A new large N reduction for Chern-Simons theory on S3
Shinji Shimasaki (Kyoto U.)
In collaboration with G. Ishiki (KEK),K. Ohta (Meiji Gakuin U.) 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
• Nonperturbative definition (regularization) of large N gauge theory (Large N reduction)
Matrix model
• Nonperturbative definition of superstring theory
[Eguchi-Kawai][Parisi][Gross-Kitazawa][Bhanot-Heller-Neuberger][Gonzalez-Arroyo – Okawa]…
[Banks-Fischler-Shenker-Susskind][Ishibashi-Kawai-Kitazawa-Tsuchiya][Dijkgraaf-Verlinde-Verlinde]
Introduction
YM on RDMatrix Model (0-dim)
☆ Can we describe curved spaces and topological invariants by matrices ?
• Description of curved spaces by matrices [Hanada-Kawai-Kimura]
• gauge theory on S1, T2, flux on T2, S2(fuzzy sphere), monopoles on S2,…
[Madore][Grosse-Madore][Grosse-Klimcik-Presnajder][Carow-Watamura – Watamura][Ishiki-SS-Takayama-Tsuchiya]…
• gauge/gravity correspondence [Lin-Lunin-Maldacena][Lin-Maldacena]
planar
In this talk, we give a new large N reduction
large N reduction for Chern-Simons theory on S3
DimensionalReduction
DimensionalReduction
large N reductionto make S1
Continuum limitof fuzzy sphere
S2
S3 = S1 on S2
point
Chern-Simons theory on S3
BF theory + mass term on S2 = YM on S2
N=1* matrix model
Reduced theories of Chern-Simons theory on S3
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 expansionwith respect to ‘t Hooft coupling
Results
In this talk, we give a new large N reduction
large N reduction for Chern-Simons theory on S3
1. Introduction
2. 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
Plan of this talk
2. Relationship between reduced theories of Chern-Simons theory on S3
Chern-Simons theory on S3
Dimensional reduction
right-invariant Killing vector on S3
: right-invariant 1-form on S3
S3 S1
S2
• Fourier expansion along the S1 fiber :
: angular momentum op. on S2
KK momenta along the S1 fiber monopole charge on S2
•
•
angular momentum op. in the presence of magnetic charge
BF theory + mass term on S2 = YM on S2
Dimensional reduction
N=1* matrix model
(cf)mass deformed superpotentialof N=4 SYM
Integrating out
• Expand around a classical solution
BF + mass term on S2 around a monopole background
N=1* matrix model
Classical relationship
Continuum limit of fuzzy sphere
fuzzy sphere
take in all monopole charge
BF + mass term on S2 around a monopole background
Planar Chern-Simons theory on S3
Classical relationship
large N reduction for nontrivial S1 fiber
reproduce all KK momentaalong the S1 fiber
=
3. Chern-Simons theory on S3
form N=1* matrix model
matrix
Diagonalize and integrate and
Use
Exact integration of N=1* matrix model[Ishiki-Ohta-SS-Tsuchiya]
The integral is decomposed into sectors which are characterizedby -dimensional representation of SU(2). (partition of )
specifies irreducible representationsand its multiplicity:
Each sector seems to be the contribution around each classical solutionof N=1* matrix model.
: irreducible rep.: multiplicity
Extract -block sector
To 2d YM on S2
partition function of SU(K) YM on S2
Equal size block configuration is dominant
and take
Set and take
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
Our matrix model - multi matrix model
Chern-Simons theory on S3 Chern-Simons matrix model
(cf)
Feynman rule for CSMM
Propagator:
Vertex: (ex)
Feynman rule for our matrix model
Propagator:
Vertex: (ex)
Planar
Nonplanar
Free energy (connected diagrams)
General connected planar diagrams of both theories are like
planar
Dashed lines ( ) should not form any loop
Correspondence between our matrix model and CSMM
our matrix model
CSMM
For planar
complete agreement !!
• Let us look at the different part between two
our matrix model
CSMM There is no correspondence for nonplanar diagrams
For nonplanar
Correspondence between our matrix model and CSMM
Wilson loop
our matrix model (great circle on S3)
CSMM (Unknot, fundamental rep.)
Wilson loop in N=1* matrix model:
For great circle on S3,
We can also see the planar correspondence for these two.
[Ishii-Ishiki-Ohta-SS-Tsuchiya]
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
Wilson loops (various contour, deformation)
We can also show that N=1* MM includes sectors corresponding to various nontrivial vacua of CS on S3/Zk.
Localization