Built on

E. Witten, “Phases of N = 2 theories in two dimensions,” Nucl. Phys., B403
1993
Earlier work this paper cites.
W. Fulton, Introduction to Toric Varieties. (AM-131) (Princeton University Press, 1993) ISBN 0691000492
1993
Earlier work this paper cites.
P. S. Aspinwall, B. R. Greene, and D. R. Morrison, “Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory,” Nucl. Phys., B416
1994
Earlier work this paper cites.
J. Distler, B. R. Greene, and D. R. Morrison, “Resolving singularities in (0,2) models,” Nucl. Phys., B481
1996
Earlier work this paper cites.
G. E. Bredon, Sheaf Theory (Springer, 1997) ISBN 0387949054
1997
Earlier work this paper cites.
R. Blumenhagen, “Target space duality for (0,2) compactifications,” Nucl. Phys., B513
1998
Earlier work this paper cites.

Similar

R. Blumenhagen, “(0,2) target-space duality, CICYs and reflexive sheaves,” Nucl. Phys., B514
1998
Cited alongside, same era.
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, “Mirror Symmetry,” (AMS Clay Mathematics Institute, 2003) Chap. Toric Geometry for String Theory
2003
Cited alongside, same era.
M. Kreuzer, “Toric Geometry and Calabi-Yau Compactifications,” (2006), arXiv:hep-th/0612307
2006
Cited alongside, same era.
S. Reffert, “The Geometer’s Toolkit to String Compactifications,” (2007), arXiv:0706.1310 [hep-th]
Original
2007
Cited alongside, same era.
F. Denef, “Les Houches Lectures on Constructing String Vacua,” (2008), arXiv:0803.1194 [hep-th]
Original
2008
Cited alongside, same era.

Then

The cohomology computations in Blumenhagen 1998 ; Blumenhagen et al. 2008 ; Blumenhagen et al. 2010 were also done using this algorithm
2010
Closest in time.
R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of Line Bundles: Applications,” (2010), arXiv:1010.3717 [hep-th]
Original
2010
Closest in time.
H. Roschy and T. Rahn, “Cohomology of Line Bundles: Proof of the Algorithm,” (2010), arXiv:1006.2392 [hep-th]
Original
2010
Closest in time.
S.-Y. Jow, “Cohomology of toric line bundles via simplicial Alexander duality,” (2010), arXiv:1006.0780 [math.AG]
Original
2010
Closest in time.
M. Cvetic, I. Garcia-Etxebarria, and J. Halverson, “Global F-theory Models: Instantons and Gauge Dynamics,” (2010), arXiv:1003.5337 [hep-th]

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2010

Cohomology of Line Bundles: A Computational Algorithm

Blumenhagen, Ralph, Jurke, Benjamin, Rahn, Thorsten et al.

Understand

We present an algorithm for computing line bundle valued cohomology classes over toric varieties.

This is the basic starting point for computing massless modes in both heterotic and Type IIB/F-theory compactifications, where the manifolds of interest are complete intersections of hypersurfaces in toric varieties supporting additional vector bundles.

Built on

E. Witten, “Phases of N = 2 theories in two dimensions,” Nucl. Phys., B403
1993
Earlier work this paper cites.
W. Fulton, Introduction to Toric Varieties. (AM-131) (Princeton University Press, 1993) ISBN 0691000492
1993
Earlier work this paper cites.
P. S. Aspinwall, B. R. Greene, and D. R. Morrison, “Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory,” Nucl. Phys., B416
1994
Earlier work this paper cites.
J. Distler, B. R. Greene, and D. R. Morrison, “Resolving singularities in (0,2) models,” Nucl. Phys., B481
1996
Earlier work this paper cites.
G. E. Bredon, Sheaf Theory (Springer, 1997) ISBN 0387949054
1997
Earlier work this paper cites.
R. Blumenhagen, “Target space duality for (0,2) compactifications,” Nucl. Phys., B513
1998
Earlier work this paper cites.

Similar

R. Blumenhagen, “(0,2) target-space duality, CICYs and reflexive sheaves,” Nucl. Phys., B514
1998
Cited alongside, same era.
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, “Mirror Symmetry,” (AMS Clay Mathematics Institute, 2003) Chap. Toric Geometry for String Theory
2003
Cited alongside, same era.
M. Kreuzer, “Toric Geometry and Calabi-Yau Compactifications,” (2006), arXiv:hep-th/0612307
2006
Cited alongside, same era.
S. Reffert, “The Geometer’s Toolkit to String Compactifications,” (2007), arXiv:0706.1310 [hep-th]
Original
2007
Cited alongside, same era.
F. Denef, “Les Houches Lectures on Constructing String Vacua,” (2008), arXiv:0803.1194 [hep-th]
Original
2008
Cited alongside, same era.

Then

The cohomology computations in Blumenhagen 1998 ; Blumenhagen et al. 2008 ; Blumenhagen et al. 2010 were also done using this algorithm
2010
Closest in time.
R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of Line Bundles: Applications,” (2010), arXiv:1010.3717 [hep-th]
Original
2010
Closest in time.
H. Roschy and T. Rahn, “Cohomology of Line Bundles: Proof of the Algorithm,” (2010), arXiv:1006.2392 [hep-th]
Original
2010
Closest in time.
S.-Y. Jow, “Cohomology of toric line bundles via simplicial Alexander duality,” (2010), arXiv:1006.0780 [math.AG]
Original
2010
Closest in time.
M. Cvetic, I. Garcia-Etxebarria, and J. Halverson, “Global F-theory Models: Instantons and Gauge Dynamics,” (2010), arXiv:1003.5337 [hep-th]

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

R. Blumenhagen, S. Moster, and E. Plauschinn, “Moduli Stabilisation versus Chirality for MSSM like Type IIB Orientifolds,” JHEP, 01

Original

2008

Cited alongside, same era.

D. A. Cox, J. B. Little, and H. Schenck, “Toric Varieties,” Unpublished book-in-progress, available at http://www.cs.amherst.edu/~dac/toric.html

Cited in the paper.

D. A. Cox, “Lectures on Toric Varieties,” Unpublished lecture notes, available at http://www.cs.amherst.edu/~dac/lectures/coxcimpa.pdf

Cited in the paper.

Another possibility to check the validity of these results is to perform the “chamber” algorithm described in Chapter 9 of Cox . We have checked that the methods of both algorithms coincide in some simple examples like ℙ 2 \mathbb{P}^{2} or d P 1 dP_{1} , but the computing time is by a factor of approximately 10 3 10^{3} shorter for our algorithm

Cited in the paper.

There is a more abstract treatment of sheaves in category theory, where the naive set-theoretic restriction used here is replaced by a general restriction mapping res U , V : ℱ ⁡ ( U ) -→ ℱ ⁡ ( V ) {\rm res}_{U,V}:\mathscr{F}(U)\relbar\joinrel\rightarrow\mathscr{F}(V)

Cited in the paper.

Note that there are a couple of mathematical fine points, like the usage of a good cover where all intersections U i ∩ U j U_{i}\cap U_{j} are contractible. See the mathematical literature for full details

Cited in the paper.

Original

2010

Closest in time.

R. Blumenhagen, A. Collinucci, and B. Jurke, “On Instanton Effects in F-theory,” (2010), arXiv:1002.1894 [hep-th]

Original

2010

Closest in time.