Type II string theory
Type II string theory refers to a pair of supersymmetric string theories—Type IIA and Type IIB—that describe the fundamental constituents of the universe as one-dimensional strings vibrating in ten-dimensional spacetime, incorporating both bosons and fermions to achieve N=2 supersymmetry with 32 supercharges.[1] These theories eliminate the tachyon instability present in bosonic string theory by including worldsheet supersymmetry, requiring a critical dimension of D=10 for anomaly cancellation and conformal invariance.[1] Both variants feature an infinite tower of massless and massive states, including gravitons, the dilaton, the Neveu-Schwarz B-field, and Ramond-Ramond (RR) p-form fields, with no tachyons in the spectrum, ensuring stability.[1] The primary distinction between Type IIA and Type IIB lies in their chirality and RR sector: Type IIA is non-chiral, with opposite-chirality fermions in left- and right-moving sectors, while Type IIB is chiral, with matching chiralities.[1] In Type IIA, the RR fields consist of a 1-form (C_1) and a 3-form (C_3), supporting stable Dp-branes for even spatial dimensions p=0,2,4,6,8, whereas Type IIB includes a 0-form (C_0), a 2-form (C_2), and a self-dual 4-form (C_4), accommodating stable Dp-branes for odd p=1,3,5,7,9.[1] This difference manifests in their low-energy effective actions, which share common terms like the Einstein-Hilbert action coupled to the dilaton and B-field but diverge in RR contributions, with Type IIB featuring a self-dual field strength.[1] T-duality relates the two theories: compactifying Type IIA on a circle transforms it into Type IIB, and vice versa, underscoring their equivalence under this symmetry and their role within the broader M-theory framework.[1] D-branes in Type II theories are BPS objects preserving half the supersymmetry (16 supercharges), sourcing RR charges and hosting the endpoints of open strings, whose low-energy dynamics are described by supersymmetric U(N) Yang-Mills theories with 16 supercharges on their worldvolumes (corresponding to N=4 super-Yang-Mills for D3-branes in Type IIB), which has profound implications for black hole entropy and AdS/CFT duality.[1] These features position Type II string theories as UV-finite, anomaly-free candidates for unifying quantum mechanics and general relativity, with applications in exploring non-perturbative phenomena through dualities and brane constructions.[1]Fundamentals
Definition and Historical Context
Type II string theory encompasses two consistent ten-dimensional superstring theories, known as Type IIA and Type IIB, both featuring maximal N=2 spacetime supersymmetry with 32 supercharges and formulated in terms of oriented closed strings.[1] These theories differ primarily in the chirality of their GSO projections: Type IIA combines left- and right-moving sectors with opposite chiralities, resulting in a non-chiral spectrum, while Type IIB uses identical chiralities for both sectors, yielding a chiral spectrum.[1] As part of the superstring framework, Type II theories eliminate the tachyon instability of bosonic string theory through worldsheet supersymmetry and are defined in the critical dimension of ten spacetime dimensions to ensure anomaly cancellation and conformal invariance.[1] The discovery of Type II string theory occurred in 1982, when Michael Green and John Schwarz classified the consistent supersymmetric string theories in ten dimensions, identifying Type I, IIA, and IIB as anomaly-free formulations that incorporate spacetime supersymmetry.[2] Building on earlier work such as the Ramond-Neveu-Schwarz model (1971) and the GSO projection (1976), which established the foundational structure for superstrings, Green and Schwarz's analysis demonstrated that these theories propagate only physical massless and massive states without ghosts or tachyons in the appropriate dimensions.[2] This classification marked a pivotal step in shifting string theory from a model of strong interactions to a candidate for a unified theory of quantum gravity. In the 1980s, key milestones included the formulation of the low-energy effective actions for Type II theories as ten-dimensional supergravity limits, with Type IIB supergravity constructed in 1983 by Schwarz and independently by Howe and West. By the mid-1980s, Type II theories were integrated into the landscape of five consistent superstring theories, alongside Type I and the two heterotic string theories, highlighting their role as non-heterotic, closed-string-based frameworks without gauge sectors from open strings.[2] The 1990s "second superstring revolution" further elevated Type II theories through the discovery of dualities, such as T-duality relating IIA and IIB on circles of varying radii, unifying the apparent multiplicity of string theories into a single underlying framework.Basic Principles and Supersymmetry
Type II string theory describes the dynamics of closed, oriented superstrings propagating on a (1+1)-dimensional worldsheet equipped with \mathcal{N}=(1,1) supersymmetry, extending the bosonic string framework to incorporate fermionic degrees of freedom while maintaining consistency at the quantum level.[3] The theory is formulated in the Ramond-Neveu-Schwarz (RNS) formalism, where the worldsheet fermions \psi^\mu satisfy specific boundary conditions that distinguish the sectors contributing to the spectrum.[3] This setup ensures the inclusion of both bosonic and fermionic excitations, leading to a unified description of matter and gravity in ten dimensions. The fundamental dynamics are governed by the Polyakov action extended to superstrings, which couples the embedding coordinates X^\mu(\sigma^a) of the worldsheet into target spacetime to Majorana-Weyl worldsheet fermions \psi^\mu. The action for Type II strings is given by S = \frac{1}{4\pi \alpha'} \int d^2\sigma \, \sqrt{h} \, h^{ab} \left( \partial_a X^\mu \partial_b X_\mu + \psi^\mu \gamma^a \partial_b \psi_\mu \right) + \text{fermionic terms}, where h_{ab} is the worldsheet metric, \alpha' is the Regge slope parameter related to the string tension by T = 1/(2\pi \alpha'), and the fermionic terms account for the supersymmetric interactions.[3] For closed strings, the action separates into independent left-moving and right-moving sectors, with \psi_L^\mu and \psi_R^\mu evolving along the respective chiralities, enabling the construction of distinct Type IIA and Type IIB variants through sector combinations.[3] The theory exhibits maximal (1,1) worldsheet supersymmetry, realized through superconformal invariance with central charge c = 15/2 in the absence of the ghost sector, which generates \mathcal{N}=2 supersymmetry in ten-dimensional spacetime upon quantization.[3] In the RNS formalism, the Neveu-Schwarz (NS) sector imposes anti-periodic boundary conditions on the worldsheet fermions (\psi^\mu(\sigma + 2\pi) = -\psi^\mu(\sigma)), yielding half-integer moded excitations that include vector representations, while the Ramond (R) sector uses periodic conditions (\psi^\mu(\sigma + 2\pi) = \psi^\mu(\sigma)), producing integer moded spinor ground states. These sectors are combined in Type II theories to form the full Hilbert space, with the Gliozzi-Scherk-Olive (GSO) projection applied to eliminate tachyonic states and enforce spacetime supersymmetry by selecting states of definite worldsheet fermion number parity.[4] Quantization of the theory requires a critical dimension of ten to cancel the conformal anomaly, ensuring the vanishing of the beta functions for the worldsheet theory and Lorentz invariance in the target space.[3] The parameter \alpha' governs the mass scale, with the string mass-squared levels given by m^2 = (N + \tilde{N} - 2)/\alpha' in the light-cone gauge, where N and \tilde{N} are the oscillator numbers for left- and right-movers, respectively.[3] The differences between Type IIA and Type IIB arise from the chirality choices in the GSO projection for the R sectors, leading to opposite or same-handed supersymmetries.[3]Type IIA String Theory
Spectrum and Fields
Type IIA string theory features a spectrum of physical states derived from closed superstrings propagating in ten dimensions, with massless modes arising primarily from the Neveu-Schwarz-Neveu-Schwarz (NS-NS) and Ramond-Ramond (RR) sectors, subject to the GSO projection that selects states of opposite chirality for left- and right-moving Ramond sectors. This opposite-chirality projection distinguishes Type IIA from Type IIB, resulting in a non-chiral theory with N=2 supersymmetry in spacetime, comprising 32 supercharges. Higher massive modes exist at levels determined by the oscillator excitations, with mass squared given by M^2 = \frac{1}{\alpha'} (N_L + N_R - 2) for the bosonic part, adjusted by fermionic contributions in the superstring, ensuring no tachyon in the spectrum due to supersymmetry.[1] The massless bosonic fields include the graviton G_{\mu\nu}, the Kalb-Ramond antisymmetric 2-form B_{\mu\nu}, and the dilaton \Phi from the NS-NS sector, alongside RR fields consisting of the 1-form potential C_1 and the 3-form C_3, with field strengths F_2 = dC_1 and F_4 = dC_3 + C_1 \wedge H_3. Fermionic partners include two Majorana-Weyl gravitinos and two dilatini of opposite chirality, completing the N=2 supersymmetry multiplet.[1] The brane content features even-dimensional D-branes (D0-, D2-, D4-, D6-, and D8-branes) as BPS saturated states sourcing the RR fields, with tensions scaling as T_p \propto 1/(g_s (2\pi l_s)^{p+1}) for p even, alongside NS5-branes and fundamental strings as sources for the NS-NS sector. These objects preserve half the supersymmetries and play a central role in non-perturbative dynamics. Via T-duality, the Type IIA spectrum relates to that of Type IIB upon circle compactification, interchanging even and odd D-branes.[1]| Sector | Bosonic Fields | Fermionic Fields | Notes |
|---|---|---|---|
| NS-NS | G_{\mu\nu} (graviton, spin-2), B_{\mu\nu} (2-form, spin-1), \Phi (dilaton, spin-0) | Two opposite-chirality Majorana-Weyl gravitinos (spin-3/2), two dilatini (spin-1/2) | Universal to Type II; 35 + 35 + 1 bosonic degrees of freedom after gauge fixing. |
| R-R | C_1 (1-form), C_3 (3-form) yielding F_2 and F_4 | - | Non-chiral due to opposite-chirality GSO; 28 + 35 degrees of freedom. |
Low-Energy Effective Theory
The low-energy effective theory of Type IIA string theory is obtained in the limit where the Regge slope parameter α' approaches zero, expanding the string worldsheet action and retaining the leading massless modes, which yield the ten-dimensional Type IIA supergravity action possessing N=2 supersymmetry—corresponding to two Majorana-Weyl supercharges of opposite chirality. This supergravity theory lacks the SL(2,ℝ) global symmetry of Type IIB and is non-chiral, distinguishing it from the chiral structure of Type IIB supergravity.[1] The bosonic sector of the Type IIA supergravity action in the string frame is given by S = \frac{1}{2\kappa_{10}^2} \int d^{10}x \sqrt{-g} e^{-2\Phi} \left[ R + 4 \partial_\mu \Phi \partial^\mu \Phi - \frac{1}{2} |H_3|^2 \right] - \frac{1}{4\kappa_{10}^2} \int \left( |F_2|^2 + |\tilde{F}_4|^2 \right) + S_{\rm CS}, where R is the Ricci scalar, H_3 = dB_2 is the NS-NS three-form field strength, F_2 = dC_1, \tilde{F}_4 = F_4 - C_1 \wedge H_3 with F_4 = dC_3, \kappa_{10}^2 = (2\pi)^7 \alpha'^4 is the ten-dimensional gravitational coupling, and the topological Chern-Simons term includes S_{\rm CS} = -\frac{1}{4\kappa_{10}^2} \int B_2 \wedge F_4 \wedge F_4 + \cdots, with higher-order terms for consistency; the action is invariant under diffeomorphisms and local supersymmetry.[1] The equations of motion derived from this action include coupled equations for the dilaton, metric, and form fields, such as the Klein-Gordon equation for Φ: \nabla^\mu (e^{-2\Phi} \partial_\mu \Phi) = sources from H3 and RR fluxes, and the Einstein equation with stress-energy from all bosonic fields. Bianchi identities govern the RR forms, e.g., dF_4 = H_3 \wedge F_2, reflecting the non-chiral nature without self-duality. These equations support solutions like compactifications on Calabi-Yau manifolds to four dimensions, generating warped geometries and moduli stabilization for phenomenological models.[1] While the supergravity action captures the perturbative low-energy regime of Type IIA string theory (α' → 0 and g_s → 0), it connects to non-perturbative aspects through dualities like T-duality to Type IIB and the broader M-theory framework upon lifting to eleven dimensions, enabling explorations of strong-coupling dynamics via branes and fluxes.[1]Type IIB String Theory
Spectrum and Fields
Type IIB string theory features a spectrum of physical states derived from closed superstrings propagating in ten dimensions, with massless modes arising primarily from the Neveu-Schwarz-Neveu-Schwarz (NS-NS) and Ramond-Ramond (R-R) sectors, subject to the GSO projection that selects states of the same chirality for both left- and right-moving Ramond sectors. This same-chirality projection distinguishes Type IIB from Type IIA, resulting in a chiral theory with (2,0) supersymmetry in spacetime, comprising 32 supercharges. Higher massive modes exist at levels determined by the oscillator excitations, with mass squared given by M^2 = \frac{1}{\alpha'} (N_L + N_R - 2) for the bosonic part, adjusted by fermionic contributions in the superstring, ensuring no tachyon in the spectrum due to supersymmetry.[1] The massless bosonic fields include the graviton G_{\mu\nu}, the Kalb-Ramond antisymmetric 2-form B_{\mu\nu}, and the dilaton \Phi from the NS-NS sector, alongside R-R fields consisting of the 0-form potential C_0, the 2-form C_2, and the 4-form C_4, whose field strength yields the self-dual 5-form F_5 = dC_4 + \cdots satisfying the self-duality condition F_5 = * F_5 in ten dimensions. The axion-dilaton system is captured by the complex scalar \tau = C_0 + i e^{-\Phi}, which transforms under the SL(2,\mathbb{Z}) duality group of the theory, ensuring invariance of the spectrum and action. Fermionic partners include two Majorana-Weyl gravitinos and two dilatini of the same chirality, completing the N=2 supersymmetry multiplet.[1][5] The brane content features odd-dimensional D-branes (D1-, D3-, D5-, D7-, and D9-branes) as BPS saturated states sourcing the R-R fields, with tensions scaling as T_p \propto 1/(g_s (2\pi l_s)^{p+1}) for p odd, alongside NS5-branes and fundamental strings as sources for the NS-NS sector. These objects preserve half the supersymmetries and play a central role in non-perturbative dynamics. Via T-duality, the Type IIB spectrum relates to that of Type IIA upon circle compactification, interchanging even and odd D-branes.[1]| Sector | Bosonic Fields | Fermionic Fields | Notes |
|---|---|---|---|
| NS-NS | G_{\mu\nu} (graviton, spin-2), B_{\mu\nu} (2-form, spin-1), \Phi (dilaton, spin-0) | Two same-chirality Majorana-Weyl gravitinos (spin-3/2), two dilatini (spin-1/2) | Universal to Type II; 35 (graviton) + 28 (B-field) + 1 (dilaton) bosonic degrees of freedom after gauge fixing. |
| R-R | C_0 (axion, part of \tau), C_2 (2-form), self-dual F_5 from C_4 (5-form strength) | - | Chiral due to same-chirality GSO; SL(2,\mathbb{Z}) acts on \tau; 1 + 28 + 35 self-dual degrees. |