spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
superalgebra and (synthetic ) supergeometry
This page contains notes to went with a seminar of the String Geometry Network meeting in October 2013.
More coherent lecture notes are meanwhile at Structure Theory for Higher WZW Terms
These notes concern the generalization of the notion of gerbes, of principal 2-bundles with principal 2-connections and generally of principal ∞-bundles with principal ∞-connection from higher differential geometry modeled on smooth manifolds to higher supergeometry modeled on supermanifolds.
A key motivation for this comes from applications to string theory and the induced higher geometry-analog of traditional spin geometry called string geometry:
Since early observations in the 1980s (Gawędzki 87) and then more prominently since (Freed-Witten 99, Carey-Johnson-Murray 02)), it is known that the B-field in string theory is mathematically a circle 2-bundle with connection and that the WZW term in the action functional of the 2-dimensional string sigma-model with such a background gauge field is the surface holonomy of this 2-connection, in higher analogy of how the gauge interaction term of the electromagnetically charged particle is the line holonomy of a circle bundle with connection over the worldline of the particle.
While this is well-explored by now, it has almost exclusively been applied to the bosonic string, or else to the bosonic sector of the superstring. Notably the anomaly cancellation between the contributions of the B-field and of the fermions (on the string‘s worldsheet and/or in spacetime) is typically considered in two independent steps: a computation in index theory gives the fermionic anomaly incarnated as a Pfaffian line bundle, and then the contribution of the B-field to the anomaly is adjusted such as to cancel the class of this anomaly bundle. The main examples of this are the Freed-Witten-Kapustin anomaly in type II string theory (incarnated as spin^c structure serving as orientation in complex K-theory) and the Green-Schwarz anomaly in heterotic string theory (incarnated as string^c structure serving as orientation in tmf-cohomology theory).
However, since in string theory all the fermionic contributions are connected by (local!) supersymmetry to the bosonic contributions, it is to be expected that on a more fundamental level there are not two independent contributions – one in spin geometry (the fermions) and one higher differential geometry (the B-field) – that happen to cancel each other, but that instead the fields arrange into one single structure (“supermultiplet”) in the higher supergeometry.
By general principles of higher geometry it is clear what higher supergeometry is to be: the theory of ∞-stacks over a site of supermanifolds, hence of smooth super ∞-groupoids. The general notions of principal ∞-bundles and of principal ∞-connections apply to this (cohesive) (∞,1)-topos as to any other and hence in principle provide all the relevant structure.
It remains to work out more examples and applications. A good testing ground is the infinitesimal approximation of smooth super ∞-groupoids by their super L-∞ algebras. It turns out (SSS 09) that that these have implicitly been recognized and used in the study of higher supergravity for a long time, starting with (D’Auria-Fré-Regge 80) and fully developed in (Castellani-D’Auria-Fré 91), namely in their dual incarnation via their Chevalley-Eilenberg algebras. In (Nieuwenhuizen 83) these super-dg-algebras have been called “free differential algebras”, abbreviated “FDA”, referring to the fact that their underlying graded superalgebra is free on a super vector space, hence is a super-Grassmann algebra. Since the differential on these Chevalley-Eilenberg algebras is crucially not free, in general, this is an unfortunate misnomer, but it did stick and is used ever since in the supergravity literature, see the references at D'Auria-Fré formulation of supergravity and at Green-Schwarz action functional.
If however one does make the homotopy theory of L-∞ algebras explicit that is hidden in the “FDA”-formulation of supergravity, then one sees that a large part of the literature has secretly been describing the infinitesimal approximation to supergeometric higher WZW terms all along (FSS 13b), namely in form of the Green-Schwarz action functionals for sigma-models of higher-dimensional branes propagating in a super spacetime target space. By (FSS 13b, last section), each of these perturbative action functionals formulated (implicitly) in terms of super L-∞ algebraic data Lie integrates to a genuine (non-perturbative) ∞-WZW model in higher supergeometry.
The present notes are aimed at spelling out class of examples of “super ∞-gerbes”, but for the most part they apply more generally and only take their motivation from this example.
$\,$
We start with traditional basics, first introducing superpoints and supermanifolds, then recalling the spin group and the classification of its spin representations, and then combining both to build $N$-supersymmetric super Minkowski spacetimes as coset spaces of the super Poincaré group of $N$ supersymmetries.
Then we introduce higher supergeometry in terms of super stacks/smooth super ∞-groupoids and finally discuss how all the exceptional L-∞ cocycles of super Minkowski spacetime and its higher L-∞ extensions yield the Green-Schwarz sigma models of the brane scan of string theory/M-theory, and in fact the whole brane bouquet of branes with “tensor multiplet” fields, such as the D-branes and notably the M5-brane.
Where ordinary differential geometry is modeled on the Cartesian spaces $\mathbb{R}^d$ with smooth functions between them, supergeometry is modeled on the super Cartesian spaces that are denoted $\mathbb{R}^{p|q}$, for $p,q \in \mathbb{N}$, where the superpoints $\mathbb{R}^{0|q}$ are characterized by the fact that their function algebra is a Grassmann algebra on $q$ generators. A supermanifold of dimension $(p|q)$ is a space locally modeled on $\mathbb{R}^{p|q}$.
Introductions to traditional discussion of supermanifolds include (Varadarajan 04, chapter 4, Hohnhold-Stolz-Teichner 11). A more detailed discussion with an eye to serious applications to super Riemann surfaces and with an emphasis on integration over supermanifolds is in (Witten 12). See also at signs in supergeometry.
In some accounts of supergeometry, going back to the influential textbook DeWitt 92, one fixes once and for all an infinite-dimensional Grassmann algebra to model the idea that one can arbitrarily probe any supermanifold by superpoints.
That fixing such an algebra is unnatural, and that the natural formulation rather is to probe by all finite-dimensional Grassmann algebras/superpoints in a way that is respects “change of odd coordinates” was realized in 1984 by Albert Schwarz and others, see (Konechny-Schwarz 98, appendix) for a review. In more abstract language this insight says that supergeometry happens in the topos over the site of superpoints, see at topos over superpoints. Besides nicely clarifying what is really going on, this perspective immediately generalizes to yield the higher supergeometry that we come to below.
The topos-theoretic perspective on supergeometry can be further enhanced by invoking the supergeometric analog of synthetic differential geometry. This synthetic differential supergeometry is developed in (Carchedi-Roytenberg 12).
By internalization all the standard notions of algebra and geometry are implemented in the super-context to yield superalgebra and supergeometry. Here we notably need some Lie theory in the super context.
By the above super-topos-perspective, one simply has that a super Lie algebra $\mathfrak{g} \in SuperLieAlg$ is a collection $(\mathfrak{g}\otimes \Lambda^q)_{even} \in LieAlg$ of ordinary Lie algebras, one for each finite dimensional Grassmann algebra $\Lambda^q$, together with compatible Lie algebra homomorphisms $(\mathfrak{g}\otimes \Lambda^{q_2})_{even} \longrightarrow (\mathfrak{g}\otimes \Lambda^{q_1})_{even}$ for each change of Grassmann coordinates $\Lambda^{q_2} \longrightarrow \Lambda^{q_1}$, hence a presheaf of ordinary Lie algebras over the site of superpoints. Via the Yoneda lemma this is equivalently super vector space equipped with a Lie bracket which is symmetric on two odd-graded elements and skew-symmetric otherwise, and which satisfies a Jacobi identity with signs depending suitably on the degree of the elements.
In precisely the same fashion one finds all super-algebraic structures such as for instance also super L-∞ algebras, which become important below in the discussion of higher supergeometry.
Similarly a super Lie group $G$ is just a system $G(\mathbb{R}^{0|q})$ of ordinary Lie groups, equipped with compatible Lie group homomorphisms $G(\mathbb{R}^{0|q_2}) \longrightarrow G(\mathbb{R}^{0|q_1})$ for each algebra homomorphisms $\Lambda^{q_2} \longrightarrow \Lambda^{q_1}$, hence a presheaf of Lie groups on the site of superpoints.
A decent exposition of this general principle of super Lie algebras and super Lie groups is for instance in (Varadarajan 04, chapter 7.1). Discussion of the classical examples is in (Varadarajan 04, chapter 7.3).
Clifford algebras, the spin group and its representations made its first appearance in Physics in 1928 when Dirac tried to describe relativistic particles moving through Minkowski spacetime: Such a particle should be described by a wave function $\psi \colon \mathbb{R}^4 \to \mathbb{C}$ (we choose $x_0$ to be the time coordinate). The dynamics of the system obeys the Schrödinger equation, which takes the form
where $H$ is the Hamilton operator measuring the energy. Now let $P_i$ be the $i$th momentum and let $m$ be the mass of the particle. Note that the above equation is of first order whereas the relativistic energy condition, $E^2 = \sum_{i=1}^3 P_i^2 + m^2$, turns out to be quadratic. Therefore, one way of obtaining a version of the above compatible with relativity is squaring the Schrödinger equation, i.e.
The resulting Klein-Gordon equation describes the kinematics of spinless scalar particles. Dirac asked the question, whether it is possible to write down a relativistically covariant first order equation. The ansatz
yields the conditions $\alpha_j\alpha_k + \alpha_k\alpha_j = \delta_{jk}\,1$, which of course cannot be satisfied, if $\alpha_j$ are complex numbers. Dirac’s conclusion was that the particle had some inner degree of freedom - the spin - which forces the wave functions $\psi$ to be vector-valued and transforming under a representation of the above algebra, which was already known as the Clifford algebra.
Let $V$ be a finite dimensional vector space over a field $k$ of characteristic $0$ and denote by $Q$ a non-degenerate quadratic form on $V$, i.e. $Q(v) = \phi(v,v)$ for some non-degenerate symmetric bilinear form $\phi \colon V \otimes V \to k$. The pair $(V,Q)$ will be called a quadratic vector space.
The Clifford algebra $Cl(V,Q)$ associated to a quadratic vector space $(V,Q)$ is the quotient of the tensor algebra $T(V) = \oplus_{r \geq 0} V^{\otimes r}$ of $V$ by the ideal $I$ generated by elements of the form $t_{x,y} = x \otimes y + y \otimes x - 2\phi(x,y)\cdot 1$. Equivalently $I$ is the ideal generated by elements $x \otimes x - Q(x) \cdot 1$.
The tensor algebra $T(V)$ is $\mathbb{Z}$-graded. Since $I$ is not homogeneous, this grading does not descend to the quotient. However, $Cl(V,Q)$ is still $\mathbb{Z}/2\mathbb{Z}$-graded and filtered by the length of tensors. This filtration leads to an important connection between the Clifford algebra associated to a quadratic vector space and its exterior algebra: The associated graded algebra of the former is isomorphic to the latter, i.e.
An isomorphism is induced by the linear map $\lambda \colon \Lambda(V) \to Cl(V)$ with $\lambda(v_1 \wedge \dots \wedge v_r) = \frac{1}{r!}\sum_{\sigma \in \Sigma_r} sign(\sigma)v_{\sigma(1)}\,\dots\,v_{\sigma(r)}$.
The following theorem is the key to all structural results about Clifford algebras. Note that tensor products are taken in the category of superalgebras.
Let $(V,Q)$ and $(V',Q')$ be two quadratic vector spaces. Then
Moreover, we have $\dim(Cl(V,Q)) = 2^{\dim(V)}$.
We drop the quadratic forms from the notation and just sketch the proof. The $k$-linear map $L \colon V \oplus V' \to Cl(V) \otimes Cl(V)$ satisfies $L(v,v')^2 = Q(v)\,1 + Q'(v')\,1$. Therefore it extends to an algebra homomorphism $Cl(V \oplus V') \to Cl(V) \otimes Cl(V')$. To construct the inverse note that the algebra inclusions $Cl(V) \to Cl(V \oplus V')$ and $Cl(V') \to Cl(V \oplus V')$ yield $Cl(V) \otimes Cl(V') \to Cl(V \oplus V')$, which is easily seen to be an algebra homomorphism inverse to the first one. By induction we obtain the formula for the dimension of $Cl(V,Q)$.
For the second statement consider the homomorphism $T(V) \to Cl(V,Q)^{op}$ sending $x_1 \otimes \dots \otimes x_k$ to $[x_k \otimes \dots \otimes x_1]$. Due to the definition of the opposite of a superalgebra, it sends $x \otimes x + Q(x)\,1$ to $0$ and the kernel is generated by these elements. Therefore it descends to a surjective map $Cl(V,-Q) \to Cl(V,Q)$. Since both sides have the same dimension, it is an isomorphism.
From this we can completely classify all complex Clifford algebras, i.e. $Cl(V,Q)$ for a complex vector space $V$. Note that in this case there is up to similarity just one quadratic form $Q(z) = \sum_{j=1}^n z_i^2$. We denote $Cl(V,Q)$ by $\mathbb{C}l(V)$ in this case.
If $V$ is a complex vector space with $\dim(V) = 2m$, then $\mathbb{C}l(V) \cong End(S)$ for $\dim(S) = 2^{m-1}|2^{m-1}$ as graded algebras.
If $V$ is a complex vector space with $\dim(V) = 2m+1$, then $\mathbb{C}l(V) \cong \mathbb{C}l(V)^+ \otimes D$, where $D = \mathbb{C}[\epsilon] \cong \mathbb{C}l(\mathbb{C}^1)$ with $\epsilon^2 = 1$. Moreover, $\mathbb{C}l(V)^+ \cong End(S_0)$ where $\dim(S_0) = 2^m$.
We can check directly that $\mathbb{C}l(\mathbb{C}^2) \cong M_2(\mathbb{C})$: Let $e_1, e_2$ be two orthogonal basis vectors of $\mathbb{C}^2$. The map $e_1 \mapsto \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$, $e_2 \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ defines an isomorphism of superalgebras. By induction we then have $\mathbb{C}l(\mathbb{C}^{2m}) \cong \mathbb{C}l(\mathbb{C}^{2m-2}) \otimes \mathbb{C}l(\mathbb{C}^{2})$. The odd dimensional case is similar.
From the above theorem we deduce that $\mathbb{C}l(V)$ has exactly two (graded) simple modules $S$ and $\Pi S$ ($S$ with the reversed grading) in the even-dimensional case and exactly one graded simple module $S = S_0 \otimes D$ in case the dimension is odd.
The classification of Clifford algebras is a little more intricate in the case of real vector spaces. Here we can without loss of generality assume that $V$ is the super vector space $\mathbb{R}^{s|t}$. The structure of the real Clifford algebras is dictated by the super Brauer group.
A superalgebra is called a super division algebra if all nonzero homogeneous elements are invertible. A real superalgebra $A$ is called central simple if $A \otimes_{\mathbb{R}} \mathbb{C} \cong M_{s|t}(\mathbb{C})$ or $A \otimes_{\mathbb{R}} \mathbb{C} \cong M_n(\mathbb{C}) \otimes_{\mathbb{C}} D$ (with $D$ as above) as superalgebras. We will write CS superalgebra for short.
There are a lot of different characterizations of CS superalgebras, see Varadarajan 04, Theorem 6.2.5. For example, each CS superalgebra over a field $k$ of characteristic $0$ is isomorphic to $M_{s|t}(k) \otimes K$ for some super division algebra $K$ over $k$ such that its supercenter is $k$.
Two CS superalgebras $A_1$ and $A_2$ are called similar if their associated super division algebras are isomorphic, i.e. if $A_i \cong M_{s_i|t_i}(k) \otimes K$ for some super division algebra $K$ over $k$. The monoid of similarity classes of CS superalgebras $sBr(k)$ with respect to the graded tensor product $\otimes_k$ turns out to be a group and is called the super Brauer group of $k$.
The following is proven in Varadarajan 04, Theorem 6.4.1.
The super Brauer group $sBr(\mathbb{R})$ is isomorphic to $\mathbb{Z}/8\mathbb{Z}$. Let $D_{\mathbb{R}}=\mathbb{R}[\epsilon]$ be the superalgebra with $\epsilon$ odd and $\epsilon^2 = 1$. If $(V,Q)$ is a quadratic vector space, then $[Cl(V,Q)] = [D_{\mathbb{R}}]^{sign(Q)}$.
This result explains the $8$-fold periodicity found in real Clifford algebras as can be seen from the following table in which $V= \mathbb{R}^{s|t}$ with dimension $d = s+t$:
$(t-s)$ mod $8$ | $Cl(s,t)$ | $N$ |
---|---|---|
$0,6$ | $M_N(\mathbb{R})$ | $2^{d/2}$ |
$2,4$ | $M_N(\mathbb{H})$ | $2^{(d-2)/2}$ |
$1,5$ | $M_N(\mathbb{C})$ | $2^{(d-1)/2}$ |
$3$ | $M_N(\mathbb{H}) \oplus M_N(\mathbb{H})$ | $2^{(d-3)/2}$ |
$7$ | $M_N(\mathbb{R}) \oplus M_N(\mathbb{R})$ | $2^{(d-1)/2}$ |
With this background about Clifford algebras at hand, we can finally define the spin group:
Let $(V,Q)$ be a quadratic vector space. The spin group $Spin(V,Q)$ is defined as
The irreducible representations of $Spin(V,Q)$ that arise from simple $Cl(V,Q)^+$-modules are called spin representations.
Note that this definition of the spin group differs from the one given in Varadarajan 04, Proposition 5.4.8, but agrees with the one from Lawson-Michelsohn 89, page 18. More precisely, the former is the connected component of the identity of the latter, i.e. for $V = \mathbb{R}^{p|q}$ we have
$Spin(p,q)$ is a double cover of $SO(p,q)$ in the sense that
is exact (Lawson-Michelsohn 89, Theorem 2.10). If $\min(p,q) = 1$ and $\max(p,q) \geq 3$, then $SO(p,q)$ and $Spin(p,q)$ have two connected components and the above restricts to a double cover of $SO(p,q)^0$ by $Spin(p,q)^0$. Hidden in the above definition is the statement that simple $Cl(V,Q)^+$-modules yield irreducible representations when restricted to the group $Spin(V,Q)$.
We have $Cl(r,s)^+ \cong Cl(r,s-1)$ as ungraded algebras. Therefore we can read off the spin representations from the above classification table. Consider for example the group $Spin(3,1) \subset Cl(3,1)^+ \cong Cl(3,0) \cong M_2(\mathbb{C})$. This has two simple modules as a real algebra, $\mathbb{C}^2$ and its conjugate.
The symmetries of $d$-dimensional Minkowski space $\mathbb{R}^{d-1|1}$ are given by the Poincaré group, which has the Lie algebra
To get the super Poincaré Lie algebra from this, we would like to extend it by an odd part $\mathfrak{g}_1$ given by a spin representation $S$. In general, a super Lie algebra can be obtained from the following data:
We know what $\mathfrak{g}_0$ and $\mathfrak{g}_1$ should be in our case. Note that $\mathfrak{so}(d-1,1) = \mathfrak{spin}(d-1,1)$, therefore $\mathfrak{g}_0$ acts on $S$ (where the translations act trivially). Therefore we need to concentrate on invariant symmetric forms $\kappa$.
One way to easily satisfy the last condition for $\kappa$ is to just look at those symmetric forms that take values in the translation part of $\mathfrak{g}_0$, i.e. the underlying vector space $\mathbb{R}^{d-1|1}$. More generally, we will summarize below the results about the existence of symmetric bilinear forms $S \otimes S \to \Lambda V$ for a quadratic vector space $(V,Q)$ and an irreducible spin representation $S$ of $Spin(V,Q)$.
We will again spell out the case of complex vector spaces in more detail and just state the results in the real case. Since we would like to use duality results to classify invariant vector-valued forms, we first need to think about scalar-valued ones. Surprisingly enough, their existence in the complex case shows an $8$-fold periodicity, depending on the dimension $d$ of the underlying complex vector space $V$:
$d$ mod $8$ | type of invariant scalar form |
---|---|
$0$ | symmetric forms on $S^{+}$ and on $S^{-}$ |
$1,7$ | symmetric form on $S$ |
$2,6$ | $S^+$ dual to $S^-$ |
$3,5$ | skew-symmetric form on $S$ |
$4$ | skew-symmetric form on $S^{+}$ and on $S^{-}$ |
This can be found in Varadarajan 04, Table 6.4.
Let $V$ be a complex vector space. If $d = \dim(V)$ is even, then $\mathbb{C}l(V) \cong End(S)$ with $S = S^+ \oplus S^-$ for irreducible spin representations $S^{\pm}$. Let $0 \leq r \leq \frac{d}{2}-1$. Then
If $d$ is odd, then $\mathbb{C}l(V)^+ \cong End(S_0)$ for an irreducible spin representation $S_0$ and
We just prove the even case: By the above table we have $S^* \cong S$ via an equivariant isomorphism. Therefore $S \otimes S \cong S^* \otimes S \cong End(S) \cong \mathbb{C}l(V) \cong \Lambda(V)$ as $Spin(V)$-modules. Now we have
and $\Lambda^r(V)$ is irreducible as $SO(V)$-module. Moreover, observe that $\Lambda^r(V) \cong \Lambda^{d-r}(V)$. The proof for the odd dimensional case is similar.
The basis in both cases is obtained from $\kappa$ defined via duality by $(\kappa(x \otimes y), v)_{\Lambda^r(V)} = (\lambda(v)x, y)_S$, where the brackets denote the invariant scalar forms on $\Lambda^r(V)$ and $S$ respectively. We can read off the parity and the symmetry of $\kappa$ from this definition and we have to restrict to $S^+ \otimes S^+$, $S^- \otimes S^-$ or $S^+ \otimes S^-$ depending on the dimension and on $r$.
This time the results are much more complex in the real case. Aside from the fact that the dimension of $V$, the signature of $Q$ and the parity of $r$ enter, the invariant forms need not be projectively unique anymore. The case most important for us, however, is that of Minkowski signature. And here - quite magically - all the ambiguities disappear and we have (Varadarajan 04, Theorem 6.7.1):
Let $V$ be a real quadratic vector space of dimension $d$ and with Minkowski signature, let $S$ be a real irreducible spin representation, then there is a projectively unique nontrivial invariant symmetric form
A class of examples of super Lie algebras of particular interest in applications to physics in general and quantum field theory in particular are super-Lie algebra extensions of the Lie algebra of infinitesimal isometries of Minkowski spacetime: the “supersymmetry” super Lie algebras that appear as local symmetries in supergravity and superstring theory and, more speculatively, as global symmetries in the MSSM.
These we turn to below, where we see that these supersymmetry super Lie algebras crucially contain spin representations and crucially depend on the subtleties of the representation theory of the spin group. Therefore here we first recall the classification and properties of spin representations in general.
A standard mathematical textbook account for this is (Lawson-Michelsohn 89, I.2, I.3, I.5), but for actual computations and notably for comparison with the bulk of the literature, it is useful to also make explicit the standard bases and matrix representations, as summarized neatly for instance in (Polchinski 01, volume II, appendix B). Decent accounts that are both mathematically satisfactory as well as geared towards the applications to supersymmetry in physics are (Varadarajan 04, chapters 5 and 6) and (Freed 99, lecture 3).
The main point of interest here is that a supersymmetry super Lie algebra for $d$-dimensional Minkowski space requires precisely a spin representation $S$ which is equipped with a linear map
which is
symmetric;
$Spin(d-1,1)$-equivariant.
(Precisely these two properties will make $\Gamma$ the odd/odd component of a super Lie bracket).
In (Varadarajan 04, section 6.6) these bilinear pairings are classified in full generality, for arbitrary spacetime signature. However it turns out that for Minkowski signature all real spinor representations (“Majorana representations”) carry an essentially unique such pairing and at the same time are the representations relevant in most applications. Therefore we concentrate below on the classification and properties of Majorana representations of the Lorentzian spin groups $Spin(d-1,1)$.
The following table lists the irreducible real representations of $Spin(V)$ (Freed 99, page 48).
$d$ | $Spin(d-1,1)$ | minimal real spin representation $S$ | $dim_{\mathbb{R}} S\;\;$ | $V$ in terms of $S^\ast$ | supergravity |
---|---|---|---|---|---|
1 | $\mathbb{Z}_2$ | $S$ real | 1 | $V \simeq (S^\ast)^{\otimes}^2$ | |
2 | $\mathbb{R}^{\gt 0} \times \mathbb{Z}_2$ | $S^+, S^-$ real | 1 | $V \simeq ({S^+}^\ast)^{\otimes^2} \oplus ({S^-}^\ast)^{\otimes 2}$ | |
3 | $SL(2,\mathbb{R})$ | $S$ real | 2 | $V \simeq Sym^2 S^\ast$ | |
4 | $SL(2,\mathbb{C})$ | $S_{\mathbb{C}} \simeq S' \oplus S''$ | 4 | $V_{\mathbb{C}} \simeq {S'}^\ast \oplus {S''}^\ast$ | |
5 | $Sp(1,1)$ | $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 8 | $\wedge^2 S_0^\ast \simeq \mathbb{C} \oplus V_{\mathbb{C}}$ | |
6 | $SL(2,\mathbb{H})$ | $S^\pm_{\mathbb{C}} \simeq S_0^\pm \otimes_{\mathbb{C}} W$ | 8 | $V_{\mathbb{C}} \simeq \wedge^2 {S_0^+}^\ast \simeq (\wedge^2 {S_0^-}^\ast)^\ast$ | |
7 | $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 16 | $\wedge^2 S_0^\ast \simeq V_{\mathbb{C}} \oplus \wedge^2 V_{\mathbb{C}}$ | ||
8 | $S_{\mathbb{C}} \simeq S' \oplus S^{\prime\prime}$ | 16 | ${S'}^\ast {S''}^\ast \simeq V_{\mathbb{C}} \oplus \wedge^3 V_{\mathbb{C}}$ | ||
9 | $S$ real | 16 | $Sym^2 S^\ast \simeq \mathbb{R} \oplus V \wedge^4 V$ | ||
10 | $S^+ , S^-$ real | 16 | $Sym^2(S^\pm)^\ast \simeq V \oplus \wedge_\pm^5 V$ | type II supergravity | |
11 | $S$ real | 32 | $Sym^2 S^\ast \simeq V \oplus \wedge^2 V \oplus \wedge^5 V$ | 11-dimensional supergravity |
Here $W$ is the 2-dimensional complex vector space on which the quaternions naturally act.
The last column implies that in each dimension there exists a linear map
which is
symmetric;
$Spin(V)$-equivariant.
This allows to form the super Poincaré Lie algebra in each of these cases. See there and see Spinor bilinear forms below for more.
Ordinary Minkowski space $\mathbb{R}^{d-1,1}$ happens to be coset space which is the quotient of (the double cover of) its own Lie group of oriented isometries, the Poincaré group $Iso(d-1,1)$ by (the spin double cover of) the Lorentz group $SO(d-1,1)$ of rotations and boosts:
The same is true already for the corresponding Lie algebras:
While for ordinary differential geometry this identification is maybe not very deep, for supergeometry it becomes crucial:
for a super Lie algebra extension $\mathfrak{siso}_N(d-1,1)$ of the Poincaré Lie algebra by a spin representation in odd degree – a super Poincaré Lie algebra – one defines the corresponding super Minkowski spacetime as the quotient
Good mathematical discussion of this construction is in (Deligne-Freed 99, section 1.1, Freed 99, lecture 3, Varadarajan 04, chapter 7). A decent summary of the standard component expressions of this construction as used in physics is in (Polchinski 01, volume II, appendix B).
Since the special orthogonal Lie algebra $\mathfrak{so}(d-1,1)$ is normal in $\mathfrak{iso}(d-1,1)$, this exhibits super Minkowski spacetime itself as a super Lie algebra, namely the super translation Lie algebra over itself. Accordingly one can ask for the Lie algebra cohomology of super Minkowski spacetime. And while that of ordinary Minkowski spacetime, regarded as the abelian translation Lie algebra, is uninteresting, super-Minkowski spacetimes – which are mildly non-abelian!, the nontrivial bracket being just the pairing $\Gamma$ from above – happen to admit a finite number of exceptional super Lie cocycles. For reasons that will become clear below, the classification of these is known as the brane scan. A decent discussion of this is in (Azcárraga-Izqierdo 95, Chryssomalakos-Azcárraga-Izquierdo-Bueno 99). This analysis of the Chevalley-Eilenberg super-dg-algebra of the super Poincaré Lie algebra goes back to (Nieuwenhuizen 83).
Since by above, super Minkowski spacetime is defined as a coset space in supergeometry, the mechanics of particles and more generally branes propagating in super Minkowski spacetimes are naturally given by super-coset models. On the Lie-algebraic level this is discussed nicely in Azcarraga-Izqierdo 95, section 8. On the other hand, this means that the geometry of curved super spacetime is most naturally thought of in terms of super-Cartan geometry. This is (somewhat implicitly) the approach to supergravity in (Castellani-D’Auria-Fré).
In order to pass from supergeometry to higher supergeometry, we proceed by the canonical route.
Above we already mentioned that ordinary supergeometry takes place in the topos $Sh(SuperPoints)$ of “super sets”. In order to model super ∞-groupoids we hence naturally pass to the (∞,1)-topos over the site of superpoints. In order to equip these with smooth structure such as to yield smooth super ∞-groupoids we form the (∞,1)-sheaf (∞,1)-topos
SmoothSuper∞Grpd$\coloneqq Sh_\infty\left(\left\{\mathbb{R}^{p|q}\right\}_{p,q \in \mathbb{N}}\right) \simeq Sh_\infty\left(SuperManifolds\right)$
over supermanifolds (see for instance FSS 13a).
As in any (∞,1)-topos, there is a canonical notion of principal ∞-bundles, of associated ∞-bundles and of ∞-gerbes in SmoothSuper∞Grpd (NSS 12). The basic fact of relevance here is that SmoothSuper∞Grpd is cohesive (S) and hence also admits refinements of these structures to differential cohomology.
In particular, due to cohesion every super ∞-group $G \in Grp(SmoothSuper\infty Grpd)$ naturally has a canonical higher Maurer-Cartan form exhibited by a cocycle
with coefficients in non-abelian de Rham hypercohomology. Since the ordinary WZW gerbe is characterized by a 3-cocycle $\mu$ in Lie algebra cohomology such that $\mu(\theta_Q) \colon G \to \Omega^3_{cl}$ is its curvature 3-form, this is a crucial ingredient for the definition of ∞-Wess-Zumino-Witten theory models. This we come to below.
We discuss how to construct circle n-bundles with connection (“bundle (n-1)-gerbes”) on super Lie groups and on super ∞-groups which generalize the familiar Wess-Zumino-Witten term that serves as the action functional for the sigma-model that describes a string propagating on a Lie group. Following the “holographic principle” we construct these ∞-Wess-Zumino-Witten theories as boundary field theories for ∞-Chern-Simons theories (which in turn are realized as boundary field theories for higher topological Yang-Mills theories).
Finally we apply this general construction of supergeometric ∞-Wess-Zumino-Witten theories to the exceptional super L-∞ algebra extended supersymmetry algebras and thus find the non-pertrubative formulation of the $p$-brane sigma-models in string theory/M-theory which constitute the brane scan/The brane bouquet.
A classical field theory/prequantum field theory is traditionally defined by an action functional: given a smooth space $\mathbf{Fields}_{traj}$ “of trajectories” of a given physical system, then the action functional is a smooth function
to the circle group. The idea of producing a quantum field theory from this is to
choose a linearization in the form of the group homomorphism $U(1) \longrightarrow GL_1(\mathbb{C})$ to the group of units of the complex numbers,
choose a measure $d\mu$ on $\mathbf{Fields}_{traj}$
and then declare that the integral (“path integral”)
is the partition function of the theory a kind of expectation value with probabilities replaced by probability amplitudes.
In order to make sense of this (for a full discussion of “motivic quantization” in this sense see (Nuiten 13), here we concentrate on the pre-quantum aspects), it is useful to allow some more conceptual wiggling room by passing to higher differential geometry. Notice that if we write $\mathbf{B}U(1)$ for the smooth universal moduli stack of circle group-principal bundles, then an action functional as above is equivalently a homotopy of the form
where on the right we used the universal property of the homotopy pullback diagram which exhibits the smooth circle group $U(1)$ as the loop space object of $\mathbf{B}U(1)$.
For instance for $X$ a smooth manifold (“spacetime”) and $\nabla \;\colon\; X \longrightarrow \mathbf{B}U(1)_{conn}$ a circle group-principal connection (“electromagnetic field on spacetime”) then for trajectories in $X$ of shape the circle, the canonical action functional (“Lorentz force gauge interaction”) is the holonomy functional
But more generally, if the trajectories have a boundary, hence if they are of the shape of an interval $I \coloneqq [0,1]$, then the holonomy functional on smooth loop space $[S^1, X]$ generalizes to the parallel transport on the path space $[I,X]$ and there it is no longer a function, but exists only as a homotopy of the form
Notice that this is a “local” description of the action functional: the data that determines it is the boundary
and from this the rest is induced by transgression.
A related class of examples are prequantized Lagrangian correspondences: Let
be a symplectic manifold. Then a symplectomorphism $f \;\colon\; X \longrightarrow X$ is a correspondence of the form
A prequantization of $(X,\omega)$ is a lift $\nabla$ in
and so a prequantized Lagrangian correspondence is
To conceptualize all this, write
for the homotopy theory obtained from the category of groupoid-valued presheaves on the category of all smooth manifolds by universally turning stalkwise equivalences of groupoids into genuine homotopy equivalences (“simplicial localization”).
This is the (2,1)-topos of smooth groupoids/smooth (moduli) stacks.
Write
for the (2,1)-category of correspondences in $\mathbf{H}$. Write $\mathbf{H}_{/\mathbf{B}U(1)_{conn}}$ for the slice (2,1)-topos over the smooth moduli stack of circle bundles with connection. Then the abovve diagrams are morphisms in $Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}})$.
The automorphism group of $\nabla \in Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}})$ is the quantomorphism group of $(X,\omega)$, hence the smooth group which is the Lie integration of the Poisson bracket Lie algebra of $(X,\omega)$.
A concrete smooth 1-parameter subgroup
is equivalently a choice $H \in C^\infty(X)$ of a smooth function and sends
where
$\exp(t \{H,-\})$ is the Hamiltonian flow induced by $H$;
$S_t = \int_0^t L$ is the Hamilton-Jacobi action functional, the integral of the Lagrangian of $H$, hence of its Legendre transform.
(see Schreiber 13).
It is now clear how to pass from this to local prequantum field theory of higher dimension.
Let now more generally
be the homotopy theory obtained from the category of Kan complex-valued presheaves on the category of all supermanifolds by universally turning stalkwise homotopy equivalences into actual homotopy equivalences.
We say that this is the (∞,1)-topos of smooth super ∞-groupoids/_supergeometric moduli ∞-stacks.
Let
be the (∞,n)-category of $n$-fold correspondences in $\mathbf{H}$. This is a symmetric monoidal (∞,n)-category under the objectwise Cartesian product in $\mathbf{H}$.
Smooth∞Grpd has the special property that it is cohesive in that it is equipped with an adjoint quadruple of adjoint (∞,1)-functors
which induce an adjoint triple of idempotent (∞,1)-monads/comonads
with $\Pi$ product-preserving, called
Here the shape modality $\Pi$ sends a simplicial manifold to the homotopy type of the fat geometric realization of the underlying simplicial topological space, hence in particular sends a smooth manifold to its homotopy type.
Write $Bord_n$ for the (∞,n)-category of framed n-dimensional cobordisms.
is equivalently a choice of object $\mathbf{Fields} \in \mathbf{H}$. It sends a cobordism $\Sigma$ to the internal hom of its shape into the higher moduli stack $\mathbf{Fields}$:
(lpqft)
Under the Dold-Kan correspondence
we have for all $n \in \mathbb{N}$ an equivalence
in $\mathbf{H}$.
Consider the induced canonical inclusion
By the above we may regard this as an action functional for an $(n+1)$-dimensional prequantum field theory with moduli stack of fields being $\mathbf{\Omega}^{n+1}_{cl}$. As such we denote it
where the subscript is supposed to refer to “universal higher topological Yang-Mills theory”.
are equivalent to objects
This sends the dual point to $\exp(- i S)$ and sends the $k$-sphere to the transgression of $\exp(i S)$ to the mapping space $[S^k , \mathbf{Fields}]$.
(lpqft)
Consider the induced canonical inclusion
By the above we may regard this as an action functional for an $(n+1)$-dimensional prequantum field theory with moduli stack of fields being $\mathbf{\Omega}^{n+1}_{cl}$. As such we denote it
where the subscript is supposed to refer to “universal higher topological Yang-Mills theory”.
Observe that by the cobordism hypothesis $Bord_n$ is the free symmetric monoidal (∞,n)-category with fully dualizable objects generated from a single object $\ast$.
Let then
the free symmetric monoidal (∞,n)-category with fully dualizable objects generated from a single object $\ast$ and a single morphism $\emptyset \longrightarrow \ast$ from the tensor unit to the generating object. By the boundary field theory/defect version of the cobordism hypothesis, this is equivalently the (∞,n)-category of cobordisms with possibly a boundary component of codimension $(n-1)$.
Hence a boundary field theory is
A boundary field theory as above is equivalently a diagram in $\mathbf{H}$ of the form
The universal boundary condition for the universal higher topological Yang-Mills theory of example is the higher moduli stack $\mathbf{B}^n U(1)_{conn}$ of circle n-bundle with connection, hence a general boundary condition for this higher topological Yang-Mills theory is a ∞-Chern-Simons theory].
The ∞-Wess-Zumino-Witten theory that we are after are boundaries of these boundary field theories, hence “corner field theories” (Sati 11, lpqft) of the higher universal topological Yang-Mills theory. This we turn to now.
The earliest and the only rigorously understood example of the holographic principle is the AdS3-CFT2 and CS-WZW correspondence between the WZW model on a Lie group $G$ and 3d $G$-Chern-Simons theory.
In (Witten 98) it is argued that all examples of the AdS-CFT duality are governed by the higher Chern-Simons theory terms in the supergravity Lagrangian on one side of the correspondence, hence that the corresponding conformal field theories] are higher dimensional analogs of the traditional [[WZW model?: that they are “∞-Wess-Zumino-Witten theory”-type models.
In particular for AdS7-CFT6 this means that the 6d (2,0)-superconformal QFT on the M5-brane worldvolume should be a 6d-dimensional WZW model holographically related to the 7d Chern-Simons theory which appears when 11-dimensional supergravity is KK-reduced on a 4-sphere:
In (Witten 96) this is argued, by geometric quantization after transgression to codimension 1, for the bosonic and abelian contribution in 7d Chern-Simons theory. (The subtle theta characteristic involved was later formalized in Hopkins-Singer 02.)
In order to formalize this in generality, one needs a general formalization of holography for local prequantum field theory as these. How are ∞-Wess-Zumino-Witten theory-models higher holographic boundaries of ∞-Chern-Simons theory? This we are dealing with now.
The datum going into ∞-Wess-Zumino-Witten theory is smooth super ∞-group $G$ which is the Lie integration of a given super L-∞ algebra $\mathfrak{g}$, and is a cocycle $\mu$ on $\mathfrak{g}$.
We discuss here how Lie integration relates $\mathfrak{g}$ to $G$ and how the Lie integration of the cocycle $\mu$ induces a smooth universal characteristic map $\exp(\mu) \colon \mathbf{B}G \longrightarrow \mathbf{B}^n (\mathbb{R}/\Gamma)$ (Fiorenza-Schreiber-Stasheff 12).
Given a (super-)L-∞ algebra $\mathfrak{g}$, its Lie integration is the ∞-stack
which assigns to a test (super Cartesian space-)Cartesian space $U$ the Kan complex whose $k$-simplices are smooth L-∞ algebra valued differential forms on $U \times \Delta^k$ which are
flat
vertical, this means that all their “legs” are along $\Delta^k$
have sitting instants: such that the boundary of the simplex has a neighbourhood such that in this neighbourhood the differential form is constant in the direction perpendicular to the boundary (this condition makes the simplicial set indeed by a Kan complex).
Let $\mathfrak{g}$ be an ordinary Lie algebra then the 1-truncation of $\exp(\mathfrak{g})$ is the delooping of the simply connected Lie group $G$ corresponding to $\mathfrak{g}$ under traditional Lie theory:
In fact also
but $\tau_3 \exp(\mathfrak{g})$ is different from $\mathbf{B}G$ if $\pi_3(G)$ is non-trivial.
For $\mathfrak{g}$ a semisimple Lie algebra and $\mathfrak{string}(\mathfrak{g})$ its string Lie 2-algebra, then
where $String(G)$ is the smooth string 2-group.
The $\exp$-construction clearly extends to a functor
An L-∞ cocycle of degree $n+1$ on $\mathfrak{g}$ is a homomorphism of L-∞ algebras of the form
Hence given a cocycle $\mu$, universal Lie integration produces a morphism of smooth super ∞-groupoids of the form
Say that the periods of $\mu$ are all values in $\mathbb{R}$ obtained by integration of the $(n+1)$-form on an $(n+1)$-sphere in the image of the biundary of an $(n+1)$-simplex under this map. Write $\Gamma_\mu \hookrightarrow \mathbb{R}$ for the subgroup of such periods
Under truncation the universal Lie integration above descents to an $(n+1)$-∞-group cocycle with coefficients in $\mathbb{R}/\Gamma_{\mu}$:
where
Proof: Use the standard simplicial (n+1)-coskeleton model for the $n$-truncation.
For $\mathfrak{g}$ a semisimple Lie algebra with its canonical 3-cocycle $\mu = \langle -,[-,-]\rangle$, then
is the smooth refinement of the fractional first Pontryagin class/second Chern class whose homotopy fiber is the delooping of the string 2-group
Observe that the ordinary WZW term
on a compact Lie group is characterized (e.g. Schweigert-Waldorf 07) in terms of the corresponding universal Chern-Simons circle 3-connection
by two pieces (the two ingredients in the homotopy fiber product-definition of ordinary differential cohomology):
the curvature of $\mathcal{L}_{WZW}$ is the value of the Chern-Simons form on the canonical Maurer-Cartan form on $G$;
the Dixmier-Douady class $\chi(\mathcal{L}_{WZW})$ is the looping of the class of the Chern-Simons circle 3-bundle.
We discuss now how this construction generalizes to higher differential geometry in general, and to higher supergeometry in particular. Then in The brane bouquet of higher super WZW terms we discuss a class of examples of this construction arising from the tower of super L-∞ extensions of super Minkowski spacetime. (FSS 13b)
Let
be a super L-∞ algebra L-∞ cocycle of degree $(n+1)$. Let
be its Lie integration in smooth super ∞-groupoids, according to (FSS 10).
Observe that the smooth ∞-group $G$ has, by cohesion, a canonical higher Maurer-Cartan form
This is a cocycle in the nonabelian de Rham hypercohomology of $G$. We want an ∞-Wess-Zumino-Witten theory model with a globally defined curvature $(n+1)$-form. Therefore consider the universal solution $\tilde G$ of making $\theta$ globally well defined, hence the homotopy pullback
Then one observes that by cohesion the pasting diagram on the right of the following exists, and hence defines a local action functional $\exp(i S_{WZW})$ by the universal factorization on the left. This is the ∞-Wess-Zumino-Witten theory induced by the L-∞ cocycle $\mu$:
This uses the following general fact about how local action functionals $\mathbf{Fields} \longrightarrow \mathbf{B}^n U(1)_{conn}$ are themselves boundary conditions for what one might call universal higher topological Yang-Mills theory (lpqft), the theory given by the local action functional
which is just the canonical inclusion of closed differential $(n+1)$-forms into the universal moduli stack of flat circle (n+1)-bundles with connection. By the universal property of ordinary differential cohomology one finds that boundary conditions for this somewhat degenerate theory are precisely differential cocycles:
(…)
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