Principles of integrability by examples/applications 18-9-2019, - PowerPoint PPT Presentation

Principles of integrability by examples/applications 18-9-2019, Pisa PRIN Kick-off Meeting Davide Fioravanti (INFN-Bologna) series of paper with M.Rossi,JE Bourgine, D.Gregori, A. Bonini, H.Poghossian,…. 1

Sketch of a PLAN in integrable words : 1)Motivations: different research topics (e.g. WL string minimal area) lead us to Thermodynamic Bethe Ansatz in the ODE/IM perspective 2) T raditional (scattering) way to TBA (I way) 3) ODE/IM and PDE/IM: functional and integral eqs. (II way to TBA) 4) OPE or Form Factor Series for null polygonal WLs re-sums to TBA: III way

Some motivations and perspectives General wall-crossing (jumping) formulae (Donaldson-Thomas invariants) e.g. by Kontsevich- Soibelman have taken a very effective form for BPS states (compactified theories) thanks to Gaiotto- Moore-Neitzke (2008) ⎡ ⎤ � ζ ′ + ζ � d ζ ′ ⎣ − 1 X γ ( ζ ) = X sf Ω ( γ ′ ; u ) ⟨ γ , γ ′ ⟩ ζ ′ − ζ log(1 − σ ( γ ′ ) X γ ′ ( ζ ′ )) ⎦ . γ ( ζ ) exp ζ ′ 4 π i ℓ γ ′ γ ′ (5.13) which are nothing but TBA EQS. In fact more that one year later, enriched perspective Note added Nov. 20, 2009 : It was pointed out to us some time ago by A. Zamolodchikov that one of the central results of this paper, equation (5.13), is in fact a version of the Thermodynamic Bethe Ansatz [45]. In this appendix we explain that remark. Another relation between four- dimensional super Yang-Mills theory and the TBA has recently been discussed by Nekrasov and Shatashvili [46]. The TBA equations for an integrable system of particles a with masses m a , at inverse temperature β , with integrable scattering matrix S ab ( θ − θ ′ ), where θ is the rapidity, are � + ∞ d θ ′ circumference R 2 π φ ab ( θ − θ ′ ) log(1 + e β µ b − � b ( θ ′ ) ) � � a ( θ ) = m a β cosh θ − (E.1) −∞ b where φ ab ( θ ) = − i ∂ ∂θ log S ab ( θ ). Here the scattering matrix is diagonal, that is, the soliton creation operators obey Φ a ( θ ) Φ b ( θ ′ ) = S ab ( θ − θ ′ ) Φ b ( θ ′ ) Φ a ( θ ). more general

Hitchin systems: the same mathematical problem as for minimal string area for gluon scattering amplitudes/Wilson loops (null, polygonal) in N=4 SYM Benefit for exchange of ideas between these fields and from integrability ideas (non-perturbative, exact, ect)which makes clear the following: The general phenomenon on the background is the so-called linear Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence (CFT s), possibly extended to linear PDE (Massive QFT s) Recently we proposed an advance (different ODE) which identifies NS (SW with one Omega background) periods with integrable quantities T ,Q: functional and integral eqs. Pandora box? I will give you a flavour. We re-summed the OPE (FF) series of Wl (collinear limit) to TBA: why? Before that, let us recall the original physics of TBA.

The Thermodynamic Bethe Ansatz I Evolution of Zamolodchikov’s idea to non-relativistic theories, where the scattering matrix does not change (as depends on difference of rapidities which are all shifted). I A cylinder (p.b.c.: torus) of very large height R (time) and circumference L (space) may be seen in the other way around: ˜ L ( space ) ↔ R ( time ) ABA ( direct ) → ABA ( mirror ) p ↔ E i.e. analytic continuation which entails the same partition function Z direct ( L , R ) = ˜ Z mirror ( L , R ) . I Advantage: asymptotic BA exact in the mirror theory at R = ∞ , then thermodynamics for minimal free energy at ’temperature’ T = 1 / L exp [ − RE 0 ( L )] = exp [ − RL ˜ f min ( L )] , R → ∞ furnishes the ground state energy of direct (string/gauge) theory E 0 ( L ) . I Infinite system of non-linear (real) integral equations and E ( L ) is a non-linear functional on the real rapidity u summed up on infinite pseudoenergies ✏ Q ( u ) (massive nodes).

Mirror (tilde) R long ˜ ABA ( mirror ) → L ABA ( direct ) Direct: T r(Z^L) or T r(….)+….

Vacuum/Excited states Thermodynamic Bethe Ansatz I Vacuum equations of the form Z dv K a , b ( u , v ) ln ( 1 + e − ✏ b ( v ) ) X ✏ a ( u ) = µ a + ˜ e a ( u ) − b with mirror energy ˜ e a ( u ) as driving term and scattering factors K a , b ( u , v ) ∝ @ v ln S a , b ( u , v ) I Excited states E ( L ) are connected to the vacuum by analytic continuation in some parameter ( e.g. µ a and L ) ⇒ additional inhomogeneous terms in the equations P i ln S a , b ( u , u i ) depending on TBA complex singularities u i : e − ✏ a ( u i ) = − 1 these are the exact Bethe roots (with wrapping). I ⇒ Delicate and massive numerical work for analytic continuation.

Excited states via the Y-system I Alternative route: for simpler integrable theories (like quantum Sine-Gordon) we proposed and checked all the states - including the ground state! - must satisfy the same functional equations, the so-called Y -system: Y a ( u ) ≡ e − ✏ a ( u ) . In a nutshell, we loose the information concerning the inhomogeneous terms as they are zero-modes of the ’TBA-operator’ (a multi-shift operator with incidence matrix), i.e. ln S a , b ( u , u i ) (sort of solution of Y -system). Universal, but we recover the specific forcing term/state by behaviour at u = ± ∞ . Besides, these terms form the Aymptotic Bethe Ansatz, once the non-linear integrals are forgotten. No true systematics. I Novelty:additional discontinuity equations on the cuts of the rapidity u -planes. We ’derived’ the dressing factor from these relations (limitation of this ’explanation’).

The Y-system It is the Y-system(not the TBA) which is encoded in a Dynkin-like diagram. I seat on a node: LHS= � � � � u − i u + i Y Q Y Q = g g RHS=Nearest neighbours products: Horizontal: � A QQ � � � = 1 + Y Q � (u) Q � � δ Q, 1 − 1 � Vertical: 1 1 + Y ( α ) ( v | Q − 1) ( u ) � , � δ Q, 1 � α 1 1 + Y ( α ) ( y | − ) ( u ) �

(w|N) (w|2) (y|+) α =1 (w|1) (v|1) (v|2) (v|N) (y| − ) Q=N Q=1 (y| − ) (v|1) (v|2) (v|N) (w|1) (y|+) α =2 (w|2) (w|N) Figure 1: The Y-system diagram corresponding to the AdS 5 / CFT 4 TBA equations. � δ Q, 1 − 1 1 � 1 + Y ( α ) � � � � u − i u + i (v | Q − 1 ) (u) � A QQ � � � � 1 + Y Q � (u) Y Q Y Q = , � δ Q, 1 g g 1 � 1 + α Q � Y ( α ) (y | − ) (u)

1 + Y ( α ) � � (v | 1 ) (u) � � � � 1 u + i u − i Y ( α ) Y ( α ) = � , (y | − ) (y | − ) 1 1 + Y ( α ) � g g � � 1 + (w | 1 ) (u) Y 1 (u) 1 �� � 1 + � δ M, 1 Y ( α ) � � � � u + i u − i (y | − ) (u) � A MN Y ( α ) Y ( α ) 1 + Y ( α ) � � = (w | N) (u) , (w | M) (w | M) 1 � � g g 1 + Y ( α ) N (y |+ ) (u) � A MN � δ M, 1 1 + Y ( α ) 1 + Y ( α ) � �� � � (v | N) (u) (y | − ) (u) � � � � u + i u − i N Y ( α ) Y ( α ) = , (v | M) (v | M) 1 1 + Y ( α ) � � g g � � 1 + (y |+ ) (u) Y M + 1 (u) where A 1 ,M = δ 2 ,M , A NM = δ M,N + 1 + δ M,N − 1 and A MN = A NM . � In the integrable model framework the Y-systems play a very central rôle. Firstly, a Y-system = Y ( α ) N � �� �� � �� � � �� 1 1 (y | − ) � � � � + ln , [ � ] ± 2 N = ∓ ln 1 + + ln 1 + � (u) = ln Y 1 (u) + 1 , Y ( α ) Y ( α ) Y ( α ) ± 2 N ± ( 2 N − M) α = 1 , 2 M = 1 (y |+ ) (y | ∓ ) (v | M) then is the function introduced ( α ) � �� Y ( α ) � � �� N � � �� 1 + 1 (y | − ) � ln = − ln , Y ( α ) Y Q ± ( 2 N − Q) (y |+ ) ± 2 N Q = 1 with N = 1 , 2 ,..., ∞ and 1 + 1 /Y ( α ) 1 + Y ( α ) � � � � (y | − ) (y | − ) ln Y ( α ) ln Y ( α ) � � � � ± 1 = ln ± 1 = ln , , (w | 1 ) (v | 1 ) 1 + 1 /Y ( α ) 1 + Y ( α ) (y |+ ) (y |+ ) where the symbol [ f ] Z with Z ∈ Z denotes the discontinuity of f (z) [ f ] Z = lim ϵ → 0 + f (u + iZ/g + i ϵ ) − f (u + iZ/g − i ϵ ), on the semi-infinite segments described by z = u + iZ/g with u ∈ ( −∞ , − 2 ) ∪ ( 2 , + ∞ ) and = + � � f (u) Z = f (u + iZ/g) − f (u ∗ + iZ/g) ⇒ function [ ] function [ f (u) ] Z is the analytic extension of the discontinuity is the analytic extension of the discontinuity (1.9) to generic complex of . To retrieve the TBA equations, the extended Y-system ha the analytic extension of the discontinuity (5.5) to

ODE/IM Correspondence: a quick review (Dorey,T ateo,BLZ,Dunning,Suzuki,Frenkel,Bender…..) Ž . half-line 0, ` . Simplest example: Schroedinger eq. on the half line (Stokes line) l q 1 x and d 2 Ž . ž l l q 1 / 2 M Ž . Ž . q x x s E c x y q c 2 2 dx x Ž . on the half-line 0, . Imposing the two possible we fix the subdominant solution such that at complex infinity 1 ž / y M r 2 M q 1 y ; x exp y x , M q 1 3 p < arg x - < . 2 M q 2 1 ž / X M r 2 M q 1 y ; y x exp y x M q 1 as x tends to infinity in any closed sector contained cally proportional Let denote the sector k 2 k p p Changing anti-Stokes sector = this solution becomes dominant Let S denote S arg x y - . k 2 M q 2 2 M q 2 From 2.2 it follows that y tends to zero Ž .