Geometry / Gauge Theory Duality and the Dijkgraaf–Vafa...

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


Outline

Introduction√

Dijkgraaf–Vafa conjecture

“Counterexample” [hep-th/0303104, 0304138]

String theory prescription [0311181]

Inclusion of flavors [0405101]

Future problems

Conclusion


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


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( )


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


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


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]


“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


“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 !


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


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


String theory prescriptionGeometric transition — general case

4IR

S3S3 S3


...N1 fluxes N2 fluxes KN fluxes

S1 S2 SK

...

4d theory: U(1)K theory with S1, . . . , SK


String theory prescriptionFlux superpotential

4IR

S3S3 S3


...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


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?


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


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).


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



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.


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


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


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


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



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



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


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 )


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


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


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


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)


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


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 ∞


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



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


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.


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.


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


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.


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