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String junctions and N 3 problem in M5-brane theory

String junctions and N 3 problem in M5-brane theory. Shotaro Shiba 2012/03/19 @ Journal Club Reference: S. Bolognesi and K. Lee, Phys. Rev. D84 (2011) 126018. H.-C. Kim, S. Kim, E. Koh , K. Lee and S. Lee, JHEP 1112 (2011) 031. N 3 problem?.

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String junctions and N 3 problem in M5-brane theory

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  1. String junctions and N3 problem in M5-brane theory Shotaro Shiba 2012/03/19 @ Journal Club Reference: S. Bolognesi and K. Lee, Phys. Rev. D84 (2011) 126018. H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, JHEP 1112 (2011) 031.

  2. N3 problem? • In string theory, the system of coincident N D-branes has ~N2degrees of freedom. • We can understand it, since the strings end on D-branes describe these d.o.f.s. • These strings become massless when D-branes are coincident. • Themodes on strings behave as massless U(N) gauge fields.

  3. N3 problem? • This can be justified also by AdS/CFT correspondence. • The low energy effective theory on D3-branes is 4-dim super Yang-Mills theory. • The near horizon geometry of D3-branes is AdS5xS5. • Its blackhole entropy shows ~N2d.o.f.s.[Klebanov-Tseytlin ’96]

  4. N3 problem? • In M-theory, the situation changes: There are no strings and no parameters. • This means the low energy effective theory on M2, M5-branes must be CFT. • The near horizon geometry of M-branes is AdS4,7 x S7,4 spacetime. • Its blackhole entropy must show the number of d.o.f.s. The result is…

  5. N3 problem? • N M2-branes: ~N3/2 • Recent researches revealed Chern-Simons gauge field on M2-branes explains it. [Aharony-Bergman-Jafferis-Maldacena ‘08] [Drukker-Marino-Putrov ‘10] • We have also studied on itin detail using the numerical calculation. [Hanada-Honda-Honma-Nishimura-SS-Yoshida ‘12] • N M5-branes: ~N3 …??

  6. M5 from D4 • M5-branes become D4-branes when they are compactified on a circle. • The low-energy worldvolume theory on D4-branes is 5-dim Yang-Mills theory. • The compactification radius determines the string/YM coupling and energy scale. • That is, the d.o.f.s increase from N2 to N3 when the energy scale becomes higher.

  7. Possible Scenario? • N3d.o.f.s are not present in 5-dim theory • Higher KK modes may explain it. • Instantons made from N instantonpartons with mass 1/NR6 in adjoint rep? • N3d.o.f.s are present but hidden! • Confinement states of BPS objects? • The states are broken in high temperature to their fundamental components? [Tong ‘10] (pair creation?) Today’s topic

  8. Possible Scenario? • A kind of analogy to QCD deconfinement transition in 4-dim gauge theory. 4-dim: the entropy jumps from N0 to N2. 5-dim: the entropy jumps from N2 to N3? • What kind of objects? 4-dim: quarks and gluons 5-dim: 1/4 BPS junctions of monopole strings? • How does it explain N2 to N3 transition...? [Lee-Yee ‘06]

  9. Plan of talk String junction is a possible scenario. (we are here) • What is a string junction? • The string junction is a solution of 5-dim theory. • Counting d.o.f.s of string junctions in 6-dim theory

  10. String junction? Ordinary • Stringjunction is a stable state if more than one kinds of strings are joined. • In M-theory, the string junction becomes a nonsingular manifold of a membrane. • conservation of charge (pi, qi) • balanced tension T=Sqrt[p^2+q^2/g^2]

  11. String junction? • In this discussion, however, we consider a string junction forming a direct product of a triangle and its dual lattice.

  12. 5d theory and junction? • 5d super Yang-Mills theory (bosonic part) • YM coupling ~ compactification radius • BPS equation (from SUSY trans. for gluino) (I=6,7,8,9,10)

  13. 5d theory and junction? • A 1/2 BPS solution: monopolestring (D2-brane connecting two D4-branes) D4 D2 The string is stretched alongthe x4 axis. D4

  14. 5d theory and junction? • A 1/2 BPS solution: monopolestring (D2-brane connecting two D4-branes) • e.o.m.: • unbroken SUSY: • tension: (M2-brane connecting two M5-branes) • This string becomes the self-dual string in the strong coupling limit with fixed. The string is stretched alongthe x4 axis.

  15. 5d theory and junction? • cf.) Other 1/2 BPS solutions (F1-string connecting two D4-branes) • 5-dim W-boson: compactified self-dual string • 5-dim instanton: KK mode along x5 direction D4 F1 The self-dual string is compactified on a circle (x5 direction). D4 (end point of F1-string) (D0-brane)

  16. 5d theory and junction? • A 1/4 BPS solution (1) • wave (massless mode) on a monopole string unbroken SUSY: • cf.) 1/4 BPS dyonicinstanton unbroken SUSY: • In 6-dim theory, both are waves on the self-dual string. left/right-moving [Lambert-Tong ’99] (for compactified self-dual string) (for wave along compactified direction)

  17. 5d theory and junction? • A 1/4 BPS solution (2) • Planar string junction: a 1/4 BPS solution of 1/16 BPS configuration equation where a,b,c,d=1,2,3,4, and we rename as , and choose a gauge . • This equation is obtained as Hamiltonian H=0 under the Gauss law (e.o.m. for A0). [Lee-Yee ‘06]

  18. 5d theory and junction? • Planar string junction solution • The configuration of monopole strings is determined by the tension balance condition. x4 x7 monopole string D4 & monopole string same tension D2 x3 x6 D4

  19. 5d theory and junction? • Low energy limit: 5-dim abelian • KK scale: 6-dim but abelian (The monopole string becomes the self-dual string but heavy.) • Hagedorn scale: deconfinement of strings (The junction and anti-junction are deconfined. dof: N2 to N3) • Nonabelian scale: 6-dim nonabelian string tension brane separation (assumption: 1/g52<<v) compact radius

  20. N3d.o.f.s and junction? • cf.) D-brane case : SU(N) gauge sym. • (i,i)-string : N-1 d.o.f.s The string ending on an identicalbrane The global U(1) is excluded. • (i,j)-string : N(N-1) d.o.f.s The string ending on distinctbranes • Total : N2-1 d.o.f.s We find the matters in the adjoint rep.

  21. N3d.o.f.s and junction? • M5-brane case: by using AN-1=SU(N) root vectors • (i,j)-wave : N(N-1) d.o.f.s The wave on self-dual string • (i,j,k)-junction : N(N-1)(N-2)/3 d.o.f.s The string junction and anti-junction • Total: (N3-N)/3 d.o.f.s up to the spin and other structures of objects • Similar discussion can be done for DN and EN groups.

  22. Concluding remarks • 1/4 BPS objects may be the fundamental ones in M5-brane worldvolume theory. • 1/4 BPS objects are the waves on self-dual strings and the junctions of them. • They seem to correctly provide the N3d.o.f.s on M5-brane worldvolume. • Note that we here discuss only in the Coulomb phase, assuming 1/g52<<v.

  23. Concluding remarks • The other approach should provide a complementary way of understanding, since it is related to the D4/F1 system. • The partition function of (i,j)-self-dual string shows ~N3d.o.f.s! [Kim2-Koh-Lee2 ‘11] (counting charged 1/4 BPS states) Fluctuations in target space Additional ~(j-i-1) d.o.f.s! Is it related to the remaining end of string junction?? i j

  24. Concluding remarks • The precise relation between this D4/F1 system and the D4/D2 system remains to be clarified at this time. • We can expect that these discussions would be important to understand the N3d.o.f.son M5-branes.

  25. Personal prospects • D-branes are described by matrices Xij. • Xij: string ending on distinct branes • Xii: string ending on an identical brane • M5-branes are by cubic matrices Xijk? • Xijk=Xjki=Xkij : junction • Xikj=Xjik=Xkji: anti-junction • Xiij=Xiji=Xjii : wave on string • Xiii : position of M-branes

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