Physlib.Particles.SuperSymmetry.N1.Basic

Given a type $\iota$ representing the indices of chiral scalars, a **chiral scalar configuration** is defined as the space of functions $\iota \to \mathbb{C}$. This configuration assigns a complex value to each chiral label and represents the physical field data of the sector.

The inductive type $\text{ChiralColor}$ represents the four possible states for an index in the $N=1$ chiral sector. It is defined as the product of two axes: 1. **Holomorphy**: Distinguishes between chiral (holomorphic) and anti-chiral (anti-holomorphic/complex conjugate) components. 2. **Variance**: Distinguishes between "up" (contravariant) and "down" (covariant) indices, representing the distinction between a module and its dual. Together, these axes allow a single index type $\iota$ to label the physical scalar fields and their conjugates across different tensor configurations.

The equality of index colors in the set $\text{ChiralColor}$ is decidable. This means that for any two colors $c_1, c_2 \in \text{ChiralColor}$, there is an algorithmic procedure to determine whether $c_1 = c_2$ or $c_1 \neq c_2$.

The dual-color involution $\tau: \text{ChiralColor} \to \text{ChiralColor}$ maps an index color to its variance dual while preserving its holomorphy. Specifically, it swaps "up" (contravariant) and "down" (covariant) labels as follows: \begin{itemize} \item $\tau(\text{chiralUp}) = \text{chiralDown}$ \item $\tau(\text{chiralDown}) = \text{chiralUp}$ \item $\tau(\text{antiUp}) = \text{antiDown}$ \item $\tau(\text{antiDown}) = \text{antiUp}$ \end{itemize} In the $N=1$ chiral sector, two indices are eligible for contraction if and only if their colors are $\tau$-related, ensuring that a holomorphic vector index $V^I$ pairs only with a holomorphic dual index $V_I$, and similarly for anti-holomorphic indices.

The function $\text{bar} : \text{ChiralColor} \to \text{ChiralColor}$ defines the complex conjugation operation on the index colors of the $N=1$ chiral sector. It interchanges the holomorphy states (mapping "chiral" to "anti" and vice versa) while preserving the variance of the index ("up" or "down"). Specifically, the mapping is defined as follows: - $\text{bar}(\text{chiralUp}) = \text{antiUp}$ - $\text{bar}(\text{antiUp}) = \text{chiralUp}$ - $\text{bar}(\text{chiralDown}) = \text{antiDown}$ - $\text{bar}(\text{antiDown}) = \text{chiralDown}$

For any index color $c \in \text{ChiralColor}$ in the $N=1$ chiral sector, the complex conjugation operation $\text{bar}$ is an involution, meaning that applying it twice returns the original color: $\text{bar}(\text{bar}(c)) = c$.

For any index color $c \in \text{ChiralColor}$ in the $N=1$ chiral sector, the complex conjugation operation $\text{bar}$ and the dual-color involution $\tau$ commute. That is, $$\text{bar}(\tau(c)) = \tau(\text{bar}(c))$$ where $\text{bar}$ inter-converts chiral and anti-chiral labels while preserving variance, and $\tau$ inter-converts up and down labels while preserving holomorphy.

For a finite set $\iota$ indexing the chiral fields, let $V = \mathbb{C}^\iota$ be the space of complex-valued functions on $\iota$. The function $\text{chiralModule}$ assigns to each color $c \in \text{ChiralColor}$ a corresponding carrier module (a complex vector space) according to its holomorphy and variance: - If $c = \text{chiralUp}$, the module is $V$. - If $c = \text{chiralDown}$, the module is the dual space $V^* = \text{Hom}_{\mathbb{C}}(V, \mathbb{C})$. - If $c = \text{antiUp}$, the module is the conjugate module $\text{ConjModule } V$, where scalar multiplication $z \cdot v$ is defined as $\bar{z} v$. - If $c = \text{antiDown}$, the module is the conjugate of the dual space $\text{ConjModule } (V^*)$.

For each color $c \in \text{ChiralColor}$, the corresponding carrier module $\text{chiralModule}(c)$—which represents the space of chiral scalars $V = \mathbb{C}^\iota$, its dual space $V^*$, or their respective complex conjugate modules—is equipped with an additive commutative group structure.

For each index color $c \in \text{ChiralColor}$, the corresponding carrier space $\text{chiralModule}(c)$ is equipped with a $\mathbb{C}$-module structure. This provides a complex vector space structure for all four possible index types: the chiral vector space $\mathbb{C}^\iota$, its dual space $(\mathbb{C}^\iota)^*$, and their respective complex conjugate modules.

For each color $c \in \text{ChiralColor}$, the function $\text{chiralRep}(c)$ defines the trivial representation of the trivial group $\text{Unit}$ on the corresponding complex carrier module $\text{chiralModule}(c)$. This implies that the chiral scalar fields carry no charge under this group action.

Given a finite indexing type $\iota$, the standard basis for the space of functions $\iota \to \mathbb{C}$ is the collection of indicator functions $\{e_i\}_{i \in \iota}$, where $e_i(j) = \delta_{ij}$ for $j \in \iota$. In the context of the $N=1$ supersymmetric chiral sector, this basis represents the standard basis for the vector carrier $\iota \to \mathbb{C}$ (the "chiral up" indices) and serves as the reference basis for the $\delta$ pairing and tensor contractions.

For each color $c \in \text{ChiralColor}$, the function $\text{chiralBasis}(c)$ defines a basis indexed by the finite set $\iota$ for the corresponding carrier module $\text{chiralModule}(c)$. Let $V = \mathbb{C}^\iota$ be the space of complex-valued functions on $\iota$ with its standard basis $\{e_i\}_{i \in \iota}$ (where $e_i(j) = \delta_{ij}$), and let $V^* = \text{Hom}_\mathbb{C}(V, \mathbb{C})$ be its dual space with the dual basis $\{e^i\}_{i \in \iota}$. The basis is assigned as follows: - For $c = \text{chiralUp}$, the basis is the standard basis $\{e_i\}$ of the holomorphic vector space $V$. - For $c = \text{chiralDown}$, the basis is the dual basis $\{e^i\}$ of the holomorphic covector space $V^*$. - For $c = \text{antiUp}$, the basis is the conjugate basis $\{\bar{e}_i\}$ of the anti-holomorphic vector space (the conjugate module $\text{ConjModule } V$). - For $c = \text{antiDown}$, the basis is the conjugate of the dual basis $\{\bar{e}^i\}$ of the anti-holomorphic covector space (the conjugate module $\text{ConjModule } V^*$).

Given a complex module $M$ with a basis $b$ indexed by a finite type $\iota$, the $\delta$ cap is the rank-2 tensor in $M \otimes_{\mathbb{C}} M$ defined by the sum $\sum_{I \in \iota} b_I \otimes b_I$. This element represents a tensor with components $\delta^{IJ}$ in the basis $b$, serving as an inverse-metric or "cap" for index contraction.

Given vector spaces $M$ and $N$ over $\mathbb{C}$ with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$ indexed by a finite set $\iota$, the function $\text{deltaBil}_2(b, b')$ is the bilinear map $M \times N \to \mathbb{C}$ defined by the dot product of the coordinate vectors. For vectors $x \in M$ and $y \in N$, the pairing is given by: $$(x, y) \mapsto \sum_{i \in \iota} x_i y_i$$ where $x_i$ and $y_i$ are the coordinates of $x$ and $y$ with respect to the bases $b$ and $b'$ respectively. This is constructed by identifying $b'$ with $b$ and using the canonical isomorphism between $M$ and its dual space $M^*$ induced by the basis $b$.

Let $M$ and $N$ be complex vector spaces with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$ indexed by a finite set $\iota$. For any vectors $x \in M$ and $y \in N$, the $\delta$-bilinear pairing evaluated at $(x, y)$ is the dot product of their coordinates: $$\text{deltaBil}_2(b, b')(x, y) = \sum_{i \in \iota} x_i y_i$$ where $x_i$ and $y_i$ denote the $i$-th coordinates of $x$ and $y$ with respect to the bases $b$ and $b'$, respectively.

Given vector spaces $M$ and $N$ over $\mathbb{C}$ equipped with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$ indexed by a finite set $\iota$, the function $\text{deltaContr}_2(b, b')$ is the linear map $M \otimes_{\mathbb{C}} N \to \mathbb{C}$ that performs a $\delta$-contraction between the two modules. For any pure tensor $x \otimes y \in M \otimes N$, the map is defined by the dot product of their coordinate vectors: $$x \otimes y \mapsto \sum_{i \in \iota} x_i y_i$$ where $x_i$ and $y_i$ are the coordinates of $x$ and $y$ with respect to the bases $b$ and $b'$, respectively. This map is obtained by lifting the bilinear pairing $\text{deltaBil}_2(b, b')$ to the tensor product.

Let $M$ and $N$ be complex vector spaces equipped with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$ indexed by a finite set $\iota$. For any vectors $x \in M$ and $y \in N$, the $\delta$-contraction map $\text{deltaContr}_2(b, b'): M \otimes_{\mathbb{C}} N \to \mathbb{C}$ evaluated on the pure tensor $x \otimes y$ is given by the dot product of their coordinates: $$\text{deltaContr}_2(b, b')(x \otimes y) = \sum_{i \in \iota} x_i y_i$$ where $x_i$ and $y_i$ denote the $i$-th coordinates of $x$ and $y$ with respect to the bases $b$ and $b'$, respectively.

Let $M$ and $N$ be vector spaces over $\mathbb{C}$ equipped with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$ respectively, where $\iota$ is a finite indexing set. For any vector $x \in M$ and any index $J \in \iota$, the $\delta$-contraction of the pure tensor $x \otimes b'_J$ in $M \otimes_{\mathbb{C}} N$ satisfies $$\text{deltaContr}_2(b, b')(x \otimes b'_J) = x_J$$ where $x_J$ denotes the $J$-th coordinate of the vector $x$ with respect to the basis $b$.

Given complex vector spaces $M$ and $N$ equipped with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$ indexed by a set $\iota$, the $\delta$-contraction of the basis elements $b_I \in M$ and $b'_J \in N$ is given by the Kronecker delta: $$\text{deltaContr}_2(b, b')(b_I \otimes b'_J) = \delta_{IJ}$$ where $\delta_{IJ}$ equals $1$ if $I = J$ and $0$ otherwise. This result expresses that the two bases are $\delta$-dual under the contraction map.

Let $\iota$ be a finite index set. Let $M$ and $N$ be complex vector spaces equipped with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$, respectively. For any vectors $x \in M$ and $y \in N$, the $\delta$-contraction of the tensor product $x \otimes y$ with respect to the pair of bases $(b, b')$ is equal to the $\delta$-contraction of the swapped tensor product $y \otimes x$ with respect to the swapped pair of bases $(b', b)$: $$\text{deltaContr}_2(b, b')(x \otimes y) = \text{deltaContr}_2(b', b)(y \otimes x)$$ where $\text{deltaContr}_2$ is defined by the sum of the products of the components of the vectors in their respective bases.

Let $\iota$ be a finite index set. Given a basis $b = \{b_I\}_{I \in \iota}$ for a complex vector space $M$ and a basis $b' = \{b'_I\}_{I \in \iota}$ for a complex vector space $N$, the two-module $\delta$-cap is the element of the tensor product $M \otimes_{\mathbb{C}} N$ defined by the sum: $$\sum_{I \in \iota} b_I \otimes b'_I$$ where $b_I$ and $b'_I$ are the basis elements corresponding to the index $I \in \iota$.

Let $\iota$ be a finite index set. Let $M$ and $N$ be complex vector spaces equipped with bases $b = \{b_I\}_{I \in \iota}$ and $b' = \{b'_I\}_{I \in \iota}$, respectively. The two-module $\delta$-cap is defined as the element $\sum_{I \in \iota} b_I \otimes b'_I$ in the tensor product $M \otimes_{\mathbb{C}} N$. This theorem states that the image of the $\delta$-cap formed by $(b, b')$ under the canonical swapping isomorphism $\tau: M \otimes_{\mathbb{C}} N \to N \otimes_{\mathbb{C}} M$ (where $\tau(x \otimes y) = y \otimes x$) is the $\delta$-cap formed by the bases in reverse order $(b', b)$: $$\tau \left( \sum_{I \in \iota} b_I \otimes b'_I \right) = \sum_{I \in \iota} b'_I \otimes b_I$$

Let $\iota$ be a finite index set. Let $M$ and $N$ be complex vector spaces equipped with bases $b = \{b_I\}_{I \in \iota}$ and $b' = \{b'_I\}_{I \in \iota}$ respectively. The two-module $\delta$-cap is defined as the element $\sum_{I \in \iota} b_I \otimes b'_I$ in the tensor product $M \otimes_{\mathbb{C}} N$. This theorem states that the $\delta$-cap formed by the bases in reverse order $(b', b)$ is equal to the image of the $\delta$-cap formed by $(b, b')$ under the canonical swapping isomorphism $\tau: M \otimes_{\mathbb{C}} N \to N \otimes_{\mathbb{C}} M$: $$\sum_{I \in \iota} b'_I \otimes b_I = \tau \left( \sum_{I \in \iota} b_I \otimes b'_I \right)$$ where $\tau(x \otimes y) = y \otimes x$.

Let $\iota$ be a finite index set. Let $M$ and $N$ be complex vector spaces equipped with bases $b = \{b_I\}_{I \in \iota}$ and $b' = \{b'_I\}_{I \in \iota}$, respectively. Let $\text{deltaCap}_2(b', b) = \sum_{I \in \iota} b'_I \otimes b_I \in N \otimes M$ be the two-module $\delta$-cap and $\text{deltaContr}_2(b, b') : M \otimes N \to \mathbb{C}$ be the $\delta$-contraction map defined by the pairing of coordinates $\text{deltaContr}_2(b, b')(x \otimes y) = \sum_{I \in \iota} x_I y_I$. For any vector $x \in M$, the snake identity holds: $$\text{lid} \left( (\text{deltaContr}_2(b, b') \otimes \text{id}_M) \left( \alpha^{-1} (x \otimes \text{deltaCap}_2(b', b)) \right) \right) = x$$ where $\alpha : (M \otimes N) \otimes M \to M \otimes (N \otimes M)$ is the canonical associativity isomorphism and $\text{lid} : \mathbb{C} \otimes M \to M$ is the left unit isomorphism $c \otimes v \mapsto cv$. In physical terms, this states that contracting a field $x$ into one leg of the $\delta$-cap returns the field $x$.

Let $\iota$ be a finite index set, and let $M$ and $N$ be complex vector spaces equipped with bases $b = \{b_i\}_{i \in \iota}$ and $b' = \{b'_i\}_{i \in \iota}$, respectively. Consider the $\delta$-cap tensors $\text{cap}(b) = \sum_{i \in \iota} b_i \otimes b_i \in M \otimes M$ and $\text{cap}(b') = \sum_{j \in \iota} b'_j \otimes b'_j \in N \otimes N$. The contraction of the second leg of $\text{cap}(b)$ with the first leg of $\text{cap}(b')$ using the $\delta$-contraction map $\text{deltaContr}_2(b, b') : M \otimes N \to \mathbb{C}$ (which maps $b_i \otimes b'_j$ to $\delta_{ij}$) results in the two-module $\delta$-cap in $N \otimes M$: $$\sum_{i \in \iota} b'_i \otimes b_i$$

Let $\iota$ be a finite index set. The `chiralTensor` is the conjugate tensor species (`ConjTensorSpecies`) for the $N=1$ SUSY chiral sector over the field $\mathbb{C}$. It organizes the vector spaces, representations, and contractions for the scalar fields and their conjugates. The structure is defined by the following components: 1. **Index Colors and Carriers**: The colors $c \in \text{ChiralColor}$ represent the four combinations of holomorphy (chiral/anti-chiral) and variance (up/down). To each color $c$, it assigns a carrier module $V_c = \text{chiralModule}(c)$ (which are the spaces $V = \mathbb{C}^\iota$, its dual $V^*$, or their complex conjugates) equipped with a basis $b_c = \text{chiralBasis}(c)$ indexed by $\iota$. 2. **Group Action**: The symmetry group is the trivial group $\text{Unit}$, which acts via the trivial representation on every carrier module. 3. **Contraction and Metric**: * The dual-color involution $\tau$ swaps the variance (up $\leftrightarrow$ down) while preserving holomorphy. * The contraction $\text{contr}_c : V_c \otimes V_{\tau(c)} \to \mathbb{C}$ is the $\delta$-pairing of the two bases: $x \otimes y \mapsto \sum_{I \in \iota} x_I y_I$. * The unit $\text{unit}_c \in V_{\tau(c)} \otimes V_c$ and the metric $\text{metric}_c \in V_c \otimes V_c$ are both defined as the $\delta$-cap tensor: $\sum_{I \in \iota} (b)_I \otimes (b')_I$. 4. **Conjugation**: The conjugation involution $\text{bar}$ swaps the holomorphy (chiral $\leftrightarrow$ anti-chiral) while preserving variance. Since all bases are indexed by the same set $\iota$, the identification between index sets of conjugated colors is the identity. The contraction maps are real in the sense that they commute with complex conjugation ($\text{star } \delta = \delta$).

For a scalar tensor $t$ (a rank-0 tensor with an empty list of indices $[]$) in the $N=1$ chiral sector, this function computes its conjugate. It applies the general species conjugation $\text{conjT}$ and uses the identity permutation to normalize the resulting color list back to the canonical empty list. Physically, this corresponds to the complex conjugation of a scalar field.

For a finite index set $\iota$, let $t$ be a holomorphic covector in the $N=1$ chiral sector, represented as a tensor of rank 1 with the color sequence $[\text{chiralDown}]$. The function `conjChiralCovector` maps $t$ to its conjugate anti-holomorphic covector with color sequence $[\text{antiDown}]$. If $t$ has components $t_I$ with respect to the holomorphic basis $\{e^I\}_{I \in \iota}$, then the resulting tensor has components $\bar{t}_I$ with respect to the conjugate basis $\{\bar{e}^I\}_{I \in \iota}$.

For any rank-0 tensor (scalar) $t$ in the $N=1$ SUSY chiral sector, let $\text{toField}(t) \in \mathbb{C}$ be its corresponding scalar value. The scalar value of the conjugated tensor $\text{conjScalar}(t)$ is equal to the complex conjugate of the scalar value of $t$: $$(\text{conjScalar } t).\text{toField} = \overline{t.\text{toField}},$$ where the overbar denotes the complex conjugate (the `star` operation in $\mathbb{C}$).

For a finite index set $\iota$ in the $N=1$ chiral sector, let $t$ be a holomorphic covector (a tensor of rank 1 with color `chiralDown`). For any index $I \in \iota$, the component of the conjugate anti-holomorphic covector $\text{conjChiralCovector}(t)$ corresponding to $I$ is the complex conjugate of the $I$-th component of $t$: $$(\text{conjChiralCovector } t)_I = \overline{t_I}$$ where the components are taken with respect to the canonical bases for the respective tensor spaces.

For any two holomorphic covectors $t_1$ and $t_2$ in the $N=1$ chiral sector (tensors of rank 1 with color $\text{chiralDown}$), the conjugation map $\text{conjChiralCovector}$ is additive: $$\text{conjChiralCovector}(t_1 + t_2) = \text{conjChiralCovector}(t_1) + \text{conjChiralCovector}(t_2)$$ where the resulting tensors are anti-holomorphic covectors with color $\text{antiDown}$.

For any complex scalar $r \in \mathbb{C}$ and any holomorphic covector $t$ in the $N=1$ chiral sector (a tensor of rank 1 with color sequence $[\text{chiralDown}]$), the conjugation operation is conjugate-linear with respect to scalar multiplication: $$\text{conjChiralCovector}(r \cdot t) = \bar{r} \cdot \text{conjChiralCovector}(t)$$ where $\bar{r}$ denotes the complex conjugate of $r$.