Nonrelativistic string theory in flat spacetime is described by a twodimensional quantum field theory with a nonrelativistic global symmetry acting on the worldsheet fields. Nonrelativistic string theory is unitary, ultraviolet complete and has a string spectrum and spacetime S-matrix enjoying nonrelativistic symmetry. The worldsheet theory of nonrelativistic string theory is coupled to a curved spacetime background and to a Kalb-Ramond two-form and dilaton field. The appropriate spacetime geometry for nonrelativistic string theory is dubbed string Newton-Cartan geometry, which is distinct from Riemannian geometry. This defines the sigma model of nonrelativistic string theory describing strings propagating and interacting in curved background fields. We also implement T-duality transformations in the path integral of this sigma model and uncover the spacetime interpretation of T-duality. We show that T-duality along the longitudinal direction of the string Newton-Cartan geometry describes relativistic string theory on a Lorentzian geometry with a compact lightlike isometry, which is otherwise only defined by a subtle infinite boost limit. This relation provides a first principles definition of string theory in the discrete light cone quantization (DLCQ) in an arbitrary background, a quantization that appears in nonperturbative approaches to quantum field theory and string/M-theory, such as in Matrix theory. T-duality along a transverse direction of the string Newton-Cartan geometry equates nonrelativistic string theory in two distinct, T-dual backgrounds. Longitudinal lightlike T-duality: Nonrelativistic string theory on a stringNewton-Cartan background is mapped to nonrelativistic string theory on a Tdual string Newton-Cartan background with a longitudinal lightlike isometry. The precise mapping between the two T-dual string Newton-Cartan background fields can be found in §3.2. 3.Transverse T-duality: Nonrelativistic string theory on a string Newton-Cartan background is mapped to nonrelativistic string theory on a T-dual string Newton-Cartan background. See §3.3 for the precise T-duality transformation rules.5 See also [1,3,4].-3 -

Nonrelativistic string theory is described by a sigma model with a relativistic worldsheet and a nonrelativistic target spacetime geometry, that is called string Newton-Cartan geometry. In this paper we obtain string Newton-Cartan geometry as a limit of the Riemannian geometry of General Relativity with a fluxless two-form field. We then apply the same limit to relativistic string theory in curved background fields and show that it leads to nonrelativistic string theory in a string Newton-Cartan geometry coupled to a Kalb-Ramond and dilaton field background. Finally, we use our limiting procedure to study the spacetime equations of motion and the T-duality transformations of nonrelativistic string theory. Our results reproduce the recent studies of beta-functions and T-duality of nonrelativistic string theory obtained from the microscopic worldsheet definition of nonrelativistic string theory. 8 Note that later, we will take the limit in the curvature constraint (3.5) with an independent spinconnection field and not in the solution (3.6) where the spin-connetion is dependent. Equivalently, one may also take the same limit directly in the second-order formalism.9 Note that only after taking the limit c → ∞ , the fields at the r.h.s. of (3.12) may be identified with the fields of string Newton-Cartan geometry. 10 We use the convention that ǫ01 = 1.

We find a Polyakov-type action for strings moving in a torsional Newton-Cartan geometry. This is obtained by starting with the relativistic Polyakov action and fixing the momentum of the string along a non-compact null isometry. For a flat target space, we show that the world-sheet theory becomes the Gomis-Ooguri action. From a target space perspective these strings are non-relativistic but their world-sheet theories are still relativistic. We show that one can take a scaling limit in which also the world-sheet theory becomes non-relativistic with an infinite-dimensional symmetry algebra given by the Galilean conformal algebra. This scaling limit can be taken in the context of the AdS/CFT correspondence and we show that it is realized by the 'Spin Matrix Theory' limits of strings on AdS 5 × S 5 . Spin Matrix theory arises as non-relativistic limits of the AdS/CFT correspondence close to BPS bounds. The duality between non-relativistic strings and Spin Matrix theory provides a holographic duality of its own and points towards a framework for more tractable holographic dualities whereby non-relativistic strings are dual to near BPS limits of the dual field theory.2 As will be clear in Sec. 2.1 we find in this paper that the TNC geometry is extended with a periodic target space direction.3 This Nambu-Goto form was also obtained in [20]. 4 The GCA was also observed in earlier work on non-relativistic limits of AdS/CFT [21]. See also Ref. [22] for useful work on representations of the GCA and aspects of non-relativistic conformal two-dimensional field theories.

Nonrelativistic string theory is a unitary, ultraviolet finite quantum gravity theory with a nonrelativistic string spectrum. The vertex operators of the worldsheet theory determine the spacetime geometry of nonrelativistic string theory, known as the string Newton-Cartan geometry. We compute the Weyl anomaly of the nonrelativistic string worldsheet sigma model describing strings propagating in a string Newton-Cartan geometry, Kalb-Ramond and dilaton background. We derive the equations of motion that dictate the backgrounds on which nonrelativistic string theory can be consistently defined quantum mechanically. The equations of motion we find from our study of the conformal anomaly of the worldsheet theory are to nonrelativistic string theory what the (super)gravity equations of motion are to relativistic string theory.

Study of the rare decays of B 0 s and B 0 mesons into muon pairs using data collected during 2015 and 2016 with the ATLAS detector The ATLAS Collaboration A study of the decays B 0 s → µ + µ − and B 0 → µ + µ − has been performed using 26.3 fb −1 of 13 TeV LHC proton-proton collision data collected with the ATLAS detector in 2015 and 2016. Since the detector resolution in µ + µ − invariant mass is comparable to the B 0 s -B 0 mass difference, a single fit determines the signal yields for both decay modes. This results in a measurement of the branching fraction B(B 0 s → µ + µ − ) = 3.2 +1.1 −1.0 × 10 −9 and an upper limit B(B 0 → µ + µ − ) < 4.3 × 10 −10 at 95% confidence level. The result is combined with the Run 1 ATLAS result, yielding B(B 0 s → µ + µ − ) = 2.8 +0.8 −0.7 ×10 −9 and B(B 0 → µ + µ − ) < 2.1×10 −10 at 95% confidence level. The combined result is consistent with the Standard Model prediction within 2.4 standard deviations in the B(B 0 → µ + µ − )-B(B 0 s → µ + µ − ) plane.