The Boundary State from Wilson Loops Online Workshop on String Field Theory and Related Aspects June 2020 Shota Komatsu, IAS
Based on two papers to appear. With Yunfeng Jiang, Amit Sever, and Edoardo Vescovi Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 1 / 26
Main Message: Boundary State = Matrix Product State Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 2 / 26
Introduction The boundary states are keys to understand the open-closed duality, both in perturbative string theory and string field theory. However they are often defined in an indirect way and some works are needed to write down their explicit expressions. Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 3 / 26
Introduction The boundary states are keys to understand the open-closed duality, both in perturbative string theory and string field theory. However they are often defined in an indirect way and some works are needed to write down their explicit expressions. For example, in perturbative string theory, they are often defined by boundary conditions: a † a † | B i = e � P n ˜ n n | 0 i � � X | B i = 0 ) n Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 3 / 26
Introduction The boundary states are keys to understand the open-closed duality, both in perturbative string theory and string field theory. However they are often defined in an indirect way and some works are needed to write down their explicit expressions. For example, in perturbative string theory, they are often defined by boundary conditions: a † a † | B i = e � P n ˜ n n | 0 i � � X | B i = 0 ) n In (open) string field theory, they are defined by solutions Ψ ⇤ to EOM: Q B Ψ ⇤ + Ψ ⇤ ⇤ Ψ ⇤ = 0 . It is not obvious how to write down | B Ψ ⇤ i for any given Ψ ⇤ . Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 3 / 26
Introduction Two proposals on | B Ψ ⇤ i in OSFT: • [Kiermaier, Okawa, Zwiebach 2008] Wilson-loop like expression for the boundary state: ✓ Z ◆� dt [ L R ( t ) + {B R , Ψ ⇤ } ] | B Ψ ⇤ i ⇠ “Tr" P exp � “Tr" is taken over the half-string Hilbert space. N insertions I * Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 4 / 26
Introduction Two proposals on | B Ψ ⇤ i in OSFT: • [Kiermaier, Okawa, Zwiebach 2008] Wilson-loop like expression for the boundary state: ✓ Z ◆� dt [ L R ( t ) + {B R , Ψ ⇤ } ] | B Ψ ⇤ i ⇠ “Tr" P exp � “Tr" is taken over the half-string Hilbert space. • [Kudrna, Maccaferi, Schnabl 2012] Overlap with closed string states from Ellwood invariants: h V| B Ψ ⇤ i ⇠ h I |V ( z = i ) | Ψ ⇤ i Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 4 / 26
Introduction Today I will address a similar question in N = 4 SYM, which is dual to closed string theory in AdS 5 ⇥ S 5 . (The analysis will be entirely within N = 4 SYM) Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 5 / 26
Introduction Today I will address a similar question in N = 4 SYM, which is dual to closed string theory in AdS 5 ⇥ S 5 . (The analysis will be entirely within N = 4 SYM) Question Can we express the Wilson loop in N = 4 SYM (at weak coupling) as a boundary state in the “closed-string" Hilbert space? Strategy Compute the analog of h V| B Ψ ⇤ i (cf [Kudrna, Maccaferi, Schnabl] ) and arrive at an expression similar to [Kiermaier, Okawa, Zwiebach] . For the 1 / 2 BPS circular Wilson loop, one can uplift the solution to finite coupling using integrability bootstrap. Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 5 / 26
Weak Coupling Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 6 / 26
⇒ Single-Trace = |V i , Wilson Loop = | B i Single-trace local operators are dual to on-shell closed strings in AdS 5 ⇥ S 5 spacetime. O ( x ) = Tr [ XXZZXZ · · · ] + · · · |V i $ Ads . Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 7 / 26
Single-Trace = |V i , Wilson Loop = | B i Single-trace local operators are dual to on-shell closed strings in AdS 5 ⇥ S 5 spacetime. O ( x ) = Tr [ XXZZXZ · · · ] + · · · |V i $ The (locally supersymmetric) Wilson loop in the fundamental rep. is dual to a disk worldsheet with the boundary ending on the contour of the WL. ✓I ◆� x � + � I Φ I | ˙ W f = Tr f d � ( iA � ˙ x | ) | B i P exp $ dads Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 7 / 26
Single-Trace = |V i , Wilson Loop = | B i Single-trace local operators are dual to on-shell closed strings in AdS 5 ⇥ S 5 spacetime. O ( x ) = Tr [ XXZZXZ · · · ] + · · · |V i $ The (locally supersymmetric) Wilson loop in the fundamental rep. is dual to a disk worldsheet with the boundary ending on the contour of the WL. ✓I ◆� x � + � I Φ I | ˙ W f = Tr f d � ( iA � ˙ x | ) | B i P exp $ The analog of h V| B i is the correlation function of O and W f : h O ( x ) W f i $ h V| B i . Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 7 / 26
In what follows, I will compute h O ( x ) W f i at weak coupling in 4 steps . Key ideas 1 Use the “generating function" of the WLs. 2 Express the WL as 1d fermion. Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 8 / 26
Step 1: Wilson Loop as Fermion Generating function of the Wilson loops in the anti-symmetric representations: ✓I ◆� x � + � I Φ I | ˙ 1 N ⇥ N + e ia P exp d � ( iA � ˙ Z ( a ) = Det x | ) = 1 + e ia W f + e 2 ia W A 2 + e 3 ia W A 3 + · · · R da e � ia Z ( a ) . One can recover the fundamental rep. by W f = Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 9 / 26
Step 1: Wilson Loop as Fermion Generating function of the Wilson loops in the anti-symmetric representations: ✓I ◆� x � + � I Φ I | ˙ 1 N ⇥ N + e ia P exp d � ( iA � ˙ x | ) Z ( a ) = Det = 1 + e ia W f + e 2 ia W A 2 + e 3 ia W A 3 + · · · R da e � ia Z ( a ) . One can recover the fundamental rep. by W f = Z ( a ) can be expressed as a path integral of 1d fermion � living on the contour of WL [cf. Gomis, Passerini 2006] : Z h x | ) � i e ia � † � e � † ( iA � ˙ x � + � I Φ I | ˙ D � † D � e � S � = Tr � Z ( a ) = Z 1 h ⇣ ⌘i x � + � I Φ I | ˙ d � � † � � � ia � iA � ˙ � S � = x | |{z} |{z} 0 ⇤ ⇤ Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 9 / 26
Step 2: Integrating out N = 4 SYM Next integrate out (free) N = 4 SYM. For simplicity we first consider the expectation value of Z ( a ) : Z Z D � † D � e � ( S � + S N = 4 ) . D A � D Φ h Z ( a ) i = Z 1 d �� † ( � � � ia � iA ) � + S � + S N = 4 = 0 2 3 Z Z 6 7 1 d 4 x Tr 6 4 � � Φ I � � Φ I � g 2 YM Φ I � I d ��� † ( � ) | ˙ x | � 4 ( x � x ( � )) 7 5 g 2 YM | {z } source term WL is a source term for the N = 4 SYM fields. Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 10 / 26
Z 1 d �� † ( � � � ia � iA ) � + S � + S N = 4 = 0 2 3 Z Z 6 7 1 d 4 x Tr 6 4 � � Φ I � � Φ I � g 2 YM Φ I � I d ��� † ( � ) | ˙ x | � 4 ( x � x ( � )) 7 5 g 2 YM | {z } source term R d ' e � ' K ' + J ' = e JK � 1 J , Integrating out N = 4 SYM using Z D � † D � e � e S � h Z ( a ) i = Z Z ⇣ ⌘ ⇣ ⌘ e � † ( � ) � ( � 0 ) � † ( � 0 ) � ( � ) d �� † ( � � � ia ) � . S � = g 2 d � d � 0 y ( � , � 0 ) + YM y ( � , � 0 ) = � I � I | ˙ x � ( � ) ˙ x ( � ) || ˙ x ( � 0 ) | � ˙ x � ( � 0 ) Yet , -i ) - ( x ( � ) � x ( � 0 )) 2 - Xt Li At E) X - Thi At E) X - Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 11 / 26
Step 3: Hubbard-Stratonovich Transf. Z Z ⇣ ⌘ ⇣ ⌘ S � = � e � † ( � ) � ( � 0 ) � † ( � 0 ) � ( � ) d �� † ( � � � ia ) � d � d � 0 y ( � , � 0 ) + N c � � ⇠ � † ( � ) � ( � 0 ) Integrating in a bilocal field � ( � , � 0 ) , Z 1 d � d � 0 � ( � , � 0 ) � ( � 0 , � 0 ) S � , � ⌘ � N c e + � y ( � , � 0 ) 0 Z ⇥ � � , � 0 ( i � � � a ) � � ( � , � 0 ) ⇤ � ( � 0 ) d � d � 0 � † ( � ) Now there is a piece of the action which scales as N c in the ’t Hooft limit. Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 12 / 26
Step 4: Integrating out � Z ⇥ � � , � 0 ( i � � � a ) � � ( � , � 0 ) ⇤ � ( � 0 ) e d � d � 0 � † ( � ) S � , � = · · · + Since the action of � is Gaussian, we can integrate them out to get ⇥ � � � , � 0 ( i � � � a ) � � ( � , � 0 ) �⇤ N c Det As a result, we get an effective action of � : ✓Z ⇤◆ ⇥ � � , � 0 ( i � � � a ) � � ( � , � 0 ) d � d � 0 � ( � , � 0 ) � ( � 0 , � ) S eff = � N c + Tr log � � y ( � , � 0 ) Shota Komatsu, IAS The Boundary State from Wilson Loops 4th June 2020 13 / 26
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