Quantum Mirror Map for Del Pezzo Geometries Yuji Sugimoto (Univ. - - PowerPoint PPT Presentation

quantum mirror map for del pezzo geometries
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Quantum Mirror Map for Del Pezzo Geometries Yuji Sugimoto (Univ. - - PowerPoint PPT Presentation

Quantum Mirror Map for Del Pezzo Geometries Yuji Sugimoto (Univ. of Science and Technology of China) with Sanefumi Moriyama Tomohiro Furukawa (Osaka City Univ.) arXiv:190?.????? Susy. Gauge Theory encoded in algebraic curve Mirror Map


slide-1
SLIDE 1

Quantum Mirror Map for Del Pezzo Geometries

Yuji Sugimoto

(Univ. of Science and Technology of China)

with Sanefumi Moriyama Tomohiro Furukawa

(Osaka City Univ.)

arXiv:190?.?????

slide-2
SLIDE 2
  • Susy. Gauge Theory

encoded in algebraic curve

Mirror Map redefining the variables

well studied not well studied A-period B-period

Free energy relating to BPS indices

ΠA(z) ∼ c1z + c2z + ...

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slide-3
SLIDE 3
  • Susy. Gauge Theory

Mirror Map redefining the variables

not well studied

ΠA(z) ∼ c1z + c2z + ...

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We need this to obtain physical quntities and reach important results.

encoded in algebraic curve A-period

slide-4
SLIDE 4

ABJM theory (U(N)k x U(N)-k)

(see also Moriyama, Kubo, Furukawa’s slides&Poster)

encoded in local P1 x P1 with quantization

b H = ⇣ b Q1/2 + b Q−1/2⌘ ⇣ b P 1/2 + b P −1/2⌘

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[Aharony-Bergman-Jafferis-Maldacena 2008]

Quantum Mirror Map

ΠA(z)

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ΠA(z, ~)

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b Q b P = q b P b Q

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  • q = ei~
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slide-5
SLIDE 5

U(N)k x U(N)-k x U(N)k x U(N)-k

(1,1,1,1) model

generalization

b H = h⇣ Q1/2 + Q−1/2⌘ ⇣ P 1/2 + P −1/2⌘i2

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ABJM theory (U(N)k x U(N)-k)

(see also Moriyama, Kubo, Furukawa’s slides&Poster)

encoded in local P1 x P1 with quantization

b H = ⇣ b Q1/2 + b Q−1/2⌘ ⇣ b P 1/2 + b P −1/2⌘

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[Aharony-Bergman-Jafferis-Maldacena 2008]

b Q b P = q b P b Q

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[Honda, Moriyama, 2014]

slide-6
SLIDE 6

A1 (SU(2)) sym.

We study D5 Del Pezzo

D5 (SO(10)) sym. → [SU(2)] sym.

3

ABJM theory (U(N)k x U(N)-k)

U(N)k x U(N)-k x U(N)k x U(N)-k

(1,1,1,1) model

generalization

b H = h⇣ Q1/2 + Q−1/2⌘ ⇣ P 1/2 + P −1/2⌘i2

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(see also Moriyama, Kubo, Furukawa’s slides&Poster)

encoded in local P1 x P1 with quantization

b H = ⇣ b Q1/2 + b Q−1/2⌘ ⇣ b P 1/2 + b P −1/2⌘

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[Aharony-Bergman-Jafferis-Maldacena 2008]

b Q b P = q b P b Q

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slide-7
SLIDE 7

A-period is similar to B-period

expressed by Weyl characters

  • coef. of characters = Integer

Result

has same reps. as those in B-period

slide-8
SLIDE 8

② ①

expressed by Weyl characters

  • coef. of characters = Integer

Plan

has same reps. as those in B-period

slide-9
SLIDE 9

b H α = e3e4 b Q−1 b P − (e3 + e4) b P + b Q b P − h−1

2 e3e4(e5 + e6) b

Q−1 + E α − (e−1

1

+ e−1

2 ) b

Q + h2

1(e1e2e7e8)−1 b

Q−1 b P −1 − h1(e1e2)−1(e−1

7

+ e−1

8 ) b

P −1 + (e1e2)−1 b Q b P −1

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D5 Del Pezzo geometry

Q = 0

<latexit sha1_base64="yGfdQ7nKLWgcKHYQb9v8obTzXYE=">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</latexit>

P = 0

<latexit sha1_base64="SghZGm+O9CFYiRz2Wz8Of5NzvrE=">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</latexit>

P = ∞

<latexit sha1_base64="Q4CsTvk7z0dAg62Av3h2DtR/w=">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</latexit>

Q = ∞

<latexit sha1_base64="aszeUDvugaVOIP7JMN3Y1S2YUo=">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</latexit>

e4

<latexit sha1_base64="qkNPC7agXJ68q4v4qax9feKqg=">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</latexit>

e3

<latexit sha1_base64="HDzKSgefJW/uzauFSY1N12Nxu34=">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</latexit>

h1e−1

8

<latexit sha1_base64="LOaFwMATVFhFqDb+qc4ZJs8PGFU=">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</latexit>

h1e−1

7

<latexit sha1_base64="1IAgu192M0dYhdGx3z/dZSquUvk=">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</latexit>

e−1

2

<latexit sha1_base64="7ia7L8ZIGNrw3aSebl8M3NUI9ro=">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</latexit>

e−1

1

<latexit sha1_base64="4mVUrj4zl14frJDRMF/DGuSOkZg=">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</latexit>

e6h−1

2

<latexit sha1_base64="vVUeIRsFZ1iO5jFcp1CovUDmr1k=">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</latexit>

e5h−1

2

<latexit sha1_base64="+2r0VA8nD78VtKAgPbY5Cgeq3s=">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</latexit>

s0

<latexit sha1_base64="OGVaEN+Oa74rOw4mjSn7DpsXiaE=">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</latexit>

s1

<latexit sha1_base64="gFHQETiu8CG+L2dqSBvJKonVPKM=">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</latexit>

s2

<latexit sha1_base64="Q8YPSzgTfP+MUAJOEs+W4s5aL8=">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</latexit>

s3

<latexit sha1_base64="rXCMdtgJBTcvMmf571hU23svzys=">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</latexit>

s4

<latexit sha1_base64="LSKWKP0c1fJPdIxVj+RXjp4n5w=">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</latexit>

s5

<latexit sha1_base64="WOTC12UYnj+o6/DA2wJmHok5YhE=">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</latexit>

D5 Weyl Group 10 parameters

(s1, s2, s3, s4, s5)

<latexit sha1_base64="pD1Stg0J8yUL6DmzMCUZF6JEy8=">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</latexit>

h2

1h2 2 = 8

Y

i=1

ei

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Kubo-Moriyama-Nosaka(2018)

slide-10
SLIDE 10

b H α = e3e4 b Q−1 b P − (e3 + e4) b P + b Q b P − h−1

2 e3e4(e5 + e6) b

Q−1 + E α − (e−1

1

+ e−1

2 ) b

Q + h2

1(e1e2e7e8)−1 b

Q−1 b P −1 − h1(e1e2)−1(e−1

7

+ e−1

8 ) b

P −1 + (e1e2)−1 b Q b P −1

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D5 Del Pezzo geometry 10 parameters - (2 + 2 + 1) parameters

( b P, b Q) ∼ (A b P, B b Q)

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h2

1h2 2 = 8

Y

i=1

ei

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5 parameters (e1, e3, e5, ¯

h1, ¯ h2)

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¯ h1 = qh1, ¯ h2 = q−1h2

  • <latexit sha1_base64="FJaG6I7USAlWheZ80Cb+oLpo+k=">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</latexit>

8 asymptotic values

Kubo-Moriyama-Nosaka(2018)

slide-11
SLIDE 11

Quantum mirror map

P[X] = Ψ[q−1X] Ψ[X] , b QΨ[X] = XΨ[X]

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" b H α + z α # Ψ[X] = 0

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solve Schrodinger eq. order by order ̈ invariant under Weyl transf.

character ??

A2 α2q−1 = e3 e1 + h1 e1 + h1e3 e1 + e3e5 h

2 2

+ e3e5 h1h

2 2

+ e2

3e5

h1h

2 2

+ e3 h2 + e3 h2e1 + e3e5 h2 + e3e5 h2e1

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χ10(e−1

i , ¯

h−1

i ) = χ10(ei, ¯

hi)

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??

ΠA(z, ~) ∼ I log P[X] X dX = Ez−1 + ✓ −E2 2 − A2 ◆ z−2 + · · ·

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10 terms

slide-12
SLIDE 12

Weyl character

A2 α2q−1 = e3 e1 + h1 e1 + h1e3 e1 + e3e5 h

2 2

+ e3e5 h1h

2 2

+ e2

3e5

h1h

2 2

+ e3 h2 + e3 h2e1 + e3e5 h2 + e3e5 h2e1

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act on

(s1, s2, s3, s4, s5)

<latexit sha1_base64="pD1Stg0J8yUL6DmzMCUZF6JEy8=">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</latexit>

(e1, e3, e5, ¯ h1, ¯ h2, α)

<latexit sha1_base64="+SJwj4jvu7R7YKE9XWiXq1rk2eE=">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</latexit>

Kubo-Moriyama- Nosaka(2018)

s2 : (e1, e3, e5, ¯ h1, ¯ h2, α) 7! ✓ e1, 1 e3 , e5, ¯ h1 e3 , ¯ h2, e3α ◆

<latexit sha1_base64="QANBRzwsE25ZSBxGo+zM0gW9LCw=">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</latexit>

E.g.)

slide-13
SLIDE 13

Weyl character

A2 α2q−1 = e3 e1 + h1 e1 + h1e3 e1 + e3e5 h

2 2

+ e3e5 h1h

2 2

+ e2

3e5

h1h

2 2

+ e3 h2 + e3 h2e1 + e3e5 h2 + e3e5 h2e1

<latexit sha1_base64="LOjc5BFqX6eDOYw6GBf6xhB2jws=">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</latexit>

acting only 5 parameters generate D5 Weyl group

act on

(s1, s2, s3, s4, s5)

<latexit sha1_base64="pD1Stg0J8yUL6DmzMCUZF6JEy8=">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</latexit>

α

<latexit sha1_base64="5vfKUC1XlKnzEhl9imHzM3/Xf2s=">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</latexit>

role of ??

(e1, e3, e5, ¯ h1, ¯ h2, α)

<latexit sha1_base64="+SJwj4jvu7R7YKE9XWiXq1rk2eE=">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</latexit>

Kubo-Moriyama- Nosaka(2018)

slide-14
SLIDE 14

Weyl character

A2 α2q−1 = e3 e1 + h1 e1 + h1e3 e1 + e3e5 h

2 2

+ e3e5 h1h

2 2

+ e2

3e5

h1h

2 2

+ e3 h2 + e3 h2e1 + e3e5 h2 + e3e5 h2e1

<latexit sha1_base64="LOjc5BFqX6eDOYw6GBf6xhB2jws=">AD13iclVG9S+RQEJ8YP1fPXbURbJZbFEFueYmKIgjqNZbqun7guHl+dYNZpOYZBc0BDsRu6srO7AQuyvsLW5f+AKu2u9Kz2wsXDyAZ4RV25C3puZ3/xmfo9RLV1zXEJuhRaxta29o7Mr1d3zoTed6etfc8y6zXiRmbpb6jU4bpm8KruTrfsGxOa6rO19W9zwG+3uC2o5nGqntg8e0a3TW0isaoiymlTyiUKjZl3rwi+16J6laVluXsftn7JPl+amQ2Qrky7uMh+WNRXDKxZzAyW8Xkm0iShjFXJv0XRXJZ9lNcClZ/Fxblv+nOjH236EJ6D3FTcGArGRyJE9Cy752pNjJQWxLZuY7lGAHTGBQhxpwMBFXwcKDn5bIAEBC3Pb4GHORk8LcQ4+pJBbxyqOFRSze3juYrQVZw2Mg5OyGY4RcfRmYWhslPcknuyQ9yRe7I45u9vLBHoOUAbzXicktJnw4WHt5l1fB2ofrMaqrZhQpMh1o1G6FmeAVLOI3Ds/uCzMrw94I+Ub+oP6v5Jbc4AuMxl92scxXzpvoUVFLsB4puYzXzpqcl8bz8vJEbm4hXlQnDMFHGMVtTMEcLMISFIEJ18Iv4U74LW6KR+KxeBKVtgxZwBemPjlCQ+rD9c=</latexit>

redundant

χ10

<latexit sha1_base64="jSRDcgLoRszd2lboD9+P4LC/GE=">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</latexit>

act on

(s1, s2, s3, s4, s5)

<latexit sha1_base64="pD1Stg0J8yUL6DmzMCUZF6JEy8=">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</latexit>

(e1, e3, e5, ¯ h1, ¯ h2, α)

<latexit sha1_base64="+SJwj4jvu7R7YKE9XWiXq1rk2eE=">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</latexit>

α = q1/2¯ h1/2

2

e1/4

1

e−1/2

3

e−1/4

5

<latexit sha1_base64="Yv2x0wTxEpatn2SQa0dM8xzr6A=">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</latexit>

A2 = √e1√e5 ¯ h2 + ¯ h2 √e1√e5 + √e1√e5 ¯ h1¯ h2 + ¯ h1¯ h2 √e1√e5 √e1e3√e5 ¯ h1¯ h2 + ¯ h1¯ h2 √e1e3√e5 + √e1 √e5 + 1 √e1√e5 + √e1 √e5 + √e5 √e1

<latexit sha1_base64="TkJgGx9U0zF1cLzeVHQEZVe/tZo=">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</latexit>

Kubo-Moriyama- Nosaka(2018)

slide-15
SLIDE 15

② ①

expressed by Weyl characters

  • coef. of characters = Integer

Plan

has same reps. as those in B-period

slide-16
SLIDE 16

Multi-covering & BPS

fractional simpler structure ?

A-period for A1 geometry has multi-covering structure

[Hatsuda-Marino-Moriyama-Okuyama(2013)]

ΠA (z, ~) ∼

X

l=1

(−1)l+1Alz−l

<latexit sha1_base64="vkW0Uhre3ySY5gUqkHxDatgXN8Q=">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</latexit>

A4 3 3 2χ54 + 5 2χ45 + 11 2 χ1

<latexit sha1_base64="zZjDK9zHXxD9krZ+BFA49/sVgPc=">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</latexit>

(coef. of n-th order) = Σ (coef. of j-th order)

multi-covering structure

j≦n

slide-17
SLIDE 17

ΠA (z, ~) ∼

X

l=1

(−1)l+1Alz−l

<latexit sha1_base64="vkW0Uhre3ySY5gUqkHxDatgXN8Q=">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</latexit>

A1 = 0 A2 = χ10 A3 = (q1/2 + q−1/2)χ16 A4 =

  • q2 + q−2

χ1 + ⇣ q3/2 + q−3/2⌘ (χ45 + 3χ1) + 3χ54 + 5χ45 + 11χ1 2

<latexit sha1_base64="50/tx1HCoNk65968pNcQIivwuTE=">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</latexit>

Multi-covering & BPS

still fractional

log z ∼ −A1z−1

eff +

  • A2 − A2

1

  • z−2

eff −

✓ A3 − 3A1A2 + 3 2A3

1

◆ z−3

eff

+ ✓ A4 − 2A2

2 − 4A1A3 − 8A2 1A2 − 8

3A4

1

◆ z−4

eff + ...

<latexit sha1_base64="Z2PNn2H+aw1Cumg+kA018wI+tVc=">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</latexit>

=: log zeff − log z

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slide-18
SLIDE 18

Multi-covering & BPS

= ✏1(q, ei, ¯ hi) = −✏2(q, ei, ¯ hi) + ✏1(q2, e2

i , ¯

h2

i )

2 = ✏3(q, ei, ¯ hi) + ✏1(q3, e3

i , ¯

h3

i )

3 = −✏4(q, ei, ¯ hi) + ✏2(q2, e2

i , ¯

h2

i )

2 + ✏1(q4, e4

i , ¯

h4

i )

4

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multi-covering structure

log z ∼ −A1z−1

eff +

  • A2 − A2

1

  • z−2

eff −

✓ A3 − 3A1A2 + 3 2A3

1

◆ z−3

eff

+ ✓ A4 − 2A2

2 − 4A1A3 − 8A2 1A2 − 8

3A4

1

◆ z−4

eff + ...

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integer !

✏1 = 0, − ✏2 = 10, ✏3 = ⇣ q1/2 + q−1/2⌘ 16 , −✏4 =

  • q2 + q−2

1 +

  • q + q−1

(45 + 31) + 41

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ΠA (z, ~) ∼

X

l=1

(−1)l+1Alz−l

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=: log zeff − log z

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slide-19
SLIDE 19

log z ∼ −A1z−1

eff +

  • A2 − A2

1

  • z−2

eff −

✓ A3 − 3A1A2 + 3 2A3

1

◆ z−3

eff

+ ✓ A4 − 2A2

2 − 4A1A3 − 8A2 1A2 − 8

3A4

1

◆ z−4

eff + ...

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Multi-covering & BPS

multi-covering structure (1,1,1,1) has multi covering structure

✏1 = 0, −✏2 = 10, ✏3 = (q

1 2 + q− 1 2 )16,

−✏4 = (q2 + q−2)1 + (q + q−1)(45 + 31) + 41, ✏5 = (q

5 2 + q− 5 2 )16 + (q 3 2 + q− 3 2 )(144 + 316) + (q 1 2 + q− 1 2 )316,

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ΠA (z, ~) ∼

X

l=1

(−1)l+1Alz−l

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=: log zeff − log z

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X

l=1

@X

n|l

(−1)n✏ l

n (qn, en

i , ¯

hn

i )

n 1 A z−l

eff

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slide-20
SLIDE 20

Multi-covering & BPS

B-period

[Moriyama-Nosaka-Yano(2017)]

same reps. as B-period in same degree

✏1 = 0, −✏2 = 10, ✏3 = (q

1 2 + q− 1 2 )16,

−✏4 = (q2 + q−2)1 + (q + q−1)(45 + 31) + 41, ✏5 = (q

5 2 + q− 5 2 )16 + (q 3 2 + q− 3 2 )(144 + 316) + (q 1 2 + q− 1 2 )316,

<latexit sha1_base64="Oq1O2v3Xl3Vw6EBsQYElEAw/E=">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</latexit>
slide-21
SLIDE 21

expressed by Weyl characters

  • coef. of characters = Integer

A-period is similar to B-period

Summary

has same reps. as those in B-period

slide-22
SLIDE 22

Future work

what happen when use SU(2) character ?

why same structure ? Q. Q. Q. what meaning of ?

α

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slide-23
SLIDE 23

✏7 = (q

11 2 + q− 11 2 )16 + (q 9 2 + q− 9 2 )(144 + 416) + (q 7 2 + q− 7 2 )(560 + 4144 + 1316)

+ (q

5 2 + q− 5 2 )(720 + 4560 + 9144 + 2516) + (q 3 2 + q− 3 2 )(3560 + 8144 + 2716)

+ (q

1 2 + q− 1 2 )(720 + 3560 + 9144 + 2716),

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before using SU(2) character after using SU(2) character

✏7 = 11

2 16 + 9 2 (144 + 316) + 7 2 (560 + 3144 + 916)

+ 5

2 (720 + 3560 + 5144 + 1216) + 3 2 (−720 − 560 − 144 + 216)

+ 1

2 (720 + 144),

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much simpler for q-dependent term & coef. but negative coef,

χn

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SU(2) character

slide-24
SLIDE 24

Future work

what about SU(2) character ? why same structure ? Q. Q. Q. what meaning of ?

α

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slide-25
SLIDE 25

expressed by Weyl characters

  • coef. of characters = Integer

A-period is similar to B-period

Summary

has same reps. as those in B-period

slide-26
SLIDE 26

without overall minus sign

with overall minus sign

−✏4 = (q2 + q−2)1 + (q + q−1)(45 + 31) + 41

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do not appear

✏4 =

  • q2 + q−2

1 +

  • q + q−1

(45 + 31) + (−54 + 45 + 31)

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(coef. of n-th order) = Σ (coef. of j-th order)

multi-covering structure

j≦n

slide-27
SLIDE 27

(1,1,1,1) model (2,2) model

(e1, e3, e5, ¯ h1, ¯ h2) = (1, 1, 1, q, q−1) = (q−1/2, q1/2, q−1/2, q, q−1)

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(effective) chemical pot. = log [ (effective) complex modulus ]

µ = log z, µeff = log zeff

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Ξk(z) =

X

N=0

zNZk(N) = Det h 1 + z b H−1i ,

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Partition fn. of N M2-branes

slide-28
SLIDE 28

from [Kubo,Moriyama,Nosaka(2018)]

Q = 0

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P = 0

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P = ∞

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Q = ∞

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e4

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e3

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h1e−1

8

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h1e−1

7

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e−1

2

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e−1

1

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e6h−1

2

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e5h−1

2

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s0

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s2

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s3

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s4

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s5

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