geometry / gauge theory duality and the dijkgraaf–vafa...
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Geometry / Gauge Theory Dualityand the Dijkgraaf–Vafa Conjecture
Masaki Shigemori
University of California, Los Angeles
Final Oral Examination & TEP Seminar
May 17, 2004
Introduction
Gauge theory: hard to study
Strongly coupled at low E
Confinement / chiral symmetry breaking
Even vacua are not known analytically
Supersymmetric gauge theory: more tractable
Sometimes exact analysis is possible
Superpotential: determines vacua
No systematic way to compute superpotential
Masaki Shigemori, Final Oral – p.1/38
IntroductionDijkgraaf–Vafa conjecture
Based on string theory duality
One can compute superpotential systematicallyusing matrix model
“Recipe”
Need to go back to string theorywhen matrix model is not enough
Sometimes counter-intuitive from gauge theoryviewpoint: e.g., glueball S for U(1), Sp(0)
Inclusion of flavors
Masaki Shigemori, Final Oral – p.2/38
Outline
Introduction√
Dijkgraaf–Vafa conjecture
“Counterexample” [hep-th/0303104, 0304138]
String theory prescription [0311181]
Inclusion of flavors [0405101]
Future problems
Conclusion
Masaki Shigemori, Final Oral – p.3/38
DV conjecture
Dijkgraaf-Vafa conjecture:
For a large class of N = 1 supersymmetric gauge theory,
i) Low E degree of freedom is glueball superfield
S ∼ tr[WαWα] = tr λαλα + . . .
ii) Effective superpotential Weff(S) which encodesnonperturbative effect can be exactly calculated bymatrix model
Masaki Shigemori, Final Oral – p.4/38
DV conjecture
Example:N = 1 U(N) theory with adjoint chiral superfield Φij
Wtree = Tr[W (Φ)],
W ′(x) = (x − a1)(x − a2) · · · (x − aK)
x=a1
x=a2 x=aK...
W x( )
Masaki Shigemori, Final Oral – p.5/38
DV conjectureClassical vacua:
Φ ∼= diag(a1, . . . , a1︸ ︷︷ ︸N1
, a2, . . . , a2︸ ︷︷ ︸N2
, . . . , aK , . . . , aK︸ ︷︷ ︸NK
)
U(N) → U(N1) × U(N2) × · · · × U(NK)
1U(N )
2U(N ) KU(N )...
W x( )
eigenvalues at the −th critical pointNi i
Masaki Shigemori, Final Oral – p.6/38
DV conjectureGlueballs
U(N) → U(N1) × U(N2) × · · · × U(NK)
...
S
S S2
1
K
W x( )
Effective glueball superpotential
Weff(S1, . . . , SK) =K∑
i=1
Ni∂F0
∂Si, F0 : MM free energy
Masaki Shigemori, Final Oral – p.7/38
DV conjecture
Systematic way to compute nonpert. superpot.
Checked for many nontrivial examples
Second part (reduction to matrix model) can beproven by superspace Feynman diagrams
Philosophy applicable to any representations[CDSW, AIVW]
Masaki Shigemori, Final Oral – p.8/38
“Counterexample”
Sp(N) with antisymmetric tensor
Breaking pattern:
Sp(N) → Sp(N1) × Sp(N2) × · · · × Sp(NK)
( )...1
2 K
Sp(N )
Sp(N ) Sp(N )
W x
Masaki Shigemori, Final Oral – p.9/38
“Counterexample”
Discrepancy
Cubic superpotential with
Sp(N) → Sp(N) × Sp(0)
One glueball S for unbrokenSp(N)
Discrepancy:
SW x
(0)Sp
Sp N ( ) ( )
WDV = Weff
(〈S〉
)6= WGT !
Masaki Shigemori, Final Oral – p.10/38
String theory prescriptionGeometric engineering of U(N) theoryBreaking pattern: U(N) → U(N1) × · · · × U(NK)
1
4
U(N )
2
...
W x( )
RI
U(N ) KU(N )
CompactificationType IIB
Non−compact Calabi−Yau
N1 N N2 KD5 D5 D5S2 S2
S2
Masaki Shigemori, Final Oral – p.11/38
String theory prescriptionGeometric transition — simple case
Open string theory on S2-blown up conifold⇐⇒ closed string theory on S3-blown up conifold
S
3 S2
S 3 S2
D5−branesN
S
2
dual
with adjoint ΦU(N) theory U(1) theory
with adjoint S
N units of RR fluxes S
S 3
A version of AdS/CFT duality
Masaki Shigemori, Final Oral – p.12/38
String theory prescriptionGeometric transition — general case
4IR
S3S3 S3
Non−compact Calabi−Yau
...N1 fluxes N2 fluxes KN fluxes
S1 S2 SK
...
4d theory: U(1)K theory with S1, . . . , SK
Masaki Shigemori, Final Oral – p.13/38
String theory prescriptionFlux superpotential
4IR
S3S3 S3
Non−compact Calabi−Yau
...N1 fluxes N2 fluxes KN fluxes
S1 S2 SK
...
Exact superpotential
Wflux(Sj) =K∑
i=1
[Ni︸︷︷︸
RR flux
Πi(Sj) − 2πiτ0︸ ︷︷ ︸NSNS flux
Si
]
Computation of Πi reduces to MM → DV conjecture
Masaki Shigemori, Final Oral – p.14/38
String theory prescriptionPhysics near critical points
More generally, for G(N) → ∏i Gi(Ni),
Wflux(Sj) ∼K∑
i=1
[N̂iSi
(1 + ln(Λ3
i /Si))− 2πiηiτ0Si
]
N̂i = (# of RR fluxes) =
{Ni U(Ni)
Ni/2 ∓ 1 SO/Sp(Ni)
⇓ focus on one critical point
Wflux(S) ∼ N̂S[1 + ln
(Λ
3
S
)]− 2πiητ0S.
S 3 S2
RR fluxesunits ofN̂S
S 3
When is S a good variable?
Masaki Shigemori, Final Oral – p.15/38
String theory prescription
N̂ = 0 case
Wflux(S) = −2πiητ0S,∂Wflux
∂S= −2πiητ0 6= 0
=⇒ No susy solutions?
Extra massless degree of freedom:
S
3 S2
S
S
as 0
D3−brane wrapping Sbecomes massless
3
3S
Masaki Shigemori, Final Oral – p.16/38
String theory prescription
N̂ = 0 caseTaking D3-brane hypermultiplet h, h̃ into account
Wflux = hh̃S − 2πiητ0S =⇒ S = 0 is a solution
For U(0), SO(2), set S = 0 from the beginning
N̂ > 0 caseD3-brane is infinitely massive with nonvanishing RRflux through it, hence there are no h, h̃.
There is a glueball forU(N > 0), SO(N > 2), Sp(N ≥ 0).
Masaki Shigemori, Final Oral – p.17/38
String theory prescriptionResolving discrepancy
Breaking pattern was
Sp(N) → Sp(N) × Sp(0)
Need glueball S even for Sp(0)!
W x
(0)Sp
Sp N ( )
S
( )S
2
1
Taking S2 into account resolves discrepancy
Masaki Shigemori, Final Oral – p.18/38
Inclusion of flavors
U(N) theory with an adjoint Φ and Nf flavors Q, Q̃
Tree level superpotential:
Wtree = Tr[W (Φ)] −Nf∑
I=1
Q̃I(Φ − zI)QI , zI = −mI
Classical vacua:i) Pseudo-confining vacua: 〈Q〉 = 〈Q̃〉 = 0
U(N) →K∏
i=1
U(Ni),K∑
i=1
Ni = N.
ii) Higgs vacua: 〈Q〉, 〈Q̃〉 6= 0
U(N) →K∏
i=1
U(Ni),K∑
i=1
Ni < N.
Masaki Shigemori, Final Oral – p.19/38
Inclusion of flavorsGeneralized Konishi anomaly formalism
Descend from CY to z-plane [CDSW,CSW]
poles
3 S3 S3
(Riemann surface)
S1 S2 SK
space
D5D5
cuts
z−plane
Calabi−Yau
double−sheeted...
z
S
S1 S2 SK...
1z=z2
z=z
Masaki Shigemori, Final Oral – p.20/38
Inclusion of flavorsHow are the vacua described?
Pseudo-confining vacua
U(N) → U(N1) × U(N2), N1 + N2 = N.
2
sheet
firstsheet
( )U N( )1 U N
second
All flavor poles are on the second sheet
Masaki Shigemori, Final Oral – p.21/38
Inclusion of flavorsHow are the vacua described?
Higgs branch
U(N) → U(N1) × U(N2), N1 + N2 < N.
Passing poles through cuts corresponds to Higgsing:
( )1 1( 1 )U N −
secondsheet
firstsheet
( )U N2U N
Masaki Shigemori, Final Oral – p.22/38
Inclusion of flavorsHow many poles can pass through a cut?
There must be a limit to this passing process,on-shell :
...U N( ) ( 1 )U N − U N −( 2 ) U ( 0 )
= 0S
At some point,The cut should close up
first sheet
poles onthe second
sheet
Masaki Shigemori, Final Oral – p.23/38
Inclusion of flavors
S = 0 solutions and matrix model
= 0S
U N = 0( )
|
| U ( 0 )
= 0Spassingpoles
S = 0 solutions should not be directly describable inmatrix model
U(N 6= 0) → U(0) is not a smooth process;# of massless photons changes discontinuously
There must be some extra charged massless DoFcondensing
Masaki Shigemori, Final Oral – p.24/38
Inclusion of flavors
Try actually passing poles through a cut!
S4
Nf poles on top of each other
O on the second sheet
zfz =
( )U N
z
Take one-cut case, solve the EOM
∂Weff(S; zf )
∂S= 0 =⇒ zf = SN/Nf + S1−N/Nf ,
and study S as a function of zf for various N, Nf
Masaki Shigemori, Final Oral – p.25/38
Inclusion of flavorsNf < N case
Typical |S| versus zf graphs (Nf = N/2):
−4 −2 2 4
0.5
1
1.5
2
2.5
1st sheet
(PS)2nd sheet
(Higgs)
| |S
fz −4 −2 2 4
0.5
1
1.5
2
2.52nd sheet(PS)
1st sheet(Higgs)
zf
S| |
−4 −2 2 4
0.5
1
1.5
2
2.5
2nd sheet(PS)
2nd sheet(PS)
S| |
zf
Three branches:i) Poles pass through and reach 1st sheet,
without obstructionii) Reverse process of i)iii) Poles get reflected back to 2nd sheet
Cut never closes up: S 6= 0
Corresponds to Higgsing U(N) → U(N − Nf )
Masaki Shigemori, Final Oral – p.26/38
Inclusion of flavorsHow can poles be reflected back to 2nd sheet?
through the cut
3) Poles proceed on the 1st sheet by a short distance
1) Poles approach from infinity on the 2nd sheet
4) Poles proceed back on the 2nd sheet
2) Poles are about to pass
Masaki Shigemori, Final Oral – p.27/38
Inclusion of flavorsN < Nf ≤ 2N case
Typical |S| versus zf graphs (Nf = 3N/2):
−1 −0.75−0.5−0.25 0.25 0.5 0.75 1
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2nd(PS)
2nd(PS)
zf
S| |
−1 −0.75−0.5−0.25 0.25 0.5 0.75 1
0.25
0.5
0.75
1
1.25
1.5
1.75
2
f 0z =
EXCLUDED
2nd(PS)
S| |
zf
Two branches:i) Poles get reflected back to 2nd sheetii) Cut closes up before poles reach it
Careful study of EOM shows that zf = 0, S = 0 is nota solution — exactly what we expected
Masaki Shigemori, Final Oral – p.28/38
Inclusion of flavorsExclusion of zf = S = 0 solution for N < Nf < 2N
Weff = S
[N + ln
(mN
A Λ2N−Nf0
SN
)]− NfS
[− ln
(zf
2+ 1
2
√z2f − 4S
mA
)
+ mAzf
4S
(√z2f − 4S
mA− zf
)+ 1
2
]+ 2πiτ0S
If zf 6= 0, this leads to zf = SN/Nf + S1−N/Nf .
If zf = 0
Weff = S(N − Nf
2
) [1 + ln
(Λ
′
0
3
S
)]+ 2πiτ0S
∂Weff
∂S =(N − Nf
2
)ln
(Λ
′
0
3
S
)+ 2πiτ0 = 0 =⇒ S 6= 0
Masaki Shigemori, Final Oral – p.29/38
Inclusion of flavorsSummary so far:
For N < Nf ≤ 2N , there are solutions with S → 0 aspoles approach the cut, but S = 0 is not a solution toEOM in the MM context.
But in the GT context (Seiberg–Witten theory),S = 0 is a solution.
This is just as expected — massless DoF is missingin the MM (glueball) framework
Nf > 2N case cannot be discussed in one-cut case,
but IR free so we must set S = 0
Nf = N case is exceptional (discussed later)
Masaki Shigemori, Final Oral – p.30/38
Inclusion of flavorsString theory interpretation: Nf = 2N
Cut closes up for zf = 0, for which
Weff = N̂S[1 + ln
(Λ
′
0
3
S
)]+ 2πiτ0S
N̂ ≡ N − Nf/2: effective # of fluxes
This is of the same form as U(N) theory w/o flavors⇓
Same mechanism as thecase w/o flavors; D3-brane wrapping S3 makesS = 0 a solution.
S
3 S2
S
S
as 0
D3−brane wrapping Sbecomes massless
3
3S
Masaki Shigemori, Final Oral – p.31/38
Inclusion of flavorsString theory interpretation: N < Nf < 2N
There is net RR flux through D3 in this case.
If there weren’t flavors, D3 would be infinitelymassive:
RR fluxes
S3
strings
wrappingD3−brane
fundamental TD3
∫d4ξ A1 ∧ HRR
3
⇒ net Fµν induced in D3
⇒ need to emanate F1
⇒ F1 extends to ∞
Masaki Shigemori, Final Oral – p.32/38
Inclusion of flavorsString theory interpretation: N < Nf < 2N
In the presence of flavor poles = noncompactD5-branes, F1 can end!
RR fluxes
S3
noncompact
fundamental
wrapping
strings
D3−brane
D5−branes(flavor poles)
( = 0)
Poles areon D3−brane
zf( = 0)|
Poles are noton D3−brane
zf
If zf = 0, the F1 are massless=⇒ D3 with F1 on it (“baryon”) is the massless DoF
Masaki Shigemori, Final Oral – p.33/38
Inclusion of flavors
String theory interpretation: N < Nf < 2N
Condensation of massless “baryon” DoF B causesU(N 6= 0) → U(0).
We don’t know the precise form of Weff(S, B), but thewhole effect should be to make S = 0 a solution
Masaki Shigemori, Final Oral – p.34/38
Inclusion of flavorsPrescription
Assume Nf poles are on the cut associated withU(N).
U N( )
Nf poleson the cut
For Nf ≥ 2N , one should set S = 0 and it’s the onlysolution.
For N < Nf < 2N , S = 0 is a physical solution,although there may be S 6= 0 solutions too.
Masaki Shigemori, Final Oral – p.35/38
Inclusion of flavorsExample
( )( )U N U N
f poleson one cut
21
N
We considered U(3) theory with cubic W (x), with allpossible breaking pattern U(3) → U(N1) × U(N2).
We put Nf poles on a cut.
We checked that WGT can be reproduced byWMM(S1, S2) by setting S1 = 0 following prescription.
Masaki Shigemori, Final Oral – p.36/38
Future problemsRefine string theory interpretation
Precise form of Weff(S, B)
Baryonic property and Nf ≥ N
Generalize to quiver theories, then descend
N = Nf case: when cut closes up, poles aren’t onthe cut
U N( )
polesN closes up
F1 has finite length, so “baryon” isn’t massless.But there must be massless DoF behind the scene
Masaki Shigemori, Final Oral – p.37/38
Conclusion
DV conjecture provides new approach to susy gaugetheories.
Geometry/gauge theory duality clarified how stringtheory treats glueballs.
Vanishing glueball S = 0 signifies existence of extramassless DoF.
String theory helps identify the DoF.
Masaki Shigemori, Final Oral – p.38/38
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