On the Non-Relativistic Expansion of Closed Bosonic Strings

We develop a novel approach to non-relativistic closed bosonic string theory that is based on a string $1/c^2$ expansion of the relativistic string, where $c$ is the speed of light. This approach has the benefit that one does not need to take a limit of a string in a near-critical Kalb-Ramond background. The $1/c^2$-expanded Polyakov action at next-to-leading order reproduces the known action of non-relativistic string theory provided that the target space obeys an appropriate foliation constraint. We compute the spectrum in a flat target space, with one circle direction that is wound by the string, up to next-to-leading order and show that it reproduces the spectrum of the Gomis-Ooguri string.

Non-relativistic string theory belongs to a growing class of string theories whose worldsheet and/or target spacetime is not described by a Lorentzian geometry. Such open and closed string theories allow us to study quantum gravity in non-Lorentzian domains, to embed non-Lorentzian field theories into a string context (e.g. via world-volume theories of branes on which non-Lorentzian open strings end), to find new non-Lorentzian examples of holographic dualities, and to study interesting limits of standard string/M-theory. In this letter we focus on the particular case of non-relativistic strings.
The study of non-relativistic (NR) string theory began in earnest with the Gomis-Ooguri string [1,2], which employs a near-critical Kalb-Ramond field to cancel a divergent term in the action and leads to a well-defined theory of strings in an infinite speed of light limit. In order for this theory to have a non-trivial spectrum, it was shown that the target space must have a circle direction that is wound by the string. Subsequent work centred on generalising the target space of NR string theory: in [3,4], NR strings were obtained via null reduction, followed by a duality transformation that replaced the null direction with a compact direction. In another direction, a theory of strings moving in a string Newton-Cartan background was developed in [5], which combined the limit approach of the original Gomis-Ooguri string with the notion of string Newton-Cartan geometry developed in [6]. The relation between the null-reduced string and the string Newton-Cartan string was clarified in [7]. Furthermore, NR strings have been shown to appear in double field theory, and doubled geometry in general turns out to include a wealth of non-Lorentzian geometries [8][9][10].
For point particles there are two ways to obtain a NR description starting from the relativistic one. Option one: we can start with the action of a massive particle and expand the geometry and the embedding scalars systematically in 1/c 2 (see [11,12]). Option two: we can place the particle in a near-critical electromagnetic field, choose the particle to be extremal by relating its charge and mass, and take a c → ∞ limit [13]. The latter approach is equivalent to performing a null reduction starting from a massless particle in one dimension higher. These two procedures in general do not lead to the same theory. In [14] option two is worked out for strings while in this letter we focus on option one.
As in the string Newton-Cartan geometry of [5,6], when performing a string 1/c 2 expansion we single out not just the time direction, but also one spatial direction called the longitudinal target space direction. The target space becomes a string Newton-Cartan geometry that admits two-dimensional Lorentzian submanifolds.
The additional spatial direction singled out in the definition of the string Newton-Cartan geometry must be compact in order that the theory has a non-trivial spectrum [1,2,15]. The circle provides a new length scale that can be compared with the string length. We will show that the non-relativistic expansion corresponds to radii that are much larger than the string length.
Our formalism allows us to formulate string theories at any given order of 1/c 2 . In this letter, we develop the formalism and demonstrate how the theory up to nextto-leading order (NLO) is related to existing NR strings, while the more elaborate next-to-next-leading (NNLO) theory will be considered in [16]. Large-c as a decompactification limit.
It is a standard result of string theory that the mass squared of a quantum closed bosonic relativistic string in a 26-dimensional target space with a compact circle is where R is the radius of the circle, n is the momentum mode and w the winding number, while N andÑ are the number operators for the right and left movers. The dimensionless parameter with respect to which we will perform the NR expansion is ǫ = α ′ /(cR 2 ). Taking cT = T eff and R/c = R eff to be independent of c, we obtain ǫ = α ′ eff c 2 R 2 eff , where α ′ eff = 1/(2πT eff ). In this way, small values of ǫ correspond to large values of R, which leads us to conclude that the large c limit in fact corresponds to a decompactification limit. More precisely, since the quantum of momentum in the compact direction is /R and R/α ′ is the mass scale of the winding string, the center of mass velocity of the string along the compact direction is v com = α ′ /R 2 , which is small compared to speed of light, v com /c ≪ 1, which we can equivalently interpret as a large R limit. Since E = M 2 c 4 + p 2 c 2 , the mass in (1) gives rise to where we have absorbed the normal ordering constant into N (0) andÑ (0) and where we have 1/c 2 -expanded the number operators and transverse momentum according is the stringy version of the large-c expansion of the point particle energy. String Newton-Cartan geometry.
Paralleling the discussion of [12] (see also [17,18]), we now show how to obtain string Newton-Cartan geometry (SNC) from D = d + 2-dimensional Lorentzian geometry. Write the Lorentzian metric G MN and its inverse as where M, N = 0, 1, . . . , d + 1 are spacetime indices, while We expand these fields according to with similar expansions for T M A and Π ⊥MN . We then find that the metric expands as where We define the strong foliation constraint in terms of the 1-forms where ω is a 1-form that is determined by solving (6) for ω. However, equation (6) does more than determining ω since it also constrains τ A . The constraint (6) played an important role in [5] in their definition of NR string theory. This condition has recently been relaxed in [19,20]. Here we will show that the 1/c 2 expansion naturally comes with its own foliation constraint [21].
The target space fields of the relativistic string obey the beta function equations involving the metric, the Kalb-Ramond 2-form and the dilaton. If we ignore the dilaton and the 2-form, these equations are simply the Einstein equations R MN = 0 to leading order in α ′ . If we expand these in 1/c 2 using (3), we find at leading order (LO) that where H ⊥MN is the leading-order component of Π ⊥MN . The above is a sum of squares for A = B = 0, 1 and thus equivalent to H ⊥QS H ⊥RT dτ A QR = 0. This in turn is equivalent to for arbitrary one-forms α A B . This is the Frobenius integrability condition for a codimension-two foliation whose leaves are d-dimensional Riemannian spaces with normal one-forms τ A . This reduces to the strong foliation con- (8) is the string NC analogue of the TTNC condition imposed in NC geometry [22] which likewise follows from the particle 1/c 2 expansion of the Einstein equations [11,23]. Expansion of the string action.
The Polyakov Lagrangian is To expand this, we must, in addition to expanding the metric G MN as in (5), expand the embedding field (that will in general depend on c) as . We also expand the worldsheet metric γ αβ as where γ (0)αβ is a Lorentzian metric with determinant − det γ (0) = e, while γ (2)αβ is a symmetric tensor. The pullback G αβ (X) = ∂ α X M ∂ β X N G MN (X) acquires the following expansion where τ αβ (x) = ∂ α x M ∂ β x N τ MN (x) is assumed to be of Lorentzian signature and where The 1/c 2 expansion of the Polyakov Lagrangian (9) is where we introduced the Wheeler-DeWitt metric G αβγδ . The reason for expanding the worldsheet metric as in (10) is so that γ (0)αβ can be related to the Lorentzian pullback metric τ αβ via the equation of motion of γ (0)αβ .
The NLO theory can be recast in a Nambu-Goto (NG) formulation. By integrating out γ (0)αβ from the P-LO Lagrangian, and by integrating out both γ (0)αβ and γ (2)αβ from the P-NLO Lagrangian, the NG Lagrangian at LO and NLO can be found to be where τ = det τ αβ . The constraints that come from integrating out γ (0)αβ and γ (2)αβ are the LO and NLO Virasoro constraints, respectively. These can also be obtained by 1/c 2 -expanding the Virasoro constraints obtained by integrating out γ αβ in (9). We remark that the constraint from integrating out γ (2)αβ at NLO is identical to that from integrating out γ (0)αβ at LO and leads to The equation of motion of the NG-LO Lagrangian for the embedding scalar x M , which features in (15), reads If we assume that the target space obeys the Frobenius condition (8), then equation (17) forces α M A A to be equal to τ M A X A for some X A that is determined by (17). A sufficient condition for (17) to be equal to zero (and hence for the y-term to drop out of L NG-NLO ) is to simply take α M A B to be traceless of which the strong foliation constraint (6) is a special case. Relation with the NR string action.
The gauge-fixed P-LO Lagrangian on flat space is The Virasoro constraints from integrating out γ (0)αβ in L P-LO are (16), which amount to τ ++ = 0 = τ −− . Without loss of generality this is equivalent to the LO constraints: In our conventions x v has dimensions of time. Since the v-direction is compact, the constraints ∂ ∓ x ± = 0 imply the following mode expansions for x ± where x ± 0 are constants, w is the winding number and R eff is the target space circle radius (in units of time).
where we have used the LO constraints and since ξ ± (0) is periodic, we can fix the residual gauge transformations by removing the non-zero modes of x ± , leaving only x ± = x ± 0 + wR eff σ ± . The relativistic energy expands in 1/c 2 as Hence, the energy at LO is the 'stringy' rest mass which matches the first term in (2). If we include a Kalb-Ramond field of the form ) and B (0)MN = 0, we produce the 'instanton term' of [1] for λ = 1/2 (see also [15]) In what follows, we will take λ = 1.
The gauge-fixed P-NLO Lagrangian on flat space is where the equations of motion for y t and y v , respectively, imply the LO equations of motion for x t and x v , while the NLO equations of x t and x v tell us that ∂ + ∂ − y t = ∂ + ∂ − y v = 0. The equation of motion for x i implies that ∂ + ∂ − x i = 0. This leads to the mode expansions: [24] x where y ± = y t ± y v , and where the momenta p (0)± are the canonical momenta, p (0)± = dσ 1 ∂LP-NLO ∂(∂0x ± ) . The constraints arising from integrating out γ (2)αβ from (14) are the same as those originating from integrating out γ (0)αβ from (13). The constraints from integrating out γ (0)αβ from (14) read We still need to fix the subleading residual gauge invariance ξ ± (2) (σ ± ), which acts infinitesimally on y ± as δy ± = wR eff ξ ± (2) (σ ± ). We fix the subleading residual gauge transformations by removing the oscillations in ∂ ± y ± .
The zero mode of each Virasoro constraint in (34) gives an expression for p (0)± = dσ 1 ∂LNLO-P ∂(∂0x ± ) = 1 2 (p (0)t ±p (0)v ), respectively, in terms of the modes of x i , where we defined the leading number operators Adding the expressions in (35), we get in agreement with the c 0 terms in (2). The momentum in the compact v-direction is quantised, p (0)v = n R eff , where n is an integer. Since p (0)v = p (0)+ − p (0)− , we obtain the level matching condition N (0) −Ñ (0) = nw. Canonically quantising the NLO theory will lead to a normal ordering constant a in the number operators N (0) = ∞ n=1 α i −n α i n − a andÑ (0) = ∞ n=1α i −nα i n − a . Standard arguments tell us that a = D−2 24 . In [16] we work out the Poisson algebra of the Noether charges of the global symmetries of the NLO Polyakov action with a flat target space. We expect the quantum theory to have the same symmetry algebra if we choose D = 26 in line with the results of [1].

Discussion.
The beta functions for NR string theory have been computed in [25][26][27] and an action has been proposed that reproduces all but one (the string analogue of the Poisson equation) of the beta functions in [20]. Based on the action for NR gravity obtained using the particle 1/c 2 expansion of GR [12,23], we expect that the string 1/c 2 expansion of NS-NS gravity could lead to an action principle for all the beta functions of NR string theory.
Recent studies of non-relativistic string theories, including other works such as [4,28], point to the existence of a landscape of string theories beyond the Lorentzian ones we are familar with. It would be of interest to study open string sectors and D-brane like objects in such theories (see e.g. [29,30]). In this light, it would be interesting to apply the 1/c 2 expansion to the study of open strings and D-brane actions.