Perturbative heterotic Calabi–Yau bundle compactification data

Let $X$ be a smooth, connected, compact complex threefold (complex dimension three) with complex structure $J$, specified Kähler form $\omega$, nowhere-zero holomorphic three-form $\mathrm{\Omega }\in H^0(X,K_X)$, complex orientation, and integral-cohomology convention. Let $G_{\mathrm{h}\mathrm{e}\mathrm{t}}$ be one chosen heterotic gauge group, either $E_8\times E_8$ or $\mathrm{Spin}(32)\mathrm{/}{\mathbb{Z}}_2$. Fix a compact Lie group $H$, meaning a compact finite-dimensional smooth manifold with smooth group multiplication and inversion, and write $\mathfrak{h}$ for its real Lie algebra. Choose a complex Lie group $H_{\mathbb{C}}$ as defined in def-cmp-heterotic-associated-vector-bundle, with Lie algebra $\mathfrak{h}{\otimes }_{\mathbb{R}}\mathbb{C}$, and an injective smooth homomorphism $j_H:H\to H_{\mathbb{C}}$ with closed image such that $d{j}_H(\mathfrak{h})$ is a real Lie subalgebra whose complexification is $\mathfrak{h}{\otimes }_{\mathbb{R}}\mathbb{C}$. Let $G_{\mathrm{h}\mathrm{e}\mathrm{t},\mathbb{C}}$ be a chosen complex Lie group, with an injective smooth homomorphism $j_G:G_{\mathrm{h}\mathrm{e}\mathrm{t}}\to G_{\mathrm{h}\mathrm{e}\mathrm{t},\mathbb{C}}$ having closed image whose Lie algebra complexifies to that of $G_{\mathrm{h}\mathrm{e}\mathrm{t},\mathbb{C}}$. Fix injective homomorphisms $\iota :H\to G_{\mathrm{h}\mathrm{e}\mathrm{t}}$ and ${\iota }_{\mathbb{C}}:H_{\mathbb{C}}\to G_{\mathrm{h}\mathrm{e}\mathrm{t},\mathbb{C}}$, with ${\iota }_{\mathbb{C}}$ holomorphic and ${\iota }_{\mathbb{C}}\circ j_H=j_G\circ \iota$. A holomorphic principal $H_{\mathbb{C}}$-bundle $P\to X$ is specified by an open cover $(U_i{)}_{i\in I}$ and holomorphic transition maps $g_{ij}:U_i\cap U_j\to H_{\mathbb{C}}$ satisfying $g_{ii}=e$ and $g_{ij}{g}_{jk}=g_{ik}$; these maps glue the local products $U_i\times H_{\mathbb{C}}$ using the right group action. A compact $H$ reduction is a principal $H$-subbundle $P_H\subset P$ whose extension of structure group to $H_{\mathbb{C}}$ is $P$. For each selected finite-dimensional holomorphic representation ${\rho }_R:H_{\mathbb{C}}\to {\mathrm{G}\mathrm{L}}_{\mathbb{C}}(W_R)$, define $V_R$ by the associated-bundle construction in def-cmp-heterotic-associated-vector-bundle. A perturbative heterotic Calabi–Yau bundle compactification datum consists of $(X,J,\omega ,\mathrm{\Omega },P)$, the representation and gauge-group embedding, and an explicitly supplied smooth principal connection $A$ on $P_H$. Require compatibility with the specified holomorphic bundle $P$: if $A_{\mathbb{C}}$ is the extended connection on $P$, its local connection form in every supplied holomorphic trivialization has $(0,1)$ part zero. Equivalently its induced $(0,1)$ covariant derivative is the given holomorphic structure; in every selected representation this requires ${\bar{\mathrm{\partial }}}_{A_R}={\bar{\mathrm{\partial }}}_{V_R}$, where $A_R$ is the induced connection on $V_R$. In the declared matrix representation its local connection form has curvature $F_A=dA+A\wedge A$; via $\iota$ regard $A$ and $F_A$ as fields in the chosen heterotic gauge algebra, and evaluate the gauge trace in its declared representation. The connection satisfies the Hermitian–Yang–Mills equations $F_A^{0,2}=0$ and ${\omega }^2\wedge F_A=0$ entrywise in the declared normalization. A supplied smooth internal three-form $H_3\in {\mathrm{\Omega }}^3(X)$ defines the tangent connection ${\mathrm{\nabla }}_M^{+}{V}^N={\mathrm{\nabla }}_M^{\mathrm{L}\mathrm{C}}{V}^N+\frac{1}{2}(H_3{)}_M{}^N{}_P{V}^P,$ whose curvature is denoted $R_{+}$. These data satisfy the local source-free Green–Schwarz relation $d{H}_3=\frac{{\alpha }^{\mathrm{\prime }}}{4}({\mathrm{Tr}}_H(F_A\wedge F_A)-{\mathrm{tr}}_H(R_{+}\wedge R_{+}))\mathrm{.}$ The traces, field basis, perturbative order in ${\alpha }^{\mathrm{\prime }}$, weak-coupling and large-radius regime, dilaton and frame, warping ansatz, and localized-source data are declared inputs; the displayed Bianchi identity is source-free on the stated patch. In the natural-unit convention with length-valued coordinates, $[{\alpha }^{\mathrm{\prime }}]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^2$, the connection components have dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$, the curvature components have dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-2}$, and $H_3$ components have dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$, so each term in $d{H}_3$ has component dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-2}$. For a rank-$r>0$ holomorphic bundle $E$, use ${\mu }_{\omega }(E)=r^{-1}{\mathrm{deg}}_{\omega }(E)$ with the slope convention of def-cmp-heterotic-slope. Stability is not substituted for an HYM connection, and this local source-free Bianchi identity does not supply global anomaly cancellation. The heterotic branch is distinct from the Type-I branch; this definition does not infer S-duality.

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