Line Defect Schur Indices, Verlinde Algebrasand Up1qr fixed points
Fei Yan

University of Texas at Austin
Strings 2018
Okinawa Institute of Science and Technology Graduate University

Joint work with Andrew Neitzke
arXiv: 1708.05323

June 25, 2018
Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

1/9

Outlook

An interesting feature of the line defect Schur index in 4d N “ 2 theories([Dimofte-Gaiotto-Gukov][Córdova-Gaiotto-Shao]):

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

2/9

Outlook

An interesting feature of the line defect Schur index in 4d N “ 2 theories([Dimofte-Gaiotto-Gukov][Córdova-Gaiotto-Shao]):
There is a close relation between
Schur index in the presence of a supersymmetric (half) line defect L,
the vevs xLy in Up1qr -invariant vacua of the theory compactified on
S 1.

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

2/9

Schur Index and Chiral Algebra

4d N “ 2 SCFT T ÝÑ 2d chiral algebra A(Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees)
Local operators contributing to the Schur index correspond tooperators in the vacuum module of A.
Ipq, ai q “ χ0 pq, ai q.

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

3/9

Schur Index and Chiral Algebra

4d N “ 2 SCFT T ÝÑ 2d chiral algebra A(Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees)
Local operators contributing to the Schur index correspond tooperators in the vacuum module of A.
Ipq, ai q “ χ0 pq, ai q.

What about characters of non-vacuum modules of A?

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 20183/9

Line Defect Schur Index

Consider supersymmetric half line defect L:extends along a ray in R4 and terminates at a point.
The endpoint support local opeators Ñ count via the line defect
Schur index ([Dimofte-Gaiotto-Gukov],[Córdova-Gaiotto-Shao]):ź fIL pq, ai q “ TrH1 p´1qF q ∆´R
ai i .
i

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 20184/9

Line Defect Schur Index and Verlinde Algebra
In various examples Córdova-Gaiotto-Shao found that:
IL pqq “ÿvL,β pqqχβ pqq,

ββ runs over a finite collection of modules closed under modular S action.

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

5/9

Line Defect Schur Index and Verlinde Algebra
In various examples Córdova-Gaiotto-Shao found that:IL pqq “ÿvL,β pqqχβ pqq,

ββ runs over a finite collection of modules closed under modular S action.
We can use the Verlinde formula to define a commutative associativealgebra V generated by rΦβ s, withγrΦα s ˆ rΦβ s “ cαβrΦγ s.

To every line defect L one could associate f pLq “řvL,β pq “ 1qrΦβ s P V.

βFei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

5/9

Line Defect Schur Index and Verlinde Algebra
In various examples Córdova-Gaiotto-Shao found that:IL pqq “ÿvL,β pqqχβ pqq,

ββ runs over a finite collection of modules closed under modular S action.
We can use the Verlinde formula to define a commutative associativealgebra V generated by rΦβ s, withγrΦα s ˆ rΦβ s “ cαβrΦγ s.

To every line defect L one could associate f pLq “řvL,β pq “ 1qrΦβ s P V.

βf : L Ñ V, where L is the commutative OPE algebra of line defects.
There is evidence that f is a homomorphism.
f is very forgetful.
Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

5/9

Up1qr -fixed Locus in 3d

Compactifying a 4d N “ 2 theory on S 1 , the Coulomb branch of thecompactified theory is hyperkähler [Seiberg-Witten].
For theories of class Srg, C s, get moduli space Mpg, C q of solutions toHitchin’s equations on C with gauge algebra g [Gaiotto-Moore-Neitzke].
Up1qr symmetry Ñ Hitchin action on Mpg, C q [Hitchin].
For Argyres-Douglas theories, the Up1qr -fixed locus consists of finitely
many isolated points. Moreover they are in 1-to-1 correspondencewith certain highest-weight modules of A.
[Fredrickson-Neitzke],[Fredrickson-Pei-W.Yan-Ye]

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

6/9

Up1qr -fixed Locus in 3d
How are the Up1qr fixed points related to V and L?

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

7/9

Up1qr -fixed Locus in 3d
How are the Up1qr fixed points related to V and L?
Denote OpF q as the algebra of functions on the set of Up1qr -fixed
points F . The modular S operator canonically diagonalizes theVerlinde algebra V. There is a canonical isomorphismh : V Ñ OpF q.
Consider the vacuum expectation values of line defects wrappedaround S 1 , these vevs are functions on the Hitchin moduli space.
Restricting these vevs to F gives a homomorphism
g : L Ñ OpF q.
g is also very forgetful.
Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

7/9

A Commutative DiagramfLVhgOpF q
L: OPE algebra of line defects
V: Verlinde-like algebra associated to the 2d chiral algebra A
OpF q: Algebra of functions on Up1qr -fixed locus F .

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 2018

8/9

Thank You!

Fei Yan (UT Austin)

Strings 2018, OIST

June 25, 20189/9