cohomCalg
line bundle cohomology on toric varieties

What is cohomCalg?

cohomCalg is a software package for the convenient computation of cohomology group dimensions of line bundles on toric varieties.

It is the result of a research collaboration with R. Blumenhagen, T. Rahn and H. Roschy where we discovered a new algorithm for this purpose—see the paper here for the details—and I developed a high-performance C++ implementation of the algorithm. You can try out cohomCalg below (with certain restrictions), see the official website for the manual.

Where can I get cohomCalg?

For the most recent version of the program see the Official cohomCalg website.

cohomCalg Test Environment

Input data: (no comments allowed, truncated to 1024 characters) vertex u1 | GLSM: ( 1, 0, 0, 1, 0, 0 ); vertex u2 | GLSM: ( 1, 0, 1, 0, 1, 0 ); vertex u3 | GLSM: ( 1, 1, 0, 0, 0, 0 ); vertex u4 | GLSM: ( 0, 1, 0, 0, 1, 0 ); vertex u5 | GLSM: ( 0, 0, 1, 0, 0, 0 ); vertex u6 | GLSM: ( 0, 0, 0, 1, 0, 0 ); vertex u7 | GLSM: ( 0, 0, 0, 0, 1, 1 ); vertex u8 | GLSM: ( 0, 0, 0, 0, 0, 1 ); srideal [u1*u2, u1*u3, u1*u6, u1*u7, u1*u8, u2*u3, u2*u4, u2*u5, u2*u7, u3*u4, u3*u5, u3*u8, u4*u5, u4*u6, u4*u7, u5*u6, u5*u8, u6*u7, u6*u8, u7*u8]; ambientcohom O( -2, -2, -2, 0, -2, 0 ); ambientcohom O( -1, 0, 0, -3, -1, -2 ); ambientcohom O( 2, 0, 1, 1, 1, 0 );

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More recently T. Rahn added a Mathematica 7 frontend for cohomCalg, which allows for the easy-to-use computation of line bundle cohomologies on hypersurfaces or complete intersection spaces. The script basically collects all the necessary line bundles, computes their cohomology using cohomCalg and then derives the requested cohomologies via exactness considerations from the Koszul sequence. T. Rahn takes care of this cohomCalg Koszul extension, while I continue to maintain the C++ core program.

What does cohomCalg cost?

Nothing. As the result of a scientific research project, it’s licensed under the GPL v3 and therefore comes completely free with full source code.

Can I use the program in my own work?

Yes. The license certainly allows for derivated work, provided that the original authors are acknowledged and the work is itself released under the GPL v3 with open source code. Therefore you should include some reference to our research paper on the algorithm at an appropiate place of your work. Further citation information is available on the official cohomCalg website. Please keep in mind that while the project will be maintained and further expanded in the foreseeable future, it’s long time support cannot be guaranteed.

Development Motivation:

Given the generators of a Stanley-Reisner ideal and the gauged linear sigma-model (GLSM) charges / degrees of the homogeneous coordinates for a toric variety, the program computes the dimensions of the sheaf cohomology groups for the twisted holomorphic line bundle / structure sheaf O(D) on the variety. Those dimensions are required frequently for the computation of numerous physically relevant quantities. The underlying idea of counting rationoms (monomials with integer exponents) occurred to R. Blumenhagen in the late ’90s. A closer inspection by us revealed the need for a refined definition of the multiplicity factors that weight the numbers of the rationoms, leading to the algorithm in its final form [1] and the subsequent proof [2]. The ease of usage for the computation of line bundle cohomology group dimensions provides a very efficient tool for string model building. Furthermore, the algorithm can—apparently—easily be used for the computation of coset spaces (i.e. equivariant cohomology groups on the original space) as well [3].

Main publications regarding the algorithm:

[1] Mar ’10: Cohomology of Line Bundles: A Computational Algorithm
presents the original algorithm conjecture
[2] Jun ’10: Cohomology of Line Bundles: Proof of the Algorithm
provides a strict mathematical proof of the algorithm
[3] Oct ’10: Cohomology of Line Bundles: Applications
details many applications in physics, explains the Koszul sequence and
conjectures an extension to equivariant cohomologies (also see this)

Related publications and material:

Jun '10: Cohomology of toric line bundles via simplicial Alexander duality
an alternative and entirely independent proof of the algorithm
Jul '11: Cox, Little, Schenck: Toric Varieties (text book)
a math book with tons of information on toric varieties and related topics

In comparison to our earlier algorithm implementation as a Mathematica 7 script, the C++ implementation cohomCalg provides a huge speedup by five to six orders of magnitude according to our measurements. For the actual geometries of interest to us, this effectivily means quasi-instantaneous computation time for all relevant examples. Besides that, cohomCalg’s usage is extremely straightforward and intuitive: data input and output is provided via a text files in a rather human-readable format, which allows for an easy integration into our day-to-day workflow. In case you are interested in some of the details of the implementation, take a look at the corresponding section in the manual contained in the package.

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