Triality in Little String Theories Stefan Hohenegger GGI Workshop: - - PowerPoint PPT Presentation
Triality in Little String Theories Stefan Hohenegger GGI Workshop: String Theory from a Worldsheet Perspective Galileo Galilei Institute, 29 Apr. 2019 based on work in collaboration with: Brice Bastian, Amer Iqbal and Soo-Jong Rey hep-th
GGI Workshop: String Theory from a Worldsheet Perspective
based on work in collaboration with: Brice Bastian, Amer Iqbal and Soo-Jong Rey
Galileo Galilei Institute, 29 Apr. 2019
hep-th 1610.07916 , hep-th 1710.02455 hep-th 1711.07921, hep-th 1807 .00186, hep-th 1810.05109, hep-th 1811.03387
very rich structure at the heart of key structures in M-theory and string theory
Strong Motivation to study supersymmetric/
(flagship example: world-volume theory of multiple M5-branes) encode topological invariants and data of underlying string geometry connection to supersymmetric gauge theories in 4 dimensions (AGT relations)
very rich structure at the heart of key structures in M-theory and string theory
Strong Motivation to study supersymmetric/
(flagship example: world-volume theory of multiple M5-branes) encode topological invariants and data of underlying string geometry connection to supersymmetric gauge theories in 4 dimensions (AGT relations)
Mathematically very involved and difficult to study using ‘traditional’ methods
typically lack of Lagrangian description (e.g. (2,0) theories — might not exist?) gauging difficult in 6 dimensions (lack of vector degrees of freedom) lack of perturbative description
very rich structure at the heart of key structures in M-theory and string theory
Strong Motivation to study supersymmetric/
(flagship example: world-volume theory of multiple M5-branes) encode topological invariants and data of underlying string geometry connection to supersymmetric gauge theories in 4 dimensions (AGT relations)
Mathematically very involved and difficult to study using ‘traditional’ methods
typically lack of Lagrangian description (e.g. (2,0) theories — might not exist?) gauging difficult in 6 dimensions (lack of vector degrees of freedom) lack of perturbative description use vast net of dualities to map the problem to a ‘tractable’ setup
very rich structure at the heart of key structures in M-theory and string theory
Strong Motivation to study supersymmetric/
(flagship example: world-volume theory of multiple M5-branes) encode topological invariants and data of underlying string geometry connection to supersymmetric gauge theories in 4 dimensions (AGT relations)
Mathematically very involved and difficult to study using ‘traditional’ methods
typically lack of Lagrangian description (e.g. (2,0) theories — might not exist?) gauging difficult in 6 dimensions (lack of vector degrees of freedom) lack of perturbative description use vast net of dualities to map the problem to a ‘tractable’ setup
Geometrically: Calabi Yau manifolds (refined) topological string Brane Configurations (Non-)compact M5/M2-systems D5-NS5-brane configurations World-sheet description M-strings
very rich structure at the heart of key structures in M-theory and string theory
Strong Motivation to study supersymmetric/
(flagship example: world-volume theory of multiple M5-branes) encode topological invariants and data of underlying string geometry connection to supersymmetric gauge theories in 4 dimensions (AGT relations)
Mathematically very involved and difficult to study using ‘traditional’ methods
typically lack of Lagrangian description (e.g. (2,0) theories — might not exist?) gauging difficult in 6 dimensions (lack of vector degrees of freedom) lack of perturbative description use vast net of dualities to map the problem to a ‘tractable’ setup
Geometrically: Calabi Yau manifolds (refined) topological string Brane Configurations (Non-)compact M5/M2-systems D5-NS5-brane configurations World-sheet description M-strings
use ‘ stringy’ tools to compute quantities in quantum field theory
Over the last decades string theory has provided insights into strongly coupled quantum systems
Over the last decades string theory has provided insights into strongly coupled quantum systems Specifically: prediction of existence of new interacting conformal field theories in dimensions D > 4
e.g.: [Seiberg 1996]
Over the last decades string theory has provided insights into strongly coupled quantum systems Specifically: prediction of existence of new interacting conformal field theories in dimensions D > 4
e.g.: [Seiberg 1996]
String theory
quantum field theory
suitable decoupling of gravity
Over the last decades string theory has provided insights into strongly coupled quantum systems Specifically: prediction of existence of new interacting conformal field theories in dimensions D > 4
e.g.: [Seiberg 1996]
String theory also predicts the existence of new ‘non-local theories’, e.g. little string theories (LSTs)
String theory
quantum field theory
suitable decoupling of gravity suitable decoupling of gravity
different approaches: [Witten 1995] [Aspinwall, Morrison 1997] [Seiberg 1997] [Intriligator 1997] [Hanany, Zaffaroni 1997] [Brunner, Karch 1997]
little string theory
Mstring
⌧ Mstring
> Mstring
effective QFT (point-like dofs) UV-comp. contains stringy dofs String theory
Over the last decades string theory has provided insights into strongly coupled quantum systems Specifically: prediction of existence of new interacting conformal field theories in dimensions D > 4
e.g.: [Seiberg 1996]
String theory also predicts the existence of new ‘non-local theories’, e.g. little string theories (LSTs)
String theory
quantum field theory
suitable decoupling of gravity suitable decoupling of gravity
different approaches: [Witten 1995] [Aspinwall, Morrison 1997] [Seiberg 1997] [Intriligator 1997] [Hanany, Zaffaroni 1997] [Brunner, Karch 1997]
little string theory
Mstring
⌧ Mstring
> Mstring
effective QFT (point-like dofs) UV-comp. contains stringy dofs
Classification of LSTs (ADE type for theories with supersymmetry )
N = (2, 0)
[Bhardwaj, Del Zotto, Heckman, Morrison, Rudelius, Vafa 2016]
String theory
Over the last decades string theory has provided insights into strongly coupled quantum systems Specifically: prediction of existence of new interacting conformal field theories in dimensions D > 4
e.g.: [Seiberg 1996]
String theory also predicts the existence of new ‘non-local theories’, e.g. little string theories (LSTs)
String theory
quantum field theory
suitable decoupling of gravity suitable decoupling of gravity
different approaches: [Witten 1995] [Aspinwall, Morrison 1997] [Seiberg 1997] [Intriligator 1997] [Hanany, Zaffaroni 1997] [Brunner, Karch 1997]
little string theory
Mstring
⌧ Mstring
> Mstring
effective QFT (point-like dofs) UV-comp. contains stringy dofs
Rich class of examples realised in 11
systems of parallel M5-branes with M2-branes stretched between them
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [SH, Iqbal, Rey 2015] [Haghighat 2015] [Haghighat, Murthy, Vafa, Vandoren 2015]
Classification of LSTs (ADE type for theories with supersymmetry )
N = (2, 0)
[Bhardwaj, Del Zotto, Heckman, Morrison, Rudelius, Vafa 2016]
String theory
M-branes (M2- and M5) are extended in objects in 11
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes
stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT
N = (2, 0)
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs many dual realisations allowing to explicitly compute quantities (e.g. partition function) String-like objects arise at the intersection of M5- and M2-branes
stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT
N = (2, 0)
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs many dual realisations allowing to explicitly compute quantities (e.g. partition function) String-like objects arise at the intersection of M5- and M2-branes
stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT
N = (2, 0)
notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds
[Morrison, Vafa 1996] [SH, Iqbal, Rey 2015] [Bhardwaj, Del Zotto, Heckman, Morrison, Rudelius, Vafa 2016] [Heckman, Morrison, Vafa 2013] [Del Zotto, Heckman, Tomasiello, Vafa 2014] [Heckman 2014] [Haghighat, Klemm, Lockhart, Vafa 2014] [Heckman, Morrison, Rudelius, Vafa 2015]
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs many dual realisations allowing to explicitly compute quantities (e.g. partition function) String-like objects arise at the intersection of M5- and M2-branes
stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT
N = (2, 0)
Little Strings depending on the details of the brane configuration, a large class of different (or their duals) can be realised and studied very explicitly notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs many dual realisations allowing to explicitly compute quantities (e.g. partition function) String-like objects arise at the intersection of M5- and M2-branes
stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT
N = (2, 0)
low energy limit associated with non-abelian supersymmetric field theories (mass deformed theories upon compactification to 4 dimensions)
N = 2∗
Little Strings depending on the details of the brane configuration, a large class of different (or their duals) can be realised and studied very explicitly notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs many dual realisations allowing to explicitly compute quantities (e.g. partition function) String-like objects arise at the intersection of M5- and M2-branes
stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT
N = (2, 0)
low energy limit associated with non-abelian supersymmetric field theories (mass deformed theories upon compactification to 4 dimensions)
N = 2∗
Little Strings depending on the details of the brane configuration, a large class of different (or their duals) can be realised and studied very explicitly Class of theories exhibits interesting (and non-expected) dualities! notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds
M-branes (M2- and M5) are extended in objects in 11
They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs many dual realisations allowing to explicitly compute quantities (e.g. partition function) String-like objects arise at the intersection of M5- and M2-branes
stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT
N = (2, 0)
low energy limit associated with non-abelian supersymmetric field theories (mass deformed theories upon compactification to 4 dimensions)
N = 2∗
Little Strings depending on the details of the brane configuration, a large class of different (or their duals) can be realised and studied very explicitly Class of theories exhibits interesting (and non-expected) dualities! notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds
in this talk: triality
Little String Theories with 16 supercharges (A-series)
IIb LST of type with supersymmetry
while 6-dimensional systems:
gst → 0 `st = fixed
AN−1
N = (2, 0)
Little String Theories with 16 supercharges (A-series)
IIb LST of type with supersymmetry
while 6-dimensional systems:
gst → 0 `st = fixed
AN−1 AN−1
N = (2, 0)
R4
Little String Theories with 16 supercharges (A-series)
IIa LST of type with supersymmetry
IIb LST of type with supersymmetry
while 6-dimensional systems:
gst → 0 `st = fixed
AN−1 AN−1 AN−1 AN−1
N = (1, 1)
N = (2, 0)
R4
R4
Little String Theories with 16 supercharges (A-series)
IIa LST of type with supersymmetry
IIb LST of type with supersymmetry
related by T-duality
while 6-dimensional systems:
gst → 0 `st = fixed
AN−1 AN−1 AN−1 AN−1
N = (1, 1)
N = (2, 0)
R4
R4
Little String Theories with 16 supercharges (A-series)
IIa LST of type with supersymmetry
IIb LST of type with supersymmetry
related by T-duality BPS states from the point of view of M5-branes correspond to M2-branes ending on them
while 6-dimensional systems:
gst → 0 `st = fixed
AN−1 AN−1 AN−1 AN−1
N = (1, 1)
N = (2, 0)
R4
R4
Little String Theories with 8 supercharges: particular class obtained as
ZM
ZN
AN−1
AM−1 N = (1, 0)
N = (1, 0)
S1 × ALEAM−1 R4/ZN
R4/ZM
while 6-dimensional systems:
gst → 0 `st = fixed
related by T-duality
Little String Theories with 8 supercharges: particular class obtained as
ZM
ZN
AN−1
AM−1 N = (1, 0)
N = (1, 0)
S1 × ALEAM−1 R4/ZN
R4/ZM
while 6-dimensional systems:
gst → 0 `st = fixed
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [SH, Iqbal, Rey 2015]
related by T-duality Explicit computation of BPS partition function using various methods
Little String Theories with 8 supercharges: particular class obtained as
ZM
ZN
AN−1
AM−1 N = (1, 0)
N = (1, 0)
S1 × ALEAM−1 R4/ZN
R4/ZM
while 6-dimensional systems:
gst → 0 `st = fixed
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [SH, Iqbal, Rey 2015]
in this talk: further dualities
related by T-duality Explicit computation of BPS partition function using various methods
The most general configuration of branes in M-theory in 11 dimensions looks like
ALEAM−1∼R4/ZM
R4
||
The most general configuration of branes in M-theory in 11 dimensions looks like
non-compact case: R
M5-branes distributed along non-comp. (6)-direction with M2-branes stretched between them
x6
. . .
tf1 tfN−1
ALEAM−1∼R4/ZM
R4
||
The most general configuration of branes in M-theory in 11 dimensions looks like
non-compact case: R
M5-branes distributed along non-comp. (6)-direction with M2-branes stretched between them M5-branes arranged on a circle
compact case: S1
x6
. . .
tf1 tfN−1
× × × × × ×
x6 tf1 tf2 tfN
. . . . . .
R6 = ρ 2πi
ALEAM−1∼R4/ZM
R4
||
The most general configuration of branes in M-theory in 11 dimensions looks like
non-compact case: R
M5-branes distributed along non-comp. (6)-direction with M2-branes stretched between them M5-branes arranged on a circle necessary for little-string interpretation
compact case: S1
x6
. . .
tf1 tfN−1
× × × × × ×
x6 tf1 tf2 tfN
. . . . . .
×
tensionful string going around S1 limit where all M5-branes form a single stack
R6 = ρ 2πi
ALEAM−1∼R4/ZM
R4
||
The most general configuration of branes in M-theory in 11 dimensions looks like
Compactification:
Compactify (0,1) to with radii and
T 2 ∼ S1 × S1
R0
R1 =: τ 2πi
ALEAM−1∼R4/ZM
T 2∼S1×S1 |
R4
||
The most general configuration of branes in M-theory in 11 dimensions looks like
Compactification: Deformations:
Compactify (0,1) to with radii and
T 2 ∼ S1 × S1
introducing complex coordinates and
(z1, z2) = (x2 + ix3, x4 + ix5)
(w1, w2) = (x7 + ix8, x9 + ix10)
there are two types of deformations with respect to the compactified (0,1)-directions
U(1)✏1 × U(1)✏2 : (z1, z2) → (e2⇡i✏1z1, e2⇡i✏2z2) and (w1, w2) → (e−i⇡(✏1+✏2)w1, e−i⇡(✏1+✏2)w2)
(0)-direct.: (1)-direct.:
U(1)m : (w1, w2) → (e2πimw1, e−2πimw2)
R0
R1 =: τ 2πi
The most general configuration of branes in M-theory in 11 dimensions looks like
Compactification: Deformations:
Compactify (0,1) to with radii and
T 2 ∼ S1 × S1
introducing complex coordinates and
(z1, z2) = (x2 + ix3, x4 + ix5)
(w1, w2) = (x7 + ix8, x9 + ix10)
gauge theory: Omega-background mass-deformation
there are two types of deformations with respect to the compactified (0,1)-directions
U(1)✏1 × U(1)✏2 : (z1, z2) → (e2⇡i✏1z1, e2⇡i✏2z2) and (w1, w2) → (e−i⇡(✏1+✏2)w1, e−i⇡(✏1+✏2)w2)
(0)-direct.: (1)-direct.:
U(1)m : (w1, w2) → (e2πimw1, e−2πimw2)
R0
R1 =: τ 2πi
[Nekrasov 2012]
For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB
m = 0
For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB
m = 0
1 2 3 4 5 6 7 8 9 D5 branes
NS5 branes
{z }
gauge theory
| {z }
transverse R3
| {z }
(p,q)−plane
For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB
m = 0
1 2 3 4 5 6 7 8 9 D5 branes
NS5 branes
{z }
gauge theory
| {z }
transverse R3
| {z }
(p,q)−plane
=1 =2 =3 =M =1 =2 =3 =M |
1
|
2
|
3
|
N
|
1
|
2
|
3
|
N
For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB
m = 0
1 2 3 4 5 6 7 8 9 D5 branes
NS5 branes
{z }
gauge theory
| {z }
transverse R3
| {z }
(p,q)−plane
=1 =2 =3 =M =1 =2 =3 =M |
1
|
2
|
3
|
N
|
1
|
2
|
3
|
N
NS5-branes
N M D5-branes
For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB
m = 0
1 2 3 4 5 6 7 8 9 D5 branes
NS5 branes
{z }
gauge theory
| {z }
transverse R3
| {z }
(p,q)−plane
=1 =2 =3 =M =1 =2 =3 =M |
1
|
2
|
3
|
N
|
1
|
2
|
3
|
N
Deformation:
= ⇒ (1,1) brane
For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB
m = 0
1 2 3 4 5 6 7 8 9 D5 branes
NS5 branes
{z }
gauge theory
| {z }
transverse R3
| {z }
(p,q)−plane
=1 =2 =3 =M =1 =2 =3 =M |
1
|
2
|
3
|
N
|
1
|
2
|
3
|
N
Deformation:
= ⇒ (1,1) brane
uplift the deformed type II configuration to M-th.
[Leung, Vafa 1997]
For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB
m = 0
1 2 3 4 5 6 7 8 9 D5 branes
NS5 branes
{z }
gauge theory
| {z }
transverse R3
| {z }
(p,q)−plane
=1 =2 =3 =M =1 =2 =3 =M |
1
|
2
|
3
|
N
|
1
|
2
|
3
|
N
Deformation:
= ⇒ (1,1) brane
uplift the deformed type II configuration to M-th.
[Leung, Vafa 1997]
topic diagram of same as deformed brane web
XN,M
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
legs N
M legs
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
web on a torus
(N, M)
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
web on a torus
(N, M)
double elliptic fibration structure with parameters (ρ, τ) · · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
ρ τ
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
web on a torus
(N, M)
different parameters representing
3NM
the area of various curves of the CY3
C
d = Z
C
ω double elliptic fibration structure with parameters (ρ, τ) · · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
ρ τ
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
web on a torus
(N, M)
different parameters representing
3NM
the area of various curves of the CY3
C
d = Z
C
ω
Kähler form
double elliptic fibration structure with parameters (ρ, τ) · · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
ρ τ
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
web on a torus
(N, M)
different parameters representing
3NM
the area of various curves of the CY3
NM
NM
NM
C
d = Z
C
ω
Kähler form
double elliptic fibration structure with parameters (ρ, τ) · · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN ρ τ
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
web on a torus
(N, M)
different parameters representing
3NM
the area of various curves of the CY3
NM
NM
NM
C
d = Z
C
ω
Kähler form
double elliptic fibration structure with parameters (ρ, τ)
NM + 2
parameters due to consistency conditions · · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN ρ τ
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
web on a torus
(N, M)
different parameters representing
3NM
the area of various curves of the CY3
NM
NM
NM
C
d = Z
C
ω
Kähler form
double elliptic fibration structure with parameters (ρ, τ)
NM + 2
parameters due to consistency conditions · · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN ρ τ
Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds XN,M
Toric Web Diagram:
web on a torus
(N, M)
different parameters representing
3NM
the area of various curves of the CY3
NM
NM
NM
C
d = Z
C
ω
Kähler form
double elliptic fibration structure with parameters (ρ, τ)
NM + 2
parameters due to consistency conditions
h h0 v v0 m m0
different possible choices for set of independent parameters
h + m = h0 + m0 v + m0 = m + v0
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 .
XN,M
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
[Aganagic, Klemm, Marino, Vafa 2003] [Iqbal, Kozçaz, Vafa 2007]
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
assign trivalent vertex to each intersection λ µ ν
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
assign trivalent vertex to each intersection
Cλµν = q
||µ||2 2
t− ||µt||2
2
q
||ν||2 2
˜ Zν(t, q) X
η
⇣q t ⌘ |η|+|λ|−|µ|
2
× sλt/η(t−ρq−ν) sµ/η(q−ρt−νt)
λ µ ν
Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
,
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
assign trivalent vertex to each intersection
Cλµν = q
||µ||2 2
t− ||µt||2
2
q
||ν||2 2
˜ Zν(t, q) X
η
⇣q t ⌘ |η|+|λ|−|µ|
2
× sλt/η(t−ρq−ν) sµ/η(q−ρt−νt)
λ µ ν
Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
,
Notation: and integer partitions
q = e2⇡i✏1 t = e−2⇡i✏2 µ , ν , λ
||µ||2 =
`
X
i=1
µ2
i
|µ| =
`
X
i=1
µi
µi
`
sµ/η
skew Schur function
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
assign trivalent vertex to each intersection
Cλµν = q
||µ||2 2
t− ||µt||2
2
q
||ν||2 2
˜ Zν(t, q) X
η
⇣q t ⌘ |η|+|λ|−|µ|
2
× sλt/η(t−ρq−ν) sµ/η(q−ρt−νt)
˜ Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
,
glue vertices according to web diagram
λ1 λ2 µ1 µ2 ν m
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
assign trivalent vertex to each intersection
Cλµν = q
||µ||2 2
t− ||µt||2
2
q
||ν||2 2
˜ Zν(t, q) X
η
⇣q t ⌘ |η|+|λ|−|µ|
2
× sλt/η(t−ρq−ν) sµ/η(q−ρt−νt)
˜ Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
,
glue vertices according to web diagram
λ1 λ2 µ1 µ2 ν m
X
ν
(−e2πim)|ν|Cµ1λ1νCµt
2λt 2νt
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
assign trivalent vertex to each intersection
Cλµν = q
||µ||2 2
t− ||µt||2
2
q
||ν||2 2
˜ Zν(t, q) X
η
⇣q t ⌘ |η|+|λ|−|µ|
2
× sλt/η(t−ρq−ν) sµ/η(q−ρt−νt)
˜ Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
,
glue vertices according to web diagram choose preferred direction
X
ν
(−e2πim)|ν|Cµ1λ1νCµt
2λt 2νt
Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
curves on the CY3 .
XN,M
Captured by topological free energy of
FN,M = ln ZN,M
XN,M
Compute the topological string partition function using the
ZN,M
refined topological vertex
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
v1 v2 vN vN+1 vN+2 v2N v2N+1 v2N+2 v3N v(M−1)N+1 v(M−1)N+2 vMN v1 v2 vN hN h1 h2 hN−1 hN h2N hN+1 hN+2 h2N−1 h2N hMN h(M−1)N+1 h(M−1)N+2 hMN−1 hMN m1 m2 mN mN+1 mN+2 m2N m(M−1)N+1 m(M−1)N+2 mMN
assign trivalent vertex to each intersection
Cλµν = q
||µ||2 2
t− ||µt||2
2
q
||ν||2 2
˜ Zν(t, q) X
η
⇣q t ⌘ |η|+|λ|−|µ|
2
× sλt/η(t−ρq−ν) sµ/η(q−ρt−νt)
˜ Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
,
glue vertices according to web diagram choose must be common to all vertices of diagram preferred direction
X
ν
(−e2πim)|ν|Cµ1λ1νCµt
2λt 2νt
preferred direction
preferred direction 3 different choices for the :
preferred direction 3 different choices for the : 1) horizontal:
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
= =
βt M βt M−1 βt 1 αM αM−1 α1 v1 v2 vM−1 vM v1 m1 mM−1 mM
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
= =
βt M βt M−1 βt 1 αM αM−1 α1 v1 v2 vM−1 vM v1 m1 mM−1 mM
W α1...αM
β1...βM ({v}, {m})
building block:
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical:
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips
= =
βt 1 βt 2 βt N α1 α2 αN h1 h2 h3 hN h1 m1 m2 mN
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block:
= =
βt 1 βt 2 βt N α1 α2 αN h1 h2 h3 hN h1 m1 m2 mN
W α1...αN
β1...βN ({h}, {m})
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block: 3) diagonal:
W α1...αN
β1...βN ({h}, {m})
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
· · · · · · · · · · · · · · · · · ·
=
1
=
2
=
M
=
1
=
2
=
M
– 1 – 2 – N – 1 – 2 – N
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block: 3) diagonal: decompose diagram into diagonal strips
W α1...αN
β1...βN ({h}, {m})
= =
α1 α2 α NM k βt 1 βt 2 βt NM k h1 h2 h3 h NM k h1 v1 v2 v NM k
k = gcd(N, M)
where
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block: 3) diagonal: decompose diagram into diagonal strips building block:
W α1...αN
β1...βN ({h}, {m})
W
α1...α NM
k
β1...β NM
k
({h}, {v})
generic form of the building block
α1 α2 α3 βt
1
βt
2
a βt
L−1
βt
L
αL a
b a1 b a2
b aL b b1 b b2
b bL S
W α1...αL
β1...βL = WL(∅) · ˆ
Z ·
L
Y
i,j=1
Jαiβj( b Qi,i−j; q, t)Jβjαi(( b Qi,i−j)−1Qρ; q, t) Jαiαj(Qi,i−j p q/t; q, t)Jβjβi( ˙ Qi,j−i p t/q; q, t)
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block: 3) diagonal: decompose diagram into diagonal strips building block:
W α1...αN
β1...βN ({h}, {m})
W
α1...α NM
k
β1...β NM
k
({h}, {v})
generic form of the building block
α1 α2 α3 βt
1
βt
2
a βt
L−1
βt
L
αL a
b a1 b a2
b aL b b1 b b2
b bL S
W α1...αL
β1...βL = WL(∅) · ˆ
Z ·
L
Y
i,j=1
Jαiβj( b Qi,i−j; q, t)Jβjαi(( b Qi,i−j)−1Qρ; q, t) Jαiαj(Qi,i−j p q/t; q, t)Jβjβi( ˙ Qi,j−i p t/q; q, t)
Jµν(x; t, q) =
∞
Y
k=1
Jµν(Qk−1
ρ
x; t, q) , Jµν(x; t, q) = Y
(i,j)∈µ
⇣ 1 − x tνt
j−i+ 1 2 qµi−j+ 1 2
⌘ × Y
(i,j)∈ν
⇣ 1 − x t−µt
j+i− 1 2 q−νi+j− 1 2
⌘
with
WL(∅) =
L
Y
i,j=1 ∞
Y
k,r,s=1
(1 − b Qi,j Qk−1
ρ
qr− 1
2 ts− 1 2 )(1 − b
Q−1
i,j Qk ρqs− 1
2 tr− 1 2 )
(1 − Qi,jQk−1
ρ
qrts−1)(1 − ˙ Qi,jQk−1
ρ
qs−1tr) , ˆ Z =
L
Y
i=1
t
||αk||2 2
q
||αt k||2 2
˜ Zαk(q, t) ˜ Zαt
k(t, q) ,
˜ Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block: 3) diagonal: decompose diagram into diagonal strips building block:
W α1...αN
β1...βN ({h}, {m})
W
α1...α NM
k
β1...β NM
k
({h}, {v})
generic form of the building block
α1 α2 α3 βt
1
βt
2
a βt
L−1
βt
L
αL a
b a1 b a2
b aL b b1 b b2
b bL S
W α1...αL
β1...βL = WL(∅) · ˆ
Z ·
L
Y
i,j=1
Jαiβj( b Qi,i−j; q, t)Jβjαi(( b Qi,i−j)−1Qρ; q, t) Jαiαj(Qi,i−j p q/t; q, t)Jβjβi( ˙ Qi,j−i p t/q; q, t)
Jµν(x; t, q) =
∞
Y
k=1
Jµν(Qk−1
ρ
x; t, q) , Jµν(x; t, q) = Y
(i,j)∈µ
⇣ 1 − x tνt
j−i+ 1 2 qµi−j+ 1 2
⌘ × Y
(i,j)∈ν
⇣ 1 − x t−µt
j+i− 1 2 q−νi+j− 1 2
⌘
with
Notation: and
b Qi,j = QS
i
Y
r=1
(QarQ−1
br ) j−1
Y
k=1
Qai−k , Qi,j = ⇢ 1 if j = L Qj
k=1 Qai−k
if j 6= L ˙ Qi,j =
j
Y
k=1
Qbi+k
Qai = e−b
ai
Qbi = e−b
bi
QS = e−S
WL(∅) =
L
Y
i,j=1 ∞
Y
k,r,s=1
(1 − b Qi,j Qk−1
ρ
qr− 1
2 ts− 1 2 )(1 − b
Q−1
i,j Qk ρqs− 1
2 tr− 1 2 )
(1 − Qi,jQk−1
ρ
qrts−1)(1 − ˙ Qi,jQk−1
ρ
qs−1tr) , ˆ Z =
L
Y
i=1
t
||αk||2 2
q
||αt k||2 2
˜ Zαk(q, t) ˜ Zαt
k(t, q) ,
˜ Zν(t, q) = Y
(i,j)∈ν
⇣ 1 − tνt
j−i+1qνi−j⌘−1
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block: 3) diagonal: decompose diagram into diagonal strips building block:
W α1...αN
β1...βN ({h}, {m})
W
α1...α NM
k
β1...β NM
k
({h}, {v})
generic form of the building block
α1 α2 α3 βt
1
βt
2
a βt
L−1
βt
L
αL a
b a1 b a2
b aL b b1 b b2
b bL S
W α1...αL
β1...βL = WL(∅) · ˆ
Z ·
L
Y
i,j=1
Jαiβj( b Qi,i−j; q, t)Jβjαi(( b Qi,i−j)−1Qρ; q, t) Jαiαj(Qi,i−j p q/t; q, t)Jβjβi( ˙ Qi,j−i p t/q; q, t)
preferred direction 3 different choices for the : 1) horizontal: decompose diagram into vertical strips
W α1...αM
β1...βM ({v}, {m})
building block: 2) vertical: decompose diagram into horizontal strips building block: 3) diagonal: decompose diagram into diagonal strips building block:
W α1...αN
β1...βN ({h}, {m})
W
α1...α NM
k
β1...β NM
k
({h}, {v})
generic form of the building block
α1 α2 α3 βt
1
βt
2
a βt
L−1
βt
L
αL a
b a1 b a2
b aL b b1 b b2
b bL S
W α1...αL
β1...βL = WL(∅) · ˆ
Z ·
L
Y
i,j=1
Jαiβj( b Qi,i−j; q, t)Jβjαi(( b Qi,i−j)−1Qρ; q, t) Jαiαj(Qi,i−j p q/t; q, t)Jβjβi( ˙ Qi,j−i p t/q; q, t)
suitable for all three expansions upon identifying: horizontal vertical diagonal b ai vi+1 + mi hi + mi vi + hi+1 b bi vi + mi hi + mi−1 hi + vi S v1 mN h1 L M N
NM k
Alternative view on the three gauge theories: Newton polygons as dual of web diagrams
Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: (N, M) = (3, 2)
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1=
2=
1=
2− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h5 h3 h1 h2 h6 h4 v1 v5 v3 v4 v2 v6
IV II VI I V III
1 4 2 5 3 6 7 10 8 11 9 12
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· · · · · · · · · · · ·
1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12
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1=
2=
1=
2− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h5 h3 h1 h2 h6 h4 v1 v5 v3 v4 v2 v6
IV II VI I V III
1 4 2 5 3 6 7 10 8 11 9 12
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· · · · · · · · · · · ·
1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12
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1=
2=
1=
2− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h5 h3 h1 h2 h6 h4 v1 v5 v3 v4 v2 v6
IV II VI I V III
1 4 2 5 3 6 7 10 8 11 9 12
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1=
2=
1=
2− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h5 h3 h1 h2 h6 h4 v1 v5 v3 v4 v2 v6
IV II VI I V III
1 4 2 5 3 6 7 10 8 11 9 12
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β1β2β3β4β5β6
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1=
2=
1=
2− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h5 h3 h1 h2 h6 h4 v1 v5 v3 v4 v2 v6
IV II VI I V III
1 4 2 5 3 6 7 10 8 11 9 12
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· · · · · · · · · · · ·
1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12
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Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: (N, M) = (3, 2)
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1=
2=
1=
2− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h5 h3 h1 h2 h6 h4 v1 v5 v3 v4 v2 v6
IV II VI I V III
1 4 2 5 3 6 7 10 8 11 9 12
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1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12 1 2 3 1 2 3 4 5 6 4 5 6 7 8 9 7 8 9 10 11 12 10 11 12
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The full partition function is obtained by gluing together the building blocks W α1...αM
β1...βM
ZN,M = X
α
⇣
M,N
Y
i=1,j=1
e−uij |αi
j|⌘
N
Y
j=1
W
α1
j···αM j
α1
j+1···αM j+1
The full partition function is obtained by gluing together the building blocks W α1...αM
β1...βM
ZN,M = X
α
⇣
M,N
Y
i=1,j=1
e−uij |αi
j|⌘
N
Y
j=1
W
α1
j···αM j
α1
j+1···αM j+1
parameters used to glue the strips together
The full partition function is obtained by gluing together the building blocks W α1...αM
β1...βM
ZN,M = X
α
⇣
M,N
Y
i=1,j=1
e−uij |αi
j|⌘
N
Y
j=1
W
α1
j···αM j
α1
j+1···αM j+1
parameters used to glue the strips together
ZN,M({h}, {v}, {m}, ✏1,2) = Zp({v}, {m}) X
~ k
e−~
k·h Z~ k({v}, {m}) = Z(N,M) hor
= Zp({h}, {m}) X
~ k
e−~
k·v Z~ k({h}, {m}) = Z(N,M) vert
= Zp({h}, {v}) X
~ k
e−~
k·m Z~ k({h}, {v}) = Z(N,M) diag
Different choices of preferred direction afford different (but equivalent) expansions:
The full partition function is obtained by gluing together the building blocks W α1...αM
β1...βM
ZN,M = X
α
⇣
M,N
Y
i=1,j=1
e−uij |αi
j|⌘
N
Y
j=1
W
α1
j···αM j
α1
j+1···αM j+1
parameters used to glue the strips together
ZN,M({h}, {v}, {m}, ✏1,2) = Zp({v}, {m}) X
~ k
e−~
k·h Z~ k({v}, {m}) = Z(N,M) hor
= Zp({h}, {m}) X
~ k
e−~
k·v Z~ k({h}, {m}) = Z(N,M) vert
= Zp({h}, {v}) X
~ k
e−~
k·m Z~ k({h}, {v}) = Z(N,M) diag
Different choices of preferred direction afford different (but equivalent) expansions:
common normalisation factor (perturbative partition function)
The full partition function is obtained by gluing together the building blocks W α1...αM
β1...βM
ZN,M = X
α
⇣
M,N
Y
i=1,j=1
e−uij |αi
j|⌘
N
Y
j=1
W
α1
j···αM j
α1
j+1···αM j+1
parameters used to glue the strips together
ZN,M({h}, {v}, {m}, ✏1,2) = Zp({v}, {m}) X
~ k
e−~
k·h Z~ k({v}, {m}) = Z(N,M) hor
= Zp({h}, {m}) X
~ k
e−~
k·v Z~ k({h}, {m}) = Z(N,M) vert
= Zp({h}, {v}) X
~ k
e−~
k·m Z~ k({h}, {v}) = Z(N,M) diag
Different choices of preferred direction afford different (but equivalent) expansions:
common normalisation factor (perturbative partition function)
Compare different series expansions with instanton partition functions of quiver gauge theories.
The full partition function is obtained by gluing together the building blocks W α1...αM
β1...βM
ZN,M = X
α
⇣
M,N
Y
i=1,j=1
e−uij |αi
j|⌘
N
Y
j=1
W
α1
j···αM j
α1
j+1···αM j+1
parameters used to glue the strips together
ZN,M({h}, {v}, {m}, ✏1,2) = Zp({v}, {m}) X
~ k
e−~
k·h Z~ k({v}, {m}) = Z(N,M) hor
= Zp({h}, {m}) X
~ k
e−~
k·v Z~ k({h}, {m}) = Z(N,M) vert
= Zp({h}, {v}) X
~ k
e−~
k·m Z~ k({h}, {v}) = Z(N,M) diag
Different choices of preferred direction afford different (but equivalent) expansions:
common normalisation factor (perturbative partition function)
Compare different series expansions with instanton partition functions of quiver gauge theories. Need to choose independent Kähler parameters of XN,M
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2)
=
1
=
2
=
1
=
2
− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
=
1
=
2
=
1
=
2
− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
b c1 b c2 b c3 b b1 b b2
ρ τ E = m1 + m2 + m3
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
a b c d 1 1 m1 m4 v1 v1 v4 e f c d 2 2 m2 m5 v2 v2 v5 3 3 v3 v3 v6 m3 m6 e f a b
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
a b c d 1 1 m1 m4 v1 v1 v4 e f c d 2 2 m2 m5 v2 v2 v5 3 3 v3 v3 v6 m3 m6 e f a b
gauge theory: U(2) × U(2) × U(2)
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
2) vertical: (τ, b c1;b b1,b b2,b b3,b b4; ρ, D)
=
1
=
2
=
1
=
2
− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
b c1 b b1 b b2 b b3 b b4
ρ τ D = m1 + m4
gauge theory: U(2) × U(2) × U(2)
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
2) vertical: (τ, b c1;b b1,b b2,b b3,b b4; ρ, D) series expansion: τ − b
c1 − → ∞ b c2 − → ∞
a b a b 1 2 3 4 5 6 1 2 3 4 5 6 m1 m2 m3 m4 m5 m6 h1 h2 h3 h3 h4 h5 h6 h6
gauge theory: U(2) × U(2) × U(2)
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
2) vertical: (τ, b c1;b b1,b b2,b b3,b b4; ρ, D) series expansion: τ − b
c1 − → ∞ b c2 − → ∞
a b a b 1 2 3 4 5 6 1 2 3 4 5 6 m1 m2 m3 m4 m5 m6 h1 h2 h3 h3 h4 h5 h6 h6
gauge theory: U(2) × U(2) × U(2) gauge theory: U(3) × U(3)
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
2) vertical: (τ, b c1;b b1,b b2,b b3,b b4; ρ, D) series expansion: τ − b
c1 − → ∞ b c2 − → ∞
3) diagonal:
=
1
=
2
=
1
=
2
− 1 − 2 − 3 − 1 − 2 − 3
m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6 b a1 b a2 b a3 b a4 b a5 F = v1 + v4 M V = m1 + (3 − 1)(h1 + h3) + (3 − 2)(v2 + h5 + v6 + h5)
(V ; b a1, b a2, b a3, b a4, b a5; M, F)
gauge theory: U(2) × U(2) × U(2) gauge theory: U(3) × U(3)
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
2) vertical: (τ, b c1;b b1,b b2,b b3,b b4; ρ, D) series expansion: τ − b
c1 − → ∞ b c2 − → ∞
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F)
series expansion: V −
→ ∞
IV V VI I II III I II III IV V VI a a h3 h4 h2 h6 h1 h5 h3 v1 v5 v3 v4 v2 v6
gauge theory: U(2) × U(2) × U(2) gauge theory: U(3) × U(3)
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
2) vertical: (τ, b c1;b b1,b b2,b b3,b b4; ρ, D) series expansion: τ − b
c1 − → ∞ b c2 − → ∞
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F)
series expansion: V −
→ ∞
IV V VI I II III I II III IV V VI a a h3 h4 h2 h6 h1 h5 h3 v1 v5 v3 v4 v2 v6
gauge theory: U(2) × U(2) × U(2) gauge theory: U(3) × U(3) gauge theory: U(6)
For each of the expansion we can choose a suitable set of independent Kähler parameters:
NM + 2
Example: (N, M) = (3, 2) 1) horizontal: (ρ,b
b1,b b2; b c1, b c2, b c3; τ, E)
series expansion: ρ − b
b1 − b b2 − → ∞
b b1 − → ∞ b b2 − → ∞
2) vertical: (τ, b c1;b b1,b b2,b b3,b b4; ρ, D) series expansion: τ − b
c1 − → ∞ b c2 − → ∞
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F)
series expansion: V −
→ ∞
IV V VI I II III I II III IV V VI a a h3 h4 h2 h6 h1 h5 h3 v1 v5 v3 v4 v2 v6
Similar sets of independent Kähler parameters proposed for generic (N, M)
[Bastian, SH, Iqbal, Rey 2017]
gauge theory: U(2) × U(2) × U(2) gauge theory: U(3) × U(3) gauge theory: U(6)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 2 s s + 1 N − 1 N 1 2 s s + 1 N − 1 N a1 ar+1 ar+2 aM a1 ar+1 ar+2 aM m
1m
2m
sm
s+1m
N−1m
Nm
rN+1m
rN+2m
rN+sm
rN+s+1m
(r+1)N−1m
(r+1)Nm
(r+1)N+1m
(r+1)N+2m
(r+1)N+sm
(r+1)N+s+1m
(r+2)N−1m
(r+2)Nm
(M−1)N+1m
(M−1)N+2m
(M−1)N+sm
(M−1)N+s+1m
MN−1m
MNv1 v2 vs vs+1 vN−1 vN vN+1 vN+2 vN+s vN+s+1 v2N−1 v2N vrN+1 vrN+2 vrN+s vrN+s+1 v(r+1)N−1 v(r+1)N v
(r+1)N+1v(r+1)N+2 v(r+1)N+s v(r+1)N+s+1 v(r+2)N−1 v(r+2)N v
(r+2)N+1v(r+2)N+2 v(r+2)N+s v(r+2)N+s+1 v(r+3)N−1 v(r+3)N v(M−1)N+1 v(M−1)N+2 v(M−1)N+s v(M−1)N+s+1 vMN−1 vMN v1 v2 vs vs+1 vN−1 vN hN h1 h2 hs−1 hs hs+1 hN−2 hN−1 hN h(r+1)N hrN+1 hrN+2 hrN+s−1 hrN+s hrN+s+1 h(r+1)N−2 h(r+1)N−1 h(r+1)N h(r+2)N h(r+1)N+1 h(r+1)N+2 h(r+1)N+s−1 h(r+1)N+s h(r+1)N+s+1 h(r+2)N−2 h(r+2)N−1 h(r+2)N hMN h(M−1)N+1 h(M−1)N+2 h(M−1)N+s−1 h(M−1)N+2 h(M−1)N+s h(M−1)N+s+1 hMN−2 hMN−1 hMN b bM−1,1 b bM−1,2 b bM−1,s b bM−1,s+1 b bM−1,N−1 b b0,1 b b0,s b b0,N−1 b c0,N b cr,N b cM−2,N b c0,1 b cr,1 b cM−2,1 b aM−1,N b aM−1,N b aM−2,1 b a0,N b aM−1,1 b aM−1,1 b aM−2,2 b a
r+1,Nb ar−1,1 b ar+2,N
b ar,1
D E ρ τ F L
hrN+s v(r+1)N+s m(r+1)N+s h(r+1)N+s mrN+s+1 v(r+1)N+s+1
b
cr,s
b
br,s
b
ar,s
Va+1 = m1+aN + ✓N k − 1 ◆ ((pL(m1+aN))1 + (pR(m1+aN))1) +
N k −2X
i=1
✓N k − 1 − i ◆ [(pL(m1+aN))2i + (pR(m1+aN))2i] +
N k −2X
i=1
✓N k − 1 − i ◆ [(pL(m1+aN))2i+1 + (pR(m1+aN))2i+1] .
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
U(2) U(2) U(2)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
U(2) U(2) U(2)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
U(2) U(2) U(2)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
U(2) U(2) U(2)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
U(2) U(2) U(2)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F) gauge theory with gauge group U(6)
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F) gauge theory with gauge group U(6)
Z(3,2)
diag
can be written as a series expansion in related to the instanton parameters
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F) gauge theory with gauge group U(6)
Z(3,2)
diag
can be written as a series expansion in related to the instanton parameters interpreted as simple, positive roots of
b a1,2,3,4,5
a5
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F) gauge theory with gauge group U(6)
Z(3,2)
diag
can be written as a series expansion in related to the instanton parameters interpreted as simple, positive roots of
b a1,2,3,4,5
a5
b a5
interpreted as imaginary root extending to
F
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F) gauge theory with gauge group U(6)
Z(3,2)
diag
can be written as a series expansion in related to the instanton parameters interpreted as simple, positive roots of
b a1,2,3,4,5
a5
b a5
interpreted as imaginary root extending to
F
Horizontal and vertical gauge theory interpretation well known in the literature
[Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013]
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F) gauge theory with gauge group U(6)
Z(3,2)
diag
can be written as a series expansion in related to the instanton parameters interpreted as simple, positive roots of
b a1,2,3,4,5
a5
b a5
interpreted as imaginary root extending to
F
Horizontal and vertical gauge theory interpretation well known in the literature Diagonal expansion leads to novel gauge theory associated with XN,M
U(2) U(2) U(2) U(3) U(3)
related to the instanton parameters
e2πib
b2
series expansion in , and
e2πi(ρ−b
b1−b b2) e2πib b1
Z(3,2)
hor
b c1,2,3 interpreted as simple, positive roots of three copies of a1
a1
b a1
2) vertical: (τ, b
c1;b b1,b b2,b b3,b b4; ρ, D) : quiver gauge theory
U(3) × U(3)
1) horizontal: : quiver gauge theory (ρ,b b1,b b2; b c1, b c2, b c3; τ, E)
U(2) × U(2) × U(2)
the instanton parameters series expansion in and related to
Z(3,2)
vert
c1)
c1
interpreted as simple, positive roots of two copies of
b b1,2,3,4
a2
a2
b a2
3) diagonal: (V ; b
a1, b a2, b a3, b a4, b a5; M, F) gauge theory with gauge group U(6)
Z(3,2)
diag
can be written as a series expansion in related to the instanton parameters interpreted as simple, positive roots of
b a1,2,3,4,5
a5
b a5
interpreted as imaginary root extending to
F
Horizontal and vertical gauge theory interpretation well known in the literature Diagonal expansion leads to novel gauge theory associated with XN,M
Triality!
[Bastian, SH, Iqbal, Rey 2017]
U(2) U(2) U(2) U(3) U(3)
Flop transition for any two curves in the diagram:
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
Cut diagram along dashed lines and re-glue
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Cut diagram along dashed lines and re-glue
h3 v1 h4 v5 h2 v3 h6 v4 h1 v2 h5 v6 h3 m1 m5 m3 m4 m2 m6 m4 m2 m6 m1 m5 m3 1 2 3 4 5 6 1 2 3 4 5 6 a a
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Cut diagram along dashed lines and re-glue
v1 h4 v5 h2 v3 h6 v4 h1 v2 h5 v6 h3 v1 m5 m3 m4 m2 m6 m1 m4 m2 m6 m1 m5 m3 1 2 3 4 5 6 2 3 4 5 6 1 a a
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
v1 h4 v5 h2 v3 h6 v4 h1 v2 h5 v6 h3 v1 m5 m3 m4 m2 m6 m1 m4 m2 m6 m1 m5 m3 1 2 3 4 5 6 2 3 4 5 6 1 a a
SL(2,Z) transformation
(1, 0) − → (1, 1) (0, 1) − → (−1, 0) (1, 1) − → (0, 1)
, ,
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for SL(2,Z) transformation
(1, 0) − → (1, 1) (0, 1) − → (−1, 0) (1, 1) − → (0, 1)
, ,
v1 v5 v3 v4 v2 v6 v1 h4 h2 h6 h1 h5 h3 m4 m2 m6 m1 m5 m3 m5 m3 m4 m2 m6 m1 1 2 3 4 5 6 5 6 1 2 3 4 a a
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
v1 v5 v3 v4 v2 v6 v1 h4 h2 h6 h1 h5 h3 m4 m2 m6 m1 m5 m3 m5 m3 m4 m2 m6 m1 1 2 3 4 5 6 5 6 1 2 3 4 a a
Flop transformation along red lines
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Flop transformation along red lines
a a 1 2 3 4 5 6 5 6 1 2 3 4 −h4 −h2 −h6 −h1 −h5 −h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 m5 + h4 + h5 m3 + h2 + h3 m5 + h4 + h5 m3 + h2 + h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 v1 + h3 + h4 v5 + h2 + h4 v3 + h2 + h6 v4 + h1 + h6 v2 + h1 + h5 v6 + h3 + h5 v1 + h3 + h4
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
a a 1 2 3 4 5 6 5 6 1 2 3 4 −h4 −h2 −h6 −h1 −h5 −h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 m5 + h4 + h5 m3 + h2 + h3 m5 + h4 + h5 m3 + h2 + h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 v1 + h3 + h4 v5 + h2 + h4 v3 + h2 + h6 v4 + h1 + h6 v2 + h1 + h5 v6 + h3 + h5 v1 + h3 + h4
SL(2,Z) transformation
, ,
(1, 0) − → (1, −1)
(0, 1) − → (0, 1) (1, 1) − → (1, 0)
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for SL(2,Z) transformation
, ,
(1, 0) − → (1, −1)
(0, 1) − → (0, 1) (1, 1) − → (1, 0)
v1 + h3 + h4 v5 + h2 + h4 v3 + h2 + h6 v4 + h1 + h6 v2 + h1 + h5 v6 + h3 + h5 v1 + h3 + h4 −h4 −h2 −h6 −h1 −h5 −h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 m5 + h4 + h5 m3 + h2 + h3 m5 + h4 + h5 m3 + h2 + h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 a a 1 2 3 4 5 6 5 6 1 2 3 4
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
v1 + h3 + h4 v5 + h2 + h4 v3 + h2 + h6 v4 + h1 + h6 v2 + h1 + h5 v6 + h3 + h5 v1 + h3 + h4 −h4 −h2 −h6 −h1 −h5 −h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 m5 + h4 + h5 m3 + h2 + h3 m5 + h4 + h5 m3 + h2 + h3 m4 + h4 + h6 m2 + h1 + h2 m6 + h5 + h6 m1 + h1 + h3 a a 1 2 3 4 5 6 5 6 1 2 3 4
Flop transformation along red lines
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Flop transformation along red lines
a a 1 2 3 4 5 6 5 6 1 2 3 4 v
1
+ h
3
+ h
4
− v
5
− h
2
− h
4
− v
3
− h
2
− h
6
− v
4
− h
1
− h
6
− v
2
− h
1
− h
5
− v
6
− h
3
− h
5
v
1
+ h
3
+ h
4
v5 + h2 v6 + h5 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 m0
5
m0
6
m0
1
m0
2
m0
3
m0
4
m0
1
m0
2
m0
3
m0
4
m0
5
m0
6
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
a a 1 2 3 4 5 6 5 6 1 2 3 4 v
1
+ h
3
+ h
4
− v
5
− h
2
− h
4
− v
3
− h
2
− h
6
− v
4
− h
1
− h
6
− v
2
− h
1
− h
5
− v
6
− h
3
− h
5
v
1
+ h
3
+ h
4
v5 + h2 v6 + h5 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 m0
5
m0
6
m0
1
m0
2
m0
3
m0
4
m0
1
m0
2
m0
3
m0
4
m0
5
m0
6
Cut diagram along dashed line and re-glue
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Cut diagram along dashed line and re-glue
a a 6 1 2 3 4 5 5 6 1 2 3 4 v1 + h3 + h4 − v5 − h2 − h4 − v3 − h2 − h6 − v4 − h1 − h6 − v2 − h1 − h5 − v6 − h3 − h5 v5 + h2 v6 + h5 v6 + h5 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 m0
5
m0
6
m0
1
m0
2
m0
3
m0
4
m0
6
m0
1
m0
2
m0
3
m0
4
m0
5
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for
a a 6 1 2 3 4 5 5 6 1 2 3 4 v1 + h3 + h4 − v5 − h2 − h4 − v3 − h2 − h6 − v4 − h1 − h6 − v2 − h1 − h5 − v6 − h3 − h5 v5 + h2 v6 + h5 v6 + h5 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 m0
5
m0
6
m0
1
m0
2
m0
3
m0
4
m0
6
m0
1
m0
2
m0
3
m0
4
m0
5
Flop transformation along red line
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Flop transformation along red line
a a 6 1 2 3 4 5 5 6 1 2 3 4 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 v1 + v6 + h3 + h4 + h5 v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 m00
5m00
6m0
1m0
2m0
3m0
4m00
6m0
1m0
2m0
3m0
4m00
5[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for SL(2,Z) transformation
, ,
(1, 0) − → (1, −1)
(0, 1) − → (0, 1) (1, 1) − → (1, 0)
a a 6 1 2 3 4 5 5 6 1 2 3 4 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 v1 + v6 + h3 + h4 + h5 v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 m00
5m00
6m0
1m0
2m0
3m0
4m00
6m0
1m0
2m0
3m0
4m00
5[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for SL(2,Z) transformation
, ,
(1, 0) − → (1, −1)
(0, 1) − → (0, 1) (1, 1) − → (1, 0)
v1 + v6 + h3 + h4 + h5 v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 m00
6
m0
1
m0
2
m0
3
m0
4
m00
5
m00
5
m00
6
m0
1
m0
2
m0
3
m0
4
a a 6 1 2 3 4 5 5 6 1 2 3 4
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Cut diagram along dashed line and re-glue
v1 + v6 + h3 + h4 + h5 v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 m00
6
m0
1
m0
2
m0
3
m0
4
m00
5
m00
5
m00
6
m0
1
m0
2
m0
3
m0
4
a a 6 1 2 3 4 5 5 6 1 2 3 4
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
X3,2 ∼ X6,1
Flop transition for any two curves in the diagram: Series of flop and SL(2,Z) transformations for Cut diagram along dashed line and re-glue
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6
m0
1
m0
2
m0
3
m0
4
m00
5
m00
6
m0
1
m0
2
m0
3
m0
4
m00
5
a a 6 1 2 3 4 5 6 1 2 3 4 5
[SH, Iqbal, Rey 2016]
hi vi m hj vj
= ⇒
hi + m vj + m −m hj + m vi + m
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
Duality leaves partiton function invariant
Z3,2({h}, {v}, {m}, ✏1,2) = Z6,1({h0}, {v0}, {m0}, ✏1,2)
[Bastian, SH, Iqbal, Rey 2017]
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
Duality leaves partiton function invariant
Z3,2({h}, {v}, {m}, ✏1,2) = Z6,1({h0}, {v0}, {m0}, ✏1,2)
Kähler parameters implied by duality transformation
[Bastian, SH, Iqbal, Rey 2017]
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
Duality leaves partiton function invariant
Z3,2({h}, {v}, {m}, ✏1,2) = Z6,1({h0}, {v0}, {m0}, ✏1,2)
Kähler parameters implied by duality transformation
[Bastian, SH, Iqbal, Rey 2017]
Vertical expansion of gives rise to a gauge theory with gauge group and part. fct.
U(6)
Z(6,1)
vert
Z6,1
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
Duality leaves partiton function invariant
Z3,2({h}, {v}, {m}, ✏1,2) = Z6,1({h0}, {v0}, {m0}, ✏1,2)
Kähler parameters implied by duality transformation
[Bastian, SH, Iqbal, Rey 2017]
Symmetry transformations do not flop any curve whose area is proportional to
V
Vertical expansion of gives rise to a gauge theory with gauge group and part. fct.
U(6)
Z(6,1)
vert
Z6,1
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
Duality leaves partiton function invariant
Z3,2({h}, {v}, {m}, ✏1,2) = Z6,1({h0}, {v0}, {m0}, ✏1,2)
Kähler parameters implied by duality transformation
[Bastian, SH, Iqbal, Rey 2017]
Symmetry transformations do not flop any curve whose area is proportional to
V
related to coupling constant of Z(3,2) diag
Vertical expansion of gives rise to a gauge theory with gauge group and part. fct.
U(6)
Z(6,1)
vert
Z6,1
v1 + v5 + h2 + h3 + h4 v3 + v5 + h2 + h4 + h6 v3 + v4 + h1 + h2 + h6 v2 + v4 + h1 + h5 + h6 v2 + v6 + h1 + h3 + h5 v1 + v6 + h3 + h4 + h5 −v1 − h3 − h4 −v5 − h2 − h4 −v3 − h2 − h6 −v4 − h1 − h6 −v2 − h1 − h5 −v6 − h3 − h5 −v1 − h3 − h4 m00
6m0
1m0
2m0
3m0
4m00
5m00
6m0
1m0
2m0
3m0
4m00
5a a 6 1 2 3 4 5 6 1 2 3 4 5
a b a b 1 2 3 1 2 3 m1 m2 m3 m4 m5 m6 h1 h2 h3 h4 h5 h6 v1 v2 v3 v4 v5 v6
Duality leaves partiton function invariant
Z3,2({h}, {v}, {m}, ✏1,2) = Z6,1({h0}, {v0}, {m0}, ✏1,2)
Kähler parameters implied by duality transformation
[Bastian, SH, Iqbal, Rey 2017]
Symmetry transformations do not flop any curve whose area is proportional to
V
Z(3,2)
diag
Z(6,1)
vert
Vertical expansion of gives rise to a gauge theory with gauge group and part. fct.
U(6)
Z(6,1)
vert
Z6,1
Duality conjectured to hold for generic (N, M)
k = gcd(N, M)
where
Duality conjectured to hold for generic (N, M)
k = gcd(N, M)
where Newton polygon (dual of the topic web diagram)
· · · · · ·
. . . . . .
N M
Duality conjectured to hold for generic (N, M)
k = gcd(N, M)
where Newton polygon (dual of the topic web diagram)
· · · · · ·
. . . . . .
N M
Fundamental domain for tiling the plane
Duality conjectured to hold for generic (N, M)
k = gcd(N, M)
where Newton polygon (dual of the topic web diagram)
· · · · · ·
. . . . . .
k , NM k )
( NM
k
+ 1, NM
k )
N M
Duality conjectured to hold for generic (N, M)
k = gcd(N, M)
where Newton polygon (dual of the topic web diagram)
· · · · · ·
. . . . . .
k , NM k )
( NM
k
+ 1, NM
k )
N M
equivalent tiling of the plane by these two lines, which visit every inequivalent point exactly once
XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Consequence: dualities between Calabi-Yau threefold (extended moduli space)
XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Consequence: dualities between Calabi-Yau threefold (extended moduli space) example: X6,5
1 2 3 4 5 6 1 2 3 4 5 6 a b c d e a b c d v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 v24 v25 v26 v27 v28 v29 v30 v1 v2 v3 v4 v5 v6 h6 h1 h2 h3 h4 h5 h6 h12 h7 h8 h9 h10 h11 h12 h18 h13 h14 h15 h16 h17 h18 h24 h19 h20 h21 h22 h23 h24 h30 h25 h26 h27 h28 h29 h30 m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 m14 m15 m16 m17 m18 m19 m20 m21 m22 m23 m24 m25 m26 m27 m28 m29 m30
XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Consequence: dualities between Calabi-Yau threefold (extended moduli space) example: X6,5 ∼ X10,3
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 a b c a b c v0
1v0
2v0
3v0
4v0
5v0
6v0
7v0
8v0
9v0
10v0
11v0
12v0
13v0
14v0
15v0
16v0
17v0
18v0
19v0
20v0
21v0
22v0
23v0
24v0
25v0
26v0
27v0
28v0
29v0
30v0
1v0
2v0
3v0
4v0
5v0
6v0
7v0
8v0
9v0
10h0
10h0
1h0
2h0
3h0
4h0
5h0
6h0
7h0
8h0
9h0
10h0
20h0
11h0
12h0
13h0
14h0
15h0
16h0
17h0
18h0
19h0
20h0
30h0
21h0
22h0
23h0
24h0
25h0
26h0
27h0
28h0
29h0
30m0
1m0
2m0
3m0
4m0
5m0
6m0
7m0
8m0
9m0
10m0
11m0
12m0
13m0
14m0
15m0
16m0
17m0
18m0
19m0
20m0
21m0
22m0
23m0
24m0
25m0
26m0
27m0
28m0
29m0
30 XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Consequence: dualities between Calabi-Yau threefold (extended moduli space) example: X6,5 ∼ X10,3
∼ X15,2 ∼ X30,1 ∼ X5,6 ∼ X3,10 ∼ X2,15 ∼ X1,30
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 a b c a b c v0
1v0
2v0
3v0
4v0
5v0
6v0
7v0
8v0
9v0
10v0
11v0
12v0
13v0
14v0
15v0
16v0
17v0
18v0
19v0
20v0
21v0
22v0
23v0
24v0
25v0
26v0
27v0
28v0
29v0
30v0
1v0
2v0
3v0
4v0
5v0
6v0
7v0
8v0
9v0
10h0
10h0
1h0
2h0
3h0
4h0
5h0
6h0
7h0
8h0
9h0
10h0
20h0
11h0
12h0
13h0
14h0
15h0
16h0
17h0
18h0
19h0
20h0
30h0
21h0
22h0
23h0
24h0
25h0
26h0
27h0
28h0
29h0
30m0
1m0
2m0
3m0
4m0
5m0
6m0
7m0
8m0
9m0
10m0
11m0
12m0
13m0
14m0
15m0
16m0
17m0
18m0
19m0
20m0
21m0
22m0
23m0
24m0
25m0
26m0
27m0
28m0
29m0
30 Summarise dualities for generic (partially conjectural): (N, M)
Summarise dualities for generic (partially conjectural): (N, M) Extended moduli space of :
XN,M XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for
intermediate K¨ ahler cone(s) that are passed through in the series of flop- and symmetry transformations connecting XN,M and XN0,M0 K¨ ahler cone of XN,M K¨ ahler cone of XN0,M0 walls of K¨ ahler cones
[SH, Iqbal, Rey 2016]
Summarise dualities for generic (partially conjectural): (N, M) Extended moduli space of :
XN,M XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for
intermediate K¨ ahler cone(s) that are passed through in the series of flop- and symmetry transformations connecting XN,M and XN0,M0 K¨ ahler cone of XN,M K¨ ahler cone of XN0,M0 walls of K¨ ahler cones
Partition function invariant (proven for )
M = 1
ZN,M({h}, {v}, {m}, ✏1,2) = ZN 0,M 0({h0}, {v0}, {m0}, ✏1,2)
[Bastian, SH, Iqbal, Rey 2017] [SH, Iqbal, Rey 2016]
Summarise dualities for generic (partially conjectural): (N, M) Extended moduli space of :
XN,M XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Partition function invariant (proven for )
M = 1
ZN,M({h}, {v}, {m}, ✏1,2) = ZN 0,M 0({h0}, {v0}, {m0}, ✏1,2)
[Bastian, SH, Iqbal, Rey 2017] [SH, Iqbal, Rey 2016]
Weak coupling regions within given Kähler cone:
m ω → ∞
h ω → ∞
v ω → ∞
Summarise dualities for generic (partially conjectural): (N, M) Extended moduli space of :
XN,M XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Partition function invariant (proven for )
M = 1
ZN,M({h}, {v}, {m}, ✏1,2) = ZN 0,M 0({h0}, {v0}, {m0}, ✏1,2)
[Bastian, SH, Iqbal, Rey 2017] [SH, Iqbal, Rey 2016]
Weak coupling regions within given Kähler cone:
m ω → ∞
h ω → ∞
v ω → ∞
quiver gauge theories with gauge groups k = gcd(N, M)
for
Ghor = [U(M)]N Gvert = [U(N)]M Gdiag = [U(NM/k)]k
Summarise dualities for generic (partially conjectural): (N, M) Extended moduli space of :
XN,M XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Partition function invariant (proven for )
M = 1
ZN,M({h}, {v}, {m}, ✏1,2) = ZN 0,M 0({h0}, {v0}, {m0}, ✏1,2)
[Bastian, SH, Iqbal, Rey 2017] [SH, Iqbal, Rey 2016]
Weak coupling regions within given Kähler cone:
m ω → ∞
h ω → ∞
v ω → ∞
quiver gauge theories with gauge groups k = gcd(N, M)
for
Ghor = [U(M)]N Gvert = [U(N)]M Gdiag = [U(NM/k)]k represent low energy limits of LSTs T-duality
Summarise dualities for generic (partially conjectural): (N, M) Extended moduli space of :
XN,M XN,M ∼ XN 0,M 0
NM = N 0M 0 gcd(N, M) = gcd(N 0, M 0)
for Partition function invariant (proven for )
M = 1
ZN,M({h}, {v}, {m}, ✏1,2) = ZN 0,M 0({h0}, {v0}, {m0}, ✏1,2)
[Bastian, SH, Iqbal, Rey 2017] [SH, Iqbal, Rey 2016]
Weak coupling regions within given Kähler cone:
m ω → ∞
h ω → ∞
v ω → ∞
quiver gauge theories with gauge groups k = gcd(N, M)
for
Ghor = [U(M)]N Gvert = [U(N)]M Gdiag = [U(NM/k)]k represent low energy limits of LSTs T-duality triality of LSTs
Web of dualities among different theories can be turned into symmetries for individual theories
[SH, Bastian 2018]
Web of dualities among different theories can be turned into symmetries for individual theories Example (N,M)=(2,1):
[SH, Bastian 2018]
Web of dualities among different theories can be turned into symmetries for individual theories Example (N,M)=(2,1):
[SH, Bastian 2018]
a a 1 2 1 2 h1 h2 h1 v1 v2 m1 m2 m1 m2 S2 S1 S2 S2 S1 S2 b a1 b a2 S R − 2S
<latexit sha1_base64="aRM3hDbdcbdVu/j+CVLnC/5+M4=">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</latexit>dual web diagrams
a a 1 2 1 2 h1 h2 h1 m2 m1 v2 v1 v2 v1 S0
2
S0
1
S0
2
S0
2
S0
1
S0
2
b a0
1
b a0
2
S0 R0 − 2S0
<latexit sha1_base64="CjkdDv3fsNTc3SveVwHYHN+cenA=">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</latexit>b a1 = v1 + h2 , b a2 = v2 + h1 , S = h2 + v2 + h1 , R − 2S = m1 − v2 .
<latexit sha1_base64="WF2iEDorm0M4yqATNCzXVvLwJR0=">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</latexit>b a0
1 = m1 + h1 ,
b a0
2 = m2 + h2 ,
S0 = h2 + m1 + h1 , R0 − 2S0 = v2 − m1 .
<latexit sha1_base64="CVuFlrti/nPAaoJ8feRohKif4cE=">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</latexit> Web of dualities among different theories can be turned into symmetries for individual theories Example (N,M)=(2,1):
[SH, Bastian 2018]
a a 1 2 1 2 h1 h2 h1 v1 v2 m1 m2 m1 m2 S2 S1 S2 S2 S1 S2 b a1 b a2 S R − 2S
<latexit sha1_base64="aRM3hDbdcbdVu/j+CVLnC/5+M4=">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</latexit>dual web diagrams
a a 1 2 1 2 h1 h2 h1 m2 m1 v2 v1 v2 v1 S0
2
S0
1
S0
2
S0
2
S0
1
S0
2
b a0
1
b a0
2
S0 R0 − 2S0
<latexit sha1_base64="CjkdDv3fsNTc3SveVwHYHN+cenA=">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</latexit>b a1 = v1 + h2 , b a2 = v2 + h1 , S = h2 + v2 + h1 , R − 2S = m1 − v2 .
<latexit sha1_base64="WF2iEDorm0M4yqATNCzXVvLwJR0=">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</latexit>b a0
1 = m1 + h1 ,
b a0
2 = m2 + h2 ,
S0 = h2 + m1 + h1 , R0 − 2S0 = v2 − m1 .
<latexit sha1_base64="CVuFlrti/nPAaoJ8feRohKif4cE=">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</latexit>Implies the following symmetry of the partition function:
B B @ b a1 b a2 S R 1 C C A = G1 · B B @ b a0
1
b a0
2
S0 R0 1 C C A
<latexit sha1_base64="LAW+Ue+jqw3j81iEv4MFyiAXPXQ=">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</latexit>with
G1 = 1 −2 1 1 −2 1 −1 1 1
<latexit sha1_base64="LCdjdSaDmDbvneMLx0KEpc2uHec=">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</latexit>det G1 = 1
<latexit sha1_base64="rtDWAOMr+vxUnjo7vHMLeD4lsoA=">ACDXicbVDLSgNBEJyNrxhfazx6GQyCBwm7UdCLEPSgxwjmAUkIs5NOMmT2wUyvJC75Bv/Aq/6AN/HqN3j3Q5w8DiaxoKGo6qa8iIpNDrOt5VaWV1b30hvZra2d3b37P1sRYex4lDmoQxVzWMapAigjAIl1CIFzPckVL3+zdivPoLSIgwecBhB02fdQHQEZ2iklp1tIAwaQOGqf0tuVeuS075+SdCegycWckR2YoteyfRjvksQ8Bcsm0rtOhM2EKRcwijTiDVEjPdZF+qGBswH3Uwmv4/osVHatBMqMwHSifr3ImG+1kPfM5s+w5e9Mbif149xs5lMxFBFCMEfBrUiSXFkI6LoG2hgKMcGsK4EuZXyntMY6mrkUFP2nUcbU4i6WsEwqhbx7li/cn+eK17OC0uSQHJET4pILUiR3pETKhJMBeSGv5M16t6tD+tzupqyZjcHZA7W1y/1nptL</latexit>G1 · G1 = 1 14×4
<latexit sha1_base64="wtf7qzBINJpa47I+8UpSEUdGcqk=">ACGnicbVC7SgNBFJ31GeNr1dLCiUGwCrsxoI0QtNAygnlANiyzk9lkyOyDmbtCXLb0L/wDW/0BO7G1sfdDnDwKk3hg4HDOvZw7x4sFV2BZ38bS8srq2npuI7+5tb2za+7tN1SUSMrqNBKRbHlEMcFDVgcOgrViyUjgCdb0Btcjv/nApOJReA/DmHUC0gu5zykBLbnm0Y1rO7QbAdbk0nYKTsF204oDPGAKVzLXLFolawy8SOwpKaIpaq7543QjmgQsBCqIUm3biqGTEgmcCpblnUSxmNAB6bG2piHROZ10/JEMn2ili/1I6hcCHqt/N1ISKDUMPD0ZEOireW8k/ue1E/AvOikP4wRYSCdBfiIwRHjUCu5ySiIoSaESq5vxbRPJKGgu5tJAT54zPK6Fnu+hEXSKJfs1L5rlKsXk0LyqFDdIxOkY3OURXdohqI4qe0At6RW/Gs/FufBifk9ElY7pzgGZgfP0CxT2fVA=</latexit>where
Generalising to include other duality transformations:
a a 1 2 1 2 h1 h2 h1 v1 v2 m1 m2 m1 m2 b a
1
b a
2
S R − 2S a a 1 2 1 2 h1 h2 h1 m2 m1 v2 v1 v2 v1 b a
1
b a
2
S0 R0 − 2S0 a a 1 2 2 1 v2 v1 v2 h1 h2 m2 m1 m1 m2 b a
2
b a
1
S00 R00 − S00 a a 1 2 2 1 m1 m2 m1 h2 h1 v1 v2 v2 v1 b a
1
b a
2
S000 R000 − S000
G1 G1 G2 G2 G3 G3
<latexit sha1_base64="sFSbrpRcqpNLJlMTQOfSm8rwOjA=">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</latexit> Generalising to include other duality transformations:
G1 = 1 −2 1 1 −2 1 −1 1 1
<latexit sha1_base64="LCdjdSaDmDbvneMLx0KEpc2uHec=">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</latexit>G2 = 1 1 1 1 −1 2 2 −4 1
<latexit sha1_base64="BfLW6Kk5QSLRbSrb6mpSqOyLJw=">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</latexit>G3 = 1 −2 1 1 −2 1 1 1 −3 1 2 2 −4 1
<latexit sha1_base64="JW+CJmV9/aKkbijiCBQA48Ycd8=">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</latexit>a a 1 2 1 2 h1 h2 h1 v1 v2 m1 m2 m1 m2 b a
1
b a
2
S R − 2S a a 1 2 1 2 h1 h2 h1 m2 m1 v2 v1 v2 v1 b a
1
b a
2
S0 R0 − 2S0 a a 1 2 2 1 v2 v1 v2 h1 h2 m2 m1 m1 m2 b a
2
b a
1
S00 R00 − S00 a a 1 2 2 1 m1 m2 m1 h2 h1 v1 v2 v2 v1 b a
1
b a
2
S000 R000 − S000
G1 G1 G2 G2 G3 G3
<latexit sha1_base64="sFSbrpRcqpNLJlMTQOfSm8rwOjA=">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</latexit> Generalising to include other duality transformations:
G1 = 1 −2 1 1 −2 1 −1 1 1
<latexit sha1_base64="LCdjdSaDmDbvneMLx0KEpc2uHec=">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</latexit>G2 = 1 1 1 1 −1 2 2 −4 1
<latexit sha1_base64="BfLW6Kk5QSLRbSrb6mpSqOyLJw=">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</latexit>G3 = 1 −2 1 1 −2 1 1 1 −3 1 2 2 −4 1
<latexit sha1_base64="JW+CJmV9/aKkbijiCBQA48Ycd8=">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</latexit>1 14×4 G1 G2 G3 1 14×4 1 14×4 G1 G2 G3 G1 G1 1 14×4 G3 G2 G2 G2 G3 1 14×4 G1 G3 G3 G2 G1 1 14×4
<latexit sha1_base64="mQrzKvVckyY065o4KhohpFWN0UA=">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</latexit>a a 1 2 1 2 h1 h2 h1 v1 v2 m1 m2 m1 m2 b a
1
b a
2
S R − 2S a a 1 2 1 2 h1 h2 h1 m2 m1 v2 v1 v2 v1 b a
1
b a
2
S0 R0 − 2S0 a a 1 2 2 1 v2 v1 v2 h1 h2 m2 m1 m1 m2 b a
2
b a
1
S00 R00 − S00 a a 1 2 2 1 m1 m2 m1 h2 h1 v1 v2 v2 v1 b a
1
b a
2
S000 R000 − S000
G1 G1 G2 G2 G3 G3
<latexit sha1_base64="sFSbrpRcqpNLJlMTQOfSm8rwOjA=">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</latexit> Generalising to include other duality transformations:
G1 = 1 −2 1 1 −2 1 −1 1 1
<latexit sha1_base64="LCdjdSaDmDbvneMLx0KEpc2uHec=">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</latexit>G2 = 1 1 1 1 −1 2 2 −4 1
<latexit sha1_base64="BfLW6Kk5QSLRbSrb6mpSqOyLJw=">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</latexit>G3 = 1 −2 1 1 −2 1 1 1 −3 1 2 2 −4 1
<latexit sha1_base64="JW+CJmV9/aKkbijiCBQA48Ycd8=">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</latexit>1 14×4 G1 G2 G3 1 14×4 1 14×4 G1 G2 G3 G1 G1 1 14×4 G3 G2 G2 G2 G3 1 14×4 G1 G3 G3 G2 G1 1 14×4
<latexit sha1_base64="mQrzKvVckyY065o4KhohpFWN0UA=">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</latexit>a a 1 2 1 2 h1 h2 h1 v1 v2 m1 m2 m1 m2 b a
1
b a
2
S R − 2S a a 1 2 1 2 h1 h2 h1 m2 m1 v2 v1 v2 v1 b a
1
b a
2
S0 R0 − 2S0 a a 1 2 2 1 v2 v1 v2 h1 h2 m2 m1 m1 m2 b a
2
b a
1
S00 R00 − S00 a a 1 2 2 1 m1 m2 m1 h2 h1 v1 v2 v2 v1 b a
1
b a
2
S000 R000 − S000
G1 G1 G2 G2 G3 G3
<latexit sha1_base64="sFSbrpRcqpNLJlMTQOfSm8rwOjA=">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</latexit>{1 14×4, G1, G2, G3} ∼ = Dih2
<latexit sha1_base64="+x5Li7e8aUgveQeRWSqyC8jCZws=">ACHXicbVBNS8NAEN34bf2KevSytQgepCS1oEdRQY8K1ha6IWy23bpZhN2J2IJ+SNe/CtePCjiwYv4b9zWHvx6MPB4b4aZeVEqhQHP+3Cmpmdm5+YXFktLyura+76xrVJMs14gyUy0a2IGi6F4g0QIHkr1ZzGkeTNaHAy8ps3XBuRqCsYpjyIaU+JrmAUrBS6dZL7pEzKfpjXCYiYG1wv9s5C31bN1j4pCEtUDxPgt5Cfin4R1kK34lW9MfBf4k9IBU1wEbpvpJOwLOYKmKTGtH0vhSCnGgSTvCiRzPCUsgHt8balito7gnz8XYF3rNLB3UTbUoDH6veJnMbGDOPIdsYU+ua3NxL/89oZdA+DXKg0A67Y16JuJjEkeBQV7gjNGcihJZRpYW/FrE81ZWADLdkQ/N8v/yXtaq/X61d1itHx5M4FtAW2ka7yEcH6AidowvUQAzdoQf0hJ6de+fReXFev1qnMnMJvoB5/0TAUygDg=</latexit>Group Structure:
Generalisation to (N,1): Symmetry group
G(N) × DihN
<latexit sha1_base64="iILGYSP1nCVkE3WnVJn94iDUmo=">ACDnicbVDLSgNBEJyNrxhfUY9eBkMgoTdKOgxqKCnEME8ILuE2ckGTL7YKZXDEu+wIu/4sWDIl49e/NvnE32oIkFDUVN91dbi4AtP8NjJLyura9n13Mbm1vZOfnevqYJIUtagQhk2yWKCe6zBnAQrB1KRjxXsJY7ukz81j2Tigf+HYxD5nhk4PM+pwS01M0Xse0RGLpufD0p1Y7sYxu4xROCHuA+IoPJ91aN18wy+YUeJFYKSmgFPVu/svuBTymA9UEKU6lhmCExMJnAo2ydmRYiGhIzJgHU19onc68fSdCS5qpYf7gdTlA56qvydi4ik19lzdmdyu5r1E/M/rRNA/d2LuhxEwn84W9SOBIcBJNrjHJaMgxpoQKrm+FdMhkYSCTjCnQ7DmX14kzUrZOilXbk8L1Ys0jiw6QIeohCx0hqroBtVRA1H0iJ7RK3oznowX4934mLVmjHRmH/2B8fkD+mubcg=</latexit> Generalisation to (N,1): Symmetry group
G(N) × DihN
<latexit sha1_base64="iILGYSP1nCVkE3WnVJn94iDUmo=">ACDnicbVDLSgNBEJyNrxhfUY9eBkMgoTdKOgxqKCnEME8ILuE2ckGTL7YKZXDEu+wIu/4sWDIl49e/NvnE32oIkFDUVN91dbi4AtP8NjJLyura9n13Mbm1vZOfnevqYJIUtagQhk2yWKCe6zBnAQrB1KRjxXsJY7ukz81j2Tigf+HYxD5nhk4PM+pwS01M0Xse0RGLpufD0p1Y7sYxu4xROCHuA+IoPJ91aN18wy+YUeJFYKSmgFPVu/svuBTymA9UEKU6lhmCExMJnAo2ydmRYiGhIzJgHU19onc68fSdCS5qpYf7gdTlA56qvydi4ik19lzdmdyu5r1E/M/rRNA/d2LuhxEwn84W9SOBIcBJNrjHJaMgxpoQKrm+FdMhkYSCTjCnQ7DmX14kzUrZOilXbk8L1Ys0jiw6QIeohCx0hqroBtVRA1H0iJ7RK3oznowX4934mLVmjHRmH/2B8fkD+mubcg=</latexit>‘ shuffling’ of roots
Generalisation to (N,1): Symmetry group
G(N) × DihN
<latexit sha1_base64="iILGYSP1nCVkE3WnVJn94iDUmo=">ACDnicbVDLSgNBEJyNrxhfUY9eBkMgoTdKOgxqKCnEME8ILuE2ckGTL7YKZXDEu+wIu/4sWDIl49e/NvnE32oIkFDUVN91dbi4AtP8NjJLyura9n13Mbm1vZOfnevqYJIUtagQhk2yWKCe6zBnAQrB1KRjxXsJY7ukz81j2Tigf+HYxD5nhk4PM+pwS01M0Xse0RGLpufD0p1Y7sYxu4xROCHuA+IoPJ91aN18wy+YUeJFYKSmgFPVu/svuBTymA9UEKU6lhmCExMJnAo2ydmRYiGhIzJgHU19onc68fSdCS5qpYf7gdTlA56qvydi4ik19lzdmdyu5r1E/M/rRNA/d2LuhxEwn84W9SOBIcBJNrjHJaMgxpoQKrm+FdMhkYSCTjCnQ7DmX14kzUrZOilXbk8L1Ys0jiw6QIeohCx0hqroBtVRA1H0iJ7RK3oznowX4934mLVmjHRmH/2B8fkD+mubcg=</latexit>G(N) ∼ = Dih3 if N = 1 , Dih2 if N = 2 , Dih3 if N = 3 , Dih∞ if N ≥ 4 .
<latexit sha1_base64="BH7MfS0czGZMYWs8Y6hu1wJE8K0=">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</latexit>‘ shuffling’ of roots
where
Generalisation to (N,1): Symmetry group
G(N) × DihN
<latexit sha1_base64="iILGYSP1nCVkE3WnVJn94iDUmo=">ACDnicbVDLSgNBEJyNrxhfUY9eBkMgoTdKOgxqKCnEME8ILuE2ckGTL7YKZXDEu+wIu/4sWDIl49e/NvnE32oIkFDUVN91dbi4AtP8NjJLyura9n13Mbm1vZOfnevqYJIUtagQhk2yWKCe6zBnAQrB1KRjxXsJY7ukz81j2Tigf+HYxD5nhk4PM+pwS01M0Xse0RGLpufD0p1Y7sYxu4xROCHuA+IoPJ91aN18wy+YUeJFYKSmgFPVu/svuBTymA9UEKU6lhmCExMJnAo2ydmRYiGhIzJgHU19onc68fSdCS5qpYf7gdTlA56qvydi4ik19lzdmdyu5r1E/M/rRNA/d2LuhxEwn84W9SOBIcBJNrjHJaMgxpoQKrm+FdMhkYSCTjCnQ7DmX14kzUrZOilXbk8L1Ys0jiw6QIeohCx0hqroBtVRA1H0iJ7RK3oznowX4934mLVmjHRmH/2B8fkD+mubcg=</latexit>G(N) ∼ = Dih3 if N = 1 , Dih2 if N = 2 , Dih3 if N = 3 , Dih∞ if N ≥ 4 .
<latexit sha1_base64="BH7MfS0czGZMYWs8Y6hu1wJE8K0=">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</latexit>‘ shuffling’ of roots
where
Explicitly
G(N) ∼ = ⌦ {G2(N), G0
2(N)
2(N))2 = (G2(N) · G0 2(N))n = 1
1} ↵
<latexit sha1_base64="x1uH53h2Tzt4P8DRlVNwufDJgXA=">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</latexit> Generalisation to (N,1): Symmetry group
G(N) × DihN
<latexit sha1_base64="iILGYSP1nCVkE3WnVJn94iDUmo=">ACDnicbVDLSgNBEJyNrxhfUY9eBkMgoTdKOgxqKCnEME8ILuE2ckGTL7YKZXDEu+wIu/4sWDIl49e/NvnE32oIkFDUVN91dbi4AtP8NjJLyura9n13Mbm1vZOfnevqYJIUtagQhk2yWKCe6zBnAQrB1KRjxXsJY7ukz81j2Tigf+HYxD5nhk4PM+pwS01M0Xse0RGLpufD0p1Y7sYxu4xROCHuA+IoPJ91aN18wy+YUeJFYKSmgFPVu/svuBTymA9UEKU6lhmCExMJnAo2ydmRYiGhIzJgHU19onc68fSdCS5qpYf7gdTlA56qvydi4ik19lzdmdyu5r1E/M/rRNA/d2LuhxEwn84W9SOBIcBJNrjHJaMgxpoQKrm+FdMhkYSCTjCnQ7DmX14kzUrZOilXbk8L1Ys0jiw6QIeohCx0hqroBtVRA1H0iJ7RK3oznowX4934mLVmjHRmH/2B8fkD+mubcg=</latexit>G(N) ∼ = Dih3 if N = 1 , Dih2 if N = 2 , Dih3 if N = 3 , Dih∞ if N ≥ 4 .
<latexit sha1_base64="BH7MfS0czGZMYWs8Y6hu1wJE8K0=">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</latexit>‘ shuffling’ of roots
where
Explicitly
G(N) ∼ = ⌦ {G2(N), G0
2(N)
2(N))2 = (G2(N) · G0 2(N))n = 1
1} ↵
<latexit sha1_base64="x1uH53h2Tzt4P8DRlVNwufDJgXA=">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</latexit>n = 3 for N = 1, 3 2 for N = 2 ∞ for N ≥ 4
<latexit sha1_base64="l9tKloRf5h2V2MdtXSWdD06gpNU=">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</latexit> Generalisation to (N,1): Symmetry group
G(N) × DihN
<latexit sha1_base64="iILGYSP1nCVkE3WnVJn94iDUmo=">ACDnicbVDLSgNBEJyNrxhfUY9eBkMgoTdKOgxqKCnEME8ILuE2ckGTL7YKZXDEu+wIu/4sWDIl49e/NvnE32oIkFDUVN91dbi4AtP8NjJLyura9n13Mbm1vZOfnevqYJIUtagQhk2yWKCe6zBnAQrB1KRjxXsJY7ukz81j2Tigf+HYxD5nhk4PM+pwS01M0Xse0RGLpufD0p1Y7sYxu4xROCHuA+IoPJ91aN18wy+YUeJFYKSmgFPVu/svuBTymA9UEKU6lhmCExMJnAo2ydmRYiGhIzJgHU19onc68fSdCS5qpYf7gdTlA56qvydi4ik19lzdmdyu5r1E/M/rRNA/d2LuhxEwn84W9SOBIcBJNrjHJaMgxpoQKrm+FdMhkYSCTjCnQ7DmX14kzUrZOilXbk8L1Ys0jiw6QIeohCx0hqroBtVRA1H0iJ7RK3oznowX4934mLVmjHRmH/2B8fkD+mubcg=</latexit>G(N) ∼ = Dih3 if N = 1 , Dih2 if N = 2 , Dih3 if N = 3 , Dih∞ if N ≥ 4 .
<latexit sha1_base64="BH7MfS0czGZMYWs8Y6hu1wJE8K0=">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</latexit>‘ shuffling’ of roots
where
Explicitly
G(N) ∼ = ⌦ {G2(N), G0
2(N)
2(N))2 = (G2(N) · G0 2(N))n = 1
1} ↵
<latexit sha1_base64="x1uH53h2Tzt4P8DRlVNwufDJgXA=">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</latexit>G2(N) = 1 1N×N . . . . . . 1 · · · 1 −1 N · · · N −2N 1
<latexit sha1_base64="8q9/C9GIDg3ryfjde7b4QPZQg=">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</latexit>with the (N + 2) × (N + 2)
<latexit sha1_base64="gD4+BufTtoJuFJNuRBZ0ukueaYc=">AB+nicbZDLSsNAFIZP6q3W6pLN4NFqAglqYIui25cSQV7gTaUyXTSDp1MwsxEKbWP4saFIm59Ene+jdM0C239YeDjP+dwzvx+zJnSjvNt5VZW19Y38puFre2d3T27uN9USIJbZCIR7LtY0U5E7Shmea0HUuKQ5/Tlj+6ntVbD1QqFol7PY6pF+KBYAEjWBurZxfLt6fVk65mIVUo5Z5dcipOKrQMbgYlyFTv2V/dfkSkApNOFaq4zqx9iZYakY4nRa6iaIxJiM8oB2DAptV3iQ9fYqOjdNHQSTNExql7u+JCQ6VGoe+6QyxHqrF2sz8r9ZJdHDpTZiIE0FmS8KEo50hGY5oD6TlGg+NoCJZOZWRIZYqJNWgUTgrv45WVoVivuWaV6d16qXWVx5OEQjqAMLlxADW6gDg0g8AjP8Apv1pP1Yr1bH/PWnJXNHMAfWZ8/ElaR7w=</latexit>matrices
G0
2(N) =
−2 1 1 1N⇥N . . . . . . −2 1 · · · −1 1 · · · 1
<latexit sha1_base64="w8aeqi7a5WENzfUI4hRGjrUK1fo=">ACpnicdVFNb9QwEHXCV9nyscCRi8uqsD10lSxIcEGq4ND2siq02620XgXHmexadZzInlRaRflp/Alu/Js62UjQFsay/TzvjT2eiQslLQbBb8+/d/Bw0dbj3vbT54+e95/8fLc5qURMBW5ys1FzC0oqWGKEhVcFAZ4FiuYxZdfG352BcbKXJ/huoBFxpdaplJwdK6o/5NlHFeCq+qwfheNh5O9z0xBikMWw1LqihvD13UlGqvp23bsj90SUsaje2wnTCqJgxlBpZONiJ2leRo/4BWezM2aFjRyZrDfvhfKtgwoJMuIWbkcoV7UX8QjILW6F0QdmBAOjuJ+r9YkosyA41CcWvnYVDgwl2KUioe6y0UHBxyZcwd1Bz96dF1Za5prvOk9A0N25qpK374iKZ9aus9gpm6La21zj/Bc3LzH9tKikLkoELTYPpaWimNOmZzSRBgSqtQNcGOlypWLFDRfoOtzRQhvf/kuOB+Pwvej8bcPg4MvXTm2yGvyhgxJSD6SA3JETsiUCG/gHXvfvVN/6E/8qT/bSH2vi3lFbpj/4xpAh8Dd</latexit>and
n = 3 for N = 1, 3 2 for N = 2 ∞ for N ≥ 4
<latexit sha1_base64="l9tKloRf5h2V2MdtXSWdD06gpNU=">ACZHicbVFBT9swFHayMaCDLQxQpqsVUMcUJW0SHBQuPCYG0AlJdVY7klo4Tma/IKIof3K3HXfhd+CWHFi7J1n63ve9/z8OS6UtBiGfz/3fu1D+sbm52PW9ufPgc7X25tXhoBQ5Gr3NzH3IKSGoYoUcF9YBnsYK7+OFirt89grEy1z+xKmCc8VTLRAqOjpoEtT5jChJkNYshlbrmxvCqZVQzYAeUIbwhHWSm8YlV2fR0YAyRv0YEnoz2kmdYLVShdL4Rc9ZqCn7XRmZDrD3iTohr1wEXQVRC3okjauJ8FvNs1FmYFGobi1oygscOyGohQKmg4rLRcPAURg5qnoEd1wuTGvrdMVPqdnJHI12wbztqnlbZbGrzDjO7LI2J/+njUpMTse1EWJoMXrRUmpKOZ07jidSgMCVeUAF0a6XamYcMFun/pOBOi5Sevgt+Lxr0+jfH3fMfrR0bZJ98I4ckIifknFySazIkgvz1r3A2/Ge/S1/197LfW9tmeX/BP+1xfWfLVc</latexit> Studied dualities in a class of Little String Orbifolds: efficiently described by dual F-theory compactification on a class of toric CY3folds
XN,M
partition function compute as topological string partition function on
XN,M
ZN,M
Kähler cone of contains three weak coupling regions in which web diagram
XN,M
decomposes into parallel strips weak coupling regions give rise to different (but equivalent) expansions of that can
ZN,M
be interpreted as instanton partition functions, realising a triality of 5dim quiver gauge th.:
Ghor = [U(M)]N ⇐ ⇒ Gvert = [U(N)]M ⇐ ⇒ Gdiag = [U( MN
k )]k
k = gcd(N, M)
for
implies (dihedral) symmetries of the partition function
Studied dualities in a class of Little String Orbifolds: efficiently described by dual F-theory compactification on a class of toric CY3folds
XN,M
partition function compute as topological string partition function on
XN,M
ZN,M
Kähler cone of contains three weak coupling regions in which web diagram
XN,M
decomposes into parallel strips weak coupling regions give rise to different (but equivalent) expansions of that can
ZN,M
be interpreted as instanton partition functions, realising a triality of 5dim quiver gauge th.:
Ghor = [U(M)]N ⇐ ⇒ Gvert = [U(N)]M ⇐ ⇒ Gdiag = [U( MN
k )]k
k = gcd(N, M)
for
Future directions: study implications of triality on W-algebras associated with AGT dual theories study extended web of dualities by considering further weak coupling regions in the extended moduli space of XN,M Generalisation to other LSTs than A-series further dualities:
[U(M)]N ⇐ ⇒ [U(M 0)]N0
NM = N 0M 0
gcd(N, M) = gcd(N 0, M 0)
for
implies (dihedral) symmetries of the partition function