Michele CiraficiCAMGSD & LARSyS, IST,

Lisbon

DEFECTS IN COHOMOLOGICAL GAUGE THEORY AND DONALDSON-

THOMAS INVARIANTS

CAMGSD@LARSyS

CAMGSD@LARSyS

based on arXiv: 1302.7297

SISSA - VBAC 2013

OUTLINE

Introduction

Defects

DT as a gauge theory

Defects and DT

An example

Conclusions

INTRODUCTION

SETTING THE STAGE

Quantum theory and geometry are deeply related

Use quantum theory to compute interesting invariants: Seiberg-Witten, Gromov-Witten, Donaldson-Thomas

Quantum theory can be modified to include defects

Broad program: new moduli spaces, new enumerative invariants. Are they interesting? useful?

Today: divisor defects in DT from gauge theory perspective

DEFECTS

DEFECTS

Defects are a fundamental modification of a quantum field theory

The simplest way to think about a defects is as a “boundary condition” (on a line, a surface...)

In physics the presence of a defect opens up a new BPS sector

Mathematically this means that we have new moduli problems

[Gaiotto-Moore-Neitzke]

SURFACE DEFECTS

For example Surface Defects are useful for certain aspects of Geometric Langlands

Consider a gauge theory: is a principal G-bundle with connection (gauge field)

We define a defect on a surface . Locally we assume . The defect is

[Gukov-Witten][Kronheimer-Mrowka]

The theory is “ramified”. parametrizes the defect

⌃

local coordinate on Poincaré dual

E �! M4

A

M4 ' ⌃⇥K

FA = 2⇡↵ �⌃ + · · ·

↵ 2 TG

z = r e i ✓ K

SURFACE DEFECTS WITH INSTANTONS

One can set up an instanton problem for gauge theory with surface defects

The moduli space of “ramified” instantons is defined via the ramified instanton equations

[Kronheimer-Mrowka]

The defect has a “topological angle”

Instanton counting on the affine space (non compact defect):

[L,↵] = 0

M0k,m

(F 0A)

+ = (FA � 2⇡↵ �⌃)+ = 0

Tr ⌘m =1

2⇡

Z

⌃Tr ⌘ FA

Z(✏1, ✏2,a;L, q, ⌘) =X

m

X

k2ZeTr ⌘m qk

I

M0k,m

1 .

[Alday-Tachikawa][Feigin-Finkelberg-Negut-Rybnikov]

GAUGE THEORY AND DT

DT FROM GAUGE THEORY

Donaldson-Thomas program: understand higher dim geometry via gauge theory (in particular CY)

Our approach is to reformulate DT theory as an instanton counting problem

Define an appropriate 6d Cohomological Gauge Theory and a “generalized instanton moduli space”

Explicit results: enumerative invariant for generic toric CY, NCDT at toric singularities, motivic invariants, cluster algebras.

[C., Sinkovics, Szabo][Nekrasov] [Nakajima]

INSTANTON COUNTING

The gauge theory is a cohomological version of 6d Yang-Mills. Contains a gauge connection on a G-bundle + other stuff. (really, coherent sheaf)

It is equivalent to study the generalized instanton moduli space of solutions of

3.2 Cohomological Yang-Mills theory in six dimensions

The problem of studying Donaldson-Thomas invariants on X is essentially a higher dimensionalinstanton problem. The associated topological gauge theory is a topological version of six dimen-sional Yang-Mills. The most economical way of thinking about this theory is via dimensionalreduction of super Yang-Mills in ten dimension. After the reduction the six dimensional fieldsare a connection A on the G-bundle E �! X, and the ad E valued complex one form Higgsfield � and the forms ⇢(3,0) and ⇢(0,3). The fermionic sector is twisted, that is the fermions canbe though of as di↵erential forms thanks to the identification between the spin bundle and thebundle of di↵erential forms

S(X) ' ⌦0,•(X) , (3.5)

which holds on any Kahler manifold. Overall the fermionic sector comprises sixteen degrees offreedom which are organized into a complex scalar ⌘, one forms 1,0 and 0,1, two forms �2,0

and �0,2 and three forms 3,0 and 0,3. The bosonic part of the action is

S =1

2

Z

X

Tr⇣

dA

� ^ ⇤dA

�+⇥

� , �⇤

2

+�

�F (0,2)

A

+ @ †A

⇢�

�

2

+�

�F (1,1)

A

�

�

2

⌘

+1

2

1

(2⇡)2

Z

X

Tr⇣

FA

^ FA

^ t+ �

3·2⇡ FA

^ FA

^ FA

⌘

, (3.6)

where dA

= d + A is the gauge-covariant derivative, ⇤ is the Hodge operator with respect tothe Kahler metric of X, F

A

= dA + A ^ A is the gauge field strength. Furthermore � is acoupling constant which in a stringy treatment of Donaldson-Thomas theory should be thoughtof as the topological string coupling constant. The gauge theory has a BRST symmetry andhence localizes onto the moduli space M inst

r

(X) of solutions of the “generalized instanton”equations

F (0,2)

A

= @ †A

⇢ ,

F (1,1)

A

^ t ^ t+⇥

⇢ , ⇢⇤

= l t ^ t ^ t ,

dA

� = 0 . (3.7)

On a Calabi-Yau we can restrict our attention to minima such that ⇢ = 0. In this case the firsttwo equations on (3.7) reduce precisely to the Donaldson-Uhlenbeck-Yau equations (3.4) andBPS states correspond to stable holomorphic vector bundles. In the following, unless explicitlystated otherwise, we will only consider bundles E such that l = 0. In the string theory picturethis corresponds to the counting of D0-D2-D6 brane bound states without D4 brane charge.Furthermore to obtain a better behaved moduli space, we will allow for more general config-urations corresponding to torsion free coherent sheaves, as is customary in instanton countingproblems (and reviewed for example in [13]); we will however sometimes switch to the more fa-miliar holomorphic bundle language to aid intuition. The moduli space of torsion free coherentsheaves M inst

r

(X) stratifies into connected components with fixed characteristic classes. We willdenote these components by M inst

n,�;r

(X) where (ch3

(E), ch2

(E)) = (n,��).The local geometry of the moduli space is captured by the instanton deformation complex

0 //⌦0,0(X, ad E) C // ⌦0,1(X, ad E)� ⌦0,3(X, ad E) DA //⌦0,2(X, ad E) //0 , (3.8)

where ⌦•,•(X, ad E) denotes the bicomplex of complex di↵erential forms taking values in theadjoint gauge bundle over X, and the maps C and D

A

represent a linearized complexified gauge

8

Generating function of DT invariants

A

3.2 Cohomological Yang-Mills theory in six dimensions

The problem of studying Donaldson-Thomas invariants on X is essentially a higher dimensionalinstanton problem. The associated topological gauge theory is a topological version of six dimen-sional Yang-Mills. The most economical way of thinking about this theory is via dimensionalreduction of super Yang-Mills in ten dimension. After the reduction the six dimensional fieldsare a connection A on the G-bundle E �! X, and the ad E valued complex one form Higgsfield � and the forms ⇢(3,0) and ⇢(0,3). The fermionic sector is twisted, that is the fermions canbe though of as di↵erential forms thanks to the identification between the spin bundle and thebundle of di↵erential forms

S(X) ' ⌦0,•(X) , (3.5)

which holds on any Kahler manifold. Overall the fermionic sector comprises sixteen degrees offreedom which are organized into a complex scalar ⌘, one forms 1,0 and 0,1, two forms �2,0

and �0,2 and three forms 3,0 and 0,3. The bosonic part of the action is

S =1

2

Z

X

Tr⇣

dA

� ^ ⇤dA

�+⇥

� , �⇤

2

+�

�F (0,2)

A

+ @ †A

⇢�

�

2

+�

�F (1,1)

A

�

�

2

⌘

+1

2

1

(2⇡)2

Z

X

Tr⇣

FA

^ FA

^ t+ �

3·2⇡ FA

^ FA

^ FA

⌘

, (3.6)

where dA

= d + A is the gauge-covariant derivative, ⇤ is the Hodge operator with respect tothe Kahler metric of X, F

A

= dA + A ^ A is the gauge field strength. Furthermore � is acoupling constant which in a stringy treatment of Donaldson-Thomas theory should be thoughtof as the topological string coupling constant. The gauge theory has a BRST symmetry andhence localizes onto the moduli space M inst

r

(X) of solutions of the “generalized instanton”equations

F (0,2)

A

= @ †A

⇢ ,

F (1,1)

A

^ t ^ t+⇥

⇢ , ⇢⇤

= l t ^ t ^ t ,

dA

� = 0 . (3.7)

On a Calabi-Yau we can restrict our attention to minima such that ⇢ = 0. In this case the firsttwo equations on (3.7) reduce precisely to the Donaldson-Uhlenbeck-Yau equations (3.4) andBPS states correspond to stable holomorphic vector bundles. In the following, unless explicitlystated otherwise, we will only consider bundles E such that l = 0. In the string theory picturethis corresponds to the counting of D0-D2-D6 brane bound states without D4 brane charge.Furthermore to obtain a better behaved moduli space, we will allow for more general config-urations corresponding to torsion free coherent sheaves, as is customary in instanton countingproblems (and reviewed for example in [13]); we will however sometimes switch to the more fa-miliar holomorphic bundle language to aid intuition. The moduli space of torsion free coherentsheaves M inst

r

(X) stratifies into connected components with fixed characteristic classes. We willdenote these components by M inst

n,�;r

(X) where (ch3

(E), ch2

(E)) = (n,��).The local geometry of the moduli space is captured by the instanton deformation complex

0 //⌦0,0(X, ad E) C // ⌦0,1(X, ad E)� ⌦0,3(X, ad E) DA //⌦0,2(X, ad E) //0 , (3.8)

where ⌦•,•(X, ad E) denotes the bicomplex of complex di↵erential forms taking values in theadjoint gauge bundle over X, and the maps C and D

A

represent a linearized complexified gauge

8

E �! X

QUIVER QUANTUM MECHANICS

Generalized ADHM construction on the affine space via instanton quiver (rep theory)

Use virtual localization wrt maximal torus

Very explicit results. I’m neglecting a lot of “virtual fundamental” subtleties

explicitly in the Coulomb branch of the theory. The formalism developed in [6] is based on ageneralized ADHM construction which parametrize ideal sheaves on C3. This is derived by anexplicit homological construction of the moduli space. As a first step one compactifies C3 to P3

by adding a divisor at infinity and then tries to parametrize the moduli space of torsion freesheaves with fixed characteristic classes on P3 and with a trivialization condition on the divisorat infinity. This is done in practice by rewriting each sheaf E via a Fourier-Mukai transformwhose kernel is the diagonal sheaf of P3⇥P3. This procedure is rather technical and the outcomeis a certain spectral sequence. Upon imposing some conditions, including the trivialization on adivisor at infinity, the spectral sequence degenerates into a four term complex characterized bya series of matrix equations. These equations give a finite dimensional parametrization of theinstanton moduli space and are called generalized ADHM equations. Based on these equationsone can construct a certain topological quantum mechanics which can be used to compute therelevant instanton integrals. This quantum mechanics can be though of as arising from thequantization of the collective coordinates around each instanton solution.

In this section we will recall the basis of this construction and show how it can be used to computeinstanton integrals. The topological quantum mechanics is given in terms of the homologicaldata of the generalized ADHM construction. The formalism is based on two vector spaces Vand W with dimC V = n and dimCW = r. Physically n represent the instanton number ofthe gauge field configuration while r is the rank of the gauge theory. The generalized ADHMformalism can be conveniently described via the auxiliary quiver diagram

V •B2 88

B1

⌫⌫

B3

EE 'ff • WIoo . (5.1)

Recall that a quiver Q = (Q0

,Q1

) is an algebraic entity defined by a set of nodes Q0

and bya set of arrows Q

1

connecting the nodes. To the arrows one can associate a set of relationsR. The path algebra of the quiver is defined as the algebra of all possible paths in the quivermodulo the ideal generated by the relations; the product in the algebra is the concatenation ofpaths whenever this makes sense and 0 otherwise. This algebra will be denoted as A = CQ/hRi.A representation of the quiver Q can be constructed by associating a complex vector space toeach node and a linear map between vector spaces for each arrow, respecting the relations R.Instanton counting is determined in terms of the representation theory of this quiver with certainrelations, the generalized ADHM equations. These facts were thoroughly reviewed in [13].

We have introduced the morphisms

(B1

, B2

, B3

,') 2 HomC(V, V ) and I 2 HomC(W,V ) . (5.2)

Here ' is a finite dimensional analogous of the field ⇢(3,0) and we will be mainly interested inrepresentations of the ADHM quiver where ' is trivial. The fields B

↵

and ' are in the adjointrepresentation of U(n) while I is a U(n)⇥U(r) bifundamental. Furthermore all fields transformunder the lift of the natural toric action of T3 on C3, to the instanton moduli space. Under thefull symmetry group U(n)⇥ U(r)⇥ T3 the transformation rules are

B↵

7�! e� i ✏↵ gU(n)

B↵

g†U(n)

,

' 7�! e� i (✏1+✏2+✏3) gU(n)

' g†U(n)

,

I 7�! gU(n)

I g†U(r)

. (5.3)

21

[B↵, B� ] +3X

�=1

✏↵��⇥B†

� , '⇤= 0 ,

3X

↵=1

⇥B↵ , B†

↵

⇤+⇥' , '† ⇤+ I I† = & ,

I† ' = 0 ,

' = 0

DEFECTS AND DT

DIVISOR DEFECT

Basic idea: study DT with a Divisor Defect

Locally we take where is a divisor and the local fiber of the normal bundle. We define the defect by

Divisor defects are classified by pairs with and , a subgroup of Levi type.

In geometrical terms we have a G-bundle whose structure group is reduced to on .

INSTANTON MODULI SPACES

Now we consider generalized instantons. Conjectural moduli space

The defect is parametrized by a natural set of topological “angles”

which measure the reduction of the structure

group on the divisor

PARABOLIC SHEAVES

Already without the defect one needs coherent sheaves to formulate the problem properly. In our case the proper sheaves to consider are parabolic sheaves.

There is a correspondence between Levi subgroups of and parabolic subgroups of

We define a parabolic structure on the sheaf over as the flag of subsheaves of

At each point this is the flag of vector spaces stabilized by the parabolic group

previous example:

[Mehta,Shesadri][Bhosle]

PARABOLIC SHEAVES

This is the analog of the reduction of the structure group for a sheaf

A more convenient definition is

The two definitions are related by

There is an alternative definition of parabolic sheaves which will be more convenient in the fol-lowing, where instead of giving a filtration for the restriction of the sheaf E on D, one constructsdirectly a filtration of sheaves over X. More precisely, one can define a torsion free parabolicsheaf E as a torsion free sheaf with the following parabolic structure over D: a filtration

F• : E = F1

(E) � F2

(E) � · · · � Fl

(E) � Fl+1

= E(�D) , (4.34)

together with a sequence of weights 0 a1

< a2

< · · · < al

1. The two definitions areequivalent and are related by the short exact sequence

0 //Fi

(E) //E //E|D

/Gi(E) //0 . (4.35)

In particular from this short exact sequence it follows the relation between the Chern charactersof the sheaves Gi(E) supported on D and the sheaves F

i

(E)

ch(Fi

(E)) = ch(E)� ch(E|D

) + ch(Gi(E)) . (4.36)

The reduction of the gauge field due to the defect is simply parametrized by the Chern classesof the sheaves F

i

via the sequence (4.35). It is therefore natural to consider a moduli space withthese characteristic classes fixed. In other words we are interested in parametrizing the modulispace of parabolic sheaves with fixed (ch

3

(Fi

(E)), ch2

(Fi

(E)), c1

(Fi

(E))) = (ni

,��i

, ui

). Notethat F

1

= E whose characteristic classes are (ch3

(E), ch2

(E), c1

(E) = n,��, u). We will denote

this moduli space with P(↵)

n,�,u;r

(X,D|{ch(Fi

(E))}), or P(↵)

n,�,u;r

for simplicity.

Note that these moduli spaces can be empty. Furthermore as in ordinary Donaldson-Thomastheory, even if they are non-empty, we don’t expect them to be well behaved. In this paper wewill make no attempt to resolve this issue. We will simply assume that, as in ordinary Donaldson-Thomas theory one can construct a meaningful intersection theory with more sophisticated tools.Indeed in the following sections we will see an explicit example where this is possible, the caseof a�ne C3 with a specific divisor operator, where the relevant moduli space is actually a fixedpoint set of the moduli space of ordinary torsion free sheaves on C3.

4.6 Summary

Finally we summarize our conjectures. We have argued that the cohomological gauge theoryproblem in the presence of a divisor operator reduces to the study of the intersection theory of

the moduli space M (↵)

n,�,u;r

. In particular the gauge theory provides a natural measure, the Euler

class of the normal bundle eul(N (↵)

n,�,u;r

), the second cohomology of the instanton deformationcomplex (3.8), when restricted to configurations obeying (4.20). As in ordinary Donaldson-Thomas theory we can construct a generating function

ZDT

(X,D)

(q,Q; r) =X

n,�, u

X

m,h,o

qk Q� vu e 2⇡ i (⌘i mi+t

Da �

io

ai +�

ini)

Z

M (↵)n,�,u;r(L;X|{m,h,o})

eul(N (↵)

n,�,u;r

) .

(4.37)

where we have for convenience parametrized the Chern classes {ch(E(L)

i

)} in terms of the integersappearing in (4.26) and have omitted the dependence on the theta angles in the partition functionZDT

(X,D)

(q,Q; r). Note that the form of the instanton action require us to work in a topological

sector where the topological numbers (m, h, o) are fixed. We have kept track of the first Chernclass u introducing a counting parameter v; however this can be safely ignored since changing u

19

Finally, one is lead to conjecture [Murayama-Yokogawa]

AFFINE SPACE

MODULI SPACE OF PARABOLIC SHEAVES

Compactify the affine space to

To define a defect on consider the flag of torsion free sheaves of rank

Framing:

discrete group action which is determined by the type of divisor operator. We expect this tobe quite a generic result. The reason is that studying the gauge theory in the complement ofthe divisor is a similar problem to blowing down the divisor to produce a singularity. This isliterally true in the four dimensional case and for certain surface operators [25]. When the resultof the blow-down is an orbifold singularity, it is natural to expect the orbifold action to select therelevant instanton configurations when the defect is removed. However it is hard to make thisconnection concrete in general and we will limit ourselves to the a�ne case. Our constructionwill be rather explicit.

Our approach is inspired by the analogous construction in four dimensional gauge theories,where (ordinary) instanton counting in the presence of a surface operator is expressed in termsof parabolic sheaves on C2 and then reformulated in terms of orbifold sheaves [31, 39]. In thefour dimensional case the instanton moduli space is obtained by compactifying C2 to P1 ⇥ P1

and the presence of a surface operator is induced by imposing a parabolic structure on one of thedivisors with P1 topology. This is equivalent to a moduli space of torsion free sheaves without

any parabolic condition but invariant with respect to an appropriate orbifold action; in otherwords the moduli space of instantons with a surface operator is the �-fixed component of themoduli space of instanton without any parabolic structure [40].

6.1 Moduli spaces of parabolic sheaves

In our case we will consider a compactification of C3 to P1 ⇥ P1 ⇥ P1 and will be interested incoherent sheaves with a certain parabolic structure on a divisor. Since the original space C3

is non-compact the proper objects to study are sheaves with a framing condition. When wewant to distinguish the three P1 we will label them by the projective coordinates as P1

ziwhere

i = 1, 2, 3. We will denote by D = P1

z1⇥ P1

z2⇥ 0

z3 the divisor corresponding to the defect, andby D1 = P1

z1⇥ P1

z2⇥ 1

z3 t P1

z1⇥ 1

z2 ⇥ P1

z3t 1

z1 ⇥ P1

z2⇥ P1

z3the divisor at infinity. We

will identify the moduli space of generalized instantons in the presence of a divisor defect as themoduli space of torsion free sheaves with a framing condition on D1, a parabolic structure onD, and fixed characteristic classes.

Let us be more precise with the definition of our moduli space of parabolic sheaves Pd, whichis based on [51, 40, 39]. For notational convenience, now we will consider only the case wherethe divisor operator is associated with the Levi subgroup T

G

or equivalent a parabolic Borelsubgroup B, and comment later on how they can be extended to the more general case. Let usfix a r-tuple of integers d = (d

0

, · · · , dr�1

) which will play the role of instanton numbers.

A parabolic sheaf F• is a flag of torsion free sheaves of rank r on P1 ⇥ P1 ⇥ P1

F0

(�D) ⇢ F�r+1

⇢ · · · ⇢ F�1

⇢ F0

. (6.1)

We will furthermore require the following conditions

(1) Framing. The sheaves in the flag are locally free on D1, together with a framing isomor-phism

F0

(�D)|D1//

'✏✏

F�r+1

|D1//

'✏✏

· · · //

'✏✏

F0

|D1

'✏✏

O�r

D1(�D) //W (1) ⌦OD1 �O�r�1

D1(�D) // · · · //W (r) ⌦OD1

(6.2)

24

withChern classes are specified by the degree

[Feigin-Finkelberg-Negut-Rybnikov],[Negut],

[Finkelberg-Rybnikov]

PARABOLIC SHEAVES AS ORBIFOLD SHEAVES

Isotypical decomposition with

One can use the covering map

to construct an isomorphism

In this case parabolic sheaves can be understood in terms of orbifold sheaves. Natural action

In particular:[Okounkov], [Biswas]

[Feigin,Finkelberg,Negut,Rybnikov]

INSTANTON COUNTING

Extend the generalized ADHM construction

These maps are associated to the following BRST transformations

QBa

↵

= r

↵

and Q a

↵

= [�, Ba

↵

]� ✏↵

Br

↵

,

Q'r = ⇠r and Q ⇠a = [�,'a]� (✏1

+ ✏2

+ ✏3

)'a ,

Q Ia = %a and Q %a = � Ia � Ia aa ,

(7.6)

where in the vector aa we have collected all the Higgs field eigenvalues al

associated with theirreducible representation ⇢

a

. Following the standard formalism of topological field theories, oneassociates to these maps two Fermi multiplets containing the anti-ghosts and the auxiliary fields,and an extra gauge multiplet to close the BRST algebra [17]. Then one proceed to construct atopological invariant action which localizes onto the critical points of the BRST operator. Thesedatas can be conveniently summarized in the generalized ADHM quiver

· · ·B

a�23 //

'

a�2

11 Va�1

•

B

a�12

⇢⇢

B

a�11

⌅⌅B

a�13 //

'

a�1

22 Va

•

B

a2

⇢⇢

B

a1

⌅⌅B

a3 //

'

a11 V

a+1

•

B

a+12

⇢⇢

B

a+11

⌅⌅B

a+13 //

'

a+1

22 · · ·

Wa�1

•

I

a�1

OO

Wa

•

I

a

OO

Wa+1

•

I

a+1

OO(7.7)

In particular to this modified quiver one associates an ideal of relations which arises from de-composing the original ADHM equations accordingly to the �–module structure. Recalling thatwe are interested in the set of minima where the field ' is set to zero, the relevant equationsare

Ba

1

Ba

2

� Ba

2

Ba

1

= 0 ,Ba+1

1

Ba

3

� Ba

3

Ba

1

= 0 ,Ba+1

2

Ba

3

� Ba

3

Ba

2

= 0 ,�

Ia+1

�†'a = 0 .

(7.8)

These equations generate the ideal of relations in the instanton quiver path algebra A�

. Their�-equivariant decomposition cuts out a certain subvariety Rep

�

(n, r;B) from the framed quiverrepresentation space

Rep�

(n, r) = Hom�

(V,Q⌦ V ) � Hom�

(V,V

3Q⌦ V ) � Hom�

(W,V ) , (7.9)

The BPS moduli space in the presence of a divisor defect is then formally defined as the quotientstack

M�

(n, r) =h

Rep�

(n, r;B).

Y

a2b�GL(n

a

,C)i

(7.10)

by the gauge group which acts as basis change automorphisms of the �-module V . As in [17], wewill think of this stack as a moduli space of stable framed representations, where every objectin the category of quiver representations with relations is 0-semistable.

We define our Donaldson-Thomas invariants in the presence of a divisor operator as the equiv-ariant volumes of these instanton moduli spaces, computed via virtual localization. In doing sowe are making explicit use of the fact that the relevant moduli space is a �-fixed componentof the moduli space of torsion free sheaves, whose toric fixed points are isolated and given byideal sheaves. Therefore all the relevant machinery of virtual localization can be applied directly

32

We can set up the DT enumerative problem as an instanton counting problem

equivariant

where

Ba1 B

a2 �Ba

2 Ba1 = 0 ,

Ba+11 Ba

3 �Ba3 B

a1 = 0 ,

Ba+12 Ba

3 �Ba3 B

a2 = 0 ,

�Ia+1

�†'a = 0 .

CONCLUSIONS

Interplay between quantum theory with defects and geometry

Divisor defects: moduli space of sheaves with parabolic condition on a divisor

Study enumerative invariants (via localization on toric CY)

Extend to other gauge theories (higher and lower dim) and other defects (higher and lower codim)