Some aspects of symmetry descent
Iñaki García Etxebarria Saghar S. Hosseini
 
Statistical Mechanics of Stochastic Quantum Control: d-adic Rényi Circuits
    Justin H. Wilson
    April 29, 2024
==========================================================================





§ INTRODUCTION



A complete definition of any given Quantum Field Theory requires
information both about local dynamics, and about its topological
sector: in general, the same set of local degrees of freedom can be
coupled to different topological sectors. From a modern point of view,
we consider any topological operator in the Quantum Field Theory a
symmetry (in a suitably generalised sense), so an equivalent
restatement of the previous remark is that theories with the same
local dynamics might have different sets of symmetries
<cit.>. These
choices of symmetries are often related in simple ways. For instance,
if we have a d-dimensional theory _d with a finite symmetry
group G,[Here we can allow for the possibility of
  group-like higher form symmetries, as in <cit.>.] we
can gauge[Perhaps after tensoring with a topological field
  theory with symmetry H. For simplicity of exposition, we will
  also, somewhat imprecisely, refer to this situation as “gauging”.]
an anomaly-free subgroup H of G and obtain a new theory _d/H,
with the same local dynamics but different symmetries. In general G
itself need not be anomaly-free, which can lead to some of the
symmetry generators in _d/H to be non-invertible, meaning
that for a given operator  there is no operator ^-1 in the
theory such that ^-1 is the identity operator.

This is far from an exotic possibility: during the last few years many
works have argued for the presence of such operators, and studied
their properties, in a large number of familiar theories. We refer the
reader to <cit.> for reviews. This process can be continued: _d/H
will have its own set of symmetries, and we can gauge a subset of
these, leading to a new theory, and so on. We refer to a choice of
representative in this set of related theories as a choice of global
form.

The discussion in the previous paragraph is somewhat unsatisfactory,
in that it started from a theory _d with “ordinary” group-like
symmetries, and we accessed more interesting symmetry structures by
sequences of gaugings. But in general, there is no requirement that a
canonical choice for _d exists, and in fact, it is possible that
none of the set of theories related by gauging operations has
group-like symmetries only. A better viewpoint is available
<cit.>: instead of considering each d-dimensional theory
with the same local dynamics as _d, we study a
(d+1)-dimensional topological field theory _d+1 (which we
call the symmetry theory, or SymTFT, as in
<cit.>) which admits a gapless boundary condition with
the same local dynamics as _d. We denote this gapless boundary
condition (which should be understood as a relative quantum field
theory <cit.>) encoding the local dynamics of _d as
Λ(_d).  All theories related by gauging of finite
symmetries lead to the same Λ(_d). This configuration in
itself should be thought of as a (d+1)-dimensional theory on a space
with boundary, but it can be turned into a definition for a
d-dimensional field theory with the same local dynamics as _d
if _d+1 admits a gapped interface ι_d to an anomaly
theory _d+1.[That is, an invertible field theory in
  d+1 dimensions encoding the anomalies of a (relative) QFT in
  d-dimensions.] Pictorially, we have:



    
    < g r a p h i c s >





We obtain a d-dimensional theory with anomaly _d+1 by
colliding ι_d with Λ(_d). Different choices of global
form for _d correspond to different choices of the pair
(_d+1,ι_d): some of the topological operators in
_d+1 will become trivial when we collapse ι_d and
Λ(_d), but some will survive as symmetries of the resulting
field theory. In this way, we have reformulated the problem of
understanding all theories related to _d and their symmetries
into two parts: first determining _d+1 given  (or
Λ(_d)) and then classifying the pairs (_d+1,ι_d)
that _d+1 can attach to.




Our focus in this paper will be on supersymmetric quantum field
theories obtained by placing string theory or M-theory at isolated
conical singularities of the form C(L), with L the base of the
cone. This class of configurations leads to superconformal field
theories living at the singular base of the cone. Generically these
superconformal theories do not admit any weakly coupled description,
so we have very poor knowledge of Λ(_d). Nevertheless, it
was argued in <cit.> (see also
<cit.> for related earlier works) that
_d+1 can be obtained (in the M-theory case) by performing a
reduction of the topological terms in the M-theory action on L,
which reduces the computation to a somewhat technical but fully
solvable problem in algebraic topology.[Ideally we would like
  to extract _d+1 from the geometry as a fully extended
  topological quantum field theory, but at the moment it is only known
  how to systematically extract the information described in the
  text.]

Our main goal in this paper is to clarify one issue in the analysis of
<cit.> that remained somewhat puzzling: while the
terms in _d+1 related to anomalies were computed via a
straightforward integration on L, there was also a BF sector,
schematically of the form

    S_BF(B,A)=  2π i N ∫ B ∧ dA    ,

that had to be computed used entirely different (and somewhat
indirect) methods. Our goal in this note is to bring the two
viewpoints closer together by explaining how this BF theory can be
derived by integration on L of an auxiliary theory in one dimension
higher.



Let us briefly review how <cit.> argues for the
existence of a BF sector in _d+1. The basic setup is
string theory or M-theory[For reasons that will become
  apparent, we are currently not able to satisfactorily apply our
  techniques to M-theory. We are hopeful that the difficulties will be
  surmountable, but we do not know how to do so at the moment.] on a
Calabi-Yau cone C(L) with Sasaki-Einstein base L times a manifold
^d:


    
    < g r a p h i c s >



In M-theory (L)=10-d and in type II string theory
(L)=9-d. We will assume that the singular locus of C(L)
consists of an isolated singularity at the tip of the cone, and that
^d is spin and torsion-free. There are light degrees of freedom
living at the singular locus of C(L), which in the examples studied
in <cit.> lead to a d-dimensional SCFT living on
^d. As argued in <cit.>, the information
given so far defines only the local data of the SCFT (namely, the
relative theory we have denoted by Λ(_d) in the
introduction) and the rest of the data necessary for fully defining a
SCFT _d are instead encoded in a choice of boundary condition at
infinity for the supergravity fields on ^d× C(L).

One of the main results of <cit.> is to reconcile this
picture, which is very natural from a string theory point of view,
with the SymTFT viewpoint described in the introduction. The basic
idea is that _d+1 arises from reducing the topological
sector of string theory over the base of the cone C(L):


    
    < g r a p h i c s >




In this proposal, we identify the tip of the cone with the gapless
boundary Λ(_d), and the boundary conditions at infinity
with the pair (_d+1,ι_d).[Although the details of correspondence need to be worked out,
  the implicit expectation in <cit.> is that
  the identification should go as follows. A choice of boundary
  conditions, given by the expectation value of a maximal isotropic
  subgroup of fluxes at infinity, as described in
  <cit.>, corresponds to a choice of
  ι_d. The partition function of the resulting string theory
  configuration is not well defined as a number, but it is rather a
  section of a line bundle over the space of boundary conditions over
  the boundary ^d× L. Restrict the supergravity fields to be
  flat asymptotically. A subset of such fields will be given by flat
  representatives of torsional cohomology groups on L times
  representatives of integral cohomology classes on ^d. Using the
  Künneth formula, which is an isomorphism in this case, these
  supergravity backgrounds can be canonically identified with
  backgrounds for the symmetries of _d on ^d. (We assume
  that there is no torsion in the cohomology of ^d here, see
  <cit.> for details.) The change in the
  partition function of string theory under gauge transformations
  within this subset of fields is precisely given by the anomaly
  theory _d+1.]

It was shown in <cit.> that a subsector of
_d+1 (leading to _d+1 for suitable choices of
boundary conditions) arises quite naturally from integrating the
Chern-Simons term in the M-theory action over L.

The BF sector in _d+1 is more subtle. For concreteness we
consider, as in <cit.>, the case of d=5 SCFTs
arising from putting M-theory on singular Calabi-Yau threefolds. As
studied in <cit.>, the local
dynamics for these SCFTs can be completed to theories with either
1-form or 2-form symmetries, and the boundary conditions for the BF
subsector of _6 determine which kind of symmetries we
have. The 1-form symmetries have generators coming from G_4 fluxes
and the 2-form symmetries from the G_7 fluxes (under the kind of
identification reviewed in footnote <ref>, and
explained in more detail in <cit.>.
These flux generators do not commute with one another <cit.>, and as a result, when choosing boundary conditions we
need to choose to which of G_4 or G_7 we give Dirichlet boundary
conditions, we cannot give Dirichlet boundary conditions to both. In
terms of the field theory we have that we cannot realise both the
1-form and 2-form symmetry at the same time. This hints at the fact
that if the BF action was to be obtained from a reduction of a
supergravity action, it should be one that contains both the G_4 and
G_7 fluxes. Ignoring some (crucial) complications from the presence
of the Chern-Simons term, the closest object is the kinetic term
∫ G_4 ∧ *G_4, if one observes that G_4 and G_7 fluxes
are dual to one another in M-theory. This expectation is borne out by
the analysis in <cit.>, which constructs the algebra
of topological operators in the BF theory from the reduction on L
of the algebra of topological operators in the M-theory background.

The discussion above explains why it is subtle to obtain the expected
BF theory from a dimensional reduction: to reproduce the
non-commutativity we would need an action that simultaneously
describes electric (G_4=dC_3) and magnetic (G_7=dC_6) fluxes. This
is closely related, by viewing the pair (C_3, C_6) as a
two-component field, to the famously difficult problem of writing an
action for a self-dual field.

There are multiple ways of approaching the problem of constructing
actions, or directly the partition function, for a self-dual field
<cit.>. In this paper, we approach this problem using a
strategy initiated by Witten <cit.>, in which we define
the partition function of a self-dual field on X in terms of
Chern-Simon theory on Y, with ∂ Y = X. Suitable choices of
boundary conditions for the Chern-Simons field (which we review in
detail below, following <cit.>) lead to the
emergence of the self-dual degrees of freedom on X. From this point
of view, our task decomposes into two (simpler) tasks: first we need
to reduce the Chern-Simons action on our internal manifold L, and
then we need to understand how the effective theory arising from
reduction on L behaves on a manifold with boundary. This is
summarised in figure <ref>.



The central result of our analysis is stated in a precise way in
eq. (<ref>) below. Intuitively, what that equation is
saying is that there is a form of inflow for the SymTFT:

    Δ∫_L _BF = δ_

where _BF and _ are suitable notions of
“Lagrangian densities” for the BF and SymTFT, Δ denotes
gauge transformations, and δ denotes the exterior
derivative. The formalism of Hopkins and Singer <cit.>
is very useful in making the notion of gauge transformations of
Lagrangian densities precise in topologically non-trivial
configurations, we review the basics in
section <ref>. A key advantage of this formalism is
that it allows us to keep control of topological aspects of the
problem in a way that is local. (In physics terms, we will be dealing
with Lagrangian densities, and not just actions.) This allows us to
make our formalism actually useful in practice: otherwise, since we
are viewing the SymTFT as arising from boundary modes, it would be
difficult to place it on spaces with boundary!

We expect that a modification of our approach (replacing ordinary
differential cohomology with twisted differential K-theory) will
allow us to recover not only the BF sector but also the anomaly
theory, at least in cases where the mathematical formalism is
understood well enough. We provide evidence for this expectation in
section <ref>.



A note on related recent literature: The papers
<cit.> appeared during the (fairly long) preparation of the
results we present here.[See also <cit.> for a
  recent analysis of the emergence of the BF theories from a
  different perspective.] They include a discussion of ideas related
to those in this paper for the derivation of the BF action, among
many other interesting results, and in particular they also start from
a theory in eleven dimensions (in the string theory case) and propose
that the SymTFT can be understood from dimensional reduction of the
eleven dimensional theory. Nevertheless, we believe that the analysis
in this paper is still useful, as it clarifies many of the subtleties,
both technical and conceptual, which we encountered in making this
picture precise (for example in giving a precise meaning to equations
like (<ref>)) and which were not addressed
in the works just mentioned. We will explain in detail which boundary
conditions one needs to take in the eleven dimensional theory for the
picture to work, how the edge modes — or in other words, the fields
in the SymTFT — relate to the dynamical fields in eleven dimensional
theory, and explain various subtle aspects of the mathematics involved
in the case of topologically non-trivial geometries.




§ DIFFERENTIAL COHOMOLOGY



Our starting point is abelian Chern-Simons theory at level k in
4n+3-dimensions. The action S for this theory can be written as
S=2π i _k[A], with (heuristically)

    _k[A] = k/2∫ A∧ dA  ,

where A is a 2n+1 form. The most familiar case is n=0, where the
self-dual field on the boundary is a self-dual
boson. Eq. (<ref>) is heuristic for three reasons.
First, we are using differential form notation for the field A, but
in the cases of interest to us A cannot be globally defined as a
differential form, but it is rather a connection on a topologically
non-trivial bundle. The second aspect of (<ref>) that
needs clarification is that, in fact, we will be interested mostly in
the case of A being a flat connection on a topologically
non-trivial bundle. (Our discussion is about IR effects, and non-flat
modes in the same topological sector encode massive excitations, which
we want to integrate out.) So, naively, dA=0. These two
complications can be dealt with by switching to the language of
differential cohomology. Below we will give a very brief review of the
main aspects of this formalism as they apply to our case. The last
subtlety in the interpretation of (<ref>) concerns the
quantisation of k. In general, this Chern-Simons theory makes sense
for arbitrary k∈, but for odd values of k there is an
implicit dependence on the Wu structure of the manifold. In the
(oriented) n=0 case the Wu structure reduces to the spin structure,
and we have the familiar statement that Chern-Simons theory at odd
values of k (or half-integral values, depending on conventions)
depends on the spin structure. We will address this final subtlety
below, after introducing some formalism we need for addressing the
first two points.

Since we will need to consider a refinement of connection A itself,
and not just the Chern-Simons action itself, we will adopt the
formalism of Hopkins–Singer <cit.>.[We will be
  working mostly with ordinary cohomology, where an equivalent cochain
  formalism was already introduced by Cheegers and Simons in
  <cit.>. The formalism of Hopkins and Singer
  allows for studying differential cochains in generalised cohomology
  theories, so with an eye towards generalisations we will refer to it
  as the Hopkins-Singer formalism.]  Our goal here is simply to set
notation and review the basic operational rules. We encourage the
reader to read <cit.> for
in-depth discussions.



Consider the cochain complex { C(n)^∙(),d} with

    C(n)^p()=
            
                C^p(;)× C^p-1(;)×Ω^p(;)    for   p≥ n 
    
                C^p(;)× C^p-1(;)    for   p<n

and differential

    d(c,h,ω)=     (δ c,ω-c-δ h,dω)  for    (c,h,ω)∈ C(n)^p()
    
            d(c,h)=   
                    (δ c,-c-δ h,0)    for   (c,h)∈ C(p)^p-1()
    
                    (δ c,-c-δ h)    otherwise.

We can alternatively think of elements of C(n)^p() for
p<n as triples (c, h, ω) with ω=0. We call elements of
C(n)^p() differential cochains, and define
differential cocycles as the closed differential cochains:

    Z(n)^p() { x∈ C(n)^p() | d x=0}  .


For notational convenience, we introduce maps I,  and R such
that for the differential cochain
a=(c,h,ω)∈ C(n)^p()

    (I( a),( a),R( a))=(c,h,ω)  ,

and define[This definition is a minor deviation from the one
  in <cit.>, but we find it slightly more convenient.]
C^p() C(p)^p(),
Z^p() Z(p)^p().

Finally, we note for future reference that the condition d a=0
for a cocycle a∈ Z(n)^p() implies that its
components satisfy δ( a) = R( a) - I( a).

The differential cohomology group H(n)^p() is then obtained
in the familiar way:

    H(n)^p()  Z(n)^p()/ d  C(n)^p-1()  .

The special case H(n)^n() coincides with the Cheeger-Simons
differential cohomology group H^n()
<cit.>. In particular, the Cheeger-Simons
differential character χ([ x])∈/ is given by

    χ([ x]) = ( x)  1  .

Here (and below) we denote elements of H^n() by
[ x], where x∈ Z(n)^n() is some
representative of the differential cohomology class, defined up to an
exact cocycle. 




The next ingredient we need is a notion of a product between
differential cochains, generalising the cup product in differential
cohomology. Given two differential cochains
a_1=(c_1,h_1,ω_1) and a_2=(c_2,h_2,ω_2)
their product a_1 · a_2 is a new cochain with
components

    (c_1 ∪ c_2,(-1)^|c_1| c_1
      ∪ h_2 + h_1 ∪ω_2 + B(ω_1, ω_2), ω_1
      ∧ω_2)   .

Here B is any natural
homotopy between ∧ and ∪

    δ(B(ω_1,
      ω_2))+B(dω_1, ω_2)+(-1)^|ω_1|B(ω_1,
      dω_2) =ω_1∧ω_2-ω_1∪ω_2  .

where we are promoting differential forms to cochains as needed. Note
that whenever ω_1=0 or ω_2=0 we can choose
B(ω_1,ω_2)=0 (since the right-hand side
of (<ref>) vanishes and this choice of homotopy is certainly
natural; or alternatively using the explicit expression given in
<cit.>). A straightforward computation shows that
for a ∈ C(p)^q(), b ∈ C(r)^s()
we have

    d( a · b) = (d a)· b + (-1)^q  a· d( b)  ,

In particular this implies that the product defined above induces a
product in differential cohomology classes, which is precisely the
product defined by Cheeger and Simons <cit.>. We
note that, for a and b cocycles, we have that
a· b and
(-1)^( a)( b) b· a are equivalent
up to gauge transformations.[This may be shown using the fact
  that cup product on cocyles is graded commutative up to the
  coboundary of cup-1 product together with the definition of
  B(ω_1,ω_2) given in <cit.> in terms
  of a sum of the cup product evaluated on subdivisions:
  B(ω_1, ω_2) - B(ω_2, ω_1) = -∑_i (ω_1
  ∪ω_2 - ω_2 ∪ω_1)(...).]



Given a fibre bundle with closed oriented fibres it is possible to
define a notion of integration along the fibre for differential
cochains <cit.>. In this note, we are only interested in
the case of trivial fibrations, namely spaces of the form
=B× F, where F is a n-dimensional, closed and
oriented.[For cohomology theories H we need F to be
  H-oriented <cit.>. In particular, if
  H is differential complex K-theory, we want F to admit a
  spin^c structure, which is always the case for any oriented
  three-manifold, the basic class of examples considered in this
  paper.] In this case, we define the integration map

    ∫_F  C(p)^p(B× F) → C(p-n)^p - n(B)

given by <cit.>

    ∫_F (c,h,ω) (c/F, h/F, ω/F )

where on the slant products <cit.> on the right
hand side we have abused notation (as we will keep doing in this
paper), and denoted by F∈ Z_n(F) the fundamental class of
F. Given a cochain c∈ C^n(B× F) and a chain v∈ C_p(F)
the (bilinear) slant product c/v ∈ C^n-p(B) satisfies
(c/v)(u)=c(u× v) for all u∈ C_n-p(B). A property of the
slant product that we will need later is

    δ(c/v) = (δ c)/v + (-1)^n-p c / (δ v)  ,

so the slant product on (co)chains descends to (co)homology.
Additionally, viewing ω as a differential form, ω/F
coincides with the usual notion of integration along F. Motivated by
this, we will sometimes abuse notation and write x/F as ∫_F x,
even when x is a cochain.



So far we have assumed that the fibre is closed. In case the fibre has
a boundary the discussion above still goes through, with some
modifications described in detail in <cit.>. A
particularly important result, in this case, is the following version
of Stokes' theorem for x ∈ C(p)^q():

    d∫_F  x = ∫_F d  x + (-1)^| x|-(F)∫_∂ F x  ,

which follows from a short computation
using (<ref>).  If F is closed then this
version of Stokes' theorem implies that integration descends to
differential cohomology

    ∫_F  H(p)^p(B× F) → H(p-n)^p-n(B)

in the obvious way.





Before we proceed any further, let us briefly discuss a few simple
examples that illuminate these techniques, and which play an important
role below. In all cases we take the base B to be a point, which we
denote by “”, so our total manifold is
× F≅ F.



  §.§.§ Example 1: U(1) theory in two dimensions


Consider first the case of an ordinary U(1) gauge 1-form connection
on a two dimensional manifold M. We take the action to be
2π i∫_M F (up to a small refinement we clarify
momentarily). This theory is trivial whenever M is closed, but it
can be useful when studying Wilson lines on ∂ M, for
instance.
We will find it useful to reformulate the action of this theory in the
language of differential cohomology as

    S_2d[ a] = 2π i (d∫_M a)  ,

with a∈ Z^2(M). Equivalently, since the action appears
in the form e^-S_2d[ a] in the path integral, we can
write, using (<ref>):

    1/2π iS_2d[ a] = χ([d∫_M  a])  1  .

Note that generically R(∫_M a)=∫_M R( a)≠ 0, so
despite the notation d∫_M a is not pure gauge, or
equivalently [d∫_M a]∈ H(1)^1() does
not necessarily vanish, and therefore the
holonomy (<ref>) is not necessarily trivial.



To see that this gives the action we are after, compute

    d∫_M a  = d  (I( a)/M, 0, R( a)/M) = (0, (R( a) - I( a))/M, 0)

where on the first equality we have used that h( a)/M=0 for
degree reasons, and similarly
δ I( a)/M = δ R( a)/M=0 on the second one. Since
I( a) is an integral cochain we find

    1/2π iS_2d[ a] = (d∫_M a) = (R( a) - I( a))/M = ∫_M F  1

as desired, where F R( a).

To see why (<ref>) is useful, take M to have
non-vanishing boundary ∂ M. By applying (<ref>) we
immediately get

    S_2d[ a] = 2π i (∫_∂ M a)  .

So we end up with the holonomy on the boundary, as expected. Note that
we have not yet quotiented by gauge transformations, so this holonomy
lives in . We can also obtain the same result starting
from (<ref>) and using the fact that a is a
cocycle (so d a=0), which then implies
R( a) - I( a) = δ( a),
so (<ref>) gives
S_2d[ a] = 2π i δ(
a)/M. Using (<ref>) and the fact that
( a)/M=0 for degree reasons, we get
S_2d[ a] = 2π i ( a)/∂ M, which agrees
with (<ref>).



Consider now gauge transformations, which act on a as
a→ a + dλ, with
λ∈ C(2)^1(M). Parameterising
λ=(c, f, 0), we have
dλ=(δ c, - c - δ f, 0), with f∈ C^0(M;)
and c∈ C^1(M;). This gauge transformation acts
on (<ref>) by

    d∫_M  a → d∫_M  a + d∫_M dλ =
      d∫_M a - ∫_∂ M dλ    = (0,((
      a)+δ f + c)/∂ M, 0)
        = (0,(( a) + c)/∂ M, 0)   .

We recognise the term δ f as the one generating small gauge
transformations. These do not affect the value of this integral (since
δ f/∂ M=δ(f/∂ M)=0 for degree reasons). On
the other hand, c is the part that generates large gauge
transformations on the boundary, and such transformations do change
the value of the action. Since c is an integral cochain, the change
is by integer multiples of 2π i, so it does not affect the
physics.



  §.§.§ Example 2: U(1)× U(1) theory in even dimensions


As a slightly more elaborate version of this example, consider the
case where B is still a point, which we denote by “”,
(F)=2n, and we want to define a theory with action

    S_k  2π i  k ∫_F F_1∧ F_2  ,

where F_1 and F_2 are field strengths of degree n for abelian
higher form fields. An important application of such an action is in
defining a discrete gauge _k theory <cit.> on
∂ F in terms of a bulk theory on F. (See also example 3
below.) As in the previous example, we reformulate this action in
differential cohomology as

    S_k = 2π i  k (d∫_F  a_1· a_2)  ,

where a_1, a_2∈ Z^n(F). The integral
of ( a_1 · a_2)∈ C^2n-1(F;) over F
vanishes for dimensional reasons and therefore

    ∫_F  a_1 · a_2 = (I( a_1 · a_2)/F, 0, R( a_2 · a_2)/F)
      ∈ C(0)^0() ≅×  .




Assume first that ∂ F=0. Then by Stokes'
formula (<ref>) the result of integration is actually closed
under d, which implies the familiar relation
I( a_1· a_2)/F = R( a_1 · a_2)/F.



More generally, if F has a boundary, (<ref>) gives
(similarly to the previous example)

    d∫_F  a_1 · a_2     = d(I( a_1 · a_2)/F, 0, R( a_2 · a_2)/F) 
        = (0, (R( a_1· a_2) - I( a_1· a_2))/F, 0)
        = (0, ( a_1· a_2)/∂ F, 0)  .

Coming back to (<ref>), we can rewrite it in more
familiar notation:

    ∫_F R( a_1)∧ R( a_2) - ∫_F I( A_1) ∪ I( a_2) = ∫_∂ F( a_1· a_2)  .

Assuming that I( a_1) is topologically trivial, then there is a
globally defined connection A_1∈Ω^1(F), and the equation can be written as

    ∫_F F_1∧ F_2 - ∫_F I( a_1) ∪ I( a_2) = ∫_∂ F A_1∧ F_2

where we have denoted the field strength R( a_i) by
F_i. Gauge transformations shift the right hand side and left hand
sides by (equal) integers, so this equation is often written as

    ∫_F F_1∧ F_2 = ∫_∂ F A_1∧ F_2  1  .

In our derivation below the more abstract
version (<ref>) of this equation will play a key
role.



It is an illuminating (and important for our later purposes) exercise
to check the variation of the integral (<ref>) under gauge
transformations of a_i. The latter are given by the ambiguity
of a_i by an exact cocycle d b_i∈ Z^n(F)
with b_i∈ C(n)^n-1(F).  Under these gauge
transformations a_i → a_i+d b_i and
(using (<ref>))

    a_1· a_2 → 
      ( a_1+d b_1)·( a_2+d b_2)
        =  a_1· a_2+d λ  ,

with
λ (-1)^n  a_1· b_2+ b_1·
a_2+(-1)^n d b_1· b_2. (There is some ambiguity in the
choice of λ, here we pick a representative that
simplifies some of the formulas later on.) Note that
R(λ)=0, since R( b_i)=0.  Thus, if
∂ F≠ 0, then by (<ref>) the gauge transformation
results in the boundary term

    ∫_∂ Fλ=(I(λ)/∂ F, 0, R(λ)/∂ F)

where again the integral (λ)/∂ F for
(λ)∈ C^2n-2(F;) vanishes for dimensional
reasons. In components, let a_i=(c_i,h_i,w_i),
b_i=(n_i,r_i,0) and so
d b_i=(δ n_i,-n_i-δ r_i,0). Then,

    I(λ)/∂ F    =I((-1)^n a_1· b_2+ b_1· a_2+(-1)^nd b_1· b_2)/∂ F
        =∫_∂ F
        (-1)^n c_1∪ n_2+n_1∪ c_2+(-1)^n δ n_1∪  n_2  ,

and R(λ)/ ∂ F=0. From here

    S_k → S_k + 2π i  k (d∫_Fdλ)     = S_k + 2π i  k (d∫_∂ Fλ) 
       = S_k - 2π i  k ∫_∂ F
      (-1)^n c_1∪ n_2+n_1∪ c_2+(-1)^n δ n_1∪ n_2  .

Assuming k∈ the variation is therefore an integer, and
e^-S_k is gauge invariant.



  §.§.§ Example 3: BF theory on a space with boundary


As our final example, we consider the theory with action
(schematically)

    S_k,BF = 2π i  k∫_F A_1∧ F_2  .

This action (which models a _k theory on F <cit.>)
is similar to the one of Chern-Simons, but we have two distinct
connections A_1 and A_2, with F_2 dA_2. It is well know that
whenever ∂ F=0 this action is gauge invariant (mod 1) and
whenever ∂ F≠ 0 the action is no longer gauge
invariant. Note that this is precisely the action that arose on the
boundary of our previous example, but we are now interested in taking
this theory and placing it on a manifold with boundary. Our interest
in this action is due to its relation to Maxwell theory: for k=1
such an action arises, for instance, as the action for the anomaly
theory for standard U(1) Maxwell theory <cit.>, where A_1 and A_2
are the background fields for the electric and magnetic 1-form
symmetries, respectively. The higher k cases arise in studying the
symmetry theory for different global forms of the U(1) theory, such
as those remaining on the IR of the Coulomb branch of =2 theories
<cit.>. We will now review how to reformulate this
action more precisely using the language of differential cohomology,
and how to see the non-gauge invariance of the action in this
language.



To formulate the BF action we promote A_1 and A_2 to
differential cohomology cocycles
a_1,  a_2∈ Z^n(F), with (F)=2n-1. It is
not difficult to repeat the analysis below for a_1 and
a_2 of generic degree (as long as they add to one more than
the dimension of F), but for the moment we focus on the case where
both differential cohomology classes are of the same degree. The
proper formulation of the BF action in differential cohomology is
then

    S_k,BF = 2π i  k (∫_F  a_1 · a_2) = k∫_F ( a_1· a_2)∈  .

Under gauge transformations a_i→ a_i+d b_i, we have

    S_k,BF→ S_k,BF + 2π i  k (∫_F dλ) = S_k,BF + 2π i  k (d∫_F λ - ∫_∂ Fλ)

with
λ (-1)^n  a_1· b_2+ b_1·
a_2+(-1)^nd b_1· b_2∈ C(2n)^2n-1 as in the
previous example. We have R(λ)=0, so

    (d∫_F λ) = (0, -I(λ)/F, 0) = -I(λ)/F ∈

so if k∈ we can drop the first term of the variation modulo
integers. We end up with

    S_k,BF→ S_k,BF - 2π i  k∫_∂ F(λ)  1  .

As promised, the action is gauge invariant (modulo 1) on closed
manifolds.

To see the gauge non-invariance on manifolds with boundary, we first
compute

    (λ)     = c_1∪ r_2 + (-1)^n-1 n_1∪ h_2 + r_1∪ω_2 + δ n_1∪ r_2 
        = c_1∪ r_2 + r_1∪ c_2 + δ n_1∪ r_2 + (-1)^n((d b_1)∪ h_2 + δ(r_1∪ h_2) )  ,

where in going to the second line we have used d a_2=0. When
integrating over ∂ F we can discard the total derivative
term, so we get the gauge variation

    S_k,BF→ S_k,BF - 2π i  k∫_∂ F c_1∪ r_2 + r_1∪ c_2
      +δ n_1∪ r_2 + (-1)^n((d b_1)∪ h_2
      ) 1  .

The fact that this gauge variation does not vanish will be crucial in
the examples we now turn to.



§ DISCRETE GAUGE THEORIES FROM COMPACTIFIED MAXWELL THEORY




A curious (and crucial for us) phenomenon in Maxwell theory, observed
in <cit.>, is that
the operators measuring electric and magnetic flux along torsional
cycles generically do not commute. A consequence of this fact is that
if we place U(1) Maxwell theory on a three-manifold with torsion,
such as the lens space S^3/_n, the resulting one dimensional
theory is non-trivially gapped: a _n discrete gauge theory
remains after integrating out the massive modes. The topological point
operators that remain are the dimensional reduction of the operators
measuring torsional flux in the S^3/_n factor. One dimensional
topological field theories are of course rather trivial, but the
techniques we introduce for analysing this case generalise
straightforwardly to higher dimensions, where the answer is more
interesting, so we find this simple example a useful starting point.

Consider the standard U(1) Maxwell theory on
X_t× (S^3/_n), where we consider _t the time
direction, and we work in the Hamiltonian picture (following
<cit.>). Electric and magnetic flux
measuring operators are labelled by elements of H^1(S^3/_n;),
with /. The short exact sequence

    1 → H^1(S^3/_n; )⊗→ H^1(S^3/_n; ) →
      H^2(S^3/_n; )→ 1

(from the universal coefficient theorem <cit.>) together with
H^∙(S^3/_n;)={, 0, _n, } implies
H^1(X;)≅ H^2(S^3/_n;)=_n. Accordingly, we
denote the electric flux measuring operators Φ_e(ζ) and the
magnetic flux measuring operators Φ_m(ξ), with
ζ,ξ∈ H^2(S^3/_n;). These operators satisfy the
commutation relation <cit.>

    Φ_e(ζ)Φ_m(ξ) = e^2π i (ζ, ξ)Φ_m(ξ)Φ_e(ζ)

with H^2(S^3/_n;)× H^2(S^3/_n;) →
the linking pairing. For a suitable choice of generator
t∈ H^2(S^3/_n;) we have (t,t) = 1/n  1, see for
example <cit.> for a computation. So, writing
ζ=pt, ξ=qt, with p,q∈_n, we have

    Φ_e(p)Φ_m(q) = e^2π i pq/nΦ_m(q)Φ_e(p)  .


When we reduce the theory on S^3/_n down to one dimension,
because there are no non-zero harmonic 1-forms in S^3/_n, there
are no massless modes arising from the KK reduction, since the mode
constant in the internal space leads to field in 1d which is pure
gauge (since both the characteristic class and curvature automatically
vanish in 1d for degree reasons). Therefore, the one dimensional
theory is gapped and one might be tempted to say that it is
trivial. This is not quite correct: the operators Φ_e(p) and
Φ_m(q) remain, with the commutation relation as above. They are
the non-trivial operators in a discrete _n gauge theory, which
admits a Lagrangian presentation with action
<cit.>

    S = 2π i n  (∫_M x · y)

where x,  y∈ Z^1(M) and M is the
one-dimensional manifold where we place the theory. Our goal in the
rest of this section is to reproduce this effective action in two
different ways, both of which prove useful when deriving the SymTFT
from a higher dimensional perspective.



 §.§ Derivation via self-dual formulations of Maxwell theory in four dimensions


The main obstruction to using standard ideas about dimensional
reduction is that the degrees of appearing in the one dimensional
action (<ref>) come from both electric and magnetic
degrees of freedom, so we would need to describe four dimensional
Maxwell theory via an action that includes both electric and magnetic
degrees of freedom. Such a description is in fact available
<cit.>,
and will lead to the correct result. Although this method will turn
out not to be quite sufficient for our purposes later in the paper, we
believe that it illuminates some non-trivial aspects of our later
general analysis, so we will briefly describe it first in the
differential cohomology formulation we are using in this paper.

The construction goes as follows. Standard Maxwell theory in Euclidean
signature is described by an action (we omit the possibility of a
θ term for simplicity)

    -S_g[ a] = ∫_^41/2g^2 R( a) ∧
      *R( a)  ,

with a ∈ Z^2(^4). In order to formulate this
expression in a way making both electric and magnetic degrees of
freedom manifest we will use, as in <cit.>, the
Hopkins-Singer reformulation of the self-dual actions in
<cit.>, also
known in the lattice field theory context as the (modified) Villain
formulation of the U(1) field
<cit.>. Our presentation is somewhat
unconventional, but it is motivated by the connection to the
Cheeger-Simons formulation later on. In this formulation, we consider
a gauge field c∈ Z^3(^4) which is pure gauge
c=d, with ∈ C(3)^2(^4). If
δ I()=0 then we can reconstruct an
a∈ Z^2(^4) by setting[At this point the
  fact that we are in the lattice is important: in the continuum we
  are taking the curvature to be a differential form, but in the
  lattice it is naturally a real cochain. We elaborate below on the
  relation between cochains and forms.]

    a()  (I(), (), I() + δ())  .

We can enforce the condition that I() is closed by
introducing a second gauge transformation parameter
∈ C(3)^2(^4) and modifying the action to be

    -S'[, ] = -S_g[ a()] + 2π i (∫_^4 d·) = -S_g[ a()] - 2π i∫_^4δ I() ∪()  .

Integration over () then implies the desired constraint,
and the theory clearly reduces to standard Maxwell theory. Note that
in this formulation there is no integration over I(), and
it does not appear anywhere in the action. As explained in detail in
<cit.> Poisson resummation on I()
implements S-duality, and leaves us with an equivalent path integral
over (), () and I(), which
we define to be the dual summation variable that arises in performing
Poisson resummation. The action in the dual path integral is

    -S_m[, ] = -S_4π^2/g[ b()] + 2π i∫_^4δ I() ∪() = -S_4π^2/g[ b()] + 2π i (∫_^4 d·)  ,

where b() is defined just as
in (<ref>). If we integrate out a in
this expression we end up with a copy of Maxwell theory with coupling
4π^2/g for the field b∈ Z^2(^4), which we
interpret as the magnetic dual field.

We now show how to use this viewpoint to
reproduce (<ref>) by compactification on a lens
space S^3/_n. We restrict to the case in which  and
 are flat, and of the form
= α· t,
= β· t, with
t∈ Z^2(S^3/_n) a flat representative of the
torsional generator of H^2(S^3/_n;)=_n, and
α,β∈ C^0(M).[For notational
  convenience we will leave implicit the pullbacks under the forgetful
  maps π_M M× S^3/_n→ M and
  π_S^3/_n M× S^3/_n→ S^3/_n.] Non-flat
choices for the S^3/_n part of  and  are of
course also present in the four-dimensional path integral, but they
lead to massive modes in the effective one dimensional theory, and we
are only interested in the behaviour at very low energies. We thus
restrict to t=(t, φ, 0). Here t is an arbitrary
cocycle representing the generator of H^2(S^3/_n;). Since
t is a cocycle, we have t = -δφ. Note that,
despite not explicitly indicating it in the notation, in this equation
we are promoting t to a cochain with  coefficients. This
promoted real cochain is trivial in cohomology, which is consistent
since t represents a torsional integral cohomology class.

Clearly S_g( a) = 0 due to flatness of t, or
equivalently S_4π^2/g( b)=0 in the magnetic formulation, so
the only relevant coupling in the IR is

    S_d[, ] = 2π i(∫_^4d·)  .

After a little bit of algebra this gives

    S_d[, ] = -2π i ∫_M× S^3/_nδα∪ t∪β∪φ = 2π i (∫_S^3/_n t∪φ)∫α∪δβ  .

The term in parenthesis is the linking pairing in S^3/_n, giving
(for some convenient choice of generator t, other choices lead to
k^2/n mod 1, where (k,n)=1)

    ∫_S^3/_n t∪φ = 1/n 1  .

Note also that due to the fact that α and β are
multiplied with t, which is n-torsional, in the path integral we
only need to sum over _n-valued α and β cochains, so
the effective path integral in 1d is

    ∫ [Dα][Dβ] exp(2π i/n∫_Mα∪δβ)

with α,β∈ C^0(M;_n). This is a well known alternative
presentation of the BF theory (<ref>), see for
example appendix B of <cit.> for a discussion.



 §.§ Derivation via BF theory in five
  dimensions


The previous formulation is quite useful for understanding the physics
of the problem, but it cannot be straightforwardly extended to the
case of a self-dual field, because in this case the initial action
analogous to (<ref>) vanishes identically. We now
rederive the results in the previous section from a different
viewpoint, namely BF theory in five dimensions.

In order to do this, first we need to explain how to construct four
dimensional Maxwell theory as an edge mode. We will do so following
(and slightly extending) the ideas
in <cit.>.[We encourage the
  readers to see also <cit.> for an application of these
  ideas to the physics of water.] The beautiful observation in
<cit.> was that self-dual fields arise as classical
solutions of Maxwell-Chern-Simons theory exponentially localised on
the boundary. The analysis below presents an essentially
straightforward extension of this argument to Maxwell-BF theory,
which leads to a self-dual formulation of Maxwell theory on the
boundary. Along the way we will elaborate on some aspects that were
left implicit in the analysis in <cit.>, in particular
how to relate it to the viewpoint in <cit.>. Other
important previous works that lead to an understanding of chiral
dynamics from a bulk Chern-Simons perspective include
<cit.>, where the boundary partition
function is characterised by its properties as a section of an
appropriate line bundle over the space of boundary fields.  There are
also many attempts to describe chiral fields without extending to an
extra dimension. However, in the latter approach, there are often
difficulties arising including non-Lorentz invariance or the need to
introduce non-polynomial dependence on an auxiliary scalar field - we
refer the reader to <cit.> for the relevant references.
We expect that it should be possible to reproduce our conclusions from
these alternative viewpoints (at least in those cases where they are
developed enough to account for non-trivial topology), but we have not
attempted to do so.



Consider the following bulk action on Y, with (Y)=2p+1 (our main
interest here is p=2):

    -S = 2π i ∫_Yi/2e^2 R( a_1)∧ * R( a_1) + i/2m^2 R( a_2)∧ * R( a_2) + k ( a_1· a_2)

where a_i ∈ Z^p+1(Y). In the limit
e^2,m^2→∞ we obtain a BF theory of the type studied above,
while if we set k=0 we obtain two decoupled copies of free p-form
Maxwell theory in 2p+1 dimensions.

Let us first work at the level of differential forms (we will extend
our analysis to topologically non-trivial differential cochains
below), by writing ( a_1) and
( a_2). More precisely, here and below, we will
assign differential forms to smooth cochains by using the Whitney map
<cit.>, having in mind the limiting case of an infinitely
refined triangulation. We refer the reader to <cit.> for a nice
explanation of this map, and an study of the convergence of the
various cochain constructions to their differential form analogues in
the limit. In terms of these differential
forms, (<ref>) becomes

    -S = 2π i ∫_Y i/2e^2 d∧ * d + i/2m^2 d∧ *d + k ∧ d  .

Under small variations →+δ and
→ + δ:

    -S    → -S + ∫_Y δ∧(2π/e^2(-1)^p d*d + 2π i k  d) + δ (-1)^p (2π/m^2 d*d - 2π i k   d)
       → - ∫_∂ Y2π/e^2δ∧ *d + 2π/m^2δ∧ *d - 2π i k δ∧  .

The bulk equations of motion are therefore

  
  
    i/e^2d* d - (-1)^p k  d    = 0  ,
    i/m^2d* d + k  d    = 0  .


or in terms of our original differential cochains

  

    i/e^2d(* R( a_1)) - (-1)^p k R( a_2)     = 0  ,
    i/m^2d(* R( a_2)) + k R( a_1)     = 0  .


These are the equations for massive fields, so far away from the
boundary at τ=0 we expect to recover the BF theory, which is
gapped. But we will now show (slightly extending <cit.>)
that these equations are also solved by massless fields localised on
the boundary obeying Maxwell's equations. In order to see how these
boundary modes arise, we parameterise a local neighbourhood of the
boundary by X× (-∞,0], with the coordinate τ
parameterising a small neighbourhood of the boundary. A sketch of the
geometry is given in figure <ref>.





Our first task is to choose boundary conditions so that the boundary
terms in (<ref>) vanish. We will impose

    (τ=0)=(τ=0)=0  ,

where we are still viewing  and  as differential forms on
Y. Our main motivation for imposing this boundary condition is to
connect with the discussion in <cit.>. We take an
ansatz for the connection near the boundary of the form

    = (1-e^ατ) F  ,
        = (1-e^ατ) G  ,

with F and G forms depending on X only. These solutions do not
vanish away from the boundary, but rather become F and G. The
equations of motion set F and G to be flat, which we can interpret
(keeping with our local point of view for the moment, so we can use
the Poincaré lemma) as demanding that  and  are pure gauge
in the near boundary region.

This ansatz is still localised on the boundary in the following way:
via a gauge transformation it can be turned into

    = d(e^ατ) ∧ A  ,

where F=dA, and similarly for . These connections do indeed
exponentially localise near the boundary. More meaningfully, the
curvature d does localise exponentially near the boundary,
assuming dF=0:

    d = -de^ατ∧ F  .

The connection (<ref>) was in fact the one
proposed in <cit.>; our
ansatz (<ref>) is gauge equivalent but it does
satisfy the boundary condition (<ref>), which is stronger
than the condition i^* = i^*=0 imposed
in <cit.>, where i∂ Y→ Y is the
inclusion of the boundary into the bulk. We emphasise that the
boundary condition (<ref>) is not gauge invariant: this is
desirable since it leads to an interpretation of the boundary modes as
gauge transformations in the near horizon region — a property
familiar from the holographic description of the Maxwell field given
in <cit.>. In the semiclassical theory this
identification of F with a gauge parameter also explains the
quantisation of the boundary U(1) connection, which is now
determined by the quantisation of the gauge transformations in the
bulk.

At any rate, our ansatz will satisfy the equations of
motion (<ref>) if

    d(*_X F)     = d(*_X G) = 0  ,
    _X F     = -i(-1)^p e^2k β/α^2 e^τ(β-α) G  ,
    _X G     = i m^2 k α/β^2 e^τ(α-β) F  .

Here *_X denotes Hodge duality on X. Consistency with
*_X(*_XF)=(-1)^p F requires m^2e^2k^2 = αβ, and the fact
that the ansatz has F,G constant in the τ direction requires
α=β. We see that as long as α,β>0 the ansatz
localises exponentially on the boundary as we increase e^2,m^2, and
the equations of motion for the boundary degrees of freedom are
precisely the Maxwell equations in vacuum, where if we identify F
with the electric field strength, G denotes the magnetic field
strength.



So far we have been working at the level of differential forms, but it
is important for our purposes to have a differential cochain
description, so that we can describe topologically non-trivial
backgrounds. We do so by promoting the
ansatz (<ref>) to the differential cocycles:

    a_1     = (-δ f, (1-e^ατ)F, -d(e^ατF)) ,
     a_2     = (-δ g, (1-e^ατ)G, -d(e^ατG))  .

Here F f+δλ, G g+δγ, with f,g
integral cochains and λ,γ real ones. We are no longer
imposing that F and G are closed, we allow them to have a
non-closed integral part (which is locally constant, so it does not
affect the analysis above). This is the right global generalisation of
the fact that F should be seen as a gauge parameter in the near
horizon region: neglecting the exponentially decaying terms we have

    a_1     = (-δ f, f+δλ, 0)  ,
     a_2     = (-δ g, g+δγ, 0)  ,

which are indeed of the form a_i=d  b_i, with b_i
gauge parameters of the form b_1 = (-f, -λ, 0) and
b_2 = (-g, -γ, 0).



A summary of the previous analysis is that, after imposing the
boundary conditions (<ref>), the equations of motion of
Maxwell-BF theory lead to localised modes on the boundary satisfying
Maxwell's equations, as encoded in (<ref>). In the
near boundary region, where the exponentially suppressed curvature is
negligible, the field strength F of the boundary Maxwell theory gets
reinterpreted as a pure gauge a_1.

This last observation allows us to connect with the picture in
<cit.>. In this
picture we send τ_β and τ_α to 0, in the notation
of figure <ref>. In terms of the couplings of the
Maxwell-BF theory we approach this limit by sending
e^2,m^2→∞. In the limit we recover pure BF theory in the
bulk. From the point of view of the bulk theory, the boundary
condition is now “screened” by the behaviour in the near boundary
region, where the a_i fields become pure gauge. That is, if we
choose to describe the bulk in terms of pure BF theory, forgetting
about the modes localised on the boundary, the boundary condition we
need to impose on the bulk fields as we approach the boundary is

    [ a_1]|_∂ Y = [ a_2]|_∂ Y = 0  .

Note that here we are only imposing that the differential cohomology
classes of a_1 and a_2 vanish, not the cocycles
themselves. The gauge transformations that connect different
representatives of the trivial cohomology class are not fixed by this
boundary condition, and as we have seen they furnish the Maxwell
degrees of freedom on the boundary. The fact that the degrees of
freedom localised on the boundary obey the Maxwell equations can be
encoded by adding to the bulk BF action a boundary term of the
form

    1/2e^2∫_∂ Y∧ *  .

This boundary action is clearly not gauge invariant, a remnant of the
fact that our original boundary conditions (<ref>) were
not local either. In the discussion below we find it useful to work in
this infinitely massive limit, so that the bulk is pure BF theory.




This might all sound somewhat exotic, but a context which might be
more familiar were something similar happens (with “boundary”
replaced by “defect”) is D-brane physics, where the
“gauge-invariant” field strength on a single D-brane is given by
F-B, with B the bulk NSNS two-form field. In this case we treat
F as a dynamical field, and B as the restriction of the bulk
field, subject to the gauge identifications F→ F+Λ,
B→ B - Λ with Λ an integrally quantised differential
form of degree two on the worldvolume of the brane. (Small gauge
transformations have Λ=dλ, with λ a 1-form
connection on a higher U(1) bundle.) Thanks to these gauge
transformations, we can trade off any choice of F by a choice of
gauge representative of B. The Maxwell action in this gauge is
precisely (<ref>). This bulk viewpoint on brane
degrees of freedom can sometimes be very useful, see for instance
<cit.>.

This point of view raises a potentially interesting connection with
recent work on non-invertible symmetries realised in string theory,
which we will only sketch: we have just argued that the worldvolume
theory on (abelian) branes can be understood in terms of the bulk
gauge transformations becoming physical. In the case of finite
symmetries, making a gauge field physical on a submanifold, or
equivalently “ungauging it”, is equivalent to gauging its dual field
on the submanifold. Gauging discrete fields on submanifolds is
precisely the operation known as “higher gauging”
<cit.> or “condensation”
<cit.>. So the discussion above suggests
that branes should encode condensation defects, at least in some
cases. This is indeed the case, as has been understood recently
<cit.>.

It is a natural question whether this understanding of branes in terms
of ungaugings can be made fully precise, and how far it goes. We will
not attempt to push it further in this paper, and only mention that a
very related idea has been recently advocated by Donagi and Wijnholt
for the case of non-abelian stacks in M-theory <cit.>.




  
Back to the SymTFT. We now have a description of Maxwell
theory in terms of BF theory on the bulk. Crucially, this
description treats electric and magnetic degrees of freedom on an
equal footing (up to a subtlety we will mention momentarily), so it is
a good candidate to make the emergence of non-commutativity between
electric and magnetic fluxes on the boundary manifest. As we argued
above, after dimensional reduction on the lens space S^3/_n this
non-commutativity effect should lead to a one-dimensional theory on
the boundary with action

    S = 2π i n ( ∫_M x· y)

with M the boundary manifold. This result follows from what we have
derived so far: we have identified the gauge fields on the boundary
with the gauge transformations b_i at the boundary, so to
understand the effective theory that we obtain after reducing the
theory on the boundary on S^3/_n we look at how the BF action
on a space with boundary changes under gauge transformations of the
form a_i→ a_i+d b_i, which was derived
in (<ref>) above. Our boundary
conditions (<ref>) require that a_1 and a_2
are always pure gauge near the boundary, so we might as well set the
initial values to zero in (<ref>) when studying
the effective action for the gauge transformations. We obtain

    S_bulk→ S_bulk + 2π i∫_∂ Yδ n_1 ∪ r_2

which is precisely the term appearing in the “almost democratic”
action (<ref>) for Maxwell theory (in
slightly different notation). The rest of the derivation now proceeds
identically to the discussion in the previous section.  Note that from
this point of view the perhaps somewhat unfamiliar “Villain”
characterisation of gauge fields in Maxwell theory as elements of
C(3)^2(^4) becomes perfectly natural, as this is precisely the
group where gauge transformations of the bulk fields live.



Note that in our discussion, it was natural to restrict to those gauge
transformations on the boundary that are constant in the integration
fibre. Since the change in the effective action depends only on the behaviour of the gauge transformation on the boundary, we are free to
choose the gauge transformation in the bulk as we wish. A convenient choice is to restrict ourselves to bulks that preserve the fibration
structure of the boundary, and to gauge transformations that are
constant along the fibre also in such bulks. If we make these choices,
then we can first integrate over the fibre in the bulk, and then study
the effect of induced gauge transformations on the resulting
theory. It is in this way that figure <ref> can
be made precise. Denoting by Ω the Chern-Simons
differential cocycle in the bulk with gauge transformation
Ω→Ω + ΔΩ, and
_ the Lagrangian (understood as a real cochain) for the
SymTFT arising after integration of the edge mode theory on L, a
precise mathematical formulation of figure <ref>
is therefore

    (∫_L ΔΩ) = δ_  .

This equation is quite reminiscent of the kind of equation that
appears when doing anomaly descent, but now it includes information
not only about the anomalies, but rather the full SymTFT. We will
expand on this point below.


As an example, let us go back to our working example of the BF
theory in five dimensions, with L=S^3/_n. The differential cocyle
representing the five dimensional BF theory (once we take its
holonomy) is

    Ω =  a_1· a_2  ,

with a_1, a_2∈ Z^3(Y). If we write
a_i=α_i· t_i, with
α_i ∈ C^1 and t_i flat representatives of
Z^2, as above, we find 

    ∫_S^3/_nΩ = (0, ℓ  I(α_1) ∪ I(α_2), 0)  1  .

Here ℓ∫_S^3/_n( t_1· t_2) is rational
number that agrees modulo one with the torsional linking pairing
between I( t_1) and I( t_2). For instance, if we choose
t_1= t_2 a flat uplift of a suitable generator of
H^2(S^3/_n;), we have ℓ=1/n mod 1. Now consider the case
in which a_i is pure gauge, namely
a_i= dλ_i, with
λ_i=β_i· t_i. Taking a_i to be
a pure gauge is equivalent to first performing the gauge
transformation a_i=  a_i+dλ_i and so
Ω→Ω + ΔΩ, and then setting
a_i=0 to be left with
ΔΩ=dλ_1· dλ_2. Therefore we can
write

    (∫_S^3/_nΔΩ) = δ(ℓβ_1∪δβ_2)  ,

and we conclude that (as expected)
_=ℓβ_1∪δβ_2.




 §.§ Inclusion of backgrounds


So far we have imposed boundary conditions
[ a_1]|_τ=0 = [ a_2]|_τ=0 = 0, which lead to
ordinary Maxwell theory on the boundary.

More generally, we can couple Maxwell theory to currents for the
electric and magnetic one-form symmetries. We do this by imposing

    [ a_1]|_τ=0 = [ j_e]      ;      [
      a_2]|_τ=0 = [ j_m]  ,

for fixed differential characters [ j_e], [ j_m]
satisfying I([ j_e]) = I([ j_m]) =0 (so that the partition
function does not vanish <cit.>). If we pick
representative cocycles j_e, j_m for [ j_e],
[ j_m], the sum over gauge transformations of a_i left
unfixed by the boundary condition becomes the path integral for
Maxwell theory in the presence of background currents j_e,
j_m for the 1-form symmetries of Maxwell. (Different choices
of representatives amount to a harmless redefinition of the Maxwell
fields in the path integral.) We refer the reader to
<cit.> for an illuminating discussion of Maxwell theory
in this formalism.

In detail, this goes as follows. Since I([ a_i])|_τ=0=0,
up to gauge transformations we can represent a_i by
differential forms B̃_i∈Ω^2(X), so that
a_i = (0, B̃_i, dB̃_i) <cit.>. So, up to gauge
transformations, there exist cochains B_i  (0,0,B̃_i) such
that a_i|_τ=0=d B_i. We interpret the differential
forms B̃_i as the background fields for the higher form symmetry. The
fields B̃_i are defined only up to gauge transformations, and in
particular the quotient by large gauge transformations makes the
physical information live in Ω^2(X)/Ω^2_(X), where
Ω^2_(X) indicates the integrally quantised differential
forms on X.

Putting the gauge transformations back into place, when doing the path
integral described above we end up with an action where the gauge
parameters F, F^D∈ C(3)^2(X) appearing the original
Maxwell kinetic term now appear shifted by the background fields. It
is customary to denote these shifted fields by
F -  B_e,
^D F^D -  B_m.

We emphasise that , ^D are not gauge parameters in
general. (Although the difference of two such fields is a gauge
parameter.) In other words, they cannot always be interpreted as
differential cocycles describing an ordinary U(1) bundle. As a
simple example, if we want to couple the Maxwell theory on the
boundary to a flat (in addition to topologically trivial)
background electric current
B^e∈Ω_closed^2(X)/Ω^2_(X) and no magnetic
current, we can take j_e = d(0,0, B̃^e)=(0, B̃^e,0)
and j_m=0, where B̃^e∈Ω^2_closed is a
differential form representative for the Wilson line B^e. For
generic choices of B^e we have that j_e is not pure gauge
(since the holonomies of a pure gauge field are integrally valued,
while the holonomies of B^e are not).

There is of course no requirement that the backgrounds B_i are flat,
only that they are topologically trivial. Whenever dB_i≠ 0, the
bulk theory that we have described can be understood as a dynamical
version of the familiar anomaly theory for the U(1)^e× U(1)^m
1-form symmetry in Maxwell theory.



§ EXAMPLE: THE BF SECTOR OF THE SYMTFT FOR THE 6D (1,1)
  THEORY



As an example of the previous discussion, we rework from our viewpoint
the BF sector of the (1,1) theories in six dimensions
<cit.>, describing the structure of discrete
1-form and 3-form symmetries of the theory. The 2-form symmetries in
6d (2,0) and (1,0) theories
<cit.> can be
analysed similarly, by studying the reduction of a Chern-Simons theory
with boundary mode the self-dual F_5 field in type IIB string
theory. For the 6d (1,1) theory constructed by putting IIA string
theory on ^2/Γ, with Γ a discrete subgroup of
SU(2), the relevant BF theory is a discrete gauge theory for the
group Γ^abΓ/[Γ,Γ], the
abelianisation of Γ. For instance, for Γ=_n we have
Γ^ab = _n. (The rest of the cases are listed in
many references, see for example <cit.>.) In the
Γ=_n case we have

    S_[X^7] = 2π i/n∫_X^7 c_2∪δ c_4

with c_i∈ C^i(X^7; _n) for i=2,4. In what follows we will
assume that X^7 has no torsion, for simplicity. From the arguments
in <cit.> it is clear that the background
fields c_2 and c_4 arise respectively from F_4 and F_6 fields
in IIA on S^3/_n× X^7. A full treatment requires that we
think of these fields in terms of K-theory. We will make more comments about this point in section <ref>.


Just as in previous examples, the type IIA fields can arise as
boundary gauge degrees of freedom of bulk fields in 11d
on S^3/_n× Y^8 with boundary
∂ (S^3/_n× Y^8)=S^3/_n× X^7. So
starting with the bulk fields as differential cocycles
a_j∈ Z^j(S^3/_n× Y^8) for j=i+3=5,7, the
action governing the dynamics of these cocycles is

    S_BF = (∫ a_5· a_7)  .

To see the boundary modes, we follow an analogous discussion to section <ref>. That is, we add kinetic terms for a_j to the above action and impose the boundary condition 

    [ a_j]|_∂=0  .



The 7d symmetry theory results from the gauge non-invariance of the
action (<ref>) on the boundary after dimensional reduction on
S^3/_n.  This gauge non-invariance arises under the gauge
transformations a_j → a_j+d b_j-1. Note the
action (<ref>) is very similar to the BF action
(<ref>) with the difference of having cocycles of two
different degrees. Thus, its gauge transformation
S_BF→ S_BF+ Δ S_BF, similarly to
(<ref>), is given by

    Δ S_BF = -(∫_∂ a_5· b_6+ b_4· a_7+ (-1)^|d b_4|d b_4· b_6) 1  .


Imposing the boundary condition (<ref>) by setting a_j=0 on the boundary, we expect the reduction of that latter action on S^3/_n to result in the 7d SymTFT, that is explicitly:

    S_[X^7] = Δ S_BF|_ a_j=0= 2π i ∫_X_7(∫_S^3/_n d b_4· b_6)   1  .

To show this, on the boundary we may write
b_4= β_2· t_1 and
b_6= β_4· t_2 with
β_i=(c̃_i,r,0). Substituting the latter in the
right-hand side of (<ref>) we find

    Δ S_BF|_ a_j=0   =
        2π i (∫_S^3/_n t_1· t_2)
        I(∫_X_7 dβ_2·β_4 )
         1
       = 2π i/n∫_X_7δc̃_2∪c̃_4 1
          .

To see the consistency with (<ref>), note that the fields 
c_i∈ C^i(X^7; _n) and c̃_i∈ C^i(X^7; ) 
are related by c_i=c̃_i  n, and so 

    δ c_2 ∪ c_4/n =δc̃_2∪c̃_4/n 1  .

Alternatively, we may represent this action in terms of U(1) fields
c_i^U(1)∈ C^i(X^7; U(1)), where c_i^U(1) = c_i/n,
and write

    S_[X^7] = 2π in∫_X^7 c^U(1)_2∪δ c^U(1)_4  .




§ QUADRATIC REFINEMENTS



So far we have been somewhat cavalier about the factor of k/2 in
front of the Chern-Simons term (<ref>). The
Chern-Simons theory relevant for describing the self-dual fields in string theory is the one at k=1, so we need to slightly
refine the discussion above. This can be done in terms of
quadratic refinements. Consider the symmetric bilinear pairing

    B_k( a,  b) = k ( ∫_Y  a· b)  1 ∈
      H^1() = /

with a,  b∈ H^2n+2(Y) and (Y)=4n+3. We
say that q_k H^2n+2(Y)→/ is a quadratic
refinement of B_k if

    B_k( a,  b) = q_k( a +  b) - q_k( a) -
      q_k( b) + q_k(0)  .

We include the q_k(0) term since it can be important when dealing
with the purely gravitational sector, but in our examples below we can
take q_k(0)=0. It is clear that whenever k is even
q_k^even  k( a)k/2 (∫_Y a· a)
is indeed a quadratic refinement of B_k, but the quadratic
refinement is more fundamental, as it also makes sense for odd k.

A way of constructing quadratic refinements was given in
<cit.>, which we now briefly
summarise. We start by introducing a differential Wu cocycle
λ=(c,h,w) ∈ Z^2n+2(E) which is such that
c=ν_2n+2 2, with [ν_2n+2] the degree-(2n+2) Wu
class. (In the special case of l=1 a choice of Wu cocycle is
equivalent to a choice of spin structure <cit.>.) We
then define the quadratic refinement of the action to be
<cit.>[In fact, we need to add
  k/8∫ (λ·λ -  L)
  to (<ref>), where L is the Hirzebruch polynomial
  <cit.>. Such a term will not affect the simple
  examples we discuss, so we omit it for conciseness, but it will be
  required in the analysis of more complicated examples.]

    𝖢𝖲_λ[ a]= k/2 (  ∫_Y   a · ( a-λ) ) 1  .

The case of most interest to us will be 4n+3=11. We assume that we
want to define our Chern-Simons term on L× M, where L is the
internal (closed) compactification space. If we assume that M is
also closed, then the manifold L× M admits an spin extension to
12 dimensions (since the 11 dimensional spin bordism group
vanishes). In this case we can view the 11d Wu class
[I(λ_6)] as a restriction of the 12-dimensional Wu
class. Luckily, the 12-dimensional Wu class is zero mod 2 for Spin
manifolds, as pointed out in <cit.>.[The Adém
  relation 1 4=5 and 2 4=6+ 5 1 give
  6=2 4+1 4 1. So for any x∈ H^6(Y;_2)
  
    ν_6∪ x=6(x)=2 (4(x))+1 (4(1(x)))  ,

  which vanishes on Spin manifolds. This is because 2 and 1,
  as maps to the top-dimensional degree, are given by multiplication
  by the second and first Wu classes, respectively, and these both
  vanish on Spin manifolds.] Thus, its integral lift and its
restriction to 11d can be taken to be trivial:
[I(λ_6)]=0 2. So in 11 dimensions, we may choose
λ to be 0, or more generally twice some differential
cocycle C. The case of interest to us is when
∂ M≠ 0, so the discussion above needs to be modified to
incorporate manifolds with corners. We will not attempt to do this in
this paper, although given that the reasoning above is mostly local
(apart from the bordism argument), it seems natural to conjecture that
λ = 0 is also a valid choice in this case.

The dependence of the Chern-Simons term on the differential
Wu cocycle is given by the formula <cit.>

    𝖢𝖲_λ-2 b[ a]=𝖢𝖲_λ[ a+ b]  .

Thus, for λ=0 we have (from (<ref>))

    𝖢𝖲_-2 b[ a]=𝖢𝖲[ a+ b]
        = k (∫_Y a· b) 
        +𝖢𝖲[ a]+𝖢𝖲[ b] 1
          .

The last equality is the statement that the Chern-Simons is a
quadratic refinement of the bilinear pairing
B( a, b)= k (∫_Y a·
  b) 1.





§ COUPLINGS




Our goal in this paper is to give a unified prescription for deriving
the SymTFT, but so far we have only discussed how to derive the BF
sector (or more generally, abelian Chern-Simons sectors). We will now
present some evidence that suggests that in favourable cases the
anomalies can also be naturally incorporated if one represents the
type II RR fields as twisted differential K-theory cocycles, and
generalises the bulk Chern-Simons theory they come from to an
H-twisted differential K-theory version of Chern-Simons theory. That
this works is not surprising: in <cit.> the anomaly
terms appeared from integrating certain topological terms in the
string theory action on the horizon of the singularity, and the
analysis of Belov and Moore <cit.> shows that the
relevant terms in the string theory action appear from the
differential K-theory version of the Chern-Simons action in eleven
dimensions. We will argue in particular that our inflow
prescription (<ref>) also reproduces the anomaly
terms.

Our analysis in this section is preliminary in two important
respects. First, we will be approximating differential K-theory by
ordinary differential cohomology, keeping track of the extra couplings
induced in the Chern-Simons action by the H-twisting. This is a
technical restriction in order to avoid introducing additional
technology, but it would certainly be interesting to do a proper
differential K-theory treatment and remove this approximation. More
importantly, we can only derive anomaly terms in this way if we know
the right geometric formulation of the “bulk” Chern-Simons theory
with our desired modes as edge modes. So at this moment we cannot do
M-theory,[Following <cit.>, a candidate for
  extending our approach to M-theory would be to study J-twisted
  cohomotopy Chern-Simons on manifolds with boundary. We have not
  explored this possibility.] or configurations that require a
formulation where H is also dynamical (for example if both the
electric and magnetic NSNS fields, B_2 and B_6, play roles in the
analysis), as H-twisted K-theory seems to be no longer a good
approximation. We refer the reader to <cit.> for a
discussion of the puzzles that arise when trying to model this more
general situation.

We emphasise that our approach and the standard anomaly inflow picture
deal with anomalies very differently: in standard anomaly inflow, one
has a Chern-Simons action S_anomaly() in
(d+1)-dimensions, where the B are background fields for the
symmetries of the d-dimensional theory. There is also a
(d+2)-dimensional action S_I(), the anomaly polynomial,
given by S_I()=δ S_anomaly( B). In all
these actions the fields  are classical. On the other hand,
in our context we have the SymTFT S_(), depending
on dynamical fields , and the (d+2)-dimensional theory is
also a dynamical (and conjecturally K-theoretical) Chern-Simons theory
S_inflow() for dynamical fields . As we
have argued above, the relation between  and  is
that  are the gauge parameters for , and
Δ S_inflow()=δ S_().



As an example, consider IIA string theory on a ^2/_n
singularity, as in section <ref>. If we choose the
SU(n) global form this theory has a _n 1-form symmetry,
measuring the n-ality of Wilson line insertions. In the string
theory construction, the Wilson lines arise from D2 branes wrapping
non-compact 2-cycles on the ^2/_n geometry, and their
remaining direction is the worldline of the Wilson line in the field
theory. We, therefore, identify the background field for the _n
1-form symmetry of this theory with F_4
<cit.>. There is also an instanton U(1)
1-form symmetry. The background for this symmetry is a degree 3
cocycle G, with connection B. There is a mixed anomaly
between the 1-form _n symmetry and the 1-form U(1) instanton
symmetry, of the schematic form

    S_anomaly = 2π i∫_X^7 dB ∪ n_inst(C_2)  ,

where C_2∈ Z^2(X^7;_n) is a bulk extension of the background
for the 1-form symmetry, and n_inst(C_2)∈/ is the
fractional part of the instanton number of the gauge bundle in the
presence of the 1-form background C_2. We will not need its precise
expression, but it can be found for example in <cit.>.
We will now show that there is a similar term in the SymTFT for the
system, with C_2 now taken to be a dynamical _n-valued
2-cochain denoted by c_2. The standard anomaly is then reproduced
when we take Dirichlet boundary conditions for c_2 on the gapped
boundary of the SymTFT. We will keep B, coming from the NSNS 2-form
field B_2, as a classical field throughout, due to the limitation of
the formalism mentioned above.



This term can indeed be argued to be present as follows. The RR fields
in IIA string theory can be modelled in terms of a self-dual RR field
F∈ K^-1, H() in H-twisted
differential K-theory <cit.>,[ For material on
  differential K-theory, we refer the reader to <cit.>,
  and <cit.> as well as the closely
  related formulations in <cit.>. Alternative descriptions of differential
  K-theory are given in <cit.>.] where H∈ H^3() is the
NS field with field strength H R( H). Similar to the
differential cohomology construction we had before, we may realise
F as the boundary mode of an 11-dimensional Chern-Simons
action on  for a field which is an element of the differential
K-theory group K^0, H(), with ∂=. We start by considering the topologically trivial case, so we
take c∈Ω^even() to be

    c=c_0+c_2+c_4+c_6+c_8+c_10  ,

a sum of differential forms of even degree, and define d_H=d-H to be
the DeRham differential twisted by H. Then, following the work of
Belov and Moore <cit.> (to which we refer
the reader for more details, see also <cit.>), we take the K-theoretical Chern-Simons action to
be

    𝖢𝖲_B[c] = 1/2∫_ c∧ (d_H^*c^*)  ,

where c_2k^*=(-1)^kc_2k and H^*=-H. One may reproduce the
supergravity equations by adding a kinetic term to this Chern-Simons
and substituting a suitable ansatz into the equation of motion.

It is important to note at this point that the full Chern-Simons is
given by a quadratic refinement, as in
section <ref>, that includes several other
terms.[See for instance <cit.> for concrete
  expressions for this quadratic refinement in differential
  K-theory.] In particular, it includes an eta invariant term which
in differential K-theory is the only term that encodes information
about the topological sector. As we have seen it is this topological
data that gives rise to a non-trivial SymTFT. Thus, to do the
computation in twisted differential K-theory, we must understand how
the eta invariant transforms under the gauge transformation of the
Chern-Simons field, and then we may apply the same general formula
(<ref>). However, to simplify the analysis we do not do
this, and we will instead recover some of the topological data by
refining (<ref>) to ordinary differential cohomology. As we
will see, doing so correctly reproduces the anomalies which we have
seen to arise from the string theory topological Chern-Simons action
as studied in <cit.>, which supports our expectation
that the proper K-theory formulation will also give the right result.






To refine the Chern-Simons (<ref>) to differential cohomology, let us first rewrite it as 

    𝖢𝖲_B[c] = 1/2∫_ c∧ dc^*+ c∧ H∧ c^*  .

As usual, we uplift the field c to the differential cohomology field a = a_1+ a_3+  a_5+ a_7+  a_9+ a_11 with a_i∈ Z^i(). Then, exactly as in the case of the Chern-Simons theory we saw before, the first term uplifts to

    1/2∫_(  a · a^*)  .

The second term is harder to make globally well defined. A naive
uplift would send of the fields c to a, which results in a
differential cocycle of degree 12. This suggests further extend 
to a 12-dimensional manifold  with boundary ∂=. A
simple choice, given that = S^3/_n× Y^8 in our case, is
to take = ^2/_n× Y^8, with
^2/_n a compactification of ^2/_n.

We should note that defining the extension  goes against our
philosophy of working locally, and makes the analysis not immediately
applicable to the (most interesting case) where  itself has a
boundary. The need to extend to 12 dimensions is indeed a drawback of
working in differential cohomology instead of differential
K-theory. However, in the spirit of testing whether our
proposal (<ref>) can in principle accommodate
anomalies, we will proceed with this preliminary analysis and leave
the refinement for future work. Then, the second term uplifts to

    -∫_(  a· H· a^*)  .

Thus, using (<ref>), we write the refined Chern-Simons action as

    𝖢𝖲_B[ a] = ( 
         1/2  d ∫_ a · a^*- ∫_ a· H· a^*)  .

The anomaly term (<ref>) results from the term

    𝖢𝖲_B[ a_5] = ∫_(  H· a_5· a_5)  .

More specifically in the symmetry inflow picture, we have

    δ( S_inflow) =
       (∫_^2/_nΔ( H· a_5· a_5)))   .

To show this, we perform the gauge transformations 
a_5 → a_5+d b_4 and find

    Δ( H· a_5· a_5)
        =  H · ( a_5· a_5+dλ)
        =  H· a_5· a_5-
        d( H ·λ)
          ,

similarly to (<ref>), with
λ (-1)^n  a_5· b_4+ b_4·
a_5+(-1)^n d b_4· b_4.  Then, imposing the boundary
condition [ a_5]|_∂=0, we find the gauge
transformed action to
be

    -∫_∂( H· d b_4· b_4) 1  .

Taking ∂=S^3/_n× Y^8, let's expand
b_4=β_2· t_1, with
β =(c̃_̃2̃,r,0), as in section
<ref>. Also, expand H=  G· 1,
with G a differential 3-cocycle on X^7 extending to Y^8
and 1 a representative of the differential uplift of the
top-dimensional cohomology class on S^3/_n. Substituting the
above expansions in action (<ref>) gives

    -∫_∂(  G· dβ_2· t_1·β_2· t_1)
           
        =-∫_S^3/_n h( t_1· t_1) ∫_Y^8 I(  G· dβ_2·β_2) 1
       = 1/2∫_S^3/_n h( t_1· t_1) ∫_X^7 I( G) ∪c̃_2∪c̃_2 1   .

To compare with (<ref>), we may identify 

    n_ins(c_2)=ℓ c̃_2∪c̃_2 1  ,

where

    ℓ1/2∫_S^3/_n h( t_1· t_1)  1  ,

and c_2∈ C^2(X^7; _n).



We thank Enrico Andriolo, Gabriel Arenas-Henriquez, Ibrahima Bah,
Federico Bonetti, Michele Del Zotto, Nabil Iqbal, Javier Magán, Sakura
Schäfer-Nameki, Tin Sulejmanpasic, Yuji Tachikawa, David Tong and
Kazuya Yonekura for helpful discussions.  I.G.E. is partially
supported by STFC grant ST/T000708/1 and by the Simons Foundation
collaboration grant 888990 on Global Categorical Symmetries. S.S.H. is
supported by WPI Initiative,
MEXT, Japan at Kavli IPMU, the University of Tokyo.



  
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