synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Traditional differential geometry is the field of mathematics that studies the geometry of smooth spaces which are equipped with a notion of differentiation. There is a refinement of this traditional theory to the general context of “higher geometry” – this is accordingly called higher differential geometry.
In higher differential geometry, geometry is paired with homotopy theory. Accordingly, smooth spaces are refined to “smooth homotopy types” – often called smooth infinity-groupoids or infinity-stacks – and function algebras of smooth functions are refined to strong homotopy algebras, often called infinity-algebras.
Here we give an informal overview of what motivates this generalization and what it is good for.
First we list some examples of how tools from higher differential geometry help to understand plain traditional differential geometry:
Besides illuminating traditional DG where it exists, higher DG naturally extends plain DG to regions where it breaks down due to singularities:
However, much of the genuine motivation for developing higher differential geometry came and increasingly comes from structures in fundamental physics, in quantum field theory and string theory, which are intrinsically of higher geometric nature, hence for which no adequate description in plain differential geometry even exists (this is in fact already true for traditional local gauge theory, see below). This we discuss in:
This section lists examples of how higher differential geometry helps with understanding plain differential geometry.
At the roots of differential geometry, prominent in the original work of Eli Cartan, is foliation theory, arising in the integration of partial differential equations.
In the context of Lie groupoid theory, foliations have a classification – a moduli stack: the Haefliger groupoid. The textbook (Moerdijk-Mrcun) discusses foliation theory from this perspective. The Haefliger theorem which classifies foliations on open manifolds is proven by way of the Haefliger groupoid.
Cartan had originally encoded the PDEs whose integration leads to foliation by “exterior differential systems”. These are equivalently (Chevalley-Eilenberg algebras) of L-∞ algebroids. The entire theory of integration is thereby part of higher Lie theory – infinity-Lie theory.
A subtopic of this is Poisson geometry, where the foliation is by symplectic leaves. A fundamental problem in Poisson geometry was the deformation quantization of Poisson manifolds.
The formal deformation quantization problem was solved by Kontsevich, which earned him a Fields medal. Cattaneo? and Felder? then showed that his formula for the star product is naturally understood as an 3-point function of the Poisson sigma model, a sigma-model QFT whose target space is the Poisson Lie algebroid of the given Poisson manifod.
The strengthening of this to the strict deformation quantization problem for a large class of cases was then established by forming the convolution algebra of sections of the prequantization of the symplectic groupoid with Lie integrates the Poisson Lie algebroid.
(In QFT the shift into higher geometry witnessed in both these proofs is a special case of what is known as the holographic principle: the 0-categorical structure of Poisson manifolds is quantized via the 1-categorical strucure of Poisson Lie algebroids and symplectic groupoids.)
The symplectic groupoid that induces the strict deformation quantization of a Poisson manifold also solves its classical desingularization problem, its symplectic realization. For any symplectic groupoid $\Sigma$ with base a Poisson manifold $P$ the target map is a symplectic realization of $P$ and the source map is a symplectic realization of the opposite structure. Thus $\Sigma$ with its symplectic structure may be regarded as a desingularization of $P$ with its Poisson structure. Since the symplectic groupoid is the Lie integration of the Poisson Lie algebroid of the Poisson manifold, symplectic realization is reduced to a problem in Lie theory.
Apart from these Poisson structures and symplectic structures, differential geometry studies many other kinds of smooth structures on smooth manifolds, such as complex structures in complex geometry. The classification of these structures in each case is infinitesimally given by deformation theory. Since seminal work referenced at model structure for L-∞ algebras it is known that deformation theory is essentially equivalently the study of L-∞ algebra in ∞-Lie theory. Classical theorems in complex differential geometry such as the discussion of period maps find their natural formulation in this context. This is amplified in (Fiorenza-Martinengo2012).
Almost as basic to traditional differential geometry as manifolds are orbifold, which incorporate non-free actions into the theory. Notably the moduli spaces arising in differential geometry tend to be orbifolds instead of manifolds.
But orbifolds are equivalently a proper étale Lie groupoid. (This relation is for some reason more famous in algebraic geometry, where the étale Lie groupoids are called Deligne-Mumford stacks.)
Differential geometry shares much of its origin and history with physics: tensor fields and Riemannian geometry grew out of the theory of gravity and Chern-Weil theory and differential cohomology out of the study of gauge theory. Maxwell's equations influenced the study of de Rham cohomology.
Out of this tradition has grown the formalization of physical fields as sections of a fiber bundle over spacetime called the field bundle. But this notion is insufficient: there cannot be a field bundle for both gauge field theory and local quantum field theory. This problem is typically not felt in perturbation theory, but it affects all attempts to go beyond.
The reason is that a gauge field, including its underlying instanton sector/charge sector, is not a section of a fiber bundle but of a fiber 2-bundle, the anolog in higher differential geometry. Detailed discussion of this is at field (physics) and in the corresponding section at geometry of physics.
Lie integration of anything more general than a finite-dimensional Lie algebra in general goes out of the real of Lie groups. Local brackets = Lie algebroids integrate to Lie groupoids and infinite-dimensional Lie algebras in general to Lie 2-groups.
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derived differential geometry allows to resolve many otherwise singular limit/intersection constructions. Modern variational calculus in terms of jet bundle (“D-geometry”) is formulated this way, as is symplectic reduction and the unification of both in the BV-BRST formalism.
Singularities arise in differential geometry if the two kinds of universal constructions fail to exist as smooth manifolds:
if an intersection (in category theory: a limit) fails to exist this is reolved by passing from critical loci to derived critical loci:
if a quotient (in category theory: a colimit) fails to exist, this is resolved by passing to homotopy quotients:
… stack … homotopy quotient … quotient stack … differentiable stack …
The category of smooth manifolds does not have many limits. Accordingly many constructions that one would like to perform in traditional differential geometry simply fail to exist. For instance the critical locus of a smooth function – of central relevance in variational calculus – rarely is again a smooth manifold and hence cannot be treated with tools of differential geometry, in general.
In the “derived differential geometry”-flavor of higher differential geometry (technically: the not-1-localic) this problem is resolved by replacig function algebras of smooth functions by simplicial algebras, which essentially amounts to refining smooth manifolds by dg-manifolds. This refinement resolves previously singular (hence: non-existing in DG) by the method of resolution familiar from homological algebra: a singular intersection is realized as the homology group of a chain complex of perfectly smooth manifolds (or rather their function algebras).
Applied the concept of critical locus this leads to the notion of derived critical locus. This is actually a construction secretly with a long tradition in differential geometry, known as the BV-BRST formalism. Higher and derived differential geometry provide a theoretical framework for handling BV-BRST formalism systematically.
To a large extent, differential geometry had been co-evolving with the descriotion of physics in terms of fields. We have already seen above in Field bundles for local gauge theory that an accurate formulation even of traditional notions of such fields requires higher differential geometry to degree 1. But modern developments in quantum field theory require notions of physical fields and quantum states that probe much more deeply into the realm of geometric homotopy theory. We indicate here some aspects. For comprehensive introductory lecture notes on this topic see at
A more technical survey of is in FSS 13.
Below we first discuss how combining the notion of local quantum field theory in its modern incarnation as extended quantum field theory with Lagranian quantum field theory necessitates higher differential geometric structures:
Then we turn to modern proposals for theories that go beyond the standard model of particle physics and which inherently contain higher geometric fields already in their non-localized formalization:
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A high point of traditional differential geometry is its impact on representation theory via the orbit method. This is actually a construction deeply rooted in quantum mechanics: given a symplectic manifold equipped with Hamiltonian action by a Lie group, regarding this as a mechanical system and then applying the differental-geometric process of geometric quantization to its yields a space of states on which the group is represented. A large class of representations arises this way and the orbit method sheds light on the corresponding representation theory.
But speaking of physics in the first place, certainly the quantum mechanics governing the orbit method is just the simplest fragment of the foundational theory of physics, which is quantum field theory in arbitrary dimension. Quantum mechanics can be thought of as quantum field theory in dimension 1 (“on the worldline”). Passig in geometric quantization/orbit method from dimension 1 to higher dimensional quantum field theory corresponds precisely, as we discuss now, to passing to higher differential geometry: “extended prequantum geometry”.
The notion of quantum field theory exists without reference to any predefined notion of configuration space of quantum fields, action functional, phase space etc.:
A quantum field theory in FQFT-axiomatization is simply a consistent assignment of spaces of quantum states, whereas in AQFT-axiomatization it is a consistent assignment of algebras of quantum observables, and that’s it.
However, most (or maybe all?) quantum field theories of interest in actual physics (as opposed to as devices of pure mathematics) are not random models of these axioms, but do arise under a process called quantization from a (local/extended) Lagrangian, hence from an action functional, defined on a configuration space of quantum fields, or else arise as holographic duals of quantum field theories that arise by quantization. Moreover, the extra information provided by the Lagrangian is commonly used (and is maybe strictly necessary) to interpret the mathematical structure of the axiomatic QFT in actual physics (though notably in AQFT there are results that re-extract at least parts of this data from the axiomatic QFT, for instance the Doplicher-Roberts reconstruction theorem which extract the global gauge group from the local net of quantum observables).
There are in turn two formalizations of the notion of quantization: algebraic deformation quantization and geometric quantization. In the latter one speaks of prequantization when referring to a precursor step to the actual quantization step, in which the symplectic form on phase space is lifted from to differential cohomology, hence to a prequantum bundle. But in the context of higher geometry and higher geometric quantization this prequantization step is already part of the data of the Lagrangian itself: an extended Lagrangian already encodes not just the action functional but also the prequantum bundle and all the prequantum (n-k)-bundles in each dimension $k$. The action functional itself is the prequantum 0-bundle in this context.
Therefore, in the refined picture of higher geometry/extended quantum field theory it makes good sense to refer in a unified way to prequantum field theory for all of the data related to Lagrangians that is not yet the final quantum field theory.
In particular, an extended prequantum field theory of dimension $n$ is a rule that assigns
to every (suitably oriented) closed manifold of dimension $k$ a prequantum (n-k)-bundle
to every (suitably oriented) compact manifold with boundary $\Sigma_k$ a section of the prequantum $(n-k+1)$-bundle assigned to the boundary $\partial \Sigma_k$ and pulled back to the space of fields over $\Sigma_k$
such that this data is related suitably under transgression.
The actual extended quantum field theory would be obtained from such a data by passing from the assignment of a given prequantum $(n-k)$-bundle to that of the (n-k)-vector space of polarized sections of a suitable associated fiber bundle.
This is summarized in the following table:
extended prequantum field theory
$0 \leq k \leq n$ | (off-shell) prequantum (n-k)-bundle | traditional terminology |
---|---|---|
$0$ | differential universal characteristic map | level |
$1$ | prequantum (n-1)-bundle | WZW bundle (n-2)-gerbe |
$k$ | prequantum (n-k)-bundle | |
$n-1$ | prequantum 1-bundle | (off-shell) prequantum bundle |
$n$ | prequantum 0-bundle | action functional |
Beyond the prequantum n-bundles involved in any extended prequantum field theory, there are various theoriesin physics that involve fields which themselves are already given by higher geometric structure, notably higher gauge fields given by higher connections on higher principal bundles. These arise notably in higher dimensional supergravity theories and their UV-completion: string theory.
The first example along these lines is the Kalb-Ramond field or B-field, which is a higher order version of the electromagnetic field. In order to fully capture the nature of these fields, one needs to describe them in higher geometry. Notably quantum anomaly cancellation conditions can lead to fairly intricate structures of the moduli infinity-stacks of such fields. For instance anomaly-free heterotic background gauge fields are given by moduli 2-stacks of twisted differential string structures.
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
The basic relation between foliation theory and Lie groupoid-theory is discussed in
The identification of orbifolds as (proper) etale Lie groupoids is due to
A discussion of how ∞-Lie theoretic deformation theory neatly captures various traditional theorems is in
For a survey of how higher differential geometry serves to capture subtle aspects of prequantum field theory see
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