a three-dimensional bridge between physics and mathematics

Tudor Dimofte Institute for Advanced Study

A three-dimensional bridge between physics and mathematics


“Mathematics provides the language for physics; physics gives life to mathematics.”

- Mina Aganagic



- Mina Aganagic

I’d like to discuss an example of this symbiosis, in a correspondence that I helped develop, which has been one of the major themes in my work.



- Mina Aganagic

I’d like to discuss an example of this symbiosis, in a correspondence that I helped develop, which has been one of the major themes in my work.

“3d-3d correspondence”


M , g Tg[M ]3-manifold A,D,E 3d (N=2) SUSY field theory

[Dimofte-Gukov-Hollands ’10]first hints:



depending only on topology of M!

[Dimofte-Gukov-Hollands ’10]first hints:




is a top-level top’l inv’t of

[Dimofte-Gukov-Hollands ’10]

Tg[M ] M

- its observables (quantities one can compute) all correspond to classical, quantum, or categorical topological invariants, some old, but many new.

first hints:






Tg[M ] M


first hints:

e.g.: hyperbolic volumeVol(M)

Mflat(M,GC)space of flat conn’sGC

[Mostow ’73]






Tg[M ] M


first hints:


Mflat(M,GC)

ZGCCS(M)

space of flat conn’sGCcomplex Chern-Simons part’n f’n

[Mostow ’73]

[Witten ’89, Reshetikhin-Turaev ’91]cf. Jones poly, WRT






Tg[M ] M


first hints:


Mflat(M,GC)

ZGCCS(M)


[Mostow ’73]

[Witten ’89, Reshetikhin-Turaev ’91]cf. Jones poly, WRTcategorification


M , g Tg[M ] [Dimofte-Gukov-Hollands ’10]first hints:


Mflat(M,GC)

ZGCCS(M)


[Mostow ’73]

[Witten ’89, Reshetikhin-Turaev ’91]cf. Jones poly, WRTcategorification

More than just theory!




“Most” , : explicit construction of g = sl2M Tg[M ][Dimofte-Gaiotto-Gukov ’11] [Cecotti-Cordova-Vafa ’11] [Dimofte-Gaiotto-v.d.Veen ’13]

g = sln : [Dimofte-Gabella-Goncharov ’13]




“Most” , : explicit construction of g = sl2M Tg[M ]

g = sln : [Dimofte-Gabella-Goncharov ’13]

Main tool: (topological) ideal triangulations + a generalization of Thurston-Neumann-Zagier gluing methods from hyperbolic geometry (’80’s) [Dimofte ’11]

[Dimofte-Gaiotto-Gukov ’11] [Cecotti-Cordova-Vafa ’11] [Dimofte-Gaiotto-v.d.Veen ’13]


M , g Tg[M ]


Results?


M , g Tg[M ]


Math:Results?

New “quantum” topological invariants,

ZGCCS(M)

for all CS levels k 2 Z [Dimofte-Gaiotto-Gukov ’11] [Dimofte ’14]

a comb’r definition of the Chern-Simons part’n functionGC


M , g Tg[M ]

Math:Results?

New “quantum” topological invariants,a comb’r definition of the Chern-Simons part’n functionGC

ZGCCS(M)


- analyzing asymptotics of (easy!)ZGCCS(M)

simple, conjectured (tested) formula for GC-twisted Reidemeister-Ray-Singer torsion of M

[Dimofte-Garoufalidis ’12]


M , g Tg[M ]

Math:Results?

New “quantum” topological invariants,

ZGCCS(M)




predictions for asymptotics of colored Jones poly’s(hard!; play a role in Volume Conjecture)[Kashaev ’97, Murakami-Murakami ’99, Gukov ‘03]



[Dimofte-Garoufalidis ’15][Dimofte-Gukov-Lenells-Zagier ’08]


M , g Tg[M ]

Math:Results?

ZGCCS(M)



Hopefully: a combinatorial definition for 3-manifold homology!

GCin progress w/ Gaiotto-Moore

(Analogous to Khovanov homology for )G



predictions for asymptotics of colored Jones poly’s(hard!; play a role in Volume Conjecture)[Kashaev ’97, Murakami-Murakami ’99, Gukov ‘03] [Dimofte-Garoufalidis ’15]

[Dimofte-Gukov-Lenells-Zagier ’08]


M , g Tg[M ]

Physics:Results?

Intuition!Properties of 3d N=2 theories are governed by the geometry of 3-manifolds!


M , g Tg[M ]

Physics:Results?

- geometric description of (IR) dualities within a large class of 3d N=2 SUSY gauge theories

(from different ways to cut/glue the same )M



M , g Tg[M ]

Physics:Results?




- systematic construction of superconformal interfacesTg[M ]

4d N=2 T 4d N=2 T’

[Dimofte-Gaiotto-v.d.Veen ’13]


M , g Tg[M ]

Physics:Results?




- systematic construction of superconformal interfacesTg[M ]

4d N=2 T 4d N=2 T’gYM g0YM ⇠ 1/gYMS-duality:


Remainder of the talk:

Tg[M ]

4d N=2 T 4d N=2 T’gYM g0YM ⇠ 1/gYMS-duality:


Remainder of the talk:- a few more details on the correspondence, and observables of Tg[M ]


- tetrahedra, formulas, and examples


- tetrahedra, formulas, and examples

- first look at homological/categorical invariants

The correspondence

Starting point: 6d (2,0) SCFT“theory ”X

[Strominger, Witten ’90’s]super-conformal-field-theory

The correspondence


[Strominger, Witten ’90’s]

(world-volume theory of M5 branes)

super-conformal-field-theory

The correspondence




labelled by an ADE symmetry algebra g


The correspondence




X on (topological twist on M)M ⇥ R3

effective theory on

labelled by an ADE symmetry algebra

Tg[M ]

g

R3

g


The correspondence





effective theory on


Tg[M ]

g

R3

g

How to describe ?Tg[M ]


The correspondence





effective theory on


Tg[M ]

g

R3

g


- direct, first-principles is hard: X has no Lagrangian


The correspondence





effective theory on


Tg[M ]

g

R3

g


- direct, first-principles is hard: X has no Lagrangian- nevertheless, can infer many properties of + its compact’nsTg[M ]


The correspondence



{vacua of on }

Basic property:

Tg[M ] R2 ⇥ S1 = {flat connections on }GC M

Mflat(M,GC)

To see this: 6d M ⇥ R2 ⇥ S1

R2

R2 ⇥ S1

Xg

Tg[M ]3d

2d

The correspondence



{vacua of on }

Basic property:


Mflat(M,GC)


M ⇥ R2

R2

R2 ⇥ S1

Xg

Tg[M ]3d

2d

5d super-YM[has a Lagrangian!]

The correspondence

{vacua of on }

Basic property:


Mflat(M,GC)


M ⇥ R2

R2

R2 ⇥ S1

Xg

Tg[M ]3d

2d

5d super-YM

look at the vacua of 5d SYM, topologically twisted on M

[has a Lagrangian!]

The correspondence

{vacua of on }

Basic property:


Mflat(M,GC)


M ⇥ R2

R2

R2 ⇥ S1

Xg

Tg[M ]3d

2d

5d super-YM


Aa,�a

(a = 1, 2, 3)

[has a Lagrangian!]

fields 2 g

The correspondence

{vacua of on }

Basic property:


Mflat(M,GC)


M ⇥ R2

R2

R2 ⇥ S1

Xg

Tg[M ]3d

2d

5d super-YM


Aa,�a Aa = Aa + i�a

(a = 1, 2, 3)

[has a Lagrangian!]

fieldsGC connection on M

2 g

The correspondence

{vacua of on }

Basic property:


Mflat(M,GC)


M ⇥ R2

R2

R2 ⇥ S1

Xg

Tg[M ]3d

2d

5d super-YM



(a = 1, 2, 3)

[has a Lagrangian!]


2 g minimize potential:dA+A ^A = 0

flat

The correspondence

{vacua on }R2 ⇥ S1 = {flat connections}GC

MTg[M ]

Mflat(M,GC)[Dimofte-Gukov-Hollands ’10]



(a = 1, 2, 3)



flat

The correspondence


MTg[M ]


quantize!



(a = 1, 2, 3)



flat

The correspondence


MTg[M ]


quantize!

Chern-Simons theory on MGC

ZCS [M ] =

ZDADA e

k+i�8⇡i ICS(A)+

k�i�8⇡i ICS(A)

ICS(A) :=

Z

MTr

⇣A ^ dA+

2

3A ^A ^A

⌘

[Witten ’91]

A gC-valued 1-form:

The correspondence


MTg[M ]


quantize!


ZCS [M ] =

ZDADA e

k+i�8⇡i ICS(A)+


ICS(A) :=

Z

MTr

⇣A ^ dA+

2

3A ^A ^A

⌘

[Witten ’91]

k 2 Z � 2 R (or C)

A gC-valued 1-form:

The correspondence


MTg[M ]


quantize!


ZCS [M ] =

ZDADA e

k+i�8⇡i ICS(A)+


ICS(A) :=

Z

MTr

⇣A ^ dA+

2

3A ^A ^A

⌘

[Witten ’91]

k 2 Z � 2 R (or C)

A gC-valued 1-form:

- classical sol’ns are flat connectionsGC

The correspondence


MTg[M ]


quantize!


ZCS [M ] =

ZDADA e

k+i�8⇡i ICS(A)+


ICS(A) :=

Z

MTr

⇣A ^ dA+

2

3A ^A ^A

⌘

[Witten ’91]

k 2 Z � 2 R (or C)

- classical sol’ns are flat connections

A gC-valued 1-form:

GC

- cf. compact CS thy: on knot complements, get Jones polysG[Witten ’89]

(combinatorial definition) [Reshetikhin-Turaev ’90, etc.]

The correspondence


MTg[M ]


quantize!


ZCS [M ] =

ZDADA e

k+i�8⇡i ICS(A)+


ICS(A) :=

Z

MTr

⇣A ^ dA+

2

3A ^A ^A

⌘

[Witten ’91]

k 2 Z � 2 R (or C)

- classical sol’ns are flat connections

A gC-valued 1-form:

GC



- combinatorial def’n missing for until recently!GC

The correspondence


MTg[M ]


ZCS [M ] =

ZDADA e

k+i�8⇡i ICS(A)+


ICS(A) :=

Z

MTr

⇣A ^ dA+

2

3A ^A ^A

⌘

[Witten ’91]

k 2 Z � 2 R (or C)





Z(k,�)CS [M ]

The correspondence


MTg[M ]


ZCS [M ] =

ZDADA e

k+i�8⇡i ICS(A)+


ICS(A) :=

Z

MTr

⇣A ^ dA+

2

3A ^A ^A

⌘

[Witten ’91]

k 2 Z � 2 R (or C)





Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]

part’n function onellipsoidally-deformed lens space

The correspondence


MTg[M ]


Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]


L(k, 1)� = S3�/Zk

' {b2|z|2 + b�2|w|2 = 1} 2 C2.(z, w) ⇠ (e

2⇡ik z, e

2⇡ik w)

b2 =k � i�

k + i�

The correspondence


MTg[M ]


Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]


L(k, 1)� = S3�/Zk

' {b2|z|2 + b�2|w|2 = 1} 2 C2.(z, w) ⇠ (e

2⇡ik z, e

2⇡ik w)b2 =

k � i�

k + i�

k=1 :[Terashima-Yamazaki ’11]![Dimofte-Gaiotto-Gukov ’11][Cordova-Jafferis ’13] — physical proof

The correspondence


MTg[M ]


Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]


L(k, 1)� = S3�/Zk

' {b2|z|2 + b�2|w|2 = 1} 2 C2.(z, w) ⇠ (e

2⇡ik z, e

2⇡ik w)b2 =

k � i�

k + i�

k=0 :

k=1 :

[Dimofte-Gaiotto-Gukov (2) ’11]

[Terashima-Yamazaki ’11]![Dimofte-Gaiotto-Gukov ’11][Cordova-Jafferis ’13]

[Lee-Yamazaki ’13]

— physical proof

— physical proof

The correspondence


MTg[M ]


Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]


L(k, 1)� = S3�/Zk

' {b2|z|2 + b�2|w|2 = 1} 2 C2.(z, w) ⇠ (e

2⇡ik z, e

2⇡ik w)

k=0 :

b2 =k � i�

k + i�

k=1 :

general k:



[Dimofte ’14]


— physical proof

— physical proof

The correspondence


MTg[M ]


Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]

k=0 : [Dimofte-Gaiotto-Gukov (2) ’11]

k=1 : [Terashima-Yamazaki ’11]![Dimofte-Gaiotto-Gukov ’11][Cordova-Jafferis ’13]

general k: [Dimofte ’14]

to tie this all together:

Zk,�CS [M ] =

X

flat ↵

Bk+i�↵ [M ]Bk+i�

↵ [M ]


— physical proof

— physical proof

The correspondence


MTg[M ]


Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]

k=0 :

k=1 :

general k:


Zk,�CS [M ] =

X

flat ↵


↵ [M ]

[Beem-Dimofte-Pasquetti ‘12]

ZTg[M ][L(k, 1)�] =X

vacua ↵


↵ [M ]

L(k, 1) ' (D2 ⇥ S1) ['2SL(2,Z) (D2 ⇥ S1)



[Dimofte ’14]


— physical proof

— physical proof

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]


Zk,�CS [M ] =

X

flat ↵


↵ [M ]

[Beem-Dimofte-Pasquetti ‘12]

ZTg[M ][L(k, 1)�] =X

vacua ↵


↵ [M ]

L(k, 1) ' (D2 ⇥ S1) ['2SL(2,Z) (D2 ⇥ S1)

So: quantum invariants of 3-manifolds! can be understood via 3d SUSY theories on lens spaces!

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]


One more step: categorify

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]



ZT [M ](S2 ⇥ S1)k=0:

is a an index= TrH(S2)(�1)F qJ+

F2

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]



ZT [M ](S2 ⇥ S1)k=0:


F2

natural vector space + differential,Tg[M ] S2Hilb. space of on , action of “Q”;

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]



ZT [M ](S2 ⇥ S1)k=0:


F2

natural vector space + differential,Hilb. space of on , action of “Q”;Tg[M ] S2

the index is its graded Euler character

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]

So: by studying more refined observables of ,! like Hilbert spaces, one obtains homological lifts of quantum inv’ts!


ZT [M ](S2 ⇥ S1)k=0:


F2



Tg[M ]

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]



ZT [M ](S2 ⇥ S1)k=0:


F2



Tg[M ]

- analogous to Khovanov homology

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]

ZT [M ](S2 ⇥ S1)k=0:


F2




Tg[M ]

- analogous to Khovanov homology- work in progress w/ Gaiotto, Moore

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]



Tg[M ]

This was the “pedestrian” version!

The correspondence


MTg[M ]

Mflat(M,GC)

Z(k,�)CS [M ]=ZTg[M ][L(k, 1)�]




Tg[M ]

Full picture: study with boundaryM

The correspondence



The right way to compactify on a space w/ bdy! is to stretch to bdy to asymptotic regions

Xg

Xg on R3⇥ M

@M1 ⇥ R+@M2 ⇥ R+

The correspondence




Xg

Xg on R3⇥

M

Tg[M ]

R3

R3 ⇥ R+R3 ⇥ R+

Tg[@M1] Tg[@M2]

@M1 ⇥ R+@M2 ⇥ R+

3d interface! between 4d (N=2)! SUSY theories[Dimofte-Gaiotto-v.d.Veen ‘13]

The correspondence




Xg

Xg on R3⇥

M

Tg[M ]

R3

R3 ⇥ R+R3 ⇥ R+

Tg[@M1] Tg[@M2]

@M1 ⇥ R+@M2 ⇥ R+

“2d-4d correspondence”[Gaiotto, Gaiotto-Moore-Nietzke ‘09]


The correspondence is functorial:

Xg on R3⇥

M

Tg[M ]

R3

R3 ⇥ R+R3 ⇥ R+

Tg[@M1] Tg[@M2]

@M1 ⇥ R+@M2 ⇥ R+




Tg : Cobordism category of 2-manifolds

objects: 2-manifoldsmorphisms: 3-cobordisms

Cat. of 4d N=2 SUSY thy’sobjects: 4d theoriesmorphisms: 3d interfaces

Xg on R3⇥

M

Tg[M ]

R3

R3 ⇥ R+R3 ⇥ R+

Tg[@M1] Tg[@M2]

@M1 ⇥ R+@M2 ⇥ R+






Can extend further,

2-morphisms: 4-cobordisms





Can extend further,

2-morphisms: 4-cobordisms 2-morphisms: 2d interfaces

cf. [Gadde-Gukov-Putrov ’13]“4d-2d correspondence”


The correspondence is effective


For a large class of 3-manifolds, can explicitly compute Tg[M ]

give an explicit 3d Lagrangian density




- includes all hyperbolic , with cusps and/or geodesic bdyM





(most 3-manifolds are hyperbolic= admit a metric of constant neg. curvature)

(given appropriate boundary conditions, the metric is unique)[Mostow ’76,…]







- method of computation: cut M into (topological) ideal tetrahedra

truncated vertices







- method of computation: cut M into (topological) ideal tetrahedra

truncated verticesTg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i


Tg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i

Remainder of the talk: g = sl2 GC = SL(2,C) (or PSL(2,C)= SL(2,C)/{±1})


Tg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i

Remainder of the talk: g = sl2 GC = SL(2,C) (or PSL(2,C)

- for simplicity, and some added intuition g = sln[Dimofte-Gabella-Goncharov ’13]

= SL(2,C)/{±1})


Tg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i

Remainder of the talk: g = sl2 GC = SL(2,C) (or PSL(2,C)

- for simplicity, and some added intuition g = sln[Dimofte-Gabella-Goncharov ’13]

- flat connections are (roughly) hyperbolic metricsPSL(2,C)

= SL(2,C)/{±1})

So: quantizes, categorifies, etc. classical hyperbolic geometry!Tg[M ]

The correspondence is effectiveSingle (ideal, hyperbolic) tetrahedron:

0

1

1

z

@H3 ' C [ {1}


0

1

1

z

@H3

- vertices at on the bdy of H3

- faces are geodesic surfaces

' C [ {1}



- faces are geodesic surfaces0

1

1

z

@H3 ' C [ {1}

- the hyperbolic structure is encoded! in 6 complexified dihedral angles

zz0

z00

z

z0

z00

z = e(torsion)+i(angle)



- faces are geodesic surfaces0

1

1

z

@H3 ' C [ {1}


zz0

z00

zz0z00 = �1

z00 + z�1 � 1 = 0

z

z0

z00

z = e(torsion)+i(angle)

equal on opposite edges,!and satisfy [W. Thurston, late ’70’s]




0

1

1

z

@H3 ' C [ {1}


zz0

z00

zz0z00 = �1

z00 + z�1 � 1 = 0

z

z0

z00z = e(torsion)+i(angle)


Flat connections: Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00) [Dimofte ’10]




0

1

1

z

@H3 ' C [ {1}


zz0

z00

zz0z00 = �1

z00 + z�1 � 1 = 0

z

z0

z00z = e(torsion)+i(angle)


Flat connections: Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

[Dimofte ’10]

⇢ ⇢

The correspondence is effectiveTetrahedron theory:

0

1

1

z

@H3

zz0

z00

zz0z00 = �1

z00 + z�1 � 1 = 0

z

z0

z00


Mflat(�, GC) = {z00 + z�1 � 1 = 0}

[Dimofte ’10]

⇢ ⇢

= single free chiral superfieldT [�]

[Dimofte-Gaiotto-Gukov ’11]


zz0z00 = �1

z00 + z�1 � 1 = 0

0

1

1

z

@H3

zz0

z00

z

z0

z00


Mflat(�, GC) = {z00 + z�1 � 1 = 0}

[Dimofte ’10]

⇢ ⇢


complex scalar, complex fermion� �, or

(function on )R3 (section of spinor bundle on )R3



Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

[Dimofte ’10]

⇢ ⇢




0

1

1

z

@H3

zz0

z00

z

z0

z00


L = |@µ�|2 + (� · @) Lagrangian:


Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

[Dimofte ’10]

⇢ ⇢




0

1

1

z

@H3

zz0

z00

z

z0

z00


This theory allows a supersymmetric mass term Z (equal for )�,

L = |@µ�|2 + (� · @) Lagrangian: +Z|�|2 + Z

(real)


Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

[Dimofte ’10]

⇢ ⇢




0

1

1

z

@H3

zz0

z00

z

z0

z00


This theory allows a supersymmetric mass term Z (equal for )�,

L = |@µ�|2 + (� · @) Lagrangian: +Z|�|2 + Z

Putting the theory on , gets complexified,! and can be identified with the hyperbolic modulus:

R2 ⇥ S1

(real)

Zz = exp(Z)


Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

⇢ ⇢


0

1

1

z

@H3

zz0

z00

z

z0

z00


R2 ⇥ S1 Zz = exp(Z)

Classical invariants?


Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

⇢ ⇢


0

1

1

z

@H3

zz0

z00

z

z0

z00




On , a standard 1-loop calculation leads toR2 ⇥ S1

Le↵ = |dfW (z)|2 fW (z) = Li2(z)

[Witten, Phases of N=2 Theories ’93]


Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

⇢ ⇢


0

1

1

z

@H3

zz0

z00

z

z0

z00





Le↵ = |dfW (z)|2 fW (z) = Li2(z)

(complex) volume of !�



Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

⇢ ⇢


0

1

1

z

@H3

zz0

z00

z

z0

z00



Le↵ = |dfW (z)|2 fW (z) = Li2(z)

(complex) volume of !�


Also, vacua of on given by T [�] R2 ⇥ S1exp

⇣zdfW (z)

dz

⌘= z00

) z00 + z�1 � 1 = 0 !


Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

⇢ ⇢


0

1

1

z

@H3

zz0

z00

z

z0

z00


⇣zdfW (z)

dz

⌘= z00

) z00 + z�1 � 1 = 0 !

Turn the crank: quantum invariants


Mflat(@�, GC) ⇡ C⇤ ⇥ C⇤ (z, z00)

Mflat(�, GC) = {z00 + z�1 � 1 = 0}

⇢ ⇢


0

1

1

z

@H3

zz0

z00

z

z0

z00


⇣zdfW (z)

dz

⌘= z00

) z00 + z�1 � 1 = 0 !

ZT [�][L(k, 1)�]The lens-space partition functionscan all be calculated explicitly — due to SUSY, the path integral!reduces to a finite-dimensional integral.

[Kapustin-Willett-Yaakov ’10][Hama-Hosomochi-Lee ’11]

[Kim ’09]

[Benini-Nishioka-Yamazaki ’11]


The correspondence is effective= single free chiral superfieldT [�]



[Kim ’09]



E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�




[Kim ’09]



E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�

depends on , phasem 2 Z |⇣| = 1 — because has a bdy�



E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�


z ⇠ qm2 ⇣

z̄ ⇠ qm2 ⇣�1



E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�


z ⇠ qm2 ⇣

z̄ ⇠ qm2 ⇣�1

is a version of a “quantum dilogarithm”ZT [�][S2 ⇥ S1]

ZT [�][S2 ⇥ S1]

� ! 1q ! 1

⇠ e�2⇡ ImLi2(z)


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�

z ⇠ qm2 ⇣

z̄ ⇠ qm2 ⇣�1


ZT [�][S2 ⇥ S1]

� ! 1q ! 1


3d SUSY thy: T [�] has a U(1) symmetrye (�, ) ! (ei✓�, ei✓ )


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�

z ⇠ qm2 ⇣

z̄ ⇠ qm2 ⇣�1


ZT [�][S2 ⇥ S1]

� ! 1q ! 1



Hilb. space is graded byH(S2) e- elec & mag charges for this U(1)(m, e) 2 Z⇥ Z

U(1)J


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�


Hilb. space is graded byH(S2) e- elec & mag charges for this U(1)

- spin(m, e) 2 Z⇥ Z

U(1)J (weight for )U(1)JJ 2 12Z


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�




U(1)J (weight for )U(1)J

- R-charge

J 2 12Z

R 2 Z (�, ) ! (�, e�i✓ )


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�




U(1)J (weight for )U(1)J

- R-charge

J 2 12Z

R 2 Z

There’s a differential Q : H(S2) ! H(S2) (one of the SUSY generators)

(�, ) ! (�, e�i✓ )

preserves m,e,J+R/2; R R+1!


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣

q = e2⇡�




(weight for )U(1)J

- R-chargeJ 2 1

2Z

R 2 Z


(�, ) ! (�, e�i✓ )


U(1)J

= TrH(S2;m)(�1)RqJ+R2 ⇣e

H•⇥H(S2;m), Q⇤

or

(definition!)


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣




(weight for )U(1)J

- R-chargeJ 2 1

2Z

R 2 Z


(�, ) ! (�, e�i✓ )


U(1)J


H•⇥H(S2;m), Q⇤

or

(definition!)

modes of

modes of �


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣


H•⇥H(S2;m), Q⇤

or

(definition!)

modes of

modes of �

Categorical/homological invariant: itselfH•⇥H(S2), Q⇤


E.g. ZT [�][S2 ⇥ S1] =

1Y

r=0

1� q1�m2 ⇣�1

1� q�m2 ⇣


H•⇥H(S2;m), Q⇤

or

(definition!)

modes of

modes of �

Categorical/homological invariant: itself

Free theory: easy TrH•

⇥H(S2;m),Q

⇤tRqj+R2 ⇣e =

1Y

r=0

1 + tq1�m2 ⇣�1

1� q�m2 ⇣

H•⇥H(S2), Q⇤

That categorifies the volume of a hyperbolic tetrahedron.

The correspondence is effectiveIn general, glue



⇥H(S2;m),Q

⇤tRqj+R2 ⇣e =

1Y

r=0

1 + tq1�m2 ⇣�1

1� q�m2 ⇣

H•⇥H(S2), Q⇤


Tg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i




⇥H(S2;m),Q

⇤tRqj+R2 ⇣e =

1Y

r=0

1 + tq1�m2 ⇣�1

1� q�m2 ⇣

H•⇥H(S2), Q⇤


Tg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i

- gluing rules come from promoting Thurston’s gluing eqs! (and symplectic properties found by )! to the level of 3d SUSY gauge theories

[Neumann-Zagier ’82]




⇥H(S2;m),Q

⇤tRqj+R2 ⇣e =

1Y

r=0

1 + tq1�m2 ⇣�1

1� q�m2 ⇣

H•⇥H(S2), Q⇤


Tg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i



- roughly, contains N chiral multiplets,! with extra gauge fields and interactions to enforce the gluing.

Tsl2 [M ]


Tg[M ] =

✓ NOi=1

Tg[�i]

◆�⇠M =

N[

i=1

�i



- roughly, contains N chiral multiplets,! with extra gauge fields and interactions to enforce the gluing.

Tsl2 [M ]

What’s in it for physics?

The correspondence is good for physicsTwo examples:

What’s in it for physics?

1. A geometric interpretation (and prediction) of dualities! in 3d SUSY theories



M =N[

i=1

�i =N 0[

j=1

�j

A 3-manifold may be glued together in many different ways



A 3-manifold may be glued together in many different ways

expect

M =N[

i=1

�i =N 0[

j=1

�j

T [M ] = ⌦T [�i]�⇠ = ⌦T [�j ]

�⇠

equivalent in the IR

The correspondence is good for physics

expect

M =N[

i=1

�i =N 0[

j=1

�j

T [M ] = ⌦T [�i]�⇠ = ⌦T [�j ]

�⇠



expect

M =N[

i=1

�i =N 0[

j=1

�j

T [M ] = ⌦T [�i]�⇠ = ⌦T [�j ]

�⇠


T [M ] = 3d SQED2 chiral multipletsU(1) gauge sym. +1 -1

�1, �2

�1

�2



�1

�2

�3 �4

�5

�1, �2

T [M ] = “XYZ model”

3 chiral multiplets �1, �2, �3

cubic superpotential W = �1�2�3

i.e. L = ...+ 1 2�3 + |�1�2|2 + ...



�1

�2

�3 �4

�5

�1, �2




i.e. L = ...+ 1 2�3 + |�1�2|2 + ...

3d SQED = “XYZ model”[Aharony-Hanany-Intriligator-Seiberg-Strassler ’97]



�1

�2

�3 �4

�5

�1, �2




i.e. L = ...+ 1 2�3 + |�1�2|2 + ...

3d SQED = “XYZ model”[Aharony-Hanany-Intriligator-Seiberg-Strassler ’97]

cf. classical Li2(x) + Li2(y) = Li2

⇣x

1� y

⌘+ Li2

⇣y

1� x

⌘+ Li2

⇣(1� x)(1� y)

xy

⌘+ logs


cf. classical Li2(x) + Li2(y) = Li2

⇣x

1� y

⌘+ Li2

⇣y

1� x

⌘+ Li2

⇣(1� x)(1� y)

xy

⌘+ logs

Two examples:

2. 3d N=2 theories on interfaces in 4d! get labelled by 3-manifolds, and gain systematic constructions



R3

R3 ⇥ R+R3 ⇥ R+

T [M ]

g2 g02 ⇠ 1/g2

E.g. electric-magnetic (S) duality in 4d maximally SUSY YM thy



R3

R3 ⇥ R+R3 ⇥ R+

T [M ]

g2 g02 ⇠ 1/g2


M = S3\(Hopf network)



R3

R3 ⇥ R+R3 ⇥ R+

T [M ]

g2 g02 ⇠ 1/g2


M = S3\(Hopf network) 4 or 5 ’s�

T [M ]

Moral

M = S3\(Hopf network) 4 or 5 ’s�

T [M ]

There is interesting structure to be discovered and developed,

Moral

There is interesting structure to be discovered and developed,both in physics and mathematics

phys

ics

math

- SUSY QFT

- partition functions- (SUSY) Hilbert spaces

- moduli spaces- topological invariants- categorification- combinatorics! of triangulations

Moral


phys

ics

math

- SUSY QFT



3d-3d

Relations like the 3d-3d correspondence allow both kinds of structure!to be developed in tandem, with double the power and intuition.

Moral


phys

ics

math

- SUSY QFT



3d-3d

Relations like the 3d-3d correspondence allow both kinds of structure!to be developed in tandem, with double the power and intuition.

I hope this type of work will find a place here at Davis.

a three-dimensional bridge between physics and mathematics

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