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Elementary question about the fibration structure of a toric CY 3-fold


some basic question about fibrationQuestion about toric idealModuli Space of elliptic fibrationCalabi-Yau manifold with fiber structureIs the nodal curve a toric variety?Question about the relation between the Weierstrass equation and weighted projective spaceRecovering data about the toric fan from minimal informationIs the quotient of a toric variety by a finite group still toricQuestion about differential forms and delta functionsWhat are the finite etale covers of a Calabi-Yau variety?













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I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper ``Branes and Toric Geometry,'' by Leung and Vafa (arXiv:hep-th/9711013) who say (on page 12, section 2.1):




We are interested in manifolds admitting a $T^n$ action, with an n-dimensional base.




This seems to suggest that a toric Calabi-Yau 3-fold (generalizing the construction in the paper) is a $T^3$ fibration over a real 3-dimensional base.



However, in the thesis ``Crystal Melting and Wall Crossing Phenomena," by Yamazaki (arXiv:1002.1709 [hep-th]), the author explicitly says (on page 16, in section 3.2):




A toric Calabi-Yau threefold $X_Delta$ is a $T^2 times mathbbR$ fibration over $mathbbR^3$, where the fibers are special Lagrangian submanifolds.




Now I understand that the phrase "special Lagrangian submanifold" implies the existence in $X_Delta$ of a special Lagrangian submanifold. But I would like to know the sense in which these two seemingly different definitions are equivalent. Any references to a further discussion about toric spaces would is very welcome! (I am a physicist learning about the underlying mathematical structures. I would not like to shy away from the formal definitions, so math references are particularly welcome.)










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    0












    $begingroup$


    I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper ``Branes and Toric Geometry,'' by Leung and Vafa (arXiv:hep-th/9711013) who say (on page 12, section 2.1):




    We are interested in manifolds admitting a $T^n$ action, with an n-dimensional base.




    This seems to suggest that a toric Calabi-Yau 3-fold (generalizing the construction in the paper) is a $T^3$ fibration over a real 3-dimensional base.



    However, in the thesis ``Crystal Melting and Wall Crossing Phenomena," by Yamazaki (arXiv:1002.1709 [hep-th]), the author explicitly says (on page 16, in section 3.2):




    A toric Calabi-Yau threefold $X_Delta$ is a $T^2 times mathbbR$ fibration over $mathbbR^3$, where the fibers are special Lagrangian submanifolds.




    Now I understand that the phrase "special Lagrangian submanifold" implies the existence in $X_Delta$ of a special Lagrangian submanifold. But I would like to know the sense in which these two seemingly different definitions are equivalent. Any references to a further discussion about toric spaces would is very welcome! (I am a physicist learning about the underlying mathematical structures. I would not like to shy away from the formal definitions, so math references are particularly welcome.)










    share|cite|improve this question









    $endgroup$














      0












      0








      0





      $begingroup$


      I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper ``Branes and Toric Geometry,'' by Leung and Vafa (arXiv:hep-th/9711013) who say (on page 12, section 2.1):




      We are interested in manifolds admitting a $T^n$ action, with an n-dimensional base.




      This seems to suggest that a toric Calabi-Yau 3-fold (generalizing the construction in the paper) is a $T^3$ fibration over a real 3-dimensional base.



      However, in the thesis ``Crystal Melting and Wall Crossing Phenomena," by Yamazaki (arXiv:1002.1709 [hep-th]), the author explicitly says (on page 16, in section 3.2):




      A toric Calabi-Yau threefold $X_Delta$ is a $T^2 times mathbbR$ fibration over $mathbbR^3$, where the fibers are special Lagrangian submanifolds.




      Now I understand that the phrase "special Lagrangian submanifold" implies the existence in $X_Delta$ of a special Lagrangian submanifold. But I would like to know the sense in which these two seemingly different definitions are equivalent. Any references to a further discussion about toric spaces would is very welcome! (I am a physicist learning about the underlying mathematical structures. I would not like to shy away from the formal definitions, so math references are particularly welcome.)










      share|cite|improve this question









      $endgroup$




      I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper ``Branes and Toric Geometry,'' by Leung and Vafa (arXiv:hep-th/9711013) who say (on page 12, section 2.1):




      We are interested in manifolds admitting a $T^n$ action, with an n-dimensional base.




      This seems to suggest that a toric Calabi-Yau 3-fold (generalizing the construction in the paper) is a $T^3$ fibration over a real 3-dimensional base.



      However, in the thesis ``Crystal Melting and Wall Crossing Phenomena," by Yamazaki (arXiv:1002.1709 [hep-th]), the author explicitly says (on page 16, in section 3.2):




      A toric Calabi-Yau threefold $X_Delta$ is a $T^2 times mathbbR$ fibration over $mathbbR^3$, where the fibers are special Lagrangian submanifolds.




      Now I understand that the phrase "special Lagrangian submanifold" implies the existence in $X_Delta$ of a special Lagrangian submanifold. But I would like to know the sense in which these two seemingly different definitions are equivalent. Any references to a further discussion about toric spaces would is very welcome! (I am a physicist learning about the underlying mathematical structures. I would not like to shy away from the formal definitions, so math references are particularly welcome.)







      algebraic-geometry complex-geometry toric-geometry string-theory






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