Post on 22-Dec-2021
Andrew Neitzke University of Texas at Austin
ABELIANIZATION INABELIANIZATION INCLASSICAL CHERN-SIMONSCLASSICAL CHERN-SIMONS
work in progress with Dan Freed
THE PLANTHE PLANBy abelianization in classical Chern-Simons I mean a relation between
classical Chern-Simons theory on classical Chern-Simons theory with defects on
๐บ โ๐ฟ๐ ๐๐บ โ๐ฟ1 ๏ฟฝฬ๏ฟฝ
where is an -fold branched cover of .๏ฟฝฬ๏ฟฝ ๐ ๐
It is a classical version of a proposal of Cecotti-Cordova-Vafa. Moregenerally, it is close to the circle of ideas around QFTs of "class R" and 3d-3dcorrespondence (Dimo๏ฟฝe, Gaiotto, Gukov, Terashima, Yamazaki, Kim, Gang,Romo, ...).
It is not the same as the abelianization of Chern-Simons studied by Beasley-Witten or Blau-Thompson.
CLASSICAL CLASSICAL CHERN-SIMONS INVARIANT CHERN-SIMONS INVARIANT๐บ๐บ โโ๐ฟ๐ฟ๐๐Given a compact spin -manifold , carrying a -connection , thereis a (level 1) classical Chern-Simons invariant (action):
3 ๐ ๐บ โ๐ฟ๐ โ
๐ถ๐(๐;โ) โ โร
If , , thenโ = ๐ + ๐ด ๐ด โ (๐; โ)ฮฉ1 ๐ค๐ฉ๐
๐ถ๐(๐;โ) = exp[ Tr (๐ด โง ๐๐ด + ๐ด โง ๐ด โง ๐ด)]1
4๐๐โซ๐
2
3
I will focus on the invariant for flat connections (critical points of theaction.)
CLASSICAL CLASSICAL CHERN-SIMONS LINE CHERN-SIMONS LINE๐บ๐บ โโ๐ฟ๐ฟ๐๐When is a manifold with boundary, is not quite a number,because the integral is not gauge invariant.
๐ ๐ถ๐(๐;โ)
Instead, it is an element of a line , determined by .๐ถ๐(โ๐;โ) โ|โ๐
If is a closed -manifold, we have a line for each over ;they fit together to Chern-Simons line bundle over the moduli space of flat
-connections over .
๐โฒ 2 ๐ถ๐( ;โ)๐โฒ โ ๐โฒ
๐บ โ๐ฟ๐ ๐โฒ
CLASSICAL CLASSICAL CHERN-SIMONS TFT CHERN-SIMONS TFT๐บ๐บ โโ๐ฟ๐ฟ๐๐The classical Chern-Simons invariant for a compact -manifold is the toppart of a -dimensional invertible spin TFT with symmetry:
3 ๐3 ๐บ โ๐ฟ๐
dim๐
3
2
โฏ
๐ถ๐(๐;โ)
โ โร
โ Lines
โฏ
We will focus just on the - part, given by a functor3 2
CS : โ LinesBord๐บ ,๐ ๐๐๐๐ฟ๐
I'm almost done reviewing, but let me recall one other fact about classicalChern-Simons theory, on -manifolds.
SHAPE PARAMETERSSHAPE PARAMETERS
3
SHAPE PARAMETERSSHAPE PARAMETERSSuppose is an ideally triangulated -manifold.๐ 3
[W. Thurston, Neumann-Zagier, ... for ][Dimo๏ฟฝe-Gabella-Goncharov, Garoufalidis-D. Thurston-Goerner-Zickert forall ]
Then (boundary-unipotent) flat -connections on can beconstructed by gluing, using
numbers per tetrahedron.
๐ โ๐ฟ๐ ๐
( โ ๐)1
6๐3
โ๐ โร
๐ = 2
๐
The have to obey algebraic equations ("gluing equations") over ,determined by combinatorics of๐ โค
๐
determined by combinatorics of .๐
SHAPE PARAMETERSSHAPE PARAMETERS
[W. Thurston]There is one case where the shape parameters have a well known geometricmeaning.
This is the case when is the connection induced by ahyperbolic structure on the ideally triangulated .
โ ๐๐๐ฟ(2,โ)๐
In this case, each tetrahedron of is isometric to an ideal tetrahedron inthe hyperbolic upper half-space.
The shape parameter is the cross-ratio of the ideal vertices lying on theboundary .
๐
4โโ
1
SHAPE PARAMETERSSHAPE PARAMETERSWhen the -connection has shape parameters , one has aformula of the sort
๐บ๐ฟ(๐,โ) โ ๐
๐ถ๐(๐;โ) = exp[ ( )]1
2๐๐โ๐
Li2 ๐
(Have to take care about the branch choices for .)Li2
[W. Thurston, Goncharov, Dupont, Neumann, ..., Garoufalidis-D. Thurston-Zickert]
SHAPE PARAMETERSSHAPE PARAMETERSOne of the aims of this talk is to explain a different geometric picture of theshape parameters , and the dilogarithmic formulas for Chern-Simonsinvariants.
๐
(Spoiler: the will turn out to be holonomies of a -connection.)
โ๐ โร ๐บ โ๐ฟ1
CHERN-SIMONS WITH DEFECTS CHERN-SIMONS WITH DEFECTS๐บ๐บ โโ๐ฟ๐ฟ11We consider Chern-Simons theory on spin manifolds carryingconnections , with two unconventional defects added.
๐บ โ๐ฟ1 ๏ฟฝฬ๏ฟฝโฬ
ie, a functor
: โ Lines๐ถ๏ฟฝฬ๏ฟฝ Bordฬ๐บ ,๐ ๐๐๐๐ฟ1
is a bordism category of spin manifolds , with -connections plus defects.Bordฬ๐บ ,๐ ๐๐๐๐ฟ1 ๏ฟฝฬ๏ฟฝ ๐บ โ๐ฟ1
CHERN-SIMONS WITH DEFECTS CHERN-SIMONS WITH DEFECTS๐บ๐บ โโ๐ฟ๐ฟ11The theory involves a codimension defect. This defect needs extra"framing" structure on its linking circle: marked points with arrows.
23
In expectation values, these codimension defects contribute a cube rootof unity for each twist of the framing, in the case where the arrows areconsistently oriented.
22๐3
CHERN-SIMONS WITH DEFECTS CHERN-SIMONS WITH DEFECTS๐บ๐บ โโ๐ฟ๐ฟ11The theory also has a codimension defect, which is a singularity of themanifold structure of : its link is a rather than .
3๏ฟฝฬ๏ฟฝ ๐2 ๐2
Each codimension defect has 4 codimension defects impinging.3 2
CHERN-SIMONS WITH DEFECTS CHERN-SIMONS WITH DEFECTS๐บ๐บ โโ๐ฟ๐ฟ11The connection does not extend over the codimension defect: it hasnontrivial holonomies over both cycles of the linking .
โฬ 3๐2
The holonomies obey a constraint:
(The signs depend on the spin structure.)
ยฑ ยฑ = 1๐๐ด ๐๐ต
ยฑ
CHERN-SIMONS WITH DEFECTS CHERN-SIMONS WITH DEFECTS๐บ๐บ โโ๐ฟ๐ฟ11For a surface , with spin structure and flat -connection ,
is the usual line of spin Chern-Simons theory.๏ฟฝฬ๏ฟฝ ๐บ โ๐ฟ1 โฬ
( ; )๐ถ๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝ โฬ
When codimension- defects impinge on , is the usual line ofspin Chern-Simons, tensored with an extra line for each defect, dependingonly on the "framing".
2 ๏ฟฝฬ๏ฟฝ ( ; )๐ถ๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝ โฬ
CHERN-SIMONS WITH DEFECTS CHERN-SIMONS WITH DEFECTS๐บ๐บ โโ๐ฟ๐ฟ11At each codimension defect we have a tiny torus boundary .3 ๐
To define the theory we need to specify an element
ฮจ โ (๐; )๐ถ๏ฟฝฬ๏ฟฝ โฬLuckily there is a natural candidate! Loosely
where is a variant of the Rogers dilogarithm,
ฮจ = ๐ exp( ๐ ( ))1
2๐๐๐๐ด
๐
๐ (๐ง) = (ยฑ๐ง) โ log(1 ยฑ ๐ง)Li21
2
THE DILOG IN THE DILOG IN CHERN-SIMONS CHERN-SIMONS๐บ๐บ โโ๐ฟ๐ฟ11The equation
ฮจ = ๐ exp( ๐ ( ))1
2๐๐๐๐ด
has two problems on its face:
The RHS is not a well defined function, because is a multivaluedfunction: requires a choice of branch.The LHS is not a well defined function, because it is an element of
: requires a choice of trivialization of the bundle underlying .
๐ (๐ง)
(๐; )๐ถ๏ฟฝฬ๏ฟฝ โฬ โฬ
These two problems cancel each other out; on both sides, we need tochoose logarithms of and , and then the transformation law of matches the WZW cocycle for .
ยฑ๐๐ด ยฑ๐๐ต ๐ (๐; )๐ถ๏ฟฝฬ๏ฟฝ โฬ
THE DILOG IN THE DILOG IN CHERN-SIMONS CHERN-SIMONS๐บ๐บ โโ๐ฟ๐ฟ11This is a fun interpretation of the dilogarithm function:
It is a section of the spin Chern-Simons line bundle over the moduli of flat -connections on the torus, restricted to the locus๐บ โ๐ฟ1
= {ยฑ ยฑ = 1} โ (๐๐ด ๐๐ต โร)2
(In fact, it is a flat section; this determines it up to overall normalization.)
THE DILOG IN THE DILOG IN CHERN-SIMONS CHERN-SIMONS๐บ๐บ โโ๐ฟ๐ฟ11From this point of view, -manifolds where is a union of tori relateto dilog identities.
3 ๏ฟฝฬ๏ฟฝ โ๏ฟฝฬ๏ฟฝ
e.g. to get the five-term identity up to a constant, contemplate for a link :
๐ = โ ๐ฟ๐3
๐ฟ
There exist flat connections on obeying at all torusboundaries. Spin Chern-Simons theory gives an element
hi h i b i f h dil id i
โฬ ๏ฟฝฬ๏ฟฝ + = 1๐๐ด ๐๐ต 5
๐ถ๐( ; ) โ ๐ถ๐( ; )๏ฟฝฬ๏ฟฝ โฬ โจ๐=1
5
๐๐ โฬ|๐๐
which is an abstract version of the dilog identity.POINT DEFECTS IN POINT DEFECTS IN CHERN-SIMONS CHERN-SIMONS๐บ๐บ โโ๐ฟ๐ฟ11
These defects have appeared before: in the open topological A model.
Suppose the -manifold is a Lagrangian submanifold of a Calabi-Yauthreefold .
We study the open topological A model on , with one D-brane on .
3 ๏ฟฝฬ๏ฟฝ๐
๐ ๏ฟฝฬ๏ฟฝ
[Witten]The open string field theory living on is Chern-Simons theory pluscorrections from holomorphic curves in ending on .
๏ฟฝฬ๏ฟฝ ๐บ โ๐ฟ1๐ ๏ฟฝฬ๏ฟฝ
POINT DEFECTS IN POINT DEFECTS IN CHERN-SIMONS CHERN-SIMONS๐บ๐บ โโ๐ฟ๐ฟ11
[Ooguri-Vafa]
The correction to the Chern-Simons action that comes from anisolated holomorphic disc is , where is the holonomy around theboundary.
๐บ โ๐ฟ1(๐)Li2 ๐
So our defects appear naturally in this context: they are the boundaries ofholomorphic discs (crushed to a point for technical convenience).
RELATING THE CHERN-SIMONS THEORIESRELATING THE CHERN-SIMONS THEORIESSo far I described two versions of classical Chern-Simons theory, given asfunctors
โ LinesBordฬ๐บ ,๐ ๐๐๐๐ฟ1
โ LinesBord๐บ ,๐ ๐๐๐๐ฟ๐
How are they related?
ABELIANIZED CONNECTIONSABELIANIZED CONNECTIONSThere is a new bordism category , fitting into a diagram:Abel
Abel
โ
โ
Bordฬ๐บ ,๐ ๐๐๐๐ฟ1
Bord๐บ ,๐ ๐๐๐๐ฟ๐
โ
โ
Lines
This diagram commutes, ie, the two functors are naturallyisomorphic.
Abel โ Lines
A morphism or object of is an abelianized connection.
This means a -connection over a manifold , with a partial gaugefixing where looks "as simple as possible".
Abel
๐บ โ๐ฟ๐ โ ๐โ
ABELIANIZED CONNECTIONSABELIANIZED CONNECTIONSIn the bulk of , parallel transports of are permutation-diagonal: theyreduce to transports of a -connection over a cover .
๐ โ๐บ โ๐ฟ1 โฬ ๏ฟฝฬ๏ฟฝ
Usually this cannot be done globally on .
(e.g. imagine = once-punctured torus: can't simultaneously diagonalizemonodromy on A and B cycles)
๐
๐
ABELIANIZED CONNECTIONSABELIANIZED CONNECTIONSWe introduce a stratification of (spectral network).๐
On codimension- strata (walls), we allow the gauge to jump by a unipotentmatrix.
1
On codimension- strata, we allow a singularity around which isbranched.
2 โ ๐๏ฟฝฬ๏ฟฝ
(Around this singularity, has holonomy , and pullback spin structuredoes not extend: need to make a twist of and the spin structure, tocancel this.)
โฬ โ1โค2 โฬ
ABELIANIZED CONNECTIONSABELIANIZED CONNECTIONSOn codimension- (points), we allow a singularity; its linking sphere lookslike:
3
There is no singularity of or here, but in the double cover and have a codimension- singularity.
๐ โ ๏ฟฝฬ๏ฟฝ โฬ3
ABELIANIZED CONNECTIONSABELIANIZED CONNECTIONSA "proof" of our equivalence, without boundaries:
When we have an abelianized connection , we can use its abelian gaugesto compute the Chern-Simons action.
โ
๐ถ๐(๐;โ) = exp[ Tr (๐ด โง ๐๐ด + ๐ด โง ๐ด โง ๐ด)]1
4๐๐โซ๐
2
3
In the bulk, just use
to reduce this to
Tr diag( , โฆ , ) = +โฏ +๐ผ1 ๐ผ๐ ๐ผ1 ๐ผ๐
๐ถ๐( ; ) = exp[ Tr (๐ผ โง ๐๐ผ)]๏ฟฝฬ๏ฟฝ โฬ1
4๐๐โซ๏ฟฝฬ๏ฟฝ
At the spectral network, this fails. Still, the walls do not contribute. Lowerstrata do contribute: they produce the codimension- and codimension-defects.
2 3
EXAMPLESEXAMPLES
Time for examples.
TRIANGULATED TRIANGULATED -MANIFOLDS-MANIFOLDS22Take a triangulated -manifold , punctured at all the vertices. We canequip it with a spectral network and a covering .
2 ๐โ ๐๏ฟฝฬ๏ฟฝ
[Fock-Goncharov, Gaiotto-Moore-N]
Using this network, a generic can be abelianized almost uniquely (infinitely many ways). The holonomies of around classes then give cluster coordinates determining .
โโฬ ๐พ โ ( ,โค)๐ป1 ๏ฟฝฬ๏ฟฝโ
TRIANGULATED TRIANGULATED -MANIFOLDS-MANIFOLDS33Now say is a triangulated -manifold.๐ 3
[Cecotti-Cordova-Vafa]Again there is a natural double cover and spectral network.โ ๐๏ฟฝฬ๏ฟฝ
The walls of the spectral network form the dual spine of the triangulation.
There is one codimension- defect in the center of each tetrahedron.3
TRIANGULATED TRIANGULATED -MANIFOLDS-MANIFOLDS33
We can reinterpret the construction of flat -connections on usingThurston's shape parameters obeying gluing equations.
๐ โ๐ฟ2 ๐๐
First, we construct a -connection over . The are itsholonomies.
๐บ โ๐ฟ1 โฬ ๏ฟฝฬ๏ฟฝ ๐
Then, we build the (unique) corresponding over .โ ๐
DILOGARITHM FORMULAS ON TRIANGULATED DILOGARITHM FORMULAS ON TRIANGULATED -MANIFOLDS-MANIFOLDS33
Our equivalence of Chern-Simons theories says
๐ถ๐(๐;โ) = ( ; )๐ถ๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝ โฬ
So, to get , we can compute in the theory on .๐ถ๐(๐;โ) ๐บ โ๐ฟ1 ๏ฟฝฬ๏ฟฝ
The line bundle underlying turns out to be globally trivial; choose atrivialization. Then,
the bulk of contributes trivially, ,
each codim- defect contributes a dilogarithm ,codim- defects can contribute third roots of .
โฬ
๏ฟฝฬ๏ฟฝ exp[ ๐ด โง ๐๐ด] = 114๐๐โซ๏ฟฝฬ๏ฟฝ
3 exp[ ๐ ( )]๐๐12๐๐ ๐๐
2 1
This recovers the dilogarithm formulas I reviewed before, for (actually a slight generalization: we don't need an "orderable"triangulation).
๐ โ๐ฟ2
ABELIANIZATION IN NATUREABELIANIZATION IN NATUREI explained that any time we have an abelianized connection we get anequivalence of Chern-Simons theories.
This statement is most interesting when we have abelianized connectionsarising in some natural way.
Natural examples are better understood in the -dimensional case than the -dimensional case, so let me start there.
23
ABELIANIZATION VIA WKBABELIANIZATION VIA WKBSuppose is a punctured Riemann surface.๐
On we can consider meromorphic Schrodinger equations, aka -opers, locally of the shape
๐ ๐๐ฟ2
[ โ ๐(๐ง)]๐(๐ง) = 0โ2๐ง โโ2
Also higher-order analogues like -opers,๐๐ฟ3
[ โ (๐ง) โ (๐ง) + (๐ง)]๐(๐ง) = 0โ3๐ง โโ2๐2 โ๐ง1
2โโ2๐โฒ2 โโ3๐3
These equations give flat connections over (with singularities atpunctures).
โโ ๐
ABELIANIZATION VIA WKBABELIANIZATION VIA WKB
[Voros, ..., Koike-Schafke]In the exact WKB method for Schrodinger operators, one studies opers bybuilding local WKB solutions, of the form:
โ
๐(๐ง) = exp[ ๐(โ) ๐๐ง]โโ1โซ๐ง
๐ง0
where has the WKB asymptotic expansion (for )๐(โ) Re โ > 0
๐(โ) = + โ + +โฏโ๐(๐ง) โ ๐1 โ2๐2
The local solutions reduce to a -connection over thespectral curve, a double cover of , branched at turning points:
๐(๐ง) โ ๐บ โ๐ฟ1 โฬ๐
= { + ๐(๐ง) = 0} โ ๐๏ฟฝฬ๏ฟฝ ๐ฆ2 ๐โ
ABELIANIZATION VIA WKBABELIANIZATION VIA WKBThese local solutions exist in domains of separated by Stokes curves;these form a spectral network.
๐
Crossing Stokes curves mixes the local solutions by unipotent changes ofbasis (WKB connection formula).
So, the structure that comes automatically from WKB analysis of an oper isthat of an abelianized connection.
ABELIANIZATION VIA WKBABELIANIZATION VIA WKBThe combinatorics of such Stokes graphs are generally rather complicated.
Except for , they are not just captured by ideal triangulations.๐บ = ๐ โ๐ฟ2
ABELIANIZATION IN CLASS ABELIANIZATION IN CLASS ๐๐A variation of this story appeared in quantum field theories of class .These are 4-dimensional theories, obtained by compactification ofsix-dimensional SCFTs on a punctured Riemann surface .
๐ = 2
(2, 0) ๐
Such a theory has a canonical surface defect preserving SUSY,whose space of couplings is .
= (2, 2)๐
[Gaiotto-Moore-N]
This defect has a flat connection in its vacuum bundle. (like ) Thisconnection is abelianized to a -connection over the Seiberg-Wittencurve . (UV-IR map). This is a powerful tool for studying the IR theory,BPS states.
โ ๐ก๐กโ
๐บ โ๐ฟ1 โฬ๏ฟฝฬ๏ฟฝ
ABELIANIZATION IN CLASS ABELIANIZATION IN CLASS ??๐ ๐
[Dimo๏ฟฝe-Gaiotto-Gukov, Cecotti-Cordova-Vafa]
There are -dimensional quantum field theories of class , associated to -manifolds instead of -manifolds.
3 ๐ 3๐ 2
One may hope that the abelianization of complex classical Chern-Simonswhich we have found has a natural interpretation here.
This could give a source of geometric examples of abelianizations over -manifolds.
3
CONCLUSIONSCONCLUSIONS
I described a relation between two versions of classical complex Chern-Simons theory:
the theory over ,the theory with defects over the branched cover .
๐บ โ๐ฟ๐ ๐๐บ โ๐ฟ1 ๏ฟฝฬ๏ฟฝ
This relation gives a new method of computing in the theory, andnaturally accounts for / extends some known facts about that theory.
๐บ โ๐ฟ๐
It still remains to find a good source of examples of a geometric nature,from class or elsewhere.๐