Black things: big and small
Atish started with some basic introduction to black hole thermodynamics in string theory. The Bekenstein-Hawking entropy is "A/4G" where "A" is the horizon area and "G" is Newton's constant. In string theory, this value can be checked for many BPS objects by microscopic calculations. One usually obtains the result in the form
- S = 2.pi.sqrt(N1.N2.N3)
where "N1,N2,N3" are three kinds of charges expressed as multiples of the elementary charge - i.e. as integers - for example the number of D1-branes, the number of D5-branes, and the momentum (or some of their dual quantities). When you consider four-charge black holes, you obtain a remarkably similar formula
- S = 2.pi.sqrt(N1.N2.N3.N4)
to the result for three charges. The cubic and quartic expressions can actually be generalized to more general values of many types of charges as long as you replace the product of three or four factors by the corresponding cubic or quartic invariant of the duality group - an invariant expressed in the representation in which the charges transform. Also, one can consider rotating black holes with angular momentum "J". In these cases, the formulae above are typically modified to have "n.w-J" instead of just "n.w" - in an example with a momentum "n" and a winding "w". Better-than-expected agreement emerges for near-extremal black holes and even some classes of non-extremal ones. In all cases, if you express these nice results - 2.pi times a square root of an integer - in terms of the geometric parameters describing the black hole, you get an agreement with the Bekenstein-Hawking entropy.
Note that in the first "Strominger-Vafa" example with three charges, you obtain zero if one of the charges is zero i.e. if there are only two charges. Indeed, the classical horizon area shrinks to zero. However, this is only the case if you neglect the stringy corrections to the Einstein-Hilbert action. Once you include higher-derivative (four-derivative) corrections, your equations of motion change and the black hole develops a horizon even for a vanishing value of the third charge. The classical horizon area is nonzero but the entropy is no longer "A/4G". You must use the full Wald's formula that extracts the entropy as a function of the Lagrangian. The formula is essentially
- Entropy = int (horizon) 4.pi. partial(Lagrangian)/partial(R_{rtrt})
where "R_{rtrt}" is a component of the Riemann tensor in the r-t plane, a plane transverse to the (D-2)-dimensional horizon; imagine that you choose coordinates at each point of the horizon that put the metric to the standard Minkowski form. If you only have the Einstein-Hilbert Lagrangian "R/16.pi.G", the integral above gives you "A/4G". If you have "R^2" corrections, you will integrate not only a constant over the horizon but also a multiple of "R".
Typically, when you do such things in "R^2" theories, the black hole entropy becomes "A/2G": one half of the result is the old Bekenstein-Hawking entropy and the other half arises from the new term of the Wald formula. However, you should keep in mind that only the final result for the entropy is physical i.e. convention-independent. When you redefine the metric "g_{mn}" by adding a small multiple of "R_{mn}", for example, this field redefinition won't be visible at infinity but it will affect the way in which the entropy "A/2G" is divided between the Bekenstein-Hawking and the subleading Wald contributions.
Heterotic black holes
Atish's name is also associated with the Dabholkar-Harvey states, supersymmetric perturbative excitations of the heterotic string. These states only have the excitations on the left-moving, bosonic side, while the right-moving, supersymmetric sector of the string is kept in its ground state which guarantees that SUSY is unbroken. Atish decided to study the CHL string. The CHL string is a "Z2" orbifold of the "E8 x E8" heterotic string in which the "Z2" also exchanges the two gauge groups, together with a shift of a circular coordinate by one half of its circumference. Visually, in the Hořava-Witten strongly coupled picture, such a procedure makes the CHL string clearly equivalent to M-theory on the Möbius strip.
Atish decided to look at the CHL string because it can still preserve the same supersymmetry as the heterotic string while allowing one to investigate more discrete choices and see new mathematical structures. In all these heterotic cases, one only encounters the vector multiplets among the states of the black hole. This means that all contributions to the index are positive, and the index can therefore be identified with the exponentiated entropy. When he switches from the heterotic string to the CHL string, the resulting formula for the degeneracy of the states changes a bit. For the heterotic string, you would get a Bessel result "I_{13}" for some argument. The CHL operation effectively identifies two groups of 8 bosons which reduces "I_{13}" to "I_{9}". However, there are different rules and normalizations for the states charged under the gauge symmetry, so that the result must be multiplied by a theta function describing "E8" current algebra at level two; that should not be surprising because by identifying two "E8" groups, we give the CHL string its level two.
The group "Sp(2,Z)" of symplectic "4 x 4" symplectic matrices plays a rather important role in these calculations, much like the analysis of its modular functions. Mathematical results from the 19th century due to Jacobi and many others are relevant in this enterprise. One can show that the resulting formula corresponds to the partition sum of the bosonic string's CFT on a genus 2 surface, as recently argued by Davide Gaiotto. Indeed, the "Sp(2,Z)" is the group of large diffeomorphisms of a genus 2 surface that rearranges its four one-cycles in a way that preserves the intersections numbers. While the occurence of a genus two Riemann surface could look like a bizarre mathematical coincidence, there is actually a chain of arguments to map the black hole states into a string network and then a genus two M2-brane.
At any rate, these heterotic examples work perfectly and the correct number of the dyonic states in vacua with 16 supercharges was counted, re-counted, and re-recounted. In a similar way, the Calabi-Yau examples based on type II string theory (OSV etc.) also work pretty well.
Natural unit of entropy
Incidentally, the general stringy microscopic formulae for the black hole entropy have the form of 2.pi times a square root of an integer, as argued above. In all those naive attempts based on "loop quantum gravity", many people wanted to obtain a brutally discrete spectrum of entropy, something like "ln(3) times integer". First of all, the "integer" is replaced by "sqrt(integer)" in the correct theory. But you may argue that this difference can be suppressed for a proper choice of a rotating black hole. However, you also see a difference in the prefactor. The chaotic, confused and inconsistent Immirzi-like prefactors are replaced by a nice number "2.pi" in string theory - this number technically arises from the application of Cardy's formula.
The numerologists among the readers could ask: is there a relation between "2.pi" and "ln(integer)"? The answer is Yes. Most people interested in mathematical physics already know that
- 1 + 2 + 3 + 4 + 5 + ... = -1/12.
This result is easily expressed as "zeta(-1)" where "zeta" is Riemann's zeta function. But can we also calculate the product of all integers?
- 1 x 2 x 3 x 4 x 5 x ... = ?
Yes. Take the logarithm of the product. You obtain
- ln(1) + ln(2) + ln(3) + ... = ?
Can we regularize this mildly divergent sum? Yes. These logarithms of integers occur in the *derivative* of "zeta(s)" with respect to "s", namely in "zeta'(s)". If you want to get rid of the power law factors from "zeta(s)", you choose "s=0". Don't forget that the logarithms appear with a minus sign if you differentiate. In summary,
- ln(1) + ln(2) + ln(3) + ... = -zeta'(0) = ln(sqrt(2.pi))
This means that
- 1 x 2 x 3 x 4 x 5 x ... = sqrt(2.pi)
Whether you can combine different versions of loop quantum gravity in such a way that you obtain the correct entropy is up to your imagination. ;-)
