Supersymmetry, T-duality and heterotic α′-corrections

Published for SISSA by Springer Received: May 10, 2021 Accepted: June 15, 2021 Published: July 15, 2021 Eric Lescano,a Carmen A. Núñeza,b and Jesús A. Rodríguezb a Instituto de Astronomía y Física del Espacio (IAFE-CONICET-UBA), Ciudad Universitaria, Pabellón IAFE, 1428 Buenos Aires, Argentina b Departamento de Física, FCEyN, Universidad de Buenos Aires (UBA), Ciudad Universitaria, Pabellón 1, 1428 Buenos Aires, Argentina E-mail: , , Abstract: Higher-derivative interactions and transformation rules of the fields in the effective field theories of the massless string states are strongly constrained by space-time symmetries and dualities. Here we use an exact formulation of ten dimensional N = 1 supergravity coupled to Yang-Mills with manifest T-duality symmetry to construct the first order α0 -corrections of the heterotic string effective action. The theory contains a supersymmetric and T-duality covariant generalization of the Green-Schwarz mechanism that determines the modifications to the leading order supersymmetry transformation rules of the fields. We compute the resulting field-dependent deformations of the coefficients in the supersymmetry algebra and construct the invariant action, with up to and including four-derivative terms of all the massless bosonic and fermionic fields of the heterotic string spectrum. Keywords: String Duality, Superstrings and Heterotic Strings, Supersymmetry and Duality ArXiv ePrint: 2104.09545 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2021)092 JHEP07(2021)092 Supersymmetry, T-duality and heterotic α0-corrections Contents 1 2 The leading order theory 2.1 Review of N = 1 supersymmetric double field theory 2.2 Parameterization and choice of section 3 4 8 3 The first order α0 -corrections 3.1 The generalized Bergshoeff-de Roo identification 3.2 Induced transformation rules on O(10, 10) multiplets 3.3 Including the heterotic gauge sector 3.4 First order corrections to N = 1 supersymmetric DFT 10 10 13 15 16 4 Transformation rules of the supergravity fields 17 5 Heterotic string effective action to O(α0 ) 23 6 Outlook and final remarks 25 A Conventions and definitions A.1 Some useful gamma function identities A.2 Leading order components of the generalized fluxes A.3 The leading order action and equations of motion 27 28 28 29 B Algebra of transformations of O(10, 10 + ng ) fields B.1 Leading order algebra B.2 First order algebra 30 30 33 C Supersymmetry of heterotic string effective action C.1 Supersymmetry algebra C.2 Invariance of the action 36 37 38 1 Introduction At low energy, or small curvature, heterotic string theory reduces to ten dimensional N = 1 supergravity coupled to super Yang-Mills [1]. Successive terms in the α0 -expansion may be expressed as higher-derivative interactions that are strongly constrained by the symmetries of string theory. There are several reasons to study the higher-order terms in the effective field theories of the massless string modes. They are needed to evaluate the stringy effects on solutions to the supergravity equations of motion [2–4], they play a central role in the –1– JHEP07(2021)092 1 Introduction –2– JHEP07(2021)092 tests of duality conjectures [5, 6], in the microstate counting of black hole entropy [7– 9] and in moduli stabilization [10].The swampland program [11] has revealed that the effective field theories of low energy physics and cosmology are limited by their couplings to quantum gravity [12–14], and together with the string lamppost principle [15], reinforces the interest in the restrictions imposed by string theory on the higher-derivative corrections to General Relativity. The first few orders of the heterotic string α0 -expansion are known explicitly. The interactions of the bosonic fields up to O(α03 ) were originally determined from the computation of scattering amplitudes of the massless string states at tree [1, 16–18] and one loop [19–22] levels in the string coupling and from conformal anomaly cancellations [23]. The contributions of the fermionic fields have been computed using supersymmetry and superspace methods [24–37]. Supersymmetry completely fixes the leading order terms [24] and it often provides an elegant underlying explanation of the higher-derivative corrections. But it holds iteratively in powers of α0 and the transformation rules of the fields demand order by order modifications that are further restricted by other string symmetries and dualities. In particular, the effective field theories for the massless string fields exhibit a global O(n, n; R) symmetry when the fields are independent of n spatial coordinates. This continuous T-duality symmetry holds to all orders in α0 [38] (see also [39–47]) and it has been explicitly displayed recently for the quadratic and some of the quartic interactions of the bosonic fields in [48, 49]. This feature motivated the construction of field theories with T-duality covariant structures, such as double field theory (DFT) [50–56] and generalized geometry [57, 58], which provide reformulations of the string (super)gravities in which the global duality invariance is made manifest. In the duality covariant frameworks, the standard local symmetries are generalized to larger groups: diffeomorphism invariance is extended to also include the gauge transformations of the two-form and the tangent space is enhanced with an extended Lorentz symmetry. Interestingly, the duality covariant gauge transformations completely determine the lowest order field interactions in string (super)gravities even before dimensional reduction (for reviews see [59–64] and references therein). Moreover, extensions of the duality group [65, 66] as well as enhancings of the gauge structure of DFT [67, 68] allowed to reproduce the four-derivative interactions of the massless bosonic heterotic string fields. Supersymmetry can be naturally incorporated in the duality covariant formulations [69–76]. A supersymmetric and manifestly O(10, 10 + ng ) covariant DFT reformulation of ten dimensional N = 1 supergravity coupled to ng abelian vector multiplets was introduced in [70–73]. Although it is formally constructed on a 20 + ng dimensional space-time, the apparent inconsistency of supergravity beyond eleven dimensions is avoided through a strong constraint that admits solutions removing the field dependence on 10 + ng coordinates, and fermions transform as spinors under the O(9, 1)L factor of the local O(9, 1)L × O(1, 9 + ng )R double Lorentz symmetry. More recently, an exact supersymmetric and manifestly duality covariant mechanism was introduced in [76], in which the global symmetry of the theory is taken to be O(D, D + k), k being the dimension of the O(1, D + k − 1) Lorentz group. To pre- 2 The leading order theory In this section we review the basic features of the DFT reformulation of N = 1 supergravity coupled to ng vector multiplets in ten dimensions that was introduced i (...truncated)