002244365 001__ 2244365
002244365 005__ 20231004101730.0
002244365 0248_ $$aoai:cds.cern.ch:2244365$$pcerncds:CERN$$pcerncds:CERN:FULLTEXT$$pcerncds:FULLTEXT
002244365 0247_ $$2DOI$$9bibmatch$$a10.1007/JHEP03(2017)152
002244365 035__ $$9arXiv$$aoai:arXiv.org:1701.08766
002244365 035__ $$9Inspire$$aoai:inspirehep.net:1511447$$d2023-10-03T07:44:11Z$$h2023-10-04T02:23:03Z$$mmarcxml$$ttrue$$uhttps://inspirehep.net/api/oai2d
002244365 037__ $$9arXiv$$aarXiv:1701.08766$$chep-th
002244365 035__ $$9Inspire$$a1511447
002244365 037__ $$aCERN-TH-2017-018
002244365 041__ $$aeng
002244365 084__ $$2CERN Library$$aTH-2017-018
002244365 100__ $$aAssel, Benjamin$$jORCID:0000-0003-0902-8360$$tGRID:grid.9132.9$$uCERN$$vTheoretical Physics Department - CERN - CH-1211 - Geneva 23 - Switzerland
002244365 245__ $$9arXiv$$aRing Relations and Mirror Map from Branes
002244365 246__ $$9arXiv$$aRing Relations and Mirror Map from Branes
002244365 269__ $$c2017-01-30
002244365 260__ $$c2017-03-29
002244365 300__ $$a62 p
002244365 500__ $$9arXiv$$a76 pages, including 41 figures. v2: discussion on the Higgs branch of non-abelian theories added, published version
002244365 520__ $$9Springer$$aWe study the space of vacua of three-dimensional $ \mathcal{N} $ = 4 theories from a novel approach building on the type IIB brane realization of the theory and in which the insertion of local chiral operators in the path integral is obtained from integrating out light modes in appropriate brane setups. Most of our analysis focuses on abelian quiver theories which can be realized as the low-energy theory of D3-D5-NS5 brane arrays. Their space of vacua contains a Higgs branch, parametrized by the vevs of half-BPS meson operators, and a Coulomb branch, parametrized by the vevs of half-BPS monopole operators. We show that the Higgs operators are inserted by adding F1 strings and D3 branes, while Coulomb operators are inserted by adding D1 strings and D3 branes, with specific orientations, to the initial brane setup of the theory. This approach has two main advantages. First the ring relations describing the Higgs and Coulomb branches can be derived by looking at specific brane setups with multiple interpretations in terms of operator insertions. This provides a new derivation of the Coulomb branch quantum relations. Secondly the map between the Higgs and Coulomb operators of mirror dual theories can be derived in a trivial way from IIB S-duality.
002244365 520__ $$9arXiv$$aWe study the space of vacua of three-dimensional $\mathcal{N}=4$ theories from a novel approach building on the type IIB brane realization of the theory and in which the insertion of local chiral operators in the path integral is obtained from integrating out light modes in appropriate brane setups. Most of our analysis focuses on abelian quiver theories which can be realized as the low-energy theory of D3-D5-NS5 brane arrays. Their space of vacua contains a Higgs branch, parametrized by the vevs of half-BPS meson operators, and a Coulomb branch, parametrized by the vevs of half-BPS monopole operators. We show that the Higgs operators are inserted by adding F1 strings and D3 branes, while the Coulomb operators are inserted by adding D1 strings and D3 branes, with specific orientations, to the initial brane setup of the theory. This approach has two main advantages. First the ring relations describing the Higgs and Coulomb branches can be derived by looking at specific brane setups with multiple interpretations in terms of operator insertions. This provides a new derivation of the Coulomb branch quantum relations. Secondly the map between the Higgs and Coulomb operators of mirror dual theories is obtained in a trivial way from IIB S-duality.
002244365 540__ $$aarXiv nonexclusive-distrib. 1.0$$barXiv$$uhttp://arxiv.org/licenses/nonexclusive-distrib/1.0/
002244365 540__ $$3preprint$$aCC-BY-4.0
002244365 540__ $$3publication$$aCC-BY-4.0$$fSCOAP3
002244365 542__ $$3preprint$$dCERN$$g2017
002244365 542__ $$3publication$$dThe Author(s)$$g2017
002244365 595__ $$aCERN-TH
002244365 65017 $$2arXiv$$ahep-th
002244365 65017 $$2SzGeCERN$$aParticle Physics - Theory
002244365 690C_ $$aCERN
002244365 690C_ $$aARTICLE
002244365 695__ $$2INSPIRE$$9bibclassify$$aoperator: Higgs
002244365 695__ $$2INSPIRE$$9bibclassify$$amonopole: operator
002244365 695__ $$2INSPIRE$$9bibclassify$$aoperator: chiral
002244365 695__ $$2INSPIRE$$9bibclassify$$ameson: operator
002244365 695__ $$2INSPIRE$$9bibclassify$$adimension: 3
002244365 695__ $$2INSPIRE$$9bibclassify$$amembrane model
002244365 695__ $$2INSPIRE$$9bibclassify$$aCoulomb
002244365 695__ $$2INSPIRE$$9bibclassify$$avacuum state
002244365 695__ $$2INSPIRE$$9bibclassify$$aD-brane
002244365 695__ $$2INSPIRE$$9bibclassify$$astring
002244365 695__ $$2INSPIRE$$9bibclassify$$amirror
002244365 695__ $$2INSPIRE$$9bibclassify$$apath integral
002244365 695__ $$2INSPIRE$$9bibclassify$$abuildings
002244365 695__ $$2INSPIRE$$9bibclassify$$aS-duality
002244365 695__ $$2INSPIRE$$9bibclassify$$aabelian
002244365 695__ $$2INSPIRE$$9bibclassify$$aduality
002244365 695__ $$2INSPIRE$$9bibclassify$$aquiver
002244365 773__ $$c152$$pJHEP$$v03$$y2017
002244365 8564_ $$s1104051$$uhttp://inspirehep.net/record/1511447/files/scoap3-fulltext.pdf?subformat=pdfa$$xpdfa$$yArticle from SCOAP3
002244365 8564_ $$81274232$$s1269161$$uhttps://cds.cern.ch/record/2244365/files/arXiv:1701.08766.pdf
002244365 8564_ $$81274231$$s4652$$uhttps://cds.cern.ch/record/2244365/files/Tabel.png$$y00019 \footnotesize{Quiver and brane configuration of the abelian quiver $T_{\rm abel}$ with $M$ abelian nodes. There are $M+1$ NS5 branes in the brane configuration.}
002244365 8564_ $$81274233$$s2290$$uhttps://cds.cern.ch/record/2244365/files/linquiv.png$$y00000 \footnotesize General linear quiver.
002244365 8564_ $$81274234$$s2243$$uhttps://cds.cern.ch/record/2244365/files/TwoSemiF1s.png$$y00008 \footnotesize{Setup for the operator $- Z^2{}_1 Z^1{}_2$ or equivalently $-Z^1{}_1 Z^2{}_2$.}
002244365 8564_ $$81274235$$s29801$$uhttps://cds.cern.ch/record/2244365/files/AppHBopD3p.png$$y00037 \footnotesize{Setups with D3' branes inserting HB operators. The insertion of the figures a,b,c and d are related by HW D3' move and insert the same operator $Y_{\alpha} \equiv \xi_{1} -\sum_{\gamma=1}^\alpha Z^\gamma{}_{\gamma} = \xi_{M+1} + \sum_{\gamma=\alpha+1}^K Z^\gamma{}_{\gamma} = \xi_i + X_i + \sum_{\gamma=\alpha+1}^{\alpha + P} Z^\gamma{}_{\gamma}$ (F-term constraints).}
002244365 8564_ $$81274236$$s16013$$uhttps://cds.cern.ch/record/2244365/files/TBCBopAndRel.png$$y00029 \footnotesize{Brane setups inserting the CB operators of theory B: a) $V^+_{[1:1]}$; b) $\varphi_3$; c) $V^+_{[1:1]}(-V^{-}_{[1:1]})$ or $(m_1-\varphi_1)(m_2-\varphi_2)(\varphi_2-\varphi_1)$; d) $V^+_{[1:1]}V^+_{[2:2]}V^+_{[3:3]}$ or $V^+_{[1:3]} (\varphi_1-\varphi_2)(\varphi_2-\varphi_3)$.}
002244365 8564_ $$81274237$$s3405$$uhttps://cds.cern.ch/record/2244365/files/AppCBopMonop.png$$y00040 \footnotesize{Setups with D1 strings inserting monopole operators. The insertions are: a) $V^+_{[i:j]}$, b) $-V^-_{[i:j]}$. The branes which might stand in the place of the dots ``..." insert additional CB operators.}
002244365 8564_ $$81274238$$s6069$$uhttps://cds.cern.ch/record/2244365/files/SQEDSemiF1_2.png$$y00016 \footnotesize{a) Setup realizing $-Z^\alpha{}_{\alpha} Z^{\alpha+1}{}_{\alpha+1}$ or $-Z^{\alpha+1}{}_\alpha Z^{\alpha}{}_{\alpha+1}$. b) Setup realizing $Z^\beta{}_\alpha \prod\limits_{\alpha<\gamma<\beta} Z^\gamma{}_\gamma$. c) Setup realizing $(-)^{\alpha+\beta+1} Z^\alpha{}_\beta \prod\limits_{\alpha<\gamma<\beta} Z^\gamma{}_\gamma$. The F1 can split along the D5s, leading to alternative interpretations of the operator insertions.}
002244365 8564_ $$81274239$$s4996$$uhttps://cds.cern.ch/record/2244365/files/OtherOpCB.png$$y00031 \footnotesize{Products of CB operators are inserted by adding several D1 strings and D3'' branes to the setups. a) $n$ semi D1s inserting $(u^+)^n$, b) $m$ D3''s inserting $\varphi^m$, c) One semi D1 and two D3''s inserting $u^+ \varphi^2$.}
002244365 8564_ $$81274240$$s3880$$uhttps://cds.cern.ch/record/2244365/files/EtaDeform.png$$y00013 \footnotesize{a) Configuration associated to the FI deformation $\eta=\xi_1-\xi_2$ ($x^{3,5}$ plane). b) Brane setup inserting the operator $Y =Z^2{}_2 + \xi_2$.}
002244365 8564_ $$81274241$$s4709$$uhttps://cds.cern.ch/record/2244365/files/SemiF1_2.png$$y00005 \footnotesize{Brane configurations for the insertion of the meson operators (a) $-Z^1{}_1$ and (b) $Z^2{}_2$.}
002244365 8564_ $$81274242$$s6034$$uhttps://cds.cern.ch/record/2244365/files/AppCBopD1.png$$y00039 \footnotesize{Setups with D1 strings inserting scalar CB operators. The insertions are: a) $\varphi_{i} - \varphi_{i-1}$, b) $\varphi_{i-1} - \varphi_{i}$, c) $\varphi_{i} - \varphi_{i-1}$, d) $\varphi_{i-1} - \varphi_{i}$, e) $ m_\alpha - \varphi_i$.}
002244365 8564_ $$81274243$$s3289$$uhttps://cds.cern.ch/record/2244365/files/FullD1.png$$y00010 \footnotesize{A D1 brane stretched between the two NS5s, crossing the D3 brane.}
002244365 8564_ $$81274244$$s3841$$uhttps://cds.cern.ch/record/2244365/files/TabelHBop0.png$$y00020 \footnotesize{Brane setups of the long mesons (a) $B$ and (b) $-\ti B$.}
002244365 8564_ $$81274245$$s5568$$uhttps://cds.cern.ch/record/2244365/files/SQEDFullD1.png$$y00018 \footnotesize{a) Single D1 crossing the D3 ($x^{3,7}$ plane). b) The same configuration shown in the $x^{3,8}$ plane. The D1 appears on top of the D3. The D5s positions along $x^8$ correspond to the (real part of the) mass parameters $m_\alpha$.}
002244365 8564_ $$81274246$$s4747$$uhttps://cds.cern.ch/record/2244365/files/TSU2Sduality.png$$y00011 \footnotesize{Brane configuration of $T[SU(2)]$. After acting with the S transformation and moving the D5s to the middle, one recovers the initial configuration.}
002244365 8564_ $$81274247$$s21486$$uhttps://cds.cern.ch/record/2244365/files/D1intoF1.png$$y00012 \footnotesize{a) Starting from the $u^+$ insertion setup and applying S-duality leads the $Z^2{}_1$ insertion setup, after brane rearrangements. b) Starting from the $\varphi$ insertion setup and applying S-duality leads to the $Z^2{}_2$ insertion setup, after brane rearrangements. }
002244365 8564_ $$81274248$$s5388$$uhttps://cds.cern.ch/record/2244365/files/SemiD1.png$$y00007 \footnotesize{a) Semi infinite D1 ending on the D3 from above, inserting the monopole operator $u^+$. b) Semi infinite D1 ending on the D3 from below, inserting the monopole operator $-u^-$. c) D3" brane crossing the D3 at an arbitrary position in $x^3$ between the two NS5s, inserting the scalar operator $\varphi$.}
002244365 8564_ $$81274249$$s5240$$uhttps://cds.cern.ch/record/2244365/files/AppCornerF1.png$$y00035 \footnotesize{Setups inserting hyper-multiplet scalar operators. The insertions are: a) $Q^a$, b) $\ti Q_a$, c) $-\ti Q_a$, d) $Q^a$, e) $q_i$, f) $-\ti q_i$.}
002244365 8564_ $$81274250$$s3814$$uhttps://cds.cern.ch/record/2244365/files/SemiF1.png$$y00004 \footnotesize{Brane configurations for the insertion of the meson operators (a) $Z^2{}_1$ and (b) $-Z^1{}_2$.}
002244365 8564_ $$81274251$$s1450$$uhttps://cds.cern.ch/record/2244365/files/AppCBopD3pp.png$$y00038 \footnotesize{Setup with a D3'' brane crossing the D3$_{(i)}$ segment, inserting $\varphi_i$.}
002244365 8564_ $$81274252$$s16812$$uhttps://cds.cern.ch/record/2244365/files/TAHBopAndRel.png$$y00028 \footnotesize{Brane setups inserting the HB operators of theory A: a) $Z^2{}_{1}$; b) $Y_3$; c) $Z^2{}_{1}(-Z^1{}_2)$ or $(\xi_1 - Y_1)(\xi_2 - Y_1)(Y_2-Y_1)$; d) $Z^4{}_3 Z^3{}_2 Z^2{}_1$ or $Z^4{}_1 (Y_1 - Y_2)(Y_2-Y_3)$.}
002244365 8564_ $$81274253$$s24573$$uhttps://cds.cern.ch/record/2244365/files/MirrorMapBrane.png$$y00026 \footnotesize{a) From the brane setup realizing the SQED monopole operator $u^+$ to the setup realizing the insertion of the long meson $W$ in $T_{\rm abel}$. b) From the brane setup realizing the SQED scalar operator $\varphi$ to the setup realizing the insertion of the meson $Y (= X'_i)$ in $T_{\rm abel}$. The letters $S$ and $HW$ denote the action of S-duality and Hanany-Witten brane moves respectively.}
002244365 8564_ $$81274254$$s3785$$uhttps://cds.cern.ch/record/2244365/files/OtherOpEx.png$$y00032 \footnotesize{Setup with D1s and D3''s. The numbers indicate the number of superposed D1s in each region. The setup can be interpreted as inserting for instance the operators $-V_{(2,0,0)} V_{(0,1,0)} V_{(0,-1,0)}\varphi_1\varphi_3$ or $(V_{(1,0,0)})^2 (\varphi_1 - \varphi_2)(\varphi_3 - \varphi_2)\varphi_1\varphi_3$.}
002244365 8564_ $$81274255$$s3796$$uhttps://cds.cern.ch/record/2244365/files/TabelHWmove.png$$y00025 \footnotesize{The local setup (a) with an external D3 segment is related to the setup (c) with a D5 brane by letting the D3 segment end on a D5 (b) and moving it between the two NS5s. The distance between the D5 and the D3 along $x^8+ix^9$ is $\varphi_{i+1} -\varphi_i$.}
002244365 8564_ $$81274256$$s2342$$uhttps://cds.cern.ch/record/2244365/files/HWmove.png$$y00002 \footnotesize{Hanany-Witten brane creation effect. The triplet of branes $(A,B,C)$ can be either $(D5,NS5,D3)$, or $(D3',D5,F1)$, or $(D3'',NS5,D1)$. As the $A$ and $B$ branes pass across each other the number of $C$ branes stretched between them goes from $N_2$ to $\ti N_2 = N_1 + N_3 - N_2 +1$.}
002244365 8564_ $$81274257$$s5101$$uhttps://cds.cern.ch/record/2244365/files/TabelD1s.png$$y00024 \footnotesize{The CB ring relations are derived from the configurations: (a) with a D1 stretched between two adjacent NS5s, realizing $-u^+_i u^-_i$ or $(\varphi_{i-1} - \varphi_{i})(\varphi_{i+1} - \varphi_{i})$, and (b) with a semi D1 stretched between two distant NS5s, ending on the D3 segments from above, realizing $\prod_{k=i}^j u^{+}_k$ or $V^{+}_{[i:j]} \prod_{k=i+1}^{j} (\varphi_{k-1} - \varphi_{k})$.}
002244365 8564_ $$81274258$$s12095$$uhttps://cds.cern.ch/record/2244365/files/SQEDSemiD1.png$$y00017 \footnotesize{a) Semi infinite D1 ending on the D3 from above, inserting the monopole operator $u^+$. b) Semi infinite D1 ending on the D3 from below, inserting the monopole operator $-u^-$. c) D3" brane crossing the D3 at an arbitrary position in $x^3$ between the two NS5s, inserting the complex scalar operator $\varphi$.}
002244365 8564_ $$81274259$$s21953$$uhttps://cds.cern.ch/record/2244365/files/SQEDSemiF1.png$$y00015 \footnotesize{Brane setups inserting the operators: a) $Z^{\alpha+1}{}_{\alpha}$; b) $-Z^\alpha{}_{\alpha+1}$; c) $-Z_{[1:\alpha]}$; d) $Z_{[\alpha+1:N_f]}$. The integers 1,2, ...,$\alpha$ indicate the number of superposed F1 strings in each region.}
002244365 8564_ $$81274260$$s5253$$uhttps://cds.cern.ch/record/2244365/files/OtherOpHB.png$$y00030 \footnotesize{Products of HB operators (mesons) are inserted by adding several F1 strings and D3' branes to the setups. a) $n$ semi F1s inserting $(Z^2{}_{1})^n$, b) $m$ D3's inserting $Y^m = (Z^2{}_2 - \xi_2)^m$, c) One semi F1 and two D3's inserting $Z^2{}_{1} Y^2$.}
002244365 8564_ $$81274261$$s6392$$uhttps://cds.cern.ch/record/2244365/files/D3pMove.png$$y00009 \footnotesize{a) Setup realizing $-Z^1{}_1$, with a D3' on the left of the configuration. b) Setup after moving the D3' inbetween the D5s. c) Setup after moving the D3' to the right, realizing $Z^2{}_2$.}
002244365 8564_ $$81274262$$s3013$$uhttps://cds.cern.ch/record/2244365/files/CornerF1.png$$y00006 \footnotesize{a) F1 string with a D3-D5 corner. b) F1 string with two D3-D5 corners.}
002244365 8564_ $$81274263$$s7250$$uhttps://cds.cern.ch/record/2244365/files/TATB.png$$y00027 \footnotesize{Quiver diagrams and brane configurations of the mirror dual theories $T_A$ and $T_B$.}
002244365 8564_ $$81274264$$s3671$$uhttps://cds.cern.ch/record/2244365/files/linquivbrane.png$$y00001 \footnotesize Brane configuration associated to a general linear quiver with three nodes.
002244365 8564_ $$81274265$$s2939$$uhttps://cds.cern.ch/record/2244365/files/SQCD.png$$y00033 \footnotesize{Quiver diagram and brane realization of $U(N)$ SQCD.}
002244365 8564_ $$81274266$$s9121$$uhttps://cds.cern.ch/record/2244365/files/AppHBopSetups.png$$y00036 \footnotesize{Setups with F1 strings inserting HB operators (mesons). The insertions are: a) $-Z^\alpha{}_{\alpha}\equiv -\ti Q_\alpha Q^\alpha$, b) $Z^\alpha{}_{\alpha}$, c) $Z^\alpha{}_{\alpha}$, d) $-Z^\alpha{}_{\alpha}$, e) $-X_i \equiv -q_i \ti q_i$, f) $Z^\beta{}_{\alpha} \equiv \ti Q_\alpha (\prod_{k=i}^j q_k) Q^\beta$, g) $-Z^\alpha{}_{\beta} \equiv (-)^{i-j} Q^\alpha (\prod_{k=i}^j \ti q_k) \ti Q_\beta$.}
002244365 8564_ $$81274267$$s6388$$uhttps://cds.cern.ch/record/2244365/files/TabelSemiD1.png$$y00023 \footnotesize{Brane setups realizing the insertion of the CB operators: a) $u^+_i$, b) $-u^-_i$, c) $\varphi_i$.}
002244365 8564_ $$81274268$$s2857$$uhttps://cds.cern.ch/record/2244365/files/SQED.png$$y00014 \footnotesize{Quiver and brane configuration of $\N=4$ SQED. There are $N_f$ D5 branes.}
002244365 8564_ $$81274269$$s2466$$uhttps://cds.cern.ch/record/2244365/files/TabelBraneRel1.png$$y00022 \footnotesize{A full F1 stretched between the two D5s and crossing the D3 segments, realizing $-W \ti W$ or $(-1)^{M+1} X_1 X_2 \cdots X_{M} X_{M+1}$.}
002244365 8564_ $$81274270$$s14571$$uhttps://cds.cern.ch/record/2244365/files/SQCDBraneRel.png$$y00034 \footnotesize{Brane setup in the 3d space $(x^3,x^7, x^8)$ with a D1 string stretched between the two NS5s and intersecting a D3 brane. In this example there are $N=5$ D3s and $N_f=4$ D5s.}
002244365 8564_ $$81274271$$s15792$$uhttps://cds.cern.ch/record/2244365/files/TabelHBop.png$$y00021 \footnotesize{Brane setups of the short mesons (a) $X'_1$, (b) $X'_{M+1}$ and (c) $X'_i$, $2 \le i \le M-1$. These brane configurations are all related by D3' brane moves along $x^3$ and are ultimately a single brane setup.}
002244365 8564_ $$81274272$$s2600$$uhttps://cds.cern.ch/record/2244365/files/TSU2.png$$y00003 \footnotesize{Quiver and brane configuration for the $T[SU(2)]$ theory.}
002244365 8564_ $$81302294$$s1104051$$uhttps://cds.cern.ch/record/2244365/files/jhep-2017-03-152.pdf$$ySpringer Open Access article
002244365 8564_ $$82334120$$s1104051$$uhttps://cds.cern.ch/record/2244365/files/scoap.pdf$$yArticle from SCOAP3
002244365 960__ $$a13
002244365 980__ $$aARTICLE