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Strings from Feynman Diagrams

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that in the half-supersymmetric sector of N=4 super Yang-Mills, every Feynman diagram maps to one specific closed-string worldsheet with an explicit embedding map, making open/closed string duality exact summand by…

desk verdict A careful, honest extension of the Strebel/Belyi program to the two-matrix model; the combinatorial dictionary is solid, but the paper's headline claim leans on a localization proof deferred to [5]. read the letter →

arxiv 2412.13397 v1 pith:KMIM5BOB submitted 2024-12-18 hep-th math-phmath.MP

Rajesh Gopakumar , Rishabh Kaushik , Shota Komatsu , Edward A. Mazenc , Debmalya Sarkar This is my paper · ORCID
classification hep-thmath-phmath.MP MSC 81T3081T1314H5714H30
keywords Feynmandiagramsopen/closedstringdualityStrebeldifferentialBelyimapstwo-matrixmodelN=4superYang-Millshalf-BPScorrelatorsAdS/CFT
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that, in the half-supersymmetric sector of N=4 super Yang-Mills, open/closed string duality can be made exact at the level of individual summands: each Feynman diagram in the perturbative expansion is literally one closed-string worldsheet configuration, not a discretized approximation to one. The claim is established concretely through a two-matrix integral whose correlators compute the protected sector. Each diagram is shown to determine a Riemann surface with a metric fixed by a Strebel differential and a holomorphic embedding into a target sphere, namely a Belyi map branched over three points. The regulated Nambu-Goto action of that string configuration reproduces the diagram's weight, so the sum over diagrams becomes a sum over worldsheets to all orders in 1/N. The paper also finds six different open-string matrix descriptions whose Feynman diagrams are graph-dual to one another and generate the same closed-string sum.

What carries the argument

The machinery has two parts. On the worldsheet side, the Strebel differential, a quadratic differential with double poles at marked points and integer-length critical graph, assigns to each Feynman diagram a unique Riemann surface with a flat metric, and decomposes that surface into open-string strips, one per edge. On the target side, the same diagram encodes a Belyi map, a holomorphic covering of the Riemann sphere branched over exactly three points, whose branching is specified by three permutations read from the two vertex types and the faces. The link between the two is the identity relating the Strebel differential to the pullback of the target-space Kähler form under the Belyi map; the regulated integral of that form is the Nambu-Goto action, which reproduces the Feynman diagram weight.

What would settle it

Compute the path integral of the proposed coset model for a four-point function in the sector and check whether it equals a sum of delta functions supported on the integer-Strebel points reconstructed from the Feynman diagrams. If the integral receives contributions from non-arithmetic Riemann surfaces, or from Belyi maps with branching not fixed by the operator insertions, the dictionary fails; likewise, finding a matrix-model correlator whose sum over diagrams cannot be reproduced by any such localized worldsheet sum would refute the claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Feynman diagram expansion of half-supersymmetric correlators of N=4 super Yang-Mills can be recast, summand by summand, as a sum over closed string configurations. Each ribbon graph yields a point on the moduli space of Riemann surfaces via the Strebel parametrization, with all edge lengths integers; the same graph yields a Belyi map from that surface to a target CP1, branched over the two operator insertion points and one midpoint. The product of the three permutations read off from the K-vertices, M-vertices, and faces is the identity, which is exactly the condition that the covering map be a Belyi map. The regulated area of the worldsheet in Strebel gauge, equal to the Nambu-Goto action evaluated on the embedding, equals the product of position-space propagators of the diagram. The paper thus claims a microscopic, order-by-order realization of open/closed duality in a topological subsector of AdS/CFT, with the closed string identified as the c=1 string at self-dual radius (equivalently, the A-twisted SL(2,R)/U(1) coset at level one).

Load-bearing premise

The argument assumes that a concrete worldsheet theory, the A-twisted SL(2,R)/U(1) coset at level one, exists and its path integral localizes onto exactly the Belyi maps constructed from the Feynman diagrams; the paper states this but defers the explicit demonstration to a companion paper.

Editorial extensions

If this is right

  • The 1/N expansion of the gauge theory becomes the genus expansion of the dual closed string, with the string coupling equal to 1/N, and the expansion truncates at finite genus because the Riemann-Hurwitz formula forbids covering maps above a maximal genus.
  • Feynman diagrams are not discretized worldsheets but discrete lattice points in string moduli space, labeled by integer Strebel lengths; the string path integral localizes to these arithmetic Riemann surfaces.
  • The weight of each diagram equals the exponential of the string action evaluated on the reconstructed configuration, matching the product of free propagators of N=4 super Yang-Mills up to overall factors.
  • The same closed-string theory admits six equivalent open-string matrix descriptions, related by partial graph duality; the F-type description is furnished by open strings on giant graviton branes.
  • The matrix model computations agree with old c=1 string theory predictions at genus zero and genus one, providing a consistency check on the identification of the dual closed string.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: if the dictionary is correct, the same Strebel-plus-Belyi reconstruction should apply to any free adjoint gauge theory, suggesting that protected sectors of other holographic pairs will exhibit similar localization to arithmetic surfaces.
  • Editorial extension: the predicted sixfold open-string multiplicity implies that the open string dual of a closed string theory is not unique; closed-string observables may be generated by any member of a family of graph-dual matrix models, with different brane interpretations.
  • Editorial extension: the localization mechanism may be testable independently of the companion paper by looking for the same integer-Strebel points in twistor-string computations of free N=4 super Yang-Mills correlators, where covering maps already appear.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a microscopic open/closed string duality for a two-matrix integral with action N/g Tr(KM), and argues that this model captures the 1/2 SUSY sector of N=4 SYM generated by 1/2-BPS operators at two boundary points. The central construction assigns to each connected Feynman diagram (i) a closed-string worldsheet via Strebel differentials, with integer edge lengths obtained by assigning unit length to each edge (§2.3–2.7), and (ii) a holomorphic Belyi covering map of CP1 whose three branch points are determined by K-vertices, M-vertices, and faces (§3.1–3.4). The two structures are tied together by showing that the Strebel metric is the pullback of a target-space Kähler form and that the regulated Nambu-Goto area reproduces the position-space propagator weight (§4.4–4.5). Section 6 derives five further matrix models related by exact integration in/out manipulations and interprets them as different open-string descriptions, including a Kontsevich-like model on giant graviton branes.

Significance. If the construction is correct, the paper gives a rare all-orders example in which each Feynman diagram of a gauge theory maps to a single closed-string configuration rather than to a discretized worldsheet, with Feynman diagrams sitting on a lattice in moduli space. The explicit four-point example is internally coherent and checkable in detail: the Strebel differential in Eq. (29), the Belyi map in Eq. (50), and the equality in Eq. (73) are explicit and consistent. The Wick-contraction derivation of the permutation triple σK, σM, σf in §3.4 is a solid combinatorial anchor, and the six equivalent matrix models in §6 are an interesting structural result independent of the string interpretation. The main caveat is that the paper's headline claim is conditional on a localization theorem for the A-twisted SL(2,R)/U(1) coset that is deferred to a companion paper; as a standalone manuscript it establishes a precise combinatorial dictionary and consistency checks rather than the full string dual.

major comments (4)
  1. [§1.1, §4.2] The abstract's claim that the Feynman expansion is 'manifestly recast' as a sum over dual closed strings is not demonstrated inside this manuscript. §4.2 states that the A-twisted SL(2,R)/U(1) coset at level k=1 will be shown in [5] to localize to Belyi maps, and explicitly postpones the branching over the third target-space point (the faces) to [5]; §1.1 likewise defers the worldsheet theory to [5]. Belyi's theorem in §4.1 shows that surfaces admitting such maps are exactly the arithmetic ones, but it does not establish that the proposed coset path integral localizes onto those maps. If the companion's localization proof fails, the present paper establishes a beautiful combinatorial correspondence but not a string dual. This is load-bearing, and the manuscript should either include the localization argument, summarize its mechanism, or state the main theorem as conditional.
  2. [§4.5, footnote 31] The claimed equality between the string action and the Feynman weight is matched only up to an overall factor. The N=4 diagram weight in Eq. (76) is (Y1·Y2/|w1-w2|^2)^E, while the string-side result in Eq. (83) is (ε_T^2/|w1-w2|^2)^E; footnote 31 explicitly concedes that Y1·Y2 and factors of 4π^2 are not accounted for and speculates about a fermionic origin. Since Y1·Y2 is assumed nonzero and encodes the R-charge alignment, this is not a trivial normalization. The statement in §4.5 that the action 'exactly reproduces' the weight should be corrected to 'reproduces the position dependence', or the missing factors should be derived.
  3. [§2.3, §2.7, §4.5] The unit-length assignment to every edge is a prescription, and the numerical weight match depends on it. If each strip is assigned width l0 rather than 1, then the cutoff relation in §4.5 becomes |X(iL_c)-w2| ∼ ε_T with L_c = (l0/4π) log(|w1-w2|^2/ε_T^2), and the regulated area is A_WS = 2 E L_c l0, giving e^{-2π A_WS} = (ε_T^2/|w1-w2|^2)^{E l0^2}. Only l0=1 reproduces Eq. (83). Footnote 24's statement that choosing l0 ≠ 1 'will not change any of the conclusions' is therefore not correct for the action computation; the unit width is an additional assumption that should be stated as such.
  4. [§5.2] The embedding into AdS5/CFT4 is presented as a sketch rather than a derivation. §5.2 states that the twistor string dual 'has not yet been derived from a first principles quantization' and that the role of the third branch point remains to be fleshed out. This is an honest caveat, but it means the paper's placement of the construction in the AdS/CFT context (abstract, §1.4) is conjectural. The distinction between the self-contained matrix-model/topological-string result and the conjectural N=4 embedding should be drawn explicitly in the abstract and introduction.
minor comments (4)
  1. [§4.5, Eq. (78)] The cross-reference 'using the transition function ... Eq.(78)' points to the wrong equation; the relevant transition functions are in Eqs. (35)–(38) and (62)–(65).
  2. [§2.5, text after Eq. (26)] The phrase 'We obtain for now two (non-trivial) families' is awkward and should be rephrased; also the third family z^{(3)}_{a,b} is dismissed by t → -t, which deserves a sentence of justification.
  3. [§5.1, Eq. (108)] The mapping g ↔ λ Y1·Y2/(4π^2|w1-w2|^2) shows that the factors missing in §4.5 are present in the matrix-model definition; footnote 31 should be connected to this equation so the reader can see precisely which factor is being matched.
  4. [§6.1, Step 2] The sentence 'We adopt a purely imaginary contour for K, which results in a delta-function for M' is terse; because this manipulation is used to derive the F-type model, a parenthetical explanation or a reference to Appendix B would help.

Circularity Check

3 steps flagged · score 6.0 of 10

Weight match and integer-lattice predictions are built into the reconstruction prescription; Belyi-map localization is deferred to companion [5].

  1. fitted input called prediction [Sec. 4.5 (Eqs. 79-83 and footnote 31)]
    "In reconstructing the worldsheet from the gauge theory Feynman diagrams, we identified every edge of the Feynman diagram with one unit-width strip. Since the Strebel metric is flat on each strip, the area of an individual strip is simply (Regulated) Area of Single Strip= height ×width = 2Lc × 1 … e^{−2πS_NG[X]} = e^{−4πEL_c} = ( ε_T^2 / |w_2−w_1|^2 )^E … We have, so far, only successfully matched the w-dependence of the propagator."

    The Feynman weight is (Y1·Y2/|w1−w2|^2)^E. The string side produces (ε_T^2/|w1−w2|^2)^E because the reconstruction set each strip width to 1, giving area = E×2L_c, and then fixed the cutoff relation L_c = (1/4π) log(|w1−w2|^2/ε_T^2) via the chosen embedding map. These are inputs to the algorithm, not outputs of an independent worldsheet action; the Y1·Y2 and 4π^2 prefactors are explicitly not reproduced (footnote 31). The 'match' is therefore a tautological consistency check of the prescription instead of an independent prediction.

  2. fitted input called prediction [Sec. 2.3 (integer length assignment) and Sec. 4.1]
    "This simple length assignment to the edges has a striking consequence: the worldsheets dual to the matrix model Feynman diagrams are all parameterized by Strebel graphs with integer lengths."

    The integer Strebel lengths, and through them the claimed 'latticization of moduli space' and localization to arithmetic Riemann surfaces, follow immediately from the earlier prescription to 'simply assign unit length to any given edge' and to add lengths when bundling homotopic edges. The paper itself concedes 'there was a certain arbitrariness to this choice', yet later presents the resulting localization as a prediction of the construction. The output (integrality) is exactly the input (unit lengths).

1 more flagged steps
  1. self citation load bearing [Sec. 1.1, Sec. 4.2, Sec. 5.2]
    "The precise worldsheet theory which gives rise to such an integrand on moduli space will be presented elsewhere [5] - a quick summary can be found in Sec. 4.2 … In [5], it will be shown explicitly, how the string theory path integral computation of correlators built out of products of these vertex operators localizes to a sum over Belyi maps."

    The central identification of the dual closed string theory (the A-twisted SL(2,R)/U(1) coset at level k=1) and the crucial property that its path integral localizes to the Belyi maps obtained from the Feynman diagrams are not derived in this paper; they are assigned to the companion paper [5], including the branching over the third point. The abstract's claim to 'manifestly recast' the gauge theory expansion as a sum over dual closed strings therefore rests on a deferred, unprovided companion argument rather than on a self-contained derivation from the Feynman diagrams.

full rationale

The Strebel-to-Riemann-surface and Feynman-to-Belyi-map constructions are self-contained mathematical translations, and the genus 0/1 checks against known c=1 string results provide independent support. However, two of the headline 'predictions' reduce to the reconstruction prescription: the integer Strebel lengths and the lattice on moduli space follow from assigning unit length to each edge, and the Nambu-Goto-area reproduction of the Feynman weight is built into the unit-width strips and the chosen cutoff/embedding, with the position-space prefactor explicitly not matched (footnote 31). The proposed worldsheet theory that localizes to Belyi maps is not established here but deferred to companion [5], so the central string-theory claim is not self-contained. These are genuine circular-by-construction elements, though the combinatorial dictionary and external c=1 checks keep the paper from being wholly circular. Score 6.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central argument rests on standard mathematical theorems (Strebel, Belyi, Riemann-Hurwitz) and on domain assumptions about N=4 SYM. The most fragile input is the assumed worldsheet theory that localizes to Belyi maps, which is deferred to [5]. The unit length edge assignment is a hand-chosen normalization, though the authors argue it does not affect the lattice of moduli points.

free parameters (1)
  • unit Strebel length per edge = 1
    Chosen by hand in Sec 2.3 following Razamat [76]; the authors argue a different common length rescales the R^n_+ fiber without changing the lattice points on moduli space, so it is a normalization rather than a fitted physical constant.
assumptions (5)
  • standard math Strebel theorem: every metrized ribbon graph with n faces corresponds to a unique point in decorated moduli space and a unique Strebel differential.
    Invoked in Secs 2.2-2.3 to reconstruct worldsheets from Feynman diagrams.
  • standard math Belyi's theorem: a compact Riemann surface admits a Belyi map iff it is arithmetic, equivalently iff it has integer Strebel lengths.
    Used in Sec 4.1 to connect integer-length Strebel graphs to localization on Belyi maps.
  • domain assumption The 1/2 SUSY correlators of N=4 SYM are protected, so the zero-coupling free field reduction to the two-matrix model is exact.
    Assumed in Sec 5.1 to derive Eq. (107) from N=4 SYM.
  • ad hoc to paper The dual closed string is the A-twisted SL(2,R)/U(1) coset at level k=1, equivalent to the c=1 string at self-dual radius, whose worldsheet theory localizes to Belyi maps.
    This is the load-bearing string theory assumption; it is asserted in Sec 4.2 and 5.2 and left to the companion paper [5].
  • ad hoc to paper Assignment of unit length to every Feynman diagram edge.
    Defined in Sec 2.3 as the prescription for translating Feynman diagrams to Strebel graphs; the authors note some arbitrariness.

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Cite this review

Pith. "Pith review of Strings from Feynman Diagrams." pith.science (2026). https://pith.science/paper/KMIM5BOB

@misc{pith2026241213397,
  author       = {Pith},
  title        = {Pith review of: Strings from Feynman Diagrams},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KMIM5BOB}},
  note         = {Machine review of arXiv:2412.13397}
}
abstract

For correlators in $\mathcal{N}=4$ Super Yang-Mills preserving half the supersymmetry, we manifestly recast the gauge theory Feynman diagram expansion as a sum over dual closed strings. Each individual Feynman diagram maps on to a Riemann surface with specific moduli. The Feynman diagrams thus correspond to discrete lattice points on string moduli space, rather than discretized worldsheets. This picture is valid to all orders in the $1/N$ expansion. Concretely, the mapping is carried out at the level of a two-matrix integral with its dual string description. It provides a microscopic picture of open/closed string duality for this topological subsector of the full AdS/CFT correspondence. At the same time, the concrete mechanism for how strings emerge from the matrix model Feynman diagrams predicts that multiple open string descriptions can exist for the same dual closed string theory. By considering the insertion of determinant operators in $\mathcal{N}=4$ SYM, we indeed find six equivalent open-string descriptions. Each of them generates Feynman diagrams related to one another via (partial) graph duality, and hence encodes the same information. The embedding of these Kontsevich-like duals into the 1/2 SUSY sector of AdS/CFT is achieved by open strings on giant graviton branes.

Figures

Figures reproduced from arXiv: 2412.13397 by the authors.

Figure 1
Figure 1. Schematic Summary of Main Result The purpose of this paper is to manifestly recast the Feynman diagram expansion of certain protected correlators in N = 4 SYM as a dual sum over closed string configurations. The top line is a connected correlation function of multiple 1/2 BPS operators in SU(N) N = 4 Super Yang Mills, inserted at two points on the boundary. The middle line shows its expansion in terms of Feynman dia… view at source ↗
Figure 2
Figure 2. Each Feynman Diagram as a Closed String Configuration We map each matrix model Feynman diagram to a specific closed string worldsheet. Each edge of the graph corresponds to one strip building up the worldsheet. The same diagram also encodes the embedding map of the string into the target space. The embedding turns out to be a covering map of the Riemann sphere, branched over precisely three points. To each edge, we … view at source ↗
Figure 3
Figure 3. Reconstructing the Worldsheet There exists a unique Strebel differential ϕS(z)dz2 on every Riemann surface. It is fully specified by a (metrized) ribbon graph. This allows us to precisely translate between gauge theory Feynman diagrams and specific closed string worldsheets, including an explicit metric on the surface. In addition, the Strebel differential defines a metric on the worldsheet, discussed in Sec. 2.4. T… view at source ↗
Figures from the paper (47 more)
Figure 4
Figure 4. Figure 4: Gluing Open String Strips The Strebel differential decomposes each Riemann surface into a collection of strips. We identify the edges of the gauge theory Feynman diagram with these strips. Precise gluing rules, discussed in Sec. 2.7, allows one to assemble the closed s…
Figure 5
Figure 5. Figure 5: Feynman Diagrams as Lattice Points on Mg,n Each Feynman diagram, contributing to a particular n-point correlator, maps onto a particular point on moduli space, labeled by a set of integers. These integers are the edge lengths of the worldsheet’s Strebel graph, related …
Figure 6
Figure 6. Figure 6: Reconstructing the Map into Target Space Each Feynman diagram encodes a specific holomorphic covering map of the Riemann sphere, XFD(z), branched over exactly three points. The vertices and faces of the diagram specify the ramification profiles over these three points.…
Figure 7
Figure 7. Figure 7: Proposed Embedding in AdS/CFT On the open string side, 1/2 BPS determinant correlators inserted at two points w1 and w2 in N = 4 SYM reduce to the two-matrix integral studied in this paper. (Left) On the closed string side, the picture we suggest is that of a tensionle…
Figure 8
Figure 8. Figure 8: Feynman Rules for the K, M Matrix Model The propagator, and the two kinds of (external) vertices, corresponding to insertions of Tr(Kn) and Tr(Mn). We can keep track of these various Wick contractions contributing to a correlators by using Feynman diagrams. These diagr…
Figure 9
Figure 9. Figure 9: The only Feynman diagram contributing to the computation of D 1 2 Tr(K2 ) 2 1 2 Tr(M2 ) 2 E c The two crossed vertices correspond to the (external) insertions of Tr K2 , while the two uncrossed ones map onto the insertions of Tr M2 . There are no internal vertices si…
Figure 10
Figure 10. Figure 10: A crash-course on the Strebel parametrization of Mg,n×R n + Every metrized ribbon graph maps onto a particular point on moduli space. The genus of the graph, defined by V − E + F = 2 − 2g, determines the genus of the Riemann surface it encodes. The number of faces mat…
Figure 11
Figure 11. Figure 11: Higher Valency Vertices in the Strebel Parametrization of Moduli Space The set of genus h metrized ribbon graphs with n faces provide a simplicial decomposition of Mg,n × R n +. Varying the lengths of the edges of a trivalent graph (keeping the n perimeters fixed) swe…
Figure 12
Figure 12. Figure 12: Strebel Graph from Horizontal Trajectories Each Riemann surface can be foliated by a set of curves known as Strebel’s horizontal trajectories (in red). Along these curves, the square root of the Strebel differential is real and defines a positive line element. They ca…
Figure 13
Figure 13. Figure 13: The Closed String Worldsheet Dual to Our Feynman Diagram An n-point correlator in the K, M-matrix model should map onto an n-point function of vertex operators in the dual closed string. The worldsheet dual to our Feynman diagram with four vertices should therefore as…
Figure 14
Figure 14. Figure 14: A Feynman Diagram Contributing to the Computation of [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]
Figure 15
Figure 15. Figure 15: Latticization of Mg,n instead of Discretization of the Worldsheet Each Feynman diagram, contributing to a particular n-point correlator, maps onto a particular point on moduli space. Since all the lengths of the corresponding Strebel graph are integers, and these serv…
Figure 16
Figure 16. Figure 16: Feynman Diagrams vs. Strebel Differentials To summarize, each vertex of the Feynman diagram corresponds to a double-pole of the Strebel differential ϕS. These reside at the marked point on the closed string worldsheet, where the associated vertex operators would be in…
Figure 17
Figure 17. Figure 17: The Explicit Worldsheet & Metric Dual to our Feynman Diagram We can explicitly reconstruct the full Strebel differential ϕW S(z)dz2 from the metrized graph data on the left. This determines the exact point on M0,4 dual to our Feynman diagram. It is a sphere with punct…
Figure 18
Figure 18. Figure 18: Strings Interact at the Vertices of the Strebel Graph Similar to light-cone gauge, the Strebel construction decomposes every Riemann surface into a collection of semi-infinite cylinders glued together along the Strebel graph. Each semi-infinite cylinder corresponds to…
Figure 19
Figure 19. Figure 19: From Vertical Trajectories to Strips There exists a second family of curves on every Riemann surface, determined by its Strebel differential, known as the vertical trajectories. (Left) For the most part, they run from marked point to marked point (in solid blue lines)…
Figure 20
Figure 20. Figure 20: Every Riemann Surface as a Collection of Strips The solid blue line is a vertical trajectory connecting two poles of the Strebel differential. The same line was also drawn in [PITH_FULL_IMAGE:figures/full_fig_p032_20.png]
Figure 21
Figure 21. Figure 21: From Strips to String Bits Each strip in the Strebel construction can be viewed as the worldsheet of a "string bit". A single trace operator built of k-matrices (in this example, k = 4) will give rise to a closed string built out of k string bits: one bit per matrix e…
Figure 22
Figure 22. Figure 22: Assembling the Closed String Worldsheet from Feynman Diagram Edges Each edge of the matrix model Feynman diagram corresponds to one strip. This closed string worldsheet will be built out of four strips since our diagram here has four edges. The dashed lines show where…
Figure 23
Figure 23. Figure 23: 1. Each Edge of the Feynman Diagram as an Open String Strip To the j-th edge of the original Feynman diagram, we associate a strip with local coordinates zj . Such an edge connects connects an M-vertex (uncrossed) and a K-vertex (crossed). The dual edge (in orange) co…
Figure 24
Figure 24. Figure 24: 2. Gluing Along (Homotopic) Edges Returning to the example of [PITH_FULL_IMAGE:figures/full_fig_p036_24.png]
Figure 25
Figure 25. Figure 25: 3. Gluing Strips at Vertices Consider the edges emanating from a common vertex of the (skeleton) Feynman diagram. Recall vertices of the K, M-model map onto the marked points of the dual worldsheet. These marked points are the conformal mapping of a semi-infinite cyli…
Figure 26
Figure 26. Figure 26: 4. Gluing Strips at Faces Consider the edges bordering a common face of the (skeleton) Feynman diagram. We "glue" their associated strips together near the center of the face (the black dot) by defining a local coordinate system ω valid in an open neighborhood shared …
Figure 27
Figure 27. Figure 27: Deriving the String Embedding Map We already reconstructed the worldsheet of the dual closed string from the Feynman diagram. Here, we derive its embedding map X(z) into the target space, also purely from the Feynman diagram. The target turns out to be the Riemann sph…
Figure 28
Figure 28. Figure 28: From Feynman Diagrams to Permutations We label all edges 1, 2, .., E, working our way around the diagram, starting from any vertex. We then read off three permutations (σK, σf , σM) ∈ SE, associated to the K-vertices, the faces and the M-vertices, respectively. There …
Figure 29
Figure 29. Figure 29: From Permutations to Embedding Maps In this string theory, the worldsheet Σg,n (here the four colored lines) wraps the entire target space CP1 (in black) multiple times. Such holomorphic covering maps are fully characterized by their branching structure. Each edge of …
Figure 30
Figure 30. Figure 30: Feynman Diagram from Target Space Interval An equivalent way to understand the encoding of the Belyi map XFD via a Feynman diagram is to view the ribbon graph as the pre-image of a target space interval. The endpoints of the interval lie at the two branchpoints X = w1…
Figure 31
Figure 31. Figure 31: The String Embedding Dual to our Feynman Diagram We can explicitly reconstruct the Belyi map, from the worldsheet into the target space, purely from the Feynman diagram data. This function is unique up to an overall SL(2, C) action, which corresponds to picking the lo…
Figure 32
Figure 32. Figure 32: "Belyi Map Bits" To understand how the reconstruction of the worldsheet and the covering map are related, we study a single edge of a Feynman diagram. We have seen how each edge is associated to one strip making up the worldsheet, which we interpreted as the worldshee…
Figure 33
Figure 33. Figure 33: Regulating the Worldsheet Action The action of the dual closed string reduces to the area of the worldsheet, computed using the Strebel metric. This area is a priori divergent, since the worldsheet geometry is that of flat semi-infinite cylinders glued to the Strebel …
Figure 34
Figure 34. Figure 34: String Worldsheet Action = Weight of Feynman Diagram The closed string action, evaluated on the embedding map X(z), reduces to the Nambu-Goto action: e −2πS[X]=e −2πAW S . While finding the explicit Strebel volume form on the worldsheet is complicated, the total (regu…
Figure 35
Figure 35. Figure 35: Three Stacks of D3 Branes and New Open Strings From a brane perspective, the insertion of determinant operators corresponds to the addition of extra D3 branes. We can think of three separate stacks lying in ten-dimensional flat space: the "original" N D3s giving rise …
Figure 36
Figure 36. Figure 36: Open-Closed-Open Triality for the 1/2 SUSY Sector of N = 4 SYM For the highly protected correlators considered in this paper, we find two open string descriptions in the form of two distinct matrix integrals. In the K, M model, single trace operators map directly to c…
Figure 37
Figure 37. Figure 37: F-Type Model as Strings on Giant Graviton Branes If we consider the back reaction of the original N D3 branes on the ambient flat spacetime, we can view the additional branes dual to determinant insertions in the K, M-model as giant graviton branes in an AdS5 × S 5 ge…
Figure 38
Figure 38. Figure 38: Yukawa Couplings in the K, M + ψ, χ Model for the open string giving rise to the Feynman diagram. a [PITH_FULL_IMAGE:figures/full_fig_p074_38.png]
Figure 39
Figure 39. Figure 39: (STEP 1) Integrating in the Fermions The two types of vertices of valency k in the original K, M model gets replaced by two new types of k-sided faces in the mixed bosonic/fermion theory. Each edge of these new faces is a fermionic propagator. The appearance of X and …
Figure 40
Figure 40. Figure 40: (STEP 2) Integrating out K, M Integrating out K, M collapses the edge connecting two different Yukawa vertices. It generates a quartic vertex, representing the ψ †ψχ†χ term in the action of the purely fermionic matrix integral of Eq.((117)), STEP 3: In some sense, the…
Figure 41
Figure 41. Figure 41: By that, we mean the effect of the Hubbard-Stratanovich transformation is to rewrite the quartic vertex as two cubic vertices, joined together by a propagator for the complex matrices S, S† . These cubic interactions arise from a new set of Yukawa-like couplings betwe…
Figure 42
Figure 42. Figure 42: (STEP 4) Integrating the Fermions Out The original faces of the K, M model are bordered by fermion propagators. Upon integrating out the fermions, the original faces collapse, going over to vertices of the S, S† model. From the point of view of the open string worldsh…
Figure 43
Figure 43. Figure 43: Steps 1 through 4 applied to our Feynman Diagram The first diagram in the top left corner is the only Feynman diagram in the K, M model contributing to log Z[tk,t¯k, sk] at order t 2 2 t¯2 2 . By integrating in the fermions, the vertices of valency two are replaced wi…
Figure 44
Figure 44. Figure 44: Feynman Diagrams of the One Fermion Description The middle diagram is the trans￾formation of our favorite Feynman diagram after step 2, as also shown in [PITH_FULL_IMAGE:figures/full_fig_p079_44.png]
Figure 45
Figure 45. Figure 45: Feynman Diagrams of the Bosonic Partial Duals Partial graph duality is a novel feature of the open-closed-open triality of the two-matrix versus one-matrix models. The A, B or C, D models generate Feynman diagrams with two types of vertices and one type of face. Their…
Figure 46
Figure 46. Figure 46: Summary of the Six Open String Descriptions & their Feynman Graph Duality By considering three stacks of branes, we find six equivalent open string descriptions in the guise of six different matrix integrals. There is a one-to-one correspondence between their Feynman …
Figure 47
Figure 47. Figure 47: Open-Closed-Open Triality on the Worldsheet The edges of the Feynman diagrams of the different (bosonic) open string descriptions map onto different trajectories of the Strebel differential on the worldsheet. They therefore all encode the same moduli of the dual Riema…
Figure 48
Figure 48. Figure 48: Graphs related via (partial) graph duality specify the same Belyi Maps The pre-image, under the same Belyi map, of two different intervals in target space connecting two branchpoints, gives rise to partial dual graphs. Viewed in reverse, we can see explicitly how the …
Figure 49
Figure 49. Figure 49: Open-Closed-Open Triality in Target Space The Feynman diagrams of the six open string descriptions, related to one another via the integrating in/out procedure of Secs. 6.1 and 6.3, can all be viewed as pre-images of various target space intervals or loops under the s…
Figure 50
Figure 50. Figure 50: All the (labelled) Feynman diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p096_50.png]

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