Anti-holomorphic Involution Research Articles

Open orbifold Gromov-Witten invariants of $${[\mathbb{C}^3/\mathbb{Z}_n]}$$ : localization and mirror symmetry

We develop a mathematical framework for the computation of open orbifold Gromov-Witten invariants of [C^3/Z_n], and provide extensive checks with predictions from open string mirror symmetry. To this aim we set up a computation of open string invariants in the spirit of Katz-Liu, defining them by localization. The orbifold is viewed as an open chart of a global quotient of the resolved conifold, and the Lagrangian as the fixed locus of an appropriate anti-holomorphic involution. We consider two main applications of the formalism. After warming up with the simpler example of [C^3/Z_3], where we verify physical predictions of Bouchard, Klemm, Marino and Pasquetti, the main object of our study is the richer case of [C^3/Z_4], where two different choices are allowed for the Lagrangian. For one choice, we make numerical checks to confirm the B-model predictions; for the other, we prove a mirror theorem for orbifold disc invariants, match a large number of annulus invariants, and give mirror symmetry predictions for open string invariants of genus \leq 2.

New examples of compact manifolds with holonomy Spin(7)

We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi-Yau 4-orbifolds. We find that different hypersurfaces within the same family of Calabi-Yau 4-orbifolds may result in different Spin(7)-manifolds.

Minimal genus problem for pseudo-real Riemann surfaces

A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).

$ \mathcal{N} = 1 $ Chern-Simons theories, orientifolds and Spin(7) cones

We construct three dimensional N=1 Chern-Simons theories living on M2 branes probing Spin(7) cones. We consider Spin(7) manifolds obtained as quotients of Calabi-Yau four-folds by an anti-holomorphic involution, following a construction by Joyce. The corresponding Chern-Simons theories can be obtained from N=2 theories by an orientifolding procedure. These theories are holographically dual to M theory solutions AdS_4 \times H, where the weak G_2 manifold H is the base of the Spin(7) cone.

REAL THETA CHARACTERISTICS AND AUTOMORPHISMS OF A REAL CURVE

AbstractLet X be a geometrically irreducible smooth projective curve defined over ℝ, of genus at least 2, that admits a nontrivial automorphism, σ. Assume that X does not have any real points. Let τ be the antiholomorphic involution of the complexification xℂ of X. We show that if the action of σ on the set 𝒮(X) of all real theta characteristics of X is trivial, then the order of σ is even, say 2k, and the automorphism $\tau \circ \widehat {\sigma }^k$ of Xℂ has a fixed point, where $\widehat {\sigma }^k$ is the automorphism of X×ℝℂ defined by σ. We then show that there exists X with a real point and admitting a nontrivial automorphism σ, such that the action of σ on 𝒮(X) is trivial, while X/〈σ〉≠ℙ1ℝ. We also give an example of X with no real points and admitting a nontrivial automorphism σ, such that the automorphism $\tau \circ \widehat {\sigma }^k$ has a fixed point, the action of σ on 𝒮(X) is trivial, and X/〈σ〉≠ℙ1ℝ.

The real topological vertex at work

We develop the real vertex formalism for the computation of the topological string partition function with D-branes and O-planes at the fixed point locus of an anti-holomorphic involution acting non-trivially on the toric diagram of any local toric Calabi–Yau manifold. Our results cover in particular the real vertex with non-trivial fixed leg. We give a careful derivation of the relevant ingredients using duality with Chern–Simons theory on orbifolds. We show that the real vertex can also be interpreted in terms of a statistical model of symmetric crystal melting. Using this latter connection, we also assess the constant map contribution in Calabi–Yau orientifold models. We find that there are no perturbative contributions beyond one-loop, but a non-trivial sum over non-perturbative sectors, which we compare with the non-perturbative contribution to the closed string expansion.

A characterization of non-tube type Hermitian symmetric spaces by visible actions

Let G/K be an irreducible Hermitian symmetric space of non-compact type, and \({G_{\mathbb{C}}/K_{\mathbb{C}}}\) its complexification by forgetting the original complex structure. Then, \({D :=G_{\mathbb{C}}/[K_{\mathbb{C}}, K_{\mathbb{C}}]}\) is a non-symmetric Stein manifold. We prove that a maximal compact subgroup of \({G_{\mathbb{C}}}\) acts on D in a strongly visible fashion in the sense of Kobayashi (Publ Res Inst Math Sci 41:497–549, 2005) if and only if G/K is of non-tube type. Our proof uses the theory of multiplicity-free representations and a construction of a slice and an anti-holomorphic involution on D.

Model building with intersecting D6-branes on smooth Calabi-Yau manifolds

We study intersecting D6-branes in Calabi-Yau manifolds that are smooth hypersurfaces in weighted projective spaces. We develop the techniques for calculating intersection numbers between special Lagrangian sub-manifolds defined as fixed loci of anti-holomorphic involutions. We present global Pati-Salam and MSSM-like models that are supersymmetric up to a decoupled hidden sector.

On Projecting Symmetries by Unbranched Regular Coverings of Riemann Surfaces

A symmetry of a Riemann surface X of genus g is an antiholomorphic involution σ of X. It is a classical result of Harnack that the set of fixed points of σ consists of k closed Jordan curves, called ovals, for some k, 0 ≤ k ≤ g + 1; when k = g or k = g+1 we say, following Natanzon [8], that σ is an (M – 1)- or an M-symmetry, respectively. Given a Riemann surface X with an M-symmetry, a Riemann surface Y and a regular covering p: X → Y, we prove that Y admits either an M- or an (M – 1)-symmetry and whenever p is unbranched we describe the groups of covering transformations of p. In the case that X is hyperelliptic we calculate as well the number of unbranched regular coverings p: X → Y in which X has an M-symmetry.

Antiholomorphic involutions of spherical complex spaces

Let X be a holomorphically separable irreducible reduced complex space, K a connected compact Lie group acting on X by holomorphic transformations, θ: K → K a Weyl involution, and μ: X → X an antiholomorphic map satisfying μ 2 = Id and μ(kx) = θ(k)μ(x) for x ∈ X, k ∈ K. We show that if 0(X) is a multiplicity free K-module, then μ maps every K-orbit onto itself. For a spherical affine homogeneous space X = G/H of the reductive group G = K C we construct an antiholomorphic map μ with these properties.

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $\sigma$ such that $\sigma |_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the action of any symmetric subgroup $H$ is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold $S$. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite and infinite dimensional cases, and for both discrete and continuous spectra.

Visible actions on irreducible multiplicity-free spaces

A holomorphic action of a Lie group $G$ on a complex manifold $D$ is called \textit{strongly visible} if there exist a totally real submanifold $S$ which meets every $G$-orbit in $D$ and an anti-holomorphic diffeomorphism $\sigma $ such that $\sigma |_{S}=\mathop{\mathrm{id}}_{S}$ and $\sigma$ preserves every $G$-orbit. In this paper, we prove that Kac’s multiplicity-free space is strongly visible, that is, if $(G_{\mathbf{C}},V)$ is an irreducible multiplicity-free space of a complex reductive Lie group $G_{\mathbf{C}}$, then a compact real form of $G_{\mathbf{C}}$ acts on $V$ in a strongly visible fashion. Furthermore, we give an explicit description of the choice of a totally real submanifold $S$ and an anti-holomorphic involution $\sigma$. This gives an evidence to Kobayashi’s conjecture , that is, $\dim_{\mathbf{R}}S$ coincides with the rank of the polynomial representation of $G_{\mathbf{C}}$ on $\mathbf{C}[V]$ in this setting.

Four-dimensional Manifolds

Four-dimensional Manifolds

The punctured torus and Lagrangian triangle groups in PU(2,1)

We embed the Teichmuller space of the once punctured torus T(1,1) into the set of conjugacy classes of groups generated by three anti-holomorphic involutions I1, I2 and I3 (Lagrangian triangle groups), acting on the complex hyperbolic plane HC. We deform this embedding, and obtain a three dimensional family E of discrete, faithful and type preserving representations of the fundamental group of the once punctured torus. AMS classification 51M10, 32M15, 22E40

Geometric cohomology frames on Hausmann–Holm–Puppe conjugation spaces

For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets). Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology. The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold X X with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of X X ), these topological cycles will give rise to a cohomology frame.

Twisted complex geometry

AbstractWe introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.

ON SYMMETRIES OF PQ-HYPERELLIPTIC RIEMANN SURFACES

A symmetry of a Riemann surface X is an antiholomorphic involution φ .T he species ofφ is the integer ek ,w herek is the number of connected components in the set Fix(φ) of fixed points of φ and e = −1 if X Fix(φ) is connected and e =1 otherwise. A compact Riemann surface X of genus g> 1 is said to be p-hyperelliptic if it admits a conformal involution ρ ,c alled ap-hyperelliptic involution, for which X/ρ is an orbifold of genus p. Symmetries of p-hyperelliptic Riemann surfaces has been studied by Klein for p =0 and by Bujalance and Costa for p> 0 . Here we study the species of symmetries of so called pq-hyperelliptic surface defined as a Riemann surface which is p -a nd q-hyperelliptic simultaneously.

Quotient spaces of anti-holomorphic involutions

Let σ be an anti-holomorphic involution on an almost complex four manifold X, a necessary and sufficient condition is given to determine weather X/σ admits an almost complex structure.

Real K3 surfaces with non-symplectic involution and applications

Classification of real K3 surfaces $X$ with non-symplectic involution $\tau$ is considered. We define a natural notion of degeneration for them. We show that the connected component of moduli of non-degenerate surfaces of this type is defined by the isomorphism class of the action of $\tau$ and the anti-holomorphic involution $\varphi$ in the homology lattice. (There are very few similar results known.) For their classification we apply invariants of integral lattice involutions with conditions that were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves $A \in | -2 K_V|$ for the classical real surfaces: $V = P^2$, hyperboloid, ellipsoid, $F_1$, $F_4$. As an application, we describe all real polarized K3 surfaces that are deformations of general real K3 double rational scrolls (the surfaces $V$ above). There are very few exceptions. For example, any non-singular real quartic in $P^3$ can be constructed in this way.