The Partition Function of ABJ Theory Masaki Shigemori (KMI, Nagoya U) Amsterdam, 28 February 2013 with Hidetoshi Awata, Shinji Hirano, Keita Nii 1212.2966, 1303.xxxx
Introduction
AdS 4 /CFT 3 M-theory on / = Type IIA on P 3,6 Chern-Simons-matter SCFT (ABJM theory) / /, / / Low E physics of M2-branes on Can tell us about M2 dynamics in principle (cf. / ) More non-trivial example of AdS/CFT 3
ABJ theory 4 Generalization of ABJM [Hosomichi-Lee 3 -Park, ABJ] CS-matter SCFT Low E dynamics of M2 and fractional M2 Dual to M on Seiberg duality with 3-form ABJ triality [Chang+Minwalla+Sharma+Yin] /, dual to Vasiliev higher spin theory?
Localization & ABJM Powerful technique in susy field theory Reduces field theory path integral to matrix model Tool for precision holography (1/ corrections) Application to ABJM [Kapustin+Willett+Yaakov] Large : reproduces ~ ~/, [Drukker+Marino+Putrov] All pert. corrections [Fuji+Hirano+Moriyama][Marino+Putrov] Wilson loops [Klemm+Marino+Schiereck+Soroush] 5
ABJ: not as well-studied as ABJM; MM harder to handle Goal: develop framework to study ABJ Rewrite ABJ MM in more tractable form Fun to see how it works Interesting physics emerges! 6
Outline & main results
ABJ theory CSM theory, Superconformal, IIB picture D3 1, 5 NS5 D3 8
Lagrangian 9
ABJ MM [Kapustin-Willett-Yaakov] Partition function of ABJ theory on : vector multiplets, matter multiplets Pert. treatment easy Rigorous treatment obstacle: 10
Strategy Analytically continue lens space ( ) MM Perturbative relation: [Marino][Marino+Putrov],, Simple Gaussian integral Explicitly doable! 11
Expression for 12 1; : -Barnes G function 1 1; 1, : partition of 1,, into two sets,,,,,,,
Now we analytically continue: It s like knowing discrete data and guessing 13
Analytically cont d 14 0, -Pochhammer symbol: 1 1 1 Analytic continuation has ambiguity criterion: reproduce pert. expansion Non-convergent series need regularization
: integral rep (1) poles from 1/sin Finite Agrees with the formal series Watson-Sommerfeld transf in Regge theory 15
: integral rep (2) Provides non-perturbative completion Perturbative (P) poles from 1/sin Non-perturbative (NP) poles, ~ 16
Comments (1) Well-defined for all No first-principle derivation; a posteriori justification: Correctly reproduces pert. expansions Seiberg duality Explicit checks for small 17
Comments (2) (ABJM) reduces to mirror expression Vanishes for Starting point for Fermi gas approach Agrees with ABJ conjecture 18
Details / Example
, : partition of 1,, into two sets,,,,,,, 20
, 1,, 1, 1,, 1, 21
How do we continue this? Obstacles: 1. What is? 2. Sum range depends on 22
-Pochhammer & anal. cont. For 1,2,, For, In particular, 1 1 1.. 1 1 1 1 1 Can extend sum range Good for any 23
Comments -hypergeometric function Normalization 1, 1! 1, 1 Γ1 Need to continue ungauged MM partition function Also need -prescription: 24
25 1 1 1 1 1 2 1 1,1 1 4 1 2 Li, 1 8 1 192 1 24 Li 1 1 Reproduces pert. expn. of ABJ MM
Perturbative expansion 26
Integral rep A way to regularize the sum once and for all: Reproduces pert. expn. Poles at, residue 1 1 2 But more! sin 12 1 Li 1 1 27
Non-perturbative completion poles at 2 zeros at 1,, 1 Im ~~ : non-perturbative P poles NP poles 1 2 NP zeros 1 Re Integrand (anti)periodic Integral = sum of pole residues in fund. domain fund. domain Contrib. from both P and NP poles 28
Contour prescription large enough: 1 2 1 smaller: 2 1 migrate 2 Continuity in fixes prescription Checked against original ABJ MM integral for 2,3 29
Similar to Guiding principle: reproduce pert. expn. Multiple -hypergeometric functions appear 30
Seiberg duality Passed perturbative test. What about non-perturbative one? 1, 5 NS5 1, 5 NS5 Cf. [Kapustin-Willett-Yaakov] 31
Seiberg duality: Partition function: duality inv. by level-rank duality 32
Odd (original) 5 2 5 5 2 5 (dual) 5 2 5 5 2 5 Integrand identical (up to shift), and so is integral P and NP poles interchanged 33
Even (original) 2 4 2 4 (dual) 2 4 2 4 Integrand and thus integral are identical P and NP poles interchanged, modulo ambiguity 34
Conclusions
Conclusions Exact expression for lens space MM Analytic continuation: Got expression i.t.o. -dim integral, generalizing mirror description for ABJM Perturbative expansion agrees Passed non-perturbative check: Seiberg duality ( 36
Future directions Make every step well-defined; understand -hypergeom More general theory (necklace quiver) Wilson line [Awata+Hirano+Nii+MS] Fermi gas [Nagoya+Geneva] Relation to Vasiliev s higher spin theory Toward membrane dynamics? 37
Thanks!