Heterotic massless matter and bundle-cohomology dictionary

Adopt the following physical input for a selected weakly coupled, large-radius perturbative heterotic compactification on a smooth compact Calabi–Yau threefold $X$ with holomorphic gauge bundle $P\to X$. Use the compact structure group $H$, its complexification $H_{\mathbb{C}}$ and embedding $\iota :H\hookrightarrow G_{\mathrm{h}\mathrm{e}\mathrm{t}}$ supplied by the compactification datum. Define the unbroken gauge group to be the centralizer $K:=\{k\in G_{\mathrm{h}\mathrm{e}\mathrm{t}}:k\iota (h)=\iota (h)k\text{ for every }h\in H\}$. Fix an actual decomposition of the complexified ten-dimensional gauge Lie algebra, under the commuting adjoint actions of $H$ and $K$, as ${\mathfrak{g}}_{\mathrm{h}\mathrm{e}\mathrm{t}}{\otimes }_{\mathbb{R}}\mathbb{C}\cong {⨁}_a(W_a{\otimes }_{\mathbb{C}}{U}_a)$. Here each $U_a$ carries an irreducible finite-dimensional complex representation $R_a$ of $K$, and each $W_a$ is its full multiplicity space carrying an $H$ representation with a specified holomorphic extension ${\rho }_a:H_{\mathbb{C}}\to \mathrm{G}\mathrm{L}(W_a)$. Summands with isomorphic $K$ representations are combined into one full multiplicity space. Select a conjugate pair of complex matter sectors with $R=R_a$ not isomorphic to its conjugate $\bar{R}$, and require the paired structure representation to be ${\rho }_a^{\vee }$; the conjugate representation acts on the conjugate vector space by complex-conjugate matrices. Self-conjugate gauge representations require their own reality constraints and are not counted as two distinct conjugate species here. Write ${\rho }_R:={\rho }_a$ as a sector label: $R$ is a representation of $K$, whereas ${\rho }_R$ is a representation of $H_{\mathbb{C}}$. Fix orientation and four-dimensional chirality convention. Let $g_{ij}$ be the holomorphic transition maps of $P$ in local trivializations; $V_R$ is the holomorphic bundle with transition maps $(x,v)\mapsto (x,{\rho }_R(g_{ij}(x))v)$. The subbundle $T_X^{0,1}\subset TX{\otimes }_{\mathbb{R}}\mathbb{C}$ is the $-i$ eigensubbundle of the complex-linear extension of the complex structure. For each $q=0,1,2,3$, let ${\mathcal{A}}^{0,q}(X,V_R)={\mathrm{\Gamma }}^{\mathrm{\infty }}(X,{\mathrm{\Lambda }}^q(T_X^{0,1}{)}^{\ast }\otimes V_R)$ be the space of smooth global $(0,q)$-forms with values in $V_R$. Its global Dolbeault operator ${\bar{\mathrm{\partial }}}_{V_R}:{\mathcal{A}}^{0,q}(X,V_R)\to {\mathcal{A}}^{0,q+1}(X,V_R)$ acts in a local holomorphic frame by ${\bar{\mathrm{\partial }}}_{V_R}({\sum }_J{s}_{J}d{\bar{z}}^J)={\sum }_{j,J}(\mathrm{\partial }{s}_J\mathrm{/}\mathrm{\partial }{\bar{z}}^j)d{\bar{z}}^j\wedge d{\bar{z}}^J$. Holomorphic transition functions make these local operators agree on overlaps, and ${\bar{\mathrm{\partial }}}_{V_R}^2=0$. Define $H_{\bar{\mathrm{\partial }}}^q(X,V_R):=\ker ({\bar{\mathrm{\partial }}}_{V_R}:{\mathcal{A}}^{0,q}\to {\mathcal{A}}^{0,q+1})\mathrm{/}\mathrm{im}({\bar{\mathrm{\partial }}}_{V_R}:{\mathcal{A}}^{0,q-1}\to {\mathcal{A}}^{0,q}),h^q(X,V_R):={\dim }_{\mathbb{C}}{H}_{\bar{\mathrm{\partial }}}^q(X,V_R),$ with ${\mathcal{A}}^{0,-1}={\mathcal{A}}^{0,4}=0$. In the library's declared chirality convention, when the endpoint groups $H_{\bar{\mathrm{\partial }}}^0$ and $H_{\bar{\mathrm{\partial }}}^3$ vanish, degree $q=1$ in $V_R$ counts multiplets in $R$ and degree $q=2$ counts multiplets in the conjugate $K$ representation $\bar{R}$, counted with opposite sign in the net chiral index; their multiplicities are $h^1(X,V_R)$ and $h^2(X,V_R)$, respectively. Both counts use the same chosen four-dimensional chiral-multiplet convention. Replacing $V_R$ by its dual selects the paired adjoint sector and exchanges the two representation assignments. This degree-to-representation assignment, like the weak-coupling and large-radius approximation, is an adopted physical input, not a theorem of pure mathematics. The postulate applies only in the stated smooth large-radius phase and does not identify all string vacua with smooth bundle compactifications.

Precisely, use natural units and a constant dimensionless dilaton ${\mathrm{\Phi }}_0$, with dimensionless coupling $g_{\mathrm{s}\mathrm{t}\mathrm{r}}=e^{{\mathrm{\Phi }}_0}>0$. Fix a smooth positive internal metric $g_0$ in dimensionless coordinates on $X$, the complex structure and holomorphic bundle data, and use the string-frame metrics $g_L=L^2{g}_0$, where $L>0$ has length dimension and ${\alpha }^{\mathrm{\prime }}>0$ has length-squared dimension. Large radius means the formal limit $\epsilon ={\alpha }^{\mathrm{\prime }}\mathrm{/}{L}^2\to 0$ at this fixed smooth shape, rather than large volume with collapsing cycles. Weak coupling means $g_{\mathrm{s}\mathrm{t}\mathrm{r}}\to 0$. Adopt the dictionary at leading tree level and in the two-derivative, linearized massless-mode approximation in this joint regime; loop, higher-derivative and nonperturbative corrections to this mode interpretation are omitted. No numerical smallness threshold, finite-parameter error bound or guarantee that an unprotected massless state persists after corrections is asserted. The finite-dimensional multiplicities and their representation assignment are assumptions of this leading-order model.

Perturbative heterotic Calabi–Yau bundle compactification data

Associated holomorphic bundle in a heterotic representation

E-valued Dolbeault operators and their L2 structure

Heterotic Type I Ten Dimensional N1 Supergravity Action

This is a postulate, not an experimentally established result or a mathematical theorem. The signed index in the next item uses the assignment fixed in the Postulate; exchanging the associated bundle with its dual reverses the sign. CHSW and Uranga use different representation-sign conventions, so a source's signed count must be translated through its stated assignment. The examples retain the absolute net-generation count when the source's representation convention is not fixed.