{"arxiv_id":"0908.4033","incoming":{"papers":25,"sentences":58,"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":4,"sentences":6,"quotes":[{"arxiv_id":"1103.0553","title":"M2-Branes and Fano 3-folds","url":"/paper/1103.0553","section":"The R-charges","sentence":"If the fugacity $s_α$ is assigned to the perfect matching $p_α$, then the Hilbert series of the mesonic moduli space can be computed by the usual Molien-Weyl formula [ref] : (s_α; X) = ∮∏^c-G-2_i=1 z_i/2πi z_i ∮∏^G-2_j=1 b_j/2 πi b_j 1/∏^c_α=1(1- s_αZ_α(z_i, b_j)), where $G$ is the number of gauge groups of the theory and $Z_α(z_i, b_j)$ represents the monomial weight of the perfect matching $p_α$ under the fugacities $z_i$, coming from linear relations among the perfect matchings, and $b_j$, arising from the D-terms.","intent_p":0.98,"contests_p":null},{"arxiv_id":"1305.3607","title":"Double Handled Brane Tilings","url":"/paper/1305.3607","section":"Conclusions and Future Directions","sentence":"By adopting the forward algorithm for Chern-Simons brane tilings [ref] [cf. forward algorithm for $4d$ quiver gauge theories in appendix app_mesonic.], the level matrix $C$ then is econc10 C=( cc U(1)_1 U(1)_2 1 1 1 -1 ) , and econc11 d = Q̃ ·P^t , Q_F = (P) , Q_D = (C)·Q̃ , Q_t = (Q_F Q_D) →G_t = (Q_t) .","intent_p":1.0,"contests_p":null},{"arxiv_id":"1010.4648","title":"Parameter spaces of massive IIA solutions","url":"/paper/1010.4648","section":"The quiver","sentence":"We introduce an integer number $n_a$ for each link in the Tiling in such a way that [equation] where $d^a_i=±1$ is the $U(1)$ charge of the field corresponding to the link [ref] .","intent_p":0.98,"contests_p":null},{"arxiv_id":"1204.6040","title":"Free Energy vs Sasaki-Einstein Volume for Infinite Families of M2-Brane Theories","url":"/paper/1204.6040","section":"Conclusions","sentence":"Hilbert Series techniques [ref] seem to be a promising direction for achieving this goal.","intent_p":0.87,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":4,"sentences":5,"quotes":[{"arxiv_id":"1108.4237","title":"On the Higgsing and UnHiggsing of Fano 3-Folds","url":"/paper/1108.4237","section":"M2-brane Theories Without a Brane Tiling Description","sentence":"These values of the CS levels mean that the theory is not the $ℬ_4$ theory of [ref] , although it has the same quiver and tiling description.","intent_p":0.98,"contests_p":null},{"arxiv_id":"1102.5481","title":"Baryonic symmetries in AdS_4/CFT_3: an overview","url":"/paper/1102.5481","section":"Global symmetries in $AdS_4/CFT_3$ and their spontaneous breaking","sentence":"While this result is strictly valid only for isolated singularities, we note it coincides with the conjecture in [ref] .","intent_p":0.93,"contests_p":null},{"arxiv_id":"1904.05344","title":"Black holes with baryonic charge and $\\mathcal{I}$-extremization","url":"/paper/1904.05344","section":"Baryonic 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background","papers":12,"sentences":12,"quotes":[{"arxiv_id":"0911.4127","title":"Chiral flavors and M2-branes at toric CY4 singularities","url":"/paper/0911.4127","section":"Introduction","sentence":"The study of $= 2$ quiver Chern-Simons theories dual to Calabi-Yau cones was initiated in [ref] , followed by a large number of works [ref] .","intent_p":0.97,"contests_p":null},{"arxiv_id":"0909.2879","title":"Probing the Space of Toric Quiver Theories","url":"/paper/0909.2879","section":"Introduction","sentence":"The works of [ref] have brought about tremendous interest in the M2-brane, and following [ref] , where the world-volume theory was realised as a Chern-Simons quiver theory in $(2+1)$-dimensions, much activity ensued (q.v. e.g. [ref] ).","intent_p":0.99,"contests_p":null},{"arxiv_id":"1009.3017","title":"Symmetries of Abelian Orbifolds","url":"/paper/1009.3017","section":"Discussions and Prospects","sentence":"By studying the world volume gauge theories of probe $D3$-branes and $M2$-branes, various toric singularities were identified and classified [ref] .","intent_p":0.98,"contests_p":null},{"arxiv_id":"1505.06308","title":"Embedding and partial resolution of complex cones over Fano threefolds","url":"/paper/1505.06308","section":"Partial Resolution method","sentence":"Examples of various toric dual quiver gauge theories (or phases) have been discussed in [ref] .","intent_p":0.9,"contests_p":null},{"arxiv_id":"1903.02586","title":"New 3d $\\mathcal{N}=2$ SCFT's with $N^{3/2}$ scaling","url":"/paper/1903.02586","section":"Introduction","sentence":"Many examples of models with the $N^3/2$ scaling have by now been found in the literature (see e.g. [ref] ).","intent_p":0.87,"contests_p":null}]}],"unlabelled_sentences":34},"outgoing":{"papers":55,"sentences":138,"groups":[{"kind":"extends","label":"Builds on","papers":3,"sentences":3,"quotes":[{"arxiv_id":"0809.1440","title":"Brane Tilings and M2 Branes","url":"/paper/0809.1440","section":"A Summary of The $𝒩=2$ Supersymmetric Chern–Simons Theory","sentence":"However, it is easy to generalise the result for $k>1$; several explicit examples are given in [ref] .","intent_p":0.83,"contests_p":null},{"arxiv_id":"0806.4212","title":"Partition Functions for Membrane Theories","url":"/paper/0806.4212","section":"A Summary of The $𝒩=2$ Supersymmetric Chern–Simons Theory","sentence":"However, it is easy to generalise the result for $k>1$; several explicit examples are given in [ref] .","intent_p":0.83,"contests_p":null},{"arxiv_id":"0808.1244","title":"Tilings, Chern-Simons Theories and M2 Branes","url":"/paper/0808.1244","section":"A Summary of The $𝒩=2$ Supersymmetric Chern–Simons Theory","sentence":"However, it is easy to generalise the result for $k>1$; several explicit examples are given in [ref] .","intent_p":0.83,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses the method or tool of","papers":10,"sentences":11,"quotes":[{"arxiv_id":"0809.1440","title":"Brane Tilings and M2 Branes","url":"/paper/0809.1440","section":"A summary of the $M^1,1,1$ theory","sentence":"Next [ref] , let us compute the Hilbert series of the divisor corresponding to $p_1$, which we will refer to as $D_1$.","intent_p":0.87,"contests_p":null},{"arxiv_id":"hep-th/0108137","title":"Quiver Theories from D6-branes via Mirror Symmetry","url":"/paper/hep-th/0108137","section":"The $SPP$ theory with $k$ units of RR 2-form fluxes on both $^1$'s","sentence":"Then, the fractional brane charges can be obtained using the method described in [ref] : ch (F_1) = (-1, 0 ,0), ch(F_2) = (0, 1, 0), ch (F_3) = (1, -1, 1) , where the 3 entries are the D4-brane charge on $^1_1$, the D4-brane charge on $^1_2$, and the D2-brane charge respectively.","intent_p":0.99,"contests_p":null},{"arxiv_id":"0811.4044","title":"M2-Branes and Quiver Chern-Simons: A Taxonomic Study","url":"/paper/0811.4044","section":"The Forward Algorithm","sentence":"We summarise the process leading to the toric diagram, which is given by the $G_t$ matrix, in the flow chart forward [ref] .","intent_p":0.88,"contests_p":null},{"arxiv_id":"0808.1244","title":"Tilings, Chern-Simons Theories and M2 Branes","url":"/paper/0808.1244","section":"A Summary of Brane Tilings","sentence":"We may assign an integer $n_i$ to the edge $i$ such that the CS level $k_a$ of the gauge group $a$ is given by[This way of representing $k_a$ is introduced in [ref] and is also used in [ref] .] k_a = ∑_i d_ai n_i , where $d$ is the incidence matrix.","intent_p":0.85,"contests_p":null},{"arxiv_id":"0808.4155","title":"Quiver Chern-Simons theories and crystals","url":"/paper/0808.4155","section":"A Summary of Brane Tilings","sentence":"We may assign an integer $n_i$ to the edge $i$ such that the CS level $k_a$ of the gauge group $a$ is given by[This way of representing $k_a$ is introduced in [ref] and is also used in [ref] .] k_a = ∑_i d_ai n_i , where $d$ is the incidence matrix.","intent_p":0.85,"contests_p":null}]},{"kind":"data_used","label":"Uses data or values from","papers":5,"sentences":6,"quotes":[{"arxiv_id":"0808.1244","title":"Tilings, Chern-Simons Theories and M2 Branes","url":"/paper/0808.1244","section":"The conifold theory with $k$ units of RR 2-form flux on $^1$","sentence":"The Hilbert series is derived in [ref] : (t, ^4/_k) = 1+t^2+2kt^k - 2kt^k+2 - t^2k - t^2k+2/(1-t^2)^3(1-t^k)^2 , where $t$ is the fugacity for R-charges (in the unit of 1/2).","intent_p":0.88,"contests_p":null},{"arxiv_id":"0903.3234","title":"Phases of M2-brane Theories","url":"/paper/0903.3234","section":"The $_0 ×$ theory from giving a VEV to one of $X^i_23$, $X^i_41$","sentence":"The mesonic Hilbert series of $×$ is given by (4.14) of [ref] : (t_1,t_2,x_1,x_2; ×) = ∑^∞_i=0 t^i_2 ∑^∞_n=0 [n;n] t^2n_1 .","intent_p":0.95,"contests_p":null},{"arxiv_id":"hep-th/0108137","title":"Quiver Theories from D6-branes via Mirror Symmetry","url":"/paper/hep-th/0108137","section":"The $_0$ theory with $k$ units of RR 4-form flux on $^1 ×^1$","sentence":"Each of the 4 nodes in the quiver diagram (f:phase1f0) corresponds to a collection of fractional branes, whose charges are given respectively as (see (4.15) of [ref] ): ch(F_1) = (-1, 1,0, 0), ch(F_2) = (1, -1, 1, -1), ch(F_3) = (1, 0, -1, 0), ch(F_4) = (-1,0, 0,0) , Let us denote by $k_1$ and $k_2$ the RR 2-form fluxes on $^1_1$ and $^1_2$, and by $k_4$ the RR 4-form flux on $^1 ×^1$.","intent_p":0.98,"contests_p":null},{"arxiv_id":"hep-th/0207006","title":"Geometric dualities in 4d field theories and their 5d interpretation","url":"/paper/hep-th/0207006","section":"The $dP_0$ (or $^3/_3$) theory with $k$ units of 4-form flux on $^2$","sentence":"The fractional brane charges associated with these 3 nodes are given in (8.2) of [ref] : ch (F_1) = (2, -1, -1/2), ch (F_2) = (-1, 1, -1/2), ch (F_3) = (-1,0,0) .","intent_p":0.99,"contests_p":null},{"arxiv_id":"0809.1440","title":"Brane Tilings and M2 Branes","url":"/paper/0809.1440","section":"Summary","sentence":"Note that the volumes of various internal manifolds can be found in [ref] .","intent_p":0.88,"contests_p":null}]},{"kind":"comparison","label":"Compares with","papers":4,"sentences":4,"quotes":[{"arxiv_id":"0903.3234","title":"Phases of M2-brane Theories","url":"/paper/0903.3234","section":"The $_0$ theory with $k$ units of RR 4-form flux on $^1 ×^1$","sentence":"Setting $k=1$, we recover the formula (6.23) of [ref] .","intent_p":0.8,"contests_p":null},{"arxiv_id":"0810.4519","title":"Master Space, Hilbert Series and Seiberg Duality","url":"/paper/0810.4519","section":"A summary of Phase II of $Q^1,1,1/_2$ (the $𝒮_2 𝒪_2$ model)","sentence":"Note that in 3+1 dimensions, these quiver and tiling correspond to Phase II of the $_0$ theory [ref] .","intent_p":0.86,"contests_p":null},{"arxiv_id":"0801.1585","title":"The Master Space of 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subsequently proposed by Aharony, Bergman, Jafferis and Maldacena (ABJM) [ref] as a model describing $N$ M2-branes in the $^4/_k$ orbifold background.","intent_p":0.95,"contests_p":null},{"arxiv_id":"hep-th/0611108","title":"Modeling Multiple M2's","url":"/paper/hep-th/0611108","section":"Introduction","sentence":"Considerable progress in understanding theories for multiple M2-branes in various backgrounds has been taking place since the works by Bagger–Lambert [ref] and Gustavsson [ref] .","intent_p":1.0,"contests_p":null},{"arxiv_id":"0709.1260","title":"Algebraic structures on parallel M2-branes","url":"/paper/0709.1260","section":"Introduction","sentence":"Considerable progress in understanding theories for multiple M2-branes in various backgrounds has been taking place since the works by Bagger–Lambert [ref] and Gustavsson [ref] .","intent_p":1.0,"contests_p":null},{"arxiv_id":"hep-th/0505211","title":"Gauge Theories from Toric Geometry and Brane Tilings","url":"/paper/hep-th/0505211","section":"Introduction","sentence":"Preceding to the developments of M2-brane theories, brane tilings have proved to be a very powerful tool in establishing the relation between $(3+1)$-dimensional gauge theories and their moduli spaces which are Calabi–Yau 3-folds [ref] (see also [ref] for further developments and [ref] for reviews).","intent_p":0.89,"contests_p":null},{"arxiv_id":"0803.4474","title":"Brane Tilings and Their Applications","url":"/paper/0803.4474","section":"Introduction","sentence":"Preceding to the developments of M2-brane theories, brane tilings have proved to be a very powerful tool in establishing the relation between $(3+1)$-dimensional gauge theories and their moduli spaces which are Calabi–Yau 3-folds [ref] (see also [ref] for further developments and [ref] for 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