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95-399: At low energies, type IIA string theory is described by type IIA supergravity in ten dimensions which is a non- chiral theory (i.e. left–right symmetric) with (1,1) d =10 supersymmetry; the fact that the anomalies in this theory cancel is therefore trivial. In the 1990s it was realized by Edward Witten (building on previous insights by Michael Duff , Paul Townsend , and others) that

190-1171: A p {\displaystyle p} -form gauge field . The 3-form gauge field has a modified field strength tensor F ~ 4 = F 4 − A 1 ∧ F 3 {\displaystyle {\tilde {F}}_{4}=F_{4}-A_{1}\wedge F_{3}} with this having a non-standard Bianchi identity of d F ~ 4 = − F 2 ∧ F 3 {\displaystyle d{\tilde {F}}_{4}=-F_{2}\wedge F_{3}} . Meanwhile, χ 1 μ {\displaystyle \chi _{1}^{\mu }} , χ 3 μ ν ρ {\displaystyle \chi _{3}^{\mu \nu \rho }} , Ψ 2 μ ν {\displaystyle \Psi _{2}^{\mu \nu }} , and Ψ 4 μ ν ρ σ {\displaystyle \Psi _{4}^{\mu \nu \rho \sigma }} are various fermion bilinears given by The first line of

285-425: A m = e ϕ δ a m {\displaystyle e_{a}^{m}=e^{\phi }\delta _{a}^{m}} and χ a = γ a ξ {\displaystyle \chi _{a}=\gamma _{a}\xi } , where ϕ {\displaystyle \phi } and ξ {\displaystyle \xi } decouple from

380-430: A Laurent series where O n {\displaystyle {\mathcal {O}}_{n}} are known as the modes and ν = 0 {\displaystyle \nu =0} or 1 / 2 {\displaystyle 1/2} depending on whether the operator is periodic or antiperiodic, respectively. The holomorphic stress-energy tensor and holomorphic supercurrent together form

475-855: A scalar field ϕ {\displaystyle \phi } . This nonchiral multiplet can be decomposed into the ten-dimensional N = 1 {\displaystyle {\mathcal {N}}=1} multiplet ( g μ ν , B μ ν , ψ μ + , λ − , ϕ ) {\displaystyle (g_{\mu \nu },B_{\mu \nu },\psi _{\mu }^{+},\lambda ^{-},\phi )} , along with four additional fields ( C μ ν ρ , C μ , ψ μ − , λ + ) {\displaystyle (C_{\mu \nu \rho },C_{\mu },\psi _{\mu }^{-},\lambda ^{+})} . In

570-433: A 3-form C n m p {\displaystyle C_{nmp}} . Since the p {\displaystyle {\text{p}}} -form gauge fields naturally couple to extended objects with p+1 {\displaystyle {\text{p+1}}} dimensional world-volume, Type IIA string theory naturally incorporates various extended objects like D0, D2, D4 and D6 branes (using Hodge duality ) among

665-417: A BRST exact state, also called a null state Q B | η ⟩ {\displaystyle Q_{B}|\eta \rangle } , being equivalent | ψ ⟩ ∼ | ψ ⟩ + Q B | η ⟩ {\displaystyle |\psi \rangle \sim |\psi \rangle +Q_{B}|\eta \rangle } . There

760-409: A closed algebra known as the N = 1 {\displaystyle N=1} super Virasoro algebra . Using a mode expansion where the stress tensor modes are given by L n {\displaystyle L_{n}} and the supercurrent modes by G r {\displaystyle G_{r}} , the algebra takes the form where c {\displaystyle c}

855-493: A model containing only bosons . Shortly after their second paper on this topic, they realized that their model can be combined with Ramond's fermionic model, which they successfully did to give rise to the Ramond–Neveu–Schwarz (RNS) model, referred to at the time as the dual pion model. This work was done with only hadronic physics in mind with no reference to strings, until 1974 when Stanley Mandelstam reinterpreted

950-427: A nonvanishing scalar potential. While the N = 2 {\displaystyle {\mathcal {N}}=2} supersymmetry transformations appear to be realised, they are actually formally broken since the theory corresponds to a D8-brane background. A closely related theory is Howe–Lambert–West supergravity which is another massive deformation of type IIA supergravity, but one that can only be described at

1045-634: A number of these sectors can give rise to consistent tachyon-free theories. In particular, the RNS model gives rise to type IIA and type IIB string theory for closed strings, while combining the open string with a modified version of the IIB string gives rise to type I string theory. Starting instead from a ( 1 , 0 ) {\displaystyle (1,0)} supergravity action gives rise to heterotic string theories. One way to classify all possible string theories that can be constructed using this formalism

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1140-455: A pair of Majorana–Weyl spinors of opposite chiralities ψ μ = ψ μ + + ψ μ − {\displaystyle \psi _{\mu }=\psi _{\mu }^{+}+\psi _{\mu }^{-}} and λ = λ + + λ − {\displaystyle \lambda =\lambda ^{+}+\lambda ^{-}} . Lastly, there

1235-402: A pair of commutating fields β {\displaystyle \beta } and γ {\displaystyle \gamma } with weight h β = 3 / 2 {\displaystyle h_{\beta }=3/2} and h γ = 1 / 2 {\displaystyle h_{\gamma }=1/2} . These have an action of

1330-1443: A physical interpretation as the coordinates of the string worldsheet embedded in spacetime, with μ {\displaystyle \mu } running over the number of spacetime dimensions. For superstring theory in flat spacetime consistency of the theory requires exactly ten dimensions. The partial derivatives are derivatives in complex coordinates ∂ = ∂ z {\displaystyle \partial =\partial _{z}} and ∂ ¯ = ∂ z ¯ {\displaystyle {\bar {\partial }}=\partial _{\bar {z}}} . Operators can be classified according to their behavior under rigid rescaling z ′ = ζ z {\displaystyle z'=\zeta z} . If they transform as O ′ ( z ′ , z ¯ ′ ) = ζ − h ζ ¯ − h ~ O ( z , z ¯ ) {\displaystyle {\mathcal {O}}'(z',{\bar {z}}')=\zeta ^{-h}{\bar {\zeta }}^{-{\tilde {h}}}{\mathcal {O}}(z,{\bar {z}})} they are said to have weight ( h , h ~ ) {\displaystyle (h,{\tilde {h}})} . The weights of

1425-506: A scalar dilaton Φ {\displaystyle \Phi } with two superpartner spinors —the dilatinos λ ± {\displaystyle \lambda ^{\pm }} , a 2- form spin-2 gauge field B μ ν {\displaystyle B_{\mu \nu }} often called the Kalb–Ramond field , a 1-form C n {\displaystyle C_{n}} and

1520-666: A string interpretation, whereby mesons behave as strings of finite length. In 1970 Pierre Ramond was working at Yale trying to extend the dual resonance models to include fermionic degrees of freedom through a generalization of the Dirac equation . This led him to constructing the first superalgebra , the Ramond superalgebra. At the same time, Andre Neveu and John Schwarz were working at Princeton to extend existing dual resonance models by adding to them anticommutating creation and annihilation operators . This originally gave rise to

1615-402: Is a scalar and the free field equation is given by d ⋆ F 10 = 0 {\displaystyle d\star F_{10}=0} , this scalar must be a constant. Such a field therefore has no propagating degrees of freedom , but does have an energy density associated to it. Working only with the bosonic sector, the ten-form can be included in supergravity by modifying

1710-517: Is automatically anomaly free since it is a non-chiral theory. RNS formalism#Ramond and Neveu–Schwarz sectors In string theory , the Ramond–Neveu–Schwarz (RNS) formalism is an approach to formulating superstrings in which the worldsheet has explicit superconformal invariance but spacetime supersymmetry is hidden, in contrast to the Green–Schwarz formalism where the latter

1805-421: Is by looking at the possible residual symmetry algebras that can arise. That is, gauge fixing does not always fully fix the entire gauge symmetry, but can instead leave behind some unfixed residual symmetry whose action keeps the gauge fixed action unchanged. The algebra corresponding to this residual symmetry is known as the constraint algebra . To give rise to a physical theory, this algebra must be imposed on

1900-538: Is compactified to acquire four-dimensional theories, this is often done at the level of the low-energy supergravity. Reduction of type IIA on a Calabi–Yau manifold yields an N = 2 {\displaystyle {\mathcal {N}}=2} theory in four dimensions, while reduction on a Calabi–Yau orientifold further breaks the symmetry down to give the phenomenologically viable four-dimensional N = 1 {\displaystyle {\mathcal {N}}=1} supergravity . Type IIA supergravity

1995-830: Is denoted by N = ( 1 , 1 ) {\displaystyle {\mathcal {N}}=(1,1)} and is known as type IIA supergravity. This theory contains a single multiplet , known as the ten-dimensional N = 2 {\displaystyle {\mathcal {N}}=2} nonchiral multiplet. The fields in this multiplet are ( g μ ν , C μ ν ρ , B μ ν , C μ , ψ μ , λ , ϕ ) {\displaystyle (g_{\mu \nu },C_{\mu \nu \rho },B_{\mu \nu },C_{\mu },\psi _{\mu },\lambda ,\phi )} , where g μ ν {\displaystyle g_{\mu \nu }}

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2090-583: Is described by type IIB supergravity in ten dimensions which is a chiral theory (left–right asymmetric) with (2,0) d =10 supersymmetry; the fact that the anomalies in this theory cancel is therefore nontrivial. In the 1990s it was realized that type IIB string theory with the string coupling constant g is equivalent to the same theory with the coupling 1/g . This equivalence is known as S-duality . Orientifold of type IIB string theory leads to type I string theory. The mathematical treatment of type IIB string theory belongs to algebraic geometry, specifically

2185-508: Is explicit. It was originally developed by Pierre Ramond , André Neveu and John Schwarz in the RNS model in 1971, which gives rise to type II string theories and can also give type I string theory . Heterotic string theories can also be acquired through this formalism by using a different worldsheet action. There are various ways to quantize the string within this framework including light-cone quantization , old canonical quantization, and BRST quantization . A consistent string theory

2280-401: Is formed by acting with r > 0 {\displaystyle r>0} modes at most once on these ground states. The Lorentz covariant , diffeomorphism invariant action for the fermionic superstring is found by coupling the bosonic and fermionic fields to two-dimensional supergravity, giving the action where e a m {\displaystyle e_{a}^{m}}

2375-719: Is formed using a ( 1 , 1 ) {\displaystyle (1,1)} superconformal field theory on the string worldsheet with an action of the form where ψ μ ( z ) {\displaystyle \psi ^{\mu }(z)} and ψ ~ μ ( z ¯ ) {\displaystyle {\tilde {\psi }}^{\mu }({\bar {z}})} are holomorphic and an antiholomorphic anticommutating fermionic fields and X μ ( z , z ¯ ) {\displaystyle X^{\mu }(z,{\bar {z}})} are bosonic fields. These bosonic fields have

2470-463: Is given by The presence of worldsheet supersymmetry gives rise to worldsheet supercurrents with the holomorphic supercurrent having weight ( 3 / 2 , 0 ) {\displaystyle (3/2,0)} and being given by Any holomorphic operator O ( z ) {\displaystyle {\mathcal {O}}(z)} with weight ( h , 0 ) {\displaystyle (h,0)} can be expanded out as

2565-590: Is inherited from the periodicity of the fermions. For open strings, the boundary condition requires that the surface term in the equations of motion vanishes which imposes the constraints Thus, there are only two sectors for open strings, the R sector and the NS sector. It is often convenient to combine the two fields into a single field with an extended range 0 ≤ σ 1 ≤ 2 π {\displaystyle 0\leq \sigma ^{1}\leq 2\pi } defined according to where now

2660-626: Is known as the Ramond (R) boundary condition and the antiperiodic case ( ν = 1 / 2 {\displaystyle \nu =1/2} ) is known as the Neveu–Schwarz (NS) boundary condition . This gives four possible ways of putting fermions on the closed string, giving rise to four sectors in the Hilbert space, the NS–NS, NS–R, R–NS, and R–R sectors. The periodicity of the supercurrents

2755-611: Is known as the Romans mass and it acts as a Lagrange multiplier for F 10 {\displaystyle F_{10}} . Often one integrates out this field strength tensor resulting in an action where M {\displaystyle M} acts as a mass term for the Kalb–Ramond field. Unlike in the regular type IIA theory, which has a vanishing scalar potential V ( ϕ ) = 0 {\displaystyle V(\phi )=0} , massive type IIA has

2850-686: Is only acquired if the spectrum of states is restricted through a procedure known as a GSO projection , with this projection being automatically present in the Green–Schwarz formalism. The discovery of the Veneziano amplitude describing the scattering of four mesons in 1968 launched the study of dual resonance models which generalized these scattering amplitudes to the scattering with any number of mesons. While these are S-matrix theories rather than quantum field theories , Yoichiro Nambu , Holger Bech Nielsen , and Leonard Susskind gave them

2945-908: Is only one copy. Closed strings are periodic in their spatial direction, a periodicity that must be respected by the fields living on the worldsheet. A Poincaré invariant theory must have periodic X μ ( σ 1 , σ 2 ) {\displaystyle X^{\mu }(\sigma _{1},\sigma _{2})} . For the fermionic fields, Lorentz invariance allows for two possible boundary condition ; periodic or antiperiodic boundary conditions ψ μ ( σ 1 + 2 π , σ 2 ) = ± ψ μ ( σ 1 , σ 2 ) {\displaystyle \psi ^{\mu }(\sigma _{1}+2\pi ,\sigma _{2})=\pm \psi ^{\mu }(\sigma _{1},\sigma _{2})} , with an analogous condition for

Type II string theory - Misplaced Pages Continue

3040-447: Is related to type IIB string theory by T-duality . Type IIA supergravity In supersymmetry , type IIA supergravity is the unique supergravity in ten dimensions with two supercharges of opposite chirality . It was first constructed in 1984 by a dimensional reduction of eleven-dimensional supergravity on a circle . The other supergravities in ten dimensions are type IIB supergravity , which has two supercharges of

3135-532: Is relevant in the low energy limit) is given by ( 8 v ⊕ 8 s ) ⊗ ( 8 v ⊕ 8 c ) {\textstyle (8_{v}\oplus 8_{s})\otimes (8_{v}\oplus 8_{c})} representation of SO(8) where 8 v {\displaystyle 8_{v}} is the irreducible vector representation, 8 c {\displaystyle 8_{c}} and 8 s {\displaystyle 8_{s}} are

3230-750: Is that type IIA theory only keeps sectors with e i π F = + 1 {\displaystyle e^{i\pi F}=+1} and e i π F ~ = ( − 1 ) α ~ {\displaystyle e^{i\pi {\tilde {F}}}=(-1)^{\tilde {\alpha }}} , while IIB theory only keeps sectors with e i π F = e i π F ~ = + 1 {\displaystyle e^{i\pi F}=e^{i\pi {\tilde {F}}}=+1} . Type I string theory can be constructed from type IIB theory that has gauged its worldsheet parity symmetry and has been combined with

3325-457: Is the central charge . The algebra is sometimes referred to as the Ramond algebra when r {\displaystyle r} , s {\displaystyle s} are integers and the Neveu–Schwarz algebra when they are half-integers. For closed strings there are two copies of this algebra, one for the holomorphic and one for the antiholomorphic side, while for open strings there

3420-403: Is the metric corresponding to the graviton , while the next three fields are the 3-, 2-, and 1-form gauge fields , with the 2-form being the Kalb–Ramond field . There is also a Majorana gravitino ψ μ {\displaystyle \psi _{\mu }} and a Majorana spinor λ {\displaystyle \lambda } , both of which decompose into

3515-504: Is the additional condition that b 0 | ψ ⟩ = L 0 | ψ ⟩ = 0 {\displaystyle b_{0}|\psi \rangle =L_{0}|\psi \rangle =0} , and for the R sector states β 0 | ψ ⟩ = G 0 | ψ ⟩ = 0 {\displaystyle \beta _{0}|\psi \rangle =G_{0}|\psi \rangle =0} . This condition truncates

3610-429: Is the corresponding charge associated with this current The physical spectrum is the set of BRST cohomology classes . This is the set of states | ψ ⟩ {\displaystyle |\psi \rangle } that are annihilated by the charge Q B | ψ ⟩ = 0 {\displaystyle Q_{B}|\psi \rangle =0} , with all states differing by

3705-405: Is the projection of the Hilbert space onto the subset of sectors that are consistent under these three conditions. One set of consistent theories that results from the projection are type 0 string theories , although these are not tachyon-free. The other set of consistent theories are type II string theories which are tachyon-free, consisting of the sectors A concise way to summarize these sectors

3800-430: Is the two-dimensional vielbein and χ a {\displaystyle \chi _{a}} is the corresponding gravitino . This has the following symmetries: The gauge symmetries of this action are diffeomorphism symmetry, Weyl symmetry, and local supersymmetry. To quantize the action, these symmetries must be gauge fixed, which is usually done through the superconformal gauge in which e

3895-462: The SO ( 8 ) {\displaystyle {\text{SO}}(8)} representation, with this being the direct product of the left-moving and right-moving representations, which decomposes into a sum over irreducible representations . There are no states where NS− is matched with NS+, R− or R+ since then the level matching condition is not meet, so the closed string theory has a single tachyon coming from

Type II string theory - Misplaced Pages Continue

3990-438: The ( 1 , 1 ) {\displaystyle (1,1)} superconformal algebra, the other allowed algebras are the ( 1 , 0 ) {\displaystyle (1,0)} , ( 1 , 2 ) {\displaystyle (1,2)} and ( 0 , 2 ) {\displaystyle (0,2)} superconformal algebras. The first of these gives rise to heterotic string theories, while

4085-414: The ( 1 , 1 ) {\displaystyle (1,1)} supergravity action down to the RNS action leaves behind a residual ( 1 , 1 ) {\displaystyle (1,1)} superconformal algebra . Physical conditions such as unitarity and a positive number of spatial dimensions limits the number of admissible constraint algebras. Besides the conformal algebra and

4180-453: The Hilbert space by projecting out unwanted states. Physical states are the ones that are annihilated by the action of this algebra on those states. For example, in bosonic string theory the original diffeomorphism × {\displaystyle \times } Weyl symmetry breaks down to a residual conformal symmetry , giving the conformal algebra whose generator is

4275-506: The Killing spinor equations and finding the supersymmetric ground states of the theory since these require that the fermionic variations vanish. Since type IIA supergravity has p-form field strengths of even dimensions, it also admits a nine-form gauge field F 10 = d C 9 {\displaystyle F_{10}=dC_{9}} . But since ⋆ F 10 {\displaystyle \star F_{10}}

4370-464: The Rarita–Schwinger action and Dirac action , respectively. The second line has the kinetic terms for the 1-form and 3-form gauge fields as well as a Chern–Simons term . The last line contains the cubic interaction terms between two fermions and a boson . The supersymmetry variations that leave the action invariant are given up to three-fermion terms by They are useful for constructing

4465-419: The chirality matrix γ ∗ {\displaystyle \gamma _{*}} behaving as just another γ {\displaystyle \gamma } matrix, except with no index. Going only up to five-index matrices, since the rest are equivalent up to Poincare duality , yields the set of central charges described by the above algebra . The various central charges in

4560-469: The deformation theory of complex structures originally studied by Kunihiko Kodaira and Donald C. Spencer . In 1997 Juan Maldacena gave some arguments indicating that type IIB string theory is equivalent to N = 4 supersymmetric Yang–Mills theory in the 't Hooft limit ; it was the first suggestion concerning the AdS/CFT correspondence . In the late 1980s, it was realized that type IIA string theory

4655-472: The path integral . The last approach starts from the Euclidean partition function where S {\displaystyle S} is the worldsheet action with some gauge symmetry group G {\displaystyle G} that represents an overcounting of the physically distinct configurations of the fields that the action depends on. This overcounting is eliminated by dividing by

4750-559: The stress-energy tensor T a b {\displaystyle T^{ab}} . The physical states | ψ ⟩ {\displaystyle |\psi \rangle } , | ψ ′ ⟩ {\displaystyle |\psi '\rangle } are then those for which ⟨ ψ | T a b | ψ ′ ⟩ = 0 {\displaystyle \langle \psi |T^{ab}|\psi '\rangle =0} . Similarly, gauge fixing

4845-540: The vacuum expectation value of e ϕ {\displaystyle e^{\phi }} , while the string length l s = α ′ {\displaystyle l_{s}={\sqrt {\alpha '}}} is related to the gravitational coupling constant through 2 κ 2 = ( 2 π ) 7 α ′ 4 {\displaystyle 2\kappa ^{2}=(2\pi )^{7}{\alpha '}^{4}} . When string theory

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4940-515: The vacuum state . Since all annihilation modes for the NS sector have r > 0 {\displaystyle r>0} , it follows that its spectrum has a unique vacuum state | 0 ⟩ NS {\displaystyle |0\rangle _{\text{NS}}} that is annihilated by all the modes The r < 0 {\displaystyle r<0} modes act as raising operators, and since they are anticommuting, each one can be acted on at most once, giving

5035-494: The 10D Majorana gravitino and the Majorana fermion ψ A ′ ∼ ( ψ a , λ ) {\displaystyle \psi _{A}'\sim (\psi _{a},\lambda )} , although the exact identification is given by where this is chosen to make the supersymmetry transformations simpler. The ten-dimensional supersymmetry variations can also be directly acquired from

5130-1118: The 11D metric decomposes into the 10D metric, the 1-form, and the dilaton as Meanwhile, the 11D 3-form decomposes into the 10D 3-form A μ ν ρ ′ → C μ ν ρ {\displaystyle A_{\mu \nu \rho }'\rightarrow C_{\mu \nu \rho }} and the 10D 2-form A μ ν 11 ′ → B μ ν {\displaystyle A_{\mu \nu 11}'\rightarrow B_{\mu \nu }} . The ten-dimensional modified field strength tensor F ~ 4 {\displaystyle {\tilde {F}}_{4}} directly arises in this compactification from F μ ν ρ σ ′ = e 4 ϕ / 3 F ~ μ ν ρ σ {\displaystyle F'_{\mu \nu \rho \sigma }=e^{4\phi /3}{\tilde {F}}_{\mu \nu \rho \sigma }} . Dimensional reduction of

5225-641: The 3-form and 1-form fields correspond to the RR states of type IIA string theory. Corrections to the type IIA supergravity action come in two types, quantum corrections in powers of the string coupling g s {\displaystyle g_{s}} , and curvature corrections in powers of α ′ {\displaystyle \alpha '} . Such corrections often play an important role in type IIA string phenomenology . The type IIA superstring coupling constant g s {\displaystyle g_{s}} corresponds to

5320-454: The D-branes (which are R {\displaystyle {\text{R}}} R {\displaystyle {\text{R}}} charged) and F1 string and NS5 brane among other objects. The mathematical treatment of type IIA string theory belongs to symplectic topology and algebraic geometry , particularly Gromov–Witten invariants . At low energies, type IIB string theory

5415-431: The NS sector spectrum. The R sector has zero modes ψ 0 μ {\displaystyle \psi _{0}^{\mu }} which map a vacuum state into another vacuum state. Under the rescaling γ μ = 2 − 1 / 2 ψ 0 μ {\displaystyle \gamma ^{\mu }=2^{-1/2}\psi _{0}^{\mu }} ,

5510-689: The NSNS 1-brane, which is equivalent to the fundamental string , while Z μ ν ρ σ δ {\displaystyle Z_{\mu \nu \rho \sigma \delta }} corresponds to the NS5-brane . The type IIA supergravity action is given up to four-fermion terms by Here H = d B {\displaystyle H=dB} and F p + 1 = d C p {\displaystyle F_{p+1}=dC_{p}} where p {\displaystyle p} corresponds to

5605-525: The NS−NS− sector. The naive RNS string Hilbert space does not give rise to a consistent string theory. There are three conditions that must be satisfied for the theory to be consistent. First, the vertex operators of the theory have to be mutually local, meaning that their OPEs have no branch cuts. Secondly, the OPEs must also closed . Lastly, the one-loop amplitudes must be modular invariant . The GSO projection

5700-559: The Neveu–Schwarz ground states are defined according to BRST quantization of the theory requires the construction of the BRST current where c {\displaystyle c} and γ {\displaystyle \gamma } are the ghosts and T B , F m , g {\displaystyle T_{B,F}^{m,g}} are the matter and ghost stress tensors and supercurrents. The BRST charge Q B {\displaystyle Q_{B}}

5795-432: The R and NS sectors correspond to a periodicity or antiperiodicity condition on this extended field. The Hilbert space of the R sector and NS sector are determined by considering the modes ψ r μ {\displaystyle \psi _{r}^{\mu }} and ψ ~ r μ {\displaystyle {\tilde {\psi }}_{r}^{\mu }} of

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5890-406: The RNS action together with a ghost action describing holomorphic and antiholomorphic ghosts that are necessary to eliminate the unphysical temporal excitations of the fields. The physical states of this theory split up into a number of sectors depending on the periodicity condition of the fermionic fields . The full theory is inconsistent and contains an unphysical tachyon, however projecting out

5985-489: The RNS model as a model for spinning strings. Joël Scherk and John Schwartz were the first to suggest that it may describe elementary particles rather than just hadrons when they showed that the spin -2 particle of the model behaves as a graviton . At the time, the main issue with the RNS model was that it contained a tachyon as the lowest energy state . It was only in 1976 with the introduction of GSO projection by Ferdinando Gliozzi , Joël Scherk, and David Olive that

6080-484: The action has the Einstein–Hilbert action , the dilaton kinetic term , the 2-form B μ ν {\displaystyle B_{\mu \nu }} field strength tensor. It also contains the kinetic terms for the gravitino ψ μ {\displaystyle \psi _{\mu }} and spinor λ {\displaystyle \lambda } , described by

6175-759: The action. Performing this gauge fixing through the Faddeev–Popov procedure leaves behind the RNS action and a BRST ghost action S → S RNS + S g {\displaystyle S\rightarrow S_{\text{RNS}}+S_{g}} . There are holomorphic and antiholomorphic ghosts in the gauge fixed superstring action. On the holomorphic side are a pair of anticommutating b {\displaystyle b} and c {\displaystyle c} fields with weight h b = 2 {\displaystyle h_{b}=2} and h c = − 1 {\displaystyle h_{c}=-1} , along with

6270-546: The algebra correspond to different BPS states allowed by the theory. In particular, the Z {\displaystyle Z} , Z μ ν {\displaystyle Z_{\mu \nu }} and Z μ ν ρ σ {\displaystyle Z_{\mu \nu \rho \sigma }} correspond to the D0, D2, and D4 branes . The Z μ {\displaystyle Z_{\mu }} corresponds to

6365-595: The anticommutating ghosts fields being periodic, while the commutating ghost fields being periodic in the R sector and antiperiodic in the NS sector. The modes satisfy the (anti)commutation relations { b m , c n } = δ n , − m {\displaystyle \{b_{m},c_{n}\}=\delta _{n,-m}} and [ γ r , β s ] = δ r , − s {\displaystyle [\gamma _{r},\beta _{s}]=\delta _{r,-s}} . The Ramond and

6460-480: The anticommutating relation for these becomes the Dirac algebra , implying that the ground state of the R spectrum forms a representation of this algebra. In ten dimensions this is a Dirac spinor , a 32 dimensional representation which can be reduced down to two Weyl representations 32 = 16 + 16 ′ {\displaystyle {\text{32}}={\text{16}}+{\text{16}}'} distinguished by their chirality . The R sector spectrum

6555-471: The antiholomorphic fields. This can concisely be summarized as where ν {\displaystyle \nu } and ν ~ {\displaystyle {\tilde {\nu }}} are independent of each other and are either 0 {\displaystyle 0} or 1 / 2 {\displaystyle 1/2} . The periodic case ( ν = 0 {\displaystyle \nu =0} )

6650-450: The context of string theory, the bosonic fields in the first multiplet consists of NSNS fields while the bosonic fields are all RR fields . The fermionic fields are meanwhile in the NSR sector. The superalgebra for N = ( 1 , 1 ) {\displaystyle {\mathcal {N}}=(1,1)} supersymmetry is given by where all terms on the right-hand side besides

6745-402: The coordinates ( z , z ¯ ) {\displaystyle (z,{\bar {z}})} defined by z = e − i w {\displaystyle z=e^{-iw}} . For the latter, a string at a given point in time is a circle around the origin in the complex plane , with smaller radii corresponding to earlier times. The RNS model

6840-452: The dimension of the irreducible representation and equivalently the number of components of the corresponding fields. The various massless fields obtained are the graviton G μ ν {\displaystyle G_{\mu \nu }} with two superpartner gravitinos ψ m ± {\displaystyle \psi _{m}^{\pm }} which gives rise to local spacetime supersymmetry,

6935-538: The eleven-dimensional ones by setting ϵ ′ = e − ϕ / 6 ϵ {\displaystyle \epsilon '=e^{-\phi /6}\epsilon } . The low-energy effective field theory of type IIA string theory is given by type IIA supergravity. The fields correspond to the different massless excitations of the string, with the metric, 2-form B {\displaystyle B} , and dilaton being NSNS states that are found in all string theories, while

7030-461: The fermionic fields. Since in the R sector the powers r {\displaystyle r} are integers, this sector has a branch cut while the NS sector has half-integer r {\displaystyle r} and so no branch cut. The operator product expansion (OPE) of the fermionic theory translate to anticommutation relations for the modes given by The states in the Hilbert space can then be built up by acting with these modes on

7125-435: The fermions must generally be done in terms of the flat coordinates ψ A ′ = e A ′ M ψ M {\displaystyle \psi _{A}'=e_{A}'^{M}\psi _{M}} , where e ′ A M {\displaystyle {e'}_{A}^{M}} is the 11D vielbein . In that case the 11D Majorana graviton decomposes into

7220-430: The first consistent tachyon-free string theories were constructed. The RNS formalism is an approach to quantizing a string by working with the string worldsheet embedded in spacetime with both bosonic and fermionic fields on the worldsheet. There are a number of different approaches for quantizing the string in this formalism. The main ones are old covariant quantization, light-cone quantization, and BRST quantization via

7315-413: The first one are the central charges allowed by the theory. Here Q α {\displaystyle Q_{\alpha }} are the spinor components of the Majorana supercharges while C {\displaystyle C} is the charge conjugation operator . Since the anticommutator is symmetric, the only matrices allowed on the right-hand side are ones that are symmetric in

7410-406: The form with a similar action for the antiholomorphic ghosts. This action gives rise to additional ghost contributions to the overall stress energy tensor T B g {\displaystyle T_{B}^{g}} and supercurrents of the theory T F g {\displaystyle T_{F}^{g}} . The ghost mode expansion is determined by their weights, with

7505-449: The ghost spectrum for kinematic reasons. It is convenient to look at the lowest energy states of this theory. Introducing the fermion number operator F {\displaystyle F} allows for the NS and R sectors to be further subdivided into NS−, NS+, R−, and R+ sectors, where the sign denotes the sign of e i π F = ± 1 {\displaystyle e^{i\pi F}=\pm 1} for

7600-1925: The irreducible representations with odd and even eigenvalues of the fermionic parity operator often called co-spinor and spinor representations. These three representations enjoy a triality symmetry which is evident from its Dynkin diagram . The four sectors of the massless spectrum after GSO projection and decomposition into irreducible representations are NS-NS :   8 v ⊗ 8 v = 1 ⊕ 28 ⊕ 35 = Φ ⊕ B μ ν ⊕ G μ ν {\displaystyle {\text{NS-NS}}:~8_{v}\otimes 8_{v}=1\oplus 28\oplus 35=\Phi \oplus B_{\mu \nu }\oplus G_{\mu \nu }} NS-R : 8 v ⊗ 8 c = 8 s ⊕ 56 c = λ + ⊕ ψ m − {\displaystyle {\text{NS-R}}:8_{v}\otimes 8_{c}=8_{s}\oplus 56_{c}=\lambda ^{+}\oplus \psi _{m}^{-}} R-NS : 8 c ⊗ 8 s = 8 s ⊕ 56 s = λ − ⊕ ψ m + {\displaystyle {\text{R-NS}}:8_{c}\otimes 8_{s}=8_{s}\oplus 56_{s}=\lambda ^{-}\oplus \psi _{m}^{+}} R-R : 8 s ⊗ 8 c = 8 v ⊕ 56 t = C n ⊕ C n m p {\displaystyle {\text{R-R}}:8_{s}\otimes 8_{c}=8_{v}\oplus 56_{t}=C_{n}\oplus C_{nmp}} where R {\displaystyle {\text{R}}} and NS {\displaystyle {\text{NS}}} stands for Ramond and Neveu–Schwarz sectors respectively. The numbers denote

7695-623: The level of the equations of motion . It is acquired by a compactification of eleven-dimensional MM theory on a circle. Compactification of eleven-dimensional supergravity on a circle and keeping only the zero Fourier modes that are independent of the compact coordinates results in type IIA supergravity. For eleven-dimensional supergravity with the graviton, gravitino, and a 3-form gauge field denoted by ( g M N ′ , ψ M ′ , A M N R ′ ) {\displaystyle (g_{MN}',\psi _{M}',A_{MNR}')} , then

7790-400: The limit of type IIA string theory in which the string coupling goes to infinity becomes a new 11-dimensional theory called M-theory . Consequently the low energy type IIA supergravity theory can also be derived from the unique maximal supergravity theory in 11 dimensions (low energy version of M-theory) via a dimensional reduction . The content of the massless sector of the theory (which

7885-623: The original action to get massive type IIA supergravity where S ~ I I A {\displaystyle {\tilde {S}}_{IIA}} is equivalent to the original type IIA supergravity up to the replacement of F 2 → F 2 + M B {\displaystyle F_{2}\rightarrow F_{2}+MB} and F 4 → F 4 + 1 2 M B ∧ B {\displaystyle F_{4}\rightarrow F_{4}+{\tfrac {1}{2}}MB\wedge B} . Here M {\displaystyle M}

7980-709: The other two give consistent but less physically interesting theories in low dimensions. Topological string theory is not found in this classification because for it the spin-statistics theorem does not hold in the conformal gauge which was required in the full argument. A string worldsheet is a two dimensional surface which can be parameterized by two coordinates ( σ 1 , σ 2 ) {\displaystyle (\sigma _{1},\sigma _{2})} where σ 2 {\displaystyle \sigma _{2}} describes Euclidean time while σ 1 {\displaystyle \sigma _{1}} parameterize

8075-427: The possible massless and tachyonic states of the RNS string. For the closed strings, the physical states are the various combinations of these four sectors as left and right-moving sectors. The resulting string has a mass-shell condition of where N {\displaystyle N} is the level, counting the creation operators used to create the state. The resulting states can again be classified according to

8170-520: The same chirality, and type I supergravity , which has a single supercharge. In 1986 a deformation of the theory was discovered which gives mass to one of the fields and is known as massive type IIA supergravity. Type IIA supergravity plays a very important role in string theory as it is the low-energy limit of type IIA string theory . After supergravity was discovered in 1976 with pure 4D N = 1 {\displaystyle {\mathcal {N}}=1} supergravity , significant effort

8265-421: The smallest spinorial representations in ten dimensions are Majorana – Weyl spinors , the supercharges come in two types Q ± {\displaystyle Q^{\pm }} depending on their chirality, giving three possible supergravity theories. The N = 2 {\displaystyle {\mathcal {N}}=2} theory formed using two supercharges of opposite chiralities

8360-475: The spinor indices α {\displaystyle \alpha } , β {\displaystyle \beta } . In ten dimensions γ μ 1 ⋯ μ p C {\displaystyle \gamma ^{\mu _{1}\cdots \mu _{p}}C} is symmetric only for p = 1 , 2 {\displaystyle p=1,2} modulo 4 {\displaystyle 4} , with

8455-468: The states. These states are classified by what spin representation of the SO ( 8 ) {\displaystyle {\text{SO}}(8)} group they belong to, which is the rotation subgroup of the ten dimensional Lorentz group SO ( 1 , 9 ) ⊃ SO ( 1 , 1 ) SO ( 8 ) {\displaystyle {\text{SO}}(1,9)\supset {\text{SO}}(1,1){\text{SO}}(8)} . In particular,

8550-712: The string at an instance in time. For closed strings σ 1 ∼ σ 1 + 2 π {\displaystyle \sigma _{1}\sim \sigma _{1}+2\pi } while for open strings σ 1 ∈ [ 0 , π ] {\displaystyle \sigma _{1}\in [0,\pi ]} . Two other coordinate systems are often employed, these being complex coordinates ( w , w ¯ ) {\displaystyle (w,{\bar {w}})} defined by w = σ 1 + i σ 2 {\displaystyle w=\sigma _{1}+i\sigma _{2}} or

8645-501: The tachyonic NS− is a singlet while the NS+ state is a vector denoted by 8 v {\displaystyle 8_{v}} . The R+ sector Majorana–Weyl spinors belong to the 8 {\displaystyle 8} representation while the R− belong to the 8 ′ {\displaystyle 8'} representation. For open strings, the NS+, NS−, R+, and R− form

8740-433: The two fermionic fields are ( 1 / 2 , 0 ) {\displaystyle (1/2,0)} and ( 0 , 1 / 2 ) {\displaystyle (0,1/2)} while that of the bosonic fields is ( 0 , 0 ) {\displaystyle (0,0)} . The holomorphic stress-energy tensor has weight ( 2 , 0 ) {\displaystyle (2,0)} and

8835-488: The volume of the gauge group V G {\displaystyle V_{G}} . BRST quantization proceeds by gauge fixing the path integral via the Fadeev–Popov procedure , which gives rise to a ghost action in addition to the now gauge fixed action. The RNS model originates from using the ( 1 , 1 ) {\displaystyle (1,1)} supergravity action which upon gauge fixing gives

8930-514: Was devoted to understanding other possible supergravities that can exist with various numbers of supercharges and in various dimensions. The discovery of eleven-dimensional supergravity in 1978 led to the derivation of many lower dimensional supergravities through dimensional reduction of this theory. Using this technique, type IIA supergravity was first constructed in 1984 by three different groups, by F. Giani and M. Pernici, by I.C.G. Campbell and P. West , and by M. Huq and M. A. Namazie. In 1986 it

9025-619: Was noticed by L. Romans that there exists a massive deformation of the theory. Type IIA supergravity has since been extensively used to study the low-energy behaviour of type IIA string theory. The terminology of type IIA, type IIB, and type I was coined by J. Schwarz , originally to refer to the three string theories that were known of in 1982. Ten dimensions admits both N = 1 {\displaystyle {\mathcal {N}}=1} and N = 2 {\displaystyle {\mathcal {N}}=2} supergravity, depending on whether there are one or two supercharges. Since

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