Worldsheet instanton corrections to five-branes and waves in double field theory

Published for SISSA by Springer Received: April 2, 2018 Accepted: June 11, 2018 Published: July 2, 2018 Tetsuji Kimura,a,1 Shin Sasakib and Kenta Shiozawab a Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan b Department of Physics, Kitasato University, Sagamihara 252-0373, Japan E-mail: , , Abstract: We make a comprehensive study on the string winding corrections to supergravity solutions in double field theory (DFT). We find five-brane and wave solutions of diverse codimensions in which the winding coordinates are naturally included. We discuss a physical interpretation of the winding coordinate dependence. The analysis based on the geometric structures behind the solutions leads to an interpretation of the winding dependence as string worldsheet instanton corrections. We also give a brief discussion on the origins of these winding corrections in gauged linear sigma model. Our analysis reveals that for every supergravity solution, one has DFT solutions that include string winding corrections. Keywords: p-branes, String Duality ArXiv ePrint: 1803.11087 1 The affiliation since April 2018: Research Institute of Science and Technology, College of Science and Technology, Nihon University, 1-8-14 Kanda Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan. Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2018)001 JHEP07(2018)001 Worldsheet instanton corrections to five-branes and waves in double field theory Contents 1 Introduction 1 3 5 5 9 11 12 13 3 Manifesting the winding corrections 3.1 H- and KK-monopoles 3.2 Q-brane 3.3 R-brane 3.4 Space-filling brane in winding space 3.5 Wave and F-string 15 15 17 19 20 21 4 Worldsheet instanton effects 4.1 Disk instantons in single-centered Taub-NUT space 4.2 Worldsheet instanton corrections to the 522 -brane geometry 22 23 25 5 Instantons in GLSM 27 6 Conclusion and discussions 29 1 Introduction Understanding the space-time structure in the Planck scale is one of the most important theme in theoretical physics. One knows that a macroscopic nature of the universe is captured by general relativity where the space-time is probed by a point particle. The geometry of space-time is realized as a Riemannian manifold whose metric satisfies the Einstein equation. One would expect that this picture fails in the Planck scale where quantum mechanical effects drastically change the structure of space-time. String theory, which is a candidate of consistent quantum gravity, provides rather intrinsic nature of the Planck scale space-time. Since space-time is probed by a one-dimensional extended object (i.e. the fundamental string (F-string)) it is therefore plausible that the conventional Riemannian geometry is modified in string theory. One of the most important features of string theory is U-duality. Consider M-theory compactified on d-dimensional torus T d , the low-energy supergravity theories in D = 11−d space-time dimensions have U-duality hidden symmetry based on the Ed(d) group [1]. –1– JHEP07(2018)001 2 DFT solutions of diverse codimensions 2.1 Five-brane solutions 2.1.1 Codimension four 2.1.2 Codimension three 2.1.3 Codimension two 2.1.4 Codimensions one and zero 2.2 Wave solutions 1 A manifestly T-duality covariant formalism is initially developed in [17–19]. –2– JHEP07(2018)001 Among other things, the existence of T-duality which interchanges the Kaluza-Klein (KK) and winding modes of wrapped strings is the most prominent difference between the theories based on point particles and strings. This makes it expressive that the T-duality interchanges the “geometrical” (or conventional) coordinates xµ , associated with the Fourier dual of the KK-modes, and the “winding” coordinates x̃µ associated with the winding modes. According to the T-duality symmetry in a setup of the flux compactification, a notion of non-geometric fluxes is revealed [2]. This is rephrased as a T-duality chain of the H-τ Q-R fluxes [3]. On the other hand, there is yet another T-duality chain of five-branes in type II string theories. The H-monopole, which is obtained by compactifying a transverse direction to the NS5-brane to S 1 , has a U(1) isometry. The T-duality transformation of the H-monopole along the isometry direction results in the KK-monopole (KK5-brane) which is geometrically described by the Taub-NUT space. As a consequence, the H-flux sourced by the H-monopole is mapped to the geometric τ -flux sourced by the KK-monopole. It is notable that we can perform further T-duality transformations. By introducing an extra isometry along a transverse direction to the KK-monopole, the second T-duality transformation becomes possible. The resulting solution is known as the Q-brane. This is a kind of exotic branes appearing non-trivially in the T-duality orbit. The conventional argument labels this as the 522 -brane [1]. Most notably, the 522 -brane is a non-geometric object in the sense that its background geometry has a monodromy in the T-duality group O(2, 2) [4, 5]. This kind of non-geometric backgrounds is called the T-fold [6]. The Q-brane is a source of the Q-flux. Indeed, exotic branes are sources of non-geometric fluxes [7–9]. A further T-duality transformation settles down to the yet unknown R-brane (532 -brane) which is the source of the R-flux. These famous T-duality chains cause an interesting observation. For example, let us focus on the T-duality relation between the H- and KK-monopoles. In [10], string worldsheet instanton corrections to the S 1 -smeared NS5-brane (i.e. the H-monopole) geometry is studied through the gauged linear sigma model (GLSM). As a result, the instanton corrections break the isometry in the S 1 and the H-monopole becomes the NS5-brane localized in the S 1 . A puzzle arises in the T-dualized KK-monopole side. Since the Taub-NUT geometry inherits the isometry, what is the corresponding worldsheet instanton effect? The worldsheet instanton corrections to the KK-monopole is analyzed [11, 12] and it is shown that the corrections break the isometry in the winding (T-dualized) space. In other words, the worldsheet instanton corrections in the KK-monopole geometry introduces the dual “winding” coordinates x̃µ dependence in the solution in addition to the space-time geometrical coordinates xµ . This originates essentially from an old question for non-isometric T-duality in the unwinding string process [13]. The same is true in further T-duality processes. Based on the GLSM introduced in [14], two of the present authors studied the worldsheet instanton corrections to the 522 -brane geometry [15] where again the winding coordinate dependence appears via the instanton effects. Double field theory (DFT) [16] is a formalism where the T-duality group O(d, d) is realized as a manifest symmetry.1 The T-duality transformation is encoded into the gen- 2 DFT solutions of diverse codimensions In this section, we discuss five-brane solutions in DFT. The f (...truncated)