USp Matrix Model Revisited

080304@KEK

originally with A. Tokura (’97)

later with A. Tsuchiya, T. Matsuo, B. Chen, H. Kihara (Osaka U.)

in recent years with H. Kihara (KIAS), R. Yoshioka (OCU)

I). Introduction

• ten years of developments in reduced matrix models

• review not as a reflection but as an opportunity for a better perspective

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Contents

I). Introduction

II). Criteria and construction [I-Tok]

III). Semi-uniqueness and loop variables [I-Tok, I-Tsuch]

IV). S-D equation [I-Tsuch]

V). Spacetime fluctuations represented by nonabelian Berry phase [I-Mats, CIK]

VI). Attempts in recent years [IY, IKY]

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II). Criteria and construction

MBFSS

by ’84

IIA IIB

←→T9

−→Ω projection

open added

type I ←→ hetero SO(32)S l W9

hetero E8×E8

16 + 16 supercharges

closed, orientable

8 + 8 supercharges

closed + open, nonorientable

large k

reduced

model

⇓ ⇓IIB matrix model

IKKT

USp matrix model

IT

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Action of IIB matrix model and USp matrix model

Let me tell you the definitions of the models first.

SIIB(vM , Ψ) =1g2

tr(

14[vM , vN ][vM , vN ]− 1

2ΨΓM [vM , Ψ]

)

Ψ : Maj-Weyl , 0 ≤ M, N ≤ 9objects with : U(2k) matrices

We also write as vm, m = 0, · · · , 3.

ΦI = 1√2(v3+I + iv6+I), I = 1, 2, 3

Ψ = (λ, 0, ψ1, 0, ψ2, 0, ψ3, 0, 0, λ, 0, ψ1, 0, ψ2, 0, ψ3)t

using 4d superfield

∴ SIIB =1

4g2tr

(∫d2θWαWα + h.c. + 4

∫d2θd2θΦ†Ie

2V ΦI

)+

1g2

(∫d2θW0 + h.c.

)

W0 =√

2tr(Φ1[Φ2, Φ3]])Requirements for the model descending from perturbative type I superstrings

i) closed (projected)

ii) nonorientable

iii) 8 + 8 susy

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• To find the matrix counterpart of the Ω projection, recall for both USp and SO.

U(2k) adj USp adj (= sym)

USp asym

U(2k) adj SO adj (= asym)

SO sym

F =(

0 I

−I 0

)

F =(

0 I

I 0

)analog of Ω

The projector ρ∓• = 12(• ∓ F−1 •t F )

In fact

XtF + FX = 0 for X ∈ usp(2k) Lie alg. ∴ X =(

M N

N∗ −M∗

)

Y tF−FY = 0 for Y ∈ antisym ∴ Y ≡ (TF )ji =

(A B

C∗ At

), Bt = −B, Ct = −C

We found out

planar diagram analysis

⇒ Chan-Paton factor of open loop all lead to usp

consistency with wv field theory

• Need to add open string degrees of freedom,

keeping 8 + 8 susy ⇒ fundamental rep. (Q•, Q•, ψQ•, ψ•Q), #(fund) = nf

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Definition of the model

V = ρ−V , Φ1 = ρ−Φ1, ΦI = ρ+ΦI, I = 2, 3

adj adj asym

SUSp =1

4g2tr

(∫d2θWαWα + 4

∫d2θd2θΦ†Ie

2V ΦIe−2V

)

+1g2

nf∑

f=1

(∫d2θd2θQ∗

(f)e2V Q(f) + Q(f)e

−2V Q∗(f)

)+

1g2

(∫d2θW (θ) + h.c.

)

W (θ) =√

2tr(Φ1[Φ2, Φ3]) +∑nf

f=1(m(f)Q(f)Q(f) +√

2Q(f)Φ1Q(f))↑

again using 4d superfield notation

also can write

SUSp = S0 + ∆S S0 : with Q(f), Q(f) set to zero

S0 = SIIB(ρb±vm, ρf±ΨA) specific projection

Parameters g2, (2k), m(f), n(f) = 16(by 6d gauge anomaly cancel.)

k →∞, scaling : g2(2k)# = const. # is difficult to determine.

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Why transparency

• Why matrices are strings

• Why F ∼ Ω

• USp not SO

• Why fundamentals needed

• nf =?

• m(f) =?

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• III). Semi-uniqueness and loop variables

• [susy, projector] = 0 in the USp case

• SIIB ; possesses 16 + 16 susy

• SUSp ; need to check ∃ 8 + 8 susy

; Is ρb∓ and ρf∓ unique?

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Projector and susyhow to have both consistently:

• susy transf.

δ(1)vM = iεΓMΨ

δ(1)Ψ = i2[vM , vN ]ΓMNε

δ(2)vM = 0

δ(2)Ψ = ξ

16 + 16 IIB case

• We have examined the conditions:[ρb∓, δ(1)(2)]vM = 0

[ρb∓, δ(1)(2)]vM = 0(∗)

To be more explicit, let

ρ(M)b∓ = Θ(M ∈M−)ρ− + Θ(M ∈M+)ρ+

ρ(A)f∓ = Θ(A ∈ A−)ρ− + Θ(A ∈ A+)ρ+

M− ∪M+ = 0, 1, 2, · · · , 9, M− ∩M+ = ∅, etc

(∗) yield eqs. w.r.t. ε, ξ, M−, M+, A−, A+

demand 8 + 8 susy

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• Solutions:

our cases 6 adj + 4asym

ρb∓ = diag(−,−,−,−,−, +, +,−, +, +)

ρf∓ = ρ−I(4) ⊗

I(2)

0

I(2)

0

+ ρ+I(4) ⊗

0

I(2)

0

I(2)

M, hetero

ρb∓ = diag(+, +, +, +,−, +, +,−, +, +)

ρf∓ = ρ−I(4) ⊗

I(2)

0

0

I(2)

+ ρ+I(4) ⊗

0

I(2)

I(2)

0

S. Rey, D. Lowe, ...

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Gauge anomalies cancellation

still somewhat mysterious

• take matrix T dual ala W. Taylor albeit being against our spirit

⇒ (zero volume limit) of 6-d. wv. gauge theory

• chirality Γ6 = Γ0Γ1Γ2Γ3Γ4Γ7

λ

0ψ1

0

, +1,

ψ2

0ψ3

0

,−1, fund,−1

nonabelian anomaly ∝ tradjF4 − trasymF 4 − nftrF 4 = (16− nf)trF 4

∴ nf = 16

• should be interpreted as a force balance.

The cases nf 6= 16 and IIB would imply an existence of residual interactions

⇒ gauge sym. breaking ??

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Closed and open loops

• Recall we have added the open string deg. of freedom,

⇒ fundamental rep. (Q, Q, ψQ, ψQ), #(fund) = nf

• To make flavor symmetry (≈ gauge sym. of strings) manifest,

Q(f) =

Q(f)

F−1Q(f−nf)

, ψQ(f)=

ψQ(f)

F−1ψQ(f−nf )

• basic operators (observables): cf. one-matrix model trelM

Φ[pM• , η; n1, n2] ≡ tr

←−Πn1

n=n0exp(−ipM

n vM − iηnΨ)

= Φ[∓pM• , η; n0, n1]

=

i.e. nonorientable

· Λ ·Π(f) ≡ (ξQ(f) + F−1ξ∗Q∗(f)) + (θψQ(f)

+ F−1θψ∗Q(f))

Ψf ′f [pM• , η•; n0, n1; Λ′, Λ] ≡ Λ′ ·Π(f ′)FU [· · · ]Λ ·Π(f)

= ∓Ψff ′[∓pM• ,∓η•; n0, n1; Λ′, Λ]

= -f

f ’f

f ’

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original(worldsheet)

C-P factor Lie algebra

− : so(2nf) ⇔ usp(2k) ⇐ our choice

+ : usp(2nf) ⇔ so(2k)

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IV). S-D equation

Schwinger-Dyson eq.

as before

Φ[(i)] ; i-th closed loop

Ψf ′f [(i)] ; i-th open loop f

f ’

Consider

0 =∫

dµ∂

∂Xrtr(U([1])T rU [(1)])Φ[(2)] · · ·Φ[(N)]Ψ[(1)] · · ·Φ[(L)]e−S

0 =∫

dµ∂

∂XrΛ(1)′·Π

f (1)′FU [(1)]T rU [(1)]Λ(1)·Πf (1)Φ[(1)] · · ·Φ[(N)]Ψ[(2)] · · ·Φ[(L)]e−S

Xr = vrM or Ψr

0 =∫

dµ∂

∂Z(f)iU [(1)]Λ(1) ·Πf (1)Φ[(1)] · · ·Φ[(N)]Ψ[(2)] · · ·Φ[(L)]e−S

Z(f)i = Q(f)i or ψQ(f)i

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• We have shown that eqs. are closed w.r.t. the loops

eq. of motion acting on the loop

→ deformation of the loop

complete set of nonorientable interactions among loops consistent with two kinds of

elementary local moves.

0 =∫

dµ∂

∂Xrtr(U [p(1)

• , η(1)• ; n(1)

2 , n(1)1 + 1]T rU [p(1)

• , η(1)• ; n(1)

1 , n(1)0 ])

Φ[(2)] · · ·Φ[(N)]Ψ[(1)] · · ·Φ[(L)]e−S

T r: generator of usp(2k), Xr: ArM or Ψr

α

0 = +

+12

(+

)

+12

(+

)

+12

(+

)

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∑2k2±kr=1 (T r) j

i (T r) lk = 1

2(δl

i δjk ∓ F−1

ik F lj)∑2k2±k

r=1 (T r) ji (T r) l

k •= 12(• ∓ F−1•tF ) = ρ∓

I) ⇒

0 = (1) kinetic term1g2〈δXΦ[(1) : Xr]Φ[(2)] · · ·Φ[(N)]Ψ[(1)] · · · 〉

+(2) splitting and twisting term #(loops) increases by 1

+(3) joining with a closed string #(loops) decreases by 1

+(4) joining with an open string #(loops) decreases by 1

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V). Spacetime fluctuations represented by nonabelian Berry phase

• variables

vM = u(M)

x(1)M

. . .

x(k)M

∓x(1)M

. . .

∓x(k)M

u(M)−1

≡ u(M)XMu(M)−1 ↑ spacetime pts.

↑

→Ψ

→ψf

→Qf

integrate → s to give

dynamics to the spacetime pts.

For simplicity, set u(M) = 1, Qf = 0

note: spacetime pts are dynamical variables

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t x

quantum mechanics parameter d.v.

QFT parameter parameter

reduced matrix model d.v. d.v.

• Rather than lnZeff[x(i)M ], we measure

Σ〈〈one-particle projector〉〉Γ for a given path Γ in x(k)M

• 〈〈1P-projector〉〉 = nonabelian Berry phase

Sfermion = SMM + Sgf + SYukawa

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=∑αl

∑

l

λlξαll ξ

αll ← fermionic eigenmodes

degeneracy l-th eigen fn.

1-P-P Plαα′ ≡ ξα

l |Ω〉〈Ω|ξαl , ξα

l =∑

A bAψαlA

b, b ∼ ψ, ψ, ψQ, ψQ∗

One way to introduce 〈〈Pαα′l 〉〉 in reduced matrix model is through infinite temperature

(or short time) limit of the corresponding matrix quantum mechanics (v0 → i ddt)

〈〈Pαα′l 〉〉Γ = lim

β→0tr

fermion

[(−)Fe−i

R β0 dβ′H(β′)Pαα′

l

]

Toss this expression to that of the first quantized Q. mechanics

= P exp[−i

∮ΓAl(xM)]αα′

Aαα′l (xM) = −i

∑A ψα†

lAdψα′lA nonabelian ← ∃degeneracy in spinor space

We will eventually consider

〈〈∑

l∈I+

∑α

Pααl 〉〉Γ

I+ : subset over all + eigenvalues ??

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Decomposition of Sfermi

XM = diag(x(1)M , · · · , x

(k)M , ρ(x(1)

M ), · · · , ρ(x(k)M ))

ρ : xµ → −xµ, µ = 0, 1, 2, 3, 4, 7xn → xn, n = 5, 6, 8, 9

• found written in terms of the three types

LI(Λ, Φ; xM , yM) = 2(Λ + Φ)ΓM(xM − yM)(Λ + Φ)

LII(Λ, Φ; xM , yM) = 2(Λ + Φ)ΓM(xM − ρ(yM))(Λ + Φ)

LIII(Λ;xµ) = ΛΓµxµΛ

Ψ =

• • × • •• • • × •• • • • ×× • • • •• × • • •• • × • •

• to LI

• to LII

× to LIII

Sfermi =12

∑

a<b

(LI(xaM , xb

M) + LI(−xaM ,−xb

M) + LII(xaM , xb

M) + LII(−xaM ,−xb

M))

+12

∑a

(LIII(xaµ − xa+k

µ ) + LIII(−xaµ + xa+k

µ ))

+∑

a,∓,f

LIII(xaµ ∓m(f)δµ,4)

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Computation of NAB

LIII ⇒ H =1g2

∑µ=1,2,3,4,7

xµγµ

eigen fn. ψα = 1NP+eα, α = 1, 4

P± = 12(14 ± xν

|x|γν), yν = xν

|x|

iA =(e1

e4

)M(e1 e4), M =

1N

P †+d1N

P+

yν: parameterizes S4. Further by a stereographic projection

yi = 2zi

1+z2, i = 1, 2, 4, 7, y3 = 1−z2

1+z2

S4

R4

zi(all read) parameterize R4

A =12

11 + z2

(zidzj − zjdzi)σij, σij =i

2E [γi, γj]Et

ASD instanton (BPST k = 1)

For us, better to view it as a point-like SU(2) monopole (Yang monopole) on R5.

Π3(SU(2)) = Z

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•LI,II ⇒ H = 1g2

∑i=1,··· ,4 Γixi similar to LIII

↑16 dim.

yi S8

zi R8

A =12

11 + z2

(zidzj − zjdzi)Σij

point-like SU(8) monopole on R9 Π7(SU(8)) = Z k = 1This eigen bundle

• mathematically generalizable to Rm+1,m = 4m = 8

Πm−1(SU(2m2 −1))

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Spacetime picture emerging from our computation

back to

〈〈∑

l∈I+

∑α

Pααl 〉〉Γ =

∑

l∈I+

trlP exp[−i

∮ΓAl(∗)]

↑arguments bellow

LI,II ⇒ SU(8) monopole point-like at entire space shared by IIB matrix model

LIII ⇒ SU(2) Yang monopole xn×4 dimensionally extended object string soliton

unique to USp matrix model

the arguments of Al

LI,II ⇒ xaM − xb

M xaM − ρ(xb

M)

LIII ⇒2xµ

xaµ ±m(f)δµ,4

⇒ ∃singular plates

These colliding singularities may act as dominant factors in the complete (bosonic)

functional integral.

It is tempting to conclude that the four dimensional structure is formed by a collection

of Yang monopoles.

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Discussion• have shown

matrices

8 + 8 susy⇒

asymmetry of the four directions

from the rest in the fluctuation

spectrum• a collection of Yang monopoles may be the seed of our spacetime structure

recent

• Considering both positive and negative eigenvalues in spinor space will lead to Yang

monoploes and anti-Yang monoploes and their octonionic generalizations.

• 〈〈∑l

∑α Pαα′

l 〉〉Γ depends on the loop Γ in xM space and would like to regard this as

a definition of matrix boundary state!

the analogy of 〈B; Γ|0〉 in 1st quantized strings.

replacing P(1)αα′ → 1 =

∑i P(i)

αα′ will lead to a complete analysis of the determinant.

• NAB and nonabelian monopoles have appeared in some recent works of SUGRA;

· G. W. Gibbons and T. K. Townsend

“Self-gravitating Yang Monopoles in all Dimensions”; hep-th/0604024

· A. Belhaj, P. Diaz and A. Segui

“On the Superstring Realization of the Yang Monopole”; hep-th/0703255

· C. Pedder, J. Sonner and D. Tong

“The Berry Phase of D0-Branes”; arXiv:0801.1813

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VI). Attempts in recent years

• Assessing further the condition [susy, projectors] = 0 in the case of possessing 4 or

8 supercharges upon Z3 orbifolding + matrix orientifolding. We have enumerated all

possibilities:# = 50

H.I. and R. Yoshioka

“Matrix orientifolding and models with four or eight supercharges”

Phys. Rev. D72 (2005) 126005; hep-th/0509146

(cf. Aoki, Iso and Suyama)

• Of course, a more systematic and hopefully exact integration of bosonic and fermionic

matrices are called for. We have developed a residue calculus and a diagrammatic

method for MNS integration [Moore, Nekrasov, Shatashvili, (cf. Hirano and Kato)] for the case

of dimensionally reduced d=4, N = 1 SYM with G arbitrary classical groups.

H.I., H. Kihara and R. Yoshioka

“Partition functions of reduced matrix models with classical gauge groups”

Nucl. Phys. B762 (2007) 285-300; hep-th/0609063

• We have converted USp matrix model into a QBRS-exact form. MNS integration yet

to be carried out completely.∃ cutoff dependence ↔ scaling ??

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