Canonical maps and integrability in T T ¯ deformed 2d CFTs
aa r X i v : . [ h e p - t h ] J a n Canonical maps and integrability in T ¯ T deformed 2d CFTs George Jorjadze a, b and Stefan Theisen ca Free University of Tbilisi,Agmashenebeli Alley 240, 0159, Tbilisi, Georgia b Razmadze Mathematical Institute of TSU,Tamarashvili 6, 0177, Tbilisi, Georgia c Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut,14496, Golm, Germany
Abstract
We study T ¯ T deformations of 2d CFTs with periodic boundary conditions. We relatethese systems to string models on R × S × M , where M is the target space of a 2dCFT. The string model in the light cone gauge is identified with the corresponding2d CFT and in the static gauge it reproduces its T ¯ T deformed system. This relatesthe deformed system and the initial one by a worldsheet coordinate transformation,which becomes a time dependent canonical map in the Hamiltonian treatment. Thedeformed Hamiltonian defines the string energy and we express it in terms of the chiralHamiltonians of the initial 2d CFT. This allows exact quantization of the deformedsystem, if the spectrum of the initial 2d CFT is known. The generalization to non-conformal 2d field theories is also discussed. Prepared for ‘Integrability, Quantization, and Geometry - Dubrovin memorial volume’, edited byI. Krichever, S. Novikov, O. Ogievetsky and S. Shlosman
Introduction
The so-called T ¯ T deformation of two-dimensional quantum field theories, which wasintroduced by Zamolodchikov in 2004 [1], has recently attracted much attention. Beinga deformation by an irrelevant operator, one would naively expect that the deformedtheory looses any of the nice properties the undeformed theory might have had andthat the UV behaviour gets completely out of control. But this is not the case. Forinstance, in [2] it was shown that if the originial theory is integrable, so is the deformedone. Another remarkable fact is that the spectrum of the deformed theory formulatedon a cylinder can be determined exactly from the one of the undeformed theory [1, 2, 3].An interesting observation first made in [3] is the connection between a deformed freeboson and string theory. More precisely, it was shown that the classical dynamics of thedeformed system is that controlled by the Nambu-Goto action with three-dimensionalMinkowski space as target space, after fixing the static gauge. In the same paper thiswas generalized to several free bosons and also to a single boson with an arbitrarypotential. Further generalizations and refinements along these lines (and beyond) wereconsidered in [4], [5] and [6], again at the classical level. The relation between T ¯ T deformed CFTs and the quantum string was studied in detail in [7].Here we also consider the connection between deformed field theories and stringtheory, mainly at the classical level. As a large part of our analysis will be within theHamiltonian framework, the next section reviews the Hamiltonian treatment of two-dimensional Lagrangian field theories. While the Lagrangian treatment is more familiarand transparent, the Hamiltonian one is more convenient for generalizations. Themain examples are non-linear sigma-models with a metric and anti-symmetric tensorbackground. Classically they are always conformally invariant. Within the context ofstring theory one needs to impose conditions on the background fields, but this will notplay a role in our classical discussion. A simple generalization, which explicitly breaksthe conformal symmetry, is adding a potential.In Section 3 we look at the T ¯ T deformation of these theories, again in the Hamil-tonian framework. A simple formula for the deformed Hamiltonian density for systemswith symmetric canonical energy-momentum tensor can be derived. This formula isvalid for arbitrary (classical) CFTs which are characterized by two independent com-ponents of the energy-momentum tensor whose Poisson brackets generate two copies ofthe centerless Virasoro algebra.The simplest conformally invariant sigma-model is a free massless scalar field on acylinder. Its deformation will be reviewed in Section 4, with emphasis on the connectionto closed string dynamics in three-dimensional space-time, where one spatial coordinateis compactified on S . When the latter is formulated in a diffeomorphism invariantway, the deformed free scalar is obtained by breaking the invariance through fixing thestatic gauge. This gauge identifies the time and one spatial coordinate of the targetspace with the worldsheet coordinates. For this reason compactification is necessary.The string energy is, up to an additive constant, equal to the Hamiltonian of thedeformed theory. If one chooses light-cone gauge instead, one reaches the undeformedtheory. We generalize the light-cone gauge treatment of a closed string dynamics with1 compactified spatial coordinate, using as space-time light-cone directions those of thecylinder. This generalization is straightforward. In particular, in this gauge the stringenergy can be computed explicitly and by using its gauge invariance one obtains anexpression for the Hamiltonian – rather than the density – of the deformed theory interms of the Hamiltonian of the undeformed theory. This result applies, in fact, to moregeneral undeformed theories than just the free massless scalar.The relation between the deformed and the undeformed theory as simply choosingdifferent gauges in the string theory, implies that the undeformed and the deformedtheory are related by a (time-dependent) canonical transformation. This will be shownin detail. The worldsheet coordinate transformation between the two gauges dependson the solutions of the equation of motion in the fixed gauge. We use the explicitform of this transformation to obtain the Hamiltonian of the deformed theory withoutresorting to the gauge invariance of the string energy.In Section 5 we show how the previous discussion extends to general conformallyinvariant sigma models and to the case when one adds a potential. A remarkableexample here is the Liouville model with a negative cosmological constant. We showthat the corresponding string model is the SL(2, R ) WZW theory with vanishing stresstensor [8]. This string model in the static and light-cone gauges coincides to the T ¯ T deformed and the initial Liouville models, respectively.Some of the results reported in this note were obtained but not published abouttwo years ago [9] and they have meanwhile appeared in various papers. We have takenthe opportunity of being asked to contribute to this volume to include them, with duereference to the existing literature. Most importantly we point out [7, 10, 11, 12, 13, 14]for extensive discussions of the relation between the T ¯ T deformed and the initial 2dfield theories in the context of worldsheet gauge transformations. We consider two-dimensional classical field theories on a cylinder with circumference2 π , described by an action S [ φ ] = 12 π Z d τ d σ L ( φ, ˙ φ, ´ φ ) . (2.1)Here, τ and σ are time and space coordinates, respectively, φ := ( φ , . . . , φ N ) denotes aset of periodic fields, φ ( τ, σ +2 π ) = φ ( τ, σ ), and we use the notation ˙ φ := ∂ τ φ , ´ φ := ∂ σ φ .The components of the canonical stress tensor ( a, b ∈ { τ, σ } ) T ab = ∂ L ∂ ( ∂ a φ k ) ∂ b φ k − δ ab L (2.2)satisfy, by Noether’s theorem, the local conservation laws ∂ a T ab = 0 . (2.3)2he first order formulation of the same dynamics is obtained from the action S [Π , φ ] = Z d τ Z π d σ π h Π k ˙ φ k − H (Π , φ, ´ φ ) i , (2.4)where Π k are the periodic canonical momenta, Π( τ, σ + 2 π ) = Π( τ, σ ). We assume thatthe Lagrangian in (2.1) is non-singular, i.e. the velocities ˙ φ k are solvable in terms ofthe momenta Π k .The stress tensor components (2.2) are T ττ = H , T τσ = Π k ´ φ k ,T στ = − ∂ H ∂ Π k ∂ H ∂ ´ φ k , T σσ = H − Π k ∂ H ∂ Π k − ´ φ k ∂ H ∂ ´ φ k , (2.5)and the conservation laws (2.3) follow from the Hamilton equations of motion˙ φ k = ∂ H ∂ Π k , ˙Π k = − ∂ H ∂φ k + ∂ σ (cid:18) ∂ H ∂ ´ φ k (cid:19) . (2.6)Note that the covariant canonical stress tensor T ab in 2d Minkowski space is symmetric( T τ σ = T σ τ ) when the Hamiltonian density satisfies the condition ∂ H ∂ Π k ∂ H ∂ ´ φ k = Π k ´ φ k . (2.7)Below we assume that (2.7) is fulfilled, without referring to 2d metric structure. We also assume that the canonical stress tensor (2.5) is traceless, i.e.ˆ V [ H ] = 2 H , where ˆ V = Π k ∂∂ Π k + ´ φ k ∂∂ ´ φ k . (2.8)In this case T ab = (cid:18) H P−P −H (cid:19) , with P := Π k ´ φ k . (2.9)The components T ττ = H and T τσ = P are interpreted as the energy and themomentum densities, respectively. They obey the Poisson bracket relations {P ( σ ) , P ( σ ) } = {H ( σ ) , H ( σ ) } = 2 π (cid:2) P ( σ ) + P ( σ ) (cid:3) δ ′ ( σ − σ ) , {P ( σ ) , H ( σ ) } = {H ( σ ) , P ( σ ) } = 2 π (cid:2) H ( σ ) + H ( σ ) (cid:3) δ ′ ( σ − σ ) , (2.10)which follow from the canonical Poisson brackets, { Π k ( σ ) , φ l ( σ ) } = 2 π δ lk δ ( σ − σ ) , (2.11) Singular Lagrangians also lead to the action (2.4) by Hamiltonian reduction, but with a reducednumber of target space fields. While we can always add improvement terms to symmetrize the energy-momentum tensor, herewe assume that the canonical one is symmetric. { T ( x ) , T ( y ) } = 2 π (cid:2) T ( x ) + T ( y ) (cid:3) δ ′ ( y − x ) , { T ( x ) , ¯ T (¯ x ) } = 0 , { ¯ T (¯ x ) , ¯ T (¯ y ) } = 2 π (cid:2) ¯ T (¯ x ) + ¯ T (¯ y ) (cid:3) δ ′ (¯ y − ¯ x ) , (2.12)with T ( x ) = 12 (cid:2) H ( x ) + P ( x ) (cid:3) , ¯ T (¯ x ) = 12 (cid:2) H ( − ¯ x ) − P ( − ¯ x )] . (2.13)The conservation laws (2.3) in terms of T and ¯ T become ∂ ¯ x T = 0 , ∂ x ¯ T = 0 , (2.14)where x = τ + σ and ¯ x = τ − σ are the chiral coordinates, and we arrive at the standardformulation of 2d CFT with zero central charge.In a more general treatment, a 2d CFT on a cylinder is provided by two periodicfunctions T ( x ) and ¯ T (¯ x ), which satisfy the Poisson bracket relations (2.12), withoutreferring to the canonical structure (2.4). Thus, the Hamiltonian density H that satisfiesthe conditions (2.7) and (2.8) corresponds to a classical 2d CFT.A standard example is the σ -model S G,B [ φ ] = 14 π Z d τ d σ h ˙ φ k G kl ( φ ) ˙ φ l − ´ φ k G kl ( φ ) ´ φ l − φ k B kl ( φ ) ´ φ l i , (2.15)where G kl ( φ ) is a target space metric tensor and B kl ( φ ) is a 2-form on the target space.This system has stress tensor T ττ = − T σσ = 12 (cid:16) ˙ φ k G kl ˙ φ l + ´ φ k G kl ´ φ l (cid:17) , T τσ = − T στ = ˙ φ k G kl ´ φ l , (2.16)and Hamiltonian density H G,B = 12 h Π k G kl Π l + ´ φ k ( G kl − B km G mn B nl ) ´ φ l i + Π k G kj B jl ´ φ l , (2.17)which indeed satisfies conditions (2.7) and (2.8).Adding a potential U ( φ ) to a 2d CFT˜ H = H + U ( φ ) , (2.18)leads to a stress tensor with non-zero trace T ab = (cid:18) H + U ( φ ) P−P −H + U ( φ ) (cid:19) . (2.19)4 T ¯ T deformation of 2d Hamiltonian systems The following analysis is usually done in the Lagrangian formulation (cf. e.g. [3, 4, 14]).Here we present a Hamiltonian version of these well-known results.We introduce the T ¯ T deformation of the system (2.4) as [1] S α [Π , φ ] = Z d τ Z π d σ π h Π k ˙ φ k − H α (Π , φ, ´ φ ) i , (3.1)with H α defined by the ‘initial’ condition H = H and the differential equation ∂ H α ∂α = 12 det[ T ( α ) ] . (3.2)Here T a ( α ) b is the canonical stress tensor obtained from (2.5) by the replacement H 7→H α .Note that det[ T ab ] = P −H = − T ¯ T for a 2d CFT. Thus, the first order correctionto the Hamiltonian density of a 2d CFT is H α = H − α T ¯ T + · · · ; (3.3)hence the name T ¯ T deformation. However, the higher order terms do not have thisstructure and are more complicated.From (3.2) and (2.5) follows that H α satisfies the equation2 ∂ H α ∂α = H α − H α ˆ V [ H α ] + P ∂ H α ∂ Π k ∂ H α ∂ ´ φ k , (3.4)and one is looking for solutions which are analytic in α at α = 0.Using (3.4), one shows by a straightforward but slightly tedious calculation that thevariable Y α = ∂ H α ∂ Π k ∂ H α ∂ ´ φ k − P satisfies the equation ∂Y α ∂α = H α Y α −
12 ˆ V ( H α Y α ) + 12 P (cid:18) ∂ H α ∂ Π k ∂Y α ∂ ´ φ k + ∂ H α ∂ ´ φ k ∂Y α ∂ Π k (cid:19) . (3.5)From the ‘initial’ condition Y α =0 = 0 then follows that Y α remains zero for all α . Hence, H α satisfies the condition ∂ H α ∂ Π k ∂ H α ∂ ´ φ k = Π k ´ φ k , (3.6)and (3.4) reduces to 2 ∂ α H α = H α − H α ˆ V [ H α ] + P . (3.7)This equation can be easily integrated if the stress tensor of the undeformed theoryis traceless. Indeed, taking into account (2.8) and ˆ V [ P ] = 2 P , one finds that H α isexpressed in terms of H and P only. Dimensional analysis suggests the ansatz H α = F α ( r H + α P ) , (3.8)5here r is a real number. Inserting it into (3.6) one finds F ′ ( u ) = ( r + 4 α u ) − .Integration, requiring the regularity condition at α = 0 and that it satisfies (3.7), leadsto [14] H α = 1 α (cid:16) √ α H + α P − (cid:17) . (3.9)The structure of the energy-momentum tensor of the deformed theory is T a ( α ) b = (cid:18) H α P−P −K α (cid:19) , (3.10)with K α = 1 α (cid:18) − α P √ α H + α P − (cid:19) = H α + α P α H α . (3.11)One also verifies Tr[ T ( α ) ] = − α det[ T ( α ) ] (3.12)and, therefore, for a 2d CFT, H α satisfies the linear equation2 α ∂ α H α + 2 H α − ˆ V [ H α ] = 0 . (3.13)The above results, in particular the form of the deformed Hamiltonian density (3.9),were derived for a particular class of conformal field theories, but one wonders howgeneral they are. If we assume that the energy-momentum tensor of the undeformedtheory is symmetric, it has only two independent components, T and ¯ T . In terms ofthose H α = 1 α (cid:18)q α ( T + ¯ T ) + α ( T − ¯ T ) − (cid:19) . (3.14)Using the algebra (2.12), which holds for any CFT, one verifies that˙ H α = { H α , H α } = ∂ σ ( T − ¯ T ) , where H α = Z π d σ π H α . (3.15)Imposing the τ -component of the conservation equation in (2.3) for the deformed theory,this shows that T στ = ¯ T − T is not deformed. Imposing instead the σ -component andrequiring symmetry of T ( α ) leads to T σ ( α ) σ = H α − ∂ H α ∂T T − ∂ H α ∂ ¯ T ¯ T . (3.16)These results are completely general for two-dimensional conformal field theories, inparticular the expression (3.14) for the Hamiltonian density.We stress that our discussion so far was classical. In particular, in the quantizedtheory the algebra (2.12) is modified by a central extension leading to the Virasoroalgebra. Even for string theory, when the contribution of the ghosts is included, theabove calculation does not go through straightforwardly because of ordering issues inthe expression for H α . 6he T ¯ T deformation of the model (2.18), with the potential U ( φ ), can be performedsimilarly. In this case ˆ V [ ˜ H ] = 2 H and ˜ H α becomes a function of H , P and U ( φ ) only.Repeating the arguments which lead to (3.9), we obtain [4]˜ H α = 1 β (cid:20)p β H + β P + α U ( φ )2 (cid:21) − α , (3.17)with β = α (cid:16) − α U ( φ ) (cid:17) . (3.18)The check of (3.6) and (3.2) is again straightforward. In this section we investigate integrability of the deformed massless free-field modelwith the undeformed Lagrangian L = 12 (cid:16) ˙ φ − ´ φ (cid:17) . (4.1)The energy and momentum densities H = 12 (cid:16) Π + ´ φ (cid:17) , P = Π ´ φ , (4.2)lead to the following deformed Hamiltonian density H α = 1 α r α (cid:16) Π + ´ φ (cid:17) + α Π ´ φ − ! . (4.3)From the related Lagrangian L α = − α (cid:18)q α ´ φ − α ˙ φ − (cid:19) , (4.4)one derives a non-linear dynamical equation which is hard to integrate directly. Fur-thermore the construction of the Hamilton operator by (4.3) seems a highly nontrivialproblem due to the non-polynomial dependence of H α on the canonical variables. How-ever, the deformed free-field theory is related to a 3d string with one compactifiedcoordinate [3]. This enables us to integrate the system both at classical and quan-tum levels. We first consider the Lagrangian approach to the compactified 3d stringdynamics and then turn to its Hamiltonian treatment.For later use we note that Π and ˙ φ of the deformed theory (4.4) are related byΠ = ˙ φ q α ´ φ − α ˙ φ , ˙ φ = Π s α ´ φ α Π , (4.5)and the energy and momentum densities in the Lagrangian formulation become H α = 1 α α ´ φ q α ´ φ − α ˙ φ − , P = ˙ φ ´ φ q α ´ φ − α ˙ φ . (4.6)7 .1 Lagrangian approach to a compactified 3d string We start with a review of the connection between the string and the deformed system[3]. The Nambu-Goto action for a closed string is S = − πα Z d τ Z π d σ q ( ˙ X ´ X ) − ( ˙ X ˙ X )( ´ X ´ X ) . (4.7) X := ( X , X , X ) is a vector in 3d Minkowski space and 1 /α is proportional to thestring tension. We use the notation ( XX ) = X µ X ν g µν with the target space metrictensor g µν = diag( − , , X on the unit circle and consider string configurations withwinding number one around this circle, i.e. we identify X ≃ X + 2 π . This enables usto parameterize X by σ . We then identify X with τ and parameterize X by √ α φ ,i.e. we use the static gauge where X µ = τσ √ α φ , ˙ X µ = √ α ˙ φ , ´ X µ = √ α ´ φ . (4.8)In this gauge the string Lagrangian in (4.7) reduces to the deformed Lagrangian (4.4),up to the additive constant 1 /α .The string energy-momentum densities obtained from the Nambu-Goto action (4.7), P µ = 1 α ˙ X µ ( ´ X ´ X ) − ´ X µ ( ˙ X ´ X ) q ( ˙ X ´ X ) − ( ˙ X ˙ X )( ´ X ´ X ) , (4.9)satisfy the (primary) constraints( ´ X P ) = 0 , α ( ´ X ´ X ) + ( P P ) = 0 . (4.10)As in the uncompactified case, the tangent vectors ´ X and ˙ X are assumed spacelike andtimelike, respectively, and X is monotonically increasing in τ, i.e.( ´ X ´ X ) > , ( ˙ X ˙ X ) < , ˙ X > . (4.11)The momentum density P µ is then timelike and P is positive. In static gauge P = 1 α α ´ φ q α ´ φ − α ˙ φ , P = − ˙ φ ´ φ q α ´ φ − α ˙ φ , P = 1 √ α ˙ φ q α ´ φ − α ˙ φ . (4.12)8omparing these expressions to (4.5)-(4.6), we find P = H α + 1 α , P = −P , P = 1 √ α Π . (4.13)Integrating the densities over σ gives the gauge invariant string energy-momentum. Inparticular, the string energy reads E str = Z π d σ π P ( σ ) = H α + 1 α , (4.14)where H α is the energy of the deformed system (3.15).Thus, the deformed system (4.4) and the compactified 3d string in the gauge (4.8)are identical dynamical systems. On the other hand, it is well known that the classicalstring dynamics is integrable in the light-cone gauge. The compactification of thecoordinate X does not destroy integrability, but rather modifies it, as we show below.The static gauge (4.8) is not a conformal one for which one requires ( ˙ X ´ X ) = 0 and( ˙ X ˙ X ) + ( ´ X ´ X ) = 0 and the equation of motion for X µ becomes the free wave equation.These two constraints have to be imposed on the solutions. We denote the conformalworldsheet coordinates by ( τ c , σ c ), to distinguish them from ( τ, σ ), and introduce thecorresponding chiral coordinates z = τ c + σ c and ¯ z = τ c − σ c . One then has ∂ z ∂ ¯ z X µ = 0,and its solutions X µ = Φ µ ( z ) + ¯Φ µ (¯ z ) (4.15)are restricted to satisfy the conformal gauge conditions(Φ ′ Φ ′ ) = 0 , ( ¯Φ ′ ¯Φ ′ ) = 0 . (4.16)The chiral functions Φ ′ µ ( z ) and ¯Φ ′ µ (¯ z ) are periodic. Therefore, similarly to the uncom-pactified case, Φ µ ( z ) and ¯Φ µ (¯ z ) obey the monodromy conditionsΦ µ ( z + 2 π ) = Φ µ ( z ) + 2 π ρ µ , ¯Φ µ (¯ z + 2 π ) = ¯Φ µ (¯ z ) + 2 π ¯ ρ µ , (4.17)where ρ µ and ¯ ρ µ are the zero modes of Φ ′ µ ( z ) and ¯Φ ′ µ (¯ z ), respectively. From theperiodicity conditions in σ one finds ρ = ¯ ρ , ρ = ¯ ρ + L , ρ = ¯ ρ , (4.18)where L is the winding number around the compactified coordinate X . For now weanalyze the case of general L , though our interest is L = 1.To find independent variables on the constraint surface (4.16), we follow the standardscheme and introduce the light-cone coordinates X ± = X ± X . Note that while oneusually chooses the space-time light-cone directions along two non-compact coordinates,our definition of X ± involves the compact direction X . The remaining freedom ofconformal transformations allows us to simplify the chiral components of X + as in theuncompactified case Φ + ( z ) = ρ + z , ¯Φ + = ¯ ρ + ¯ z . (4.19) The conditions (4.19) require ρ + > ρ + >
0. We will see in (4.27) that these conditions areindeed fulfilled. ρ + Φ ′ − ( z ) = α F ′ ( z ) , ¯ ρ + ¯Φ ′ − (¯ z ) = α ¯ F ′ (¯ z ) , (4.20)where X is rescaled similarly to (4.8), i.e. Φ ( z ) = √ α F ( z ) and ¯Φ (¯ z ) = √ α ¯ F (¯ z ).As a result, one obtains the following parameterization of the string coordinates X µ = (cid:2) ρ + z + Φ − ( z ) + ¯ ρ + ¯ z + ¯Φ − (¯ z ) (cid:3) (cid:2) ρ + z − Φ − ( z ) + ¯ ρ + ¯ z − ¯Φ − (¯ z ) (cid:3) √ α (cid:2) F ( z ) + ¯ F (¯ z ) (cid:3) . (4.21)The functions F ( z ) and ¯ F (¯ z ) have the mode expansions F ( z ) = q + pz √ X m =0 a n n e − i nz , ¯ F (¯ z ) = q + p ¯ z √ X n =0 ¯ a n n e − i n ¯ z , (4.22)with p = √ α ρ , and Φ − ( z ) and ¯Φ − (¯ z ) are obtained from (4.20) (see Appendix A). Inparticular, one has ρ − = α hρ + , ¯ ρ − = α ¯ h ¯ ρ + , (4.23)where h and ¯ h are the chiral free-field Hamiltonians h = Z π d z π F ′ ( z ) = p X n> | a n | , ¯ h = Z π d¯ z π ¯ F ′ (¯ z ) = p X n> | ¯ a n | . (4.24)Note that we set ¯ p = p in (4.22), due to the third relation in (4.18). The other tworelations of (4.18), in terms of the light-cone variables, read ρ + + ρ − − ¯ ρ + − ¯ ρ − = 0 , ρ + − ρ − − ¯ ρ + + ¯ ρ − = 2 L . (4.25)For L = 0 this leads to differences for the compactified case as compared to the non-compact one.Indeed, for L = 0, the solution of (4.23)-(4.25) is ρ + = ¯ ρ + , ρ − = ¯ ρ − = α hρ + = α ¯ h ¯ ρ + , h = ¯ h . (4.26)Here, ρ + is a free dynamical variable. The condition h = ¯ h becomes, after quantization,the level matching condition in the zero winding sector.When L = 0, we obtain instead the following solution of (4.23)-(4.25) ρ ± = 12 (cid:16) α E L ± αL (¯ h − h ) ± L (cid:17) , ¯ ρ ± = 12 (cid:16) α E L ± αL (¯ h − h ) ∓ L (cid:17) , (4.27)with E L = 1 L α q L + 2 L α ( h + ¯ h ) + α ( h − ¯ h ) . (4.28)10ere, solving quadratic equations, we choose the positive roots, since they correspondto the physical solutions for which ρ ± > ρ ± > L = 0, the string solutions (4.21) are completely parametrized by thechiral free fields F ( z ) , ¯ F (¯ z ). We now find that the level matching condition is modifiedto L ( ρ + ¯ ρ ) = α (¯ h − h ) . (4.29)According to (4.9), the string energy density in the conformal gauge is given by α ∂ τ c X , and from (4.27) we obtain the string energy for winding number LE ( L )str = 12 α (cid:0) ρ + + ρ − + ¯ ρ + + ¯ ρ − (cid:1) = E L . (4.30)For winding number one, which corresponds to the deformed system, this yields E str = 1 α q α ( h + ¯ h ) + α ( h − ¯ h ) , (4.31)and, due to the gauge invariance of the string energy, we obtain from (4.14) [5] H α = 1 α (cid:18)q α ( h + ¯ h ) + α ( h − ¯ h ) − (cid:19) . (4.32)This expression for the Hamiltonian should be contrasted with (3.14). There theHamiltonian density of the deformed theory was expressed in terms of the energy-momentum densities of the undeformed theory while here the relation is between theintegrated densities. Furthermore, this expression can be easily quantized as h and ¯ h are diagonal in the Fock-space of the undeformed theory.In Section 5.1 we will briefly discuss generalizations to general CFTs. In this casethe expression for H α is straightforwardly generalized by replacing ( h, ¯ h ) by ( L , ¯ L ) ofthe undeformed theory. In fact, many of the expressions in the following discussion aregeneralized if one replaces in the expression in Appendix A the L n of the free field bythe generators of the Virasoro algebra of a general CFT.In Appendix B we derive (4.32) directly (without referring to the gauge invariance),using the map that relates the worldsheet coordinates and the fields in two differentgauges. We will now analyze this map in detail.Comparing the string coordinates in the gauges (4.8) and (4.21), we find the mapfrom the coordinates ( z, ¯ z ) to ( τ, σ ) τ = 12 (cid:2) ρ + z + Φ − ( z ) + ¯ ρ + ¯ z + ¯Φ − (¯ z ) (cid:3) ,σ = 12 (cid:2) ρ + z − Φ − ( z ) + ¯ ρ + ¯ z − ¯Φ − (¯ z ) (cid:3) , (4.33)and we also express the solutions of the deformed system by the undeformed one φ ( τ, σ ) = F ( z ) + ¯ F (¯ z ) . (4.34) Recall that z = τ c + σ c and ¯ z = τ c − σ c . τ, σ and using (4.20), we obtain˙ z = ρ + ( α ¯ F ′ − ¯ ρ + 2 ) α (cid:2) ( ρ + ¯ F ′ ) − ( ¯ ρ + F ′ ) (cid:3) , ´ z = ρ + ( α ¯ F ′ + ¯ ρ + 2 ) α (cid:2) ( ρ + ¯ F ′ ) − ( ¯ ρ + F ′ ) (cid:3) , ˙¯ z = − ¯ ρ + ( αF ′ − ρ + 2 ) α (cid:2) ( ρ + ¯ F ′ ) − ( ¯ ρ + F ′ ) (cid:3) , ´¯ z = − ¯ ρ + ( αF ′ + ρ + 2 ) α (cid:2) ( ρ + ¯ F ′ ) − ( ¯ ρ + F ′ ) (cid:3) . (4.35)A similar differentiation of (4.34), with the help of (4.35), gives˙ φ = α ¯ F ′ F ′ + ¯ ρ + ρ + α (cid:0) ρ + ¯ F ′ + ¯ ρ + F ′ (cid:1) , ´ φ = α ¯ F ′ F ′ − ¯ ρ + ρ + α (cid:0) ρ + ¯ F ′ + ¯ ρ + F ′ (cid:1) , (4.36)and they lead to 1 + α ´ φ − α ˙ φ = (cid:0) ρ + ¯ F ′ − ¯ ρ + F ′ (cid:1) (cid:0) ρ + ¯ F ′ + ¯ ρ + F ′ (cid:1) . (4.37)The left hand side here defines the determinant of the induced worldsheet metric instatic gauge and for regular surfaces it has to be positive. Thus, for regular surfaces,the expressions ρ + ¯ F ′ ± ¯ ρ + F ′ have no zeros. Note that these expressions have the samesign for a sufficiently large zero mode p . Assuming this, we get q α ´ φ − α ˙ φ = ρ + ¯ F ′ − ¯ ρ + F ′ ρ + ¯ F ′ + ¯ ρ + F ′ . (4.38)From (4.5) then follows Π = α ¯ F ′ (¯ z ) F ′ ( z ) + ¯ ρ + ρ + α (cid:2) ρ + ¯ F ′ (¯ z ) − ¯ ρ + F ′ ( z ) (cid:3) , (4.39)and using (4.35) we obtain12 (cid:16) ´ φ + Π (cid:17) = ´ z F ′ ( z ) , (cid:16) ´ φ − Π (cid:17) = ´¯ z ¯ F ′ (¯ z ) . (4.40)Equation (4.33), for a fixed τ , defines z and ¯ z as functions of σ . For example, whenthe non-zero modes of F ′ and ¯ F ′ are not excited, z = τ p α p + σ , ¯ z = τ p α p − σ . (4.41)In general, writing these functions as z = ζ ( σ ), ¯ z = ¯ ζ ( − σ ), we find that they aremonotonic ζ ′ ( x ) >
0, ¯ ζ ′ (¯ x ) > ζ ( x + 2 π ) = ζ ( x ) + 2 π , ¯ ζ (¯ x + 2 π ) = ¯ ζ (¯ x ) + 2 π , (4.42)related to diffeomorphisms of a circle. In the next subsection we show that (4.40)realizes a time dependent canonical map between the two gauges.12oncluding this subsection we express the energy-momentum density componentsin the static gauge (4.12) in terms of the light-cone gauge variables, using (4.35), (4.36)and (4.38). With (4.13), P is obtained from (4.39) and P = ( α ¯ F ′ + ¯ ρ + 2 )( αF ′ + ρ + 2 ) α (cid:2) ( ρ + ¯ F ′ ) − ( ¯ ρ + F ′ ) (cid:3) = ´ z (cid:18) ρ + α + F ′ ρ + (cid:19) = − ´¯ z (cid:18) ¯ ρ + α + ¯ F ′ ¯ ρ + (cid:19) , P = − ( α ¯ F ′ F ′ + ¯ ρ + ρ + )( α ¯ F ′ F ′ − ¯ ρ + ρ + ) α (cid:2) ( ρ + ¯ F ′ ) − ( ¯ ρ + F ′ ) (cid:3) (4.43)= ´ z (cid:18) ρ + α − F ′ ρ + (cid:19) − α = − ´¯ z (cid:18) ¯ ρ + α − ¯ F ′ ¯ ρ + (cid:19) + 1 α . We will use these relations in the next section to relate the static and light-cone gaugesin the Hamiltonian formulation.
We now consider the Hamiltonian treatment of the same system. In the first orderformulation of 3d string dynamics the action is S = Z d τ Z π d σ π h P µ ˙ X µ − λ C − λ C i , (4.44)where λ , λ are Lagrange multipliers and C , C are the Virasoro constraints C = ( P ´ X ) , C = 12 h α ( P P ) + ( ´ X ´ X ) i . (4.45)The compact coordinate X has the expansion (for L = 1) X = σ + X n ∈ Z q n e − i n σ , (4.46)with q − n = q ∗ n , while the canonical momenta P µ and the coordinates ( X , X ) remainperiodic. They have the standard mode expansion without the σ term in (4.46).It follows from the canonical Poisson brackets on the extended phase space {P µ ( σ ) , X ν ( σ ) } = 2 π δ νµ δ ( σ − σ ) , (4.47)that the Poisson brackets of the constraints (4.45) form the algebra (2.10) {C ( σ ) , C ( σ ) } = 2 π (cid:2) C ( σ ) + C ( σ ) (cid:3) δ ′ ( σ − σ ) , {C ( σ ) , C ( σ ) } = 2 π (cid:2) C ( σ ) + C ( σ ) (cid:3) δ ′ ( σ − σ ) , {C ( σ ) , C ( σ ) } = 2 π α (cid:2) C ( σ ) + C ( σ ) (cid:3) δ ′ ( σ − σ ) , (4.48)and one has to complete these first class constraints by gauge fixing conditions in orderto eliminate non-physical degrees of freedom.13his can be done by the Faddeev-Jackiw reduction in static gauge X = τ, X = σ .For this one computes P µ ˙ X µ on the constrained surface C = C = 0 in this gauge. Theaction (4.44) then reduces to S | st.g. = Z d τ Z π d σ π (cid:16) P + P ˙ X (cid:17) , (4.49)where P becomes a function of the reduced canonical variables ( P , X ). Hence, P = −P plays the role of the Hamiltonian density.In order to relate the reduced Hamiltonian system to the deformed model, we rescalethe canonical variables, P = Π √ α , X = √ α φ , (4.50)and rewrite the constraints (4.45) as C = Π φ ′ + P = 0 , C = α (Π + ´ φ ) + α P − α P + 1 = 0 . (4.51)These equations define the remaining phase space variables P = − Π φ ′ , P = − α r α (cid:16) Π + ´ φ (cid:17) + α (cid:16) Π ´ φ (cid:17) (4.52)and we finally obtain S | st.g. = Z d τ Z π d σ π (cid:20) Π ˙ φ − (cid:18) H α + 1 α (cid:19)(cid:21) . (4.53) H α is the Hamiltonian density of the deformed model (4.3). Thus, the Faddeev-Jackiwreduction of the compactified 3d string in the static gauge leads to the deformed free-field model.We now consider Hamiltonian reduction of (4.44) in light-cone gauge. Introducingthe light-cone coordinates X ± = X ± X , P ± = 12 ( P ± P ) , (4.54)the string action (4.44) and the constraints become S = Z d τ Z π d σ π h P + ˙ X + + P − ˙ X − + P ˙ X − λ C − λ C i , (4.55)with C = P + ´ X + + P − ´ X − + P ´ X , C = 12 h α P + ´ X − α P + P − − ´ X + ´ X − i . (4.56)Using the gauge freedom, we can eliminate the non-zero modes of P − ( σ ) and X + ( σ ),similarly to the uncompactified case. Taking into account that X has winding numberone, the light-cone gauge condition reads X + ( σ ) = − α P − τ + σ , ´ P − ( σ ) = 0 . (4.57)14his provides ´ X + ( σ ) = 1 and P − ( σ ) = p − , where p − is the zero mode of P − ( σ ).Rescaling then the canonical variables similarly to (4.50) P = Π √ α , X = √ α Φ , (4.58)the constraints (4.56) can be written as C = P + + p − ´ X − + P = 0 , C = 2 α H − α p − P + − ´ X − = 0 , (4.59)with P = Π ´ Φ , H = 12 (cid:16) Π + ´ Φ (cid:17) . (4.60)By (4.59) one finds P + and ´ X − in terms of ( Π, Φ ) and the zero mode p − P + = − α p − H + P − α p − , ´ X − = 2 α H + 4 α p − P − α p − . (4.61)The zero modes of the constraints (4.59) satisfy( p + − p − ) + P = 0 , α H + (1 − α p − p + ) = 0 , (4.62)where p + is the zero mode of P + and P = Z π d σ π P , H = Z π d σ π H . (4.63)The string energy then becomes E str = − ( p + + p − ) = 1 α √ α H + α P . (4.64)Faddeev-Jackiw reduction of the action (4.55) by the constraints (4.56)-(4.57) yields S | l-c. g. = Z d τ Z π d σ π h Π ( σ ) ˙ Φ ( σ ) − α p + ( p − + ˙ p − τ ) + p − ˙ x − i , (4.65)where x − is the zero mode of the periodic part of X − ( σ ), and we have used the rescaledvariables (4.58). Neglecting the total derivative term dd τ ( − α p + p − τ ) in (4.65), weobtain S | l-c. g. = Z d τ Z π d σ π h Π ( σ ) ˙ Φ ( σ ) + p − ˙ q − − α p + p − i . (4.66)with q − = x − + 2 α p + p − τ . Using (4.62) and neglecting also the constant term 1 / (2 α ),we end up with the action S | l-c. g. = Z d τ Z π d σ π h Π ˙ Φ + p − ˙ q − − H i , (4.67) Note that the pairs (Π , φ ) and (
Π, Φ ) differ from each other, though they denote the same variablesin the initial extended phase space. H is the free-field Hamiltonian density (4.60) and p − is obtained from (4.62) p − = 12 (cid:18) P − α √ α H + α P (cid:19) . (4.68)The situation here is similar to the uncompactified case, where instead of (4.68) onehas the level matching condition P = h − ¯ h = 0. Further Hamiltonian reduction inboth cases is inconvenient. One has to quantize the free-field model together with theparticle ( p − , q − ) and impose the condition (4.68) at the quantum level. Note that theright hand side in (4.68) is a well defined operator in the Fock space of the free-fieldtheory.We now discuss the relation between the static and light-cone gauges in the Hamil-tonian approach. In general, reduced Hamiltonian systems obtained in two differentgauges are related to each other by a canonical transformation generated by the con-straints of the initial gauge invariant system. Our aim is to describe the canonical mapbetween the light-cone and the static gauges of the compactified 3d string.First note that the Virasoro constraints (4.45) can be represented in the form C ( σ ) := f µ ( σ ) f µ ( σ ) = 0 , ¯ C ( σ ) := ¯ f µ ( σ ) ¯ f µ ( σ ) = 0 , (4.69)with f µ ( σ ) = 12 √ α (cid:16) α P µ ( σ ) + ´ X µ ( σ ) (cid:17) , ¯ f µ ( σ ) = 12 √ α (cid:16) α P µ ( − σ ) − ´ X µ ( − σ ) (cid:17) . (4.70)From the canonical Poisson brackets (4.47) follows {C ( σ ) , f µ ( σ ) } = 2 π ∂ σ [ f µ ( σ ) δ ( σ − σ )] , { ¯ C ( σ ) , ¯ f µ ( σ ) } = 2 π ∂ σ [ ¯ f µ ( σ ) δ ( σ − σ )] , {C ( σ ) , ¯ f µ ( σ ) } = { ¯ C ( σ ) , f µ ( σ ) } = 0 . (4.71)The corresponding infinitesimal transformations f µ ( σ ) f µ ( σ ) + ∂ σ [ ǫ ( σ ) f µ ( σ )] , ¯ f µ ( σ ) ¯ f µ ( σ ) + ∂ σ (cid:2) ¯ ǫ ( σ ) ¯ f µ ( σ ) (cid:3) , (4.72)lead to the global ones f µ ( σ ) ζ ′ ( σ ) f µ ( ζ ( σ )) , ¯ f µ ( σ ) ¯ ζ ′ ( σ ) ¯ f µ ( ¯ ζ ( σ )) , (4.73)where ζ ( σ ) , ¯ ζ ( σ ) are diffeomorphisms of the unit circle. Note that, in general, the groupparameters ǫ ( σ ) , ¯ ǫ ( σ ) could be functions on the phase space, since the transformationsare on-shell. In the Hamiltonian formulation the light-cone gauge is not a complete gauge fixing for the closedstring. The constraint corresponding to the remaining gauge freedom is the level matching condition.After complete gauge fixing one arrives at a conformal gauge and the Hamiltonian formulation is thenequivalent to the Lagrangian formulation in light cone gauge [17, 18]. f µ and ¯ f µ f µ st .g. ( σ ) = 12 √ α P ( σ ) √ α P ( σ ) + √ α Π( σ ) + φ ′ ( σ ) , ¯ f µ st .g. ( σ ) = 12 √ α P ( − σ ) √ α P ( − σ ) − √ α Π( − σ ) − φ ′ ( − σ ) , (4.74)where P and P are given by (4.52).The light-cone gauge parameterization of f µ and ¯ f µ is obtained from (4.57)-(4.61) f µ l-c.g. ( σ ) = 12 ρ + √ α + √ α [ Π ( σ )+ Φ ′ ( σ )] ρ + ρ + √ α − √ α [ Π ( σ )+ Φ ′ ( σ )] ρ + Π ( σ ) + Φ ′ ( σ ) , ¯ f µ l-c .g. ( σ ) = 12 ¯ ρ + √ α + √ α [ Π ( − σ ) − Φ ′ ( − σ )] ρ + ¯ ρ + √ α − √ α [ Π ( − σ ) − Φ ′ ( − σ )] ρ + Π ( − σ ) − Φ ′ ( − σ ) , (4.75)where we have used 2 ρ + = 1 − α p − , ρ + = − − α p − . (4.76)Based on (4.73), we introduce the relations f µ st.g. ( σ ) = ζ ′ ( σ ) f µ l-c.g. ( ζ ( σ )) , ¯ f µ st.g. ( σ ) = ¯ ζ ′ ( σ ) f µ l-c.g. ( ¯ ζ ( σ )) , (4.77)which by (4.74)-(4.75) are equivalent to α (cid:2) P ( σ ) + P ( σ ) (cid:3) + 1 = 2 ρ + ζ ′ ( σ ) ,α (cid:2) P ( σ ) − P ( σ ) (cid:3) − α ρ + ζ ′ ( σ )[ Π ( ζ ( σ )) + ´ Φ ( ζ ( σ ))] , Π( σ ) + ´ φ ( σ ) = ζ ′ ( s ) (cid:16) Π ( ζ ( σ )) + ´ Φ ( ζ ( σ )) (cid:17) , (4.78)and similarly for the anti-chiral part α (cid:2) P ( − σ ) + P ( − σ ) (cid:3) − ρ + ¯ ζ ′ ( σ ) ,α (cid:2) P ( − σ ) − P ( − σ ) (cid:3) + 1 = α ρ + ¯ ζ ′ ( σ )[ Π ( − ¯ ζ ( σ )) + ´ Φ ( − ¯ ζ ( σ ))] , Π( − σ ) + ´ φ ( − σ ) = ¯ ζ ′ ( σ ) (cid:16) Π ( − ¯ ζ ( σ )) + ´ Φ ( − ¯ ζ ( σ )) (cid:17) . (4.79)The integration in (4.78) over σ provides the relations α (cid:0) E str + P (cid:1) + 1 = 2 ρ + , α (cid:0) E str − P (cid:1) − αhρ + , (4.80)which for the string energy leads again to (4.32). The same result is obtained for theantichiral part (4.79).Equations (4.78)-(4.79) are equivalent to (4.40) and (4.43) with z ( σ ) = ζ ( σ ) and¯ z ( σ ) = ¯ ζ ( − σ ), which indicates that they define a canonical map between the two gauges.17he direct computation with the help of (4.33)-(4.34) shows that this map preservesthe canonical symplectic form Z π d σ π δ Π( σ ) ∧ δφ ( σ ) = Z π d σ π δΠ ( σ ) ∧ δΦ ( σ ) = Z π d x π δF ( x ) ∧ δF ′ ( x ) + Z π d¯ x π δ ¯ F (¯ x ) ∧ δ ¯ F ′ (¯ x ) + 12 δp ∧ (cid:2) δF (0) + δ ¯ F (0) (cid:3) . (4.81) In this section we first generalize the scheme described in Section 4.2 to other 2d CFTs.Recall that starting from the free field model we had arrived at the T ¯ T deformedaction. This was identified with the Nambu-Goto action of a 3d string in static gauge.We then wrote the unfixed NG action in Hamiltonian form and fixed the light-conegauge. Faddeev-Jackiw reduction of the gauge fixed action lead to the original free fieldHamiltonian.Guided by this, starting from a 2d CFT with a canonical description, specified by aHamiltonian density H (Π , φ, ´ φ ), we will devise a first order system such that after goingto static gauge we recover the deformed Hamiltonian while when working in light-conegauge we arrive at the undeformed Hamiltonian H . We then apply the same schemeto the model (2.18) with a potential, which explicitly breaks conformal symmetry.Relevant references for this section are [11, 12, 13]. We introduce a constrained Hamiltonian system with a string type action S = Z d τ Z π d σ π h P ˙ X + P ˙ X + Π k ˙ φ k − λ C − λ C i , (5.1)where C = P ´ X + P ´ X + P , C = 12 h α (cid:0) P − P (cid:1) + (cid:16) ´ X − ´ X (cid:17)i + α H (Π , φ, ´ φ ) . (5.2) H and P are the Hamiltonian and momentum densities of a 2d CFT. We assume thatthe conditions (2.7)-(2.8) are fulfilled. Because of (2.10) the Poisson brackets of theconstraints (5.2) satisfy (4.48).The system is reparametrization invariant (with the appropriate transformationproperties of λ , [16]). This enables us to introduce the static gauge, where again X isa compact coordinate. Doing this and applying the Faddeev-Jackiw reduction one findsthat the action (5.1) reduces to the T ¯ T -deformed system (3.1) with the Hamiltoniandensity H α + 1 / (2 α ), where H α as in eq. (3.9).18f we fix instead light-cone gauge (4.57) and use the definitions (4.54), we arrive againat (4.67). The equations (4.59)-(4.67) are trivially generalized with the replacements Π ˙ Φ Π k ˙ Φ k , Π ´ Φ Π k ´ Φ k , (cid:16) Π + ´ Φ (cid:17)
7→ H ( Π, Φ, ´ Φ ) . (5.3) We now generalize the above discussion to theories with a conformal symmetry breakingpotential U ( φ ). To this end we introduce a string like dynamical system such that instatic gauge it reduces to the deformed theory specified by the Hamiltonian density(3.17).Consider an action of the type (5.1), where the constraint C is the same as in (5.2)but with a modified C of the form C = 12 h g ( P − P ) + 1 − g b g ( ´ X − ´ X ) − b g ( P ´ X + P ´ X ) + 2 H i . (5.4)This has the structure of the Hamiltonian density (2.17) which guarantees that theconstraints C and C satisfy the algebra (4.48). The matrices G and B in the spacespanned by ( X , X ), are G kl = g (cid:18) − (cid:19) , B kl = b (cid:18) − (cid:19) . (5.5)As before the system is reparametrization invariant and we can fix either static orlight-cone gauge.The Faddeev-Jackiw reduction in the static gauge is again straightforward. If weidentify [11, 13] g = β , b = α U ( φ )2 β , β = α (cid:16) − α U (cid:17) (5.6)it leads to the Hamiltonian system (2.4) with the deformed Hamiltonian (3.17).We now turn to the reduction in light-cone gauge. The precise form of this gaugechoice is less obvious in the non-conformal case and to find it we rewrite the first ordersystem in second order Lagrangian form as a sigma-model with target space coordinates( X , X , φ k ): S = S [ φ ] + 12 π Z dτ dσ (cid:16) − β ( φ ) ∂ z X + ∂ ¯ z X − + 1 α (cid:0) ˙ X ´ X − ´ X ˙ X (cid:1)(cid:17) (5.7)where the first term is the 2d CFT action and the last term does not contribute to theequations of motion. For the light-cone fields X ± they are ∂ z (cid:18) β ( φ ) ∂ ¯ z X − (cid:19) = 0 , ∂ ¯ z (cid:18) β ( φ ) ∂ z X + (cid:19) = 0 , (5.8)19hich can be integrated once1 β ( φ ) ∂ ¯ z X − = ρ − (¯ z ) , β ( φ ) ∂ z X + = ρ + ( z ) . (5.9) ρ + and ρ − transform as one-forms under reparametrizations of the circle. Assuming thatthey have constant sign, which poses a restriction on the potential, one can gauge awaythe non-constant (oscillator) parts. In light-cone gauge ρ ± are (arbitrary) constants.If we insert this into the equation of motion for φ , we obtain δδφ k S [ φ ] + 14 α ρ + ρ − ∂∂φ k U ( φ ) = 0 . (5.10)For appropriate choice for ρ ± these are the equations of motion of the undeformedtheory. In the case of a single scalar field φ with a free action and potential U ( φ ) = 2 − e φ (5.11)equation (5.10) becomes the Liouville equation.For the same choice of potential and α = 1, the action (5.7) (before gauge fixing)and ignoring the boundary term is the SL (2) WZW-model [19, 20]. Acknowledgements
We would like to thank Harald Dorn and Alessandro Sfondrini for useful discussions.G.J. thanks MPI for Gravitational Physics in Potsdam for warm hospitality during hisvisits in 2018 and 2019 and S.T. is grateful to the Mathematical Institute of TSU forsupporting a visit in Tbilisi.
A Solution for the light-cone chiral fields
Due to (4.22), the Fourier mode expansions of F ′ ( z ) and ¯ F ′ (¯ z ) F ′ ( z ) = X n ∈ Z L n e − i nz , ¯ F ′ (¯ z ) = X n ∈ Z ¯ L n e − i n ¯ z , (A.1)define L n and ¯ L n as the Virasoro generators in the standard free-field form L n = 12 X n ∈ Z a m a n − m , ¯ L n = 12 X n ∈ Z ¯ a m ¯ a n − m , (A.2)with a = ¯ a = p . The solution of (4.20) can then be written asΦ − ( z ) = ρ − z + i αρ + X n =0 L n n e − i nz , ¯Φ − (¯ z ) = ¯ ρ − z + i α ¯ ρ + X n =0 ¯ L n n e − i n ¯ z , (A.3)where ρ − and ¯ ρ − are given by (4.23). We neglect the constant zero modes of Φ − ( z )and ¯Φ − (¯ z ); they correspond to translations of X and X .20 String energy in the static and light-cone gauges
The integration of (4.43) over σ , for a fixed τ , yields1 α Z π d σ π α ´ φ q α ´ φ − α ˙ φ = 1 α Z π d z π (cid:18) ρ + + αF ′ ( z ) ρ + (cid:19) = 1 α ( ρ + + ρ − )= 1 α Z π d¯ z π (cid:18) ¯ ρ + + α ¯ F ′ (¯ z )¯ ρ + (cid:19) = 1 α ( ¯ ρ + + ¯ ρ − ) . (B.1)According to (4.14), the left hand side of this equation is the string energy in thestatic gauge and the right hand sides correspond to the string energy in the light-conegauge (4.31). This straightforward calculation confirms the validity of (4.32), withoutreferring to the gauge invariance of the string energy.A similar calculation for the string momentum P by (4.43) yields P = Z π d σ π − ˙ φ ´ φ q α ´ φ − α ˙ φ = 1 α Z π d z π (cid:18) ρ + − αF ′ ( z ) ρ + (cid:19) =1 α ( ρ + − ρ − −
1) = ¯ h − h. (B.2) References [1] A. B. Zamolodchikov, “Expectation value of composite field T anti-T in two-dimensional quantum field theory,” hep-th/0401146.[2] F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum fieldtheories,” Nucl. Phys. B (2017) 363 [arXiv:1608.05499 [hep-th]].[3] A. Cavaglia, S. Negro, I. M. Szcsnyi and R. Tateo, “ T ¯ T -deformed 2D QuantumField Theories,” JHEP (2016) 112 [arXiv:1608.05534 [hep-th]].[4] G. Bonelli, N. Doroud and M. Zhu, “ T ¯ T -deformations in closed form,” JHEP (2018) 149 [arXiv:1804.10967 [hep-th]].[5] R. Conti, L. Iannella, S. Negro and R. Tateo, “Generalised Born-Infeld models, Laxoperators and the TT perturbation,” JHEP (2018) 007 [arXiv:1806.11515[hep-th]].[6] E. A. Coleman, J. Aguilera-Damia, D. Z. Freedman and R. M. Soni, “ T T -deformedactions and (1,1) supersymmetry,” JHEP (2019) 080 [arXiv:1906.05439 [hep-th]].[7] N. Callebaut, J. Kruthoff and H. Verlinde, “ T ¯ T deformed CFT as a non-criticalstring,” arXiv:1910.13578 [hep-th]. 218] B. Sundborg, “Mapping pure gravity to strings in three-dimensional anti-de Sittergeometry,” arXiv:1305.7470 [hep-th].[9] G. Jorjadze and S. Theisen, “Hamitonian approach to T ¯ T deformed 2d CFTs,”Talk presented at school and workshop: ”Joint FAR/ANSEF-ICTP and RDP-VWsummer school in theoretical physics”, July 2-7, 2018, Yerevan, Armenia.[10] M. Baggio and A. Sfondrini, “Strings on NS-NS Backgrounds as Integrable Defor-mations,” Phys. Rev. D (2018) no.2, 021902 [arXiv:1804.01998 [hep-th]].[11] S. Frolov, “TTbar deformation and the light-cone gauge,” arXiv:1905.07946 [hep-th].[12] S. Frolov, “ T T , e J J , J T and e J T deformations,” arXiv:1907.12117 [hep-th].[13] A. Sfondrini and S. J. van Tongeren, “ T ¯ T deformations as TsT transformations,”arXiv:1908.09299 [hep-th].[14] R. Conti, S. Negro and R. Tateo, “Conserved currents and T ¯T s irrelevant defor-mations of 2D integrable field theories,” JHEP (2019) 120 [arXiv:1904.09141[hep-th]].[15] L. D. Faddeev and R. Jackiw, “Hamiltonian Reduction of Unconstrained and Con-strained Systems,” Phys. Rev. Lett. (1988) 1692.[16] V. F. Mukhanov and A. Wipf, “On the symmetries of Hamiltonian systems,” Int.J. Mod. Phys. A (1995) 579 [hep-th/9401083].[17] C. B. Thorn, “Introduction to the Theory of Relativistic Strings,” in Santa Barbara1985, Proceedings, Unified String Theories.[18] G. Arutyunov, “Lectures on String Theory” (unpublished).[19] J. Balog, L. O’Raifeartaigh, P. Forgacs and A. Wipf, “Consistency of String Prop-agation on Curved Space-Times: An SU(1,1) Based Counterexample,” Nucl. Phys.B (1989) 225.[20] O. Coussaert, M. Henneaux and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,” Class. Quant.Grav.12