Chiral Quantum Moment Map

• Chiral quantum moment map is an algebraic construct arising from the quantum deformation of classical chiral superspaces, ensuring supersymmetric invariance in noncommutative settings.
• It employs the quantum super Grassmannian, quantum matrix superalgebra, and explicit coactions from quantum supergroups to model chiral Minkowski and conformal superspaces.
• This framework underpins robust invariant structures for noncommutative supersymmetric field theories, facilitating consistent quantum deformations with preserved group symmetries.

A chiral quantum moment map is an algebraic structure that arises from the quantum deformation of chiral supersymmetric spaces and encodes the quantum-group symmetry actions in noncommutative extensions of classical supergeometry. The construction is rooted in the quantization of chiral Minkowski and conformal superspaces, modeled as quantum homogeneous superspaces with explicit coactions from quantum supergroups. This framework is fundamental for creating a mathematically consistent platform for noncommutative supersymmetric field theories, particularly those involving chiral superfields, and ensures the persistence of underlying supersymmetry and group invariance in quantum deformed environments (Cervantes et al., 2011).

1. Classical and Quantum Chiral Superspaces

The classical chiral conformal superspace is represented by the super Grassmannian $\mathrm{Gr}(2|0, 4|1)$, viewed as the space of $(2|0)$-planes in $\mathbb{C}^{4|1}$ and embedded projectively via (super) Plücker coordinates. These coordinates, denoted $q_{ij}$ and an odd generator $a_{55}$, satisfy the super Plücker relations such as $q_{12}q_{34} - q_{13}q_{24} + q_{14}q_{23} = 0$, ensuring the geometric consistency of the Grassmannian structure.

Quantization proceeds via introducing Manin’s quantum matrix superalgebra $$\mathcal{M}_q(m|n) = \mathbb{C}_q \langle a_{ij} \rangle / I_M$$, with generators $a_{ij}$ that obey graded noncommutative relations:

$$a_{ij}a_{il} = (-1)^{p(a_{ij})p(a_{il})}q^{\pm 1}a_{il}a_{ij}, \quad (j < l)$$

Here, $p(a_{ij})$ denotes the parity and the sign/power of $q$ depends on the ordering of indices. The quantum general linear supergroup is defined as $GL_q(4|1) = \mathcal{M}_q(4|1)[D_1^{-1}, D_2^{-1}]$, with $D_1$, $D_2$ quantum determinants for even blocks.

2. Quantum Super Grassmannian and Big Cell

The quantum super Grassmannian, denoted $\mathrm{Gr}_q$, models the quantum chiral conformal superspace and is generated by quantum minors (superdeterminants) in $GL_q(4|1)$:

• $D_{ij} = a_{i1}a_{j2} - q^{-1}a_{i2}a_{j1}$, for $1 \leq i < j \leq 4$
• $D_{i5} = a_{i1}a_{52} - q^{-1}a_{i2}a_{51}$, for $i=1,\ldots,4$
• $D_{55} = a_{51}a_{52}$

These generators must satisfy quantum Plücker relations, e.g., $$D_{12}D_{34} - q^{-1}D_{13}D_{24} + q^{-2}D_{14}D_{23} = 0$$.

The quantum Minkowski superspace is realized as the big cell by localizing at an invertible minor:

$$\mathcal{O}_q(U_{12}) = \mathrm{Gr}_q[D_{12}^{-1}]_0$$

This identifies the affine piece corresponding to chiral Minkowski superspace within the noncommutative framework.

3. Quantum Supergroup Coaction

The quantum supergroup $GL_q(4|1)$, particularly its lower parabolic subgroup, coacts on the quantum Grassmannian via the comultiplication restriction:

$$\Delta_{\mathrm{Gr}_q}: \mathrm{Gr}_q \to GL_q(4|1) \otimes \mathrm{Gr}_q$$

After localization to $\mathcal{O}_q(U_{12})$, the coaction is transferred:

$$\widehat{\Delta}: \mathcal{O}_q(U_{12}) \to \mathcal{O}(\mathrm{Pl}, q) \otimes \mathcal{O}_q(U_{12})$$

where $\mathcal{O}(\mathrm{Pl}, q)$ is the quantum coordinate ring for the parabolic subgroup preserving the big cell. Explicit transformation formulas (Paper Eqns. (12), (7.5)) show coactions on quantum coordinates such as

$$\widetilde{t}_{ij} = t_{ij} \otimes 1 + y_{ia} S(x)_{bj} \otimes (\text{other generators})$$

This coaction defines the quantum symmetry underpinning the noncommutative superspace structure.

4. Emergence and Role of the Chiral Quantum Moment Map

Embedding classical Minkowski superspace into chiral conformal superspace enables quantum deformation through the ring of the quantum super Grassmannian, utilizing projective embedding and localization. The coaction structure from the quantum group translates into a chiral quantum moment map: a dual algebraic object that encodes how quantum chiral superfields retain consistent symmetry properties under quantum group actions. In classical settings, a moment map translates group actions into functions on phase space; in the quantum case, the chiral moment map provides the corresponding algebraic mapping in a noncommutative environment.

5. Algebraic Structure and Physical Implications

In the quantum environment, chiral superfields are represented as functions on $\mathcal{O}_q(U_{12})$, obeying the noncommutative relations set by quantum supergroup symmetry. The chiral quantum moment map, realized as the restriction of the quantum group coaction to the coordinate ring, ensures that supersymmetric invariance is preserved. This encoding is crucial for formulating supersymmetric field theories on noncommutative superspaces—previously, deformations of the odd directions led to covariance breaking, but the quantum group framework here maintains full symmetry, providing a mathematically robust setting.

Such a construction allows for noncommutative chiral superfield theories characterized by invariant quantum algebraic structures, with explicit formulas for generators, relations, and symmetry coactions. For example, the quantum matrix superalgebra, quantum minors, Big Cell ring, and coaction are all specified in closed form:

• $$\mathcal{M}_q(m|n) = \mathbb{C}_q\langle a_{ij} \rangle / I_M$$
• $GL_q(4|1) = \mathcal{M}_q(4|1)[D_1^{-1}, D_2^{-1}]$
• $D_{ij}, D_{i5}, D_{55}$, as above
• $$D_{12}D_{34} - q^{-1}D_{13}D_{24} + q^{-2}D_{14}D_{23} = 0$$
• $$\mathcal{O}_q(U_{12}) = \mathrm{Gr}_q[D_{12}^{-1}]_0$$
• $$\Delta: \mathcal{O}_q(U_{12}) \to \mathcal{O}(\mathrm{Pl}, q) \otimes \mathcal{O}_q(U_{12})$$

This setup ensures symmetry preservation, algebraic consistency, and physical applicability for noncommutative chiral superfield theories.

6. Extension to Supersymmetric Field Theories

The chiral quantum moment map framework leads directly to quantum analogues of classical supersymmetric field theories, such as quantum chiral superfield models. The quantum homogeneous superspace with the canonical quantum group coaction supplies the correct invariance structure, facilitating the definition and analysis of quantum chiral superfields. It provides an avenue for future research, including the extension to other types of quantum homogeneous superspaces, exploration of quantum corrections, and formulation of invariant field theories on quantized backgrounds. The chiral quantum moment map thus stands as a central construct for operationalizing quantum symmetry in noncommutative supersymmetric geometry.