**M&m and little strings**It is amazing how efficiently String theorists build on previous work of their colleagues constructing step by step in a consistent way this beautiful framework of ideas and tools we call String theory.

There is an interesting troika of papers [1], [2], [3] following the pioneered work of Vafa et al [4], [5], [6] on M-strings (intersections of open M2 and M5 branes, refer to figure 2) which relate the M-strings of 6d SCFTs, the monopole Strings of 5d Supersymmetric Yang-Mills theories (coined m-strings) and the 6d little strings.

The basic result is that there is a direct correspondence between these type of Strings in the sense that monopole and little strings are basically bound states of M-strings!

In [1] it is proposed that the BPS degeneracies of bound-state of M-strings provide the elliptic genus of the moduli space of m-strings in the Nekrasov-Shatashvili limit.

This is based on the fact that BPS states of the 5d gauge theory are related via S-duality to magnetic monopole m-strings of the same theory but also via the appropriate compactification scheme to the M-strings in 6d.

The partition function of the 5d Supersymmetric Yang-Mills theory corresponds to an index that counts the degeneracies of BPS bound-states of W-bosons with instanton particles. Then the S-dual m-string picture can be shown to be the elliptic genus of certain moduli spaces.

The authors of [1] manage to match the partition function of the M-strings with this elliptic genus and thus establishing the correspondence.

To calculate the M-strings partition function they rely on the original work of Vafa et al [4], [5], [6] and use the powerful topological vertex formalism to calculate the topological string partition function of certain elliptically fibered Calabi-Yau 3-folds.

The key point here is that if we manage to geometrically engineer the 5d theory by compactifying M-theory on this CY the topological string partition function will give the partition function of the gauge theory. The gauge theory partition function on the other hand appears in the world-volume of certain (p,q) 5 branes of IIB which are associated to the corresponding M-theory branes configurations where the M-strings appear. From the IIB we can uplift to M-theory in 11 dimensions and thus find the appropriate compactification on the corresponding CY (it is well known that IIB (p,q) brane web diagrams are associated to CY toric diagrams in the context of toric geometry-branes correspondence).

Thus via the 5d theory the topological String partition function on the CY captures the BPS degeneracies of M-strings. This was suggested in the quite famous troika of the "parent" papers [4], [5], [6] where the partition function of the 5d theory was derived via three different methods, the topological string partition function, localization techniques in the 5d field theory and the (2,0) elliptic genus of the M-Strings (refer to figure 1).

Similar arguments are made in [3].

Now how the little strings come into the picture in [2]?

This can be achieved by compactifying the dimension along which the M5 branes are separated. Then even in case the M5 branes coincide (and thus M-strings become tensionless) there can always be M2 branes suspended between the first and last M5 and thus tensile strings; these are the little strings (little string theories are six-dimensional non-local quantum theories with non-gravitational string excitations which at energies far below the string scale flow to the SCFTs counterparts).

Another way for getting LSTs is in the world-volume of NS5 branes in IIA and IIB. The two LSTs (IIa and IIb respectively) are related by T-duality much like the parent string theories. This T-duality is reflected in the M5-M2 branes configuration setup by exchanging suitable compactification circles and in the topological string partition function computation by a fiber-base duality of the corresponding elliptically fibered CY.

There is a illuminating analogy that the authors make with Dp branes and open strings. Here is the relevant excerpt:

They compare multiple M5-branes on a transverse circle with multiple Dp-branes on a transverse circle and they interpret the M-strings (i.e. open M2-branes) as noncritical counterparts of open fundamental strings, while a little string ground state is the noncritical counterpart of a closed fundamental string. In the same way as multiple open fundamental strings on the Dp-branes can form a closed string and move freely in ambient ten-dimensional bulk spacetime, multiple open M2-branes ending on M5-branes can form a closed M2-brane and move freely in eleven dimensional spacetime but what makes the little strings very different from fundamental strings is that, in the decoupling limit (i.e. g-->0) the little strings are confined inside the five-brane worldvolume, i.e. the six dimensional spacetime the little string theories live in.

Now, by checking the compact partition functions (corresponding to little strings) to their non-compact counterparts (corresponding to M-strings) studied in [1], it is found that the counting function of compact BPS configurations can fully be constructed as a linear superposition of the non-compact ones. This implies that little strings can be viewed as bound-states of M-strings.

The overall result is stunning. As the authors state the fact that the little strings can be viewed as bound-states of M-strings means that "for the purpose of BPS counting of IIb little strings at least, one only needs to know BPS excitations of the (2,0) superconformal field theory, which is just the low-energy limit of the IIb little string theory".

As far as I know this is the first time that such relation between little strings and M-strings is proposed. In my mind this will have far reaching implications for the study of the elusive 6d SCFT where M-strings become tensionless.

Much more and other beautiful insights inside.

Figures were taken from [6] plus a bonus, a rare picture of uncle Albert giving a lecture on M5/M2 branes, M-strings and 6d SCFT.

[1] M String, Monopole String and Modular Forms

http://arxiv.org/abs/1503.06983[2] Instanton-Monopole Correspondence from M-Branes on S1 and Little String Theory

http://arxiv.org/abs/1511.02787[3] From strings in 6d to strings in 5d

http://arxiv.org/abs/1502.06645v2[4] M-strings

http://arxiv.org/abs/1305.6322[5] On orbifolds of M-Strings

http://arxiv.org/abs/1310.1185[6] M-strings, Elliptic Genera and N=4 String Amplitudes

http://arxiv.org/abs/1310.1325