USp Matrix Model Revisited 080304@KEK originally with A. Tokura - - PowerPoint PPT Presentation

USp Matrix Model Revisited

080304@KEK

• riginally 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

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]

II). Criteria and construction

M

BFSS by ’84 IIA IIB ← → T9 − → Ω projection

• pen added

type I ← → hetero SO(32) S

W

9

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

Action of IIB matrix model and USp matrix model

Let me tell you the definitions of the models first. SIIB(vM, Ψ) = 1 g2tr 1 4[vM, vN][vM, vN] − 1 2 ¯ ΨΓM[vM, Ψ]

• Ψ : Maj-Weyl ,

0 ≤ M, N ≤ 9

• bjects 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

• d2θW αW α + h.c. + 4
• d2θd2¯

θΦ†

Ie2V ΦI

• + 1

g2

• 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

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

U(2k) adj ր USp adj (= sym) ց USp asym U(2k) adj ր SO adj (= asym) ց SO sym F = I −I

• F =

I I

• analog of Ω

The projector ˆ ρ∓• = 1

2(• ∓ 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)j

i =

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

Definition of the model

V = ˆ ρ−V , Φ1 = ˆ ρ−Φ1, ΦI = ˆ ρ+ΦI, I = 2, 3 adj adj asym SUSp = 1 4g2tr

• d2θW αWα + 4
• d2θd2¯

θΦ†

Ie2V ΦIe−2V

• + 1

g2

nf

• f=1
• d2θd2¯

θQ∗

(f)e2V Q(f) + ˜

Q(f)e−2V ˜ Q∗

(f)

• + 1

g2

• d2θW(θ) + h.c.
• W(θ) =

√ 2tr(Φ1[Φ2, Φ3]) + nf

f=1(m(f) ˜

Q(f)Q(f) + √ 2 ˜ Q(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(ˆ ρb±vm, ˆ 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.

Why transparency

• Why matrices are strings
• Why F ∼ Ω
• USp

not SO

• Why fundamentals needed
• nf =?
• m(f) =?

• 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?

Projector and susy

how to have both consistently:

• susy transf.

             δ(1)vM = i¯ ǫΓMΨ δ(1)Ψ = i

2[vM, vN]ΓMNǫ

δ(2)vM = 0 δ(2)Ψ = ξ 16 + 16 IIB case

• We have examined the conditions:
• [ˆ

ρb∓, δ(1)(2)]vM = 0 [ˆ ρb∓, δ(1)(2)]vM = 0 (∗) To be more explicit, let ˆ ρ(M)

b∓ = Θ(M ∈ M−)ˆ

ρ− + Θ(M ∈ M+)ˆ ρ+ ˆ ρ(A)

f∓ = Θ(A ∈ A−)ˆ

ρ− + Θ(A ∈ A+)ˆ ρ+ M− ∪ M+ = {{0, 1, 2, · · · , 9}}, M− ∩ M+ = ∅, etc (∗) yield eqs. w.r.t. ǫ, ξ, M−, M+, A−, A+ demand 8 + 8 susy

• Solutions:
• ur cases

6 adj + 4asym                    ˆ ρb∓ = diag(−, −, −, −, −, +, +, −, +, +) ˆ ρf∓ = ˆ ρ−I(4) ⊗        I(2) I(2)        + ˆ ρ+I(4) ⊗        I(2) I(2)        M, hetero                    ˆ ρb∓ = diag(+, +, +, +, −, +, +, −, +, +) ˆ ρf∓ = ˆ ρ−I(4) ⊗        I(2) I(2)        + ˆ ρ+I(4) ⊗        I(2) I(2)       

• S. Rey, D. Lowe, ...

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 ψ1      , +1,      ψ2 ψ3      , −1, fund, −1 nonabelian anomaly ∝ tradjF 4 − trasymF 4 − nftrF 4 = (16 − nf)trF 4 ∴ nf = 16

• should be interpreted as a force balance.

The cases nf = 16 and IIB would imply an existence of residual interactions ⇒ gauge sym. breaking ??

Closed and open loops

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

⇒ fundamental rep. (Q, ˜ Q, ψQ, ψ ˜

Q), #(fund) = nf

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

Q(f) =

• Q(f)

F −1 ˜ Q(f−nf) , ψQ(f) =    ψQ(f) F −1ψ ˜

Q(f−nf )

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

trelM Φ[pM

• , η; n1, n2]

≡ tr← − Π n1

n=n0 exp(−ipM n vM − i¯

ηnΨ) = Φ[∓pM

• , η; n0, n1]

=

i.e. nonorientable · Λ · Π(f) ≡ (ξQ(f) + F −1ξ∗Q∗

(f)) + (θψQ(f) + F −1¯

θψ∗

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 ’

• riginal(worldsheet)

C-P factor Lie algebra − : so(2nf) ⇔ usp(2k) ⇐ our choice + : usp(2nf) ⇔ so(2k)

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

M or Ψr

0 =

• dµ

∂ ∂Z(f)i U[(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

• 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 ∂

∂Xr{tr(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: Ar

M or Ψr α

= + + 1 2

• +
• +

1 2

• +
• +

1 2

• +

2k2±k

r=1

(T r) j

i (T r) l k = 1 2(δ l i δj k ∓ F −1 ik F lj)

2k2±k

r=1

(T r) j

i (T r) l k •= 1 2(• ∓ F −1•tF) = ˆ

ρ∓ I) ⇒ = (1) kinetic term 1 g2δXΦ[(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

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

t x quantum mechanics parameter d.v. QFT parameter parameter reduced matrix model d.v. d.v.

• Rather than ln Zeff[x(i)

M ], we measure

Σone-particle projectorΓ for a given path Γ in x(k)

M

• 1P-projector = nonabelian Berry phase

Sfermion = SMM + Sgf + SYukawa

=

• αl
• l

λl¯ ξαl

l ξαl l

← fermionic eigenmodes ր տ degeneracy l-th eigen fn. 1-P-P ˆ Plαα′ ≡ ¯ ξα

l |ΩΩ|ξα l ,

ξα

l = A bAψα lA

b,¯ b ∼ ψ, ¯ ψ, ψQ, ψQ∗ One way to introduce ˆ Pαα′

l

in reduced matrix model is through infinite temperature (or short time) limit of the corresponding matrix quantum mechanics (v0 → i d

dt)

ˆ Pαα′

l

Γ = lim

β→0 trfermion

• (−)Fe−i

R β

0 dβ′H(β′) ˆ

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

ˆ Pαα

l

Γ I+ : subset over all + eigenvalues ??

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, 7 xn → 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 =1 2

• a<b
• LI(xa

M, xb M) + LI(−xa M, −xb M) + LII(xa M, xb M) + LII(−xa M, −xb M)

• + 1

2

• a
• LIII(xa

µ − xa+k µ

) + LIII(−xa

µ + xa+k µ

)

• +
• a,∓,f

LIII(xa

µ ∓ m(f)δµ,4)

Computation of NAB

LIII ⇒ H = 1 g2

• µ=1,2,3,4,7

xµγµ eigen fn. ψα = 1

NP+eα,

α = 1, 4 P± = 1

2(14 ± xν |x|γν),

yν = xν

|x|

iA = e1 e4

• M(e1 e4),

M = 1 N P †

+d 1

N P+ yν: parameterizes S4. Further by a stereographic projection yi =

2zi 1+z2,

i = 1, 2, 4, 7, y3 = 1−z2

1+z2

S4 R

4

zi(all read) parameterize R4 A = 1 2 1 1 + 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

• LI,II

⇒ H = 1

g2

• i=1,··· ,4 Γixi

similar to LIII ↑ 16 dim. yi S8 zi R8 A = 1 2 1 1 + z2(zidzj − zjdzi)Σij point-like SU(8) monopole on R9 Π7(SU(8)) = Z k = 1 This eigen bundle

• mathematically generalizable to Rm+1,

m = 4 m = 8 Πm−1(SU(2

m 2 −1))

Spacetime picture emerging from our computation

back to

• l∈I+
• α

ˆ Pαα

l

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 ⇒ xa

M − xb M

xa

M − ρ(xb M)

LIII ⇒ 2xµ xa

µ ± 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

• f Yang monopoles.

Discussion

• have shown

matrices 8 + 8 susy ⇒ asymmetry

• f 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 ˆ

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 ˆ P(1)

αα′ → 1 = i ˆ

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

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

• f 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

??