Tying together instantons and anti-instantons NIKITA NEKRASOV - PowerPoint PPT Presentation

Path integral as period The action in e − S/ � � � 1 S = − i p d q + T ds H ( p ( s ) , q ( s )) γ 0 The fields: F = L P is the space of parametrized loops ϕ : S 1 → P ϕ ( s ) = ( p ( s ) , q ( s ) ) ∈ P , ϕ ( s + 1) = ϕ ( s ) .

Complexify the classical picture • Complex phase space ( P C , ̟ C ) , ̟ C = d p C ∧ d q C • Holomorphic Darboux coordinates ( p C , q C )

Now contour is in the complexified loop space

Contour in the complexified loop space

Complex Saddle Points: qualitative picture

Complex Saddle Points: qualitative picture The complexified phase space is C 2 ≈ R 4 now

Complex Saddle Points: qualitative picture The complex energy level space is now an elliptic curve E ≈ T 2

Complex Saddle Points: qualitative picture The complex energy level space is now an elliptic curve E ≈ T 2 compactified

Complex Saddle Points: qualitative picture Our old friends real energy levels are the real slices of that T 2

Complex Saddle Points: qualitative picture

Complex Saddle Points: qualitative picture

Complex Saddle Points: qualitative picture Instanton gas Maps to piecewise linear paths on the torus:

Torus cycles : winding (3 , 4) ↑↑↑ This is not a critical point!

Torus cycles : winding (3 , 4) ↑↑↑ The gradient flow moves towards a critical point!

Torus cycles : winding (3 , 4) It moves . . .

Torus cycles : winding (3 , 4) And moves . . .

Torus cycles : winding (3 , 4) And moves . . .

Torus cycles : winding (3 , 4) And moves further down . . .

Torus cycles : winding (3 , 4) Until we reach the critical point

Where are the instantons?

Where are the instantons and anti-instantons?

What are the critical points ϕ a ’s in general?

With an additional assumption of ”algebraic integrability” P C fibers over B C ⊂ C r

The critical points are : rational windings on tori T 2 r - complex tori (abelian varieties)

Two winding vectors n , m ∈ Z r

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 we can solve for the Complex Saddle Points δS = 0 ⇔ i d p ds = − T ∂H i d q ds = T ∂H ∂ q , ∂ p

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

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

Complex Saddle Points Pass to action-angle variables dφ ds = i T ∂H d a ∂ a , ds = 0 = ⇒ the critical loop ϕ a = [ γ ( s )] sits in a particular fiber T 2 r b , b ∈ B C where the motion is a straight line in the angle variables φ ( s ) = φ (0) + Ω s Ω = i T ∂H ∂ a The fiber b is fixed by φ (0) = φ (1) up to the periods φ (1) = φ (0) + n + τ · m

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

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

So, now we are facing the next question : Where are the critical points of the superpotential W n , m ?

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

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

Another curious 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