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Introduction Intersecting surface defects Summary Open problems Intersecting surface defects and 2d Conformal Field Theory Yiwen Pan Uppsala University [1610.03501] Gomis, Le Floch, YP, Peelaers [1612.xxxxx] YP, Peelaers Nov 17 2016

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects and 2d Conformal Field Theory

Yiwen Pan

Uppsala University [1610.03501] Gomis, Le Floch, YP, Peelaers [1612.xxxxx] YP, Peelaers Nov 17 2016 Oviedo

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 1 / 27

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Introduction Intersecting surface defects Summary Open problems

Outline

Introduction: class-S, CFT, partition functions, AGT Surface defects and their intersection

Construction Two simplest intersecting defect systems Partition functions, correlators, dualities Higgsing

Summary, conjectures Open problems

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 2 / 27

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Introduction Intersecting surface defects Summary Open problems

Introduction

class-S theories Liouville/Toda AGT surface defect (class-S construction) AGT with surface defect

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 3 / 27

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Introduction Intersecting surface defects Summary Open problems

Motivations

QFTs are well studied on smooth spaces (Sn, Sn × S1, ...), spaces with boundaries (Dn, Rk, . . .) Explore QFTs on intersecting spaces, e.g., R2

x1,x2=0 ∪ R2 x3,x4=0 ⊂ R4,

R2

x1,x2=0 ∩ R2 x3,x4=0 = (0, 0, 0, 0)

Enrich the family of surface defects in four dimensions Generalize AGT correspondence to include intersecting surface defects Explore new dualities

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 4 / 27

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Introduction Intersecting surface defects Summary Open problems

Class S of type Anf [Gaiotto]

4d N = 2 theories on M 4 Labeled by punctured Riemann surfaces Σg,n ⇒ Tg,n on M 4 M 4 unrelated to Σ Some of them in weak coupling regime ⇒ quiver gauge theories

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 5 / 27

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Introduction Intersecting surface defects Summary Open problems

Class S of type Anf [Gaiotto]

4d N = 2 theories on M 4 Labeled by punctured Riemann surfaces Σg,n ⇒ Tg,n on M 4 M 4 unrelated to Σ Some of them in weak coupling regime ⇒ quiver gauge theories Examples: consider Riemann spheres

  • n2

f Free hypers nf nf

  • SU(nf) SQCD

nf nf nf Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 5 / 27

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Introduction Intersecting surface defects Summary Open problems

Partition functions ZM

For QFTs T on space M who have Lagrangians Defined formally as path integral ZM(T ) ≡

  • D [fields] e−SM [fields]

For some theories of class-S, simplified to ordinary integrals/sums ZM(T ) =

  • Zcl(Φ)Z1−loop(Φ)Zinstanton(Φ)

Examples will be shown later

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 6 / 27

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Introduction Intersecting surface defects Summary Open problems

Liouville/Toda CFT [Teschner, ‘95; Zamolodchikov, Zamolodchikov, ‘96;]

Liouville theory: 2d CFT on Σg,n; Toda, the generalized version Depend on a param b (↔ central charge) Liouville ↔ W2 ∼ V irasoro, Toda ↔ Wnf Vertex operators Vα(x) :

Location: x Momentum: α

Special ones: degenerate vertex op. Vαdeg(x)

Pick R: irrep of su(nf)

  • deg. momentum αdeg ∝ ΩR

Insert Vα(x) at the punctures Correlation functions Vα0(0)Vβ1(x1)...Vβn(xn)Vα1(1)Vα∞(∞)

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 7 / 27

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Introduction Intersecting surface defects Summary Open problems

AGT relation [Alday, Gaiotto, Tachikawa]

S4

b -partition functions (of Tg,n) = Liouville/Toda correlators (on Σg,n);

ZS2⊂S4

b

  • nf

nf

  • = |x|2γ0|1 − x|2γ1
  • Vα0 (0)

Vα∞ (∞)

  • Vα1 (1)
  • ZS2⊂S4

b

  • nf

nf nf

q

  • = |x|2γ0|1 − x|2γ1
  • Vα0 (0)

Vα∞ (∞)

  • Vα1 (1)
  • Vα(q)
  • Yiwen Pan (Yiwen Pan)

Intersecting defects & 2d CFT 2016 Nov 8 / 27

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Introduction Intersecting surface defects Summary Open problems

Surface defect (class-S construction) [Gomis, Le Floch; Gadde, Gukov; Gaiotto, Kim, ...]

Insert degenerate puncture(s)/vertex operator(s); Labeled by a representation R of su(nf) with highest weight ΩR;

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 9 / 27

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Introduction Intersecting surface defects Summary Open problems

Surface defect (class-S construction) [Gomis, Le Floch; Gadde, Gukov; Gaiotto, Kim, ...]

Insert degenerate puncture(s)/vertex operator(s); Labeled by a representation R of su(nf) with highest weight ΩR; Examples (before)

  • n2

f Free hypers nf nf

  • SU(nf) SQCD

nf nf nf Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 9 / 27

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Introduction Intersecting surface defects Summary Open problems

Surface defect (class-S construction) [Gomis, Le Floch; Gadde, Gukov; Gaiotto, Kim, ...]

Insert degenerate puncture(s)/vertex operator(s); Labeled by a representation R of su(nf) with highest weight ΩR; Examples (after), R determines 2d quiver (inside dashed box)

  • ×

R

n2

f Free hypers + defect nf nf nν . . .

2d

  • ×

R

SU(nf) SQCD + defect

nf nf nf nν

. . .

2d

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 9 / 27

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Introduction Intersecting surface defects Summary Open problems

AGT relation with one defect [Gomis, Le Floch; ...]

S2 ⊂ S4

b -partition function = Liouville/Toda deg. correlators; Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 10 / 27

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Introduction Intersecting surface defects Summary Open problems

AGT relation with one defect [Gomis, Le Floch; ...]

S2 ⊂ S4

b -partition function = Liouville/Toda deg. correlators;

Example: αdeg = −bΩsymmn = −nbh1, x ∝ e−2πξFI, ZS2⊂S4

b

  • nf

nf n

2d, ξFI

  • = |x|2γ0|1 − x|2γ1
  • Vα0 (0)

Vα∞ (∞)

  • Vα1 (1)

×

Vαdeg (x)

  • Yiwen Pan (Yiwen Pan)

Intersecting defects & 2d CFT 2016 Nov 10 / 27

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects

One surface defect (QFT construction) Intersecting surface defect (QFT construction) Couplings The example Higgsing

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 11 / 27

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Introduction Intersecting surface defects Summary Open problems

One surface defect: as 4d-2d system

QFT Construction

4d bulk space M4, N = 2 theory T 4d .

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27

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Introduction Intersecting surface defects Summary Open problems

One surface defect: as 4d-2d system

QFT Construction

4d bulk space M4, N = 2 theory T 4d . 2d subspace D ⊂ M4, N = (2, 2) theory T 2d . T 4d, T 2d couple supersymmetrically.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27

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Introduction Intersecting surface defects Summary Open problems

One surface defect: as 4d-2d system

QFT Construction

4d bulk space M4, N = 2 theory T 4d . 2d subspace D ⊂ M4, N = (2, 2) theory T 2d . T 4d, T 2d couple supersymmetrically.

M 4, T 4d D, T 2d

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27

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Introduction Intersecting surface defects Summary Open problems

One surface defect: as 4d-2d system

QFT Construction

4d bulk space M4, N = 2 theory T 4d . 2d subspace D ⊂ M4, N = (2, 2) theory T 2d . T 4d, T 2d couple supersymmetrically.

M 4, T 4d D, T 2d

Physical quantity: the partition function of the 4d-2d coupled system

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 12 / 27

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects: as 4d-2d-0d system

QFT Construction

bulk M4, N = 2 theory T 4d.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects: as 4d-2d-0d system

QFT Construction

bulk M4, N = 2 theory T 4d. DL ⊂ M4, N = (2, 2) theory T 2d

L

. DR ⊂ M4, N = (2, 2) theory T 2d

R

.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects: as 4d-2d-0d system

QFT Construction

bulk M4, N = 2 theory T 4d. DL ⊂ M4, N = (2, 2) theory T 2d

L

. DR ⊂ M4, N = (2, 2) theory T 2d

R

. P ≡ DL ∩ DR ⊂ M4, 0d N = 2 theory T 0d (0d N = 2 Fermi or chiral).

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects: as 4d-2d-0d system

QFT Construction

bulk M4, N = 2 theory T 4d. DL ⊂ M4, N = (2, 2) theory T 2d

L

. DR ⊂ M4, N = (2, 2) theory T 2d

R

. P ≡ DL ∩ DR ⊂ M4, 0d N = 2 theory T 0d (0d N = 2 Fermi or chiral). T 4d, T 2d

L,R and T 0d couple supersymmetrically. Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects: as 4d-2d-0d system

QFT Construction M 4, T 4d DR, T 2d

R

DL, T 2d

L

P, T 0d

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27

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Introduction Intersecting surface defects Summary Open problems

Intersecting surface defects: as 4d-2d-0d system

QFT Construction M 4, T 4d DR, T 2d

R

DL, T 2d

L

P, T 0d Physical quantity: the partition function of the 4d-2d-0d coupled system

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 13 / 27

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Introduction Intersecting surface defects Summary Open problems

Coupling across dimensions

Two basic operations: gauging and superpotential.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 14 / 27

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Introduction Intersecting surface defects Summary Open problems

Coupling across dimensions

Two basic operations: gauging and superpotential. Decompose: f4d N =2(x1, . . . , x4) → fN =(2,2)

x3,x4

(x1, x2) → f0d N =2

x1,x2,x3,x4

Examples (R4 ⊃ R2

x3,x4 ⊃ {0}):

A4d N =2 → AN =(2,2)

x3,x4

⊕ ΦN =(2,2)

x3,x4

, QN =2 → ΦN =(2,2)

x3,x4

⊕ ˜ ΦN =(2,2)

x3,x4

AN =(2,2) → A0d N =2

x1,x2

⊕ Φ0d N =2

x1,x2

, ΦN =(2,2) → Φ0d N =2

x1,x2

⊕ F 0d N =2

x1,x2 Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 14 / 27

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Introduction Intersecting surface defects Summary Open problems

Coupling across dimensions

Two basic operations: gauging and superpotential. Decompose: f4d N =2(x1, . . . , x4) → fN =(2,2)

x3,x4

(x1, x2) → f0d N =2

x1,x2,x3,x4

Examples (R4 ⊃ R2

x3,x4 ⊃ {0}):

A4d N =2 → AN =(2,2)

x3,x4

⊕ ΦN =(2,2)

x3,x4

, QN =2 → ΦN =(2,2)

x3,x4

⊕ ˜ ΦN =(2,2)

x3,x4

AN =(2,2) → A0d N =2

x1,x2

⊕ Φ0d N =2

x1,x2

, ΦN =(2,2) → Φ0d N =2

x1,x2

⊕ F 0d N =2

x1,x2

Higher-dim vector multiplet gauge lower-dim global symmetries Bulk and defect fields superpotential ⇒ relations between R-charges and masses.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 14 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: the quivers

M 4 = S4

b , DL = S2 L, DR = S2 R, {N, S} = S2 L ∩ S2 R Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 15 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: the quivers

M 4 = S4

b , DL = S2 L, DR = S2 R, {N, S} = S2 L ∩ S2 R

S2

L

S2

R

N S

S4

b

S4

b :

ℓ−2(x2

1+x2 2)+˜

ℓ−2(x2

3+x2 4)+r−2x2 5=1

S2

L:

ℓ−2(x2

1+x2 2)+

+r−2x2

5=1

S2

R:

+˜ ℓ−2(x2

3+x2 4)+r−2x2 5=1

b2 = ℓ/˜ ℓ

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 15 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: the quivers

M 4 = S4

b , DL = S2 L, DR = S2 R, {N, S} = S2 L ∩ S2 R

4d-2d-0d coupled system: Fermi-type and Chiral-type quivers

nL nR nf nf S2

R

S2

L

ξL

FI

ξR

FI

S4

b

N S n′ n nf nf S2

R

S2

L

ξ′

FI

ξFI S4

b

N S

4d free hypers 2d chirals 0d Fermis 0d chirals

  • n S4

b

  • n S2

L, S2 R respectively

  • n N, S
  • n N, S

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 15 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: partition function

The full partition function reads ZS2

L∩S2 R⊂S4 b = Z

S4

b

freehyper

  • BL,R
  • dσL,R
  • α=L,R

Z

S2

α

SQCD(A)(σα, Bα)

× ZN

F/C(σ, B)ZS F/C(σ, B) Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 16 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: partition function

The full partition function reads ZS2

L∩S2 R⊂S4 b = Z

S4

b

freehyper

  • BL,R
  • dσL,R
  • α=L,R

Z

S2

α

SQCD(A)(σα, Bα)

× ZN

F/C(σ, B)ZS F/C(σ, B)

the 0d Fermi/chiral contribution at N- and S-poles ZN/S

F/C(σ, B) =

     nR

a=1

nL

b=1 ∆N/S ab

nR

a=1

nL

b=1

  • ± (∆N/S

ab

± (b2 + b−2)/2)

−1

where ∆N/S

ab

≡ b−1(iσL

a±BL a /2) − b(iσR b ±BR b /2). Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 16 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: compare with Liouville [Fateev, et.al, ’09; Zamolodchikov, Zamolodchikov;]

Liouville CFT on Σg=0 = S2; parameter b

su(2) fund. (= symm1 = ∧1), highest weight Ω = 1/2 .

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 17 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: compare with Liouville [Fateev, et.al, ’09; Zamolodchikov, Zamolodchikov;]

Liouville CFT on Σg=0 = S2; parameter b

su(2) fund. (= symm1 = ∧1), highest weight Ω = 1/2 .

Consider 3 generic vertex op., 1 degenerate vertex, αdeg = −b/2 − b−1/2

  • Vα0(0)V−b/2−b−1/2(x, ¯

x)Vα1(1)Vα∞(∞)

  • Yiwen Pan (Yiwen Pan)

Intersecting defects & 2d CFT 2016 Nov 17 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: compare with Liouville [Fateev, et.al, ’09; Zamolodchikov, Zamolodchikov;]

Liouville CFT on Σg=0 = S2; parameter b

su(2) fund. (= symm1 = ∧1), highest weight Ω = 1/2 .

Consider 3 generic vertex op., 1 degenerate vertex, αdeg = −b/2 − b−1/2

  • Vα0(0)V−b/2−b−1/2(x, ¯

x)Vα1(1)Vα∞(∞)

  • 5th

= 1 AF(x, ¯ x)ZS2

L∪S2 R⊂S4 b

  

1 1 2 2

  

5th

= 1 AC(x, ¯ x)ZS2

L∪S2 R⊂S4 b

  

1 1 2 2

  

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 17 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: compare with Liouville [Fateev, et.al, ’09; Zamolodchikov, Zamolodchikov;]

Liouville CFT on Σg=0 = S2; parameter b

su(2) fund. (= symm1 = ∧1), highest weight Ω = 1/2 .

Consider 3 generic vertex op., 1 degenerate vertex, αdeg = −b/2 − b−1/2

  • Vα0(0)V−b/2−b−1/2(x, ¯

x)Vα1(1)Vα∞(∞)

  • 5th

= 1 AF(x, ¯ x)ZS2

L∪S2 R⊂S4 b

  

1 1 2 2

  

5th

= 1 AC(x, ¯ x)ZS2

L∪S2 R⊂S4 b

  

1 1 2 2

   nf = 2, ξFI = ξ′

FI = ξR FI = −ξL FI ≡ ξ. Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 17 / 27

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Introduction Intersecting surface defects Summary Open problems

Example: compare with Liouville [Fateev, et.al, ’09; Zamolodchikov, Zamolodchikov;]

Liouville CFT on Σg=0 = S2; parameter b

su(2) fund. (= symm1 = ∧1), highest weight Ω = 1/2 .

Consider 3 generic vertex op., 1 degenerate vertex, αdeg = −b/2 − b−1/2

  • Vα0(0)V−b/2−b−1/2(x, ¯

x)Vα1(1)Vα∞(∞)

  • 5th

= 1 AF(x, ¯ x)ZS2

L∪S2 R⊂S4 b

  

1 1 2 2

  

5th

= 1 AC(x, ¯ x)ZS2

L∪S2 R⊂S4 b

  

1 1 2 2

   nf = 2, ξFI = ξ′

FI = ξR FI = −ξL FI ≡ ξ.

Left-right mass relations b−1mi = bm′

i + i 2(b2 − b−2), etc: superpotential

CFT R, R′ = x α′s Quiver n = n′ = ... = 1 e−2πξ masses

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 17 / 27

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Introduction Intersecting surface defects Summary Open problems

Seiberg-like duality

Hint from previous slide: quivers of F-type and C-type have equal partition function 1 AF(x, ¯ x)Z      

nL nR nf nf S2

R

S2

L

ξL

FI

ξR

FI

S4

b

N S

     

2nd

= 1 AC(x, ¯ x)Z      

n′ n nf nf S2

R

S2

L

ξ′

FI

ξFI S4

b

N S

     

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 18 / 27

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Introduction Intersecting surface defects Summary Open problems

Seiberg-like duality

Hint from previous slide: quivers of F-type and C-type have equal partition function 1 AF(x, ¯ x)Z      

nL nR nf nf S2

R

S2

L

ξL

FI

ξR

FI

S4

b

N S

     

2nd

= 1 AC(x, ¯ x)Z      

n′ n nf nf S2

R

S2

L

ξ′

FI

ξFI S4

b

N S

      Parameters relations

n = nR, n′ = nf − nL ξFI = ξ′

FI = ξR FI = −ξL FI ≡ ξ

Mass relations mj + i/2 = mR

j , m′ j + i/2 = ˜

mL

j , etc. Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 18 / 27

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Introduction Intersecting surface defects Summary Open problems

Symmetry enhancement

Round sphere limit b → 1. αdeg = −b/2 − b−1/2 → −b ∼ highest weight of R ≡ symm2 su(2). By [Gomis, Le Floch], intersecting defect

b→0

− − − → single defect on one S2: ZS2

L∪S2 R⊂S4 b

  • 1

1 2 2 S2

R

S2

L

N S

  • b→1

− − − → ZS2⊂S4

  • 2

2 2 S2

  • Yiwen Pan (Yiwen Pan)

Intersecting defects & 2d CFT 2016 Nov 19 / 27

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Introduction Intersecting surface defects Summary Open problems

Higgsing

Procedure of constructing surface defects [Gaiotto, Razamat, Rastelli; Gaiotto, Kim;]. T N =2

UV Higgsing

− − − − → T N =2

IR

+ surface defects C-type Quiver from Higgsing, I.P.F. with surface defects from Higgsing Factorization [Pan, Peelaers, to appear]: Y → Y L, Y R, Zinst(Y ) ∼ Zvortex(Y L)Zvortex(Y R)Z0dchiral(Y L, Y R) Example:

nf nf nf

Higgsing

− − − − →

n′ n nf nf N S nf nf nf nf

Higgsing

− − − − →

n′ n nf nf nf N S Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 20 / 27

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Introduction Intersecting surface defects Summary Open problems

Summary/conjectures

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 21 / 27

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Introduction Intersecting surface defects Summary Open problems

The conjecture: ingredients

Players

4d-2d-0d coupled quiver gauge theories : F- and C-type Degenerate Liouville/Toda correlators

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 22 / 27

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Introduction Intersecting surface defects Summary Open problems

The conjecture: ingredients

Players

4d-2d-0d coupled quiver gauge theories : F- and C-type Degenerate Liouville/Toda correlators

Input: two reps R, R′ of su(nf):

determine 2d part of the quivers; determine the deg. momentum: αdeg = −bΩR − b−1ΩR′.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 22 / 27

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Introduction Intersecting surface defects Summary Open problems

The conjecture: ingredients

Players

4d-2d-0d coupled quiver gauge theories : F- and C-type Degenerate Liouville/Toda correlators

Input: two reps R, R′ of su(nf):

determine 2d part of the quivers; determine the deg. momentum: αdeg = −bΩR − b−1ΩR′.

Other input: ξ ↔ x, m, ˜ m ↔ α0, α∞

ξ: 2d FI parameter; m, ˜ m: 2d masses; x: position of the deg. vertex operator; α0, α∞ momenta of other vertex operators.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 22 / 27

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Introduction Intersecting surface defects Summary Open problems

The conjecture: inputs R, R′

R, R′ determines 2d part of the F-type quivers:

ν′ nf−n′

1

nf−n′

2+n′ 1

... nf−n′

ν′−1+n′ ν′−2

nf−n′

ν′+n′ ν′−1

ν nν−nν−1 nν−1−nν−2 ... n2−n1 n1

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 23 / 27

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Introduction Intersecting surface defects Summary Open problems

The conjecture: inputs R, R′

R, R′ determines 2d part of the F-type quivers:

S4

b

S2

R

S2

L

0d Fermi

nf nf n′

ν′

n′

ν′−1

· · · n′

1

nν nν−1 · · · n1

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 23 / 27

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Introduction Intersecting surface defects Summary Open problems

The conjecture: inputs R, R′

R, R′ determines 2d part of the F-type quivers:

S4

b

S2

R

S2

L

0d Fermi

nf nf n′

ν′

n′

ν′−1

· · · n′

1

nν nν−1 · · · n1

R, R′ determines the degenerate momentum: αdeg = −bΩR − b−1ΩR′:

  • ˆ

Vα0(0) ˆ Vαdeg(x, ¯ x) ˆ Vα1(1) ˆ Vα∞(∞)

  • Yiwen Pan (Yiwen Pan)

Intersecting defects & 2d CFT 2016 Nov 23 / 27

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Introduction Intersecting surface defects Summary Open problems

The conjectures

Full (S4 ⊃ S2

L ∪ S2 R)-partition function of the F-type quiver gauge theory =

Liouvlle/Toda degenerate correlator with ˆ V−bΩR−b−1ΩR′ A special set of F-type quivers are Seiberg-dual to C-type quivers:

nL nR nf nf S2

R

S2

L

ξL

FI

ξR

FI

S4

b

N S Seiberg-like duality n′ n nf nf S2

R

S2

L

ξ′

FI

ξFI S4

b

N S

where nL + n′ = nf, n = nR.

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 24 / 27

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Introduction Intersecting surface defects Summary Open problems

Open problems

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 25 / 27

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Introduction Intersecting surface defects Summary Open problems

Open problems

F-type quiver from Higgsing ? Seiberg-like, hopping dualities when T 4d is interacting ? Brane realization of the Seiberg-like duality (C-type ↔ F-type)? Generalize to intersecting Levi-type defects, and Wρ correlators? What is the low energy effective theories for intersecting GLSM’s? NLSM’s on intersecting worldsheets?

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 26 / 27

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Introduction Intersecting surface defects Summary Open problems

And ...

Thank you for your attention!

Yiwen Pan (Yiwen Pan) Intersecting defects & 2d CFT 2016 Nov 27 / 27