Non-perturbative Study on Duality andPhase Structure in Minimal String Theory
Hirotaka Irie (YITP, Kyoto Univ.)!

March 7th, 2014
JSPS/RFBR Workshop @ Ritsumeikan Univ.
!

based on collaborations withChuan-Tsung Chan (Tunghai Univ.) and Chi-Hsien Yeh (NCTS, Taiwan)
Main Reference[CIY4], “Analytic Study for the String Theory Landscapes via Matrix Models,”
Phys.Rev. D86 (2012) 126001 [arXiv:1206.2351 [hep-th] ][CIY5], “Duality Constraints on String Theory I: spectral networks and instantons,”
arXiv:1308.6603 [hep-th][CIY6], “Duality Constraints on String Theory II: Vacuum Connection Formula and Bootstrapfor String Theory,” in progress

“Non-perturbative Study of Strings”
Non-perturbative Completionsa	  number	  of	  comple(onsF(g) '1X

g2n 2n=0Informa(on	  of	  Exact	  Theory

Fn + θ✓⇥pg exp

1h1 X

g n=0

(inst)
Fni+ ···

perturbative string theory

1) Instanton spectrum to guarantee the exact theory2) Behavior of F(g)g 2 C (Phase Structure in g)3) Physics of Dualities at a Full Non-perturbative Level
4) Capturing the Landscape of (Meta-)stable Vacua

“Non-perturbative Study of Strings”
Non-perturbative Completionsa	  number	  of	  comple(onsF(g) '1
X

n=0Informa(on	  of	  Exact	  Theory

g2n 2

Fn + θ✓⇥pg exp

1h1 X

g n=0(inst)
Fni+ ···

perturbative string theory

Additional Principles are Necessary(# several principles)
Non-perturbative Duality ← Duality Constraints [CIY5 ‘13]cf)	  also	  [Imamura-­‐san’s	  talk]Today,	  we	  discuss	  duality	  constraints	  and	  its	  beyond	  (especially	  in	  (2,3);	  (2,5)	  minimal	  strings,	  as	  examples)Plan of the talk1) What is “Duality Constraints”?
2) Stokes phenomena in Isomonodromy Systems3) Phase Structure in g, (and the landscapes)
4) Summary

Non-perturbative Duality
Two-matrix model

Z=Z

dXdY eIntegrate Y

N tr[V1 (X)+V2 (Y ) XY ]Integrate X

X-systemY-system
spectral (p-q) dualNOTE）X
Y describe
dual notspacetimes
These twoandsystems
are generally
the sameat the levelDoubleof non-perturbativecompletions
Field Theory

Spectral Curves
X-systemY-systemResolvent of X
R(x) =⌧

Resolvent of Y

11tr
N x XeR(y)
=

⌧

1
1
tr
N y Y

They are given by the same algebraic equation:
F (x, R) = f x, R

That is,e =f Re
Fe (y, R)0V1 (x)

V20 (y), y = 0

We can identify

f (x, y) = 0
if y is spacetime

if x is spacetime

X-system=0

p-q dual

Y-systemFrom spectral curve to full amplitudes

f (x, y) = 0

if x is spacetime

X-system

Isomonodromy descriptions@g= Q(t; x)@x@g= B(t; x)@t

1. . det yQ(t; x)

pp matrix-valued

rational function in x

(y has p branches)/ f (x, y) = 0g!0

2. isomonodromic τ = MM partition function
Z(t)⌧X (t)(But, we have to be careful!)Sequence of models
Iso-Monodromy Systems (IMS) are given by spectral curves

We focus on the casef (x, y) = 0

a polynomial in (x,y)(p,q) minimal string theory
pf (x, y) = y + a1 y(p,q)=(2,3)(p,q)=(2,5)

Other IMSʼs:p 1q+ · · · + b0 x + b1 xq 1+ ···(3,2): Pure-gravityPainleve I (PI )(5,2): Yang-Lee edge, …

(p,q) MST = a single (but higher) singularity at x,y=Multiple singularities → [Marshakov-san,Gavrylenko-san]• Multiple singularities + qIMS → [Yamada-san]• System of Toda equations → [Ito-san,Locke-san]•Universal framework for Matrix models/Gauge theory

Duality in Isomonodromy Systemsf (x, y) = 0
if y is spacetime

if x is spacetime

X-systemY-system@g= Q(t; x)@x@g= B(t; x)@t@ee y) eg= Q(t;@y@ee y) eg= B(t;@tdet y

Q(t, x)/ f (x, y) = 0g!0

p-q dual

det xe y)Q(t,/ f (x, y) = 0g!0

p-q duality and T-duality: References• [Fukuma-Kawai-Nakayama ʼ92]
[P,Q]=1 <=> [Q,-P]=1• [Kharchev-Marshakov ʼ92]
Kontsevich MM: from (p,1) to (p,q)• [Bertola-Eynard-Harnad ʼ01-04]
Two-matrix models, saddle point analysis• [Eynard-Orantin ʼ07 - …]
x-y symmetry: All-order perturbative check• [Asatani-Kuroki-Okawa-Sugino-Yoneya ʼ96]
Kramers-Wannier duality in Random Surfacs• [Kuroki-Sugino ʼ07]
D-instanton fugacity

“Equivalence” of the duality in the literature@g= Q(t; x)@x@g= B(t; x)@tp-q dual@ee y) eg= Q(t;@y@ee y) eg= B(t;@tstring equation (diﬀ. eqn for ⌧ (t) ) [Fukuma-Kawai-Nakayama ʻ92]

⇥g@x

Q(t; x), g@t⇤

B(t; x) = 0 (equivalent equations)⇥⇤e y), g@t B(t;e y) = 0, g@y Q(t;E.g) pure-gravity (p,q)=(2,3)
Both systems give(3,2)u(t) = @t2 ln ⌧ (t)PI : u00 (t) = 6(u2 + t)

However, this only guarantees the equivalence
under formal (asymptotic) solutions (i.e. perturbation theory)Non-perturbative duality v.s. perturbative equivalence00

2PI : u (t) = 6(u + t)u(t) 'pt

1
X

un t

4n518+ ✓ ⇥ ( t) ep

8 35g (5t) 4

n=0 integration constant+ ···separately described by these two systems

X-systemp-q dual⌧X (t) = ⌧Y (t)Y-system
Y-Matrix Model

X-Matrix Model⌧X (t) = Z(t)⌧X (t) = Z(t) = ⌧Y (t)solution space⌧Y (t) = Z(t)solution space

Only a part of solutions are realized by matrix models
Duality can even give a constraint on matrix models [CIY5ʼ13]

So, we will brieﬂy discuss

1) What is “solution space” in IMS
2) What is the matrix-model solutions

1) Solution Space and Stokes Phenomena
Solu%on	  spaces	  of	  String	  Equa%on	  	  	  ↔	  	  	  Stokes	  Phenomena	  of	  ΨSpectral Networks [Deift-Zhouʼ93,…](⇣+ ) =

ζ=x
K=Knbranch cuts(⇣ )Stokes jump graphh(g; ⇣) ' Ip +1X

n=1ng Z(n)i(⇣) Zcl (⇣)e1g '(⇣)(⇣ )Gn⇣ 2 Kn(⇣+ )Gn

g ! 0; fixed ⇣ 2 C \ K#K•Every	  line	  Kn	  is	  associated	  with	  a	  pxp	  matrix	  	   Gnn=1•Junc(on	  sa(sﬁes	  the	  matrix	  conserva(on	  law

Gn1 Gn2 · · · GnL = Id
If G is properly chosen, Ψ satisﬁes this expansion in ⇣ 2 C \ K

Spectral Networks in (p,q) Minimal String Theory [CIY4 ʼ12; CIY5 ʼ13]
(there are several consistency conditions)

Results(Here we only show the two cases)(p,q)=(2,3)
X-system✓ii◆(p,q)=(2,5)
X-system

✓ii

◆◆1ip

✓1µ

✓

1i1µ2

◆1i

◆

1i

1

✓µ1✓p41i(more generally)
✓◆

◆

✓ii

◆

11d'(⇣) = ± p (4⇣ 22pqpµ) ⇣ + µ

1ip✓✓p1d'(⇣) = ± p (2⇣217 p◆

1µ11+i1p
417 p◆µp2 µ⇣qpµ) ⇣ + µ
✓ii

◆p1µp

✓1i

✓1ip11◆1µ41i17 p◆1(more generally)

↵11pi

◆

µ ✓◆
1↵2◆

1✓✓µ1+p417 pµ2) “Spectral Networks” = Matrix-Model Contours [CIY4 ʻ12]
Potential in (2,9) minimal string theoryVe↵ (x)

Instantons

Condensationof other eigenvaluesxMatrix-model contours

Z(t) 'Z

1g Veff (x)dxe
CSpectral Networks(Stokes Phenom)R(x)e'(2,1) (x)xe'(1,2) (x)Spectral Networks in (p,q) Minimal String Theory [CIY4 ʼ12; CIY5 ʼ13]
(there are several consistency conditions)

Results(Here we only show the two cases)(p,q)=(3,2)
Y-system(p,q)=(5,2)
Y-system<5,3><2,1><2,5><4,3>

yc

<5,1>
<4,3><1,5><5,4>[5,3][2,1][2,5][4,3]<5,4><3,1>

<1,5>
<4,5>

<5,3><2,1><2,5><4,3>

Duality Check of (3,2)(2,3) Models [CIY6 ʻ14]

We can check the phase structure in g -> 0 (or μ -> )(µ =Y-systemt)

Potential of (2,3) Minimal StringVe↵ (x)D-instanton②③Large
inst

Small	  inst
inﬁnite # of polesµ ⇠g45Small
inst

X-system①No	  inst

pole free(Pert. Vac.)

No	  inst④(Non-Pert. Vac.)⑤

3-truncated solutions [Its-et.al.,…]Duality Check of (5,2)(2,5) Models [CIY6 ʻ14]

We can check the phase structure in g -> 0 (or μ -> )Potential of (2,5) Minimal StringY-system<5,3><2,1><2,5><4,3>

ycVe↵ (x)<5,1><4,3><1,5><5,4>[5,3][2,1][2,5][4,3]<5,4><3,1>

D-instanton

<1,5>
<4,5>

<5,3><2,1>x<2,5><4,3>pµµ1p417 pµ1+p417 pµX-systempole free(Pert. Vac.)pole freeinﬁnite # of poles

BUT New Pert. Vac)

(Non-Pert. Vac.)

2) Condition for matrix-model solutions
One-cut Boundary Condition [CIY2 ʻ11]
Resolvent⌧matrix models

11tr
N x X

There should be a single cut

So far, there is no inconsistency with this(p,q)=(2,3)

pµY-system

p-q dualpµ2X-system<5,3><2,1><2,5><4,3>

yc

<5,1>

(p,q)=(2,5)

p1<4,3><1,5><5,4>µp-q dual
p417 pµ1+p417 p[5,3][2,1][2,5][4,3]<5,4><3,1>

<1,5>

µ

<4,5>
<5,3><2,1><2,5><4,3>

BUT, a non-trivial situation happens if
Potential of (2,5) Minimal String

Potential of (2,5) Minimal StringVe↵ (x)Ve↵ (x)

xp

µ1

p
4

17 p

µ1+

p
4

x

V.S.
17 p

µp

µ1p
417 p

µ1+p
4

17 p

µ

See large instanton modes
F ' Fpert + e+ g1 SI+ ···[CIY5 ʻ13]

Large instantons → observed on both sides
Potential of (2,5) Minimal StringVe↵ (x)<5,3><2,1><5,1><2,5><4,3><42>

cD-instanton<4,3><1,5><5,4>[5,3][2,1]x[2,5][4,3]<5,4><3,1>

<1,5>
<4,5>

<5,3><2,1><2,5><4,3>

picked up by vertical contour

large D-instanton

One-cut Boundary condition<5,3><2,1><42><2,5><4,3>[CIY5 ʻ13]

c

<5,1>
<4,3><1,5><5,4>[5,3][2,1][2,5][4,3]<5,4><3,1>

<1,5>

eR(y)

<4,5>
<5,3><2,1><42><2,5><4,3>This model cannot satisfy One-cut BCeR(y)yy<5,3><2,1>p<2,5><4,3>

yc

<5,1>

µ1

p-q dualp417 p1+µp
417 p⌧X (t) = ⌧Y (t)µ<4,3><1,5><5,4>[5,3][2,1][2,5][4,3]<5,4><3,1>

<1,5>
<4,5>

<5,3><2,1>

p-q dual

X-system<2,5><4,3>

Y-system⌧X (t) = ⌧Y (t)Y-Matrix Model⌧Y (t) = Z(t)X-Matrix Model⌧X (t) = Z(t)solution spacesolution space<5,3><2,1>

p<5,1>µ1

p-q dualp417 pµ1+p417 pµ⌧X (t) = ⌧Y (t)<2,5><4,3><4c<4,3><1,5><5,4>[5,3][2,1][2,5][4,3]<5,4><3,1>

<1,5>
<4,5>

<5,3><2,1><2,5><4,3>

Summary• We consider Duality in Non-perturbative Completion• Matrix models are known to possess non-perturbative[contour] ambiguity (which is the ambiguity of Dinstanton chemical potentials θ).

• Duality may be broken non-perturbatively• Therefore, if one requires “string duality acts nonperturbatively,” as a principle, then it provides aconstraint on non-perturbative ambiguity of string
theoryThis is the first quantitative observation
on non-perturbative principle of string theory

Non-perturbative Phase Structure in complex g [CIY6 ʻ14]and change of Scenery in String Theory LandscapesVacuum	  Connec%on	  Formulatachyonic

meta-­‐stable

②
DKK2

DKK2③

DKK1

DKK1

VV②②

tachyonic

③③
Large
inst

DKK1Vµ ⇠ g 5/4

DKK2

Small	  inst

V

DKK2

tachyonic
V
DKK2

DKK1

Small
inst

tachyonic①
No	  inst④V①VNo	  inst

DKK1⑤④⑤

DKK1

Thank you for your attention!