Learning M-theory: Gauge theory of membranes, brane intersections, and the self-dual string

I’ve been learning a lot about M-theory. It’s such a broad topic that, when people ask me ‘what is M-theory?’, I continue to struggle to know where to start. Right now, much of my learning is textbook and I have more questions than answers. I naturally take the approach of first wanting as broad and general of a picture as possible. In some sense, it is like starting with the general and working toward the particular. Or, in another way, it’s like when being introduced to a new landscape and wanting, at the outset, a broad orientation to its general geographical features, except in this case we are speaking in conceptual and quantitative terms. I may not ever be smart enough to grasp M-theory in its entirety, but what is certain is that I am working my hardest.

In surveying its geographical features and charting my own map, if I may continue the analogy, obtaining a better sense of the fundamental objects of M-theory is a particular task; but my main research interest has increasingly narrowed to the study and application of gauge theory and higher gauge theory. This can be sliced down further in that I am very interested in the relationship between string and gauge theory, and furthermore in studying the higher dimensional generalisation of gauge theory. This interest naturally follows from the importance of gauge theory in contemporary physics, and then how we may understand it from the generalisation of point particle theory to string theory and then to other higher dimensional extended objects (i.e., branes). We’ve talked a bit in the past about how the dynamics on the D-brane worldvolume is described by a gauge theory. We’ve also touched on categorical descriptions, and how in p-brane language when we study the quantum theory the resemblance of the photon can be seen as a p-dimensional version of the electromagnetic field (by the way, we’re going to start talking about p-branes in my next string note). That is to say, we obtain a p-dimensional analogue of Maxwell’s equations. More advanced perspectives from the gauge theory view, or in this case higher gauge theory view in M-theory, illuminate the existence of new objects like self-dual strings.

There is so much here to write about and explore, I look forward to sharing more as I progress through my own studies and thinking. In this post, though, I want to share some notebook reflections on things I’ve been learning more generally in the context of M-theory: some stuff about membranes, 11-dimensional supergravity, and the self-dual string. This post is not very technical; it’s just me thinking out loud.

11-dimensional supergravity

The field content of 11-dimensional supergravity consists of the metric $g_{\mu \nu}$, with 44 degrees of freedom; a rank 3 anti-symmetric tensor field $C_{\mu \nu \rho}$, with 84 degrees of freedom; and these are paired off with a 32 component Majorana gravitino $\Psi_{\alpha \mu}$, with 128 degrees of freedom. Although much has progressed since originally conceived, the Lagrangian for the bosonic sector is similar to as it was originally written [3]

$S_{SUGRA} = \frac{1}{2k_{11}^2} \int_{M_{11}} \sqrt{g} \ (R - \frac{1}{48}F^{2}_{4}) - \frac{1}{6} F_{4} \wedge F_{4} \wedge C_3. \ \ (1)$

The field strength is $F_4 = dC_3$ and $k_{11}$ is the 11-dimensional coupling constant. The field strength is defined conventionally,

$\mid F_n \mid^2 = \frac{1}{n !} G^{M_1 N_1} G^{M_2 N_2} ... G^{M_n N_n}F_{M_{1}M_{2} ... M_{n}}F_{N_1 N_2 ... N_n}. \ \ (2)$

The 11-dimensional frame field in the metric combination is $G_{MN} = \eta_{AB}E^{A}_{M}E^{B}_{N}$, where we have the elfbeins $E^{B}_{N}$, $M,N$ are indices for curved base-space vectors, and $A,B$ are indices for tangent space vectors. The last term in (2) is the Cherns-Simons structure. This is a topological dependent term independent of the metric. We see this structure in a lot of different contexts.

Although, from what I presently understand, the total degrees of freedom of M-theory are not yet completely nailed down, we can of course begin to trace a picture in parameter space. As we’ve discussed before on this blog, it can be seen how 10-dimensional type IIA theory in the strong coupling regime behaves as an 11-dimensional theory whose low-energy limit is captured by 11-dimensional supergravity. Reversely, compactify 11-dimensional supergravity on a circle of fixed radius in the $x^{10} = z$ direction, from the 11-dimensional metric we then obtain the 10-dimensional metric, a vector field and the dilaton. The 3-form potential leads to both a 3-form and a 2-form in 10-dimensions. The mysterious 11-dimensional theory can also be seen to give further clue at its parental status given how supergravity compactified on unit interval ${\mathbb{I} = [0,1]}$, for example, leads to the low-energy limit of $E8 \times E8$ heterotic theory.

Non-renomoralisability of 11-dimensional SUGRA

One thing that I’ve known about for sometime but I have not yet studied in significant detail concerns precisely how 11-dimensional supergravity is non-renormalisable [4,5,6]. Looking at the maths, what I understand is that above two-loops the graviton-graviton scattering is divergent. Moreover, as I still have some questions about this, what I find curious is that in the derivative expansion in 11-dimensional flat spacetime (using a 1PI/quantum effective Lagrangian approach) the generating functional for the graviton S-matrix is non-local. But due to supersymmetry, low order terms in the derivative expansion can be separated into local terms, such as $t_8 t_8 R^4$, and non-local (or global) terms that correspond to loop amplitudes. But what happens is that, at 2-loops, a logarithmic divergence that is cut off at the Planck scale mixes with a local term of the schematic form $D^{12}R^4$, where $R^4$ is the supersymmetrised vertex. In the literature, one will find a lot of discussion about this $R^4$ vertex. But like I said, I really need more time looking at this.

In short, the important mechanism in string theory that allows us to avoid UV divergences is absent, or appears absent, in maximal supergravity. What could the UV regulator be? As in any supergravity, from what I understand, it is not clear that a Lagrangian description is sufficient at the Planck scale.

The facts of 11-dimensional supergravity and how it relates to 10-dimensional string theory are textbook and well-known. Going beyond dualities relating different string theories, an obvious question concerns what M-theory actually constitutes. One thing that is known is that M-theory reduces to 11-dimensional SUGRA at low-energies, as we touched on, and it is known that fundamental degrees of freedom are 2-dimensional and 5-dimensional objects, known as M2-branes and M5-branes. Study of these non-perturbative states offer several intriguing hints. There are also solutions to classical supergravity known as F1 – the fundamental string – and its magnetic dual, the NS5-brane. As it relates to the story of the five string theories, the M-branes realize all D-branes, and this is why D-branes are considered consistent objects in quantum gravity.

The way that M-theory sees D-branes is via the net of dualities. All of the D-branes and the NS5 brane are solutions to type II theories, both A and B. So, when you reduce M-theory on a circle, in that you get back to Type IIA, the M2-branes and M5-branes reduce to the various D-branes such that under S-duality from the D5-brane you get the NS5.

The worldvolume theory of the M5-brane is always strongly coupled, which can be seen in moduli space (its parameters are simply a point). So there is no Lagrangian for this theory, and it suggests something deep is needed or is missing. It is expected that its worldvolume theory will be a 6-dimensional superconformal field theory, typically known as the 6d(2,0) theory. The worldvolume theory for M2-branes (on an orbifold) has been found to be a 3-dimensional superconformal Chern-Simons theory with classical $\mathcal{N} = 6$ supersymmetry.

If one considers a single M5-brane, a theory can be formulated in terms of an Abelian (2,0)-tensor multiplet, consisting of a self-dual 2-form gauge field, 5 scalars, and 8 fermions, but it is not known how to generalise the construction to describe multiple M5-branes. To give an example, using AdS/CFT [7] it is described how the worldvolume theory for a stack of $N$ M5-branes is dual to M-theory on $AdS7 \times S4$ with $N$ units of flux through the 4-sphere, which reduces to 11-dimensional SUGRA on this background in the limit large $N$ limit.

Brane intersections and stacks

The existence of branes is one of the most fascinating things about quantum gravity. There is a lot to unpack when learning about D1-branes, D3-branes, D5-branes, M2-branes, and M5-branes, as well as how they may intersect and what sort of consistent solutions have already been found [8,9, 10, 11, 12].

For example, an M2-brane, or a stack of coincident M2-branes, can end on a D5-brane. This is similar to the more simplified story of how D-branes, coincident D-branes, can intersect in string theory. Typically, D1-D3 systems in Type IIB string theory are studied because this system relates to the M2-M5 system by dimensional reduction and T-duality.

Self-dual string

For a membrane to end on a D5-brane, the membrane boundary must carry the charge of the self-dual field $B$ on the five-brane worldvolume. There are different solutions to the field equations of $B$. For instance, a BPS solution was found [10] by looking at the supersymmetry transformation.

The linearised supersymmetry equation is

$\delta_{\epsilon} \Omega^{j}_{\beta} = \epsilon^{\alpha i}(\frac{1}{2} (\gamma^{a})_{\alpha \beta}(\gamma_{b^{\prime}})^{j}_{i}\partial_a X^{b^{\prime}} - \frac{1}{6}(\gamma^{abc})_{\alpha \beta}\delta^{j}_i h_{abc}) = 0. \ \ (3)$

Here $b^{\prime}$ labels transverse scalars, a indices label worldvolume directions, $\alpha, \beta$ denote spinor indices of spin(1,5), and i,j are spinor indices of $USp(4)$. The solution balances the contribution of the 3-form field strength h with a contribution from the scalars. Additionally, the worldvolume of the string soliton can be taken to be in the 0,1 directions with all fields independent of $x^0$ and $x^1$. An illustration of the solution is given below, showing an M2-brane ending on an M5-brane with a cross section $S^3 \times \mathbb{R}$.

As I am still trying to understand the calculation, I am currently looking at the following string solution

$H_{01m} = \pm \frac{1}{4} \partial_m \phi,$

$H_{mnp} = \pm \frac{1}{4} \epsilon_{emnpq}\delta^{qr}\partial_r \phi,$

$\phi = \phi_0 + \frac{2Q}{\mid x - x_0 \mid^2}, \ \ (4)$

where $\phi$ may be replaced by a more general superposition of solutions. We denote $\pm Q$ as the magnetic and electric charge. There is a conformal factor in the full equations of motion which guarantees that they are satisfied even at $x = x_0$, which means the solution is solitonic. This string soliton is said to possess its own anomalies that require cancellation (I assume Weyl, Lorentz). What is neat is that this string can be dimensionally reduced to get various T-duality configurations, which is something that would be fun to look into at some point down the road.

References

[1] D. Fiorenza, H. Sati, and U. Schreiber, The rational higher structure of m-theory. Fortschritte der Physik, 67(8-9):1910017, May 2019. [arXiv:1903.02834 [hep-th]].

[2] E. Witten, String theory dynamics in various dimensions. Nuclear PhysicsB, 443(1):85 – 126, 1995.

[3] E. Cremmer, B. Julia, and J. Scherk, Supergravity Theory in 11-dimensions. Phys. Lett. B76, No. 4, (409-412) 19 June 1978.

[4] S. Chester, S. Pufu, and X Yin, The M-Theory S-Matrix from ABJM: Beyond 11D supergravity. (2019). [arXiv:1804.00949v3 [hep-th]].

[5] A. Tseytlin, R4 terms in 11 dimensions and conformal anomaly of (2,0) theory. (2005). [arXiv:hep-th/0005072v4 [hep-th]].

[6] G. Russo, and A. Tseytlin, One-loop four-graviton amplitude in eleven-dimensional supergravity. (1997). [arXiv:hep-th/9707134v3 [hep-th]].

[7] P. Heslop, and A. Lipstein, M-theory Beyond The Supergravity Approximation. (2017). [arXiv:1712.08570 [hep-th]].

[8] P.K. Townsend, D-branes from M-branes. (1995). [arXiv:hep-th/9512062 [hep-th]].

[9] A. Strominger, \textit{Open p-branes}. Phys. Lett. B 383 (1996) 44. [arXiv:hep-th/9512059 [hep-th]].

[10] P.S. Howe, N.D. Lambert, and P.C. West, The self-dual string soliton. Nucl. Phys. B 515 (1998) 203. [arXiv:hep-th/9709014 [hep-th]].

[11] M. Perry and J.H. Schwarz, Interacting chiral gauge fields in six dimensions and Born-Infeld theory. Nucl. Phys. B 489 (1997) 47. [arXiv:hep-th/9611065 [hep-th]].

[12] D.S. Berman, Aspects of M-5 brane world volume dynamics. Phys. Lett. B 572 (2003) 101. [arXiv:hep-th/0307040 [hep-th]].

[13] J. Huerta, H. Sati, and U. Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes. (2018). [arXiv:1805.05987 [hep-th]].

[14] N. Copland, Aspects of M-Theory Brane Interactions and String Theory Symmetries. [https://arxiv.org/abs/0707.1317].

[15] S. Palmer, Higher gauge theory and M-theory. [https://arxiv.org/abs/1407.0298].