivial CP 2 CP 3 subspaces. [Gutíerrez, Lozano & Rodríguez-Gómez, [hep-th]] [JM & AP: [hep-th]]

23 Restricted Schur polynomials in ABJM Restricted Schur polynomials constructed naïvely from composite scalars in ABJM are not orthogonal!... but we can alter the definition (still using labels R, {r}) to write down a complete, orthonormal basis for the holomorphic sector: O R,{r} n 1 n χ R,{r} (σ) (A 1) a i α i (A 2) aj α j (τ) α 1 α n β 1 β n σ S n i=1 j=1+n 1 n 11 i=1 (B 1 )β i a σ(i) n 1 (B 2 )βi a σ(i) i=1+n 11 n 1 +n 21 (B 1 )βi a σ(i) i=1+n 1 n (B 2 )βi a σ(i) i=1+n 1 +n 21 with τ the following element of the group algebra of S n : τ = R d R n! f R [ RdMK, BM, JM, AP: 1202:4925 [hep-th] ] σ S n χ R (σ 1 ) σ.

24 How do we construct D4-brane giant gravitons embedded into CP 3 in type IIA string theory on AdS 4 CP 3? Originally, we made use of an analogy to giant gravitons embedded into T 11 in the Klebanov-Witten correspondence. But it turns out to be easier to go another route... We look first at M5-brane giant gravitons embedded into S 7 in M-theory on AdS 4 S 7 and then study their descendants under the compactification to AdS 4 CP 3.

25 Giant gravitons in S 7 C 4 Again, a holomorphic surface C in the complex manifold C 4 with coordinates w a allows us to construct an M5-brane giant graviton embedded into S wa = C spherical S 5 S 7 giant graviton 1 4 waw b = C non-spherical giant graviton 1 8 waw bw c = C or w aw b w cw d = C non-spherical giant graviton which must once more be boosted into motion in the S 7. [ Mikhailov: hep-th/ ]

26 CP 3 comes from a compactification on the Hopf fibre of S 7 Redefine (w 1, w 2, w 3, w 4 ) = (z 1, z 2, z 3, z 4 ) with z a = r a e iχa complex coordinates of C 4. We can write as new z 1 = r 1 e iτ A 1 z 2 = r 2 e i(τ+ϕ1) A 2 z 3 = r 3 e i(τ χ) B 1 z 4 = r 4 e i(τ χ ϕ2) B 2 with τ the overall phase and the Hopf fibre. M-theory on AdS 4 S 7 becomes IIA string theory on AdS 4 CP 3 after a compactification on this 11th dimension τ.

27 Giant gravitons in CP 3 S 7 C 4 We can study the D4-brane descendants of the M5-brane giant gravitons under the compactification from S 7 to CP BPS: z1 = C -BPS: z1z2 = C or z1 z3 = C -BPS: z1z2 z3 = C or z1 z3 z4 = C or z1z2 z3 z4 = C which move in the complex projective space CP 3. These giant graviton ansätze survive the reduction to CP 3 unchanged! What happens to the other giant gravitons?. [Herrero, Lozano and Picos: [hep-th] ] [AP, JM, YL, MP: work in progress ]

28 The 1 4 -BPS giant graviton associated with z 1 z 3 = C will reduce to a non-spherical giant graviton embedded into CP 3 and associated with a Schur polynomial χ R (φ 11 ) = χ R (A 1 B 1 ). We constructed this configuration in [DG, JM, AP: [hep-th] ]. It is energetically degenerate with the point graviton

29 An explicit formula for the energy and momentum can be obtained { H = P χ = N α ( ) ( )} 1 α0 2 1 α0 ln, α 0 with α 0 the constant size parameter (plotted in units of N kr4 2π 2 ).

30 Key observation This dynamically stable D4-brane giant graviton in CP 3 has a non-spherical changing shape! As the size α 0 increases, the worldvolume of the giant graviton pinches off, factorizing at maximum size into two topologically stable D4-branes wrapped on different CP 2 CP 3 subspaces. Its fluctuation spectrum was found to exhibit a dependence on its size α 0 and shape - this must be linked to a geometric dependence in the non-bps excitations of the dual Schur polynomial operator.

31 Outline 1 Motivation 2 3 4

32 What are the other descendants of M5-brane giant gravitons embedded into S 7 under the compactification to CP 3 Can we apply what we have learned in the ABJM model to study non-spherical giant gravitons in MSYM theory? Can we observe the geometric dependence of the giant s fluctuation spectrum in the anomalous dimensions of non-bps excitations of the dual operators? What about trying to observe topology as well as geometry?