Advanced String Theory
Heterotic Strings
Okay, yeah, this isn't technically math, but it's close enough, and I really need the time. It will expand a bunch on the stuff we've seen in the introduction. A very quick recap of the introduction to string theory is as such:
The bosonic string is fixed in its gauge onto the flat worldsheet, parametrized as (τ, σ), where τ the axis the worldsheet moves through time on, and σ is its phase. The string action is written
where each sector has some conformal anomaly. For the ghost sector it's -26, and for the fields it's +1. The criticality of the action is described as the total conformal invariance (-25). As such, the dimension of the problem is 26, and μ is indexed from 0 to 25. The equations of motion lead to a definition for the fields X, which gives the level matching conditions for excitation numbers N and normal-ordering constant a.
The normal-ordering constants for these fields are equal to 1/24, and for the ghost, it will have to multiply that with the dimension, with the degrees of freedom removed. The ghost normal-ordering constant is 1.
The superstring replaces the fields X with majorana-weyl terms, so that
with c = -15. The differentiation of the majorana-weyl terms vanish. It splits the action into a left and right-moving part, along with symmetry as the boundary condition. From the normal ordering emerge the two different forms of sectors: NS (a = 1/2) and R (a = 0). The spectra of these sectors consist of the tachyon, massless and massive part (NS), or the massless part only (R). These need to be combined into a consistent theory. This is done through the GSO-projection, which tries to find a valid combination among the type-IIB/A options.
The heterotic string consists of different left/right-moving closed string theories. The bosonized idea has a bosonic string moving left, and a superstring moving right. These of course come with their own ghost-systems. This falls into the N = (0, 1) supersymmetry on the world-sheet. The fermionic description is equivalent. Either way, the 16 extra dimensions of the fields to arrive at the desired 10-dimensional string theory. It fixes the action
with the global symmetry of S: SO(1, 9) ⨉ SO(32) where the latter are the internal global rotational symmetries. The boundary conditions for λ are a form of 2π-periodicity used to classify the GSO-projections. It leaves 9 inequivalent heterotic theories in 10 dimensions. It's either SO(32), E(8) ⨉ E(8), SO(16) ⨉ SO(16) and 6 unstable theories. The global SO(32) WS symmetry is a gauge symmetry in spacetime. It'll couple the super-gravity to the SO(32) symmetry. The E(8) ⨉ E(8) symmetry has 248 dimensions. The classification of simple Lie-groups comes in a 4 series (A, D, B, C) and 5 outliers where the E-outliers are considered exceptional:
Effective Action & Green-Schwarz
The low-energy spectrum for typeII-A/B superstrings with 10d super-gravity area given by 32 supercharges. It's usually denoted as IIA: (16 + 16') and IIB: (16 + 16), to better showcase their T-duality. TypeII-A supergravity is related to 11d supergravity through S¹ compactification from ℝ(1, 10) to ℝ(1, 9). N = 1 supergravity in ℝ(1, 10) has the highest amount of spacetime symmetry which doesn't require massless states with spin > 2. This is interesting, because such states are incompatible with flat ℝ(1, 3) with Lorentz covariance. Compactification on a torus to 4d preserves all supercharges. Together with the previous argument, this caps the possible supersymmetry to 32 real supercharges at the most.
The matter-content of 11d supergravity is derived from the gravity multiplet, consisting of 2⁸ = 256 states, split similarly to the supercharges into equal, but dual halves. The bosonic gravity multiplet is described through a metric gravity tensor with a 3-form gauge potential. The tensor contributes 9P2 + 0 -1 = 44 states, and the potential contributes the remaining 9P3 = 84. The fermionic gravity multiplet is simply contained in the gravitino, hence gets its own particle. This shouldn't immediately be a surprise, since most pure fermions are massless. The gravitino is described by two indices, describing a vector-spinor of SO(9) with its trace. The trace contributes 16 degrees of freedom, so it has 9*16 - 16 = 128 states. The supersymmetry determines the action with bosonic part:
This features the gauge symmetries:
The general kinetic term features a topological component called the "Chern-Simons term" which is invariant under said gauge-symmetries. From this invariance, we can generalize
The dimensional reduction to 10d into ℝ(1,9) ⨉ S(1) with some radius R is done through construction of a new gravitational tensor. The indices M, N run from 0 to 10 and the indices ν, μ run from 0 to 9. The rest follows from the volume elements.
The bosonic field content is straight-forwardly described by the massless bosonic content of type-IIA.
The fermionic field content is a touch more complicated to describe and arrive at. The 128 elements split into the T-dual sums with N = 11 benefit from redefinition through the gravity tensor. This gives expressions that can be plugged into the type-IIA action straight-forwardly, which at the moment consists of the Neveu-Schwarz term, Ramond term, and Chern-Simons term.
For 11d, the gauge invariance and diffeomorphism properties can be used to identify the integral scaling factor.
The electro-magnetic duality acts on gauge invariant field strengths for magnetic duals for a formulation exhausting the field content derived from 11d supergravity.
The non-dynamical contents follow from duality with type IIB, but doesn't need any propagating degrees of freedom. It's instead associated with an energy density. It describes a massive type IIA supergravity.
Type IIB Supergravity and Type I Heterotic Strings
In the type IIB supersymmetry, the NSNS sectors are identical to the ones in the type IIA sector, while the RR sector sees the F(n)-forms (n = {1, 3, 5}) as derivatives of C(n-1)-forms. For Fermions, this yields an N = (2, 0) supergravity, inheriting self-duality of F(5) on the level of e.o.m. and SL(2, ℝ) invariance of the classical action. Because of this theory only including the NSNS and RR modes, it's invariant under exchange of left/right movers. These qualities can be used in the characterization of the type I heterotic string.
Type I string theory could be considered type IIB/Ω ⊕ 32 D9-branes, where Ω is the worldsheet-parity transformation σ to l - σ. The quotient of the worldsheet parity yields an orientifold. Using closed oscillators on the string field the the quotient defines an unoriented theory, in which the RR mode of the bosonic sector behaves fermionically under worldsheet parity. The C(0) + C(4) form is Ω-odd. Since the B-tensor has index-exchange parity, it's also Ω-odd and has to be projected out of the NSNS-mode.
This means, the bosonic type I string sees the C(0) and C(4)-form projected out of the RR mode, and the B-tensor out of the NSNS mode. An open bosonic string is represented by the vector boson A. In the fermionic sector, only those forms remain that adhere to this symmetry. Through the diagonal combination of the 2 counter-propagating copies of the string shows that that this doesn't pose a problem. The fermionic string doesn't change from the type IIB construction, and the string action is of course still the sum of closed and open action. The Ω-projection is consistent at interaction-level.
via the Chern-Simons forms. ω is the spin connection, which transforms similarly to the vector boson.
Making the string heterotic reintroduces the B-tensor, but removes the entire RR-contents. It's otherwise best written using vectors in E₈, noted down using H-forms.
Under observation of 1-loop amplitudes shows that type I strings can't be closed due to IR divergences of the 1-loop partition function. This divergence can be cancelled if the 10D open string's degrees of freedoms are adapted to the system. The massless level spans a sypersymmetry between the vector boson A and the gaugino Ψ, forming a multiplet that needs to be included 496 times to cancel the divergence. 10D open string inclusions automatically imply D9-brane inclusion. They are oriented and carry gauge groups U(N). Extra constraints given by the worldsheet parity projection act on the CP-factors, leading to the bosonic theory. The absence of IR divergences requires adding 32 D9-branes with Ω-action leading to the gauge group SO(32). This is the only 10D brane configuration respecting 10D Poincare invariance.
Anomalies in Gauge Theory
A massive action coupling to an external gauge field, assumed to be non-dynamical, with Euclidean quantum effective action
The gauge anomaly for non-invariant Γ[A] under a gauge transformations of A or φ. Such a transformation is
This gauge anomaly signals a non-conservation of J at the operator level. Generalized to theories with gravity, viewed as a gauge theory with connection ω, so that R = dω + [ω, ω] and ω → ω + Dθ. The resulting non-local polynomial Γ then has perturbative contributions at 1-loop only. It's characterized through the n-point loop functions known from QFT (or QFT II). With contributions of the (anti-)chiral Weyl fermions, spin-3/2 fields (gravitinos) and (anti-)self-dual tensors F(d/2), d = 4k+2. No anomalies are possible if d is odd.
If a gauge anomaly satisfies the "Weiss-Zumino consistency" (WZC)
it's considered a "consistent anomaly". This is true for Γ[A]. Up to normalization, the WZC determines the form of the anomaly up to "trivial anomalies", those that can be cancelled by locally gauge variant counterterms in the actions. The general form of Γ[A], as determined by WZC is
with Ω₂ₙ₊₂ given compactly by characteristic classes. In 10 dimensions with N=1 SUGRA coupled to SYM with some gauge group G, the chiral field content consists of gaugino (8), gravitino (56) and dilatino (8'). If the space is Minkowski, the induced gauge-gravitational anomaly is
descending from Ω₁₂. Through explicit computation, one finds that the anomaly is non-trivial, but a constraint may be imposed to make sure it vanishes. To fulfil this constraint, introduce the Green-Schwarz counterterms to the action.
This constraint is possible for dim G = 496, and if Tr F⁶ is expressible by lower traces in such a way, that the fist term in Ω₁₂ vanishes. This is only true for E₈ ⨉ E₈, SO(32), U(1)⁴⁹⁶, E₈ ⨉ U(1)²⁴⁸. The first two already show up in 10d string theory, and string computation at 1-loop sees type I and heterotic strings that are free of gravitational gauge-anomalies. The non-physical degrees of freedom decouple in interactions, and the result hinges on either "modular invariance" (heterotic) or "tadpole cancellation" (type I). The effective action contains a counterterm
The other two choices are inconsistent as SUGRA theories.
D-Branes as BPS objects
The choice of vacuum of a Dp-brane can have spontaneous partial breaking of Poincare invariance, which stems from the anticommutator of the SUSY generators. The super-Poincare generators are associated with the Noether charges associated with global worldsheet currents (spacetime momentum P, spacetime angular momentum J).
u is the spacetime spin polarization of the spinor, φ is the worldsheet field participating in bosonisation of super-ghosts. The right/left-movers are defined as
The OPE on the worldsheet reproduce the spacetime (anti-)commutator relations of the SUSY algebra and the precise SUSY generator action. The necessary condition for SUSY is that the left/right-movers can be transformed into one another using a parity projection operator. Type IIA has Dp-branes with even p, and Type IIB has Dp-branes with odd p for preserving SUSY.
Dp-branes preserve a modified form of their supercharges. One can think of this as a form of energy conservation. For a parallel Dp and Dp' brane, the preserved SUSY is a solution to the spinor equation using the parity operators for the Dp and Dp' brane:
A Dp-Dp' system can only be SUSY for p - p' ∈ 4ℤ, due to the properties of the spacetime angular momentum and its effect on the number of dimensions that are fixed to the system. D6p-branes rotated at an angle function essentially the same, though the projection operators are reduced to products of certain β-forms. The common SUSY corresponds to
If Φ is generic, then all SUSY is broken. If the sum of all three fulfills the condition, then 1/4 of the 16 SUSYs are preserved, so that in 4d, there are 4 real supercharges. If only Φ₂ + Φ₃ = 0 mod 2π, then the Φ₄ = 0, and 8 real supercharges are preserved in 4d. Branes at angles in n complex directions preserving some SUSY implies that the underlying manifold has SU(n) symmetry.
BPS property of D-Branes
First regard general BPS states. Extended N = k SUSY algebra cointains the central charges Z as pairwise commuting generators of the algebra. Hence, the associated charges must be constant on all elements of the irreducible representation Z=Z(R). A Bogomolnyi-Prasad-Sommerfield bound (BPS) state satisfies that the state of mass M and central charge Z are equal (generally, M ≥ Z). BPS states are annihilated by the one of their negative charges, meaning they're invariant under half their SUSY in N=2. BPS states construct an irreducible representation under the remaining SUSY subalgebra, so they organize in short multiplets. A state that is BPS at weak coupling, sees M and Z receive corrections at increasing coupling. The number of states inside a multiplet can't jump as g varies continuously, so short multiplets can't turn into long multiplets. This preserves the BPS property up to strong coupling. This means that M = Z at all g, if they are equal at weak (or no) coupling.
Consider type II SuGra in Minkowski-metric and right/left-moving zero-modes in absence of string winding modes. The compactification along compact points at S1-topology introduces charges with respect to U(1) gauge-fields A and B. The string couples using currents, fixing the charges
This allows for replacing the anti-commutation relation with
From the BPS property then, mass maps onto tension, and charge onto the central charge, here noted as RR charge.
Brane Actions & Anomaly inflow Mechanism
Localized, chiral, massive fermions at brane intersections lead to gauge-gravitational anomalies along the manifold Σ₁₂ = Σ₁ ∩ Σ₂ = ∪ᵣ Σʳ₁₂, which is self-consistent by coupling to the anomalous brane action where the gauge variance cancels the resulting anomalies. This is considered the "anomaly inflow". For D5-branes, the democratic formulation of type IIB SUGRA leads to the anomalous brane action
where α is a free parameter. For C₆, the action introduces the 4-form δ⁴(Σᵢ), which is Poincare-Dual to Σᵢ, so that all 6-forms α₆ act as:
So D5 is an electric source for C₆ and magnetic source for C₂. Similarly, for F₃, via C₂ and the Bianch identity, derive a gauge-invariant field strength
This cancels the anomaly for α = lₛ⁴. So far, the gravitational anomaly has been ignored. Including it again is the minimal effort of adding back in the action of the tangent bundle to Σ, and with k even/odd in type IIB/A.
D-brane Bound States and Solitons
Mutually non-SUSY branes can form BPS bound states, minimizing their energy. For example, using (p,q)-strings, meaning p fundamental strings (F-strings) and q D1-branes (D-strings), parallel along some x-axis and at rest. The charges of both strings, along with the anticommutator of the supercharges give the eigenvalues, from which the BPS bound condition is derived.
Said condition is satisfied for (p, q) = (1, 0) for F-strings and (p, q) = (0, 1) for D-strings. Other (p, q) break SUSY. BPS bound states has the F1-strings break up somewhere in the middle, and the separation points ending on D1. These endpoints carry opposite U(1) charge on D1, meaning they repel each other and moving away from one another (to infinity). The electric flux along D1 is meant to guarantee conservation of NS-charge. It's not accounted for here. Generally, (p, q)-strings exist as BPS bound states with a tension τ for coprime p, q.
In D-branes, quantum-fluctuations of Dp-branes are associated with perturbative open string excitations for small interaction constants. Independently, SuGra gives p-branes soliton-characteristics in its wave functions. The general (p+1)-form potential with field strength Fₚ₊₂ and string frame action is valid as an effective action at small interactions and slow field variations (compared to l). Fₚ₊₂ is analogous to the electric field strength, and F₈₋ₚ. The resulting electrically charged object is a p-brane, also writable as a p+1-dimensional soliton with Poincare symmetry. Magnetically charged objects with respect to Fₚ₊₂ are electrically charged objects with respect to F₈₋ₚ.
A unique solution to the action with required symmetries for each pair (M, Q) exists. Define the mass M = (p+1)-Volume * Tension of the p-brane with the tension τ. There is a critical value for M/Q, at which the solution has a horizon and a singularity behind it. Consider it a "black p-brane", i.e. the black hole in the normal (q - p) directions. These are often BPS bound. For larger M/Q, the solution has 2 horizons and no naked singularities. If it's smaller, the solution would produce naked singularities, though these solutions are usually forbidden by BPS-bound. The effective action is only valid for very small interactions, and the characteristic distances are much larger than l. The SuGra solution is valid for very large units of electric charge for many branes on top of each other. The validity of stringy-interaction as a Dp-brane, the expansion parameter for open string perturbation theory on N Dp-branes is gN. If this value is very small, the brane is best described as a Dp-brane, if it's very large, it's best described as a BPS soliton in SuGra.
SL(2, ℤ) duality of Type IIB
The classical type IIB SuGra was constructed to be invariant under SL(2, ℤ). If this is correct beyond the classical SuGra action, it's implied that type IIB is the same at strong and weak coupling, only with B₂ and C₂ exchanged. B₂ couplies electrically to the F-string, and C₂ to the D-string. This means that the F- and D-strings are also exchanged at strong/weak coupling.
At weak coupling, the massless modes of a D-string are best obtained through quantizing the F-string ending on a D-string. It gives 8 transverse and 2 parallel bosonic (massless) modes. These are the Goldstone bosons of SO(1,9) → SO(1,1) ⨉ SO(8). The R vacuum adheres to the Dirac equation, which gives the structure for the open string ground state 16 → (1/2, 8) ⊕ (-1/2, 8') (Goldstinos). These are associated with the breaking of 16 out of the 32 SUSYs of the D1-brane. A long F-string then sees 8 transverse Foldstones for bosonic excitations, and the BPS algebra will give the Goldstinos from there.
The physical tensions can be computed perturbatively at small interactions, both of which are BPS, and subsequently will be BPS at all interaction strengths. The tensions are always related to the central charge.
For weak interactions, F1 is much lighter than D1. The inverse is true for large interactions. Solving for length scales of F1 and D1 reveals that for small interactions, the F1-length scale is reached before gravity becomes relevant, and for large interactions, the D1-length scale is reached before gravity becomes relevant.
These observations motivate the conjecture that the full type IIB string is invariant under the subgroup of SL(2, ℤ) transformations.
Exchanging the B₂ and C₂ via the S-duality implies the exchange of the electric and magnetic sources of F1 and D1. The magnetic source of C₂ is a D5-brane with tension τ₅ , and the magnetic source of B₂ is a IIB NS5-brane, defined as a BPS solution in string frame.
At small interactions, the NS5 is much heavier than the D5-brane, and it's non-perturbative. The backreaction on the geometry then is naturally much stronger, and not suppressed by the effects of interaction.
M-theory
The strong coupling limits are determined by the lowest-dimensional brane within a theory, due to the length scale increasing on smaller dimensions. As such, the energy scale drops at large interactions with lower dimensions. For type IIA, this means that D0-branes become relevant. n D0-branes layered on top of one another are BPS and together can be considered a bound-state at threshold with a tension of nτ₀. The BPS state forms by giving a VEV to scalar field excitations from strings between the branes. As the interaction increases, the BPS approaches the light limit. The result is equivalent to a KK tower for S1-reduction with a fixed radius, so type IIA string theory becomes effectively 11-dimensional, which then allows 11D-SuGra by adding a KK U(1) gauge-potential.
The D0-charge is then analogous to the momentum along S¹. At the same time, 11d SuGra only requires 1 dimensionful coupling constant (reduced 11d Planck mass)
M-theory is the hypothetical 11d theory with low-energy effective action generated by 11D SuGra. It's primarily defined through encoding within BPS object, and unknown fundamentally.
The 11D A3 couples electrically to an M2-membrane as a (2+1)-dimensional object, while coupling magnetically to a M5-brane for a (5+1)-dimensional solution. The type IIA BPS follow by dimensional reduction, by which the D0-brane is analogous to the state of the KK momentum along S1, and the IIA F-string corresponds to the M2-brane wrapped around S1 once. M2-branes not wrapped around S2 can be considered regular D2-branes. The tensions relate as follows
To equate M2 and D2 branes, their degrees of freedom need to match. D2 has 7 transverse massless normal scalars, and 1 massless parallel vector. The vector is considered the magnetic dual to a scalar. By this relation, D2 can be described via (7+1) massless scalars, which matches the 8 transverse massless scalars of the usual M2 brane. By similar mechanism, the D4-brane is the same as the M5 brane along S1. The type IIA NS5-brane corresponds to the magnetic monopole to H3 in type IIA, so as M5-branes not wrapped S1. The NS5-brane doesn't carry the vector field, but a 2-form potential instead. Take a type IIB D1-brane ending on a IIB NS5, which, by T-duality is parallel to NS5, is normal to D1, so comes up as a D2 ending on a IIA NS5. M5 carries a 2-form potential. By the M5-solution and the contents of massless degrees of freedom shows that T2 is self-dual. Both of these 5D-branes preserve (2, 0) SUSY in 6 dimensions. The lowest 6d (2,0) multiplet contains 1 self-dual tensor and 5 scalars.
The D6-brane under M-theory is magnetically dual to the D0-brane, where the D6-brane is magnetically charged with respect to U(1), and the D0-brane is electrically charged with respect to U(1). This makes the D6-brane a KK-monopole of M-theory, corresponding to a solution of the 11D SuGra, where
which is a monopoly with respect to A.
Type IIB, and Type I - SO(32) Heterotic Duality
S1-spheres can be compactified in type IIA SUSY within M-theory on ℝ(1, 8) ⨉ S¹ (though on a different S1-sphere, it can't be compactified onto itself). T-duality sends S1 onto type IIB SUSY on the same M-theory.
At the vanishing limit of the torus volume, 10D type IIB arises, where the interaction is geometrized in terms of complex structure for the torus. In general,
The physical type IIB SL(2, ℤ) maps dually to the geometric SL(2, ℤ). The duality is part of the diffeomorphisms of 11D M-theory. This is to be expected, since all global symmetries in quantum gravity are gauged by construction. From this duality arises F-theory, focusing on type IIB SUSY with varying axio-dilaton τ(z) under influence of F-branes. This theory is dual to M-theory in the sense, that it assumes vanishing limit of the torus under type IIB. This is the Newton-Witten law.
By the strong-weak-coupling duality between the effective actions of type I and SO(32) heterotic actions, there is no reason to assume that this extends to the full theory through BPS states. The massless BPS excitation on D1 strings match the massless fields of the SO(32) heterotic string worldvolume. The 8 massless scalars from normal fluctuations map onto the heterotic bosonic fields, the 8 massless fermionic right-movers onto the right-moving fermions, and the 32 fermionic left movers onto the 32 left-moving fermions.
Within M-theory for type IIA, there is only space for the missing string theory on 1 more topology, for compactification of 1 dimension. The ℤ2 action on S1 breaks 16 of 32 SUSYs, and as R₁₀ vanishes, the 10D theory with 16 SUSYs leaves the gravitational sector anomalous and requiring the introduction of gauge degrees of freedom at the ℤ2 fixed points. 10D E₈ ⨉ E₈ SYM couples to 11D SuGra as an effective theory. The heterotic string takes the place of M2 branes stretched between both planes along the interval S1/ℤ2. Strong coupling separates the E₈ theories to the point at which they only interact gravitationally, and at weak coupling they coincide and recivers the original E₈ ⨉ E₈ heterotic string (with perturbations).
What remains is a non-perturbative defintion of M-theory with 11D SuGra at low-energy.
Calabi-Yau, Heterotic Compactifications & Type II Orientifolds
Kaehler Manifolds have holonomy group U(n) and there can be defined a Ricci (1,1)-form, so that the Ricci-class directly defines the first Chern class.
By the Calabi-Yau theorem, A Kaehler n-fold admits a unique Ricci-flat metric for given Kaehler class iff the first Chern class is 0. CYₙ has several topological properties, which can be fixed in a Hodge diamond.
Deformations of the internal CY-metric retaining Ricci-flatness have a corresponding scalar moduli field. Such deformations are 2-fold. One group is the volume moduli, in which the Kaehler moduli measures volumes.
The other is such, that the entries of the Hodge diamond give the massless complex scalars
The traditional way to obtain 4d N=1 compactifications with gauge dynamics starts from the heterotic string in CY₃. Assume gaugino variation on G either SO(32) or E₈ ⨉ E₈
For the spin connection, SU(3) is a gauge field and the structure group of TM. SU(3) then needs to be embedded into G and G breaks to a SU(3)-commutant.
Applying the SUSY condition from the gaugino variation will give hermitian Young-Mills equations as equivalent formulation of the condition. They imply that F is the curvature of a holomorphic vector bundle V, and one of the equations can be solved iff V is stable with respect to the slope.
The vacuum on the target space then is additionally defined by the choice of said slope-stable vector bundle by the second Chern-form. Such a choice could break G again to a commutant of the structure group of V within G. The interactions in 4D follow from 10D SYM interactions by dimensional reductions. The couplings in 4D correspond to the internal wave function overlapping on the target space.
type IIA/B on ℝ(1, 3) ⨉ M⁶, M⁶ = CY₃ → 4D N=2. Spacetime-filling Dp-branes with worldvolume
is subject to stability, supersymmetry at KK scale, and Gauss' law, which is expressed here through the tadpole cancellation condition. These reinforce a Poincare-duality between the k-th homology group and the k-th cohomology group, fix 4d N=1 SUSY at KK-scale, and give the Bianchi-identity for the relevant F-forms
On compact spaces then, the total charge vanishes, since the flux lines can't escape to infinity. To avoid instability, take several branes along Σ, which is incompatible with SUSY at first glance. To re-enable this construction, SUSY objects of negative RR-charge are required, which exist in the type II orientifolds. These objects fix WS-parity Ω and holomorphic involution σ in type IIB SUSY. type IIA orientifolds take the anti-holomorphic involution instead. The construction then is
We also require a gauge group ΠU(N) where the total rank is constrained by the size of the Oₚ-planes. At this point, embedding SU(3) ⨉ SU(2) ⨉ U(1) into the gauge group in configuration with 3 chiral massless generation of standard model matter yields the standard model, along with extra hidden sectors generated by other branes and moduli/axions.