M5-branes and Wilson Surfaces! in AdS7/CFT6 Correspondence Hironori Mori (Osaka Univ.) based on arxiv:1404.0930 with Satoshi Yamaguchi (Osaka Univ.) 014/05/8, Holographic vistas on Gravity and Strings @ YITP
AdS7/CFT6
AdS7/CFT6 M5-branes ear-horizon limit Low energy M-theory on AdS 7 S 4? 6D (,0) SCFT
6D (,0) theory equivalent = [Douglas 10] [Lambert, Papageorgakis, Schmidt-Sommerfeld 10] S 1 compactification 5D maximally SYM Exact results on curved backgrounds [Källén, Zabzine 1] [Hosomichi, Seong, Terashima 1] [Källén, Qiu, Zabzine 1] [Kim, Kim 1] [Imamura 1] [Kim, Lee 1] [Fukuda, Kawano, Matsumiya 1] [Kim, Kim, Kim 1] [Qiu, Zabzine 13] [Kim, Kim, Kim, Lee 13] [Schmude 14]
? AdS7/CFT6 M-theory on AdS 7 S 4 6D (,0) SCFT (Boundary = S 5 S 1 ) on S 5 S 1 M-branes! Bubbling geometry? Wilson surfaces = Wilson loops W R 5D MSYM on S 5
Summary M-branes M-brane wrapping AdS 3 AdS 7 AdS7/CFT6 [Minahan, edelin, Zabzine, 13] Wilson surfaces Fundamental rep W exp M5-brane wrapping AdS 3 S 3 ew Symmetric rep W k exp k 1+ k AdS 7 M5-brane wrapping AdS 3 S3 AdS 7 S 4 Anti-symmetric rep W k.. exp k 1 k
Plan 1. Review: Wilson surface!. CFT side! 3. Gravity side: From M5-branes! 4. Summary
1. Review Wilson surface! Maya diagram
6d (,0) SCFT : Effective theory on M5-branes (but no Lagrangian) M5-brane
Wilson surface : on-local operator! extending on -dimensional space M5-brane Boundary of a M-brane on M5-brane M-brane
Maya diagram
Bubbling geometry : A class of classical half-bps solutions of 11D supergravity M 11 = AdS 3 S 3 S3 AdS 3 [Lin, Lunin, Maldacena, 04] [Yamaguchi, 06] [Lunin, 06] [D Hoker, Estes, Gutperle, Krym, 08] S 3 S 3 S3 y S 3 x y =0
Bubbling geometry is labeled by
Bubbling geometry is labeled by Maya diagram The dual field theory may have! this eigenvalue distribution of the matrix model.
Plan 1. Review: Wilson surface!. CFT side! 3. Gravity side: From M5-branes! 4. Summary
. CFT side! Chern-Simons matrix model! Calculations of Wilson surfaces!
6D (,0) theory on S 5 S 1 equivalent = R 6 = g YM 8 (R 6 : radius of S 1 ) 5D maximally SYM on S 5 Chern-Simons matrix model [Källén, Qiu, Zabzine 1] [Kim, Kim 1] Z i=1 d i exp i=1 i + i=j ln sinh ( i j) raduius of S 5 : r, = g YM r
6D (,0) theory on S 5 S 1 Wilson surface (S 1 S 1 ) equivalent = 5D maximally SYM on S 5 Wilson loop (S 1 ) W R = Tr R e Chern-Simons matrix model [Källén, Qiu, Zabzine 1] [Kim, Kim 1] Z i=1 d i exp i=1 i + i=j ln sinh ( i j) raduius of S 5 : r, = g YM r
Chern-Simons matrix model O( 3 ) Z i=1 d i exp i=1 i + i=j ln sinh ( i j) Large''limit'('''''fixed) Z i=1 d i exp i=1 i + i=j i j Saddle point equations 0= i + j,i=j sign ( i j )
Chern-Simons matrix model O( 3 ) Z i=1 d i exp i=1 i + i=j ln sinh ( i j) Large''limit'('''''fixed) Z i=1 d i exp i=1 i + i=j i j Eigenvalue distribution consistent with bubbling geometry! i = 1 i ( 1 > > > ) / /
Symmetric rep ( 1 > > > ) Tr e = exp k i l leading exp [k 1 ] k 1 i 1 i k l=1 changes': 1 1 = 1+ k W exp 1 + 1 j + k 1 + (terms independent of k) k j= saddle point exp k 1+ k 1+ k / /
Anti-symmetric rep ( 1 > > > ) Tr e = k.. 1 i 1 < <i k exp k l=1 i l leading exp [( 1 + + k )] i do'not'change': i = 1 i ( 1 > > > ) W k.. exp [( 1 + + k )] saddle point exp k 1 k 1 k / /
Summary for W R Symmetric : W k exp k 1+ k 1+ k / / Anti-symmetric : W k.. exp k 1 k 1 k / /
Plan 1. Review: Wilson surface!. CFT side! 3. Gravity side: From M5-branes! 4. Summary
3. Gravity side! Calculations of Wilson surfaces from probe M5-branes
Summary for W R Symmetric : W k exp k 1+ k 1+ k / / Anti-symmetric : W k.. exp k 1 k 1 k / /
Summary for W R Symmetric : W k exp k 1+ k Bubbling! geometry 1+ k M5-brane AdS 3 S 3 / / Anti-symmetric : W k.. exp k 1 k 1 k M5-brane AdS 3 S3 / /
SUGRA on AdS 7 S 4 Metric ds = L cosh d + d + sinh d 5 + L 4 d + sin d 3 Boundary = S 5 S 1 Identification : + R 6 r Wilson surface r R 6 S 5 S 1
PST action [Pasti, Sorokin, Tonin 97] ''on a single M5-brane S M5 = T 5 d 6 x g L + 1 4 H mn H mn + T 5 C 6 1 C 3 H 3 otice: We must add the boundary term S bdy to regularize S M5. We carefully determine the boundary term S bdy.
AdS 7 AdS 3 S 3 Poincare coordinate : ds = L y dy + dr 1 + r 1d + dr + r d 3 + L 4 d 4 Ansatz : r = y, y = y( ) Flux quantization : = k, k Z PST action S M5 (Volume of the M5-brane on the boundary) = Boundary term S bdy regularized PST action S reg M5 = S M5 + S bdy =0
AdS 7 AdS 3 S 3 Poincare coordinate : ds = L y dy + dr 1 + r 1d + dr + r d 3 + L 4 d 4 Ansatz : r = y, y = y( ) Flux quantization : = k, k Z PST action S M5 (Volume of the M5-brane on the boundary) = Boundary term S bdy regularized PST action S reg M5 = S M5 + S bdy =0
AdS 7 AdS 3 S 3 Global coordinate : ds = L cosh u(cosh wd + dw + sinh wd )+du + sinh ud 3 + L 4 d 4 Flux quantization : sinh u k = k S M5 = 4 R 6 r k 1+ k sinh w 0 Boundary term sinh w 0 cosh w 0 R 6 = g YM 8 = g YM r S reg M5 = k 1+ k agree W k
AdS 7 S 4 AdS 3 S3 Global coordinate : ds = L cosh d + d + sinh d 5 + L 4 d + sin d 3 Flux quantization : cos k =1 k k S 3 S M5 = 4 R 6 r k 1 k sinh 0 S 4 Boundary term sinh 0 cosh 0 R 6 = g YM 8 = g YM r S reg M5 = k 1 k agree W k..
4. Summary
Summary M-branes M-brane wrapping AdS 3 AdS 7 AdS7/CFT6 [Minahan, edelin, Zabzine, 13] Wilson surfaces Fundamental rep W exp M5-brane wrapping AdS 3 S 3 ew Symmetric rep W k exp k 1+ k AdS 7 M5-brane wrapping AdS 3 S3 AdS 7 S 4 Anti-symmetric rep W k.. exp k 1 k