Hyperholomorphic Mirror Partner

• Hyperholomorphic mirror partners are dual objects in mirror symmetry that maintain hyperholomorphicity across multiple complex structures in geometry and analysis.
• Techniques like Fourier–Mukai transforms and SYZ fibrations enable explicit construction of these partners by linking algebraic invariants with dual geometric frameworks.
• They also appear in operator theory and slice hyperholomorphic function analysis, offering insights into symmetry and duality in advanced mathematical physics.

A hyperholomorphic mirror partner is an object—either a geometric space, a line bundle, a sheaf, a vector bundle, or a functional system—that emerges in duality constructions where hyperholomorphicity is preserved or transferred by mirror symmetry. Hyperholomorphicity refers to the compatibility of an object (function, bundle, operator) with all complex structures available in a hyperkähler manifold or, more generally, to function theories in quaternionic or Clifford settings where specific “slice” conditions hold. Mirror partners typically arise in string theory, algebraic geometry, or operator theory as dual entities whose algebraic, analytic, or geometric invariants correspond, often via deep symmetry properties, to those of their original objects.

1. Hyperholomorphicity and Mirror Symmetry: Foundational Concepts

Mirror symmetry classically relates two spaces (manifolds, often Calabi–Yau or their generalizations) X and Y such that Hodge numbers and deeper invariants are interchanged, as in $h^{1,1}(X)$ with $h^{2,1}(Y)$. In the hyperkähler context, such as in Borcea–Voisin's construction, mirror symmetry can exchange intricate geometric quantities, particularly when K3 surfaces with non-symplectic involutions are involved (Rohde, 2010). Hyperholomorphicity in this setting requires the curvature (or the analytic structure) of bundles to remain of type (1,1) with respect to every complex structure (I, J, K) in the quaternionic family.

Hyperholomorphic sheaves, line bundles, and analytic function spaces are defined by stringent invariance conditions. For example, a holomorphic bundle $E$ on a hyperkähler manifold is hyperholomorphic if $c_1(E),c_2(E)$ are SU(2)-invariant and are of type $(p,p)$ for all induced complex structures (Gukov, 2010, Hitchin, 2013).

2. Constructions and Examples of Mirror Partners

Mirror partners in the hyperholomorphic setting are constructed via several frameworks:

• Borcea–Voisin construction of Calabi–Yau 3-manifolds: Family of K3 surfaces (with non-symplectic involutions) and elliptic curves lead to Calabi–Yau 3-folds whose mirror is formed by interchanging geometries, resulting in explicit Hodge number formulas:

$$h^{1,1}(X) = 11 + 5N - N', \quad h^{2,1}(X) = 11 + 5N' - N.$$

Mirror families and subfamilies carry dense sets of fibers with complex multiplication (CM), implying rich arithmetic and physical properties (Rohde, 2010).

• SYZ and Fourier–Mukai transforms: In torus models and K3 surfaces, SYZ (Strominger–Yau–Zaslow) mirror symmetry and Fourier–Mukai transforms construct the mirror by dualizing Lagrangian torus fibrations or translating sheaf-theoretic data. The mirror coisotropic brane becomes a (B,B,B) brane—a hyperholomorphic sheaf—whose invariants can be computed explicitly:

$$ch(\widetilde{\mathcal{C}_c}) = \frac{1}{\hbar} + \frac{1}{\hbar^2} b_1 \wedge b_2 - f_1 \wedge f_2 - \frac{1}{\hbar} b_1 \wedge b_2 \wedge f_1 \wedge f_2.$$

Rank and Chern character formulas are given by integrals over SYZ fibers,

$$\mathrm{rank}(\widetilde{\mathcal{C}_c}) = \int_F \frac{F^n}{n!}.$$

(Gukov, 2010, Kobayashi, 2019).

• Hyperholomorphic line bundles: On hyperkähler manifolds with circle action, the hyperholomorphic line bundle's curvature is $F = \omega_1 + dd^c p$. Under hyperkähler quotients (e.g., Kronheimer ALE spaces), the bundle descends naturally and preserves hyperholomorphicity (Hitchin, 2013).

3. Functional Analysis and Operator-Theoretic Mirror Partners

Hyperholomorphic mirror partners appear in noncommutative function theory, especially in the paper of slice hyperholomorphic functions on quaternionic and Clifford domains:

• Slice Hyperholomorphic Schur Functions and Interpolation: The boundary Nevanlinna–Pick interpolation problem is solved with slice hyperholomorphic functions, with solutions given by Möbius-type fractional transformations and Blaschke products (Abu-Ghanem et al., 2014). Reproducing kernel Hilbert spaces, fundamental matrix inequalities, and duality properties imply that RKHS structures manifest mirroring phenomena in analytic data, paralleling dual geometry.
• Duality in Function Spaces: BMO (Bounded Mean Oscillation) and VMO (Vanishing Mean Oscillation) spaces of slice hyperholomorphic functions mirror classical complex BMOA/VMOA spaces, with duality relations

$$\left(\mathrm{VMOSH}(\mathbb{D})\right)^* \cong H(\mathbb{D}), \qquad \left(H(\mathbb{D})\right)^* \cong \mathrm{BMOSH}(\mathbb{D}),$$

and invariance under slice hyperholomorphic Möbius transformations (Gantner et al., 2016). Carleson measure characterizations further reinforce the analogy with complex function theory.

• Functional Calculus and Operator Theory: For right linear Clifford operators, the $H^{\infty}$-functional calculus for right slice hyperholomorphic functions is defined via regularization:

$f(T) := (f e)(T) e(T)^{-1},$

formulated with closure in the sense of multivalued operators. Independence from the regularizer and the product rule

$(fg)(T) = f(T)g(T)$

are established for intrinsic function classes (Colombo et al., 5 May 2025).

4. Algebraic and Geometric Mirror Symmetry: LG Models and Moduli Spaces

Mirror symmetry is realized categorically and explicitly in several contexts:

• Landau–Ginzburg (LG) models: The Berglund–Hübsch–Krawitz (BHK) mirror symmetry originally addresses diagonal symmetries, but recent advances enable mirror maps for nonabelian symmetry groups via explicit technical conditions—Diagonal Scaling Condition (DSC) and Equivariant $\Phi$ condition—which guarantee a bidegree-preserving isomorphism between LG A-model and B-model state spaces:

$\mathcal{A}_{W, G} \cong \mathcal{B}_{W^T, G^\vee}$

when permutation parts are cyclic of odd prime order (Clawson et al., 2023).

• Homological Mirror Symmetry for Hypersurfaces: For hypersurfaces $H \subset (\mathbb{C}^*)^n$ defined by Laurent polynomials, the fiberwise wrapped Fukaya category of the toric Landau–Ginzburg mirror $(Y, W)$ is constructed. For distinguished admissible Lagrangians $L \subset Y$, Floer cohomology rings satisfy:

$HW(L, L) \cong \mathbb{K}[x_1, \ldots, x_n]/(f).$

This correspondence embeds the derived category of coherent sheaves on $H$ into the Fukaya category of the mirror (Abouzaid et al., 2021).

• GL(2) Higgs Moduli Spaces and Brane Duality: Lagrangian branes supported on U(1,1) Higgs subspaces correspond—via functional equations of equivariant indices—to even exterior powers of a hyperholomorphic Dirac bundle (“mirror branes”) on SL(2) Higgs moduli spaces. The matching of index formulas satisfies mirror symmetry predictions and illustrates the role of hyperholomorphic objects in Langlands program connections (Hausel et al., 2017).

5. The Role of Complex Multiplication, Representation Theory, and Theoretical Physics

A dense set of complex multiplication (CM) fibers is a recurring feature of hyperholomorphic mirror partners:

• Arithmetic implications: In Calabi–Yau families over Shimura varieties, the property that both original and mirror families have dense sets of CM fibers (i.e., the Hodge group is a torus, $Hg(X)$ abelian) implies abundance of algebraic cycles and large commutative endomorphism algebras (Rohde, 2010), formula

$X\text{ has CM } \iff Hg(X)\text{ is a torus}.$

• Physical consequences: Work by Gukov and Vafa shows that rational conformal field theories (RCFTs) arise precisely when both mirror fibers admit CM, thereby linking deep symmetries in algebraic geometry with properties of quantum field theories.
• Langlands duality and representation theory: Quantization via mirror symmetry leads, in the hyperkähler setting, to computation of Ext-groups between mirror branes, reproducing Verlinde numbers and mapping representation-theoretic data (dimensions, fusion rules) to geometric invariants of hyperholomorphic sheaves on dual moduli spaces (Gukov, 2010).

6. Analytical, Operator-Theoretic, and Representation-Theoretic Extensions

Mirror symmetry principles extend to diverse analytic and operator-theoretic frameworks:

• Infinite order differential operators: On spaces of entire hyperholomorphic or monogenic functions with exponential bounds, continuity and spectral properties are ensured, with applications to quantum mechanical evolution and superoscillation phenomena (Alpay et al., 2020).
• Quaternionic and Clifford analysis: Both left and right $\psi$-hyperholomorphic functions in complex quaternion settings are analytic in the Cartan basis, enabling explicit solutions to Cauchy–Fueter equations and representations in holomorphic variables, which reflects dualities underpinning “mirror partner” structure (Kuzmenko et al., 2023).
• Function-theoretic duality: RKHS techniques, boundary interpolation, and star-product calculus in the theory of slice hyperholomorphic functions suggest frameworks for constructing “mirror partners” as dual entities within analytic function spaces (Abu-Ghanem et al., 2014).

7. Challenges and Research Directions

Ambiguities in transition functions for vector bundles on tori, intricacies of noncommutative operator regularization, and combinatorial structures of LG or Higgs moduli spaces pose prominent challenges in establishing bijections and correspondences between mirror partners (Kobayashi, 2019, Colombo et al., 5 May 2025). Future directions include:

• Extending mirror symmetry correspondence to bundles and objects with categorical and derived structures in higher dimensions.
• Refining operator-theoretic functional calculi for broader classes of hyperholomorphic functions.
• Deepening connections between arithmetic CM geometry, representation-theoretic invariants, and physical (quantum field) dualities.
• Systematic development of the “mirror partner” theory across analytic, algebraic, and geometric contexts, ensuring invariance of key spectral and cohomological data.

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In summary, the notion of a hyperholomorphic mirror partner encapsulates duality structures across geometric, analytic, operator-theoretic, and categorical frameworks where hyperholomorphicity or related invariance properties are preserved. Its realization requires sophisticated mechanisms (Fourier–Mukai transforms, SYZ fibrations, functional calculi, RKHS duality, arithmetic conditions) that ensure deep symmetries and correspondences not only in complex geometry but also in extended domains of hypercomplex analysis and quantum theory.