Counting versus Integration Anton Gerasimov (ITEP/TCD/HMI) SCGP - PowerPoint PPT Presentation

Counting versus Integration Anton Gerasimov (ITEP/TCD/HMI) SCGP Workshop November 2012

Examples of counting: 1. Holomorphic maps of compact curves Σ → X 2. Vortexes/Monopoles/Instantons 3. D-branes / sheaves of various types 4. Summing perturbative series in String Theory

In many examples Z ( t ) = ∑ Z k ( t ) ∼ Ψ ( t ) k is a wave function of another (dual) quantum system. The wave function has an infinite-dimensional integral representation via the Hartle-Hawking representation in the dual system. Sometimes it also has a nice finite-dimensional integral representation (or at least with a lower number of integration variables). We discuss a possibility of the Hartle-Hawking type representation of the wave function in the original theory capturing counting sum Z ( t ) .

An old example of instanton counting for N = 4 d = 4 SYM (Vafa-Witten 94’) Z ( t ) = ∑ Z k ( t ) ∼ Ψ ( t ) k where Ψ ( t ) is naturally a conformal block in some CFT. Many examples for N = 2 case.

Example: counting holomorphic maps P 1 → P ℓ Counting holomorphic curves in homogeneous spaces such as projective spaces, flag spaces et cet after Givental. Recall the case of the target space X = P ℓ . We are interested in calculation of the sum Z ( x ) ∼ ∑ Z d ( x ) d of G = S 1 × U ℓ + 1 -equivariant volumes of the spaces of of degree d holomorphic maps P 1 → P ℓ : � e x ω G ( λ ) Z d ( x , λ ) = ( P 1 → P ℓ ) d where S 1 acts on P 1 by rotations, U ℓ + 1 acts on target space P ℓ following the tautological representation U ℓ + 1 → End ( C ℓ + 1 ) .

The space of holomorphic maps shall be properly compactified. One way to do it is to use the space of quasi-maps . A quasi-map φ ∈ QM d ( P ℓ ) of degree d is a collection ( a 0 ( y ) , a 1 ( y ) , . . . a ℓ ( y )) of homogeneous polynomials a i ( y ) in variables y = ( y 1 , y 2 ) of degree d d 1 y d − j a k , j y j ∑ a k ( y ) = , k = 0, . . . , ℓ 2 j = 0 considered up to the multiplication of all a i ( y ) ’s by a nonzero complex number.

Example: Rational maps f : P 1 → P 1 , f ( z ) = p ( z ) q ( z ) , deg p ( z ) = deg q ( z ) = d When polynomials have common zero the degree of the map drops by one. Thus the space of degree d -maps is non-compact. One shall consider instead the space of pairs of polynomials modulo action of C ∗ .

The space QM d ( P ℓ ) is a non-singular projective variety P ( ℓ + 1 )( d + 1 ) − 1 with the action of ( λ , g ) ∈ C ∗ × GL ℓ + 1 on QM d ( P ℓ ) is induced by ( y 1 , y 2 ) − → ( λ y 1 , y 2 ) λ : � � ℓ + 1 ℓ + 1 ( a 0 , a 1 , . . . , a ℓ )) − → ∑ ∑ g : g 1, k a k − 1 , . . . , g ℓ + 1, k a k − 1 k = 1 k = 1

Thus we shall calculate the following integral � P ( ℓ + 1 )( d + 1 ) − 1 e x ω G ( λ ,¯ h ) Z d ( x , λ , ¯ h ) = where ω G is G = S 1 × U ℓ + 1 -equivariant extension of the generator of H 2 ( P ℓ , Z ) . Here λ = ( λ 1 , . . . , λ ℓ + 1 ) is an elements of the h is a generator Lie ( S 1 ) such diagonal subalgebra of u ℓ + 1 and ¯ that the S 1 × U ℓ + 1 -equivariant cohomology ring of P ( ℓ + 1 )( d + 1 ) − 1 is given by H ∗ S 1 × U ℓ + 1 ( P ( ℓ + 1 )( d + 1 ) − 1 , C ) = C [ γ , ¯ h ] ⊗ C [ λ 1 , . . . , λ ℓ + 1 ] S ℓ + 1 / ℓ + 1 d h ] ⊗ C [ λ 1 , . . . , λ ℓ + 1 ] S ℓ + 1 ∏ ∏ ( γ − λ j − ¯ / hm ) C [ γ , ¯ j = 1 m = 0

Recall that for U ℓ + 1 -equivariant cohomology of P ℓ realized as H ∗ U ℓ + 1 ( P ℓ , C ) = C [ γ ] ⊗ C [ λ 1 , . . . , λ ℓ + 1 ] S ℓ + 1 ℓ + 1 ( γ − λ j ) C [ γ ] ⊗ C [ λ 1 , . . . , λ ℓ + 1 ] S ℓ + 1 ∏ / j = 1 we have an integral representation for the pairing of cohomology classes with the U ℓ + 1 -equivariant fundamental cycle [ P ℓ ] � 1 P ( γ , λ ) d γ � P , [ P ℓ ] � = P ∈ H ∗ U ℓ + 1 ( P ℓ , C ) , ∏ ℓ + 1 2 π ı j = 1 ( γ − λ j ) C where the integration contour C encircles the poles.

Taking P = e x ω G and generalizing to the case of the action of S 1 × U ℓ + 1 on QM d = P ( C ( ℓ + 1 )( d + 1 ) ) we obtain integral formula for equivariant volume of QM d � e ı x γ d γ 1 Z d ( x , λ , ¯ h ) = ∏ ℓ + 1 j = 1 ∏ d 2 π ı m = 0 ( γ − λ j − ¯ hm ) C

Taking the limit d → ∞ Givental proposed to consider the limiting space QM d ( P ℓ ) , d → + ∞ as a substitute of the universal cover of the space � L P ℓ + + of � of holomorphic disks in P ℓ . The algebraic version L P ℓ L P ℓ + is defined as a set of collections of regular series a i ( z ) = a i ,0 + a i ,1 z + a i ,2 z 2 + · · · , 0 ≤ i ≤ ℓ modulo the action of C ∗ . This space inherits the action of G = S 1 × U ( ℓ + 1 ) defined previously on QM d ( P ℓ ) .

Let us take the limit d → + ∞ on the level of cohomology groups H ∗ ( QM d ( P ℓ )) . In the limit d → ∞ we obtain � ℓ + 1 ∞ 1 d γ e x γ / ¯ h h ) ∼ ∏ ∏ Z ∗ ( x , λ , ¯ hn . γ − λ j − ¯ j = 1 n = 0 and we shall replace arising infinite products by Γ -functions � � λ k − γ � ℓ + 1 λ k − γ γ x ∏ Z ∗ ( x , λ , ¯ h ) = d γ e ¯ h Γ , h h ¯ ¯ ¯ h k = 1 This finite-dimensional integral is equal to the infinite-dimensional one � e x ω G / ¯ S 1 × U ℓ + 1 ( L P ℓ h , ω ∈ H 2 Z ∗ ( x , λ , ¯ h ) = + , C ) L P ℓ +

The resulting function is a solution of the parabolic version of the Toda open chain � ℓ + 1 � � − e x � h ∂ ∏ λ j − ¯ Z ∗ ( z , λ , ¯ h ) = 0 ∂ x j = 1 Note that solutions of this equation can be also written as matrix elements in infinite-dimensional representations of GL ℓ + 1 ( R ) . It is known that counting function of Gromov-Witten invariants of P ℓ satisfies this equation and various solutions are distinguished by a choice of the particular two-point function (so we actually work on moduli space M 0,2 ).

Direct derivation of an integral representation It is instructive to directly calculate the infinite-dimensional integral. The integral is an integral over a toric manifold (limit of a projective spaces) i.e. modulo some divisors it is a product of a torus on a polyhedron. This allows to define an analog of angle-action variables. Integrand does not depend on the angle variables and integrating over angles one obtains the integral over a projection of the toric variety under the momentum map. For finite d the resulting integral can be written in the following form � ℓ + 1 d d � ℓ + 1 d d � ( λ i + n ) t i , n ∑ Z ( d ) ( x , λ , ¯ ∏ ∏ ∑ ∑ ∏ h ) ∼ t i , n − x dt i , n δ e n = 0 i = 1 n = 0 i = 1 n = 0 i = 1 � � d � e λ i T i Ξ d ( T i ) ∏ = T 1 + . . . + T ℓ + 1 − x dT 1 . . . dT ℓ + 1 δ i = 1 where Ξ d ( T ) is S 1 -equivariant volume of P ( C [ z ] / z d + 1 C [ z ]) � � 1 − e T � d n dt n e ∑ d n = 0 nt n δ ( ∑ t n − x ) = ∏ Ξ d ( T ) = j = 0

Using renromalization x → x − ( ℓ + 1 ) ln d and taking the limit d → ∞ we obtain � ℓ + 1 λ i T i × ∑ Z ( x , λ , ¯ h ) = dT 1 . . . dT ℓ + 1 e i = 1 R ℓ + 1 + ℓ + 1 ℓ + 1 ∑ ∏ × δ ( x − T i ) Ξ ∞ ( T i ) , i = 1 i = 1 where � � d 1 − e T / d ∼ e − e T Ξ ∞ ( T ) = lim d → ∞ is an equivariant volume of P ( C [ z ]) .

Thus we arrive at the following Givental/Hori-Vafa integral representation of P ℓ -parabolic Whittaker function: � e λ 1 T 1 − e T 1 + ... + λ ℓ + 1 T ℓ + 1 − e T ℓ + 1 Z ∗ ( x , λ ) = T ∈ R ℓ + 1 | ∑ j T j = x

QFT realization of the limit d → ∞ One can show that the equivariant volume of the space of holomorphic maps of the disk D into P ℓ can be identified with a correlation function in type A topologically twisted linear gauged sigma model on a disk. This interpretation allows to make the previous considerations more natural and in particular to use mirror symmetry to obtain a finite-dimensional integral representation from the infinite-dimensional one. In the dual type B topologically twisted Landau-Ginzburg theory on a disk the corresponding correlation function is given by a finite-dimensional integral derived before � e λ 1 T 1 − e T 1 + ... + λ ℓ + 1 T ℓ + 1 − e T ℓ + 1 Z ∗ ( x , λ ) = T ∈ R ℓ + 1 | ∑ j T j = x

Note that we have derived mirror symmetric description A-model on P ℓ via the Landau-Ginzburg model with superpotential ℓ + 1 e T j | ∑ ℓ + 1 ∑ W 0 ( T ) = j = 1 T j = x j = 1

Lessons to learn from counting of holomorphic maps: 1. There is a way to replace the sum of the integrals over finite-dimensional moduli spaces of compact holomorphic curves by an integral over an infinite-dimensional space ( universal moduli space of curves ). 2. This universal moduli space of curves obtained by taking the degree of the map d → ∞ can be interpreted as a space of maps of non-compact curves (disks). 3. This approach allows straightforward derivation of mirror symmetry map.