ABJ Partition function Wilson Loops and Seiberg Duality

1. KIAS Pre-Strings 2013 with H. Awata, K. Nii (Nagoya U) & M. Shigemori (YITP) (1212.2966 & to appear soon) ABJ Partition function Wilson Loops and Seiberg Duality ShinjiHirano (University of the Witwatersrand)

2. ABJ(M) ConjectureAharony-Bergman-Jefferis-(Maldacena) M-theory on AdS4 x S7/Zk with (discrete) torsion C3 II N=6 U(N1)k x U(N1+M)-kCSM theory for large N1 and finitek

3. Discrete torsion ( fractional M2 = wrapped M5 ) • IIA regime large N1 and large k with λ = N1/k fixed S7/ZkCP3 & C3B2

4. Higherspinconjecture(Chang-Minwalla-Sharma-Yin) N = 6parity-violating Vasiliev’shigherspintheory on AdS4 II N = 6 U(N1)k x U(N2)-kCSM theory with large N1 and k with fixedN1/k and finiteN2 where

5. Why ABJ(M)? Integrabilitygoesboth ways and deals with non-BPS but large N Localizationgoesthis way and deals only with BPS but finiteN • Weareused to the idea • Localization of ABJ(M) theory ClassicalGravity StronglyCoupled Gauge Theory @ large N StronglyCoupled Gauge Theory @ finiteN “QuantumGravity”

6. Progress to date AdS radius shift Leading • The ABJM partition function ( N1 = N, M = 0 ) Perturbative “Quantum Gravity” Partition Function II AiryFunction A mismatch in 1/N correction

7. Why ABJ? • DoesAirypersist with the AdS radius shift with B field ? (presumablyyes) • A prediction on the AdS4higherspin partition function • A study of Seibergduality

8. In this talk • Study ABJ partition function & Wilson loops and theirbehaviors under Seibergduality • Do not answer Q1 & Q2 but makeprogress to the point thattheseanswersarewithin the reach • Answer Q3 with reasonablesatisfaction

9. ABJ Partition Function

10. OurStrategy • U(N1) x U(N2) Lens space matrix model • perform all the eigenvalue integrals (Gaussian!) • Analyticcontinuation • rank N2 - N2 ABJ Partition Function/Wilson loops

11. ABJ(M) Matrix Model one-loop • Localizationyields (A = Φ = 0, D = - σ) gs = -2πi/k where

12. Lens spaceMatrix Model

13. Change of variables CoshSinh Vandermonde

14. Gaussian integrals N=N1+N2 CompletelyGaussian!

15. The lens space partition function multiple q-hypergeometric function

16. (q-Barnes G function) (q-Gamma) (q-number) • (q-Pochhammer)

17. U(1) x U(N2) case • U(2) x U(N2) case q-hypergeometricfunction (q-ultrasphericalfunction) Schur Q-polynomial double q-hypergeometric function

18. AnalyticContinuation Lens spaceMM  ABJ MM

19. ABJ Partition Function U(N1) x U(N2) = U(N1) x U(N1+M) theory U(M) CS Note: ZCS(M)k = 0 for M > k (SUSY breaking)

20. Integral Representation • The sum is a formal series not convergent, not well-defined at for evenk

21. The following integral representation renders the sum well-defined • regularized & analyticallycontinued in the entire q-plane (“non-perturbativecompletion”) Ppoles NPpoles

22. s perturbative non-perturbative integration contourI

23. U(1)kx U(N)-kcase (abelian Vasiliev on AdS4) • This is simple enough to study the higherspin limit

24. ABJ Wilson Loops

25. 1/6 BPS Wilson loops with windingn

26. Wilson loop results for N1 < N2

27. for N1 < N2

28. 1/2 BPS Wilson loop with windingn

29. s perturbative non-perturbative integration contourI

30. SeibergDuality

31. U(N1)k x U(N1+M)-k = U(N1+k-M)k x U(N1)-k

32. Partition function (Example)

33. The partition functions of the dual pair More generally

34. Fundamental Wilson loops • 1/6 BPS Wilson loops • 1/2 BPS Wilson loops

35. Discussions • The Seibergdualitycanbeproven for general N1 and N2 • Wilson loops in general representations • The Fermi gas approach to the ABJ theory (non-interacting & only simple change in the density matrix) • Interesting to study the transition from higherspinfields to strings

36. The End