Parametrised Homotopy Theory and Gauge Enhancement

TL;DR

This paper explores the derivation of Type IIA D-brane charges from M-theory using super rational homotopy theory, providing a mathematical framework for understanding gauge enhancement.

Contribution

It introduces a first-principles derivation of D-brane charges within a homotopy-theoretic approach, connecting M-theory and Type IIA string theory.

Findings

Derivation of D-brane charges from M-theory

Application of super rational homotopy theory to string theory

Mathematical framework for gauge enhancement

Abstract

We review a first-principles derivation of Type IIA D-brane charges from M-theory degrees of freedom in the approximation of super rational homotopy theory.

Equations

$${\Sigma }_{p+1}⟶X\text{leavevmode}.$$

$$32\otimes 32$$

$$(\psi ,\phi )$$

$$\mathrm{d}:\{\begin{array}{ll}{e}^a& ⟼\overline{\psi }{\Gamma }^a\psi \\ {\psi }^{\alpha }& ⟼0\end{array}$$

$${\mu }_{M2}$$

$${\mu }_{M5}$$

$$\mathrm{d}{\mu }_{M5}=-\frac{1}{2}{\mu }_{M2}\wedge {\mu }_{M2}\text{leavevmode}.$$

$$\mathrm{CE}(\text{mathbbm}{R}^{10,1|32})\stackrel{{\mu }_{M2/M5}}{\leftarrow }\mathfrak{l}{S}^4\text{leavevmode},$$

$$\text{mathbbm}{R}^{10,1|32}\stackrel{{\mu }_{M2/M5}}{\to }{S}^4\text{leavevmode},$$

$$\text{NS5}.$$

$${\mu }_{F1}$$

$${\mu }_{D0}$$

$${\mu }_{D2}$$

$${\mu }_{D4}$$

$${\mu }_{NS5}$$

$${\mu }_{D6}$$

$${\mu }_{D8}$$

$$\mathrm{d}{\mu }_{F1}$$

$$\mathrm{d}{\mu }_{D0}$$

$$\mathrm{d}{\mu }_{\mathrm{D}(2p+2)}$$

$$\mathrm{d}{\mu }_{NS5}$$

$${\mathrm{Spaces}}_{/B{S}^1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\text{mathbbm}{R}^{10,1|32}\cong \mathrm{Ext}\left({\mu }_{D0}:\text{mathbbm}{R}^{9,1|16+\overline{16}}\to B{S}^1\right)$$

$$\text{mathbbm}{R}^{9,1|16+\overline{16}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$S^1$$

$$(Y\to X)⟼({\Omega }_X^{\infty }{\Sigma }_X^{\infty }Y\to X)$$

$$S^4//S^1⟶\mathrm{Cyc}(S^4)$$

$$\text{mathbbm}{R}^{9,1|16+\overline{16}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\widehat{{\mu }_{M2/M5}}:\text{mathbbm}{R}^{9,1|16+\overline{16}}\to {\Omega }_{B^{2}U(1)}^{\infty }{\Sigma }_{B^{2}U(1)}^{\infty }{S}^4//S^1$$

$${\Omega }_{B^{2}U(1)}^{\infty }{\Sigma }_{B^{2}U(1)}^{\infty }{S}^4//S^1$$

$$\big{(}\Omega^{\infty-2}_{B^{2}U(1)}ku\big{)}/\!\!/BU(1)$$

Full text

Parametrised Homotopy Theory and Gauge Enhancement

LMS/EPSRC Durham Symposium on Higher Structures in M-Theory

Vincent Braunack-Mayer111Corresponding author e-mail: vincent.braunack-mayer@uni-hamburg.de aa Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146 Hamburg, Germany

\shortabstract

In the article [1], we apply two universal constructions in super rational homotopy theory in order to solve the problem of gauge enhancement of super M-branes (recalled below). We prove that the topological charge structure of fundamental super M-branes produces twisted K-theory charges of Type IIA D-branes under double dimensional reduction when all torsion effects are ignored222here torsion refers to torsion subgroups of abelian groups, rather than the perhaps more familiar (super-)torsion tensor.. Our result is based upon Sati’s conjecture [2] that the topological M-brane charge is classified by degree-$4$ cohomotopy, and our result in turn contributes to a growing body of evidence for this conjecture (see [3, 4, 5, 6, 7, 8, 9, 10]).

1 Super $p$-branes and cocycles

We recall that a fundamental (or Green–Schwarz) $p$-brane333as opposed to a supergravity black brane or a boundary CFT D-brane—fundamental branes are fundamental in the sense that they generalise the fundamental Green–Schwarz superstring. propagating in a spacetime $X$ is described by a $(p+1)$-dimensional sigma model defined on the space of maps

[TABLE]

In this schematic, $\Sigma_{p+1}$ is the abstract worldvolume of the $p$-brane; a $(p+1)$-manifold encoding a $p$-dimensional extended object dynamically evolving in time. For fundamental super $p$-branes propagating on supergravity backgrounds, the worldvolume $\Sigma_{p+1}$ is taken to be a $(p+1)$-dimensional supermanifold. Compatibility with local supersymmetry on the background $X$ enforces a strict topological constraint: the super volume form on $\Sigma_{p+1}$ must locally trivialise a non-trivial $(p+2)$-cocycle $\mu_{p+2}$ in the supersymmetry super Lie algebra cohomology of super-spacetime [11]. In this manner, fundamental super $p$-branes are in one-to-one correspondence with such super $(p+2)$-cocycles. Analysing such Green–Schwarz sigma models from the point of view of their controlling super Lie algebra cocycles, hence from the point of view of super rational homotopy theory, provides a powerful mathematical toolbox for elucidating many previously elusive aspects of M-theory.

In the case of $d=11$, $\mathcal{N}=1$ supergravity, the local structure of super-spacetime is controlled by the supertranslation super Lie algebra $\mathbbm{R}^{10,1|\mathbf{32}}$ (a classical treatment using different terminology is in [12]). The even piece of this super Lie algebra is spanned by the standard basis $\{e_{a}\}_{a=0}^{10}$ of Minkowski space $\mathbbm{R}^{10,1}$, the odd piece is determined by the irreducible real $\mathrm{Spin}(10,1)$-representation $\mathbf{32}$, and the only non-trivial component of the Lie bracket is the odd-odd superbracket

[TABLE]

determined by the $\mathrm{Spin}(10,1)$-invariant spinor-to-vector pairing (here, as throughout this article, summation over repeated indices is understood). The correspondingChevalley–Eilenberg algebra $\mathrm{CE}(\mathbbm{R}^{10,1|\mathbf{32}})$ is the super differential graded algebra with underlying algebra structure free on the even degree-$1$ generators $\{e^{a}\}_{a=0}^{10}$ and odd degree-$1$ generators $\{\psi^{\alpha}\}_{\alpha=1}^{32}$, corresponding to dual bases of $\mathbbm{R}^{10,1}$ and $\mathbf{32}$ respectively. The differential ${\rm d}$ on $\mathrm{CE}(\mathbbm{R}^{10,1|\mathbf{32}})$ is obtained by dualising the super Lie bracket; it is defined on generators by

[TABLE]

and extended to all of $\mathrm{CE}(\mathbbm{R}^{10,1|\mathbf{32}})$ as a graded derivation.

Under the correspondence between cocycles andGreen–Schwarz sigma models recalled above, the fundamental M2 and M5-branes correspond respectively to the elements

[TABLE]

in the algebra $\mathrm{CE}(\mathbbm{R}^{10,1|\mathbf{32}})$. The exceptional Fierz identities for $\mathrm{Spin}(10,1)$ imply the relation

[TABLE]

In these algebraic terms, Sati’s conjecture on M-brane charge is reflected in the observation that $\mu_{M2}$ and $\mu_{M5}$ combine, via (5), to define a map of (super) commutative differential graded algebras

[TABLE]

where $\mathfrak{l}S^{4}$ is the minimal model of the rational homotopy type of the sphere $S^{4}$ (see [13] for a survey of rational homotopy theory). Viewed through the lens of rational homotopy theory, the combined cocycle $\mu_{M2/M5}$ thus encodes the torsion-free part of a map of (super) homotopy types

[TABLE]

controlling the local cohomological charge structure of fundamental M2 and M5-branes propagating in $11$-dimensional supergravity.

2 Gauge enhancement

2.1 Double dimensional reduction …

Insofar as it exists, M-theory is supposed to be the joint non-perturbative completion of perturbative string theory and $11$-dimensional supergravity. In particular, a complete M-theory must reproduce all of the known structure of perturbative string theory as we pass to various limits. For instance, dimensional reduction of 11-dimensional supergravity along an $S^{1}$-fibration yields 10-dimensional Type IIA supergravity equipped with fields corresponding to the Fourier modes in the $S^{1}$-fibre. In the presence of M2 and M5-branes, the brane worldvolumes either extend along (“wrap”) the $S^{1}$-fibres or they do not. If, say, the M2-brane worldvolume wraps $S^{1}$-fibres, then it produces a fundamental string in 10-dimensional Type IIA supergravity; whereas if it does not wrap it produces a D2-brane. This is the process of double dimensional reduction; the dimension of the ambient super-spacetime as well as possibly the brane worldvolume is reduced by one. Figure 1 is a schematic of this process.

According to this picture, under double dimensional reduction M-theory readily produces the fundamental string F1, the NS5-brane and the D2 and D4-branes of Type IIA string theory (whereas the D0-brane is also obtained as the Chern class which classifies the 11-dimensional background as an $S^{1}$-fibration over the 10-dimensional background). We emphasise that this schematic is a heuristic only; although there are many hints for this double dimensional reduction procedure in the literature [14, 15, 16, 17], there is no derivation beyond the rational approximation [5, 6].

The problem of gauge enhancement of super M-branes appears within this context of double dimensional reduction. Perturbative Type IIA string theory admits fundamental $p$-branes in dimensions $p=0,1,2,4,5,6$ and $8$; namely the D0, F1, D2, D4, NS5, D6, and D8-branes respectively. Moreover, the D$p$-branes are widely expected to carry a combined charge in K-theory twisted by the B-field carried by the fundamental string [18, 19, 20, 21, 22]. However, the double dimensional reduction schematic (Figure 1) makes no mention of the missing D6 and D8 branes, nor of the combined D-brane charge in twisted K-theory. The gauge enhancement problem is therefore:

How does M-theory give rise to the fundamental D6 and D8 branes, and how does it exhibit the combined D-brane charge in twisted K-theory?

In [1] we provide a full answer to this question in the torsion-free approximation, using new techniques in super rational homotopy theory from [23]. To sketch this solution we first observe that, similarly to our previous discussion, the Green–Schwarz sigma models for fundamental $p$-branes in Type IIA string theory are controlled by elements in the Chevalley–Eilenberg complex $\mathrm{CE}(\mathbbm{R}^{9,1|\mathbf{16}+\overline{\mathbf{16}}})$ of the $d=10$, $\mathcal{N}=(1,1)$ supertranslation super Lie algebra. Explicitly, these elements are:

[TABLE]

controlling the fundamental string, D0, D2, D5, NS5, D6 and D8-branes respectively. These elements satisfy various compatibility conditions amongst themselves:

[TABLE]

In particular, the F1 and D$p$-brane cocycles together define a cocycle in the rational image of twisted K-theory.

The first universal construction appearing in our solution of the gauge enhancement problem is an adjunction

[TABLE]

implementing double dimensional reduction. Here, the left adjoint $\mathrm{Ext}$ sends a map of spaces $\tau=(X\to BS^{1})$ to the total space of the $S^{1}$-bundle on $X$ classified by $\tau$ (hence the extension of $X$ by the cocycle $\tau$). The right adjoint $\mathrm{Cyc}$ sends a space $Y$ to the (homotopy) quotient of the free loop space of $Y$ by the action of rigid rotations of loops: $\mathrm{Cyc}(Y)=[S^{1},Y]/\!\!/S^{1}$. We implement a version of this adjunction in super rational homotopy theory, where one finds that the $d=11$, $\mathcal{N}=1$ supertranslation super Lie algebra is the central extension of the $d=10$, $\mathcal{N}=(1,1)$ supertranslation super Lie algebra by the D0-brane $2$-cocycle:

[TABLE]

(this is an algebraic shadow of D0-brane condensation). Taking the combined M2/M5-brane cocycle, we apply the $\mathrm{Cyc}$ functor and compose with the unit of the $(\mathrm{Ext}\dashv\mathrm{Cyc})$-adjunction to obtain a diagram of super rational homotopy types fibred over $BS^{1}$:

[TABLE]

By direct calculation, we see that the cocycle $\widetilde{\mu_{\rm M2/M5}}$ so obtained reproduces precisely the F1, D2, D4 and NS5-brane cocycles. The D0-brane cocycle is encoded in the map $\mathbbm{R}^{9,1|\mathbf{16}+\overline{\mathbf{16}}}\to BS^{1}$, hence as the rational image of the Chern character.

2.2 …and gauge enhancement

In order to complete this picture and produce the D6 and D8-brane cocycles, two additional points are in order. Firstly, recent results [10] provide an equivariant enhancement of the combined M2/M5-brane cocycle at ADE subgroups of $SU(2)$, establishing a homotopy-theoretic underpinning for the black brane scan. The original setting of gauge enhancement of Chan–Paton factors at $\mathbbm{Z}_{n}$-orbifold points [24] corresponds to equivariant enhancement of the $\mu_{\rm M2/M5}$ cocycle at A-series subgroups of $SU(2)$. Considering A-series actions in the limit as $n\to\infty$, the $A_{n}\sim\mathbbm{Z}_{n+1}$-actions exhaust an $S^{1}$-action, so that we are naturally led to expect the existence of an $S^{1}$-equivariant cocycle such as the dotted arrow in the diagram:

[TABLE]

However, we find that the existence of such an $S^{1}$-equivariant cocycle is obstructed by nonvanishing of the D4-brane cocycle.

This leads us to our second main point: homotopical perturbation theory. Homotopy theory is extremely rich but very computationally demanding. There is a tower of increasingly accurate approximations to full homotopy theory provided by the Goodwillie calculus of functors (see [25] for a review), which is analogous to analysing smooth functions via their Taylor series expansions. The first-order approximation of a space $X$ in the Goodwillie calculus is $\Omega^{\infty}\Sigma^{\infty}X$, the underlying space of the free infinite loop space on $X$. This homotopical linearisation assignment $X\mapsto\Omega^{\infty}\Sigma^{\infty}X$ is analogous to “linearising” a set $S$ by passing to the free abelian group $\mathbbm{Z}[S]$ and then forgetting the group structure. There is also such a homotopical linearisation procedure in the relative setting of fibred spaces, where we have an assignment

[TABLE]

that applies homotopical linearisation fibrewise to a space $Y$ fibred over a base space $X$. Working modulo torsion, the main result of the author’s thesis [23] provides algebraic models for parametrised stable rational homotopy types. In particular, this allows for straightforward algebraic calculations of homotopical linearisations in the torsion-free approximation.

Returning to the gauge enhancement problem, we obtain a solution by passing to homotopical perturbation theory. We had previously found that passing to the limit over A-series actions does not descend to an $S^{1}$-equivariant cocycle in $d=10$, $\mathcal{N}=(1,1)$ supergravity. However, there is a natural comparison map

[TABLE]

that fibres rationally over $B^{2}U(1)$. Passing to the fibrewise homotopical linearisations now yields a diagram of super rational homotopy types as in Figure 2

and we can ask if the dotted arrow in that diagram exists. This is equivalent to asking whether $\mu_{\rm M2/M5}$ descends to an $S^{1}$-equivariant cocycle up to first order in homotopical perturbation theory. Using the algebraic models of [23], we find that

1. i)

there exists a morphism

[TABLE]

rendering the diagram of Figure 2 commutative; 2. ii)

the rational homotopy type of

[TABLE]

contains the rational homotopy type of gerbe-twisted connective K-theory

[TABLE]

as a factor; and 3. iii)

the extended cocycle $\widehat{\mu_{\rm M2/M5}}$ hits precisely this twisted K-theory factor, and is moreover unique up to homotopy for this specification.

Put differently, by passing to the first-order approximation in the Goodwillie calculus of functors we obtain an unique-up-to-homotopy extension $\widehat{\mu_{\rm M2/M5}}$ of the cocycle $\widetilde{\mu_{M2/M5}}$ of (12) which

1. i)

recovers the Type IIA D0, D2, F1, D4, D6 and D8-brane cocycles444though the NS5-brane cocycle disappears—it is “unstable” in homotopical perturbation theory.; and 2. ii)

exhibits the combined twisted K-theory charge carried by Type IIA D-branes in the torsion-free approximation.

This is the main result of [1]: the full derivation of gauge enhancement in M-theory, proceeding from M-brane charges in rational cohomotopy.

3 Conclusion

We conclude with some comments on how our result fits into the broader M-theory literature. To begin with, there are all manner of twisted spectra which look like twisted K-theory in the torsion-free approximation of rational homotopy theory. Our result only applies to this rational approximation, so we have not obtained a derivation of twisted K-theory coefficients from M-theory. Indeed, despite wideheld beliefs within the string theory community, there is as of yet no definitive mathematically rigorous argument for twisted K-theory being the right coefficients for D-brane charge: the purported derivation of [26] is rather a consistency check on the sign of the partition function over K-theory fields—a priori many other (twisted) cohomology theories could yield the same sign rule; the arguments of [18] are differential form computations, hence cannot capture any torsion information; and there are moreover persistent conceptual issues with the proposal that D-brane charge is classified by twisted K-theory (see for example [27, 28]).

The main result of [1] fills an important gap in the literature by providing the first full mathematical derivation of rational D-brane charge from M-theory degrees of freedom. Additionally, since we argue using universal constructions in homotopy theory, this derivation already contains hints to its own completion beyond the rational approximation.