Derivation of theIntriligator-Leigh-Seiberg
linearity principle fromDijkgraaf-Vafa
Yuji Tachikawa(Univ. of Tokyo, Hongo)

December 20, 2002
based on hep-th/0211274

1.
2.
3.
4.
5.

Introduction
Brief review of ILS
Brief review of DGLVZILS from DVConclusions

1. Introduction to DV
d = 4, N = 1 DUALITYOld
SYM theories⇐⇒
matrix models• Gauge Theory : Nc Finite and Fixed
Φ: adjoint, Nc × Nc
S : gaugino condensate tr W αWα
Wα: (λα (gaugino), Fµν + (field strength))• Matrix Model : Planar Limit
Φ: M × M matrix,
S : ’t Hooft coupling M → ∞ with S fixedGaugeMatrix

Bare superpotential
Wtree(Φ)⇓Eff. superpotential
Weff(S)

ActionWtree(Φ)⇓Free energy
F (S)• The Correspondence :Weff(S) = Nc

∂F∂S
conjectured to be EXACT

First Indication — Factorization? Matrix Model Side
Example:·µ¶¸ZM 11dΦ exp −m tr Φ2 + g tr Φ3S 23• (Propagators)−1 and couplings ∝ M/SE+h= |{z}χ•
V −|{z}|{z}|{z}
vertices

propagators

diagram ∝ M hµindex loops⇓¶V −EM
S

Euler#= M χS V −E;

M → ∞, planars dominate,
correlation functions factorize:
in hABi,hAihBi

h=5;
M 4S 2hABiconn.

h=3;
M 2S 2(E = 5, V = 4)

?Gauge theory side
Chiral VEVs factorize:hΦ1(x)Φ2(y)i = hΦ1(x)ihΦ2(y)i
Because• ∂µΦ1(x) = {D̄, [D, Φ1(x)]} is D̄ -exact.
• Hence,∂µhΦ1(x)Φ2(y)i = h{D̄, [D, Φ1(x)]}Φ2(y)i= h[D, Φ1(x)]{D̄, Φ2(y)}i=0• Next, Take |x − y| → ∞• Finally use the Cluster Property.

Stringy ideas behind the proposal
DV arrived this proposal
by chasing the following dualities:

N = 1 SYM∩
D5 wrapped on!!!
←→matrix modelkB brane on

S 2 in CY

twist

S 2 in CY=⇒largeN
dual lNl largedual

IIB on CY

B model on CY

with flux in S 3

twist

=⇒

with flux in S 3

Checking the proposal

Wtree = 0

Known Exact Results(ADS superpotential,SW curve...)⇓Use the (conjectured)
ILS linearity!
(more on this in section 2.)⇓Wtree 6= 0

Weff(S)⇑Use Matrix Model!

⇑Wtree(Φ)

Field-theoretic derivations
There are two:

• Dijkgraaf-Grisaru-Lam-Vafa-Zanon
Perturbation in an external gaugino(more on this in section 3.)• Cachazo-Douglas-Seiberg-Witten ,Seiberg
Factorization and generalized anomaly⇓
We can now use DV to study SYM!
An example:
the ILS linearity principle from DV

2. Brief Review of ILS
From the bare lagrangianZ
Lbare = L0 + d2θWtree(Φ) + c.c.,Pwhere Wtree =giOi⇓
To the low energy
superpotentialZd2θ Weff(Φ)

holomorphy
Coupling constants gi as the vevs of chiral superfields;
Weff(Φ) depends only on gi , not on gi.
symmetry• Many classical symmetries on chiral superfields and coupling constants.

• Even anomalous symmetries are usefulonce one assigns charges to the dynamical scale Λ appropriately.

classical and perturbative limit• In the classical limit Λ → 0 , Weff(Φ)
must approach Wtree(Φ).
• If one takes some coupling gi → 0 ,Weff should smoothly become Wtree.
⇓Weff does not contain negative powersof Λ and gi.
Strong constraint for in Weff(Φ) ,fixing completely in some simple cases .
Example.
SU (Nc) SYM with Nf < Nc pairs of quarks
Qi and Q̃i.
• Gauge invariants are
Tij = QiQ̃j .
• Wtree = mij Tij as the bare superpotential• SU (Nf ) × SU (Nf ) flavor symmetry onquarks and antiquarks.

• R-charges:
θ Q Q̃ m Wα
R-charge 1 0 0 2
1• The anomaly fromq → e−iφq,

λ → e+iφλ,leads to:

θ → θ + 2(Nc − Nf )φ.
BecauseΛ

3Nc−Nf3Nc−Nfexp
∼ Λ0Ã

8π 2

− 2 + iθg0!

,

the R-charge of Λ3Nc−Nf is 2(Nc − Nf ) .

⇓Weff = cÃ

Λ

3Nc−Nfdet Tij!1/(Nc−Nf )+ mij Tij ,| {z }%

the coeff. is fixed by m → 0 limit.

the linearity principle• The result can be summarized as
Weff =Wn.p.
+Wtree| {z }
non-perturbative,
independent ofcoupling constants• Various other models have this form.
ILS proposed this linearity as an organizing principle.
• However, symmetry arguments are notalways strong enough.
Another Example• Consider N = 1 SU (Nc) SYM with oneadjoint Φ.

• The R-charge of Λ is zero .
• Introduce the bare superpotential
m tr Φ2/2.
• Symmetry alone cannot exclude the corrections of the form mΛ2.

3. Brief Review of DGLVZHow to derive Dijkgraaf-Vafa?
⇓Directly equate Feynman rules in thegauge theory and in the matrix model .
Strategy
1. Introduce an external gauge superfield. In particular, introduce a spacetime constant gaugino.
2. Perturbatively integrate out the matter superfields. Winst(S)3. Make the gauge superfield dynamical.
Weff(S) =W (S)+Winst(S)| VY{z }dynamical effectfrom the gauge superfieldFeynman rules for step 2=Feynman rules for Matrix Modelthe propagatorm̄

wherehΦΦi = 2p + mm̄ + W απαp : bosonic momentum,
W α : field strength superfield,
πα : fermionic momentum.
Integration of loop momenta:!
ÃZZ
l4Y
d pa
2π
da4(2π)a=1=⇒ Every l-loop diagram carries a factorof Wα2l .

=⇒ if Wα = 0, no correction to F terms.
This is the Perturbative
Non-Renormalization Theorem!
What if Wα 6= 0?

Origin of Planarity• tr Wαm is D̄ -exact for m ≥ 3, i.e. ∼ zeroin the F terms. (CDSW)• Therefore, Weff is a function of32πS = tr W αWα.

;
To give Wα2l requires at leastloops.

l index• # of index loops h satisfies
h = l + 1 − 2 × genus.

l=V3

h=4

l=
W3

h=2

Only GENUS 0 diagrams contribute!

• Further tricks reveal the resulting Feynman rules are the same as those of thematrix models in the planar limit.

Making the vector superfield dynamicalThis introduces the Veneziano-Yankielowicz
termS
WVY = NcS(1 − log 3 ).
Λ(Λ: the dynamical scale of SQCD)
There are arguments against further corrections.

The total effective superpotential:
Weff(S) = WVY(S) + Winst(S).

• hSi is determined by the extremalization∂Weff/∂S = 0.
P
• For Wtree =giOi, the vevs are
hOii =∂

Weff(hSi, gi)∂gi¯¯¯∂Weff ¯¯∂Weff ¯¯∂hSi ∂Weff ¯¯=.
=+¯¯¯∂gi S∂gi
∂S gi
∂gi S

4. ILS from DGLVZ• Introduce Wtree =PgiOi.

• From DV, calculate Weff and hWtreei in
Weff = Wn.p. + hWtreei.
• Is Wn.p. independent of gi as a functionof Λ , gi and hOii ? This is the linearity
principle.
Note:= WVY+Winst−hWtreei=∂W∂S = 0Weff⇓hWeffiWn.p.

Calculate separately

Restriction on the matter rep.
We impose:• Vector-like , i.e. gauge-invariant massterms can be given to all fields.

• No U (1) factor left in low energy.
• Each gauge invariants Oi is a polynomial of Fa’s. e.g.
Fi = tr Φi, (i = 1, . . . , Nc)
O = FiFj Fk · · · = (tr Φi)(tr Φj ),etc.
• Either the R-charge of Λ vanishes, or
satisfies the Restriction (♣) :no dynamical constraints
among hFai
P (hF1i, hF2i, · · · ) = 0.

on constraints...
(♣) may sounds a serious restriction, but
it is NOT!
Example: SU (N c) SYM with Nf pairs offundamentals Qi, Q̃i.
Tij = Qa
Gauge invariants:i Q̃ja,b Qc · · ·Bijkl... = ²abc...Qa
Qi j kb Q̃c · · ·Q̃
B̃ijkl... = ²abc...Q̃ai j k

Nc > Nf : No constraints.
Nc = Nf : One constraint:classicalquantum⇒

det Tij = B B̃det Tij − B B̃ = Λ2Nf

However, R-charge of Λ is zero.
; need not impose (♣)!

Nc < N f :classicalquantum⇒

det Tij = B B̃Dissolves!

Satisfies (♣)!

• The SO , Sp cases are quite similar.
• (Non-)existence of constraints can bechecked by DV. (more about this later...)

How to prove the ILS from the DV?
Strategy• Step 1. Shows Wn.p. = (Nc − Nf )hSi.
• Step 2. Shows, if there’s no dynamical constraint (♣),

hSi = f (hFii, Λ, ×)gi
Do not depend explicitly on gi.

;• If the R-charge of Λ vanishes, Wn.p. = 0
from Step 1.
• If it does not vanish, by the restriction(♣) and Step 2, Wn.p. ∝ S is independentof the coupling constants.
This is the linearity principle.

Q.E.D.

Step 1.
We show Wn.p. = (Nc − Nf )S .

•−V|{z}#(vertices)E|{z}+#(propagators)

L|{z}=1#(loops)• V , E and L can be counted bygj ∂/∂gj ,µ1−S∂∂S¶− mi∂/∂mi,S∂/∂S⇓Ã!
XX
∂
∂+gj
D=miD.
∂mi∂gj{z}|
the contrib. of D to hWtreei• One loop diagrams ∝ Nf S log m⇒
Another contrib. Nf S to hWtreei⇒¶µ∂ X
DhWtreei = Nf S + 1 − S
∂SX
∂Weff
∂
• Because=(WVY +D) = 0,∂S
∂SX
∂WVYhWtreei =D+S.
∂S• Hence Wn.p. = (Nc − Nf )S.

Note Nc − Nf is half the R-charge of ΛβStep 2.
We show S is independent of gi when
expressed as a function of Λ and hOii.
⇑The same as inquiring whether S does
not change when gi’s are varied as longashO i remain fixed.
| i{z}
by the factorization,
the same as fixing hFai
Take O1 = F1, O2 = F2, . . . , Or = Fr , andOi (i > r ) be polnomials<DERIVATION>• From δ(∂Weff/∂S) = 0,−∂ 2Weff∂S∂SδhSi =i• Hence δhFj i =Gij =∂ 2Weff∂gi∂gjXP−δgi∂ 2Weff∂gi∂S.

i δgiGij where∂ 2Weff ∂ 2Weff . ∂ 2Weff∂gi∂S ∂S∂gj ∂S∂S
∂hOii ∂ 2Weff . ∂ 2Weff∂hOii−=.
∂gj∂S ∂S∂gj ∂S∂S• From factorization,
Gij = h∂Oi
∂FaiGaj .

HenceδhFj i =X

a|{z}

µδgih

∂Oi
∂Fa

¶

i Gaj

This sum only runs over 1, 2, . . . , r• Similarly, one obtains
µ¶
2X
∂Oi
∂hFai
∂ WeffδhSi =δgihi−∂S∂S∂F∂Saa• The restriction (♣) ensures that therank of Gij is maximal.

Thus, δhFii = 0 implies δhSi = 0.

N.B.
When the rank of Gij is not maximal, Gij
contains the info of the dynamical constraints. Further, Gij can be computedperturbatively!
⇓The dynamical constraints, if any, can bereadily studied in this approach.

5. Conclusions
Summary• We derived the ILS linearity principle in
the framework of Dijkgraaf-Vafa

Future work
Directly related to this work:• Lifting the restriction (♣)
More generally, interesting directions aroundthe DV framework are:
• Direct topological twisting of N = 1SYM theories• Extending to matter superfields not ina vector-like representation.