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The auxiliary-metric brane action is a quadratic embedding action with an independent worldvolume metric. Its embedding equation is ${\partial }_{\mu }(\sqrt{-\gamma }{\gamma }^{\mu \nu }{\partial }_{\nu }{X}^a)=0$. Its metric equation is $h_{\mu \nu }-{\gamma }_{\mu \nu }({\gamma }^{\rho \sigma }{h}_{\rho \sigma }-(p-1))/2=0$. For $p\ne 1$, tracing determines ${\gamma }^{\rho \sigma }{h}_{\rho \sigma }=p+1$ and hence $\gamma =h$; eliminating the auxiliary field gives twice the invariant worldvolume volume, before including the overall brane tension. The Weyl-invariance exception for the string among branes prevents unique elimination at $p=1$.

For a spatially flat, tuned one-brane background without energy transfer to the bulk, the Israel junction conditions produce the displayed modified Friedmann equation. Positive brane tension $\lambda$ gives ordinary expansion when $\rho \ll \lambda$, while the quadratic density correction enhances expansion at high energy. Big Bang nucleosynthesis, early acoustic evolution and primordial spectra can constrain this change. Spatial curvature, a residual cosmological constant and bulk energy exchange require their additional terms rather than this special formula.

The M5-brane has five spatial dimensions and carries the magnetic charge dual to the M2-brane's electric three-form charge. Wrapping it on a compact cycle lowers its effective spatial dimension and multiplies its brane tension by the cycle volume.

An NS5-brane has five spatial dimensions and is magnetically charged under the Neveu–Schwarz two-form. Its string-frame brane tension scales as $g_s^{-2}$.

A supersymmetric type-II D-brane carries a Ramond–Ramond potential charge and saturates the corresponding generalized BPS bound in supersymmetry: its brane tension equals the charge-dependent central/tensorial-charge bound in the relevant normalization. It is therefore a BPS state, preserving sixteen of the thirty-two spacetime supercharges. Identical parallel branes with the same orientation impose the same supersymmetry projector. This statement does not apply to arbitrary unstable non-BPS branes or to the bosonic branes of the preceding parts.

One direct demonstration of the parallel D-brane no-force identity is the annulus vacuum amplitude. For separation $r$, its oscillator-dependent part is proportional toHereThe first two terms describe the projected Neveu–Schwarz sector, and the last is the oppositely signed Ramond sector contribution. The Jacobi abstruse identity gives $f_3^8-f_4^8-f_2^8=0$ for every $t$. ThereforeIn the closed-string channel, attractive graviton and dilaton exchange cancel repulsive Ramond–Ramond potential exchange. The classical probe calculation in the D-brane supergravity solution gives the same cancellation. The common supersymmetry allows separation to remain a flat modulus; relative angles, reversed RR charge or supersymmetry-breaking backgrounds can remove this protection. For a brane–antibrane pair the RR interaction reverses and does not cancel the NS-NS attraction. The charge and half-supersymmetry interpretation is established in Dirichlet-Branes and Ramond-Ramond Charges.

A D-brane is a localized RR-charged source whose long-distance fields solve supergravity. The extremal charged $p$-brane solutions carry the same RR charges, brane tension and preserved supersymmetry as the open-string construction. For the standard asymptotically flat cases $p<7$, the D-brane supergravity solution in the string frame isThe function $H$ is a positive harmonic function away from the source, in $9-p$ transverse dimensions. In an electric gauge, $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$, with its orientation chosen to match the source. Magnetic branes use the dual flux; for the D3-brane, the five-form includes both electric and magnetic pieces and is self-dual. Killing spinors obey one brane projector, leaving sixteen supersymmetries. Aligned multicenter harmonic functions describe several branes with no static binding force.

To check this identification directly, insert an equally oriented static probe into the solution. Put ${\mu }_p=(2\pi {)}^{-p}{\alpha }^{\prime -(p+1)/2}$, so the asymptotic tension is ${\mu }_p/g_s$. The induced-volume factor is $H^{-(p+1)/4}$; combined with $e^{-\Phi }$, its DBI action density is $-{\mu }_p{g}_s^{-1}{H}^{-1}$. Its Wess-Zumino brane coupling contributes $+{\mu }_p{g}_s^{-1}(H^{-1}-1)$. Hence the total static Lagrangian density is the constant $-{\mu }_p/g_s$, independently of position. This gives the parallel D-brane no-force identity in the spacetime description.

The supergravity background is the back-reacted, coarse long-distance description, while the boundary-condition description exposes the light open-string degrees of freedom. Controlled use of supergravity requires curvature small in string units and suitable local coupling. Singular cores and the special codimension-two or domain-wall cases are not automatically resolved by the elementary harmonic ansatz.

The exterior derivative is a linear map $d:{\Omega }^k(M)\to {\Omega }^{k+1}(M)$, agrees with the ordinary differential on functions, satisfies $d^2=0$, and obeys the graded Leibniz ruleIn local coordinates it differentiates the coefficient functions:Antisymmetry cancels the second partial derivatives, proving $d^2=0$ locally. These properties determine the coordinate-independent derivative.

For any smooth $\varphi :N\to M$, the pullback of a differential form is defined byThere is no requirement that $\varphi$ be invertible or an immersion. Pullback preserves wedge products and sends $df$ to $d(f\circ \varphi )$ by the chain rule. Locally every form is a sum of $fd{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}$; applying the preceding identities and $d^2{\varphi }^i=0$ proves

Describe the brane by an embedding $X:\Sigma \to M$, with nondegenerate induced worldvolume metric $h=X^{\ast }g$. Locally the Poincaré lemma gives $F=dA$ because $F$ is a closed differential form. For a timelike worldvolume, the geometric area and Wess-Zumino brane coupling give the action$T$ is the brane tension and $q$ the charge in this normalization. The geometric term is the higher-dimensional version of the Nambu–Goto action. Both integrals are invariant under changes of worldvolume coordinates.

The additive gauge transformation is $A\mapsto A+d\Lambda$. Its change of action isThe last step is the Generalized Stokes theorem. It vanishes for a closed worldvolume, or with suitable boundary restrictions. An open worldvolume can instead have a compensating boundary coupling, as in gauge invariance of an open-brane coupling.

More directly, for a variation vector $V$ of the embedding, Cartan's magic formula givesThus the interior force depends on $F$, not the gauge potential. Define the trace of the second fundamental form byThe area variation is $-\int \sqrt{|h|}g(V,K)$ plus a boundary term. For compactly supported interior variations the resulting equations areThey exhibit the field-strength force on a charged brane and are gauge invariant.

Two global qualifications are necessary. First, replacing $A$ by $d\Lambda$ alone, as printed, gives $d{A}^{\prime }=0$ and would erase a nonzero $F$; the intended transformation must be the additive shift above. Second, closedness of $F$ only guarantees a potential locally. If a global potential does not exist and the worldvolume bounds a filling $B$, its charged term can be defined as $q{\int }_{B}F$. Different fillings differ by a period of $F$; independence of the quantum phase requires $q{\int }_{C}F\in 2\pi \hslash \mathbb{Z}$ for the corresponding closed cycles. Without the required global potential or patching/filling data, the given closed form still defines the local variational equations, but not automatically a single globally defined real-valued action.

In the one-brane Randall–Sundrum model, gravity localization in a noncompact warped dimension produces a normalizable massless graviton despite the infinite proper fifth dimension. At long distances its zero mode reproduces four-dimensional gravity; bulk Kaluza-Klein modes give corrections. This localization mechanism, rather than a large compact volume, is the reason an infinite dimension need not destroy the usual Newton law, as established in the original single-brane warped-gravity construction. Its observational tests include:

For an expanding brane, its motion in the bulk and the Israel junction conditions modify the four-dimensional cosmological expansion. In the tuned low-energy regime it approaches ordinary cosmology, but high-energy braneworld expansion has a density-squared correction when the matter density approaches the brane tension. Faster early expansion changes weak-interaction freeze-out and light-element yields, so Big Bang nucleosynthesis is a sensitive test. It also changes time-temperature relations, recombination distances and acoustic scales in the Cosmic microwave background anisotropy.

The bulk can supply dark radiation on a cosmological brane, a projected-Weyl contribution that scales like radiation and is related to a bulk black-hole mass in the simplest vacuum construction. Nucleosynthesis and CMB measurements constrain its contribution to the expansion. Energy transferred from a hot brane into bulk gravitons can further modify the radiation history. None of these terms must be identified with an ordinary thermal particle component without a model.

Inflation on the brane can alter scalar and tensor production at high energy; the gravitational-wave zero-mode normalization and coupling to bulk modes can change primordial spectra. CMB temperature/polarization, primordial gravitational waves and later structure growth can test these effects. Projected bulk perturbations may introduce nonlocal corrections to the brane's effective perturbation equations, so a background expansion law alone does not determine all observables. Late-time measurements of expansion and structure also test residual departures, although the basic tuned model does not automatically predict dark-energy acceleration.

All proposed signals depend on the curvature scale, fundamental gravitational scale, brane tension, reheating history and bulk state. The minimal single-brane construction has no second-brane separation radion; an arbitrary radion fifth-force prediction from a two-brane compactification cannot simply be imported here. The main probes are corrected gravity, accessible bulk radiation, and high-energy/dark-radiation changes to cosmological evolution.

The three identifications areThey match the same lower-dimensional theory, including its moduli of a string compactification, charged states and extended branes. Different weakly coupled descriptions occupy different limits of its moduli space.

For the M-theory circle duality, write ${\ell }_s=\sqrt{{\alpha }^{\prime }}$ and let ${\ell }_p$ be the eleven-dimensional Planck length. The standard length dictionary isThe circle radius is the type IIA superstring theory coupling modulus. The eleven-dimensional metric supplies the ten-dimensional metric, dilaton and Ramond–Ramond potential $C_1$ through Kaluza-Klein theory; the three-form supplies $C_3$ and the Neveu–Schwarz two-form $B_2$ by $A_3=C_3+B_2\wedge dy$. Thus the field content also agrees with type IIA supergravity.

The M2-brane and M5-brane tensions are $T_2=(2\pi {)}^{-2}{\ell }_p^{-3}$ and $T_5=(2\pi {)}^{-5}{\ell }_p^{-6}$. Wrapping one spatial direction multiplies the effective brane tension by $2\pi R_{11}$. Consequently:

These tension and charge relations explain why increasing $g_A$ reveals an eleventh dimension. The simple circle dictionary describes massless type IIA supergravity; a D8-brane involves its massive extension and Romans mass.

For the M-theory torus duality, the complex structure of the torus is the IIB axion-dilatonIts area supplies the remaining radius modulus. In a common nine-dimensional mass normalization, an M2-brane wrapped on the entire torus has mass $T_2\mathcal{A}$, identified with $1/R_B$; hence $R_B$ is inversely related to the torus area in that normalization. Radii quoted in different string or Einstein metrics also include their Weyl rescaling. A rectangular torus with cycle radii $R_{10},R_{11}$ has $g_B=R_{11}/R_{10}$. Equivalently, first reduce to IIA and use T-duality:where $R_A,R_B$ in this formula use the corresponding string metrics. Shrinking the torus area at fixed $\tau$ decompactifies the IIB circle. The torus mapping class group $SL(2,\mathbb{Z})$ becomes IIB S-duality.

An M2-brane wrapped on a primitive $(p,q)$ one-cycle gives a (p,q) string, whose tension is the membrane tension times the cycle length before conversion to the IIB metric. Momentum on the torus matches winding of these strings on the IIB circle. An M5-brane wrapped on the whole torus gives an unwrapped D3-brane; an unwrapped M2-brane gives a D3-brane wrapped on the IIB circle. An M5-brane wrapped on one torus cycle gives a (p,q) five-brane wrapped on the IIB circle. The unwrapped M5-brane is the magnetic dual of the IIB circle's Kaluza-Klein mode, represented by its Kaluza-Klein monopole five-brane. These identifications match effective spatial dimensions as well as electric and magnetic charges.

For the M-theory K3 duality, the common noncompact spacetime has seven dimensions and sixteen supersymmetry generators. A Ricci flat K3 surface has a positive three-plane in its second real cohomology, equipped with the K3 intersection lattice of signature $(3,19)$. The unit-volume moduli space of Ricci-flat K3 metrics has local formwith a discrete lattice identification and one additional positive volume modulus. This is the heterotic Narain moduli space on $T^3$, together with the seven-dimensional coupling: its torus metric has six parameters, its antisymmetric tensor three, and its sixteen gauge Wilson lines forty-eight, giving $6+3+48=57$ before the dilaton.

The coupling–volume relation can be checked dimensionally rather than guessed. A wrapped M5-brane gives a heterotic fundamental string with tension $T_{5}V$, so its squared string length is proportional to ${\ell }_p^6/V$. The seven-dimensional gravitational coupling is proportional to ${\ell }_p^9/V$. Their dimensionless ratio givesThus the large-volume M-description corresponds to strong heterotic coupling. Overall constants depend on the normalization of the lower-dimensional coupling.

There are twenty-two gauge fields from reducing $A_3$ on the twenty-two harmonic two-forms of K3, matching the heterotic sixteen gauge vectors plus three metric and three two-form vectors. The remaining seven-dimensional three-form is dual to the heterotic two-form. A wrapped M2-brane gives an electrically charged particle in the shared charge lattice, matching heterotic momentum, winding and gauge charges; an M5-brane on a two-cycle gives its magnetic three-brane. An M5-brane wrapping all of K3 gives the heterotic string, while an unwrapped M2-brane corresponds to a heterotic NS5-brane wrapped on $T^3$. At suitable collapsing two-cycles, charged membrane states become massless and yield ADE gauge enhancement. The geometry accounts for both generic abelian charges and nonabelian enhancement. There is no extra K3 two-form modulus in the M-description: its fundamental potential is a three-form, and $b_3(K3)=0$.

The hierarchy problem concerns the enormous separation between the electroweak scale and a possible fundamental ultraviolet scale such as the reduced Planck mass: $v\simeq 246\mathrm{GeV}$ whereas $M_{\mathrm{Pl}}\simeq 2.4\times 10^{18}\mathrm{GeV}$. For an elementary Higgs field, heavy states or a physical high-energy ultraviolet cutoff generically generateHere $\kappa$ represents the appropriate dimensionless couplings. Maintaining a small physical Higgs mass then requires a very accurate cancellation against its bare mass parameter. The problem is sensitivity to actual high-energy physics, rather than merely the presence of a quadratic divergence in one choice of regulator.

The cosmological constant problem concerns the much smaller gravitationally observable vacuum energy. In the Einstein equation, a constant term in the effective action contributes to the cosmological constant, with ${\Lambda }_{\mathrm{eff}}={\Lambda }_{\mathrm{bare}}+{\rho }_{\mathrm{vac}}/M_{\mathrm{Pl}}^2$ in the usual normalization. Generic zero-point contributions scale as ${\Lambda }_{\mathrm{UV}}^4$, and vacuum transitions also change ${\rho }_{\mathrm{vac}}$. The dark-energy density is only of order $10^{-47}{\mathrm{GeV}}^4\sim (\text{a few meV}{)}^4$. The puzzle is why the total remains so small despite much larger individual contributions. A constant shift of the nongravitational Hamiltonian cannot remove its gravitational effect.

Supersymmetry relates bosonic and fermionic degrees of freedom, including their interactions. Their opposite loop signs cancel the quadratic contributions to a scalar field mass in the unbroken theory. With soft supersymmetry breaking, the surviving correction is controlled by the mass splitting, schematicallyThus a sufficiently low soft-breaking scale protects a small electroweak scale against a much higher ultraviolet scale. This is the useful supersymmetric response to the hierarchy problem.

Exact global supersymmetry also cancels vacuum energy: the supersymmetry algebra expresses the Hamiltonian as a positive sum of anticommutators, so a vacuum annihilated by every supercharge has zero energy. Equivalently, with canonical kinetic terms,vanishes in a supersymmetric vacuum. However, ordinary particles and their superpartners cannot have the identical mass spectrum of unbroken supersymmetry, so a useful phenomenological theory needs supersymmetry breaking. Its vacuum energy is generically of order the fourth power of the breaking scale; even a scale of order $10^2\mathrm{GeV}$ gives about $10^8{\mathrm{GeV}}^4$, enormously above the required value. In supergravity, a negative term can cancel positive breaking contributions:but arranging a tiny residual is a new tuning. Consequently supersymmetry can protect the hierarchy after soft breaking, while its exact vacuum-energy cancellation does not by itself survive realistic breaking.

One higher-dimensional proposal uses large extra dimensions. Take $n$ flat compact dimensions of common radius $R$, with gravity propagating in the bulk and Standard Model fields confined to a four-dimensional brane. Reducing the gravitational Einstein-Hilbert action givessoA fundamental gravitational scale $M_{\ast }$ near a TeV then produces apparently weak four-dimensional gravity through dilution into the large compact volume. For $n=2$ and $M_{\ast }=1\mathrm{TeV}$, $R=M_{\mathrm{Pl}}/(2\pi M_{\ast }^2)$ is about $0.075\mathrm{mm}$, using $1{\mathrm{GeV}}^{-1}\simeq 1.97\times 10^{-16}\mathrm{m}$. At distances much smaller than $R$, the gravitational potential scales as $r^{-(n+1)}$; at distances much larger than $R$ it scales as $r^{-1}$. Confining charged Standard Model fields to the brane avoids a light tower of charged Kaluza-Klein modes associated with the large radius. The primary explanatory challenge is radius stabilization: one must generate and stabilize $R{M}_{\ast }\gg 1$, rather than simply postulate a huge volume and transfer the original hierarchy into a modulus.

A second proposal is the Randall–Sundrum model. Compactify on an interval, equivalently an orbifold with branes at $y=0$ and $y=L$, and use the warped metricA negative bulk cosmological constant and suitably related brane tensions support this geometry. Integrating over the full orbifold interval $[-L,L]$ givesFor a scalar field $h$ on the brane at $L$, its induced metric gives the kinetic coefficient $e^{-2kL}$ and mass coefficient $e^{-4kL}{m}_0^2$. The canonical field is $h_c=e^{-kL}h$, henceWith $k$ and $M_5$ of the same fundamental order, an exponent $kL\simeq 35$--$37$ converts a fundamental mass into a weak-scale mass, without requiring a large proper compactification volume.

Again the separation must be selected dynamically. The original warped solution leaves a massless radion, so merely choosing $L$ does not complete the explanation. For example, the Goldberger–Wise stabilization mechanism introduces a bulk scalar field with boundary values $v_0,v_L$ and a small mass $m_{\chi }$. Its static equation and solution areFor small backreaction and large $kL$, the leading radius-dependent potential is proportional to $e^{-4kL}(v_L-v_0{e}^{-ϵkL}{)}^2$. Its leading minimum obeys $kL\simeq {ϵ}^{-1}\log (v_0/v_L)$, so a modest boundary-value ratio and a small $ϵ$ can give the required exponent. Corrections and the residual four-dimensional cosmological constant must still be controlled. Both proposals need a stable dynamical origin for the compactification geometry; their volume or warp relations alone do not supply it.

Let $h_{\mu \nu }={\partial }_{\mu }{X}^a{\partial }_{\nu }{X}_a$ be the induced worldvolume metric and write $H={\gamma }^{\mu \nu }{h}_{\mu \nu }$. Varying the embedding in the auxiliary-metric brane action and integrating by parts givesThe boundary contribution, including its relative factor, isAt initial and final times one fixes the endpoint configurations or uses variations of compact support. At a spatial boundary, free target directions require Neumann boundary conditions $n_{\mu }{\gamma }^{\mu \nu }{\partial }_{\nu }{X}^a=0$; fixed directions require Dirichlet boundary conditions $\delta X^a=0$. Mixed conditions must make this boundary pairing vanish. A closed brane has no spatial boundary. Since the auxiliary worldvolume metric enters without derivatives, its variation produces no extra boundary term.

Using $\delta \sqrt{-\gamma }=-(1/2)\sqrt{-\gamma }{\gamma }_{\mu \nu }\delta {\gamma }^{\mu \nu }$, variation of the inverse worldvolume metric givesTaking the trace yields $(p-1)(H-p-1)=0$. For $p\ne 1$, this forces $H=p+1$, and substitution gives ${\gamma }_{\mu \nu }=h_{\mu \nu }$. For a nondegenerate timelike embedding, eliminating the auxiliary worldvolume metric therefore gives twice the worldvolume area, with the overall physical brane tension supplied by the action normalization.

For $p=1$, the traced metric equation is an identity, and the remaining equation only says $h_{\mu \nu }=(H/2){\gamma }_{\mu \nu }$. It fixes the worldsheet metric up to a Weyl transformation, rather than determining it uniquely. Indeed $\sqrt{-\gamma }{\gamma }^{\mu \nu }$ is Weyl invariant precisely in two worldvolume dimensions, and the constant term vanishes precisely at $p=1$. This is the Weyl-invariance exception for the string among branes. In a nondegenerate interior it gives the familiar classical equivalence of Polyakov and Nambu–Goto actions; degeneracies at a free string endpoint must be treated through the original equations.

For the open string, choose conformal gauge ${\gamma }_{\mu \nu }=\mathrm{diag}(-1,1)$ on $0\le \sigma \le \pi$. The embedding equation is the wave equation, and NN means ${\partial }_{\sigma }{X}^a=0$ at both spatial endpoints. For these operator formulas restore the conventional overall normalization $S=-TI/2$, with $T=(2\pi \alpha \prime {)}^{-1}$; the overall factor does not change the preceding classical equations. Its open-string mode expansion isReality requires $({\alpha }_n^a{)}^{†}={\alpha }_{-n}^a$. Canonical quantization givesThe metric equation is the vanishing of the worldsheet stress tensor: $({\partial }_{\tau }X+{\partial }_{\sigma }X{)}^2=({\partial }_{\tau }X-{\partial }_{\sigma }X{)}^2=0$. The two endpoint-compatible mode expansions contain the same string oscillator family. Their quadratic coefficients are the classical Virasoro constraints.

For the operators, use normal ordering with positive-index string oscillators as annihilators:Thus $({\partial }_{\tau }X\pm {\partial }_{\sigma }X{)}^2$ has modes $4{\alpha }^{\prime }{L}_m$ before the quantum ordering correction. The physical-state Virasoro conditions for an open string areOne imposes only the positive modes on kets, with the adjoint conditions on bras, as in Gupta-Bleuler quantization. Requiring every positive and negative mode to annihilate the same state would conflict with the Virasoro central extension. The intercept $a$ is the zero-mode ordering constant. The standard critical bosonic string theory has $d=26$ and $a=1$; in light-cone gauge in string theory the transverse zero-point energy gives $a=(d-2)/24$, while full anomaly-free Lorentz or BRST quantization fixes the critical values. The mass constraint is then ${\alpha }^{\prime }{M}^2=N-a$, with $N={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$.

To compute the Virasoro algebra, commute a quadratic generator with one string oscillator:These identities and the Jacobi identity imply that $[L_m,L_n]-(m-n)L_{m+n}$ commutes with every string oscillator and with the center-of-mass coordinates and momenta. Mode number permits a scalar term only for $m+n=0$. Its coefficient follows from the formal zero-momentum Fock vacuum, on which $L_0=0$. For $m>0$,The two possible string oscillator contractions giveThe timelike target coordinate still contributes one to this central charge: its two metric signs cancel in ${\eta }_{ab}{\eta }^{ab}=d$. ConsequentlyThis is the free-boson Virasoro central term. It vanishes for the three global conformal modes $m=0,\pm 1$.

To keep the intercept convention separate, define ${\mathcal{L}}_m=L_m-a{\delta }_{m,0}$. The Virasoro zero-mode shift changes the displayed central term toThe matter central charge here is $d$, not $d-2$. Covariant worldsheet ghost fields contribute $-26$; their inclusion cancels the anomaly at $d=26$, while the intercept is handled by the appropriate zero-mode and physical-state convention.

The first term uses the induced worldvolume metric from the supervielbein; the second is the Wess-Zumino brane coupling to the super-three-form. Its fermionic dependence is retained before bosonic truncation. The relative charge magnitude equals the brane tension, while $q=\pm 1$ sets the orientation and matching kappa symmetry projector sign. In an on-shell eleven-dimensional supergravity superspace, this action has kappa symmetry.