[Paper Review] Blowups of Heterotic Orbifolds using Toric Geometry

Stefan Groot Nibbelink|ArXiv.org|Aug 14, 2007

Algebraic structures and combinatorial models3 references5 citations

TL;DR

This paper develops a unified framework to construct heterotic string compactifications on resolved orbifold singularities using both explicit blowup geometry and toric geometry. It demonstrates exact agreement between the spectra of heterotic $b{C}^3/b{Z}_3$ and $b{C}^3/b{Z}_4$ orbifolds and their blowup counterparts, showing that all twisted states—including anomalous U(1) sectors—are consistently reproduced or reinterpreted as non-universal axions, with full anomaly cancellation confirmed in all cases.

ABSTRACT

Heterotic orbifold models are promising candidates for models with MSSM like spectra. But orbifolds only correspond to a special place in moduli space, the bigger picture is described by the moduli space of Calabi-Yau spaces. In this talk we will make explicit connections between both points of view. To this end we study blowups of orbifold singularities using both explicit constructions and toric geometry techniques. We show that matching of all orbifold models in blowups are possible.

Motivation & Objective

• To bridge the gap between heterotic orbifold compactifications and smooth Calabi-Yau compactifications by resolving orbifold singularities.
• To systematically construct explicit blowups of $b{C}^n/b{Z}_n$ singularities with U(1) gauge bundles satisfying Hermitian Yang-Mills equations.
• To verify that the particle spectra of heterotic orbifolds match those of their resolved smooth counterparts, including twisted states and anomalous U(1) sectors.
• To extend the method to more complex orbifolds like $b{C}^3/b{Z}_4$ using toric geometry where explicit constructions are difficult.
• To confirm anomaly freedom in all resolved models through explicit computation of gauge and gravitational anomalies.

Proposed method

• Constructing the Kähler potential for the blowup of $b{C}^n/b{Z}_n$ using an SU(n)-invariant variable $X = (1 + ar{z}z)^n |x|^2$, with resolution parameter $r$.
• Deriving the curvature 2-form from the Kähler potential, showing it mimics a regularized delta function and localizes curvature at the exceptional divisor.
• Defining a U(1) gauge background via $i{rak F}_V = ig( rac{r}{r+X} ig)^{1 - 1/n} ig( ar{e}e - rac{n-1}{n^2} rac{ar{ ho} ho}{r+X} ig) H_V$, satisfying the Hermitian Yang-Mills condition.
• Computing topological invariants via integrals over $b{CP}^{p-1} times b{C}$ and $b{CP}^p$, yielding $ ext{tr}rak R^2$ and $ ext{tr}(i{rak F}/2ar au)^p$ with exact values: $n(n+1)$ and $1$ respectively.
• Using toric geometry to describe resolutions of $b{C}^3/b{Z}_4$, with linear equivalence relations among divisors $D_i$, $E_1$, $E_2$.
• Computing intersection numbers from the toric diagram, such as $D_1 E_1 E_2 = 1$, $E_1^3 = 8$, $E_2^3 = 2$, and $E_2^2 E_1 = -2$, to compute gauge and gravitational anomalies.

Experimental results

Research questions

• RQ1 Can explicit blowups of $b{C}^n/b{Z}_n$ orbifolds with U(1) gauge bundles be constructed such that the resulting geometry and gauge background satisfy the Hermitian Yang-Mills equations?
• RQ2 To what extent do the spectra of heterotic $b{C}^3/b{Z}_3$ orbifolds match those of their resolved smooth compactifications?
• RQ3 How are anomalous U(1) gauge fields in the orbifold model reinterpreted in the blowup geometry, and is anomaly cancellation preserved?
• RQ4 Can toric geometry techniques be used to compute spectra and anomalies in cases where explicit blowups are not known, such as $b{C}^3/b{Z}_4$?
• RQ5 Which orbifold models fail to have a blowup counterpart, and why?

Key findings

• Exact agreement is found between the spectra of the heterotic $b{C}^3/b{Z}_3$ orbifold and its blowup, with all twisted states reproduced, and the missing states reinterpreted as non-universal axions.
• For $b{C}^3/b{Z}_4$, the blowup model reproduces all orbifold spectra except model 4, which lacks a first twisted sector and thus cannot be blown up.
• The resolution of $b{C}^3/b{Z}_4$ yields intersection numbers $D_1 E_1 E_2 = D_2 E_1 E_2 = 1$, $E_1^3 = 8$, $E_2^3 = 2$, and $E_2^2 E_1 = -2$, which are essential for anomaly computation.
• The gauge background on the blowup is expanded as ${rak F}_V = -rac{1}{2} E_1 H_1 - rac{1}{4}(E_1 + 2E_2) H_2$, with $H_1 = V_1^I H_I$, $H_2 = V_2^I H_I$, and the $V_I$ satisfying Bianchi identities on exceptional divisors.
• All blowup models have anomaly-free spectra, confirmed via vanishing of Bianchi identities on $E_1$, $E_2$, and the $b{C}^2/b{Z}_2$ resolution, with $V_1^2 + V_1 ullet V_2 = 4$, $V_1 ullet V_2 = -2$, and $V_2^2 = 6$.
• The method successfully extends to complex orbifolds like $b{Z}_6$-II, and toric geometry enables analysis even when explicit blowups are not available.