In 11-dimensional supergravity the brane electrically charged under the supergravity C-field is the M2-brane/membrane. The dual under electric-magnetic duality is the M5-brane.
As a Green-Schwarz sigma-model: BLNPST 97
As a black brane solution of 11-dimensional supergravity the M5-brane is given (Gueven 92) by the spacetime $\mathbb{R}^{5,1} \times (\mathbb{R}^5-\{0\})$ with pseudo-Riemannian metric given by
for $H = 1 + \frac{1}{r}$ and $r$ the distance in $\mathbb{R}^5$ from the origin, and with field strength of the supergravity C-field being
This is a $1/2$-BPS state of 11-dimensional supergravity.
The near horizon geometry of this spacetime is AdS7$\times$S4. For more on this see at AdS-CFT.
1/2 BPS black branes in supergravity: D-branes, F1-brane, NS5-brane, M2-brane, M5-brane
(table taken from Blumenhagen-Lüst-Theisen 13, Chapter 18.5)
More generally for 1/2 BPS black M5-branes, the near horizon geometry is $AdS_7 \times S^4/G$, where $G$ is a finite subgroup of SU(2) (ADE subgroup) acting by left multiplication on the quaternions $\mathbb{H}$ in the canonical way, under the identitfication $S^4 \simeq S(\mathbb{R}^5) \simeq S(\mathbb{R}\oplus \mathbb{H})$ (MFF 12, section 8.3).
While this geometric discussion in MFF 12, section 8.3 works for all the finite subgroups of SU(2), folklore has it that in M-theory the M5-branes appear only at A-type singularities, while the more general 6d (2,0)-superconformal field theories for all possible ADE-singularities appear only after passage to F-theory (ZHTV 14, p. 3).
On the other hand, when placing the M5 at an MO5-orientifold (Witten 95) its worldvolume theory breaks from $(2,0)$ to $(1,0)$-supersymmetry and all ADE-singularities should be allowed.
(…)
the worldvolume theory of the M5-brane is the 6d (2,0)-superconformal QFT.
This worldvolume theory involves self-dual higher gauge theory of the nonabelian kind (Witten07, Witten09): the fields are supposed to be connections on a 2-bundle($\sim$ gerbe), presumably with structure 2-group the automorphism 2-group $AUT(G)$ of some Lie group $G$.
For instance in the proposal of (SSW11) one sees in equation (2.1) almost the data of an $\mathfrak{aut}(\mathfrak{g})$-Lie 2-algebra valued forms.
The M5-brane admits two solitonic excitations ($p$-branes within branes)
$p = 1$: the self-dual string
$p = 3$: the 3-brane in 6d (see there for more)
See also
On dimensional reduction of 11-dimensional supergravity on a circle the M5-brane turns into the NS5-brane and the D4-brane of type II string theory.
The compactification of the 5-brane on a Riemann surface yields as worldvolume theory N=2 D=4 super Yang-Mills theory. See at N=2 D=4 SYM – Construction by compactification of 5-branes.
The AdS/CFT correspondence for the 5-brane is $AdS_7/CFT_6$ and relates the 6d (2,0)-superconformal QFT to 7-dimensional supergravity obtained by reduction of 11-dimensional supergravity on a 4-sphere to an asymptotically 7d anti de Sitter spacetime.
The self-dual 2-connection-field (see there for more details) on the 6-dimensional worldvolume M5-brane is supposed to have a holographic description in terms of a 7-dimensional Chern-Simons theory (Witten 1996). We discuss the relevant “fractional” quadratic form on ordinary differential cohomology that defines the correct action functional.
Let $\hat G$ be the circle 3-bundle with connection on a 7-dimensional manifold $X$ with boundary the M5-brane, thought of as the compactification of the supergravity C-field from 11-dimensional supergravity down to 7-dimensional supergravity.
As discussed there, the 7-dimensional Chern-Simons theory action functional on these 3-connections is
where
$\exp(i \int_X (-))$ is the higher holonomy / fiber integration in ordinary differential cohomology from $X$ to the point
of the Beilinson-Deligne cup product 7-connection $\hat G_4 \cup \hat G_4$.
The space of states of this 7d theory on the M5 worldvolume $\partial X$ would be the space of conformal blocks of the 6d (2,0)-supersymmetric QFT on the worldvolume.
Except, that it turns out that the first Chern class of the corresponding prequantum line bundle is twice that required from geometric quantization.
Therefore the above action functional is not yet the correct one, but only a fractional version of it is. However, the class $G_4 \cup G_4$ in integral cohomology has in general no reason to be divisible by 2.
This is related to the fact that as a quadratic form on the ordinary differential cohomology group $\hat H^4(X)$, the above is not a quadratic refinement of
but of twice that. In (Witten 1996) it was argued, and later clarified in (Hopkins-Singer), that instead the action functional should be replaced by a proper quadratic refinement.
This is accomplished by shifting the center of the quadratic form by a lift $\lambda \in H^4(X, \mathbb{Z})$ of the degree-4 Wu class $\nu_4 \in H^4(X, \mathbb{Z}/2)$ from 0 to $\frac{1}{2}\lambda$.
(For that to make sense in integral cohomology, either the Wu class $\lambda$ happens to be divisible by 2 on $X$, or else one has to regard it itself as a twisted differential character of sorts, as explained in (Hopkins-Singer). For the moment we will assume that $X$ is such that $\lambda$ is divisbible by 2.)
Since $X$, being a spacetime for supergravity, admits (and is thought to be equipped with) a spin structure, by the discussion at Wu class it follows that $\lambda$ is the first fractional Pontryagin class $\frac{1}{2}p_1$
By the very definition of Wu class, it follows that for any $\hat \alpha \in \hat H^4(X)$ the combination
is divisible by 2.
Therefore define then the modified quadratic form
(see differential string structure for the definition of the differential refinement $\hat \mathbf{\lambda} = \frac{1}{2}\hat \mathbf{p}_1$), where, note, we have included a global factor of 2, which is now possible due to the inclusion of the integral lift of the Wu class.
Notice that where the equations of motion of the original action functional are $\hat a = 0$, those of this shifted one are $\hat a = - \frac{1}{2}\hat \mathbf{\lambda}$. One may therefor calls $-\frac{1}{2}\lambda$ here a background charge for the 7-d Chern-Simons theory.
This is now indeed a quadratic refinement of the intersection pairing:
To express the correct action functional for the 7d Chern-Simons theory it is useful to define the shifted supergravity C-field
which the object whose equations of motion with respect to the 7d Chern-Simons theory are still $\hat a = 0$.
Then in terms of the original $\hat G_4$ the action functional for the holographic dual 7d Chern-Simons theory reads
This is the action as it appears in (Witten96, (3.6)).
In terms of twisted differential c-structures we may summarize the outcome of this reasoning as follows:
The divisibility of the action functional requires a $2(G_4 - a)$-twisted Wu structure in $\mathbb{Z}/2$-cohomology. Its lift to integral cohomology is the $2(G_4 - a)$-twisted differential string structure known as the “Witten quantization condition” on the supergravity C-field.
We discuss the conditions on the restriction of the supergravity C-field on the ambient 11-dimensional supergravity spacetime to the M5-brane.
This is similar to the analogous situation in type II string theory. The Freed-Witten anomaly cancellation condition demands that the restriction of the B-field $\hat H_3 \hat H^3(X)$ on spacetime $X$ to an oriented D-brane $Q \hookrightarrow X$ has to trivialize, up to torsion, relative to the integral Stiefel-Whitney class $W_3 = \beta(w_2)$, where $\beta$ is the Bockstein homomorphism induced from the short exact sequence $\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \to \mathbb{Z}_2$:
thus defining a twisted spin^c-structure on the D-brane.
The analog of this for the M5-brane is discussed in (Witten00, section 5). There it is argued that there is a class
on the 5-brane such that under the Bockstein homomorphism $\beta'$ induced by the short exact sequence $\mathbb{Z} \to \mathbb{R} \to U(1)$ we have for the supergravity C-field $\hat G \in \hat H^4(X)$ the condition
By the above quantization condition, this may also be thought of as witnessing a twisted string structure on the 5-brane (Sati).
This condition reduces to the above one for the $B$-field under double dimensional reduction on the circle.
See at M5-brane charge
Consider an 11-dimensional spin-manifold $X^{(11)}$ and a 2-parameter family of 6-dimensional submanifolds $Q_{M5} \hookrightarrow X^{(11)}$. When regarded as a family of worldvolumes of an M5-brane, the family of normal bundles $N_X Q_{M5}$ of this inclusion carries a characteristic class
where
the first summand is the class of the chiral anomaly of chiral fermions on $Q_{M5}$ (Witten 96, (5.1)),
the second term the class of the quantum anomaly of a self-dual higher gauge field (Witten 96, (5.4))
Moreover, there is the restriction of the I8-term (see there) to $Q_{M5}$, hence to the tangent bundle of $X^{11}$ to $Q_{M5}$ (the “anomaly inflow” from the bulk spacetime to the M5-brane)
The sum of these cohomology classes, evaluated on the fundamental class of $Q_{M5}$ is proportional to the second Pontryagin class of the normal bundle
This result used to be “somewhat puzzling” (Witten 96, p. 35) since consistency of the M5-brane in M-theory should require its total quantum anomaly to vanish. But $p_2(N_{Q_{M5}})$ does not in general vanish, and the right conditions to require under which it does vanish were “not clear” (Witten 96, p. 37).
(For more details on computations involved this and the following arguments, see also Bilal-Metzger 03).
A resolution was proposed in (Freed-Harvey-Minasian-Moore 98), further clarified in (Monnier 13), see also (Bah-Bonetti-Minasian-Nardoni 18). There it is asserted that
the correct bulk anomaly inflow is not just that from $I_8$ itself, but includes also a contribution from the class $G_4$ of the supergravity C-field, as per (4) below (Monnier 13, around (3.11));
for $G^{M5}_4$ the “restriction” of the class of the supergravity C-field to $Q_{M5}$, the term $I^{M5}_{C}$ in (1) should have a further summand $-\tfrac{1}{2}\big( G_4^{M5} \big)^2$ (Monnier 13, around (3.7), using Monnier 14b, (2.13))
for 11d spacetime a 4-sphere-fiber bundle,
as befits the near horizon geometry of a black M5-brane, the supergravity C-field should be taken to be of the form (Monnier 13, (3.12))
with $\tfrac{1}{2}\chi$ the degree-4 Euler class, whose integral over the 4-sphere fiber is unity (this Prop.), reflecting the presence of a single M5.
By this proposal (also Bah-Bonetti-Minasian-Nardoni 18 (5), BBMN 19 (2.9) and appendix A.4, A.5), the anomaly inflow from the bulk would not be just $I_8$, as in (2) but would be all of the following fiber integration
Here we used this Prop to find that
which would cancel against the first term $\tfrac{1}{24} p_2$ in (4). Hence with this proposal, the remaining M5-brane anomaly (3) would be canceled.
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$ | compactificationon elliptic fibration followed by T-duality |
7-dimensional supergravity | |
$\;\;\;\;\downarrow$ topological sector | |
7-dimensional Chern-Simons theory | |
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality | |
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-brane worldvolume theory with type IIB S-duality |
$\;\;\;\;\; \downarrow$ topological twist | |
topologically twisted N=2 D=4 super Yang-Mills theory | |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | |
A-model on $Bun_G$, Donaldson theory |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
type II string theory |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$ |
$\;\;\;\; \downarrow$ topological sector |
5-dimensional Chern-Simons theory |
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality |
N=4 D=4 super Yang-Mills theory |
$\;\;\;\;\; \downarrow$ topological twist |
topologically twisted N=4 D=4 super Yang-Mills theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface |
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence |
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
The history as of the 1990 is reviewed in
Further reviews and general accounts include
Robbert Dijkgraaf, The mathematics of fivebranes (pdf)
Hisham Sati, Geometric and topological structures related to M-branes (2010)
The M5 was first found as a black brane of 11-dimensional supergravity (the black fivebrane) in
That this metric, as well as that of every black $p$ brane for odd $p$, is completely non-singular was observed in
Classification of more general M5-brane ADE-singularities is in
also to some extent in
but see p. 3 of
Identification of the $\mathcal{N} = (2,0)$ black M5-brane sitting at the A-type singularity of an $MO5$ $\mathbb{Z}/2$-orientifold locally of the form $\mathbb{R}^{5,1} \times ( \mathbb{R}^5 \sslash (\mathbb{Z}/2) )$ is due to
Discussion in terms of E11-U-duality and current algebra is in
Hirotaka Sugawara, Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra, International Journal of Modern Physics A, Volume 32, Issue 05, 20 February 2017 (arXiv:1701.06894)
Shotaro Shiba, Hirotaka Sugawara, M2- and M5-branes in E11 Current Algebra Formulation of M-theory (arXiv:1709.07169)
The Green-Schwarz action functional-type sigma-model of the (single) M5-brane of was found in covariant form in
Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for a D=11 Five-Brane with the Chiral Field, Phys. Lett. B398 (1997) 41 (arXiv:hep-th/9701037)
Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory, Phys. Rev. Lett. 78 (1997) 4332-4334 (arXiv:hep-th/9701149)
generally following
Paul Townsend, section 3.3 of D-branes from M-branes, Phys. Lett. B373 (1996) 68-75 (arXiv:hep-th/9512062)
(which did not yet have the Hopf-Wess-Zumino term)
and using the covariant mechanism for self-dual higher gauge fields from
based on the non-covariant form of the self-duality mechanism (Perry-Schwarz action) due to
Malcolm Perry, John Schwarz, Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory, Nucl. Phys. B489 (1997) 47-64 (arXiv:hep-th/9611065)
John Schwarz, Coupling a Self-Dual Tensor to Gravity in Six Dimensions, Phys. Lett. B395:191-195, 1997 (cds:317663, doi:10.1016/S0370-2693(97)00094-4)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, World-Volume Action of the M Theory Five-Brane, Nucl.Phys. B496 (1997) 191-214 (arXiv:hep-th/9701166)
Discussion of the equivalence of these superficially different action functionals is in
The equations of motion in super spacetime were derived in
and using the superembedding approach in
see
A variant adapted to a 3+3-dimensional split in
The computation of the small fluctuations of this GS-type sigma-model around a solution embedding as the asymptotic boundary of the AdS-spacetime near-horizon geometry of a black 5-brane as above, and the proof, to low order, that the result is the 6d (2,0)-supersymmetric QFT appearing in AdS7-CFT6 duality is due to
A review with emphasis on the coupling to the M2-brane is in
Further developments include
The original article suggesting the description of the self-dual higher gauge theory on the 5-brane holographically by a dual higher dimensional Chern-Simons theory is
A precise mathematical formulation of the proposal made there is given in
A discussion that embeds this argument into the larger context of AdS-CFT duality is in
Discussion of S-duality in 6d self-dual higher gauge theory via non-commutative-deformation:
See also the references at 6d (2,0)-supersymmetric QFT.
The double dimensional reduction to the D4-brane D=5 super Yang-Mills theory and the relation to Khovanov homology is discussed in
with further comments in
A proposal for a construction as a higher gauge theory for string 2-connections is due to
based on
The higher WZW term of the M5-brane (Hopf-Wess-Zumino term) was first proposed in
and had been settled by the time of
The resemblence of the first summand of the term to the Whitehead integral formula for the Hopf invariant was noticed in
which hence introduced the terminology “Hopf-Wess-Zumino term”. Followup to this terminology includes
Jussi Kalkkinen, Kellogg Stelle, Section 3.2 of: Large Gauge Transformations in M-theory, J. Geom. Phys. 48 (2003) 100-132 (arXiv:hep-th/0212081)
Shan Hu, Dimitri Nanopoulos, Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted, JHEP 1110:054, 2011 (arXiv:1110.0861)
Alex Arvanitakis, Section 4.1 of Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid (arXiv:1804.07303)
More on the relation to the Hopf invariant in
Discussion of the full 6d WZ term is in
The original computation of the total M5-brane anomaly due to
left a remnant term of $\tfrac{1}{24} p_2$. It was argued in
Dan Freed, Jeff Harvey, Ruben Minasian, Greg Moore, Gravitational Anomaly Cancellation for M-Theory Fivebranes, Adv.Theor.Math.Phys.2:601-618, 1998 (arXiv:hep-th/9803205)
Jeff Harvey, Ruben Minasian, Greg Moore, Non-abelian Tensor-multiplet Anomalies, JHEP9809:004, 1998 (arXiv:hep-th/9808060)
Adel Bilal, Steffen Metzger, Anomaly cancellation in M-theory: a critical review, Nucl.Phys. B675 (2003) 416-446 (arXiv:hep-th/0307152)
that this term disappears (cancels) when properly taking into account the singularity of the supergravity C-field at the locus of the black M5-brane.
A more transparent version of this argument was offered in
based on a refined discussion of the quantum anomaly of the self-dual higher gauge field on the M5-brane in
Samuel Monnier, The anomaly line bundle of the self-dual field theory, Comm. Math. Phys. 325 (2014) 41-72 (arXiv:1109.2904)
Samuel Monnier, The global gravitational anomaly of the self-dual field theory, Comm. Math. Phys. 325 (2014) 73-104 (arXiv:1110.4639, pdf slides)
This formulation via an anomaly 12-form is (re-)derived also in
Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Class $\mathcal{S}$ Anomalies from M-theory Inflow, Phys. Rev. D 99, 086020 (2019) (arXiv:1812.04016)
Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Anomaly Inflow for M5-branes on Punctured Riemann Surfaces (arXiv:1904.07250)
The relation of the M5-brane to the D4-brane and the D=5 super Yang-Mills theory in its worldvolume theory by double dimensional reduction:
Eric Bergshoeff, Mees de Roo, Tomas Ortin, The Eleven-dimensional Five-brane (pdf)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, Section 6 of Dual D-Brane Actions, Nucl. Phys. B496 (1997) 215-230 (arXiv:hep-th/9702133)
Neil Lambert, Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills, JHEP 1101:083 (2011) (arXiv:1012.2882)
Chong-Sun Chu, Sheng-Lan Ko, Non-abelian Action for Multiple Five-Branes with Self-Dual Tensors, (arXiv:1203.4224) JHEP05(2012)028
Neil Lambert, Miles Owen, Charged Chiral Fermions from M5-Branes (arXiv:1802.07766)
See also (Witten 11).
Discussion of open M5-branes ending on M9-branes in a Yang monopole is in
The fact that the worldvolume theory of the M5-brane should support fields that are self-dual connections on a 2-bundle ($\sim$ a gerbe) is discussed in
as well as sections 3 and 4 of
Proposals for how to implement this are for instance in
Chong-Sun Chu, A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian Gauge Symmetry $G \times G$ (arXiv:1108.5131)
Henning Samtleben, Ergin Sezgin, Robert Wimmer, (1,0) superconformal models in six dimensions (arXiv:1108.4060)
A formal proposal is here.
Edward Witten, Duality relations among topological effects in string theory, J. High Energy Phys. 0005 (2000) 031 (arXiv:hep-th/9912086)
Hisham Sati, Geometric and topological structures related to M-branes II: Twisted String and String^c structures (arXiv:1007.5419)
Hisham Sati, Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures (arXiv:1109.4461)
Last revised on September 10, 2019 at 14:49:30. See the history of this page for a list of all contributions to it.