algebra of the infrared
DESCRIPTION
Algebra of the Infrared. SCGP, October 15, 2013. Gregory Moore, Rutgers University. … work in progress …. collaboration with Davide Gaiotto & Edward Witten. Three Motivations . Two-dimensional N=2 Landau- Ginzburg models. . 2. Knot homology. . - PowerPoint PPT PresentationTRANSCRIPT
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Gregory Moore, Rutgers University
SCGP, October 15, 2013
collaboration with Davide Gaiotto & Edward Witten
…work in progress ….
Algebra of the Infrared
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Three Motivations
1. Two-dimensional N=2 Landau-Ginzburg models.
2. Knot homology.
3. Categorification of 2d/4d wall-crossing formula.
(A unification of the Cecotti-Vafa and Kontsevich-Soibelman formulae.)
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D=2, N=2 Landau-Ginzburg Theory
X: Kähler manifold W: X C Superpotential (A holomorphic Morse function)
Simple question:
Answer is not simple!
What is the space of BPS states on an interval ?
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Witten (2010) reformulated knot homology in terms of Morse complexes.
This formulation can be further refined to a problem in the categorification of Witten indices in certain LG models (Haydys 2010, Gaiotto-Witten 2011)
Gaiotto-Moore-Neitzke studied wall-crossing of BPS degeneracies in 4d gauge theories. This leads naturally to a study of Hitchin systems and Higgs bundles.
When adding surface defects one is naturally led to a “nonabelianization map” inverse to the usual abelianization map of Higgs bundle theory. A “categorification” of that map should lead to a categorification of the 2d/4d wall-crossing formula.
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5
OutlineIntroduction & Motivations
Web Constructions with Branes
Supersymmetric Interfaces
Summary & Outlook
Landau-Ginzburg Models & Morse Theory
Web Representations
Webs, Convolutions, and Homotopical Algebra
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Definition of a Plane Web
We show later how it emerges from LG field theory.
Vacuum data:
2. A set of weights
1. A finite set of ``vacua’’:
Definition: A plane web is a graph in R2, together with a labeling of faces by vacua so that across edges labels differ and if an edge is oriented so that i is on the left and j on the right then the edge is parallel to zij = zi – zj .
We begin with a purely mathematical construction.
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Useful intuition: We are joining together straight strings under a tension zij. At each vertex there is a no-force condition:
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Deformation TypeEquivalence under translation and stretching (but not rotating) of strings subject to no-force constraint defines deformation type.
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Moduli of webs with fixed deformation type
(zi in generic position)
Number of vertices, internal edges.
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Rigid, Taut, and Sliding
A rigid web has d(w) = 0. It has one vertex:
A taut web has d(w) = 1:
A sliding web has d(w) = 2
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Cyclic Fans of VacuaDefinition: A cyclic fan of vacua is a cyclically-ordered set
so that the rays are ordered clockwise
Local fan of vacua at a vertex v:
and at
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Convolution of Webs
Definition: Suppose w and w’ are two plane webs and v V(w) such that
The convolution of w and w’ , denoted w *v w’ is the deformation type where we glue in a copy of w’ into a small disk cut out around v.
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The Web RingFree abelian group generated by oriented deformation types of plane webs.
``oriented’’: Choose an orientation o(w) of Dred(w)
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The taut element
Definition: The taut element t is the sum of all taut webs with standard orientation
Theorem:
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Extension to the tensor algebra
• vanishes unless there is some ordering of the vi so that the fans match up.
• when the fans match up we take the appropriate convolution.
Define an operation by taking an unordered set {v1, … , vm} and an ordered set {w1,…, wm} and saying
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Convolution Identity on Tensor Algebra
satisfies L relations
Two-shuffles: Sh2(S)
This makes W into an L algebra
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Half-Plane Webs Same as plane webs, but they sit in a half-plane H.
Some vertices (but no edges) are allowed on the boundary.
Interior vertices
time-ordered boundary vertices.
deformation type, reduced moduli space, etc. ….
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Rigid Half-Plane Webs
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Taut Half-Plane Webs
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Sliding Half-Plane webs
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Half-Plane fansA half-plane fan is an ordered set of vacua,
are ordered clockwise:
such that successive vacuum weights:
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Convolutions for Half-Plane Webs
Free abelian group generated by oriented def. types of half-plane webs
There are now two convolutions:
Local half-plane fan at a boundary vertex v:
Half-plane fan at infinity:
We can now introduce a convolution at boundary vertices:
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Convolution Theorem
Define the half-plane taut element:
Theorem:
Proof: A sliding half-plane web can degenerate (in real codimension one) in two ways: Interior edges can collapse onto an interior vertex, or boundary edges can collapse onto a boundary vertex.
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Tensor Algebra Relations
Sum over ordered partitions:
Extend tH* to tensor algebra operator
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Conceptual Meaning
WH is an L module for the L algebra W
There is an L morphism from the L algebra W to the L algebra of the Hochschild cochain complex of WH
WH is an A algebra
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Strip-Webs
Now consider webs in the strip
Now taut and rigid strip-webs are the same, and have d(s)=0.
sliding strip-webs have d(s)=1.
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Convolution Identity for Strip t’s
Convolution theorem:
where for strip webs we denote time-concatenation by
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Conceptual Meaning
WS : Free abelian group generated by oriented def. types of strip webs.
+ … much more
W S is an A bimodule
There is a corresponding elaborate identity on tensor algebras …
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33
OutlineIntroduction & Motivations
Web Constructions with Branes
Supersymmetric Interfaces
Summary & Outlook
Landau-Ginzburg Models & Morse Theory
Web Representations
Webs, Convolutions, and Homotopical Algebra
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Web RepresentationsDefinition: A representation of webs is
a.) A choice of Z-graded Z-module Rij for every ordered pair ij of distinct vacua.
b.) A degree = -1 pairing
For every cyclic fan of vacua introduce a fan representation:
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Web Rep & Contraction
Given a rep of webs and a deformation type w we define the representation of w :
by applying the contraction K to the pairs Rij and Rji on each edge:
There is a natural contraction operator:
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L -algebras, again
Now,
Rep of the rigid webs.
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Half-Plane ContractionsA rep of a half-plane fan:
(u) now contracts
time ordered!
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The Vacuum A Category
Objects: i V.
Morphisms:
(For the positive half-plane H+ )
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Hint of a Relation to Wall-Crossing
The morphism spaces can be defined by a Cecotti-Vafa/Kontsevich-Soibelman-like product as follows:
Suppose V = { 1, …, K}. Introduce the elementary K x K matrices eij
phase ordered!
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Defining A Multiplications
Sum over cyclic fans:
Interior amplitude:
Satisfies the L ``Maurer-Cartan equation’’
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Proof of A Relations
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Hence we obtain the A relations for :
and the second line vanishes.
Defining an A category :
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Enhancing with CP-FactorsCP-Factors: Z-graded
module
Enhanced A category :
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Example: Composition of two morphisms
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Boundary AmplitudesA Boundary Amplitude B (defining a Brane) is a solution of the A MC:
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47
OutlineIntroduction & Motivations
Web Constructions with Branes
Supersymmetric Interfaces
Summary & Outlook
Landau-Ginzburg Models & Morse Theory
Web Representations
Webs, Convolutions, and Homotopical Algebra
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Constructions with Branes
Strip webs with Brane boundary conditions help answer the physics question at the beginning.
The Branes themselves are objects in an A category
Given a (suitable) continuous path of data we construct an invertible functor between Brane categories, only depending on the homotopy class of the path. (Parallel transport of Brane categories.)
(“Twisted complexes”: Analog of the derived category.)
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Convolution identity implies:
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Interfaces webs & Interfaces Given data
These behave like half-plane webs and we can define an Interface Amplitude to be a solution of the MC equation:
Introduce a notion of ``interface webs’’
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Composite webs Given data
Introduce a notion of ``composite webs’’
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Composition of Interfaces
Defines a family of A bifunctors:
Product is associative up to homotopyComposition of such bifunctors leads to categorified parallel transport
A convolution identity implies:
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53
OutlineIntroduction & Motivations
Web Constructions with Branes
Supersymmetric Interfaces
Summary & Outlook
Landau-Ginzburg Models & Morse Theory
Web Representations
Webs, Convolutions, and Homotopical Algebra
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Physical ``Theorem’’
Finitely many critical points with critical values in general position.
• Vacuum data. • Interior amplitudes. • Chan-Paton spaces and boundary amplitudes. • “Parallel transport” of Brane categories.
(X,): Kähler manifold (exact)
W: X C Holomorphic Morse function
Data
We construct an explicit realization of above:
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Vacuum data: Morse critical points i
Actually,
Connection to webs uses BPS states:
Semiclassically, they are solitonic particles.
Worldlines preserving “-supersymmetry”are solutions of the “-instanton equation”
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Now, we explain this more systematically …
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SQM & Morse Theory(Witten: 1982)
M: Riemannian; h: M R, Morse function
SQM:
MSW complex:
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1+1 LG Model as SQM
Target space for SQM:
Recover the standard 1+1 LG model with superpotential: Two –dimensional -susy algebra is manifest.
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Boundary conditions for
Boundaries at infinity:
Boundaries at finite distance: Preserve -susy:
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Lefshetz ThimblesStationary points of h are solutions to the differential equation
If D contains x -
The projection of solutions to the complex W plane sit along straight lines of slope
If D contains x +
Inverse image in X defines left and right Lefshetz thimbles
They are Lagrangian subvarieties of X
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Solitons For D=R
For general there is no solution.
But for a suitable phase there is a solution
Scale set by W
This is the classical soliton. There is one for each intersection (Cecotti & Vafa)
(in the fiber of a regular value)
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MSW Complex
(Taking some shortcuts here….)
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InstantonsInstanton equation
At short distance scales W is irrelevant and we have the usual holomorphic map equation.
At long distances the theory is almost trivial since it has a mass scale, and it is dominated by the vacua of W.
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Scale set by W
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The Boosted Soliton - 1
Therefore we produce a solution of the instanton equation with phase if
We are interested in the -instanton equation for a fixed generic
We can still use the soliton to produce a solution for phase
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The Boosted Soliton -2
Stationary soliton
Boosted soliton
These will define edges of webs…
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Path integral on a large disk
Consider a cyclic fan of vacua I = {i1, …, in}.
Consider the path integral on a large disk:
Choose boundary conditions preserving -supersymmetry:
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Ends of moduli space
This moduli space has several “ends” where solutions of the -instanton equation look like
Path integral localizes on moduli space of -instantons with these boundary conditions:
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Label the ends by webs w. Each end produces a wavefunction (w) associated to a web w.
The total wavefunction is Q-invariant
L identities on the interior amplitude
The wavefunctions (w) are themselves constructed by gluing together wavefunctions (r) associated with rigid webs r
Interior Amplitude From Path Integral
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Half-Line SolitonsClassical solitons on the right half-line are labeled by:
MSW complex:
Grading the complex: Assume X is CY and that we can find a logarithm:
Then the grading is by
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Scale set by W
Half-Plane Instantons
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The Morse Complex on R+ Gives Chan-Paton Factors
Now introduce Lagrangian boundary conditions L :
define boundary conditions for the -instanton equation:
Half-plane fan of solitons:
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Boundary Amplitude from Path Integral
Again Q=0 implies that counting solutions to the instanton equation constructs a boundary amplitude with CP spaces
Construct differential on the complex on the strip.
Construct objects in the category of Branes
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A Natural ConjectureFollowing constructions used in the Fukaya category, Paul Seidel constructed an A category FS[X,W] associated to a holomorphic Morse function W: X to C.
Tw[FS[X,W]] is meant to be the category of A-branes of the LG model.
But, we also think that Br[Vac[X,W]] is the category of A-branes of the LG model!
Tw[FS[X,W]] Br[Vac[X,W]]
So it is natural to conjecture an equivalence of A categories:
“ultraviolet” “infrared”
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Solitons On The Interval
The Witten index factorizes nicely:
But the differential
is too naïve !
Now consider the finite interval [xl, xr] with boundary conditions Ll, Lr
When the interval is much longer than the scale set by W the MSW complex is
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Instanton corrections to the naïve differential
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80
OutlineIntroduction & Motivations
Web Constructions with Branes
Supersymmetric Interfaces
Summary & Outlook
Landau-Ginzburg Models & Morse Theory
Web Representations
Webs, Convolutions, and Homotopical Algebra
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Families of Theories
Now consider a family of Morse functions
Let be a path in C connecting z1 to z2.
View it as a map z: [xl, xr] C with z(xl) = z1 and z(xr) = z2
C
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Domain Wall/Interface
From this construction it manifestly preserves two supersymmetries.
Using z(x) we can still formulate our SQM!
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Parallel Transport of Categories
To we associate an A functor
To a homotopy of 1 to 2 we associate an equivalence of A functors. ( Categorifies CVWCF.)
To a composition of paths we associate a composition of A functors:
(Relation to GMN: “Categorification of S-wall crossing”)
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84
OutlineIntroduction & Motivations
Web Constructions with Branes
Supersymmetric Interfaces
Summary & Outlook
Landau-Ginzburg Models & Morse Theory
Web Representations
Webs, Convolutions, and Homotopical Algebra
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Summary
1.We gave a viewpoint on instanton corrections in 1+1 dimensional LG models based on IR considerations.
2. This naturally leads to L and A structures.
3. As an application, one can construct the (nontrivial) differential which computes BPS states on the interval.
4. When there are families of LG superpotentials there is a notion of parallel transport of the A categories.
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Outlook1. Finish proofs of parallel transport statements.
2. Relation to S-matrix singularities?
4. Generalization to 2d4d systems: Categorification of the 2d4d WCF.
5. Computability of Witten’s approach to knot homology? Relation to other approaches to knot homology?
3. Are these examples of universal identities for massive 1+1 N=(2,2) QFT?