We exactly compute the partition function for $U(2)_k\times U(2)_{-k}$ ABJM
theory on $\mathbb S^3$ deformed by mass $m$ and Fayet-Iliopoulos parameter
$\zeta $. For $k=1,2$, the partition function has an infinite number of
Lee-Yang zeros. For general $k$, in the decompactification limit the theory
exhibits a quantum (first-order) phase transition at $m=2\zeta $.Comment: 11 pages, 1 figure. v2: references adde

Russo, Jorge G.

Silva, Guillermo A.

Journal of High Energy Physics

English

arXiv

Abstract: We exactly compute the partition function for U(2)k × U(2)− k ABJM theory on S3 deformed by mass m and Fayet-Iliopoulos parameter ζ. For k = 1, 2, the partition function has an infinite number of Lee-Yang zeros. For general k, in the decompactification limit the theory exhibits a quantum (first-order) phase transition at m = 2ζ.Fil: Russo, Jorge Guillermo. Universidad de Barcelona; España. Institució Catalana de Recerca i Estudis Avancats; España. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Silva, Guillermo Ariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Física; Argentina. Abdus Salam International Centre for Theoretical Physics; Itali

Russo, Jorge Guillermo

Silva, Guillermo Ariel

CONICET Digital

Exact partition function in U(2) × U(2) ABJM theory deformed by mass and Fayet-Iliopoulos terms

We exactly compute the partition function for U(2)k × U(2)-k ABJM theory on S3 deformed by mass m and Fayet-Iliopoulos parameter ζ. For k = 1, 2, the partition function has an infinite number of Lee-Yang zeros. For general k, in the decompactification limit the theory exhibits a quantum (first-order) phase transition at m = 2ζ.Facultad de Ciencias ExactasInstituto de Física La Plat

El Servicio de Difusión de la Creación Intelectual

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

JHEP12(2015)092Published for SISSA by SpringerReceived: October 21, 2015Accepted: November 29, 2015Published: December 15, 2015Exact partition function in U(2)×U(2) ABJM theorydeformed by mass and Fayet-Iliopoulos termsJorge G. Russoa,b and Guillermo A. Silvac,daInstitució Catalana de Recerca i Estudis Avançats (ICREA),Pg. Lluis Companys, 23, 08010 Barcelona, SpainbDepartment ECM, Institut de Ciències del Cosmos,Universitat de Barcelona, Mart́ı Franquès, 1, 08028 Barcelona, SpaincInstituto de F́ısica La Plata, CONICET & Departamento de F́ısica,Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, ArgentinadAbdus Salam International Centre for Theoretical Physics, Associate Scheme,Strada Costiera 11, 34151 Trieste, ItalyE-mail: jorge.russo@icrea.cat, silva@fisica.unlp.edu.arAbstract: We exactly compute the partition function for U(2)k × U(2)−k ABJM theoryon S3 deformed by mass m and Fayet-Iliopoulos parameter ζ. For k = 1, 2, the partitionfunction has an infinite number of Lee-Yang zeros. For general k, in the decompactificationlimit the theory exhibits a quantum (first-order) phase transition at m = 2ζ.Keywords: Matrix Models, Supersymmetric gauge theory, AdS-CFT Correspondence,Chern-Simons TheoriesArXiv ePrint: 1510.02957Open Access, c© The Authors.Article funded by SCOAP3.doi:10.1007/JHEP12(2015)092JHEP12(2015)092Contents1 Introduction 12 Residue integration 23 Summary of results and limits 54 The special case m2 = 0 81 IntroductionThe dynamics of two coincident M2 branes on the orbifold R8/Zk is described by ABJMtheory, three-dimensional U(2)k × U(2)−k supersymmetric Chern-Simons theory with bi-fundamental matter [1]. For this particular gauge group, the ABJM theory has N = 8superconformal symmetry and is in fact equivalent to Gustavsson-Bagger-Lambert the-ory [2, 3]. The partition function for the theory on S3 can be computed by supersymmetriclocalization [4, 5]. This theory can be deformed, preserving N = 4 supersymmetry, byadding mass and Fayet-Iliopoulos (FI) parameters m, ζ, and the localization techniquethen reduces the full supersymmetric functional integral to the matrix integral [5]Z =14∫d2µ(2π)2d2ν(2π)2sinh2 µ1−µ22 sinh2 ν1−ν22∏i,jcosh(µi−νj+m2)cosh(µi−νj−m2) e ik4π ∑i (µ2i−ν2i )− ik2π ζ∑i (µi+νi) (1.1)where i, j = 1, 2. The parameter ζ represents a Fayet-Iliopoulos parameter for the diagonalU(1) subgroup, whereas m corresponds to a mass for the chiral multiplets. The partitionfunction should be understood as a function Z(2ζ,m; k), but for ease of presentation wewill omit its arguments unless needed. For k = 1, the theory is mirror dual to N = 4 super-symmetric super Yang-Mills theory with gauge group U(2) coupled to a single fundamentalhypermultiplet and a single adjoint hypermultiplet [5].By shifting the integration variables, x ≡ µ − ζ, y ≡ ν + ζ, the partition functionbecomesZ =14∫d2x(2π)2d2y(2π)2sinh2 x1−x22 sinh2 y1−y22∏i,jcoshxi−yj+m12 coshxi−yj−m22eik4π∑i(x2i−y2i ), (1.2)where m1, m2 arem1 = m+ 2ζ and m2 = m− 2ζ . (1.3)Note that ζ has dimension of mass. We are using units where the radius R of the three-sphere is R = 1.– 1 –JHEP12(2015)092The purpose of this note is to explicitly carry out the integration in (1.2). In them = ζ = 0 case, the integral was computed in [6] (a discussion of the partition functionin the more general ABJ case can be found in [7]). On the other hand, the m, ζ-deformedABJM theory was studied in [8] using the Fermi-gas formulation [9] and at at large N forthe U(N)k×U(N)−k gauge group in [10] (with ζ = 0) and in [11] (with general m, ζ 6= 0),where phase transitions in the complex parameter space generated by m1, m2 and g = 2πi/kwere investigated. Our explicit formula will uncover some interesting physical propertiesof the mass-deformed system with gauge group U(2)k ×U(2)−k.The partition function (1.2) manifests the m1 ↔ m2 symmetry or, equivalently,ζ → −ζ. A less obvious symmetry is m2 → −m2, or [8, 11]Z(2ζ,m; k) = Z(m, 2ζ; k) . (1.4)For the k = 1 case, this symmetry already appeared in [5], where it was also explained bythe fact that the corresponding brane configuration is self-mirror. The symmetry implies,in particular, that a FI-deformation ζ on the massless theory is equivalent to a mass-deformation m = 2ζ in the theory with vanishing FI-parameter. The case m = 2ζ —representing a fixed point of this symmetry — is special, as we shall shortly see. In thedual N = 4 supersymmetric super Yang-Mills theory, m2 = 0 corresponds to coupling thetheory to a massless adjoint hypermultiplet.2 Residue integrationThe partition function for the m, ζ-deformed ABJM theory with U(N)k × U(N)−k gaugegroup can be written in the following form [5, 11]Z(2ζ,m; k) =∑ρ(−1)ρ 1N !∫dNτe−ikm2∑i τi∏i cosh(kπτi) cosh(π(τi − τρ(i))− m12) , (2.1)where the sum goes over permutations. The derivation uses a trigonometric identity,Fourier integrations and only holds for opposite Chern-Simons levels (see section 2 in [11]for details). For N = 2, the formula (2.1) then leads to the following expressionZ =12(Z1 − Z2) , (2.2)withZ1 =∫dτ1dτ2e−ikm2(τ1+τ2)cosh(πkτ1) cosh(πkτ2) cosh2(m12) , (2.3)andZ2 =∫dτ1dτ2e−ikm2(τ1+τ2)cosh(πkτ1) cosh(πkτ2) cosh(π(τ1 − τ2)− m12)cosh(π(τ1 − τ2) + m12) ,(2.4)Using the identity1cosh2 m12− 1cosh(πτ − m12)cosh(πτ + m12) = sech2 m12 sinh2 πτcosh(πτ − m12)cosh(πτ + m12) (2.5)– 2 –JHEP12(2015)092and the formula for the Fourier transform [11]∫due−ikm2ucosh(πk2 (u+ v))cosh(πk2 (u− v)) = 4 sin(km2v)k sinh(πkv) sinhm2, (2.6)we obtainZ =1k2 sinh(m2) cosh2 m12∫dusin(m2u) sinh2 πuksinh(πu) cosh(πuk −m12)cosh(πuk +m12) . (2.7)In the limit m2 → 0, the partition function becomesZ∣∣m2=0=1k2 cosh2 m12∫duu sinh2 πuksinh(πu) cosh(πuk −m12)cosh(πuk +m12) . (2.8)In the following, we compute the integrals (2.7), (2.8) by residue integration.To compute (2.7) we follow the ideas in [6], where the partition function was computedin the case m = ζ = 0.Thus we start by writing the integrand as the product of two even functions f, gZ =1k2 sinh(m2) cosh2 m12∫duf(u)g(u) , (2.9)withf(u) =sinm2usinhπu, g(u) =sinh2 πukcosh(πuk −m12)cosh(πuk +m12) . (2.10)Under the shift u→ u+ ik these functions transform asf(u)→ (−)k cosh(m2k)f(u) + odd function , (2.11)g(u)→ g(u)These properties imply that the integral in (2.9) along the curve u = x + ik with x ∈ Rwill differ from the integration along the real axis by the factor (−)k cosh(m2k). Therefore,the rectangular contour composed by the real axis, two vertical segments and the displacedreal axis u = x+ ik becomes appropriate for residue computation in the case m2 6= 0 (seefigure 1).1The residues encircled by the contour comprise the ones arising from the poles of f(z)located at z = in with n = 1, . . . , k and those of g(z) located at z± = ±m1k2π + ik2 . Thepole located at z = ik does not contribute due to a double zero in the numerator of g(z).Calling C the closed rectangular contour described above and F(z) = f(z)g(z) one finds∮CdzF(z) =(1− (−)k cosh(m2k)) ∫duF(u)= 2πi[ k−1∑n=1Resz=inF(z) + Resz=z± F(z)]1It is easily seen that the vertical contours do not contribute when we push them to infinity.– 3 –JHEP12(2015)092ii(k − 1)02ii(k − 2)bz...ikz− z+...Figure 1. Rectangular contour for residue computation. The poles on the imaginary axis z = inwith n = 1, . . . , k − 1 arise from the f function, while those at z± = ±m1k2π + ik2 follow from the gfunction.which gives∫duF(u) = 2πi1− (−)k cosh(m2k)[− iπk−1∑n=1(−)nsin2(nπk)sinh(m2n)cosh(m12 −inπk)cosh(m12 +inπk) + Rk](2.12)whereRk =(−)k2ikπcothm12sinhkm22sinhkm12cos km1m22π , k even(−)k+12ikπcothm12coshkm22coshkm12sin km1m22π , k odd(2.13)Case m2 = 0, k odd. It is evident from (2.12) that the m2 → 0 limit of (2.9) is smooth,the result isZ∣∣m2=0=1k2 cosh2m[ k−1∑n=1(−)nn sin2(nπk)cosh(m− inπk)cosh(m+ inπk) − (−) k+12 k2m cothmπ cosh km], k odd(2.14)where we have used m1 = 2m.Case m2 = 0, k even. The factor multiplying the bracket in (2.12) prevents taking m2 → 0in the even k case. To compute the integral in (2.8) we considerI =∫duf̃(u)g(u) , (2.15)with g(u) as in (2.10) andf̃(u) =ik(u− ik/2)2sinhπu.– 4 –JHEP12(2015)092Upon integration, the odd piece in f̃ vanishes against g(u) and therefore the partitionfunction (2.8) can be written asZ∣∣m2=0=1k2 cosh2mI (2.16)The shift u→ u+ ik in f̃(u) givesf̃(u)→ (−)k+1f̃(−u) .As discussed below (2.11), this property makes the rectangular contour in figure 1 appro-priate for computing I by residues.For the residues analysis we should now consider the pole in f̃(z) at the origin z = 0but a zero in g(z) eliminates it; along the same lines the residue from z = ik/2 is absentsince a zero appears for f̃ . Calling F̃(z) = f̃(z)g(z) one finds∮Cdz F̃(z) = 2I ,on the other hand∮Cdz F̃(z) = 2πi[ k−1∑n=0Resz=in F̃(z) + Resz=z± F̃(z)]= 2πi[ikπk−1∑n=1(−)n(k2− n)2 sin2 (nπk )cosh(m− inπk)cosh(m+ inπk) + R̃k] . (2.17)whereR̃k = (−)k22i(mk)2π3coth(m) sinhmkcosh(2mk)− 1The n = k2 term in the sum vanishes as expected. The final result isZ∣∣m2=0= − 1k cosh2m·[ k−1∑n=1(−)n(nk− 12)2 sin2 (nπk )cosh(m− inπk)cosh(m+ inπk) + (−) k2 2m2kπ2coth(m) sinhmkcosh(2mk)−1](2.18)3 Summary of results and limitsThus we have obtainedZ =2k2 sinh(m2)11− (−1)k cosh(m2k)(J1 − J2) (3.1)whereJ1 =1cosh2(m12) k−1∑n=1(−1)nsin2(nπk)sinh(m2n)cosh(m12 −inπk)cosh(m12 +inπk) (3.2)– 5 –JHEP12(2015)092andJ2 =(−)k22k sinhkm22sinh(m1) sinhkm12cos km1m22π , k even(−)k+122k coshkm22sinh(m1) coshkm12sin km1m22π , k odd(3.3)Using21 + coshα=1cosh2(α2) , 21− coshα= − 1sinh2(α2) , (3.4)we can finally put the partition function in the formZ∣∣k even= − 1k2 sinh(m2) sinh2(km22) (J1 − J2) (3.5)Z∣∣k odd=1k2 sinh(m2) cosh2(km22) (J1 − J2) (3.6)In the formulas (3.5)–(3.6), the symmetry m1 ↔ m2 — which is manifest in the integralform (1.2) — is hidden. Interestingly, this symmetry is only recovered upon summationover n. On the other hand, the symmetry m2 → −m2 is manifest.Note that Z is real. While this is expected in a unitary theory, it is not generally thecase in Chern-Simons theories (for a discussion, see [12]). In the present case, it is relatedto the fact the theory is a combination of two Chern-Simons theory with opposite levels.2Consider, as particular examples, the important cases k = 1, 2. The partition functionstake the formZ∣∣k=1=2sinh(m1) sinh(m2) cosh(m12)cosh(m22) sin(m1m22π), (3.7)Z∣∣k=2=2sinh2(m1) sinh2(m2)sin2(m1m22π). (3.8)Now the symmetry m1 ↔ m2 has become manifest.Note that the partition functions for k = 1, 2 have zeros. Restoring the R dependence,the zeros are located atm1m2R2 = 2π2n , n = ±1,±2, . . . (3.9)They represent Lee-Yang zeros (see, for example, [13]). In the infinite volume, R→∞,the zeros condense in a certain line, and a phase transition should emerge. The fact thatthe partition function has zeros seems to be related to the fact that the coupling, g = 2πi/k,is imaginary for real k. Indeed, from the general expressions (3.2)–(3.3) we see that thearguments of the sine and cosine functions in (3.7), (3.8) contain a factor π/k. If thecoupling g is (unphysically) continued to the real line by taking k → ik, the partitionfunction zeros would then lie on the imaginary g-axis, in accordance with the Lee-Yangtheorem (see [11] for a related discussion).For the undeformed ABJM theory, the k = 1 case is of special interest, since it isconjectured to describe the dynamics of two M2 branes in eleven-dimensional Minkowski2We thank Miguel Tierz for comments on this point.– 6 –JHEP12(2015)092spacetime. An interesting question is what is the origin of these Lee-Yang singularities inthe brane realization.The partition function Z(2ζ,m; k) does not have any zeros for k > 2. For higher valuesof k, the partition function becomes more involved, below we quote explicitly the k = 3and k = 4 casesZ∣∣k=3=232− sin(3m1m22π)csch(m12)csch(m22)(coshm1 + cosh 2m1)(coshm2 + cosh 2m2)(3.10)Z∣∣k=4=1− sech(m1)− sech(m2) + cos(2m1m2π)sech(m2) sech(m1)8 sinh2m1 sinh2m2(3.11)Note that the symmetry under the exchange m1 ↔ m2 is manifest.Asymptotic formulas. Let us consider the limit of a large sphere, mR 1, at fixed k.Assuming m1 > 0, m2 > 0 and restoring the R dependence, we findZ∣∣k=1∼ 32 e−32(m1+m2)R sin(m1m2R22π), (3.12)Z∣∣k=2∼ 32 e−2(m1+m2)R sin2(m1m2R22π), (3.13)Z∣∣k>2∼ 64k2e−2(m1+m2)R sin2(πk). (3.14)The general asymptotic formula with arbitrary sign for m2 and m2 6= 0, is obtained byreplacing m2 by |m2|.The absolute value implies a discontinuity in the first derivative of F = − lnZ. Thisindicates a first-order phase transition in the parameter m2 at m2 = 0, i.e., when the twomass scales m, 2ζ cross. Explicitly, at large R, we haveF = 2(|m1|+ |m2|)R+O(1) , k > 1 . (3.15)Henced∆Fdm2∣∣∣∣m2=0= 4R , ∆F ≡ Fm2>0 − Fm2<0 . (3.16)For k = 1 the discontinuity in the first derivative of ∆F is equal to 3R, as can be seenfrom (3.12).For the general theory with gauge group U(N)k ×U(N)−k, large N phase transitionsin the complex parameter Ng = 2πiN/k were studied in [10, 11]. These phase transi-tions require taking infinite volume and, at the same time, a strong coupling limit withfixed kR — a limit that already appeared in the context of supersymmetric U(N) Chern-Simons theory with massive fundamental matter in [14, 15]. It should be noted that suchdecompactification limit is different from the present (more physical) limit of large R atfixed k.Another interesting aspect of (3.14) is that it is in a form suitable for a weak couplingexpansion in powers of 1/k:Z∣∣k>2∼ −32k2e−2(m1+m2)R∞∑n=1(−1)n(2n)!(2πk)2n. (3.17)– 7 –JHEP12(2015)092The perturbative expansion has an infinite radius of convergence. However, the originaltheory on the three-sphere of finite radius R has an asymptotic perturbative expansion,with 2n! asymptotic behavior for the 1/k2n term. This can be seen by using the integralform (2.7) and generalizing the study of [16, 17] on the resurgence properties of the per-turbation series of ABJM theory. Now, expanding the integrand in (2.7), one finds a serieswith finite radius of convergence determined by the poles of sech(πu/k ± m1/2) in thecomplex u-plane. The integral over u then adds an extra (2n)!, leading to an asymptotic(but Borel summable) perturbation series.4 The special case m2 = 0The m2 = 0 case is special and must be considered separately. In particular, it representsthe critical point in the phase transitions that arise in the decompactification limit. Insection 2 we have obtained the following formulas:Odd k:Z∣∣m2=0=1k2 cosh2mk−1∑n=1(−)nn sin2 πnkcosh(m+ iπnk)cosh(m− iπnk) + (−) k−12 2mπ cosh(km) sinh(2m).(4.1)Even k:Z∣∣m2=0=1k cosh2mk−1∑n=1(−)n+1(nk− 12)2 sin2 (nπk )cosh(m− inπk)cosh(m+ inπk)+ (−)k2+1 4m2π2sinhmksinh(2m)(cosh(2mk)− 1) (4.2)In particular,Z∣∣k=1=2mπ cosh(m) sinh(2m),Z∣∣k=2=2m2π2 sinh2(2m). (4.3)Note that the partition function does not have zeros in this case.Asymptotic formulas m2 = 0. Consider again the limit of a large sphere, mR  1,at fixed k, but now with m2 = 0. We findZ∣∣k=1∼ 8mRπe−3mR, (4.4)Z∣∣k=2∼ 8π2m2R2 e−4mR, (4.5)Z∣∣k>2∼ 4k2e−4mR tan2πk. (4.6)– 8 –JHEP12(2015)092Note that these formulas differ from the asymptotic formulas (3.12)–(3.14) given abovefor Z(m1,m2) at m2 = 0. This is expected, since the latter were obtained by assuming|m1R|, |m2R| → ∞.Unlike the m2 6= 0 case, the perturbation series for this flat-theory limit has now finiteradius of convergence |π/k| < π/2, therefore perturbation series is convergent for all k > 2,where the formula applies. On the other hand, just like the general m2 6= 0 case, the theoryon a finite-radius S3 has an asymptotic perturbation series with 2n! asymptotic behavior.Finally, it would be interesting to study supersymmetric Wilson loops in the presentmass/FI deformed theory, along the lines of [18].AcknowledgmentsWe would like to thank Miguel Tierz for useful comments. GAS acknowledge support fromICTP and would like to thank Department ECM of Universitat de Barcelona for warmhospitality during the early stages of this work. We acknowledge financial support fromprojects FPA2013-46570, 2014-SGR-104, PIP0595/13 CONICET and X648 UNLP.Open Access. This article is distributed under the terms of the Creative CommonsAttribution License (CC-BY 4.0), which permits any use, distribution and reproduction inany medium, provided the original author(s) and source are credited.References[1] O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091[arXiv:0806.1218] [INSPIRE].[2] A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66[arXiv:0709.1260] [INSPIRE].[3] J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes,Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [INSPIRE].[4] A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformalChern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].[5] A. Kapustin, B. Willett and I. Yaakov, Nonperturbative tests of three-dimensional dualities,JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].[6] K. Okuyama, A note on the partition function of ABJM theory on S3,Prog. Theor. Phys. 127 (2012) 229 [arXiv:1110.3555] [INSPIRE].[7] H. Awata, S. Hirano and M. Shigemori, The partition function of ABJ theory,Prog. Theor. Exp. Phys. 2013 (2013) 053B04 [arXiv:1212.2966] [INSPIRE].[8] N. Drukker and J. Felix, 3d mirror symmetry as a canonical transformation,JHEP 05 (2015) 004 [arXiv:1501.02268] [INSPIRE].[9] M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. (2012) P03001[arXiv:1110.4066] [INSPIRE].– 9 –JHEP12(2015)092[10] L. Anderson and K. Zarembo, Quantum phase transitions in mass-deformed ABJM matrixmodel, JHEP 09 (2014) 021 [arXiv:1406.3366] [INSPIRE].[11] L. Anderson and J.G. Russo, ABJM theory with mass and FI deformations and quantumphase transitions, JHEP 05 (2015) 064 [arXiv:1502.06828] [INSPIRE].[12] C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact terms,unitarity and F-maximization in three-dimensional superconformal theories,JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].[13] C. Itzykson and J.M. Drouffe, Statistical field theory. Vol. 1: From Brownian motion torenormalization and lattice gauge theory, Cambridge University Press, Cambridge U.K.(1989) [INSPIRE].[14] A. Barranco and J.G. Russo, Large-N phase transitions in supersymmetric Chern-Simonstheory with massive matter, JHEP 03 (2014) 012 [arXiv:1401.3672] [INSPIRE].[15] J.G. Russo, G.A. Silva and M. Tierz, Supersymmetric U(N) Chern-Simons-matter theoryand phase transitions, Commun. Math. Phys. 338 (2015) 1411 [arXiv:1407.4794] [INSPIRE].[16] J.G. Russo, A note on perturbation series in supersymmetric gauge theories,JHEP 06 (2012) 038 [arXiv:1203.5061] [INSPIRE].[17] I. Aniceto, J.G. Russo and R. Schiappa, Resurgent analysis of localizable observables insupersymmetric gauge theories, JHEP 03 (2015) 172 [arXiv:1410.5834] [INSPIRE].[18] S. Hirano, K. Nii and M. Shigemori, ABJ Wilson loops and Seiberg duality,Prog. Theor. Exp. Phys. 2014 (2014) 113B04 [arXiv:1406.4141] [INSPIRE].– 10 –

Jorge G. Russo

Guillermo A. Silva

Crossref

https://ri.conicet.gov.ar/bitstream/11336/54209/2/CONICET_Digital_Nro.58fecc9b-f650-476b-a1ce-ad0627b7084e_A.pdf

Exact partition function in $U(2)\times U(2)$ ABJM theory deformed by mass and Fayet-Iliopoulos terms

Abstract

Similar works

Full text

Available Versions

CONICET Digital

El Servicio de Difusión de la Creación Intelectual

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Crossref

Exact partition function in U(2)×U(2)U(2)\times U(2)U(2)×U(2) ABJM theory deformed by mass and Fayet-Iliopoulos terms

Abstract

Similar works

Full text

Available Versions

CONICET Digital

El Servicio de Difusión de la Creación Intelectual

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Crossref

Exact partition function in $U(2)\times U(2)$ ABJM theory deformed by mass and Fayet-Iliopoulos terms