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We show that on shell DDF amplitudes are on shell lightcone amplitudes and that Mandelstam maps emerge naturally with a precise normalization and are intrinsic to the DDF states. Off shell DDF and Mandelstam amplitudes \`a la Kaku-Kikkawa differ. Underway we give a very explicit formula for the conformal transformation of a generic vertex in the form of a compact generating function for free theories., Comment: 38 pages, 3 figures

We provide a compact expression for the generating function of correlators involving an arbitrary number of bosonic open string DDF states. The explicit correlators for $M$ DDF states can then be obtained by differentiating this generating function with respect to the DDF polarization tensors. The generating function depends on single and double complex integrals centred around the punctures corresponding to the insertion points of the DDF vertices on the real axis in the upper half-plane. We have explicitly evaluated these integrals for an arbitrary number of DDF states. To check our results, we computed some massive scalars and spin-2 amplitudes for $M=3$ and $M=4$, verifying that these amplitudes have the form expected from Lorentz invariance. Moreover, for the first excited levels, we also show that the DDF amplitudes with generic polarizations can be reassembled into Lorentz-covariant amplitudes, with emergent covariant polarizations that match the expressions obtained by comparing the string states in both the DDF and covariant formalisms., Comment: Latex, 83 pages

Educators are more than workers within educational systems; they are stewards of educational systems. They must analyze student performance data, identify patterns that inform targeted interventions and personalized learning plans, continuously develop the curriculum, set ambitious learning goals and use up-to-date pedagogical theory to adapt instructional strategies, act as advocates for educational policies that promote inclusivity and equity, and much more. Most educators deeply care about the learning and wellbeing of their students and colleagues. Given the chance, they will do whatever they can to make improvements to these ends. In this role as architects of change, educators deal with conflicting definitions of success, multiple stakeholders, complex causal relationships, ambiguous data, and intricate human factors. Amid all this, most educators and the educational systems around them are strained to the capacity of what their time, training, and budgets allow. The problem is not merely that they must perform demanding tasks, but more so that they must constantly implement improvements and interventions amid the complex challenges of the organizations in which they work. These challenges can be especially difficult in implementation of related education technology, which is continuously developing at sometimes rapid pace. Whether the context is an individual classroom, a school district, or a postsecondary institution, implementing beneficial human-technology partnerships requires attending to the needs and constraints of these classrooms, districts, institutions, and so forth as organizations and engaging in this work as a partnership with educators. This chapter lays out the principles and processes of developing successful educator-technology partnerships including key considerations for each step and an example protocol for engaging in this endeavor., Comment: Book Chapter, 24 pages, 6 tables, 1 figure

We compute the irreps and their multiplicities of bosonic string spectrum up to level 10 and we give explicitly the on shell lightcone states which make the irreps. For scalars and vectors we compute the multiplicity up to level 22 and 19 respectively. The first scalar at odd level appears at level 11. For the bosonic string in non critical dimensions we argue that at level $N$ there are always states transforming as tensors with $ s\ge N/2$ indices. Only in critical dimensions there are states with $s\le N/2$. Looking at the explicit coefficients of the combinations needed to make the irreps from the lightcone states we trace the origin of the chaotic behavior of certain cubic amplitudes considered in literature to the extremely precise and sensitive mixtures of states. For example the vectors at level $N=19$ are a linear combinations of states and when the coefficients are normalized to be integer some of them have more than 1200 figures., Comment: In ancillary files there are the data in lisp files (which can be loaded in the CAS maxima) and the explicit results of the analysis in separated tex and pdf for levels from 3 to 10

We define the framed DDF operators by introducing the concept of local frames in the usual formulation of DDF operators. In doing so it is possible to completely decouple the DDF operators from the associated tachyon and show that they are good zero-dimensional conformal operators. This allows for an explicit formulation of the general solution of the Virasoro constraints both on-shell and off-shell. We then make precise the realization of the intuitive idea that DDF operators can be used to embed light-cone states in the covariant formulation. This embedding is not unique, but depends on a coset. This coset is the little group of the embedding of the light-cone and is associated with a frame. The frame allows us to embed the $SO(D-2)$ light-cone physical polarizations into the $SO(1,D-1)$ covariant ones in the most general way. The solution to the Virasoro constraints is not in the gauge that is usually used. This happens since the states obtained from DDF operators are generically the sum of terms which are partially transverse due to the presence of a projector but not traceless and terms which are partially traceless but not transverse. To check the identification, we verify the matching of the expectation value of the second Casimir of the Poincar'e group for some light-cone states with the corresponding covariant states built using the framed DDFs., Comment: 37 pages (main) + 21 pages (appendix)

We discuss how the perturbative particle paradigm fails in certain background with space-like singularity but asymptotically flat which should admit a S-matrix. The Feynman approach relies on the interaction picture. This approach means that we can interpret interactions as exchanges of particles. Particles are the modes of the quadratic part of the Lagrangian. In certain backgrounds with space-like singularity the interaction Hamiltonian is well defined but the perturbative expansion of the evolution operator through the singularity and the perturbative $S$ matrix do not exist. On the other hand, relying on minisuperspace approximation we argue that the non perturbative evolution operator does exist. The complete breakdown of the perturbative expansion explains why the perturbative computations in the covariant formalism in string theory in temporal orbifold fail, at least at the tree level., Comment: 38 pages, 5 double figures

We discuss what is light-cone quantization on a curved spacetime also without a null Killing vector. Then we consider as an example the light-cone quantization of a scalar field on a background with a Killing vector and the connection with the second quantization of the particle in the same background. It turns out that the proper way to define the light-cone quantization is to require that the constant light-cone time hypersurface is null or, equivalently, that the particle Hamiltonian is free of square roots. Moreover, in order to quantize the scalar theory it is necessary to use not the original scalar rather a scalar field density, i.e. the Schr\"odinger wave functional depends on a scalar density and not on the original field. Finally we recover this result as the second quantization of a particle on the same background, where it is necessary to add as input the fact that we are dealing with a scalar density., Comment: 25+1 pages

We consider time-dependent orbifolds in String Theory and we show that divergences are not associated with a gravitational backreaction since they appear in the open string sector too. They are related to the non existence of the underlying effective field theory as in several cases fourth and higher order contact terms do not exist. Since contact terms may arise from the exchange of string massive states, we investigate and show that some three points amplitudes with one massive state in the open string sector are divergent on the time-dependent orbifolds. To check that divergences are associated with the existence of a discrete zero eigenvalue of the Laplacian of the subspace with vanishing volume, we construct the Generalized Null Boost Orbifold where this phenomenon can be turned on and off., Comment: 43 pages

We consider a two-dimensional fermion on the strip in the presence of an arbitrary number of zero-dimensional boundary changing defects. We show that the theory is still conformal with time dependent stress-energy tensor and that the allowed defects can be understood as excited spin fields. Finally we compute correlation functions involving these excited spin fields without using bosonization., Comment: 40 pages, 4 figures

We consider the classical instantonic contribution to the open string configuration associated with three D-branes with relative rotation matrices in SO(4) which corresponds to the computation of the classical part of the correlator of three non Abelian twist fields. We write the classical solution as a sum of a product of two hypergeometric functions. Differently from all the previous cases with three D-branes, the solution is not holomorphic and suggests that the classical bosonic string knows when the configuration may be supersymmetric. We show how this configuration reduces to the standard Abelian twist field computation. From the phenomenological point of view, the Yukawa couplings between chiral matter at the intersection in this configuration are more suppressed with respect to the factorized case in the literature., Comment: 41 pages, 8 figures

String theory gives S matrix elements om which is not possible to read any gauge information. Using factorization we go off shell in the simplest and most naive way and we read which are the vertices suggested by string. To compare with the associated Effective Field Theory it is natural to use color ordered vertices. The alpha'=0 color ordered vertices suggested by string theory are more efficient than the usual ones since the three gluon color ordered vertex has three terms instead of six and the four gluon one has one term instead of three. They are written in the so called Gervais-Neveu gauge. The full Effective Field Theory is in a generalization of the Gervais-Neveu gauge with alpha') corrections. Moreover a field redefinition is required to be mapped to the field used by string theory. We also give an intuitive way of understanding why string choose this gauge in terms of the minimal number of couplings necessary to reproduce the non abelian amplitudes starting from color ordered ones., Comment: 37 pages, 9 figures

We consider the simplest possible setting of non abelian twist fields which corresponds to $SU(2)$ monodromies. We first review the theory of hypergeometric function and of the solutions of the most general Fuchsian second order equation with three singularities. Then we solve the problem of writing the general solution with prescribed $U(2)$ monodromies. We use this result to compute the classical string solution corresponding to three $D2$ branes in $R^4$. Despite the fact the configuration is supersymmetric the classical string solution is not holomorphic. Using the equation of motion and not the KLT approach we give a very simple expression for the classical action of the string. We find that the classical action is not proportional to the area of the triangle determined by the branes intersection points since the solution is not holomorphic. Phenomenologically this means that the Yukawa couplings for these supersymmetric configurations on non factorized tori are suppressed with respect to the factorized case., Comment: 51 pages, 16 figures

We study the canonical quantization of a bosonic string in presence of N twist fields. This generalizes the quantization of the twisted string in two ways: the in and out states are not necessarily twisted and the number of twist fields N can be bigger than 2. In order to quantize the theory we need to find the normal modes. Then we need to define a product between two modes which is conserved. Because of this we need to use the Klein-Gordon product and to separate the string coordinate into the classical and the quantum part. The quantum part has different boundary conditions than the original string coordinates but these boundary conditions are precisely those which make the operator describing the equation of motion self adjoint. The splitting of the string coordinates into a classical and quantum part allows the formulation of an improved overlap principle. Using this approach we then proceed in computing the generating function for the generic correlator with L untwisted operators and N (excited) twist fields for branes at angles. We recover as expected the results previously obtained using the path integral. This construction explains why these correlators, Comment: 41 pages, 8 figures

We compute the generic correlator with L untwisted operators and N (excited) twist fields for branes at angles on T^2 and show that it is given by a generalization of the Wick theorem. We give also the recipe to compute efficiently the generic OPE between an untwisted operator and an excited twisted state., Comment: 49 pages, 6 figures

We investigate epsilon-deformed N=2 superconformal gauge theories in four dimensions, focusing on the N=2* and Nf=4 SU(2) cases. We show how the modular anomaly equation obeyed by the deformed prepotential can be efficiently used to derive its non-perturbative expression starting from the perturbative one. We also show that the modular anomaly equation implies that S-duality is implemented by means of an exact Fourier transform even for arbitrary values of the deformation parameters, and then we argue that it is possible, perturbatively in the deformation, to choose appropriate variables such that it reduces to a Legendre transform., Comment: 30 pages, LeTeX2e. V2: references added, appendix B expanded, a few typos corrected

We quantize the bosonic part of the D1 string with closed boundary conditions on the light cone and we consider the U(1) worldsheet gauge field a dynamical variable. We compute also 3-Reggeon vertex by the overlapping technique. We find that the Fock space is the sum of sectors characterized by the momentum of the U(1) Wilson line and that these sectors do not interact among them. Each sector has exactly the same spectrum of the usual bosonic string when expressed in properly sector dependent rescaled variables. Rescaling is forced by factorization of the string amplitudes. We are also able to determine the relative string coupling constant of the different sectors. It follows a somewhat unexpected picture in which the effective action is always the same independently on the sector but string amplitudes are only the same when expressed in sector dependent rescaled variables., Comment: 17 pages, 2 figures

We study the non-perturbative properties of N=2 super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic epsilon-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov's approach. These results allow to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly epsilon-dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory., Comment: 33 pages, no figures, LaTeX2e

We compute the Green functions and correlator functions for N twist fields for branes at angles on T^2 and we show that there are N-2 different configurations labeled by an integer M which is roughly associated with the number of obtuse angles of the configuration. In order to perform this computation we use a SL(2,R) invariant formulation and geometric constraints instead of Pochammer contours. In particular the M=1 or M=N-1 amplitude can be expressed without using transcendental functions. We determine the amplitudes normalization from N -> N-1 reduction without using the factorization into the untwisted sector. Both the amplitudes normalization and the OPE of two twist fields are unique (up to one constant) when the \epsilon <-> 1-\epsilon symmetry is imposed. For consistency we find also an infinite number of relations among Lauricella hypergeometric functions., Comment: 40 pages, 13 figures; V2 published version with misprints and a minor error corrected

We show that the generating function of all amplitudes with N twisted and M untwisted states, i.e. the Reggeon vertex for magnetized branes on R^2 can be computed once the correlator of N non excited twisted states and the corresponding Green function are known and we give an explicit expression as a functional of the these objects, Comment: 26 pages

We quantize the open string in an arbitrary constant magnetic field with a non factorized metric on a torus. We then discuss carefully the vertexes which describe the emission of dipole open strings and closed strings in the non compact limit. Finally we compute various stringy form factors which in the compact case induces a Kaehler and complex structure dependence and suppression of some amplitudes with KK states., Comment: 45 pages

We construct the six-dimensional Lagrangian for the massless twisted open strings with one end-point ending on a stack of D5 and the other on a stack of D9 branes, interacting with the gauge multiplets living respectively on the D5 and D9 branes. It is first obtained by uplifting to six dimensions the four-dimensional Lagrangian of the N=2 hypermultiplet and manifestly exhibits an SU(2) symmetry. We show by an explicit calculation that it is N=1 supersymmetric in six dimensions and then we check various terms of this Lagrangian by computing string amplitudes on the disk. Finally, starting from this Lagrangian and assuming the presence of non-zero magnetic fluxes along the extra compact dimensions, we determine the spectrum of the Kaluza-Klein states which agrees with the corresponding one obtained from string theory in the field theory limit., Comment: 54 pages, Latex

The F-theory background of four D7 branes in a type I' orientifold was conjectured to be described by the Seiberg-Witten curve for the superconformal SU(2) gauge theory with four flavors. This relation was explained by considering in this background a probe D3 brane, which supports this theory with SU(2) realized as Sp(1). Here we explicitly compute the non-perturbative corrections to the D7/D3 system in type I' due to D-instantons. This computation provides both the quartic effective action on the D7 branes and the quadratic effective action on the D3 brane; the latter agrees with the F-theoretic prediction. The action obtained in this way is related to the one derived from the usual instanton calculus \`a la Nekrasov (or from its AGT realization in terms of Liouville conformal blocks) by means of a non-perturbative redefinition of the coupling constant. We also point out an intriguing relation between the four-dimensional theory on the probe D3 brane, which has a SO(8) flavor symmetry, and the eight-dimensional dynamics on the D7 branes. On the latter, SO(8) represents a gauge group and the flavor masses correspond to the vacuum expectation values of an adjoint scalar field m: what we find is that the exact effective coupling in four dimensions is obtained from its perturbative part by taking into account in its mass dependence the full quantum dynamics of the field m in eight dimensions., Comment: 44 pages, 2 figures, PdfLaTeX. V2: Typos fixed, a few references added, Eq. (2.28) corrected, improved discussion in Section 3.4, a couple of sentences reformulated in section 7. Version published on JHEP

We discuss carefully the vertices which describe the dipole open strings and closed strings on a D-brane with magnetic flux on a torus. Translation invariance along closed cycles forces surprisingly closed string vertices written in open string formalism to acquire Chan-Paton like matrices. Moreover the one loop amplitudes have a single trace for the part of gauge group with the magnetic flux. These peculiarities are also required by consistency of the action of T-duality in the open string sector. In this way we can show to all orders in perturbation theory the equivalence of the T-dual open string theories, gravitational interactions included. We provide also a new and direct derivation of the bosonic boundary state in presence of constant magnetic and Kalb-Ramond background based on Sciuto-Della Selva-Saito vertex formalism., Comment: 63 pages, 7 figures

We compute the partition function for the exotic instanton system corresponding to D-instantons on D7 branes in Type I' theory. We exploit the BRST structure of the moduli action and its deformation by RR background to fully localize the integration. The resulting prepotential describes non-perturbative corrections to the quartic couplings of the gauge field F living on the D7's. The results match perfectly those obtained in the dual heterotic theory from a protected 1-loop computation, thus providing a non-trivial test of the duality itself., Comment: 42 pages, 3 figures

We consider the D7/D(--1) system in Type I' as a prototypical "exotic" brane instanton. With respect to systems such as the D3/D(-1) ones, which correspond to gauge instantons in four dimensions, exotic systems lack the bosonic mixed moduli w of the ADHM construction, related to the instanton size, and their possible field-theoretical interpretation as classical solutions is an important open question. For the system at hand, we propose that it corresponds to the point-like limit of the eight-dimensional so-called SO(8) instanton solution. This configuration is a solution of the quartic term of the non-abelian D7 action, i.e., the term which stays finite in the limit in which alpha' goes to zero with the string coupling fixed that preserves the D(-1) effects. As a necessary consistency condition, we check that the next order term in the non-abelian effective action vanishes on the SO(8) solution so that the limit we take is well-defined., Comment: 39 pages, 3 figures. Some references added, and minor corrections: a more precise statement of eq. (3.18) and a few coefficients in eq. (3.10)