Archimedean L-factors and Topological Field Theories II
aa r X i v : . [ h e p - t h ] D ec Archimedean L -factors andTopological Field Theories II Anton Gerasimov, Dimitri Lebedev and Sergey Oblezin
Abstract . In the first part of this series of papers we propose a functional integral repre-sentation for local Archimedean L -factors given by products of the Γ-functions. In particularwe derive a representation of the Γ-function as a properly regularized equivariant symplecticvolume of an infinite-dimensional space. The corresponding functional integral arises in thedescription of a type A equivariant topological linear sigma model on a disk. In this paper weprovide a functional integral representation of the Archimedean L -factors in terms of a type B topological sigma model on a disk. This representation leads naturally to the classical Eulerintegral representation of the Γ-functions. These two integral representations of L -factors interms of A and B topological sigma models are related by a mirror map. The mirror symmetryin our setting should be considered as a local Archimedean Langlands correspondence betweentwo constructions of local Archimedean L -factors. Introduction
In [GLO1] we propose a framework of topological quantum field theory as a proper way to de-scribe arithmetic geometry of Archimedean places of the compactified spectrum of global numberfields. In particular we provide a functional integral representation of local Archimedean L -factorsas correlation functions in two-dimensional type A equivariant topological sigma models. Thisrepresentation implies that local Archimedean L -factors are equal to properly defined equivariantsymplectic volumes of spaces of holomorphic maps of a disk into complex vector spaces. Thus theequivariant infinite-dimensional symplectic geometry (in the framework of a topological quantumfield theory) appears as the Archimedean counterpart of the geometry over non-Archimedean localfields.The construction of local Archimedean L -factors in terms of type A equivariant topologi-cal sigma models should be considered as an analog of “arithmetic” construction of local non-Archimedean L -factors in terms of representations of local non-Archimedean Galois group. Thereis another, “automorphic” construction of the non-Archimedean L -factors which uses a theory ofinfinite-dimensional representations of reductive groups. For Archimedean places this provides arepresentation of the corresponding L -factors as products of classical Euler’s integral representationsof the Γ-functions. In [GLO1] we conjecture that this finite-dimensional integral representation of L -factors naturally arises in a type B topological sigma model which is mirror dual to the type A topological sigma model considered in [GLO1]. This would lead to an identification of localArchimedean Langlands correspondence between “arithmetic” and “automorphic” constructionsof L -functions with a mirror symmetry between corresponding type A and type B equivarianttopological sigma models. In this note we propose the type B topological sigma model dual to1he one considered in [GLO1] and identify a particular set of correlation functions on a disk withArchimedean L -functions. As expected the resulting functional integral representation of the L -factors is reduced to a product of the Euler integral representations of Γ-functions.The type B equivariant topological sigma model considered below is an S -equivariant sigmamodel on a disk D with the target space X = C ℓ +1 and a non-trivial superpotential W . Weimply that S acts by rotations of the disk D . A particular superpotential W corresponding to themirror dual to the type A equivariant topological sigma model with target space C ℓ +1 is well-known[HV]. However our considerations have some new interesting features. At first, the S -equivarianceprovides a new solution of the so-called Warner problem in topological theories on non-compactmanifolds. The standard way to render the theory consistent is to introduce a non-trivial boundaryinteraction corresponding to a collection of D -branes in the target space [K], [KL], [O]. We showthat in the case of S -equivariant sigma model on the disk D there is a universal boundary termleading to a consistent topological theory. Another not quite standard feature of our approach isa choice of a real structure on the space of fields of the topological theory. One can construct atopological quantum field theory starting with an N = 2 SUSY quantum field theory and usinga twisting procedure (see e.g. [W1], [W2]). This provides a particular real structure on the spaceof fields. Another approach is to construct directly topological theory combining (equivariant)topological multiplets of quantum fields. Although this approach produces topological field theoriesclosely related with those obtained by the twisting procedure the resulting real structure may bedifferent (for a discussion of an example see e.g. [W1]). In our considerations we use a real structurewhich is different from the one appeared in twisted N = 2 SUSY two-dimensional sigma models.We also comment on an explicit mirror map of type A and type B topological sigma models. Weprovide a heuristic derivation of the B -model superpotential W by applying Duistermaat-Heckmanlocalization formula to an infinite-dimensional projective space. The sum over fixed points canbe related to the sum over instantons used in the previous derivations of the superpotential [HV].We also consider an explicit change of variables in the functional integral transforming A -modelinto B -model. Although these considerations are heuristic they reveal interesting features of thetopological theories discussed in this note and in [GLO1].The plan of the paper is as follows. In Section 1 we provide a construction of a S -equivarianttype B topological sigma model on a disk D . In Section 2 we identify a particular correlation func-tion of the topological sigma model with a product of Γ-functions thus providing a new functionalintegral representation of local Archimedean L -factors. In Section 3 we give heuristic constructionsof a mirror map of type A topological sigma model considered in [GLO1] to a type B topologicalsigma model considered in Section 2. In Section 4 we conclude with some general remarks anddiscuss further directions of research. Acknowledgments : The research was partly supported by Grants RFBR-08-01-00931-a, 09-01-93108-NCNIL-a. The research of SO was partly supported by RF President grant MK-544.2009.1.AG was also partly supported by Science Foundation Ireland grant. B Topological sigma-models
We start by recalling the standard construction of a topological sigma model associated with aK¨ahler manifold with trivial canonical class supplied with holomorphic superpotential. For generaldiscussion of the two-dimensional topological sigma models see e.g. [CMR] and reference therein.Let X be a K¨ahler manifold of complex dimension ( ℓ + 1) with trivial canonical class and let2 ∈ H ( X, O ). Let M (Σ , X ) = M ap (Σ , X ) be the space of maps Φ : Σ → X of a compactRiemann surface Σ into X . Let ( z, ¯ z ) be local complex coordinates on Σ. We pick a hermitianmetric h on Σ and denote √ h d z the corresponding measure on Σ. The complex structure onΣ defines a decomposition d = ∂ + ∂ , ∂ = dz ∂ z , ∂ = d ¯ z ∂ ¯ z of the differential d acting on thedifferential forms on Σ. Let K and ¯ K be canonical and anti-canonical bundles over Σ. Let ω and g be the K¨ahler form and the K¨ahler metric on X and T C X = T , X ⊕ T , X be a decomposition ofthe complexified tangent bundle of X . We choose local complex coordinates ( φ j , ¯ φ j ) on X . LocallyLevi-Civita connection Γ and the corresponding Riemann curvature tensor R are given byΓ ijk = g i ¯ n ∂ j g k ¯ n , R i ¯ jk ¯ l = g m ¯ j ∂ ¯ l Γ mik . (1.1)Now let us specify the standard field content of the type B topological sigma model associatedwith a pair ( X, W ). Denote Π the parity change functor. Thus Π E is a bundle E with the oppositeparity of the fibers. Let η , θ be sections of Φ ∗ (Π T , X ), ρ be a section of ( K ⊕ ¯ K ) ⊗ Φ ∗ (Π T , X ).We also introduce the fields ¯ G and G given by sections of Φ ∗ ( T , X ) and K ⊗ ¯ K ⊗ Φ ∗ ( T , X )respectively. The BRST transformation δ is defined as follows: δ ¯ φ ¯ i = ¯ η ¯ i , δ ¯ η ¯ i = 0 , δθ ¯ i = ¯ G ¯ i − Γ ¯ i ¯ j ¯ k ¯ η ¯ j θ ¯ k , δ ¯ G ¯ i = − Γ ¯ i ¯ j ¯ k ¯ G ¯ j ¯ η ¯ k , (1.2) δρ i = − dφ i , δφ i = 0 , δG i = dρ i + Γ ijk dφ j ∧ ρ k + 12 R ijk ¯ l ¯ η ¯ l ρ j ∧ ρ k . Straightforward calculations show that δ = 0. One can define new variables¯ G ¯ i = ¯ G ¯ i − Γ ¯ i ¯ j ¯ k ¯ η ¯ j θ ¯ k , G i = G i + 12 Γ ijk ρ j ∧ ρ k , (1.3)such that the action of δ has the following canonical form: δ ¯ φ ¯ i = ¯ η ¯ i , δ ¯ η ¯ i = 0 , δθ ¯ i = ¯ G ¯ i , δ ¯ G ¯ i = 0 , (1.4) δρ i = − dφ i , δφ i = 0 , δ G i = dρ i . Here the property δ = 0 is obvious. The advantage of (1.2) is that the fields G i and ¯ G ¯ j arecovariant with respect to diffeomorphisms of the target space X .Consider a topological sigma model with the action given by S = S + S ¯ W + S W , (1.5)where S = Z Σ ( g i ¯ j dφ i ∧ ∗ d ¯ φ ¯ j + g i ¯ j ρ i ∧ ∗ D ¯ η ¯ j − g i ¯ j θ ¯ j Dρ i + g i ¯ j G i ¯ G ¯ j − R i ¯ lk ¯ j ¯ η ¯ l θ ¯ j ρ i ∧ ρ k ) , (1.6) S ¯ W = Z Σ d z √ h (cid:16) D ¯ i ∂ ¯ j ¯ W ( ¯ φ ) ¯ η ¯ i θ ¯ j + ¯ G ¯ i ∂ ¯ i ¯ W ( ¯ φ ) (cid:17) , (1.7) S W = Z Σ (cid:18) − D i ∂ j W ( φ ) ρ i ∧ ρ j + G i ∂ i W ( φ ) (cid:19) , (1.8)3nd D i ∂ j W ( φ ) = ∂ i ∂ j W − Γ kij ∂ k W, D ¯ η ¯ j = d ¯ η ¯ j + Γ ¯ j ¯ k ¯ ℓ dφ ¯ k η ¯ ℓ . The Hodge ∗ -operator acts on one forms as follows ∗ dz = ıdz , ∗ d ¯ z = − ıd ¯ z .The parts S and S ¯ W are δ -exact as it follows from δ = 0 and the following representation S = Z Σ δ V , S ¯ W = Z Σ d z √ h δ V ¯ W , where V = − g i ¯ j ρ i ∧ ∗ d ¯ φ ¯ j + G i θ i , V ¯ W = θ ¯ j ∂ ¯ j ¯ W ( ¯ φ ) , (1.9)and θ i = g i ¯ j θ ¯ j . The variation of S W is given by δS W = Z Σ d ( ρ i ∂ i W ( φ )) , (1.10)and thus is trivial on a compact surface Σ. Note that the action S W is δ -closed but does not δ -exact.In this paper we consider a particular case of an equivariant type B topological sigma modelon a non-compact two-dimensional manifold Σ. Let Σ be a disk D = { z ∈ C | | z | ≤ } . We fix aflat metric h on Dh = 12 ( dzd ¯ z + d ¯ z dz ) = ( dr ) + r ( dσ ) , r ∈ [0 , , σ ∈ [0 , π ] , (1.11)where z = re ıσ . This metric is obviously invariant with respect to the rotation group S acting by σ → σ + α .We would like to consider an S -equivariant version of the type B topological linear sigmamodel on a disk D with a superpotential W . To construct an S -equivariant extension of thetopological field theory we modify the δ -transformations taking into account an interpretation of δ as the de Rham differential in the infinite-dimensional setting. Let us first recall a constructionof an algebraic model of S -equivariant cohomology. Let M be a 2( ℓ + 1)-dimensional manifoldsupplied with an action of S . The Cartan algebraic model of S -equivariant de Rham cohomology H ∗ S ( M ) is the following equivariant extension (Ω ∗ S ( M ) , d S ) of the standard de Rham complex(Ω ∗ ( M ) , d ): Ω ∗ S ( M ) = (Ω ∗ ( M )) S ⊗ C [ ~ ] , d S = d + ~ ι v , (1.12)where (Ω ∗ ( M )) S is an S -invariant part of Ω ∗ ( M ), ~ is a generator of the ring H ∗ ( BS ) and v isa vector field on M corresponding to a generator of Lie( S ). We have d S = ~ L v , L v = d ι v + ι v d, (1.13)where L v is the Lie derivative along the vector field v . The equivariant differential d S satisfies d S = 0 when acting on Ω ∗ S ( M ). The cohomology groups H ∗ S ( M ) of the complex (1.12) have anatural structure of modules over H ∗ S (pt) = C [ ~ ].The S -equivariant version of the BRST transformations (1.4) is a direct generalization of theexpression (1.12) for the equivariant differential to the infinite-dimensional setting. Taking intoaccount an induced action of S on the space of fields we have δ S ¯ φ ¯ i = ¯ η ¯ i , δ S ¯ η ¯ i = ~ ι v d ¯ φ ¯ i , δ S θ ¯ i = ¯ G ¯ i , δ S ¯ G ¯ i = ~ ι v dθ ¯ i , S G i = dρ i , δ S ρ i = − dφ i − ~ ι v G i , δ S φ i = ~ ι v ρ i . Obviously we have δ S = ~ L v .In terms of the variables G i and ¯ G i we have the following transformations: δ S ¯ φ ¯ i = ¯ η ¯ i , δ S ¯ η ¯ i = ~ ι v d ¯ φ ¯ i , δ S θ ¯ i = ¯ G ¯ i − Γ ¯ i ¯ j ¯ k ¯ η ¯ j θ ¯ k ,δ S ¯ G ¯ i = − Γ ¯ i ¯ j ¯ k ¯ η ¯ j ¯ G ¯ k + ~ ι v (cid:0) Dθ ¯ i ) + ~ ∂ l Γ ¯ i ¯ j ¯ k ( ι v ρ l )¯ η ¯ j θ ¯ k ,δ S G i = dρ i + Γ ijk dφ j ∧ ρ k + R ijk ¯ l ¯ η ¯ l ρ j ∧ ρ k + ~ Γ ijk ( ι v G j ) ∧ ρ k ,δ S ρ i = − dφ i − ~ ι v G i − ~ Γ ijk ( ι v ρ j ) ρ k , δ S φ i = ~ ι v ρ i . Now the S -equivariant version of (1.6) and (1.7) on a disk Σ = D is obtained by applying modified δ S to V and V ¯ W given by (1.9). The action S W given by (1.8) is not δ S invariant on the diskand needs a correction boundary term. Proposition 1.1
The following modified action functional of a type B topological sigma model S = Z D (cid:18) g i ¯ j (cid:0) dφ j + ~ ι v G j (cid:1) ∧ ∗ d ¯ φ ¯ j + g i ¯ j ρ i ∧ ∗ D ¯ η ¯ j − g i ¯ j θ ¯ j Dρ i + g i ¯ j G i ¯ G ¯ j − R i ¯ lk ¯ j ¯ η ¯ l θ ¯ j ρ i ∧ ρ k (cid:19) + Z D d z √ h (cid:16) D ¯ i ∂ ¯ j ¯ W ( ¯ φ )¯ η ¯ i θ ¯ j + ∂ ¯ i ¯ W ( ¯ φ ) ¯ G ¯ i (cid:17) + Z D (cid:18) − D i ∂ j W ( φ ) ρ i ∧ ρ j + ∂ i W ( φ ) G i (cid:19) − ~ Z S = ∂D dσ W ( φ ) (1.14) is δ S -invariant.Proof . Direct calculation shows that δ S -variation of the sum of the integrals over D in (1.14) isgiven by the boundary term δ S S = Z ∂D ρ i ∂ i W ( φ ) . The δ S -variation of the boundary term in (1.14) precisely cancels this contribution. ✷ Remark 1.1
The action (1.14) does not have a smooth limit ~ → . This is a so called “Warnerproblem” in the type B topological sigma model with a non-trivial superpotential W ∈ H ( X, O ) on non-compact surface Σ . In non-equivariant setting it is resolved by imposing special boundaryconditions corresponding to a collection of D -branes on the target space X [K], [KL], [O]. Remark-ably the S -equivariant setting discussed above allows a construction of a universal δ S -invariantboundary condition by adding boundary term (1.14) . Remark 1.2
The relation between the boundary term in (1.14) and the variation (1.10) is a partic-ular instance of a general descent relation between various observables in topological field theories. Linear sigma model on a disk
In this Section we calculate a particular correlation function of the S -equivariant type B linearsigma model on the disk D with the target space C ℓ +1 and a generic superpotential W . The δ S -transformations in the case of X = C ℓ +1 are given by δ S ¯ φ ¯ i = ¯ η ¯ i , δ S ¯ η ¯ i = ~ ι v d ¯ φ ¯ i , δ S θ ¯ i = ¯ G ¯ i , δ S ¯ G ¯ i = ~ ι v dθ ¯ i , (2.1) δ S ρ i = − dφ i − ~ ι v G i , δ S φ i = ~ ι v ρ i , δ S G i = d ρ i . The action (1.14) in this case is reduced to S = ℓ +1 X j =1 Z D (cid:0) ( dφ j + ~ ι v G j ) ∧ ∗ d ¯ φ j + ρ j ∧ ∗ d ¯ η j − θ j dρ j + G j ¯ G j (cid:1) (2.2)+ ℓ +1 X i,j =1 Z D d z √ h (cid:0) ∂ i ∂ j ¯ W ( ¯ φ )¯ η i ¯ θ j + ∂ i ¯ W ( ¯ φ ) ¯ G i (cid:1) + Z D (cid:18) − ∂ i ∂ j W ρ i ∧ ρ j + ∂ i W G i (cid:19) − ~ Z S = ∂D dσW ( φ ) . Topological linear sigma model (2.2) allows a non-standard real structure. This means the following.Let us consider the fields φ i , ¯ φ i , θ i , ¯ θ i , ¯ η i , η i , ρ i ¯ ρ i , G i , ¯ G i as independent complex fields. Thesubspace of the fields entering the description of the topological theory with the action (2.2) isdefined as a subspace invariant with respect to an involution acting as follows:( φ i ) † = ¯ φ i , ( θ i ) † = ¯ θ i , (¯ η i ) † = η i , ( ρ i ) † = ¯ ρ i , ( G i ) † = ¯ G i . (2.3)The involution defines a real structure on the space of fields. One can however consider anotherreal structure defined by the reality conditions( φ i ) † = φ i , ( ¯ φ i ) † = − ¯ φ i , ( θ i ) † = − θ i , (2.4)(¯ η i ) † = − ¯ η i , ( ρ i ) † = ρ i , ( G i ) † = G i , ( ¯ G i ) † = − ¯ G i . Thus for example the fields φ i and ı ¯ φ i are real independent fields. To distinguish the real fieldsin the sense (2.4) let us introduce new notations φ i + , φ i − , G i + , G i − for φ i , ı ¯ φ i , G i , ı ¯ G i . Similarlywe redefine the fields ¯ η and θ by multiplying them on ı and considering the resulting fields as realones. The S -equivariant BRST operator can be defined on the new set of real fields as follows: δ S φ i − = η i , δ S η i = ~ ι v dφ i − , δ S θ i = G i − , δ S G i − = ~ ι v dθ i , (2.5) δ S ρ i = − dφ i + − ~ ι v G i + , δ S φ i + = ~ ι v ρ i , δ S G i + = d ρ i , where now the fields η i and θ i are odd real zero-form valued fields, ρ i are odd real one-form valuedfields, G i − are even real zero-form valued fields and G i + are even real two-form valued fields. Theaction of the sigma model for the new real structure is now given by S = − ı ℓ +1 X j =1 Z D (cid:16) ( dφ j + + ~ ι v G j + ) ∧ ∗ dφ j − + ρ j ∧ ∗ dη j − θ j dρ j + G j + G j − (cid:17) (2.6)6 ℓ +1 X i,j =1 Z D d z √ h − ∂ W − ( φ − ) ∂φ i − ∂φ j − η i θ j − ı ∂W − ( φ − ) ∂φ i − G i − ! + ℓ +1 X i,j =1 Z D − ∂ W + ( φ + ) ∂φ i + ∂φ j + ρ i ∧ ρ j + ∂W + ( φ + ) ∂φ i + G i + ! − ~ Z S = ∂D dσW + ( φ + ) . Here W + and W − are arbitrary independent regular functions on R ℓ +1 . Thus defined action is δ S -closed. Remark 2.1
Our choice of the real structure is such that the constructed type B topological sigmamodel is a mirror dual to the type A topological sigma model considered in [GLO1]. In Section 3.3we demonstrate that the mirror correspondence applied to the type A topological sigma models from[GLO1] leads to the real structure of type (2.4) . Note also that the construction of the topologicalYang-Mills theories using an equivariant setting [W1] also leads to the non-standard real structureanalogous to the one we use.
In the following we consider the case of W − ( φ − ) = 0. Thus we have S = − ı ℓ +1 X j =1 Z D (cid:16) ( dφ j + + ~ ι v G j + ) ∧ ∗ dφ j − + ρ j ∧ ∗ dη j − θ j dρ j + G j + G j − (cid:17) (2.7)+ ℓ +1 X i,j =1 Z D − ∂ W + ( φ + ) ∂φ i + ∂φ j + ρ i ∧ ρ j + ∂W + ( φ + ) ∂φ i + G i + ! − ~ Z S = ∂D dσW + ( φ + ) . Given an observable O ( z, ¯ z ) on the disk D we define its correlation function as a functional integralbelow (cid:10) O ( z, ¯ z ) (cid:11) W + := Z Dµ O ( z, ¯ z ) e − S ∗ ,Dµ = ℓ +1 Y i =1 [ Dφ i + ][ Dφ i − ][ Dη i ][ Dθ i ][ Dρ i ][ DG i + ][ DG i − ] . (2.8) Lemma 2.1
The following observable inserted at the center z = 0 of the disk D O ∗ (0) := O ∗ ( z, ¯ z ) | z =0 = ℓ +1 Y i =1 δ ( φ i − ( z, ¯ z )) η i ( z, ¯ z ) | z =0 (2.9) is δ S -invariant.Proof. We have δ S O ∗ ( z, ¯ z ) = ℓ +1 X m =1 η m ( z, ¯ z ) Y j = m δ ( φ j − ) ℓ +1 Y i =1 η i ( z, ¯ z )+ ℓ +1 X m =1 Y j δ ( φ j − )( − m η ...η m − ( ~ ι v η m ) η m +1 ....η ℓ +1 . The first term is equal to zero since for odd variables η = 0. The second term vanishes since thecenter of the disk z = 0 is a fixed point of the S -action so that ι v ( η m ) (cid:12)(cid:12) z =0 = 0. ✷ heorem 2.1 The correlation function of the observable (2.9) in the type B topological S -equivariantlinear sigma model (2.7) is given by hO ∗ (0) i W + = Z R ℓ +1 ℓ +1 Y j =1 dt j e ~ W + ( t ) . (2.10) Proof . Firstly we make an integration over G i − : Z [ DG − ] exp n ı Z D ℓ +1 X i =1 G i + ( z ) G i − ( z ) o = ℓ +1 Y i =1 δ ( G i + ) . The integration over G j + is then equivalent to the substitution of G j + = 0. Thus we should calculatethe following functional integral: Z = Z [ Dφ + ] [ Dφ − ] O (0) exp n ı Z D ℓ +1 X i =1 dφ i + ∧ ∗ dφ i − − ~ Z S dσW + ( φ + ) o Z f ( φ + ) , (2.11)where Z f ( φ + ) = Z [ Dρ ] [ Dθ ][ Dη ] O (0) exp n ı Z D ℓ +1 X i =1 (cid:0) ρ i ∧ ∗ dη i − θ i dρ i ) + 12 Z D ℓ +1 X i,j =1 ∂ W + ∂φ i + ∂φ j + ρ i ∧ ρ j o , and O (0) = ℓ +1 Y j =1 δ ( φ i − (0)) , O (0) = ℓ +1 Y j =1 η i (0) . Let us first integrate over θ in Z f . We have Z f ( φ + ) = Z [ Dρ ] [ Dη ] O (0) ℓ +1 Y j =1 δ ( dρ j ) exp n ı Z D ℓ +1 X j =1 ρ j ∧ ∗ dη j + 12 Z D ℓ +1 X i,j =1 ∂ W + ∂φ i + ∂φ j + ρ i ∧ ρ j o . One-forms allow the following decomposition: ρ i = df i + ∗ df i = ∂ z ¯ F i dz + ∂ ¯ z F i d ¯ z, F i = f i − ıf i . (2.12)It is easy to check (using for example series expansions) that for given ρ i the solutions f , f of(2.12) always exist and are unique up to addition to F i a holomorphic function. Therefore we makethe following change of variables ρ i → ( f i , f i ) / ∼ where the equivalence relation is generated byaddition to f j and f j of real and imaginary parts of a holomorphic function g ( z ) f i ∼ f i + Re( g i ( z )) , f i ∼ f i + Im( g i ( z )) . (2.13)Thus we have [ Dρ ] = [ Df ] [ Df ][ Dg ] J ac − , where Jacobian is given by the determinant of the operator( d ⊕ ∗ d ) : ( f i , f i ) → ρ i = df i + ∗ df i , A orth ⊂ A ( D ) orthogonal to its kernel. We define a determinant of an operator actingbetween different spaces as a square root of the determinant of the product of the operator and itsconjugated J ac = | det ′A orth ⊕A orth ( d + ∗ d ) | := (cid:16) det ′A orth ⊕A orth ( d + ∗ d ) (cid:17) = det ′A orth ∆ , where ∆ = ( d + d ∗ ) acting in the space of functions A . We have δ ( dρ i ) = δ ( d ( df i + ∗ df i )) = δ ( d ∗ df i ) , and thus Z f ( φ + ) = Z [ Dη ] [ Df ] [ Df ][ Dg ] 1det ′A orth ∆ O (0) ℓ +1 Y i =1 δ ( d ∗ d f i ) × exp n ı Z D ℓ +1 X i =1 ( df i + ∗ d f i ) ∧ ∗ dη i + 12 Z D ℓ +1 X i,j =1 ∂ W + ∂φ i + ∂φ j + ( df i + ∗ d f i ) ∧ ( df j + ∗ d f j ) o . Let us fix a representative for the equivalence relation (2.13) by the condition that f i is in thesubspace orthogonal to the space of harmonic functions on the disk. This leaves a freedom to addto f i a real constant (indeed Im( g i ( z )) = 0 implies g i ( z ) = a i ∈ R ). We denote by [ Df ] ′ theinduced measure on this subspace. The integration over f i gives Z f ( φ + ) = Z [ Dη ] [ Df ] ′ O (0) exp n ı Z D ℓ +1 X i =1 df i ∧ ∗ dη i + 12 Z D ℓ +1 X i,j =1 ∂ W + ∂φ i + ∂φ j + df i ∧ df j o , where the determinant in the denominator is canceled by the determinant appearing from theintegration of the delta-function.We split the space of functions A ( D ) on a disk on the space A h of harmonic functions and thespace A N of functions that have zero normal derivative on the boundary: f i = f ih + f iN , f ih ∈ A h , f iN ∈ A N , ∆ f ih = 0 , ∂ n f iN | S = 0 . The subspace A h can be identified with the space Fun( S ) of functions on the boundary S = ∂D .This is not an orthogonal decomposition with respect to the natural scalar product on the space offunctions on the disk. Thus we have a non-trivial Jacobian in the integration measure:[ Df ] = [ Df h ] [ Df N ] J ac − , which is a some constant. Note that the following relation holds: Z D ℓ +1 X i =1 df i ∧ ∗ dη i = Z D ℓ +1 X i =1 η iN ∗ ∆ f i ,N − Z S ℓ +1 X i =1 η ih ∗ df i ,h . Taking integral over η i ,N and η i ,h we obtain Z f ( φ + ) = 1 J ac Z [ Df ] ′ ℓ +1 Y i =1 δ (∆ f i ,N ) δ ( ∗ df i ,h )9 exp n Z D ℓ +1 X i,j =1 ∂ W + ∂φ i + ∂φ j + d ( f i ,N + f ,h ) ∧ d ( f j ,N + f ,h ) o = 1 J ac det ′A N ∆ det ′ Fun( S ) ( ∗ d ) . Now let us calculate the functional integral (2.11). The calculation is basically the same as in thecase of Z f . The only difference (apart of the fact that Jacobins and determinants appear inverse)is that the integral over constant mode of φ j − is present and is eaten up by the delta-functioninsertion. On the other hand the integral over constant mode of φ j + remains. Taking into accountthe cancelation of the Jacobians and determinants for fermions and bosons the total integral isequal to Z = Z R ℓ +1 ℓ +1 Y j =1 dt j e ~ W + ( t ) , where t j are constant modes of the fields φ j + . ✷ Corollary 2.1
The correlation function of the observable (2.9) in the type B topological S -equivariant linear sigma model (2.7) with the superpotential W (0)+ ( φ + ) = ℓ +1 X j =1 ( λ j φ j + − e φ j + ) , (2.14) is given by the following product of the Γ -functions hO ∗ (0) i W (0)+ = ℓ +1 Y j =1 ~ λj ~ Γ (cid:18) λ j ~ (cid:19) . (2.15) Proof . Using the result of the previous Theorem for the superpotential (2.14) we straightforwardlyhave hO ∗ (0) i W (0)+ = Z R ℓ +1 ℓ +1 Y j =1 dt j e ~ P ℓ +1 j =1 ( λ j t j − e tj ) = ℓ +1 Y j =1 ~ λj ~ Γ (cid:18) λ j ~ (cid:19) . ✷ The expression (2.15) is equivalent to the one obtained in type A topological sigma modelconsidered in [GLO1]. The coincidence of a particular correlation functions in type A modelconsidered in [GLO1] and the correlation function from Corollary 2.1 is a manifestation of themirror symmetry between two underlying sigma models. Without taking into account the involved S -equivariance, the mirror correspondence between the two models follows from the results of [HV].In particular the exponential terms in the superpotential (2.14) are attributed to the summationover instantons in type A sigma model. In the following Section we provide heuristic arguments forthe mirror symmetry between the topological theory considered in this note and the one consideredin [GLO1]. A and B topological sigma models As it was demonstrated in the previous Section the Euler integral representation of the Γ-functionΓ( s ) = Z + ∞−∞ dx e xs e − e x , Re( s ) > , (3.1)10aturally arises as a particular correlation function in a certain S -equivariant type B topologicalsigma model on the disk D . In [GLO1] it was argued that this integral representation is dual tothe representation of the Γ-function as an equivariant symplectic volume of an infinite-dimensionalspace. The natural framework for this duality is a mirror symmetry. Below we establish a directrelation of the Euler integral representation (3.1) of the Γ-function with the representation of the Γ-function as an equivariant symplectic volume of an infinite-dimensional space proposed in [GLO1].We also discuss an explicit mirror map between the type A equivariant topological linear sigmamodel considered in [GLO1] and the type B equivariant topological sigma model considered in theprevious Sections. Finally we elucidate the appearance of the non-standard real structure (2.4) ina simple example of the mirror map for a sigma model on P with the target space being an infinitecylinder C ∗ = R × S . In this Subsection we derive the Euler integral representation of the Gamma-function (3.1) applyingthe Duistermaat-Heckman fixed point formula to the infinite-dimensional integral representationfor the Gamma function proposed in [GLO1].Let us start with recalling the functional integral representation of the Γ-function as an equiv-ariant symplectic volume from [GLO1]. Let M ( D, C ) be a space of holomorphic maps of the disk D = { z ∈ C | | z | ≤ } into the complex plane C . An element of M ( D, C ) can be described as acomplex function ϕ ( z, ¯ z ) on D , satisfying the equation ∂ ¯ z ϕ ( z, ¯ z ) = 0 . (3.2)We denote the complex conjugated function by ¯ ϕ ( z, ¯ z ). Define a symplectic form on the space M ( D, C ) as follows Ω = ı π Z π δϕ ( σ ) ∧ δ ¯ ϕ ( σ ) dσ, (3.3)where ϕ ( σ ), ¯ ϕ ( σ ) are restrictions of ϕ ( z, ¯ z ), ¯ ϕ ( z, ¯ z ) to the boundary ∂D = S and σ is a coordinateon the boundary such that σ ∼ σ + 2 π . The symplectic form (3.3) is invariant with respect to theaction of the group S of loop rotations and to the action of U (1) induced from the standard actionof U (1) on C ϕ ( z ) −→ e ıα ϕ ( z ) , ¯ ϕ (¯ z ) −→ e − ıα ¯ ϕ (¯ z ) , e ıα ∈ U (1) , (3.4) ϕ ( z ) −→ ϕ ( e ıβ z ) , ¯ ϕ (¯ z ) −→ ¯ ϕ ( e − ıβ ¯ z ) , e ıβ ∈ S . (3.5)Let ~ and λ be generators of the Lie algebras of S and U (1) correspondingly. The action of S × U (1) on ( M ( D, C ) , Ω) is Hamiltonian and the corresponding momenta are given by H S = − ı π Z π ¯ ϕ ( σ ) ∂ σ ϕ ( σ ) dσ, H U (1) = 14 π Z π | ϕ ( σ ) | dσ. (3.6)The S × U (1)-equivariant volume of M ( D, C ) is defined formally as follows [GLO1]. Let χ ( z, ¯ z )and ¯ χ ( z, ¯ z ) be a pair of complex conjugated odd functions satisfying the equations ∂ ¯ z χ ( z, ¯ z ) = 0 , ∂ z ¯ χ ( z, ¯ z ) = 0 . (3.7)11he functions ( χ ( z, ¯ z ) , ¯ χ ( z, ¯ z )) can be considered as a section of the odd tangent bundle Π T M ( D, C )to M ( D, C ). Using the standard correspondence between differential forms on a manifold X andthe functions on the odd tangent bundle Π T X one can write down the symplectic form (3.3) asfollows: Ω = ı π Z π dσ χ ( σ ) ¯ χ ( σ ) . Below we freely use the equivalence between differential forms and functions on superspaces withoutfurther notice.The S × U (1)-equivariant volume of the space of holomorphic maps M ( D, C ) is given by thefollowing functional integral: Z ( λ, ~ , µ ) = Z Π T M ( D, C ) dm ( ϕ, χ ) e µ ( λH U (1) + ~ H S +Ω) , Re( µ ) < , (3.8)where H S , H U (1) are given by (3.6), and dm ( ϕ, χ ) is a canonical integration measure on thesuperspace Π T M ( D, C ) defined in [GLO1]. The integral (3.8) is an infinite-dimensional Gaussianintegral and is understood using the zeta-function regularization. Note that in general, regularizedinfinite-dimensional integrals depend on auxiliary parameters defined by a particular choice of aregularization scheme. For the integral (3.8) this leads to the following general dependence on aregularization scheme [GLO1]: Z ( λ, ~ , µ ) = A ( µ ) B ( µ ) λ ~ Γ (cid:18) λ ~ (cid:19) , (3.9)where A ( µ ) and B ( µ ) are some λ -independent functions. Thus taking into account the dependenceon a choice of a regularization scheme it is natural to consider the S × U (1)-equivariant volumeof the space of holomorphic maps M ( D, C ) (and thus in particular the Gamma-function) as a R ∗ × R + -torsor. The regularization scheme we use below leads to a particular choice of A and B .In [GLO1] the integral (3.8) was expressed in terms of infinite-dimensional determinant andno obvious relation with the Euler integral representation (3.1) was given. Below we consider aheuristic derivation of (3.9) using an infinite-dimensional version of the Duistermaat-Heckman fixedpoint formula [DH]. In this derivation the Euler integral representation (3.1) appears in a naturalway.To proceed let us first recall a construction of a projective space P N as the Hamiltonian reductionof a symplectic manifold ( C N +1 , ω C N +1 ) by the Hamiltonian action of the group U (1). Here thesymplectic form ω C N +1 is given by ω C N +1 = ı N +1 X j =1 dz j ∧ d ¯ z j , (3.10)and the U (1) action e ıα : z j −→ e ıα z j , e ıα ∈ U (1) , j = 1 , . . . , N + 1 , (3.11)is generated by the vector field v = N +1 X i =1 ı n z i ∂∂z i − ¯ z i ∂∂ ¯ z i o . H U (1) corresponding to the Hamiltonian action (3.11) is defined by the equation ι v ω = − dH U (1) and is given by H U (1) = P N +1 j =1 | z j | . Projective space P N can be realized as aHamiltonian reduction of ( C N +1 , ω C N +1 ) by U (1) P N = n z ∈ C N +1 (cid:12)(cid:12)(cid:12) H U (1) ( z, ¯ z ) = 12 r o. U (1) , r ∈ R . (3.12)Thus constructed P N has a canonical symplectic structure ω P N proportional to the Fubini-Studyform. In terms of inhomogeneous coordinates w j = z j /z N +1 , z N +1 = 0 it is given by ω P N = ır P Ni =1 | w i | ) P Nj =1 dw j ∧ d ¯ w j − P Ni,j w i ¯ w j dw j ∧ d ¯ w i (1 + P Ni =1 | w i | ) . (3.13)The symplectic space ( C N +1 , ω C N +1 ) allows also the Hamiltonian action of the group U (1) N +1 z i z i e ıα i , e ıα i ∈ U (1) i , i = 1 , . . . , N + 1 , (3.14)generated by vector fields v i = ı n z i ∂∂z i − ¯ z i ∂∂ ¯ z i o , i = 1 , . . . , N + 1 . Solving the equations ι v i ω C ℓ +1 = − dH i we find the corresponding momenta H i = 12 | z i | , i = 1 , . . . , N + 1 . The action of U (1) N +1 descents to the Hamiltonian action on ( P N , ω P N ) with the correspondingmomenta H P N j = r | w j | P Nj =1 | w j | , j = 1 , . . . N, (3.15)and H P N N +1 = r P Nj =1 | w j | . (3.16) Lemma 3.1
The following identity holds: πµ Z C N +1 δ (cid:16) H U (1) − r / (cid:17) e µ ( ω C N +1 + P N +1 j =1 λ j H j ) = Z P N e µ ( ω P N + P N +1 j =1 λ j H P Nj ) , (3.17) where ω P N is given by (3.13) and the reduced Hamiltonians H P N j are given by (3.15) and (3.16) .Proof. Let us introduce new variables w j = z j /z N +1 , j = 1 , . . . , N and t = | z N +1 | , θ = ı ln z N +1 ¯ z N +1 ,so that z N +1 = √ t e ıθ . Then we have µ N π (cid:16) ı (cid:17) N +1 Z C N +1 N +1 ^ i =1 dz i ∧ d ¯ z i δ (cid:16) N +1 X i =1 | z i | − r (cid:17) e µ P N +1 j =1 λ j H j = µ N r N π (cid:16) ı (cid:17) N Z π dθ Z ∞ dt t N Z C N N V n =1 ( dw n ∧ d ¯ w n )1 + P | w n | δ (cid:16) t − r P | w n | (cid:17) e µ P N +1 j =1 λ j H j = µ N r N (cid:16) ı (cid:17) N Z C N N V n =1 ( dw n ∧ d ¯ w n ) (cid:0) P | w n | (cid:1) N +1 e µ P N +1 j =1 λ j H P Nj . ω N P N N ! = r N (cid:16) ı (cid:17) N N V n =1 ( dw n ∧ d ¯ w n ) (cid:0) P | w n | (cid:1) N +1 , we obtain the identity (3.17). ✷ We shall use an infinite-dimensional analog of the identity (3.17) to calculate the integral (3.8).Let us rewrite the integral (3.8) as follows: Z ( λ, ~ , µ ) = Z + ∞−∞ dt e µλt Z t ( ~ , µ ) , Z t ( ~ , µ ) = Z M ( D, C ) e µ ( ~ H S +Ω) δ ( t − H U (1) ) . (3.18)Now taking into account (3.17) we can interpret Z t ( ~ , µ ) as an integral over the infinite-dimensionalprojective space P M ( D, C ) Z t ( ~ , µ ) = 2 πµ Z P M ( D, C ) e µ ( ~ ˜ H S +Ω( t )) , (3.19)where Ω( t ) is an induced symplectic form on P M ( D, C ) and ˜ H S is a momentum corresponding tothe S -action on P M ( D, C ). We should stress that the integral in (3.19) is an infinite-dimensionalone and thus requires a proper regularization which will be discussed below.To calculate the integral (3.19) we use an infinite-dimensional version of the Duistermaat-Heckman formula [DH] (for a detailed introduction into the subject see e.g. [Au]). Let M be a2 N -dimensional symplectic manifold with the Hamiltonian action of S having only isolated fixedpoints. Let H be the corresponding momentum. The tangent space T p k M to a fixed point p k ∈ M S has a natural action of S . Let v be a generator of Lie( S ) and let ˆ v be its action on T p k M . Thenthe following identity holds: Z M e µ ( ~ H + ω ) = X p k ∈ M S e µ ~ H ( p k ) det T pk M ~ ˆ v/ π . (3.20)Let us formally apply (3.20) to the integral (3.19). A set of fixed points of S acting on P M ( D, C )can be easily found using linear coordinates on M ( D, C ) (considered as homogeneous coordinateson P M ( D, C )). Let ϕ ( z ) be a holomorphic map of D to C . It represents an S -fixed point on P M ( D, C ) if rotations by S can be compensated by an action of U (1) e ıα ( β ) ϕ ( e ıβ z ) = ϕ ( z ) , β ∈ [0 , π ] . (3.21)It is easy to see that solutions of (3.21) are enumerated by non-negative integers and are given by ϕ ( n ) ( z ) = ϕ n z n , ϕ n ∈ C ∗ n ∈ Z ≥ . (3.22)The tangent space to M ( D, C ) at an S -fixed point ϕ ( n ) has natural linear coordinates ϕ m /ϕ n , m ∈ Z ≥ , m = n where coordinates ϕ k , k ∈ Z ≥ are defined by the series expansion of ϕ ∈ M ( D, C ) ϕ ( z ) = ∞ X k =0 ϕ k z k . After identification of ~ in (3.19) with a generator of Lie( S ) its action on the tangent space at thefixed point is given by a multiplication of each ϕ m /ϕ n on ( m − n ). Thus to define an analog of the14enominator in the right hand side of the Duistermaat-Heckman formula (3.20) one should providea meaning to the infinite product Q ∞ m =0 ,m = n ~ ( m − n ) / π . We use a ζ -function regularization (seee.g. [H] and also Appendix in [GLO1])ln h Y m ∈ Z ≥ ,,m = n ~ π ( m − n ) i a := − ∂∂s n X m =1 e − ıπs ( a ~ m/ π ) s + ∞ X m =1 a ~ m/ π ) s !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s → , (3.23)where a is a normalization multiplier. The introduction of a is to take into account a multiplicativeanomaly det( AB ) = det A · det B appearing for generic operators A and B . We specify a at thefinal step of the calculation of (3.19). Lemma 3.2
The regularized product (3.23) is given by hQ m ∈ Z ≥ ,m = n ~ ( m − n ) / π i a = ( − n ( a ~ / π ) − n n ! √ a ~ π . (3.24) Proof . Using the Riemann ζ -function ζ ( s ) = ∞ X n =1 n s , one can express the right hand side of (3.23) as follows:ln h Y m ∈ Z ≥ ,,m = n ~ π ( m − n ) i a = ( ζ (0) + n ) ln a ~ / π + ln n ! − ζ ′ (0) + ıπn. Taking into account ζ (0) = − and ζ (0) ′ = − ln 2 π we obtain (3.24). ✷ Let us now calculate the difference of the values of S -momentum map ˜ H S at two S -fixedpoints ϕ ( n ) , ϕ (0) ∈ P M ( D, C ). Consider an embedded projective line P ⊂ P M ( D, C ), containing ϕ ( n ) and ϕ (0) . Let us choose homogeneous coordinates [ z : z ] on P such that ϕ (0) = [1 : 0] and ϕ ( n ) = [0 : 1]. The action of S on P M ( D, C ) descends to the embedded P via the vector field V = ın n w ∂∂w − ¯ w ∂∂ ¯ w o , w = z /z . (3.25)The pull back of the symplectic form Ω( t ) is given by ω P = ıt dw ∧ d ¯ w (1 + | w | ) . The action of the vector field (3.25) on P is the Hamiltonian one. Let H ( n ) S be the correspondingmomentum given by a restriction of the momentum ˜ H S for S -action P M ( D, C ). From thedefinition of the momentum map we have H ( n ) S ( ϕ ( n ) ) − H ( n ) S ( ϕ (0) ) = Z [0:1][1:0] dH ( n ) S = − Z [0:1][1:0] ι V ω P . (3.26)A momentum defined as a solution of the equation i V ω = − dH is unique up an additive constant.To fix this constant we normalize the momentum ˜ H S ( ϕ ) so that H S ( ϕ (0) ) = 0. Thus we obtainthe following: H ( n ) S ( ϕ ( n ) ) = nt Z [0:1][1:0] wd ¯ w + ¯ wdw (1 + | w | ) = − nt h | w | ) i ∞ = nt . (3.27)15ubstituting (3.27) and (3.24) into (3.20) for M = P M ( D, C ) we obtain Z t ( ~ , µ ) = 2 πµ s a ~ (2 π ) ∞ X n =0 ( − n e ntµ ~ ( a ~ / π ) n n ! = µ √ a ~ exp n − πa ~ e µ ~ t o , (3.28)where the dependence on the normalization constant a reflects an ambiguity of the regularizedinfinite-dimensional integral. Taking into account (3.18), the regularized S × U (1)-equivariantsymplectic volume of M ( D, C ) is given by Z reg ( λ, ~ , µ ) = Z ∞ dt e µλt Z t ( ~ , µ ) = µ √ a ~ Z ∞ dt e µλt e − πa ~ e µ ~ t = (cid:16) a ~ (cid:17) / (cid:18) a ~ π (cid:19) λ ~ Z + ∞− ln( a ~ / π ) du e λ ~ u e − e u , (3.29)where u = µ ~ t − ln( a ~ / π ). To get rid of the renormalization ambiguity we take the limit a → + ∞ in the following way: Z ( λ, ~ ) = lim a → + ∞ (cid:16) a ~ (cid:17) − / (cid:16) a π (cid:17) − λ ~ Z reg ( M ; λ, ~ ) = ~ λ ~ Γ (cid:16) λ ~ (cid:17) . (3.30)Thus we show that the formal application of the Duistermaat-Heckman formula to the infinite-dimensional integral (3.8) in the form (3.18) leads to the Euler integral representation (3.1) of theΓ-function and reproduces the results of Section 2. C In this Subsection we consider an explicit mirror map of the type A topological sigma modelconsidered in [GLO1] to the type B topological sigma model considered in Section 1.In the previous Sections we take into account the action (3.4) of U (1) on the symplectic space( M ( D, C ) , Ω) of holomorphic maps of the disk D into the complex plane C . Now we introducea larger infinite-dimensional group acting on ( M ( D, C ) , Ω) in a Hamiltonian way. The space( M ( D, C ) , Ω) supports the Hamiltonian action of a commutative Lie algebra G = Map( S , R ) ofreal functions on S given by α · ϕ ( σ ) = ı [ α ( σ ) ϕ ( σ )] + , α · ¯ ϕ ( σ ) = − ı [ α ( σ ) ¯ ϕ ( σ )] − , where α ( σ ) ∈ G and ϕ ( σ ), ¯ ϕ ( σ ) are restrictions of ϕ ( z ), ¯ ϕ (¯ z ) to the boundary S = ∂D . Theprojectors [ ] ± are defined as follows:[ e ınσ ] + = e ınσ , n ≥ , [ e ınσ ] + = 0 , n < , [ e ınσ ] − = e ınσ − [ e ınσ ] + . Given a Hamiltonian action of G one can define corresponding momentum map of M ( D, C ) intothe dual to the Lie algebra G . The value of the momentum on the element α ( σ ) of the Lie algebra G is given by H G ( α ) = Z π dσ α ( σ ) H G ( ¯ ϕ ( σ ) , ϕ ( σ )) , H G ( ¯ ϕ ( σ ) , ϕ ( σ )) = 14 π | ϕ ( σ ) | . (3.31)Note that the subalgebra u (1) ⊂ G corresponding to α ( σ ) = const coincides with the Lie alge-bra of the group U (1) considered in the previous Subsection. The momenta (3.31) motivate anintroduction of a new parametrization of M ( D, C ) ϕ ( σ ) = τ / ( σ ) e ıφ ( σ ) , ¯ ϕ ( σ ) = τ / ( σ ) e − ıφ ( σ ) , τ ( σ ) = | ϕ ( σ ) | , φ ( σ ) = − ı (cid:18) ϕ ( σ )¯ ϕ ( σ ) (cid:19) . (3.32)Note that thus defined τ ( σ ) is constraint by the condition to be a restriction to the boundary S of the square module of a holomorphic function on D . Also let us stress that φ ( σ ) given by (3.32)is not single-valued. Indeed let ϕ ( n ) ( z ) = p n ( z ) ϕ (0) ( z ) be a holomorphic function on D such that p n ( z ) = Q nj =1 ( z − a j ), a j ∈ D is a polynomial of degree n and ϕ (0) ( z ) is a holomorphic functionwithout zeroes inside D . Then we have for the corresponding function φ ( σ ) φ ( n ) ( σ + 2 π ) = φ ( n ) ( σ ) + 2 πn, n ∈ Z ≥ . (3.33)Hence the space of holomorphic maps has the following decomposition (modulo subspaces of non-zero codimension): M ( D, C ) = ∪ ∞ n =0 M ( n ) ( D, C ) , (3.34)where M ( n ) ( D, C ) includes holomorphic maps ϕ ( z ) such that for the corresponding function φ therelation (3.33) holds. We would like to reformulate the integral (3.8) using new variables (3.32) andthe decomposition (3.34). Let us decompose the space of fields τ ( σ ) on the subspace of constantmodes τ ( σ ) = 2 t and the orthogonal subspace of τ ∗ ( σ ) such that R S dσ τ ∗ ( σ ) = 0.For ϕ ∈ M ( n ) ( D, C ) the momenta (3.31) for U (1)- and S -actions in the new variables ( τ, φ )are given by H U (1) = 14 π Z π dσ τ ( σ ) = t, H S = 14 π Z π dσ τ ( σ ) ∂ σ φ ( σ ) = − π Z π dσ ∂ σ τ ( σ ) φ ( σ ) + nt, where we take into account (3.33). Thus we have the following equivalent representation for (3.8) Z ( λ, ~ , µ ) = + ∞ X n =0 Z M ( n ) ( D, C ) dt [ Dτ ∗ ] [ Dφ ] J ( τ ∗ + t ) e − µ π R S dσ ~ ∂ σ τ ∗ φ + µt ( ~ n + λ ) , (3.35)where J ( τ ∗ + t ) is a Jacobain of the transformation from the variables ( ϕ, ¯ ϕ ) to the variables ( τ, φ ).The integration over φ leads to a delta-function with a support on the space of solutions of theequation ∂ σ τ ( σ ) = 0 , τ ( σ ) = | ϕ ( z ) | | z = e ıσ , (3.36)where ϕ ( z ) is a holomorphic function on the disk D . The solutions of (3.36) are given by ϕ ( n ) ( z ) = ϕ n z n , n ∈ Z ≥ (3.37)and coincide with the fixed points (3.22) of the S -action on P M ( D, C ). Thus the sum over n for afixed t is an analog of the sum over S -fixed points entering Duistermaat-Heckman formula appliedto P M ( D, C ). It remains to integrate the delta-function δ ( ∂ σ τ ) in the vicinity of each solution(3.37) taking into account that τ ( σ ) is a square of a holomorphic function such that the integral R π dστ ( σ ) = 2 t is fixed. Actually we already evaluated this integral which is equivalent to theregularized product (3.24) entering the Duistermaat-Heckman formula. Thus we obtain Z ( λ, ~ , µ ) reg = µ √ a ~ ∞ X n =0 Z + ∞ dt ( − n ( a ~ / π ) n n ! e tµ ( ~ n + λ ) = µ √ a ~ Z + ∞ dt e µtλ − πa ~ e µ ~ t . (3.38)17ote that to make the integral (3.38) well-define we should sum the series for an appropriaterange of the variables µ and a . The integral (3.38) reproduces the regularized integral (3.29).Taking appropriate limit (3.30) we recover the expression obtained using the Duistermaat-Heckmanformula.Using the evaluation of the integral (3.35) near the solutions (3.37) and summing the series onecan rewrite (3.35) in the following form: Z ( λ, ~ , µ ) reg = Z ∞ dt Z [ Dτ ∗ ] det ∆ δ (∆ τ ∗ ) δ ( ∂ σ τ ∗ | S ) e µtλ − πa ~ e µ ~ t , (3.39)where ∆ is a Laplace operator on the disk D and now the functional integral is taken over the spaceof real functions on the disk orthogonal to the subspace of constant functions. It is easy to see thatthe integral over τ ∗ reduces to an additional t -independent factor for Z ( λ, ~ , a ) reg . Combining thevariables t and τ ∗ into a new variable τ = τ ∗ + t − ~ − ln( a ~ / π ) and taking the limit a → + ∞ weobtain the following: Z ( λ, ~ , µ ) = 1 ~ lim a →∞ C ( a, ~ ) a − λ/ ~ Z ( λ, ~ , a ) reg (3.40)= Z [ Dτ ] det ∆ δ (∆ τ ) δ ( ∂ σ τ | S ) e π R π dσ ( µλτ ( σ ) − e ~ µτ ( σ ) ) , where C ( a, ~ ) is an appropriate function. Let us note that the integral representation (3.40) can bedirectly derived from (3.35) in the limit a → + ∞ . Indeed, in the limit a → ∞ (taking into accountthe shift t → t − ~ − ln( a ~ / π )) the Jacobain becomes field independent and the condition on thefunction τ to be the square of a holomorphic function reduces to the harmonicity condition on τ due to the expansion∆ ln( τ − ~ − ln a ~ / π ) = − ~ ln a ~ / π ∆ τ ∗ + · · · , a → + ∞ . The summation over n with the weight factor obtained by a proper integration over n zeroes of τ leads to the exponential term in (3.40).To make a contact with the representation of the equivariant volume integral (3.8) in termsof type B topological sigma model described in Section 1 we note that the condition ∂ σ τ | S = 0imposed on restrictions of harmonic functions to the boundary S = ∂D is equivalent to thecondition ∂ n τ | S = 0 where ∂ n is a normal derivative to the boundary of D . Therefore we have Z ( λ, ~ , µ ) = Z [ Dτ ] det ∆ δ (∆ τ ) δ ( ∂ n τ | S ) e π R π dσ ( µλτ ( σ ) − e µ ~ τ ( σ ) ) . (3.41)The δ -functions can be replace by an integral over an auxiliary field κ ( σ ). Thus we obtain thefollowing integral representation: Z ( λ, ~ , µ ) = Z [ Dτ ] [ Dκ ] det ∆ e R D ı dκ ∧∗ dτ + R S dσ ( µλτ ( σ ) − e ~ µτ ( σ ) ) δ ( κ (0)) . (3.42)This functional integral is equivalent to the one entering the formulation of the Corollary 2.1 for ℓ = 0 with τ = φ + and κ = φ − . This can be demonstrated by integrating over the fields η , θ and ρ in the type B model considered in previous Section.18 .3 T -duality for target space C ∗ Finally we clarify the appearance of the non-standard real structure in the topological type B -modelproposed in Section 1 as a mirror dual to the topological type A -model considered in [GLO1]. Toelucidate this issue we consider a simple example of the bosonic sigma model on P with the targetspace C ∗ = R × S . The mirror symmetry in this case is straightforwardly realized as a T -dualitywith respect to S . We will demonstrate below that starting with a sigma model similar to the oneconsidered in [GLO1] we obtain after T -duality the topological sigma model with the real structureon the space of fields considered in Section 1.Let us given the following action functional S = Z P (cid:18) t F ∧ ∗ F + F ∧ ∂ ¯ ϕ − F ∧ ∂ϕ (cid:19) (3.43)= Z P (cid:16) t F ∧ ∗ F − ıF ∧ ∗ dτ − ıF ∧ dφ (cid:17) , (3.44)where ϕ = τ + ıφ is a complex coordinate on the cylinder R × S , φ ∼ φ + 2 π and F = ¯ F z dz + F ¯ z d ¯ z is a real valued one-form. We imply that P is supplied with the K¨ahler metric associated with thestandard K¨ahler form ω = ı dz ∧ d ¯ z (1 + | z | ) . Note that (in the classical theory) the action (3.43) does not depend on the choice of the two-dimensional K¨ahler metric. This action (3.43) is a part of an action of the topological sigma modelconsider in [GLO1] adopted to the case of the target space C ∗ . Indeed, the integration over F givesthe standard functional integral for the sigma-model Z = Z [ DF ][ Dϕ ] e − S = Z [ DF ¯ z ][ D ¯ F z ][ Dϕ ] exp n − Z P d z (cid:0) t ¯ F z F ¯ z − ıF ¯ z ∂ z ¯ ϕ − ı ¯ F z ∂ ¯ z ϕ (cid:1) o = C ( t ) Z [ Dϕ ] exp n − t − Z P d z ∂ z ¯ ϕ ∂ ¯ z ϕ o , (3.45)where d z = ıdz ∧ d ¯ z and C ( t ) is a function of t .The standard way to implement T -duality is to introduce an auxiliary field B = B z dz + B ¯ z d ¯ z and κ and consider a theory with the following action: S = ı Z P dκ ∧ B + Z P (cid:18) t F ∧ ∗ F − ıF ∧ ∗ dτ − ıF ∧ B (cid:19) . (3.46)Indeed integrating over κ leads to a constraint B = dφ where φ is a real valued field and thus wecome back to the action (3.44). On the other hand, integration over B leads to the action S = Z P (cid:18) t F ∧ ∗ F − ıF ∧ ∗ dτ (cid:19) , with the constraint F = dκ. (3.47)19hus the integration over F with the constraint (3.47) gives S = Z P (cid:18) t dκ ∧ ∗ dκ − ıdκ ∧ ∗ dτ (cid:19) . (3.48)In [GLO1] we consider a sigma-model without F ∧ ∗ F -term (i.e. we imply that t = 0). Taking t = 0 in (3.48) we obtain S = − ı Z P dκ ∧ ∗ dτ. (3.49)This action is precisely the two-derivative term in (2.2) where the role of κ and τ is played by thefields φ + and φ − . Thus the non-standard real structure on the fields in (2.2) is a consequence oftaking a limit t → t = 0 in (3.44) and integrating out φ . Let us finally notethat the action (3.49) arising in the limit t → To conclude this note we briefly outline some directions for future research. The constructionsof [GLO1] and of this note allow several straightforward generalizations. For instance one canconsider an equivariant type A topological sigma model on a disk D with a compact target spacebeing (partial) flag manifolds. Their mirror dual type B topological theories are also know [Gi].Simple examples are provided by projective spaces P ℓ and more generally Grassmannian spaces Gr ( m, ℓ + 1). Such topological sigma models can be described in terms of a twisting of N = 2SUSY gauged linear sigma models [W3], [MP]. For instance in the case of the target space X = P ℓ the corresponding linear sigma model has target space C ℓ +1 gauged by the diagonal action of U (1).For its mirror dual see for example [HV]. An analog of the correlation functions considered in[GLO1] but for the target space P ℓ should be equal to a degenerate gl ℓ +1 -Whittaker function givenby Ψ λ ,...,λ j +1 ( x ) = Z C dγ e ıγx ℓ +1 Y j =1 Γ (cid:18) γ − λ j ~ (cid:19) . (4.1)For a detailed discussion of the relation of (4.1) to Toda chains see [GLO2]. The same expressionshould be equal to an analog of the correlation function in mirror dual type B equivariant topologicalsigma model with the target space C ℓ and a superpotential W ( φ ) = P ℓj =1 (( λ j − λ ℓ +1 ) φ j − e φ j ) − e x − P ℓk =1 φ k . The structure of the integral (4.1) is quite transparent. The product of Γ-functionsis a correlation function in the type A topological sigma model with the target space C ℓ +1 of thetype considered in [GLO1] (as well as a correlation function in the mirror dual type B theory) andthe integral over γ is a projection corresponding to an integration over the fields in the topological U (1)-gauge multiplet (over dual scalar topological multiplet in the mirror dual type B theory).Similar reasoning can be applied to the case of the Grassmannian target space [W4]. We willprovide a detailed discussion of these cases in [GLO3]. The case of general partial flag manifolds isa bit more complicated but accessible by the technique developed in [GLO4] and will be discussedelsewhere.Let us stress that the discussed examples of explicit calculations of particular correlation func-tions in topological theories on non-compact manifolds is not restricted to the case of dimension20wo. 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Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia;School of Mathematics, Trinity College, Dublin 2, Ireland;Hamilton Mathematics Institute, Trinity College, Dublin 2, Ireland;
D.L.
Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia ; E-mail address : [email protected] S.O.