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Duong H. Phong

Publications and source records attributed to Duong H. Phong.

44 records · Page 3Linked to original sources
arXiv2017

On estimates for the Fu-Yau generalization of a Strominger system

We study an equation proposed by Fu and Yau as a natural $n$-dimensional generalization of a Strominger system that they solved in dimension $2$. It is a complex Hessian equation with right hand side depending on gradients. Building on the methods of Fu and Yau, we obtain $C^0$, $C^2$ and $C^{2,α}$ a priori estimates. We also identify difficulties in extending the Fu-Yau arguments for non-degeneracy from dimension $2$ to higher dimensions.

math.CV↗
arXiv2017

Geometric flows and Strominger systems

A geometric flow on $(2,2)$-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established, using Hamilton's version of the Nash-Moser implicit function theorem.

math.DG↗
arXiv2017

The Fu-Yau equation with negative slope parameter

The Fu-Yau equation is an equation introduced by J. Fu and S.T. Yau as a generalization to arbitrary dimensions of an ansatz for the Strominger system. As in the Strominger system, it depends on a slope parameter $α'$. The equation was solved in dimension $2$ by Fu and Yau in two successive papers for $α'>0$, and for $α'<0$. In the present paper, we solve the Fu-Yau equation in arbitrary dimension for $α'<0$. To our knowledge, these are the first non-trivial solutions of the Fu-Yau equation in any dimension strictly greater than $2$.

math.CV↗
arXiv2015

Pseudo-locality for a coupled Ricci flow

Let $(M,g,ϕ)$ be a solution to the Ricci flow coupled with the heat equation for a scalar field $ϕ$. We show that a complete, $κ$-noncollapsed solution $(M,g,ϕ)$ to this coupled Ricci flow with a Type I singularity at time $T<\infty$ will converge to a non-trivial Ricci soliton after parabolic rescaling, if the base point is Type I singular. A key ingredient is a version of Perelman pseudo-locality for the coupled Ricci flow.

math.DG↗
arXiv2015

Higher Order Deformations of Complex Structures

Deformations of complex structures by finite Beltrami differentials are considered on general Riemann surfaces. Exact formulas to any fixed order are derived for the corresponding deformations of the period matrix, Green's functions, and correlation functions in conformal field theories with vanishing total central charge. The stress tensor is shown to give a simple representation of these deformations valid to all orders. Such deformation formulas naturally enter into the evaluation of superstring amplitudes at two-loop order with Ramond punctures, and at higher loop order, in the supergravity formulation of the RNS superstring.

hep-th↗
arXiv2015

The Super Period Matrix with Ramond Punctures in the supergravity formulation

In a very recent preprint, Witten showed how to construct a $g|r \, \times \, g|r$ super period matrix for super Riemann surfaces of genus $g$ with $2r$ Ramond punctures, which is symmetric in the ${\bf Z}_2$ graded sense. He also showed how it can be applied to analyze supersymmetry breaking in string compactifications which are supersymmetric at tree-level. Witten's construction is in the purely holomorphic formulation of super Riemann surfaces. In this paper, a construction is given in the formulation of two-dimensional supergravity. The variations of the super period matrix with respect to supermoduli deformations are also given, as well as an explicit illustration of how the super period matrix with two Ramond punctures would emerge from a degeneration of the super period matrix without punctures in higher genus.

hep-th↗
arXiv2013

Two-loop vacuum energy for Calabi-Yau orbifold models

A precise evaluation of the two-loop vacuum energy is provided for certain Z_2 x Z_2 Calabi-Yau orbifold models in the Heterotic string. The evaluation is based on the recent general prescription for superstring perturbation theory in terms of integration over cycles in supermoduli space, implemented at two-loops with the gauge-fixing methods based on the natural projection of supermoduli space onto moduli space using the super-period matrix. It is shown that the contribution from the interior of supermoduli space (computed with the procedure that has been used in previous two-loop computations) vanishes identically for both the E_8 x E_8 and Spin (32)/Z_2 Heterotic strings. The contribution from the boundary of supermoduli space is also evaluated, and shown to vanish for the E_8 x E_8 string but not for the Spin (32)/Z_2 string, thus breaking supersymmetry in this last model. As a byproduct, the vacuum energy in Type II superstrings is shown to vanish as well for these orbifolds.

hep-th↗