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Maximal super Yang-Mills theories on curved background with off-shell supercharges. 総合研究大学院大学 藤塚 理史. 共同研究者: 吉田 豊 氏 (KEK), 本多 正純 氏 ( 総研大 /KEK). b ased on M. F, M. Honda and Y. Yoshida, arxiv: 1209.4320[hep-th]. 2012. 10.24. String Advanced Lectures (SAL) at KEK. Our motivation. - PowerPoint PPT Presentation
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Maximal super Yang-Mills theories on curved background with off-shell supercharges総合研究大学院大学藤塚 理史
共同研究者: 吉田 豊 氏 (KEK), 本多 正純 氏 ( 総研大 /KEK)
based on M. F, M. Honda and Y. Yoshida, arxiv:1209.4320[hep-th]
2012. 10.24. String Advanced Lectures (SAL) at KEK
Our motivationGauge/Gravity correspondence
SUGRA solution for N Dp-branes
[Maldacena ‘97]
The well known example:
dual !?
Different views of low energy effective theory on D-branes (or M-branes)
(p+1)-dim. U(N) SYM
・ N D3-branes
at near horizon
In non-conformal field theory, the correspondence is also expected.
ex.)
[Itzhaki-Maldacena-Sonnenschein-Yankielowicz ‘98]
Our motivationGauge/Gravity correspondence
SUGRA solution for N Dp-branes
[Maldacena ‘97]
dual !?
Different views of low energy effective theory on D-branes (or M-branes)
(p+1)-dim. U(N) SYMat near horizon
This duality is a strong/weak duality:
strong coupling region weak coupling region
“Localization” method
Localization
In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories.
・ a Supercharge Q such that
・ V such that
Deformation of the expectation value:
Then we note
Localization
In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories.
・ a Supercharge Q such that
Then we note
Off-shell
Localization
In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories.
・ a Supercharge Q such that
Off-shell
If we consider it on flat space,
“infrared effect” and “flat directions” divergence !
compact space mass terms
Localization
If we consider it on flat space,
“infrared effect” and “flat directions” divergence !
compact space mass terms
Ex) 4D N=4 SYM on [Pestun ‘07]
etc…
[Pestun ‘07]
[Kapustin-Willet-Yaakov ‘09]
[Hosomichi –Seong-Terashima ‘12]
[Hama-Hosomichi-Lee ‘11], [Imamura-Yokoyama ‘11]
[Hama-Hosomichi ‘12]
[Imamura ‘12]
[Gang ‘09]
[Hama-Hosomichi-Lee ‘10], [Jafferis ‘10]
[Imamura-Yokoyama ‘12]
Many off-shell SUSY theories on curved space have been studied in these days:
Round sphere
Squashed sphere
Others
4D N=1 [Festuccia-Seiberg ‘11][Dumitrescu-Festuccia-Seiberg ‘12]
The more the number of SUSY and dimension grows, the more difficult we construct the off-shell SUSY theories on curved space generally.
It has been constructed partially by Berkovits. [Berkovits ‘93]
However its general formalism has not been known.
Off-shell SUSY
ex.) off-shell maximal SYM on flat space
Rigid SUSY on curved space
ex.)
We can construct off-shell maximal SYM on curved space on which a Killing spinor exists.
Our research purpose
Maximal SYM It’s important for gauge/gravity duality.
Off-shell formulation on curved space
Localization
Main result
Off-shell maximal SYM on flat sp.[Berkovits ‘93]
Contents
1. Off-shell maximal SYM on flat sp.
2. Off-shell maximal SYM on curved sp.
3. Some examples
4. Summary and discussions
SUSY tr.
Notation
where
Berkovits method [Berkovits ‘93]
on-shell SYM on flat space:
where is a 16 components Majorana-Weyl spinor, and .
maximal SYMdred
Charge conjugation matrix:
Note
where is a constant bosonic spinor.
Notation
where
Berkovits method [Berkovits ‘93]
on-shell SYM on flat space:
where is a 16 components Majorana-Weyl spinor, and .
Charge conjugation matrix:
In off-shell,
7-(bosonic) auxiliary fields
where is a (bosonic) pure imaginary auxiliary field.
where depends on , and is (bosonic) spinor satisfying
Off-shell maximal SYM on flat space
SUSY tr.
For any nonzero , there exist which satisfy above constraint.
solution
The number of off-shell supercharges
[Berkovits ‘93][Evans ‘94]
Given any , we can construct which solves the constraints.
linear in
conventional SUSY
d.o.f of the number of off-shell supercharges
We can construct the solution which has 9 off-shell supercharges at least.
more than 9 ??
16-components
The number of off-shell supercharges
We impose the restriction to as
(1). 8 off-shell supercharges
By using the ,
Reduce “d.o.f of “ to 1/2 8 off-shell supercharges
Next in the case of 9 off-shell charges solution…
We have to introduce concrete notation.
16-components
the solution of 8 the solution of 9
eigenspinors of
In this representation,
the solution with 8 off-shell charges:
where is the anti-symmetric matrix satisfying
Notation
eigenspinors of
In this representation,
the solution with 8 off-shell charges:
(2). 9 off-shell supercharges
Note in 8 off-shell charges,
We can construct a solution in which is nonzero:
(2). 9 off-shell supercharges
Note in 8 off-shell charges,
We can construct a solution in which is nonzero:
Introduce a matrix:
Then,
9 off-shell supercharges
On curved space
SUST tr.Same as the flat one
Constant spinor doesn’t exist on the curved space in general.
Note
On curved space
SUST tr.Same as the flat one
Constant spinor doesn’t exist on the curved space in general.
Note
Then,
Parallel spinors
The condition for the invariance is
Existence of the above spinors can be characterized by the holonomy group.
[Hitchin ‘74][Wang ‘89]
For example these don’t include spheres.
is the odd product of internal gamma matrices.
Killing spinors extension
where is a constant that depends on a space, and
The above eq. implies
Next we consider whether SUSY theories can be constructed on curved space on which Killing spinor exists.
On curved space
Then,
The condition for invariance is
So the action is not invariant.
Deformation of the action and SUSY tr.
Class 1 (d=4)
We modify the action and transformation in the following way,
SUSY tr.
Using the Killing spinor eq.
Note that this is the equivalent to the well known theory on conformally flat space.
[Pestun ‘07] etc.ex.)
Thus, the action
is invariant under the transformation
SUSY algebra
We consider the square of the SUSY tr. of the each field,
The dilatation vanishes automatically in this class because of anti-symmetry of .
BMN matrix model [Berenstein-Maldacena-Nastase ‘02]
If we integrate out, this is the on-shell BMN matrix model.
We take d=1 and in class 2, then
SUSY tr.
BMN matrix model [Berenstein-Maldacena-Nastase ‘02]
β-function and Wilson loop of
・ Non-perturbative formulation of
[Ishiki-Shimasaki-Tsuchiya ‘11]
conformal map
[Ishii-Ishiki-Shimasaki-Tsuchiya ‘08]
・ Gravity dual corresponding to theory around each vacuum [Lin-Maldacena ‘06]
flat direction
Large-N equivalence
3D N=8 SYM on
There are 2 ways of constructing this theory:
(1). Applying to the class 2 directly
i.e. we take d=3 and in class 2.
(2). Dimensional reduction of the class 1 on to
These theories are different!
main difference
R-symmetry:
reduction from 4D.
(1) (2)
・ We have constructed off-shell maximal SYM on curved space on which a Killing spinor exists.
・ This class of the space contains and so on.
・ We have also constructed the different maximal SYM with same number of supercharges on same space.
Ex.) d=3, N=8 SYM on
Summary
Future work
・ Gauge/Gravity duality
ex.)
・ Localization of BMN matrix model
Non-perturbative verification of the large-N equivalence
・ Extending to more larger class of curved space
ex.) spaces which include a connection and so on.
The rewriting of Killing spinor eq.
Killing spinor eq.
We can decompose as
Then we can rewrite the Killing spinor eq.
where D is the Dirac op.
Therefore the existence of the Killing spinors can be characterized by the holonomy group similarly:
Killing spinors
Killing spinor eq.
Here we introduce “cone” over
Then the Killing spinor eq. can be rewritten as
where is covariant derivative on the cone.
-action: subgroup of isometry .
Since Killing spinors on are constants along , so the former is also the Killing spinors on .
Therefore the number of off-shell supercharges is 4 at least.
Also there are orbifolds in which exist Killing spinors :
Also the solution of the Killing spinor eq. is
where is any constant spinor.
The solution of the Killing spinor eq. is
The solution for Killing spinor eq.
We give the metric:
Also the solution of the Killing spinor eq. is
where is any constant spinor.
The solution of the Killing spinor eq. is
We give the metric:
Furthermore we can construct another class of maximal SYM on with off-shell SUSY :
d=3, N=8 SYM on
dred