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80ac38e7c20418238a3df501c038a5f33dd4d4ad | subsection | 213 | 319 | Finding solutions. | It is easy to check numerically in the asymptotic finite-volume solution that \log \textbf {B} and \log \textbf {H} satisfy the requirements mentioned above in both the undeformed and deformed case. We assume this extends beyond the asymptotic solution, yielding a solution of the finite-difference equation (REF ). In p... | {
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c6b35b41fcdf897cd2e76cabd48c84b6583ef2bf | subsection | 214 | 319 | Constructing | Now we have constructed an upper-band gauge with good properties we can continue building good gauges for the other two wings of the T hook. We will focus on the construction of a good gauge \mathbb {T}_R (simply written as \mathbb {T} to avoid heavy notation) for the right hook, since the construction for the left and... | {
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0d38a1da5582352097a8295913cc934a46db52f9 | subsection | 215 | 319 | Defining | We directly define the \mathbb {T} gauge from the \textbf {T} gauge through the transformation\mathbb {T}_{a,s} = (-1)^{as} \textbf {T}_{a,s} \left(\textbf {T}_{0,1}^{[a+s-1]}\right)^{\frac{a-2}{2}}.One can check directly that this gauge satisfies the Hirota equation on the T hook using the fact that \textbf {T} does. ... | {
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b7ece7e04053a1d5f921822414b0a217d53f33e9 | subsection | 216 | 319 | Defining | Indeed, the ratior= \frac{1+1/Y_{2,2}}{1+Y_{1,1}} = \frac{\mathcal {T}_{2,2}^+\mathcal {T}_{2,2}^-\mathcal {T}_{0,1}}{\mathcal {T}_{1,1}^+\mathcal {T}_{1,1}^-\mathcal {T}_{2,3}}is expressed in right-band \mathcal {T} functions only and satisfies r(u+i\epsilon )=1/r(u-i\epsilon ) for u \in \check{Z}_{0} due to eqn. (REF... | {
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3c88216d1b2cea433025ec15b522bab1644dcc3d | subsection | 217 | 319 | Short-cutted | First of all, we can consider short-cutted versions of the \mathcal {T} gauge. These functions can be found by continuation and are given by\hat{\mathcal {T}}_{0,s} &= 1, \\\hat{\mathcal {T}}_{1,s} &= s + \hat{G}^{[+s]}-\hat{G}^{[-s]}, \\\hat{\mathcal {T}}_{2,s} &= \left(1 + \hat{G}^{[s+1]}-\hat{G}^{[s-1]}\right)\left(... | {
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340c89893e6f9083c4eeb7299d4879bea6ce4e61 | subsection | 218 | 319 | Step 3: the | Having defined the \mathcal {T} gauge we can see how it relates to the \mathbb {T} gauge defined before, since two gauges parametrising the same set of Y functions are related by a gauge transformation, as in eqn. (). This will show all the wanted properties of the \mathbb {T} gauge.The first thing to check is that the... | {
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25d3c5320890946b335a68d3c0ccaceef7019071 | subsection | 219 | 319 | Step 3: the | If we want to apply the conjugate bar to the shifts we write \overline{f^+}.\mathbb {T}_{1,s} = \left( \left(\frac{f_1}{f_1^{--}}\right)^{[s]} \times \left(\frac{\overline{f_1}}{\overline{f_1}^{++}}\right)^{[-s]} \right) \mathcal {T}_{1,s} \text{ for } s\ge 1,which shows that \mathbb {T}_{1,s} \in \mathcal {A}_s, as de... | {
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167d27878e0a94c6a8dc961270aba80e969949ab | subsection | 220 | 319 | Step 4.1: is | The real difficulty is to prove that the \mathbb {T} gauge is actually \mathbb {Z}_4-symmetric. We will go through the following steps: first we will prove that the product \hat{\mathbb {T}}_{1,1}\hat{\mathbb {T}}_{1,-1} has only two cuts at Z_{\pm 1}. Second we will show that we can force each of these functions to ha... | {
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f5a4fe94b1267992b07193d4849ea1a9a5d11e30 | subsection | 221 | 319 | Step 4.1: is | We use the analyticity of the \textbf {T} and \mathbb {T} gauges to telescope the sums in the expression above:\left[\log \left(\frac{\mathbb {T}_{1,1}^+ \mathbb {T}_{1,-1}^+ }{\dot{\mathbb {T}}_{1,1}^+ \dot{\mathbb {T}}_{1,-1}^+} \frac{\textbf {T}_{0,1}\textbf {T}_{0,-1}}{\textbf {T}_{1,1}^+\textbf {T}_{1,-1}^+}\right... | {
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13c5f1ae6bdf84f44f7a63fed1c986fefe65ed3e | subsection | 222 | 319 | Step 4.1: is | Going through the analytic and group-theoretical properties of \textbf {T} one finds that nothing changes after transforming them as\textbf {T}_{a,s} \rightarrow \left(\prod _{k=-(|s|-1)/2}^{(|s|-1)/2} \frac{e^{i \phi ^{[a+2k]}}}{e^{i \phi ^{[-a+2k]}}} \right)^{\text{sgn}(s)} \textbf {T}_{a,s},as long as \phi is real w... | {
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c9357f2fd5b92ad273542b8c0e674cbecfc8d0de | subsection | 223 | 319 | Step 4.1: is | Now, since we want \hat{\mathbb {T}}_{1,1}^{\text{new}} to have only two Z_{\pm 1}-cuts we should cancel any extra cuts in \hat{\mathbb {T}}_{1,1} through \phi :i \left[\,\hat{\phi } \, \right]_{-2N}= - \left[ \log \hat{\mathbb {T}}_{1,1}\right]_{-1-2N},\quad i \left[\,\bar{\hat{\phi }} \, \right]_{2N}= \left[ \log \ha... | {
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79bf0c9042150ba94812216ebcf2ee5537de21c9 | subsection | 224 | 319 | Step 4.2: actually proving | The gauge transformation (REF ) can be written like\mathbb {T}_{1,s} = h^{[+s]}\bar{h}^{[-s]}\mathcal {T}_{1,s}where h is a function analytic in the upper half-plane. Translating this equation for s=1 to the short-cutted sheet reads\hat{\mathbb {T}}_{1,1} = \hat{h}^{+}\bar{\hat{h}}^{-}\hat{\mathcal {T}}_{1,1}.Knowing t... | {
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bac0721ad248cd06454b18b11ad4e9fbfaa17971 | subsection | 225 | 319 | Step 4.2: actually proving | Consider u\in \hat{Z}_0, then we have\mathbb {T}_{2,3}=\hat{\mathbb {T}}_{2,3},\quad \mathbb {T}_{2,2}(u\pm i c)=\hat{\mathbb {T}}_{2,2}(u\pm i c\mp i\epsilon ),\quad \mathbb {T}_{1,1}(u\pm i c)=\hat{\mathbb {T}}_{1,1}(u\pm i c\mp i\epsilon ).Additionally we find from the gauge transformation that\hat{\mathbb {T}}_{2,s... | {
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7f8ffd3c70dfa844651068c24518313fad5e15c8 | subsection | 226 | 319 | Step 4.2: actually proving | (REF ) and then using these properties we get successively\mathbb {T}_{1,1}^+\mathbb {T}_{1,1}^--\mathbb {T}_{1,0}\mathbb {T}_{1,2} = \frac{\mathbb {T}_{2,2}^+\mathbb {T}_{2,2}^--\mathbb {T}_{3,2}\mathbb {T}_{1,2}}{\mathbb {T}_{2,3}},\hat{\mathbb {T}}_{1,1}^{[+1-\epsilon ]}\hat{\mathbb {T}}_{1,1}^{[-1+\epsilon ]}+\left... | {
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851734e49a8f7e8e003bc17d25b239da8c1e64fa | subsection | 227 | 319 | Step 4.2: actually proving | Using the regularity requirement that our \hat{\mathbb {T}}s do not have any poles except possibly at the branch points we want \hat{h} to be pole and zero free. For the undeformed case numerical analysis showed in that \hat{h} could grow at most polynomially as u \rightarrow \infty , implying that \bar{\hat{h}}/\hat{h... | {
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853d0593c25c91d30eff92773231e11eae0f4edf | subsection | 228 | 319 | A step back | Let's take a step back and see what we have achieved so far: we started out with the analytic Y-system and constructed four gauges {T}, \textbf {T}, \mathcal {T} and \mathbb {T}. The first two gauges have good analyticity properties (see table ) in the upper band of the T hook , and the \textbf {T} gauge has particular... | {
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deecfe3d26c826fcef4315f34af5b33ecbb5506d | subsection | 229 | 319 | Wronskian parametrisation | As anticipated in the previous section we can use further simplify the \hat{\mathbb {T}} gauge by noticing that the analytical continuation discussed in section REF can also be applied to the \hat{\mathbb {T}} gauge, resulting in a solution to the Hirota equation on a horizontal infinite band of width two. The general ... | {
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826acc2796f0f5a2fa01bbc4846995301665accf | subsection | 230 | 319 | Wronskian parametrisation | One can check that there also exists a scalar symmetryp_i \rightarrow A p_i,\quad q_i \rightarrow A^{-1} q_i,where A has the same periodicity properties as the components of H. The more general n-band solution has another scalar symmetry, but that is trivial for n=2.To find an explicit solution for our case, we will ha... | {
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fd108b1098d25805c20b0c84557127d75267c678 | subsection | 231 | 319 | Wronskian parametrisation | From the parametrisation we see that the q_i should satisfy a system of Baxter equations\hat{q}^{[2r-1]}\hat{\mathbb {T}}_{1,1} & = \hat{q}^+\hat{\mathbb {T}}_{1,r}^{[r-1]} - \hat{q}^-\hat{\mathbb {T}}_{1,r-1}^{[r]} \\\hat{q}^{[-2r+1]}\hat{\mathbb {T}}_{1,1} & = \hat{q}^-\hat{\mathbb {T}}_{1,r}^{[-r+1]} - \hat{q}^+\hat... | {
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8cf3c3954f292548813413bc9bff688b7a3e80ce | subsection | 232 | 319 | Wronskian parametrisation | The simplest choice for this kernel isK_c(u) = \zeta (u) -\zeta (u+2\pi ) +\zeta (2\pi ),with \zeta the quasi-elliptic Weierstraß function with quasi-halfperiods (2\pi , ic). This kernel is elliptic and has exactly two simple poles in its fundamental parallelogram with residues 1 and -1. If we take discontinuities on ... | {
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048912c39d1c1659e04b8014e7127d6f7d7eef43 | subsection | 233 | 319 | Conclusions | In this chapter we have found basic building blocks for the TBA Y-functions in the form of four T-gauges. In particular, the \mathbb {T} and \textbf {T} gauges have good properties and parametrise the entire T-hook in an efficient way. Moreover, after discovering that the short-cutted version of the left and right \mat... | {
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b3289a5b26015d74655230409ce46d4d7cebc2b9 | subsection | 234 | 319 | The middle nodes and | Following the group-theoretical properties we see that the middle nodes (0,s) of the \textbf {T} gauge are all generated from a single function \textbf {T}_{0,0}, which by itself is also 2ic-periodic, such that there are only two unique entries on the middle line of the \textbf {T} hook: \textbf {T}_{0,0} = \textbf {T}... | {
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6b37250dafa2bfd601eae91653223c80de3e0183 | subsection | 235 | 319 | Switching conventions. | We would like to switch from the lower-half-plane conventions to the upper-half-plane conventions: this can be easily achieved by flipping the sign of u, u\rightarrow -u. This changes only:\check{f} = \hat{f} \text{ in the strip } 0> \text{Im}(u)>-c\quad &\Longrightarrow \quad \check{f} = \hat{f} \text{ in the strip } ... | {
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c66eebffaa9a69b0d74a804bb53c282edc83e195 | subsection | 236 | 319 | Introducing extra | As mentioned before, to constrain the \mathbf {P} functions and \mu _{12} we need to extract information from the analyticity of the upper band . Using the gauge transformation (REF ) we find for example that\textbf {T}_{2,1} = \left(\tilde{\mathbf {P}}_1 \mathbf {P}_{2}^{[-2]} - \tilde{\mathbf {P}}_2 \mathbf {P}_{1}^{... | {
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1c7c1313edea05434e37b9d0b5521362eed6b029 | subsection | 237 | 319 | Introducing extra | We can reinterpret these equations as those expressing the analyticity of four new \mathbf {P} functions \mathbf {P}^1, \mathbf {P}^2, \mathbf {P}_3 and \mathbf {P}_4 on all the Z_n for n\ne 0, defined as follows: first consider 2 ic-periodic solutions \mu _{\alpha 3} and \mu _{\alpha 4} (for a = 1,2) of the following ... | {
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99f793ed395a78f1de8889faeeed503f6345b847 | subsection | 238 | 319 | Introducing extra | This analytic structure is illustrated in fig. REF .
[Figure: The analytic structure of the \mathbf {P}_a in the undeformed (a) and the deformed (b) case. The thick line indicates the only branch cut on the first sheet. The continuation of \mathbf {P}_a from the first sheet to the second sheet is indicated by a solid p... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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48ad4befc2e998e0b38133aee4755c73abe0d1e2 | subsection | 239 | 319 | Solving the Riemann-Hilbert problem. | In order for the above construction to work we need to know that the newly introduced \mu functions actually exist. For that we should show that the Riemann-Hilbert problem (REF ) has a solution in both the undeformed and the deformed case. Note that these equations are of the form\tilde{f}-f= g,where in the deformed c... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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0adaeaca1f19b40722f9892ad7bf3684ccbd0019 | subsection | 240 | 319 | Combining the relations. | So far we have introduced eight \mathbf {P} functions \mathbf {P}_a and \mathbf {P}^a with a=1,\ldots ,4 and five \mu functions \mu _{12}, \mu _{1 3}, \mu _{23}, \mu _{14} and \mu _{24}, which are constrained by the equations (REF ), (REF ), (REF ) and (REF ). We can introduce one more \mu function to fill the upper tr... | {
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"source_ref_... | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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959df159e5f4a0492f102f2fa95870b834e739c7 | subsection | 241 | 319 | Combining the relations. | It is because of this that the \mathbf {P}\mu system is identical for both the undeformed and the deformed case.By introducing the inverse of the \mu matrix with upper indices as \mu _{ab} \mu ^{bc} = \delta ^c_a we also find the upper-index form of the \mathbf {P}\mu system\tilde{\mu }^{ab}-\mu ^{ab} = -\mathbf {P}^a ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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a805da97cc7683ca810e1052e84334bc2f54fef5 | subsection | 242 | 319 | Conjugation properties of | We can use the \mathbf {P}\mu system to derive the conjugation properties of \mu : first we show that on \mathbf {P} functions conjugation and continuation by the tilde commute, by assuming that all \mathbf {P} functions can be expressed as a real function times a convergent series in the x function. To show then that ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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0f069441c7b04d7956bf0a88ecfe9817efcde70a | subsection | 243 | 319 | Information from TBA | Our task is to find some property of the QSC functions \mathbf {P}, \mathbf {Q}, \mu and \omega that depends on the quantum numbers of the state we would like to compute the energy of.For good measure, let us mention here that in the undeformed case the AdS/CFT correspondence implies that this is the same as computing ... | {
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superstring | [
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2fe9df2d3da0dfcb0071d9f58766574dcc01b31a | subsection | 244 | 319 | Connecting the | The ratio Y^{(\alpha )}_+ / Y^{(\alpha )}_- does not depend on \alpha , as we see from the TBA equations (REF ):\frac{Y_+^{(\alpha )}}{Y_-^{(\alpha )}} = \exp \left( \Lambda _P \star K^{Py}\right).In the following we will omit \alpha and find an expression in the \mathbf {P}\mu system that equals these two quantities. ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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6c21a25e3cfc7bb812d09b3ef93f67729ae2dc35 | subsection | 245 | 319 | Connecting the | We see here for example that we get inequivalent results if we had chosen the short-cutted \mu _{12}: in that case
so\mathbb {T}_{3,2} = \mathbb {T}_{2,3}\mu _{12}^{[+4]} = \mathbb {T}_{2,3}\mu _{12}instead of \hat{\mu }_{12}^{[+4]}, which does not have a clear simplification on the short-cutted sheet. Now we find\frac... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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0bb486a2b14c1ac45b6ce79da33704bd9aa3765b | subsection | 246 | 319 | Connecting the | Now we can find \mathbb {T}_{1,0}:\mathbb {T}_{1,1}^+\mathbb {T}_{1,1}^- &= \mathbb {T}_{0,1}\mathbb {T}_{2,1}+\mathbb {T}_{1,0}\mathbb {T}_{1,2} = \mathbb {T}_{2,1}+\mathbb {T}_{1,0}\mathbb {T}_{1,2} \\\mathbb {T}_{1,0} &= \frac{\mathbb {T}_{1,1}^+\mathbb {T}_{1,1}^- - \mathbb {T}_{2,1}}{\mathbb {T}_{1,2}} = \frac{\ti... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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bd4a63263a4efe15f9af91483a10cd4d826a9309 | subsection | 247 | 319 | Connecting the | We therefore find in terms of short cuts that&\hat{\mathbb {T}}_{1,0} = \mu _{12}+\\&\frac{\left( \mathbf {P}_1^{[+2]}\tilde{\mathbf {P}}_2-\tilde{\mathbf {P}}_1\mathbf {P}_2^{[+2]} \right)\left(\mathbf {P}_1\mathbf {P}_2^{[-2]}-\mathbf {P}_1^{[-2]}\mathbf {P}_2 \right)- \left(\mathbf {P}_1^{[+2]}\mathbf {P}_2-\mathbf ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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9c9b2b7a36a519f1b2996a9cd13759a521cb3760 | subsection | 248 | 319 | Sensitivity to conventions. | An important aspect of this derivation is that it is sensitive to where we say long- and short-cutted versions of functions should coincide: slightly above or slightly below the real axis. In the derivation above we chose the upper-half-plane convention that functions agree just above the real axis. Let us see what cha... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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cf004b61f7760064157a83077dbf56fb9714a75c | subsection | 249 | 319 | Sensitivity to conventions. | Note first now that the equation relating \mu _{12} to \tilde{\mu }_{12} gets deduced from the absence of a discontinuity for \textbf {T}_{2,1} on Z_0 and this condition gets changed as a result of our choice, taking the form\mu _{12}-\tilde{\mu }_{12}= \mathbf {P}_1\tilde{\mathbf {P}}_2-\tilde{\mathbf {P}}_1\mathbf {P... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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"Rob Klabbers"
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fab90c5f6b4848c44dab5ccd3a1a26950f1b9c62 | subsection | 250 | 319 | Undeformed derivation | The previous sections have shown that\frac{\mu _{12}^{\pm \pm }}{\mu _{12}} = \exp \left( \Lambda _P \star K^{Py}\right) = \exp \left( \Lambda _P \star \left( K(v+icQ,u ) - K(v-i cQ,u )\right)\right),with the sign depending on the chosen conventions. For now we will choose the lower-half-plane conventions, as the kerne... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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7be13272c3976ab3a908291d7349e8b04ba33e40 | subsection | 251 | 319 | Undeformed derivation | This implies that
\begin{equation}\log \frac{\mu _{12}^{--}}{\mu _{12}} = -\frac{2iM}{g u} +\mathcal {O}\left(u^{-2}\right)
\end{equation}and after comparison with eqn. (\ref {eq:KPyundeformedexpanded}) we find that M = E. So we find that 12 uE. Can we do something similar in the deformed case? | {
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62c8bb13ccbe2082373279424e0b5609666d79e9 | subsection | 252 | 319 | Now the deformed case | The deformed case is more subtle: first off, it is not clear in which direction we should consider asymptotics, since taking the same limit u\rightarrow \infty does not make sense on a cylinder. Really, the only two asymptotic directions available to us are up and down the cylinder, i.e. u \rightarrow \pm i \infty . In... | {
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4ea1261b7b56b967146c53fabe2f9b64c18ba4d0 | subsection | 253 | 319 | Now the deformed case | In short, an ansatz for the asymptotics of \mu _{12} would be\mu _{12} \simeq m(u) e^{i u \alpha },with \simeq meaning that\lim _{u\rightarrow -i \infty }\frac{m(u) e^{i u \alpha }}{ \mu _{12}(u)} = 1,where the limit has to be taken avoiding branch cuts, i.e. here |\text{Re}(u)| > \theta with \alpha \in having positive... | {
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3f4ea77d215f34cf9613b2e25b6890343831ea8c | subsection | 254 | 319 | Remark. | Note that this expression is very similar to its undeformed counterpart \mu _{12} \sim u^{E}, although the way we obtain this expression is subtly different. In the deformed case, we do not find energy and momentum as separate coefficients in the expansion of K^{Py}, but instead they come together already at the lowest... | {
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332a33bf27d7d9fc92d2af933c62d09a95a70089 | subsection | 255 | 319 | Comments on the periodicity of the deformed | It is important to note the factor of 1/2 in the exponent in eqn. (REF ), as it implies that for odd integer E the \mu _{12} function is at most anti-periodic instead of periodic. This is exactly a possible situation at lowest order in perturbation theory, where quantum corrections play no role yet. Indeed, at lowest o... | {
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dec3e790d911d69dad876cc62b9175c8136d408e | subsection | 256 | 319 | Comments on the periodicity of the deformed | Instead, E becomes a measure of the jump of \mu _{12} over the line Re(u)=\pi very far in the lower half-plane. This is illustrated in fig. REF . | {
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38ae18766bd173a410e4596a6a425d1f9ffcf106 | subsection | 257 | 319 | Switching to the upper-half-plane conventions | Now that we have performed the analysis in the previous section we can switch permanently to the upper-half-plane conventions. We do this as before (see section REF ), where now we have the additional changes\tilde{\hat{\mu }}_{ab} = \hat{\mu }_{ab}^{--} \quad &\Longrightarrow \quad \tilde{\hat{\mu }}_{ab} = \hat{\mu }... | {
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cfff1c2e6bcc91b1f568725cd215babf037858f4 | subsection | 258 | 319 | Anticipating the result | Our actual aim in this chapter is to find whether the basic QSC functions \mathbf {P} and \mathbf {Q} depend on the six quantum numbers (REF ): conjecturing that this is the case states that the QSC as we have derived it can be extended to describe any state in the system, provided we pick up the right “boundary condit... | {
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29f17030f30bdc10ff16545ffb023d0635f47132 | subsection | 259 | 319 | Deriving the asymptotics | We can deduce the above asymptotics from consistency of the QSC and the asymptotic behaviour we found for \mu _{12}. For the deformed case we can also use the fact that its asymptotics should limit to the undeformed asymptotics, whereas for the undeformed case we can use the knowledge of the classical spectral curve. I... | {
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6ba05e8fb6b7b6026a6beee73beaf1067bed5cd9 | subsection | 260 | 319 | Undeformed asymptotics. | Starting from the fact that \mu _{12} \sim u^E as u\rightarrow \infty We use \sim instead of \simeq to allow for an normalisation constant. we postulate that all the \mathbf {P} and \mathbf {Q} functions have interesting asymptotics in this limit:We refrain from putting an extra label “und" on the quantities in this se... | {
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79ed1fc5d6cb6fb96ccba4312341ef132e44b0a9 | subsection | 261 | 319 | Undeformed asymptotics. | (REF ) for the product B_{j_0}B^{j_0}.We can also find asymptotics for \omega and \mu : Since \omega ^{++} = \omega we find that \omega \simeq 1, as this is the only 2i/g-periodic function with power-like asymptotics . Choosing the right \mathbf {Q}_i basis we can choose it to be such that it is anti-symmetric with \om... | {
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058077f23d57a669a5700101aec9e6cf5c2694c7 | subsection | 262 | 319 | Deformed asymptotics. | This time we postulate that all the \mathbf {P}_a and \mathbf {Q}_i have interesting asymptotics as we send u\rightarrow i \infty outside the branch cut strip:\mathbf {P}_a \simeq A_a z^{-\tilde{M}_a}, \quad \mathbf {Q}_i \simeq B_i z^{\hat{M}_i}\quad \mathbf {P}^a \simeq A^a z^{\tilde{M}^a}, \quad \mathbf {Q}^i \simeq... | {
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f4343ab0b3c87261e5dc18c4a16e314f74d3340a | subsection | 263 | 319 | Deformed asymptotics. | (REF ) should hold (which in fact it does). Solving these equations again we findA_{a_0}A^{a_0} =2\frac{\prod _{j}\sinh \left( c\frac{\tilde{M}_{a_0}- \hat{M}_{j}}{2}\right)}{\prod _{b\ne a_0} \sinh \left( c\frac{ \tilde{M}_{a_0}- \tilde{M}_{b}}{2}\right)} \qquad \text{ for } a_0=1,\cdots ,4.Performing a similar analys... | {
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bf7b3ff46090e2f59e7ec9d032c97939e715e3ae | subsection | 264 | 319 | Dependence of | Our remaining task now is to find the dependency of the M's on the global charges. In order to deduce this we should consider the following set of constraints on the asymptotics:The \mu _{12} asymptotics have a specified form as follows from the TBA.
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b449f3cd199156f0b662cdfc1ac8bfac5b7315eb | subsection | 265 | 319 | Deriving the deformed charges. | Obtaining the deformed charges is less straightforward, simply because the classical spectral curve is not known for the \eta -deformed model. We therefore proceed by imposing the constraints listed in the previous paragraph and additionally impose that the dependence on the charges is linear and that the \mathbf {P} a... | {
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42dd3f87297dbac432bb971cbdd6980e27226b39 | subsection | 266 | 319 | Deriving the deformed charges. | Hence the subleading term must become leading in the undeformed limit.Since the \mathbf {P}_{3,4} are auxiliary variables anyway, their asymptotics are fixed by consistency of the QSC, hence uniquely in terms of the well-defined undeformed limit of the \mathbf {P}_{1,2}. From this reasoning we find the following compar... | {
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100f14e47504525ff11a8c6e8e5cd8602d1251f3 | subsection | 267 | 319 | Putting together the results | For the readers convenience let us summarise all the results. We went through the complete derivation of the quantum spectral curve for the {\rm AdS}_5\times {\rm S}^5 superstring and its \eta deformation. The final form of the \mathbf {P}\mu - and the \mathbf {Q}\omega -system is the same for both models and are\tilde... | {
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1ff134f1f9e913c6ad61acf013e51c5cecea571f | subsection | 268 | 319 | Putting together the results | This can be interpreted as the trigonometric version of the phenomenon in the undeformed case that \mu _{ab} develops a branch point at infinity for non-zero coupling.Although we started from the ground-state TBA-equations we ended up with a proposed set of boundary conditions for the \mathbf {P} and \mathbf {Q} functi... | {
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95e25491fb0940385316651c59c6e418982b8411 | subsection | 269 | 319 | Putting together the results | Lastly, due to the presence of trigonometric functions instead of rational functions as in the undeformed case we can rightfully call the \eta -deformed QSC to be a trigonometric QSC. | {
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16eef1161cd13d9f018f1a74c25f57dfa7262c7c | subsection | 270 | 319 | Checking using weak coupling | We can check the derived quantum spectral curve for the \eta -deformed model and asymptotics by comparing to the Bethe-Yang equations in (REF ). We will zoom in on the \mathfrak {sl}_2 sector of the theory, similar to the analysis done for the undeformed case in . In this sector only three of the global charges (J_1 = ... | {
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7a8adafd7e2894cd321f1b45c0e85cb282e940b4 | subsection | 271 | 319 | Checking using weak coupling | This splits the \mathbf {P}\mu system into two parts as in the undeformed case and using some algebra we obtain the following TQ-like equation for Q=\mu _{12}^+:-T Q +\frac{1}{\left(\mathbf {P}_2^-\right)^2}Q^{--}+\frac{1}{\left(\mathbf {P}_2^+\right)^2}Q^{++}=0,where T is given by the following rational function of \m... | {
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5e97077daeb01946abae610f107895f9cef5c2f2 | subsection | 272 | 319 | Checking using weak coupling | Using its asymptotics and reality we can use the lemma in appendix to restrict the series to be a trigonometric polynomial. In fact, assuming as in the undeformed case that the pole at the origin has order J/2, the trigonometric polynomial trivialises to a constant, leaving us with\mathbf {P}_2(u) \sim S(u)^{-J/2}.This... | {
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7625adb6e6c38205585b3436d04ecd509ed6d4fb | subsection | 273 | 319 | Intermezzo: why was the road this long? | A very tempting question to ask is why it was not possible to use the undeformed QSC more directly to find its deformed analogue: already on the level of the TBA equations it is clear that the analytic structure of the two cases are very similar: both have square root branch cuts, which come in two types and both deal ... | {
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4f5050f9bfa950867c0132133b13ec4a8fdaa277 | subsection | 274 | 319 | Implications from mirror symmetry | As we already explored in chapter , the family of \eta -deformed strings is actually closed under the associated double Wick rotation. Namely, taking \theta \rightarrow \pi - \theta at fixed c is equivalent to a double Wick rotation. To explicitly match the parametrisation this should be combined with a shift u \righta... | {
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7f9b297d8e068c9f219590b3a13160a64d9bad72 | subsection | 275 | 319 | Implications from mirror symmetry | In this limit, the cut structure of the QSC is as expected: the \mathbf {P} and \mu functions now have long cuts with branch points \pm 2g,These are the branch points in the conventions of ; we have tacitly continued our discussion in the TBA conventions, where the branch points are located at \pm 2. while the \mathbf ... | {
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3de768c582690de8159acfffb9d3289d226a2911 | subsection | 276 | 319 | Conclusions and future work | In this thesis we have derived the quantum spectral curve for the \eta -deformed superstring in a language that allowed us to simultaneously review the construction of the QSC for the {\rm AdS}_5\times {\rm S}^5 superstring. The resulting description of the spectral problem is arguably much simpler than the starting po... | {
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74234d4a0dc856ff7d45b723dc18b720345faf74 | subsection | 277 | 319 | Conclusions and future work | This is very similar to the usual description of energies in quantum-integrable models using Bethe equations: the equations are well-behaved, but finding explicit solutions might not be easy. | {
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88ad60b3917fd991010183597d24a243e2907236 | subsection | 278 | 319 | Future directions | Now that we have derived the \eta -deformed QSC there is a whole host of possible further directions of study. | {
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48a9d7b908548bff796dd0e8b877ef36fab7ba1b | subsection | 279 | 319 | Finding deformed string energies. | Of course, the prime usage of the deformed QSC is the computation of string energies for the \eta -deformed string theory. The spectrum of the deformed theory should interpolate from the undeformed string spectrum to the undeformed mirror theory spectrum, each extreme having a string theory interpretation. The deformat... | {
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b8a3a60d70bd4c454be5095f65d159f4eb0c00ce | subsection | 280 | 319 | Studying thermodynamics. | As mentioned in section , it would also be very interesting to use the trigonometric QSC to look at the thermodynamics of our models, in particular the undeformed string. Here it would be great to understand how to incorporate chemical potentials in the \eta -deformed QSC, analogous to how this was done in the undeform... | {
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9b421a0b2a22b224a13588990e09f49c9b824730 | subsection | 281 | 319 | Other deformed QSCs. | We saw that the \eta -deformed {\rm AdS}_5\times {\rm S}^5 string “contains” many other integrable deformations of the {\rm AdS}_5\times {\rm S}^5 string, fitting into the class of homogeneous Yang-Baxter deformations , , , , , . Namely, by considering particular coupled infinite boost and q\rightarrow 1 limits, it is ... | {
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59742a9dcc6ff2faeb677d51ea211c215f4eabdd | subsection | 282 | 319 | The root of unity case. | Another interesting direction to investigate is the model defined by the exact q-deformed S-matrix for q a root of unity, i.e. q=e^{i \pi /k}, instead of real. The representation theory of quantum algebras is quite different when q is a root of unity, which will presumably reflect itself in the QSC – as it does in the ... | {
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1bc9aaf2f8140b515914d222992b8384e82fc7ed | subsection | 283 | 319 | Elliptic QSC. | Finally, even more speculatively, one might wonder whether it is possible to find an elliptic deformation of the {\rm AdS}_5\times {\rm S}^5 string, or at least its exact S-matrix. The resulting theory should constitute the “xyz" cousin of the {\rm AdS}_5\times {\rm S}^5 (xxx) and \eta -deformed (xxz) string theories, ... | {
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def5db9c870938d4e6615813c8dfbf7fd3ec9aee | subsection | 284 | 319 | Characterisation | The x functions are solutions to the shortening condition for the fundamental representations of the manifest symmetry algebra of the S matrix, \mathfrak {psu}_q(2|2)^{\otimes 2} (with q=1 for the undeformed case) when viewed as a functional equation: the deformed shortening condition ise^{iu} = \frac{x(u) + \frac{1}{x... | {
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53de8f75832b1a553da6f4fb847e01fa32825506 | subsection | 285 | 319 | Characterisation | Finally, just like in our discussion about analytic continuation in section REF the two solutions x(u) and 1/x(u) are related by analytic continuation.We can characterise the solutions of the shortening condition that we find most convenient uniquely by the following set of properties:it has a short or long cut between... | {
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77f149350fbdda249005535bb3963fe52e9b55c3 | subsection | 286 | 319 | Conjugation. | The undeformed x-functions are both real analytic and as such conjugation of these functions is simple: for both x_s and x_m we find\overline{x^{\text{und}}(u)} = x^{\text{und}}\left(\overline{u}\right).The deformed case is more complicated:\overline{x_s(u)} = \frac{x_s\left( \overline{u}\right)+\xi }{x_s\left( \overli... | {
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095e2e3e94f0e93613bba71e88245fd039069ef0 | subsection | 287 | 319 | Continuation. | By construction both sets of x functions coincide on the lower half-plane and are inverse on the upper half-plane, following the lower-half-plane conventions. Following the definition of the tilde continuation this implies that in both cases\widetilde{x^{\text{und}}(u)} = \frac{1}{x^{\text{und}}\left(u\right)}, \qquad ... | {
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43711d94b083e1243005977e91e08ab608a35b43 | subsection | 288 | 319 | Commutativity of conjugation and continuation. | An interesting and non-trivial property that survives deformation is the fact that conjugation and continuation using the tilde of the x functions commutes. In the undeformed case this is quite trivial, as\widetilde{\overline{x^{\text{und}}(u)}} = \widetilde{x^{\text{und}}\left(\overline{u}\right)} = \frac{1}{x^{\text{... | {
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9cb6dd2b6c2650ac78ed52d1002631e7f2e28909 | subsection | 289 | 319 | Deformed kernels | The kernels that appear in the \eta -deformed TBA-equations (REF ) are defined as follows:K_M (u) = \frac{1}{2\pi i} \frac{d}{du} \log S_M(u) = \frac{1}{2\pi } \frac{ \sinh Mc}{\cosh Mc- \cos u},withS_{Q}(u) = \frac{S^{[-Q]}}{S^{[+Q]}} = \frac{\sin \frac{1}{2} (u-i Q c)}{\sin \frac{1}{2} (u+i Q c)},using our definition... | {
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d0fb1f5631e723cf2b302221c0192fbf6aa3fb3e | subsection | 290 | 319 | Deformed kernels | They are given byK_{xv}^{QM}(u,v) &=& \frac{1}{2\pi i} \frac{d}{du} \log S_{xv}^{QM} (u,v), \\
K_{vwx}^{MQ}(u,v) &=& -\frac{1}{2\pi i} \frac{d}{du} \log S_{xv}^{QM} (v,u), \\
K_{\beta }^{Qy}(u,v) &=& \frac{1}{2\pi i} \frac{d}{du} \log S_{\beta }^{Qy} (u,v), \\
K_{\beta }^{yQ}(u,v) &=& \beta \frac{1}{2\pi i} \frac{d}{du... | {
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f1a4b1072025def9d4537c50cbf550241e68ef85 | subsection | 291 | 319 | Deformed kernels | We effectively go back to the |q|=1 conventions in our definition of K^{\Sigma }_{QM} below, to get kernels and TBA equations analogous to the undeformed model.-i\log \Sigma ^{QM}(y_1,y_2) =& \, \Phi (y_1^+,y_2^+)-\Phi (y_1^+,y_2^-)-\Phi (y_1^-,y_2^+)+\Phi (y_1^-,y_2^-) \\ &\, -\frac{1}{2}\left(\Psi (y_1^+,y_2^+)+\Psi ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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a5607f55c3d59befb5fa6faf00e30fc2b624bd92 | subsection | 292 | 319 | Deformed kernels | These expressions are necessary to derive the simplified expression for the dressing phase contribution to the simplified TBA-equations, as discussed in appendix .Some important identities for the kernels areK_{Qy}(u,v) &:=& K_{-}^{Qy}(u,v)-K_{+}^{Qy}(u,v) = K(u+ i Q c,v) -K(u- i Qc,v) \\
K_{yQ}(u,v) &:=& K_{-}^{yQ}(u,... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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0e6c27cd1cfbd083e994ba235a63da7e6930a1e1 | subsection | 293 | 319 | Undeformed quantities. | After letting q\rightarrow 1, taking x= x^{\text{und}}_m (defined in (REF )) and replacing the shift distance c\rightarrow 1/g the kernels defined above are the undeformed TBA-kernels. In particular this means we use S^{\text{und}} instead of S. The undeformed mirror energy and momentum are given by\tilde{p}_{\text{und... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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aa2ebf2a08795af8d6ee14ed234f017432d531af | subsection | 294 | 319 | Dressing-phase simplification | An important role in the derivation of the analytic Y-system from the TBA equations is played by the dressing-phase kernel K_{\mathfrak {sl}(2)}^{PQ}. Its explicit form is very convoluted, making its analysis tricky. In order to derive the analytic Y-system we need to know the discontinuities that arise because of the ... | {
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76edae93255a0ce9a08f9d6eb871762e678a625a | subsection | 295 | 319 | Dressing-phase simplification | There are no particular subtleties in taking the \epsilon \rightarrow 0 limit and direct evaluation gives-i \log \hat{\Sigma }_Q = & \, \Phi (y_1^-,x)- \Phi (y_1^-,1/x)- \Phi (y_1^+,x)+ \Phi (y_1^+,1/x)\\
& \, + \frac{1}{2} \left( \Psi (y_1^-,x)- \Psi (y_1^-,1/x)+ \Psi (y_1^+,x)- \Psi (y_1^+,1/x)\right)\\
&\, + \Psi (x... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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1ee377194fd2b39ca2f4d61e4c783a73fc277eac | subsection | 296 | 319 | Dressing-phase simplification | We recognise this combination of x functions as\frac{\partial }{\partial t} \log S(s,t), \qquad S(s,t) = \frac{x(s)-x(t)}{x(s)-1/x(t)},such that, with v=u(y),\oint _{\mathcal {C}} \frac{dz}{2 \pi i} \frac{1}{z-y} f(z) = -\int _{\hat{Z}_0} \frac{dt}{2\pi i} \left(\frac{\partial }{\partial t} \log S(v,t)\right) f(x(t)).W... | {
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9a2ceb98ae7b5f729d56890c24d1de65aa1737bf | subsection | 297 | 319 | Dressing-phase simplification | Integrating by parts in the t_1 integral, noting that K_{Qy} vanishes at \pm \theta , and taking out factors of 2 \pi i to define kernels, we end up with\frac{1}{2\pi }\frac{\partial }{\partial u} \Delta \Phi = 2 K_{Qy} \,\hat{\star }\, K_{q\Gamma }^{[2]} \,\hat{\star }\, \bar{K},where K_{q\Gamma }^{[Q]}(u) = \tfrac{1}... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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d79b623be1d9d59fee07b923da88999d380929e4 | subsection | 298 | 319 | Dressing-phase simplification | By the same propertyS^{[Q+2]}_{q\Gamma }(u,t) =S^{[Q]}_{q\Gamma }(u,t)/(e^{i(u-t)}S_Q(u-t))We hence find\frac{1}{2\pi } \frac{\partial }{\partial u} \Delta \Psi = (2 K_{q\Gamma }^{[Q]}-K_Q)\, \hat{\star }\,\bar{K} + \frac{Q-2}{2\pi } \star \bar{K}.The next line gives, upon integration by parts again\frac{1}{2\pi } \fra... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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7692bf474407c7c0788fd0a00c590f0b040e784e | subsection | 299 | 319 | Dressing-phase simplification | Note that this expression is manifestly compatible with the relation K^{[Q+2]}_{q\Gamma } = K^{[Q]}_{q\Gamma } - K_Q -1 encountered above. Using these identities we can finally simplify \hat{K}_Q^\Sigma to\hat{K}_Q^\Sigma & \, = \sum _{N=1}^{\infty } \left(K_{Qy}\,\hat{\star }\, \bar{K}_{2N} + \bar{K}_{Q+2N}\right). | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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06a84312fb9f4971cce837d7ccb8de35ffc39f26 | subsection | 300 | 319 | Contribution in the TBA. | The contribution of the improved mirror dressing factor to \Delta appearing in the TBA equations is given by\Delta ^\Sigma (u) = 2 \sum _Q L_Q \star \hat{K}^\Sigma _Q (u).This function has a short cut on the real line. We would like to continue it from the upper half-plane to have long cuts. Using the simplified expres... | {
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b431bab5c5262177ae020772475714bff3abf8df | subsection | 301 | 319 | Deriving a local discontinuity equation for | We are looking for a local discontinuity equation that we can impose on the Y-system equations instead of the non-local discontinuity (REF ). As an example of how these computations are carried out in practice we show it in detail here. We compute (simultaneously for the deformed and undeformed case)\left[\log \frac{Y_... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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"Rob Klabbers"
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a9f07c4fbdf729fb1a0b3dacd82fd5c56b163d25 | subsection | 302 | 319 | Deriving the TBA equations for | We present a full derivation of Y_{(v)w} TBA-equations (see section REF in the main text for a summary of this derivation).We can write the left-hand side of the TBA equation as a contour integral as\log Y_{M|(v)w}(u) =\sum _{\tau }\sum _{l=0}^{\infty } \int _{Z_0} \frac{dz}{2\pi i} H(z+2i\tau N c-u) \left[ \log Y_{M|(... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
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3db92169ad8f415616c90747343c3239fca01183 | subsection | 303 | 319 | Deriving the TBA equations for | Applying it once yields& &\left[ \log Y_{M|vw} \right]_{M+2l} = \\
&=& \sum _{N=1}^{\infty } A_{MN} \left[L_{N|vw}\right]_{M+2l -1} + \sum _{N=1}^{\infty } A_{M-1,N} \left[L_{N|vw}\right]_{M-1+2l -1} -\left[ \log Y_{M|vw} \right]_{M+2l-2}\\
&+& \left[\log Y_{M-2|vw}\right]_{M+2l-2} +\left[\log Y_{M|vw}\right]_{M+2(l-1)... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
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ab1759da199274741785825b9b774c2542d2ab04 | subsection | 304 | 319 | Deriving the TBA equations for | Using the line (REF ) with first term \left[ \log Y_{M+1|vw} \right]_{M+2l-1} (So M\rightarrow M+1,l\rightarrow l-1) gives\left[ \log Y_{M+1|vw} \right]_{M+2l-1} = \left[\log Y_{M+2|vw}\right]_{M+2l-2} +\sum _{Q=2}^{M+1} \sum _{N=1}^{\infty } A_{QN} \left[L_{N|vw}\right]_{Q+2l -3}\\+ \left[\log Y_{1|vw}\right]_{2l-1}-\... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
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a3b906ef7bb4b4e2da487bda7266bb7e3d35a0c3 | subsection | 305 | 319 | Deriving the TBA equations for | The triple sum can be put in a more suggestive form:& &\sum _{J=0}^l\sum _{Q=1+J}^{M+J} \sum _{N=1}^{\infty } A_{QN} \left[L_{N|vw}\right]_{Q+2l -1-2J} \\&=& \sum _{J=1}^l\sum _{Q=1+J}^{M+J} \left( \left[L_{Q+1|vw}\right]_{Q+2l -1-2J}+\left[L_{Q-1|vw}\right]_{Q+2l -1-2J}\right)+ \sum _{Q=1}^{M} \sum _{N=1}^{\infty } A_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1857,
"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",
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6d8489b10c21652dd6e519a33b87f578fa54d01d | subsection | 306 | 319 | Deriving the TBA equations for | REF and fig. respectively, we can rewrite this expression for Y_-'s as\left(\Lambda _--\Lambda _+\right) \hat{\star } K_M.[Figure: The contour \gamma _0 in the undeformed (a) and in the deformed (b) case. The dotted line indicates that the lower half of the contour runs on the second sheet.]Collecting all the remaining... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
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921d6fa4fa35cb7ad534485ffda0f6f12b0862f2 | subsection | 307 | 319 | Deriving the TBA equations for | \sum _{Q=1+l}^{M+l} \Lambda _Q(z+\tau Qci)\right).These terms excluding the \Lambda _Q terms in the last line together form the K_{PQ} kernel:&+& \sum _{Q=1}^{M-1} L_{Q|vw}\star K_{M-Q}- \sum _{\tau }\sum _{l=1}^{\infty }\tau \int _{Z_0 -i\tau \epsilon }H(z+\tau (M+2l)ci-u)\cdot \\& &\qquad \left(L_{l|vw}(z+\tau li c)+... | {
"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"
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556acd81ec68e84efa6ba27b4975e56fd3412744 | subsection | 308 | 319 | Deriving the TBA equations for | Our total right-hand side for the the TBA-equation of Y_{M|vw} now reads\sum _{N=1}^{\infty } L_{N|vw} \star K_{NM} + \left(\Lambda _--\Lambda _+\right) \hat{\star } K_M-\sum _{Q=1}^{\infty } \sum _{l=\max (M-Q,0)}^{M-1} \Lambda _{Q+1}\star K_{Q+1+2l-M}.Rewriting the \Lambda _{\beta }'s into L_{\beta }'s and using the ... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
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81268cecfeac03b9c05219e047a7db4ac731f485 | subsection | 309 | 319 | Deriving discontinuities for | For the derivation of the Y_Q TBA-equation in section from the analytic Y-system we need to derive discontinuities of the form\left[ \log Y_{M} \right]_{\pm (2M+l)},where M,l\in \mathbb {N}. We will derive them from the Y-system in a spirit very similar to the corresponding derivation in the Y_{(v)w} case.This means in... | {
"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"
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42d4ea5a95cdfcfb61049c5444978c3c99a2bb62 | subsection | 310 | 319 | Deriving discontinuities for | Taking discontinuity brackets at M+2l-1 for M\ge 2 leads to\left[ \log Y_{M} \right]_{M+2l} &=& -\sum _{N=1}^{\infty } A_{MN} \left[L_{N}\right]_{M+2l -1} + \left[L_{vw|M-1}^{(\alpha )}\right]_{M+2l -1} - \left[ \log Y_{M} \right]_{M+2l-2} \\
&=& -\sum _{N=1}^{\infty } A_{MN} \left[\Lambda _{N}\right]_{M+2l -1} + \left... | {
"cite_spans": []
} | 1804.06741 | Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$
superstring | [
"Rob Klabbers"
] | [
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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4924bfb1d4615d6e738f47f962ab0f13a4ace86a | subsection | 311 | 319 | Deriving discontinuities for | When combining the statements, these terms will cancel. We can repeatedly do this to end up with& &\left[ \log Y_{M} \right]_{M+2l} = \left[ \log Y_{M+l+1} \right]_{M+l-1} +\sum _{J=1}^{l+1}\sum _{Q=J}^{M+J-2} \left[L_{vw|Q}^{(\alpha )}\right]_{Q+2l-2J+2} \\&-&\sum _{J=0}^l\sum _{Q=1+J}^{M+J} \sum _{N=1}^{\infty } A_{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 | [
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d95d6bdd0e4d5c9684248879c6a2b14a3bd59a21 | subsection | 312 | 319 | Deriving discontinuities for | We find that, while being careful about the implications of changing M+2l\rightarrow \pm (M+2L),\left[ \log Y_{M} \right]_{M+2l} &= \sum _{J=1}^{l+1}\sum _{Q=1+J}^{M+J-1} \left[L_{vw|Q-1}^{(\alpha )}\right]_{Q+2l-2J+1} \\&-\sum _{J=0}^l\sum _{Q=1+J}^{M+J} \sum _{N=1}^{\infty } A_{QN} \left[\Lambda _{N}\right]_{Q+2l -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"
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