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The Partition Function of ABJ Theory

This paper investigates the partition function of ABJ theory within the framework of AdS4/CFT3. We emphasize its significance in understanding M2-brane dynamics through a non-perturbative approach and analytic continuation techniques. By employing powerful methods from supersymmetric field theory, we establish connections between ABJ theory and several advanced concepts including dualities and localization. Our findings offer a pathway for further exploration into the dynamics of M2 and fractional M2-branes, providing a clearer framework for studying ABJ and its holographic implications.

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The Partition Function of ABJ Theory

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  1. Masaki Shigemori (KMI, Nagoya U) The Partition Functionof ABJ Theory with Hidetoshi Awata, Shinji Hirano, Keita Nii • 1212.2966, 1303.xxxx Amsterdam, 28 February 2013

  2. Introduction

  3. AdS4/CFT3 • Low E physics of M2-branes on • Can tell us about M2 dynamics in principle (cf. ) • More non-trivial example of AdS/CFT Chern-Simons-matter SCFT (ABJM theory) M-theory on = Type IIA on ,

  4. ABJ theory • Generalization of ABJM [Hosomichi-Lee3-Park, ABJ] • CS-matter SCFT • Low E dynamics of M2 and fractional M2 • Dual to M on with 3-form Seiberg duality “ABJ triality” [Chang+Minwalla+Sharma+Yin] • dual to Vasiliev higher spin theory?

  5. Localization & ABJM • Powerful technique in susy field theory • Reduces field theory path integral to matrix model • Tool for precision holography ( 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]

  6. ABJ: not as well-studied as ABJM; MM harder to handle Goal: develop framework to study ABJ • Rewrite ABJ MMin more tractable form • Fun to see how it works • Interesting physics emerges!

  7. Outline &main results

  8. ABJ theory • CSM theory • Superconformal • IIB picture D3 5 NS5 D3

  9. Lagrangian

  10. ABJ MM [Kapustin-Willett-Yaakov] • Pert. treatment — easy • Rigorous treatment — obstacle: Partition function of ABJ theory on : vector multiplets matter multiplets

  11. Strategy “Analytically continue” lens space () MM Perturbative relation: [Marino][Marino+Putrov] Simple Gaussian integral Explicitly doable!

  12. Expression for • : -Barnes G function • partition of into two sets

  13. Now we analytically continue: It’s like knowing discrete data and guessing

  14. Analytically cont’d • Analytic continuation has ambiguitycriterion: reproduce pert. expansion • Non-convergent series — need regularization , -Pochhammer symbol:

  15. : integral rep (1) • Finite • “Agrees” with the formal series • Watson-Sommerfeldtransf in Regge theory poles from

  16. : integral rep (2) • Provides non-perturbative completion Perturbative (P) poles from Non-perturbative (NP) poles,

  17. Comments (1) • Well-defined for all • No first-principle derivation;a posteriori justification: • Correctly reproduces pert. expansions • Seiberg duality • Explicit checks for small

  18. Comments (2) • (ABJM) reduces to “mirror expression” • Vanishes for — Starting point for Fermi gas approach — Agrees with ABJ conjecture

  19. Details / Example

  20. partition of into two sets

  21. How do we continue this ? Obstacles: • What is • Sum range depends on

  22. -Pochhammer & anal. cont. For For , In particular, Can extend sum range Good for any 

  23. Comments • -hypergeometric function • Normalization Need to continue “ungauged” MM partition function Also need -prescription:

  24.  Reproduces pert. expn. of ABJ MM

  25. Perturbative expansion

  26. Integral rep • Reproduces pert. expn. A way to regularize the sum once and for all: Poles at , residue “ ” • But more!

  27. Non-perturbative completion • poles at • zeros at : non-perturbative • Integrand (anti)periodic • Integral = sum of pole residues in “fund. domain” • Contrib. fromboth P and NP poles NP poles NP zeros P poles fund. domain

  28. Contour prescription large enough: smaller: migrate • Continuity in fixes prescription • Checked against original ABJ MM integral for

  29. Similar to • Guiding principle: reproduce pert. expn. • Multiple -hypergeometric functions appear

  30. Seiberg duality Passed perturbative test. What about non-perturbative one? 5 NS5 5 NS5 Cf. [Kapustin-Willett-Yaakov]

  31. Seiberg duality: Partition function: duality inv. by“level-rank duality”

  32. Odd (original) (dual)  • Integrand identical (up to shift), and so is integral • P and NP poles interchanged

  33. Even (original) (dual)  • Integrand and thus integral are identical • P and NP poles interchanged, modulo ambiguity

  34. Conclusions

  35. 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!

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