<jats:p>
The thermal partition function,
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>Z</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>Z</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, of a
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>CFT_d</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mi>F</mml:mi>
<mml:msub>
<mml:mi>T</mml:mi>
<mml:mi>d</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
on
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>S^{d-1}</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msup>
<mml:mi>S</mml:mi>
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mo>−</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
is parameterized by the inverse temperature
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\beta</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>β</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
along with
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\lfloor d/2\rfloor</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mo stretchy="false" form="prefix">⌊</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>/</mml:mi>
<mml:mn>2</mml:mn>
<mml:mo stretchy="false" form="postfix">⌋</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
angular velocities
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\omega_i</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
. In this paper, we investigate the behaviour of this partition function when
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>n</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>n</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of the
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\omega_i</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
are scaled to unity (the largest allowed value) at fixed values of the other
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>(\lfloor d/2\rfloor-n)</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mo stretchy="false" form="prefix">(</mml:mo>
<mml:mo stretchy="false" form="prefix">⌊</mml:mo>
<mml:mi>d</mml:mi>
<mml:mi>/</mml:mi>
<mml:mn>2</mml:mn>
<mml:mo stretchy="false" form="postfix">⌋</mml:mo>
<mml:mo>−</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo stretchy="false" form="postfix">)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
angular velocities. We argue that
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\ln Z</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mrow>
<mml:mi mathvariant="normal">ln</mml:mi>
<mml:mo></mml:mo>
</mml:mrow>
<mml:mi>Z</mml:mi>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
develops a simple pole in
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>(1-\omega_i)</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mo stretchy="false" form="prefix">(</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>−</mml:mo>
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo stretchy="false" form="postfix">)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
for each
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\omega_i</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msub>
<mml:mi>ω</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\beta</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>β</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>n</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>n</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
special angular momenta
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>J_1\ldots J_n</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:msub>
<mml:mi>J</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mi>…</mml:mi>
H. Anand, N. Benjamin, Vipul Kumar et al.· SciPost Physics· 8 citations· ⚡3
<jats:p>
We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$c$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>c</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, using Fourier analysis on
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$\mathrm{SL}_{2}(\mathbb{Z}/c\mathbb{Z})$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>SL</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo>(</mml:mo>
<mml:mi>Z</mml:mi>
<mml:mo>/</mml:mo>
<mml:mi>c</mml:mi>
<mml:mi>Z</mml:mi>
<mml:mo>)</mml:mo>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and an amplification argument with non-abelian characters. For sums of length
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$\sqrt{c}$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msqrt>
<mml:mi>c</mml:mi>
</mml:msqrt>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, our method produces a non-trivial bound for all moduli except near-primes, saving
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$c^{-1/12}$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mrow>
<mml:mo>−</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>12</mml:mn>
</mml:mrow>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the Pólya–Vinogradov range for all moduli. We give applications to moments of twisted cuspidal
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$L$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>L</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-functions, and to large sieve inequalities for exceptional cusp forms with composite levels.
</jats:p>
Alexandru Pascadi· Geometric and Functional Ana...· 0 citations
<jats:p>
The periodic Temperley–Lieb category consists of connectivity diagrams drawn on a ring with
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>N</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>N</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>N'</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:msup>
<mml:mi>N</mml:mi>
<mml:mo>′</mml:mo>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\mathsf{M}(N)</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mi mathvariant="sans-serif">𝖬</mml:mi>
<mml:mo stretchy="false" form="prefix">(</mml:mo>
<mml:mi>N</mml:mi>
<mml:mo stretchy="false" form="postfix">)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
over the enlarged periodic Temperley–Lieb algebra for varying values of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>N</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mi>N</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, endowed with an action
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\mathsf{M}(N') \to \mathsf{M}(N)</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mrow>
<mml:mi mathvariant="sans-serif">𝖬</mml:mi>
<mml:mo stretchy="false" form="prefix">(</mml:mo>
<mml:msup>
<mml:mi>N</mml:mi>
<mml:mo>′</mml:mo>
</mml:msup>
<mml:mo stretchy="false" form="postfix">)</mml:mo>
<mml:mo>→</mml:mo>
<mml:mi mathvariant="sans-serif">𝖬</mml:mi>
<mml:mo stretchy="false" form="prefix">(</mml:mo>
<mml:mi>N</mml:mi>
<mml:mo stretchy="false" form="postfix">)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>\frac12</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mn>2</mml:mn>
</mml:mfrac>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative.
</jats:p>
Y. Ikhlef, Alexi Morin-Duchesne· SciPost Physics Core· 2 citations
<jats:p>
In this paper, we introduce the notion of generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules respect to some subclass
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, extending the classical notion of Gorenstein projective modules. By exploiting the correspondence between projective modules over the endomorphism ring
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{End}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>End</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of a module
<jats:italic>C</jats:italic>
and elements of its additive closure
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}=\textrm{Add}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>W</mml:mi>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>Add</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, we establish a fundamental correspondence between Gorenstein projective
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{End}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>End</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-modules and generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{Add}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>Add</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules. This result refines existing relative homological settings and provides a natural extension of well-known results in Gorenstein homological algebra. We explore key properties, such as closure under direct summands and sums, and identify conditions under which the class of generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules coincides with other classes of modules, like Gorenstein projective modules.
</jats:p>
Driss Bennis, C. Lomp, Abderrazak Nassir· Algebras and Representation...· 0 citations
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