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

Yukawa Institute Workshop Strings and Fields Nov. 9, 2015@YITP 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 1

Backgrounds Sugino-K. 0-dim. SUSY matrix model: Endres-K.-Sugino-Suzuki 𝑇 = 𝑂tr 1 2 𝐶 2 + 𝑗𝐶 𝜚 2 − 𝜈 2 + 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!! 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 origin: instanton in MM − 32 ～ isolated eigenvalue of 𝜚 : 𝐺 ∼ 𝑓 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 more evidence necessary!! computing the VEV of SUSY inv. operator (order parameter) → higher order , multi-pt. In particular, 𝐺 = 0 in all order in 1/𝑂 -expansion reflecting this talk perturbative SUSY, but nonzero nonperturbatively origin: instanton in MM − 32 ～ isolated eigenvalue of 𝜚 : 𝐺 ∼ 𝑓 3𝑕𝑡 counterpart in superstring side? (D-brane?) 4

Perturbation series & nonperturbative effect (bosonic string) 𝑑 = 0 noncritical string (2D pure gravity) Brezin-Kazakov '90 ( 𝑢 − 5 𝑣 2 + 1 Gross-Migdal ’90 2 𝐺 : 2 𝑣 = 𝑢 4 ∼ 𝑕 𝑡 : C.C. or string coupling) 𝑣 = 𝜖 𝑢 6 𝜖 𝑢 Douglas-Shenker '90 1 2 − 5 2 ℎ = ℎ≥0 𝑣 ℎ 𝑕 𝑡 −2+2ℎ → 𝑣 ℎ ∼ 𝐷 −2ℎ 2ℎ ! : perturbative soln.: 𝑣 = ℎ≥0 𝑣 ℎ 𝑢 large order behavior  characteristic of string (vol. of moduli sp. of Riemann surface) cf. particle: 𝑜!  nonperturbative ambiguity: 𝑣 ℎ 1 −2+2ℎ 𝑡 2ℎ ~ 𝑣 𝑡 = ℎ≥0 Borel transf.: 𝑣 → 2ℎ ! 𝑕 𝑡 𝑡 2 𝑡 𝐷/𝑕 𝑡 1− 𝐷 /𝑕𝑡 ∞ ∞ 1 𝑣 𝑡 𝑓 −𝑡 𝑒𝑡 ∼ 2 𝑓 −𝑡 𝑒𝑡 inverse: 𝑣 = 0 0 𝑡 1− 𝐷 /𝑕𝑡 − 𝐷 → ambiguity ∼ 𝑓 𝑕𝑡 , where 𝐷: instanton action in MM = tension of D-brane 5

Perturbation series & nonperturbative effect (SUSY DWMM) 𝑇 = 𝑂tr 1 2 𝐶 2 + 𝑗𝐶 𝜚 2 − 𝜈 2 + 𝜔(𝜚𝜔 + 𝜔𝜚) perturbative SUSY → 𝐺 = 0 in all order in 1/𝑂 2 ~𝑕 𝑡 2 -expansion, but ∃ nonpert. effect − 32 3𝑕𝑡 ≠ 0 & SUSY → in SUSY theory, no relation between LOB & nonpert. effect?? 𝐺 ~ 𝑓 genus ℎ quantities: 𝒫(1/𝑂 2ℎ )~𝒫(𝑕 𝑡 2ℎ−2 ) : 1 𝑂 tr 𝜚 2 − 𝜈 2 1 : SUSY protected (Note: 𝑗𝑅 tr 𝜖 𝜈 2 𝐺 ℎ = −𝑗 𝑂 tr 𝐶 ∼ 𝜔 = tr𝐶 ) ℎ ℎ − 32 3𝑕𝑡 → no prediction perturbatively zero 𝐺 ~ 𝑓 → due to huge cancellation 1 𝑂 tr 𝜚 2𝑙+1 → non-SUSY inv. operator: → stringy LOB? instanton action? ℎ 6

All order result for non-SUSY operator 𝐶, 𝜔, 𝜔 : Gaussian → MM only for 𝜚 → eigenvalue density 1 1 1 1 𝜍 ℎ (𝑨) = 𝑂 tr 𝜀 𝑨 − 𝜚 = 𝜌 Im 𝑂 tr 𝑨−𝑗𝜗−𝜚 ℎ ℎ 1 𝑂 tr 𝜚 2𝑙+1 = 𝑒𝑦𝜍 ℎ 𝑦 𝑦 2𝑙+1 → difficult to ℎ compute!! trick 2 𝑨 1 1 𝜍 ℎ (𝑨) = 𝜌 Im 𝑂 tr : SUSY protected, 𝑨−𝑗𝜗−𝜚 2 ℎ Nicolai mapping [Haagerup-Thorbjørnsen ’10 [ math.PR]] → GUE: all order result in 1/𝑂 -expansion known! 7

ℎ = 8 3𝜌 − 1 6ℎ − 3 ! ℎ! 3ℎ − 2 ! 4 − 𝑦 − 𝜈 2 2 1 𝜍 ℎ 𝑦 2 12 :alternating series (Borel summable, no ambiguity) reflecting SUSY protected 1 𝑂 tr𝜚 2𝑙+1 = 𝑒𝑦 𝜍 ℎ 𝑦 𝑦 2𝑙+1 ℎ ℎ− 𝑙+2 3 𝑙+2 −1 𝑙+1 Γ 𝑙+ 3 ℎ 3ℎ−𝑙−3 ! = 𝑂 − 2 2 1 𝑕 𝑡 3 2 3 12 ℎ! 64 2𝜌 2 non-SUSY op. → positive term series 2 , standard expansion in terms of 𝑕 𝑡 → non-Borel summable (obtained from 𝜍 ℎ ) LOB: (2ℎ)! → stringy! 𝑡 32 𝑙+ 5 − 32 2 𝑓 −𝑡 𝑒𝑡 ∼ 𝑓 ∞ 1 − 𝑡 2 3𝑕 𝑡 inverse Borel: 3𝑕𝑡 16∙64 0 2 9𝑕𝑡 :exact agreement with instanton action! 8

Conclusions 1-pt. function of Nicolai mapping SUSY op. resolvent of GUE: all order in 1/𝑂 -exp. Im Im eigenvalue density : Borel summable all order • MM is a superstring theory • spontaneous SUSY proved integration → FIRST EX. established 1-pt. function of • non-Borel summable non-SUSY op. • stringy LOB :all order 9 • reproduce instanton action exactly

Discussions  Superstring counterpart of instanton which triggers SUSY breaking likely D-brane → D-brane generation triggers spont. SUSY breaking!  1-pt. function of 𝜚 2𝑙+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 of supermoduli K-Sugino, work in progress  Application to SUSY gauge theories? LOB of non- SUSY op. existence of 𝜍(𝑦) is essential localization works? nonperturbative isolated eigenvalue effect (brane) (instanton) ABJM: different story? 10