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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 $b_3(K3)=0$.

Take an open neighbourhood $\tilde{U}$ on which $p$ is one-to-one and a homeomorphism onto its image. By invariance of domain for topological surfaces, that image is open. Shrink $\tilde{U}$ so the image lies in a complex chart $(V,z)$ of $R$, and use $z\circ p$ as a chart of $\tilde{R}$. These charts cover $\tilde{R}$. On any overlap their transition is the transition between the original charts of $R$, restricted to an open set, so is holomorphic with holomorphic inverse. They define the desired complex structure, and in these charts $p$ is locally the identity. Thus $p$ is holomorphic. The openness supplied by invariance of domain is important when the hypothesis is phrased only as a homeomorphism onto the image.

Let $f:\mathbb{C}\to Y$ be holomorphic and let $\pi :\mathbb{D}\to Y$ be the stated covering. Choose a lift of $f(0)$. The covering-space lifting theorem says that a map from a simply connected, locally path-connected domain has a unique lift after fixing this initial point: lift paths from zero, and homotopy lifting makes their endpoint independent of path. Since $\mathbb{C}$ is simply connected, this constructs $\tilde{f}:\mathbb{C}\to \mathbb{D}$ with $\pi \circ \tilde{f}=f$. On a small neighbourhood, the lift is a biholomorphic local inverse of $\pi$ composed with $f$, so it is holomorphic. Its modulus is less than one, and Liouville theorem therefore makes it constant. Hence every holomorphic map $\mathbb{C}\to Y$ is constant.

The complex structure on a complex manifold splits its complexified cotangent bundle into the $+i$ and $-i$ eigenspaces, locally spanned by $d{z}^j$ and $d{\bar{z}}^j$. A differential form of type (p, q) is a smooth section of ${\bigwedge }^p(T^{1,0}X{)}^{\ast }\otimes {\bigwedge }^q(T^{0,1}X{)}^{\ast }$, so in holomorphic coordinates it has the expressionHere the increasing multi-indices avoid duplicating alternating factors. The exterior derivative has just two type components: the conjugate Dolbeault operator $\partial$ and the Dolbeault operator $\bar{\partial }$. On such a local expression they areHolomorphic changes of coordinates preserve these types, so the definitions are intrinsic. Comparing type components of $d^2=0$ gives ${\partial }^2={\bar{\partial }}^2=0$ and $\partial \bar{\partial }+\bar{\partial }\partial =0$.

A real (p, p)-form satisfies $\bar{\eta }=\eta$, where complex conjugation acts on both coefficients and differential factors. Conjugation exchanges the two Dolbeault operators, soThus $\bar{\partial }\eta =0$ forces $\partial \eta =0$ as well. Conversely, if $d\eta =0$, its distinct $(p+1,p)$ and $(p,p+1)$ components both vanish. ThereforeThe assertion remains valid in top degree, when the indicated type components are zero automatically.

For a holomorphic function $f$, $\bar{\partial }f=0$ and $\partial \bar{f}=0$. Writing its real part as $u=(f+\bar{f})/2$ givesHence the real part of a holomorphic function is a pluriharmonic function.

For the local exactness assertion, work on a smaller polydisc whose closure lies in the original one. The allowed one-variable integral is the Cauchy-Green operator $T_j$ in coordinate $z_j$. Applied coefficientwise, it satisfies ${\partial }_{{\bar{z}}_j}{T}_{j}a=a$ on a smaller disc and commutes with differentiation in all other coordinates; the smooth parameter dependence follows by applying the integral to a smooth compactly supported extension in the integration variable. This extension can be made with a cutoff equal to one on the smaller disc.

Begin with a Dolbeault operator closed form $\phi$ and decompose it uniquely as $\phi =d{\bar{z}}^1\wedge \alpha +\beta$, where neither $\alpha$ nor $\beta$ contains $d{\bar{z}}^1$. Set ${\psi }_1=T_1\alpha$. The $d{\bar{z}}^1$ component of $\bar{\partial }{\psi }_1$ is exactly $d{\bar{z}}^1\wedge \alpha$, sohas no $d{\bar{z}}^1$ factor. It is still $\bar{\partial }$-closed. The $d{\bar{z}}^1$ component of $\bar{\partial }{\phi }_1=0$ now shows that every coefficient of ${\phi }_1$ is holomorphic in $z_1$.

Apply the same construction in $z_2$ to remove its antiholomorphic differential factor. The operator $T_2$ preserves holomorphicity in $z_1$, since it commutes with ${\partial }_{{\bar{z}}_1}$. Consequently the second subtraction introduces no previously removed factor. Continue through all $n$ coordinates, shrinking the polydisc when necessary. The final remainder has positive antiholomorphic degree but contains none of the $d{\bar{z}}^j$, and hence is zero. This coordinate elimination for local Dolbeault primitives constructsIf $q>n$, the original form is already zero, so one can take $\psi =0$.

Finally let $\phi$ be a smooth $(0,1)$-form on the complex projective line. Choose $0<r<R$ and the two coordinate neighborhoods $U=\{|z|<R\}$ and $V=\{|z|>r\}\cup \{\infty \}$, the latter expressed in $w=1/z$. They cover the sphere. The one-variable construction, integrating over slightly larger closed discs in each coordinate chart, gives smooth functions $a$ on $U$ and $b$ on $V$ with $\bar{\partial }a=\phi {|}_U$ and $\bar{\partial }b=\phi {|}_V$.

Their difference $h=a-b$ is a holomorphic function on the annulus $r<|z|<R$. Its Laurent series splits asThe first series extends holomorphically throughout $U$, while the second extends throughout $V$, including infinity, where it has value zero. Thus $a-h_{+}$ and $b+h_{-}$ agree on the overlap and glue to a global smooth function $F$. Both corrections are holomorphic, soThis Laurent gluing of Dolbeault primitives on the projective line uses no Hodge theorem.

Let $g$ be the Riemannian metric underlying the Hermitian metric and $J$ its compatible complex structure. The fundamental form of a Hermitian manifold is $\omega (u,v)=g(Ju,v)$. It is a real alternating form of type $(1,1)$. At a point choose an orthonormal real coframe $x^1,y^1,\dots ,x^n,y^n$ adapted to $J$, with $x^j+i{y}^j$ of type $(1,0)$. ThenThe orientation here is the complex orientation. In coordinates for which $g={\sum }_j((d{x}^j{)}^2+(d{y}^j{)}^2)$, the same convention reads $\omega =(i/2){\sum }_{j}d{z}^j\wedge d{\bar{z}}^j$.

Use the complex Hodge star operator, the complex-linear extension of the real Hodge star operator. With the pointwise Hermitian inner product linear in its first variable, its defining identity isfor forms of the same total degree. It maps type $(p,q)$ to $(n-q,n-p)$ and satisfies ${\ast }^2=(-1{)}^{p+q}$ in real dimension $2n$. It commutes with complex conjugation. These conventions distinguish it from the conjugate-linear Hodge star $\alpha \mapsto \ast \bar{\alpha }$.

Write a top holomorphic-type form as $\eta =a{\zeta }^1\wedge \cdots \wedge {\zeta }^n$, where ${\zeta }^j=x^j+i{y}^j$. On each oriented two-plane, $\ast x^j=y^j$ and $\ast y^j=-x^j$, so the two-dimensional star of ${\zeta }^j$ is $-i{\zeta }^j$. In taking the star of their product, regrouping complementary one-form factors across the $n$ two-planes contributes $(-1{)}^{n(n-1)/2}$. Thus the Hodge star on top holomorphic forms isThis is pointwise and requires neither holomorphicity of the coefficient $a$ nor closedness of $\omega$.

To compute the formal adjoint, take compactly supported forms $\alpha$ of degree $k-1$ and $\beta$ of degree $k$. The Stokes theorem givesSince the degree of $d(\ast \bar{\beta })$ is $2n-k+1$, the identity for ${\ast }^2$ makes the right side equal to ${\int }_X\alpha \wedge \ast \overline{(-\ast d\ast \beta )}$. Consequently the formal adjoint of $d$ in even real dimension is $d^{\ast }=-\ast d\ast$. Splitting by type, $-\ast \partial \ast$ lowers antiholomorphic degree, while $-\ast \bar{\partial }\ast$ lowers holomorphic degree. Orthogonality of distinct types therefore yieldsThe argument applies locally with compact support, and globally without support restrictions on a compact manifold without boundary.

The Dolbeault Laplacian is ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$. A form is $\bar{\partial }$-harmonic when it lies in its kernel. On a compact Hermitian manifold, the adjoint identities implyThus harmonicity is equivalent to both first-order equations vanishing. The Dolbeault Hodge decomposition on a compact Hermitian manifold states that the harmonic space ${\mathcal{H}}_{\bar{\partial }}^{p,q}$ is finite-dimensional and that smooth forms have the orthogonal direct-sum decomposition

To deduce the identification with Dolbeault cohomology, let $\bar{\partial }\alpha =0$ and decompose $\alpha =h+\bar{\partial }u+{\bar{\partial }}^{\ast }v$. The closedness of $\alpha$ makes it orthogonal to the entire image of ${\bar{\partial }}^{\ast }$. Taking its inner product with ${\bar{\partial }}^{\ast }v$ and using the orthogonality of the decomposition gives $\Vert {\bar{\partial }}^{\ast }v{\Vert }^2=0$. Hence $\alpha =h+\bar{\partial }u$, so each cohomology class has a harmonic representative. If a harmonic form equals $\bar{\partial }u$, its squared norm is $⟨{\bar{\partial }}^{\ast }h,u⟩=0$, so the representative is unique. We have proved

For the complementary types, define $C\alpha =\ast \bar{\alpha }$. It maps $(p,q)$ to $(n-p,n-q)$ and is conjugate-linear. If $\alpha$ is harmonic, then $\bar{\partial }\alpha =0$ and $\partial (\ast \alpha )=0$ by the adjoint formula. Conjugating the latter equation gives $\bar{\partial }(\ast \bar{\alpha })=0$. Also, with $k=p+q$,since $\partial \bar{\alpha }=\overline{\bar{\partial }\alpha }$. Therefore $C$ carries harmonic forms to harmonic forms, and $C^2=(-1{)}^k$ makes it a bijection. This proves complementary Dolbeault harmonic types on a Hermitian manifold:The metric supplies a natural conjugate-linear isomorphism. If a complex-linear isomorphism is desired, choose a basis $h_j$ of the first space and send it to the basis $C{h}_j$ of the second, extending linearly. This last choice is not canonical; no Kähler assumption is needed for either the dimension equality or the harmonic bijection.

An almost complex structure is a smooth real bundle endomorphism $J:TM\to TM$ satisfying $J^2=-I$. In particular the real dimension is even, say $2m$. After complexifying, the type decomposition of the complexified tangent bundle isThe corresponding differential forms of type (p, q) are smooth sections of ${\Lambda }^p(T^{1,0}M{)}^{\ast }\otimes {\Lambda }^q(T^{0,1}M{)}^{\ast }$, regarded as alternating forms by the wedge product of differential forms. A form of type $(p,q)$ vanishes unless its arguments include $p$ vectors of type $(1,0)$ and $q$ of type $(0,1)$.

Besides vanishing of the Nijenhuis tensor, two equivalent descriptions of an integrable almost complex structure are: the smooth sections of $T^{0,1}M$ are closed under Lie bracket; and every point has complex coordinates in which $J$ is multiplication by $i$, with holomorphic changes of coordinates. The equivalence with existence of these coordinates is the Newlander-Nirenberg theorem for smooth $J$. A further equivalent condition is that the exterior derivative has only the two type components $d=\partial +\bar{\partial }$, with no components of bidegree $(2,-1)$ or $(-1,2)$. Indeed evaluation of $d$ of a $(1,0)$-form on two $(0,1)$ vector fields detects the $(1,0)$ component of their bracket; complex conjugation detects the other component.

For the almost complex structure induced by a complex atlas, write $z^j=x^j+i{y}^j$ and put $J{\partial }_{x^j}={\partial }_{y^j}$, $J{\partial }_{y^j}=-{\partial }_{x^j}$. The differentials of holomorphic transition maps are complex linear, so these local definitions agree. In these coordinates $J$ has constant coefficients and all coordinate vector fields commute. The expression defining the Nijenhuis tensor is tensorial, so evaluating it on this frame proves it is zero everywhere. Thus the induced structure is integrable.

Here is a direct proof for the smooth submanifolds with invariant complex tangent spaces are complex assertion. At a point of $Y$, its tangent space is a complex-linear subspace, of real dimension $2k$. Make a complex-linear change of ambient holomorphic coordinates so that this tangent space is ${\mathbb{C}}^k\times 0$. Projection onto the first $k$ coordinates has invertible real differential on $Y$. The real inverse function theorem states that a smooth map between equal-dimensional manifolds with invertible differential is a diffeomorphism on sufficiently small neighbourhoods. It therefore writes $Y$ locally as a smooth graph $w=F(z)$ over an open set in ${\mathbb{C}}^k$.

A tangent vector to this graph is $(v,dF(v))$. Its image under $J$ is $(iv,idF(v))$. Tangent-space invariance impliesso $F$ satisfies the Cauchy-Riemann equations. Smooth functions satisfying these equations are holomorphic: restrict to each coordinate complex line, use the one-variable criterion, and apply the iterated Cauchy integral formula on a small polydisc to get jointly holomorphic coefficients. Hence the graph parametrization is holomorphic. Restrictions of ambient holomorphic coordinates supply holomorphic transition maps between these graph charts. The restricted structure on $Y$ is an integrable complex structure. This also proves the induced structure agrees with the original restriction, rather than merely producing some complex structure on $Y$.

A nonsingular analytic subvariety is a closed analytic subset which near each of its points is the zero set of a holomorphic submersion of constant rank; equivalently, it is locally a coordinate subspace in holomorphic coordinates. The holomorphic implicit function theorem says that if a holomorphic map to ${\mathbb{C}}^r$ has differential of rank $r$ at a zero, its nearby zero set is a complex submanifold of codimension $r$.

The homogeneous equation defines a subset of Complex projective space because multiplying a vector by $\lambda \ne 0$ multiplies its quadratic value by ${\lambda }^2$. Its gradient on ${\mathbb{C}}^4$ is $2Qz$, which is nonzero for $z\ne 0$. More explicitly, in an affine chart set $z_i=1$. If all derivatives of the restricted equation vanished at one of its zeros, all components of $2Qz$ except possibly the $i$th would vanish. The Euler homogeneous function theorem gives ${\sum }_j{z}_j{\partial }_{j}p=2p=0$, forcing the remaining component to vanish as well. This contradicts invertibility of $Q$. The holomorphic implicit function theorem thus proves $S$ is a smooth complex surface.

To see its geometry, reduce the nondegenerate symmetric bilinear form to diagonal form over $\mathbb{C}$. There is a vector of nonzero self-pairing, since otherwise polarization would make the entire form zero. Normalize such a vector using a square root of its self-pairing. Its orthogonal complement remains nondegenerate, so induction gives an orthonormal basis. Apply the same argument to the nondegenerate form $w_0{w}_3-w_1{w}_2$: the two forms are equivalent under an invertible linear change of variables. The allowed projective linear transformation therefore reduces $S$ toView its coordinates as the nonzero rank-one matrix $A=\left(\begin{array}{cc}{w}_0& w_1\\ w_2& w_3\end{array}\right)=u{v}^T$. At $[A]$, holding $[u]$ fixed and varying $[v]$ gives a projective line, and holding $[v]$ fixed and varying $[u]$ gives another. They are distinct and meet at $[A]$. In fact these are the only lines through it: if a second rank-one matrix is $a{b}^T$, the entire pencil has determinant zero only if $\det (u,a)\det (v,b)=0$, so one projective factor is fixed. These are the two rulings of a smooth quadric surface.

The Segre embedding is the holomorphic mapEvery nonzero rank-one matrix has such a factorization, unique up to reciprocal rescaling, so $\Phi$ is bijective. Its inverse is holomorphic on each open set where a matrix entry $a_{ij}\ne 0$: the projective column $[a_{0j}:a_{1j}]$ recovers $[u]$, and the projective row $[a_{i0}:a_{i1}]$ recovers $[v]$. These formulas agree on overlaps. Thus the Segre description of a smooth quadric surface gives the promised stronger conclusion

Let $g$ be the real Riemannian metric underlying the Hermitian metric, and let $J$ be the complex structure. The fundamental form of a Hermitian manifold isThe identity $g(Ju,Jv)=g(u,v)$ implies $g(Ju,v)=-g(u,Jv)$, so this is an alternating real two-form. Moreover $\omega (Ju,Jv)=\omega (u,v)$. A two-form with this invariance has type $(1,1)$: on two vectors of type $(1,0)$, applying $J$ to both multiplies its value by $i^2=-1$, forcing that value to vanish; the same argument applies to type $(0,1)$. Hence $\omega$ is a real $(1,1)$-form.

Use the complex-linear convention for the complex Hodge star operator, extending the real star. The complex orientation and metric define $dV={\omega }^n/n!$, andHere the Hermitian inner product is linear in the first argument. With this convention, on the even-dimensional underlying real manifold the formal adjoints are $d^{\ast }=-\ast d\ast$, ${\bar{\partial }}^{\ast }=-\ast \partial \ast$ and ${\partial }^{\ast }=-\ast \bar{\partial }\ast$. A conjugate-linear star is another convention, but then the adjoint formulas must change accordingly.

The Hodge Laplacian and Dolbeault Laplacian areAlso write ${\Delta }_{\partial }=\partial {\partial }^{\ast }+{\partial }^{\ast }\partial$. They are nonnegative, since for exampleThe requested Dolbeault Hodge decomposition on a compact Hermitian manifold is a compactness statement: on a compact Hermitian manifold without boundary,The harmonic space is finite-dimensional, and taking the harmonic representative induces an isomorphism ${\mathcal{H}}_{\bar{\partial }}^{p,q}\cong H_{\bar{\partial }}^{p,q}(X)$. Compactness is essential to this formulation; it is not asserted on an arbitrary noncompact Hermitian manifold.

Now suppose $X$ is compact and Kähler, so $d\omega =0$. The permitted Kähler Laplacian identity is ${\Delta }_d=2{\Delta }_{\bar{\partial }}=2{\Delta }_{\partial }$, making the three harmonic spaces identical in each bidegree. A $d$-closed pure-type form has $\partial \alpha =\bar{\partial }\alpha =0$, since these two terms have different types. We will construct a Green operator potential for ddbar exactness, rather than invoking the ddbar lemma as the conclusion to be proved.

Let $G$ be the inverse of ${\Delta }_{\bar{\partial }}$ on the orthogonal complement of its harmonic forms, zero on the harmonic space. It is twice the Green operator of the Hodge Laplacian. The Hodge decomposition theorem and elliptic inversion give ${\Delta }_{\bar{\partial }}G=I-H$, with $H$ harmonic projection. It preserves bidegrees. The operators $\partial$ and $\bar{\partial }$ commute with ${\Delta }_{\bar{\partial }}$: the first follows from ${\Delta }_{\bar{\partial }}={\Delta }_{\partial }$ and ${\partial }^2=0$, and the second directly from ${\bar{\partial }}^2=0$. They and their formal adjoints annihilate harmonic forms, so they preserve the orthogonal complement of the harmonic space. Consequently they commute with $G$, by uniqueness of the inverse on the harmonic complement. Thus $u=G\alpha$ and $v=G^2\alpha$ are killed by both differential operators.

The assumed orthogonality makes $H\alpha =Hu=0$. Applying the two Laplacians givesWe also need the mixed anticommutator, which follows from the permitted Laplacian identity rather than an extra unproved lemma. Expand ${\Delta }_d$ using $d=\partial +\bar{\partial }$ and $d^{\ast }={\partial }^{\ast }+{\bar{\partial }}^{\ast }$. Since ${\Delta }_d={\Delta }_{\partial }+{\Delta }_{\bar{\partial }}$, its mixed terms sum to zero. Their two bidegrees $(1,-1)$ and $(-1,1)$ are distinct, so each vanishes separately, in particular ${\bar{\partial }}^{\ast }\partial +\partial {\bar{\partial }}^{\ast }=0$. HenceWe have obtained the explicit answerThe positive assumptions on $p,q$ ensure these degrees are available. Conversely such a form is orthogonal to harmonic forms by integration by parts, so the same proof also explains the harmonic orthogonality criterion for ddbar exactness.

For injectivity of the Lefschetz operator on a Hermitian manifold on real one-forms, work at one point in a metric-orthonormal coframe $(e^1,f^1,\dots ,e^n,f^n)$ with $\omega ={\sum }_j{e}^j\wedge f^j$. If $a\ne 0$, a unitary change of coframe makes $a=c{e}^1$, $c\ne 0$. Thensince the displayed basis three-forms are linearly independent. Therefore $L(a)=a\wedge \omega$ has zero kernel. Injectivity on complex one-forms follows by taking real and imaginary parts, since $\omega$ is real. This pointwise proof of injectivity of powers of the Lefschetz operator for degree one does not require the hard Lefschetz theorem.

Closedness of $\omega$ gives $\bar{\partial }L=L\bar{\partial }$. Using the permitted Kähler identities, specifically $[{\bar{\partial }}^{\ast },L]=i\partial$, givesMultiplying by two proves $[{\Delta }_d,L]=0$. In particular the Lefschetz operator preserves harmonic forms and the injective map on one-forms restricts to an injection ${\mathcal{H}}_d^1\hookrightarrow {\mathcal{H}}_d^3$. The de Rham Hodge theorem identifies these real harmonic spaces with the corresponding real cohomology groups. Taking dimensions yields

Choose a Riemannian metric $h$ on $\Sigma$ compatible with its complex structure $j$, and use $g_J=\omega (\cdot ,J\cdot )$ on the target. The Dirichlet energy of a map isIt is independent of the particular conformal representative $h$: rescaling $h$ multiplies the squared differential norm by the inverse factor and the area element by the same factor. For a J-holomorphic curve, the Energy identity for a J-holomorphic curve isTo prove it, take an oriented $h$-orthonormal frame $e_1,e_2=j{e}_1$ and put $a=du(e_1)$, $b=du(e_2)$. The J-holomorphic curve equation says $b=Ja$. The pointwise energy density is therefore $\frac{1}{2}(|a{|}_{g_J}^2+|Ja{|}_{g_J}^2)=|a{|}_{g_J}^2$, while the pulled-back area density is $\omega (a,Ja)=|a{|}_{g_J}^2$. Integrating proves the identity.

More generally, the same frame givesWith the full tensor norm of ${\bar{\partial }}_{J}u$, its two frame components are $(a+Jb)/2$ and $(b-Ja)/2$, so their squared norms sum to $\frac{1}{2}|b-Ja{|}^2$. Hence the full identity isThis also fixes the normalization of the error term. The energy is nonnegative and vanishes exactly when $du=0$. The formulas apply whenever the relevant integrals are defined, in particular on compact source surfaces.

The unheaded definitions use the complex structure $J$ on the real tangent bundle. Compatibility means $g(Ju,Jv)=g(u,v)$. Its fundamental Hermitian form is $\omega (u,v)=g(Ju,v)$; the compatibility identity makes this real, alternating and of type $(1,1)$. The metric is a Kähler metric precisely when $d\omega =0$. In complex dimension one a real three-form is zero, so every compatible metric is a Kähler metric.

Now assume closedness and prove the Kähler normal holomorphic coordinates condition. Start with holomorphic coordinates $w$ centered at the point and make a complex-linear change so that the positive Hermitian coefficient matrix satisfies $h(0)=I$. Closedness of the $(1,1)$ form givesDefine $a_{i\bar{j},k}={\partial }_{w_k}{h}_{i\bar{j}}(0)$; it is symmetric in $i,k$. Choose new coordinates implicitly byThe derivative of this holomorphic map at zero is the identity, so the holomorphic inverse function theorem makes $z$ valid local coordinates. In the new coordinates,At zero, $h^{\prime }=I$, and differentiating givesHermitian symmetry makes all antiholomorphic first derivatives zero as well. Taylor's theorem for a smooth function now yieldsThus condition (a) implies condition (b), with the exact factor $i/2$ in the fundamental-form convention retained.

Take an oriented closed string in flat, critical bosonic string theory, with ${\alpha }^{\prime }=1/(2\pi T)$ and a mostly-plus target Minkowski metric. After continuation of the worldsheet to Euclidean signature, the Polyakov path integral sums over embeddings $X$ and worldsheet metrics, divided by worldsheet diffeomorphisms and Weyl transformations. In a flat target its kinetic action isTarget-time continuation or analytic continuation of external momenta defines the Lorentzian scattering amplitude; a naive real Euclidean Gaussian for timelike $X^0$ would not be convergent.

An external tachyon is represented by the tachyon vertex operator $V_p(z,\bar{z})=:e^{ip\cdot X}:$, with conformal weights $h=\bar{h}={\alpha }^{\prime }{p}^2/4$. The physical integrated vertex has $(h,\bar{h})=(1,1)$, so $p^2=4/{\alpha }^{\prime }$. Schematically its tree amplitude isThe Faddeev-Popov determinant from conformal gauge is represented by the worldsheet ghost fields. At tree level the worldsheet is a Riemann sphere, whose unpunctured complex structure has no moduli. Its residual conformal automorphisms are the Möbius transformations, $PSL(2,\mathbb{C})$. Fix three insertion points, accompanying their unintegrated vertices by the required $c\tilde{c}$ ghost factors. The remaining $N-3$ complex insertion positions are integrated over the sphere; equivalently one integrates all positions and divides by the residual conformal group.

The embedding fields are free, withTheir zero-mode integral gives momentum conservation, and their nonzero-mode Gaussian integral gives the Koba-Nielsen factorThis explains the sphere tachyon position integral without needing its evaluation. For four tachyons the resulting Virasoro–Shapiro amplitude displays the exchanged string spectrum directly.

Use all-incoming external momenta and introduce $u=-(p_1+p_4{)}^2/(8\pi T)$ alongside the two printed dimensionless Mandelstam variables. Since $p_a^2=8\pi T$ and ${\sum }_a{p}_a=0$, one obtains $s+t+u=-4$. The physical center-of-mass energy squared in the $s$ channel is $s_{\mathrm{phys}}=8\pi Ts=4s/{\alpha }^{\prime }$. The $t$ channel measures the analogous crossed momentum transfer, with sign determined by the mostly-plus convention. Rewriting the Gamma factors in a symmetric form givesThe Gamma function poles imply, at generic fixed values of the other invariant,The denominator Gamma factors can remove residues at special intersecting channel kinematics; the statement concerns a generic single-channel limit. An $s$-channel pole occurs when the intermediate momentum $p_1+p_2$ satisfies the mass-shell condition for a closed-string state:The pole at $s=-1$ exchanges the ground-state tachyon, the pole at $s=0$ exchanges massless states, and the positive integer poles exchange the infinite massive tower. The $t$-channel interpretation is the crossed version. Factorization means that each residue is a sum of products of couplings to intermediate physical states that couple to the chosen external particles. It need not expose every representation at that mass.

For example, the Gamma function recurrence and Gamma function residue at a nonpositive integer give the dimensionless Virasoro–Shapiro amplitude pole residueThe residue polynomial has degree $2n$, consistent with maximum spin $2n$ in the exchanged level. The amplitude also has the corresponding $u$-channel poles by crossing symmetry.

The massless fields can be treated as target backgrounds rather than separate asymptotic insertions. Write the target metric as $G_{mn}={\eta }_{mn}+h_{mn}$, introduce a Kalb–Ramond field $B_{mn}$, and a dilaton $\Phi$. In conventional Euclidean signs their string nonlinear sigma model action isHere ${ϵ}^{ab}$ is the antisymmetric tensor density. The three backgrounds correspond to the graviton, antisymmetric tensor and scalar states at closed-string level one. Expanding the vacuum functional in $h,B,\Phi$, then Fourier expanding the backgrounds, produces exactly their integrated string vertex operators. Its functional derivatives therefore generate the amplitudes with massless external strings. The connected vacuum functional organizes connected amplitudes; the spacetime effective action organizes the corresponding vertices after treating massless propagation consistently.

At momenta small compared with $1/\sqrt{{\alpha }^{\prime }}$, massive string propagators can be expanded in powers of momenta over their masses. The analytic part of the amplitudes consequently determines local higher-derivative interactions, ordered by powers of ${\alpha }^{\prime }$. Massless exchange poles are retained through propagation of the massless fields, rather than expanded into local contact terms. Up to field redefinitions, the leading massless-sector action in the string-frame metric isThe terms denoted $O({\alpha }^{\prime })$ contain additional derivatives, including curvature-squared terms in the bosonic theory. The expansion concerns the massless sector around the perturbative bosonic background; the tachyon instability remains and is not cured by omitting its field from this displayed action. Thus this is a formal perturbative effective description, not a claim of a stable bosonic vacuum.

Finally split the dilaton into a constant and its variation, $\Phi ={\Phi }_0+\varphi$. By the Gauss-Bonnet theorem, its dilaton Euler-characteristic weighting on a connected closed oriented surface follows fromDefine the string coupling by $g_s=e^{{\Phi }_0}$. A genus-$g$ path integral is weighted byThe sphere carries $g_s^{-2}$, the torus carries $g_s^0$, and each extra handle adds $g_s^2$. At higher genus one integrates over complex-structure moduli as well as insertion points, with the associated antighost insertions supplying the correct moduli measure. With canonically normalized external vertices an $N$-point genus-$g$ amplitude scales as $g_s^{2g-2+N}$.

Summing connected worldsheets of every genus of a surface yields the string-loop effective action expansionEach coefficient has its own low-energy ${\alpha }^{\prime }$ expansion. The two parameters have different roles: ${\alpha }^{\prime }$ resolves finite string size through higher derivatives, while $g_s^2$ counts additional string loops.