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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
– Typeset by FoilTEX – 1
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]
– Typeset by FoilTEX – 2
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
– Typeset by FoilTEX – 3
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
– Typeset by FoilTEX – 4
• 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
– Typeset by FoilTEX – 5
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.
– Typeset by FoilTEX – 6
Why transparency
• Why matrices are strings
• Why F ∼ Ω
• USp not SO
• Why fundamentals needed
• nf =?
• m(f) =?
– Typeset by FoilTEX – 7
• 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?
– Typeset by FoilTEX – 8
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
– Typeset by FoilTEX – 9
• 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, ...
– Typeset by FoilTEX – 10
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 ??
– Typeset by FoilTEX – 11
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 ’
– Typeset by FoilTEX – 12
original(worldsheet)
C-P factor Lie algebra
− : so(2nf) ⇔ usp(2k) ⇐ our choice
+ : usp(2nf) ⇔ so(2k)
– Typeset by FoilTEX – 13
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
– Typeset by FoilTEX – 14
• 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
(+
)
– Typeset by FoilTEX – 15
∑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
– Typeset by FoilTEX – 16
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
– Typeset by FoilTEX – 17
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
– Typeset by FoilTEX – 18
=∑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 ??
– Typeset by FoilTEX – 19
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)
– Typeset by FoilTEX – 20
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
– Typeset by FoilTEX – 21
•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))
– Typeset by FoilTEX – 22
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.
– Typeset by FoilTEX – 23
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
– Typeset by FoilTEX – 24
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 ??
– Typeset by FoilTEX – 25