Dimensional Base Space Research Articles

Moduli of Tango Structures and Dormant Miura Opers

The purpose of the present paper is to develop the theory of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers (for a semisimple Lie algebra $\mathfrak{g}$) defined on pointed stable curves in positive characteristic. A (pre-)Tango structure is a certain line bundle of an algebraic curve in positive characteristic, which gives some pathological (relative to zero characteristic) phenomena. In the present paper, we construct the moduli spaces of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers respectively, and prove certain properties of them. One of the main results of the present paper states that there exists a bijective correspondence between the (pre-)Tango structures (of prescribed monodromy) and the dormant generic Miura $\mathfrak{s} \mathfrak{l}_2$-opers (of prescribed exponent). By means of this correspondence, we achieve a detailed understanding of the moduli stack of (pre-)Tango structures. As an application, we construct a family of algebraic surfaces in positive characteristic parametrized by a higher dimensional base space whose fibers are pairwise non-isomorphic and violate the Kodaira vanishing theorem.

Multi-centered black hole flows

We describe the systematical construction of the first order formalism for multi-centered black holes with flat three dimensional base-space, within the so-called $T^{3}$ model of N=2, D=4 ungauged Maxwell-Einstein supergravity. The three possible flow classes (BPS, composite non-BPS and almost-BPS) are analyzed in detail, and various solutions, such as single-centered (static or under-rotating) and all known multi-centered black holes, are recovered in this unified framework. We also consider the possibility of obtaining new solutions. The almost-BPS class is proved to split into two general sub-classes, corresponding to a positive or negative value of the duality-invariant polynomial for the total charge; the well known almost BPS system is shown to be a particular solution of the second sub-class.

Cosmological Einstein-Maxwell instantons and euclidean supersymmetry: beyond self-duality

We construct new supersymmetric solutions to the Euclidean Einstein-Maxwell theory with a non-vanishing cosmological constant, and for which the Maxwell field strength is neither self-dual or anti-self-dual. We find that there are three classes of solutions, depending on the sign of the Maxwell field strength and cosmological constant terms in the Einstein equations which arise from the integrability conditions of the Killing spinor equation. The first class is a Euclidean version of a Lorentzian supersymmetric solution found in [1, 2]. The second class is constructed from a three dimensional base space which admits a hyper-CR Einstein-Weyl structure. The third class is the Euclidean Kastor-Traschen solution.

Description of1/4BPS configurations in minimal type IIB supergravity

In this paper we present an effort to extend the LLM construction of $1/2$ BPS states in minimal IIB supergravity to configurations that preserve $1/4$ of the total number of supersymmetries. Following the same techniques we reduce the problem to that of a single scalar which satisfies a nonlinear equation. In particular, the scalar is identified to be the Kahler potential with which a four dimensional base space is equipped.

New class ôN of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable S matrix

Statistical models corresponding to a new class of braid matrices (ôN;N⩾3) presented in a previous paper are studied. Indices labeling states spanning the Nr dimensional base space of T(r)(θ), the rth order transfer matrix are so chosen that the operators W (the sum of the state labels) and (CP) (the circular permutation of state labels) commute with T(r)(θ). This drastically simplifies the construction of eigenstates, reducing it to solutions of relatively small number of simultaneous linear equations. Roots of unity play a crucial role. Thus for diagonalizing the 81 dimensional space for N=3, r=4, one has to solve a maximal set of five linear equations. A supplementary symmetry relates invariant subspaces pairwise [W=(r,Nr) and so on] so that only one of each pair needs study. The case N=3 is studied fully for r=(1,2,3,4). Basic aspects for all (N,r) are discussed. Full exploitation of such symmetries lead to a formalism quite different from, possibly generalized, algebraic Bethe ansatz. Chain Hamiltonians are studied. The specific types of spin flips they induce and propagate are pointed out. The inverse Cayley transform of the YB matrix giving the potential leading to factorizable S matrix is constructed explicitly for N=3 as also the full set of R̂tt relations. Perspectives are discussed in a final section.

Geometric phase forN-level systems through unitary integration

Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin $1∕2$, the Bloch sphere ${S}^{2}$, together with a U(1) phase, provides a complete SU(2) description. We generalize to $N$-level systems and $\mathrm{SU}(N)$ in terms of a $2(N\ensuremath{-}1)$-dimensional base space and reduction to a $(N\ensuremath{-}1)$-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an $N$-level system and gives $(N\ensuremath{-}1)$ geometric phases explicitly. A complete analytical construction of an ${S}^{4}$ Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).

Thermodynamics of Taub-NUT/bolt black holes in Einstein-Maxwell gravity

First, we construct the Taub-NUT/bolt solutions of $(2k+2)$-dimensinal Einstein-Maxwell gravity, when all the factor spaces of $2k$-dimensional base space $\mathcal{B}$ have positive curvature. These solutions depend on two extra parameters, other than the mass and the NUT charge. These are electric charge $q$ and electric potential at infinity $V$. We investigate the existence of Taub-NUT solutions and find that in addition to the two conditions of uncharged NUT solutions, there exist two extra conditions. These two extra conditions come from the regularity of vector potential at $r=N$ and the fact that the horizon at $r=N$ should be the outer horizon of the NUT charged black hole. We find that the NUT solutions in $2k+2$ dimensions have no curvature singularity at $r=N$, when the $2k$-dimensional base space is chosen to be ${\mathbb{C}\mathbb{P}}^{2k}$. For bolt solutions, there exists an upper limit for the NUT parameter which decreases as the potential parameter increases. Second, we study the thermodynamics of these spacetimes. We compute temperature, entropy, charge, electric potential, action and mass of the black hole solutions, and find that these quantities satisfy the first law of thermodynamics. We perform a stability analysis by computing the heat capacity, and show that the NUT solutions are not thermally stable for even $k$'s, while there exists a stable phase for odd $k$'s, which becomes increasingly narrow with increasing dimensionality and wide with increasing $V$. We also study the phase behavior of the 4 and 6 dimensional bolt solutions in canonical ensemble and find that these solutions have a stable phase, which becomes smaller as $V$ increases.

Taub-NUT/bolt black holes in Gauss-Bonnet-Maxwell gravity

We present a class of higher-dimensional solutions to Gauss-Bonnet-Maxwell equations in $2k+2$ dimensions with a $U(1)$ fibration over a $2k$-dimensional base space $\mathcal{B}$. These solutions depend on two extra parameters, other than the mass and the Newman-Unti-Tamburino charge, which are the electric charge $q$ and the electric potential at infinity $V$. We find that the form of metric is sensitive to geometry of the base space, while the form of electromagnetic field is independent of $\mathcal{B}$. We investigate the existence of Taub-Newman-Unti-Tamburino/bolt solutions and find that in addition to the two conditions of uncharged Newman-Unti-Tamburino solutions, there exist two other conditions. These two extra conditions come from the regularity of vector potential at $r=N$ and the fact that the horizon at $r=N$ should be the outer horizon of the black hole. We find that for all nonextremal Newman-Unti-Tamburino solutions of Einstein gravity having no curvature singularity at $r=N$, there exist Newman-Unti-Tamburino solutions in Gauss-Bonnet-Maxwell gravity. Indeed, we have nonextreme Newman-Unti-Tamburino solutions in $2+2k$ dimensions only when the $2k$-dimensional base space is chosen to be ${\mathbb{C}\mathbb{P}}^{2k}$. We also find that the Gauss-Bonnet-Maxwell gravity has extremal Newman-Unti-Tamburino solutions whenever the base space is a product of 2-torii with at most a 2-dimensional factor space of positive curvature, even though there a curvature singularity exists at $r=N$. We also find that one can have bolt solutions in Gauss-Bonnet-Maxwell gravity with any base space. The only case for which one does not have black hole solutions is in the absence of a cosmological term with zero curvature base space.

NUT-charged black holes in Gauss-Bonnet gravity

We investigate the existence of Taub-NUT/bolt solutions in Gauss-Bonnet gravity and obtain the general form of these solutions in $d$ dimensions. We find that for all non-extremal NUT solutions of Einstein gravity having no curvature singularity at $r=N$, there exist NUT solutions in Gauss-Bonnet gravity that contain these solutions in the limit that the Gauss-Bonnet parameter $\alpha$ goes to zero. Furthermore there are no NUT solutions in Gauss-Bonnet gravity that yield non-extremal NUT solutions to Einstein gravity having a curvature singularity at $r=N$ in the limit $% \alpha \to 0$. Indeed, we have non-extreme NUT solutions in $2+2k$ dimensions with non-trivial fibration only when the $2k$-dimensional base space is chosen to be $\mathbb{CP}^{2k}$. We also find that the Gauss-Bonnet gravity has extremal NUT solutions whenever the base space is a product of 2-torii with at most a 2-dimensional factor space of positive curvature. Indeed, when the base space has at most one positively curved two dimensional space as one of its factor spaces, then Gauss-Bonnet gravity admits extreme NUT solutions, even though there a curvature singularity exists at $r=N$. We also find that one can have bolt solutions in Gauss-Bonnet gravity with any base space with factor spaces of zero or positive constant curvature. The only case for which one does not have bolt solutions is in the absence of a cosmological term with zero curvature base space.

F-theory from Dirichlet 3-branes

Starting from the type IIB Dirichlet 3-brane action, we obtain a Nambu-Goto action. It is interpreted as the world volume action of a fundamental 3-brane, and its target space theory as F-theory. The target space is twelve dimensional, with signature (11, 1). It is an elliptic fibration over a ten dimensional base space. The SL(2, Z) symmetry of type IIB string has now an explicit geometric interpretation. Also, one gets a glimpse of the conjectured self-duality of M-theory.

Dimensional reduction in gauge theory

We consider a gauge theory with a general non-Abelian structure group and an (n+4)-dimensional base space withn compact dimensions. We show that integrating out the compact dimensions gives an effective four-dimensional theory with mass matrices for both the gauge bosons and for the matter fields. More interesting, we show that we still get the usual conserved Noether current in four dimensions, which is so necessary for renormalizability.