D=11 massless superparticle covariant quantization, pure spinor BRST charge and hidden symmetries
Igor A. Bandos
备注：
40 pages, no figures. V2: some references and comments on them added. V.3: misprints corrected. V.4: reference on 1985 preprint by M.J. Duff ([62]) and notice on it added. Published in Nucl. Phys. B
We consider the covariant quantization of the D=11 massless superparticle (M0-brane) in the spinor moving frame or twistor-like Lorentz harmonics formulation. The action involves the set of 16 constrained 32 component Majorana spinors, the spinor Lorentz harmonics parametrizing (as homogeneous coordinates, modulo gauge symmetries) the celestial sphere S9. There presence allows us to separate covariantly the first and the second class constraints of the model. After taking into account the second class constraints by means of Dirac brackets and after further reducing the first class constraints algebra, the system is described in terms of a simple BRST charge associated to the d=1, n=16 supersymmetry algebra. The study of the cohomology of this BRST operator requires a regularization by complexifying the bosonic ghosts for the kappa-symmetry and further reduction of the regularized cohomology problem to the one for a simpler complex BRST charge which is essentially the pure spinor BRST operator by Berkovits, but with a composite pure spinor. This exhibits a possible origin of the complexity (non-hermiticity) characteristic of the Berkovits pure spinor approach.
The simple structure of the nontrivial cohomology of the M0-brane BRST charge finds explanation in the properties that the superparticle action exhibits in the so-called `covariantized light-cone' basis. The the covariant quantization in this basis hints possible hidden symmetries of D=11 supergravity. Besides SO(16), we discuss also some indirect arguments in favor of the possible E8 symmetry.
The ten-dimensional superparticle is covariantly quantized by constructing a BRST operator from the fermionic Green-Schwarz constraints and a bosonic pure spinor variable. This same method was recently used for covariantly quantizing the superstring, and it is hoped that the simpler case of the superparticle will be useful for those who want to study this quantization method. It is interesting that quantization of the superparticle action closely resembles quantization of the worldline action for Chern-Simons theory.
This paper is devoted to the quantization of the second-ilk superparticle using the Batalin-Vilkovisky method. We show the full structure of the master action. By imposing gauge conditions on the gauge fields rather than on coordinates we find a gauge-fixed quantum action which is free. The structure of the BRST charge is exhibited and the BRST cohomology yields the same physical spectrum as the light- cone quantization of the usual superparticle.
We present a novel twistor formulation of the ten-dimensional massless superparticle. This formulation is based on the introduction of pure spinor variables through a field redefinition of another model for the superparticle, and in the new description we find that the super-Pauli-Lubanski three-form naturally arises as a constraint. Quantization is studied in detail for both models and they are shown to correctly describe the $D = 10$ super-Yang-Mills states.
Ten dimensional supersymmetric Yang-Mills theory may be described, in the light-cone gauge, in terms of either a vector or spinor superfield satisfying certain projection conditions (type I or II). These have been presented in a $ SO(9,1) $ form, and used to construct spinning superparticle theories in extended spaces. This letter presents the covariant quantisation of a "spinor-twin" type II superparticle theory by using the standard techniques of Batalin and Vilkovisky. The quantum action defines a quadratic field theory, whose ghost-independent BRST cohomology class gives the spectrum of N=…
The $D=11$ pure spinor formulation of the superparticle allows a simple realization of covariant quantization, unlike the $D=11$ Brink-Schwarz-like superparticle. We explicitly show the equivalence between the cohomologies of these two models in the context of two different group decompositions: $SO(10,1) \rightarrow SO(1,1)\times SO(9)$ and $SO(10,1) \rightarrow SO(3,1)\times SO(7)$. We also carry out a light-cone analysis of the pure spinor cohomology, and show that it correctly reproduces the $SO(9)$ equations of motion for $D=11$ linearized supergravity.
I present a toy model for the Berkovits pure spinor superparticle. It is a D=1, N=2 superparticle with no physical degrees of freedom. We study the cohomology in various ways, in particular finding an explicit expression for the `b'-field. Finally, we construct the topological string B-model from a straightforward generalization of the system.
The spinorial local world-line $κ$-symmetry of the covariant Brink-Schwarz formulation of the 4-$D$ superparticle is abelian in an off-shell phase-space formulation. The result is shown to generalize to the extended spinorial transformations of the spinning superparticle.
The $D=11$ pure spinor superparticle has been shown to describe linearized $D=11$ supergravity in a manifestly covariant way. A number of authors have proposed that its correlation functions be used to compute amplitudes. The use of the scalar structure of the eleven-dimensional pure spinor top cohomology introduces a natural measure for computing such correlation functions. This prescription requires the construction of ghost number one and zero vertex operators. In these notes, we construct explicitly a ghost number one vertex operator but show the incompatibiliy of a ghost number zero verte…
We develop an approach based on the Noether method to construct nilpotent BRST charges and BRST-invariant actions. We apply this approach first to the holomorphic part of the flat-space covariant superstring, and we find that the ghosts b, c_z which we introduced by hand in our earlier work, are needed to fix gauge symmetries of the ghost action. Then we apply this technique to the superparticle and determine its cohomology. Finally, we extend our results to the combined left- and right-moving sectors of the superstring.
{ This letter discusses the BRST cohomology of superparticles type I and II. It was used an extended super-space to construct $S0(9,1)$ superparticle actions that lead to super-wave functions whose spinor components satisfy $S0(9,1)$ covariant constraints. Their BRST charges were found by using BV methods, since the models present a large number of symmetries and only close on-shell. It is shown that the zero ghost-number cohomology class of both models reproduce the same spectrum as that of N=1 ten dimensional super-Yang-Mills theory. }
We consider firstly simple D=4 superalgebra with six real tensorial central charges $Z_{μν}$, and discuss its possible realizations in massive and massless cases. Massless case is dynamically realized by generalized Ferber-Shirafuji (FS) model with fundamental bosonic spinor coordinates. The Lorentz invariance is not broken due to the realization of central charges generators in terms of bosonic spinors. The model contains four fermionic coordinates and possesses three kappa-symmetries thus providing the BPS configuration preserving 3/4 of the target space supersymmetries. We show that the phy…
We revisit the theory of background fields constructed on the BRST-algebra of a spinning particle with $\mathcal{N}=4$ worldline supersymmetry, whose spectrum contains the graviton but no other fields. On a generic background, the closure of the BRST algebra implies the vacuum Einstein equations with a cosmological constant that is undetermined. On the other hand, in the "vacuum" background with no metric, the cohomology is given by a collection of free scalar- and vector fields. Only certain combinations of linear excitations, necessarily involving a vector field, can be extended beyond the l…
We present a modification of the Berkovits superparticle. This is firstly in order to covariantly quantize the pure spinor ghosts, and secondly to covariantly calculate matrix elements of a generic operator between two states. We proceed by lifting the pure spinor ghost constraints and regaining them through a BRST cohomology. We are then able to perform a BRST quantization of the system in the usual way, except for some interesting subtleties. Since the pure spinor constraints are reducible, ghosts for ghosts terms are needed, which have so far been calculated up to level 4. Even without a co…
On the basis of the Berkovits pure spinor formalism of covariant quantization of supermembrane, we attempt to construct a M(atrix) theory which is covariant under $SO(1,10)$ Lorentz group. We first construct a bosonic M(atrix) theory by starting with the first-order formalism of bosonic membrane, which precisely gives us a bosonic sector of M(atrix) theory by BFSS. Next we generalize this method to the construction of M(atrix) theory of supermembranes. However, it seems to be difficult to obtain a covariant and supersymmetric M(atrix) theory from the Berkovits pure spinor formalism of supermem…
We classify the admissible types of constraint (hermitian, holomorphic, with reality conditions on the bosonic sectors, etc.) for generalized supersymmetries in the presence of complex spinors. We further point out which constrained generalized supersymmetries admit a dual formulation. For both real and complex spinors generalized supersymmetries are constructed and classified as dimensional reductions of supersymmetries from {\em oxidized} space-times (i.e. the maximal space-times associated to $n$-component Clifford irreps). We apply these results to sistematically construct a class of model…
In a previous paper, the BRST cohomology in the pure spinor formalism of the superstring was shown to coincide with the light-cone Green-Schwarz spectrum by using an SO(8) parameterization of the pure spinor. In this paper, the SO(9,1) Lorentz generators are explicitly constructed using this SO(8) parameterization, proving the Lorentz invariance of the pure spinor BRST cohomology.
We construct multiloop superparticle amplitudes in 11d using the pure spinor formalism. We explain how this construction reduces to the superparticle limit of the multiloop pure spinor superstring amplitudes prescription. We then argue that this construction points to some evidence for the existence of a topological M theory based on a relation between the ghost number of the full-fledged supersymmetric critical models and the dimension of the spacetime for topological models. In particular, we show that the extensions at higher orders of the previous results for the tree and one-loop level ex…
The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra $\Cl(9,1)$. The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a $3 \times 3$ Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in …
Gauge invariant complex covariant actions for superparticles are derived from the field equations for the chiral superfields in a precise manner. The massive and massless cases in four dimensions are treated both free and in interaction with an external super Maxwell field. By means of a generalized BRST quantization these complex actions are related to real actions with second class constraints which are new in some cases.
The classical pure spinor version of the heterotic superstring in a supergravity and super Yang-Mills background is considered. We obtain the BRST transformations of the world-sheet fields. They are consistent with the constraints obtained from the nilpotence of the BSRT charge and the holomorphicity of the BRST current.
To exhibit the possible origin of the inner complexity of the Berkovits's pure spinor approach, we consider the covariant BRST quantization of the D=11 massless superparticle (M0-brane) in its spinor moving frame or twistor-like Lorentz harmonics formulation. The presence of additional twistor-like variables (spinor harmonics) allows us to separate covariantly the first and the second class constraints. After taking into account the second class constraints by means of Dirac brackets and after further reducing the first class constraints algebra, the dynamical system is described by the cohomo…
We show that new BRST charges in RNS superstring theory with nonstandard ghost numbers, constructed in our recent work, can be mapped to deformed pure spinor (PS) superstring theories, with the nilpotent pure spinor BRST charge $Q_{PS}=\oint{λ^α}d_α$ still retaining its form but with singular operator products between commuting spinor variables $λ^α$. Despite the OPE singularities, the pure spinor condition $λγ^mλ=0$ is still fulfilled in a weak sense, explained in the paper. The operator product singularities correspond to introducing interactions between the pure spinors. We conjecture that …
We discuss an analysis presented by N. Berkovits in [arXiv:0712.0324] in the context of classical mechanics and perform its extension to the case of pure spinor string in general background.
All the BRST-invariant operators in pure spinor formalism in $d=10$ can be represented as BRST commutators, such as $V=\lbrace{Q_{brst}},{θ_{+}\over{λ_{+}}}V\rbrace$ where $λ_{+}$ is the U(5) component of the pure spinor transforming as $1_{5\over2}$. Therefore, in order to secure non-triviality of BRST cohomology in pure spinor string theory, one has to introduce "small Hilbert space" and "small operator algebra" for pure spinors, analogous to those existing in RNS formalism. As any invariant vertex operator in RNS string theory can also represented as a commutator $V=\lbrace{Q_{brst}},LV\rbr…
The pure spinor heterotic string in a generic super Yang-Mills and supergravity background is considered. We determine the one-loop BRST anomaly at the cohomological level. We prove that it can be absorbed by consistent corrections of the classical constraints due to Berkovits and Howe, in agreement with the Green-Schwarz cancelation mechanism.
In this paper, we outline a method to compute supersymmetric one-loop integrands in ten-dimensional SYM theory. It relies on the constructive interplay between their cubic-graph organization and BRST invariance of the underlying pure spinor superstring description. The five- and six-point amplitudes are presented in a manifestly local form where the kinematic dependence is furnished by BRST-covariant expressions in pure spinor superspace. At five points, the local kinematic numerators are shown to satisfy the BCJ duality between color and kinematics leading to supergravity amplitudes as a bypr…
We describe the complete cohomology of the Berkovits BRST operator for the superparticle. It is non-zero at eight ghost numbers, splitting into two quartets, the members of each quartet being completely isomorphic. Based only on considerations of the isomorphisms of the cohomology, and using only the standard inner product, we derive the inner product appropriate for string amplitudes. It is in agreement with Berkovits' conjectured prescription, which is one element of an equivalence class. We discuss the Chern-Simons style action for D=10 super Yang-Mills, which is now manifestly superspace c…
We present a derivation of the scattering amplitude prescription for the pure spinor superstring from first principles, both in the minimal and non-minimal formulations, and show that they are equivalent. This is achieved by first coupling the worldsheet action to topological gravity and then proceeding to BRST quantize this system. Our analysis includes the introduction of constant ghosts and associated auxiliary fields needed to gauge fix symmetries associated with zero modes. All fields introduced in the process of quantization can be integrated out explicitly, resulting in the prescription…
We replace our earlier condition that physical states of the superstring have non-negative grading by the requirement that they are analytic in a new real commuting constant t which we associate with the central charge of the underlying Kac-Moody superalgebra. The analogy with the twisted N=2 SYM theory suggests that our covariant superstring is a twisted version of another formulation with an equivariant cohomology. We prove that our vertex operators correspond in one-to-one fashion to the vertex operators in Berkovits' approach based on pure spinors. Also the zero-momentum cohomology is equa…
A manifestly super-Poincaré covariant formalism for the superstring has recently been constructed using a pure spinor variable. Unlike the covariant Green-Schwarz formalism, this new formalism is easily quantized with a BRST operator and tree-level scattering amplitudes have been evaluated in a manifestly covariant manner. In this paper, the cohomology of the BRST operator in the pure spinor formalism is shown to give the usual light-cone Green-Schwarz spectrum. Although the BRST operator does not directly involve the Virasoro constraint, this constraint emerges after expressing the pure spino…
The grading Becchi-Rouet-Stora-Tyutin (BRST) method gives a way to construct the integer $W_{2,s}$ strings, where the BRST charge is written as $Q_B=Q_0+Q_1$. Using this method, we reconstruct the nilpotent BRST charges $Q_{0}$ for the integer $W_{2,s}$ strings and the half-integer $W_{2,s}$ strings. Then we construct the exact grading BRST charge with spinor fields and give the new realizations of the half-integer $W_{2,s}$ strings for the cases of $s=3/2$, 5/2, and 7/2.
After adding a pair of non-minimal fields and performing a similarity transformation, the BRST operator in the pure spinor formalism is expressed as a conventional-looking BRST operator involving the Virasoro constraint and (b,c) ghosts, together with 12 fermionic constraints. This BRST operator can be obtained by gauge-fixing the Green-Schwarz superstring where the 8 first-class and 8 second-class Green-Schwarz constraints are combined into 12 first-class constraints. Alternatively, the pure spinor BRST operator can be obtained from the RNS formalism by twisting the ten spin-half RNS fermions…
In this paper, we construct the nilpotent Becchi-Rouet-Stora-Tyutin($BRST$) charges of spinor non-critical $W_{2,s}$ strings. The cases of $s=3,4$ are discussed in detail, and spinor realization for $s=4$ is given explicitly. The $BRST$ charges are graded.
The transition amplitude is obtained for a free massive particle of arbitrary spin by calculating the path integral in the index-spinor formulation within the BFV-BRST approach. None renormalizations of the path integral measure were applied. The calculation has given the Weinberg propagator written in the index-free form with the use of index spinor. The choice of boundary conditions on the index spinor determines holomorphic or antiholomorphic representation for the canonical description of particle/antiparticle spin.
We report on progresses on the derivation of pure spinor constraints, BRST algebra and BRST invariant sigma models a la pure spinors from the algebraic structure of the FDA underlying supergravity.
Recently, the superstring was covariantly quantized using the BRST-like operator $Q = \oint λ^αd_α$ where $λ^α$ is a pure spinor and $d_α$ are the fermionic Green-Schwarz constraints. By performing a field redefinition and a similarity transformation, this BRST-like operator is mapped to the sum of the Ramond-Neveu-Schwarz BRST operator and $η_0$ ghost. This map is then used to relate physical vertex operators and tree amplitudes in the two formalisms. Furthermore, the map implies the existence of a $b$ ghost in the pure spinor formalism which might be useful for loop amplitude computations.
In this paper a multiparticle generalization of linearized ten-dimensional super Yang--Mills superfields is proposed. Their equations of motions are shown to take the same form as in the single-particle case, supplemented by contact terms. A recursive construction of these superfields is inspired by the iterated OPEs among massless vertex operators in the pure spinor formalism. An enlarged set of BRST-covariant pure spinor blocks is then defined in a streamlined fashion and combined to multiparticle vertex operators. The latter can be used to universally describe tree-level subdiagrams in the …
Covariant first and second quantization of the free d=4 massless superparticle are implemented with the introduction of purely gauge auxiliary spinor Lorentz harmonics. It is shown that the general solution of the condition of maslessness is a sum of two independent chiral superfields with each of them corresponding to finite superspin. A translationally covariant, in general bijective correspondence between harmonic and massless superfields is constructed. By calculation of the commutation function it is shown that in the considered approach only harmonic fields with correct connection betwee…
Using an elementary perturbative open string field theory solution involving a twistor-like parameter, we study the cohomology of new nilpotent BRST charge corresponding to the space-time background defined by this solution. The BRST cohomology of the deformed background automatically cuts off the GSO-odd spectrum in RNS superstring theory and keeps the GSO-even spectrum intact, without a need of GSO-projection. The on-shell constraints in the GSO-even sector get deformed in the new background, corresponding to BRST type transformations of the low-energy effective action with the ghost-like co…
We consider the superparticle models invariant under the supersymmetries with tensorial central charges, which were not included in D=4 Haag-Lopuszanski-Sohnius (HLS) supersymmetry scheme. We present firstly a generalization of D=4 Ferber-Shirafuji (FS) model with fundamental bosonic spinors and tensorial central charge coordinates. The model contains four fermionic coordinates and possesses three kappa- symmetries thus providing the BPS configuration preserving 3/4 of the target space supersymmetries. We show that the physical degrees of freedom (8 real bosonic and 1 real Grassmann variable) …
We construct the manifestly Lorenz-invariant formulation of the N=1 D=4 massive superparticle with tensorial central charges. The model contains a real parameter k and at $k\ne 0$ possesses one $κ$-symmetry while at k=0 the number of $κ$-symmetry is two. The equivalence of the formulations at all $k\ne 0$ is obtained. The local transformations of $κ$-symmetry are written out. It is considered the using of index spinor for construction of the tensorial central charges. It is obtained the equivalence at classical level between the massive D=4 superparticle with one $κ$-symmetry and the massive D…
In this Thesis we study first- and second-quantized approaches describing $D=11$ supergravity using pure spinor variables. We introduce the so-called $D=11$ pure spinor superparticle through BRST cohomology arguments starting from the semi-light-cone gauge $D=11$ Brink-Schwarz-like superparticle. After performing a light-cone gauge analysis of the pure spinor BRST cohomology at ghost number three, we find the linearized equations of motion of $D=11$ supergravity in $D=9$ superspace. In addition, we construct a BRST-closed, ghost number one vertex operator made out of worldline fields and $D=11…
Notes taken by Oscar A. Bedoya of lectures of Nathan Berkovits in June 2009 at the Galileo Galilei Institute School "New Perspectives in String Theory". Topics include the pure spinor superparticle and superstring, tree and loop amplitudes, Ramond-Ramond backgrounds, and open problems.
A twistorial formulation of the N=1 D=4 superparticle with tensorial central charges describing massive and massless cases in uniform manner is given. The twistors resolve energy-momentum vector whereas the tensorial central charges are written in term of spinor Lorentz harmonics. The model makes possible to describe states preserving all allowed fractions of target-space supersymmetry. The full analysis of the number of conserved supersymmetries in models with N=1 D=4 superalgebra with tensorial central charges has been carried out.
We construct the manifestly Lorentz-invariant twistorial formulation of N=1 D=4 superparticle with tensorial central charges which describes massive and massless cases in a uniform manner. The tensorial central charges are realized in terms of even spinor variables and central charge coordinates. The full analysis of the number of conserved supersymmetries has been carried out. In the massive case the superparticle preserves 1/4 or 1/2 of target-space supersymmetries whereas the massless superparticle preserves two or three supersymmetries.
In this work we investigate connections between superalgebras and their realizations in terms of particles, branes and field theory models. We start from Poincaré superalgebras with brane charges and study its representations. The existence of new supermultiplets in different dimensions including an ultra short supermultiplet in D=11 different from the supergravity multiplet is shown. Generalizations of superalgebras containing brane charges, including those in D$>$11 are considered. The realization of these algebras at the level of relativistic particle models and, upon quantization, at the l…
We present the massless six-point one-loop amplitudes in the open and closed superstring using BRST cohomology arguments from the pure spinor formalism. The hexagon gauge anomaly is traced back to a class of kinematic factors in pure spinor superspace which were recently introduced as BRST pseudo-invariants. This complements previous work where BRST invariance arguments were used to derive the non-anomalous part of the amplitude. The associated worldsheet functions are non-singular and demonstrated to yield total derivatives on moduli space upon gauge variation. These cohomology considerations…
We describe a pure spinor BRST cohomology framework to compactly represent ten-dimensional one-loop amplitudes involving any number of massless open- and closed-string states. The method of previous work to construct scalar and vectorial BRST invariants in pure spinor superspace signals the appearance of the hexagon gauge anomaly when applied to tensors. We study the systematics of the underlying BRST anomaly by defining the notion of pseudo-cohomology. This leads to a rich network of pseudo-invariant superfields of arbitrary tensor rank whose behavior under traces and contractions with extern…
On-shell supergravity vertex operators in an $AdS_5\times S^5$ background are described in the pure spinor formalism by the zero mode cohomology of a BRST operator. After expanding the pure spinor BRST operator in terms of the $AdS_5$ radius variable, this cohomology is computed using N=4 harmonic superspace variables and explicit superfield expressions are obtained for the behavior of supergravity vertex operators near the boundary of $AdS_5$.
We compute the one-loop beta function for the Type II superstring using the pure spinor formalism in a generic supergravity background. It is known that the classical pure spinor BRST symmetry puts the background fields on-shell. In this paper we show that the one-loop beta functions vanish as a consequence of the classical BRST symmetry of the action.
In this work, some aspects of the ambitwistor pure spinor string are investigated. The $b$ ghost is presented and its main properties are derived in a simple way, very similar to the usual pure spinor $b$ ghost construction. The heterotic case is also addressed with a new proposal for the BRST charge. The BRST cohomology is shown to correctly describe the heterotic supergravity spectrum and a semi-composite $b$ ghost is constructed.
We study the zero mode cohomology of the sum of two pure spinors. The knowledge of this cohomology allows us to better understand the structure of the massless vertex operator of the Type IIB pure spinor superstring.
The character of holomorphic functions on the space of pure spinors in ten, eleven and twelve dimensions is calculated. From this character formula, we derive in a manifestly covariant way various central charges which appear in the pure spinor formalism for the superstring. We also derive in a simple way the zero momentum cohomology of the pure spinor BRST operator for the D=10 and D=11 superparticle.
After adding an RNS-like fermionic vector ψ^m to the pure spinor formalism, the non-minimal b ghost takes a simple form similar to the pure spinor BRST operator. The N=2 superconformal field theory generated by the b ghost and the BRST current can be interpreted as a "dynamical twisting" of the RNS formalism where the choice of which spin half ψ^m variables are twisted into spin 0 and spin 1 variables is determined by the pure spinor variables that parameterize the coset SO(10)/U(5).
The pure spinor superstring in a type II curved background is considered. In order to define reparametrization ghosts, non-minimal pure spinor variables have to be present in the formalism. The BRST transformations of the non-minimal variables are obtained. It is found that the BRST transformations of a set of world-sheet variables have the form of the corresponding transformations in flat spacetime, up to a field-dependent Lorentz rotation. This simplify the construction of the reparametrization $b$ ghosts.
A $b$-ghost was constructed for the $D=11$ non-minimal pure spinor superparticle by requiring that $\{Q , b\} = T$ where $Q = Λ^αD_α + R^α\bar{W}_α$ is the usual non-minimal pure spinor BRST operator. As was done for the $D=10$ $b$-ghost, we will show that the $D=11$ $b$-ghost can be simplified by introducing an $SO(10,1)$ fermionic vector $\barΣ^{i}$ constructed out of the fermionic spinor $D_α$ and pure spinor variables. This simplified version will be shown to satisfy $\{Q, b\} = T$ and $\{b , b\} =$ BRST - trivial.
In these lecture notes presented at the 2015 Villa de Leyva Summer School, we give an introduction to superstring theory. We begin by studying the particle and superparticle in order to get a better understanding on the superstring side. Afterwards, we review the pure spinor formalism and end by computing the scattering amplitude for three gravitons at tree-level.
Although it is not known how to covariantly quantize the Green-Schwarz (GS) superstring, there exists a semi-light-cone gauge choice in which the GS superstring can be quantized in a conformally invariant manner. In this paper, we prove that BRST quantization of the GS superstring in semi-light-cone gauge is equivalent to BRST quantization using the pure spinor formalism for the superstring.
