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In a six-dimensional cubic scalar field theory, three labelled box Feynman diagrams give the one-loop local quartic vertex at zero external momentum. With an effective-action term $g_4{\varphi }^4/4!$, their contribution is $\delta g_4=-3g^4{I}_4$. To check its sign and multiplicity, expand the one-loop scalar effective action as $\frac{1}{2}\mathrm{Tr}\log (D+g\varphi )$, where $D=-{\partial }^2+m^2$. Its fourth-order term is $-g^4\mathrm{Tr}(D^{-1}\varphi {)}^4/8$, giving the stated coefficient. The associated amputated connected diagram insertion has the opposite sign.

For a cubic graph with $E$ external legs, $3V=2I+E$ and $L=I-V+1$. Its superficial degree of divergence is therefore $6L-2I=6-2E$. Divergences in vacuum, one-, two-, and three-point functions can be absorbed into vacuum energy, a linear term, mass, wave-function renormalization, and cubic-coupling counterterms. Higher-point proper diagrams are superficially convergent after subtraction of divergent subgraphs. This establishes perturbative renormalizability, without asserting a nonperturbative positive measure for the unstable real cubic potential.

For a reduced plane algebraic curve, an ordinary double point is a point whose lowest nonzero local homogeneous term has degree $2$ and factors into two distinct linear factors over an algebraic closure. The two factors give distinct tangent directions. In characteristic not $2$, $x^2+y^2+\text{higher terms}$ is an example.

In an attractor with one adiabatic clock, the leading squeezed primordial bispectrum is the response to a coordinate dilation. Holding a halo's physical mass and smoothing scale fixed cancels this dilation in its abundance response. Thus the consistency-relation term does not generate the physical $k^{-2}$ scale-dependent halo bias from local non-Gaussianity. For example, a top-hat filter of fixed physical radius has coordinate radius $R_c=e^{-{\zeta }_L}{R}_0$, while the short density contrast is ${\delta }_s(\mathbf{x})={\delta }_s^{(0)}(e^{{\zeta }_L}\mathbf{x})$. The substitution $\mathbf{y}=e^{{\zeta }_L}\mathbf{x}$ shows that its normalized average over $|\mathbf{x}|<R_c$ equals the original average over $|\mathbf{y}|<R_0$. Genuine local effects start at two spatial derivatives of the long mode. Non-attractor backgrounds or nonstandard initial states need not satisfy this argument.

In an augmented chain complex of a simplex with vertices $v_0,\dots ,v_n$, put $s(1)=[v_0]$ and $s[v_{i_0},\dots ,v_{i_q}]=[v_0,v_{i_0},\dots ,v_{i_q}]$ when $v_0$ is absent, and zero when it is present. Direct expansion of the boundary operator gives $ds+sd=1$. This proves that every positive-degree chain cycle is a chain boundary, without invoking cellular homology.

For the conventional Euclidean dilaton coupling $+(4\pi {)}^{-1}\int \sqrt{h}R\phi$ and no antisymmetric background, the metric worldsheet Weyl anomaly vanishes at first order in ${\alpha }^{\prime }$ when $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$. The one-loop Ricci term follows from the background field expansion of a string sigma model. Under a Weyl transformation, the dilaton coupling varies as $-(2\pi {)}^{-1}\int \sqrt{h}\omega {\square }_h\phi$; the embedding equations replace this by the target Hessian contribution. Full Weyl invariance also requires the dilaton curvature coefficient, including the central-charge deficit, to vanish. See the Lorentzian dilaton coupling sign convention when translating an action with a specified Lorentzian sign.

The conventional coefficient agrees with the background equations in Tong's string theory notes, Section 7.2.3.

For the conventional positive Euclidean dilaton curvature coupling, with no antisymmetric background field, the leading scalar Weyl coefficient is the displayed expression. The matter/ghost central charge gives $(D-26)/6$. In a geodesic background field expansion of a string sigma model, contracting $\frac{1}{2}{\nabla }_a{\nabla }_b\phi Y^a{Y}^b$ in the curvature coupling produces the counterterm $\delta \phi =-{\alpha }^{\prime }\square \phi /(2\epsilon )$ and hence the Laplacian term. The full embedding equation ${\square }_h{X}^a+{\Gamma }^a{}_{bc}\partial X^b\partial X^c=({\alpha }^{\prime }/2)R^{(2)}{\nabla }^a\phi$ supplies the gradient-square improvement in the dilaton Weyl variation. Setting this coefficient to zero fixes the constant in the dilaton equation from contracted Bianchi identity; the metric equation alone does not. The Lorentzian dilaton coupling sign convention determines the named-field translation.

The noncritical coefficient is also given in Tong's string theory notes, Section 7.4.4.

For an abelian group $A$, the constant sheaf has sections over an open set given by its locally constant $A$-valued functions. It is the sheafification of the constant presheaf, and its sections on a disconnected open set need not have a single common value.

An exact sequence of sheaves of abelian groups is a sequence of sheaf homomorphisms for which the image of each map equals the kernel of the next. Exactness can be checked on every stalk of a sheaf. Exactness of the associated sequences of sections on every open set is a stronger condition: a surjective sheaf homomorphism need only admit lifts locally.

A sheaf of abelian groups is injective if every morphism to it from a subsheaf extends to the containing sheaf. Every such sheaf embeds in an injective sheaf. Injective sheaves are flasque: the inclusion of extension by zero constant-integer sheaves associated to $U\subseteq V$ turns the extension property into surjectivity of restriction from $V$ to $U$.

If $0\to \mathcal{F}\to {\mathcal{A}}^0\to {\mathcal{A}}^1\to \cdots$ is an exact sequence of sheaves of abelian groups on $X$ and $H^q(X,{\mathcal{A}}^j)=0$ for every $q>0$ and $j\ge 0$, then sheaf cohomology is computed by the cochain complex of global sections:In particular a flasque resolution computes sheaf cohomology. Acyclicity is required on the space on which cohomology is being computed.

For a coherent sheaf on a Noetherian scheme, its sheaf cohomology vanishes in degrees greater than its support dimension. In particular, a zero-dimensional projective scheme has no positive-degree coherent cohomology.

For an ample line bundle $A$ on a projective scheme and a coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F}\otimes A^m)=0$ for every $i>0$ and all sufficiently large $m$.

On a proper scheme over a field, the Euler characteristic of a coherent sheaf is $\chi (X,\mathcal{F})={\sum }_i(-1{)}^i\dim H^i(X,\mathcal{F})$. The sum is finite by Grothendieck vanishing, and long exact sequences in sheaf cohomology make it additive in short exact sequences of sheaves.

An injective resolution is an exact sequence starting with a sheaf and continuing with injective sheaves. It is constructed by successively embedding each quotient in an injective sheaf. Taking global sections and then cohomology computes sheaf cohomology; in particular an injective sheaf has zero higher cohomology.

A coherent sheaf $\mathcal{F}$ on ${\mathbb{P}}^N$ is $r$-regular if $H^i({\mathbb{P}}^N,\mathcal{F}(r-i))=0$ for every $i>0$. It is then $(r+1)$-regular and $\mathcal{F}(r)$ is globally generated. Persistence follows by induction with a hyperplane avoiding the associated points and its restriction exact sequence. A sheaf on an embedded projective scheme is tested after pushforward to projective space.

A compact real form of a complex semisimple Lie algebra $\mathfrak{g}$ is a real Lie subalgebra $\mathfrak{u}$ whose complexification of a Lie algebra is $\mathfrak{g}$ and whose Killing form is negative definite. Its simply connected Lie group is compact. The positive internal inner product used in unitary gauge theory is therefore $-\kappa$. For ${\mathfrak{sl}}_2(\mathbb{C})$ the SU(2) Lie algebra is a compact real form, whereas ${\mathfrak{sl}}_2(\mathbb{R})$ is a different real form with indefinite Killing form.

An adapted càdlàg process is a semimartingale exactly when it is a good integrator: on each finite horizon, for every sequence of bounded elementary predictable processes with stopping-time intervals $H^n$ whose deterministic uniform bounds tend to zero, the elementary terminal integrals $(H^n\cdot X{)}_T$ tend to zero in probability. The elementary integral is a finite sum of measurable coefficients times subsequent increments; the characterization says precisely when this operation extends continuously to stochastic integration. The theorem is used here as a standard characterization, not proved from first principles.

If $X$ is a semimartingale under $P$ and $Q\ll P$, then it is a semimartingale under $Q$. The Bichteler-Dellacherie theorem gives a short proof: terminal elementary integrals associated with uniformly small elementary predictable processes with stopping-time intervals converge to zero in $P$-probability. Uniform convergence on compacts in probability under an absolutely continuous measure change, or its single-random-variable proof, transfers convergence to $Q$. The same good-integrator characterization now applies under $Q$. Adaptedness and càdlàg paths are preserved because $P$-null sets remain $Q$-null; if necessary use the usual $Q$-completion of the same filtration. If the integrands are bounded only outside $Q$-null sets, clip their coefficients to the specified deterministic bounds before applying the $P$-criterion; this preserves their $Q$-versions. Completion adds only equivalent versions of measurable coefficients. This theorem does not preserve the local martingale property.

A subsemigroup is a nonempty subset of a semigroup closed under its associative operation. In a compact Hausdorff space carrying a left-topological semigroup, descending chains of nonempty compact subsemigroups have a nonempty compact subsemigroup as their intersection. This supplies the minimal object used in the Ellis–Numakura lemma.