chunk_uid stringlengths 40 40 | chunk_type stringclasses 2
values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
c7d4ad73e53093700a5b711618e618d8b9f1b19d | subsection | 113 | 319 | Central charges. | The central charges can be expressed in terms of the Chevalley basis as\mathbf {K} &= - \tfrac{1}{2}\mathbf {H}_1-\mathbf {H}_2- \tfrac{1}{2}\mathbf {H}_3, \\\mathbf {C} &= \left\lbrace \left[\mathbf {E}_1,\mathbf {E}_2 \right] , \left[\mathbf {E}_3,\mathbf {E}_2 \right] \right\rbrace , \\\mathbf {D} &= \left\lbrace \l... | {
"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 | [
0.05487166345119476,
0.027084873989224434,
-0.040802791714668274,
-0.015083604492247105,
-0.006252409424632788,
-0.05920524522662163,
0.038422372192144394,
0.0503549762070179,
-0.02015724964439869,
0.011429053731262684,
0.010421954095363617,
-0.010460101999342442,
-0.010971281677484512,
0.... | |
37e431f69d3df9a991e0ad870373bef858a891a6 | subsection | 114 | 319 | The quantum group U | Now we are ready to introduce the q deformation of U\left( \mathfrak {psu}(2|2)_{\text{c.e.}}\right) known as the quantum group U_q\left( \mathfrak {psu}(2|2)\right)_{\text{c.e.}}, which we will denote as \mathfrak {psu}_q(2|2)_{\text{c.e.}} or simply \mathfrak {h}. The deformation will manifest itself as deformations ... | {
"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 | [
-0.026785915717482567,
0.04725310206413269,
-0.04609314352273941,
-0.02301604673266411,
-0.007906042039394379,
-0.013209805823862553,
0.04774150624871254,
-0.0070437039248645306,
-0.0035638241097331047,
0.05528124421834946,
0.015934186056256294,
0.021703459322452545,
-0.015247367322444916,
... | |
5c67b5957e42c02a2fc4be811f0f0d5a6831d6d0 | subsection | 115 | 319 | Deformed commutation relations. | Not all the commutation relations get deformed under the q deformation, but we will list all of them for completeness: for j,k=1,2,3 we have the undeformed relations\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, \mat... | {
"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 | [
-0.029219981282949448,
0.02581734210252762,
-0.029265757650136948,
-0.03182918205857277,
-0.013457976281642914,
-0.005393871106207371,
0.009048276580870152,
0.007312624715268612,
0.0040167937986552715,
0.00047372799599543214,
0.005050555802881718,
-0.0018357854569330812,
-0.00655733002349734... | |
79f474ddb50411dadc01b8bd8eed6055f020ab8d | subsection | 116 | 319 | Deformed Serre relations. | The Serre relations (REF ) get deformed as well, but we will not need their explicit form here. By dropping the Serre relations (REF ) which do not get q deformed we obtain the centrally extended quantum group \mathfrak {psu}_q(2|2)_{\text{c.e.}} containing three central charges. | {
"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 | [
-0.01245026383548975,
0.041378818452358246,
-0.025419287383556366,
-0.030378032475709915,
-0.01293850876390934,
-0.02453434281051159,
0.05080806091427803,
-0.008262031711637974,
-0.019392505288124084,
0.018492303788661957,
0.02416815795004368,
0.02888278104364872,
-0.03912068158388138,
0.0... | |
57c33c7cd043c23fb1e51cfd6f46afb6c74c0efe | subsection | 117 | 319 | Braided coproduct. | Following our discussion in section REF we will straightforwardly introduce a non-standard coproduct, based on a new abelian generator \mathbf {U} that associates non-trivial charges \lbrace 2,1,-1,-2\rbrace to the generators \lbrace \mathbf {C},\mathbf {E}_2,\mathbf {F}_2,\mathbf {D}\rbrace respectively, whereas other... | {
"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 | [
-0.02636714093387127,
0.008392319083213806,
-0.05596913769841194,
-0.004287712275981903,
-0.02713007852435112,
0.019241299480199814,
0.023010212928056717,
0.046722330152988434,
-0.005836476571857929,
0.033538758754730225,
-0.01570126600563526,
0.006576526444405317,
0.0037326745223253965,
0... | |
9eab903ffa7bcd3c7ab76e3684010a0086318524 | subsection | 118 | 319 | Counit. | The counit \varepsilon : \mathfrak {h} \rightarrow is defined as
\begin{equation}
\varepsilon (1) = 1,\quad \varepsilon \left( \mathbf {U} \right) = 1, \quad \varepsilon \left(\mathbf {M}\right) = 0
\quad \text{ for all } \mathbf {M} \in \lbrace \mathbf {H}_j,\mathbf {E}_j,\mathbf {F}_j\rbrace _{j=1,2,3}.
\end{equation... | {
"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 | [
-0.0036293023731559515,
0.019239308312535286,
-0.05169133469462395,
-0.006320287939161062,
-0.02692892774939537,
-0.07189184427261353,
0.06523971259593964,
-0.005732886493206024,
0.0026623783633112907,
0.013777236454188824,
-0.039455074816942215,
-0.023068860173225403,
-0.007243347819894552,... | |
c9b439c70b8fa39ac5a6d7eb01972637d8c83ce9 | subsection | 119 | 319 | Counit. | Hermiticity is consistent with the identification
\begin{equation}
\mathbf {H}_j^{\dagger } = \mathbf {H}_j, \quad \mathbf {E}_j^{\dagger } = q^{-\mathbf {H}_j} \mathbf {F}_j,
\end{equation}
implying that the central charges are related as
\begin{equation}
\mathbf {K}^{\dagger } = \mathbf {K}, \quad \mathbf {C}^{\dagge... | {
"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 | [
-0.022344015538692474,
0.028448937460780144,
-0.02072620950639248,
0.006604762747883797,
-0.027624772861599922,
-0.07069499790668488,
0.05772203952074051,
0.013369779102504253,
-0.003411125158891082,
0.022847671061754227,
-0.014422878623008728,
-0.030448298901319504,
0.024618098512291908,
... | |
40957eed09578431cad41b85aad5941cf1a9fc91 | subsection | 120 | 319 | Fundamental representation | The representation theory of \mathfrak {h} for real q is structurally very similar to the representation theory of \mathfrak {su}(2|2) : in the following we will only need the structure of short multiplets, which are characterised by the central-charge eigenvalues satisfying the shortening condition\left[K \right]_q^2 ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 446,
"openalex_id": "",
"raw": "N. A. Ky and N. I. Stoilova, “Finite dimensional representations of the quantum superalgebra U-q(gl(2/2)). 2. Nontypical representations at generic q”, J. Math. Phys. 36, 5979 (1995), hep-th/9411098... | 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 | [
-0.07020628452301025,
0.01072172075510025,
-0.013682594522833824,
-0.016498476266860962,
-0.02843354642391205,
-0.038766078650951385,
0.06410139054059982,
-0.019321991130709648,
-0.012186895124614239,
0.05030433088541031,
0.003245132975280285,
0.015811676159501076,
0.03785034641623497,
0.0... | |
365b569cc06d5b50d74e04e59cee5aa28ab6e293 | subsection | 121 | 319 | Fundamental representation | The most general action of the Chevalley generators on this basis is specified by\begin{aligned}\mathbf {H}_1 \mathopen | \phi ^1 \mathclose \rangle &= -\mathopen | \phi ^1 \mathclose \rangle , \\
\mathbf {H}_1 \mathopen | \phi ^2 \mathclose \rangle &= +\mathopen | \phi ^2 \mathclose \rangle , \\
\mathbf {H}_3 \mathope... | {
"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 | [
0.01684163138270378,
0.014477090910077095,
-0.06388836354017258,
0.009282728657126427,
-0.02233346737921238,
-0.01621617190539837,
0.02254704013466835,
0.04060908034443855,
0.009191198274493217,
0.03179163113236427,
-0.014355050399899483,
-0.024469181895256042,
0.012349003925919533,
0.0235... | |
86334796c5cb5439ab9775181cea972a74498b8c | subsection | 122 | 319 | Fundamental representation | The other actions follow straightforwardly from others by using the relations between generators. All the basis states are eigenvectors of \mathbf {U} with eigenvalue U. Not all these parameters are independent: the closure of the algebra requires that these parameters are related to the central charges asad = \left[ K... | {
"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 | [
0.0286782905459404,
0.04627598449587822,
-0.05515877529978752,
-0.025488421320915222,
-0.014415457844734192,
-0.0027567942161113024,
0.06776562333106995,
0.005177812650799751,
0.00699786888435483,
-0.0050366343930363655,
-0.00998932495713234,
-0.03293653577566147,
-0.006219480186700821,
0.... | |
d722a854eb64a45a159b47d1457f543e4e5759a5 | subsection | 123 | 319 | Requiring cocommutativity. | Another constraint on the representation ultimately descends from the fact that we are interested in Hopf algebras that allow for a non-trivial R matrix. As we have seen in section REF , this at least requires us to consider only cocommutative Hopf algebras for which there exists an R matrix \mathbb {R} that satisfies ... | {
"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 | [
-0.016978994011878967,
-0.008314062841236591,
-0.03746667504310608,
0.007314849644899368,
-0.005224892869591713,
-0.02456080913543701,
0.038839638233184814,
0.02982383966445923,
0.025598160922527313,
0.009763303212821484,
-0.03115103952586651,
-0.02556765079498291,
0.008626793511211872,
0.... | |
57ee7ec06538364bea76f34f99faa89bc2f13aed | subsection | 124 | 319 | Solving the shortening condition. | For a fixed coupling constant h and deformation parameter q a fundamental short representation corresponds to a point on a torus, which uniformises the shortening condition (REF ):
this torus has real period 2\omega _1 ( \kappa ) = 4K(m) and imaginary period 2\omega _2 ( \kappa ) = 4 iK(1-m)-4K(m), where K(m) is the el... | {
"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 | [
0.0016318648122251034,
0.014077338390052319,
-0.041140925139188766,
-0.025728337466716766,
-0.012925208546221256,
-0.05206707492470741,
0.034975890070199966,
-0.0068326592445373535,
0.02531631849706173,
0.04596308246254921,
-0.017701584845781326,
0.001231290167197585,
0.042422764003276825,
... | |
0fdc7bcaa827bb8eba978e8d241d8f01035a7793 | subsection | 125 | 319 | Solving the shortening condition. | This is illustrated in fig. \ref {fig:xfunctions}.
\begin{}[!t]
\centering \begin{}{6cm}
\includegraphics [width=6cm]{Pictures/deformedxs.pdf}
{x_s}
\end{} \quad \begin{}{7cm}
\includegraphics [width=7cm]{Pictures/deformedxm.pdf}
{x_m}
\end{}
\end{}\caption {The analytic structure of the deformed x-functions: the branc... | {
"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 | [
-0.03803670406341553,
-0.015736693516373634,
-0.03483136370778084,
-0.028283311054110527,
-0.019384676590561867,
0.013286897912621498,
0.01487430464476347,
-0.012226083315908909,
0.010531831532716751,
0.030618630349636078,
0.006200806703418493,
-0.022773178294301033,
0.0011457182699814439,
... | |
39a9ca1866560751bde9c62df50b0f2e66cd3e8a | subsection | 126 | 319 | Undeformed limit. | We will often consider the relation between our \eta -deformed expressions and the corresponding quantity in the undeformed {\rm AdS}_5\times {\rm S}^5 superstring. The undeformed limit that takes us from \eta deformed to undeformed consists of a rescaling and a limit and its exact form depends on which conventions one... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 658,
"openalex_id": "",
"raw": "G. Arutyunov and S. Frolov, “String hypothesis for the {\\rm AdS}_5\\times {\\rm S}^5 mirror”, JHEP 0903, 152 (2009), arxiv:0901.1417.",
"source_ref_id": "22efe2c4381709ee58366f0aa2313abf3cd39... | 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 | [
-0.014561682939529419,
0.028558900579810143,
-0.027490992099046707,
-0.0049390727654099464,
-0.00903144758194685,
-0.003352466970682144,
0.011563913896679878,
0.014218427240848541,
0.04177044332027435,
0.03298309072852135,
-0.012875914573669434,
0.020473314449191093,
-0.02068689651787281,
... | |
24923751f05bd309f57f1dc422f22aeb76ac8a6e | subsection | 127 | 319 | On the | We are now in a good position to understand the construction of the S matrix: we exhibited the Hopf algebra structure present in \mathfrak {h} and restricted the central charges as to no longer obstruct cocommutativity. Moreover we have found an efficient description of the shortening condition in terms of x^{\pm } and... | {
"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 | [
-0.009038691408932209,
0.015041602775454521,
-0.02904585376381874,
0.02269970066845417,
0.014866168610751629,
-0.03362240642309189,
0.004813008010387421,
0.03383598104119301,
0.045033279806375504,
0.051593005657196045,
-0.008771725930273533,
-0.01780278980731964,
0.010892195627093315,
0.03... | |
2022b78e04026735eaf5591513dac49811077aed | subsection | 128 | 319 | On the | The most general ansatz now is\mathbb {S}_{12}(p_1,p_2)=\sum _{k=1}^{10}a_k(p_1,p_2)\mathbf {\Lambda }_k\, ,where the ten \mathfrak {su}(2) \times \mathfrak {su}(2)-invariants are given by Did the page get separated properly?\begin{aligned}\mathbf {\Lambda }_1=&\mathbb {E}_{1111}+\frac{q}{2}\mathbb {E}_{1122}+\frac{1}{... | {
"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 | [
-0.022780628874897957,
0.027144499123096466,
-0.04830774664878845,
-0.048460330814123154,
-0.01518962811678648,
0.009406736120581627,
0.027937930077314377,
0.014930237084627151,
0.02442852407693863,
0.03118794597685337,
-0.01644843816757202,
0.048460330814123154,
-0.007125621661543846,
0.0... | |
dc94dc546a75d266a2142b1d14184d18d029c4e1 | subsection | 129 | 319 | On the | \end{aligned}Up to an overall factor the unknown functions a_j are completely fixed by the cocommutativity requirement for the generators \mathbf {E}_2 and \mathbf {F}_2: we set a_1 =1 yielding\begin{aligned}a_2=&-q+\frac{2}{q}\frac{x^-_1(1-x^-_2x^+_1)(x^+_1-x^+_2)}{x^+_1(1-x^-_1x^-_2)(x^-_1-x^+_2)},\\
a_3=&\frac{U_2V_... | {
"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 | [
-0.02511545456945896,
0.031615566462278366,
-0.07360690087080002,
-0.009544177912175655,
-0.00841505080461502,
-0.04492095857858658,
0.010391023941338062,
0.0004417806339915842,
0.00882702972739935,
0.02236892841756344,
0.0029601447749882936,
0.00741943484172225,
-0.007472839672118425,
0.0... | |
b724977e4a7d05fc226ca1e4795ce39c82a47bf0 | subsection | 130 | 319 | Yang Baxter equation and crossing symmetry. | As we discussed in section REF for the R matrix to give rise to an S matrix that defines a consistent scattering theory it should satisfy both the Yang-Baxter equation and the crossing equation. We might need to expand on this. By checking every element it was shown that the S matrix indeed satisfies the Yang-Baxter eq... | {
"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 | [
-0.05162603035569191,
-0.0014845916302874684,
-0.05571461841464043,
0.004984871484339237,
0.014020189642906189,
-0.023906026035547256,
-0.015278803184628487,
0.032251011580228806,
0.005606550257652998,
0.04454728588461876,
-0.014149865135550499,
0.00824582390487194,
0.035088613629341125,
0... | |
20791d128949cddb3b47b7d939c1adbdb6117ee6 | subsection | 131 | 319 | Yang Baxter equation and crossing symmetry. | This solution is not the unique solution of the deformed crossing equation (\ref {eq:deformedcrossing}), but at present it is the only explicitly-known solution. It was constructed using knowledge of the dressing phase for the undeformed case, where the ``physical" solution was found by comparing with other data, comin... | {
"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 | [
0.006233445834368467,
0.03161738067865372,
-0.019821899011731148,
0.017227809876203537,
-0.015266982838511467,
-0.04400797560811043,
0.04171907156705856,
0.04370278865098953,
0.007248192559927702,
0.03531014174222946,
-0.03820941969752312,
-0.004665547050535679,
0.02432340942323208,
0.0032... | |
a0dd1a8bc0d49d1b93fb5674967af393ae8b33ff | subsection | 132 | 319 | Matching with the classical theory | In the previous section we have reviewed the bootstrapping of the \eta -deformed S matrix, which defines the (factorised) scattering theory of some integrable QFT with \mathfrak {psu}(2|2)^{\otimes 2} off-shell symmetry. Indeed, a priori this is the only thing we can be certain about and relating it to the classical \e... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1103,
"openalex_id": "",
"raw": "G. Arutyunov and S. Frolov, “Foundations of the {\\rm AdS}_5\\times {\\rm S}^5 Superstring. Part I”, J.Phys. A42, 254003 (2009), arxiv:0901.4937.",
"source_ref_id": "6d3f25e7c5baec1c90ad35c03... | 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 | [
0.003905738703906536,
0.03814198076725006,
0.005015670321881771,
-0.011663817800581455,
0.0037493566051125526,
-0.06255284696817398,
0.008543803356587887,
0.013708227314054966,
0.02186298370361328,
0.048180948942899704,
0.012571596540510654,
0.04271901771426201,
0.0048936158418655396,
-0.0... | |
02849098ce09c434069a6e1cd2c5782949714467 | subsection | 133 | 319 | Unitarity. | One subtle issue of the S matrix is that of unitarity. We saw that the ZF algebra directly implies that \mathbb {R} satisfies the braiding unitarity relation (REF ). However, for a physical system the property we really should impose is matrix unitarity of the R matrix, i.e. \mathbb {R}_{12}\mathbb {R}_{12}^{\dagger } ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1751-8113/41/25/255204",
"end": 844,
"openalex_id": "https://openalex.org/W2172007900",
"raw": "N. Beisert and P. Koroteev, “Quantum Deformations of the One-Dimensional Hubbard Model”, J. Phys. A41, 255204 (2008), arxiv:0802.0777.",... | 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 | [
-0.04177248850464821,
0.047722551971673965,
-0.02633284032344818,
0.015866832807660103,
-0.04018580541014671,
-0.05959216505289078,
-0.00285107153467834,
0.024425769224762917,
0.0006827315082773566,
0.006457343231886625,
-0.04116222634911537,
-0.025890398770570755,
0.009031889960169792,
0.... | |
fcfaf7b9c58129b392c68947f14bdc40ab06b5ea | subsection | 134 | 319 | Renormalisation of | The identification of the deformation parameter q with the deformation parameter \eta and the string tension g as in eqn. (REF ) and the unitarity bound (REF ) together pose an interesting question, as satisfying both simultaneously is not straightforward: the exact quantum theory described by the S matrix is parametri... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1093,
"openalex_id": "",
"raw": "G. Arutyunov, M. de Leeuw and S. J. van Tongeren, “The exact spectrum and mirror duality of the (\\text{AdS}_5{\\times }S^5)_\\eta superstring”, Theor. Math. Phys. 182, 23 (2015), arxiv:1403.6104."... | 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 | [
-0.014657909981906414,
0.029224256053566933,
-0.01713777519762516,
-0.01889275759458542,
0.004116576164960861,
-0.0520390160381794,
0.021059777587652206,
0.0003023527970071882,
0.008591778576374054,
0.02058669552206993,
-0.02212802693247795,
0.02560746856033802,
0.023165754973888397,
0.031... | |
0e8e9fa76e1b3696518bc426f4f38318464271e4 | subsection | 135 | 319 | Labelling of states | Thus far our discussion of the quantum theory has been mostly algebraic: we only used the classical action once to find the perturbative S-matrix to check the exact S-matrix. Moreover, the exact S-matrix was bootstrapped in such a way as to ensure that the quantum theory has \mathfrak {psu}_q(2|2)^{\otimes 2} on-shell ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1259,
"openalex_id": "",
"raw": "N. Gromov, V. Kazakov, S. Leurent and D. Volin, “Quantum spectral curve for arbitrary state/operator in AdS_{5}/CFT_{4}”, JHEP 1509, 187 (2015), arxiv:1405.4857.",
"source_ref_id": "e4006b448... | 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 | [
-0.034690383821725845,
0.05018969625234604,
-0.01783335953950882,
0.003129228251054883,
0.002212007762864232,
-0.048389580100774765,
0.05589514970779419,
-0.007143259979784489,
-0.02321845479309559,
0.056261274963617325,
0.010815955698490143,
0.03346996754407883,
-0.0068190861493349075,
0.... | |
c8545b89ec78f07766f0ac9a09087fe375148745 | subsection | 136 | 319 | Energy and dispersion. | Before we can start discussing bound states one particularly important identification that we have to make is which (combination) of the central charges in the fundamental representation corresponds to the world-sheet excitation energy and momentum. As in the undeformed case, we associate the world-sheet excitation ene... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1731,
"openalex_id": "",
"raw": "G. Arutyunov, M. de Leeuw and S. J. van Tongeren, “The exact spectrum and mirror duality of the (\\text{AdS}_5{\\times }S^5)_\\eta superstring”, Theor. Math. Phys. 182, 23 (2015), arxiv:1403.6104."... | 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 | [
-0.0502004474401474,
0.008437947370111942,
-0.0025081150233745575,
0.0030269038397818804,
-0.02127034030854702,
-0.021026205271482468,
0.030990002676844597,
0.025680046528577805,
0.04012984037399292,
0.027862010523676872,
-0.024474624544382095,
0.00938397366553545,
-0.0006184458616189659,
... | |
8ced56c33420061b48399e7102bfc748434c1b54 | subsection | 137 | 319 | Mirror duality | The relation between dispersion relations at different values of \theta was found to extend to other aspects of the theory. The mirror transformation (REF ) can be recast in the language of the x functions: all real momenta and energies satisfying eqn. (REF ) can be parametrised by real rapidities through the x_s funct... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2426,
"openalex_id": "",
"raw": "G. Arutyunov, M. de Leeuw and S. J. van Tongeren, “The exact spectrum and mirror duality of the (\\text{AdS}_5{\\times }S^5)_\\eta superstring”, Theor. Math. Phys. 182, 23 (2015), arxiv:1403.6104."... | 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 | [
-0.06142103299498558,
0.013950999826192856,
-0.02578034996986389,
0.007345644291490316,
-0.01996488869190216,
-0.001953750615939498,
0.003295046044513583,
0.0415782555937767,
0.061543144285678864,
0.01395863201469183,
0.016286343336105347,
0.020315952599048615,
0.0008905401336960495,
0.015... | |
cc2e1f402628d12bc71d54918f5bd0ba2569b5a8 | subsection | 138 | 319 | Spectrum and thermodynamics. | One important aspect of mirror duality that might not be immediately obvious is that for the models under consideration it provides a relation between thermodynamics and the spectral problem: we will use the thermodynamic Bethe ansatz on the mirror theory to obtain the spectrum, but the primary result of this method is... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 609,
"openalex_id": "",
"raw": "T. Harmark and M. Wilhelm, “The Hagedorn temperature of AdS5/CFT4 via integrability”, arxiv:1706.03074.",
"source_ref_id": "384328eb279687de3438e6eded5a7842b52c24b7",
"start": 529
}
... | 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 | [
-0.06437798589468002,
0.019496461376547813,
-0.025674916803836823,
0.00906936265528202,
-0.011563627980649471,
0.0007136688800528646,
-0.012723042629659176,
0.004801651928573847,
0.03658255934715271,
0.026910606771707535,
0.0028565824031829834,
0.023234045132994652,
0.0005196386482566595,
... | |
fa3a0e152de5379ed15590f1c7ba319d628c167d | subsection | 139 | 319 | Undeformed | The mirror duality of the the \eta -deformed theory does not have an undeformed analogue. Although the mirror transformation in that case can also be achieved by replacing x_s\rightarrow x_m, one cannot achieve this result by shifting the parameters a finite amount: the undeformed x functions have fundamentally differe... | {
"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 | [
-0.0530535914003849,
0.04050754755735397,
-0.020253773778676987,
-0.0022264651488512754,
-0.06169235333800316,
0.046734780073165894,
-0.009188222698867321,
0.04774212837219238,
0.01839170791208744,
0.000639130943454802,
0.016605956479907036,
-0.020879549905657768,
0.010767925530672073,
-0.... | |
7817c7a6e559438e4e8417f939fd9af079bfcb48 | subsection | 140 | 319 | TBA for the | We have now collected all the basic ingredients to consider the central question in this thesis:What is the simplest set of equations we can find that describe the spectrum of the \eta -deformation of the {\rm AdS}_5\times {\rm S}^5 superstring?At this point in the derivation we have found the fundamental two-body S-ma... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1539,
"openalex_id": "",
"raw": "A. B. Zamolodchikov, “Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models”, Nucl. Phys. B342, 695 (1990).",
"source_ref_id": "a308c23539ec801bc14f... | 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 | [
-0.025900864973664284,
0.03397485241293907,
-0.016270078718662262,
0.0056891851127147675,
0.0036268080584704876,
-0.05064176395535469,
0.02489352412521839,
-0.0010464513907209039,
0.06489715725183487,
0.036172688007354736,
0.012545970268547535,
0.023077258840203285,
-0.027320299297571182,
... | |
cfb3a27ff455d44804c3acb56a777415b1c7083b | subsection | 141 | 319 | Finding the spectrum from the mirror model | A first step to finding the spectrum of the \eta -deformed model is to find the ground-state energy E_0. One way to characterise E_0 is as the leading term in the low-temperature expansion of the (Euclidean) partition functionZ(\beta , L) = \sum _{j\ge 0} e^{-\beta E_j},where \beta = 1/T and L is the volume our theory ... | {
"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 | [
-0.03478739783167839,
0.029904955998063087,
0.005015945993363857,
0.029508257284760475,
0.022230368107557297,
-0.05132666975259781,
0.03031691163778305,
-0.015806905925273895,
0.039090048521757126,
0.04247724264860153,
0.0158831924200058,
0.044155582785606384,
0.0017326946835964918,
0.0152... | |
044038f0c25fd38bc9d84eb17dfa263db8886294 | subsection | 142 | 319 | Finding the spectrum from the mirror model | Fermionic fields, however, are anti-periodic, which means that in the mirror theory we should not compute the partition function Z but rather Witten's index
Z_W = \text{Tr}\left( (-1)^F e^{-\beta \tilde{H}}\right),
where F is the fermion number operator.
What should worry us much more is that generically we have no ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 859,
"openalex_id": "",
"raw": "S. J. van Tongeren, “Integrability of the {\\rm Ad}{{{\\rm S}}_{5}}\\times {{{\\rm S}}^{5}} superstring and its deformations”, J. Phys. A47, 433001 (2014), arxiv:1310.4854.",
"source_ref_id": ... | 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 | [
-0.05698386952280998,
0.02667589858174324,
0.02484459988772869,
0.035771340131759644,
-0.005615978501737118,
-0.04135679826140404,
0.03799942135810852,
0.007111538201570511,
0.020342661067843437,
0.041417840868234634,
-0.006565963849425316,
0.0200984887778759,
-0.013147188350558281,
0.0207... | |
91406ba22f4ecdf9c5a0dbcf4ee926ec5ab43398 | subsection | 143 | 319 | Asymptotic and thermodynamic Bethe ansatz | With Zamolodchikov's mirror trick we have turned the problem of finding the ground-state energy of a QFT in finite volume into finding the Helmholtz free energy density of the mirror theory in infinite volume but at finite temperature. To find out the quantisation conditions on the momenta of the scattering we employ t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2604,
"openalex_id": "",
"raw": "S. J. van Tongeren, “Integrability of the {\\rm Ad}{{{\\rm S}}_{5}}\\times {{{\\rm S}}^{5}} superstring and its deformations”, J. Phys. A47, 433001 (2014), arxiv:1310.4854.",
"source_ref_id":... | 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 | [
-0.032288290560245514,
0.01800575666129589,
-0.007114563137292862,
-0.010124423541128635,
0.01080345455557108,
0.0008306681411340833,
0.0018139274325221777,
-0.01175714936107397,
0.055878885090351105,
0.043335892260074615,
0.013138099573552608,
0.015533780679106712,
-0.010833973065018654,
... | |
54bca1ef8f504654983f0946fc80e6feb996c5ab | subsection | 144 | 319 | Asymptotic and thermodynamic Bethe ansatz | The trace is taken in the auxiliary space and the product of matrices is ordered left to right. Plugging in one of the momenta of the particles p^{\prime }=p_k the transfer matrix reproduces the right-hand side of the Bethe-Yang equations (REF ) with a minus sign. Moreover, by virtue of the Yang-Baxter equation the tra... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1225,
"openalex_id": "",
"raw": "L. Faddeev, E. Sklyanin and L. Takhtajan, “The Quantum Inverse Problem Method. 1”, Theor.Math.Phys. 40, 688 (1980).",
"source_ref_id": "7dbe7e397264e6674ebd0c8b4119e3ccf4372ce5",
"start... | 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 | [
-0.011593361385166645,
0.009022991172969341,
-0.024315550923347473,
0.02022736519575119,
0.011387426406145096,
0.03703773766756058,
0.021874843165278435,
0.021798569709062576,
0.023674864321947098,
0.05281081050634384,
-0.0440242663025856,
0.010380635038018227,
0.00721534201875329,
0.04283... | |
e55ce77e67450b32a2c1ed7c4ac3f2b52fe7465e | subsection | 145 | 319 | Matching of the excitation parameters. | The K counting excitation numbers can be matched with four out of the six quantum numbers specifying a multiplet in the \eta -deformed theory. The precise relation with the quantum numbers (REF ) is Check this!q_{\alpha } = \tilde{K}^{II}_{\alpha }-2\tilde{K}^{III}_{\alpha }, \quad s_{\alpha } = \tilde{K}^{I} - \tilde{... | {
"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 | [
-0.0010131013114005327,
0.015797512605786324,
-0.022757574915885925,
0.016133304685354233,
-0.017125418409705162,
-0.01614856906235218,
0.045637257397174835,
-0.029061317443847656,
0.006528875324875116,
0.018148060888051987,
-0.0064983488991856575,
0.028740787878632545,
-0.03321293368935585,... | |
d83ade5b20afac1da0798c54675e1fb75bab6b3d | subsection | 146 | 319 | Including bound states | In order to find the infinite-volume thermodynamics of the mirror model we wish to compute the mirror free energy f = e- T s, where e is the energy density and s is the entropy density. Considering the infinite-volume limit R\rightarrow \infty we note that the thermodynamic behaviour of the system will be dominated by ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2569,
"openalex_id": "",
"raw": "S. J. van Tongeren, “Integrability of the {\\rm Ad}{{{\\rm S}}_{5}}\\times {{{\\rm S}}^{5}} superstring and its deformations”, J. Phys. A47, 433001 (2014), arxiv:1310.4854.",
"source_ref_id":... | 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 | [
-0.03416862711310387,
0.017999544739723206,
-0.009747211821377277,
-0.0013347120257094502,
0.01902155391871929,
-0.018152084201574326,
0.036853305995464325,
-0.004919367376714945,
0.020455416291952133,
0.04594460129737854,
-0.011570046655833721,
0.0006840399582870305,
0.01095989253371954,
... | |
48f97f182d9847c82cdf2990f23aa84153b19d36 | subsection | 147 | 319 | From the Bethe-Yang equations to TBA | The true thermodynamics of our model we find only when taking the thermodynamic limit, i.e. taking the length to infinity (R\rightarrow \infty ) as well as the number of particles. We argued that we can use the knowledge of string solutions to take this limit properly. Note, however, that the analysis we followed in th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/0022-3719/15/31/015",
"end": 1102,
"openalex_id": "https://openalex.org/W2055175236",
"raw": "F. Woynarovich, “On the eigenstates of a Heisenberg chain with complex wavenumbers not forming strings”, Journal of Physics C: Solid State... | 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 | [
-0.030207062140107155,
0.021587369963526726,
-0.02317400462925434,
0.030847817659378052,
0.008039960637688637,
-0.032800596207380295,
0.029566306620836258,
0.008184893056750298,
0.031030891463160515,
0.04149657115340233,
-0.007803490851074457,
0.012365062721073627,
0.010755544528365135,
0.... | |
ca48fd2fc3ea4e5fa8782f341670b08048a1934e | subsection | 148 | 319 | Branch cuts and convolutions. | Branch cuts will play a prominent role in our further analysis. They will almost exclusively be of square-root typeThe only exception being the logarithmic branch cut responsible for the reconstruction of the driving term in section REF . and for the \eta -deformed case are located on one of the following line segments... | {
"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 | [
-0.03759657219052315,
-0.0027751121670007706,
-0.012084612622857094,
0.026519009843468666,
0.01301537174731493,
0.01754710078239441,
-0.013038258999586105,
0.011863366700708866,
0.033904049545526505,
0.029799554497003555,
-0.0068471841514110565,
-0.010734247975051403,
0.0020121948327869177,
... | |
6a8a57f371b7e61394e1ff6ed84b490fadc6b87a | subsection | 149 | 319 | TBA equations. | With these definitions we can introduce the TBA equations for the \eta -deformed model:\begin{aligned}\log Y_{Q} =&\, -J \tilde{E}_Q +\Lambda _{P} \star K^{PQ}_{\mathfrak {sl}(2)} + \sum _\alpha \Lambda ^{(\alpha )}_{M|vw}\star K^{MQ}_{vwx} + \sum _\alpha L^{(\alpha )}_{\beta }\, \hat{\star } \,K_{\beta }^{yQ}, \\
\log... | {
"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 | [
0.012221242301166058,
0.05105153098702431,
-0.0033738105557858944,
-0.03554993122816086,
-0.0009278455981984735,
-0.011694859713315964,
0.017683420330286026,
0.015036247670650482,
0.03201019763946533,
0.040554385632276535,
0.01103878766298294,
0.044490814208984375,
0.010283542796969414,
0.... | |
f4fa5b8108cdf4dbd3633d779b64c2f5f87f07b8 | subsection | 150 | 319 | TBA equations. | Repeated indices are summed over, M,N,\ldots \in \mathbb {N}, \beta =\pm , and \alpha = l,r distinguishes a so-called left and right set of Y functions. All Y functions are periodic with period 2\pi , and have branch cuts of square-root type on some \check{Z}_Ns as illustrated in fig. REF .
[Figure: The cut structures ... | {
"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 | [
-0.02059336006641388,
0.0038936706259846687,
-0.037739239633083344,
0.02558152936398983,
0.0011231008684262633,
0.031179873272776604,
-0.01618485525250435,
0.0028010783717036247,
0.021371331065893173,
0.0332239530980587,
-0.0010487359249964356,
0.012188218533992767,
0.023400159552693367,
0... | |
a4aefed1cb8ad31abf68e14c26db4e81b6e706ba | subsection | 151 | 319 | Ground-state energy. | A solution of the TBA equations is a set of functions\lbrace Y_Q, Y_{M|w}^{(\alpha )}, Y_{M|vw}^{(\alpha )}, Y_{\beta }^{(\alpha )}\rbrace ,which implicitly depend on the length J present in front of the driving term in the Y_Q TBA-equation. The energy corresponding to a solution of these equations is given byE(J) &=&-... | {
"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 | [
0.008637137711048126,
0.015542269684374332,
-0.040866173803806305,
-0.008980486541986465,
-0.003670020494610071,
-0.017793113365769386,
0.06024632602930069,
0.012940446846187115,
0.022859420627355576,
0.02273734100162983,
-0.021898044273257256,
0.011734909377992153,
0.02496529556810856,
0.... | |
6c4e7804a6bf89ff99c41a6d8b33ee68bedd6e00 | subsection | 152 | 319 | Undeformed TBA-equations. | Since we will want to compare our own analysis with that done for the undeformed case it will be useful to have access to the undeformed TBA-equations, i.e. the TBA equations for the undeformed {\rm AdS}_5\times {\rm S}^5 superstring as well. It is both lucky and unlucky that the undeformed TBA equations can be written... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1126-6708/2007/12/024",
"end": 981,
"openalex_id": "https://openalex.org/W3098196523",
"raw": "G. Arutyunov and S. Frolov, “On String S-matrix, Bound States and TBA”, JHEP 0712, 024 (2007), arxiv:0710.1568.",
"source_ref_id": ... | 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 | [
-0.003099540015682578,
0.037110552191734314,
-0.01182593684643507,
0.001894057379104197,
0.02986239828169346,
0.006962043698877096,
0.053651608526706696,
0.0007534266333095729,
0.04062018543481827,
0.0609455406665802,
-0.01263467874377966,
0.005912968423217535,
-0.002971743466332555,
0.038... | |
f33f7368410c27acefc8020ce8954d96992d0bcf | subsection | 153 | 319 | Using the TBA equations | The TBA equations are derived originally to compute the ground-state energy for the original model in finite volume. As these equations are extremely complicated, in most cases numerical approaches are the only viable way forward: forgetting about the current context of the mirror model, this usually yields the free en... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 546,
"openalex_id": "",
"raw": "R. Klabbers, “Thermodynamics of Inozemtsev's Elliptic Spin Chain”, Nucl. Phys. B907, 77 (2016), arxiv:1602.05133.",
"source_ref_id": "5ab41b8707591de5fb6c941b1b708c1dc5bbd204",
"start": ... | 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 | [
0.012428052723407745,
0.030669596046209335,
-0.03378232941031456,
0.014045454561710358,
0.023803267627954483,
-0.02323870360851288,
0.05056668817996979,
-0.0063093919306993484,
0.04531776160001755,
0.035796452313661575,
-0.020324328914284706,
0.02101096138358116,
-0.017516763880848885,
0.0... | |
0c33105148cf0b7b480089c5bc52be959f1ceccb | subsection | 154 | 319 | Analytic continuation. | It turns out, however, that in practice this is just the beginning: using other techniques it is possible to get access to the energies of other states as well. The most important mathematical tool for this purpose is analytic continuation. Since this will also prove to be a very important tool for us, let us spend som... | {
"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 | [
-0.013032637536525726,
0.02572954073548317,
-0.017168287187814713,
0.026370489969849586,
-0.003225730499252677,
-0.037602365016937256,
0.07196946442127228,
0.006047052331268787,
-0.0387316569685936,
0.00978210847824812,
0.0009304258273914456,
-0.010781684890389442,
-0.0010844635544344783,
... | |
95dd80ee75da7f1de6d1f275f907bc23d8465394 | subsection | 155 | 319 | Analytic continuation in the TBA. | This technique can be applied in more complicated settings as well. In particular, one can use analytic continuation to find excited-state energies from the ground-state TBA-equations , although mathematically the problem looks a little different. Consider a non-linear integral equation resembling the TBA equations for... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/s0550-3213(96)00516-0",
"end": 248,
"openalex_id": "https://openalex.org/W3102347813",
"raw": "P. Dorey and R. Tateo, “Excited states by analytic continuation of TBA equations”, Nucl.Phys. B482, 639 (1996), hep-th/9607167.",
"... | 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 | [
-0.006767272017896175,
-0.011077879928052425,
-0.0031890864484012127,
0.0390624962747097,
0.009384154342114925,
0.0017595288809388876,
0.0396728478372097,
0.01429748348891735,
0.011276244185864925,
0.0300750695168972,
-0.005168914329260588,
0.0025711057242006063,
0.004257201682776213,
0.01... | |
c0f183e59a24c4b33ef5d5ecab358931b5fc40a6 | subsection | 156 | 319 | Beyond the TBA | This completes our review of the derivation of the ground-state TBA equations all the way from the first definitions of the (\eta -deformed) model. These equations and all the related machinery we have introduced will form the basis of our analysis in the rest of this thesis. We will venture down one of the possible pa... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0378-4371(92)90149-k",
"end": 574,
"openalex_id": "https://openalex.org/W2054172906",
"raw": "A. Klümper and P. Pearce, “Conformal weights of RSOS lattice models and their fusion hierarchiess”, Physica A183, 304 (1992).",
"sou... | 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 | [
0.0011034257477149367,
0.03025885298848152,
-0.03347852826118469,
0.008453557267785072,
0.005443694535642862,
-0.007263345643877983,
0.0324103906750679,
0.021362781524658203,
0.030945513397455215,
0.05838142707943916,
0.019775832071900368,
-0.010742426849901676,
-0.016876596957445145,
0.01... | |
00a850e098b931cf3937cc399552db354ac0d5aa | subsection | 157 | 319 | Analytic | Our goal in this chapter is to derive the analytic Y-system, a set of functional equations for the Y functions featuring in the TBA equations (REF ) along with an additional set of analytic constraints. We will treat the derivation of the undeformed and deformed case simultaneously. To avoid needless repetition and unn... | {
"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 | [
-0.022060640156269073,
0.000929205329157412,
-0.025493312627077103,
0.010969295166432858,
0.017758358269929886,
0.00808585062623024,
-0.007826493121683598,
0.019039887934923172,
0.019345015287399292,
0.02801060490310192,
-0.00644198153167963,
0.005435064435005188,
-0.02441011369228363,
0.0... | |
69b49c76817f264760ce2941e570842f2369f020 | subsection | 158 | 319 | Setup | A lot of the model's properties is encoded in the analytic properties of the Y functions, so our first task is to derive some of them from the TBA equations. We will see that these properties, when interpreted correctly, are very similar for the undeformed and deformed Y-functions. The fundamental difference between th... | {
"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 | [
-0.01830001175403595,
0.011225737631320953,
-0.04044623300433159,
0.008295292966067791,
0.028556566685438156,
-0.02710660733282566,
0.0017466285498812795,
0.016086917370557785,
0.028083421289920807,
0.03705790638923645,
-0.022665152326226234,
-0.0020833625458180904,
0.005013806279748678,
0... | |
202d3819b0a20ee73c251e3db6fdb24f8a024ffb | subsection | 159 | 319 | Analyticity strips | The region around the real axis plays a special role in our analysis. In particular, the region with |\text{Im}(u)| < c we will call the physical strip. Generically, the Y functions are especially nicely behaved on a strip with |\text{Im}(u)| < cM for some M\in \mathbb {N}, which can be derived directly from the TBA eq... | {
"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 | [
-0.014186033047735691,
0.006165849510580301,
-0.02734951116144657,
-0.020863158628344536,
0.005295915529131889,
-0.03711719438433647,
-0.0189706701785326,
-0.015735125169157982,
0.030981868505477905,
0.04908260330557823,
-0.03479737043380737,
0.008500936441123486,
-0.0005413242033682764,
0... | |
48d827b35a5c0de00149ec6d12c2e48ea8b9f9c1 | subsection | 160 | 319 | Analyticity strips. | The solutions of the TBA equations we are interested in are Y functions which are analytic in a wider strip around the real axis. To write this succinctly we introduce some notation: the functions which are analytic in the strip\left\lbrace u \in | \, |\text{Im}(u)| < Mc\right\rbrace ,possibly with the exception of a f... | {
"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 | [
-0.00985702034085989,
-0.004375387914478779,
-0.02712969295680523,
0.0029658798594027758,
0.040374111384153366,
-0.04659959673881531,
-0.0011586957843974233,
0.029830455780029297,
0.04202203452587128,
0.043151166290044785,
-0.02436789683997631,
0.016326643526554108,
0.015914663672447205,
0... | |
49ced86cbd118aab49d017e1d29b7e13c775bd5d | subsection | 161 | 319 | Branch points and discontinuities | All the Y functions have branch cuts: the behaviour of these Ys around the branch cuts can be captured in discontinuities. We can therefore use these discontinuities to pick up exactly the solution to the Y-system equations that we are interested in. In order to understand how they arise from the TBA equations, we will... | {
"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 | [
0.006923980545252562,
0.00437565054744482,
-0.03344873711466789,
0.03906422108411789,
-0.0023079935926944017,
-0.0036183998454362154,
0.017426304519176483,
-0.0051691425032913685,
0.03454741835594177,
0.035524025559425354,
-0.029801728203892708,
-0.02424728311598301,
-0.012337274849414825,
... | |
17e60ee0c35827e40485c40b2deb68f3fa552c49 | subsection | 162 | 319 | Discontinuity from an incomplete interval | To understand the origin of branch points from an incomplete-interval convolution let us discuss some general theory. We will focus on the non-periodic case, it is quite straightforward to extend these results to the periodic case.Consider a function f : defined as
\begin{equation}
f(u) = \int _{[a,b]} dz g(z) K(z-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 | [
0.015476628206670284,
-0.006182260811328888,
-0.021876279264688492,
0.0043325405567884445,
0.016765711829066277,
-0.009923653677105904,
0.03063289262354374,
-0.020777888596057892,
0.013699371367692947,
0.05119720473885536,
-0.01064828597009182,
0.006937404163181782,
-0.03478236868977547,
0... | |
5fc80dccd84c7cef9b4ce86a61ec3d70cd212a67 | subsection | 163 | 319 | Discontinuity from an incomplete interval | At the left end point a of the integration contour the continuations are given by
\begin{eqnarray}f\Big (
u_{\hspace{-2.0pt}\begin{}
[->] (0.15,-0.03) arc (350:0:0.15cm);
\node at (0,0) {\scalebox {0.75}{a}};
\end{}
}
\Big ) = f(u)+2\pi i g(u)\mbox{Res}(K)(u), \\
f\Big (
u_{\hspace{-2.0pt}\begin{}
[<-] (0.15,-0.03) ... | {
"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 | [
-0.017973441630601883,
-0.013579255901277065,
-0.013548740185797215,
-0.0012053496902808547,
0.008086523041129112,
0.00878074299544096,
0.05404238402843475,
-0.003549296874552965,
0.0020883826073259115,
0.03117125667631626,
-0.0013064311351627111,
0.013884407468140125,
0.010520108975470066,
... | |
9080463cb001df085f7f2461cb401e13ec6f5bc9 | subsection | 164 | 319 | Discontinuity from an incomplete interval | \end{equation}Now we can continue f and end up with
\begin{equation}f\Big (
u_{\hspace{-2.0pt}\begin{}
[<-] (0.15,-0.03) arc (350:0:0.15cm);
\node at (0,0) {\scalebox {0.75}{b}};
\end{}
}
\Big ) = \int _{[a,b]} dz g(z) K\Big (z,
u_{\hspace{-2.0pt}\begin{}
[<-] (0.15,-0.03) arc (350:0:0.15cm);
\node at (0,0) {\scalebo... | {
"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 | [
-0.005554770585149527,
0.017717277631163597,
-0.035312470048666,
-0.011620763689279556,
0.01907544955611229,
0.021700231358408928,
0.02214278094470501,
-0.01979268528521061,
-0.00041250657523050904,
0.012635577470064163,
-0.0017978610703721642,
0.011780996806919575,
-0.014802548103034496,
... | |
6dca9503a545696567dffbb802831987fcbd4706 | subsection | 165 | 319 | Discontinuity from an incomplete interval | Similarly, for the point a we get
\begin{equation}f(u) = \int _{U} dz g(z) K(z,u)+2\pi i \mbox{Res}(K) g(u),
\end{equation}which after continuation becomes
\begin{equation}f\Big (
u_{\hspace{-2.0pt}\begin{}
[->] (0.15,-0.03) arc (350:0:0.15cm);
\node at (0,0) {\scalebox {0.75}{a}};
\end{}
}
\Big ) = \int _{[a,b]} dz 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 | [
0.004261225461959839,
0.037813130766153336,
-0.05029543116688728,
-0.002767698373645544,
0.013291055336594582,
-0.0026799559127539396,
0.02433895878493786,
0.010521450079977512,
0.008087554015219212,
0.013230017386376858,
-0.00915572140365839,
0.005504877306520939,
-0.015946215018630028,
0... | |
8915ac8562b4c89eda7d14066097f3503801f5a3 | subsection | 166 | 319 | Discontinuity from an incomplete interval | \end{eqnarray}Continuing f twice in the same direction leads to
\begin{equation}
f\Big (u_{
\begin{}
[<-] (0.15,-0.03) arc (350:0:0.15cm);
\node at (0,0) {\scalebox {0.75}{b}};
\end{}
\begin{}
[<-] (0.15,-0.03) arc (350:0:0.15cm);
\node at (0,0) {\scalebox {0.75}{b}};
\end{}
}
\Big ) = f(u) + (2\pi i) \text{Res}(K) \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 | [
-0.0006831080536358058,
0.02063911035656929,
-0.04295619949698448,
0.013248387724161148,
0.024849306792020798,
0.009045818820595741,
0.030783241614699364,
0.0013890976551920176,
0.005441982764750719,
0.03773921728134155,
-0.012401771731674671,
0.004233077168464661,
-0.026740843430161476,
0... | |
cb0471da568324cb8fc5a50a00a0b6a6ef264011 | subsection | 167 | 319 | Discontinuity from an incomplete interval | In turn, the integration domain becomes closed on the cylinder and the points - and , which are identified on the cylinder, are not special: we can follow the deformation from the previous paragraph to show that a continuation around those points is trivial. The other results from this section do carry over to the peri... | {
"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 | [
-0.014566748403012753,
0.03442910686135292,
-0.021533453837037086,
0.023456355556845665,
-0.020404130220413208,
-0.028355179354548454,
0.03220098465681076,
-0.0028881686739623547,
0.033421874046325684,
0.03583313152194023,
-0.013879997655749321,
-0.013078788295388222,
0.016329407691955566,
... | |
3b6ff8470c337f8d21142b8ee24751a7dbd44b61 | subsection | 168 | 319 | Discontinuities, continuations and short- and long-cutted functions | The analytic continuation of a function can be encoded in a lot of ways and can be subtle to define. In this thesis we will often talk about both analytic continuation and discontinuities or jump of functions, so we should make sure that our definitions are clear and consistent. Although the notation introduced in the ... | {
"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 | [
-0.04795160889625549,
-0.0008632281678728759,
-0.001561441458761692,
0.0437394380569458,
-0.011446119286119938,
-0.04593709111213684,
0.0018733482575044036,
0.03064507618546486,
-0.010034431703388691,
0.004952354356646538,
-0.006199981551617384,
-0.023212730884552002,
-0.02200707234442234,
... | |
1a1bd87f902a1d1cb0e06e2a8e8306a7ec11816f | subsection | 169 | 319 | Discontinuity. | For a function f: we define its discontinuity across its cut on ZN as
\begin{equation}
\left[ f \right]_N := \lim _{\epsilon \rightarrow 0^+} f(u+ i N c+i \epsilon ) - f(u+ i N c-i \epsilon ),
\end{equation}where u lies on the cut.\footnote {The undeformed versions of all these definitions follow directly by replacing ... | {
"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 | [
-0.05576596409082413,
-0.0022320121061056852,
-0.037116263061761856,
0.012667146511375904,
0.03479649871587753,
-0.003189679002389312,
0.044808123260736465,
0.029821209609508514,
0.021244177594780922,
0.027318302541971207,
-0.004036698956042528,
-0.028356093913316727,
-0.028523970395326614,
... | |
5dc93c5979104c95cf45f747a8017f9b3b93ebdf | subsection | 170 | 319 | Short- and long-cutted functions. | Basically all the branch points in the spectral problem are of square-root type and they come in pairs. This creates the freedom how to connect these branch points with a branch cut, which we have resolved by choosing them to be “long" or “short". Whenever we want to emphasise the choice of branch cuts we will write \h... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1191,
"openalex_id": "",
"raw": "A. Cavagli`a, D. Fioravanti and R. Tateo, “Extended Y-system for the AdS_5/CFT_4 correspondence”, Nucl.Phys. B843, 302 (2011), arxiv:1005.3016.",
"source_ref_id": "ed7f9a203b2f38f3ce07ed3c8c3... | 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 | [
0.001325220917351544,
-0.011982284486293793,
-0.0315459780395031,
0.004793676547706127,
0.002509339014068246,
-0.03911213204264641,
0.05003423988819122,
0.00822209008038044,
0.03670194372534752,
0.03337650001049042,
-0.006513603497296572,
-0.008046665228903294,
-0.008298361673951149,
0.023... | |
dde682d081283841a60130af80bfa0c32fc9a58f | subsection | 171 | 319 | Simplified TBA-equations | The (canonical) TBA-equations (REF ) can be simplified to the appropriately named simplified TBA-equations, by acting on both sides of the TBA equations with the operator(K+1)^{-1}_{PQ} :=\delta _{P,Q} - (\delta _{P,Q+1}+ \delta _{P,Q-1}) \mathbf {s},where \mathbf {s} is the doubly-periodic analogue of the undeformed k... | {
"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 | [
-0.026359334588050842,
0.050033923238515854,
-0.026038995012640953,
-0.00544576533138752,
0.014438143000006676,
0.01337034534662962,
0.0658373236656189,
0.006425850559026003,
0.024513570591807365,
0.06321358680725098,
-0.03911188244819641,
0.01028898824006319,
0.004057629033923149,
0.03566... | |
2e317bd34be26c78f9a6d8a4e59266a4d394bf9a | subsection | 172 | 319 | Simplified TBA-equations | This gives the set of simplified TBA equations presented in \begin{aligned}
\log Y_1 = \, & \sum _\alpha L_{-}^{\alpha } \, \hat{\star }\, \mathbf {s}- L_2 \star \mathbf {s}-\hat{\Delta }_e\,\check{\star }\, \mathbf {s}\, , \\
\log Y_Q = \, & -(L_{Q-1} + L_{Q+1})\star \mathbf {s}+ \sum _\alpha L_{Q-1|vw}^{(\alpha )} \s... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1636,
"openalex_id": "",
"raw": "G. Arutyunov, M. de Leeuw and S. J. van Tongeren, “The exact spectrum and mirror duality of the (\\text{AdS}_5{\\times }S^5)_\\eta superstring”, Theor. Math. Phys. 182, 23 (2015), arxiv:1403.6104."... | 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 | [
0.011116516776382923,
0.027222877368330956,
-0.05459835007786751,
0.010239098221063614,
0.015282349660992622,
-0.00991864874958992,
0.02969490922987461,
0.02145480178296566,
0.0191964004188776,
0.027940072119235992,
-0.028352078050374985,
0.02705502323806286,
-0.01821979507803917,
0.026276... | |
422f5b6c362a2eae5e23d108cbb842cff1976cf7 | subsection | 173 | 319 | Simplified TBA-equations | We refer to for more details on the simplified TBA-equations. | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 62,
"openalex_id": "",
"raw": "S. J. van Tongeren, “Integrability of the {\\rm Ad}{{{\\rm S}}_{5}}\\times {{{\\rm S}}^{5}} superstring and its deformations”, J. Phys. A47, 433001 (2014), arxiv:1310.4854.",
"source_ref_id": "... | 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 | [
0.009884608909487724,
0.03901980072259903,
-0.05110098794102669,
0.02669454738497734,
0.02699962630867958,
-0.013774385675787926,
0.0907919630408287,
0.02388780564069748,
0.004217738285660744,
0.05064336955547333,
-0.01799975149333477,
-0.002253782469779253,
-0.0015244493260979652,
0.04823... | |
97e83930e2a9f4e13f03123f8a455981c7bfdd79 | subsection | 174 | 319 | Deriving the | To get to the Y system we define the operator \mathbf {s}^{-1} by its action on a function f asf \circ \mathbf {s}^{-1}(u) = \lim _{\epsilon \rightarrow 0} f(u + i c- i\epsilon ) + f(u - i c+ i\epsilon )\, ,so that it acts as a right inverse of f\mapsto f\star \mathbf {s}:(f \star \mathbf {s}) \circ \mathbf {s}^{-1}(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 | [
-0.03689286857843399,
0.020857132971286774,
-0.05538507178425789,
0.03313949704170227,
0.04903791472315788,
-0.00191864266525954,
0.006247985642403364,
0.05047212541103363,
0.03927305340766907,
0.030484676361083984,
-0.03957820311188698,
-0.0063357166945934296,
-0.01768355444073677,
0.0402... | |
a4a3930dca05ad411495debf50e512971173ff90 | subsection | 175 | 319 | Deriving the | We can use the Y system to find the other expression: We have for |Re(u)|<\theta that\log Y_1(u +ic) Y_1(u-ic) = \sum _{\alpha }L_-^{(\alpha )}(u)-L_2(u).We can analytically continue this equation to approach the branch cut on \check{Z}_1 from above:\log Y_1(u +ic+i\epsilon ) Y_1(u-ic+i\epsilon ) = \sum _{\alpha }L_-^{... | {
"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 | [
-0.03255794197320938,
0.03390054032206535,
-0.04973706603050232,
-0.011068785563111305,
0.011251866817474365,
-0.012045218609273434,
0.0004348178918007761,
0.04296306148171425,
0.02229013852775097,
0.0005950139602646232,
-0.03161202371120453,
-0.00315433694049716,
-0.03051353618502617,
0.0... | |
e7f1be6b62a52541c8ca9c788b30751663377f27 | subsection | 176 | 319 | Deriving the | However, since our tools are more suited to analyse long-cutted functions we consider a new function that coincides with \hat{\Delta } on the upper half-planeThis is indeed an unfortunate choice to work in the upper-half-plane conventions, although in line with the work for the undeformed case. It is immaterial for the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 296,
"openalex_id": "",
"raw": "A. Cavagli`a, D. Fioravanti and R. Tateo, “Extended Y-system for the AdS_5/CFT_4 correspondence”, Nucl.Phys. B843, 302 (2011), arxiv:1005.3016.",
"source_ref_id": "ed7f9a203b2f38f3ce07ed3c8c3d... | 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 | [
-0.06788906455039978,
0.01912432536482811,
-0.05351147800683975,
-0.0028255311772227287,
-0.030891966074705124,
0.020070619881153107,
0.012225526385009289,
0.028404125943779945,
-0.008898229338228703,
0.03458557277917862,
0.0021825844887644053,
-0.011538699269294739,
-0.009325589053332806,
... | |
184736d2adeeb3d25e2b7e30113b9c7fb68c7033 | subsection | 177 | 319 | Deriving the | (\ref {eq:simplifieddressing}).}
\begin{align}
2 L_Q \star K_Q^{\Sigma } = \sum _\alpha \oint _{\gamma _x} \log Y^{(\alpha )}_-(z) \left(\sum _{N=1}^{\infty } K(z + 2 i N c, u) + K(z - 2 i N c, u)\right).
\end{align}
discussed in appendix \ref {app:dressing}. Putting together these ingredients we find the following dis... | {
"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 | [
-0.035213652998209,
0.06044908985495567,
-0.05358334258198738,
-0.0023210039362311363,
-0.004123262595385313,
-0.012518545612692833,
-0.0010899374028667808,
0.021680504083633423,
0.018857918679714203,
0.03237581253051758,
-0.019330892711877823,
0.02070404216647148,
-0.016798194497823715,
0... | |
c041af6f16906a8dbcff04c6328265a627ea392a | subsection | 178 | 319 | Deriving the | Since it does not help us in our current analysis we refrain from this rewriting.}
\begin{equation}
\begin{aligned}\frac{Y_1^+ Y_1^-}{Y_2} & = \frac{\prod _{\alpha }
\left(1-\frac{1}{Y^{(\alpha )}_{-}}\right)}{1+Y_2}\, , \, \, \, \, \, \mbox{for} \, \,
u\in \hat{Z}_0\, ,\\
\frac{Y_Q^+ Y_Q^-}{Y_{Q+1}Y_{Q-1}} & = \frac{... | {
"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 | [
-0.015060498379170895,
0.011428888887166977,
-0.03945942223072052,
0.017730800434947014,
0.0016222079284489155,
-0.013664312660694122,
-0.002443325472995639,
0.06414826214313507,
0.031677402555942535,
-0.008918805979192257,
-0.007190739270299673,
-0.0011406000703573227,
-0.026123175397515297... | |
640d0af0bb96e7b54ff34a128c0d5be87828cf2c | subsection | 179 | 319 | Deriving the | They come in the form of discontinuity equations:
\begin{equation}
\begin{aligned}\left[\log Y_{1|w}^{(\alpha )}\right]_{\pm 1}(u) &= \left[L_{-}^{(\alpha )}\right]_0(u), \\
\left[\log Y_{1|vw}^{(\alpha )}\right]_{\pm 1}(u) &= \left[\Lambda _{-}^{(\alpha )}\right]_0(u),\\
\left[\log \frac{Y_-}{Y_+} \right]_{\pm 2N}(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 | [
-0.025976654142141342,
-0.0037202169187366962,
-0.02162686176598072,
-0.009073513559997082,
0.018467538058757782,
-0.03263107314705849,
0.013820129446685314,
0.03543936088681221,
0.0426432266831398,
0.04917554557323456,
-0.014720612205564976,
0.005544077139347792,
-0.007814363576471806,
0.... | |
8aa737d01eb6e8344579944d1c1c7e8165806d33 | subsection | 180 | 319 | Deriving the | The resulting equations in the analytic Y-system make no reference to a specific state (apart from the J appearing in the logarithmic branch cut), however, which are characterised by zeroes, poles and asymptotic behaviour of the Y functions. We will continue our reduction of the TBA equations keeping this in mind, but ... | {
"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 | [
-0.013686231337487698,
0.01618850789964199,
-0.05395152047276497,
0.03188876807689667,
0.017348099499940872,
-0.031171651557087898,
0.01763799786567688,
0.011504368856549263,
0.008208687417209148,
0.03597785159945488,
-0.0440644770860672,
0.0025270702317357063,
0.0019262951100245118,
0.031... | |
7e82212438b01b31197afebc60db6eca862dc44b | subsection | 181 | 319 | Deriving the | The process of finding these expressions can be easily programmed, see appendix A of .Next we construct the \mathbf {Q}\omega system from the \mathbf {P}\mu system. Define the 4\times 4-matrix U asU^b_a = \delta ^b_a + \mathbf {P}_a \mathbf {P}^band consider the finite-difference equationX_a^- = U^b_a X_b^+for the unkn... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 86,
"openalex_id": "",
"raw": "N. Gromov, V. Kazakov, S. Leurent and D. Volin, “Quantum spectral curve for arbitrary state/operator in AdS_{5}/CFT_{4}”, JHEP 1509, 187 (2015), arxiv:1405.4857.",
"source_ref_id": "e4006b448c8... | 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 | [
0.011088217608630657,
0.04355921968817711,
-0.048870574682950974,
0.013018925674259663,
0.005960012320429087,
-0.028418803587555885,
0.02779304049909115,
0.02677045203745365,
0.017475580796599388,
0.01334706973284483,
0.006154609378427267,
-0.0024038462433964014,
0.029227716848254204,
0.02... | |
3bbde5f57a84742fb775cac95fa9b44b2e720f2d | subsection | 182 | 319 | Deriving the | We can find that\left(\mathcal {Q}^{a|i}\right)^- = V_b^a\left(\mathcal {Q}^{a|i}\right)^+where V = U^{-1} is simply given by V_b^a = \delta _b^a -\mathbf {P}_b \mathbf {P}^a. This shows directly that \mathbf {Q}^i also has only one \check{Z}_0 cut and we can write\mathbf {P}_a = -\mathbf {Q}^i \mathcal {Q}_{a|i}^{\pm ... | {
"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 | [
-0.05689303204417229,
0.03032362088561058,
-0.04105210676789284,
-0.02452443726360798,
0.015390724875032902,
-0.012254588305950165,
0.038915567100048065,
0.055092234164476395,
0.015398356132209301,
0.05368822067975998,
-0.023028859868645668,
-0.0027011982165277004,
-0.00212509511038661,
0.... | |
0b121f30bba8f2192a1e7a6b25cf51a34cc5e792 | subsection | 183 | 319 | Q particles | \frac{Y_1^+ Y_1^-}{Y_2} & = \frac{\prod _{\alpha }
\left(1-\frac{1}{Y^{(\alpha )}_{-}}\right)}{1+Y_2}\, , \, \, \, \, \, \mbox{for} \, \,
u\in \hat{Z}_0\, ,\\
\frac{Y_Q^+ Y_Q^-}{Y_{Q+1}Y_{Q-1}} & = \frac{\prod _{\alpha } \Bigg (
1+\frac{1}{Y_{Q-1|vw}^{(\alpha )}}\Bigg )}{(1+Y_{Q-1})(1+Y_{Q+1})} \, . | {
"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 | [
-0.02498278208076954,
-0.0014097645180299878,
-0.027790864929556847,
0.031468842178583145,
-0.0055055213160812855,
0.003550164634361863,
-0.019442923367023468,
0.06617308408021927,
0.014078568667173386,
-0.0014593637315556407,
-0.017474211752414703,
0.011560450308024883,
-0.04132765531539917... | |
bf8ec061ccf33ac4c01c4c64de12e5b69f705edd | subsection | 184 | 319 | Discontinuity equations | Our goal is to find enough equations such that when imposed on the Y-system equations we have enough information to rederive the TBA equations from them, thereby showing equivalence of the two systems. Naturally, the discontinuities of the Y functions are related through the Y-system equations. This allows us to only i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 521,
"openalex_id": "",
"raw": "A. Cavagli`a, D. Fioravanti and R. Tateo, “Extended Y-system for the AdS_5/CFT_4 correspondence”, Nucl.Phys. B843, 302 (2011), arxiv:1005.3016.",
"source_ref_id": "ed7f9a203b2f38f3ce07ed3c8c3d... | 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 | [
-0.009771034121513367,
0.036704305559396744,
-0.022791452705860138,
0.02720024064183235,
-0.010175300762057304,
0.026467984542250633,
0.050464607775211334,
0.014736642129719257,
0.010693981312215328,
0.03871801123023033,
-0.013233992271125317,
-0.014988354407250881,
0.011525396257638931,
0... | |
9ed6ceb339de99f350306b98a2d318f8ff7958f2 | subsection | 185 | 319 | A local equation for | Finding a local equation can be done by analysing the non-local one we just derived: we simply compute the Z_{2N} discontinuity of the second and third expression in the non-local equation (REF ): for N\in \mathbb {N} we find\left[ \log Y_-/Y_+ \right]_{\pm 2N} = \left[ -\Lambda _P \star K_{Py} \right]_{\pm 2N} = - \su... | {
"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 | [
-0.037011902779340744,
0.04464007541537285,
0.009527589194476604,
-0.0020348154939711094,
-0.02363208308815956,
0.01347135566174984,
0.039910607039928436,
-0.008657977916300297,
0.02299131639301777,
0.02770552784204483,
-0.014928337186574936,
0.04982723295688629,
-0.04479263722896576,
0.03... | |
85f6f9b33f754b1762120b6189b40e8b5c6ae7d5 | subsection | 186 | 319 | Showing it is equivalent to the non-local equation | To make sure that the local discontinuity equation (REF ) for Y_- can be imposed on the Y-system equations instead of the non-local equation (REF ) we consider whether they are equivalent while assuming the Y-system equations. We have already shown that the local equation can be derived from the non-local one in the pr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 480,
"openalex_id": "",
"raw": "A. Cavagli`a, D. Fioravanti and R. Tateo, “Extended Y-system for the AdS_5/CFT_4 correspondence”, Nucl.Phys. B843, 302 (2011), arxiv:1005.3016.",
"source_ref_id": "ed7f9a203b2f38f3ce07ed3c8c3d... | 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 | [
-0.04602283239364624,
0.03601256012916565,
0.01966957189142704,
-0.009567744098603725,
-0.012848548591136932,
0.008980250917375088,
0.04745722934603691,
0.039125509560108185,
0.04455791413784027,
0.015732605010271072,
-0.01727382093667984,
0.008240162394940853,
-0.03534113988280296,
0.0310... | |
55f9b2ee7a2b41bbae48e14970118315ee28b03e | subsection | 187 | 319 | Two approaches. | We can treat both the deformed and undeformed case simultaneously. The algebraic details of the deformed and undeformed derivations are almost identical, the crucial differences sit in the analytical properties of the functions involved: the basic idea of this derivation is to consider the function F(u)=\log \frac{Y_-}... | {
"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 | [
-0.029574278742074966,
0.03226007521152496,
-0.009934394620358944,
-0.020723359659314156,
0.0313139446079731,
-0.002168856794014573,
0.03103925846517086,
-0.01500841323286295,
-0.0018808203749358654,
0.06970862299203873,
-0.004398754332214594,
-0.019121037796139717,
-0.009163754060864449,
... | |
1a0eec4ba3efa522ae6845cbece11c46b0f563f1 | subsection | 188 | 319 | Two approaches. | REF and REF .
[Figure: Undeformed case: the integration contours \gamma (in (a)) and \Gamma (in (b)) are indicated by the dashed lines, which extend to \pm \infty . The dots in (b) indicate that the contour continues to \pm \i \infty . The solid lines are the branch cuts connecting 2+ i N/g to -2+ i N/g for N\in \mathb... | {
"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 | [
-0.020716097205877304,
-0.004892995581030846,
-0.0066015394404530525,
-0.013920057564973831,
0.002068940084427595,
-0.013149687089025974,
0.03676420822739601,
0.02658921666443348,
-0.002505610231310129,
0.05067663639783859,
-0.00648712832480669,
-0.000508177443407476,
-0.01156318187713623,
... | |
e794fba0b2ca8b4a4cd33aa15f7bf1822be857c1 | subsection | 189 | 319 | Rewriting the result. | With these definitions we can proceed. First, we define the functionB(z,u) =P(z-u)G(z) = G(u) 2\pi i K(z,u),where B is 2\pi -periodic for the deformed case. A simple rewriting shows that&-&\sum _{Q=1}^{\infty } \int _{Z_0} \frac{dz}{2\pi i} \Lambda _Q(z)\left(B(z-iQ c,u) -B(z+iQc,u) \right) \\
&=& \sum _{Q=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 | [
-0.043767087161540985,
0.020159106701612473,
-0.025897052139043808,
-0.0060736751183867455,
0.017107009887695312,
0.010018511675298214,
0.03134504705667496,
0.0016834228299558163,
-0.007988866418600082,
0.06873324513435364,
0.0030921567231416702,
0.007172430399805307,
0.024157356470823288,
... | |
a1fb81fd7df2605f85a255ad216782001b2a1cf7 | subsection | 190 | 319 | Rewriting a contour integral. | We can write the left-hand side of eqn. (REF ) asG(u) \log \frac{Y_-}{Y_+}(u) = \oint _{\gamma } \frac{dz}{2\pi i}\log \frac{Y_-}{Y_+}(z)B(z,u),since \log \frac{Y_-}{Y_+}(z)G(z) is analytic on the inside of \gamma and the integrand has a first-order pole with residue 1. We can deform \gamma into \Gamma (see fig. REF an... | {
"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 | [
-0.0264928936958313,
0.04099073261022568,
0.015215100720524788,
-0.005776244215667248,
0.012697055004537106,
0.009667769074440002,
0.009423594921827316,
0.010652096010744572,
0.027286458760499954,
0.04642360657453537,
-0.02646237052977085,
0.04276099428534508,
-0.0056732334196567535,
0.006... | |
0c3c6a7ecacec1c9995a0fe9c71b530e89748771 | subsection | 191 | 319 | Rewriting a contour integral. | This shows that& &\oint _{\gamma }\frac{dz}{2\pi i}\log \left(Y_-/Y_+\right)(z) B(z,u) \\&=& -\sum _{N,\tau }\left(\int _{Z_{-2N\tau }+i\epsilon }- \int _{Z_{-2N\tau } -i\epsilon } \right)\frac{dz}{2\pi i} \sum _{P=1}^{N} \Lambda _P (z+i \tau P c)B(z,u).Using the fact that \Lambda _P and B do not have any poles except ... | {
"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 | [
-0.04438268393278122,
0.03833882510662079,
-0.01157643273472786,
0.010782795026898384,
-0.008752912282943726,
0.015353845432400703,
0.017978955060243607,
0.007470882032066584,
0.004914451390504837,
0.03653787821531296,
-0.05286850780248642,
0.046427831053733826,
-0.009867058135569096,
0.00... | |
6e8028f381bcac9ea874951aae8de0e395c1bba8 | subsection | 192 | 319 | Rewriting a contour integral. | After recognising the left-hand side as G(u)\log \left(Y_-/Y_+\right)(u) we see that equation (REF ) is equivalent to\log \frac{Y_-}{Y_+}(u) = -\Lambda _P \star K_{Py},which is what we were after. In conclusion, this proves that the local discontinuities (REF ) for Y_- contain exactly the same amount of information as ... | {
"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 | [
-0.06742698699235916,
-0.009175866842269897,
-0.008779237046837807,
0.0028545870445668697,
-0.011845487169921398,
-0.0008738239412195981,
0.029106490314006805,
0.05531453341245651,
0.05720615014433861,
0.013637946918606758,
-0.01022083219140768,
0.03118116594851017,
-0.015315993689000607,
... | |
49ebab9bb726bc9e97b42b7474132d1a5b875b17 | subsection | 193 | 319 | Asymptotics | The undeformed and deformed case are different when it comes to the final pieces of analyticity data. The natural direction for asymptotics in the undeformed case is to consider u\rightarrow \infty , usually just above the real axis.The direction u\rightarrow -\infty would be natural as well, but lead to additional min... | {
"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 | [
-0.03186860308051109,
-0.001303056487813592,
-0.030998626723885536,
0.02695400081574917,
-0.011683627963066101,
-0.019292104989290237,
-0.001262037898413837,
0.010172616690397263,
0.03882841393351555,
0.040537841618061066,
0.0031384010799229145,
0.006303512025624514,
0.02321462891995907,
0... | |
a6c3ac7f83dbb33b547f0f91e259cad52de3fa5b | subsection | 194 | 319 | Other assumptions | To make the entire rederivation as clear as possible we list the other assumptions we make: we assume the Y functions haveno poles other than possibly at branch points.
the analyticity strips as we derived at the beginning of this chapter: Y_Q, Y_{Q|(v)w}^{(\alpha )} \in \mathcal {A}^{\text{p}}_Q and Y_-^{(\alpha )} a... | {
"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 | [
-0.030634067952632904,
-0.007776751182973385,
-0.010419854894280434,
0.009161233901977539,
0.005526489578187466,
-0.02152623049914837,
0.007826332934200764,
0.021861862391233444,
0.044883180409669876,
-0.002313192468136549,
-0.006960554514080286,
-0.021831350401043892,
-0.008505226112902164,... | |
1c818a5f80d4df3aa9ff478ea7ec7666df7b4025 | subsection | 195 | 319 | General strategy | To rederive the TBA equations (REF ) we will follow (almost) the same strategy for all four sets of equations:We write the left-hand side of the TBA equations as a contour integral,
we deform the contour using our knowledge of the branch cut structure of the integrand to yield a sum over discontinuities,
We use the Y... | {
"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 | [
-0.011977077461779118,
0.023694777861237526,
0.0065988353453576565,
0.012114393524825573,
-0.003627452300861478,
0.04253769665956497,
0.047786250710487366,
0.04497888311743736,
0.05004434660077095,
0.04308696463704109,
-0.04821345582604408,
0.03927260637283325,
-0.0017908400623127818,
0.04... | |
95dcf7cc412a8435f35ec8e16408c26fe71cba2a | subsection | 196 | 319 | Deriving discontinuities from the | Seeing the details of a computation at least once can help in understanding the concepts involved, which is why we will show a detailed derivation of the discontinuities (REF ). Those who are more interested in the general approach can skip this section and continue with section REF .The Y-system equation for Y_{\pm } ... | {
"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 | [
-0.059880033135414124,
-0.007370554376393557,
-0.01229188684374094,
-0.008186961524188519,
-0.020814567804336548,
-0.022874660789966583,
0.004780178423970938,
0.018052516505122185,
0.012185067869722843,
0.02040254883468151,
-0.014435909688472748,
0.025453591719269753,
-0.016450222581624985,
... | |
bed6e4bd5859d056dbed8186c3a1f9c84dd7b3e2 | subsection | 197 | 319 | Deriving discontinuities from the | Using the Y-system equationsY_{1|vw}^+Y_{1|vw}^- = \frac{1+Y_{2|vw}}{1+Y_2}\frac{1-Y_-}{1-Y_+}, \quad Y_{1|w}^+Y_{1|w}^- = (1+Y_{2|w})\frac{1-Y^{-1}_-}{1-Y^{-1}_+}we immediately see that\left[\log Y_{1|vw}/Y_{1|w}\right]_{\pm (2N-1)} &=& -\left[\log Y_{1|vw}/Y_{1|w}\right]_{\pm (2N-3)}- \left[\Lambda _2+L_{2|vw} - L_{2... | {
"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 | [
-0.035606879740953445,
0.04784197360277176,
-0.0017429674044251442,
0.011823192238807678,
-0.010213712230324745,
-0.009107671678066254,
0.012242725118994713,
0.02683483250439167,
0.014492945745587349,
-0.0005787643021903932,
-0.00985520239919424,
0.013623368926346302,
-0.013341137208044529,
... | |
e13d17fe89e155081cd93db8773d91d78106249c | subsection | 198 | 319 | Deriving discontinuities from the | Note that we introduced the infinite sums here only to have convenient notation, all of these sums have a finite number of non-zero terms.If we define A_{1J} = \delta _{2,J} we can rewrite equation (REF ) as& &\left[\log Y_{1|vw}/Y_{1|w}\right]_{\pm (2N-1)}= -\left[\log Y_{1|vw}/Y_{1|w}\right]_{\pm (2N-3)}- \left[\Lamb... | {
"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 | [
-0.05547187104821205,
0.05266471207141876,
-0.0008171650697477162,
-0.011060810647904873,
-0.043846577405929565,
0.023769300431013107,
0.03191616013646126,
0.009489412419497967,
0.008162115700542927,
0.015744490548968315,
-0.015286802314221859,
0.05275624990463257,
-0.0077997781336307526,
... | |
91362a15b8b54cbee768cfc13fe169104b9d06fd | subsection | 199 | 319 | Deriving discontinuities from the | (REF ) one last time to the final term here gives& &\left[\log Y_{N|vw}/Y_{N|w} \right]_{\pm N} =-\left[\log Y_{N|vw}/Y_{N|w} \right]_{\pm (N-2)} +\sum _{J=1}^{\infty } A_{NJ}\left[L_{J|vw} - L_{J|w}\right]_{\pm (N-1)} \\
&-&\left[\Lambda _{N+1} \right]_{\pm (N-1)} + \left[\log Y_{N+1|vw}/Y_{N+1|w} \right]_{\pm (N-1)}+... | {
"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 | [
-0.0266733355820179,
0.043458618223667145,
0.006672149058431387,
-0.0329601876437664,
-0.028184011578559875,
-0.0019522425718605518,
0.0016670834738761187,
0.02267538756132126,
0.0020371226128190756,
0.031525809317827225,
0.026978522539138794,
0.03114432469010353,
-0.013779189437627792,
0.... | |
d260c0bf4b50c640d1918705a5c81e8fce79161c | subsection | 200 | 319 | Deriving discontinuities from the | \\Plugging in the assumed discontinuity relations we get for the original relations (REF )& &\left[\log Y_- Y_+ \right]_{\pm 2N} +\left[\log Y_- Y_+\right]_{\pm (2N-2)} \\
&=& 2\left(\left[ L_{1|vw} - L_{1|w}-\Lambda _1\right]_{\pm (2N-1)}\right) - \left[\log Y_-/Y_+\right]_{\pm (2N-2)} -\sum _{Q=2}^{N}\left[\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 | [
-0.05759976804256439,
0.014773668721318245,
-0.014781296253204346,
0.008275390602648258,
-0.04475575312972069,
-0.01273723691701889,
0.008641490712761879,
0.04539642855525017,
0.0018047596095129848,
0.005502943880856037,
-0.009015217423439026,
0.03453545272350311,
-0.03749476373195648,
0.0... | |
3d1c1708869a5b7c9ec6a583c42db30336821f0c | subsection | 201 | 319 | Rederiving the | With the result for the discontinuities (REF ) in hand we can finally rederive the Y_- TBA-equation. Plugging in the discontinuities (REF ) in the integral expression (REF ) yields\log Y_- Y_+(u) = \sum _{N=1}^{\infty }\sum _{\tau } \int \frac{dz}{2\pi i}H(z+2i \tau N c-u) \sum _{J=1}^N \left[ \left( 2L_{J|vw} - 2L_{J|... | {
"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 | [
-0.039396967738866806,
0.042387597262859344,
-0.020522432401776314,
0.02491682767868042,
0.01133692730218172,
0.03323260694742203,
0.022780662402510643,
0.02107173204421997,
0.010619786567986012,
0.045530810952186584,
-0.033049508929252625,
0.015075214207172394,
-0.010772369801998138,
0.03... | |
45b2824a1ac361a82fb76b9a9bc39e3f411248bc | subsection | 202 | 319 | Rederiving the | Plugging this into our partial result (REF ) we find, employing the kernel identity (REF ) that\log Y_- &=& (L_{J|vw} - L_{J|w})\star K_J -1/2\sum _{J=1}^{\infty } \Lambda _J\star K_J -1/2\Lambda _Q\star \left(K_-^{Qy} -K_+^{Qy} \right) \\
&=& (L_{J|vw} - L_{J|w})\star K_J -\Lambda _J\star K_-^{Qy},which is precisely t... | {
"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 | [
0.0027504446916282177,
0.0687878206372261,
-0.015526087954640388,
0.011070444248616695,
-0.017364803701639175,
0.02652023732662201,
0.0415351465344429,
0.01643400266766548,
0.020447133108973503,
0.03668276593089104,
-0.0009622741490602493,
0.012321686372160912,
-0.017196955159306526,
0.020... | |
6f8614bdd7a7bf572a1dd667268ada27b1513f63 | subsection | 203 | 319 | Checking the assumptions. | First we note that for the application of this lemma we check that there areno poles and zeroes in the strip with |Im(u)<(a-1)c, and 1+Y_{a,0} \in \mathcal {A}_a, for a\ge 2, where we use the notation introduced just below (REF ).
no poles and zeroes in the physical strip for 1+Y_1.
that 1+Y_a does not attain negativ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 601,
"openalex_id": "",
"raw": "J. Balog and A. Hegedus, “{\\rm AdS}_5\\times {\\rm S}^5 mirror TBA equations from Y-system and discontinuity relations”, JHEP 1108, 095 (2011), arxiv:1104.4054.",
"source_ref_id": "ddd4bb06b7... | 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 | [
-0.026933744549751282,
0.012879365123808384,
-0.014741074293851852,
0.0016394867561757565,
0.0098960530012846,
-0.05432528629899025,
-0.00394087191671133,
0.016923241317272186,
0.013596581295132637,
0.01577874831855297,
-0.02701004408299923,
0.003227471373975277,
0.01333716232329607,
0.028... | |
2db0ce27482212f496122e402cc75cd931eafd78 | subsection | 204 | 319 | Applying the chain lemma. | Assuming the above requirements are met, we can follow the chain lemma and write down the suspected solution\sigma _a = {T}_{a,0} = \exp \left(\sum _{j =1}^{\infty } \Lambda _j \star l_a^j \right),withl_a^j = \sum _{n=0}^{j-1} K_{a+1-j+2n},using the convention that we expand K_N to non-positive integers by demanding K_... | {
"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 | [
-0.03552253171801567,
0.05801403522491455,
-0.031189028173685074,
0.0012321496615186334,
-0.010581991635262966,
0.016311679035425186,
0.02455144189298153,
0.013687161728739738,
0.00159740773960948,
0.0008988398476503789,
-0.017181431874632835,
0.02359013631939888,
-0.029953064396977425,
0.... | |
05c29f6504f7bda24c4eb87401579c963c12f70a | subsection | 205 | 319 | Applying the chain lemma. | Only if j=a do we get a contribution from the pole of K_1: in order to make use of the identity (REF ) we must push the contours past the pole, leading to a Cauchy integral. For small positive imaginary part the pole with negative residue is picked up using a clockwise contour, whereas for small negative imaginary part... | {
"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 | [
-0.044467248022556305,
0.051334187388420105,
0.019731005653738976,
0.016709553077816963,
-0.016267016530036926,
-0.014145894907414913,
0.028352828696370125,
0.0004825932264793664,
0.02958887815475464,
0.02719307877123356,
0.00992654263973236,
0.020524518564343452,
0.015885518863797188,
0.0... | |
e214c1903646aa1527a21d54f7193605aa77b857 | subsection | 206 | 319 | Extending the solution. | The solution (REF ) solves eqn. () at least in a neighbourhood of the physical strip, but we can in most cases extend this: the solution already implies that the {T}_{a,0} do not have a cut at Z_{\pm 1}. For the rest we use induction: suppose we have shown that for all a\le k {T}_{a,0} has no cuts until possibly Z_{\pm... | {
"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 | [
-0.03493392840027809,
0.029243815690279007,
-0.007024922873824835,
0.006338448263704777,
0.017817828804254532,
-0.017848338931798935,
-0.015163459815084934,
0.01546856015920639,
0.02715388312935829,
0.009450466372072697,
-0.009122484363615513,
-0.002591441385447979,
0.008306342177093029,
0... | |
effbc98c712917d2bcc1a00a89aa7fc779c41f6c | subsection | 207 | 319 | Use | From the {T} gauge we can construct a gauge with even better properties, known as the \textbf {T} gauge which will be our final gauge for T functions in the upper band. In addition to the analyticity properties listed in table the \textbf {T}s can be made to also satisfy the following identities, which were dubbed “gro... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/jhep07(2012)023",
"end": 485,
"openalex_id": "https://openalex.org/W2092559872",
"raw": "N. Gromov, V. Kazakov, S. Leurent and D. Volin, “Solving the AdS/CFT Y-system”, JHEP 1207, 023 (2012), arxiv:1110.0562.",
"source_ref_id"... | 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 | [
-0.012194006703794003,
0.027470853179693222,
-0.03290397673845291,
0.03659728169441223,
0.040901049971580505,
-0.0330260694026947,
-0.015124230645596981,
0.02962273731827736,
0.0017340976046398282,
0.02895122766494751,
-0.02815762534737587,
-0.00792839378118515,
-0.01182772871106863,
0.024... | |
824861690921c6c3196916b14f052a8e8372276e | subsection | 208 | 319 | Use | On f_1,f_2 we will impose that\mathbf {B} = \frac{\left(f_1 f_2\right)^-}{\left(f_1 f_2\right)^+}.If we can find f_1,f_2 satisfying this constraint, then the \textbf {T}s satisfy the identities\frac{\textbf {T}_{3,\pm 2}\textbf {T}_{0,\pm 1}}{\textbf {T}_{2,\pm 3}\textbf {T}_{0,0}^-} = 1 =
\frac{\textbf {T}_{3,\pm 2}\t... | {
"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 | [
-0.012657301500439644,
0.025604521855711937,
-0.006965711712837219,
0.023193608969449997,
0.012008795514702797,
-0.008804415352642536,
0.013916164636611938,
0.03576698526740074,
-0.007152634207159281,
0.02961762808263302,
-0.05032402649521828,
-0.01614397205412388,
-0.009643658064305782,
0... | |
52d9e77ed6797eab0fa737b021fbec9a2a02da09 | subsection | 209 | 319 | Solving periodic difference equations | To find the gauge transformation in the previous section we are supposed to solve the finite-difference equations\log \textbf {B} &= \log \left(F^-\right)-\log \left(F^+\right), \\- \log \textbf {H} &= \log \left(G^-\right)-\log \left(G^+\right),where F=f_1 f_2 and G= f_1/f_2.Let us consider the general finite-differen... | {
"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 | [
-0.012271171435713768,
0.01826939359307289,
-0.001965066185221076,
-0.00701318820938468,
-0.011828554794192314,
-0.01779625006020069,
0.009554419666528702,
0.035958804190158844,
0.010042823851108551,
0.03266207128763199,
-0.026083869859576225,
-0.008554715663194656,
-0.01800992712378502,
-... | |
6f12629e9bb95eeb30fe756c0c09059ba8cc5ccf | subsection | 210 | 319 | Non-periodic case. | In the undeformed case we are looking for non-periodic solutions of eqn. (REF ). If \lim _{u\rightarrow \infty } \Omega = 0 we can write the solution \zeta in the form\zeta (u) = -\frac{1}{2\pi i} \int _{Z_0} dv \frac{\rho _{\Omega }(v) }{u-v},for Im(u)>0. The spectral density \rho _{\Omega } is defined as\rho _{\Omega... | {
"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 | [
-0.023700406774878502,
-0.006386748515069485,
-0.0014850907027721405,
-0.013300003483891487,
-0.009072693064808846,
-0.017611248418688774,
0.052101217210292816,
0.0143301235511899,
0.045264266431331635,
0.03421526774764061,
-0.014139360748231411,
0.03748113289475441,
0.00036721894866786897,
... | |
3962362d647ad07ba0dc51eb024dff40f50a4563 | subsection | 211 | 319 | Periodic case. | We provide the proof of the periodic case here in details, by proving the following lemma.Lemma. Let \Omega : be a 2-periodic function regular in the upper half-plane obeying
\begin{equation}\lim _{u\rightarrow i \infty } \Omega (u) \in \mathbb {R},
\end{equation}and which converges uniformly to an L1 function on [-,].... | {
"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 | [
-0.017733683809638023,
0.026341082528233528,
-0.03400227800011635,
0.0005861310637556016,
0.01994657889008522,
-0.029088124632835388,
0.008714227937161922,
-0.002531475620344281,
0.020175497978925705,
0.023517733439803123,
-0.024173971265554428,
0.010209839791059494,
0.00423502316698432,
0... | |
160c83fb43314478ec5807768df6cd4b678d248b | subsection | 212 | 319 | Periodic case. | Around ui we find
\begin{equation}\int _{[-\pi +\pi ]+\text{Im}(u)i+i\epsilon } \frac{\Omega (v)}{\tan \left( u-v+i \epsilon \right)}dv \sim i \int _{[-\pi +\pi ]+\text{Im}(u)i+i\epsilon } dv\Omega (v)\text{Im}\left(\cot \left( u-v\right)\right),
\end{equation}which is purely imaginary as long as the contribution of is... | {
"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 | [
-0.0030701172072440386,
0.02813066355884075,
-0.011365154758095741,
-0.011182092130184174,
-0.015270496718585491,
-0.024728745222091675,
0.016307853162288666,
0.014973019249737263,
0.05183731019496918,
0.022852351889014244,
0.003079651854932308,
0.02570508047938347,
-0.020335236564278603,
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.