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Modular functions & elliptic genus

Several interesting debates we've had today can't be quite revealed because of copyright-like issues. Nevertheless, some of them can.

Finally, I understood the details how Xi Yin - together with Davide Gaiotto and Andy Strominger - have actually calculated the modified elliptic genus of the MSW (4,0) conformal field theory describing the 1+1-dimensional dynamics of M5-branes wrapped on a four-cycle of a Calabi-Yau three-fold. When I say it, it will probably sound simple and the reader won't know why I didn't get the whole line of reasoning earlier.

At any rate, you consider the trivially dual, type IIA string theory on a Calabi-Yau three-fold with D4-branes wrapped on a four-cycle "P" with some possible D2-brane and D0-brane charge. The intersection of "P" with itself is a curve (which means a complex curve - a real surface) whose genus is a linear function of the triple self-intersection number of "P". The textbook example that they look at is the most popular Calabi-Yau manifold, namely the quintic hypersurface in CP^4, and the hyperplane as the four-cycle. The hyperplane is the intersection of the quintic hypersurface and a CP^3 given by a linear equation inside the CP^4.

You ask what is the generalized index - called the modified elliptic genus

  • Trace (-1)^F (J_R)^2 q^{L0} qBAR^{L0BAR}

It is modified because "(J_R)^2" had to be inserted to counteract the effect of some right-moving fermionic zero modes in order to get a non-zero result. It is an elliptic genus and not an index because it depends on "q" while an index should be a universal constant. However, the elliptic genus should formally be independent of "qBAR". At the end, you find out a dependence on "qBAR", too, but it is simple. This dependence appears because some short multiplets have a non-zero value of "L0BAR".

This trace is a function of "q=exp(2.pi.i.tau)" where "tau" is the complex structure of the torus. It is a modular function whose weights can be determined.

Recall that the holomorphic modular functions (without singularities) are polynomials of the modular functions of weight 4 and 6, respectively. One of the appendices of Green, Schwarz, Witten tells you why. When you allow poles, you get many more choices. For example, the allowed weight zero modular functions include "G4^3 / G6^2" and similar expressions as long as you know that "4 x 3" as well as "6 x 2" are equal to "12".

It turns out that Xi et al. can compute a few coefficients of the modified elliptic genus - expanded into powers of "q" and "qBAR" - by counting the number of bound states of D4-branes with various combinations of D2-branes and D0-branes. These degeneracies can only be counted up to some values of "L0" but it is actually more than enough: they can determine a couple of coefficients of some singular terms in the modified elliptic genus as well as some coefficients of non-singular terms that can be used as a consistency check.

For example, if you study all possible weight zero meromorphic modular forms with certain singularities, you will find out that the coefficients of the singular terms that behave like negative powers for small "q" - i.e. for "tau" going to "i.infinity" - fully determine the coefficients in terms of functions such as "G4^3 / G6^2" mentioned previously. You should see that the number of unknown coefficients only grows linearly with the degree of the pole: it is parameterized by a power of "G4" while the power of "G6" is fixed by the total weight. Once you reconstruct these coefficients, you may determine the full modular form, including infinitely many coefficients in front of higher powers of "q".

The number of the coefficients that you can determine is actually higher than what you need to determine the modular forms: so you have some consistency check and it works. Inversely speaking, if Xi et al. had found and trusted their calculation 15 years ago - when he was in the kindergarden - without any consistency check, they could have determined the number of twisted rational cubics in the quintic - and it seems that they would have obtained this result without mirror symmetry.

All readers of The Elegant Universe by Brian Greene who have a good memory must remember that the physicists in 1991 were able to compute the number of these complex curves inside the quintic to be 317,206,375 while the mathematicians obtained a wrong result, namely 2,682,549,425.

Here I will assume that everyone, including Bright Simons, knows the trivial fact that the number of conics, or degree two curves, in the quintic hypersurface is equal to 609,250 and it is only the number of the twisted rational cubics that can be slightly difficult for the reader. ;-)

Two weeks after the joint conference of the mathematicians and physicists, the mathematicians had found an error in their code. After they corrected it, the result was clear: 317,206,375 & physics won. ;-)

In 1991, the physicists used mirror symmetry to solve a problem that went beyond the skills of the mathematicians. Mirror symmetry is a typical example of the power of physics that can be used in mathematics. Why is mirror symmetry physics? For example, some time ago, Jonathan Bolton informed me that the biggest physics mystery among the professors of humanities at Harvard is the question why the mirrors exchange the left and right side but don't flip the image upside-down. It seems as a discrimination of the vertical dimension and I hope many bright readers will get the answer! ;-)

Xi Yin et al. could have obtained this number above 300 million from modular invariance of a partition sum of a conformal field theory that describes D4-branes or, more precisely, M5-branes in M-theory wrapped on a four-cycle. Again, a physical picture gives you great tools to solve otherwise difficult problems in enumerative geometry.

Oh, Bright Simons has just informed me that he has calculated the number of the degree 4 curves in the quintic to be 242,467,530,000. Very well. Paul Boutin wanted a better result and counted the degree 10 curves: 704,288,164,978,454,686,113,488,249,750.

Cheap issues about maths vs. physics

Some general comments about similar problems in physics: arguably, it is more mathematics than physics. But it is still very important and physically-flavored mathematics. The papers are written in the physics fashion, not the mathematics fashion. More importantly, the key features of particle physics reasoning are used in these considerations. While it is true that it is difficult to create physical conditions in which the properties of the quintic hypersurface playing the role of extra dimensions can be directly measured, it is probably not impossible in principle.

Nevertheless, even if you decided that some parts of string theory are more mathematics than physics, it won't change the fact that they are cool, important, and they will play a very important role in the thinking of scholars in 2050. The fact that these insights are closer to mathematics than most facts about condensed matter physics are cannot imply that they should not be intensely investigated.

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