Tying together instantons and anti-instantons in O ( N ) and CP N - PowerPoint PPT Presentation

Algebraic integrability : action variables � � a i = p d q , a D,i = B i p d q A i 2 r variables on r -dimensional space: non-independent a D d a = d F F -prepotential of the effective low-energy N = 2 action

Algebraic integrability : action variables � � a i = p d q , a D,i = p d q A i B i Well-defined on � B C \ Σ Monodromy in Sp (2 r, Z )

Algebraic integrability : action variables near degeneration locus Σ Complex codimension 1 stratum: one vanishing cycle 1 a → 0 , a D = 2 π i a log( a ) + . . .

Algebraic integrability : Feature of complex angle variables: Double periodicity � � ∂ 2 F ̟ j = δ i j , B i ̟ j = τ ij = ∂a i ∂a j A i r � n i , m k ∈ Z τ ij m j , φ i ∼ φ i + n i + j =1

Now solve for Saddle Points δS = 0 ⇔ i d p ds = − ∂H i d q ds = ∂H ∂ q , ∂ p

Now solve for Saddle Points δS = 0 ⇔ i d p ds = − ∂H i d q ds = ∂H ∂ q , ∂ p = ⇒ the critical loop ϕ a = [ γ ( s )] sits in a particular fiber T 2 r u , u ∈ B C

Complex Saddle Points Pass to action-angle variables dφ ds = i ∂H d a ∂ a , ds = 0 = ⇒ the critical loop ϕ a = [ γ ( s )] sits in a particular fiber T 2 r u , u ∈ B C where the motion is a straight line in the angle variables φ ( s ) = φ (0) + Ω s Ω = i ∂H ∂ a

Complex Saddle Points The motion is a straight line in the angle variables φ ( s ) = φ (0) + Ω s Ω = i ∂H ∂ a The fiber u is fixed by periodicity: φ (1) = φ (0) + n + τ · m

Superpotential for Complex Critical Points Ω = n + τ · m = i ∂H ∂ a for some integer vectors n , m ∈ Z r ⇔ d W n , m = 0 W n , m ( u ) = n · a ( u ) + m · a D ( u ) − H ( u ) Well-defined on � B C \ Σ

Landau-Ginzburg description! Supersymmetric d = 2 N = 2 LG model for integer vectors n , m ∈ Z r d W n , m = 0 W n , m ( u ) = n · a ( u ) + m · a D ( u ) − H ( u )

Where are the critical points of the superpotential W n , m ?

Picard-Lefschetz theory : In the limit where T → ∞ u → u ∗ ∈ Σ degeneration locus codim C = 1 stratum: one vanishing cycle a D ∼ 2 S i + 1 a ∼ T 0 ( u − u ∗ ) → 0 , 2 π i a (log( a ) − 1) + . . . ∂a ∂a D ∂u ∼ T 0 ∂u → T 0 , 2 π ilog ( T 0 ( u − u ∗ )) + . . . can make estimates . . .

In the limit T → ∞ the complex energy u is thus fixed n e − 2 πT u ∼ u m,n = u ∗ + u 0 e − 2 π i m nT 0 , Two quantum numbers! n = 1 , 2 , . . . , and m = 0 , 1 , . . . , n − 1

Doubling of quantum numbers: emergent topology! For ( m, n ) = (0 , 1) these are BI-ons of G.Dunne and M.Unsal’13-15 Also, G.Dunne,R.Dabrowski, G.Basar, M.Unsal, M.Shifman, . . . First examples: J. L. Richard and A. Rouet, 1981!

Complex energy In the limit T → ∞ n e − 2 πT u ∼ u m,n = u ∗ + u 0 e − 2 π i m nT 0

Fine structure of the saddle points

Where are the instantons/antiinstantons?

Algebraic integrability r = 1 , one degree of freedom, examples 2 p 2 + U ( x ) H = 1 ̟ = dp ∧ dx , Mathieu, Heun, Higgs

Another curious quantum-mechanical example Probe particle in a black hole background

Another curious quantum-mechanical example Probe particle in a mass M Schwarzschild black hole background Fixed energy E , fixed angular momentum L = ⇒ elliptic curve in the complexified phase space � L � 2 � � � � 1 + L 2 dr 1 − 2 M = E 2 − r 2 r 2 dϕ r � 1 + z 2 � dϕ = Ldz p 2 = E 2 − (1 − 2 Mz ) , p

Next steps • Zero-modes: the whole abelian variety. Only middle-dimensional cycle contributes to T a

Next steps • Zero-modes: the whole abelian variety. Only middle-dimensional cycle contributes to T a • Non-zero modes: Evaluate the one-loop determinants

Next steps • Zero-modes: the whole abelian variety. Only middle-dimensional cycle contributes to T a • Non-zero modes: Evaluate the one-loop determinants • Relative phases of ϕ a contributions: the imprint of the “negative” modes • Set up perturbation theory to include � -corrections • Recognize in the asymptotic nature of � -expansion the influence of different ϕ a ’s, e.g. • in the poles of the Borel transforms

Resurgence connects perturbative and non-perturbative physics

Resurgence, perturbative/non-perturbative relations J. Ecalle’81 A. Voros’81-04 F .Pham’83-97 A. Vainshtein’64 C. Bender and T. Wu’69 J.J. Duistermaat and V.W. Guillemin’75 L. Lipatov’77 B. Malgrange’79 M. Shifman, A. Vainshtein, V. Zakharov’83 E Bogomolny, J. Zinn-Justin’84 M.V. Berry and C.J. Howls’94 P . Argyres, M. Unsal’12 M. Kontsevich and Y. Soibelman’??

Origin of the superpotential Bethe/gauge correspondence Gauge theories with N = (2 , 2) d = 2 super-Poincare invariance ⇔ Quantum integrable systems ♦

QIS ≈ Bethe Ansatz soluble

Bethe/gauge correspondence NN, S.Shatashvili, circa 2007 Supersymmetric vacua (in finite volume) of gauge theory ⇔ Stationary states of the QIS

Quantum mechanics from 4d gauge theory Four dimensional theories e.g. N = 2 super-Yang-Mills theory in four dimensions Viewed as two dimensional theories with SO (2) R -symmetry rotations of two extra dimensions

Quantum mechanics from 4d gauge theory Four dimensional N = 2 theory Compactified onto D � × S 1 × R 1 (cigar × circle × time axis) θ -angular coordinate on D � → D µ φ + � F µθ With Ω -deformation along the cigar D = D µ φ −

Quantum mechanics from 4d gauge theory Four dimensional N = 2 theory Compactified onto D � × S 1 × R 1 (cigar × circle × time axis) With Ω -deformation along the cigar D At low energy

Quantum mechanics from 4d gauge theory Four dimensional N = 2 theory Compactified onto D � × S 1 × R 1 (cigar × circle × time axis) × S 1 × R 1 at low energy ↓ × R 1 Becomes 2d sigma model on R + × R 1

Quantum mechanics from 4d gauge theory Four dimensional N = 2 theory Compactified onto D � × S 1 × R 1 (cigar × circle × time axis) × S 1 × R 1 at low energy ↓ × R 1 Becomes 2d sigma model on R + × R 1 = ⇒ deformation quantization introduced in 1978 by F. Bayen, L. Boutet de Monvel, M. Flato, C. Fronsdal, A. Lichnerowicz et D. Sternheimer’78, NN, E.Witten’2009 existence of formal def.quant. shown by M. Kontsevich in 1999 Using A.Kapustin,D.Orlov’s branes’2003 sigma model explored by A. Cattaneo and G. Felder’99

Quantum mechanics from 4d gauge theory Partition function of the quantum system � Tr H qis e − 1 k τ k � H k �

Quantum mechanics from 4d gauge theory Partition function of the quantum system � � Tr H qis e − 1 k τ k � H k = Tr H vac e − 1 k τ k O k � � with τ k the set of “times” - generalized Gibbs ensemble with O k the basis of the twisted chiral ring

Quantum mechanics from 4d gauge theory Partition function of the quantum system � � � Tr H qis e − 1 k τ k � H k = Tr H vac e − 1 k τ k O k = Tr H vac ( − 1) F e − 1 k τ k O k � � � assuming all vacua are bosonic