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

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