Gökçe Başar. University of Maryland. July 25, Resurgence in Gauge and String Theories, Lisboa, 2016

Transcription

1 Resurgence, exact WKB and quantum geometry Gökçe Başar University of Maryland July 25, 2016 Resurgence in Gauge and String Theories, Lisboa, 2016 based on: with G.Dunne, 16xx.xxxx with G.Dunne, M. Ünsal related: Monte-Carlo dynamics, Lefschetz thimbles and the sign problem , , , , with A. Alexandru, P. Bedaque, G. Ridgway, N. Warrington

2 Many expansions in physics are asymptotic: f ( ) c n n, c n n! n=0 some examples: (beware! highly incomplete list) quartic/cubic oscillator, Mathieu, Zeeman, Stark,... Euler-Heisenberg, QFT in ds/ads background, large N,... genus expansion in string theory (c g (2g)!) [Shenker] hydrodynamics [Heller,Spalinski; GB, Dunne; Aniceto, Spalinski]

3 f ( ) = k 1 Resurgence c n,k,l n }{{} n=0 k=0 l=1 perturbative fluctuations ( [ exp c ]) ( [ k ln ± 1 ]) l }{{ } }{{} k instantons quasi-zero-modes resurgence: c n,k,l s are related: large order terms of perturbative series low order terms of non-perturbative series

4 Punchline of this talk: Beyond resurgence [Dunne Ünsal; GB, Dunne] For certain Schrödinger equations (relevant for SUSY QFTs) in addition to the large order - low order relations between perturbative and non-perturbative expansions, there is a surprising low order - low order relation between them. It can be understood in terms of the geometry of the spectral curve.

5 Mathieu equation [GB, Dunne; ] 2 2 d 2 ψ + cos(z) ψ = u(n, ) ψ dz2 NS limit of the N = 2, SU(2) theory, u tr Φ 2, moduli space coord. [see talks by Hatsuda, Kashani-Poor, Russo] Wilson loops in N = 4 (via AdS/CFT and Pohlmeyer Reduction) [Kruczenski et. al]... more generally, ODE 2D integrable models [Dorey, Tateo; Voros; Bazhanov, Fateev, Lukyanov, Zamolodchikov;...]

6 Strong coupling expansion: N := λ 1 u(ħ) Weak coupling expansion: λ 1 ħ

7 Trans-series near u 1, tightly bound states, tunneling exponentially suppressed [ u(n, ) 1 + N + 1 ] [ ( 2 N + 1 ) ] [ ( N e S inst ) n f n (N) cos θ n } {{ } 1 instanton +... ( N + 1 2) ]... + e 2S inst n g n (N, θ) n } {{ } 2 instanton trans-monomials: n (perturbative fluctuations), e k S inst (multi instantons), log( 1/ ) l (quasi zero modes)

8 c n (N = 0) n! 2S n I Resurgence relations large order growth of perturbative series: ( n n(n 1)... ) instanton anti-instanton fluctuations: (leading ambiguity) ( Im u(0, ) π e 2S inst/ 1 5 ( ) ( ) )

9 Beyond resurgence In addition to the large order - low order relations between perturbative and non-perturbative expansions, there is a surprising low order - low order relation between them! allows one to fully construct the non-perturbative fluctuations from perturbative data. valid everywhere in the spectrum

10 WKB expansion ψ e i Q(z,u; ) Q 2 + i Q 2(u V (z)) = 0 (Ricatti eqn.) Q(z) 2(u n Q n (z, u) = V )dz + n Q n (z, u) n=0 WKB actions: a(u; ) = 1 2π a D (u; ) = 1 2π n=1 [Dunham] A B Q dz Q dz a n (u) 2n n=0 an D (u) 2n n=0 perturbative : a(u; ) = 2 (N + 1/2) u pt.(n) non-perturbative (tunneling): u = 2 π a(u) and a D (u) are related order by order in! u pt. N e 2π Im[aD ]

11 WKB expansion ψ e i Q(z,u; ) Q 2 + i Q 2(u V (z)) = 0 (Ricatti eqn.) Q(z) 2(u n Q n (z, u) = V )dz + n Q n (z, u) n=0 WKB actions: a(u; ) = 1 2π a D (u; ) = 1 2π n=1 [Dunham] A B Q dz Q dz a n (u) 2n n=0 an D (u) 2n n=0 perturbative : a(u; ) = 2 (N + 1/2) u pt.(n) non-perturbative (tunneling): u = 2 π u pt. N e 2π Im[aD ] a(u) and a D (u) are related order by order in! P = NP

12 Geometry and WKB Set = 0 for now. Classically the (complex) phase space can be identified with the moduli space of complex tori. u moduli space parameter u = p2 2 + cos z x cos z, y = ẋ 2 y 2 = (x 2 1)(x u) genus-1 elliptic curve

13 Geometry and WKB WKB actions: integrals of abelian differentials over the two independent cycles of torus 2 2 u x a 0 (u) = u V (z) dz = dx 2π A π A y 2 2 a0 D u x (u) = u V (z) dz = dx 2π π y B Seiberg-Witten differentials for SU(2) [Gorsky, Krichever, Marshakhov, Mironov, Morozov] B

14 Geometry and WKB a 0 and a D 0 are related via Riemann bilinear identity da0 D a 0 du da 0 ad 0 du = i S inst 2 T T = 2π =period of the harm. oscll. at the bottom of the well a 0, a0 D : satisfy a Picard-Fuchs equation Bilinear identity Wronskian a 0 1 (u) 4(1 u 2 ) a 0(u) = 0 alternatively: a D 0 (u) = τ 0(u) a 0 (u) i S inst ω 0(u) where ω 0 = a 0, modular parameter: τ 0 = ω D 0 /ω 0

15 Geometry and WKB: Quantum corrections a(u; ) a n (u) 2n, a D (u; ) n=0 an D (u) 2n n=0 All higher order actions are encoded in the lowest order (classical) action a n (u) = p n (u)a 0 (u) + q n (u)a 0(u) a D n (u) = p n (u)a D 0 (u) + q n (u)a D 0 (u) p n, q n : rational functions that can be derived from Schrödinger eqn.

16 Geometry and WKB: Quantum corrections quantum corrections to the bilinear identity ( a a ) a D u [GB, Dunne] (a D ad ) a u = 2i π connects the perturbative expansion to non-perturbative fluctuations order by order valid everywhere in the spectrum SUSY inspired proof via Matone s relation [Gorsky, Milekhin]

17 quantum corrections to the Picard Fuchs equation: [GB, Dunne, Ünsal, in prep] a (u) + F (u)a (u) + G(u)a(u) = 0 F (u) := n f n (u), G(u) := n=0 n g n (u) n=0 quantum corrections: higher order poles 1 f 0(u) = 0, g 0 = 8( 1 + u) 1 8(1 + u) 1 f 1(u) = 96(u + 1) (u 1) 2, g 1 = no new singularities! 1 96(u + 1) (u + 1)

18 P = NP perturbative expansion: [ u pt. (N, ) 1 + N + 1 ] [ ( N ) ] band width (non-perturbative, 1-instanton+fluctuations) : u 1 inst. (N, ) = upt. N es inst 0 ( d u pt. (N, ) 3 N + 2 (N+ 2 1 ) )2 S inst checked up to 3 loops via explicit calculation [Escobar-Ruiz, Shuryak, Turbiner]

19 Strong coupling expansion: N 1 u(ħ) ħ u pert. (N, ) 2 8 ( N (N 2 1) ( ) N (N 2 1) 3 (N 2 4) ( ) )

20 Strong coupling (λ 1) gauge theory detour [Alday, Gaiotto, Tachikawa; Marshakov et. al.;...] Z inst. (a; ɛ 1, ɛ 2 ) = ( Λ 2 n=0 ɛ 1 ɛ 2 ) 2n Q 1 ([1n ], [1 n ]), Q (Y, Y ) = L Y L Y ( ) from AGT: = 1 ɛ 1 ɛ 2 a 2 (ɛ 1+ɛ 2 ) 2 4 ɛ 2 0 limit, twisted superpotential: W inst. NS (a; ɛ 1) ɛ 1 4πi identify ɛ 1 =, a = N /2, c = 1 6(ɛ 1+ɛ 2 ) 2 ɛ 1 ɛ 2 [Nekrasov, Shatashvili] lim ɛ 2 log ( Z inst. (a, ɛ 1, ɛ 2 ) ) ɛ 2 0 u = iπ 2 Λ Winst. NS Λ = 2 8 ( 8Λ 4 (N 2 1) 4 + 8Λ 8 ( 5N ) ) (N 2 4) (N 2 1)

21 Strong coupling (λ 1) back to QM: level splitting u pert (N, ) is not the whole story In the limit N, λ 1 there are exponentially small gaps in the spectrum u pert (N, ) determines the center of the gap u gap N 2 4 N 2 2π gap width: 1 (2 N 1 (N 1)!) 2 ( e ) 2N N ( ) 2 2N [ ( 1 + O 4 )]

22 Strong coupling: complex instantons Is there a semi-classical interpretation of the exponentially small gaps at strong coupling, similar to the instantons for the case of exponentially small bands at weak coupling? YES! complex instantons For u > 1, the turning points are complex. a D goes around these complex turning points. when 1 and N 1 (u 1) semi-classically: u gap N 2 π u pert. N 2π e Im ad N 2 ( e ) 2N 2π N

23 Strong coupling (λ 1) A physical analogy: Schwinger effect in monochromatic electric field E cos(ωt) Pair production rate behaves differently for different ωs Keldysh adiabaticity parameter: γ m ω E γ 1 constant field, γ 1 multi-photon limit In our analogy: 4ω2 E, N m ω, λ = 2γ P QED = e m2 π E g(γ) m2 π e E, γ 1 e m2 π E 4 πγ log(4γ) = ( E 4m ω ) 4m/ω, γ 1 in the worldline formalism: γ 1 real instantons, γ 1 complex instantons

24 Fluctuations around complex instantons quantum bilinear identity relates u pert. (N, ) to u gap N u(n, ) 2 8 N 1 n=1 ( ) 2n P n(n) 4 n k=1 (N2 k 2 ) 2 n k ± (2 N 1 (N 1)!) ( ) 2N 1 N 1 2 n=1 ( ) 2n R n(n) 4 n k=1 (N2 k 2 ) 2 n k 2 The level splitting term (gap width) has the same structure with the leading perturbative expansion. P n (N), R n (N) are related! [GB, Dunne, Ünsal, in prep] New results for Mathieu equation!!

25 How general is the P- NP connection? Mathieu (classical, = 0) modular parameter: τ 0 (u) = ωd 0 (u) ( ω 0 (u) = i 2 F 1 1 2, 1 ( 2 1 2F 1 2, 1 2 ) 1 u, 1; 2 ), 1; 1+u 2 τ 0 satisfies a Schwarzian equation: {τ 0, u} + Q 0 (u) = 0 {τ 0, u} := τ 0 τ ( ) where: τ 2 0 1, Q τ 0 0(u) = (u 1) 2 8(u+1) 4(u+1) 2 8(u 1) spectrum can be obtained by inversion (q 0 := e iπτ 0 ): u(q 0 ) = 1 + λ(q 0 ) = q 0 256q q

26 How general is the P- NP connection? Mathieu (quantum, 0) quantum correction to the Schwarzian equation: {τ, u} + Q(u) = 0 where: Q(u) = n=0 2n Q n(u) }{{} poles at u=±1 inversion spectrum: u(q) = 1 + λ(q) + 2n f n (q) n=1

27 How general is P = NP? consider the generalization: (classical) modular parameter: τ 0 (u) = i ωd 0 (u) ( ω 0 (u) = 2 F 1 1 M, 1 1 M, 1; 1 u) ( 1 2F 1 M, 1 1 M, 1; u) a Picard-Fuchs equation: 0(u) M 1 1 M 2 u(1 u) a 0(u) = 0 corresponding potential: V (x) = T 2 n (x), n = M/(M 2) n = 2 : double well n = 3 : triple well n = 4 : quadruple well etc...

28 How general is P = NP? consider the generalization: (classical) modular parameter: τ 0 (u) = i ωd 0 (u) ( ω 0 (u) = 2 F 1 1 M, 1 1 M, 1; 1 u) ( 1 2F 1 M, 1 1 M, 1; u) a Picard-Fuchs equation: 0(u) M 1 1 M 2 u(1 u) a 0(u) = 0 corresponding potential: V (x) = T 2 n (x), n = M/(M 2) A B n = 2 : double well n = 3 : triple well n = 4 : quadruple well etc...

29 4 special cases: M = 2, 3, 4, 6 4 cos 2 (π/m) is an integer (Mathieu, triple well, double well, cubic oscl.) inversion formula u(τ): Ramanujan s elliptic functions [Berndt, Ramanujan s notebooks vol. II] remarks: appear in number theory [Borwein 2, Berndt, Bhargava, Garvan, Chan,... ] only cases with arithmetic Hecke groups [Shen] related to superconformal N = 2 theories [Minahan,Nemeschansky;Lerda et. al.,... ]

30 4 special cases: M = 2, 3, 4, 6 4 cos 2 (π/m) is an integer (Mathieu, triple well, double well, cubic oscl.) inversion formula u(τ): Ramanujan s elliptic functions [Berndt, Ramanujan s notebooks vol. II] remarks: appear in number theory [Borwein 2, Berndt, Bhargava, Garvan, Chan,... ] only cases with arithmetic Hecke groups [Shen] related to superconformal N = 2 theories [Minahan,Nemeschansky;Lerda et. al.,... ]

31 only these 4 cases have the same modifications upon quantization: quantum bilinear identity: ( a a ) a D u Schwarzian & Picard-Fuchs equations: {τ, u} + Q (M) (u) = 0 ) (a D ad a u = is inst 2π a (u) + F (M) (u)a (u) + G (M) (u)a(u) = 0 Q (M) (u) := n=0 n Q n (M) (u), etc... 0 F n (M) (u), G n (M) (u), Q n (M) (u): sum over higher order poles at the same locations as the classical curve

32 examples with more complicated P = NP relations: generic genus-1: 2 nd order Picard-Fuchs eqn. for a 0(u) Lamé equation V (z) = P(z; t) related to N = 2 SU(2) [GB, Dunne; Kashani-Poor, Troost,... ] V (z) = cos(z) + 2m 1m 2 + (m 1 m 2 ) 2 cos(z)+1 sin 2 (z) related to N = 2, SU(2), N f = 2 Double sine gordon V (z) = sin 2 (z) + µ sin(z) Asymmetric double well V (z) = (z 2 1) 2 + µz...

33 Conclusions In an infinite class of QM systems in addition to the standard resurgence relations there is a low order -low order relation between perturbative and non-perturbative sectors Classically it is related to the topology of the spectral curve It is valid everywhere in the spectrum even though the series are drastically different (asymptotic vs. convergent) in different regions Quantization preserves this P = NP relation 4 examples such that P = NP is particularly simple (Mathieu, triple well, double well, cubic oscl.)