Full Moduli Space Research Articles

Periodic monopoles from spectral curves

We consider $\text{SU}(2)$ Bogomolny equations on $\mathbb{R}^2\times\hat{S}^1$ and use the spectral curve defined by the holonomy in the periodic direction to approximate the fields in the limit of large size to period ratio. Symmetries of the Nahm transform allow a study of the effective two dimensional dynamics, which is compared with known results on the full moduli space. The techniques are applied to systems of higher charge and higher rank gauge group, allowing a direct comparison to other periodic Yang-Mills systems.

Towards large volume big divisor D3/D7 “ μ-split supersymmetry” and Ricci-flat Swiss-cheese metrics, and dimension-six neutrino mass operators

We show that it is possible to realize a “ μ-split SUSY” scenario (Cheng and Cheng, 2005) [1] in the context of large volume limit of type IIB compactifications on Swiss-cheese Calabi–Yau orientifolds in the presence of a mobile space–time filling D3-brane and a (stack of) D7-brane(s) wrapping the “big” divisor. For this, we investigate the possibility of getting one Higgs to be light while other to be heavy in addition to a heavy higgsino mass parameter. Further, we examine the existence of long lived gluino that manifests one of the major consequences of μ-split SUSY scenario, by computing its decay width as well as lifetime corresponding to the three-body decays of the gluino into either a quark, a squark and a neutralino or a quark, squark and goldstino, as well as two-body decays of the gluino into either a neutralino and a gluon or a goldstino and a gluon. Guided by the geometric Kähler potential for Σ B obtained in Misra and Shukla (2010) [2] based on GLSM techniques, and the Donaldsonʼs algorithm (Barun et al., 2008) [3] for obtaining numerically a Ricci-flat metric, we give details of our calculation in Misra and Shukla (2011) [4] pertaining to our proposed metric for the full Swiss-cheese Calabi–Yau (the geometric Kähler potential being needed to be included in the full moduli space Kähler potential in the presence of the mobile space–time filling D3-brane), but for simplicity of calculation, close to the big divisor, which is Ricci-flat in the large volume limit. Also, as an application of the one-loop RG flow solution for the higgsino mass parameter, we show that the contribution to the neutrino masses at the EW scale from dimension-six operators arising from the Kähler potential, is suppressed relative to the Weinberg-type dimension-five operators.

The Master Space of Supersymmetric Gauge Theories

We give a short review on the study of the moduli space and the spectrum of chiral operators for gauge theories living on branes at singularities. We focus on theories with four real supercharges in 3+1 and 2+1 dimensions. The theories are holographically dual to AdS5 × H5 or AdS4 × H7 backgrounds, in Type‐IIB or ‐M theory, respectively. We demonstrate that most of the information on the moduli space and spectrum of the quiver gauge theories is encoded in the concept of the “Master Space”, which is roughly the full moduli space for one brane, consisting of mesonic and baryonic degrees of freedom. We summarize the relevant information in generating functions for chiral operators, which can be computed using plethystics techniques and the language of complex geometry.

Extended supersymmetric moduli space and a SUSY/non-SUSY duality

We study N = 1 supersymmetric U ( N ) gauge theories coupled to an adjoint chiral field with superpotential. We consider the full supersymmetric moduli space of these theories obtained by adding all allowed chiral operators. These include higher-dimensional operators that introduce a field-dependence for the gauge coupling. We show how Feynman diagram/matrix model/string theoretic techniques can all be used to compute the IR glueball superpotential. Moreover, in the limit of turning off the superpotential, this leads to a deformation of N = 2 Seiberg–Witten theory. In the case where the superpotential drives the squared gauge coupling to a negative value, we find that supersymmetry is spontaneously broken, which can be viewed as a novel mechanism for breaking supersymmetry. We propose a new duality between a class of N = 1 supersymmetric U ( N ) gauge theories with field-dependent gauge couplings and a class of U ( N ) gauge theories where supersymmetry is softly broken by nonzero expectation values for auxiliary components of spurion superfields.

Gauge theory on fuzzyS2×S2and regularization on noncommutative Bbb R4

We define U(n) gauge theory on fuzzy S^2_N x S^2_N as a multi-matrix model, which reduces to ordinary Yang-Mills theory on S^2 x S^2 in the commutative limit N -> infinity. The model can be used as a regularization of gauge theory on noncommutative R^4_\theta in a particular scaling limit, which is studied in detail. We also find topologically non-trivial U(1) solutions, which reduce to the known "fluxon" solutions in the limit of R^4_\theta, reproducing their full moduli space. Other solutions which can be interpreted as 2-dimensional branes are also found. The quantization of the model is defined non-perturbatively in terms of a path integral which is finite. A gauge-fixed BRST-invariant action is given as well. Fermions in the fundamental representation of the gauge group are included using a formulation based on SO(6), by defining a fuzzy Dirac operator which reduces to the standard Dirac operator on S^2 x S^2 in the commutative limit. The chirality operator and Weyl spinors are also introduced.

Moduli of quantum Riemannian geometries on ⩽4 points

We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously [S. Majid, Commun. Math. Phys. 225, 131 (2002)]. The full moduli space is found for ⩽3 points, and a restricted moduli space for 4 points. Generalized Levi–Cività connections and their curvatures are found for a variety of models including models of a discrete torus. The topological part of the moduli space is found for ⩽9 points based on the known atlas of regular graphs. We also remark on aspects of quantum gravity in this approach.

Orbits of SU(2)-representations and minimal isometric immersions

In contrast to all known examples, we show that in the case of minimal isometric immersions of \(S^3\) into \(S^N\) the smallest target dimension is almost never achieved by an \(SU(2)\)-equivariant immersion. We also give new criteria for linear rigidity of a fixed minimal isometric immersion of \(S^3\) into \(S^N\). The minimal isometric immersions arising from irreducible SU(2)-representations are linearly rigid within the moduli space of SU(2)-equivariant immersions. Hence the question arose whether they are still linearly rigid within the full moduli space. We show that this is false by using our new criteria to construct an explicit SU(2)-equivariant immersion which is not linearly rigid. Various authors [GT], [To3], [W1] have shown that minimal isometric immersions of higher isotropy order \(d \ge 1\) play an important role in the study of the moduli space of all minimal isometric immersions of \(S^3\) into \(S^N\). Using a new necessary and sufficient condition for immersions of isotropy order \(d \ge 1\), we derive a general existence theorem of such immersions.

On four-dimensional compactifications of F-theory

Branches of moduli space of F-theory in four dimensions are investigated. The transition between two branches is described as a 3-brane-instanton transition on a 7-brane. A dual heterotic picture of the transition is presented and the F-theory-heterotic theory map is given. The F-theory data, complex structure of the Calabi-Yau fourfold and the instanton bundle on the 7-brane is mapped to the heterotic bundle on the elliptic Calabi-Yau threefold CY 3. The full moduli space has a web structure which is also found in the moduli space of semi-stable bundles on CY 3. Matter content of the four-dimensional theory is discussed in both F-theory and heterotic theory descriptions.

Semistable Bundles over an Elliptic Curve

We identify the moduli space Mn, d of semistable bundles of rank n and degree d on an elliptic curve C as a symmetric product of the curve, its dimension being the greatest common divisor of n and d. Under this identification the determinant map det: Mn, d corresponds to the Abel-Jacobi map α: ShC → Jh, so that the fixed-determinant moduli space is identified with a projective space. We then describe the Brill-Noether loci, theta divisors, and Picard groups and compute the space of theta functions for both the full moduli space and the fixed-determinant moduli space of any rank and any degree. We extend to elliptic curves a theorem of Drezet and Narasimhan on the Picard group of moduli spaces, as well as two formuulas of Beauville, Narasimhan, and Ramanan on theta functions of level one. Our results verify in particular for an elliptic curve a formula of E. Verlinde, and an equality first noticed by Bott and Szenes relating theta functions over the full moduli space and over the fixed-determinant moduli space.