Monodromies revisited with twisted homology and BCJ Piotr Tourkine, - - PowerPoint PPT Presentation

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Monodromies revisited with twisted homology and BCJ Piotr Tourkine, - - PowerPoint PPT Presentation

Oct 27, 2023 482 likes •997 views

Monodromies revisited with twisted homology and BCJ Piotr Tourkine, CNRS & LPTHE, Sorbonne Universits QCD meets Gravity V, dec 2019, UCLA, Mani L. Bhaumik Institute for Theoretical Physics [arXiv:1910.08514] [Work in progress] +

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Monodromies revisited with twisted homology and BCJ

Piotr Tourkine, CNRS & LPTHE, Sorbonne Universités QCD meets Gravity V, dec 2019, UCLA, Mani L. Bhaumik Institute for Theoretical Physics

[arXiv:1910.08514] Monodromy relations from twisted homology

  • E. Casali, S. Mizera, P. Tourkine

[arXiv:1608.01665] Phys.Rev.Lett. 117 (2016) 211601 Higher-loop amplitude monodromy relations in string and gauge theory

  • P. Tourkine, P. Vanhove

[arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts

  • A. Ochirov, P. Tourkine, P. Vanhove

[Work in progress]

  • E. Casali, S. Mizera, P. Tourkine

+

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previously

B 1 B i B g B 1 B i B g A 1 A 1 A 1 A i A i A g A g A 1 A i A i A g A g d r a
  • 1
c
  • 1
b ( 1 ) 1 b ( 1 ) r 1 + 1 a
  • i
c
  • i
b ( i ) 1 b ( i ) r i + 1 a
  • g
c
  • g
b ( g ) r g + 1 b ( g ) 1 c + g a + g b ( g ) 1 b ( g ) r g + 1 a + 1 c + 1 b ( 1 ) 1 b ( 1 ) r + 1 a + i c + i b ( i ) r i + 1 b ( i ) 1

C+

d r
  • 1
d 2 d 1

C

β 1 β 2 β i β i + 1 β g α r + 1 α ˜ r 1 α ˜ r i−1 + 1 α ˜ r i Q

z

2

z

3

z

4

z

n − 1

z

n

α

3

α

4

α

n − 1

α

n

d

1

d

2

d

3

d

n − 1

d

1

C

−

C

+

d

4

d

n − 2

Set of linear relationship between open string theory loop integrands

[arXiv:1608.01665] Phys.Rev.Lett. 117 (2016) 211601 Higher-loop amplitude monodromy relations in string and gauge theory

  • P. Tourkine, P. Vanhove

[arXiv:1702.04963] Nucl.Phys. B925 (2017) 63-134 Monodromy Relations in Higher-Loop String Amplitudes

  • S. Hohenegger, S. Stieberger

[arXiv:1910.08514] Monodromy relations from twisted homology

  • E. Casali, S. Mizera, P. Tourkine

[arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts

  • A. Ochirov, P. Tourkine, P. Vanhove

2

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Will dissect the relations in field theory

now

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motivations

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motivations

  • string theoretic formalisms can often teach us a lot about

generic properties of perturbative QFT

  • because the worldsheet is nicer than the worldline (true as

well for CHY)

  • and because of non-pertubative physics + dualities +

supersymmetry (but susy also on worldline, see Green Bjornsson)

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  • The KLT relations between open (=gauge) and closed (=gravity)

strings is somehow at the roots of this conference.

  • Looking at the monodromies, we will try to draw some conclusions

concerning two related and interconnected questions

  • the “labeling problem” (BCJ around the loop)
  • some more speculative aspects on the double copy
  • pen open

closed

× =

motivations

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BCJ around the clock

1 2 3 4

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BCJ around the clock

  • =

ns nt nu

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BCJ around the clock

  • Do successive BCJ moves

around the loop.

  • First and last diagrams do not

match: there is a loop momentum shifting ambiguity.

  • Generalises to higher loops

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BCJ around the clock

  • Do successive BCJ moves

around the loop.

  • First and last diagrams do not

match: there is a loop momentum shifting ambiguity.

  • Generalises to higher loops

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BCJ around the clock

  • Do successive BCJ moves

around the loop.

  • First and last diagrams do not

match: there is a loop momentum shifting ambiguity.

  • Generalises to higher loops

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Double-copy

  • works amazingly well
  • up to 4 loops
  • 5 loops : needs to be modified. Why ?

[arXiv:1708.06807] Phys.Rev. D96 (2017) 126012 Five-loop four-point integrand of N=8 supergravity as a generalized double copy

  • Z. Bern, J. J. M. Carrasco, W. Chen, H. Johansson, R. Roiban, M. Zeng

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  • In this talk, we will see how much we can extract from the

monodromy relations and twisted homology to attack these two questions

  • Will require a closer look at the field theory limit of the

monodromy relations

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Outline

  • Loop momentum in string theory
  • Monodromy relations
  • Tree-level : sum and difference
  • Loop-level : same

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The field theory limit

  • One topology in string theory (genus = number of loops)

descends to all possible graphs in the field theory limit

  • Scherk 70’s : tree-level
  • Bern-Kosower string based

rules 90’s : one-loop

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The loop momentum in string theory

  • String theory has a uniform definition of the loop momentum
  • zero-mode of momentum field

:

  • Descends to all the topologies of graphs generated in the field

theory limit. Example :

Pμ = ∂tXμ ℓμ = ∮ ∂tXμdσ

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field theory limit of the monodromy relations

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warm-up : tree-level

z2 z3 z4 zn−1 zn α3 α4 αn−1 αn d1 d2 d3 dn−1 d1

C− C+

d4 dn−2 18

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warm-up : tree-level

eiπα′sA(s, u) + A(s, t) + e−iπα′tA(t, u) = 0 e−iπα′sA(s, u) + A(s, t) + e+iπα′tA(t, u) = 0

valid for complex kinematics so not quite just complex conjugation

z2 z3 z4 1 ∞ z1

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warm-up : tree-level

eiπα′sA(s, u) + A(s, t) + e−iπα′tA(t, u) = 0 e−iπα′sA(s, u) + A(s, t) + e+iπα′tA(t, u) = 0

Sum and difference gives cosines and sines :

A(s, u) = sin(α′πt) sin(α′πs) A(t, u) cos(πα′s)A(s, u) + A(s, t) + cos(πα′t)A(t, u) = 0

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warm-up : tree-level

A(s, u) + A(s, t) + A(t, u) = 0

As , one is left with

α′ → 0

sA(s, u) + tA(t, u) = 0 eiπα′sA(s, u) + A(s, t) + e−iπα′tA(t, u) = 0 e−iπα′sA(s, u) + A(s, t) + e+iπα′tA(t, u) = 0

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loop-level

  • generic mechanism
  • boundary effects

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One-loop

C− C+

d1 d2 zm z2 z3 zm zm−1 zm−2 dm−2 dm−1 a− a+ c− c+ α2 α3 αm−2 αm−1 β

B B A0 A00 A0 A00

zm+1 zn−1 zn zm+2 αm+1 αm+2 αn−1 αn zm+1 zm+2 zn−1 zn b1 b2 bn−m bn−m+1 b1 b2 bn−m bn−m+1

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One-loop

C− C+

d1 d2 zm z2 z3 zm zm−1 zm−2 dm−2 dm−1 a− a+ c− c+ α2 α3 αm−2 αm−1 β

B B A0 A00 A0 A00

zm+1 zn−1 zn zm+2 αm+1 αm+2 αn−1 αn zm+1 zm+2 zn−1 zn b1 b2 bn−m bn−m+1 b1 b2 bn−m bn−m+1

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Loop monodromies

4 2 3

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Loop monodromies

3 4 2 1 3 4 2 1 3 4 2 1

4 2 3

3 4 2 1

eiπα′k1⋅k2 eiπα′k1⋅(k2+k3) eiπα′k1⋅ℓ

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3 4 2 1 3 4 2 1 3 4 2 1

4 2 3

3 4 2 1 4 1 2 3

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4 1 2 3

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Relations

I(1234) + eiπα′k1⋅k2I(2134) + eiπα′k1⋅(k2+k3)I(2314) −eiπα′k1⋅ℓI(234|1) + Jup − Jdown = 0 Sum and difference, plus field theory limit gives KK and fundamental BCJ Let’s look at the ’s in the formula above

I

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I(1234) + eiπα′k1⋅k2I(2134) + eiπα′k1⋅(k2+k3)I(2314) − eiπα′k1⋅ℓI(234|1)

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ℓ · k1 1 2 4 3 + (ℓ + k2) · k1 2 1 4 3 + (ℓ + k23) · k1 2 3 4 1

  • I(1234) + eiπα′k1⋅k2I(2134) + eiπα′k1⋅(k2+k3)I(2314) − eiπα′k1⋅ℓI(234|1)

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ℓ · k1 1 2 4 3 + (ℓ + k2) · k1 2 1 4 3 + (ℓ + k23) · k1 2 3 4 1

  • 2 · k1 = ( + k1)2 − 2

2( + k2) · k1 = ( + k1 + k2)2 − ( + k2)2 2( + k2 + k3) · k1 = ( + k1 + k2 + k3)2 − ( + k2 + k3)2

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  • 2 · k1 = ( + k1)2 − 2

2( + k2) · k1 = ( + k1 + k2)2 − ( + k2)2 2( + k2 + k3) · k1 = ( + k1 + k2 + k3)2 − ( + k2 + k3)2

1 2 4 3 + 2 1 4 3 + 2 3 4 1

+

  • +

+

  • 32
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  • The graphs that appear in the monodromy relations are

grouped by BCJ triplets.

  • What happens at the boundary ?

[arXiv:1608.01665] Phys.Rev.Lett. 117 (2016) 211601 Higher-loop amplitude monodromy relations in string and gauge theory

  • P. Tourkine, P. Vanhove

[arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts

  • A. Ochirov, P. Tourkine, P. Vanhove

33

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I(1234) + eiπα′k1⋅k2I(2134) + eiπα′k1⋅(k2+k3)I(2314) −eiπα′k1⋅ℓI(234|1) + Jup − Jdown = 0 Now let’s look at the J’s

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[arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts

  • A. Ochirov, P. Tourkine, P. Vanhove

Now let’s look at the J’s

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“Triangles”

  • Bern-Kosower rules (90’s) : explain which integrands

generate these triangles when z1 → zn

n 1 2 n − 1 n 1 2 n − 1 + n 1 2 n − 1 + . . .

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New graphs : “exotic triangles” [work in progress]

[arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts

  • A. Ochirov, P. Tourkine, P. Vanhove

Now let’s look at the J’s

zm = it z1 m 1 ` z1 zm = it m 1 `

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4 2 3

<div style: pen and paper>

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Field theory limit of one- loop monodromies

  • There are new diagrams (exotic triangles, contact terms)

that come from the boundary terms, absent from the

  • riginal rules
  • Bottom triangles drop out of these relations (one-loop)
  • The boundary terms trivialise the boundary Jacobi’s
  • Hold in loops

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  • pen open

closed

× =

= K-L-T, Dotsenko, Bjerrum-Bohr Sondergaard Vanhove Damgaard

  • Somehow, at the roots of this conference, are the

relations between open (=gauge) and closed (=gravity) strings.

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  • pen open

closed

× =

The momentum kernel

[arXiv:1706.08527] JHEP 1708 (2017) 097 Combinatorics and Topology of Kawai- Lewellen-Tye Relations

  • S. Mizera

↳ intersection number between twisted cycles

[arXiv:1910.08514] Monodromy relations from twisted homology

  • E. Casali, S. Mizera, P. Tourkine

⇒ gives bases

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towards KLT

  • In theory, because we have a basis, we know that we can

write something like

  • which should be a closed string like amplitude
  • intersection numbers of the cycles in the twisted
  • homology. Can also compute in the cohomology.

Basically, the (inverse of the) momentum kernel.

M = ∑

a,b

nab∫Γa ω∫Γb ˜ ω na,b =

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Bases of cycles ?

B1 Bi Bg B1 Bi Bg A0

1

A00

1

A0

1

A0

i

A0

i

A0

g

A0

g

A00

1

A00

i

A00

i

A00

g

A00

g

D

Physical n-cycles, e.g. on the boundary of an annulus : z1 < z2 < … < zn Twisted cycle : Twisted n-cycle : constructed in the 
 same fashion (fibration)

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  • The bulk cycles are twisted cycles, one

would expect them to square in a one- loop KLT formula.

  • Motivates a posteriori the “generalized”

double copy ?

  • Suggests a very of generalized double

copy with only those kind of terms to be used for contact terms.

d1 z2 zm dm−1 a+ c+ b1 zm−1 zm+1 zn ka+ kc+ . . . m+1 1 ` − km ` m m − 1 2 (ct)

towards KLT

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Summary, outlook

  • Twisted homology is a robust formalism to understand the

monodromies of string amplitudes.

  • In particular, it allows to come up with a counting and a

basis of integrands.

  • Those bases are to be used for a KLT formula.
  • In the field theory limit, contact arise that should play a

role in the squaring, in accordance with the generalized double copy procedure. Prescriptive way ?

50