in supersymmetric matrix model Tsunehide Kuroki (National Institute - - PowerPoint PPT Presentation

Large order behavior and instanton action in supersymmetric matrix model

Tsunehide Kuroki (National Institute of Technology, Kagawa College)

Collaboration with Fumihiko Sugino (OIQP) JHEP 1403 (2014) 006, Nucl. Phys. B876 (2013) 758 + recent progress

Yukawa Institute Workshop Strings and Fields

• Nov. 9, 2015@YITP

1

Backgrounds

0-dim. SUSY matrix model: 𝑇 = 𝑂tr 1 2 𝐶2 + 𝑗𝐶 𝜚2 − 𝜈2 + 𝜔(𝜚𝜔 + 𝜔𝜚) nilpotent SUSY:

 (predicted to be) nonperturbative formulation of a two-dim.

superstring theory in RR background

 two SUSY’s preserved in all order in 1/𝑂-expansion  SUSY’s are broken nonperturbatively and spontaneously

(if established) the first example of nonpert. superstring theory realizing spont. SUSY breaking!!

Sugino-K. Endres-K.-Sugino-Suzuki

2

 Evidence

perturbative: reproduce several kinds of and infinitely many

two-pt. functions at the tree level in the two-dim. superstring theory (agreement of functional form)

nonperturbative: prove spont. SUSY breaking in MM side by

computing the VEV of SUSY inv. operator (order parameter) In particular, 𝐺 = 0 in all order in 1/𝑂-expansion reflecting perturbative SUSY, but nonzero nonperturbatively

• rigin: instanton in MM

～isolated eigenvalue of 𝜚: 𝐺 ∼ 𝑓

− 32

3𝑕𝑡

counterpart in superstring side? (D-brane?)

3

 Evidence

perturbative: reproduce several kinds of and infinitely many

two-pt. functions at the tree level in the two-dim. superstring theory (agreement of functional form)

nonperturbative: prove spont. SUSY breaking in MM side by

computing the VEV of SUSY inv. operator (order parameter) In particular, 𝐺 = 0 in all order in 1/𝑂-expansion reflecting perturbative SUSY, but nonzero nonperturbatively

• rigin: instanton in MM

～isolated eigenvalue of 𝜚: 𝐺 ∼ 𝑓

− 32

3𝑕𝑡

counterpart in superstring side? (D-brane?)

4

more evidence necessary!! → higher order , multi-pt. this talk

Perturbation series & nonperturbative effect (bosonic string)

𝑑 = 0 noncritical string (2D pure gravity) 𝑣 = 𝜖𝑢

2𝐺:

(𝑢−5

4 ∼ 𝑕𝑡: C.C. or string coupling)

perturbative soln.: 𝑣 = ℎ≥0 𝑣ℎ 𝑢

1 2−5 2ℎ = ℎ≥0 𝑣ℎ 𝑕𝑡

−2+2ℎ→ 𝑣ℎ ∼ 𝐷−2ℎ 2ℎ !:

 characteristic of string (vol. of moduli sp. of Riemann surface) cf. particle: 𝑜!  nonperturbative ambiguity:

Borel transf.: 𝑣 →

𝑣 𝑡 = ℎ≥0

𝑣ℎ 2ℎ ! 𝑕𝑡 −2+2ℎ𝑡2ℎ~ 1 1−

𝑡 𝐷 /𝑕𝑡 2

inverse: 𝑣 =

∞

𝑣 𝑡 𝑓−𝑡𝑒𝑡 ∼

∞ 1 1−

𝑡 𝐷 /𝑕𝑡 2 𝑓−𝑡𝑒𝑡

→ ambiguity ∼ 𝑓

− 𝐷

𝑕𝑡, where 𝐷: instanton action in MM = tension of D-brane

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𝑡

𝐷/𝑕𝑡

Brezin-Kazakov '90 Gross-Migdal ’90 Douglas-Shenker '90

𝑣2 +

1 6 𝜖𝑢 2𝑣 = 𝑢

large order behavior

Perturbation series & nonperturbative effect (SUSY DWMM)

𝑇 = 𝑂tr 1 2 𝐶2 + 𝑗𝐶 𝜚2 − 𝜈2 + 𝜔(𝜚𝜔 + 𝜔𝜚) perturbative SUSY → 𝐺 = 0 in all order in 1/𝑂2~𝑕𝑡

2-expansion, but ∃nonpert. effect

𝐺 ~ 𝑓

− 32

3𝑕𝑡 ≠ 0 & SUSY → in SUSY theory, no relation between LOB & nonpert. effect??

genus ℎ quantities: 𝒫(1/𝑂2ℎ)~𝒫(𝑕𝑡

2ℎ−2):

𝜖𝜈2𝐺

ℎ = −𝑗 1 𝑂 tr 𝐶 ℎ

∼

1 𝑂 tr 𝜚2 − 𝜈2 ℎ

: SUSY protected (Note: 𝑗𝑅 tr 𝜔 = tr𝐶) perturbatively zero 𝐺 ~ 𝑓

− 32

3𝑕𝑡 → no prediction

→ due to huge cancellation → non-SUSY inv. operator:

1 𝑂 tr 𝜚2𝑙+1 ℎ

→ stringy LOB? instanton action?

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All order result for non-SUSY operator

𝐶, 𝜔, 𝜔: Gaussian → MM only for 𝜚 → eigenvalue density

𝜍ℎ(𝑨) =

1 𝑂 tr 𝜀 𝑨 − 𝜚 ℎ

=

1 𝜌 Im 1 𝑂 tr 1 𝑨−𝑗𝜗−𝜚 ℎ

→

1 𝑂 tr 𝜚2𝑙+1 ℎ

= 𝑒𝑦𝜍ℎ 𝑦 𝑦2𝑙+1

trick

𝜍ℎ(𝑨) =

2 𝑨 𝜌 Im 1 𝑂 tr 1 𝑨−𝑗𝜗−𝜚2 ℎ

: SUSY protected, Nicolai mapping → GUE: all order result in 1/𝑂-expansion known!

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difficult to compute!!

[Haagerup-Thorbjørnsen ’10 [math.PR]]

𝜍ℎ 𝑦 = 8 3𝜌 − 1 12

ℎ

6ℎ − 3 ! ℎ! 3ℎ − 2 ! 4 − 𝑦 − 𝜈2 2 1

2

:alternating series (Borel summable, no ambiguity) reflecting SUSY protected

1 𝑂 tr𝜚2𝑙+1 ℎ

= 𝑒𝑦 𝜍ℎ 𝑦 𝑦2𝑙+1

= 𝑂−2

3 𝑙+2 −1 𝑙+1 Γ 𝑙+3 2

2𝜌

3 2

1 12 ℎ 3ℎ−𝑙−3 ! ℎ! 𝑕𝑡

2

64 ℎ−𝑙+2

3

standard expansion in terms of 𝑕𝑡

2,

LOB: (2ℎ)! → stringy! inverse Borel:

∞ 1 − 𝑡2

16∙64 9𝑕𝑡 2

𝑙+5

2 𝑓−𝑡𝑒𝑡 ∼ 𝑓

− 32

3𝑕𝑡

:exact agreement with instanton action!

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non-SUSY op. → positive term series → non-Borel summable (obtained from 𝜍ℎ)

𝑡

32 3𝑕𝑡

Conclusions

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1-pt. function of SUSY op. resolvent of GUE: all order in 1/𝑂-exp. eigenvalue density : all order 1-pt. function of non-SUSY op. :all order Im integration Nicolai mapping Im Borel summable

• non-Borel summable
• stringy LOB
• reproduce instanton action exactly
• MM is a superstring theory
• spontaneous SUSY proved

→ FIRST EX. established

Discussions

 Superstring counterpart of instanton which triggers SUSY breaking

likely D-brane → D-brane generation triggers spont. SUSY breaking!

 1-pt. function of 𝜚2𝑙+1 in higher order in 1/𝑂-expansion

=1-pt. function of RR field at higher genus: problem of supermoduli MM side: completely well-defined → further perturbative confirmation of our claim with solving the problem

• f supermoduli

 Application to SUSY gauge theories?

existence of 𝜍(𝑦) is essential localization works? ABJM: different story?

10

K-Sugino, work in progress

LOB of non- SUSY op. nonperturbative effect (brane) isolated eigenvalue (instanton)