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675e0b4d3b5605d41632719903cd6b1857e21b64 | subsection | 13 | 319 | Body | In particular, writing now \mathfrak {g} = \mathfrak {su}(2,2|4) we see that \mathfrak {g}^{(0)} can be identified with the subalgebra \mathfrak {so}(4,1)\oplus \mathfrak {so}(5)\subset \mathfrak {su}(2,2)\oplus \mathfrak {su}(4) and that the central element i\mathbb {I}_8\in \mathfrak {g}^{(2)}.The presence of \kappa ... | {
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d5e43d263eab17928589dc60c5cc8702e1ce643f | subsection | 14 | 319 | Body | The main strength of this approach, which can be characterised as a deformation of the Poisson structure, is that it maintains the integrability of the model manifestly during the deformation procedure.The deformation is governed by an r matrix, being a skew-symmetric non-split solution of the modified classical Yang-B... | {
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da25b68e4337ef5bdb2ff3d7592179f6fbb78147 | subsection | 15 | 319 | Body | We saw that the undeformed off-shell S-matrix was invariant under \mathfrak {psu}(2|2)_{\text{c.e.}}^{\otimes 2} and could be bootstrapped from this knowledge completely. There is a natural construction of the q deformation of centrally extended \mathfrak {su}(2|2) that defines the quantum group U_q\left( \mathfrak {su... | {
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8aaccf0c1857ed8111be5a4dbcfd0e88fc251920 | subsection | 16 | 319 | Body | It also discusses an alternative approach to the analysis of the quantum deformed representation theory using an affinisation based on doubling the fermionic generators. Its value will not be important in the remainder, but let us mention that to follow the undeformed case it is convenient to set\gamma = \sqrt{-iq^{1/2... | {
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d0edb2f366e42b6fe36faf607a41ef0752337581 | subsection | 17 | 319 | Body | In particular, we will restrict to \theta \in (-\pi ,\pi ] which will ensure unitarity of the S matrix later.For the \eta -deformed mirror model the resulting equations were found in and are1 = e^{i\tilde{p}_j R} \prod _{{k=1 \\ k\ne j}}^{\tilde{K}^{\mathrm {I}}} S_{\mathfrak {sl}(2)}(\tilde{p}_j,\tilde{p}_k)\prod _{\a... | {
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7f4fe76dbedc7fe74ca97c2b45b737fa490d5da7 | subsection | 18 | 319 | Body | (REF ):S_{\mathfrak {sl}(2)}\left( - \tilde{p}_1 , - \tilde{p}_2 \right)\big |_{\theta =\theta _0} =
S_{\mathfrak {su}(2)}\left( p_1 , p_2 \right)\big |_{\theta = \theta _0+\pi },using the identification of momenta as in eqn. (REF ). The undeformed Bethe-Yang equations follow by plugging in the undeformed x-function an... | {
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23a87320023e491a027862748e08b9adf41f92db | subsection | 19 | 319 | Body | THe relation between the TBA and Y system has been analysed in great detail in the recent work .Y_{1|w}^+ Y_{1|w}^- & =(1+Y_{2|w})\left(\frac{1-Y_-^{-1}}{1-Y_+^{-1}}\right)^{\vartheta (\theta -|u|)}, \\
Y_{M|w}^+ Y_{M|w}^- & =(1+Y_{M-1|w})(1+Y_{M+1|w})\,,Y_{1|vw}^+ Y_{1|vw}^- & =\frac{1+Y_{2|vw}}{1+Y_2}\left(\frac{1-Y_... | {
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2158178845e63f25d73dc475cf19c0d728debb10 | subsection | 20 | 319 | Body | Since Y_+ and Y_- are related by analytic continuation, the discontinuity of \log Y_{1|w}^{(\alpha )} can be written as\left[\log Y_{1|w}\right]_{\pm 1}(u) =\left[L_{-}\right]_0(u).The derivation for Y_{1|vw}^{(\alpha )} is very similar, but relies on kernel identities to first rewrite the TBA equation for Y_{1|vw}^{(\... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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16778bd8e32e28c5e33535832e67fdf41f6403ba | subsection | 21 | 319 | Body | Continuing with the other terms and using the equation (REF ) derived below we find\left(\Lambda _Q \star K_{Qy}\right)\hat{\star } K_1 - \Lambda _Q \star K_{xv}^{Q1} &=&\Lambda _Q \star \left(K_{Qy}\hat{\star } K_1\right) - \Lambda _Q \star K_{xv}^{Q1} \\
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d1031595ee24173ac802e617e8bcec2b2474ca51 | subsection | 22 | 319 | Body | So without computation we can use the non-local equation\log \frac{Y_-}{Y_+}(u) = -\Lambda _P \star K_{Py}.Now we see that\log Y_- = 1/2 \left( \log Y_-Y_+ + \log Y_-/Y_+ \right) = 1/2 \left( \log Y_-Y_+ -\Lambda _Q\star \left(K_-^{Qy} -K_+^{Qy} \right)\right).This already yields part of the TBA equation for Y_-^{(\alp... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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bae9c2bafb306f2c0cc667e92b4f16feeb572159 | subsection | 23 | 319 | Body | The result is\left[\log Y_- Y_+ \right]_{\pm 2N} = 2\sum _{J=1}^N \left[L_{J|vw} - L_{J|w}\right]_{\pm (2N-J)}-\sum _{Q=1}^{N}\left[\Lambda _Q \right]_{\pm (2N-Q)},as we will derive in the next section.Our next task is to prove that we can rederive the TBA equations of the Y_{w} and Y_{vw} functions. Since their TBA eq... | {
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3d4732df18135144c39af4f3b8bf45a3350ab6ba | subsection | 24 | 319 | Body | The relevant Y-system equations areY_{1|w}^+Y_{1|w}^- &=& (1+Y_{2|w}) \left(L_--L_+\right) \\
Y_{M|w}^+Y_{M|w}^- &=& (1+Y_{M+1|w})(1+Y_{M-1|w}),\\Y_{1|vw}^+Y_{1|vw}^- &=& (\frac{1+Y_{2|vw}}{1+Y_2}) \left(\Lambda _--\Lambda _+\right) \\
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76e169f94c03b93b86324f04d8f47fef2bcf78db | subsection | 25 | 319 | Body | Repeated application, plugging in the Y_{1|vw} discontinuity equation and some rewriting ultimately yield\left[ \log Y_{M|vw} \right]_{(M+2l)\tau } = \left[D^{M|vw}_{(M+2l)\tau } -\delta _{l,0}\Lambda _-\right]_0,where we have defined a set of D functions as& &D^{M|vw}_{(M+2l)\tau }(u) =\left(\Lambda _--\Lambda _+ \rig... | {
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1eb9236ad803830f513097a4bbd5b62c180dc258 | subsection | 26 | 319 | Body | To do this most transparently we first rewrite the term\left(L_-^{(\alpha )}-L_+^{(\alpha )}\right) \hat{\star } K_M &=& \left(\Lambda _-^{(\alpha )}-\Lambda _+^{(\alpha )}\right) \hat{\star } K_M -\log Y_-^{(\alpha )}/Y_+^{(\alpha )} \hat{\star } K_M \\&=& \left(\Lambda _-^{(\alpha )}-\Lambda _+^{(\alpha )}\right) \ha... | {
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1d2b76b3488947b977fbcf0eca6b8d3349c658da | subsection | 27 | 319 | Body | (REF ) holds, leading to the following non-linear integral equation for the density \rho :\frac{1+1/Y_{2,2}}{1+Y_{1,1}}= \frac{\left(1+\mathcal {K}_1^+\hat{{\star }}\rho -\rho /2 \right)\left(1+\mathcal {K}_1^-\hat{{\star }}\rho -\rho /2 \right)}{\left(1+\mathcal {K}_1^+\hat{{\star }}\rho +\rho /2 \right)\left(1+\mathc... | {
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7a7b3042b4740404469a39b7c385f81b5be963dd | subsection | 28 | 319 | Body | Nevertheless, we are formally required to assume that the equations (REF ) can be solved so we will do so.From the objects defined so far we can now in fact define the entire QQ system containing 256 functions \mathcal {Q}_{A|I} with multi-indices A and I: define the basic functionsThe unimodularity constraint \mathcal... | {
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754a46eaab594fa318e23d3fee3829f2a22234f7 | subsection | 29 | 319 | Body | However, we will see that for most applications this amount of H symmetry will suffice.To analyse this a bit further, let us consider the conjugation properties of our basic functions \mathbf {P}_a,\mathbf {Q}_i and \mu _{ab}: it is not clear from our construction that we can ensure nice conjugation properties, but we ... | {
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e675f8687247d3de142c6f502c2b5929a831eb46 | subsection | 30 | 319 | Spectral problem. | It is in the search for scaling dimensions – usually dubbed the spectral problem – that the presence of integrability proved to be of vital importance: the gauge-invariant single-trace operators built up from a fixed amount L of only two of the scalars have a very simple structure, being a trace over products of these ... | {
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ecd195d9ef4c7406c52ff50b2c820cdf4b3eb727 | subsection | 31 | 319 | String theory. | The most commonly used formulation of the type IIB string theory on {\rm AdS}_5\times {\rm S}^5 is in the Green-Schwarz formalism, which allows one to actually write down the action of the model in a compact form . In contrast, the presence of a self-dual Ramond-Ramond five-form flux makes it unclear how to follow the ... | {
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a57fedbd772df4e3eab3fe6b52b50038ebe613a4 | subsection | 32 | 319 | Thermodynamic Bethe ansatz. | The name of the thermodynamic Bethe ansatz (TBA) method suggests it is meant to obtain information about the thermodynamics of physical systems. Indeed, the original application of the TBA method by Yang and Yang to find the free energy of the Lieb-Liniger gas had exactly this purpose and has since been applied to many... | {
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c7d68be54cdb1b0344cc3a0ccbf89c07d18a40c9 | subsection | 33 | 319 | Beyond the TBA. | For many systems the TBA equations provide the simplest form of the spectral problem. Although it is fairly straightforward to write the equations in the form of a Y system , , , a system of functional difference equations, these do not provide a true simplification: the Y-system equations have many solutions and selec... | {
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90c2107bd134b09c9fffcabb67b26d43baa91964 | subsection | 34 | 319 | Beyond the TBA. | However, starting from the fact that the T functions can be decomposed into Q functions as follows from the Wronskian solution of the Y system a further (and most likely final) major simplification of the spectral problem is possible: transferring the analytic properties of the T functions from the FiNLIE to the langua... | {
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19591ab0ce4878b8b26e7c7ce7d20888a46b831e | subsection | 35 | 319 | Beyond the TBA. | The presence of OSp(4|6) symmetry compared to the PSU(2,2|4) symmetry in the {\rm AdS}_5\times {\rm S}^5 case at first glance changes the algebraic structure of the QSC significantly. Closer inspection shows, however, that after the proper identification of functions the algebraic structure can be made to match exactly... | {
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58962ee93a6143ce50e7a536b3e8a787e87edbef | subsection | 36 | 319 | Deformations. | The fact that the spectral problem for {\rm AdS}_5\times {\rm S}^5 could be simplified to the QSC is such a great achievement, that one can also rightly ask how unique the QSC's existence is. A way to find out is is to look at deformations, alterations of the original theory continuously parametrised by a parameter suc... | {
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9820ceb0582e7345d2504a6a7afa741912dd57ae | subsection | 37 | 319 | Hopf-twisted deformations. | On the gauge theory side perhaps the most natural thing to look for are exactly marginal deformations of the lagrangian, since after all the \mathcal {N}=4 theory is conformal. The existence of \mathcal {N}=1 marginal deformations was proven in and a particular three-dimensional family of deformations, known as Leigh-S... | {
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798c1c8267e2be5c44b38ca1a3e4322a8bd5a3cc | subsection | 38 | 319 | TsT-based deformations. | The existence of the AdS/CFT correspondence immediately induces two questions about the CFT deformations: is there a gravity dual for the found integrable deformations and – more generally – how can we describe deformations in the language of string theory? The nice framework of non-linear sigma models has helped a lot... | {
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d8063da07738ef8b7d13c00d3e370f067cd20746 | subsection | 39 | 319 | Yang-Baxter deformations. | These deformations are obtained by deforming the underlying Poisson structure of the Lax formulation of the model , , . The input for these deformations are anti-symmetric solutions (or r matrices) of the modified classical Yang-Baxter equation, thereby allowing for a classification of these integrable deformations thr... | {
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74e4bc79f4761fa9f23ae6808b79158d18f3af33 | subsection | 40 | 319 | Quantum group deformations. | A very important example where this turns out to be possible is the real-q deformation of the {\rm AdS}_5\times {\rm S}^5 superstring (commonly called the \eta deformation) , , which follows and builds upon earlier work on deforming the principal chiral model , . This particular deformation breaks all the supersymmetry... | {
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4c073387b89a0f02a96de0d73778d6c0b5adda23 | subsection | 41 | 319 | Even more deformations. | Other ways to deform either the \mathcal {N}=4 gauge or string theory have been considered. One large class we have not mentioned yet are the orbifoldings: taking a discrete subgroup of the R-symmetry group of the gauge theory one can define a projection of the fields dependent on this subgroup. The projected fields ar... | {
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43056ba51932201d951030046eff8fe7a50efe0f | subsection | 42 | 319 | Aim of this thesis and summary. | In this thesis we consider the spectral problem for the \eta -deformed {\rm AdS}_5\times {\rm S}^5 superstring, ultimately culminating in the construction of the \eta -deformed quantum spectral curve, a one-parameter deformation of the quantum spectral curve constructed for the {\rm AdS}_5\times {\rm S}^5 superstring. ... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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ab3b7b3a9de3ce82edb2296c56593417e3356465 | subsection | 43 | 319 | A note on notions and notation. | We have tried to stay close to the literature in our choice of notation, which should allow for easy comparison. As this thesis builds on work by many others we have had to make some choices which conventions to stick to.The quantum-deformation parameter of the \eta -deformed model has been denoted in this thesis as c ... | {
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d60806e1eaac03c972311aa88f889a0ec6c61243 | subsection | 44 | 319 | Classical | To put our explorations on a firm foundation, we now first introduce the {\rm AdS}_5\times {\rm S}^5 superstring theory in the Green-Schwarz formalism, starting from the superconformal algebra \mathfrak {psu}(2,2|4). | {
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b1829d36c19273128fafcbfa85311ac002349d0d | subsection | 45 | 319 | Coset description of the Green-Schwarz superstring | These are all the elements we need to introduce the coset description of the Green-Schwarz string: we view the string as the embedding of a two-dimensional world sheet \Sigma \cong \mathbb {R}\times S^1 with coordinates (\tau , \sigma ) into a target space given by the coset\frac{\text{PSU}(2,2|4)}{\text{SO}(4,1) \time... | {
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0e05f2236c37ba288129029cb42c25a5a239f8ce | subsection | 46 | 319 | Coset description of the Green-Schwarz superstring | An easy way to implement the modding out of the U(1) subgroup is to enforce tracelessness of A^{(2)}. The isometry group of the lagrangian is now given by PSU(2,2|4), which acts by left multiplication. The form of the lagrangian density (REF ) is not the most convenient for our deforming purposes, but in order to rewri... | {
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f9119658fd652cc3c4723cdade39f1c9c63bb236 | subsection | 47 | 319 | Bosonic action. | For later convenience we introduce the Polyakov action describing the bosonic part of the model above. With target-space coordinates X^M =
\lbrace t,\rho ,\zeta ,\psi _i\rbrace \cup \lbrace \phi ,r,\xi ,\phi _i \rbrace for the AdS_5 and the S^5 spaces and target-space metric G_{MN} it can be written in the standard for... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.008939971216022968,
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-0.005507388152182102,
-0.013806610368192196... | |
fd3cbe2f1e563ac8f2d88ace5308370b5b46b2c8 | subsection | 48 | 319 | Equations of motion. | Varying the lagrangian with respect to \mathbf {g} gives the equation of motion\partial _{\alpha }\left(\gamma ^{\alpha \beta } A_{\beta }^{(2)}\frac{\kappa }{2} \epsilon ^{\alpha \beta }\left(A_{\beta }^{(1)}-A_{\beta }^{(3)} \right) \right) -\left[ A_{\alpha }, \left(\gamma ^{\alpha \beta } A_{\beta }^{(2)}\frac{\kap... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.03299776837229729,
-0.010569358244538307... | |
aeaf02451a39557fb4a08268e59574306322a4cd | subsection | 49 | 319 | Hamiltonian formalism. | In order to understand how to derive the \eta deformation we consider the {\rm AdS}_5\times {\rm S}^5 superstring in the hamiltonian formalism .The author is indebted to Gleb Arutyunov for his exposition of this topic. We start from the loop group \hat{G} = C\left( S^1, G\right) consisting of continuous maps from the c... | {
"cite_spans": [
{
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"doi": "",
"end": 145,
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"raw": "F. Delduc, M. Magro and B. Vicedo, “Derivation of the action and symmetries of the q-deformed AdS_{5} \\times S^{5} superstring”, JHEP 1410, 132 (2014), arxiv:1406.6286.",
"source_ref_id": "b9... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.... | |
d996bbf2576e287aa3f259be1098b9b464d79df0 | subsection | 50 | 319 | Hamiltonian formalism. | (REF )):\mathbf {A} = - \mathbf {g}^{-1} \partial _{\sigma } \mathbf {g} \in G, \quad \Pi = - \mathbf {g}^{-1}X\mathbf {g} \in \mathfrak {g},which obey the following Poisson brackets for its projected components\left\lbrace A_1^{(i)}(\sigma ) , A_2^{(j)}(\sigma ^{\prime }) \right\rbrace &= 0, \\\left\lbrace A_1^{(i)}(\... | {
"cite_spans": [
{
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"doi": "",
"end": 1111,
"openalex_id": "",
"raw": "F. Delduc, M. Magro and B. Vicedo, “Derivation of the action and symmetries of the q-deformed AdS_{5} \\times S^{5} superstring”, JHEP 1410, 132 (2014), arxiv:1406.6286.",
"source_ref_id": "b... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.0... | |
770e7b3d9b959d3dcc8c56a7722d42a28fcef307 | subsection | 51 | 319 | Hamiltonian formalism. | The integrability of this model follows due to the existence of a Lax pair \left(L,M\right) (see also ):L(z) &= A^{(0)} + \frac{1}{4} \left( z^{-3} + 3 z\right) A^{(1)} + \frac{1}{2} \left( z^{-2} + z^2\right) A^{(2)} + \frac{1}{4} \left(3 z^{-1}+ z^3\right)A^{(3)} \\&+ \frac{1}{2} \left(1 - z^{4}\right) \Pi ^{(0)} + \... | {
"cite_spans": [
{
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"doi": "",
"end": 874,
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"raw": "I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the \\mathit {AdS}_{5}\\times \\mathit {S}^5 superstring”, Phys. Rev. D69, 046002 (2004), hep-th/0305116.",
"source_ref_id": "582ad5... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.00553... | |
74b9204924d8eb5199192a41f123089e77a0f3ba | subsection | 52 | 319 | Hamiltonian formalism. | The Lax matrix L satisfies the Poisson brackets
\begin{equation}
\left\lbrace L_1(z_1, \sigma ), L_2(z_2, \sigma ^{\prime }) \right\rbrace = \left[ \mathcal {R}_{12}, L_1(z_1,\sigma ) \right] \delta _{\sigma \sigma ^{\prime }} -
\left[ \mathcal {R}_{21}, L_2(z_2,\sigma ^{\prime }) \right] \delta _{\sigma \sigma ^{\pri... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
0.018428124487400055,
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-0.017756901681423187,
-... | |
f984347a7c7544890d20fce4a32e60a7408c0a94 | subsection | 53 | 319 | Retrieving | One particular consequence of this construction is that we can find the fields \mathbf {g} and \mathbf {X} by analysing the behaviour of the Lax matrix around z=1, which is the pole of the twist function \phi _S:L(z) = A - 2(z-1) \Pi + \mathcal {O}\left((z-1)^2\right).We introduce the gauge transformationL^{\mathbf {g}... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.0579187385737896,
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0.01... | |
2987a1711ed3043611ac3d6595950072d4653c2d | subsection | 54 | 319 | Retrieving | Now we can work on massaging the integral I_{C_2}, defined as
\begin{align}
I_{C_2} &= \oint _{C_2} \frac{dz}{2\pi i} \Delta (z)B(z,u)\\&= \sum _{N=1}^{\infty } \sum _{\tau } \oint _{\gamma _x} \frac{dz}{2\pi i} \Delta (z+\tau i 2N c)B(z+\tau i 2N c,u),
\end{align}
with \gamma _x as in fig. \ref {fig:gammax}.
\begin{}[... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
ea2ffa948bf71ec7b064ae0423dbe2666e3b2777 | subsection | 55 | 319 | Retrieving | The first term can be treated as follows: we deform \gamma _x using the analyticity of the integrand into \gamma . Now we can rewrite the first term using the analyticity of L_{\pm } in the {\mbox{\small lower}\\ \mbox{\small upper}} half-plane and using that K(z_*,u) =-K(z,u):
\begin{align}
& G(u) \sum _{N=1}^{\infty ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.03375638276338577,
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0.00817203614860773,
0.0033859391696751118,
0... | |
eb9882da0d018b59020e4a121ba97721a035a084 | subsection | 56 | 319 | Retrieving | \\[5mm]
The third term can immediately be seen to give
\begin{equation}G(u) \sum _{\alpha }\oint _{\gamma _x}dz \log Y_-^{(\alpha )}(z)\sum _{N=1}^{\infty } \left( K(z+ i 2N c,u)-K(z- i 2N c,u)\right),
\end{equation}which matches the expression (\ref {eq:simplifieddressing}) for the dressing phase factor. So we see tha... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.009857057593762875,
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0.04931580647826195,
0.01197037473320961,
... | |
7f21ce39b1b7e4ac54a0ebc14cd49e674961c823 | subsection | 57 | 319 | Retrieving | Writing \gamma ^{\pm } for the parts of \gamma in the upper and lower half-plane respectively we now find
\begin{align}
\log Y_Q &= \oint _{\gamma } \frac{dz}{2\pi i} \log Y_{Q}(z) H(z-u) \\&= \int _{\gamma ^+} \frac{dz}{2\pi i} \left( \log Y_{Q}(z) +cQJ - cQ J \right)H(z-u) \\&+ \int _{\gamma ^-} \frac{dz}{2\pi i} \le... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.04364533722400665,
-0.01078924909234047,
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0.006214882247149944,
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0.00005120649075252004,
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0.03964705765247345,
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0.018953673541545868,
-0.004894839599728584,
0... | |
eff7614b31713153700794f9ed7511b3dfc4753d | subsection | 58 | 319 | Retrieving | The result is given in terms of D functions, which are defined for l\ge 0 as
\begin{align}
D_{\tau (Q+2l)}^Q(u) &= \sum _{J=1}^{l+1}\sum _{M=J}^{Q+J-2} L_{vw|M}^{(\alpha )}(u+\tau (M+2l-2J+2)ic) + L_-^{(\alpha )}(u+2\tau l i c) \\&-\sum _{J=1}^l \left( 2 \sum _{M=1}^{Q-1} \Lambda _{M+J}(u+\tau (M+2l-J)ic) + \Lambda _{Q... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.050429817289114,
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0.008738217875361443,
0.03809710219502449,
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0.027718083932995796,
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-0.002850414253771305,
-0.007643078453838825,
0.0... | |
15b226fabc22dab5d4b963bdca8ade3aa3e309a9 | subsection | 59 | 319 | Retrieving | As a first step we simplify the contributions coming from the D functions and \log Y_1 separately. The D-function contribution can be written as
\begin{align}
&\sum _{\tau }\sum _{l=0}^{\infty } \int _{Z_0}\frac{dz}{2\pi i} \left[D_{\tau (Q+2l)}^Q\right]_0 (u) H(z+\tau (Q+2l)ci-u) \\&=- \sum _{\tau }\sum _{l=1}^{\inft... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.014267432503402233,
0.04312274232506752,
-0.010475500486791134,
-0.001348539488390088,
0.0031949130352586508,
-0.006767493672668934,
0.004284044727683067,
0.025406712666153908,
0.011124019511044025,
0.02314833737909794,
-0.009567572735249996,
-0.015053286217153072,
0.018158551305532455,
... | |
b77e478560a96f7adb20d68ca999e0882b95c24f | subsection | 60 | 319 | Retrieving | (\ref {eq:Y1contribution}):
\begin{align}
&\sum _{\tau }\sum _{l=0}^{\infty } \int _{Z_0}\frac{dz}{2\pi i} \left(\left[L_-^{(\alpha )}\right]_{\tau 2l}(u) -\delta _{l,0}\delta _{\tau ,+1}\left[L_-^{(\alpha )} \right]_0(u) \right) H(z+\tau (Q+2l)ci-u)\\&=-\oint _{\gamma _x}\frac{dz}{2\pi i} L_-^{(\alpha )}(u)H(z+Qci-u) ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.0531313382089138,
0.0553896501660347,
-0.01132207177579403,
-0.019729701802134514,
-0.019012534990906715,
0.023880111053586006,
0.026107903569936752,
0.009758038446307182,
0.012611445039510727,
0.005413079168647528,
-0.04565449804067612,
0.014968938194215298,
0.00265313358977437,
0.0385... | |
1385043668ee0cd2a1cc990936b347074db2adde | subsection | 61 | 319 | Retrieving | \end{align}
The terms in the previous expression containing Y_Q functions can be simplified to\\ -\sum _{M=1}^{\infty } \Lambda _M \star K_{M Q}, whereas the Y_{vw} functions can be simplified to the term
\begin{align}
&- \sum _{\tau }\sum _{l=1}^{\infty }\tau \int _{Z_0 -i\tau \epsilon } \sum _{M=1+l}^{Q+l-1} L_{vw|M}... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.018269019201397896,
0.05228877067565918,
-0.02561020478606224,
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0.012019092217087746,
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0.02716696262359619,
0.038277946412563324,
0.031287793070077896,
0.04313137009739876,
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0.0025526261888444424,
0.010164717212319374,
0.... | |
659a362ea6991d4c95ff7a2689db5c0e6a47e8f0 | subsection | 62 | 319 | Retrieving | \end{align}
The next step is using our knowledge of \Delta to rewrite the first contour integral:
\begin{align}
&\oint _{\gamma _x}\log Y_{1}(u+ic) K_Q(z-u) = \int _{\check{Z}_0}\hat{\Delta }(u) K_Q(z-u)
\\&= \frac{1}{2}\oint _{\gamma _x} \left(\Delta -\sum _{\alpha }L_-^{(\alpha )}\right)(u) K_Q(z-u)
+ \int _{\check{... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.00520640192553401,
0.042932793498039246,
-0.013189551420509815,
0.004180378280580044,
-0.020306874066591263,
0.014364330098032951,
0.0439702570438385,
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0.0343584381043911,
0.030513711273670197,
0.005114860832691193,
0.018659135326743126,
0.011267188005149364,
0.000... | |
d4a99d0dc916158a44023c73ba8333da3cdbf8fb | subsection | 63 | 319 | Retrieving | However, incorporating the branch cuts is quite straightforward:
\begin{align}
\oint _{\gamma _x}dz { K_Q(z-u)&= {(u+i Qc)- {(u-i Qc)+ \left( \int _{i \mathbb {R}-\epsilon }-\int _{i \mathbb {R}+\epsilon }\right) dz {(u)K_Q(z-u) \\&= -2\tilde{E}_Q
+\int _{i \mathbb {R}}2\pi i K_Q(z-u)dz = -2\tilde{E}_Q- 2 cQ,
}
where ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.047313231974840164,
0.06794790923595428,
-0.013629263266921043,
-0.013201918452978134,
0.0015434034867212176,
0.005170115269720554,
0.04737428203225136,
-0.011645160615444183,
0.04133039712905884,
0.03669064864516258,
-0.002142450073733926,
-0.03003627248108387,
-0.0004375997232273221,
... | |
a3eb1d0a0a9164ec3fd85c2d98fef1049357a984 | subsection | 64 | 319 | Retrieving | (\ref {fsimple}): if f has poles itself, the right-hand side of eqn. (\ref {fsimple}) will also feature the poles of f. In this case
\begin{equation}f(z) = \frac{d}{dt} \log \left(x(t+i M c) -x(z)\right)\left(x(t-i M c) -x(z)\right),
\end{equation}which gives for the L_{M|vw} contribution
\begin{align}
& \frac{1}{2\pi ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
-0.04595000296831131,
0.03517951816320419,
-0.047292497009038925,
-0.016277773305773735,
0.0030454080551862717,
0.013081715442240238,
0.03383702039718628,
0.02822294272482395,
0.021907106041908264,
0.017132088541984558,
-0.029473906382918358,
0.03043501079082489,
0.0014588210033252835,
0.0... | |
59c7ad0d863f6e2b4c48bed4a5fb8937f5c1398e | subsection | 65 | 319 | Retrieving | (\ref {eq:y1expl}) to the TBA equation can be summed up as
\begin{align}
& \frac{1}{2}\oint _{\gamma _x}\sum _{\alpha } \left( L^{(\alpha )}_{-} + L^{(\alpha )}_{+} \right) \hat{\star } K(z) K_Q(z-u) \\&- \int _{\check{Z}_0} \left( \sum _{\alpha }\check{L}_-^{(\alpha )}(z)\right) K_Q(z-u) + \int _{Z_0+i \epsilon } L_-^... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
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d4d568b4200c59461658d6c0c959f3f39925dd63 | subsection | 66 | 319 | Retrieving | Using the TBA equation for Y_- we then obtain
\begin{equation}\Delta ^{\Sigma }(u) = 2 \Lambda _P \star \oint _{\gamma _x} ds K^{Py}_-(s) \left( \oint _{\gamma _x} dt K_{q\Gamma }^{[2]}(s-t)K(t,u)-K_{q\Gamma }^{[2]}(s-u) \right) \text{ for } u \notin \check{Z}_0.
\end{equation}We can recognise the right-hand side of th... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
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eddc5603dcdd389e8484dae8bf78f2734737177e | subsection | 67 | 319 | Retrieving | (\ref {prelres}) we get
\begin{equation}\log Y_Q(u) = -J \tilde{E}_Q + \sum _{\alpha }\left( \sum _{M=1}^{\infty } L_{M|vw}^{(\alpha )}\star K^{MQ}_{vwx}(u)+ L_{\beta }^{\alpha } \hat{\star } K_{\beta }^{yQ}\right) + \Lambda _P \star K_{\mathfrak {sl}(2)}^{PQ},
\end{equation}which indeed coincides with the TBA equation... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
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c0378d58ac23baa64120d83d1404ca928fb50b10 | subsection | 68 | 319 | Retrieving | The standard argument uses the Banach fixed-point theorem on the integral operator that relates the left- and right-hand sides of the TBA equation as follows: we can schematically write the TBA equation as
\begin{equation}
\mathbf {Y} = L\left( \mathbf {Y} \right),
\end{equation}
where L is some integral operator and ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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252154acf37063b2d3ee29430acf2bacba6dedf9 | subsection | 69 | 319 | Retrieving | The parametrisation of Y functions in terms of T functions has a gauge freedom that will allow us to tune these properties to some extent, but will make the discussion technically involved. It turns out that it seems impossible to define a set of T functions such that all its T functions have nice analytic properties. ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
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36e7fc0b4a36b32707f3b2c069313455b0f78947 | subsection | 70 | 319 | Retrieving | \begin{}
\begin{}
\begin{}[t]{ l | l}
For a\ge |s| & For s\ge a \\
\hline \textbf {T}_{a,0} \in \mathcal {A}_{a+1} & \mathbb {T}_{0, \pm s } = 1 \\
\textbf {T}_{a,\pm 1} \in \mathcal {A}_{a} & \mathbb {T}_{1, \pm s } \in \mathcal {A}_s \\
\textbf {T}_{a,\pm 2} \in \mathcal {A}_{a-1} & \mathbb {T}_{2, \pm s } \in \mathc... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
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745d9d85ebf0526733ac22e2dcc6b807db5d33c7 | subsection | 71 | 319 | Retrieving | This puts the Y functions on the \emph {Y hook}, illustrated in fig. \ref {fig:Yhook}.
\begin{}[t]
\centering \includegraphics [width=10cm]{Pictures/Thook.pdf}
\caption {The T hook organising all the T functions. The \emph {upper}, \emph {left} and \emph {right} band are separated at the red lines. The T functions live... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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a65ac46c0683bf971bb6274b41a3c4a62bbc9ef0 | subsection | 72 | 319 | Retrieving | What is more, this gauge freedom is also present in the Hirota equation: if \lbrace T_{a,s}\rbrace is a solution to the Hirota equation, so is
\begin{displaymath}
\left\lbrace g_1^{[a+s]}g_2^{[a-s]}g_3^{[-a+s]}g_4^{[-a-s]}T_{a,s}\right\rbrace .
\end{displaymath}
Therefore this gauge freedom is a genuine gauge freedom o... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
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1be309f59433e284c24f56ae37a0a8f715306506 | subsection | 73 | 319 | Retrieving | Also, \xi _j do not have poles and zeroes near the real axis or take negative real values and only \xi _1 is allowed to have discontinuities on the real axis with bounded branch points there. In those cases we assume that only \sigma _1 has discontinuities on the first lines Z_{\pm 1}. The wanted solution is given by\s... | {
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"source_ref_id": "ddd4bb06b7... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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791f2b55ee1cf85b4e97d26fb898e33ce9ec8e6b | subsection | 74 | 319 | Lagrangian. | One can bypass the entire construction from the Poisson structure and directly postulate the lagrangian density for the deformed theory and a Lax pair exhibiting its integrability. For \eta \in [0,1) it is given by (compare with eqn. (REF ))\mathcal {L}_{\eta } = -\frac{g}{2}\left(1+\eta ^2 \right) P_-^{\alpha \beta }\... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
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d9b950e916db52a75f281f17c2a55f133d523620 | subsection | 75 | 319 | Polyakov form. | The bosonic part of the action becomes (compare with eqn. (REF ))S^b_{\eta } &= -\frac{1}{2}\left(\frac{1+\eta ^2}{1-\eta ^2} g \right) \int d\sigma d\tau \left( \gamma ^{\alpha \beta } \partial _{\alpha } X^M \partial _{\beta } X^N G_{MN}^{\eta }
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"source_ref_id": "23ba831a79b0247560bd0bdeed1d... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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c02de767566aa713381e31e28728b1c12d5ae30a | subsection | 76 | 319 | Polyakov form. | Note that as \eta \rightarrow 0 one recovers the {\rm AdS}_5\times {\rm S}^5 action (REF ). | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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f2f35fa85ec1075a61d5f46e862838059a4b473e | subsection | 77 | 319 | Symmetries. | This lagrangian has the following symmetries:It has \mathfrak {psu}_q(2,2|4) :=U_q\left(\mathfrak {psu}(2,2|4)\right) symmetry with q \in \mathbb {R}. Note however that the realisation of this symmetry is far from obvious: in the undeformed case the target space manifestly carried the \mathfrak {psu}(2,2|4) symmetry as... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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0af9fbd6be9a99ac68d5dec285bd8c1259992854 | subsection | 78 | 319 | Construction from the Poisson structure. | Now let us find out how to construct the \eta deformation from the undeformed sigma model: as anticipated we consider the undeformed model in the hamiltonian formalism as discussed in section REF . The dynamics of this integrable model is completely defined in terms of the Lax matrix L, the fields A and \Pi and their P... | {
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"raw": "F. Delduc, M. Magro and B. Vicedo, “Derivation of the action and symmetries of the q-deformed AdS_{5} \\times S^{5} superstring”, JHEP 1410, 132 (2014), arxiv:1406.6286.",
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superstring | [
"Rob Klabbers"
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f21914dd3843aa39e8a27565c966a25808dca329 | subsection | 79 | 319 | Construction from the Poisson structure. | In the undeformed case we had a clear recipe (see REF ) to find these variables from an expansion of the Lax matrix, which however changed due to the deformation. So we should search for a generalisation of this recipe.The generalisation found by Delduc, Magro and Vicedo considers the complexified group G^{ whose Iwasa... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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a4e8cb788987c037b33147bc02040aca16fb12c0 | subsection | 80 | 319 | Construction from the Poisson structure. | (\ref {eq:deformedLaxX}) is indeed eqn. (\ref {eq:gaugeL2}) as required. This limit also shows that eqn. (\ref {eq:deformedLaxX}) can be regarded as a ``finite-difference derivative". The final thing we need to do is find the explicit form of the action. In order to do this we use that for every \mathbf {X} \in \mathfr... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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f1862b2281972c37340c2e7d68a15d16a452cad0 | subsection | 81 | 319 | Construction from the Poisson structure. | \end{equation}\paragraph {Maximal deformation limit.} The \eta -deformed theory constitutes a one-parameter family, deforming the {\rm AdS}_5\times {\rm S}^5 superstring theory more and more as \eta increases from zero to one. The point \eta =1 cannot be accessed directly from the langrangian as it is singular in that ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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44c4b7c3904d9b4b79f1f0030092fc84622b512c | subsection | 82 | 319 | Construction from the Poisson structure. | The presence of symmetry in general only implies that the background is a solution to a set of generalised supergravity equations \cite {Arutyunov:2015mqj,Wulff:2016tju}, which do not directly imply Weyl invariance. This breakdown is worth investigating: the deformation was introduced such as to manifestly maintain int... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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8412bcbef0788fde4d804d9c63164cacb8ad3d10 | subsection | 83 | 319 | Related work | The construction of the \eta deformation of the {\rm AdS}_5\times {\rm S}^5 string theory has sparked a widespread interest, which has led to explorations of many possible extensions and generalisations of the \eta -deformed theory. For example, many classical solutions have been found, see for example , , , , , . Also... | {
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a15807fa508a1ed2bb975144d5eb82880a76e490 | subsection | 84 | 319 | The world-sheet quantum field theory | Having gathered all the relevant information from the classical theory we are ready to consider the quantum version of these models. Even though both the undeformed and deformed models are integrable and have a lot of symmetry it is impossible to consider quantisation directly. In order to quantise we first have to get... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
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74b927d214f8319c35d20c0ed4330e7687042441 | subsection | 85 | 319 | Light-cone gauge | The presence of reparametrisation as well as \kappa symmetry implies that not all the degrees of freedom of the classical string theory are physical. This is an obstruction for the quantisation of this theory and should therefore be removed, which can be done by choosing a particular gauge known as the light-cone gauge... | {
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"raw": "G. Arutyunov and S. Frolov, “Integrable Hamiltonian for classical strings on AdS(5) x S**5”, JHEP 0502, 059 (2005), hep-th/0411089.",
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superstring | [
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65754ba4bb73eebaf0297a9f5ec03f251f40fe73 | subsection | 86 | 319 | Light-cone gauge | It is furthermore important to note that the light-cone directions are isometry directions and as such give rise to conserved quantities. Indeed, we find the string energy E and the angular momentum J along \phi asE= \int _{-L/2}^{L/2} d\sigma p_t, \quad J= \int _{-L/2}^{L/2} d\sigma p_{\phi },which in light-cone coord... | {
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"source_ref_id": "6d3f25e7c5baec1c90ad35c03... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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d7668b71e8483733d082353794e5fe4a3a6f2ea2 | subsection | 87 | 319 | Fermions. | We have illustrated light-cone gauge fixing above by considering the bosonic part of the theories only. To obtain the S matrix it is necessary, however, to have a good grasp on the fermions after gauge fixing as well. Moreover, in principle the presence of fermions might spoil the procedure described above. As it turns... | {
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8afeaa1a5f1d225faa51e03645e0ca70f4230f5c | subsection | 88 | 319 | Finding the perturbative | With the hamiltonian containing all the terms quadratic in fermions we can proceed to derive the S matrix describing scattering of the quantised model on the world sheet through the method of perturbative quantisation: by expanding the hamiltonian in inverse powers of the string tension g keeping only the leading order... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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7d337a58b4f9057c1990106383e234ac594a790b | subsection | 89 | 319 | Finding the perturbative | We can define in and out operators a_{\text{in},\text{out}} that evolve freely, i.e. their time evolution is governed by the free part H_{\text{free}} of the hamiltonian only:\partial _{\tau } a_{\text{in},}^k (p,\tau ) &= i\left[ H_{\text{free}}\left( a_{\text{in}}^{\dagger },a_{\text{in}}\right), a_{\text{in}}^k(p,\t... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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83afe68b950eae102d59de97ad652c5bcaf79d24 | subsection | 90 | 319 | Finding the perturbative | Out of these operators we can build a unitary operator \mathbb {S} – known as the S matrix – that maps out states to in states\mathopen | p_1,p_2,\ldots ,p_n \mathclose \rangle ^{\text{in}}_{k_1,k_2,\ldots k_n} = \mathbb {S} \mathopen | p_1,p_2,\ldots ,p_n \mathclose \rangle ^{\text{out}}_{k_1,k_2,\ldots k_n}and is giv... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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46255fdc5adad188a3f328564d2242f7bcbcf9a4 | subsection | 91 | 319 | Bootstrapping the exact | For a generic QFT, finding the S matrix perturbatively as discussed in the previous section is the best tool we have available to find the spectrum. For integrable field theories on the other hand we can use much more powerful techniques to obtain an exact S-matrix. Unfortunately, it is very difficult to establish the ... | {
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5fc3ddb2b642d5a920db095c10823abee68eecb7 | subsection | 92 | 319 | Integrable field theories in two dimensions | To understand what is so special about the theories we consider, let us analyse scattering in two-dimensional integrable theories in more generality: integrability is due to the existence of infinitely many symmetries, giving rise to an infinite set of commuting charges \lbrace \mathbb {Q}_j \rbrace which all mutually ... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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75387764699b79c938344adcf22331f49a2eee15 | subsection | 93 | 319 | Integrable field theories in two dimensions | As the particles are constrained on a line all pairs of particles will necessarily scatter at some time \tau . When two particles with momenta p_i>p_j meet and scatter, the conservation of charges dictates that the resulting scattering state must be proportional to \mathopen | p_j,p_i \mathclose \rangle _{k_2^{\prime }... | {
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"raw": "A. B. Zamolodchikov and A. B. Zamolodchikov, “Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models”, Annals Phys. 120, 253 (1979).",
"so... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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81ce68128fad8452aa6890b213eec6f11b5c4680 | subsection | 94 | 319 | Zamolodchikov-Faddeev algebra | The ideas in the previous subsection can be captured by a special type of creation and annihilation operators A_k^{\dagger }(p) and A^k(p), which form the Zamolodchikov-Faddeev algebra. The A_k^{\dagger }(p) create in-going or out-going particles with definite flavour and momentum from the vacuumA^k(p) \mathopen | \mat... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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a7294425e250099e7ede172226b01c832656cba0 | subsection | 95 | 319 | Zamolodchikov-Faddeev algebra | Combining this with the action of the ZF operators on two-particle states (REF ) givesA^{\dagger }_{k_1}(p_1)A^{\dagger }_{k_2}(p_2) = (-1)^{\epsilon (k_3)\epsilon (k_4)} S^{k_3,k_4}_{k_1,k_2}(p_1,p_2)A^{\dagger }_{k_2}(p_2)A^{\dagger }_{k_1}(p_1).We can streamline the notation by considering the two-particle states as... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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"Rob Klabbers"
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a6979ad8df293c683c0e4e9f279f41f158d896f2 | subsection | 96 | 319 | Zamolodchikov-Faddeev algebra | The approach we would like to follow here is a bit more algebraic however, as the \eta deformation can be more naturally formulated in the context of Hopf algebras.Before we move to the Hopf algebra setting, let us use the ZF algebra for some final observations: first of all the ZF algebra relations directly imply that... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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0d3a9cecbcbf89761ef242168f12799fe18ff7ae | subsection | 97 | 319 | Symmetry action on scattering states | The formalism in the previous section gives us a natural description of scattering states and the S matrix. It cannot by itself, however, constrain the S matrix far enough to bootstrap it completely, since the S matrix depends on the symmetry algebra of the theory under consideration. Therefore, to determine the S matr... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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fafacf372665cfc296d9a7e9406035487d1c741b | subsection | 98 | 319 | Symmetry action on scattering states | Consistent with our previous formalism, let us note that for the action on two-particle states we can write\mathbb {J}_{12}(p_1,p_2;C_1,C_2) = \mathbb {J}\left(p_1;C_1\right)\otimes \mathbb {I} + \mathbb {I}^g \left( \mathbb {I} \otimes \mathbb {J}\left(p_2;C_2\right)\right)\mathbb {I}^g,where \mathbb {I}^g is the grad... | {
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0e516938b2cb93c513d8bb28e294d27b03e0f60d | subsection | 99 | 319 | Hopf algebra | Our motivation to study Hopf algebras is twofold: they arise naturally from the scattering theory we have discussed and are also the most natural framework to discuss quantum groups, which form the basis for the \eta -deformed theory that is central in this thesis. The fact that quantum groups were in fact Hopf algebra... | {
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2a0809eceda73e5f7de4b2e5041a37c061d96ee3 | subsection | 100 | 319 | Construction of | The construction starts with a Lie (super)algebra, in our case the symmetry algebra \mathfrak {J}, and we consider its universal enveloping algebra \mathfrak {H} = U\left(\mathfrak {J}\right): this is defined as the quotient of the tensor algebraT\left(\mathfrak {J}\right) = \bigoplus _{n\ge 0}^{\infty } \mathfrak {J}^... | {
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85f64858687c854eb121cc4eedaa3b3ca47d4edf | subsection | 101 | 319 | Construction of | As we will not really need these axioms we refer the interested reader to \cite {Chari:1995gqg}. In particular, the definition of the counit can be derived from the consistency axioms. | {
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e8b2f5e82138ca47d8d8843811fa73e314139cb0 | subsection | 102 | 319 | Into a Hopf algebra. | To turn the bialgebra \mathfrak {H} into a Hopf algebra all we need to do is define one more map \mathcal {S} : \mathfrak {H}\rightarrow \mathfrak {H}, called the antipode, that should not be confused with the S matrix. It is defined by \mathcal {S}(x) =-x for all x\in \mathfrak {J} and is such that the diagram in fig.... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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560e244ca8a7fc196f67875b99a7f43a4b71963a | subsection | 103 | 319 | Quasitriangularity. | In the structure we have defined so far we have not seen a role for either the S or R matrix, confer eqn. (REF ). There is, however, a very natural way for the R matrix to occur: consider the graded permutation operator \tau on \mathfrak {H} \otimes \mathfrak {H} defined by \tau \left( x \otimes y \right) = (-1)^{\epsi... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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33697fcc7f3fab79e578426d35a75615745d1182 | subsection | 104 | 319 | Quasitriangularity. | The second relation is called the crossing equation and plays an important role in bootstrapping the S matrixJust to be explicit: in the present context the S matrix is related to \mathbb {R}. The antipode \mathcal {S} plays a different role entirely.: apart from certain analyticity requirements it is the only constrai... | {
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{
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"raw": "N. Beisert, M. de Leeuw and R. Hecht, “Maximally extended sl(2|2) as a quantum double”, J. Phys. A49, 434005 (2016), arxiv:1602.04988.",
"source_ref_id": "2b8677444bf28c9714c4b9d27bfe2c89d086... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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be8bb31dfab3df45bf1d0484d42183ba571e658f | subsection | 105 | 319 | Braiding. | As it turns out, the Hopf algebra we just constructed does not accurately describe the scattering theory of the {\rm AdS}_5\times {\rm S}^5 world-sheet QFT, due to our choice of the coproduct \Delta . This does not take into account the effect of “length changing"This nomenclature comes from the spin chain picture on t... | {
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{
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"raw": "J. Plefka, F. Spill and A. Torrielli, “On the Hopf algebra structure of the AdS/CFT S-matrix”, Phys. Rev. D74, 066008 (2006), hep-th/0608038.",
"source_ref_id": "b5a512f7620d284ea6327e34e761a7... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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9fcfe58ff8f2c63175449c11394a17099a7f98ca | subsection | 106 | 319 | The | The matrix form of the R matrix corresponding to fundamental short representations was determined in , , and its overall scalar factor known as the dressing factor , , , , was determined by solving the crossing equation. This then defines the S matrix of the {\rm AdS}_5\times {\rm S}^5 world-sheet QFT, thereby defining... | {
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"doi": "10.4310/atmp.2008.v12.n5.a1",
"end": 222,
"openalex_id": "https://openalex.org/W3105431322",
"raw": "N. Beisert, “The su(2|2) dynamic S-matrix”, Adv. Theor. Math. Phys. 12, 948 (2008), hep-th/0511082.",
"source_ref_id": "6644ee8f25802... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
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d720f30b4374083a78157e2329efd9e08bff293f | subsection | 107 | 319 | The universal enveloping algebra U | The most convenient point to start is the centrally extended Ideally we connect this more strongly with the original description of PSU(2,2|4) as a matrix quotient group by giving the expressions for these generators for example.algebra \mathfrak {g} =\mathfrak {su}(2|2)_{\text{c.e.}}, which is generated by the \mathfr... | {
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} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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ee41619e6189d8c41c2cb1aed445d167c03246b3 | subsection | 108 | 319 | The universal enveloping algebra U | The algebra relations are given by the following nonvanishing Lie brackets:\begin{aligned}
\left[\mathbf {R}^a_{\,\,\,b}, \mathbf {R}^c_{\,\,\,d} \right] &= \delta ^c_b \mathbf {R}^a_{\,\,\,d}-\delta ^a_d \mathbf {R}^c_{\,\,\,b}, \\
\left[\mathbf {R}^a_{\,\,\,b}, \mathbf {Q}^{\gamma }_{\,\,\,d} \right] &= -\delta ^a_d... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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07c38f9a52e87226f7de04cd7f25cb502511a85b | subsection | 109 | 319 | The universal enveloping algebra U | We furthermore have\lbrace \mathbf {Q}^{\alpha }_{\,\,\, b}, \mathbf {Q}^{\gamma }_{\,\,\, d} \rbrace = \epsilon ^{\alpha \gamma } \epsilon _{b d} \mathbf {C},\quad \lbrace \mathbf {Q}^{\dagger a}_{\,\,\,\,\,\beta }, \mathbf {Q}^{\dagger c}_{\,\,\,\,\, \delta } \rbrace = \epsilon ^{a c } \epsilon _{\beta \delta } \math... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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e019da73271505a8d33a30823412bac596d7efb2 | subsection | 110 | 319 | Chevalley basis. | We do not need all the original generators to describe U \left( \mathfrak {g}\right). We can define a Chevalley basis of U \left( \mathfrak {g}\right) by three Cartan generators \mathbf {H}_j, three simple positive roots \mathbf {E}_j and three simple negative roots \mathbf {F}_j. Expressed in the Lie algebra generator... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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0dc507e10f2fa0cfdf5d3d674cbb73e41e22f361 | subsection | 111 | 319 | Commutation relations. | In this basis the commutation relations can be written compactly as follows: for j,k=1,2,3\left[ \mathbf {H}_j, \mathbf {H}_k \right] = 0, \quad \left[ \mathbf {H}_j, \mathbf {E}_k \right] = A_{jk} \mathbf {E}_k, \quad \left[ \mathbf {H}_j, \mathbf {F}_k \right] = - A_{jk} \mathbf {F}_k.The commutators between positive... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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"hep-th",
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4787b02a18d5d2db39894f5fc1c759d63f1fabab | subsection | 112 | 319 | Serre relations. | The Serre relations, which are the usual result of imposing consistency of higher order algebra relations, put additional restrictions on positive and negative simple roots\left[ \mathbf {E}_1, \mathbf {E}_3 \right] &= \mathbf {E}_2 \mathbf {E}_2 = \left[ \mathbf {E}_1, \left[ \mathbf {E}_1, \mathbf {E}_2\right] \right... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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