Surface Defects and the BPS Spectrum of 4d

Surface Defects and the BPS Spectrum of 4d

Surface Defects and the BPS Spectrum of 4d N=2 Theories Penn Strings-Math, June 11, 2011 Gregory Moore, Rutgers University Davide Gaiotto, G.M. , Andy Neitzke Wall-crossing in Coupled 2d-4d Systems: 1103.2598 Framed BPS States: 1006.0146 Wall-crossing, Hitchin Systems, and the WKB Approximation: 0907.3987 Four-dimensional wall-crossing via three-dimensional Field theory: 0807.4723

A Motivating Question Given an arbitrary four-dimensional field theory with N=2 supersymmetry, is there an algorithm for computing its BPS spectrum? A basic question, and a good litmus test to see how well we understand these theories We describe techniques which should lead to such an algorithm for

``Ak theories of class S Outline 1. Review of some N=2,d=4 theory 2. Theories of Class S a. 6d (2,0) and cod 2 defects b. Compactified on C c. Related Hitchin systems

d. BPS States and finite WKB networks 3. Line defects and framed BPS States 4. Surface defects a. UV and IR b. Canonical surface defects in class S c. 2d4d BPS + Framed BPS degeneracies 5. 2d/4d WCF 6. Algorithm for theories of class S 7. Overview of results on hyperkahler geometry

1 Preliminaries: 1/4 Basic Data: Special Kahler manifold , the ``Coulomb branch. Local system of lattices -> , equipped with integral

antisymmetric form. Preliminaries: 2/4 Charges of unbroken flavor symmetries Symplectic lattice of (elec,mag) charges of IR abelian gauge theory Self-dual IR abelian gauge field

Preliminaries:3/4 BPS States and Indices Seiberg-Witten Moduli Space Hyperkahler target space of 3d sigma model from compactification on R3 x S1

Seiberg & Witten The six-dimensional theories Claim, based on string theory constructions: There is a family of stable field theories, T[k] with six-dimensional (2,0) superconformal symmetry, and certain characterizing properties. They are not free field theories for k>1. (Witten; Strominger; Seiberg). These theories have not been constructed even by physical standards - but the characterizing properties of these hypothetical theories can be deduced

from their relation to string theory and M-theory. These properties will be treated as axiomatic. Later they should be theorems. Related free field theories with six-dimensional (2,0) superconformal symmetry can be rigorously constructed although this has not been done in full generality, and itself presents some nontrivial issues in topology and physics.

1. The theory is defined on oriented spin 6-folds with integral lift of w4 and is valued in a 7D topological field theory. 2. On R6 there is a moduli space of vacua (Rk x R5)/Sk 3. The low energy dynamics around points on this moduli space are described by abelian tensormultiplets valued in u(k). 4. In addition there are finite tension strings, charged under the abelian tensormultiplets, with charges given by roots of u(k) 5. There are so(5) multiplets of chiral BPS operators of dimensions given by the exponents of u(k)

6. There are supersymmetric defects labeled by (, ). In the IR, far out on the moduli space of vacua these are well-approximated by a sum over the weights of of the B-field holonomy + supersymmetrizing terms. 7. There are half-BPS codimension two defects with nontrivial global symmetries. They induce half-BPS boundary conditions on maximal SYM theory in 5 and 4 dimensions. 2

Theories of Class S Consider nonabelian (2,0) theory T[g] for ``gauge algebra g The theory has half-BPS codimension two defects D(m) Compactify on a Riemann surface C with D(ma) inserted at punctures za Witten, 1997 GMN, 2009

Gaiotto, 2009 Twist to preserve d=4,N=2 Seiberg-Witten = Hitchin 5D g SYM -Model:

Digression: Puncture Zoo Regular singular points: Irregular singular points: Physics depends on the closure of the Gc orbit of:

Seiberg-Witten Curve SW differential For g= su(k) is a k-fold branch cover Local System of Charges

determines A local system over a torsor for spaces of holomorphic differentials BPS States: Geometrical Picture BPS states come from open M2 branes stretching between sheets i and j. Here i,j, =1,, k. This leads to a nice geometrical picture with string networks:

Klemm, Lerche, Mayr, Vafa, Warner; Mikhailov; Mikhailov, Nekrasov, Sethi, Def: A WKB path on C is an integral path Generic WKB paths have both ends on singular points za Finite WKB Networks 1/3 But at critical values of =* ``finite WKB networks appear:

Hypermultiplet Finite WKB Networks 2/3 Closed WKB path Vectormultiplet Finite WKB Networks 3/3

At higher rank, we get string networks at critical values of : A ``finite WKB network is a union of WKB paths with endpoints on branchpoints or such junctions. These networks lift to closed cycles in and represent BPS states with 3

Line Defects & Framed BPS States A line defect (say along Rt x {0 } ) is of type preserves the susys: Example:

if it Framed BPS States saturate this bound, and have framed protected spin character: Piecewise constant in and u, but has wall-crossing across ``BPS walls (for () 0): Particle of charge binds to the line defect:

Similar to Denefs halo picture Wall-Crossing for Across W(h) Denefs halo picture leads to: constructed from Wall-Crossing for

Consistency of wall crossing of framed BPS states implies the Kontsevich-Soibelman ``motivic WCF for We simply compare the wall-crossing associated to two different paths relating (u1,1) and

6D theory T[g] has supersymmetric surface defects S(, ) For T[g,C,m] consider Line defect in 4d labeled by isotopy class of a closed path and

k=2: Drukker, Morrison, Okuda Complex Flat Connections (A, ) solve Hitchin equations iff is a complex flat connection on C

On R3 x S1 line defects become local operators in the 3d sigma model: = holomorphic function on M 4 Surface defects

Preserves d=2 (2,2) supersymmetry subalgebra Twisted chiral multiplet : IR DESCRIPTION Finite set of vacua Effective Solenoid

, ARE NOT QUANTIZED Torsor of Effective Superpotentials A choice of superpotential = a choice of gauge = a choice of flux i

Extends the central charge to a - torsor i Canonical Surface Defect in T[g,C,m] For z C we have a canonical surface defect Sz It can be obtained from an M2-brane ending at x1=x2=0 in R4 and z in C In the IR the different vacua for this M2-brane are the different sheets in the fiber of the SW curve over z.

Therefore the chiral ring of the 2d theory should be the same as the equation for the SW curve! Alday, Gaiotto, Gukov, Tachikawa, Verlinde; Gaiotto Example of SU(2) SW theory Chiral ring of the CP1 sigma model.

2d-4d instanton effects Twisted mass Gaiotto Superpotential for Sz in T[g,C,m] xi

xj z NEED TO JUSTIFY Homology of an open path on joining xi to xj in the fiber over Sz New BPS Degeneracies:

2D soliton degeneracies. Flux: For Sz in T[su(k),C,m], is a signed sum of open finite BPS networks ending at z New BPS Degeneracies: Flux:

Degeneracy: Supersymmetric Interfaces: 1/2 UV: IR: Flux:

Supersymmetric Interfaces: 2/2 Our interfaces preserve two susys of type and hence we can define framed BPS states and form: Susy interfaces for T[g,C,m]:1/2 Interfaces between Sz and Sz are labeled by open paths on C

So: framed BPS states are graded by open paths ij on with endpoints over z and z Susy interfaces for T[g,C,m]:2/2 Wrapping the interface on a circle in R3 x S1 compactification: Framed BPS Wall-Crossing

Across BPS W walls the framed BPS degeneracies undergo wall-crossing. Now there are also 2d halos which form across walls As before, consistency of the wall-crossing for the framed BPS degeneracies implies a general wall-crossing formula for unframed degeneracies and . Framed Wall-Crossing for

T[g,C,m] WE CAN ALSO STUDY WC IN z: The separating WKB paths of phase on C are the BPS walls for 5 Formal Statement of 2d/4d

WCF 1.Four pieces of data 2.Three definitions 3.Statement of the WCF 4.Relation to general KSWCF 5.Four basic examples 2d-4d WCF: Data A. Groupoid of vacua, V :

Objects = vacua of S: i = 1,, k & one distinguished object 0. Morphism spaces are torsors for , and the automorphism group of any object is isomorphic to : 2d-4d WCF: Data B. Central charge Z Hom(V, C) :

Here a, b are morphisms i ij ; valid when the composition of morphisms a and b, denoted a+b, is defined. C. BPS Data: D. Twisting function: &

when a+b is defined 2d-4d WCF: 3 Definitions A. A BPS ray is a ray in the complex plane: IF IF B. The twisted groupoid algebra C[V]:

2d-4d WCF: 3 Definitions C. Two automorphisms of C[V]: CV-like: KS-like: 2d-4d WCF: Statement Fix a convex sector:

The product is over the BPS rays in the sector, ordered by the phase of Z WCF: is constant as a function of Z, so long as no BPS line enters or leaves the sector Four ``types of 2d-4d WCF-A A. Two 2d central charges sweep past each other:

Cecotti-Vafa Four ``types of 2d-4d WCF - B B. Two 4d central charges sweep past each other: Four ``types of 2d-4d WCF - C C. A 2d and 4d central charge sweep past each other:

Four ``types of 2d-4d WCF - D D. Two 2d central charges sweep through a 4d charge: 6 A The Algorithm Fix a phase . On the UV curve C draw the separating WKB paths of phase : These

begin at the branch points but end at the singular points (for generic ) : Massive NemeschanskyMinahan E6 theory, realized as a trinion theory a la Gaiotto. B

Label the walls with the appropriate S factors these are easily deduced from wall-crossing. Now, when a ij-line intersects a jk-line, new lines are created. This is just the CV wall-crossing formula SSS = SSS. C: Iterate this process. Conjecture: It will terminate after a finite number

of steps (given a canonical structure near punctures). Call the resulting structure a ``minimal Swall network (MSWN) D: Now vary the phase . This determines the entire 2d spectrum for all The MSWN will change isotopy class precisely when

an S-wall sweeps past a K-wall in the - plane. Equivalently, when an (ij) S-wall collides with an (ij) branch point: E: Finally, use the 2d/4d WCF to determine the 4d BPS spectrum:

Concluding slogan for this talk The 2D spectrum controls the 4D spectrum. Spectrum Generator? Can we work with just one ? Perhaps yes, using the notion of a

``spectrum generator and ``omnipop This worked very well for T[su(2),C,m] to give an algorithm for computing the BPS spectrum of these theories. Very recently: We have made this work and have an algorithm for the BPS spectrum of Ak theories of class S. Hyperkahler Summary - A

1. Hyperkahler geometry: A system of holomorphic Darboux coordinates for SW moduli spaces can be constructed from a TBA-like integral equation, given . 2.

From these coordinates we can construct the HK metric on . 3. Hyperkahler Summary - B 4. 5.

For T[su(2),C,m], turn out to be closely related to FockGoncharov coordinates We are currently exploring how the coordinates for T[su(k),C,m] are related to the ``higher Teichmuller theory of Fock & Goncharov Hyperkahler Summary - C

6. For T[su(2),C,m] the analogous functions: associated to are sections of the universal bundle over , and allow us moreover to construct hyper-holomorphic connections on this bundle.

7. Explicit solutions to Hitchin systems (a generalization of the inverse scattering method)

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