Physlib.Mathematics.Calculus.Wirtinger.Hessian

For a complex-valued function $f: \mathbb{C}^\iota \to \mathbb{C}$ and a point $u \in \mathbb{C}^\iota$, the **mixed Wirtinger Hessian** is the $\iota \times \iota$ matrix whose entries are given by the mixed second-order Wirtinger derivatives: $$(H_{f,u})_{IJ} = \partial_I \bar{\partial}_J f(u)$$ where $\bar{\partial}_J$ is the anti-holomorphic partial Wirtinger derivative with respect to the $J$-th coordinate and $\partial_I$ is the holomorphic partial Wirtinger derivative with respect to the $I$-th coordinate. In the context of physics, if $f$ is a Kähler potential, this matrix represents the Kähler metric $g_{I\bar{J}}$.

For a real-valued function $K: \mathbb{C}^\iota \to \mathbb{R}$ and a point $u \in \mathbb{C}^\iota$, the **mixed Wirtinger Hessian** is the $\iota \times \iota$ matrix whose entries are given by the mixed second-order Wirtinger derivatives of $K$ (treated as a complex-valued function): $$(H_{K,u})_{IJ} = \partial_I \bar{\partial}_J K(u)$$ where $\bar{\partial}_J$ is the anti-holomorphic partial Wirtinger derivative and $\partial_I$ is the holomorphic partial Wirtinger derivative. In physics, when $K$ represents a real Kähler potential, this matrix corresponds to the Kähler metric $g_{I\bar{J}}$.

For any real-valued function $K: \mathbb{C}^\iota \to \mathbb{R}$ and any point $u \in \mathbb{C}^\iota$, if $K$ is twice continuously differentiable ($C^2$) at $u$, then its mixed Wirtinger Hessian matrix $H$ (often representing the Kähler metric $g_{I\bar{J}} = \partial_I \bar{\partial}_J K(u)$) is Hermitian. This property is expressed as $(g_{J\bar{I}})^* = g_{I\bar{J}}$ for all indices $I, J \in \iota$.

For a given sector $k \in K$, this definition represents the continuous $\mathbb{R}$-linear map that restricts a global field configuration $q: (\sum_{k \in K} \iota_k) \to \mathbb{C}$ to the components belonging to that specific sector. The map is defined by the restriction $q \mapsto q \circ \sigma_k$, where $\sigma_k: \iota_k \to \sum_{j \in K} \iota_j$ is the canonical injection $a \mapsto (k, a)$, such that the result is a function in $\iota_k \to \mathbb{C}$.

For any sector $k \in K$ and any global configuration of complex scalar fields $q: \left(\sum_{j \in K} \iota_j\right) \to \mathbb{C}$, the restriction of $q$ to sector $k$ is given by the composition $q \circ \sigma_k$, where $\sigma_k: \iota_k \to \sum_{j \in K} \iota_j$ is the canonical injection $a \mapsto (k, a)$.

Let $K$ be a collection of sectors and for each $k \in K$, let $\iota_k$ be the set of indices for the fields within that sector. Let $\text{restrict}_k$ be the continuous $\mathbb{R}$-linear map that restricts a global field configuration $q: \left( \sum_{j \in K} \iota_j \right) \to \mathbb{C}$ to the $k$-th sector, such that $(\text{restrict}_k q)(a) = q(\langle k, a \rangle)$. For any sector $k \in K$, index $a' \in \iota_k$, and complex value $z \in \mathbb{C}$, let $\text{single}_i z$ be the function that evaluates to $z$ at index $i$ and $0$ otherwise. Then the restriction of a global configuration concentrated at $\langle k, a' \rangle$ to sector $k$ is given by: $$\text{restrict}_k (\text{single}_{\langle k, a' \rangle} z) = \text{single}_{a'} z$$

For any two distinct sectors $k, k' \in K$ (with $k \neq k'$), let $a' \in \iota_{k'}$ be an index in sector $k'$ and $z \in \mathbb{C}$ be a complex value. Let $\delta_{(k', a'), z}$ be the field configuration defined on the total space $\sum_{j \in K} \iota_j$ that takes the value $z$ at the index $(k', a')$ and $0$ otherwise. Then the restriction of this configuration to sector $k$, denoted as $\text{restrict}_k(\delta_{(k', a'), z})$, is the zero function on $\iota_k$.

Let $K$ be a collection of sectors, and for each $k \in K$, let $\iota_k$ be the set of indices for the fields within that sector. Let $g: (\iota_k \to \mathbb{C}) \to \mathbb{C}$ be a function that depends only on the fields of sector $k$. For a global field configuration $u: \left( \sum_{j \in K} \iota_j \right) \to \mathbb{C}$, let $u|_k$ denote its restriction to sector $k$, defined by $u|_k(a) = u(\langle k, a \rangle)$. If $g$ is real-differentiable at $u|_k$, then the holomorphic partial Wirtinger derivative of the "lifted" function $f(q) = g(q|_k)$ with respect to the coordinate $\langle k, a \rangle$ at the point $u$ is equal to the holomorphic partial Wirtinger derivative of $g$ with respect to the coordinate $a$ at the point $u|_k$: $$\partial_{\langle k, a \rangle} f(u) = \partial_a g(u|_k)$$

Let $K$ be a collection of sectors, and for each $k \in K$, let $\iota_k$ be the set of indices for the fields within that sector. Let $g: (\iota_{k'} \to \mathbb{C}) \to \mathbb{C}$ be a function that depends only on the fields of sector $k'$. For a global field configuration $u: \left( \sum_{j \in K} \iota_j \right) \to \mathbb{C}$, let $u|_{k'}$ denote its restriction to sector $k'$, defined by $u|_{k'}(a') = u(\langle k', a' \rangle)$. If $k \neq k'$ and $g$ is real-differentiable at $u|_{k'}$, then the holomorphic partial Wirtinger derivative of the "lifted" function $f(q) = g(q|_{k'})$ with respect to the coordinate $\langle k, a \rangle$ (where $a \in \iota_k$) at the point $u$ is zero: $$\partial_{\langle k, a \rangle} f(u) = 0$$

Let $K$ be a collection of sectors, and for each $k \in K$, let $\iota_k$ be the set of indices for the fields within that sector. Let $g: \mathbb{C}^{\iota_k} \to \mathbb{C}$ be a function that depends only on the fields of a specific sector $k$. For a global field configuration $u: \left( \sum_{j \in K} \iota_j \right) \to \mathbb{C}$, let $u|_k$ denote its restriction to sector $k$, defined by $u|_k(a) = u(\langle k, a \rangle)$ for $a \in \iota_k$. If $g$ is real-differentiable at $u|_k$, then the anti-holomorphic partial Wirtinger derivative of the "lifted" function $f(q) = g(q|_k)$ with respect to the coordinate $\langle k, a \rangle$ at the point $u$ is equal to the anti-holomorphic partial Wirtinger derivative of $g$ with respect to the coordinate $a$ at the point $u|_k$: $$\bar{\partial}_{\langle k, a \rangle} f(u) = \bar{\partial}_a g(u|_k)$$

Let $K$ be a type of sector indices, and for each $k \in K$, let $\iota_k$ be a type indexing the fields in sector $k$. Let $k, k' \in K$ be distinct sector indices ($k \neq k'$). Let $g: \mathbb{C}^{\iota_{k'}} \to \mathbb{C}$ be a function of the fields in sector $k'$. For any coordinate index $a \in \iota_k$ and any point $u \in \mathbb{C}^{\sum_{j \in K} \iota_j}$, if $g$ is real-differentiable at the projection $u \circ \text{Sigma.mk } k'$, then the anti-holomorphic Wirtinger derivative of the function $q \mapsto g(q \circ \text{Sigma.mk } k')$ with respect to the coordinate $\langle k, a \rangle$ vanishes at $u$: $$\bar{\partial}_{\langle k, a \rangle} (g(q|_{k'})) = 0$$

Let $K$ be a collection of sectors, and for each $k \in K$, let $\iota_k$ be the set of field indices for that sector. For each $k$, let $f_k : (\iota_k \to \mathbb{C}) \to \mathbb{C}$ be a function. Given a point $u$ in the total field space $(\sum_{k \in K} \iota_k) \to \mathbb{C}$, let $u_k$ denote the restriction of $u$ to the $k$-th sector (formally, $u_k = u \circ \sigma_k$ where $\sigma_k(a) = \langle k, a \rangle$). If every $f_k$ is real-differentiable at $u_k$, then the holomorphic partial Wirtinger derivative of the sum of sector functions at $u$ with respect to the coordinate index $\langle k_0, a \rangle$ is equal to the holomorphic partial Wirtinger derivative of the $k_0$-th sector function $f_{k_0}$ at $u_{k_0}$ with respect to its $a$-th coordinate: $$\partial_{\langle k_0, a \rangle} \left( \sum_{k \in K} f_k(q \circ \sigma_k) \right)(u) = \partial_a f_{k_0}(u_{k_0})$$ where $\sigma_k : \iota_k \to \sum_{j \in K} \iota_j$ is the natural inclusion map.

Suppose we have a collection of functions $f_k: \mathbb{C}^{\iota_k} \to \mathbb{C}$ indexed by sectors $k \in K$, where each $\iota_k$ is the index set for fields in sector $k$. Let $u \in \mathbb{C}^{\sum_{k \in K} \iota_k}$ be a point in the total field space, and let $u \circ \text{mk}_k$ denote the restriction of $u$ to the coordinates of sector $k$. If each $f_k$ is $\mathbb{R}$-differentiable at $u \circ \text{mk}_k$, then the anti-holomorphic partial Wirtinger derivative of the sum of these sector functions with respect to the coordinate $\langle k_0, a \rangle$ (where $k_0 \in K$ and $a \in \iota_{k_0}$) is given by the anti-holomorphic partial Wirtinger derivative of $f_{k_0}$ with respect to $a$: $$\bar{\partial}_{\langle k_0, a \rangle} \left( \sum_{k \in K} f_k(q \circ \text{mk}_k) \right)(u) = \bar{\partial}_a f_{k_0} (u \circ \text{mk}_{k_0})$$ where $\bar{\partial}_I$ denotes the anti-holomorphic partial Wirtinger derivative (`dWirtingerAntiCoord`).

Let $K$ be a set of sectors. For each $k \in K$, let $\iota_k$ be a finite index set and $f_k: \mathbb{C}^{\iota_k} \to \mathbb{C}$ be a complex-valued function. Let $u \in \mathbb{C}^{\sum_{k \in K} \iota_k}$ be a point in the total field space, and let $u_k = u \circ \sigma_k \in \mathbb{C}^{\iota_k}$ denote the restriction of $u$ to the coordinates of the $k$-th sector (where $\sigma_k: \iota_k \to \sum_{j \in K} \iota_j$ is the natural inclusion). If for every $k \in K$, the function $f_k$ is twice continuously $\mathbb{R}$-differentiable at $u_k$, then the mixed Wirtinger Hessian of the sum of sector functions $F(q) = \sum_{k \in K} f_k(q \circ \sigma_k)$ at the point $u$ is equal to the block-diagonal matrix whose $k$-th block is the mixed Wirtinger Hessian of $f_k$ at $u_k$: $$\text{Hess}\left( \sum_{k \in K} f_k \right)(u) = \text{diag}\left( \text{Hess}(f_k)(u_k) \right)_{k \in K}$$ where the $(I, J)$ entry of each Hessian matrix is given by the mixed Wirtinger derivative $\partial_I \bar{\partial}_J$.

Given a bijection $\varepsilon : \iota \simeq \iota'$, the term `precompEquiv ε` is the continuous $\mathbb{R}$-linear map from the space of complex-valued functions $(\iota \to \mathbb{C})$ to the space $(\iota' \to \mathbb{C})$ that performs a relabelling of coordinates. Specifically, it maps a function $q$ to $q \circ \varepsilon^{-1}$, where $\varepsilon^{-1}$ is the inverse of the bijection $\varepsilon$.

Let $\varepsilon : \iota \simeq \iota'$ be a bijection between two index sets (representing field labels). The coordinate relabeling map $\text{precompEquiv}_\varepsilon$ maps a configuration $q: \iota \to \mathbb{C}$ to the configuration $q \circ \varepsilon^{-1} : \iota' \to \mathbb{C}$. For any index $c \in \iota$ and complex value $z \in \mathbb{C}$, let $\delta_{c, z}$ be the function that takes the value $z$ at index $c$ and $0$ otherwise. This theorem states that: $$\text{precompEquiv}_\varepsilon(\delta_{c, z}) = \delta_{\varepsilon(c), z}$$ In other words, the relabeling map sends a basis vector (or a single-component configuration) at coordinate $c$ to the corresponding basis vector at coordinate $\varepsilon(c)$.

Let $\iota$ and $\iota'$ be index sets and $\varepsilon : \iota \simeq \iota'$ be a bijection between them. Let $H : \mathbb{C}^{\iota'} \to \mathbb{C}$ be a complex-valued function that is twice continuously differentiable at the point $u \circ \varepsilon^{-1} \in \mathbb{C}^{\iota'}$, where $u \in \mathbb{C}^\iota$ is a coordinate configuration. Then the mixed Wirtinger Hessian of the function $f(q) = H(q \circ \varepsilon^{-1})$ evaluated at $u$ is obtained by re-indexing the mixed Wirtinger Hessian of $H$ evaluated at $u \circ \varepsilon^{-1}$ using $\varepsilon$. Specifically, for any $i, j \in \iota$: $$\left( \text{hessianMatrixOf}(H(q \circ \varepsilon^{-1}), u) \right)_{i,j} = \left( \text{hessianMatrixOf}(H, u \circ \varepsilon^{-1}) \right)_{\varepsilon(i), \varepsilon(j)}$$ where the mixed Wirtinger Hessian is defined by the matrix of second-order derivatives $\partial_I \bar{\partial}_J$.

The continuous $\mathbb{R}$-linear map from the space of configurations $\iota_1 \oplus \iota_2 \to \mathbb{C}$ to the space $\iota_1 \to \mathbb{C}$ that restricts a configuration $q$ to the indices in the first sector $\iota_1$. It is defined by the assignment $q \mapsto q \circ \text{inl}$, where $\text{inl}: \iota_1 \to \iota_1 \oplus \iota_2$ is the natural injection into the first summand.

The continuous $\mathbb{R}$-linear map from the space of complex-valued functions on $\iota_1 \oplus \iota_2$ to the space of complex-valued functions on $\iota_2$, which restricts a function $q$ to the second sector via the mapping $q \mapsto q \circ \text{inr}$, where $\text{inr}$ is the canonical injection into the second summand of the disjoint union.

For any configuration $q: \iota_1 \oplus \iota_2 \to \mathbb{C}$ of complex fields indexed by the disjoint union of two sectors $\iota_1$ and $\iota_2$, applying the continuous $\mathbb{R}$-linear restriction map $\text{restrictInl}$ to $q$ is equal to the composition $q \circ \text{inl}$, where $\text{inl}: \iota_1 \to \iota_1 \oplus \iota_2$ is the canonical injection into the first summand.

For any function $q : \iota_1 \oplus \iota_2 \to \mathbb{C}$ representing a configuration of complex scalar fields, the application of the restriction map $\text{restrictInr}$ to $q$ is equal to the composition $q \circ \text{inr}$, where $\text{inr} : \iota_2 \to \iota_1 \oplus \iota_2$ is the canonical injection into the second sector.

Let $f_1: \mathbb{C}^{\iota_1} \to \mathbb{C}$ and $f_2: \mathbb{C}^{\iota_2} \to \mathbb{C}$ be complex-valued functions that are twice continuously $\mathbb{R}$-differentiable at $u \circ \text{inl}$ and $u \circ \text{inr}$ respectively, where $u: \iota_1 \oplus \iota_2 \to \mathbb{C}$ is a point in the joint field space. The mixed Wirtinger Hessian of the sum $f(q) = f_1(q \circ \text{inl}) + f_2(q \circ \text{inr})$ evaluated at $u$ is the block-diagonal matrix: $$\text{Hess}(f)(u) = \begin{pmatrix} \text{Hess}(f_1)(u \circ \text{inl}) & 0 \\ 0 & \text{Hess}(f_2)(u \circ \text{inr}) \end{pmatrix}$$ where $\text{Hess}(g)$ denotes the matrix of mixed Wirtinger derivatives $(\partial_I \bar{\partial}_J g)_{IJ}$, and $\text{inl}$ and $\text{inr}$ are the natural injections of the two sectors into the sum type $\iota_1 \oplus \iota_2$.