aa r X i v : . [ m a t h - ph ] A ug HEISENBERG-PICTURE QUANTUM FIELD THEORY
THEO JOHNSON-FREYD
Abstract.
This paper discusses what we should mean by “Heisenberg-picture quantum field the-ory.” Atiyah–Segal-type axioms do a good job of capturing the “Schr¨odinger picture”: these axiomsdefine a “ d -dimensional quantum field theory” to be a symmetric monoidal functor from an ( ∞ , d )-category of “spacetimes” to an ( ∞ , d )-category which at the second-from-top level consists of vectorspaces, so at the top level consists of numbers. This paper argues that the appropriate parallel no-tion “Heisenberg picture” should also be defined in terms of symmetric monoidal functors from thecategory of spacetimes, but the target should be an ( ∞ , d )-category that in top dimension consistsof pointed vector spaces instead of numbers; the second-from-top level can be taken to consist ofassociative algebras or of pointed categories. The paper ends by outlining two sources of suchHeisenberg-picture field theories: factorization algebras and skein theory. Contents
1. Introduction and motivation 12. Modulation 33. The point of pointings 54. Non-affine field theory 85. Extended affine field theory 106. Extended non-affine field theory 137. A (non-)dualizability result 168. From factorization algebra to Heisenberg-picture field theory 189. From skein theory to Heisenberg-picture field theory 22References 271.
Introduction and motivation
Open the nearest book called
Introduction to Quantum Mechanics — you probably have onelying around somewhere. Almost certainly somewhere in it is a discussion of the two so-called“pictures” of quantum mechanics, named after Erwin Schr¨odinger and Werner Heisenberg. The“Schr¨odinger picture” has a natural generalization to quantum field theory via Atiyah–Segal-typeaxioms. My goal in this paper is to motivate, propose, and expound upon axioms for a “Heisenbergpicture” of quantum field theory. Let us, then, begin by learning what we can from that
QuantumMechanics textbook. The discussion of the Schr¨odinger picture translates well into a mathematicaldefinition:
Definition 1.1. A Schr¨odinger-picture quantum mechanical system consists of the following data:(1) A vector space V (over some ground field K ) called the space of states . Date : August 25, 2015.I would like to thank Alexandru Chirvasitu, Owen Gwilliam, and Claudia Scheimbauer for many ongoing discus-sions about these and related ideas, and I am grateful to Stephan Stolz and Peter Teichner for sharing with me theconstruction in Section 8. This work is supported by the grant DMS-1304054. (2) For each “time” t ∈ R > , a linear map u t : V → V called time evolution . These shouldsatisfy a group law u t + t = u t ◦ u t .(3) Some distinguished states v i ∈ V depending on some labeling set I which our laboratoryfriends know how to prepare, and some distinguished costates w j : V → K that we knowhow to “post-pare” (or is it “post-pair”?). There may also be some distinguished observables a k ∈ End( V ) that we know how to test by making some manipulation while the experimentruns. Remark 1.2.
The axioms for a group (a set with a binary operation satisfying . . . ) don’t commu-nicate what questions one might want to know about a group. Moreover, the axioms for a groupare often too liberal: groups “in nature” are usually Lie or algebraic or finite or hyperbolic orotherwise more structured than the axioms would suggest. Similarly, Schr¨odinger-picture quantummechanical systems are usually defined over C specifically, the spaces of states are usually Hilbertspaces, time evolution operators are usually unitary, the sets of distinguished states and costatesusually agree, and observables are usually self-adjoint.But none of these extra conditions are what that textbook of yours claims is the essence ofquantum mechanics, which is the linearity illustrated by the two-slit experiment. So I won’t putit in the definition. Indeed, Hilbert structures and unitarity are closely related to time-reversalsymmetry (a recent discussion is available in [JF15c]); our axiomatics should accommodate non-symmetric examples as well. General questions one might want to ask about a Schr¨odinger-picturequantum mechanical system include the spectrum of time evolution and the (perhaps asymptotic)values of various compositions of the data. Remark 1.3.
The discussion of the Schr¨odinger picture in that textbook of yours probably alsoincludes an important comment emphasizing that the space V itself isn’t “physical”: only itsprojectivization P V = ( V r { } ) / K × is. It is often tempting to ignore this comment. The discussion of the Heisenberg picture begins by emphasizing that in physics, what’s mostphysical are the observables in a given system, and that in quantum mechanics these form an as-sociative but generally noncommutative algebra. (Just like how physically-interesting Schr¨odinger-picture quantum mechanics deals with Hilbert spaces rather than general vector spaces, physically-interesting Heisenberg-picture quantum mechanics deals with algebras equipped with extra struc-ture — C-star or von Neumann, for example — but we should not build such structure into thebasic axiomatics.) Let us recall the main examples and then try to extract a definition:
Example 1.4.
Let (
V, u t , . . . ) be a Schr¨odinger-picture quantum mechanical system such that thetime evolution operators u t are all isomorphisms. The corresponding Heisenberg-picture quantummechanical system consists of the following data:(1) The associative algebra A = End( V ) of all endomorphism of V is the algebra of observables of the system.(2) For each t ∈ R > , evolution for time t is encoded by the “conjugate by u t ” algebra auto-morphism a u t au − t .(3) The distinguished observables a k ∈ A already do not reference V . To remove V from thedata of the distinguished states and costates, we can encode them as ideals: given v i ∈ V ,remember the left ideal Ann( v i ) = { a ∈ A s.t. av i = 0 } ; for w j : V → K , use the right idealAnn( w j ) = { a ∈ A s.t. w j ◦ a = 0 } .This construction is discussed in [Sch09], which does not answer the question of what to do whenthe operators u t are not isomorphisms. I will propose an answer in Proposition 3.4. Remark 1.5.
Some contravariance in the constructions later in this paper is unavoidable. Justto fix a convention, let’s declare (as is most standard) that End( V ) acts on V from the left. Fora general category C and object C ∈ C therein, the algebra End C ( C ) of endomorphisms of C has EISENBERG-PICTURE QUANTUM FIELD THEORY 3 multiplication f g = f ◦ g = ( C g → C f → C ). My hope is that the reader will largely ignore left/rightquestions. This example illustrates already one feature of the Heisenberg picture: it solves the problemfrom Remark 1.3 that only P V is physical, not V . Indeed, multiplying u t by a non-zero constantdoes not change the conjugation map a u t au − t , and similarly the ideals Ann( v i ) and Ann( w j )depend only on the lines spanned by v i and w j . Moreover, in many situations a vector or Hilbertspace V can be recovered up to non-unique isomorphism from the algebra End( V ); the ambiguityin recovering V is exactly the group of invertible scalars, so that End( V ) and P V encode exactlythe same information.The following example illustrates the other main feature of the Heisenberg picture: it is flex-ible enough to accommodate classical as well as quantum mechanical systems (and systems in-termediate between these extremes, corresponding to non-commutative algebras that are not asfully-noncommutative as End( V )): Example 1.6. A classical mechanical system consists of a symplectic manifold X and a fam-ily of symplectomorphisms { ϕ t : X → X } t ∈ R > satisfying a group law, and perhaps some otherdistinguished data. Such a system can be encoded in the Heisenberg picture as follows:(1) The algebra of observables is the commutative algebra O ( X ) of smooth functions on M .(2) Time evolution is encoded by the algebra isomorphisms f t = ϕ ∗ t : O ( X ) → O ( X ).(3) One type of distinguished data might be submanifolds of “boundary conditions.” Thesecan be encoded by ideals in O ( X ). Examples 1.4 and 1.6 suggest the following definition of the Heisenberg picture:
Definition 1.7 (Tentative) . A Heisenberg-picture quantum mechanical system consists of:(1) An associative algebra of observables A .(2) For each time t ∈ R > , a time evolution homomorphism f t : A → A . These should satisfya group law f t + t = f t ◦ f t .(3) Some distinguished right ideals encoding initial states/boundary conditions, some distin-guished left ideals encoding terminal costates/boundary conditions, and some distinguishedelements of A encoding available experimental manipulations. Definition 1.7 is only tentative, and will be revised in the next section. For instance, it is notclear how (or whether?) to accommodate Schr¨odinger-picture systems with non-invertible timeevolution. 2.
Modulation
The Schr¨odinger picture of Definition 1.1 generalizes naturally to an Atiyah–Segal style axiomat-ics for quantum field theory. Indeed, Definition 1.1, and in particular the group law, already definesquantum mechanical systems as a type of functor:
Definition 2.1.
Let
I, J, K be label sets. The category
QMSpacetimes of quantum mechanicalspacetimes with “point defect” label sets I, J, K has:
Objects:
Two objects, called { pt } and ∅ . Generating morphisms:
For each t ∈ R > , there is a morphism u t : { pt } → { pt } , whichcan be thought of as a metrized interval of length t . These are required to satisfy the grouplaw relation u t ◦ u t = u t + t .There are also morphisms v i : ∅ → { pt } for each i ∈ I , w j : { pt } → ∅ for each j ∈ J , and a k : { pt } → { pt } , called point defects . We impose no relations on these.The full category then consists of compositions of the generating morphisms (modulo the grouplaw relation). THEO JOHNSON-FREYD
Let
Vect denote the category of K -vector spaces and linear maps. Definition 1.1 can thenbe rephrased to state that a Schr¨odinger-picture quantum mechanical system is a functor V : QMSpacetimes → Vect along with an isomorphism V ( ∅ ) ∼ = K . Remark 2.2.
Vect and
QMSpacetimes are examples of pointed categories : categories equippedwith distinguished objects (namely K ∈ Vect and ∅ ∈
QMSpacetimes ). Then V is precisely a strong pointed functor from Vect to QMSpacetimes (compare Remark 4.3).One can alternately freely generate a symmetric monoidal category from
QMSpacetimes underthe condition that ∅ becomes the monoidal unit. Calling the monoidal structure ⊔ , the objectsof this freely-generated category are finite sets — formal disjoint unions of copies of { pt } — andits morphisms are disjoint unions of directed metrized intervals with “point defects” labeled byelements of I , J , and K . This functorial point of view can be applied to higher-dimensional spacetimes:
Definition 2.3 ([Ati88, Seg04]) . The d -dimensional non-extended spacetime category for geometry G is the symmetric monoidal category Spacetimes G d − ,d whose objects are ( d − G and whose morphisms are isomorphism types of d -dimensional cobordisms equipped with geometry of type G .A Schr¨odinger-picture ( d -dimensional, non-extended, for geometry G ) quantum field theory is asymmetric monoidal functor Spacetimes G d − ,d → Vect . Remark 2.4.
The notion of “geometry” should be interpreted quite liberally: it could include met-rics, background fields, labeled defects, etc. For some discussion and examples of geometric cobor-dism categories, see [Aya08, Res10, ST11]. When G is sufficiently topological, Spacetimes G d − ,d can be built from the category Bord top d − ,d of unstructured (i.e. topological) spacetimes as in [Lur09,Section 3.2], elaborated upon in [JF15c]. Ignoring for the moment the boundary-condition ideals and distinguished manipulations (item (3)in Definition 1.7), a Heisenberg-picture quantum mechanical system is also a functor: its source isthe category
QMSpacetimes of metric segments from Definition 2.1, and its target is the category
Alg homo whose objects are associative algebras and whose morphisms are algebra homomorphisms(we will soon introduce a different category called
Alg , hence the name for this one). Let
Vect inv denote the maximal subgroupoid of
Vect : it has the same objects, but morphisms must be iso-morphisms. There is a functor End :
Vect inv → Alg homo which sends a vector space V to itsendomorphism algebra End( V ) and an isomorphism f : V → W to the algebra homomorphism a f af − : End( V ) → End( W ). Example 1.4 then consists of taking in a Schr¨odinger-picturequantum mechanical system V : QMSpacetimes → Vect inv and producing the composition A = End ◦ V : QMSpacetimes → Alg homo . This functor is used in [Sch09] to turn Schr¨odinger-picture quantum field theories valued in
Vect inv into nets of algebras in the sense of “axiomatic”or “algebraic” quantum field theory.The question then arises: can we similarly compose arbitrary Schr¨odinger-picture quantum fieldtheories with this End functor to produce Heisenberg-picture quantum field theories? My goal,of course, is to explain that the answer is “yes.” But the main obstruction, hinted at alreadyin Example 1.4 and after Definition 1.7, must be confronted: in most Schr¨odinger-picture quan-tum field theories, the linear maps associated to spacetimes are not invertible. To resolve thisobstruction, let us seek guidance from another example:
Example 2.5.
Typical classical field theories correspond to partial differential equations. Let D be some partial differential equation for fields on d -dimensional G -geometric manifolds. Then theclassical field theory F assigns to a ( d − N the space F ( N ) = { germsof solutions to the PDE D on a small d -dimensional neighborhood of N } and to a d -dimensional EISENBERG-PICTURE QUANTUM FIELD THEORY 5 cobordism M the space F ( M ) = { solutions to D on M } . These data package into a span of spaces :to any cobordism ( M : N → N ) ∈ Spacetimes G d − ,d , we get the span F ( N ) ← F ( M ) → F ( N ).The corresponding Heisenberg-picture TQFT should assign to N the algebra O ( F ( N )) of func-tions on F ( N ). If the span F ( N ) ← F ( M ) → F ( N ) were the graph of a function f M : F ( N ) ←F ( N ), then we could associative in the Heisenberg picture the homomorphism f ∗ M : O ( F ( N )) →O ( F ( N )). But typically it is not the graph of a function. Applying O to all pieces produces acospan of algebras O ( F ( N )) → O ( F ( M )) ← O ( F ( N )). This is about as far as abstract nonsensecan take us. Perhaps we should follow Example 2.5 and expect that Heisenberg-picture TQFTs should assignto cobordisms cospans of associative algebras? This is a tempting answer, but has a few problems.It doesn’t fully accommodate the “boundary ideals” from item (3) of Definition 1.7 — if these weretwo-sided ideals, we could instead use the quotient rings, but generically they are only one-sided.Moreover, the canonical way to compose cospans is via a push-out square. These exist in thecategory of associative algebras, but are too large (about as large as a free associative algebra).Considering further the classical field theory of Example 2.5, one notes that the composition of spansof spaces, given by a pullback of spaces, corresponds to the pushout of commutative algebras, whichis also the tensor product. Given cospans of associative algebras A → B ← A → B ← A ,one can give B a right A -module structure and B a left A -module structure, and consider thecomposition B ⊗ A B . But this generically is not an associative algebra.On the other hand, it is a module, as are ideals. And cospans of commutative algebras areexamples of bimodules. Perhaps (bi)modules are the appropriate target of Heisenberg-picture fieldtheory? This will not quite be our final definition, but it is sufficiently important as to merit itsown name: Definition 2.6.
The
Morita bicategory
Mor = Mor ( Vect K ) over the field K has: Objects:
Associative algebras over K . A 1-morphism between associative algebras A and B is an A - B -bimodule A M B . These compose by tensor product. A 2-morphism is a homomorphism of bimodules.This bicategory is symmetric monoidal with the usual tensor product over K [Shu07, Shu10].A (non-extended) Morita-picture quantum field theory for the spacetime category Spacetimes is a symmetric monoidal functor
Spacetimes → Mor . How can we translate a field theory in the sense of Definition 1.7 into a Morita-picture fieldtheory? It’s not yet clear how to accommodate the distinguished elements of A from item (3), butthe distinguished ideals are already modules, hence easy to handle. As for time evolution, one canalways turn homomorphisms into modules: Definition 2.7.
Let f : A → B be a homomorphism of associative algebras. The modulation of f is the A - B -bimodule M ( f ) = f B , which as a left B -module is just B , and has a right A -actionvia f : a ⊲ b ′ ⊳ b = f ( a ) b ′ b . One may easily check that modulation defines a functor M : Alg homo → Mor . The name isfrom [TWZ07]. 3.
The point of pointings
Unfortunately, the modulation functor of Definition 2.7 loses too much information:
Lemma 3.1.
Let f, g : A → B be homomorphisms of associative algebras. Then M ( f ) ∼ = M ( g ) ∈ Mor if and only if there exists an invertible element b ∈ B such that for each a ∈ A , f ( a ) = b − g ( a ) b . THEO JOHNSON-FREYD
Proof.
Given such a b ∈ B , the map M ( f ) → M ( g ) given by multiplication by b on the left is anisomorphism of A - B -bimodules. The converse is an easy exercise for the reader. (cid:3) In particular, one can fully recover from its modulation the Heisenberg-picture encoding of aclassical mechanical system in the sense of Example 1.6. However, if one starts with a Schr¨odingerpicture quantum mechanical system, applies Example 1.4, and then modulates the output, allinformation is lost: the functor
QMSpacetimes → Mor produced in this way does not depend(up to isomorphism) on the time evolution operators U t .To fix this requires breaking the multiplication by b in the proof of Lemma 3.1. A minimal wayto do this is to remember one extra bit of data: which element of M ( f ) corresponds to 1 B ∈ B .With this extra information, homomorphisms f : A → B can be recovered. Indeed, given the A - B -bimodule M ( f ) = f B and the vector 1 B ∈ f B , the element f ( a ) ∈ B for a given a ∈ A is theunique solution to the equation a ⊲ B = 1 B ⊳ f ( a ). This suggests that we revise Definition 2.7: theoutput of modulation is not just a bimodule, but a pointed bimodule. Definition 3.2.
The bicategory
Alg = Alg ( Vect K ) of algebras and pointed bimodules has: Objects:
An object of
Alg is an associative algebra over K . A 1-morphism from A to B is an A - B -bimodule A M B along with a pointing M ∈ M . The composition of ( A M B , M ∈ M ) with ( B N C , N ∈ N ) is the A - C -bimodule M ⊗ B N pointed by the class of 1 M ⊗ N . A 2-morphism ( A M B , M ∈ M ) → ( A N B , N ∈ N ) is a bimodule homomor-phism f : M → N such that f (1 M ) = 1 N .Like Mor , this bicategory is symmetric monoidal with the usual tensor product over K .There is an obvious forgetful functor Alg → Mor which forgets all pointings. The modulationfunctor factors through it: abusing notation, we let M : Alg homo → Alg denote the modulationfunctor that sends a homomorphism f : A → B to the pointed bimodule ( f B, B ). Consider modulating the Heisenberg-picture quantum mechanical system from Example 1.4 cor-responding to a Schr¨odinger-picture system in which time evolution u t is invertible. By Lemma 3.1,the modulation of a u t au − t is isomorphic in Mor to the modulation of the identity M (id A ) = A A A , where A = End( V ). They are not isomorphic in Alg when u t = 1. The isomorphism in Mor consists of multiplication by u t . It follows that: Lemma 3.3.
Given V ∈ Vect and u : V ∼ → V an isomorphism, let A = End( V ) . The modulation M ( a uau − ) of conjugation by u is isomorphic in Alg to the trivial bimodule A A A pointed notby A but by the element u ∈ A . This suggests that even when time evolution is not an isomorphism, we can nevertheless encodeit as a pointed bimodule: the identity bimodule, pointed by time evolution. Indeed:
Proposition 3.4.
There is a contravariant functor
End :
Vect → Alg taking a vector space V toits endomorphism algebra End( V ) and taking a linear map f ∈ hom( V, W ) to the pointed bimodule (cid:0) hom( V, W ) , f (cid:1) , where End( V ) and End( W ) act on hom( V, W ) by pre- and post-composition.This functor is symmetric monoidal when restricted to finite-dimensional vector spaces or when Vect (and, correspondingly,
Alg ) is replaced by an appropriate category of topological vectorspaces. For example,
End is symmetric monoidal if
Vect is replaced by the category of Hilbertspaces and bounded operators,
End( V ) is topologized as a von Neumann algebra in the usual way,and Alg consists of von Neumann algebras and pointed Hilbert-space bimodules with the von Neu-mann tensor product. Remark 3.5.
The functor End is contravariant because, following Remark 1.5, hom(
V, W ) carriesa left action by End( W ) and a right action by End( V ). EISENBERG-PICTURE QUANTUM FIELD THEORY 7
We leave checking details to the reader. Lemma 3.3 assures that the functor End extends to allof
Vect the composition “conjugate, then modulate” from
Vect inv via
Alg homo to Alg . WithProposition 3.4 in place, we define:
Definition 3.6. A (non-extended, affine) Heisenberg-picture quantum field theory is a symmetricmonoidal functor Spacetimes → Alg . Example 3.7.
Any Schr¨odinger-picture quantum field theory Z : Spacetimes → Vect deter-mines a Heisenberg-picture quantum field theory End ◦Z : Spacetimes → Alg via Proposition 3.4,provided the values of the functor Z are in some subcategory (e.g. finite-dimensional vector spacesor Hilbert spaces) for which the functor End : Vect → Alg is symmetric monoidal. Example 3.8.
Let
Spans denote the category of spans of spaces. Then a classical field theory as inExample 2.5 is a symmetric monoidal functor F : Spacetimes → Spans . Any such classical fieldtheory determines a Heisenberg-picture quantum field theory
O ◦ F : Spacetimes → Alg , where O ( X ) is the algebra of functions on the space X . When N is an object of Spacetimes (i.e. a ( d − O ( F ( N )) as an associative algebra.When M is a morphism in Spacetimes (i.e. a d -dimensional manifold with appropriate geometry)we regard O ( F ( M )) just as a pointed vector space (pointed by its unit element 1 O ( F ( M )) ). Finally we can update Definition 1.7:
Definition 3.9. A Heisenberg-picture quantum mechanical system is a Heisenberg-picture quantumfield theory for the spacetime category
QMSpacetimes of one-dimensional spacetimes with pointdefects from Definition 2.1 (or, rather, the symmetric monoidal envelope thereof).Given a system as in Definition 1.7, we define the contravariant symmetric monoidal functor H : QMSpacetimes → Alg by declaring:(1) H ( { pt } ) = A . H ( ∅ ) = K by symmetric monoidality.(2) H ( u t ) = M ( f t ) is the modulation of time evolution.(3) Given a distinguished experimental manipulation a ∈ A , the corresponding point defectis sent to ( A A A , a ), the identity bimodule pointed by a . Given a distinguished left ideal A I ⊆ A A , the corresponding point defect is sent to the left module A/I , pointed by theclass of 1 A . Given a distinguished right ideal J A ⊆ A A , the corresponding point defect issent to the right module J \ A , pointed by the class of 1 A .The main thing to note is the treatment of the distinguished ideals and elements in item (3): alldetermine pointed (bi)modules. Remark 3.10.
I have discussed pointed modules as a way of assigning algebras of quantum ob-servables to codimension-one spaces without loosing too much information about the algebra. Butpointed modules themselves have a direct interpretation in quantum field theory as the “purelyalgebraic part” of path integrals.Indeed, suppose we are given an n -dimensional affine variety X of “field configurations over M ”along with a polynomial “action” functional s ∈ O ( X ) and a volume form on X . The Feynman pathintegral invites us to consider the values of “oscillating” integrals h f i = R X f e s dVol for polynomial“observables” f . This integral is insufficiently defined: to define it requires choosing a contour in X along which integrals against the measure e s dVol converge; up to homotopies of contours thatleave all integrals unchanged, the space of contours is parameterized by the relative cohomologygroup H n ( X ; {ℜ ( s ) ≪ } ).The purely algebraic part of integration is the calculation of the class of f in the quotient O ( X ) / (total derivatives), as the integrals of total derivative vanish on any contour. This vectorspace is naturally pointed by the class of 1, and is isomorphic (via multiplication by dVol) to the n th cohomology group of the twisted de Rham complex for s . In good situations, the class of f in O ( X ) / (total derivatives) (or of f dVol in the twisted de Rham complex) can be calculated using THEO JOHNSON-FREYD homotopy algebra [JF15a]. In general Heisenberg-picture quantum field theories, Z ( M ) can beinterpreted as “ O ( X ) / (total derivatives)” for an ill-defined path integral. Non-affine field theory
Do you still have that
Introduction to Quantum Mechanics textbook? In one of its more philo-sophical sections, it is likely to discuss the following basic premise of experimental science: theuniverse consists only of things that are in principle measurable; if no experiment can distin-guish two states, then those states are equal. The tautologous version of this philosophy is themathematicians’ Yoneda lemma. But there is a non-tautologous version, which asserts that by“measurement” and “experiment” we should mean “element of the algebra of observables.” Forclassical phase spaces, for example, the non-tautologous version asserts that points can be sepa-rated by real-valued functions — that all spaces are affine in the sense of algebraic geometry. Thisassertion is true for many types of spaces (smooth manifolds; locally compact Hausdorff spaces)but by no means all spaces: there are many roles in physics for non-affine schemes and stacks.A piece of philosophy that probably is not in your textbook is that “most of 0-algebraic geometryis 1-affine.” Said another way, although schemes and stacks usually are not determined by theiralgebras (0-categories) of global functions, they are often determined by their symmetric monoidal(1-)categories of quasicoherent sheaves of modules — such a category should be understood as“the algebra of
Vect -valued global functions.” (In general, one could call a space k -affine if it isdetermined by its symmetric monoidal k -category of global maps to the k -categorical analogue of Vect .) Most algebrogeometric objects one comes across are known to be 1-affine [Lur09, BZFN10,Bra11, CJF13, BC14, HR14, BCJF14]; [HR14] also records a few objects which are known to benon-1-affine.It’s not my intention to develop here the theory of 1-affine algebraic geometry, but it’s worthmaking a few remarks. The Gabriel–Rosenberg theorem [Gab62, Ros98b] reconstructs a schemeup to isomorphism from its category of quasicoherent modules with no extra structure: only thecategory itself is used. Here is a baby case of this result: let A be a commutative ring and Mod A thecategory of right A -modules. Then A can be reconstructed as the algebra of natural endomorphismsof the identity functor id : Mod A → Mod A .This has suggested to many workers in “noncommutative algebraic geometry” that “abeliancategory” is a good definition of “noncommutative scheme” (e.g. [Ros98a]). I would argue, however,that this misunderstands the variability of categories. Consider, for example, the stacks { pt } ⊔ { pt } and { pt } / ( Z / Z /
2. Provided we work over a ringin which 2 is invertible, these have equivalent categories of modules. But they are honestly differentas stacks. To fully recover { pt } ⊔ { pt } and { pt } / ( Z /
2) from their categories of modules it sufficesto remember additionally the symmetric monoidal structures on those categories. Moreover, thenatural homomorphisms of abelian categories are the exact functors, but these do not have directgeometric meaning as morphisms of schemes. Thus the papers [Gab62, Ros98b] do not reconstructa scheme functorially from its category of modules. For comparison, the papers [Lur09, Bra11,CJF13, BC14, HR14, BCJF14] do provide functorial reconstruction of various algebrogeometricobjects by remembering their module categories’ symmetric monoidal structures and demandingthat functors be symmetric monoidal.By a similar token, an associative algebra A is not determined by the equivalence type of thecategory Mod A , which encodes only the class of A in the Morita bicategory Mor . But A can berecovered if Mod A is equipped with a pointing — a distinguished object — in Mod A , namely therank-one free module A A . I therefore propose: Definition 4.1. A noncommutative 1-affine stack X over K is a K -linear cocomplete category Qcoh ( X ) equipped with a distinguished object X = O X ∈ Qcoh ( X ). EISENBERG-PICTURE QUANTUM FIELD THEORY 9
Let
Cocomp K denote the bicategory whose objects are K -linear cocomplete categories, whose 1-morphisms are cocontinuous K -linear functors, and whose 2-morphisms are natural transformations.It is symmetric monoidal for a version of Deligne’s tensor product ⊠ ; the monoidal unit is Vect [Kel05, Section 6.5]. Noncommutative 1-affine stacks are the objects of a bicategory which we willcall
Alg lax0 ( Cocomp K ). It has: Objects:
An object of
Alg lax0 ( Cocomp K ) is a noncommutative stack, i.e. a pair ( C , C ∈ C )where C ∈
Cocomp K is a cocomplete K -linear category and C is a pointing thereof. A 1-morphism ( A , A ) → ( B , B ) is a pair ( F, F ) where F : A → B is a K -linear cocontinuous functor and 1 F : B → F ( A ) is a homomorphism in B . A 2-morphism ( F, F ) → ( G, G ) is a natural transformation η : F → G suchthat η A ◦ F = 1 G : B → G ( A ).The bicategory Alg lax0 ( Cocomp K ) is symmetric monoidal for ⊠ . Remark 4.2.
Recall that a category is cocomplete if it is closed under colimits, and a functor is cocontinuous if it preserves colimits. For set theoretic reasons it is often preferable to work justwith locally presentable categories rather than all cocomplete categories. Actually, [Kel05, Section6.5] works with small categories closed under some small set of colimit shapes, and so to extractthe tensor product ⊠ on Cocomp K requires the type of judicious Grothendieck-universe jumpingstandard in category theory. Remark 4.3.
The 1-morphisms ( F, F ) : ( A , A ) → ( B , B ) in Alg lax0 ( Cocomp K ) are lax ho-momorphisms of pointed categories , a.k.a. lax pointed functors , as opposed to the strong pointedfunctors of Remark 2.2. The use of “lax” here is consistent with the general notion of “lax homo-morphism” in [JFS15]. Example 4.4.
The Eilenberg–Watts theorem [Eil60, Wat60] asserts that the functor
Mor → Cocomp K sending an algebra A to the category Mod A of right A -modules and a bimodule A M B to the functor ( − ) ⊗ A M : Mod A → Mod B is a fully faithful inclusion of bicategories in the sensethat it induces an equivalence of categorieshom Cocomp K ( Mod A , Mod B ) ∼ = hom Mor ( A, B ) . Given an algebra A , consider the pointed category ( Mod A , A A ) ∈ Alg lax0 ( Cocomp K ). What isthe category of homomorphisms ( F, f ) : (
Mod A , A A ) → ( Mod B , B B )? By the Eilenberg–Wattstheorem, the data of a cocontinuous linear functor F : Mod A → Mod B consists (up to canonicalisomorphism) of an A - B -bimodule A F B . What about the homomorphism f : B B → F ( A ) ∼ = A A ⊗ A A F B ? It is nothing but an element of the underlying vector space of F .Thus the Eilenberg–Watts inclusion Mor ֒ → Cocomp K lifts to an inclusion EW : Alg ( Vect K ) ֒ → Alg lax0 ( Cocomp K ) sending A ( Mod A , A A ) . It is reasonable to declare therefore that a non-commutative stack is (0-)affine if it is in the essential image of EW.It is worth noting that EW is symmetric monoidal. Remark 4.5.
The composition EW ◦M : Alg homo → Alg lax0 ( Cocomp K ) of the Eilenberg–Wattsand modulation functors picks out precisely those 1-morphisms ( F, F ) : ( A , A ) → ( B , B ) whichare strong pointed functors in the sense that 1 F : B → F ( A ) is an isomorphism. With Definition 4.1 in place, we may study quantum field theories valued in “noncommutativestacks”:
Definition 4.6. A (non-extended) 1-affine Heisenberg-picture quantum field theory over K is acontravariant symmetric monoidal functor Spacetimes → Alg lax0 ( Cocomp K ). Any 0-affine quantum field theory
Spacetimes → Alg ( Vect K ) provides an example of a1-affine quantum field theory, by composing with the Eilenberg–Watts inclusion from Example 4.4. Remark 4.7.
One-affine Heisenberg-picture quantum field theory is one possible formalization of twisted [ST11] or relative [FT12] quantum field theory: Schr¨odinger-picture quantum field theoryvalued in a categorified Schr¨odinger-picture quantum field theory. Indeed, given a Heisenberg-picture quantum field theory Z : Spacetimes → Alg lax0 ( Cocomp K ), one can forget to a “cate-gorified” quantum field theory Z : Spacetimes → Cocomp K . The extra data of the field theory Z is a symmetric monoidal oplax natural transformation from the trivial field theory to Z in thesense of [JFS15]. Example 4.8.
Given a vector space V ∈ Vect K , there are two natural ways to produce an objectof Alg lax0 ( Cocomp K ):(1) Following Proposition 3.4 and Example 4.4, apply End : Vect → Alg ( Vect K ) to pro-duce the algebra A = End( V ) and then apply EW : Alg ( Vect K ) ֒ → Alg lax0 ( Cocomp K )to produce the pointed category (cid:0) Mod
End( V ) , End( V ) End( V ) (cid:1) .(2) Recognize that ( Vect K , V ) is already a pointed category, pointed by V rather than K .Both constructions are contravariantly functorial. Functoriality of the first we have already ad-dressed. For the second, given a linear map f : V → W , the pair (id Vect , f ) is a pointed func-tor (
Vect K , W ) → ( Vect K , V ). (The Eilenberg–Watts theorem identifies cocontinuous functors Vect → Vect with vector spaces. So a general pointed functor (
Vect K , V ) → ( Vect K , W )consists of a pair ( X, f ) where X ∈ Vect is a vector space and f ∈ hom( W, V ⊗ X ) is a linearmap.)But these constructions are not honestly different for the most important V . Suppose that V = 0is finite-dimensional. Then End( V ) is Morita-equivalent to K : there is an equivalence of categories Mod
End( V ) ≃ Vect K . The choice of V provides a canonical such equivalence ⊗ End( V ) V . Under thisequivalence, the object V ∗ End( V ) ∈ Mod
End( V ) is identified with K ∈ Vect K and the rank-one freemodule End( V ) End( V ) ∈ Mod
End( V ) is identified with V ∈ Vect K . So, at least for non-zero finite-dimensional V , the pointed categories (cid:0) Mod
End( V ) , End( V ) End( V ) (cid:1) and ( Vect K , V ) are equivalentin Alg lax0 ( Cocomp K ). Similar remarks apply when V is a separable Hilbert space and one worksin an analytic context in which the algebra of bounded operators on V is Morita-equivalent to K .The construction V ( Vect K , V ) is fully symmetric monoidal whereas, as mentioned inProposition 3.4, V End( V ) is only symmetric monoidal when dim V < ∞ . So in some senseconstruction (2) above is the more natural one: it turns any Schr¨odinger-picture quantum field the-ory, independent of dimension, into a Heisenberg-picture one. But it also explains construction (1):most Schr¨odinger-picture quantum field theories (including all topological ones) are valued in vectorspaces V for which ( Vect K , V ) is (0-)affine in the sense of Example 4.4. Extended affine field theory
Atiyah [Ati88] and Segal [Seg04] introduced their functorial axioms for quantum field theory inan attempt to capture the locality of physics. In the decades since, it has become clear that localityis stronger than a functor that takes values on ( d − d -dimensional“spacetimes”: a quantum field theory should also assign algebraic data to k -dimensional manifoldsfor lower k , and to spacetimes with such corners, since manifolds can be cut and glued along suchmanifolds [Law93, Fre94, BD95].The modern consensus (see e.g. [Lur09, CS15a]) is that to fully capture locality, the source Spacetimes d of a d -dimensional quantum field theory should not be just a (symmetric monoidal)category, but a (symmetric monoidal) ( ∞ , d )-category: k -dimensional morphisms for k ≤ d shouldbe k -dimensional manifolds with corners, equipped with the appropriate geometric structure;“higher” morphisms for k > d should be isomorphisms of spacetimes and isotopies (of isotopiesof . . . ) thereof. The target of a quantum field theory must also be an ( ∞ , d )-category. EISENBERG-PICTURE QUANTUM FIELD THEORY 11
Definition 5.1. A delooping of a symmetric monoidal ( ∞ , n )-category C is a choice of ( ∞ , n + 1)-category D along with an equivalence C ∼ = hom( D , D ), where D is the unit object in D .A d -dimensional fully-extended Schr¨odinger-picture quantum field theory over K is a symmetricmonoidal functor Spacetimes G d → d Vect , where: • Spacetimes G d is the symmetric monoidal ( ∞ , d )-category whose d -morphisms are d -dimensionalcobordisms with corners equipped with geometric structure of type G ; • d Vect K is some ( d − Vect K .For example, the bicategories Mor K and Cocomp K are reasonable choices for K . Remark 5.2.
Although the modern consensus is that
Spacetimes G d should be a symmetric monoidalhigher category, there is some evidence that the usual notions of “higher category” (e.g. those in[BSP11]) may not provide the correct framework in which to organize cobordisms, and that otherrelated versions are needed [MW11].That said, when G is sufficiently topological, Spacetimes G d can be built from the “topological”bordism category Bord smooth d following [Lur09, Section 3.2] or [SP11, Section 3.3] (elaboratedupon in [JF15c]). A detailed outline of the construction of Bord smooth d itself is given in [Lur09]and clarified in [CS15b]. For a similar generalization of Heisenberg-picture quantum field theories, we should look forinteresting deloopings of
Alg ( Vect K ) or Alg lax0 ( Cocomp K ). Various options are available, butwe will use one which has a natural interpretation in terms of the pictures drawn in quantum fieldtheory of insertion of local observables. To wit, consider choosing a vector space V and placing on R (considered just as an oriented manifold) some “beads” labeled by vectors in V . These “beads” canslide back and forth but cannot pass through each other. Further assume that the beads can “fuse”in a nuclear reaction. Imposing linearity in the labels, this “fusion” is an operation V ⊗ V → V . Atmicroscopic scales, fusion isn’t really a discrete process: instead, when beads become very close toeach other, they can become bonded and behave like a single particle. But suppose that all physicsof beads and fusion is “topological” in the sense that it is independent of distances on R . Thenthe fusion operation V ⊗ V → V is associative. The “invisible bead” — no bead at all — is a unitfor this fusion. So such a system is the same as an associative algebra structure on V . The beadsare nothing but “local observables” drawn from the “algebra of observables” V . The theory oflocal observables for a general quantum field theory has been formulated in terms of factorizationalgebras in [CG14], and restricts to “beads on R ” in the case of one-dimensional topological theories.More generally, consider dividing R into intervals separated by “point defects.” One can imaginea situation in which the different regions follow different physical laws: one might be filled withwater, for example, and another air. Each defect might also have its own physics: perhaps there arewaves that live only where water and air meet. Each region and each defect has its own vector spaceof observables. As observables move around, they can fuse, and we will require that the universebe topological. An observable in a region can move onto a defect, but not otherwise. There shouldbe “invisible observables” that can be inserted at any point. These rules together comprise (1) anassociative algebra assigned to each region, and (2) a pointed bimodule assigned to each defect.These are nothing but the objects and morphisms of Alg ( Vect K ).The natural generalization is to consider systems of regions and defects on higher-dimensionalspaces — vector spaces of observables assigned to each stratum, with operations describing the wayspoints can collide. Let’s impose a topological condition. The Eckmann–Hilton argument [EH62] then says that the algebra assigned to any two-dimensional region is commutative: vw fusion = v w = v w = vw = vw fusion = wv What about a one-dimensional defect separating regions whose commutative algebras of functionsare A and B ? Its local observables form an associative ( A ⊗ B )-algebra.Commutative algebras are mildly disappointing because they are not particularly “quantum.”Fortunately, there is more to the Eckmann–Hilton argument than the answer “the algebra is com-mutative.” Indeed, the Eckmann–Hilton argument actually gives two proofs of the commutativityof multiplication, the above one and: vw fusion = v w = v w = vw = vw fusion = wv Linear maps are either equal or unequal, but in a homotopical or higher categorical world, how two operations are “the same” is a type of data. One can therefore define a nontrivial notion of n -algebra in a symmetric monoidal category S to be an object of S with operations parameterizedby labeled configurations of points in R n such that homotopies between configurations correspondto homotopies between operations. (A precise definition, along with much discussion, is in [Lur14,Chapter 5], where n -algebras are called E n -algebras .) For example, a 1-algebra is a homotopy-coherent version of an associative algebra. A 2-algebra in the ( ∞ , pointed object , i.e. an object X ∈ S along with amap → X , where ∈ S is the monoidal unit.The bicategory Alg ( Vect K ) of associative algebras and pointed bimodules can then be gener-alized to higher algebra by working, as discussed above, with systems of algebras of observables thatare topological but vary at prescribed lower-dimensional strata in R n . The following summarizesthe main results of [CS15a]: Theorem 5.3 ([CS15a]) . Let S be a symmetric monoidal ( ∞ , -category admitting filtered colimitsand such that the symmetric monoidal structure distributes over filtered colimits. (1) For each n ∈ N , there is a symmetric monoidal ( ∞ , n ) -category Alg n ( S ) with: Objects: n -algebras in S . ( n − -algebra bimodules between n -algebras. ( n − -algebra bimodules between ( n − -algebras, for which the variousactions of the ambient n -algebras are compatible. . . . : . . . EISENBERG-PICTURE QUANTUM FIELD THEORY 13 n -morphisms: pointed bimodules between -algebras, for which the various actions oflower-dimensional morphisms are compatible. ( n + 1) -morphisms: equivalences of n -morphisms. ( n + 2) -morphisms: equivalences of equivalences. . . . : . . . (2) Two n -algebras in S are equivalent as objects in Alg n ( S ) if and only if they are equivalentas n -algebras (i.e. via homomorphisms, rather than via bimodules). (3) Let
Bord fr d denote the “spacetime” ( ∞ , d ) -category of framed cobordisms between smoothmanifolds. For each n -algebra X in S , there is a unique (up to contractible choices) sym-metric monoidal functor Bord fr d → Alg n ( S ) assigning X to the standard-framed point { pt } ∈ Bord fr d . This functor is called factorization homology or topological chiral homol-ogy with coefficients in X , written M R M X . Remark 5.4.
Part (3) of Theorem 5.3 is proved by explicitly constructing the functor R X , ratherthan by appealing to Lurie’s celebrated classification theorem from [Lur09]. Lurie called [Lur09]an “outline,” and while it seems the consensus among experts is that it is in every important waycorrect and complete, it also seems best to prove results without appealing to an “outline.” As I said earlier, it is mildly disappointing that 2-algebras in
Vect are automatically commu-tative. But there is a close cousin to
Vect that admits honestly non-commutative n -algebras forall n . The category DGVect of chain complexes of vector spaces (“derived” vector spaces) hasa model category structure making it into an ( ∞ , Definition 5.5. A d -dimensional fully-extended derived affine Heisenberg-picture quantum fieldtheory based on the spacetime category Spacetimes G d is a contravariant symmetric monoidal func-tor of ( ∞ , d )-categories Spacetimes G d → Alg d ( DGVect ). Then part (3) of Theorem 5.3 asserts that each d -algebra (among chain complexes) defines a d -dimensional fully-extended framed topological derived affine Heisenberg-picture quantum fieldtheory. 6. Extended non-affine field theory
Of course, category theory accommodates affine non -derived quantum field theories — functorsto
Alg d ( Vect ) — but these will not exhibit truly “quantum” behavior except in codimensions 0and 1. If the derived world is to be avoided, another option for capturing honestly “quantum”examples is to give up on affineness. (Indeed, as we will see in Section 9, important examples arenot affine.) Let S be a symmetric monoidal ( ∞ , k )-category, say the bicategory Cocomp K . Thenotion of “ d -algebra in S ” never uses non-invertible 2- or higher morphisms, and so depends only onthe maximal ( ∞ , S . Thus one can throw away that data and define Alg d ( S )as in Theorem 5.3. But the higher morphisms in S can be used to enriched Alg d ( S ): Theorem 6.1 ([JFS15]) . Let S be a symmetric monoidal ( ∞ , k + 1) -category satisfying conditionsanalogous to those of Theorem 5.3 (for the details, see [JFS15] ). Then the ( ∞ , n ) -category Alg n ( S ) from Theorem 5.3 is the n -dimensional truncation of an ( ∞ , n + k + 1) -category Alg lax n ( S ) . Thisextended version has the same - through n -morphisms as its non-extended cousin. In particular,the n -morphisms are pointed objects of S which are acted upon by their sources and targets. The ( n + 1) -morphisms are lax homomorphisms of pointed bimodules; the ( n + 2) -morphisms are laxhomomorphisms thereof; etc. Example 6.2.
Let S be a symmetric monoidal ( ∞ , k + 1)-category with unit object ∈ S .Generalizing Definition 4.1, the ( ∞ , k + 1)-category Alg lax0 ( S ) has: Objects:
An object of
Alg lax0 ( S ) consists of a pair ( X, X ) where object X ∈ S and 1 X : → X is a pointing of X . X X A 1-morphism ( X, X ) → ( Y, Y ) is a lax homomorphism of pointed objects.It consists of a pair ( F, F ) where F : X → Y is a 1-morphism in S and 1 F : 1 Y → F ◦ X is a 2-morphism. XY X Y F F A 2-morphism ( F, F ) → ( G, G ) is pair ( T, T ) where T : F → G is a 2-morphism in S and 1 T : 1 G → T ◦ F is a 3-morphism. XY T ⇛ X Y G G F F T . . . : . . . Example 6.3.
Up to a contractible space of choices, one can identify objects of
Alg lax1 ( Cocomp K )with monoidal K -linear cocomplete categories C , by which I mean that the monoidal functor is K -linear and cocontinuous in each variable, so that it extends to a functor ⊗ : C ⊠ K C → C . Recallthat to be monoidal, in addition to the monoidal functor, we should have distinguished “associator”and “unitor” natural isomorphisms satisfying standard “pentagon” and “triangle” axioms. I willgenerally suppress these auxiliary data.Let C be a monoidal K -linear cocomplete category and X a K -linear cocomplete category. A leftaction of C on X consists of a 1-morphism (i.e. K -linear cocontinuous functor) ⊲ : C ⊠ K X → X and an associator ( C ⊗ C ) ⊲ X ∼ → C ⊲ ( C ⊲ X ) and a unitor C ⊲ X ∼ → X (natural in X ∈ X and C , C ∈ C , of course) satisfying the appropriate pentagon and triangle equations. A rightaction is similar: there should be a functor ⊳ : X ⊠ K C → X and an associator and a unitor.Suppose that ( B , ⊗ B ) and ( C , ⊗ C ) are monoidal K -linear cocomplete category and X is a K -linearcocomplete category equipped with a left action ⊲ : B ⊠ K X → X and a right action X ⊠ K C → X .A compatibility between these actions consists of an associator α B,X,C : (
B ⊲ X ) ⊳ C ∼ → B ⊲ ( X ⊳ C )that satisfies the appropriate pentagon and triangle equations. Such an X is called a B - C -bimodule .A 1-morphism in Alg lax1 ( Cocomp K ) between monoidal K -linear cocomplete categories B and C is a pointed B - C -bimodule, i.e. a B - C -bimodule X with a distinguished object X ∈ X . Let ( X , X )be a pointed B - C -bimodule and ( Y , Y ) a pointed C - D -bimodule. The balanced tensor product X ⊠ C Y is the K -linear cocomplete category which is universal for the following: • There is a 1-morphism P : X ⊠ K Y → X ⊠ C Y . EISENBERG-PICTURE QUANTUM FIELD THEORY 15 • There is an “associator” α X,C,Y : P (( X ⊳ C ) ⊠ Y ) ∼ → P ( X ⊠ ( C ⊲ Y ))depending naturally on X ∈ X , C ∈ C , and Y ∈ Y . • The associator α satisfies triangle and pentagon equations. Suppressing the associators andunitors from X , C , Y , these say that α X, ,Y = id and α X,A ⊗ B,Y = α X,A,B⊲Y ◦ α X⊳A,B,Y .That the balanced tensor product exists follows from [Kel05, Theorem 6.23], which implies (moduloRemark 4.2) that
Cocomp K contains all colimits. Indeed, if if we were to ignore the units, X ⊠ C Y would be the colimit of the following diagram: X ⊠ YX ⊠ C ⊠ YX ⊠ C ⊠ C ⊠ Y ⊳ ⊠ id id ⊠ ⊲⊳ ⊠ id ⊠ id id ⊠ id ⊠ ⊲ id ⊠ ⊗ ⊠ id The three 2-cells are: = associator for X , = associator for Y , = the commutativity (id ⊠ ⊲ ) ◦ ( ⊳ ⊠ id ⊠ id) = ( ⊳ ⊠ id) ◦ (id ⊠ id ⊠ ⊲ ) . Including the unit axioms simply makes X ⊠ C Y into a slightly more complicated colimit.The category X ⊠ C Y is naturally a B - D -bimodule, and the balanced tensor product is coherentlyassociative and unital. If X is pointed by X and Y by Y , then X ⊠ C Y is pointed by P ( X ⊠ Y ).This categorifies the 0- and 1-morphisms of the bicategory Alg ( Vect K ) from Definition 3.2.Let ( X , X ) and ( Y , Y ) now be two B - C -bimodules. A 2-morphism ( X , X ) → ( Y , Y ) in Alg lax1 ( Cocomp K ) is a lax pointed bimodule homomorphism , which is to say a K -linear cocontinuousfunctor F : X → Y , natural transformations
B ⊲ F ( X ) → F ( B ⊲ X ) and F ( X ) ⊳ C → F ( X ⊳ C )intertwining the various associators and unitors, and a homomorphism 1 F : Y → F ( X ) in Y .A 3-morphism ( F, F ) → ( G, G ) is a natural transformation η : F → G intertwining the variousnatural transformations and homomorphisms. All together this makes Alg lax1 ( Cocomp K ) into aweak 3-category. Remark 6.4.
I know of no examples of monoidal categories appearing in quantum field theory thatare not generated under colimits by their (1-)dualizable objects (see Definition 7.1). Suppose that B and C are monoidal K -linear cocomplete categories generated under colimits by their (1-)dualizableobjects, and that X and Y are B - C -bimodules. Then in fact all lax bimodule homomorphisms F : X → Y are strong , which is when the natural transformations
B ⊲ F ( X ) → F ( B ⊲ X ) and F ( X ) ⊳ C → F ( X ⊳ C ) are isomorphisms; the proof is the same as that of [DSPS14, Lemma 2.10].The pointing 1 F : Y → F ( X ) is not necessarily an isomorphism. Suppose that S is a k -fold delooping of Vect — “the ( ∞ , k + 1)-category of cocomplete K -linear categories,” for example. Then Alg lax n ( S ) is an ( n + k )-fold delooping of Alg ( Vect ); i.e. indimension ( n + k ), Alg lax n ( S ) consists of pointed vector spaces. This is what a fully-extended ( n + k )-dimensional Heisenberg-picture quantum field theory would assign to top-dimensional spacetimes.Therefore a d -dimensional fully-extended k -affine Heisenberg-picture quantum field theory , for d ≥ k ,should be a symmetric monoidal functor Spacetimes d → Alg lax d − k ( S ), where S is a k -fold deloopingof Vect . For the “derived” version, replace
Vect by DGVect . My favorite delooping of
Vect K is Cocomp K , and so I generally choose: Definition 6.5. A G -geometric d -dimensional fully-extended -affine Heisenberg-picture quan-tum field theory is a symmetric monoidal functor Spacetimes G d → Alg lax d − ( Cocomp K ), where Spacetimes G d is an ( ∞ , d )-category of 0- through d -dimensional manifolds equipped with geomet-ric structures of type G . A (non-)dualizability result
Let us focus on the case of d -dimensional fully-extended framed topological quantum field the-ories, which are based on the spacetime category Spacetimes = Bord fr d . The famous cobordismhypothesis , whose proof is outlined in detail in [Lur09], asserts that for any symmetric monoidal( ∞ , N )-category C , symmetric monoidal functors Bord fr d → C are the same as d -dualizable objectsin C . (It is not too hard to convince oneself that every such functor picks out a d -dualizable object.The deep part of [Lur09] is that each d -dualizable object determines uniquely such a functor, whichis in turn a statement about the topology of manifolds.) Definition 7.1 ([Lur09]) . A 1-morphism f in an ( ∞ , N )-category C is if it has leftand right adjoints in the homotopy bicategory h C , each of which have left and right adjoints,ad infinitum. This homotopy bicategory has the same 0- and 1-morphisms as has C , but its 2-morphisms are the equivalence classes of 2-morphisms in C . (Any data from non-invertible 3-and higher morphisms in C is discarded.) Thus 1-dualizability asserts the existence of various“evaluation” and “coevaluation” 2-morphisms in C , which must satisfy certain relations up to non-unique equivalence.For k >
1, let f be a k -morphism in C with source and target ( k − x and y . Then x and y are parallel : they have the same source and target ( k − f is if it is 1-dualizable in the ( ∞ , N − k )-category whose objects are all ( k − x and y . Thus 1-dualizability of a k -morphism asserts the existence of various“evaluation” and “coevaluation” ( k + 1)-morphisms in C , which must satisfy certain relations upto non-unique equivalence.For d >
1, a k -morphism f is d -dualizable if it is 1-dualizable and the evaluation and coevaluation( k + 1)-morphisms witnessing such 1-dualizability are themselves ( d − C is a symmetric monoidal ( ∞ , N )-category, an object X ∈ C is d -dualizable if the functor ⊗ X : C → C is d -dualizable as a 1-morphism in the ( ∞ , N + 1)-category with a single object ⋆ andhom( ⋆, ⋆ ) = C and composition given by ⊗ . In general, given a symmetric monoidal ( ∞ , N )-category C , it is interesting to ask what are the d -dualizable objects for various values of d . First, a trivial observation: 1-dualizable 1-morphismsin an ( ∞ , N + 1)-dualizable objects in a symmetric monoidal( ∞ , N )-category are invertible. Thus to have non-invertible examples, we should let d ≤ N . Well-known examples include: Example 7.2.
The 1-dualizable objects in
Vect are the finite-dimensional vector spaces. The2-dualizable objects in
Mor are the finite-dimensional semisimple algebras. These examples are typical of deloopings of
Vect , in which such “full” dualizability is a strong“finiteness” condition which nevertheless admits interesting examples. In terms of field theories,
EISENBERG-PICTURE QUANTUM FIELD THEORY 17 this suggests that d -dimensional fully-extended Schr¨odinger field theories are fairly rigid but can bequite nontrivial. Similar results about 3-dimensional Schr¨odinger-picture topological field theoriesare in [DSPS13].What about Heisenberg-picture field theories? Theorem 5.3 implies that, for any ( ∞ , k + 1)-category S and any n , every object of the ( ∞ , n + k + 1)-category Alg lax n ( S ) is n -dualizable. Thus0-affine Heisenberg-picture topological field theories (when n = d and k = 0) are quite flexible. Butthey are not “fully dualizable”: even when S is an ( ∞ , Alg lax n ( S ) is an ( ∞ , n + 1)-category. In fact, asking for any more dualizablity collapses the whole enterprise: Theorem 7.3.
Let S be a symmetric monoidal ( ∞ , k +1) -category. Then the groupoid of ( n + k +1) -dualizable objects in Alg lax n ( S ) is contractible — every ( n + k + 1) -dualizable object is canonicallyequivalent to the unit object .Proof. I will give the idea of the proof, as providing all details would require working in too muchdetail for this paper with some particular model of ( ∞ , n )-categories.We begin by proving the claim when k = 0. Let ( X, X ) be a 1-dualizable object in Alg lax0 ( S )with dual ( X, X ) ∗ = ( Y, Y ). Denote the evaluation and coevaluation maps by:( F, F ) : ( , id ) → ( X, X ) ⊗ ( Y, Y ) ∼ = ( X ⊗ Y, X ⊗ Y ) , ( G, G ) : ( Y ⊗ X, Y ⊗ X ) ∼ = ( Y, Y ) ⊗ ( X, X ) → ( , id ) . The compatibility conditions between the evaluation and coevaluation necessary for 1-dualizabilityassert that two equations, the first reading: XXX ⊗ Y ⊗ X X X ⊗ Y ⊗ X F ⊗ id X F ⊗ id X X id X ⊗ G id X ⊗ G = XX X X id X id X Considering just
X, Y, F, G , one finds that X and Y are duals in S . Unpacking the extra conditionscoming from 1 F and 1 G gives two commuting triangles, the first reading:1 X X ◦ (cid:0) G ◦ (1 Y ⊗ id X ) (cid:1) ◦ X X ⊗ (cid:0) G ◦ (1 Y ⊗ X ) (cid:1) X (id X ⊗ G ) ◦ ( F ⊗ id X ) ◦ X ∼ = ∼ = id X ⊗ G F ⊗ id X id X These triangles precisely assert that 1 X : → X and G ◦ (1 Y ⊗ id X ) : X → are dual 1-morphisms.In particular, 1 X is 1-dualizable.An induction implies more generally that a d -dualizable object ( X, X ) ∈ Alg lax0 ( S ) consists ofa d -dualizable object X ∈ S along with a d -dualizable 1-morphism 1 X : → X . But, as remarked above, an ( n + 1)-dualizable 1-morphism in an ( ∞ , n + 1)-category is necessarily invertible. Thusthe pointing 1 X furnishes a canonical equivalence between ( X, X ) and ( , id ).Now let k be arbitrary. I will describe a canonical symmetric monoidal functor Alg lax n ( S ) → Alg lax0 ( Alg lax n ( S )) which splits the forgetful map Alg lax0 ( Alg lax n ( S )) → Alg lax n ( S ). (This splittingis a version of the Eilenberg–Watts functor considered in Example 4.4.) With such a functor inhand, any ( n + k + 1)-dualizable object X ∈ Alg lax n ( S ) determines an ( n + k + 1)-dualizable objectin Alg lax0 ( Alg lax n ( S )), which by above is trivial, and so X is trivial.First, let M be an m -morphism in Alg lax n ( S ) for m < n ; i.e. an ( n − m )-algebra in S , with n − m ≥
1, with some compatible actions by some ( n − m + 1)-algebras. We map it to thepointed m -morphism ( M, M M ), where M M is the “right regular M -module,” which is M treatedas an ( n − m − M -action. In terms of the factorization algebrapictures used in [CS15a] to define Alg lax n ( S ), the m -morphism M is described by a factorizationalgebra on R n which is locally constant with respect to the stratification given by the subspaces { x = 0 } , { x = x = 0 } , . . . , { x = · · · = x m = 0 } . The pointed m -morphism ( M, M M ) is givenby pushing M forward along the constructible map ~x ( ~x if x + · · · + x n ≥ x , . . . , x n − , − x − · · · − x n − ) if x + · · · + x n ≤ . Let M now by an n -morphism with source S and target T . Then S and T are 1-algebras and M = ( S M T , M ) is a pointed S - T -bimodule (subject to compatibility conditions coming from thecommon source and target of S and T ). The composition M ( S S ) = S S ⊗ S S M T is the module M T in given by forgetting the S -action. The pointing 1 M then determines a (strong, and in particularlax) pointed T -module map T T → M T given by 1 T M . Thus ( M, M ) “is” an n -morphism in Alg lax0 ( Alg lax n ( S )). The construction M ( M, M ) extends naturally to lax pointed bimodulemorphisms, and completes the description of the splitting Alg lax n ( S ) → Alg lax0 ( Alg lax n ( S )). (cid:3) In summary, the interesting dualizability questions related to k -affine Heisenberg-picture topo-logical quantum field theory are about d -dualizability in Alg n ( S ) for S a k -fold delooping of Vect and n + 1 ≤ d ≤ n + k .8. From factorization algebra to Heisenberg-picture field theory
I would like now to explain an example which is based on unpublished work of Dwyer, Stolz, andTeichner. I will outline my interpretation of some parts of their construction, but details will needto wait for future work.
Definition 8.1.
Let G be a not-necessarily-topological local geometry for d -dimensional manifolds,and Emb G d the symmetric monoidal category of possibly-open d -dimensional G -geometric manifolds,with symmetric monoidal structure given by disjoint union. Given a target category S , a G -geometric prefactorization algebra valued in S is a symmetric monoidal functor F : Emb G d → S .In particular, for every M ∈ Emb G d , F ( M ) is pointed by F ( ∅ ֒ → M ) : S → F ( M ), and eachembedding M ⊔ M ֒ → M determines a “multiplication” map F ( M ) ⊗ F ( M ) → F ( M ). Aprefactorization algebra is a factorization algebra if it is local for the Weiss topology (see [Gin13,CG14, CS15a] for details). We will need two variations of the notion of “factorization algebra.” Let X be a topological space.A factorization algebra on X is a precosheaf F on X , local for the Weiss topology, such that if U and V are disjoint opens, then F ( U ⊔ V ) ∼ = F ( U ) ⊗ F ( V ). Let ( X, ∗ ∈ X ) be a pointed topologicalspace. An unpointed point-module on ( X, ∗ ∈ X ) is similar, but whereas in a factorization algebrathere are maps F ( U ) → F ( V ) for every inclusion U ⊆ V of open sets in X , in an unpointedpoint-module we do not ask for such maps in the special case where U ∋ ∗ but V
6∋ ∗ . These arecalled “unpointed” because in the special case when X = {∗} , an unpointed point-module is just EISENBERG-PICTURE QUANTUM FIELD THEORY 19 an object of S , whereas a factorization algebra is a pointed object. The restriction of a G -geometricfactorization algebra to a space with G -geometry is a factorization algebra on that space; given afactorization algebra on a pointed space, one can forget to an unpointed point-module.When G is a not-necessarily-topological “rigid supergeometry,” a construction of the unextendedspacetime category Spacetimes G d − ,d is described in [ST11]. In that construction, the objects are( d − N equipped with germs of one-sided collars diffeomorphic to N × [0 , ǫ ) with G -structures on N × (0 , ǫ ). A morphism from N to N is a d -dimensional manifold M with boundary ∂M ∼ = N ⊔ N and a germ of an extension of M past N : N N M Given a G -geometric factorization algebra F , we will build a Heisenberg-picture field theory Z F : Spacetimes G d − ,d → Alg ( Cocomp K ) for this version of Spacetimes G d − ,d . Definition 8.2.
Fix a G -geometric factorization algebra F : Emb G d → Vect K . Let M be a(possibly open) G -manifold with boundary. Let M/∂M denote the quotient (in topological spaces)in which ∂M is contracted to a point; it is naturally pointed by that point. The category BC F ( M )of boundary conditions for F on M is BC F ( M ) = (cid:8) unpointed point-modules ˜ F on M/∂M s.t. ˜ F | M r ∂M = F | M r ∂M (cid:9) . The free boundary condition is the boundary condition given by pushing forward of F | M r ∂M along M r ∂M → M/∂M . Lemma 8.3.
The category of boundary conditions for F on M depends only on the germ of a G -manifold around ∂M .Proof. This follows from descent/locality of factorization algebras [CS15a]. Indeed, suppose that M ′ ֒ → M is a submanifold with boundary such that the inclusion maps ∂M ′ ∼ → ∂M . One canrestrict unpointed point-modules on M/∂M to unpointed point-modules on M ′ /∂M ′ . Conversely,suppose we are given an unpointed point-module ˜ F on M ′ /∂M ′ whose restriction to M ′ r ∂M ′ is F | M ′ r ∂M ′ . Then it can be canonically extended to an unpointed point-module on M/∂M by gluingwith F | M r ∂M . (cid:3) Proposition 8.4.
Let M be a manifold with boundary. Its category BC F ( M ) of boundary condi-tions is cocomplete and K -linear. The proof will show indeed that BC F ( M ) is locally presentable. Proof.
I will give an alternate characterization of BC F ( M ), closer to construction of Dwyer, Stolz,and Teichner. Consider the poset P ( M ) whose objects are open neighborhoods of ∂M , ordered byinclusion. We build a linear category C F ( M ) whose objects are the elements of P ( M ) and whosemorphisms are: hom( A, B ) = ( , if A BF ( B r A ) , if A ⊆ B. Note in particular that hom(
A, A ) = F ( ∅ ) = K . The composition is given by the factorizationalgebra structure maps F ( A r B ) ⊗ F ( B r C ) ∼ = F (( A r B ) ∪ ( B r C )) → F ( A r C ).Let Fun K ( A , B ) denote the category of K -linear functors between K -linear categories A and B .There is a fully faithful embedding BC F ( M ) ֒ → Fun K ( C F ( M ) , Vect K ) that sends each unpointed point-module ˜ F to the functor A ˜ F ( A r ∂M ) . Functoriality is given by the factorization algebrastructure. Since
Fun K ( C F ( M ) , Vect K ) is K -linear, so is BC F ( M ).The essential image of this embedding consists of those K -linear functors C ( N ) → Vect K sat-isfying an appropriate “locality” condition coming from the locality condition for factorizationalgebras. This locality condition consists of a series of assertions of the following form: ˜ F ( A ) arisesas a weighted colimit of a diagram each term of which is ˜ F ( B ) for some B ( A , with the weightingsbeing tensor products of F ( A r B )s.Thus BC F ( M ) is the category of models of a colimit sketch structure on C F ( M ), and is thereforelocally presentable and in particular cocomplete. (cid:3) With Lemma 8.3 and Proposition 8.4 in hand, we may define:
Definition 8.5.
Given an object N ∈ Spacetimes G d − ,d and a factorization algebra F : Emb G d → Vect K , we set Z F ( N ) = BC F ( N × [0 , ǫ )) ∈ Alg lax0 ( Cocomp K ), where the pointing is via the freeboundary condition. Lemma 8.6.
The assignment N
7→ Z F ( N ) is symmetric monoidal in the sense that there arecanonical equivalences Z F ( ∅ ) = Vect K and Z F ( N ⊔ N ) = Z F ( N ) ⊠ Z F ( N ) .Proof. The category of boundary conditions for the empty set is the category of unpointed point-modules on { pt } , which is easily seen to be Vect K . Assume N and N are nonempty. Given K -linear categories A and B , define their naive tensor product A⊗ K B to be the category with objectset ob( A ) × ob( B ) and morphisms hom(( A , B ) , ( A , B )) = hom A ( A , A ) ⊗ hom B ( B , B ). Thenthere is a natural equivalence Fun K ( A , Vect K ) ⊠ Fun K ( B , Vect K ) = Fun K ( A ⊗ K B , Vect K ).From this and the description of boundary conditions given in the proof of Proposition 8.4 one mayderive the natural equivalence Z F ( N ⊔ N ) = Z F ( N ) ⊠ Z F ( N ). (cid:3) Definition 8.7.
We now extend Z F to morphisms in Spacetimes G d − ,d . Let M : N → N be aone-morphism, where we have chosen representative collars N × [0 , ǫ ) around N and N × [0 , ǫ )around N . Since Heisenberg-picture field theories are assumed to be contravariantly functorial,we want to build a functor Z F ( M ) : Z F ( N ) → Z F ( N ).By Lemma 8.3, Z F ( N ) = BC F ( M ⊔ N N × [0 , ǫ )), as the inclusion N × [0 , ǫ ) → M ⊔ N N × [0 , ǫ ) is an isomorphism near the boundary. Given a boundary condition ˜ F ∈ BC F ( M ⊔ N N × [0 , ǫ )), we may push it forward along the map ( M ⊔ N N × [0 , ǫ )) /N → ( N × [0 , ǫ )) /N that isthe identity on N × (0 , ǫ ) and collapses M to the point. The resulting pushed-forward factorizationalgebra then restricts over N × (0 , ǫ ) to F | N × (0 ,ǫ ) and so defines a boundary condition on N × [0 , ǫ ).We set Z F ( M ) : Z F ( N ) → Z F ( N ) to be this push-forward operation.Cocontinuity of Z F ( M ) follows from Remark 8.8. Its structure as a lax pointed functor comesfrom the canonical maps F ( A r N ) → F (( M r N ) ⊔ N A ), where A ⊆ N × [0 , ǫ ) is anopen neighborhood of the boundary N . This defines the functor Z F : Spacetimes G d − ,d → Alg lax0 ( Cocomp K ); functoriality follows from the functoriality of push-forward of factorizationalgebras, and symmetric monoidality can be proven by extending Lemma 8.6. Remark 8.8.
One can give Z F ( N ) a much more explicit description, completing the comparisonto the construction described by Dwyer, Stolz, and Teichner. I will continue with the notation fromthe proof of Proposition 8.4.Set M = N × [0 , ǫ ). For each object A ∈ C F ( M ), consider the full subcategory C F ( M ) ( A of C F ( M ) on the objects B with B ( A . There are restriction functors Fun K ( C F ( M ) ( A , Vect K ) → Fun K ( C F ( M ) ( B , Vect K ) for B ( A whose left adjoints define extension functors Fun K ( C F ( M ) ( B , Vect K ) → Fun K ( C F ( M ) ( A , Vect K ) . Unpacking the locality condition for factorization algebras reveals Z F ( N ) = BC F ( M ) = lim A → ∂M Fun K ( C F ( M ) ( A , Vect K ) , EISENBERG-PICTURE QUANTUM FIELD THEORY 21 where the limit is taken over P ( M ) along the extension functors. It is worth noting that the colimitalong restriction functors vanishes because hom( A, B ) = 0 if A B .Let M now be a G -geometric cobordism between N and N . There is a canonical inclusion C F ( N × [0 , ǫ )) ֒ → C F ( M ) given by B M ⊔ N B , and hence a restriction functor Fun K ( C F ( M ⊔ N N × [0 , ǫ )) ( M ⊔ N B , Vect K ) → Fun K ( C F ( N × [0 , ǫ )) ( B , Vect K ) . Composing this with the projection Z F ( N ) = BC F ( M ⊔ N N × [0 , ǫ )) → Fun K ( C F ( M ⊔ N N × [0 , ǫ )) ( M ⊔ N B , Vect K ) gives a sequence of cocontinuous functors Z F ( N ) → Fun K ( C F ( N × [0 , ǫ )) ( B , Vect K ) . One may directly check that these commute with the extension functors for
B ֒ → B ′ , and sodefine a cocontinuous functor Z F ( N ) → Z F ( N ), which is nothing but the functor Z F ( M ) fromDefinition 8.7.As a final remark, note that limits can be computed by passing to cofinal subcategories, andso Z F ( N ) can be presented as a limit over categories just for those opens that contract onto N . Moreover, consider pushing forward F | N × (0 ,ǫ ) along the projection N × (0 , ǫ ) → (0 , ǫ ) toproduce a factorization algebra on (0 , ǫ ). One can then consider the category of boundary conditionson [0 , ǫ ) for this factorization algebra. By comparing descriptions in terms of limits of functorcategories along extension functors, one can show that this category is nothing but Z F ( N ). Withthis description and some careful study of limits of locally presentable categories, one can showin particular that when F is locally constant, Z F ( N ) = Mod R N F , and so the construction abovematches the factorization homology functor from [CS15a]. The construction of Z F ( N ) did not require that N was compact. Suppose that N is an open( n − G -structure on N × [0 , ǫ ); then there is stilla category BC F ( N × [0 , ǫ )) of boundary conditions for F | N × (0 ,ǫ ) , and it is still described as in theproof of Proposition 8.4. Proposition 8.9.
The assignment N
7→ Z F ( N ) defines a symmetric monoidal precosheaf valuedin Cocomp K on the category Emb G d − of ( d − -dimensional manifolds equipped with germs of G -structures. In fact, Z F is a factorization algebra, but the proof will await future work. Proof.
Suppose that N ֒ → N is an inclusion compatible with germs of G -structures. The corre-sponding map N × [0 , ǫ ) ֒ → N × [0 , ǫ ) extends to an inclusion (cid:0) N × [0 , ǫ ) (cid:1) ⊔ N × (0 ,ǫ ) (cid:0) N × (0 , ǫ ) (cid:1) ֒ → (cid:0) N × [0 , ǫ ) (cid:1) , which in turn descends to a continuous function (cid:0) N × [0 , ǫ ) /N (cid:1) ⊔ N × (0 ,ǫ ) (cid:0) N × (0 , ǫ ) (cid:1) ֒ → (cid:0) N × [0 , ǫ ) /N (cid:1) which is a bijection on points, although not in general a homeomorphism.Any boundary condition on N × [0 , ǫ ) extends canonically to an unpointed point-module on (cid:0) N × [0 , ǫ ) /N (cid:1) ⊔ N × (0 ,ǫ ) (cid:0) N × (0 , ǫ ) (cid:1) by gluing with F | N × (0 ,ǫ ) . Pushing forward along the bijectionabove defines the functor Z F ( N ) → Z F ( N ).Symmetric monoidality of Z F follows from Lemma 8.6. (cid:3) Nowhere in the construction of Z F did we need that the factorization algebra F took valuesin Vect K — any locally presentable target category would have sufficed. Let Pres K denote thebicategory of K -linear locally presentable categories; it is a full subbicategory of Cocomp K . Pres K seems in many ways like it is a “locally presentable bicategory”: it is cocomplete [Bir84]; everyobject has a presentation [AR94, Theorem 1.46]. The theory of locally presentable higher categories is under active development. Eventually,one should expect to be able to iterate the above construction F ; Z F to produce an extendedfield theory Z Z F valued in “locally presentable Pres K -linear bicategories.” Iterating further wouldproduce a fully extended Heisenberg-picture field theory from a factorization algebra.9. From skein theory to Heisenberg-picture field theory
In this final example I would like to describe a family of topological Heisenberg-picture field the-ories which one can prove are not affine. The construction is my interpretation of the Reshetikhin–Turaev invariants of knots and links [RT90]. The extension to a local field theory is inspired byWalker’s work [Wal06, MW11]. It is the three-dimensional extension of the two-dimensional quan-tum field theories studied in [BZBJ15] and is expected to match the answer given by [AFR15]. Fulldetails will appear in [JF15b]. Recall some standard definitions:
Definition 9.1.
Let C = ( C , ⊗ , . . . ) be a small K -linear monoidal category. (The “ . . . ” denote theauxiliary data of an associator, unit, and unitors that I will generally suppress.) A strong monoidalfunctor ( F, f ) consists of a functor F and a natural isomorphism f : F ( − ) ⊗ F ( − ) ∼ → F ( − ⊗− ) expressing compatibility with the monoidal structure, which must itself be compatible withassociators (and also an isomorphism expressing compatibility with the units, but that isomorphismcan always be canonically strictified to an identity, and so I will suppress it from the notation). Let ⊗ op denote the opposite monoidal structure on C .The monoidal category C is braided if it is equipped with a strong monoidal functor (id C , β ) :( C , ⊗ , . . . ) → ( C , ⊗ op , . . . ) whose underlying functor is the identity functor id C . (This is equivalent tothe usual hexagon relations for β .) A full twist for C is is an isomorphism θ : (id C , β − ) ∼ → (id C , β )of monoidal functors between ( C , ⊗ ) and ( C , ⊗ op ), or equivalently an isomorphism of monoidalendofunctors θ : (id C , id) ∼ → (id C , β ) of ( C , ⊗ ). Spelled out, a full twist consists of a naturalautomorphism θ X : X ∼ → X such that β X,Y ◦ ( θ X ⊗ θ Y ) = β − Y,X ◦ θ X ⊗ Y for all X, Y ∈ C .The name “full twist” comes from interpreting θ X as a 360 ◦ twist of a ribbon labeled by X :=Categories equipped with a full twist are called balanced in [JS91], but I will not use this word so asnot to conflict with the notion of “balanced tensor product” from Example 6.3 and Definition 9.4.A small monoidal category C has duals if for every object X ∈ C the functor X ⊗ : C → C hasa right adjoint of the form ∗ X ⊗ and a left adjoint of the form X ∗ ⊗ for some objects X ∗ , ∗ X ∈ C .(These objects are unique up to unique isomorphism if they exist, in which case ⊗ X ∗ and ⊗ ∗ X are the left and right adjoints respectively to ⊗ X . Compare Definition 7.1.) If C has duals, thenthere is a canonically defined double dual functor X X ∗∗ which is a monoidal equivalence( C , ⊗ , . . . ) → ( C , ⊗ , . . . ). Suppose that C = ( C , ⊗ , β, . . . ) is a small K -linear braided monoidalcategory with duals. The braiding determines isomorphisms τ β : ( − ) ∗∗ ∼ → (id , β ) and τ β − : ( − ) ∗∗ ∼ → (id , β − ) EISENBERG-PICTURE QUANTUM FIELD THEORY 23 of monoidal functors
C → C , via X ∗∗ X ∗∗ ⊗ X ⊗ X ∗ X ⊗ X ∗∗ ⊗ X ∗ X unit of adjunction counit of adjunction β X ∗∗ ,X ⊗ id X ∗ β − X,X ∗∗ ⊗ id X ∗ τ β ( X ) τ β − ( X ) with inverses X X ∗ ⊗ X ∗∗ ⊗ X X ∗ ⊗ X ⊗ X ∗∗ X ∗∗ unit of adjunction counit of adjunctionid X ∗ ⊗ β − X,X ∗∗ id X ∗ ⊗ β X ∗∗ ,X τ − β ( X ) τ − β − ( X ) A braided monoidal category C with duals is ribbon if it is equipped with a full twist θ satisfyingthe quadratic equation θ = τ − β − ◦ τ β : =Let ( C , ⊗ , β, θ, . . . ) be a ribbon category. A C -labeled ribbon tangle in [0 , is a piecewise-smoothembedding (except for as described in item (2)) into [0 , of a finite collection of rectangles subjectto the following conditions:(1) The rectangles come in two kinds, “long thin” ribbons and “short squat” coupons . The bottom end of a ribbon is the subset [0 , ×{ } of its boundary, and the top end if [0 , ×{ } .The bottom side of a coupon is the subset [0 , × { } of its boundary, and the top side is[0 , × { } .(2) Each bottom end of each ribbon is a subset either of the bottom of the cube [0 , × { } orof a top side of a coupon; in the latter case the map [0 , × { } ֒ → [0 , × { } mapping theend of the ribbon into the top or bottom of the coupon is orientation-preserving. The topend of each ribbon is a subset either of the top of the cube [0 , × { } or of a bottom sideof a coupon; again in the latter case orientations should be preserved. These are the onlyintersections of ribbons and coupons with each other or with the boundary of [0 , .(3) Each ribbon is labeled with an object of C . Each coupon is labeled with a morphism in C compatibly with the ribbons that end on it: if the ribbons ending on its bottom are labeledin order X , . . . , X m and the ribbons ending on its top are labeled in order Y , . . . , Y n , thenthe coupon itself is labeled with an element of hom( X ⊗ · · · ⊗ X m , Y ⊗ · · · ⊗ Y n ). The main coherence theorem about ribbon categories is the following:
Theorem 9.2 ([RT90]) . There is a well-defined interpretation map from C -labeled ribbon tangles in [0 , to morphisms in C . The source and target of the morphism depend only on the intersectionsof the tangle with the bottom and top, respectively, of [0 , . The interpretation of a tangle dependsonly on the isotopy-rel-boundary class of the tangle. Suppose that [0 , ֒ → [0 , is an orientation-preserving nesting of a smaller cube into theinterior of a larger cube. Suppose further that we are given two C -labeled ribbon tangles in thelarger cube such that (i) the two tangles agree outside the smaller cube, (ii) the tangles intersectthe boundary of smaller cube only along its top and bottom, and (iii) the intersections of the twotangles with the smaller cube have the same interpretation. Then the two tangles have the sameinterpretation in the larger cube. Theorem 9.2 allows us to use C to study tangles in more complicated manifolds. The followingconstruction is particularly important: Definition 9.3.
Let C be a ribbon category and let N be a possibly-open oriented surface admittinga finite cover by disks. The skein category R N C is the K -linear category described as follows: objects: configurations of disjoint “short” oriented intervals in N , each labeled by an objectof C .There is a natural extension of the notion of “ribbon tangle labeled by C ” to tangles in N × [0 , N × { } and with N × { } are naturallyobjects of R N C . morphisms: the space of morphisms from X to X is spanned by the set of C -labeled ribbontangles that intersect N × { i } at X i , modulo the following relation: for every orientation-preserving embedding [0 , ֒ → N × [0 ,
1] of a “small” cube into the interior of N × [0 , P T a if (i) all the T a s are equal outside thesmall cube, (ii) they intersect the small cube only along its top and bottom sides, and theintersection is transverse, and (iii) P interpret( T a ∩ (small cube)) = 0.It follows from Theorem 9.2 that isotopic-rel-boundary tangles represent the same morphism in R N C . Composition is by stacking of tangles in N × [0 , ∪ N N × [0 , ≃ N × [0 , R N C is naturally pointed by the empty object. If N has boundary, we set R N C = R ˚ N C . Definition 9.4.
Let A be a small K -linear monoidal category with a right module X and a leftmodule Y . Recall from the proof of Lemma 8.6 that the naive tensor product of X with Y isthe category X ⊗ K Y with object set ob( X ) × ob( Y ) and morphisms hom(( X , Y ) , ( X , Y )) =hom X ( X , X ) ⊗ hom Y ( Y , Y ). The naive balanced tensor product X ⊗ A Y is the K -linear categoryformed from X ⊗ K Y by adding, for each object ( X, A, Y ) ∈ X ⊗ K A ⊗ K Y , a natural-in-( X, A, Y )isomorphism α X,A,Y : (
X ⊳ A ) ⊗ Y ∼ → X ⊗ ( A ⊲ Y ), subject to a pentagon relation (and also addingsome unitor relations that we will not write down). Compare Example 6.3. It is not hard to prove the following:
Proposition 9.5.
The skein category R N ⊔ N C of a disjoint union is the naive tensor product ofskein categories ( R N C ) ⊗ K ( R N C ) .The construction R C : { surfaces } → { categories } is functorial on embeddings. It follows thatfor any oriented one-manifold P , R P C = R P × [0 , C is naturally a monoidal category and for anyoriented zero-manifold P , R P C = R P × [0 , C is naturally a braided monoidal category. It follows alsothat if N is a surface with boundary ∂N , then R N C is naturally a module category for R ∂N C . Theinterpretation map from Theorem 9.2 provides a braided monoidal equivalence from R { pt } C to C .Moreover, the functor R C : { surfaces } → { categories } satisfies the following version of excision .Suppose that N is a surface with a decomposition as N = N ⊔ P N along a one-manifold P . Then R N C is naturally a right R P C -module and R N C is naturally a left module, and R N C is equivalentto the naive balanced tensor product Z N C ≃ (cid:18)Z N C (cid:19) ⊗ R P C (cid:18)Z N C (cid:19) . EISENBERG-PICTURE QUANTUM FIELD THEORY 25
Combining results from [AF12, CS15a] gives:
Corollary 9.6.
The assignment N R N C defines a symmetric monoidal functor Bord or2 → Alg lax2 ( Cat K ) , where Bord or2 is the fully-extended oriented 2-bordism category and
Cat K is thebicategory of small linear categories and linear functors. This is not the preferred target of a Heisenberg-picture field theory — as in Definition 6.5, Iwould much rather land in
Alg lax2 ( Cocomp K ). But this is not a problem. The following is provedin [Kel05]: Lemma 9.7.
Let X be a small K -linear category. The free cocompletion b X of X is Fun K ( X op , Vect K ) .It satisfies the following universal property: for any E ∈
Cocomp K , there is a natural equivalence Fun K ( X , E ) ≃ hom Cocomp K ( b X , E ) . Free cocompletion is symmetric monoidal: \ X ⊗ K Y ≃ b X ⊠ K b Y .The essential image of d ( − ) consists of those cocomplete categories generated under colimits by someset of compact projective objects. There is a natural equivalence hom Cocomp K ( b X , b Y ) ≃ Fun K ( X ⊗ K Y op , Vect K ) . Linear functors
X ⊗ K Y op → Vect K should be thought of as bimodules between the categories X and Y , where X and Y themselves are “many object associative algebras.” Corollary 9.8.
Free cocompletion takes naive balanced tensor products to balanced tensor products: \ X ⊗ A Y ≃ b X ⊠ b A b Y if A is a small K -linear monoidal category with a left module X and a right module Y .Proof. Both sides satisfy the same universal property. (cid:3)
It follows that N dR N C defines a symmetric monoidal functor Bord or2 → Alg lax2 ( Cocomp K ).This is not yet a Heisenberg-picture field theory: it assigns pointed linear categories, not pointedvector spaces, to top-dimensional manifolds. To build a Heisenberg-picture field theory requiresextending this functor to three-manifolds. Definition 9.9.
Let M be a compact three-dimensional manifolds with boundary decomposed as ∂M = N bot ⊔ P ×{ } P × [0 , ⊔ P ×{ } N top , where P is a one-manifold and N bot and N top are the“bottom” and “top” parts of the boundary. Such manifolds are the three-morphisms in Bord or3 .Given a ribbon category C , the skein module R M C of M is (the span of) ribbon tangles in M modulolocal relations for little cubes [0 , ֒ → M just as in the morphisms of Definition 9.3. By stackingon cylinders over N bot and N top , is naturally an R N bot C – R N top C –bimodule, i.e. a functor (cid:18)Z N bot C (cid:19) ⊗ K Z N top C ! op → Vect K , and so defines a cocontinuous functor [ Z M C : \ Z N bot C → \ Z N top C . The empty tangle gives [ R M C the structure of a lax pointed functor. In [JF15b], I will prove:
Theorem 9.10.
When P is nonempty, for each object X ∈ R P C , there is a natural operation thattakes a tangle in M and outputs its union with the identity tangle over X . This operation makes [ R M C into a lax module functor between the dR P C -modules \ R N bot C and \ R N top C . The assignment dR C defines an oriented topological fully extended 1-affine Heisenberg-picture field theory Bord or3 → Alg lax2 ( Cocomp K ) . A braided monoidal cocomplete category
A ∈
Alg lax2 ( Cocomp K ) is of the form b C for C a smallribbon category if and only if it satisfies all of the following: • The unit object A ∈ A is compact projective. • Every object of A is a colimit of dualizable objects. • A is equipped with a full twist whose restriction makes the full subcategory A fd on thedualizable objects into a ribbon category. • A fd is equivalent to a small category.For such A , A = d A fd . In general, ( b C ) fd is the Karoubi envelop or Cauchy completion of C : it’s whatyou get when you take C , add all finite direct sums, and then split all idempotents.These conditions are satisfied, for example, for A = Rep integrable C ( q ) ( U q g ) the category of ind-finite-dimensional representations of a quantum group at generic quantum parameter q , or for A = Rep algebraic C ( G ) the category of algebraic representations of a reductive group when C ; the du-alizable objects are then the finite-dimensional modules. The conditions fail for (non-semisimplified)quantum groups at roots of unity and for finite groups over fields of characteristic dividing the orderof the group. Example 9.11.
Let’s say that a trivial ribbon category is the category
Mod fd A of finitely generatedprojective modules over a commutative K -algebra A with ⊗ = ⊗ A , or some ribbon subcategorythereof. The monoidal category ( Mod fd A , ⊗ A ) admits a unique braiding and unique full twist, sinceit is generated under direct sums and passing to direct summands by the monoidal unit. All ribbonsubcategories of Mod fd A determine the same Heisenberg-picture field theory valued in Cocomp K ,since they have canonically identified free cocompletions. The 1-affine Heisenberg-picture fieldtheory dR C is 0-affine if and only if C is trivial. Example 9.12. If G is a reductive group over K = C and N is a surface, then \ R N Rep fd C ( G ) ≃ QCoh (Loc G ( N )), where QCoh is the category of quasicoherent sheaves of O -modules and Loc G ( N )is the stack of G -local systems, presented by the groupoid hom( π ( N ) , G ) /G [BZBJ15]. This canbe proven by showing that N QCoh (hom( π ( N ) , G ) /G ) satisfies excision.An object X = ⊔{ X i } ∈ R N Rep fd ( G ) represents the following sheaf on Loc G ( N ): given a localsystem on N , restrict it at each of the small intervals X i : [0 , → N in X ; tensor each restrictionwith the corresponding G -module to get a flat vector bundle on the i th interval; tensor togetherthe spaces of flat sections of each of these bundles.Suppose that M is a three-manifold with boundary ∂M , read as a morphism M : ∅ → ∂M . Then [ R M C : Vect K → \ R ∂M C ≃
QCoh (Loc G ( ∂ )) is the pushforward of the structure sheaf O (Loc G ( M ))along the restriction map Loc G ( M ) → Loc G ( ∂M ). Example 9.13.
Let C = TL be the well-known Temperley–Lieb category, which is a ribbon cate-gory defined over K = Z [ q, q − ] whose C -linearization is the monoidal subcategory of Rep C ( U q sl )generated by the defining module. The Heisenberg-picture field theory R TL packages together allof Kauffman-bracket skein theory. Remark 9.14.
Using the same skein-theoretic technology, one can show: • Every K -linear braided monoidal locally presentable category in which the unit object iscompact projective and every object is a colimit of dualizable objects defines a framedHeisenberg-picture field theory Bord fr3 → Alg lax2 ( Cocomp K ). • Every K -linear monoidal locally presentable category in which the unit object is compactprojective and every object is a colimit of dualizable objects defines a framed Heisenberg-picture field theory Bord fr2 → Alg lax2 ( Cocomp K ). • Every K -linear monoidal locally presentable category in which the unit object is compactprojective and every object is a colimit of dualizable objects, which is additionally equipped EISENBERG-PICTURE QUANTUM FIELD THEORY 27 with a “pivotal” structure, defines an oriented Heisenberg-picture field theory
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