of Yang–Mills **theory** **and** **the** NLSM model, ref. [36] offered **the** first double-copy real- ization of self-dual Born–Infeld [37] scattering amplitudes. **The** idea that there can exist a duality between electric **and** magnetic field densities is as old as gauge **theory**. Sat- isfied by sourceless Maxwell electrodynamics, this natural duality has inspired analysis **and** generalizations that have been key to understanding aspects of supersymmetry, sym- metry breaking, **and** **string** **theory**, starting with perhaps most famously **the** Born–Infeld non-linear generalization of electromagnetics [38]. **The** emergence of duality invariance in **the** form of Born–Infeld scattering due to a double-copy interplay between YM **and** **the** low-energy limit of abelian Z-**theory** is remarkable. In concordance with **the** structure of open-**string** amplitudes given as a double copy between Yang–Mills constituents **and** Z- **theory** disk-integrals [2], **the** double-copy representation of Born–Infeld amplitudes as its

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nevertheless gather valuable information from such quantum gravity interactions by properly dealing with **the** singularities.
It is in fact not just an aesthetic question to find a quantum description of grav- ity or to find a unified framework describing all four fundamental interactions but it happens to be, for modern physics, an immediate necessity because both Stan- dard Model **and** **the** concordance model of cosmology have their own deficiencies. To mention a few for **the** Standard Model, this **theory** requires almost 20 parame- ters whose values are needed to be fixed by experimental input. Some of these pa- rameters are needed to be fine-tuned upto very non-practical degrees of accuracy, which is **the** so-called hierarchy problem of Standard Model. In addition, **the** Stan- dard Model does not give satisfactory answers to neutrino mass spectrum **and** **the** strong CP problem. An ingenious way to answer **the** hierarchy problem is to appeal to **the** supersymmetry, which puts both bosonic **and** fermionic degrees of freedoms in terms of an underlying symmetry of **the** quantum **theory**. This set-up provides bosons **and** fermions of same quantum characters thereby adding elements to solve for **the** hierarchy problem. Moreover, **the** localization of supersymmetry gives rise to a supersymmetric **theory** of quantum gravity, called supergravity which is however again crippled with non-renormalizability issues, but its degree is milder to that of naive quantum gravity. In **the** observable world, supersymmetry is not manifest **and** thus this symmetry should be broken at **the** phenomenological energy scale. Thus instead of discouraging, it cues to look forward for **the** physics at **the** energy scales higher than that of common interactions which might reveal these interesting sym- metry structures.

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In fact, SL(2, Z) invariance of **the** type IIB **string** gives rise to some tension between **the** different four-point results collected in Table 1. For IIB strings, **the** only difference between **the** quartic NSNS contributions at tree level **and** one loop is **the** dilaton factor e −2φ . Other than that, **the** kinematic structures are identical. This means that **the** only way of completing **the** purely NSNS expressions to SL(2, Z) invariant ones is by making each term invariant **and** multiplying **the** entire expression quartic in fields by **the** SL(2, Z) function E 3/2(τ, ¯ τ ). In particular this means that **the** local U (1) symmetry of type IIB supergravity is respected by **the** four-particle interactions. This is indeed consistent with **the** results of [33,34] showing that **the** U (1)-violating contributions start at **the** level of five-particle interactions. From **the** other side, **the** four-point result including RR fields given in [18] cannot be completed to an SL(2, Z) invariant expression without modular forms that transform under weights ±1, **and** are hence U(1)-violating. We shall return to this issue in subsection 4.2.

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Regular word functions extend **the** robust family of regular languages, preserving many of its characterizations **and** algorithmic properties. A word function maps words over a finite input alphabet to words over a finite output alphabet. Regular word functions have been studied in **the** early seventies, in **the** form of (deterministic) two-way finite state automata with output [1]. Engelfriet **and** Hoogeboom [8] later showed that monadic second-order definable graph transductions, restricted to words, are an equivalent model — this justifies **the** notation “regular” word functions, in **the** spirit of classical results in automata **theory** **and** logic by Büchi, Elgot, Rabin **and** others. Recently, Alur **and** Cerný [2] proposed an enhanced version of one-way transducers called streaming transducers, **and** showed that they equivalent to **the** two previous models. A streaming transducer processes **the** input word from left to right, **and** stores (partial) output words in finitely many, write-only registers. A variant of streaming transducers extended by stacks has been introduced in [3] **and** shown to capture precisely **the** monadic-second order definable tree transductions.

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ture that replaces **the** horizon of a black hole: we have directly constructed this structure in supergravity. As we emphasized in [46], this approach could have failed at many different stages throughout its development. **The** most recent hurdle has been to show that supergravity has structures that might contain enough states to count **the** entropy of **the** black hole. In [5] we have argued that this can happen if **string** **theory** contains three-charge **superstrata** solutions that can be parameterized by arbitrary continuous functions of two variables. **The** present paper shows explicitly that these solutions exist **and** furthermore that they are smooth in **the** duality frame where **the** black hole has D1,D5 **and** momentum charges. (It was **the** successful clearing of this latest hurdle that led to our somewhat celebratory title for this paper.) Though most of **the** recent literature on **the** information paradox has focused on “Alice-**and**-Bob” Gedankenexperiments, we believe that general quantum information arguments about physics at a black-hole horizon will always fall short of resolving **the** paradox: failure is inevitable without a mechanism to support structure at **the** horizon scale. It is remarkable that **string** **theory** can provide a natural **and** beautiful solution to this essential issue **and**, as was shown in [47], microstate geometries provide **the** only possible gravitational mechanism **and** so must be an essential part of **the** solution to **the** paradox.

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5.2 Systematic construction of a large family of smooth four-center solutions
In this section, we present a systematic construction of **the** largest known family of scaling four- center smooth horizonless solutions that have **the** same charges as large black holes. Our construction allows us to easily build scaling four-center BPS solutions with any aspect ratios between **the** centers. Moreover, we focus on solutions which are asymptotically R 1,4 but **the** method can be adapted to any asymptotics. **The** main idea is to start with BPS solutions with three collinear supertube centers in Taub-NUT. As we will see, defining a parameter space where those solutions satisfy **the** Denef equations, **the** no-CTC condition **and** **the** scaling condition is rather easy. Then, **the** next step is to regularize **the** solutions at **the** supertube centers. As detailed in Section 4.1.3.3, this can be done by performing two or three generalized spectral flows (4.1.30). Each generalized spectral flow transforms a singular magnetic charge of its corresponding species of Supertube to a smooth KKm charge **and** then transforms a species of Supertube to a smooth GH center. Thus, with two spectral flows we obtain a solution with three GH centers **and** a Supertube. This solution is smooth in **the** D1-D5-P frame as explained in Section 4.1.3.3. With three spectral flows, we have a solution of four smooth GH centers in five-dimensions. Because generalized spectral flows also change **the** asymptotics, **the** last step consists in applying gauge transformations **and** change of background moduli to have asymptotically flat solutions.

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Music **theory** has been developed empirically down **the** ages until it achieved status as a recognized discipline. It now provides us with **the** means for describing **the** underlying mechanisms **and** subtle interaction between musicians as they perform, based on complex combinatory rules relating to rhythm, harmony **and** melody. Computer science also aims to describe **the** subtle organization **and** treatment of data. It therefore follows that **the** study of modeling in **the** field of music might lead to **the** discovery of concepts, abstract tools **and** modeling principles which are applicable to modern computer engineering.

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Classification of elliptically fibered Calabi-Yau manifolds
It has been shown by Gross [30] that **the** number of distinct topological types of elliptically fibered Calabi-Yau threefolds is finite up to birational equivalence. Finiteness of **the** set of topologically distinct elliptically-fibered Calabi-Yau threefolds is shown in [6] using minimal surface **theory** **and** **the** fact that **the** Weierstrass form for an elliptic fibration over a fixed base has a finite number of possible distinct singularity structures. These arguments, however, do not give a clear picture of how such compactifications can be sys- tematically classified. A complete mathematical classification of elliptically fibered Calabi-Yau threefolds would be helpful in understanding **the** range of F-**theory** compactifications. **The** analogue of this question for four dimen- sions, while probably much more difficult, would be of even greater interest, since at this time we have very little handle on **the** scope of **the** space of four dimensional supergravity theories which can be realized through F-**theory** compactifications on Calabi-Yau fourfolds.

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These experimental results can be compared to those obtained from **the** **theory**. Experi- mentally, we found two modes which mostly involve **string** 31 at 123.59 Hz **and** at 123.78 Hz. With **the** vibratory model, **the** **string** mode associated to **string** 31 is found at 123.48 Hz. Note that **the** model takes only one **string** polarization into account. **The** agreement between model **and** measurement is, as expected, very good since **the** tension value of each **string** of **the** simplified harp has been fixed in such a way that **the** eigenfrequencies of uncoupled strings correspond to those of **the** real instrument.

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X N S5 −→ X N S5
def = R (1|5) × (SL 2 (R) N /U(1) N × SU(2) N /U(1) N ) /Z N ,
with k = N + 2. Hence, **the** interest in NS5-branes provides strong motivation to inves- tigate **string** **theory** on **the** cigar. Let us also note that there exists a rich **and** interesting class of compactifications of NS5-branes on Calabi-Yau spaces that, by **the** same line of reasoning, involve **the** cigar as a central building block [76] (see also [7]). We should finally note that these backgrounds certainly involve some amount of supersymmetry which we suppress in our discussion here. As in **the** case of **the** analogous compact coset theories, adding supersymmetry has relatively minor effects on **the** world-sheet **theory**. Since we are more interested in **the** qualitative features of our non-compact coset model, we shall neglect **the** corrections that supersymmetry brings about, even though they are certainly important in concrete applications. For treatments of **the** supersymmetric models, we can refer **the** reader to a number of interesting recent publications [77]- [87].

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1.3 Observations **and** consequences
Today, we have a few quasi-certainties about cosmology thanks to observa- tions. First **the** existence of dark matter was suggested by Zwicky who measured **the** velocity dispersions of **the** galaxies in **the** Coma cluster **and** found that some of **the** gravitational mass was missing [4]. Another solid evidence of “unseen” mass is **the** flatness of **the** rotation curves of observed galaxies that can’t be ex- plained by **the** visible mass of **the** galaxies alone [5]. **The** general idea is that **the** rotation velocity is a function of **the** inner mass (virial theorem). **The** missing matter is to be found in **the** dark matter halo of **the** galaxy. Another additional proof of **the** existence of dark matter is strong lensing, **the** light coming from distant galaxies is deviated by **the** presence of dark matter. Numerical simula- tions [6] highlight **the** necessity to include dark matter, otherwise **the** formation of large-scale structures does not start at **the** right time. Nevertheless, dark mat- ter has not been detected yet **and** some alternatives to DM theories have been developed. **The** famous MOND (MOdified Newtonian Gravity) **theory** tends to explain **the** rotation curves by a modification of Newtonian gravity for very small accelerations. This was first proposed by Milgrom [7]. But **the** observations of **the** bullet cluster [8] (see figure 1.1) are in contradiction with such theories **and** are strong evidence in favor of dark matter.

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model **and** show that for highly extensible molecules (as characterised by large values of **the** FENE extensibility parameter b > 10 3 ) there is a long intermediate elastic regime
(which may extend for times of 10 20 ) in which neither **the** initial response of **the** solvent nor **the** nite length of **the** molecules is important **and** **the** chains indeed extend eectively as innitely extensible dumbbells with a single characteristic relaxation time. This has also been veried experimentally using a homologous series of polystyrene test
uids with solutes of increasing molecular weight (Anna & McKinley (2001)). In **the** present study, we therefore do not consider **the** nal stages of breakup where **the** nite length of **the** polymers begins to aect **the** necking process. This nal asymptotic regime has been considered for **the** FENE model by Entov & Hinch (1997), **and** more recently for **the** Giesekus model by Fontelos & Li (2004). In each case **the** extensional viscosity of **the**
uid is bounded **and** **the** lament radius ultimately decreases linearly in time. **The** initial onset of this regime can also be observed in **the** very last stages of **the** experimental measurements when **the** thread radius has reduced to O(1 10 m); however we do not include this data in our comparison between **theory** **and** experimental observation.

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charge c of **the** corresponding worldsheet **theory** are related by
α ′ M 0 2 = h − c/24 (2)
where α ′ ≡ (2πσ) −1 is **the** Regge slope, **and** h is **the** conformal dimension of that
primary field in **the** worldsheet **theory** which produces **the** ground state. In most cases this primary field is merely **the** identity field with h = 0, so that **the** coefficient of **the** pseudo-Coulomb term directly yields **the** corresponding central charge. In all other cases, however, **the** coefficient of this term yields information concerning only **the** difference h − c/24 ≡ −˜c/24. This is dramatically illustrated in **the** Ramond **string**: here h = c/24 = (D − 2)/16, whereupon ˜c = 0, **the** ground state is massless, **and** **the** long-range pseudo-Coulomb term is absent.

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The Fock module of a single string of momentum p is obtained by the action of the matter, ghost and antighost oscillators on the (ghost number one) highest weight vec[r]

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gauge group, Type I SO(32) **and** Type II (A **and** B) theories. **The** Heterotic theories are theories with closed strings whereas Type I **and** Type II contain both closed **and** open strings. Let us mention here a few important features of **string** **theory** which make it a good candidate for a **theory** of all interactions. Closed strings have a spin 2 mass- less mode which can be identified with **the** graviton, moreover **the** **string** perturbation **theory** is UV-finite thus making **string** **theory** a consistent **theory** of quantum gravity. Standard model-like interactions can arise from **string** **theory**. Ideas like grand unifica- tion **and** supersymmetry can (naturally) be incorporated in **string** **theory**. Indeed, **the** low energy effective field **theory** arising from strings is a supergravity **theory**. Another important feature of **string** theories is that they are well-defined only in ten space-time dimensions, hence they predict six extra dimensions. In order for **string** **theory** to make sense as a fundamental (or effective) **theory** of Nature **the** extra dimensions have to be small (usually compact). There are indeed solutions which allow for a spacetime of **the** form

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