Canonical Analysis of Non-Relativistic String with Non-Relativistic World-Sheet
aa r X i v : . [ h e p - t h ] J a n Preprint typeset in JHEP style - HYPER VERSION
Canonical Analysis of Non-Relativistic String withNon-Relativistic World-Sheet
J. Klusoˇn
Department of Theoretical Physics and AstrophysicsFaculty of Science, Masaryk UniversityKotl´aˇrsk´a 2, 611 37, BrnoCzech RepublicE-mail: [email protected]
Abstract:
We perform canonical analysis of non-relativistic string theory with non-relativistic world-sheet gravity. We determine structure of constraints and symplecticstructure of canonical variables. ontents
1. Introduction and Summary 12. Hamiltonian Analysis of SMT String 3
3. Singular Case 10
1. Introduction and Summary
AdS/CFT correspondence is the most known example of holographic duality [1]. Thiscorrespondence, in its strongest form, claims that SU ( N ) N = 4 SYM theory in fourdimensions is equivalent to type IIB theory on AdS × S at any values of N and ’tHooftcoupling λ . On the other hand understanding this duality at the strongest form is stilllacking and hence we should restrict to some limits of this correspondence.Recently such an interesting limit was suggested in [2] and it is known as Spin Ma-trix Theory (SMT) and describes near BPS limit of AdS/CFT. It is quantum mechanicaltheory with Hamiltonian given as sum of harmonic oscillator operators that transformboth in adjoin representation of SU ( N ) and in a particular spin subgroup G s of the globalsuperconformal P SU (2 , |
4) symmetries of N = 4.One can ask the question what is the dual description of this quantum mechanicalmodel. It was suggested [3] and further studied in [6, 5, 7] that dual theory in the bulkcorresponds to non-relativistic string theory with non-relativistic world-sheet known asSMT string. These special non-relativistic theories should be considered in the broadercontext of non-relativistic string theories that were studied recently in [3, 5, 6, 7] and also[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. This development is related to thegeneralization of Newton-Cartan geometry [23] to the stringy Newton-Cartan geometry [8]and torsional Newton-Cartan geometry. Moreover, SMT string was derived in [3, 5, 6] byspecific non-relativistic limit on the world-sheet of non-relativistic string in torsional NCbackground. Recently this SMT string was very intensively studied in [7] where particularclass of backgrounds for SMT string, known as flat-fluxed backgrounds, was analysed. Inthese backgrounds SMT string reduces to a free theory. These world-sheet theories areanalogues of the Polyakov action on Minkowski target space-time.The next step would be to analyse properties of SMT string in general background. Inorder to do this we should certainly study classical dynamics as for example its Hamiltonianform. The aim of this paper is to find such a formulation in the most general case.– 1 –et us be more explicit. We start with the action for SMT string that was foundin [6] and perform canonical analysis of this theory. As opposite to Polyakov form of therelativistic string now the action is formulated using vierbein e aα where α = 0 , a = 0 , e aα is invertible matrix with inverse θ αa . Now it is crucial that the quadratic termwith ∂ α x µ ∂ β x ν h µν is multiplied with θ α θ β as opposite to the relativistic case when thisterm has the form θ αa θ βb η ab . Then it is necessary to distinguish two cases. In the firstcase we presume that θ = 0. Then the relation between momenta and time derivativeof x µ is invertible. As a result we obtain Hamiltonian together with set of the primaryconstraints that follow from the structure of the theory. Careful analysis of the preservationof the primary constraints gives two secondary constraints that are first class constraintsthat reflect the fact that the theory is invariant under world-sheet diffeomorphism. Wealso identify four additional second class constraints and Poisson brackets between them.Finally we determine symplectic structure for canonical variables which is given in terms ofthe Dirac brackets. We identify that in this case the Dirac brackets coincide with Poissonbrackets.The situation is different when θ = 0. In this case it is not possible to expresstime derivative of x µ using canonical variables. Instead we get new d − constraints where d − is number of dimensions labelled with x µ . Then the canonical analysis is slightly morecomplicated than in previous case. However we again find two first class constraints thatreflect invariance of the action under reparameterization. We further identify second classconstraints and Poisson brackets between them. The presence of these constraints then im-ply non-trivial symplectic structure between canonical variables x µ which confirms analysispresented in [7].Let us outline our results and suggest further directions of research. We found Hamil-tonian formulation of SMT string and we identified structure of constraints. We discussedtwo cases when in the first one we were able to invert relation between time derivative of x µ and canonical momenta. In fact, this is the most general situation where all componentsof θ α are non-zero. On the other hand the second case when θ = 0 deserves separatetreatment. This fact suggests that the spatial gauge as was used in [7] cannot be reachedfrom the general Hamiltonian. It is instructive to compare this situation with the standardrelativistic Lagrangian where the relation between momenta and ∂x µ contains expression θ a η αb η ab ∂ α x µ that can be certainly inverted even if we impose condition θ = 0. On theother hand when we studied the situation when θ = 0 separately we found theory withnon-trivial symplectic structure as in [7].Certainly this work can be extended in many directions. It would be nice to studythe most general form of the string with the non-relativistic world-sheet and study itsconsistency from canonical point of view. It would be also extremely interesting to studysupersymmetric generalization of this two dimensional theory.This paper is organized as follows. In the next section (2) we review basic propertiesof non-relativistic string and we perform canonical analysis it the most general case. Wealso determine symplectic structure of given theory. In section (3) we separately discussthe case θ = 0 and we determine corresponding Hamiltonian and symplectic structure.– 2 – . Hamiltonian Analysis of SMT String We begin with the Polyakov form of the action for SMT string that was introduced in [6]and that has the form S = − T Z d σ (2 ǫ αβ m α ∂ β η + eθ α θ β h αβ + ωǫ αβ e α τ β + ψǫ αβ ( e α ∂ β η + e α τ β )) . (2.1)Let us explain meaning of various symbols that appear in (2.1). The world-sheet is labelledby σ , σ ≡ σ and T is string tension. Further, m µ , h µν and τ µ are target space-timeNewton-Cartan fields that obey conditions τ µ h µν = 0 , v µ h µν = 0 , τ µ v µ = − , h µν h νρ − τ µ v ρ = δ ρµ . (2.2)The world-sheet metric is defined with the help of zwiebein e aα , a = 0 , θ αa that obey e aα θ αb = δ ab , e aα θ βa = δ βα . (2.3)As was argued in [6] the world-sheet theory is non-relativistic since e aα play different role inthe action. This can be already seen from (2.1) since zweibein inverse θ αa does not appearin Lorentz invariant way θ αa θ βb η ab but instead there is an expression θ α θ β . This fact hasan important consequence for the structure of this theory. Note also that e = det e aα (2.4)and m α = m µ ∂ α x µ , h αβ = h µν ∂ α x µ ∂ β x ν , τ α = τ µ ∂ α x µ , (2.5)where x µ label embedding of the string into target space-time. Finally η is scalar fielddefined on world-sheet.We should stress that the theory is manifestly invariant under world-sheet diffeomor-phism σ ′ α = f α ( σ ) where world-volume fields transform as x ′ µ ( σ ′ ) = x µ ( σ ) , η ′ ( σ ′ ) = η ( σ ) , e ′ bβ ( σ ′ ) = e bα ( σ ) ∂σ α ∂σ ′ β . (2.6)Our goal is to find Hamiltonian formulation of this theory in order to investigate possiblenon-relativistic nature of it. First of all we start with the definition of conjugate momenta.From (2.1) we obtain π αb = ∂ L ∂ ( ∂ e aα ) ≈ , p ψ = ∂ L ∂ ( ∂ ψ ) ≈ , π ω = ∂ L ∂ ( ∂ ω ) ≈ ,p η = ∂ L ∂ ( ∂ η ) = T m + T ψe ,p µ = ∂ L ∂ ( ∂ x µ ) = − T m µ ∂ η − T eθ θ β h µν ∂ β x ν + T ωe τ µ + T ψe τ µ . (2.7)– 3 –t is clear that definition of p η implies following primary constraintΣ ≡ p η − T m − T ψe ≈ . (2.8)In this section we will presume that θ is non-zero and hence we can express time derivativeof x µ as function of p µ . On the other hand there is another primary constraint that followsfrom the definition of p µ given in (2.7)Σ ≡ v µ p µ + T v µ m µ ∂ η + T ωe + T ψe ≈ v µ h µν = 0 , v µ τ µ = − H B = p µ ∂ x µ + p η ∂ η − L == − T eθ θ [ p µ h µν p ν + 2 T p µ h µν m ν ∂ η + T ∂ ηm µ h µν m ν ∂ η ] −− θ θ θ θ ( p µ ∂ x µ + p µ v µ τ + ∂ ηm + ∂ ηm µ v µ τ ) + T ωe τ + T ψ ( e τ + e ∂ η ) . (2.10)As is well known from the theory of systems with constraints the time evolution is gov-erned by extended Hamiltonian that incorporates bare Hamiltonian together with set ofall primary constraints. Explicitly we have H E = H B + Ω Σ + Ω Σ + Ω aα π αa + Ω ψ p ψ + Ω ω p ω , (2.11)where Ω , Ω , Ω aα , Ω ψ and Ω ω are Lagrange multipliers.Now we should analyse condition of the preservation of all primary constraints π αa ≈ , p ω ≈ , p ψ ≈ , Σ ≈ , Σ ≈
0. To do this we need following canonical Poisson brackets n e aα ( σ ) , π βb ( σ ′ ) o = δ βα δ ab δ ( σ − σ ′ ) , (cid:8) ψ ( σ ) , p ψ ( σ ′ ) (cid:9) = δ ( σ − σ ′ ) , (cid:8) ω ( σ ) , p ω ( σ ′ ) (cid:9) = δ ( σ − σ ′ ) . (2.12)First of all we have that Σ , are second class constraints together with p ψ , p ω as followsfrom Poisson brackets (cid:8) p ψ ( σ ) , Σ ( σ ′ ) (cid:9) = T e ( σ ) δ ( σ − σ ′ ) , (cid:8) π ( σ ) , Σ ( σ ′ ) (cid:9) = T ψ ( σ ) δ ( σ − σ ′ ) , (cid:8) p ψ ( σ ) , Σ ( σ ′ ) (cid:9) = − T e ( σ ) δ ( σ − σ ′ ) , (cid:8) p ω ( σ ) , Σ ( σ ′ ) (cid:9) = − T e ( σ ) δ ( σ − σ ′ ) , (cid:8) π ( σ ) , Σ ( σ ′ ) (cid:9) = − T ω ( σ ) δ ( σ − σ ′ ) , (cid:8) π ( σ ) , Σ ( σ ′ ) (cid:9) = − T ψ ( σ ) δ ( σ − σ ′ ) . – 4 – Σ ( σ ) , Σ ( σ ′ ) (cid:9) = − T v µ m µ ( σ ′ ) ∂ σ ′ δ ( σ − σ ′ ) − T m ν ( σ ) ∂ σ δ ( σ − σ ) v µ ( σ ′ ) == T v µ ∂ ν m µ ∂ σ x ν δ ( σ − σ ′ ) , { Σ ( σ ) , Σ ( σ ) } = (cid:8) Σ ( σ ) , Σ ( σ ′ ) (cid:9) = 0 (2.13)using the fact that f ( σ ′ ) ∂ σ δ ( σ − σ ′ ) = f ( σ ) ∂ σ δ ( σ − σ ) + ∂ σ f ( σ ) δ ( σ − σ ′ ) . (2.14)We see that there is non-zero Poisson bracket between π , π and Σ , which makesanalysis slightly complicated. In order to resolve this issue let us introduce ˜ π as a specificlinear combinations of primary constraints that has vanishing Poisson brackets with Σ , Σ .Explicitly, we have ˜ π = π − e ψp ψ − e ωp ω + π e e (2.15)that obeys (cid:8) ˜ π , Σ (cid:9) = 0 , (cid:8) ˜ π , Σ (cid:9) = 0 . (2.16)In the same way we introduce ˜ π defined as˜ π = π − e ψp ω (2.17)that clearly obeys (cid:8) ˜ π , Σ (cid:9) = 0 , (cid:8) ˜ π , Σ (cid:9) = 0 . (2.18)In the same way we have (cid:8) ˜ π , p ψ (cid:9) ≈ , (cid:8) ˜ π , p ω (cid:9) ≈ , (cid:8) ˜ π , p ψ (cid:9) ≈ , (cid:8) ˜ π , p ω (cid:9) ≈ , (cid:8) ˜ π , π αa (cid:9) ≈ , (cid:8) ˜ π , π αa (cid:9) ≈ . (2.19)Note that π ≈ , π are unchanged. Then clearly ˜ π ≈ , ˜ π ≈ π ≈ , π ≈ p ω ≈ ∂ p ω = { p ω , H E } = − T e τ − Ω T e = 0 , (2.20)where H E = R dσ H E . Note that (2.20) can be solved for Ω asΩ = − τ e e . (2.21)Further, condition of the preservation of the constraint p ψ ≈ ∂ p ψ = { p ψ , H E } = − T e τ + e ∂ η ) + Ω T e − Ω T e = 0 (2.22)– 5 –hat can be solved for Ω as Ω = − ee τ + e ∂ η . (2.23)Let us finally analyse conditions of preservation of constraints Σ ≈ ≈
0. In caseof Σ ≈ ∂ Σ ( σ ) = { Σ ( σ ) , H E } = Z dσ ′ ( (cid:8) Σ ( σ ) , H B ( σ ′ ) (cid:9) + Ω ψ (cid:8) Σ ( σ ) , p ψ ( σ ′ ) (cid:9) ++Ω (cid:8) Σ ( σ ) , Σ ( σ ′ ) (cid:9) ) = 0 (2.24)which is equation for Ω ψ . In the same way requirement of the preservation of the constraintΣ ( σ ) ≈ ∂ Σ ( σ ) = { Σ ( σ ) , H E } = Z dσ ′ ( (cid:8) Σ ( σ ) , H B ( σ ′ ) (cid:9) + Ω ψ (cid:8) Σ ( σ ) , p ψ ( σ ′ ) (cid:9) ++Ω ω (cid:8) Σ ( σ ) , p ω ( σ ′ ) (cid:9) + Ω (cid:8) Σ ( σ ) , Σ ( σ ′ ) (cid:9) ) = 0 (2.25)that, using the fact that we know Ω and Ω ψ allows us to solve for Ω ω . These results areconsequence of the fact that Σ , Σ and p ω , p σ are second class constraints.As the final step we study the question of preservation of the constraints˜ π ≈ , ˜ π ≈ , π ≈ , π ≈ . (2.26)First of all we use the fact that θ αa has following components θ αa = θ θ θ θ ! = 1 e e − e − e e ! (2.27)so that H B is equal to H B = − e T e e [ p µ h µν p ν + 2 T p µ h µν m ν ∂ η + T ∂ ηm µ h µν m ν ∂ η ] ++ e e ( p µ ∂ x µ + p µ v µ τ + T ∂ ηm + T ∂ ηm µ v µ τ ) ++ T ωe τ + T ψ ( e τ + e ∂ η ) . (2.28)To proceed further we use the fact that (cid:8) π αa ( σ ) , e ( σ ′ ) (cid:9) = n π αa ( σ ) , det e bβ ( σ ′ ) o = − θ αa e ( σ ) δ ( σ − σ ′ ) . (2.29)Then we start with the requirement of the preservation of constraint π and we obtain ∂ π = (cid:8) π , H E (cid:9) = e T e e [ p µ h µν p ν + 2 T p µ h µν m ν ∂ η + T ∂ ηm µ h µν m ν ∂ η ] − – 6 – e ( p µ ∂ x µ + p µ v µ τ + T ∂ ηm + T ∂ ηm µ v µ τ ) − T ωτ − T ψ∂ η == e T e e [ p µ h µν p ν + 2 T p µ h µν m ν ∂ η + 2 T p η τ − T m τ + T ∂ ηm µ h µν m ν ∂ η ] −− e ( p µ ∂ x µ + p η ∂ η ) + Σ (cid:18) − e + e e e (cid:19) + 1 e Σ (2.30)using the fact that ψ = 2 T e ( − Σ + p η − T m ) ,ω = 2 T e (Σ + Σ e e − v µ p µ − T v µ m µ ∂ η − e e p η + e e T m ) (2.31)as follows from the definition of the primary constraints Σ , Σ .In the same way we can proceed with the time evolution of constraint π and we get ∂ π = (cid:8) π , H E (cid:9) == − T e [ p µ h µν p ν + 2 T p µ h µν m ν ∂ η + 2 T p η τ − T m τ + T ∂ ηm µ h µν m ν ∂ η ] + 1 e Σ . (2.32)In case of ˜ π ≈ ∂ ˜ π = (cid:8) ˜ π , H E (cid:9) == e T e e [ p µ h µν p ν + 2 T p µ h µν m ν ∂ η + 2 T p η τ − T m τ + T ∂ ηm µ h µν m ν ∂ η ] − e e e Σ . (2.33)In the same way we can proceed with ˜ π and we obtain that all constraints (2.26) arepreserved when we introduce two secondary constraints H = p µ h µν p ν + 2 T p µ h µν m ν ∂ η + 2 T p η τ − T m τ + T ∂ ηm µ h µν m ν ∂ η ≈ , H = p η ∂ η + p µ ∂ x µ ≈ . (2.34)Note also that using these secondary constraints the Hamiltonian density H B can be writtenas H B = − e T e e H + e e H + + e e τ (Σ + e e Σ ) − Σ e ( e τ + e ∂ η ) . (2.35)– 7 –e see that Hamiltonian is linear combinations of constraints. As the last step we shouldanalyse Poisson brackets between constraints H and H . Since they contain spatial deriva-tives of x µ it is convenient to introduce their smeared form multiplied by arbitrary functions N , M and N , M . Explicitly, we have T , ( N , ) ≡ Z dσN , H , , T , ( M , ) = Z dσM , H , . (2.36)Then using standard Poisson brackets we obtain (cid:8) T ( N ) , T ( M ) (cid:9) = 0 , (cid:8) T ( N ) , T ( M ) (cid:9) = T ( N ∂ M − M ∂ N ) . (2.37)Finally we determine Poisson bracket between generator of spatial diffeomorphism T ( N )and H and we obtain (cid:8) T ( N ) , H ( σ ) (cid:9) = − ∂ N H − N ∂ H ≈ H is tensor density. These results show that H ≈ , H ≈ H ≈ , H ≈ H , and Σ , Σ do not vanish. Insteadwe know that Σ , Σ have non-zero Poisson brackets between p ψ , ψ ω so that they can beinterpreted as second class constraints. Let us denote these second class constraints asΨ A = ( p ω , Σ , p ψ , Σ ) with following structure of Poisson brackets (cid:8) Ψ A ( σ ) , Ψ B ( σ ′ ) (cid:9) = △ AB ( σ, σ ′ ) , (2.39)where △ AB = e − e − e v µ ∂ m µ − e e − v µ ∂ m µ e T δ ( σ − σ ′ ) (2.40)with inverse matrix △ AB = 2 T − e − v µ ∂ m µ e e e − e e e v µ ∂ m µ e e e e e e − e δ ( σ − σ ′ ) . (2.41)Let us then introduce modified constraints ˜ H i , i = 1 , H i = H i − Ψ A △ AB (cid:8) Ψ B , H i (cid:9) , (2.42)– 8 –here summation over A includes also integration over σ implicitly. Using the fact that {H i , H j } ≈ n ˜ H i , ˜ H j o ≈ . (2.43)Then we have n ˜ H i , Ψ A o = n ˜ H i , Ψ A o + (cid:8) Ψ A , Ψ C (cid:9) △ CB (cid:8) Ψ B , H i (cid:9) ≈ H i have vanishing Poisson brackets with all constraints. On the other handsince Ψ A are second class constraints that vanish strongly in the end of the procedure wefind that ˜ H i coincide with H i . Of course, this can be done on condition that we replaceordinary Poisson brackets by Dirac brackets whose structure will be studied in the nextsection. We saw above that Ψ A are second class constraints with the matrix of Poisson bracketsgiven in (2.40) and its inverse given in (2.41). In order to determine Dirac brackets betweencanonical variables we firstly calculate Poisson brackets between canonical variables andsecond class constraints Ψ A (cid:8) x µ ( σ ) , Ψ A ( σ ′ ) (cid:9) = (0 , , , v µ ) δ ( σ − σ ′ ) , (cid:8) p µ ( σ ) , Ψ A ( σ ′ ) (cid:9) = (0 , T ∂ µ m ν ∂ x ν δ ( σ − σ ′ ) + T m µ ( σ ′ ) ∂ σ ′ δ ( σ − σ ′ ) , , − ∂ µ v ν p ν δ ( σ − σ ′ ) − T ∂ µ ( v ν m ν ) ∂ ηδ ( σ − σ ′ )) , (cid:8) η ( σ ) , Ψ A ( σ ′ ) (cid:9) = (0 , , , − T v µ m µ ( σ ′ ) ∂ σ ′ δ ( σ − σ ′ )) , (cid:8) p η ( σ ) , Ψ A ( σ ) (cid:9) = (0 , δ ( σ − σ ′ ) , , . (2.45)Then we find following form of Dirac brackets between canonical variables (cid:8) η ( σ ) , p η ( σ ′ ) (cid:9) D = (cid:8) η ( σ ) , p η ( σ ′ ) (cid:9) −− Z dσ dσ (cid:8) η ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , p η ( σ ′ ) (cid:9) = δ ( σ − σ ′ ) , (cid:8) η ( σ ) , η ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) η ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , η ( σ ′ ) (cid:9) = 0 , (cid:8) p η ( σ ) , p η ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) p η ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , p η ( σ ′ ) (cid:9) = 0 (cid:8) x µ ( σ ) , p ν ( σ ′ ) (cid:9) D = (cid:8) x µ ( σ ) , p ν ( σ ′ ) (cid:9) −− Z dσ dσ (cid:8) x µ ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , p ν ( σ ′ ) (cid:9) = δ µν δ ( σ − σ ′ ) , (cid:8) x µ ( σ ) , x ν ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) x µ ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , x ν ( σ ′ ) (cid:9) , (cid:8) p µ ( σ ) , p ν ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) p µ ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , p ν ( σ ′ ) (cid:9) = 0 . (2.46)– 9 –inally we determine mixed Dirac brackets (cid:8) x µ ( σ ) , η ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) x µ ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , η ( σ ′ ) (cid:9) = 0 , (cid:8) x µ ( σ ) , p η ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) x µ ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , p η ( σ ′ ) (cid:9) = 0 , (cid:8) p µ ( σ ) , η ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) p µ ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , η ( σ ′ ) (cid:9) = 0 , (cid:8) p µ ( σ ) , p η ( σ ′ ) (cid:9) D = − Z dσ dσ (cid:8) p µ ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , p η ( σ ′ ) (cid:9) = 0 . (2.47)These results show that Dirac brackets between p µ , x µ , p η , η have the same form as Poissonbrackets. In the next section we consider situation when θ = 0.
3. Singular Case
Canonical analysis performed in previous section was valid on condition that θ = 0 orequivalently e = 0. However spatial gauge that was imposed in [6, 7] is valid on conditionwhen e = 0. In other words this gauge fixing cannot be reached in previous analysis anddeserves separate treatment. We call this case as singular since, as we will see below, itwill not be possible to express time derivative of x µ as function of canonical variables.To see this explicitly we start with the action (2.1) from which we determine followingconjugate momenta π αb = ∂ L ∂∂ e aα ≈ , p ψ ≈ , , π ω ≈ , p η = T m ,p µ = ∂ L ∂∂ x µ = − T m µ ∂ η + T ψe τ µ (3.1)that implies an existence of primary constraintsΣ ≡ p η − T m , Σ µ ≡ p µ + T m µ ∂ η − T ψe τ µ ≈ . (3.2)For further purposes we introduce following linear combination of constraints that wedenote as H : H ≡ ∂ x µ Σ µ + Σ ∂ η = p µ ∂ x µ + p η ∂ η − T ψe τ ≈ H B = p µ ∂ x µ + p η ∂ η − L = T eθ θ h + T ωe τ + T ψ ( e ∂ η + e τ ) . (3.4)– 10 –et us now proceed to the analysis of preservation of primary constraints. We introduceextended Hamiltonian as H E = Z dσ ( H B + Ω Σ + Ω µ Σ µ + v ψ p ψ + v ω p ω ) . (3.5)We observe that we can always write Ω = ˜Ω ∂ η Σ so that when we use (3.3) we canexpress ∂ η Σ with the help of H and hence extended Hamiltonian density H E can bewritten in the form H E = T e e e h + T ωe τ + T ψ ( e ∂ η + e τ ) ++ v ψ p ψ + v ω p ω + ˜Ω H + ˜Ω µ Σ µ , (3.6)where we introduced ˜Ω µ as ˜Ω µ = Ω µ − ˜Ω ∂ x µ . Then in what follows we will omit tilde onΩ ′ s . Now we are ready to analyse requirement of the preservation of all constraints. In caseof p ω we get ∂ p ω = { p ω , H E } = − T e τ ≡ − T e Σ IIω ≈ , (3.7)where Σ IIω = τ ≈ ∂ x µ = 0 however this is very strong condition. We should rather presume that thebackground has non-zero component τ only so that this constraint is equal to Σ IIω ≡ ∂ x ≈
0. As a consequence H is standard spatial diffeomorphism constraint which is thefirst class constraint.Now using the fact that τ = 0 , τ i = 0 we haveΣ = p + T m ∂ η − T ψe τ , Σ i = p i + T m i ∂ η . (3.8)For further purposes we calculate Poisson brackets between primary constraints (cid:8) p ψ ( σ ) , Σ ( σ ′ ) (cid:9) = T e τ δ ( σ − σ ′ ) , (cid:8) Σ i ( σ ) , Σ j ( σ ′ ) (cid:9) = − T ( ∂ i m j − ∂ j m i ) ∂ ηδ ( σ − σ ′ ) ≡ −F ij δ ( σ − σ ′ ) . (3.9)In the same way we denote Poisson bracket between Σ and Σ i as (cid:8) Σ i ( σ ) , Σ ( σ ′ ) (cid:9) = −F i δ ( σ − σ ′ ) . (3.10)Let us now study the requirement of the preservation of constraint p ψ ∂ p ψ = { p ψ , H E } = − T e ∂ η + Ω T e τ = 0 (3.11)that has solution Ω = e τ e ∂ η . (3.12)– 11 –n other words, Σ ≈ , p ψ ≈ p ψ and ψ . We return to this problem below. Instead we focus on the time evolution ofconstraint Σ ≈ ∂ Σ = (cid:8) Σ , H E (cid:9) = Z dσ (cid:18)(cid:8) Σ , H E (cid:9) − T e τ δ ( σ − σ ′ ) v ψ + F i Ω i (cid:19) = 0 (3.13)which can be solved for v ψ . Finally, the requirement of the preservation of constraintsΣ i ≈ ∂ Σ i = (cid:8) Σ i , H E (cid:9) = Z dσ ′ ( (cid:8) Σ i , H B ( σ ′ ) (cid:9) − F i δ ( σ − σ ′ )Ω + F ij δ ( σ − σ ′ )Ω j ) = 0 . (3.14)Since F ij is non-singular by definition we can solve the equation above for Ω i .Let us analyse requirement of the preservation of constraints π αa . Following analysispresented in section (2) we replace π with ˜ π defined as˜ π = π − ψe p ψ (3.15)that has vanishing Poisson bracket with Σ ≈
0. Further, requirement of the preservationof π has the form ∂ π = (cid:8) π , H E (cid:9) = − e (cid:20) T h + 1 τ ( p − T m ∂ η ) ∂ η (cid:21) + 1 e τ Σ ≈ T ψ = 1 e τ ( p − T m ∂ η − Σ ) (3.17)and also that e is equal to e = det e aα = e e . We see that in order to obey equation(3.16) we should introduce secondary constraint H defined as H = T h + 1 τ ( p − T m ∂ η ) ∂ η ≈ . (3.18)On the other hand requirement of the preservation of the constraint ˜ π ≈ ∂ ˜ π = (cid:8) ˜ π , H E (cid:9) = e ( e ) H − τ ( e ) Σ ≈ . (3.19)Clearly (cid:8) H ( σ ) , H ( σ ′ ) (cid:9) = 0 , (cid:8) H ( σ ) , H ( σ ′ ) (cid:9) ≈ , (cid:8) H ( σ ) , H ( σ ′ ) (cid:9) ≈ .1 Symplectic structure In this section we study symplectic structure of the theory studied in previous section. Forsimplicity of our analysis we will consider partial fixed theory with fixed spatial diffeomor-phism constraint H ≈
0. This can be done by introducing gauge fixing function G : η − σ ≈ {G ( σ ) , H ( σ ′ ) } = δ ( σ − σ ′ ), H and G are second class constraints that stronglyvanish. From H = 0 we express p η as p η = − p µ ∂ x µ . (3.22)Further, as we argued in previous section, we have second class constraints Ψ A = ( p ψ , Σ , Σ i )with following matrix of Poisson brackets (cid:8) Ψ A ( σ ) , Ψ B ( σ ′ ) (cid:9) = T e τ − T e τ F j −F i −F ij δ ( σ − σ ′ ) . (3.23)For simplicity we will presume that F j = 0. Then the matrix inverse to △ AB is equal to △ AB = − T e τ T e τ −F ij , (3.24)where F ij is matrix inverse to F ij . Further, we have Poisson brackets (cid:8) x i ( σ ) , Ψ A ( σ ′ ) (cid:9) = (0 , , δ ij ) δ ( σ − σ ′ ) (3.25)and hence (cid:8) x i ( σ ) , x j ( σ ′ ) (cid:9) D == − Z dσ dσ (cid:8) x i ( σ ) , Ψ A ( σ ) (cid:9) △ AB ( σ , σ ) (cid:8) Ψ B ( σ ) , x j ( σ ′ ) (cid:9) = −F ij δ ( σ − σ ′ ) . (3.26)We see that there is non-trivial symplectic structure which is in agreement with the obser-vation presented in [7]. Then the equation of motion for x i have the form ∂ x i = (cid:8) x i , H (cid:9) D = F ik ∂ [ λh kl ∂ x l ] − λ F ik ∂ k h mn ∂ x m ∂ x n , (3.27)where we used the fact that the Hamiltonian is equal to H = λ H , H = T h ij ∂ x i ∂ x j + 1 τ ( p − T m ) , (3.28)where λ is Lagrange multiplier and where m and τ do not depend on x i .To conclude, we derived symplectic structure for SMT string in the gauge when e = 0and we showed that it is non-trivial and depend on the field m µ .– 13 – cknowledgments This work was supported by the Grant Agency of the Czech Republic under the grantGA20-04800S.
References [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,”
Int.J. Theor. Phys. (1999), 1113-1133 doi:10.1023/A:1026654312961 [arXiv:hep-th/9711200[hep-th]].[2] T. Harmark and M. Orselli, “Spin Matrix Theory: A quantum mechanical model of theAdS/CFT correspondence,” JHEP (2014), 134 doi:10.1007/JHEP11(2014)134[arXiv:1409.4417 [hep-th]].[3] T. Harmark, J. Hartong and N. A. Obers, “Nonrelativistic strings and limits of the AdS/CFTcorrespondence,” Phys. Rev. D (2017) no.8, 086019 doi:10.1103/PhysRevD.96.086019[arXiv:1705.03535 [hep-th]].[4] T. Harmark, J. Hartong, L. Menculini, N. A. Obers and Z. Yan, “Strings withNon-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence,” JHEP (2018), 190 doi:10.1007/JHEP11(2018)190 [arXiv:1810.05560 [hep-th]].[5] T. Harmark, J. Hartong, L. Menculini, N. A. Obers and G. Oling, “Relating non-relativisticstring theories,” JHEP (2019), 071 doi:10.1007/JHEP11(2019)071 [arXiv:1907.01663[hep-th]].[6] T. Harmark, J. Hartong, L. Menculini, N. A. Obers and Z. Yan, “Strings withNon-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence,” JHEP (2018), 190 doi:10.1007/JHEP11(2018)190 [arXiv:1810.05560 [hep-th]].[7] T. Harmark, J. Hartong, N. A. Obers and G. Oling, “Spin Matrix Theory String Backgroundsand Penrose Limits of AdS/CFT,” [arXiv:2011.02539 [hep-th]].[8] R. Andringa, E. Bergshoeff, J. Gomis and M. de Roo, “’Stringy’ Newton-Cartan Gravity,” Class. Quant. Grav. (2012), 235020 doi:10.1088/0264-9381/29/23/235020[arXiv:1206.5176 [hep-th]].[9] J. Gomis, Z. Yan and M. Yu, “T-Duality in Nonrelativistic Open String Theory,” [arXiv:2008.05493 [hep-th]].[10] J. Gomis, Z. Yan and M. Yu, “Nonrelativistic Open String and Yang-Mills Theory,” [arXiv:2007.01886 [hep-th]].[11] J. Klusoˇn, “Unstable D-brane in Torsional Newton-Cartan Background,” JHEP (2020),191 doi:10.1007/JHEP09(2020)191 [arXiv:2001.11543 [hep-th]].[12] D. Hansen, J. Hartong and N. A. Obers, “Non-Relativistic Gravity and its Coupling toMatter,” JHEP (2020), 145 doi:10.1007/JHEP06(2020)145 [arXiv:2001.10277 [gr-qc]].[13] Z. Yan and M. Yu, “Background Field Method for Nonlinear Sigma Models in NonrelativisticString Theory,” JHEP (2020), 181 doi:10.1007/JHEP03(2020)181 [arXiv:1912.03181[hep-th]].[14] J. Klusoˇn, “T-duality of Non-Relativistic String in Torsional Newton-Cartan Background,” JHEP (2020), 024 doi:10.1007/JHEP05(2020)024 [arXiv:1909.13508 [hep-th]]. – 14 –
15] E. A. Bergshoeff, J. Gomis, J. Rosseel, C. S¸im¸sek and Z. Yan, “String Theory and StringNewton-Cartan Geometry,”
J. Phys. A (2020) no.1, 014001 doi:10.1088/1751-8121/ab56e9[arXiv:1907.10668 [hep-th]].[16] A. D. Gallegos, U. G¨ursoy and N. Zinnato, “Torsional Newton Cartan gravity fromnon-relativistic strings,” JHEP (2020), 172 doi:10.1007/JHEP09(2020)172[arXiv:1906.01607 [hep-th]].[17] J. Gomis, J. Oh and Z. Yan, “Nonrelativistic String Theory in Background Fields,” JHEP (2019), 101 doi:10.1007/JHEP10(2019)101 [arXiv:1905.07315 [hep-th]].[18] J. Klusoˇn, “ ( m, n ) -String and D1-Brane in Stringy Newton-Cartan Background,” JHEP (2019), 163 doi:10.1007/JHEP04(2019)163 [arXiv:1901.11292 [hep-th]].[19] J. Klusoˇn, “Note About T-duality of Non-Relativistic String,” JHEP (2019), 074doi:10.1007/JHEP08(2019)074 [arXiv:1811.12658 [hep-th]].[20] J. Klusoˇn, “Nonrelativistic String Theory Sigma Model and Its Canonical Formulation,” Eur.Phys. J. C (2019) no.2, 108 doi:10.1140/epjc/s10052-019-6623-9 [arXiv:1809.10411[hep-th]].[21] E. Bergshoeff, J. Gomis and Z. Yan, “Nonrelativistic String Theory and T-Duality,” JHEP (2018), 133 doi:10.1007/JHEP11(2018)133 [arXiv:1806.06071 [hep-th]].[22] J. Klusoˇn, “Remark About Non-Relativistic String in Newton-Cartan Background and NullReduction,” JHEP (2018), 041 doi:10.1007/JHEP05(2018)041 [arXiv:1803.07336 [hep-th]].[23] E. Cartan, “Sur les vari´et´es `a connexion affine et la th´eorie de la relativit´e g´en´eralis´ee.(premi`ere partie),” Annales Sci. Ecole Norm. Sup. (1923) 325.(1923) 325.