Mithat Unsal North Carolina State University Some of the work - PowerPoint PPT Presentation

Semi - classics, adiabatic continuity and resurgence in quantum theories Mithat ¨ Unsal North Carolina State University Some of the work presented here is done in collaboration with : Y a ff e, Shifman, Argyres, Poppitz, Schaefer, Dunne, Cherman, Sulejmanpasic, Tanizaki

0 ) Thanks for nice talks to Tatsu Misumi and Gerald Dunne 1 ) Critical points at infinity vs. real/complex bions in QM 2 ) Coupling a TQFT to QM 3 ) Adiabatic continuity and deformed Y ang - Mills 4 ) Coupling a TQFT to YM 5 ) Critical points at infinity and magnetic/neutral bions in QFT I will review some ideas and some new, and will tell you two parallel stories with the hope to merge them. Will also describe some ( yet unresolved ) puzzles.

Part 1 Critical points at infinity and real/complex bions in QM

QM with Grassmann valued fields I will consider first the following QM systems. ( Many parallels with the saddles in semi - classical QFTs with fermions. ) S = 1 Z x 2 + 1 2 ( W 0 ) 2 + 1 ⇣ ψ i − ˙ ⌘ 2 ( ¯ ψ i ˙ ¯ 2 W 00 [ ¯ 1 ψ i ψ i ) + 1 2 ˙ ψ i , ψ i ] i = 1 , . . . , N f . dt , g N f =1 SUSY QM N f >1 related to QES systems. If exp [ +W ] or exp [- W ] is normalizable, the lowest N f states are exactly solvable! These systems are called Quasi - Exactly Solvable ( QES ) ( Turbiner, Ushveridze 87 ) , and to my mind, not less interesting than supersymmetric QM ( very likely more. ) Quantizing the fermions, ( or integrating them out exactly ) , we end up with N f M � N f � b deg( H ( N f ,k ) ) b H = deg( H ( N f ,k ) ) = H ( N f ,k ) , k k =0

⇣ ( W 0 ) 2 + ζ gW 00 ⌘ p 2 + 1 H ( N f ,k ) = g ˆ 2 b ζ = 2 k − N f , k = 0 , . . . , N f . 2 g L ζ = 1 ⇣ x 2 + ( W 0 ) 2 ± ζ gW 00 ⌘ ˙ 2 g Note that the potential has a classical and quantum part. The tilting is a one - loop quantum e ff ect, induced by integrating out fermions. If the tilting is rendered classical, the story changes quite a bit. But such quantum induced potential appears naturally by integrating out fermions both in QM and QFT, it is worthwhile to discuss this system for its own right.

Examples of exactly solvable states

Basics of instantons - 1 � � Z Z Z x 2 + 1 2 ( W 0 ( x )) 2 = � � xW 0 � x ⌥ W 0 ( x )) 2 1 1 2 ˙ 2 ( ˙ ± ˙ � = | W ( x 2 ) � W ( x 1 ) | dW � � � | {z } � 0 x = ± W 0 ( x ) . ˙ Instanton equation ⇣ x ⌘ W ( x ) = 4 cos = ⇒ x I ( τ ) = 4 arctan (exp[ τ − τ c ]) , Instanton solution 2 6 5 4 3 2 1 - 10 - 5 5 10

Basics of instantons - 2 The instanton amplitude: " c # � 1 2 det M I I ≡ ξ = J τ c e � S I P I ( g ) , det M 0 • The overall amplitude: density of the instantons. Characteristic separa- tion between instantons: ∼ e + S I , dilute instanton gas. p • J t c = S I / (2 π ): Jacobian associated with the bosonic zero mode. 2.0 • M I = − d 2 d τ 2 + V 00 ( x ) | x = x I ( t ) = − d 2 d τ 2 + 1 − 2 sech 2 ( τ − τ c ) , : quadratic 1.5 fluctuation operator in the background of the instanton. (P¨ oschl-Teller 1.0 form). Exact zero mode is given by 0.5 2 Ψ 0 ( τ ) = ˙ x I ( τ ) = cosh( τ − τ c ) - 10 - 5 5 10 - 0.5 The “hat”: the zero mode has to be removed, and det M 0 is a normal- ization factor, which we take to be the corresponding free fluctuation - 1.0 operator. • Perturbative expansion around instanton: X 1 b I,n g n , P I ( g ) = n =0 which is a formal asymptotic series, which is in general not Borel summable. • The determinant of the instanton fluctuation operator can be computed using the Gel’fand-Yaglom (GY) method. (See Marino’s book).

Remark: Do instantons always contribute to physical observables? In almost all books and texts, you will see the discussion of double - well or periodic potential, but not a more generic potential with harmonic degenerate minima as shown in figure. Why not? 2.0 1.5 1.0 0.5 - 2.0 - 1.5 - 1.0 - 0.5 0.5 1.0 Despite the fact that there are exact instanton solutions, for generic potential of this type, they typically do not contribute to the spectrum at the exp [- S ] order, rather, the first NP contribution appears at order exp [- 2S ] , related to the concept of critical point at infinity ( which I will explain ) . The reason instantons do not contribute at leading order is that the determinant of fluctuation operator is infinite unless the frequency in two consecutive well are the same. Therefore, in QM, instanton contributing to spectrum is exception instead of being a rule.

Perturbation theory by Bender - Wu method Bender-Wu Mathematica package written by Tin Sulejmanpasic: https://library.wolfram.com/infocenter/MathSource/9479/. Description The BenderWu package allows for analytic computation of the perturbative series in 1D quantum mechanics around a harmonic minimum of the potential. The code is based on the method pioneered by Bender and Wu. ∞ X E pert ( N, g ) ∼ ~ n a n ( N ) n =0 "✓ # "✓ ◆# ◆ 2 ◆ 3 − g 2  N + 1 � N + 1 + 1 N + 1 + 3 ✓ N + 1 − g ∼ 2 16 2 4 16 2 2 4 2 " # ◆ 4 ◆ 2 − g 3 ✓ ✓ 5 N + 1 + 17 N + 1 + 9 16 3 2 2 4 2 32 " ◆# ◆ 5 ◆ 3 − g 4 ✓ ✓ ✓ 33 N + 1 + 205 N + 1 + 405 N + 1 − . . . 16 4 4 2 8 2 64 2 2 2 N Γ ( n + 2 N + 1) Large - order factorial growth for harmonic level N a n ( N ) ∼ − π ( N !) 2 (2 S I ) n +2 N +1 2 · (2 S I ) 1 8 · (2 S I ) 2 ✓ ◆ a n ( N = 0) ∼ − 1 n ! 1 − 5 − 13 Large - order factorial growth for n ( n − 1) + . . . (2 S I ) n +1 ground state. n π

Instanton interactions Since instanton equations and Euclidean eq of motion are non - linear, two instanton configurations is not a solution at finite separation. x II ( τ ) = x I ( τ − τ 1 ) + x I ( τ − τ 2 ) , I ( τ ) = x I ( τ − τ 1 ) − x I ( τ − τ 2 ) , x I ¯ S II ( τ 12 ) = 2 S I + A repulsive , g e − τ 12 , I ( τ 12 ) = 2 S I − A attractive S I ¯ g e − τ 12 , Attractive/repulsive are just words, inheritance from old literature. Caused too much confusion in past. This formula just means that these combos are not exact solution for finite separation. That is all. Tau direction is called quasi - moduli space.

Cluster expansion For particle on a circle with unique minimum on the circle ( for simplicity ) In the β → ∞ limit, we can write Z as d τ 1 + ξ 2 d τ 1 d τ 2 e − V 12 + ξ 3 ✓ ◆ Z Z Z 1 + ξ d τ 1 d τ 2 d τ 3 e − V 123 + . . . Z = e − β E 0 P 0 ( g ) . 1! 2! 3! whee ξ ∼ e − S I is the instanton amplitude. II ] ± − [ I 3 ] − [ I 2 ¯ Z dilute = e − β ( E 0 P 0 ( g ) − [ I ] − [¯ I ] − [ I 2 ] − [¯ I 2 ] − [ I ¯ I ] ± − [¯ I ] ... ) ! 0 1 0 1 I [ I ¯ I ] n I ¯ ∞ ∞ ∞ β n I ¯ β n I [ I ] n I β n ¯ I [ I ] n ¯ I I X X X ± = e − β E 0 P 0 ( g ) A . . . @ A @ n I ! I ! I ! n ¯ n I ¯ n I =0 I =0 I =0 n ¯ n I ¯

Compactify R → S 1 β in order to study Z ( β ) = Tr [ e − β H ]. The interaction between two events is modified in a fairly obvious way into: S ( τ ) = ± A ⇣ e − τ + e − ( β − τ ) ⌘ g Z β ! 1 g ( e − τ + e − ( β − τ ) ) − β / 2 A [ I ¯ [ I ][¯ I ] = d τ e I ] 2 0 τ τ 1 τ 0 τ 0 τ − 1 The Lefschetz thimbles for the I ¯ I saddle, showing the downward flows (blue curves) connecting τ 0 to τ ± 1 when g → g e i θ with θ → 0 + . The directions are flipped about the imaginary axis for θ → 0 − .

Borel - Ecalle summability in bosonic theory ✓ ✓ A ◆ ◆ [ I ¯ [ I ][¯ I ] ± = ⌥ i π � γ � log + . . . I ] g ✓ ✓ A ◆ ◆ ✓ ◆ 1 � 5 2 · g � 13 8 · g 2 . . . [ I ¯ e − (2 S I ) /g I ] ± ⇠ ⌥ i π � γ � log + . . . g 2 · (2 S I ) 1 8 · (2 S I ) 2 ✓ ◆ a n ( N = 0) ∼ − 1 n ! 1 − 5 − 13 n ( n − 1) + . . . (2 S I ) n +1 n π up to O ( e − 4 S I ) Im B 0 , θ =0 ± + Im [ II ] θ =0 ± = 0 , The leading terms ( structures ) obtained in Bogomolny and Zinn - Justin early 80s, but not su ffi ciently appreciated. The interesting thing is, B - ZJ was not an unknown work. The problem was that their methods in the derivation did not su ffi ciently convince people. ( Otherwise, they would held this conference in ~1985 ) . I was personally fascinated by what they did, and was convinced that their main claim was correct. The overall structure was obtained in 2014, in Gerald Dunne and MU.

SUSY , QES and in between: parametric resurgence S = 1 Z x 2 + 1 2 ( W 0 ) 2 + 1 ⇣ ψ i − ˙ ⌘ 2 ( ¯ ψ i ˙ ¯ 2 W 00 [ ¯ 1 ψ i ψ i ) + 1 2 ˙ ψ i , ψ i ] i = 1 , . . . , N f . dt , g L ζ = 1 x 2 + ( W 0 ) 2 ± ζ gW 00 ⌘ ⇣ ˙ 2 g Instanton interactions in the presence of fermions or quantum tilting z }| { ⇣ e − τ + e − ( β − τ ) ⌘ + A z}|{ S II ( τ ) = + ζ τ . g ⇣ e − τ + e − ( β − τ ) ⌘ I ( τ ) = − A + S I ¯ ζ τ g |{z} | {z } quantum classical