A new large N reduction for Chern-Simons theory on S 3

1. A new large N reduction for Chern-Simons theory on S3 Shinji Shimasaki (Kyoto U.) In collaboration with G. Ishiki (KEK), K. Ohta (Meiji GakuinU.) and A. Tsuchiya (Shizuoka U.) (ref.) Ishiki-Ohta-SS-Tsuchiya, PLB 672 (2009) 289. arXiv:0811.3569[hep-th] Ishiki-Ohta-SS-Tsuchiya, to appear

2. Introduction • Matrix model • Nonperturbative definition (regularization) of large N gauge theory • (Large N reduction) [Eguchi-Kawai][Parisi][Gross-Kitazawa] [Bhanot-Heller-Neuberger][Gonzalez-Arroyo – Okawa]… YM on RD Matrix Model (0-dim) planar • Nonperturbative definition of superstring theory [Banks-Fischler-Shenker-Susskind][Ishibashi-Kawai-Kitazawa-Tsuchiya] [Dijkgraaf-Verlinde-Verlinde] ☆ Can we describe curved spaces and topological invariants by matrices ? [Madore][Grosse-Madore] [Grosse-Klimcik-Presnajder] [Carow-Watamura – Watamura] [Ishiki-SS-Takayama-Tsuchiya]… • gauge theory on • S1, T2, flux on T2, S2(fuzzy sphere), • monopoles on S2,… • gauge/gravity correspondence [Lin-Lunin-Maldacena][Lin-Maldacena] • Description of curved spaces by matrices [Hanada-Kawai-Kimura]

3. In this talk, we give a new large N reduction large N reduction for Chern-Simons theory on S3 • Reduced theories of Chern-Simons theory on S3 Chern-Simons theory on S3 Dimensional Reduction large N reduction to make S1 S3 = S1 on S2 BF theory + mass term on S2 = YM on S2 S2 Continuum limit of fuzzy sphere Dimensional Reduction N=1* matrix model point

4. In this talk, we give a new large N reduction large N reduction for Chern-Simons theory on S3 • Results • A particular sector of N=1* matrix model reproduce • the planar limit of Chern-Simons theory on S3. • Planar free energy and Wilson loop (unknot) of CS on S3 • is reproduced from our matrix model • This is the first explicitly shown large N reduction on S3. • Interesting application to topological field theory • Alternative regularization of CS on S3 All order correspondence for perturbative expansion with respect to ‘t Hooft coupling

5. Plan of this talk • Introduction • Relationships between reduced theories of • Chern-Simons theory on S3 • 3. Chern-Simons theory on S3 • from N=1* matrix model • 4. Summary and Outlook

6. 2. Relationship between reduced theories of Chern-Simons theory on S3

7. Dimensional reduction S1 S3 • Chern-Simons theory on S3 S2 : right-invariant 1-form on S3 right-invariant Killing vector on S3 : angular momentum op. on S2 • Fourier expansion along the S1 fiber : angular momentum op. in the presence of magnetic charge KK momenta along the S1 fiber monopole charge on S2

8. Dimensional reduction • BF theory + mass term on S2= YM on S2 Integrating out • N=1* matrix model (cf) mass deformed superpotential of N=4 SYM

9. Classical relationship • N=1* matrix model • Expand around a classical solution fuzzy sphere Continuum limit of fuzzy sphere • BF + mass term on S2 around a monopole background

10. Classical relationship • BF + mass term on S2 around a monopole background large N reduction for nontrivial S1 fiber take in all monopole charge = reproduce all KK momenta along the S1 fiber • Planar Chern-Simons theory on S3

11. 3. Chern-Simons theory on S3 form N=1* matrix model

12. Exact integration of N=1* matrix model [Ishiki-Ohta-SS-Tsuchiya] matrix • Diagonalize and integrate and • Use

13. The integral is decomposed into sectors which are characterized by -dimensional representation of SU(2). (partition of ) specifies irreducible representations and its multiplicity: : irreducible rep. : multiplicity Each sector seems to be the contribution around each classical solution of N=1* matrix model.

14. To 2d YM on S2 Extract -block sector and take Equal size block configuration is dominant Set and take partition function of SU(K) YM on S2

15. To Chern-Simons on S3 Extract the following sector We expect that in the limits the planar limit of the partition function of CS on S3 is reproduced. In

16. Our matrix model - multi matrix model Chern-Simons theory on S3 Chern-Simons matrix model (cf)

17. Feynman rule for CSMM Propagator: Vertex: (ex)

18. Feynman rule for our matrix model Propagator: Vertex: (ex)

19. Free energy (connected diagrams) Planar Nonplanar

20. General connected planar diagrams of both theories are like planar Dashed lines ( ) should not form any loop

21. Correspondence between our matrix model and CSMM • Let us look at the different part between two For planar our matrix model complete agreement !! CSMM

22. Correspondence between our matrix model and CSMM For nonplanar our matrix model There is no correspondence for nonplanar diagrams CSMM

23. Wilson loop Wilson loop in N=1* matrix model: [Ishii-Ishiki-Ohta-SS-Tsuchiya] For great circle on S3, our matrix model (great circle on S3) CSMM (Unknot, fundamental rep.) We can also see the planar correspondence for these two.

24. 4. Summary and Outlook • We give a new type of the large N reduction extended • to curved space, S3, and its application to CS theory. • In the planar limit, a particular sector of N=1* matrix model • reproduce the planar Chern-Simons theory on S3. • Free energy and Wilson loop are reprodeced • We can also show that N=1* MM includes sectors • corresponding to various nontrivial vacua of CS on S3/Zk. • Wilson loops (various contour, deformation) • Localization