Double Field Theory and Geometric Quantisation
PPrepared for submission to JHEP
QMUL-PH-21-04
Double Field Theory and Geometric Quantisation
Luigi Alfonsi and David S. Berman
Centre for Research in String Theory, School of Physics and Astronomy,Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK
E-mail: [email protected] , [email protected] Abstract:
We examine various properties of double field theory and the doubled stringsigma model in the context of geometric quantisation. In particular we look at T-dualityas the symplectic transformation related to an alternative choice of polarisation in theconstruction of the quantum bundle for the string. Following this perspective we adopt avariety of techniques from geometric quantisation to study the doubled space. One applica-tion is the construction of the "double coherent state" that provides the shortest distancein any duality frame and a "stringy deformed" Fourier transform.
Keywords:
Double Field Theory, geometric quantisation, T-duality, background indepen-dence, symplectic geometry, non-commutative geometry a r X i v : . [ h e p - t h ] F e b ontents Geometric quantisation provides an approach to quantisation that is underpinned by thesymplectic geometry of phase space. Its emergence in the 1970s from the work of Kostantand Souriau has produced a geometric approach to quantisation that provides numerous– 1 –nsights into the quantisation procedure. In particular, it showed how the symplectic sym-metry of phase space is broken in naive quantisation methods even though the physics is leftinvariant and how the underlying symplectic symmetry maybe restored (or even extendedto the metaplectic group). In more mundane language, classical Hamiltonian physics is in-variant under canonical transformations and yet the wavefunctions of quantum mechanicsare functions of just half the coordinates of phase space and thus not symplectic represen-tations. A key part of quantum mechanics is that physics cannot depend on the choice ofbasis of wavefunctions. We can transform between the coordinate and momentum basisand the physics is invariant. In fact, the coordinate and momentum representations aremutually non-local and to move between different bases requires a nonlocal transformation(this is the Fourier transform, in a free theory). This is highly analogous to duality sym-metries in field theories. To take the example of Maxwell theory, electromagnetic dualityis manifest in the Hamiltonian form of the theory, the transformation between electric andmagnetic variables maybe generated by symplectic transformations and yet from the La-grangian perspective (where the duality is not manifest) the electric and magnetic variablesare mutually non-local and related via a Fourier transform [1].The recovery of manifest symplectic symmetry in quantisation was an early motivator tointroduce new quantisations methods. First, Weyl [2] in 1927 and then Groenewold [3] andMoyal [4] in the 1940s produced reformulations of quantum mechanics to make the sym-plectic symmetry of the classical Hamiltonian form a manifest symmetry in quantisation.Out of this came the ideas of deformation quantisation where the algebra of functions isdeformed.From a contemporary perspective the emergence of duality in string theory has thrown upa similar set of challenges. As is well known, the Hamiltonian of the string in toroidalbackgrounds has a symmetry that is not manifest in the Lagrangian formulation. T-dualitymay be generated by canonical transformations of the string phase space, see [5, 6] but fromthe traditional Lagrangian world sheet perspective the world sheet fields and their T-dualsare mutually non-local.Double Field theory, first introduced by Siegel [7, 8] and then as a truncation of string fieldtheory by Hull and Zwiebach [9], reformulates the string background spacetime in a wayto make T-duality a manifest symmetry. It does so, as the eponymous title of the theorysuggests, by doubling the dimensions of space time to include both duality perspectivesinto one object. See [10–12] for reviews of this approach. From the perspective of thispaper, the doubling will be just returning to the original phase space of the string. This isrelated to the work of Tseytlin [13–16] and Duff [17–19] where the world sheet theory wasreformulated in a doubled space and T-duality a manifest symmetry.Here we will examine string quantisation using the ideas and techniques of geometric quan-tisation. This will produce an infinite dimensional phase space associated to the loop spaceof the string. The zero modes of the loop space will give us the doubled space of DFTand the choice of polarisation in the quantisation will provide the T-duality frame. Wewill construct transformations between T-dual descriptions based on the transformationsinduced by different choices of polarisation. This will lead to the idea of a coherent statein the doubled that saturates the uncertainty bound on distance in the doubled theory i.e.– 2 –he shortest distance in any duality frame.This quantisation procedure will produce a noncommutative algebra associated to the dou-bled phase space. As usual the noncommutative nature of position and momentum iscontrolled by the dimensionful deformation parameter (cid:126) . Now though in addition we willalso have an additional deformation parameter given by (cid:126) α (cid:48) which controls the noncommu-tativity of coordinates x i and their "duals" ˜ x j . α (cid:48) is the square of the string length scale.Thus the space has two types of noncommutativity or quantisations, one has its origins inthe usual phase space that that requires the traditional choice of polarisation to picking theLagrangian submanifold of phase space and another which requires picking a Lagrangiansubmanifold in the doubled space or in simpler language picking the "duality frame".To make the paper as self contained as possible we will first review the aspects of geometricquantisation that we will use for the string and hopefully foreshadow aspects double fieldtheory.We will then move on to the string loop space description and its phase space followed bythe quantisation and the identification of the doubled space. We will then be equipped touse more of the machinery of geometric quantisation and produce the double coherent stateand the transforms between T-dual frames.We will end with a discussion on further directions and the implications for objects likeT-folds.Note that there has been other related work looking at double field theory from a worldsheet perspective in [20–23] and also [24]. Of particular overlap with the approach in thispaper is the work [25, 26]. Much of this can be found in any one of the books by [27–30] or the recent review by [31].
Hamiltonian mechanics.
Let us recall that a symplectic manifold ( P , ω ) is defined asa smooth manifold P equipped with a closed non-degenerate -form ω ∈ Ω ( M ) , calledthe symplectic form. In Hamiltonian mechanics, a classical system ( P , ω, H ) is definedby a symplectic manifold ( P , ω ) , describing the phase space of the system, and a smoothfunction H ∈ C ∞ ( P ) , called the Hamiltonian . In Newtonian terms, the phase space andthe Hamiltonian encode respectively the kinematics and the dynamics of a classical system.The equations of motion are described Hamilton’s equation as follows: ι X H ω = d H. (1.1)A vector field X H ∈ X ( P ) which solves the Hamilton equation is called Hamiltonian vector for the Hamiltonian H . The flow of a Hamiltonian vector fields describes the motion of theclassical system on the phase space. This means if we choose a starting point γ ∈ M inphase space, the motion of the classical system will be given by the path γ : R −→ P τ (cid:55)−→ γ ( τ ) = e τX H γ (1.2)where τ ∈ R is a -dimensional parameter.– 3 –ocally, on an simply connected open subset U ⊂ P , we can apply Poincaré lemma to thesymplectic form and find ω = d θ , where the local -form θ ∈ Ω ( U ) is called Liouvillepotential. The definition of the Liouville potential is gauge dependent, meaning that anyother choice of potential θ (cid:48) = θ + d λ with λ ∈ C ∞ ( U ) equally satisfies ω = d θ (cid:48) . Now, givena path γ : R → P on the phase space, we define the Lagrangian L H ∈ Ω ( R ) by L H = γ ∗ θ − H d τ . (1.3)Where we denote the pull-back of the Liouville one-form θ to the curve γ by γ ∗ θ . Theaction S H [ γ ( τ )] associated to such a Lagrangian will be given by S H [ γ ( τ )] = (cid:90) R ( γ ∗ θ − H d τ ) . (1.4)For future use, let us notice that we can rewrite γ ∗ θ = ι X H θ d τ , when restricted on thepath γ . Thus note that the choice of Liouville potential effects the Lagrangian description. Classical algebra of observables.
An observable is defined as a smooth function f ∈C ∞ ( P ) of the phase space. Crucially, a symplectic manifold ( P , ω ) is canonically also aPoisson manifold ( P , {− , −} ) , where the Poisson bracket is given as it follows: { f, g } := ω ( X f , X g ) (1.5)for any pair of observables f, g ∈ C ∞ ( P ) . In other words, this means that the observablesof a symplectic manifold ( M, ω ) constitute a Poisson algebra: [ X f , X g ] = X { f,g } . (1.6)Thus, the Hamiltonian vector fields on a symplectic manifold ( P , ω ) constitute a Lie algebra,which we will denote as ham ( P , ω ) . Prequantum geometry.
Let us consider the Lie group U (1) (cid:126) := R / π (cid:126)Z . The pre-quantum bundle Q (cid:16) P is defined as the principal U (1) (cid:126) -bundle, whose first Chern class c ( Q ) ∈ H ( M, Z ) is the image of the element [ ω ] ∈ H ( M, R ) of the de Rham cohomologygroup. We can now define the associated bundle E (cid:16) P to the prequantum bundle withfibre C , i.e. E := Q × U (1) (cid:126) C , (1.7)where the natural action U (1) (cid:126) × C → C is given by the map ( φ, z ) (cid:55)→ e i (cid:126) φ z . Now, the prequantum Hilbert space of the system is defined by H pre := L ( P , E ) , (1.8)i.e. the Hilbert space of L -integrable sections of the bundle E on the base manifold P .Whenever the first Chern class of Q is trivial, then the bundle E = P × C is trivial andthe prequantum Hilbert space reduces to H pre = L ( P ; C ) , i.e. the Hilbert space of L -integrable complex functions. – 4 – uantum algebra of observables. In geometric quantisation, given a classical observ-able f ∈ C ∞ ( P ) , we define a quantum observable ˆ f ∈ Aut( H pre ) by the expression ˆ f := i (cid:126) ∇ V f + f, (1.9)where V f ∈ ham ( P , ω ) is the Hamiltonian vector of the Hamiltonian function f ∈ C ∞ ( P ) and ∇ is the connection on T P given by the Liouville potential θ . By using equations (1.5)and (1.6), we find that the commutator of two quantum observables closes. In particular,given ˆ f and ˆ g , we obtain the commutator [ ˆ f , ˆ g ] = i (cid:126) (cid:92) { f, g } (1.10)where the classical observable of (cid:92) { f, g } is the Poisson bracket { f, g } of the classical observ-ables f and g . Thus, we can use this fact to define the Heisenberg Lie algebra heis ( P , ω ) of quantum observables on our phase space ( P , ω ) . Quantum geometry.
Denote the tangent bundle of phase space by T P . A polarisation ofthe phase space ( P , ω ) is an involutive Lagrangian subbundle L ⊂ T P , i.e. an n -dimensionalsubbundle of T P such that ω | L = 0 and [ V, W ] ⊂ L for any pair of vectors V, W ∈ L .The square root bundle of a line bundle B (cid:16) M is defined as a complex line bundle, whichwe will denote as √B (cid:16) M , equipped with a bundle isomorphism √B ⊗ √B (cid:39) −−→ B whichsends sections √ s ∈ Γ( M , √B ) to √ s ⊗ √ s (cid:55)→ s ∈ Γ( M , B ) .Let us consider the determinant bundle det( L ) := ∧ n L ∗ C of a Lagrangian subbundle L ⊂ T P of our phase space, where n = rank( L ) . We need now to consider the square root bundle (cid:112) det( L ) of the determinant bundle, which comes equipped with the isomorphism (cid:112) det( L ) ⊗ (cid:112) det( L ) (cid:39) −−→ det( L ) (1.11)The choice of square root bundle (cid:112) det( L ) is also related to the metaplectic correction whichleads to the quantum theory forming a representation of the metaplectic group rather thanthe symplectic group.. The quantum Hilbert space is defined by the following space ofsections: H := (cid:110) ψ ∈ L (cid:0) P , E ⊗ (cid:112) det( L ) (cid:1) (cid:12)(cid:12)(cid:12) ∇ V ψ = 0 ∀ V ∈ L (cid:111) . (1.12)If the Lagrangian subbundle L is integrable, we can write L = T M for some n -dimensionalsubmanifold M ⊂ P of the phase space. Then, quantum states | ψ (cid:105) ∈ H can be uniquelychosen of the form | ψ (cid:105) = ψ ⊗ (cid:112) vol M , (1.13)where ψ ∈ L ( M , E ) is a polarised section and √ vol M ∈ Γ( P , (cid:112) det( L )) is the half-formwhose square is a fixed volume form vol M ∈ Ω n ( M ) . The inner product of the Hilbertspace is given by the integral (cid:104) ψ | ψ (cid:105) = (cid:90) M ψ ∗ ψ vol M (1.14)for any couple of quantum states | ψ (cid:105) = ψ ⊗ √ vol M and | ψ (cid:105) = ψ ⊗ √ vol M ∈ H .– 5 –rucially, the Hilbert space defined in (1.12) does not depend on the choice of Lagrangiansubbundle L . If we call H L the quantum Hilbert space polarised along the Lagrangiansubbundle L and H L (cid:48) the one along another Lagrangian subbundle L (cid:48) , we have a canonicalisomorphism H L ∼ = H L (cid:48) . At the end of this section we will explain why this is the case. Example: wave-functions of QM.
To illustrate the ideas in this section lets look at asimple exmaple with ( M, ω ) = ( R n , d p µ ∧ d x µ ) , where { p µ , x µ } are Darboux coordinateson R n . We can now choose the gauge θ = p µ d x µ for the Liouville potential. We havetwo perpendicular polarisations defined by the Lagrangian fibrations L p := Span (cid:16) ∂∂p µ (cid:17) and L x := Span (cid:0) ∂∂x µ (cid:1) . Recall that the covariant derivative is related to the Liouville potentialby ∇ V = V − i (cid:126) ι V θ . In our case, this implies ∇ ∂∂xµ = ∂∂x µ − i (cid:126) p µ ∇ ∂∂pµ = ∂∂p µ . (1.15)Therefore, for the polarisation L p and L x , we obtain respectively the sections | ψ (cid:105) = ψ ( p ) e − ip µ x µ ⊗ (cid:112) d n p | ψ (cid:105) = ψ ( x ) ⊗ √ d n x, (1.16)where √ d n p is the half form such that √ d n p ⊗ √ d n p = d n p and analogously for √ d n x . Let us recall that a symplec-tomorphism between two manifolds ( P , ω ) f −→ ( P (cid:48) , ω (cid:48) ) is a diffeomorphism f : P → P (cid:48) which maps the symplectic form of the first manifold into the symplectic form of the secondone, i.e. such that it satisfies ω = f ∗ ω (cid:48) . According to [32], what in Hamiltonian physicsis known under the name of canonical transformation with generating function F is equiv-alently a symplectomorphism f : ( P , ω ) → ( P (cid:48) , ω (cid:48) ) such that the Liouville potential isgauge-transformed by θ − f ∗ θ (cid:48) = d F . However, this first formalisation can be significantlyrefined. Lagrangian correspondence.
Following [33], there exists a powerful way to formalisea canonical transformation by using the notion of
Lagrangian correspondence . To define aLagrangian correspondence we first need to introduce the graph of a symplectomorphism f : ( P , ω ) → ( P (cid:48) , ω (cid:48) ) , which is the submanifold of the product space P × P (cid:48) given by Γ f := (cid:8) ( a, b ) ∈ P × P (cid:48) (cid:12)(cid:12) b = f ( a ) (cid:9) . (1.17)– 6 –et us call ι : Γ f (cid:44) → P × P (cid:48) the inclusion in the product space. Now, a Lagrangiancorrespondence is defined a correspondence diagram of the form ( P × P (cid:48) , π ∗ ω − π (cid:48)∗ ω (cid:48) )( P , ω ) ( P (cid:48) , ω (cid:48) ) π (cid:48) π f (1.18)where f is a symplectomorphism and π, π (cid:48) are the canonical projections of P ×P (cid:48) onto P , P (cid:48) respectively. The submanifold Γ f ⊂ P × P (cid:48) can be immediately recognised as a Lagrangiansubmanifold of ( P ×P (cid:48) , π ∗ ω − π (cid:48)∗ ω (cid:48) ) , i.e. the total symplectic form vanishes when restrictedon Γ f . In other words, we have ι ∗ (cid:0) π ∗ ω − π (cid:48)∗ ω (cid:48) (cid:1) = 0 . (1.19)To formalise a canonical transformation, we need to add another condition: the corre-spondence space ( P × P (cid:48) , π ∗ ω − π (cid:48)∗ ω (cid:48) ) must be symplectomorphic to a symplectic manifold ( T ∗ M , ω can ) for some manifold M , where ω can ∈ Ω ( T ∗ M ) is just the canonical symplecticform of the cotangent bundle.This implies that we can write the combination of Liouville potentials π ∗ θ − π (cid:48)∗ θ (cid:48) as theLiouville 1-form on P × P (cid:48) ∼ = T M . Since Γ f is Lagrangian, the Liouville potential can betrivialised on T Γ f . In other words, we have the equation π ∗ θ − π (cid:48)∗ θ (cid:48) = d( F ◦ Π) on T Γ f , (1.20)where Π : T ∗ M (cid:16) M is the canonical projection and where the function F ∈ C ∞ ( M ) canbe interpreted as the generating function of the canonical transformation associated to thesymplectomorphism f . Example: canonical transformations.
For clarity, let us consider a simple example.Let us start from symplectic manifolds which are cotangent bundles of configuration spaces,i.e. P = T ∗ M and P (cid:48) = T ∗ M (cid:48) . Thus we can write the Liouville potential as p µ d x µ − p (cid:48) µ d x (cid:48) µ = d F (1.21)in local coordinates on the correspondence space P × P (cid:48) = T ∗ ( M × M (cid:48) ) . We immediatelynotice that, in the notation of the previous paragraph, we have M := M × M (cid:48) . Now thegenerating function F = F ( x, x (cid:48) ) of the canonical transformation of type can be properlyseen as the pullback of a function of the product manifold M × M (cid:48) . Equation (1.21) canbe equivalently written as p µ = ∂F∂x µ , p (cid:48) µ = − ∂F∂x (cid:48) µ . (1.22)In particular, If we choose M, M (cid:48) = R d and F ( x, x (cid:48) ) = δ µν x µ x (cid:48) ν , we recover the symplecticlinear transformation ( x, p ) (cid:55)→ f ( x, p ) = ( p, − x ) .– 7 – anonical transformation on the Hilbert space. So far we formalised canonicaltransformations as symplectomorphisms. Now, we need to show how these symplectomor-phisms give rise to isomorphisms of the corresponding quantum Hilbert spaces.First of all, we must fix a symplectomorphism f : P → P (cid:48) , then we must choose twopolarizations L ⊂ T P and L (cid:48) ⊂ T P (cid:48) which satisfy L = f ∗ ( L (cid:48) ) . Let us call H L and H L (cid:48) the quantum Hilbert spaces corresponding respectively to the L and L (cid:48) polarisations of thephase space.Now, notice that T Γ f ⊂ T ( P × P (cid:48) ) is a Lagrangian submanifold of the Lagrangian corre-spondence space ( P × P (cid:48) , π ∗ ω − π (cid:48)∗ ω (cid:48) ) . As observed by [27], The T Γ f -polarised Hilbertspace H T Γ f of the correspondence space ( P × P (cid:48) , π ∗ ω − π (cid:48)∗ ω (cid:48) ) is isomorphic to the topolog-ical tensor product H T Γ f ∼ = H L (cid:98) ⊗ H ∗ L (cid:48) , which is nothing but the space of Hilbert–Schmidtoperators H L (cid:48) −→ H L . Then, we will obtain the following diagram: H T Γ f H L H L (cid:48) π (cid:48)∗ π ∗ f ∗ (1.23)Now we can lift sections ψ ∈ H L and ψ (cid:48) ∈ H L (cid:48) to the Hilbert space H T Γ f and considertheir products (cid:104) π ∗ ψ | π (cid:48)∗ ψ (cid:48) (cid:105) in this space. This is then naturally defines a pairing (( · , · )) : H L × H L (cid:48) → C between the two polarised Hilbert spaces given by (( · , · )) := (cid:10) π ∗ · (cid:12)(cid:12) π (cid:48)∗ · (cid:11) (1.24)But any such pairing is equivalently a linear isomorphism f ∗ : H L (cid:48) ∼ = −−→ H L such that (( · , · )) = (cid:104) · | f ∗ · (cid:105) (1.25)where this time the product on the right hand side is the hermitian product of the firstHilbert space H L .Let us workout what this means in coordinates. Recall that on T Γ f ⊂ T ( P × P (cid:48) ) we havethe gauge transformation π (cid:48)∗ θ (cid:48) = π ∗ θ − d F , where F ( x, x (cid:48) ) is the generating function ofthe symplectomorphism. Therefore, wave-functions ψ ( x ) of H L will be lifted by ψ ( x ) (cid:55)→ ψ ( x, x (cid:48) ) = ψ ( x ) and wave-functions ψ (cid:48) ( x (cid:48) ) of H L (cid:48) will be lifted by ψ (cid:48) ( x (cid:48) ) (cid:55)→ ψ (cid:48) ( x, x (cid:48) ) = ψ ( x ) e − i (cid:126) F ( x,x (cid:48) ) to wave-functions of H T Γ f . Thus, the pairing will be given by (( ψ, ψ (cid:48) )) = (cid:90) M d n x d n x (cid:48) ψ † ( x ) ψ (cid:48) ( x (cid:48) ) e − i (cid:126) F ( x,x (cid:48) ) (1.26)where we called M the manifold such that T ∗ M ∼ = P × P . Finally the isomorphism f ∗ : H L → H L (cid:48) induced by the diffeomorphism f will be given in coordinates by ( f ∗ ψ (cid:48) )( x ) = (cid:90) M (cid:48) d n x ψ (cid:48) ( x (cid:48) ) e − i (cid:126) F ( x,x (cid:48) ) (1.27)where we called M (cid:48) the manifold such that T ∗ M (cid:48) ∼ = P (cid:48) . Therefore we have a natural iso-moprhism H L ∼ = H L (cid:48) any time there is a canonical transformation mapping the Lagrangian– 8 –ubbundle L into L (cid:48) and thus we are allowed to write just H for the Hilbert space of aquantum system, without specifying the polarization. We will write just | ψ (cid:105) ∈ H (1.28)for an abstract element of the Hilbert space, independent from the polarization. Example: quantum canonical transformations.
To give some intuition for this idea,let us consider a simple example. Choose
M, M (cid:48) = R n and let the symplectomorphism f : ( R n , d p µ ∧ d x µ ) → ( R n , d p (cid:48) µ ∧ d x (cid:48) µ ) be the linear transformation f ( x, p ) = ( p, − x ) . Thisis generated by generating function F ( x, x (cid:48) ) = δ µν x µ x (cid:48) ν . Thus, if we substitute ( x (cid:48) , p (cid:48) ) = f ( x, p ) = ( p, − x ) , we recover that ( f ∗ ) − is exactly the Fourier transformation of wave-functions: ( f ∗ ψ (cid:48) )( x ) = (cid:90) M (cid:48) d n p ψ (cid:48) ( p ) e − i (cid:126) p µ x µ (cid:0) ( f ∗ ) − ψ (cid:1) ( p ) = (cid:90) M d n x ψ ( x ) e i (cid:126) p µ x µ (1.29)Thus the same quantum state | ψ (cid:105) ∈ H can be represented as a wave-function (cid:104) x | ψ (cid:105) = ψ ( x ) or as its Fourier transform (cid:104) p | ψ (cid:105) = ψ ( p ) in the two basis (cid:8) (cid:104) x | (cid:9) x ∈ M and (cid:8) (cid:104) p | (cid:9) p ∈ M (cid:48) givenby the Lagrangian correspondence. σ -models. The fields X µ ( σ, τ ) are embeddings from asurface Σ into a target space M , i.e. smooth maps C ∞ (Σ , M ) , denoted by X µ : Σ (cid:44) −→ M ( σ, τ ) (cid:55)−→ X µ ( σ, τ ) (2.1) The configuration space of the closed string.
Consider a surface of the form Σ (cid:39) R × S with coordinates σ ∈ [0 , π ) and τ ∈ R . The fields X µ ( σ, τ ) of the σ -model cannow be seen as curves C ∞ ( R , L M ) on the free loop space L M := C ∞ ( S , M ) of the originalmanifold M . This will be denoted as follows: X µ ( σ ) : R (cid:44) −→ L Mτ (cid:55)−→ X µ ( σ, τ ) (2.2)where X µ ( σ, τ ) is a loop for any fixed τ ∈ R . In other words we have C ∞ ( R , L M ) ∼ = C ∞ (Σ , M ) (2.3)This is why the configuration space for the closed string can be identified with the free loopspace L M of the spacetime manifold M .Fortunately, for any given smooth manifold M , the free loop space L M is a Fréchet manifoldand this assures that there will be a well-defined notion of differential geometry on it.– 9 –or any loop X ( σ ) : S → M of L M we can consider the space of sections Γ( S , X ∗ T M ) .This is homeomorphic to the loop space L R n where n = dim( M ) , which, thus, plays therole analogous to a local patch.Since the points of the loop space are loops X ( σ ) in M , a smooth function F ∈ C ∞ ( L M ) can be identified with a functional F [ X ( σ )] . Similarly, a vector field V ∈ T ( L M ) will begiven by a functional operator of the form V [ X ( σ )] = (cid:73) d σ V µ [ X ( σ )]( σ ) δδX µ ( σ ) . (2.4)For a wider and deeper exploration of use of loop spaces to formalise some kinds of pathintegrals in physics, see [34]. The transgression functor.
For any n -form ξ = n ! ξ µ ...µ n d x µ ∧ · · · ∧ d x µ n , given inlocal coordinates { x µ } of M , there exist a map, named transgression functor , from thecomplex of differential forms on M to the one of the differential forms on the loop space L M : T : Ω n ( M ) −→ Ω n − ( L M ) ξ (cid:55)−→ (cid:73) d σ n − ξ µ ...µ n ( X ( σ )) ∂X µ ( σ ) ∂σ δX µ ( σ ) ∧ · · · ∧ δX µ n ( σ ) (2.5)Crucially, it satisfies the following functorial property: δ T = T d (2.6) The phase space of the closed string.
Thus the choice for phase space of a stringon spacetime M will be the free loop space of T ∗ M . By definition, this can be used as adefinition of the cotangent bundle of L M , i.e. T ∗ L M := L ( T ∗ M ) (2.7)i.e. the smooth space of loops ( X ( σ ) , P ( σ )) in the cotangent bundle of T ∗ M . This spacecomes equipped with a canonical symplectic form: Ω := (cid:73) d σ δP µ ( σ ) ∧ δX µ ( σ ) ∈ Ω ( T ∗ L M ) (2.8)In the next paragraph we will illustrate that the phase space of the closed string is exactlythe infinite-dimensional symplectic manifold ( T ∗ L M, Ω) we just described.We can now define a Liouville potential Θ such that its derivative is the canonical symplecticform Ω ∈ Ω ( T ∗ L M ) . Thus we have Θ := (cid:73) d σ P µ ( σ ) δX µ ( σ ) ∈ Ω ( T ∗ L U ) (2.9)We can verify that Ω = δ Θ by calculating: δ Θ = (cid:73) d σ (cid:18) δX µ ( σ ) ∧ δ Θ δX µ ( σ ) + δP µ ( σ ) ∧ δ Θ δP µ ( σ ) (cid:19) = (cid:73) d σ (cid:73) d σ (cid:48) δ ( σ − σ (cid:48) ) δP µ ( σ ) ∧ δX µ ( σ (cid:48) )= (cid:73) d σ δP µ ( σ ) ∧ δX µ ( σ ) = Ω (2.10)– 10 – he closed string as classical system. Let us start from the action of the closed string S [ X ( σ, τ ) , P ( σ, τ )] = 12 (cid:90) d τ (cid:73) d σ (cid:16) P µ ˙ X µ + − g µν ( X ) (cid:0) P µ − B µλ ( X ) X (cid:48) λ (cid:1)(cid:0) P ν − B νλ ( X ) X (cid:48) λ (cid:1) + g µν ( X ) X (cid:48) µ X (cid:48) ν (cid:17) . (2.11)Recall that the σ -model of a closed string (cid:0) X ( σ, τ ) , P ( σ, τ ) (cid:1) : Σ (cid:39) R × S → T ∗ M canbe equivalently expressed as a path R → T ∗ L M on the cotangent bundle of the loopspace. Recall also that the Lagrangian density L ∈ Ω ( R ) of a classical system, consistingof a phase space ( T ∗ L M, Ω) and a Hamiltonian function H ∈ C ∞ ( T ∗ L M ) , is given by L = ( ι V H Θ − H )d τ , where V H ∈ ham ( T ∗ L M, Ω) is the Hamiltonian vector of H . Thismeans that the action will be S [ X ( σ, τ ) , P ( σ, τ )] = (cid:90) τ τ d τ (cid:0) ι V H Θ − H (cid:1) (2.12)where Θ is the Liouville potential of the symplectic form Ω . Now we want to check thatthe symplectic structure Ω ∈ Ω ( T ∗ L M ) of the phase space of the closed string is exactlythe canonical symplectic structure (2.8) on T ∗ L M . To do that, we can assume that theHamiltonian vector field is just the translation along proper time. In other words we impose V H := dd τ = (cid:73) d σ (cid:18) ˙ X µ ( σ ) δδX µ ( σ ) + ˙ P µ ( σ ) δδP µ ( σ ) (cid:19) (2.13)By putting together definition (2.13) and equation (2.12) we immediately get the equation ι V H Θ = (cid:73) d σ ˙ X µ ( σ ) P µ ( σ ) , (2.14)which is solved by the Liouville potential Θ = (cid:73) d σ P µ ( σ ) δX µ ( σ ) ∈ Ω ( T ∗ L U ) . (2.15)Its differential is, indeed, exactly the canonical symplectic form (2.8), i.e. Ω = δ Θ= (cid:73) d σ δP µ ( σ ) ∧ δX µ ( σ ) (2.16)Moreover, by combining the equation (2.12) with the action (2.11), we can immediatelyfind the Hamiltonian of a closed string: H [ X ( σ ) , P ( σ )] = (cid:73) d σ (cid:16) g µν ( X ) (cid:0) P µ − B µλ ( X ) X (cid:48) λ (cid:1)(cid:0) P ν − B νλ ( X ) X (cid:48) λ (cid:1) + g µν ( X ) X (cid:48) µ X (cid:48) ν (cid:17) We can formally pack together the momentum P ( σ ) and the derivative X (cid:48) ( σ ) in the fol-lowing doubled vector: P M ( σ ) := (cid:32) X (cid:48) µ ( σ ) P µ ( σ ) (cid:33) (2.17)– 11 –ith M = 1 , . . . , n . Notice that P M ( σ ) is uniquely defined at any given loop ( X ( σ ) , P ( σ )) in the phase space. Thus, we can rewrite the Hamiltonian of the string as H [ X ( σ ) , P ( σ )] = (cid:73) d σ P M ( σ ) H MN ( X ( σ )) P N ( σ ) (2.18)where the matrix H MN is defined by H MN := (cid:32) g µν − B µλ g λρ B ρν B µλ g µν − g µλ B λµ g µν (cid:33) . (2.19)In conclusion, by putting everything together, we can see that a closed string is a classicalsystem ( T ∗ L M, Ω , H ) , where Ω is the canonical symplectic form on T ∗ L M and the Hamil-tonian H is given by definition (2.18). We now see the appearance of the generalised metric,described by matrix (2.19). This metric is a representative of an O ( d, d ) /O ( d ) × O ( d ) coset,and defines the generalised metric of generalised geometry. As such the Hamiltonian (2.18)has a manifest O ( d, d ) symmetry. This is of course the T-duality symmetry of the string.As discussed in the introduction, the Hamiltonian will often exhibit the symmetries notpresent in the Lagrangian and T-duality is one of these symmetries. In the geometric quan-tisation of an ordinary particle we have, in local Darboux coordinates, a local Liouvillepotential given by θ = p µ d x µ , where p µ is the canonical momentum. In presence of an elec-tromagnetic field with a minimally coupled -form potential A , the canonical momentum p µ which is defined from the Lagrangian perspective by p µ = ∂ L ∂ ˙ q µ is given by: p µ = k µ + eA µ .(We have used k µ to denote the naive non-canonical momentum, also sometimes called thekinetic momentum).Then the Liouville potential can be rewritten as θ = k µ d x µ + eA , with A is the pullback ofthe electromagnetic potential to the phase space. Consequently the symplectic form takesthe form ω = d k µ ∧ d x µ + eF . Let us call the -form ω A =0 := d k µ ∧ d x µ . Thus, thegeometric prequantisation condition [ ω ] = [ ω A =0 ] + e [ F ] ∈ H ( T ∗ M, Z ) of the symplecticform on the phase space implies the Dirac quantisation condition e [ F ] ∈ H ( M, Z ) of theelectromagnetic field on spacetime.Notice that, in canonical coordinates, we have a Hamiltonian H = g µν ( p µ − eA µ )( p ν − eA ν ) and the commutation relations [ˆ x µ , ˆ x ν ] = 0 , [ˆ p µ , ˆ x ν ] = i (cid:126) δ νµ , [ˆ p µ , ˆ p ν ] = 0 . (2.20)On the other hand, in terms of the kinetic non-canonical coordinates, we have the Hamil-tonian H = g µν k µ k ν and commutation relations [ˆ x µ , ˆ x ν ] = 0 , [ˆ k µ , ˆ x ν ] = i (cid:126) δ νµ , [ˆ k µ , ˆ k ν ] = i (cid:126) eF µν . (2.21)It is worth observing that the space coordinates do not commute anymore and the non-commutativity term is proportional to the field strength of the electromagnetic field. Asimilar picture will hold for strings. – 12 – inetic coordinates for a charged string. Similarly to the charged particle, for astring we require our Ω defined by equation (2.8) to be quantised as [Ω] ∈ H ( T ∗ L M, Z ) .We will now see that this implies, similarly to the electromagnetic field, the quantisationof the Kalb-Ramond field flux [ H ] ∈ H ( M, Z ) .Let us recall that an abelian gerbe with Dixmier-Douady class [ H ] ∈ H ( M, Z ) on the basemanifold M is encoded by the following patching conditions: H = d B α ∈ Ω ( M ) B α − B β = dΛ αβ ∈ Ω ( U α ∩ U β )Λ αβ + Λ βγ + Λ γα = d G αβγ ∈ Ω ( U α ∩ U β ∩ U γ ) (2.22)(see [35] and [36] for details). By using the properties of the transgression functor from M to its loop space L M , we immediately obtain the new patching conditions T H = δ ( T B α ) ∈ Ω ( L M ) T B α − T B β = δ ( T Λ αβ ) ∈ Ω ( L U α ∩ L U β ) (2.23)on L M . Therefore, the transgression functor sends a gerbe on a manifold M to a circlebundle on its loop space L M , i.e. in other words T : Gerbes( M ) ∼ = −−−→ U (1)Bundles( L M ) , (2.24)where the first Chern class of the circle bundle is T H ∈ H ( L M, Z ) .Now we can decompose the canonical symplectic form of the phase space of the closed string ( T ∗ L M, Ω) by Ω = Ω B =0 + T H. (2.25)We can write Ω B =0 := (cid:72) d σ δK µ ( σ ) ∧ δX µ ( σ ) , so we find that the symplectic form can beexpressed by Ω = (cid:73) d σ δ (cid:16) K ν ( σ ) + B µν (cid:0) X ( σ ) (cid:1) X (cid:48) µ ( σ ) (cid:17) ∧ δX ν ( σ ) (2.26)where K ν ( σ ) := P ν ( σ ) − B µν (cid:0) X ( σ ) (cid:1) X (cid:48) µ ( σ ) is the non-canonical momentum of the stringand P µ ( σ ) is its canonical momentum. The relevance of the transgression of gerbes to theloop space in dealing with T-duality and, more generally, with Double Field Theory wasunderlined by [37].For a clarification on the relation between the phase space of a closed string, seen as a loopspace, and the Courant algebroids of supergravity, see [38]. The machinery of geometric quantisation can now be applied on the phase space ( T ∗ L M, Ω) of the closed string. See also [39] for a different quantisation approach on a loop space.Given our choice of gauge for the Liouville potential Θ = (cid:72) d σP µ ( σ ) δX µ ( σ ) , we can nowdetermine the algebra heis ( T ∗ L M, Ω) of quantum observables defined by ˆ f = − i (cid:126) ∇ V f + f (2.27)– 13 –or any classical observable f ∈ C ∞ ( T ∗ L M ) . For the classical observables corresponding tothe canonical coordinates X µ ( σ ) , P µ ( σ ) of the phase space of the closed string, we have thefollowing quantum observables: ˆ P µ ( σ ) = − i (cid:126) δδX µ ( σ ) , ˆ X µ ( σ ) = i (cid:126) δδP µ ( σ ) + X µ ( σ ) (2.28)If we choose the polarisation determined by the Lagrangian subbundle L = T L M with thecorresponding basis (cid:8) | X ( σ ) (cid:105) (cid:9) X ( σ ) ∈L M , we have the following operators acting on wave-functional Ψ[ X ( σ )] = (cid:104) X ( σ ) | Ψ (cid:105) (cid:68) X ( σ ) (cid:12)(cid:12)(cid:12) ˆ P µ ( σ ) (cid:12)(cid:12)(cid:12) Ψ (cid:69) = − i (cid:126) δδX µ ( σ ) Ψ[ X µ ( σ )] (cid:68) X ( σ ) (cid:12)(cid:12)(cid:12) ˆ X µ ( σ ) (cid:12)(cid:12)(cid:12) Ψ (cid:69) = X µ ( σ )Ψ[ X µ ( σ )] (2.29)The commutation relations of these operators will then as follows: (cid:2) ˆ P µ ( σ ) , ˆ X ν ( σ (cid:48) ) (cid:3) = 2 πi (cid:126) δ νµ δ ( σ − σ (cid:48) ) , (cid:2) ˆ X µ ( σ ) , ˆ X ν ( σ (cid:48) ) (cid:3) = 0 , (cid:2) ˆ P µ ( σ ) , ˆ P ν ( σ (cid:48) ) (cid:3) = 0 . (2.30)These define the Heisenberg algebra heis ( T ∗ L M, Ω) of quantum observables on the phasespace of the closed string.On the other hand, if we use the non-canonical, kinetic coordinates ( K ( σ ) , X ( σ )) , we obtaincommutation relations of the form (cid:2) ˆ K µ ( σ ) , ˆ X ν ( σ (cid:48) ) (cid:3) = 2 πi (cid:126) δ νµ δ ( σ − σ (cid:48) ) , (cid:2) ˆ X µ ( σ ) , ˆ X ν ( σ (cid:48) ) (cid:3) = 0 , (cid:2) ˆ K µ ( σ ) , ˆ K ν ( σ (cid:48) ) (cid:3) = H µνλ (cid:0) X ( σ ) (cid:1) X (cid:48) λ ( σ ) δ ( σ − σ (cid:48) ) . (2.31) Let us consider a closed string propagating in the back-ground M = T n with constant metric g ij and constant Kalb-Ramond field B ij . As explainedby [40], the compactification condition x i = x i + 2 π of a torus target space is background-independent, i.e. it does not depend on the bosonic background fields g ij and B ij . In fact,the coordinates x i are periodic with radius π and the physical radii of the compactificationare given by R i := (cid:82) S i √ g ii d x i = √ g ii .Consider the phase space T ∗ L M ∼ = L ( T n × R n ) of a closed string on a torus background.Let us call the matrix E := g + B , so that its transpose is E T = g − B . Let us also definethe generalised metric by the constant matrix H MN = (cid:32) g ij − B ik g k(cid:96) B (cid:96)j − g ik B kj B ik g kj g ij (cid:33) . (2.32)– 14 –e can, now, explicitly expand position and momentum in σ as it follows: X i ( σ ) = x i + α (cid:48) w i σ + (cid:88) n ∈ N \{ } n (cid:16) α in ( H ) e inσ + ¯ α in ( H ) e − inσ (cid:17) P i ( σ ) = p i + (cid:88) n ∈ N \{ } (cid:16) E T ij α jn ( H ) e inσ + E ij ¯ α jn ( H ) e − inσ (cid:17) (2.33)where we used the notation α in = α in ( H ) and ¯ α in = ¯ α in ( H ) to indicate that the higher-modes depend on the background, encoded by the generalised metric. The operators ˆ α in and ˆ¯ α in are the creation and annihilation operators for the excited states of the string, whichdepend on the background. However, ss pointed out by [40], the operators ˆ X i ( σ ) and ˆ P i ( σ ) must be thought as background-independent objects. From the Fourier expansion, we alsoimmediately obtain that the zero-modes coordinate p i and w i are integers because of theperiodicity of x i .The T-dual coordinates (cid:101) X i ( σ ) are defined by, (cid:101) X (cid:48) i ( σ ) = P i ( σ ) , (2.34)and so we must have (cid:101) X i ( σ ) = ˜ x i + (cid:82) σ d σ (cid:48) P i ( σ (cid:48) ) . Therefore, we obtain the expressions (cid:101) X i ( σ ) = ˜ x i + α (cid:48) p i σ + (cid:88) n ∈ N \{ } n (cid:16) − E T ij α jn ( H ) e inσ + E ij ¯ α jn ( H ) e − inσ (cid:17)(cid:101) P i ( σ ) = w i + (cid:88) n ∈ N \{ } n (cid:16) α in ( H ) e inσ + ¯ α in ( H ) e − inσ (cid:17) (2.35)Notice that the T-dual coordinate (cid:101) X i ( σ ) is also periodic with period π . Moreover thenew coordinates (cid:101) X ( σ ) and (cid:101) P ( σ ) are independent from the background. The commutationrelations [ ˆ X i ( σ ) , ˆ P j ( σ (cid:48) )] = iδ ij δ ( σ − σ (cid:48) ) (2.36)become, on zero and higher modes, [ˆ x i , ˆ p j ] = iδ ij [ ˆ α in ( H ) , ˆ α jm ( H )] = nδ n + m, g ij (2.37)As we have seen in the first section, the action is given on the phase space by the equation S := (cid:72) d σ ˙ X µ ( σ, τ ) P µ ( σ, τ ) − H [ X ( σ, τ ) , P ( σ, τ )] . We can, thus, rewrite the action of theclosed string on a torus as S [ X ( σ, τ ) , P ( σ, τ )] = (cid:90) d τ (cid:73) d σ (cid:18) ˙ X µ P µ − P M H MN P N (cid:19) (2.38) We will now show that it is possible to interpretT-duality as a symplectomorphism of phase spaces of two closed strings of the form f : ( T ∗ L M, Ω) −→ ( T ∗ L (cid:102) M , (cid:101) Ω) (cid:0) X µ ( σ ) , P µ ( σ ) (cid:1) (cid:55)−→ (cid:0) (cid:101) X µ ( σ ) , (cid:101) P µ ( σ ) (cid:1) . (2.39)– 15 –n fact, in [5], it was firstly argued that T-duality can be seen a canonical transformation.However, a canonical transformation with generating functional F [ X ( σ ) , (cid:101) X ( σ )] is nothingbut the symplectomorphism f associated to the following Lagrangian correspondence of theform (1.18) (cid:0) T ∗ L ( M × (cid:102) M ) , π ∗ Ω − (cid:101) π ∗ (cid:101) Ω (cid:1) ( T ∗ L M, Ω) ( T ∗ L (cid:102) M , (cid:101) Ω) (cid:101) ππ f (2.40)which satisfies the following trivialisation condition for Liouville potential: π ∗ Θ − (cid:101) π ∗ (cid:101) Θ = δF. (2.41)We can immediately check that, if we substitute the expression for the Liouville potentials,we get the expression (cid:73) d σ (cid:16) P µ ( σ ) δX µ ( σ ) − (cid:101) P µ ( σ ) δ (cid:101) X µ ( σ ) (cid:17) = (cid:73) d σ (cid:32) δFδX µ ( σ ) δX µ ( σ ) + δFδ (cid:101) X µ ( σ ) δ (cid:101) X µ ( σ ) (cid:33) and hence we recover exactly the equations of the canonical transformation P µ ( σ ) = δFδX µ ( σ ) , (cid:101) P µ ( σ ) = − δFδ (cid:101) X µ ( σ ) . (2.42)By considering the generating functional F [ X ( σ ) , (cid:101) X ( σ )] := (cid:90) D, ∂D = S d (cid:101) X µ ∧ d X µ = 12 (cid:73) d σ (cid:0) X (cid:48) µ ( σ ) (cid:101) X µ ( σ ) − X µ ( σ ) (cid:101) X (cid:48) µ ( σ ) (cid:1) (2.43)which was originally proposed by [5, 41], we obtain exactly T-duality on the phase space: P µ ( σ ) = (cid:101) X (cid:48) µ ( σ ) , (cid:101) P µ ( σ ) = X (cid:48) µ ( σ ) . (2.44)The Lagrangian correspondence space in (2.40) is then the loop space of the doubled spaceof DFT. We can notice that, in this simple case, the doubled space can be identified withthe correspondence space of a topological T-duality [42–46] over a base point. A similarobservation was made in [47]. Relation with the symplectic form on the doubled space.
Let us consider the sym-plectic -form (cid:36) := d x µ ∧ d (cid:101) x µ ∈ Ω ( M × (cid:102) M ) on the product space, where { x µ , ˜ x µ } are localcoordinates on M × (cid:102) M . Notice that, for such a symplectic form, we can choose a Liouvillepotential of the form (˜ x µ d x µ − x µ d˜ x µ ) . (This is like the choice in Weyl quantisation orwhen we construct a Fock space). Now, we can immediately recognise that the generatingfunctional (2.43) is nothing but the transgression of this Liouville potential to the loopspace L ( M × (cid:102) M ) , i.e. we have F [ X ( σ ) , (cid:101) X ( σ )] = 12 T (˜ x µ d x µ − x µ d˜ x µ ) . (2.45)– 16 –y using the functorial property δ T = T d of the transgression functor, we can rewrite thetrivialisation condition (2.41) of T-duality by π ∗ Θ − (cid:101) π ∗ (cid:101) Θ = T ( (cid:36) ) . (2.46)Therefore, T-duality on the phase space is associated to the symplectic -form (cid:36) on the doubled space M × (cid:102) M . Notice that this -form is a particular and simple case of thefundamental -form considered by [48–50]. Background independence.
Since the two loop phase spaces T ∗ L M and T ∗ L (cid:102) M aresymplectomorphic, they can be effectively considered the same symplectic ∞ -dimensionalFréchet manifold. In this "passive" symplectomorphism perspective, the T-duality from ( X ( σ ) , P ( σ )) to ( (cid:101) X ( σ ) , (cid:101) P ( σ )) can be interpreted as a change of coordinates on the phasespace of the closed string. The Hamiltonian formulation of the closed string on the phasespace is thus T-duality invariant. T-duality as isomorphism of classical systems.
T-duality, seen as a symplectomor-phism f : ( T ∗ L M, Ω) → ( T ∗ L (cid:102) M , (cid:101) Ω) of the phase space of the closed string, does alsopreserve the Hamiltonian of the closed string, i.e. we have f ∗ H = H. (2.47)In other words a T-duality is not just a symplectomorphism of our phase space ( T ∗ L M, Ω) ,but also an isomorphism of the classical system ( T ∗ L M, Ω , H ) of the closed string.In general we can T-dualise the Hamiltonian of the closed string by applying a transforma-tion O ∈ O ( n, n ; Z ) to the doubled metric H . Notice that the Hamiltonian functional doesnot change under such transformations. We have, in fact, H [ (cid:101) X, (cid:101) P ] = (cid:73) d σ
12 ( O P ) M ( O T HO ) MN ( O P ) N = (cid:73) d σ P M H MN P N = H [ X, P ] T-duality as change of basis on the Hilbert space.
The Lagrangian correspondence(2.56) induces a diagram of quantum Hilbert spaces H T Γ f H L H (cid:101) Lπ (cid:48)∗ π ∗ f ∗ (2.48)where H L and H (cid:101) L are respectively polarised along the Lagrangian subbundles L = T ( L M ) and (cid:101) L = T ( L (cid:102) M ) . Now, as we have seen in (1.23), the map ( f ∗ ) − H L ∼ = H (cid:101) L is anisomorphism H L ∼ = H (cid:101) L of Hilbert spaces. Therefore we can use just the notation H for theabstract quantum Hilbert space.Any quantum state | Ψ (cid:105) ∈ H can be expressed in the two basses defined by the two differentpolarisations: | Ψ (cid:105) = (cid:90) D X ( σ ) Ψ[ X ( σ )] | X ( σ ) (cid:105) , | Ψ (cid:105) = (cid:90) D (cid:101) X ( σ ) (cid:101) Ψ[ (cid:101) X ( σ )] | (cid:101) X ( σ ) (cid:105) (2.49)– 17 –here we called (cid:104) X ( σ ) | Ψ (cid:105) =: Ψ[ X ( σ )] , (cid:104) (cid:101) X ( σ ) | Ψ (cid:105) =: (cid:101) Ψ[ (cid:101) X ( σ )] . (2.50)The expansions in different basses will be then related by the Fourier-like transformation ( f ∗ ) − of string wave-functionals, given by (cid:101) Ψ[ (cid:101) X ( σ )] = (cid:90) L M D X ( σ ) e i (cid:126) F [ X ( σ ) , (cid:101) X ( σ )] Ψ[ X ( σ )] (2.51)in accord with [5]. We can also explicitly write the matrix of the change of basis on H by (cid:104) X ( σ ) | (cid:101) X ( σ ) (cid:105) = e i (cid:126) F [ X ( σ ) , (cid:101) X ( σ )] (2.52)Interestingly, this isomorphism is naturally defined by lifting the polarised wave functionals Ψ[ X ( σ )] ∈ H L and (cid:101) Ψ[ (cid:101) X ( σ )] ∈ H (cid:101) L to wave-functionals Ψ[ X ( σ )] on the doubled space andby considering their Hermitian product in the Hilbert space of the doubled space. In doublefield theory solving the so called strong constraint provides the choice of polarisation. Herethe quantisation procedure itself demands a polarisation choice and the strong constraint issolved automatically. It is interesting to consider the weak constraint from this perspectivebut this is beyond the goals of this paper. T-duality invariant dynamics.
The dynamics of the quantised closed string is encodedby the background independent equation i (cid:126) ∂∂τ | Ψ (cid:105) + ˆ H | Ψ (cid:105) = 0 . (2.53)Let us consider, for simplicity, that we are starting from a Minkowski flat backgroundwith g µν = η µν and B µν = 0 . Then we will have a trivial doubled metric H MN = δ MN .Therefore, the equation of motion can be expressed in the basis (cid:8) | X ( σ ) (cid:105) (cid:9) X ( σ ) ∈L M by i (cid:126) ∂∂τ Ψ[ X ( σ )] + (cid:73) d σ (cid:18) − (cid:126) δ δX ( σ ) + X (cid:48) ( σ ) (cid:19) Ψ[ X ( σ )] = 0 , (2.54)but immediately also in the T-dual basis (cid:8) | (cid:101) X ( σ ) (cid:105) (cid:9) X ( σ ) ∈L (cid:102) M by i (cid:126) ∂∂τ (cid:101) Ψ[ (cid:101) X ( σ )] + (cid:73) d σ (cid:32) (cid:101) X (cid:48) ( σ ) − (cid:126) δ δ (cid:101) X ( σ ) (cid:33) (cid:101) Ψ[ (cid:101) X ( σ )] = 0 . (2.55) T-duality as a symplectomorphism for torus bundles.
Let us conclude this sectionby considering a slightly more general class of examples: (geometric) T-duality of torusbundles.Let M (cid:16) N and (cid:102) M (cid:16) N be two principal T n -bundles on a common base manifold N .T-duality can be still seen as a symplectomorphism between loop phase spaces T ∗ L M → – 18 – ∗ L (cid:102) M and we can still employ the machinery of Lagrangian correspondence (1.18). Now,the Lagrangian correspondence of the T-duality on the phase space of the closed string is (cid:0) T ∗ L ( M × N (cid:102) M ) , π ∗ Ω − (cid:101) π ∗ (cid:101) Ω (cid:1) ( T ∗ L M, Ω) ( T ∗ L (cid:102) M , (cid:101) Ω) π (cid:48) π f (2.56)where the fiber product M × N (cid:102) M can be naturally seen as the doubled torus bundle of theduality. For the torus fibration, we have the following equation for the Liouville potential: π ∗ Θ − (cid:101) π ∗ (cid:101) Θ = T ( (cid:36) ) , (2.57)where T ( (cid:36) ) is the transgression to the loop space of the fundamental -form (cid:36) ∈ Ω ( M × N (cid:102) M ) , which lives on the doubled torus bundle M × N (cid:102) M and it is given by (cid:36) = (d x i + A i ) ∧ (d (cid:101) x i + (cid:101) A i ) ∈ Ω ( M × N (cid:102) M ) (2.58)If the bundle M × N (cid:102) M (cid:16) N is trivial, then we recover the symplectic form (cid:36) = d x i ∧ d (cid:101) x i from the previous paragraphs. Notice that, by moving to the generalised coordinates, wecan easily recover the equation characterising topological T-duality by [45], i.e. we have H − (cid:101) H = d( A i ∧ (cid:101) A i ) (2.59)on the doubled torus bundle M × N (cid:102) M . In this class of cases, we notice that the doubledspace can be identified with the correspondence space M × N (cid:102) M of a topological T-duality[42–46] over a base manifold M . To describe doubled strings, we introducenew coordinates (cid:101) X µ ( σ ) which satisfy the equation P µ ( σ ) = (cid:101) X (cid:48) µ ( σ ) . Let us define thefollowing doubled loop-space vectors: X M ( σ ) := (cid:32) X µ ( σ ) (cid:101) X µ ( σ ) (cid:33) , P M ( σ ) := (cid:32) X (cid:48) µ ( σ ) P µ ( σ ) (cid:33) = X (cid:48) M ( σ ) . (3.1)Therefore, for a doubled string the doubled momentum P M ( σ ) coincides with the derivativealong the circle of the doubled position vector X M ( σ ) . Thus, instead of encoding the σ -model of the closed string by an embedding ( X µ ( σ ) , P µ ( σ )) into the phase space, we canencode it by an embedding X M ( σ ) = ( X µ ( σ ) , (cid:101) X µ ( σ )) into a doubled position space. Ourobjective is, then, be able to reformulate a string wave-functional Ψ[ X µ ( σ ) , P µ ( σ )] in termsof doubled fields as a wave-functional of the form Ψ (cid:2) X M ( σ ) (cid:3) .– 19 –owever, notice that, since the new coordinates (cid:101) X µ ( σ ) are the integral of the momenta ofthe string, specifying (cid:101) X µ ( σ ) is a stronger statement than specifying P µ ( σ ) = (cid:101) X (cid:48) µ ( σ ) . Thisobservation is crucial when considering the possible boundary conditions of the doubledstring σ -model.Let us define the following zero-modes of the doubled loop-space vectors: x M := 12 π (cid:73) d σ X M ( σ ) , p M := 12 πα (cid:48) (cid:73) d σ X (cid:48) M ( σ ) , (3.2)which, in components, read x M = (cid:32) x µ ˜ x µ (cid:33) , p M = (cid:32) ˜ p µ p µ (cid:33) ≡ (cid:32) w µ ˜ w µ (cid:33) (3.3)By using the new coordinate (cid:101) X µ , we can rewrite the action of a closed string by S string [ X ( σ, τ ) , P ( σ, τ )] = 12 πα (cid:48) (cid:90) d τ (cid:73) d σ (cid:18) ˙ X µ (cid:101) X (cid:48) µ − X (cid:48) M H MN X (cid:48) N (cid:19) (3.4)Let us use the following notation for the derivatives ˙ X ( σ, τ ) := ∂ X ( σ, τ ) ∂τ X (cid:48) ( σ, τ ) := ∂ X ( σ, τ ) ∂σ (3.5) The generalised boundary conditions.
Since in the action of the closed string thefield X M ( σ ) never appears, but only its derivatives X (cid:48) M ( σ ) , we only need to require thatthe latter are periodic, i.e. X (cid:48) M ( σ + 2 π ) = X (cid:48) M ( σ ) (3.6)This implies that the generalised boundary conditions are X M ( σ + 2 π, τ ) = X M ( σ, τ ) + 2 πα (cid:48) p M ( τ ) , (3.7)where the quasi-period p M ( τ ) can, in general, be dynamical and depend on proper time.Let us define the quasi-loop space L Q M of a manifold M as it follows: L Q M := (cid:8) X : [0 , π ) → M (cid:12)(cid:12) d X (2 π ) = d X (0) (cid:9) , (3.8)where we used the simple identity X (cid:48) M ( σ ) d σ = d X M ( σ ) .The phase space of the doubled string will be a symplectic manifold ( L Q M , Ω ) where thesymplectic form Ω ∈ Ω ( L Q M ) will be determined in the following subsection. A remark on the global geometry of the doubled space.
Given local coordinates x M on the doubled space we can express the vector X (cid:48) ( σ ) by X (cid:48) M ( σ ) d σ = d X M ( σ ) = X ∗ (cid:0) d x M (cid:1) , (3.9)where X ∗ (cid:0) d x M (cid:1) denotes the pullback of d x M to the quasi-loop space. Therefore, the re-quirement that X (cid:48) ( σ ) is periodic can be immediately recasted as the requirement that thepullback of d x M is periodic. References [35] and [36] explore the idea that the doubled– 20 –pace M is globally not a smooth manifold, but a more generalised geometric object (seethere for details). In particular, in the references, it is derived that the patching con-ditions for local coordinate patches U ( α ) and U ( β ) of the doubled space M should be ofthe form x M ( β ) = x M ( α ) + Λ M ( αβ ) + ∂ M φ ( αβ ) , where we have an additional gauge-like transfor-mation φ ( αβ ) on the overlap of patches. Notice that this implies the patching conditions d x M ( β ) = d x M ( α ) + dΛ M ( αβ ) and therefore the gauge transformations φ ( αβ ) do not appear for thedifferential. Since the Čech cocycle condition dΛ M ( αβ ) + dΛ M ( βγ ) + dΛ M ( γα ) = 0 is satisfied, wedo not encounter problems for X ∗ (cid:0) d x M (cid:1) = d X ( σ ) being periodic. In other words, a doubledstring can naturally live on a doubled space M that is patched in a more general way thana manifold (like the proposal by [35] and [36]) exactly because X ( σ ) does not appear in theaction, but only X (cid:48) ( σ ) does. Recall that the Lagrangian density is related to the Liouville potential Θ by L H = ( ι V H Θ − H )d τ. (3.10)Therefore, we can find the symplectic structure Ω = δ Θ on the phase space of the doubledstring from its full Lagrangian. One should also note here that the different choices ofLiouville potential will give different results corresponding to either a particular choice ofduality frame or a duality symmetric frame. The Tseytlin action.
The doubled string σ -model as first constructed by Tseytlin is bynow well known [51, 52] and there are many routes one might take to its construction. Here,since we already have the doubled perspective in place for the Hamiltonian, the immediatemethod is to take the action as given by (3.4) and then allow the dual variables to alsobe dynamical by augmenting the term ˙ X µ P µ term in the action with its dual equivalent: ˙ (cid:101) X µ (cid:101) P µ . This is like picking a duality symmetric choice for the Liouville potential. Oncethis term is included then one can simply substitute the expressions for the doubled vectorsinto the action. It is these ˙ XP terms that produce the Legendre transformation betweenthe Hamiltonian and the Lagrangian. They are sometimes called the abbreviated action andfrom now on we will adopt this nomenclature. The duality augmented abbreviated actionis then: S abb = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ (cid:16) ˙ X µ P µ + ˙ (cid:101) X µ (cid:101) P µ (cid:17) (3.11)which using the expressions for the doubled vectors becomes (up to total derivative termsof which we will discuss more later) the O ( d, d ) manifestly symmetric term: S abb = (cid:90) d τ (cid:73) d σ πα (cid:48) (cid:16) ˙ X M η MN X (cid:48) N (cid:17) . (3.12)Combining this with the Hamiltonian to produce the total action S = S abb − H gives theTseytlin action [51, 52]: S Tsey [ X ( σ, τ )] = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ (cid:16) ˙ X M η MN X (cid:48) N − X (cid:48) M H MN X (cid:48) N (cid:17) . (3.13)– 21 –his action has been the subject of much study and we will return to the quantum equiv-alence to the usual string action later. Let us first examine this action taking care withthe important property that the fields are quasi-periodic in σ with quasi-period p M whichimplies X M ( σ + 2 π, τ ) = X M ( σ, τ ) + 2 πα (cid:48) p M ( τ ) . This means that although the world sheetis periodic and has no boundary the total derivative terms of such quasi periodic fields cancontribute to the action. These contributions to the Hamiltonian produce the importantzero mode contributions to the Hamiltonian from the winding and momenta. What followsis an analysis of the doubled abbreviated action with quasi-periodic fields. The total derivative contributions to the Tseytlin action.
When integrating withrespect to σ we may write (cid:72) d σ as (cid:82) σ +2 πσ d σ . Then the integral of a total derivativeis (cid:82) d σ dd σ f ( σ ) = f (2 π + σ ) − f ( σ ) . For any periodic function f ( σ ) this then vanishesas it should. However for a quasi-periodic function this integral will be non zero e.g. p M = πα (cid:48) (cid:72) d σ X (cid:48) M ( σ ) . Note, that this is still independent of σ as it should be since σ is an entirely arbitrary choice of coordinate origin in the loop.Let us write the doubled abbreviated action explicitly with the manifest dependence on σ as follows: S Tsey ( σ ) = 14 πα (cid:48) (cid:90) d τ (cid:90) σ +2 πσ d σ ˙ X M ( σ, τ ) η MN X (cid:48) N ( σ, τ ) . (3.14)Then taking the derivative with respect to σ produces: d S Tsey d σ = 14 πα (cid:48) (cid:90) d τ (cid:16) ˙ X M ( σ + 2 π, τ ) − ˙ X M ( σ , τ ) (cid:17) η MN X (cid:48) N ( σ , τ )= 12 (cid:90) d τ ˙ p M ( τ ) η MN X (cid:48) N ( σ , τ ) . (3.15)To get to the second line we have used the periodicity of X (cid:48) N ( σ + 2 π, τ ) = X (cid:48) N ( σ, τ ) andthe quasi-periodicity of X N ( σ ) . We thus have an anomaly. The action now depends on onthe arbitrary choice of σ when we allow quasi-periodic fields to encode the zero modes inthe doubled space. The proposal in [53] is then to add an explicit "boundary term" to theTseytlin action to cancel this piece as follows: S ∂ Tsey [ p ( τ ) , X ( σ , τ )] := − (cid:90) d τ ˙ p M ( τ ) η MN X N ( σ , τ ) (3.16)The full action, therefore, does not depend on the choice of σ , i.e. dd σ ( S Tsey + S ∂ Tsey ) = 0 (3.17)and thus diffeomorphism-invariance of the doubled string is restored.Putting together all these terms we obtain the action S [ X ( σ, τ )] = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ (cid:16) ˙ X M ( σ, τ ) η MN X (cid:48) N ( σ, τ ) − X (cid:48) M ( σ, τ ) H MN X (cid:48) N ( σ, τ ) (cid:17) − (cid:90) d τ p M ( τ ) η MN ˙ X N (0 , τ ) . (3.18)– 22 –his doubled action is now world sheet diffeomorphism invariant for quasi-periodic fieldsbut how does it relate to the original string action? Recall that the in writing down theTseytlin action total derivative terms were neglected. We will now examine the relationshipbetween the Tseytlin string and ordinary string with quasi-periodic fields. The Relation between Tseytlin string and ordinary string.
Let us begin with theusual abbreviated action for the ordinary string, and then integrate by parts keeping the σ total derivatives, we may neglect the total derivatives: in τ since there is no quasi periodicityin this variable: πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X µ P µ = 12 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X µ (cid:101) X (cid:48) µ = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ( ˙ X µ (cid:101) X (cid:48) µ + ˙ X µ (cid:101) X (cid:48) µ )= 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ( ˙ X µ (cid:101) X (cid:48) µ + X (cid:48) µ ˙ (cid:101) X µ + dd σ ( ˙ X µ (cid:101) X µ ))= 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M η MN X (cid:48) N + 14 πα (cid:48) (cid:90) d τ (cid:16) ˙ X µ (2 π + σ ) (cid:101) X µ (2 π + σ ) − ˙ X µ ( σ ) (cid:101) X µ ( σ ) (cid:17) (3.19)Here we remark that we have also here made a choice in boundary total derivative term for σ . We are free to exchange it with: − dd σ ( X µ ˙ (cid:101) X µ ) . (3.20)This choice is simply related by a neglected total derivative in τ . A natural possibility isalso the duality symmetric combination:
12 dd σ (cid:16) ˙ X µ (cid:101) X µ − X µ ˙ (cid:101) X µ (cid:17) . (3.21)Now, the quasi-periodicity of the fields is: X I (2 π + σ ) = X I ( σ ) + 2 πα (cid:48) p I (3.22)so that we may evaluate the final term in (3.19) as follows: ˙ X (2 π + σ ) (cid:101) X (2 π + σ ) − ˙ X ( σ ) (cid:101) X ( σ ) = 2 πα (cid:48) (cid:16) ˙˜ p (cid:101) X ( σ ) + p ˙ X ( σ ) + 2 πα (cid:48) ˙˜ pp (cid:17) . (3.23)Substituting this into (3.19) produces: πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X µ P µ = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M η MN X (cid:48) N + (cid:90) d τ (cid:18)
12 ˙˜ p (cid:101) X ( σ ) + 12 p ˙ X ( σ ) + πα (cid:48) ˙˜ pp (cid:19) (3.24)Now, ˙ X (2 π ) = ˙ X (2 π ) + 2 πα (cid:48) ˙˜ p is quasi-periodic, so the initial term (cid:82) d τ (cid:72) d σ ˙ XP suffersfrom the same anomaly as we described in the previous section. We need to also add a– 23 –boundary term" for (cid:82) d τ (cid:72) d σ ˙ XP so that there is no dependence on σ . This implies thatthe abbreviated action on the LHS of (3.19) must be changed to S abb := 12 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X µ P µ − (cid:90) d τ ˙˜ p µ (cid:101) X µ ( σ ) (3.25)so there is no overall σ dependence. Then including the same term to the RHS of (3.19)will produce: S abb = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M η MN X (cid:48) N + (cid:90) d τ (cid:18) −
12 ˙˜ p µ (cid:101) X µ ( σ ) + 12 p µ ˙ X µ ( σ ) + πα (cid:48) ˙˜ p µ p µ (cid:19) (3.26)We may now write this, using integration by parts and deleting total derivatives in τ , orby using the duality symmetric choice of total derivative terms in σ , to give a dualitysymmetric action S abb = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M η MN X (cid:48) N + (cid:90) d τ (cid:18) + 12 ˜ p ˙ (cid:101) X ( σ ) + 12 p ˙ X ( σ ) + πα (cid:48) pp − ˜ p ˙ p ) (cid:19) (3.27)that we may write in a using doubled vectors in an O ( d, d ) -invariant form, to give S abb = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M η MN X (cid:48) N + (cid:90) d τ (cid:18) p M η MN ˙ X N ( σ ) + πα (cid:48) p M ω MN p N (cid:19) (3.28)We may now identify the (cid:82) d τ p M η MN ˙ X N (0) term as the "boundary term" (3.16) weintroduced earlier to make the Tseytlin action independent of σ . Thus when the dustsettles we see that the abbreviated action of the string, including the "boundary" piece,produces the Tseytlin string (including the doubled "boundary" term) and a correctionterm from the final term in (3.28). Thus, the relation between S abb of the usual string andthe Tseytlin abbreviated action S Tsey , abb (including "boundary" pieces) is S abb = S Tsey , abb + (cid:90) d τ πα (cid:48) p M ω MN p N . (3.29) The dual picture.
Let us define the abbreviated action of the T-dual string, includingthe "boundary" piece, as follows: (cid:101) S abb := 12 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ (cid:101) X µ (cid:101) P µ − (cid:90) d τ ˙ p µ X µ ( σ ) . (3.30)Now, we can repeat all this procedure with this "dual" abbreviated action. Then thedifference between the two duality related frames for the action will be the difference ofthese abbreviated actions (the Hamiltonian being invariant), thus: S abb − (cid:101) S abb = πα (cid:48) (cid:90) d τ ˙ p M ω MN p N . (3.31)The implication is that ordinary string and its dual are related by a phase shift: exp (cid:18) i (cid:126) S abb (cid:19) = exp (cid:18) iπα (cid:48) (cid:126) (cid:90) d τ ˙ p M ω MN p N (cid:19) exp (cid:18) i (cid:126) (cid:101) S abb (cid:19) . (3.32)– 24 –et us make a sanity check. For an ordinary toroidal space p is constant in τ so it is zerofor ordinary strings on a torus. The reader at this point may feel frustrated that aftersome considerable care with total derivatives and quasi-periodic fields we have generated aterm that vanishes. However, crucially this term will not vanish for strings in backgroundswhere the winding number is not conserved. Such a situation is exactly where double fieldtheory is most useful captures the dynamic nature of winding. For example this occurswith a string in a Kaluza-Klein monopole background; a set up that was first considered in[54] and studied using double field theory in [55]. This term will also make a contributionif there is no globally defined duality frame and so one needs to form a good cover overthe space and choose a duality frame in each patch. Such spaces with no globally definedT-duality frame are called T-folds. The above phase shift will then be part of the transitionfunction between different patches acting on the string wavefunction. From the geometricquantisation perspective this is reminiscent of the Maslov correction. The Hull "topological" term and its role.
In addition to the discussion above, Hullproposed the addition of a "topological" to Tseytlin action based on global requirementsfor a gauging procedure [56]. The importance of this term for the partition function wasemphasized in [57, 58]. This term is given by: S top [ X ( σ, τ )] = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M ω MN X (cid:48) N . (3.33)Now, we can use Stokes’ theorem as it follows: S top [ X ( σ, τ )] = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M ω MN X (cid:48) N = 12 πα (cid:48) (cid:90)(cid:73) Σ d X µ ∧ d (cid:101) X µ = 12 πα (cid:48) (cid:90) d τ (cid:16) X µ ( σ + 2 π, τ ) ˙ (cid:101) X µ ( σ + 2 π, τ ) − X µ ( σ , τ ) ˙ (cid:101) X µ ( σ , τ ) (cid:17) = 12 (cid:90) d τ (cid:16) ˙˜ p (cid:101) X ( σ ) + p ˙ X ( σ ) + 2 πα (cid:48) ˙˜ pp (cid:17) (3.34)where, without any loss of generality, we have chosen the gauge X µ d (cid:101) X µ for the potentialof the -form d X µ ∧ d (cid:101) X µ .Just like for the Tseytlin action, we can define a boundary term for the topological term S ∂ top [ X (0 , τ ) , p ( τ )] = − (cid:90) d τ ˙ p M ( τ ) ω MN X N ( σ , τ ) (3.35)to remove the dependence on the cut σ .Recall the equation (3.24) relating the Tseytlin abbreviated action of a doubled stringwith the term (cid:82) d τ (cid:72) d σ ˙ X µ P µ . By combining equation (3.24) with equation (3.34), weimmediately obtain the relation πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X µ P µ = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M ( η MN + ω MN ) X (cid:48) N . (3.36)– 25 –t this point, we can provide both sides of the equation with the boundary term, so thatwe have S abb = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M ( η MN + ω MN ) X (cid:48) N − (cid:90) d τ p M ( η MN + ω MN ) ˙ X N ( σ ) (3.37)where we recall the definition (3.25) for the abbreviated action (including the boundaryterm) of the ordinary string: S abb := 12 πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X µ P µ − (cid:90) d τ ˙˜ p µ (cid:101) X µ ( σ ) (3.38)This equation can be interpreted as the fact that we need to add the boundary term − (cid:82) d τ ˙˜ p µ (cid:101) X µ ( σ ) to the usual abbreviated action πα (cid:48) (cid:82) d τ (cid:72) d σ ˙ X µ P µ of an ordinary stringto obtain the manifestly O ( n, n ) -covariant abbreviated action (3.37). The total action.
Finally, putting all this together, the total action of the doubled stringwill be given as follows S [ X ( σ, τ )] = 14 πα (cid:48) (cid:90) d τ (cid:73) d σ (cid:16) ˙ X M ( σ, τ )( η MN + ω MN ) X (cid:48) N ( σ, τ ) − X (cid:48) M ( σ, τ ) H MN X (cid:48) N ( σ, τ ) (cid:17) − (cid:90) d τ p M ( η MN + ω MN ) ˙ X N ( σ ) . (3.39) Fourier expansion of the kinetic term.
Now, recall that the Hamiltonian of the dou-bled string is the functional H [ X ( σ )] = 14 πα (cid:48) (cid:73) d σ X (cid:48) M ( σ ) H MN (cid:0) X ( σ ) (cid:1) X (cid:48) N ( σ ) . (3.40)We can now expand X M ( σ ) in σ by X M ( σ ) = x M + α (cid:48) σ p M + (cid:88) n ∈ Z \{ } n α Mn e inσ (3.41)where α Mn satisfies the identity α M − n = ¯ α Mn and it can be decomposed as α Mn = ( α µn , (cid:101) α nµ ) with (cid:101) α nµ = (cid:40) − E T µν α νn , n > E µν α νn , n < (3.42)The coordinates of the phase space can be taken as { x M , p M , α µk , ¯ α µk } k ∈ N \{ } . Now we canexplicitly express the kinetic part of the action of the doubled string in these coordinates.Firstly, we calculate the mode expansion of the abbreviated Tseytlin action together withthe topological term: πα (cid:48) (cid:90) d τ (cid:73) d σ ˙ X M ( η MN + ω MN ) X (cid:48) N = 12 (cid:90) d τ p µ ˙ x µ + πα (cid:48) ˙˜ p µ p µ + ˙˜ p µ (cid:88) n ∈ Z \{ } n ˙˜ α nµ + i (cid:88) n ∈ Z \{ } n ω MN ˙ α M − n α Nn (3.43)– 26 –hen, we expand the boundary terms: ( S ∂ Tsey + S ∂ top )[ X (0 , τ ) , p ( τ )] = − (cid:90) d τ p M ( τ )( η MN + ω MN ) ˙ X N ( σ , τ )= − (cid:90) d τ ˜ p µ ˙˜ x µ + ˜ p µ (cid:88) n ∈ Z \{ } n ˙˜ α nµ (3.44)By adding these terms together, we obtain the the mode expansion of the abbreviatedaction of the doubled string, which is the following: S [ X ( σ, τ )] + (cid:90) d τ H [ X ( σ, τ )] = (cid:90) d τ p M ˙ x M − πα (cid:48) ω MN p M ˙ p N + i (cid:88) n ∈ Z \{ } n ω MN ˙ α M − n α Nn (3.45) The symplectic structure of the doubled string.
Now we can use the equation S [ X ( σ, τ )] + (cid:90) d τ H [ X ( σ, τ )] = (cid:90) d τ ι V H Θ (3.46)to determine the Liouville potential Θ on the phase space of the doubled string and, hence,the its symplectic structure. To solve the equation, we choose again the Hamiltonian vector V H associated to the time flow, which, in the new coordinates { x M , p M , α µk , ¯ α µk } n ∈ Z \{ } ofthe phase space, takes the form V H = dd τ = ˙ x M ∂∂ x M + ˙ p M ∂∂ p M + (cid:88) n> (cid:18) ˙ α µn ∂∂α µn + ˙¯ α µn ∂∂ ¯ α µn (cid:19) (3.47)Now, by solving the equation (3.46) we obtain the following Liouville potential: Θ = p M d x M − πα (cid:48) ω MN p M d p N + i (cid:88) n ∈ Z \{ } n ω MN d α M − n α Nn . (3.48)By calculating the differential Ω = δ Θ , we finally obtain the symplectic form Ω = d p M ∧ d x M − πα (cid:48) ω MN d p M ∧ d p N + i (cid:88) n ∈ N \{ } n ω MN d α M − n ∧ d α Nn . (3.49)This is, therefore, the symplectic form of the phase space ( L Q M , Ω ) of the doubled string.Notice that we can also rewrite this symplectic form as Ω = (cid:73) d σ ω MN δ X M ( σ ) ∧ δ X (cid:48) N ( σ ) + d p M ∧ d X M ( σ ) (3.50)Notice that, if we ignore the second term encoding the boundary, the first term can beimmediately given by a potential (cid:72) d σ ω MN δ X M ( σ ) X (cid:48) N ( σ ) , which is nothing but thetransgression of a symplectic form (cid:36) := ω MN d x M ∧ d x N = d x µ ∧ d˜ x µ defined on thedoubled space. Notice that this is still a particular example of the fundamental -formwhich appears in Born geometry in [48–50] for a background without fluxes.– 27 – .3 Algebra of observables We want to determine the algebra heis ( L Q M , Ω ) of quantum observables of the phase spaceof the doubled string. ˆ f = − i (cid:126) ∇ V f + f (3.51)We, thus, obtain the following commutation relations: (cid:2) ˆ X M ( σ ) , ˆ X N ( σ (cid:48) ) (cid:3) = iπ (cid:126) α (cid:48) ω MN − i (cid:126) η MN ε ( σ − σ (cid:48) ) (3.52)where the function ε ( σ ) is the quasi-periodic function defined by ε ( σ ) := σ − i (cid:88) n ∈ Z \{ } e inσ n (3.53)and it satisfies the following properties: firstly, its derivative ε (cid:48) ( σ ) = δ ( σ ) is the Dirac comb;secondly, it satisfies the boundary condition ε ( σ + 2 πn ) = ε ( σ ) + 2 πn and, finally, it is anodd function, i.e. ε ( − σ ) = − ε ( σ ) .The fact that the operators associated with X µ ( σ ) and (cid:101) X µ ( σ (cid:48) ) do not commute by a term ∝ (cid:15) ( σ − σ (cid:48) ) was already observed as far as in [59]. However, the commutation relations (3.52)contain a new term: the skew-symmetric constant matrix ∝ π (cid:126) α (cid:48) ω MN , originating fromthe topological term (3.33) in the total action of the doubled string and totally analogousof the one recently observed by [60, 61].We can also easily derive the commutation relations for the higher modes [ ˆ α µn , ˆ α νm ] = ng µν δ m + n, = (cid:2) ˆ¯ α µn , ˆ¯ α νm (cid:3) . (3.54) Limits of the algebra of observables
It is worth remarking the role of the dimensionfulconstants (cid:126) and α (cid:48) in providing the deformation to the classical algebra. We are used toseeing (cid:126) as a quantum deformation parameter but here we also see (cid:126) α (cid:48) as another quantumdeformation parameter. This suggests interesting limits. The classical limit is the obviouslimit given by (cid:126) → . The particle limit is (cid:126) fixed but α (cid:48) → . This algebra suggests a newlimit: (cid:126) → α (cid:48) → ∞ ; α (cid:48) (cid:126) fixed . (3.55)This would be a classical stringy limit where we keep the stringy deformation but removethe quantum deformation. It would be interesting to study the system further in this limitto identify the pure string deformation based effects. Recall that we can expand the fields X M ( σ ) of our doubled string σ -model by X M ( σ ) = x M + α (cid:48) σ p M + (cid:88) n ∈ Z \{ } n α Mn e inσ (4.1)– 28 –here the coordinates of the phase space of a doubled string are { x M , p M , α µk , ¯ α µk } , withthe former { x M , p M } co-ordinatising the zero-modes of the string and the latter { α µk , ¯ α µk } co-ordinatising its higher-modes. A zero-mode truncated doubled string is a doubled stringwhere we are neglecting the higher-modes and it can be seen as a simple embedding of theform X M ( σ ) = x M + α (cid:48) σ p M . (4.2)The zero-modes of a doubled string X M ( σ, τ ) can be thought as a particle in a doubledphase space ( x M ( τ ) , p M ( τ )) . Similarly we expect that the wave-functional Ψ[ X ( σ )] at zeromodes is just a wave-function ψ ( x , p ) on the doubled phase space of zero modes: Ψ[ X ( σ )] modes −−−−−−→ ψ ( x , p ) , Ω modes −−−−−−→ ω (4.3)The phase space of the zero-modes of a doubled string is, therefore, a n -dimensionalsymplectic manifold ( P , ω ) with symplectic form ω = η MN d p M ∧ d x N − πα (cid:48) ω MN d p M ∧ d p N (4.4)and underlying smooth manifold P = R n . Notice that this -form, obtained by a Hamil-tonian treatment of the total action of a doubled string σ -model, exactly agrees with thesymplectic form found by [60, 61] by starting from vertex algebra arguments.Now, we can apply the machinery of geometric quantisation to this symplectic manifold ( P , ω ) to quantise the zero-modes of a doubled string. Kinetic coordinates for the doubled phase space.
Let us change the canonicalmomentum coordinates with the untwisted non-canonical momentum coordinates k M =( e − B ) MN p N . Given a doubled string σ -model X ( σ, τ ) , these will be related by k µ ( σ, τ ) = (cid:73) d σ g µν ˙ X ν ( σ, τ ) , ˜ k µ ( σ, τ ) = (cid:73) d σ g µν ˙ (cid:101) X ν ( σ, τ ) . (4.5)We can rotate the doubled coordinates, accordingly x M (cid:55)→ ( e − B ) MN x N to the untwistedframe. We can now rewrite the symplectic form in the kinetic coordinates { x M , k M } andhave ω = η MN d k M ∧ d x N − πα (cid:48) ω ( B ) MN d k M ∧ d k N (4.6)where we called the matrix ω ( B ) MN = (cid:32) B µν δ νµ − δ µν (cid:33) . (4.7)Then, we can choose the following gauge for the Liouville potential: θ = η MN k M d x N − πα (cid:48) ω ( B ) MN k M d k N . (4.8)– 29 – he action of the zero-mode string. As we remarked, in geometric quantisation theLagrangian density L ∈ Ω ( γ ) of a particle is related to the Liouville potential θ ∈ Ω ( P ) by the equation L H = ( ι V H θ − H )d τ. (4.9)We can then immediately use it, in the form S [ x ( τ ) , k ( τ )] = (cid:90) γ d τ (cid:0) ι V H θ − H (cid:1) with V H = ˙ x M ∂∂ x M + ˙ k M ∂∂ k M , (4.10)to find the action of the zero-mode doubled string: S [ x ( τ ) , k ( τ )] = (cid:90) γ d τ (cid:18) η MN k M ˙ x N − πα (cid:48) ω ( B ) MN k M ˙ k N − H (0) MN k M k N (cid:19) (4.11)where we called the matrix H (0) MN = (cid:32) g µν g µν (cid:33) . (4.12) Recall that in geometric quantisation a quantum observable ˆ f ∈ Aut( H ) is a linear automor-phism of the Hilbert space, obtained from the corresponding classic observable f ∈ C ∞ ( P ) by the following identification: ˆ f := i (cid:126) ∇ V f + f, (4.13)where the vector V f ∈ X ( P ) is the Hamiltonian vector with Hamiltonian function f , i.e.the vector which solves the Hamilton equation ι V f ω = d f. (4.14)In this subsection we want to determine the Lie algebra of quantum observables heis ( P , ω ) on the doubled phase space. Hamiltonian vector fields.
Let us first solve the Hamilton equation (4.14) for a genericHamiltonian function f ∈ C ∞ ( P ) . We expand the vector V f ∈ X ( P ) in the kinetic coordi-nates V f = V Mf, x ∂∂ x M + V Mf, k ∂∂ k M . (4.15)Hence, the Hamilton equation (4.14), in coordinates, becomes η MN ( V Mf, k d x N − V Mf, x d k N ) − πα (cid:48) ω ( B ) MN V Mf, k d k N = ∂f∂ x M d x M + ∂f∂ k M d k M . (4.16)Therefore, the Hamiltonian vector field V f with Hamiltonian f is given by V f = (cid:16) η MN ∂f∂ x N (cid:17) ∂∂ k M + (cid:16) − η MN ∂f∂ k N − πα (cid:48) ω MN ( B ) ∂f∂ x N (cid:17) ∂∂ x M (4.17)where we called ω MN ( B ) := η ML ω ( B ) LP η P N . In particular the Hamiltonian vector fields corre-sponding to the classical observables of the kinetic coordinates x M and k M are V f = x N = η MN ∂∂ k M − πα (cid:48) ω MN ( B ) ∂∂ x M V f = k N = − η MN ∂∂ x M . (4.18)– 30 – on-commutative Heisenberg algebra. By applying the definition (4.13) of quantumobservable, we find that the operators associated to the kinetic coordinates are the following: ˆ x M = i (cid:126) η NM ∂∂ k N − i (cid:126) πα (cid:48) ω NM ( B ) ∂∂ x N + x M ˆ k M = − i (cid:126) η NM ∂∂ x N (4.19)Therefore the commutation relations between the coordinates operators are the following: [ˆ x M , ˆ x N ] = πi (cid:126) α (cid:48) ω MN ( B ) , [ˆ x M , ˆ k N ] = i (cid:126) η MN , [ˆ k M , ˆ k N ] = 0 . (4.20)Thus, the n -dimensional Lie algebra heis ( P , ω ) can be regarded as a non-commutativeversion of the usual Heisenberg algebra, where the position operators do not generallycommute.Explicitly, in undoubled notation, we have the following commutation relations: [ˆ x µ , ˆ x ν ] = 0 , [ˆ x µ , ˆ˜ x ν ] = πi (cid:126) α (cid:48) δ µν , [ˆ˜ x µ , ˆ˜ x ν ] = − πi (cid:126) α (cid:48) B µν , [ˆ k µ , ˆ k ν ] = 0 , [ˆ k µ , ˆ˜ k ν ] = 0 , [ˆ˜ k µ , ˆ˜ k ν ] = 0 , [ˆ x µ , ˆ˜ k ν ] = [ˆ˜ x µ , ˆ k ν ] = 0 , [ˆ x µ , ˆ k ν ] = i (cid:126) δ µν , [ˆ˜ x µ , ˆ˜ k ν ] = i (cid:126) δ νµ . (4.21)Examining this algebra from the perspective of the limits we discussed earlier we see that (cid:126) controls the noncommutativity of the position with the momentum and (cid:126) α (cid:48) the noncom-mutativity of the coordinates and their duals. Finally, α (cid:48) B the noncommutativity of thespacetime coordinates. Thus when the B-field is included we have three noncommutativityparameters. Uncertainty principle on the doubled space.
Following standard text book tech-niques applied to the commutation relations (4.21), we can immediately show that anyposition coordinate x µ and its dual ˜ x µ satisfy the following uncertainty relation: ∆ x ∆˜ x ≥ π (cid:126) α (cid:48) . (4.22)This means that x µ and ˜ x µ cannot be measured with absolute precision at the same time,but there will be always a minimum uncertainty proportional to the area (cid:126) α (cid:48) . This providessupport to the intuition of a minimal distance scale in string theory. The standard lore isthat for small distances one goes to the T-dual frame and the distances will always be largerthan the string scale.In addition, both the couples ( x, p ) and (˜ x, ˜ p ) satisfy the usual uncertainty relation betweenposition and momentum: ∆ x ∆ p ≥ (cid:126) , ∆˜ x ∆˜ p ≥ (cid:126) . (4.23)However, it is worth noticing that the momentum and its dual can be measured at the sametime: ∆ p ∆˜ p ≥ . (4.24)– 31 – amiltonian. Notice that the Hamiltonian operator of the zero-mode doubled string willbe given by ˆ H = H (0) MN ˆ k M ˆ k N = − (cid:126) H MN (0) ∂∂ x M ∂∂ x N (4.25)where we called H MN (0) := η ML η NP H (0) LP . Non-commutative Heisenberg algebra in canonical coordinates.
In the zero-modestring canonical coordinates { x M , p M } we obtain the following operators: ˆ x M = i (cid:126) ∂∂ p M − i (cid:126) πα (cid:48) ω NM ∂∂ x N + x M ˆ p M = − i (cid:126) ∂∂ x M (4.26)Therefore the commutation relations between the canonical coordinates observables are [ˆ x M , ˆ x N ] = πi (cid:126) α (cid:48) ω MN , [ˆ x M , ˆ p N ] = i (cid:126) δ MN , [ˆ p M , ˆ p N ] = 0 . (4.27) Relation with the symplectic structure of the doubled space.
Let us now focus onthe subalgebra generated by the operators ˆ x µ and ˆ˜ x µ . This will be given by the followingcommutation relations: [ˆ x µ , ˆ x ν ] = 0 , [ˆ x µ , ˆ˜ x ν ] = πi (cid:126) α (cid:48) δ µν , [ˆ˜ x µ , ˆ˜ x ν ] = 0 (4.28)Notice that this can be seen as an ordinary n -dimensional Heisenberg algebra h (2 n ) . Thismeans that such an algebra is immediately given by a symplectic manifold ( M , (cid:36) ) with M ∼ = R n and symplectic form (cid:36) := π (cid:126) α (cid:48) d x µ ∧ d˜ x µ . This symplectic structure on thedoubled space is exactly the one introduced by [62]. In ordinary quantum mechanics we choose to represent the wavefunctions in either the po-sition or the momentum basis and it is the Fourier transform that maps the wavefunctionin one basis to the other basis. From the persepective of geometric quantisation this is thetransformation between elements of the Hilbert spaces constructed with different choicesof Lagrangian submanifold ie. different polarisations. T-duality is a change in our choiceof polarisation. We can then follow the pairing construction used in [27–30] to constructthe transformation for the string wavefunction moving between different duality frames.This will produce a string deformed Fourier transform (that reduces to the usual Fouriertransform in the α (cid:48) → limit). From the geometric quantisation perspective these trans-formations are known as Blatter-Kostant-Sternberg [27] kernel’s. Polarisations.
In the geometric quantisation of a symplectic space ( P , ω ) , a polarisationcorresponds to a choice of an integrable Lagrangian subspace L ⊂ P . Since P is a vectorspace, the first Chern class of the prequantum U (1) -bundle whose curvature is the symplec-tic form ω ∈ Ω ( P ) , is necessarily trivial. In geometric quantisation, this implies that the– 32 –ilbert space of the quantised system is defined by the space of the complex L -functionson the Lagrangian submanifold L ⊂ P , i.e. by H := L ( L, C ) . (4.29)Remarkably, this does not depend on the choice of polarisation L ⊂ P and it is possible toprove that, for any other Lagrangian subspace L (cid:48) ⊂ P , we would have an isomorphism ofHilbert spaces H ∼ = L ( L (cid:48) , C ) . T-duality as a change of polarisation.
Let us rewrite the symplectic form ω ∈ Ω ( P ) in canonical coordinates ( x M , p M ) , i.e. ω = η MN d p M ∧ d x N − πα (cid:48) ω MN d p M ∧ d p N , (4.30)and let us recall that the momenta doubled vector can be interpreted as the doubled vectorof winding numbers p M = ( w µ , ˜ w µ ) . It is now immediate that the vector spaces L := Span( x µ , w µ ) , (cid:101) L := Span(˜ x µ , ˜ w µ ) (4.31)are Lagrangian subspaces of the symplectic space ( P , ω ) . This means that we will have twopolarisations corresponding to the two T-duality frames ( x µ , w µ ) and (˜ x µ , ˜ w µ ) . We can thusdefine two basis {| x, w (cid:105)} ( x,w ) ∈ L and {| ˜ x, ˜ w (cid:105)} (˜ x, ˜ w ) ∈ (cid:101) L for our Hilbert space H . If we considera generic state | ψ (cid:105) ∈ H of our Hilbert space, we can now express it in the basis associatedto both the T-duality frames by ψ w ( x ) := (cid:104) x, w | ψ (cid:105) , (cid:101) ψ ˜ w (˜ x ) := (cid:104) ˜ x, ˜ w | ψ (cid:105) . (4.32)Now, we want to explicitly find the isomorphism L ( L ; C ) ∼ = L ( (cid:101) L ; C ) between wave-functions in the two T-duality frames. Let us expand our zero-mode truncated string by X µ ( σ ) = x µ + α (cid:48) σ ˜ p µ and (cid:101) X µ ( σ ) = ˜ x µ + α (cid:48) σp µ and call X M ( σ ) = (cid:0) X µ ( σ ) , (cid:101) X µ ( σ ) (cid:1) . A zero-mode truncated string X M ( σ ) = x M + α (cid:48) σ p M is represented by a point ( x M , p M ) ∈ P of thephase space of the zero-mode doubled string. Thus, if we truncate to the zero-modes thegenerating functional (2.43) of the symplectomorphism encoding T-duality, we will obtainthe following generating function on the phase space of the zero-mode doubled string: F [ X ( σ ) , (cid:101) X ( σ )] = (cid:73) d σ X M ( σ ) ω MN X (cid:48) N ( σ )= X µ (2 π ) (cid:101) X µ (2 π ) − X µ (0) (cid:101) X µ (0)= ˜ p µ ˜ x µ − p µ x µ + πα (cid:48) p µ ˜ p µ . (4.33)In other words, we can express the symplectomorphism f : P → P encoding T-duality onthe phase space of the zero-mode doubled string by the generating function F ( x , p ) = ˜ p µ ˜ x µ − p µ x µ + πα (cid:48) p µ ˜ p µ = ω MN p M x N + πα (cid:48) η MN p M p N . (4.34)– 33 –uch a symplectomorphism is simply the O ( n, n ) transformation of the doubled coordinatesand momenta by ( x M , p M ) (cid:55)→ ( η MN x N , η MN p N ) . Now, by applying the machinery ofgeometric quantisation, the matrix of the change of basis on the Hilbert space H will begiven by the generating function (4.34) as it follows: (cid:104) x, w | ˜ x, ˜ w (cid:105) = exp i (cid:126) (cid:0) p µ x µ − ˜ p µ ˜ x µ + πα (cid:48) p µ ˜ p µ (cid:1) = exp i (cid:126) (cid:18) ω MN p M x N + πα (cid:48) η MN p M p N (cid:19) , (4.35)where w µ ≡ ˜ p µ and ˜ w µ ≡ p µ . Therefore we can equivalently rewrite the transformation (cid:104) ˜ x, ˜ w | ψ (cid:105) = (cid:90) L d n x d n w (cid:104) ˜ x, ˜ w | x, w (cid:105) (cid:104) x, w | ψ (cid:105) (4.36)as the following, stringy Fourier transformation : (cid:101) ψ ˜ w (˜ x ) = (cid:90) L d n x d n w exp i (cid:126) (cid:18) ω MN p M x N + πα (cid:48) η MN p M p N (cid:19) ψ w ( x ) . (4.37)This is the transformation between the wavefunctions in different duality frames. Math-ematically it is the isomorphism L ( L, C ) ∼ = L ( (cid:101) L, C ) . In undoubled coordinates we canexplicitly rewrite such a stringy Fourier transformation as it follows: (cid:101) ψ ˜ w (˜ x ) = (cid:90) L d n x d n w exp i (cid:126) (cid:0) ˜ w µ x µ − w µ ˜ x µ + πα (cid:48) ˜ w µ w µ (cid:1) ψ w ( x ) , (4.38)where we used the identities w µ ≡ ˜ p µ and ˜ w µ ≡ p µ . Notice that, even if the form ofthe symplectomorphism f is particularly simple, the transformation for wave-function ismore complicated than just a Fourier transform. The difference from the usual Fouriertransformation is given by the additional πα (cid:48) η MN p M p N term. This then will reduces to astandard Fourier transform in the limit α (cid:48) → .In terms of basis, this transformation can be also be expressed by | ˜ x, ˜ w (cid:105) = (cid:90) L d n x d n w exp i (cid:126) (cid:18) ω MN p M x N + πα (cid:48) η MN p M p N (cid:19) | x, w (cid:105) . (4.39) A phase term in the change of polarisation.
Finally, notice that, if we restrict ourgeneralised winding to ordinary integer winding w, ˜ w ∈ Z n , we will obtain a change ofpolarisation of the form (cid:101) ψ ˜ w (˜ x ) = (cid:88) w ∈ Z n e i (cid:126) πα (cid:48) ˜ w µ w µ (cid:90) M d n x e i (cid:126) ( ˜ w µ x µ − w µ ˜ x µ ) ψ w ( x ) . (4.40)In this context, as firstly noticed with different arguments by [61], T-duality does not simplyact as a "double" Fourier transformation of the wave-function of a string, because therewill be an extra phase contribution given by exp (cid:0) iπ α (cid:48) (cid:126) ˜ w µ w µ (cid:1) for any term with w, ˜ w (cid:54) = 0 .Since we are restricting now to the case where w, ˜ w are integers and (cid:112) (cid:126) /α (cid:48) is just theunit of momentum, we immediately conclude that the only possible phase contributions are exp (cid:0) iπ α (cid:48) (cid:126) ˜ w µ w µ (cid:1) ∈ { +1 , − } , depending on the product ˜ w µ w µ ≡ p µ w µ being even or odd.Notice that the presence of the topological term in the action induces a very similar phaseterm in the partition function of a string with an analogous role, as seen by [63].– 34 – arboux coordinates for the zero-mode string. Let us find the Darboux coordinateson the manifold P for the symplectic form ω ∈ Ω ( P ) . If we define the new coordinates q µ := x µ ˜ q µ := ˜ x µ − πα (cid:48) p µ (4.41)and we pack them together as q M := ( q µ , ˜ q µ ) , we can rewrite the symplectic form simplyas ω = d p M ∧ d q M (4.42)with p M = η MN p N = ( p µ , ˜ p µ ) . Therefore the conjugate variable on the phase to thecanonical momenta p M of the zero-mode string is the new coordinate q M . Notice that thisvariable is not the proper position x M on the doubled space, but a mix of position andmomentum. This change of coordinates is intimately related to what is known as Bopp’sshift in non-commutative quantum mechanics. The non-commutativity we are exploring follows that in [53], but is different from (thoughclose to) the one introduced by [64] and further explored by [65], where the non-commutativityof the doubled space is induced by the presence of fluxes. For a more recent account see [66]and [38]. The notion of non-commutativity we are considering is completely independent bythe presence of fluxes and characterises even flat and topologically trivial doubled spaces.As we will see in the next section, the non-commutativity between a physical coordinateand its T-dual is intrinsic and linked to the existence of a minimal length (cid:96) s = √ (cid:126) α (cid:48) on thedoubled space.The link between the two notions of non-commutativity is provided by [53]. The presenceof flux implies monodromies for the generalised metric of the form H ( x + 2 π ) = OH ( x ) O T (4.43)with monodromy matrix O ∈ O ( n, n ) . Thus, as explained by [53], we need to consider thefurther generalised boundary conditions X M ( σ + 2 π ) = O MN X N ( σ ) + 2 π p M (4.44)for our doubled string σ -model. When we write the Tseytlin action, we then have togeneralise its boundary term accordingly. As seen by [53], the new action produces thenon-commutativity given by the fluxes on the doubled phase space. Non-commutative quantum mechanics was introduced as far as in 1947 by [67]. The fun-damental idea, at the time, was to quantize flat spacetime by introducing a minimal lengthand generalising the uncertainty principle to make it fuzzy.– 35 –s we saw in the previous section, the zero-mode truncation of the pre-quantised wave-functional Ψ[ X ( σ )] of a doubled string can be seen as a conventional pre-quantised wave-function ψ ( x , p ) of a particle in a doubled space. In other words, the zero-modes of stringsbehave like particles in a double space. However, such a doubled space is intrinsically non-commutative. As we derived, indeed, the commutation relations of the position operatorsare of the form [ˆ x M , ˆ x N ] = iϑ MN with ϑ MN := π (cid:126) α (cid:48) ω MN . (5.1)This means that the Quantum Mechanics of the string zero-modes will be non-commutative(NCQM). Let us choose units where we only require c = 1 . This way the two physicaldimensions of length and energy are explicitly parametrised by the two universal constantsas it follows: (cid:2) (cid:126) α (cid:48) (cid:3) = length , (cid:20) (cid:126) α (cid:48) (cid:21) = energy (5.2)and thus the string scale must be expressed as (cid:96) s = √ (cid:126) α (cid:48) . We notice that any couple ofT-dual coordinates fails in commuting by an area which is proportional to the string scale,i.e. we have (cid:2) ˆ x µ , ˆ˜ x µ (cid:3) = iπ(cid:96) s (5.3)for any fixed µ = 1 , . . . , n . In this context π(cid:96) s can be interpreted as a minimal area of thedoubled space. Let us start from the non-commutative Heisenberg algebra heis ( ω , P ) of the phase space ( ω , P ) of the zero-modes truncated doubled string: [ˆ x M , ˆ x N ] = πi (cid:126) α (cid:48) ω MN , [ˆ x M , ˆ p N ] = i (cid:126) δ MN , [ˆ p M , ˆ p N ] = 0 . (5.4)Notice that the subspace of P = R n spanned by ( x µ , ˜ x µ ) is not a Lagrangian subspace,therefore there is no well-defined notion of wave-function of the form ψ ( x µ , ˜ x µ ) . In quantummechanical terms, since T-dual coordinates [ˆ x µ , ˆ˜ x ν ] (cid:54) = 0 do not generally commute, thereexists no basis (cid:8) | x µ , ˜ x µ (cid:105) (cid:9) (cid:54)⊂ H of eigenstates of the doubled position operators. Coherent states.
However, we can define the following annihilation and creation oper-ators ˆ z µ = 1 √ π (cid:126) (cid:16) ˆ x µ + i ˆ˜ x µ (cid:17) ˆ z † µ = 1 √ π (cid:126) (cid:16) ˆ x µ − i ˆ˜ x µ (cid:17) (5.5)whose commutator (cid:104) ˆ z µ , ˆ z † ν (cid:105) = α (cid:48) δ µν (5.6)satisfies the commutation relations of the Fock algebra. Thus the non-commutative quan-tum configuration space is a bosonic Fock space F cs := (cid:75) k ∈ N C n = C ⊕ C n ⊕ ( C n (cid:12) C n ) ⊕ ( C n (cid:12) C n (cid:12) C n ) ⊕ . . . (5.7)– 36 –enerated by vectors of the form | (cid:105) , ˆ z † µ | (cid:105) , √ z † µ ˆ z † µ | (cid:105) , √
3! ˆ z † µ ˆ z † µ ˆ z † µ | (cid:105) , . . . (5.8)where the vacuum state | (cid:105) is defined by the equation ˆ z µ | (cid:105) = 0 for all µ = 1 , . . . , n .The important aspect of working with the creation and annihilation operators is that thereexist eigenstates (cid:12)(cid:12) z , · · · , z n (cid:11) for all the operators ˆ z µ with µ = 1 , . . . , n . These satisfy thefollowing defining properties: ˆ z µ (cid:12)(cid:12) z , · · · , z n (cid:11) = z µ (cid:12)(cid:12) z , · · · , z n (cid:11)(cid:68) z , · · · , z d (cid:12)(cid:12)(cid:12) ˆ z † µ = (cid:68) z , · · · , z d (cid:12)(cid:12)(cid:12) ¯ z µ (5.9)for eigenvalues ( z , · · · , z n ) ∈ C n . These states are called coherent states .Let us now use the compact notation | z (cid:105) := (cid:12)(cid:12) z , · · · , z n (cid:11) for coherent states. A normalisedcoherent state can be expressed by | z (cid:105) := exp (cid:18) − δ µν α (cid:48) z µ ¯ z ν (cid:19) exp (cid:18) − δ µν α (cid:48) z µ ˆ z † ν (cid:19) | (cid:105) (5.10)where the vacuum state | (cid:105) ∈ F cs is defined as previously. These states constitute a completebasis on the Fock space space F cs , since they satisfy the property (cid:90) C n d n z d n ¯ z | z (cid:105) (cid:104) z | = 1 (5.11)There is an isomorphism between the non-commutative quantum configuration space F cs and the quantum Hilbert space H , i.e. F cs ∼ = H . (5.12)Such an isomorphism will be explicitly presented in equation (5.23). Mean position of a coherent state.
The expectation value of the non-commutativeposition operators on a coherent state | z (cid:105) can be found by (cid:104) z | ˆ x µ | z (cid:105) = √ π (cid:126) Re ( z µ ) =: x µ , (cid:104) z | ˆ˜ x µ | z (cid:105) = √ π (cid:126) Im ( z µ ) =: ˜ x µ (5.13)The doubled vector x M = ( x µ , ˜ x µ ) is the mean position of the coherent state | z (cid:105) on thedoubled space R d , also known as quasi-coordinate vector. It is important to remark that x M are not coordinates, i.e. they are not eigenvalues of the operators ˆ x M . Thus, by workingwith coherent states | z (cid:105) with mean position x M , we can bypass the problem of not being ableto work with eigenstates of the position operators. In general, any operator ˆ f (ˆ x M ) can beexpressed as a function of the mean positions of a coherent state by F ( x M ) := (cid:104) z | ˆ f (ˆ x M ) | z (cid:105) . Minimal uncertainty.
Coherent states minimize the uncertainty between a coordinateoperator of the doubled space and its T-dual, i.e. ∆ x µ ∆˜ x ν = π(cid:96) s δ µν (5.14)The coherent states of the quantum configuration space can then be interpreted as stateswhich are approximately localised at a point x M of the doubled space, the mean position.– 37 – .2 Free particles on the doubled spacePlane waves. The mean value of a plane wave operator on a coherent state is given by (cid:28) z (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) i (cid:126) p M ˆ x M (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) z (cid:29) = exp (cid:18) i (cid:126) p M x M − πα (cid:48) (cid:126) δ MN p M p N (cid:19) (5.15)This can be immediately proved by defining the complex momentum operators ˆ p + µ = (cid:114) π (cid:126) (cid:16) ˆ p µ + i ˆ˜ p µ (cid:17) ˆ p − µ = (cid:114) π (cid:126) (cid:16) ˆ p µ − i ˆ˜ p µ (cid:17) (5.16)and by applying the Baker–Campbell–Hausdorff formula as it follows: exp (cid:18) i (cid:126) p M ˆ x M (cid:19) = exp (cid:16) i p + µ ˆ z † µ + i p − µ ˆ z µ (cid:17) = exp (cid:16) i p + µ ˆ z † µ (cid:17) exp (cid:0) i p − µ ˆ z µ (cid:1) exp (cid:18) − πα (cid:48) (cid:126) δ MN p M p N (cid:19) . (5.17) The Hilbert space of a free particle.
The subspace L p = Span( p M ) ⊂ P is La-grangian. This can be immediately understood by writing the symplectic form in Darbouxcoordinates as ω = d p M ∧ d q M . Therefore we can express our Hilbert space H as the spaceof complex L -functions on L p , i.e. as H ∼ = L ( L p ; C ) (5.18)Since the doubled momenta commute, i.e. [ˆ p M , ˆ p N ] = 0 , we can define a basis of eigenstates {| p (cid:105)} p ∈ L p ⊂ H of the doubled momentum operators ˆ p M without the problems encounteredwith the doubled position operator. These satisfy ˆ p M | p (cid:105) = p M | p (cid:105) for any M = 1 , . . . , d .If we choose the basis of coherent states {| z (cid:105)} z ∈ C n , a doubled momentum eigenstate canthen be expressed by (cid:104) z | p (cid:105) = exp (cid:18) i (cid:126) p M x M − πα (cid:48) (cid:126) δ MN p M p N (cid:19) (5.19)where x M is the mean position of the coherent state | z (cid:105) . This can be interpreted as theexpression of a free particle state on the doubled space, where we are using the meanposition as a variable. Strings are waves.
Interestingly, if we choose the basis | x, w (cid:105) with w µ = ˜ p µ , whichdiagonalizes the commuting operators of the physical position ˆ x µ and winding ˆ˜ p µ , a doubledmomentum eigenstate can be expressed just as a free particle in the wave-function on thephysical space (cid:104) x, w | p (cid:105) = exp (cid:18) i (cid:126) p µ x µ (cid:19) (5.20)– 38 –here now x µ are proper eigenvalues of the position operator. Analogously, in the T-dualframe | ˜ x, ˜ w (cid:105) with ˜ w µ = p µ , we recover a free particle on the T-dual space (cid:104) ˜ x, ˜ w | p (cid:105) = exp (cid:18) i (cid:126) ˜ p µ ˜ x µ (cid:19) (5.21)The interpretation of this fact is that a free particle, i.e. a plane wave, on the doubled spacewith fixed doubled momentum p M = ( p µ , ˜ p µ ) can be quantum-mechanically interpreted as• a free string on the physical space with fixed momentum p µ and winding w µ = ˜ p µ ,• a free string on the T-dual space with fixed momentum ˜ p µ and winding ˜ w µ = p µ .As classically derived in [68], this implies that a plane wave with doubled momentum p M =( p µ , on the doubled space reduces to a plane wave with momentum p µ on the physicalspace and one with p M = (0 , ˜ p µ ) reduces to a standing string with winding w µ = ˜ p µ . Probability distribution.
Let us calculate the probability distribution of the wave-function (5.19) of a free particle on the doubled space: |(cid:104) z | p (cid:105)| = (cid:18) α (cid:48) (cid:126) (cid:19) n exp (cid:18) − πα (cid:48) (cid:126) δ MN p M p N (cid:19) . (5.22)Hence, the probability of measuring the doubled momentum p M (or equivalently a stringwith momentum p µ and winding ˜ p µ ) is not uniform, but it exponentially decays far fromzero. Coherent state as superposition of strings.
Now, the eigenstate | p (cid:105) can be inter-preted as a free string for which we know with certainty the momentum p µ and the windingnumber w µ = ˜ p µ . The equation (5.19) can be immediately interpreted as the expansion ofa coherent state | z (cid:105) in the basis | p (cid:105) , i.e. | z (cid:105) = (cid:90) d n p (2 π ) n exp (cid:18) i (cid:126) p M x M − πα (cid:48) (cid:126) δ MN p M p N (cid:19) | p (cid:105) (5.23)where x M is the mean doubled position of the coherent state | z (cid:105) . Hamiltonian as number operator.
Observe that the Hamiltonian operator is given by ˆ H = H MN ˆ p M ˆ p N (5.24)Let us consider the simple case where the generalised metric is trivial, i.e. H MN = δ MN .We can then define a number operator ˆ N µ := ˆ p − µ ˆ p + µ for any fixed µ = 1 , . . . , n and thetotal number operator as a sum ˆ N = (cid:80) nµ =1 ˆ N µ . What we obtain is that the Hamiltonianis proportional to the number operator by ˆ H = π (cid:126) ˆ N .– 39 – .3 Minimal scale of the doubled spaceNon-commutative Fourier transform. Let us consider a general string state | ψ (cid:105) ∈ H .We can express this state as a wave function ψ ( p ) = (cid:104) p | ψ (cid:105) on the momentum space. Thus,if we want to express it in the coherent states basis | z (cid:105) , we need to use equation (5.23) asit follows: (cid:104) z | ψ (cid:105) = (cid:90) d n p (2 π ) n exp (cid:18) i (cid:126) p M x M − πα (cid:48) (cid:126) δ MN p M p N (cid:19) (cid:104) p | ψ (cid:105) . (5.25)Now we can transform wave-functions ψ ( p ) := (cid:104) p | ψ (cid:105) on the doubled momentum space towavefunctions ψ ( x ) := (cid:104) z | ψ (cid:105) expressed in the basis of the coherent states. (Here x M denotesthe mean position of | z (cid:105) and is not a coordinate.) This is effectively a non-commutativeversion of the Fourier transform.Let us now choose the free particle state | ψ (cid:105) = | p (cid:105) , which will have wave-function ψ ( p ) = 1 on the doubled momentum space. The change of basis (5.23) gives ψ ( x ) = (cid:90) d n p (2 π ) n exp (cid:18) i (cid:126) p M x M − πα (cid:48) (cid:126) δ MN p M p N (cid:19) = exp (cid:32) − (cid:12)(cid:12) x M (cid:12)(cid:12) π (cid:126) α (cid:48) (cid:33) (5.26)which is a Gaussian distribution on the doubled space and not a delta function. This meansthat, even if the doubled momentum is maximally spread, the uncertainty on the doubledcoordinates cannot be zero. This is because each couple of T-dual coordinates can shrinkonly to a minimal area proportional to (cid:96) s = (cid:126) α (cid:48) . Thus α (cid:48) is the parameter which controlsthe fuzziness of doubled space between physical and T-dual coordinates. Amplitude between coherent states.
Let us consider two coherent states | z (cid:105) and | z (cid:105) , respectively with mean position x M and x M . We want now to calculate the scatteringamplitude (cid:104) z | z (cid:105) between such states. We can use that fact that {| p (cid:105)} p ∈ L p ⊂ H is acomplete basis for our Hilbert space and write (cid:104) z | z (cid:105) = (cid:90) d n p (2 π ) n (cid:104) z | p (cid:105) (cid:104) p | z (cid:105) . (5.27)By using equation (5.19) we obtain the integral (cid:104) z | z (cid:105) = (cid:90) d n p (2 π ) n exp (cid:18) i (cid:126) p M ( x M − x M ) − πα (cid:48) (cid:126) δ MN p M p N (cid:19) . (5.28)From this we get the result that the amplitude between two coherent states with differentmean positions x M and x M is given by (cid:104) z | z (cid:105) = exp (cid:32) − (cid:12)(cid:12) x M − x M (cid:12)(cid:12) π (cid:126) α (cid:48) (cid:33) (5.29)Thus the ordinary Dirac delta function is replaced by a Gaussian probability distributionwhose width is proportional to string length scale (cid:96) s = √ (cid:126) α (cid:48) . Physically, this means thatthe probability is high if the distance between the mean positions x and x of the respectivecoherent states | z (cid:105) and | z (cid:105) is smaller than √ π(cid:96) s .– 40 – he α (cid:48) → limit. If we take the limit α (cid:48) → the fuzziness of the doubled spacedisappears. The non-commutative Heisenberg algebra of quantum observables reduces toan ordinary commutative n -dimensional Heisenberg algebra, whose commutation relationsare given by lim α (cid:48) → [ˆ x M , ˆ x N ] = 0 , lim α (cid:48) → [ˆ x M , ˆ p N ] = i (cid:126) δ MN , lim α (cid:48) → [ˆ p M , ˆ p N ] = 0 . (5.30)Consequently the minimal uncertainty in measuring a coordinate and its dual vanishes. Thebasis of coherent states | z (cid:105) reduces to a basis of eigenstates | x, ˜ x (cid:105) of the position operator ˆ x , which are now well-defined. Moreover, the scattering amplitudes shrink to lim α (cid:48) → (cid:104) z | z (cid:105) = δ ( x − x ) . (5.31)In the limit α (cid:48) → , the quantum mechanics on the doubled space becomes ordinary com-mutative quantum mechanics on a n -dimensional spacetime. Let us consider on our Hilbert space the basis {| x, w (cid:105)} ( x,w ) ∈ L ⊂ H , which corresponds tothe T-duality frame given by the Lagrangian subspace L ⊂ P with coordinates ( x µ , w µ ) .Recall that, given a string state | ψ (cid:105) ∈ H , we can express it as a wave-function on theLagrangian subspace L ⊂ P by ψ w ( x ) := (cid:104) x, w | ψ (cid:105) , where w µ := ˜ p µ is the generalisedwinding number. Thus | ψ w ( x ) | can be interpreted as the probability of measuring a stringat the point x µ on physical spacetime with winding number w µ . Now, we want express acoherent state | z (cid:105) in this basis. In other words we want to calculate ψ coh w ( x ) := (cid:104) x, w | z (cid:105) . (5.32)To do that, we can use the fact that | p (cid:105) are a complete basis for the Hilbert space H andwrite (cid:104) x, w | z (cid:105) = (cid:90) d n p (cid:48) (2 π ) n (cid:10) x, w (cid:12)(cid:12) p (cid:48) (cid:11) (cid:10) p (cid:48) (cid:12)(cid:12) z (cid:11) . (5.33)Let us now use the notation (cid:104) x M (cid:105) := (cid:104) z | ˆ x M | z (cid:105) for the mean position of a coherent sate | z (cid:105) and again ( x µ , w µ ) for the coordinates of the Lagrangian subspace L ⊂ P where thepolarised wave-function lives. Let us use the expressions (5.19) and (5.20) of a free particlein the doubled space to calculate the intermediate terms (cid:10) x, w (cid:12)(cid:12) p (cid:48) (cid:11) = exp (cid:18) i (cid:126) p (cid:48) µ x µ (cid:19) δ ( w − ˜ p (cid:48) ) (cid:10) p (cid:48) (cid:12)(cid:12) z (cid:11) = exp (cid:18) − i (cid:126) p (cid:48) M (cid:10) x M (cid:11) − πα (cid:48) (cid:126) δ MN p (cid:48) M p (cid:48) N (cid:19) . (5.34)Hence the integral (5.33) becomes a Fourier transform in the physical momentum p (cid:48) µ only (cid:104) x, w | z (cid:105) = (cid:18)(cid:90) d n p (cid:48) (2 π ) n exp (cid:18) − i (cid:126) p (cid:48) µ ( x µ − (cid:104) x µ (cid:105) ) − πα (cid:48) (cid:126) (cid:12)(cid:12) p (cid:48) µ (cid:12)(cid:12) (cid:19)(cid:19) exp (cid:18) i (cid:126) w µ (cid:104) ˜ x µ (cid:105) − πα (cid:48) (cid:126) | w µ | (cid:19) – 41 –hus we obtain the following wave-function: ψ coh w ( x ) = exp (cid:32) − | x µ − (cid:104) x µ (cid:105)| π (cid:126) α (cid:48) (cid:33) exp (cid:18) i (cid:126) w µ (cid:104) ˜ x µ (cid:105) − πα (cid:48) (cid:126) | w µ | (cid:19) (5.35)We notice that the first term of this wave-function is a Gaussian on the physical positionspace and that the second term contains an exponential cut-off for large winding numbers. Probability distribution.
The probability of measuring a string with position x µ andwinding number w µ , for a given coherent state | z (cid:105) with mean doubled position (cid:10) x M (cid:11) , willbe immediately given by (cid:12)(cid:12)(cid:12) ψ coh w ( x ) (cid:12)(cid:12)(cid:12) = exp (cid:18) − π (cid:126) α (cid:48) | x µ − (cid:104) x µ (cid:105)| − πα (cid:48) (cid:126) | w µ | (cid:19) (5.36)This probability distribution exponentially decays by going away from the mean position (cid:104) x µ (cid:105) on the physical position space and from zero on the winding number space. Limit α (cid:48) → . It is immediate to notice that, in the limit α (cid:48) → , the probabilitydistribution spreads in the winding space and localizes in the physical position space. Inother words, our probability distribution shrinks to a Dirac delta (cid:12)(cid:12) ψ coh w ( x ) (cid:12)(cid:12) = δ ( x − (cid:104) x (cid:105) ) . Change of T-duality frame.
Now we want to find the T-dual wave-function of (5.35).To do so, we only need to express the same coherent state | z (cid:105) ∈ H in another basis of ourHilbert space, the basis | ˜ x, ˜ w (cid:105) corresponding to the complementary Lagrangian subspace (cid:101) L ⊂ P . In other words we must calculate (cid:101) ψ coh˜ w (˜ x ) := (cid:104) ˜ x, ˜ w | z (cid:105) . We immediately obtainthe following wave-function on the dual Lagrangian subspace (cid:101) L ⊂ P (cid:101) ψ coh˜ w (˜ x ) = exp (cid:32) − | ˜ x µ −(cid:104) ˜ x µ (cid:105)| π (cid:126) α (cid:48) (cid:33) exp (cid:18) i (cid:126) ˜ w µ (cid:104) x µ (cid:105) − πα (cid:48) (cid:126) | ˜ w µ | (cid:19) . (5.37)where the role of the physical and T-dual coordinates is exchanged. Let us focus on the doubled space M . We observed that it comes, at least locally,equipped with a canonical symplectic form (cid:36) . Let now assume that ( M , (cid:36) ) is simplya n -dimensional symplectic manifold and L ⊂ T M be a Lagrangian subbundle. The metaplectic structure.
The metaplectic group
M p (2 n, R ) is the universal doublecover of the symplectic group Sp (2 n, R ) . It is, thus, given by a group extension of the form Z M p (2 n, R ) Sp (2 n, R ) . (6.1)A metaplectic structure on a symplectic manifold ( M , (cid:36) ) is defined as the lift of the struc-ture group Sp (2 n, R ) of the bundle T M along the group extension M p (2 n, R ) (cid:16) Sp (2 n, R ) .There is a lemma (see [27]) which states that T M admits a metaplectic structure if andonly if L admits a metalinear structure. Another result states [27] that the existence of a– 42 –etalinear structure on a bundle E is equivalent to the existence of the square root bundle (cid:112) det( E ) . By putting these two lemmas together we obtain that T M admits a metaplecticstructure if and only if (cid:112) det( L ) exists.The existence of a metaplectic structure is intimately linked to the definition of the canonical Spin( d, d ) spinor bundle S M = ∧ • (cid:101) L ⊗ (cid:112) det( L ) (6.2)If the Lagrangian subbundle L is integrable, there exists a submanifold M ⊂ M such that L = T M , i.e. the physical spacetime. In this case, the canonical
Spin( d, d ) spinor bundleis isomorphic to the spinor bundle of generalised geometry on M , which is defined in [69].Thus, the isomorphism L ⊕ L ∗ by a B-shift L ⊕ (cid:101) L e − B −−→ L ⊕ L ∗ can be immediately extendedto an isomorphism S M ∼ = ∧ • T ∗ M ⊗ (cid:112) det( T M ) (6.3)given by the untwist Φ (cid:55)→ e − B ∧ Φ on polyforms Φ ∈ ∧ • (cid:101) L ⊗ (cid:112) det( L ) . Notice that thisrecovers a construction which is analogous to [70]. The quantum Hilbert space.
The physical necessity for the existence of (cid:112) det( L ) isthat it is this measure that is used to construct the quantum Hilbert space. In half-formquantisation, one thinks of a state as the combination of the wavefunction with the half-formused to construct its norm.Thus, the quantum Hilbert space of this symplectic manifold, which will be: H = (cid:26) ψ ∈ Γ (cid:0) M , E ⊗ (cid:112) det( L ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ∇ V ψ = 0 ∀ V ∈ L (cid:27) . (6.4)Let us now consider sections of the form e − φ √ vol M ∈ Γ (cid:0) M , (cid:112) det( L ) (cid:1) , where the top formis the Riemannian volume form vol M := (cid:112) det( g ) d x ∧ · · · ∧ d x n and φ ∈ C ∞ ( M ) is just afunction. Any section | ψ (cid:105) ∈ H can be uniquely written as: | ψ (cid:105) = ψe − φ ⊗ (cid:112) vol M . (6.5)With ψ and φ obeying the polarisation condition.For A ∈ GL ( n ; R ) acting on the bundle L , we have that sections of the square root bundletransform accordingly by e − φ (cid:112) vol M (cid:55)−→ (cid:112) det( A ) ( e − φ (cid:112) vol M ) (6.6)Consider a state | ψ (cid:105) ∈ H . Let us call simply ψ the corresponding wave-function. We, thus,have a Hilbert product given by (cid:104) ψ | ψ (cid:105) = (cid:90) M ψ † ψ (cid:112) det( g ) e − φ d x ∧ · · · ∧ d x n (6.7)where (cid:112) det( g ) e − φ is nothing but the string frame measure and it is T-duality invariant.By following the literature we can, define a T-duality invariant dilaton by d := φ −
12 ln det( g ) , (6.8)– 43 –o that we can rewrite the measure as (cid:112) det( g ) e − φ = e − d . Now, notice that a Hilbertproduct (cid:104) ψ | ψ (cid:105) does not depend on the choice of polarisation and, therefore, it must beinvariant under change of T-duality frame. Under the symplectomorphism encoding T-duality we have the volume half form transforming by e − d √ d x ∧ · · · ∧ d x n (cid:55)→ e − d √ d˜ x ∧ · · · ∧ d˜ x n (6.9)Thus, we can express the same state | ψ (cid:105) ∈ H as an (cid:101) L -polarised section ˜ ψe − ˜ φ ⊗ (cid:112) vol (cid:102) M ,where the dual measure is vol (cid:102) M := (cid:112) det(˜ g ) d˜ x ∧ · · · ∧ d˜ x n . In this T-duality frame theHilbert product will immediately have the following form: (cid:104) ψ | ψ (cid:105) = (cid:90) (cid:102) M ˜ ψ † ˜ ψ (cid:112) det(˜ g ) e − φ d˜ x ∧ · · · ∧ d˜ x n (6.10)Thus the dilaton transformation arises from the transformation of the measure in the half-form quantisation of the string. The Metaplectic correction to observables
There is one further effect associated tothe Metaplectic structure of quantisation. When we move to the representation of observ-ables the operators now act on states in H i.e. ψe − φ ⊗ √ vol M not just on the wavefunctions ψ . Practically that means there may be in additional contribution to an operator given bythe Lie derivative generated by the vector field associated to the observable acting on thehalf form. Contributions of this type occur with holomorphic polarizations in which casethe Hamiltonian operator is shifted by / . For the simple harmonic oscillator in quantummechanics this is just the usual "zero-point" energy shift. In this context, the Hamiltonianconstructed in section 5.24 would receive a zero-point shift. This would be relevant forT-fold type configurations where the space time moves between x and ˜ x spaces. Of course,we have only dealt with the bosonic string, it is a open question as to whether Fermioniccontributions might cancel this shift for the full superstring. The Maslov correction
A related effect is the Maslov quantisation condition [71] (alsoknown as Einstein–Brillouin–Keller quantisation) is π (cid:73) γ θ = (cid:126) (cid:18) n + µ ( γ )4 (cid:19) , (6.11)where n ∈ Z and µ ( γ ) is the Maslov index of the loop γ . Notice that the prequantisationcondition [ ω ] ∈ H ( P , Z ) alone implies only that π (cid:72) γ θ = (cid:126) n for some integer n ∈ Z . TheMaslov quantisation condition adds an explicit correction to the quantisation proceduredepending on the Maslov index of the loop.These metaplectic/Maslov type corrections really only appear when the polarisation is non-trivial by which we mean in the double field theory context a spacetime that moves between x and ˜ x spaces. One expects such a description is needed for a T-fold where no global T-duality frame exists. These subtle "quantum" effects will then change the string spectrum inthe T-fold background. We leave the detailed study of the metaplectic/Maslov correctionsfor T-folds for future work. – 44 – Discussion
This paper follows the approach of geometric quantisation for strings and links to the dou-bled space in double field theory. A key result is the identification of the stringy effects linkedto the noncommutativity of the doubled space controlled by the string length. The choiceof polarisation in quantisation then becomes the choice of duality frame. Transformationsbetween frames is then given geometrically by changing polarisations and constructing thenon-local transforms acting on wavefunctions. The constuction of a double coherent stategives a minimal distance state which we can examine from the point of view of traditionalpolarisations. Finally, the subtle metaplectic effects may have important consequences forquantising strings on T-folds.All of this leads to some further questions far outside the scope of this paper. Exceptionalfield theory is the extension of double field theory to M-theory where the U-duality groupbecomes a manifest symmetry. See [72, 73] for a recent reviews. Usually the properties ofdouble field theory are shared with exceptional field theory. Here though seems a mystery.If double field theory is just phase space and its subsequent quantisation then what isexceptional field theory. Is there some sense in which it can be thought of as a moregeneral "quantisation" with the generalised "phase space" being related to the extendedspace. Spacetime would no longer be a Lagrangian submanifold. Perhaps some clue isavailable in the construction of the basic states of theory as given in [74] where the braneswere again momentum states in the extended space but now also combined with a typeof generalised monopole to give a self-dual configuration. Other mysterious properties ofM-theory phase space have been noticed in [75]. Other exotica that would be curious toexplain from the phase space perspective would be the recently discovered non-Riemannianphase to double and exceptional field theory as discussed in [76–81]; this is also somewhatof a mystery from the quantisation perspective. Any insight into such backgrounds fromthe quantisation approach developed here would be very interesting and we leave for futurework.
Acknowledgments
The authors would like to thank Chris Blair, Laurent Friedel, Emanuel Malek, MalcolmPerry, Franco Pezzella, Paul Townsend and Alan Weinstein for fruitful discussions. Alsoin particular Chris Blair for sharing unpublished notes on the noncommutativity of thedoubled space through world sheet quantisation. DSB is supported by the UK Scienceand Technology Facilities Council (STFC) with consolidated grant ST/L000415/1, StringTheory, Gauge Theory and Duality.
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