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Invariant Homology on Standard Model Manifolds

Burt A. Ovrut, Tony Pantev, Rene Reinbacher

TL;DR

This work constructs torus-fibered Calabi–Yau threefolds $Z$ with fundamental group ${ m Z}_2 imes{ m Z}_2$ as quotients $Z=X/({ m Z}_2 imes{ m Z}_2)$ of fiber products $X=B imes_{{f P}^1}B'$, where $B,B'$ are restricted rational elliptic surfaces with freely acting ${ m Z}_2 imes{ m Z}_2$ automorphisms. A central technical goal is to compute the ${ m Z}_2 imes{ m Z}_2$-invariant subspace of the fourth homology, $H_4(Z,Z)$, by first working through $H_2(B,Z)$ under the involutions, using a second ${f P}^1$-fibration to obtain a concrete basis of generators. The paper explicitly determines the actions of the involutions $(-1)_B$, $ ho_B$, and translations $t_{e_6},t_{e_4}$, constructs the invariant generators on $B$ (and hence on $X$), and identifies the seven independent ${ m Z}_2 imes{ m Z}_2$-invariant classes descendent to $Z$, yielding $ ext{rank}igl(H_4(Z,Z)igr)=7$ and $h^{1,1}_Z=h^{2,1}_Z=7$. These invariant classes underpin the stable, holomorphic ${ m SU}(4)$ instantons and the anomaly-free, three-family standard-like models with suppressed nucleon decay in heterotic string/M-theory contexts.

Abstract

Torus-fibered Calabi-Yau threefolds Z, with base dP_9 and fundamental group pi_1(Z)=Z_2 X Z_2, are reviewed. It is shown that Z=X/(Z_2 X Z_2), where X=B X_{P_1} B' are elliptically fibered Calabi-Yau threefolds that admit a freely acting Z_2 X Z_2 automorphism group. B and B' are rational elliptic surfaces, each with a Z_2 X Z_2 group of automorphisms. It is shown that the Z_2 X Z_2 invariant classes of curves of each surface have four generators which produce, via the fiber product, seven Z_2 X Z_2 invariant generators in H_4(X,Z). All invariant homology classes are computed explicitly. These descend to produce a rank seven homology group H_4(Z,Z) on Z. The existence of these homology classes on Z is essential to the construction of anomaly free, three family standard-like models with suppressed nucleon decay in both weakly and strongly coupled heterotic superstring theory.

Invariant Homology on Standard Model Manifolds

TL;DR

This work constructs torus-fibered Calabi–Yau threefolds with fundamental group as quotients of fiber products , where are restricted rational elliptic surfaces with freely acting automorphisms. A central technical goal is to compute the -invariant subspace of the fourth homology, , by first working through under the involutions, using a second -fibration to obtain a concrete basis of generators. The paper explicitly determines the actions of the involutions , , and translations , constructs the invariant generators on (and hence on ), and identifies the seven independent -invariant classes descendent to , yielding and . These invariant classes underpin the stable, holomorphic instantons and the anomaly-free, three-family standard-like models with suppressed nucleon decay in heterotic string/M-theory contexts.

Abstract

Torus-fibered Calabi-Yau threefolds Z, with base dP_9 and fundamental group pi_1(Z)=Z_2 X Z_2, are reviewed. It is shown that Z=X/(Z_2 X Z_2), where X=B X_{P_1} B' are elliptically fibered Calabi-Yau threefolds that admit a freely acting Z_2 X Z_2 automorphism group. B and B' are rational elliptic surfaces, each with a Z_2 X Z_2 group of automorphisms. It is shown that the Z_2 X Z_2 invariant classes of curves of each surface have four generators which produce, via the fiber product, seven Z_2 X Z_2 invariant generators in H_4(X,Z). All invariant homology classes are computed explicitly. These descend to produce a rank seven homology group H_4(Z,Z) on Z. The existence of these homology classes on Z is essential to the construction of anomaly free, three family standard-like models with suppressed nucleon decay in both weakly and strongly coupled heterotic superstring theory.

Paper Structure

This paper contains 14 sections, 228 equations, 13 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The Calabi-Yau Threefolds $Z$
  3. Rational Elliptic Surfaces
  4. $B$ as a Double Cover of $\bar{Q}={\mathbb P}^{1}\times {\mathbb P}^{1}$
  5. $B$ as a Ruled Surface
  6. Generators of $H_2(B,{\mathbb Z})$
  7. Transformations under $(-1)_B$
  8. The Action of $\alpha_B$
  9. The Action of $t_{e_6}$ and $t_{e_4}$
  10. The Action of $t_{e_6}$ and $t_{e_4}$ on Fiber Components
  11. Transformations of the Canonical Generators under $t_{e_6}$
  12. Transformations of the Canonical Generators under $t_{e_4}$
  13. The $\tau_{B1}$, $\tau_{B2}$ Action on $H_2(B,{\mathbb Z})$ and Invariant Classes
  14. The Homology of $X$ and $Z$

Figures (13)

  • Figure 1: The bidegree $(2,4)$ curve $M=T\cup r$ in $Q={\mathbb P}^{1}\times {\mathbb P}^{1}$. The projections $p_i: Q \to \mathbb{P}_i^1, i=1,2$ are explicitly shown.
  • Figure 2: A schematic representation of curve $M=t^{"}\cup s\cup i \cup j \cup r$ in $Q={\mathbb P}^{1}\times {\mathbb P}^{1}$.
  • Figure 3: The reducible fiber $I_2$, where $o$ denotes the proper transform and $n$ the exceptional divisor.
  • Figure 4: The elliptic fibration $W_M$ with the four distinguished sections.
  • Figure 5: Three different ways to consider $B$ as a degree four cover of $Q$.
  • ...and 8 more figures